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Full text of "Applied mechanics"

------- -^^^ 

^ 

REESE LIBRARY 

OF THK 

UNIVERSITY OF CALIFORNIA. -; 
Glass 



APPLIED MECHANICS. 



BY 



GAETANO LANZA, S.B., C. &M.E., 

ii 

PROFESSOR OF THEORETICAL AND APPLIED MECHANICS, MASSACHUSETTS 
INSTITUTE OF TECHNOLOGY. 



NINTH EDITION, REVISED. 
FIRST THOUSAND. 



UNIVERSITY 

OF 



NEW YORK: 

JOHN WILEY & SONS. 

LONDON : CHAPMAN & HALL, LIMITED. 

1905. 



REESE 



c 

COPYRIGHT, 1885, 1900, 1905, 

BY 
GAETANO LANZA. 



RObBKT DRUMMOND, 1RIN1BK, NBW YORK. 



PREFACE. 



THIS book is the result of the experience of the writer 
in teaching the subject of Applied Mechanics for the last 
twelve years at the Massachusetts Institute of Technology. 

The immediate object of publishing it is, to enable him to 
dispense with giving to the students a large amount of notes. 
As, however, it is believed that it may be found useful by 
others, the following remarks in regard to its general plan 
are submitted. 

The work is essentially a treatise on strength and stabil- 
ity ; but, inasmuch as it contains some other matter, it was 
thought best to call it " Applied Mechanics," notwithstanding 
the fact that a number of subjects usually included in trea- 
tises on applied mechanics are omitted. 

It is primarily a text-book ; and hence the writer has endeav- 
ored to present the different subjects in such a way as 
seemed to him best for the progress of the class, even though 
it be at some sacrifice of a logical order of topics. While 
no attempt has been made at originality, it is believed that 
some features of the work are quite different from all pre- 



147GG3 



iv PREFACE. 



vious efforts ; and a few of these cases will be referred to, 
with the reasons for so treating them. 

In the discussion upon the definition of "force," the object 
is, to make plain to the student the modern objections to the 
usual ways of treating the subject, so that he may have a 
clear conception of the modern aspect of the question, rather 
than to support the author's definition, as he is fully aware 
that this, as well as all others that have been given, is open 
to objection. 

In connection with the treatment of statical couples, it 
was thought best to present to the student the actual effect 
of the action of forces on a rigid body, and not to delay this 
subject until dynamics of rigid bodies is treated, as is usually 
done. 

In the common theory of beams, the author has tried to 
make plain the assumptions on which it is based. A little 
more prominence than usual has also been given to the longi- 
tudinal shearing of beams. 

In that part of the book that relates to the experimental 
results on strength and elasticity, the writer has endeavored 
to give the most reliable results, and to emphasize the fact, 
that, to obtain constants suitable for use in practice, we 
must deduce them from tests on full-size pieces. This prin- 
ciple of being careful not to apply experimental results to 
cases very different from those experimented upon, has long 
been recognized in physics, and therefore needs no justifica- 
tion. 

The government reports of tests made at the Watertown 
Arsenal have been extensively quoted from, as it is believed 



PREFACE. 



that 'they furnish some of our most reliable information on 
these subjects. 

The treatment of the strength of timber will be found to 
be quite different from what is usually given ; but it speaks 
for itself, and will not be commented upon here. 

In the chapter on the " Theory of Elasticity," a combina- 
tion is made of the methods of Rankine and of Grashof. 

In preparing the work, the author has naturally consulted 
the greater part of the usual literature on these subjects ; and, 
whenever he has drawn from other books, he has endeavored to 
acknowledge it. He wishes here to acknowledge the assist- 
ance furnished him by Professor C. H. Peabody of the Massa- 
chusetts Institute of Technology, who has read all the proofs, 
and has aided him materially in other ways in getting out the 
work. 

GAETANO LANZA. 

MASSACHUSETTS INSTITUTE OF TECHNOLOGY, 
April, 1885. 



PREFACE TO THE FOURTH EDITION. 

THE principal differences between this and the earlier 
editions consist in the introduction of the results of a large 
amount of the experimental work that has been done during 
the last five years upon the strength of materials. 

The other changes that have been made in the book are not 
a great many, and have been suggested as desirable by the 
author's experience in teaching. 

September, 1890. 



PREFACE TO THE SEVENTH EDITION. 



THE principal improvements in this edition consist in the 
introduction, in Chapter VII, of the results of a considerable 
amount of the experimental work on the strength of materials 
that has been done during the last six years. A few changes 
have also been made in other parts of the book. 

October, 1896. 



PREFACE 'TO THE EIGHTH EDITION. 



IN this edition a considerable number of additional results 
of recent tests, especially upon full-size pieces, have been 
introduced, some of the older ones having been omitted to 
make room for them. 

September, 1900. 



PREFACE TO THE NINTH EDITION. 



THE principal improvements in the Ninth Edition consist 
in very extensive changes in Chapter VII, in order to bring 
the account of the experimental work that has been performed 
in various places up to date. 

Some changes have also been made in the mathematical 
portion of the book, especially in the Theory of Columns. 



TABLE OF CONTENTS. 



CHAPTER I. 

COMPOSITION AND RESOLUTION OF FORCES . , 



CHAPTER II. 
DYNAMICS . . 75 

CHAPTER III. 
ROOF-TRUSSES **,* 138 

CHAPTER IV. 
BBIDGE-TRUSSES 184 

CHAPTER V. 
CENTRE OF GRAVITY 221 

CHAPTER VI. 
STRENGTH OF MATERIALS 240 

CHAPTER VII. 
STRENGTH OF MATERIALS AS DETERMINED BY EXPERIMENT ..... 350 



viii TABLE OF CONTENTS. 

CHAPTER VIII. 
CONTINUOUS GIRDERS 743 

r 
CHAPTER IX. 

EQUILIBRIUM CURVES. ARCHES AND DOMES . . 779 

CHAPTER X. 
THEORY OF ELASTICITY, AND APPLICATIONS .... < 852 




APPLIED MECHANICS, 



CHAPTER I. 
COMPOSITION AND RESOLUTION OF FORCES. 

i. Fundamental Conceptions. The fundamental con- 
ceptions of Mechanics are Force, Matter, Space, Time, and 
Motion. 

2. Relativity of Motion. The limitations of our natures 
are such that all our quantitative conceptions are relative. 
The truth of this statement may be illustrated, in the case of 
motion, by the fact, that, if we assume the shore as fixed in 
position, a ship sailing on the ocean is in motion, and a ship 
moored in the dock is at rest ; whereas, if we assume the sun 
as our fixed point, both ships are really in motion, as both par- 
take of the motion of the earth. We have, moreover, no means 
of determining whether any given point is absolutely fixed in 
position, nor whether any given direction is an absolutely fixed 
direction. Our only way of determining direction is by means 
of two points assumed as fixed ; and the straight line joining 
them, we are accustomed to assume as fixed in direction. 
Thus, it is very customary to assume the straight line joining 
the sun with any fixed star as a line fixed in direction ; but if 
the whole visible universe were in motion, so as to change the 
absolute direction of this line, we should have no means of 
recognizing it. 



APPLIED MECHANICS. 



3. Rest and Motion. - In order to define rest and 
motion, we have the following ; viz., 

When a single point is spoken of as having motion or rest, 
some other point is always expressed or understood, which is 
for the time being considered as a fixed point, and some direc- 
tion is assumed as a fixed direction : and we then say that the 
first-named point is at rest relatively to the fixed point, when 
the straight line joining it with the fixed point changes neither 
in length, nor in direction; whereas it is said to be in motion 
relatively to the fixed point, when this straight line changes in 
length, in direction, or in both. 

If, on the other hand, we had considered the first-named 
point as our fixed point, the same conditions would determine 
whether the second was at rest, or in motion, relatively to the 
first. 

A body is said to be at rest relatively to a given point and 
to a given direction, when all its points are at rest relatively to 
this point and this direction. 

4. Velocity and Acceleration. When the motion of 
one point relatively to another, or of one body relatively to 
another, is such that it describes equal distances in equal times, 
however small be the parts into which the time is divided, 
the motion is said to be uniform and the velocity constant. 

The velocity, in this case, is the space passed over in a unit 
of time, and is to be found by dividing the space passed over in 
any given time by the time ; thus, if s represent the space 
passed over in time /, and v represent the velocity, we shall 
have 



When the motion is not uniform, if we divide the time into 
small parts, and then divide the space passed over in one of 
these intervals by the time, and then pass to the limit as these 
intervals of time become shorter, we shall obtain the velocity 



FORCE. 



Thus, if A.y represent the space passed over in the interval of 
time A^, then we shall have 

v = limit of as A/ diminishes, 
A/ 

or 

ds 



In this case the rate of change of velocity per unit of time 
is called the Acceleration, and if we denote it by/, we have 



5. Force. We shall next attempt to obtain a correct defi- 
nition of force, or at least of what is called force in mechanics. 

It may seem strange that it should be necessary to do this ; 
as it would appear that clear and correct definitions must have 
been necessary in order to make correct deductions, and there- 
fore that there ought to be no dispute whatever over the mean- 
ing of the word force. Nevertheless, it is a fact in mechanics, 
as well as in all those sciences which attempt to deal with the 
facts and laws of nature, that correct definitions are only gradu- 
ally developed, and that, starting with very imperfect and often 
erroneous views of natural laws and phenomena, it is only after 
these errors have been ascertained and corrected by a long 
range of observation and experiment, and an increased range of 
knowledge has been acquired, that exactness and perspicuity 
can be obtained in the definitions. 

Now, this is precisely what has happened in the case of 
force. 

In ancient times rest was supposed to be the natural state 
of bodies ; and it was assumed that, in order to make them 
move, force was necessary, and that even after they had been 
set in motion their own innate inertia or sluggishness would 
cause them to come to rest unless they were constantly urgea 



APPLIED MECHANICS. 



on by the application of some force, the bodies coming to rest 
whenever the force ceased acting. 

It was under the influence of these vague notions that such 
terms arose as Force of Inertia, Moment of Inertia, Vis Viva 
or Living Force, etc. 

A number of these terms are still used in mechanics; but 
in all such cases they have been re -defined, such new mean- 
ings, having been attached to them as will bring them into 
accord with the more advanced ideas of the present time. 
Such definitions will be given in the course of this work, as 
the necessity may arise for the use of the terms. 

NEWTON'S FIRST LAW OF MOTION. 

Ideas becoming more precise, in course of time there was 
framed Newton's first law of motion ; and this law is as fol- 
lows : 

A body at rest will remain at rest, and a body in motion will 
continue to move uniformly and in a straight line, unless and 
until some external force acts upon it. 

The assumed truth of this law was based upon the observed 
facts of nature ; viz., 

When bodies were seen to be at rest, and from rest passed 
into a state of motion, it was always possible to assign some 
cause ; i.e., they had been brought into some new relationship, 
either with the earth, or with some other body: and to this 
cause could be assigned the change of state from rest to motion. 
On the other hand, in the case of bodies in motion, it wa<> seen, 
that, if a body altered its motion from a uniform rectilinear 
motion, there was always some such cause that could be 
.assigned. Thus, in the case of a ball thrown from the hand, 
the attraction of the earth and the resistance of the ait soon 
caused it to come to rest. In the case of a ball rolled along 
the ground, friction (i.e., the continual contact and collision with 
the ground) gradually destroyed its motion, and brought it to 



FORCE. 



rest ; whereas, when such resistances were diminished by rolling 
it on glass or on the ice, the motion always continued longer : 
hence it was inferred, that, were these resistances entirely 
removed, the motion would continue forever. 

In accordance with these views, the definition of force 
usually given was substantially as follows : 

Force is that which causes, or tends to cause, a body to change 
its state from rest to motion, from motion to rest, or to change its 
motion as to direction or speed. 

Under these views, uniform rectilinear motion was recog- 
nized as being just as much a condition of equilibrium, or of 
the action of no force or of balanced forces, as rest ; and the 
recognition of this one fact upset many false notions, destroyed 
many incorrect conclusions, and first rendered possible a science 
of mechanics. Along with the above-stated definition of force 
is ordinarily given the following proposition ; viz., 

Forces are proportional to the velocities that they impart, in a 
unit of time (i.e. to the accelerations that they impart), to the 
same body. The reasoning given is as follows : 

Suppose a body to be moving uniformly and in a straight 
line, and suppose a force to act upon it for a certain length of 
time t in the direction of the body's motion : the effect of the 
force is to alter the velocity of the body ; and it is only by this 
alteration of velocity that we recognize the action of the force. 
Hence, as long as the alteration continues at the same rate, we 
recognize the same force as acting. 

If, therefore,/ represent the amount of velocity which the 
force would impart in one unit of time, the total increase in 
the velocity of the body will be //; and, if the force now stop 
acting, the body will again move uniformly and in the same 
direction, but with a velocity greater by//. 

Hence, if we are to measure forces by their effects, it will 
follow that 

The velocity which a force will impart to a given (or standard) 



APPLIED MECHANICS. 



body in a unit of time is a proper measure of the force. And 
we shall have, that two forces, each of which will impart the 
same velocity to the same body in a unit of time, are equal to 
each other ; and a force which will impart to a given body twice 
the velocity per unit of time that another force will impart to 
the same body, is itself twice as great, or, in other words, 

Forces are proportional to the velocities that they impart, in a 
unit of time (i.e. to the accelerations that they impart), to the 
same body. 

MODERN CRITICISM OF THE ABOVE. 

The scientists and the metaphysicians of the present time 
are recognizing two other facts not hitherto recognized, and the 
result is a criticism adverse to the above-stated definition of 
force. Other definitions have, in consequence, been proposed ; 
but none are free from objection on logical grounds, and at the 
same time capable of use in mechanics in a quantitative way. 

The two facts referred to are the following ; viz., 

i. That all our ideas of space, time, rest, motion, and even 
of direction, are relative. 

2. That, because two effects are identical, it does not follow 
that the causes producing those effects are identical. 

Hence, in the light of these two facts, it is plain, that, inas- 
much as we can only recognize motion as relative, we can only 
recognize force as acting when at least two bodies are con- 
cerned in the transaction ; and also that if the forces are simply 
the causes of the motion in the ordinary popular sense of the 
word cause, we cannot assume, that, when the effects are equal, 
the causes are in every way identical, although we have, of 
course, a perfect right to say that they are identical so far as 
the production of motion is concerned. 

I shall now proceed, in the light of the above, to deduce a 
definition of force, which, although not free from objection, 
seems as free as any that has been framed. 

It is one of the facts of nature, that, when bodies are by any 



FORCE. 



means brought under certain relations to each other, certain 
tendencies are developed, which, if not interfered with, will 
exhibit themselves in the occurrence of certain definite phe- 
nomena. What these phenomena are, depends upon the nature 
of the bodies concerned, and on the relationships into which 
they are brought. 

As an illustration, we know that if an apple is placed at a 
certain height above the surface of the earth, there is developed 
between the two bodies a tendency to approach each other ; 
and if there is no interference with this tendency, it exhibits 
itself in the fall of the apple. If, on the other hand, the apple 
were hung on the hook of a spring balance in the same posi- 
tion as before, the spring would stretch, and there would be 
developed a tendency of the spring to make the apple move 
upwards. This tendency to make the apple move upwards 
would be just equal to the tendency of the earth and apple to 
approach each other. This would be expressed by saying that 
the pull of the spring is just equal and opposite to the weight 
of the apple. 

As other illustrations of these tendencies developed in 
bodies when placed in certain relations to each other, we have 
the following cases : 

(a} When two bodies collide. 

(b} When two substances, coming together, form a chemical 
union, as sodium and water. 

(c) When the chemical union is entered into only by raising 
the temperature to some special point. 

Any of these tendencies that are developed by bringing 
about any of these special relationships between bodies might 
properly be called a force ; and the term might properly be, and 
is, used in the same sense in the mental and moral world, as 
well as in the physical. In mechanics, however, we have to 
deal only with the relative motion of bodies ; and hence we 
give the name force only to tendencies to change the relative 



8 APPLIED MECHANICS. 

motion of the bodies concerned ; and this, whether these ten- 
dencies are unresisted, and exhibit themselves in the actual 
occurrence of a change of motion, or whether they are resisted 
by equal and opposite tendencies, and exhibit themselves in 
the production of a tensile, compressive, or other stress in the 
bodies concerned, instead of motion. 

DEFINITION OF FORCE. 

Hence our definition of force, as far as mechanics has to 
deal with it or is capable of dealing with it, is as follows; 
viz., - 

Force is a tendency to change the relative motion of the two 
bodies between which that tendency exists. 

Indeed, when, as in the illustration given a short time ago, 
the apple is hung on the hook of a spring balance, there still 
exists a tendency of the apple and the earth to approach each 
other ; i.e., they are in the act of trying to approach each other ; 
and it is this tendency, or act of trying, that we call the force of 
gravitation. In the case cited, this tendency is balanced by 
an opposite tendency on the part of the spring ; but, were the 
spring not there, the force of gravitation would cause the apple 
to fall. 

Professor Rarikine calls force "an action between two bodies, 
either causing or tending to cause change in their relative rest 
or motion ;" and if the act of trying can be called an action, my 
definition is equivalent to his. 

For the benefit of any one who wishes to follow out the 
discussions that have lately taken place, I will enumerate the 
following articles that have been written on the subject : 

(a) " Recent Advances in Physical Science," by P. G. Tait, 
Lecture XIV. 

(b) Herbert Spencer, "First Principles of Philosophy* 
(certain portions of the book). 



MEASURE OF FORCE. 



() Discussion by Messrs. Spencer and Tait, " Nature," Jan. 
2, 9, 1 6, 1879. 

(d] Force and Energy, "Nature," Nov. 25, Dec. 2, 9, 16, 
1880. 

6. External Force. We thus see, that, in order that a 
force may be developed, there must be two bodies concerned 
in the transaction ; and we should speak of the force as that 
developed or existing between the two bodies. 

But we may confine our attention wholly to the motion or 
condition of one of these two bodies ; and we may refer its 
motion either to the other body as a fixed point, or to some 
body different from either; and then,' in speaking of the force, 
we should speak of it as the force acting on the body under 
consideration, and call it an external force. It is the tendency 
of the other body to change the motion of the body under con- 
sideration relatively to the point considered as fixed. 

7. Relativity of Force. In adopting the above-stated 
definition of force, we acknowledge our incapacity to deal with 
it as an absolute quantity ; for we have defined it as a tendency 
to change the relative motion of a pair of bodies. Hence it is 
only through relative motion that we recognize force; and hence 
force is relative, as well as motion. 

8. Newton's First Law of Motion. In the light of 
the above discussion, we might express Newton's first law of 
motion as follows : 

A body at rest, or in uniform rectilinear motion relatively to 
a given point assumed as fixed, will continue at rest, or in uni- 
form motion in the same direction, unless and until some external 
force acts either on the body in question, or on the fixed point, 
or on the body which furnishes us our fixed direction. This law 
is really superfluous, as it has all been embodied in the defini- 
tion. 

9. Measure of Force. We next need some means of 
comparing forces with each other in magnitude ; and, subse- 



10 APPLIED MECHANICS. 

quently, we need to select one force as our unit force, by means 
of which to estimate the magnitude of other forces. 

Let us suppose a body moving uniformly and in a straight 
line, relatively to some fixed point ; as long as this motion 
continues, we recognize no unbalanced force acting on it ; 
but, if the motion changes, there must be a tendency to change 
that motion, or, in other words, an unbalanced force is acting 
on the body from the instant when it begins to change its 
motion. 

Suppose a body to be moving uniformly, and a force to be 
applied to it, and to act for a length of time /, and to be so applied 
as not to change the direction of motion of the body, but to 
increase its velocity; the result will be, that the velocity will be 
increased by equal amounts in equal times, and if f represent 
the amount of velocity the force would impart in one unit of 
time, the total increase in velocity will 'be//. This results 
merely from the definition of a force ; for if the velocity pro- 
duced in one (a standard) body by a given force is twice as 
great as that produced by another given force, then is the ten- 
dency to produce velocity twice as great in the first case as in 
the second, or, in other words, the first force is twice as great 
as the second. Hence 

Forces are proportional to the velocities which they will impart 
to a given (or standard} body in a unit of time. 

We may thus, by using one standard body, determine a 
set of equal forces, and also the proportion between different 
forces. 

10. Measure of Mass. After having determined, as 
shown, a set of equal (unit) forces, if we apply two of them 
to different bodies, and let them act for the same length of time 
on each, and find that the resulting velocities are unequal, these 
bodies are said to have unequal masses ; whereas, if the result- 
ing velocities are equal, they are said to have equal masses. 

Hence we have the following definitions : 



RELATION BETWEEN FORCE AND MOMENTUM. II 

I . Equal forces are those which, by acting 1 for equal times 
on tJie same or standard body, impart to it equal velocities. 

2. Equal masses are those masses to which equal forces 
will impart equal velocities in equal times. 

11. Suppose two bodies of equal mass moving side by 
side with the same velocity, and uniformly, let us apply to 
one of them a force F in the direction of the body's motion : 
the effect of this force is to increase the velocity with which the 
body moves ; and if we wish, at the same time, to increase 
the velocity of the other, so that they will continue to move 
side by side, it will be necessary to apply an equal force to that 
also. 

We are thus employing a force 2F to impart to the two 
bodies the required increment of velocity. 

If we unite them into one, it still requires a force 2F to 
impart to the one body resulting from their union the re- 
quired increment of velocity : hence, if we double the mass 
to which we wish to impart a certain velocity, we must double 
the force, or, in other words, employ a force which would 
impart to the first mass alone a velocity double that required. 
Hence 

Forces are proportional to the masses to which they will impart 
the same velocity in the same time. 

12. Momentum. The product obtained by multiplying 
the number of units of mass in a body by its velocity is called 
the momentum of the body. 

13. Relation between Force and Momentum. The 
number of units of momentum imparted to a body in a unit of 
time by a given force, is evidently identical with the number 
of units of velocity that would be imparted by the same force, 
in the same time, to a unit mass. Hence 

Forces are proportional to the momenta (or velocities per unit 
of mass) which they will generate in a unit of time. 



12 APPLIED MECHANICS. 

Hence, if F represent a force which generates, in a unit of 
time, a velocity/" in a body whose mass is m, we shall have 



and, inasmuch as the choice of our units is still under our con- 
trol, we so choose them that 

F = mf; 

i.e., the force F contains as many units of force as mf contains 
units of momentum ; in other words, 

The momentum generated in a body in a unit of time by a. 
force acting in the direction of the body's motion, is taken as 
a measure of the force. 

14. Statical Measure of Force. When the forces are 
prevented from producing motion by being resisted by equal 
and opposite fofces, as is the case in that part of mechanics 
known as Statics, they must be measured by a direct comparison 
with other forces. An illustration of this has already been 
given in the case of an apple hung on the hook of a spring 
balance. In that case the pull of the spring is equal in magni- 
tude to the weight of the apple : indeed, it is very customary 
to adopt for forces what is known as the gravity measure, in 
which case we take as our unit the gravitation, or tendemy to 
fall, of a given piece of metal, at a given place on the surface 
of the earth ; in other, words, its weight at a given place. 

The gravity unit may thus be the kilogram, the pound, or 
the ounce, etc. 

It is evident, moreover, from our definition of force, and the 
subsequent discussion, that whatever we take as our unit of 
mass, the statical measure of a force is proportional to its 
dynamical measure ; i.e., the numbers representing the magni- 
tudes of any two forces, in pounds, are proportional to the 
momenta they will impart to any body in a unit of time. 

15. Gravity Measure of Mass. If we assume one 
pound as our unit of force, one foot as our unit of length, and 



NEWTON'S SECOND LAW OF MOTION 13 

one second as our unit of time, the ratio between the number 
of pounds in any given force and the momentum it will impart 
to a body on which it acts unresisted for a unit of time, will 
depend on our unit of mass ; and, as we are still at liberty to fix 
this as we please, it will be most convenient so to choose it 
that the above-stated ratio shall be unity, so that there shall be 
no difference in the measure of a force, whether it is measured 
statically or dynamically. Now, it is known that a body falling 
freely under the action of its own weight acquires, every second, 
a velocity of about thirty-two feet per second : this number is 
denoted by g t and varies for different distances from the centre 
of the earth, as does also the weight of the body. 

Now, if W represent the weight of the body in pounds, and 
m the number of units of mass in its mass, we must have, in 
order that the statical and dynamical measures may be equal, 

W = mg. 
Hence 

m.y, 

g 

i.e., the number of units of mass in a body is obtained by divid- 
ing the weight in pounds, by the value of g at the place where 
the weight is determined. 

The values of W and of g vary for different positions, but 
the value of m remains always the same for the same body. 

UNIT OF MASS. 

If m = I, then W g; or, in words, 

The weight in pounds of the unit of mass (when the gravity 
measure is used} is equal to the value of g in feet per second for 
the same place. 

16. Newton's Second Law of Motion. Newton's 
second law of motion is as follows : 



14 APPLIED MECHANICS. 

" Change of momentum is proportional to the impressed mov- 
ing f rce > an d occurs along the straight line in which the force is 
impressed" 

Newton states further in his " Principia :" 

" If any force generate any momentum, a double force 
will generate a double, a triple force will generate a triple, 
momentum, whether simultaneously and suddenly, or gradually 
and successively impressed. And if the body was moving 
before, this momentum, if in the same direction as the motion, 
is added; if opposite, is subtracted; or if in an oblique direc- 
tion, is annexed obliquely, and compounded with it, according 
to the direction and magnitude of the two." 

Part of this law has reference to the proportionality between 
the force and the momentum imparted to the body ; and this 
has been already embodied in our definition of force, and illus- 
trated in the discussion on the measure of forces. 

The other part is properly a law of motion, and may be 
expressed as follows : 

If a body have two or more velocities imparted to it simulta- 
neously, it will move so as to preserve them all. 

The proof of this law depends merely upon a proper con- 
ception of motion. To illustrate this law when two velocities 
are imparted simultaneously to a body, let us suppose a man 
walking on the deck of a moving ship : he then has two motions 
in relation to the shore, his own and that of the ship. 

Suppose him to walk in the direction of motion of the 
ship at the rate of 10 feet per second, while the ship moves at 
25 feet per second relatively to the shore : then his motion in 
relation to the shore will be 25 -|- 10 = 35 feet per second. 
If, on the other hand, he is walking in the opposite direction at 
the same rate, his motion relatively to the shore will be 25 
10 15 feet per second. 

Suppose a body situated at A (Fig. i) to have two motions 
imparted to it simultaneously, one of which would carry it to B 




POLYGON OF MOTIONS 15 

in one second, and the other to C in one second ; and that it is 
required to find where it will be at the end of one second, and 
what path it will have pursued. c 

Imagine the body to move in obedience 
to the first alone, during one second : it 
would thus arrive at B ; then suppose the 
second motion to be imparted to the body, 
instead of the first, it will arrive at the end of the next sec- 
ond at D, where BD is equal and parallel to AC. When 
the two motions are imparted simultaneously, instead of suc- 
cessively, the same point D will be reached in one second, 
instead of two; and by dividing AB and AC into the same 
(any) number of equal parts, we can prove that the body will 
always be situated at some point of the diagonal AD of the 
parallelogram, hence that it moves along AD. Hence follows 
the proposition known as the parallelogram of motions. 

PARALLELOGRAM OF MOTIONS. 

If there be simultaneously impressed on a body two velocities, 
which would separately be represented by the lines AB and AC, 
the actual velocity will be represented by the line AD. which is 
the diagonal of the parallelogram of which AB and AC are the 
adjacent sides. 

17. Polygon of Motions. In all the above cases, the 
point reached by the body at the end of a second when the 
two motions take place simultaneously is the same as that which 
would be reached at the end of two seconds if the motions took 
place successively ; and the path described is the straight line 
joining the initial position of the body, with its position at the 
end of one second when the motions are simultaneous. 

The same principle applies whatever be the number of 
velocities that may be imparted to a body simultaneously. 
Thus, if we suppose the several velocities imparted to be 
(Fig. 2) AB, AC, AD, AE, and AF, and it be required to 




1 6 APPLIED MECHANICS. 

determine the resultant velocity, we first let the body move 
with the velocity AB for one second ; at the 
end of that second it is found at B ; then let 
it move with the velocity AC only, and "at 
the end of another second it will be found 
at c ; then with AD only, and at the end of 
the third second it will be found at d; at the 
end of the fourth at e; at the end of the fifth 
at /. Hence the resultant velocity, when all 
are imparted simultaneously, is Af, or "the 

closing side of the polygon. 

This proposition is known as the polygon of motions. 

POLYGON OF MOTIONS. 

If there be simultaneously impressed on a body any number 
of velocities, the resulting velocity will be represented by the 
closing side of a polygon of which the lines representing tJie 
separate velocities form the other sides. 

1 8. Characteristics of a Force A force has three 

characteristics, which, when known, determine it ; viz., Point 
of Application, Direction, and Magnitude. These can be repre- 
sented by a straight line, whose length is made proportional to 
the magnitude of the force, whose direction is that of the 
motion which the force imparts, or tends to impart, and one end 
of which is the point of application of the force ; an arrow-head 
being usually employed to indicate the direction in which the 
force acts. 

19. Parallelogram of Forces. 

PROPOSITION. If two forces acting simultaneously at the 
same point be represented, in point of application, direction, 
and magnitude, by two adjacent sides of a parallelogram, their 
resultant will be represented by the diagonal of the parallelo- 
gram, drawn from the point of application of the two forces. 

PROOF. In the last part of 16 was proved the propo* 



PARALLELOGRAM OF FORCES. I/ 

sition known as the Parallelogram of Motions, for the state- 
ment of which the reader is referred to the close of that 
section. 

We have also seen that forces are proportional to the velo- 
cities which they impart, or tend to impart, in a unit of time, 
to the same body. 

Hence the lines representing the two impressed forces are 
coincident in direction with, and proportional to, the lines repre- 
senting the velocities they would impart in a unit of time to 
the same body ; and moreover, since the resultant velocity is 
represented by the diagonal of the parallelogram drawn with 
the component velocities as sides, the resultant force must coin- 
cide in direction with the resultant velocity, and the length of 
the line representing the resultant force will bear to the result- 
ant velocity the same ratio that one of the component forces 
bears to the corresponding velocity. Hence it follows, that the 
resultant force will be represented by the diagonal of the paral 
lelogram having for sides the two component forces. 

20. Parallelogram of Forces : Algebraic Solution. 

PROBLEM. Given two forces F and F, acting at the same 
point A (Fig. 3), and inclined to each other at an angle ; required 
the magnitude and direction of the resultant 
force. 

Let AC represent F, AB represent F t , 
and let angle BAG ; then will R = AD A 
represent in magnitude and direction the 
resultant force. Also let angle DAC a; then from the tri~ 
angle DAC we have 

AD 2 = AC 2 + CD 2 - 2AC. CDcosACD. 
But 

ACD = 180 - .'. cosACD = -cos* 

.'. R 2 = F 2 + F 2 + 2FF, cos (9 




+ F 2 -f 2FF, cos (9. 



i8 



APPLIED MECHANICS. 



This determines the magnitude of R. To determine its direc- 
tion, let angle CAD a. .'. angle BAD = a, and we 
shall have from the triangle DAC 



or 



and similarly 



CD : AD = sin CAD : smACD, 
F t : R = sin a : sin 

T? 

.*. sin a = -sin0, 
R 



sin(0-a) = sin0. 
R 



EXAMPLES. 



. Given F = 47-34, 



75.46, = 73 14' 21"; find R and a. 



2. Given ^ = 5.36, F l = 4.27, = 32 10' ; find R and a. 
3. Given F = 42.00, F t = 31.00, = 150 ; find R and a. 

4. Given F = 47.00, F t 75.00, 6 = 253 ; find R and a. 



21. Parallelogram of Forces when 6 = 90. When 
the two given forces are at right angles to each other, the for- 
mulae become very much simplified, since the parallelogram 
becomes a rectangle. 

From Fig. 4 we at once deduce 




R = V^F* + ^;, 

sin a = ^, 
R 

COS a = . 



1. Given ^ = 

2. Given ^ = 

3. Given ^ = 

4. Given /? = 



3.0, ^ = 

3.0, F t = 

5.0, F l = 

23.2, F t = 



5.0 ; find ^ and a. 

5.0 ; find i? and a. 

12.0 ; find ^ and a. 

21.3 ; find R and a. 



DECOMPOSITION OF FORCES IN ONE PLANE. 19 

22. Triangle of Forces. If three forces be represented* 
in magnitude and direction, by the three sides of a triangle taken 
in order, then, if these forces be simultaneously applied at one 
point, they will balance each other. 

Conversely, three forces which, when simultaneously applied 
at one point, balance each other, can be correctly represented in 
magnitude and direction by the three sides of a triangle taken in 
order. 

These propositions, which find a very extensive application, 
especially in the determination of the stresses in roof and 
bridge trusses, are proved as follows : 

If we have two forces, AC and AB (see Fig. 3), acting at the 
point A, their resultant is, as we have already seen, AD ; and 
hence a force equal in magnitude and opposite in direction to 
AD will balance the two forces AC and AB. Now, the sides of 
the triangle AC DA, if taken in order, represent in magnitude 
and direction the force AC, the force CD or AB, and a force 
equal and opposite to AD ; and these three forces, if applied at 
the same point, would balance each other. Hence follows the 
proposition. 

Moreover, we have 

AC : CD \ DA = sinAUC : sin CAD : smACZ>, 
or 

F-.F, \R = sin(0 -a) : sin a : sintf; 

or each force is, in this case, proportional to the sine of the 
angle between the other two. 

23. Decomposition of Forces in one Plane. It is 
often convenient to resolve a force into two components, in two 
given directions in a plane containing the force. Thus, suppose 
we have the force R = AD (Fig. 3), and we wish to resolve it 
into two components acting respectively in the directions AC 
and AB ; i.e., we wish to find two forces acting respectively in 
these directions, of which AD shall be the resultant : we 



20 



APPLIED MECHANICS. 



determine these components graphically by drawing a parallelo- 
gram, of which AD shall be the diagonal, and whose sides shall 
have the directions AC and AB respectively. The algebraic 
values of the magnitudes of the compo- 
nents can be determined by solving the 
triangle ADC. In the case when the 
directions of the components are at right 
angles to each other, let the force R 
(Fig. 5), applied at O, make an angle a 
with OX. We may, by drawing the rect- 
angle shown in the figure, decompose R 

into two components, F and F u along OX and O Y respectively ; 
and we shall readily obtain from the figure, 



F = R cos a, Fi = R sin a. 




FIG. 6. 



EXAMPLES. 

i. The force exerted by the steam upon the piston of a steam-engine 

at the moment when it is in the position shown in the figure is AB = 

1000 Ibs. The resistance of the 

guides upon the cross-head DE is 

vertical. Determine the force acting 

along the connecting-rod AC and 

the pressure on the guides ; also 

resolve the force acting along the connecting-rod into 
two components, one along, and the 
other at right angles to, the crank OC. 
2. A load of 500 Ibs. is placed at 
the apex C of the frame ACB ': find 

the stresses in AC and CB respectively. 

3. A load of 4000 Ibs. is hung at C, on the crane 

ABC: find the pressure in the boom BC, and the pull 

on the tie AC, where BC makes an angle of 60 with the horizontal, 

and AC an angle of 15. 





COMPOSITION OF FORCES IN ONE PLANE. 21 

4. A force whose magnitude is 7 is resolved into two forces whose 
magnitudes are 5 and 3 : find the angles they make with the given; 
force. 

24. Composition of any Number of Forces in One 
Plane, all applied at the Same Point. 

(a) GRAPHICAL SOLUTION. Let the forces be represented 
(Fig. 2) by AB, AC, AD, AE, and AF respectively. Draw Be 
|| and = AC, cd || and = AD, de || and = AE, and ef j| and = 
AF; then will Af represent the resultant of the five forces. 
This solution is to be deduced from 

17 in the same way as 19 is deduced 
from 1 6. c, 

(b) ALGEBRAIC SOLUTION. Let 
the given forces (Fig. 9), of which B, 
three are represented in the figure, be 

F, F t) F 2 , Fy F 4 , etc. ; and let the angles l 

made by these forces with the axis OX o 1 ^ < jj 

be a, a,, 02, a 3 , a 4 , etc., respectively. FlG -9- 

Resolve each of these forces into two components, in the 

directions OX and OY respectively. We shall obtain for the 

components along OX 

OA = Fcosa, OB = ^cosa,, OC F 2 cosa 2 , etc.; 
and for those along OY 

OA, = Fs'ma, OB, == ^sino,, OC, = J? 2 sma 2 , etc. 




These forces are equivalent to the following two ; viz., a 
force Fcos a -f F, cos a, + F 2 cos a 2 -f- F 3 cos a 3 + etc. along OX, 
and a force .Fsin a + F\ $ m a i + F* siri 2 + F z sin a 3 -f- etc. along 
OY. The first may be represented by ^Fcosa, and the second 
by ^Fsina, where 2, stands for algebraic sum. There remains 
only to find the resultant of these two, the magnitude of which 
is given by the equation 

R = V(2^cosa)2 -j- 



22 



APPLIED MECHANICS. 



and, if we denote by a^ the angle made by the resultant with 
OX, we shall have 



COS a r = 



sin OT 



R 



EXAMPLES. 





a 3 = 112 



Find the result- 
ant force and 
its direction. 



Solution. 



F. 


a. 


COS a. 


sin a. 


F COS a. 


F sin a. 


47 


21 


0-93358 


o.35 8 37 


43.87826 


16.84339 


73 


4 8 


0.66913 


o-743i5 


48.84649 


54-24995 


43 


82 


O.I39I7 


0.99027 


5-9843 1 


42.58161 


23 


112 


-0.37461 


0.92718 


-8.6l603 


21.32414 










90.09303 


134.99909 



*. 2^ cos a = 90.09303, 
-. R 



a = 134.99909, 



log ^F COS a = 1.954691 

= 2.210331 



+ (2F sin a) 2 = 162.2976. 



log COS Or = 9.744360 

Or = 56 I/- 

OBSERVATION. It would be perfectly correct to use the minus sign 
in extracting the square root, or to call R = 162.2976 ; but then we 
should have 



050,= 90.09303 



or 



162.2976 
i8o -f 56 



- 134.99909 ? 
162.2976 

= 23 6 - 7'; 



COMPOSITION OF FORCES APPLIED AT SAME POINT. 2$ 

a result which, if plotted, would give the same force as when we call 
R = 162.2976 and a* == 56 if. 

Hence, since it is immaterial whether we use the plus or the minus sign 
in extracting the square root provided the rest of the computation be 
consistent with it, we shall, for convenience, use always plus. 

* = 77> 



3. 




a,= 82, 
a 2 = 163, 

S= 275- 



a, = o, 

2 = 90. 



25. Polygon of Forces. If any number of forces be 
represented in magnitude and direction by the sides of a polygon 
taken in order, then, if these forces be simultaneously applied at 
one point, they will balance each other. 

Conversely, any number of forces which, when simultaneously 
applied at one point, balance each other, can be correctly repre- 
sented in magnitude and direction by the sides of a polygon taken 
in order. 

These propositions are to be deduced from 24 (a) in the 
same way as the triangle of forces is deduced from the parallelo- 
gram of forces. 

26. Composition of Forces all applied at the Same 
Point, and not confined to One Plane. This problem can 
be solved by the polygon of forces, since there is nothing in 
the demonstration of that proposition that limits us to a plane 
rather than to a gauche polygon. 

The following method, however, enables us to determine 
algebraic values for the magnitude of the resultant and for its 
direction. 



2 4 



APPLIED MECHANICS. 




FIG. 10. 



We first assume a system of three rectangular axes, OX, 

OY, and OZ (Fig. 10), whose origin 
is at the common point of the given 
forces. Now, let OE = F be one 
of the given forces. First resolve 
it into two forces, OC and OD, the 
first of which lies in the z axis, and 
the second perpendicular to OZ, 
x or, as it is usually called, in the z 
plane ; the plane perpendicular to 
OX being the x plane, and that 
perpendicular to OY the y plane. 
Then resolve OD into two com- 
ponents, OA along OX, and OB along OY. We thus obtain 
three forces, OA, OB, and OC respectively, which are equivalent 
to the single force OE. These three components are the edges 
of a rectangular parallelepiped, of which OE = Fis the diagonal. 
Let, now, 

angle EOX = a, EOY = (3, and EOZ = y ; 

and we have, from the right-angled triangles EOA, EOB, and 
EOC respectively, 

OA = Fcosa, OB = Fcosp, OC = Fcosy. 
Moreover, 

OA 2 + OB 2 = OD 2 and OD 2 + OC 2 = OE 2 
.'. OA 2 + OB 2 + OC 2 = OE 2 , 

and by substituting the values of OA, OB, and OC, given above, 
we obtain 

COS 2 a -j- COS 2 (3 + COS 2 y = I ; 

a purely geometrical relation existing between the three angles 
that any line makes with three rectangular co-ordinate axes. 

When two of the angles a, /3, and y are given, the third can 
be determined from the above equation. 



COMPOSITIOA r OF- FORCES APPLIED AT SAME POINT. 2$ 



Resolve, in the same way, each of the given forces into 
three components, along OX, OY, and OZ respectively, and we 
shall thus reduce our entire system 
of forces to the following three 
forces : 



i. A single force 2/7 cos a along OX. 
2. A single force 2/7 cos ft along OY. 
3. A single force 2/7 cosy along OZ. 

We next proceed to find a sin- 
gle resultant for these three forces. 
Let (Fig. ii) 

OA = 2/7 cos a 
OB = 2/7 cos ft 
OC = 




FIG. xx. 



Compounding OA and OB, we find OD to be their resultant ; 
and this, compounded with OC, gives OE as the resultant of 
the entire system. Moreover, 

OE 2 = OD 2 -4- OC 2 = OA 2 + OB 2 + OC 2 , 
or 

fc = (2/7 cos a)* 4- (S^cos^) 2 -h (2/? cosy)* 



( 2.F cos 0) 



and if we let BOX a r , EOY = ft, and 
have 



(2/7 cosy)*; 

= y r , we shall 



cos a " = 



OA 
OE 



R 



2/7 cos 8 2/7 co 

r = Y~^, and cosy r = ^ - 



This gives us the magnitude and direction of the resultant. 

The same observation applies to the sign of the radical for 
R as in the case of forces confined to one plane. 



26 



APPLIED MECHANICS. 



DETERMINATION OF THE THIRD ANGLE FOR ANY ONE FORCE. 

When two of the angles a, /3, and y are given, the cosine of 
the third may be determined from the equation, 

cos 2 a + cos 2 /? + cos 2 y = i ; 

but, as we may use either the plus or the minus sign in extract* 
ing the square root, we have no means of knowing which of 
the two supplementary angles whose cosine has been deduced 
is to be used. 

Thus, suppose a = 45, (3 = 60, then 



cosy = i -- i - J = f 
/. y = 60, or 1 20 ; 

but which of the two to use we have no means of deciding. 

This indetermination will be more clearly seen from the fol- 
lowing geometrical considerations : 

The angle a (Fig. 12), being given as 45, locates the line 

representing the force on a right 
circular cone, whose axis is OX, 
and whose semi-vertical angle is 
AOX-BOX = 4$. On the other 
hand, the statement that (3 = 60 
locates the force on another right 
circular cone, having O Y for axis, 
and a semi-vertical angle of 60; 
both cones, of course, having their 
vertices at O. Hence, when a and 
(3 are given, we know that the line 

representing the force is an element of both cones ; and this is 
all that is given. 

(a) Now, if the sum of the two given angles is less than 
90, the cones will not intersect, and the data are consequently 
inconsistent. 




DETERMINATION OF THE THIRD ANGLE. 2/ 

(b) If, on the other hand, one of the given angles being 
greater than 90, their difference is greater than 90, the cones 
will not intersect, and the data are again inconsistent. 

(c) If a + /? = 90, the cones are tangent to each other, 
and 7 = 90. 

(d) If a -f- J3 > 90, and a /? or /3 < 90, the cones 
intersect, and have two elements in common ; and we have no 
means of determining, without more data, which intersection 
is intended, this being the indetermination that arises in the 
algebraic solution. 



I. Given 



F = 



EXAMPLES. 



63 a = 53 

49 a = 8 7 
2 = 70 



ft = 42' 
7 = 72' 
7 = 45' 



Find the magnitude 
and direction of 
the resultant. 



Solution. 



p 


a. 


p. 


Y- 


COS a. 


cosp. 


COS Y . 


F COS a. 


/^COS/3. 


F COS y. 


63 

49 

2 


53 
87 


42 
700 


7 2 

45 


0.60182 
0.05234 
0.6l888 


0.74314 
0.94961 
0.34202 


0.29250 
0.30902 
0.70711 


37.91466 
2.56466 
1.23776 


46.81782 
46.53089 
0.68404 


18.42750 
15.14198 
I.4I422 












41.71708 

2/^cos a 


94-03275 
2/^cos 3 


34-98370 

2.F cos y 



R = V(S^cosa) 2 -j- (XF cos/3) 2 + (S/? cosy) 2 = 108.6569. 

log 2/^cosa = 1.620314 log S^cos/^ = 1.973279 log 2/^cosy = 1.543866 
log j? = 2.036057 log R = 2.036057 log R = 2.036057 



log cos a r =9-584257 Iogcosj8 r =9.937222 log cos y r =9.507809 
a r = 67 25' 20 X/ (3 r = 30 4' i4 /x =71 13' 5" 



28 



APPLIED MECHANICS. 





F. 


a. 


0- 




F. 


a. 





V- 


2. 


4-3 


47 2' 


65 7' 


3- 


5 


9 


90 






37.5 


88 3' 


10 5 ' 




7 











6.4 


68 4' 


8 3 2' 




4 



















75 


73 




45 



27. Conditions of Equilibrium for Forces applied at a 
Single Point. 

i. When the forces are not confined to one plane, we have 
already found, for the square of the resultant, 



But this expression can reduce to zero only when we have 
a = o, S/^cos (3 = o, and 2/^cos y = o ; 



for the three terms, being squares, are all positive quantities, 
and hence their sum can reduce to zero only when they are 
separately equal to zero. 

Hence : If a set of balanced forces applied at a single point 
be resolved into components along three directions at right angles 
to each other, the algebraic sum of the components of the forces 
along each of the three directions must be equal to zero, and con- 
versely. 

2. When the forces are all confined to one plane, let that 
plane be the z plane ; then y = 90 in each case, and 

/. (3 = 90 - a 

/. cos (3 = sin a 

/. fc = (^F cos a) 2 4- 



Hence, for equilibrium we must have 

cos a) 2 4- CSJ? sin a) 2 = o; 



STATICS OF RIGID BODIES. 29 

and, since this is the sum of two squares, 

o, and S/^sina = o. 



Hence : If a set of balanced forces, all situated in one plane \ 
and acting at one point, be resolved into components along two 
directions at right angles to each other, and in their own plane, 
the algebraic sum of the components along each of the tzvo given 
directions must be equal to zero respectively; and conversely. 

28. Statics of Rigid Bodies. A rigid body is one that 
does not undergo any alteration of shape when subjected to 
the action of external forces. Strictly speaking, no body is 
absolutely rigid ; but different bodies possess a greater or less 
degree of rigidity according to the material of which they are 
composed, and to other circumstances. When a force is ap- 
plied to a rigid body, we may have as the result, not merely a 
rectilinear motion in the direction' of the force, but, as will be 
shown later, this may be combined with a rotary motion ; in 
short, the criterion by which we determine the ensuing motion 
is, that the effect of the force will distribute itself through the 
body in such a way as not to interfere with its rigidity. 

What this mode of distribution is, we shall discuss here- 
after ; but we shall first proceed to some propositions which can 
be proved independently of this consideration. 

29. Principle of Rectilinear Transferrence of Force in 
Rigid Bodies. If a force be applied to a rigid body at the 
point A (Fig. 13) in the direction AB, 
whatever be the motion that this force 
would produce, it will be prevented from 
taking place if an equal and opposite 
force be applied at A, B, C, or D, or at FlG - I3 - 

any point along the line of action of the force : hence we have 
the principle that 

The point of application of a force acting on a rigid body, 
may be transferred to any other point which lies in the line of 




APPLIED MECHANICS. 



action of the force, and also in the body, without altering the 
resulting motion of the body, although it does alter its state of 
stress. 

30. Composition of two Forces in a Plane acting at 
Different Points of a Rigid Body, and not Parallel to Each 
Other. Suppose the force F (Fig. 14) to be applied at A, and 
F, at B t both in the plane of the paper, and acting on the rigid 
body abcdef. Produce the lines of direction of the forces till 
they meet at <9, and suppose both F and F, to act at O. Con- 
struct the parallelogram ODHE, where OD = F and OE = F t ; 

then will OH R rep^ 
resent the resultant 
force in magnitude and 
in direction. Its point 
of application may be 
conceived at any point 
along the line OH, as 



at C, or any other 
point ; and a force 
equal and opposite to 
OH, applied at any point of the line OH, will balance F at A, 
and F, at B. 

The above reasoning has assumed the points A, B, C and 
O, all within the body : but, since we have shown, that when 
this is the case, a force equal and opposite to R at C will bal- 
ance Fat A, and F t at B, it follows, that were these three forces 
applied, equilibrium would still subsist if we were to remove 
the part bafeghc of the rigid body ; or, in other words, 

The same construction holds even when the point O falls out- 
side the rigid body. 

31. Moment of a Force with Respect to an Axis Per- 
pendicular to the Force. 

DEFINITION. The moment of a force with respect to an 
axis perpendicular to the force, and not intersecting it, is the 




FIG. 14. 



EQUILIBRIUM OF THREE PARALLEL FORCES. 




FIG. 15. 



product 'of the force by the common perpendicular to (shortest 
distance between) the force and the axis. 

Thus, in Fig. 15 the moment of F about 
an axis through O and perpendicular to the 
plane of the paper is F(OA). The sign of 
the moment will depend on the sign attached 
to the force and that attached to the perpen- 
dicular. These will be assumed in this book 
in such a manner as to render the following true ; viz., 

The moment of a force with respect to an axis is called posi- 
tive when, if the axis were supposed fixed, the force would cause 
the body on which it acts to rotate around the axis in the direc- 
tion of the hands of a watch as 
seen by the observer looking at 
the face. It will be called nega- 
tive when the rotation would take 
place in the opposite direction. 

32. Equilibrium of Three 
Parallel Forces applied at 
Different Points of a Rigid 
Body. Let it be required to 
find a force (Fig. 16) that will 
balance the two forces F at A, 
and -F t at B. Apply at A and B 
respectively, and in the line AB, 
the equal and opposite forces Aa 
and Bb. Their introduction will 
produce no alteration in the 
body's motion. 

The resultant of F and Aa 
is Af, that of F, and Bb is Bg. 
Compound these by the method 
of 30, and we obtain as result- 
ant ce. A force equal in magnitude and opposite in direction 




FIG. 16. 



32 APPLIED MECHANICS. 

to cej applied at any point of the line cC, will be the force 
required to balance Fat A and F, at B ; and, as is evident from 
the construction, this line is in the plane of the two forces. 
Moreover, by drawing triangle fKl equal to Bbg, we can readily 
prove that triangles oce and Afl are equal : hence the angle oce 
equals the angle fAl, and R is parallel to /^and F t . Also 

R = ce = ch + he = ,4 AT + A7 = F + ^ 



__ _ 

AC fK Ad 
and 

CL~. M. =- -5.- 

BC~ Bb~ Bb' y 
.'. since ^4# = ^ 



BC F " BC AC AB 

where qr is any line passing through C. 

Hence we have the following propositions ; viz., 
If three parallel forces balance each other, 
1. They must lie in one plane. 

2. The middle one must be equal in magnitude and opposite 
in direction to the sum of the other 
two. 

3. Each force is proportional to the 

fcj IB o distance between the lines of direction 
of the other two as measured on any 
line intersecting all of them. 

The third of the above-stated con- 
ditions may be otherwise expressed, 
thus : 

FIG. 17. The algebraic sum of the moments 

of the three forces about any axis perpendicular to the forces 
must be zero. 



RESULTANT OF A PAIR OF PARALLEL FORCES. 33 

PROOF. Let F, F a and R (Fig. 17) be the forces ; and let 
the axis referred to pass through O. Draw OA perpendicular 
to the forces. Then we have 

F(OA) + Ft (OB) = F(OC + CA) + F t (OC - BC) 
= (F+F l )OC + F(AC) - 

But, from what we have already seen, 

F + F, = -R 

and 

JL^JH 

BC AC 

.-. F(AC) = ^(^C) 
.-. F(OA) + Ft(OB) = -R(OC) -f o 

F,(OB) + tf(0C) = o, 



or the algebraic sum of the moments of t\\Q forces about the 
axis through O is equal to zero. 

33. Resultant of a Pair of Parallel Forces. In the 
preceding case, the resultant of any two of the three forces 
F y F iy and R, in Fig. 16 or Fig. 17, is equal and opposite to the 
third force. Hence follow the two propositions : 

I. If two parallel forces act in the same direction, their 
resultant lies in the plane of the forces, is equal to their sum, 
acts in the same direction, and cuts the line joining their points 
of application, or any common perpendicular to the two forces, 
at a point which divides it internally into two segments in- 
versely as the forces. 

II. If two unequal parallel forces act in opposite directions, 
their resultant lies in the plane of the forces, is equal to their 
difference, acts in the direction of the larger force, and cuts the 
line joining their points of application, or any common perpen- 
dicular to them, at a point which (lying nearer the larger force) 



34 APPLIED MECHANICS. 

divides it externally into two segments which are inversely as 
the forces. 

Another mode of stating the above is as follows : 

i. The resultant of a pair of parallel forces lies in the plane 
of the forces. 

2. It is equal in magnitude to their algebraic sum, and coin- 
cides in direction with the larger force. 

3. The moment of the resultant about an axis perpendicu- 
lar to the plane of the forces is equal to the algebraic sum of 
the moments about the same axis. 

EXAMPLES. 

1. Find the length of each arm of a balance such that i ounce at 
the end of the long arm shall balance i pound at the end of the short 
arm, the length of beam being 2 feet, and the balance being so propor- 
tioned as to hang horizontally when unloaded. 

2. Given beam =28 inches, 3 ounces to balance 15. 

3. Given beam = 36 inches, 5 ounces to balance 25 ounces. 



MODE OF DETERMINING THE RESULTANT OF A PAIR OF PARALLEL 
FORCES REFERRED TO A SYSTEM OF THREE RECTANGULAR 
AXES. 

Let both forces (Fig. 18) be parallel to OZ ' ; then we have, 
from what has preceded, 

F = = F_F> = 
be ab ac a 

But from the figure 



or 



.'. Fx 2 Fxt = FjX F^ 2 




RESULTANT OF NUMBER OF PARALLEL FORCES. 35 



and similarly we may prove that 



or 



i. The resultant of two parallel forces is parallel to the 
forces and equal to their algebraic sum. 



R=F+F, 




FIG. 18. 

2. The moment of the resultant with respect to OX is 
equal to the algebraic sum of their moments with respect to 
OX ; and likewise when the moments are taken with respect 
to OY. 

34. Resultant of any Number of Parallel Forces. 
Let it be required to find the resultant of any number of paral- 
lel forces. 

In any such case, we might begin by compounding two of 
them, and then compounding the resultant of these two with a 
third, this new resultant with a fourth, and so on. Hence, for 
the magnitude of any one of these resultants, we simply add 
to the preceding resultant another one of the forces ; and for 
the moment about any axis perpendicular to the forces, we add 



APPLIED MECHANICS. 



to the moment of the preceding resultant the moment of the 
new force. 

Hence we have the following facts in regard to the resultant 
of the entire system : 

I . The resultant will be parallel to the forces and equal to 
their algebraic sum. 

2. The moment of the resultant about any axis perpendicular 
to the forces will be equal to the algebraic sum of the moments 
of the forces about the same axis. 

The above principles enable us to determine the resultant 
in all cases, except when the algebraic sum of the forces is 
equal to zero. This case will be considered later. 

35. Composition of any System of Parallel Forces 
Y when all are in One Plane. 
Refer the forces to a pair of rect- 
angular axes, OX, OY (Fig. 19), 
and assume OY parallel to the 
forces. 

The forces and the co-ordinates 
of their lines of direction being as 
indicated in the figure, if we denote 
by R the resultant, and by X Q the 
co-ordinate of its line of direction, 
we shall have, from the preceding, 

R = ^F; ( i ) 

and if moments be taken about an 
axis through O, and perpendicular 



F, F, 



FIG. 19. 



to the plane of the forces, we shall also have 

Rx = -S.FX. 
Hence 

R = ^F and x (t = 



(2) 



determine the resultant in magnitude and in line of action, 
.except when %F = o, which case will be considered later. 



EQUILIBRIUM OF ANY SET OF PARALLEL FORCES. $? 

36, Composition of any System of Parallel Forces not 
confined to One Plane. Refer the forces to a set of rect- 
angular axes so chosen that OZ is parallel to their direction. 
If we denote the forces by F iy F 2 , F y F 4 , etc., and the co-ordinates 
of their lines of direction by (* 7,), (x 2J jj> 2 ), etc., and if we 
denote their resultant by R, and the co-ordinates of its line of 
direction by (x m j^ ), we shall have, in accordance with what has 
been proved in 34, 

1. The magnitude of the resultant is equal to the algebraic 
mm of the forces , or 

R = 2F. 

2. The moment of the resultant about OY is equal to the 
mm of the moments of the forces about OY, or 



3. The moment of the resultant about OX is equal to the 
of the moments about OX, or 



Hence 



determine the resultant in all cases, except when 2<F = o. 

37. Conditions of Equilibrium of any Set of Parallel 
Ferces. If the axes be assumed as before, so that OZ is 
parallel to the forces, we must have 

^F = o, ^Fx = o, and ^Fy o. 



To prove this, compound all but one of the forces. Then equilib- 
rium will subsist only when the resultant thus obtained is equal 
and directly opposed to the remaining force ; i.e, it must be 
equal, and act along the same line and in the opposite direction. 
Hence, calling R a the resultant above referred to, and (x a , y a ) 
the co-ordinates of its line of direction, and calling F H the 



38 APPLIED MECHANICS. 

remaining force, and (x w y^ the co-ordinates of its line of direc- 
tion, we must have 

Ra = ~Fn, *a = *n, J^ = JK, 

. ' . R a + F n = O, R a X a + F n X n O, ^JFa + F n y n = O, 

.-. 2F = o, *ZFx =o, ^Fy = o. 




When the forces are all in one plane, the conditions become 
2F = o, ^Fx = o. 

38. Centre of a System of Parallel Forces. The 

resultant of the two parallel forces F and F t (Fig. 20), ap- 
plied at A and B respectively, is a force R = F -\- F lt whose 
line of action cuts the line AB at a point C, 
which divides it into two segments inversely as 
the forces. If the forces F and F, are turned 
through the same angle, and assume the posi- 
tions AO and BO l respectively, the line of 
action of the resultant will still pass through 
C, which is called the centre of the two parallel 
forces F and /v Inasmuch as a similar reasoning will apply 
in the case of any number of parallel forces, we may give the 
following definition : 

The centre of a system of parallel forces is the point through 
which the line of action of the resultant always passes, no matter 
how the forces are turned, provided only 

i. Their points of application remain the same. 
2. Their relative magnitudes are unchanged. 
3. They remain parallel to each other. 

Hence, in finding the centre of a set of parallel forces, we 
may suppose the forces turned through any angle whatever, and 
the centre of the set is the point through which the line of 
action of the resultant always passes. 



DISTRIBUTED FORCES. 



39 



39- Co-ordinates of the Centre of a Set of Parallel 
Forces. Let F l (Fig. 21) be one of the forces, and (x lt y u zj 
the co-ordinates of its point 
of application. Let F 2 be 
another, and (x 2t y 2t z 2 ) co- 
ordinates of its point of 
application. Turn all the 
forces around till they are 
parallel to OZ, and find the 
line of direction of the re- 
sultant force when they are 
in this position. The co- 
ordinates of this line are 

FIG. 21. 




and, since the centre of the system is a point on this line, the 
above are two of the co-ordinates of the centre. Then turn 
the forces parallel to OX, and determine the line of action of 
the resultant. We shall have for its co-ordinates 



y* = 



Hence, for the co-ordinates of the centre of the system, we 
have 



y = 



When 2F = o the co-ordinates would be oo, therefore such 
a system has no centre. 

40. Distributed Forces While we have thus far as- 
sumed our forces as acting at single points, no force really acts 
at a single point, but all are distributed over a certain surface 



40 APPLIED MECHANICS. 

or through a certain volume ; nevertheless, the propositions 
already proved are all applicable to the resultants of these 
distributed forces. We shall proceed to ' discuss distributed 
forces only when all the elements of the distributed force are 
parallel to each other. As a very important example of such a 
distributed force, we may mention the force of gravity which 
is distributed through the mass of the body on which it acts. 
Thus, the weight of a body is the resultant of the weights of 
the separate parts or particles of which it is composed. As 
another example we have the following : if a straight rod be 
subjected to a direct pull in the direction of its length, and if 
it be conceived to be divided into two parts by a plane cross- 
section, the stress acting at this section is distributed over the 
surface of the section. 

41. Intensity of a Distributed Force. Whenever we 
have a force uniformly distributed over a certain area, we obtain 
its intensity by dividing its total amount by the area over which 
it acts, thus obtaining the amount per unit of area. 

If the force be not uniformly distributed, or if the intensity 
vary at different points, we must adopt the following means 
for rinding its intensity. Assume a small area containing the 
point under consideration, and divide the total amount of force 
that acts on this small area by the area, thus obtaining the 
mean intensity over this small area : this will be an approxima- 
tion to the intensity at the given point ; and the intensity is the 
limit of the ratio obtained by making the division, as the area 
used becomes smaller and smaller. 

Thus, also, the intensity, at a given point, of a force which 
is distributed through a certain volume, is the limit of the 
ratio of the force acting on a small volume containing the 
given point, to the volume, as the latter becomes smaller and 
smaller. 

42. Resultant of a Distributed Force. i. Let the 
force be distributed over the straight line AB (Fig. 22), and 



RESULTANT OF A DISTRIBUTED FORCE. 




let its intensity at the point E where AE = x, be represented 
by EF ' = p <(*), a function of x ; 
then will the force acting on the por- 
tion Ee = A^r of the line be/A^r: and 
if we denote by R the magnitude of 
the resultant of the force acting on the 
entire line AB, and by x the distance 
of its point of application from A, we shall have 

R = 3/A.x approximately, 
or 

R = fpdx exactly ; 

and, by taking moments about an axis through A perpendicular 
to the plane of the force, we shall have 

XoR = ^x(pkx) approximately, 
or 

x R = fpxdx exactly ; 

whence we have the equations 



R = fpdx, 



Spdx 



Let the force be distributed over a plane area EFGff 

(Fig. 23), let this area be re- 
ferred to a pair of rectangular 
axes OX and OV, in its own 
plane, and let the intensity 
of the force per unit of area 
at the point P, whose co- 
B ordinates are x and y, be 
p = $(x, y) ; then will p&x&y 
be approximately the force act- 
ing on the small rectangular 
area A^rAj/. Then, if we rep- 
resent by R the magnitude of 
the resultant of the distributed force, and by x m y m the co-ordi- 




DC 
FIG. 23. 



42 APPLIED MECHANICS. 

nates of the point at which the line of action of the resultant 
cuts the plane of EFGH, we shall have 

R 2/A^cAy approximately, 
x R = 



or, as exact equations, we shall have 

R = fSpdxdy, 

ffpxdxdy = ffpydxdy 

~ ' ~ 



3. Let the force be distributed through a volume, let this 
volume be referred to a system of rectangular axes, OX, O Y, 
and OZ, let A V represent the elementary volume, whose co- 
ordinates are x, y, z t and let p = <j>(x, y y z] be the intensity of 
the force per unit of volume at the point (x, y, z) ; then, if we 
represent by R the magnitude of the resultant, and by x , y , z m 
the co-ordinates of the centre of the distributed force, we shall 
have, from the principles explained in 38 and 39, the approx- 
imate equations 

R = 



and these give, on passing to the limit, the exact equations 



R - MV - - SpydV - 

~ Jpd x ~' y ~' ~ 



43. Centre of' Gravity. The weight of a body, or system 
of bodies, is the resultant of the weight of the separate parts 
or particles into which it may be conceived to be divided ; and 
the centre of gravity of the body, or system of bodies, is the 
centre of the above-stated system of parallel forces, i.e., the 
point through which the resultant always passes, no matter how 
the forces are turned. The weight of any one particle is the 
force which gravity exerts on that particle : hence, if we repre- 



FORCE APPLIED TO CENTRE OF STRAIGHT ROD. 43 

sent the weight per unit of volume of a body, whether it be 
the same for all parts or not, by w, we shall have, as an 
approximation, 



and as exact equations, 



fwxdV fwydV fwzdV 

> (0 



where W denotes the entire weight of the body, and x ot y m z , 
the co-ordinates of its centre of gravity. 

If, on the other hand, we let M = entire mass of the body, 
dM mass of volume dV t and m = mass of unit of volume, 
we shall have 

W = Mg, w = mg, wdV mgdV = gdM. 
Hence the above equations reduce to 

fxdM fydM fzdM 



Equations (i) and (2) are both suitable for determining the 
centre of gravity; one of the sets being sometimes most con- 
venient, and sometimes the other. 

44. Centre of Gravity of Homogeneous Bodies __ If 
the body whose centre of gravity we are seeking is homogeneous, 
or of the same weight per unit of volume throughout, we shall 
have, that w ==. a constant in equations (i) ; and hence these 
reduce to 



45. Effect of a Single Force applied at the Centre of 
a Straight Rod of Uniform Section and Material. If a 
straight rod of uniform section and material have imparted to it 



44 APPLIED MECHANICS. 

a motion, such that the velocity imparted ima unit of time to 
each particle of the rod is the same, and if we represent this 
velocity by/, then if at each point of the rod, we lay off a line 
xy (Fig. 24) in the direction of the motion, 
and representing the velocity imparted to that 
point, the line bounding the other ends of 
the lines xy will be straight, and parallel to the 
rod. If we conceive the rod to be divided 
into any number of small equal parts, and 




denote the mass of one of these parts by <\M, then will 
contain as many units of momentum as there are units of force 
in the force required to impart to this particle the velocity 
f in a unit of time ; and hence f&M is the measure of this 
force. 

Hence the resultant of the forces which impart the velocity 
f to every particle of the rod will have for its measure 

fM, 

where M is the entire mass of the rod ; and its point of applica- 
tion will evidently be at the middle of the rod. 

It therefore follows that 

The effect of a single force applied at the middle of a straight 
rod of uniform section and material is to impart to the rod a 
motion of translation in the direction of the force, all points of , 
the rod acquiring equal velocities in equal times. 

46. Translation and Rotation combined. Suppose that 
we.have a straight rod AB (Fig. 25), and suppose that such a 
force or such forces are applied to it as will impart to the point 
A in a unit of time the velocity Aa, and to the point B the 
(different) velocity Bb in a unit of time, both being perpendicu- 
lar to the length of the rod. It is required to determine the 
motion of any other point of the rod and that of the entire 
rod. 



TRANSLATION AND ROTATION COMBINED. 



45 




FIG. 25. 



Lay off Aa and Bb (Fig. 25), and draw the line ab t and pro- 
duce it till it meets AB produced 
in O : then, when these velocities 
Aa and Bb are imparted to the 
points A and B, the rod is in the 
act of rotating around an axis 
through O perpendicular to the plane of the paper ; for when a 
body is rotating around an axis, the linear velocity of any point 
of the body is perpendicular to the line joining the point in 
question with the axis (i.e., the perpendicular dropped from the 
point in question upon the axis), and proportional to the dis- 
tance of the point from the axis. 

Hence : If the velocities of two of the points in the rod are 
given, and if these are perpendicular to the rod, the motion 
of the rod is fixed, and consists of a rotation about some axis 
at right angles to the rod. 

Another way of considering this motion is as follows : Sup- 
pose, as before, the velocities of the points A and B to be 

represented by Aa and Bb respec- 
tively, and hence the velocity of 
any other point, as x (Fig. 26), to 
be represented by xy, or the length 
of the line drawn perpendicular to 
FIG. 26. AB, and limited by AB and ab. 

Then, if we lay off Aa, Bb, = \(Aa + Bb) = Cc, and draw 
a,b,, and if we also lay off Aa 2 a,a, and Bb 2 = bjb, we shall 
have the following relations ; viz., 

Aa = Aa, Aa 2 , 
Bb = Bb, + Bb 2 , 
xy = X y, xy 2 , etc., 

or we may say that the actual motion imparted to the rod in a 
unit of time may be considered to consist of the following two 
parts : 




46 APPLIED MECHANICS. 

i. A velocity of translation represented by Aa J} the mean 
velocity of the rod ; all points moving with this velocity. 

2. A varying velocity, different for every different point, 
and such that its amount is proportional to its distance from 
Cy the centre of the rod, as graphically shown in the triangles 
Aa 2 CBb 2 . In other words, the rod has imparted to it two 
motions : 

i. A translation with the mean velocity of the rod. 

2. A rotation of the rod about its centre. 

47. Effect of a Force applied to a Straight Rod of 
Uniform Section and Material, not at its Centre. If the 
force be not at right angles to the rod, resolve it into two com- 
ponents, one acting along the rod, and the other at right angles 
to it. The first component evidently produces merely a trans- 
lation of the rod in the direction of its length : hence the second 
component is the only one whose effect we need to study. 

To do this we shall proceed to show, that, when such a rod 
has imparted to it the motion described in 46, the single re- 
A cd B sultant force which is required to impart 

this motion in a unit of time is a force 
acting at right angles to the rod, at a point 
different from its centre ; and we shall de- 

(/ 

FIG. 27. termine the relation between the force and 

the motion imparted, so that one may be deduced from the 
other. 

Let A be the origin (Fig. 27), and let 

Ac = x, cd = dx. 

AB I = length of the rod. 

ce =f= velocity imparted per unit of time at distance x 
from A. 

Aa = / Bb = f 2 . 

w weight per unit of length. 

m mass per unit of length = ^!. 

g 




EFFECT OF FORCE APPLIED TO A STRAIGHT ROD. 47 

W = entire weight of rod. 

M = entire mass of rod . 

g 
R = single resultant force acting for a unit of time to 

produce the motion. 
x distance from A to point of application of R. 

Then we shall have, 



Hence, from 42, 

AabB) = ^(/ +/,)/ = ^(/ + / 2 ). (i) 

2 2 

(2) 



. I /. + '/. , (-) 

" 3 /, +/, 

We thus have a force R, perpendicular to AB, whose mag- 
nitude is given by equation (i), and whose point of application 
is given by equation (3) ; the respective velocities imparted by 
the force being shown graphically in Fig. 27. 



EXAMPLES. 

i. Given Weight of rod = W = 100 Ibs., 
Length of rod = 3 feet, 

Assume g = 32 feet per second, 

Force applied = R = 5 Ibs., 
Point of application to be 2.5 feet from one end; 

determine the motion imparted to the rod by the action of the force for 
one second. 



48 APPLIED MECHANICS. 

Solution. 
Equation (i) gives us, 

5 = ( 

Equation (2) gives, 

< 2 '5)(5) - (^p) (3)C/ + /), or/ + 2/ 2 = 8 

.-. / 2 = 4.8, / = -1.6. 

Hence the rod at the end nearest the force acquires a velocity of 4.8 
feet per second, and at the other end a velocity of 1.6 feet per 
second. The mean velocity is, therefore, 1.6 feet per second; and we 
may consider the rod as having a motion of translation in the direc- 
tion of the force with a velocity of 1.6 feet per second, and a rotation 
about its centre with such a speed that the extreme end (i.e., a point 
| feet from the centre) moves at a velocity 4.8 1.6 = 3.2 feet per 

second. Hence angular velocity = ^ = 2.14 per second = 122. 6 

per second. , 

2. Given JF== 50 Ibs., /= 5 feet. It is desired to impart to it, 
in one second, a velocity of translation at right angles to its length, of 5 
feet per second, together with a rotation of 4 turns per second : find the 
force required, and its point of application. 

3. Assume in example 2 that the velocity of translation is in a 
direction inclined 45 to the length of the rod, instead of 90. Solve 
the problem. 

4. Given a force of 3 Ibs. acting for one-half a second at a distance 
of 4 feet from one end of the rod, and inclined at 30 to the rod : 
determine its motion. 

5. Given the same conditions as in example 4, and also a force 
of 4 Ibs., parallel and opposite in direction to the 3-lb. force, and acting 
also for one-half a second, and applied at 3 feet from the other end : 
determine the resulting motion. 



MOMENT OF THE FORCES CAUSING ROTATION. 49 

6. Given two equal and opposite parallel forces, each acting at right 
angles to the length of the rod, and each equal to 4 Ibs., one being 
applied at i foot from one end, and the other at the middle of the rod ; 
find the motion imparted to the rod through the joint action of these 
forces for one-third of a second. 

48. Moment of the Forces causing Rotation. Re- 
ferring again to Fig. 26, and considering the motion of the 
rod as a combination of translation and rotation, if we take 
moments about the centre C, and compare the total moment 
of the forces causing the rotation alone, whose accelerations 
are represented by the triangles aajbj), with the total moment 
of the actual forces acting, whose accelerations are represented 
by the trapezoid AabB, we shall find these moments equal to 
each other ; for, as far as the forces represented by the rectangle 
are concerned, every elementary force nt(xy^dx on one side of 
the centre C has its moment (Cx)\m(xy^dx\ equal and opposite 
to that of the elementary force at the same distance on the other 
side of C : hence the total moment of the forces represented 
graphically by the rectangle AaJb^B is zero, and hence 

The moment about C of those represented by the trapezoid 
equals the moment of those represented by the triangles. 

Hence, from the preceding, and from what has been pre- 
viously proved, we may draw the following conclusions : 

i. If a force be applied at the centre of the rod, it will 
impart the same velocity to each particle. 

2. If a force be applied at a point different from the centre, 
and act at right angles to its length, it will cause a translation 
of the rod, together with a rotation about the centre of the rod. 

3. In this latter case, the moment of the forces imparting 
the rotation alone is equal to the moment of the single resultant 
force about the centre of the rod, and the velocity of translation 
imparted in a unit of time is equal to the number of units of 
force in the force, divided by the entire mass of the rod. 



APPLIED MECHANICS. 



49. Effect of a Pair of Equal and Opposite Parallel 
Forces applied to a Straight Rod of Uniform Section and 
Material. Suppose the rod to be AB (Fig. 28), and let the 
two equal and opposite parallel forces be Dd and Ee, each equal 
to F, applied at D and E respectively. 
The mean velocity imparted in a unit 

F 



of time by either force will be ; and, 

from what we have already seen, the trap- 
ezoid AabB will furnish us the means of 
representing the actual velocity imparted 
to any point of the rod by the force Dd. 
The relative magnitudes of Aa and Bb, the 
accelerations at the ends, will depend, of 

course, on the position of D ; but we shall 

77 

always have Cc l(Aa -f- Bb) = , a 

M 

quantity depending only on the magnitude 
of the force. So, likewise, the trapezoid AaJb^B will represent 
the velocities imparted by the force Ee ; and while the relative 
magnitude of Aa l and Bb l will depend upon the position of E, 

we shall always have Cc l = \(Aa l + Bb,) = . Hence, since 

Cc = Cc,, the centre C of the rod has no motion imparted to it 
by the given pair of forces, hence the motion of the rod is one 
of rotation about its centre C. 

The resulting velocity of any point of the rod will be the 
difference between the velocities imparted by the two forces ; 
and if these be laid off to scale, we shall have the second 
figure. Hence 

A pair of equal and opposite parallel forces, applied to a 
straight rod of uniform section and material, produce a rota- 
tion of the rod about its centre. Also, 

Such a rotation about the centre of the rod cannot be pro- 




FIG. 28. 



EFFECT OF STATICAL COUPLE ON STRAIGHT ROD. 51 

duced by a single force, but requires a pair of equal and op- 
posite parallel forces. 

50. Statical Couple. A pair of equal and opposite 
parallel forces is called a statical couple. 

51. Effect of a Single Force applied at the Centre of 
Gravity of a Straight Rod of Non-Uniform Section and 
Material. In the case of a straight rod of non-uniform sec- 
tion and material, we may consider the rod as composed of a 
set of particles of unequal mass : and if we imagine each par- 
ticle to have imparted to it the same velocity in a unit of time,, 
then, using the same method of graphical representation as 
before (Fig. 24), the line ab, bounding the other ends of the 
lines representing velocities, will be parallel to AB ; but if we 
were to represent by the lines xy, not the velocities imparted, 
but the forces per unit of length, the line bounding the other 
ends of these forces would not, in this case, be parallel to AB. 
Moreover, since these forces are proportional to the masses, and 
hence to the weights of the several particles, their resultant 
would act at the centre of gravity of the rod. Hence 

A force applied at the centre of gravity of a straight rod will 
impart the same velocity to each point of the rod ; i.e., will im- 
part to it a motion of translation only. 

52. Effect of a Statical Couple on a Straight Rod of 
Non-Uniform Section and Material. Let such a rod have 
imparted to it only a motion of rotation about its centre of 
gravity, and let us adopt the same modes of graphical repre- 
sentation as before. 

Let the origin be taken at O (Fig. 29), 
the centre of gravity of the rod. 

Let Aa = /, = velocity imparted to A. 
Bb = / 2 = velocity imparted to B. 
OA = a, OB = b, OC = x. 
CD = f = velocity imparted to C. 
dM = elementary mass at C. 




52 APPLIED MECHANICS, 

Then, from similar triangles, we have 

/_4*-4r, 

a b 

and hence for the force acting on dM we have 
dF=(CE)dx = ^xdM. 



Hence the whole force acting on AO, and represented graph- 
ically by Aa^Oy is 

f (*x---a 

J - \ xdM, 
aj x = o 

and that acting on OB, and represented by B0b t , is 

/ f*x = f (*x = o 

-V 2 / xdM = J - I xdM. 
bjx = -b ajx = -b 

Hence for the resultant, or the algebraic sum, of the two, we 
have 



But from 43 we have for the co-ordinate x of the centre of 
gravity of the rod 



f, 



x = a 

xdM 



M 
but, since the origin is at the centre of gravity, we have 

X = O, 

and hence 

\xdM = o .-. R = o. 

Jx=-6 

Hence the two forces represented by Aa,O and Bb,O are equal 
in magnitude and opposite in direction : hence the rotation 
about the centre of gravity is produced by a Statical Couple. 



MEASURE OF THE ROTATORY EFFECT. 53 

Now, a train of reasoning similar to that adopted in the case 
of a rod of uniform section and material will show that a single 
force applied at some point which is not the centre of gravity 
of the rod will produce a motion which consists of two parts ;. 
viz., a motion of translation, where all points of the rod have 
equal velocities, and a motion of rotation around the centre of 
gravity of the rod. 

53. Moment of a Couple. The moment of a statical 
couple is the product of either force by the perpendicular dis- 
tance between the two forces, this perpendicular distance being 
called the arm of the couple. 

54. Measure of the Rotatory Effect. Before proceed- 
ing to examine the effect of a statical couple upon any rigid 
body whatever, we will seek a means of measuring its effect in 
the cases already considered. 

The measure adopted is the moment of the couple ; and, in 
order to show that it is proper to adopt this measure, it will be 
necessary to show 

That the moment of the couple is proportional to the angu- 
lar velocity imparted to the same rod in a unit of time ; and 
from this it will follow 

That two couples in the same plane with equal moments will 
balance each other if one is right-handed and the other left-handed 

If we assume the origin of co-ordinates at C (Fig. 30), the 
centre of gravity of the rod, and if we 
denote by a the angular velocity imparted 
in a unit of time by the forces F and F, 
and let CD * CE = X M then we have 
for the linear velocity of a particle situated 
at a distance x from C the value 

CLT. FIG. 30. 

The force which will impart this velocity in a unit of time to 

the mass dM is 

axdM. 



54 APPLIED MECHANICS. 

The total resultant force is 

afxtfM, 

which, as we have seen, is equal to zero. The moment of the 
elementary force about C is 



and the sum of the moments for the whole rod is 



and this, as is evident if we take moments about C, is equal t* 
Fxi - Fx 2 = F(x, - x,) = F(DE). 

Now, fx z dM is a constant for the same rod : hence any quan- 
tity proportional to F(DE) is also proportional to a. 

The above proves the proposition. 

Moreover, we have 

F(DE) = a 




whence it follows, that when the moment of the couple is given, 
and also the rod, we can find the angular velocity imparted in 
a unit of time by dividing the former by fx z dM. 

55. Effect of a Couple on a Straight Rod when the 

Forces are inclined to the Rod. We shall next show that 

the effect of such a couple is the same as that of a couple of 

equal moment whose forces are perpen- 

r% =^^.^ dicular to the rod. 

/} ^"-^^^ In this case let AD and BC be the 

forces (Fig. 31). The moment of this 
couple is the product of AD by the per- 
c * pendicular distance between AD and BC, 
the graphical representation of this being 
the area of the parallelogram ADBC. 



EFFECT OF A STATICAL COUPLE ON A RIGID BODY. 55 

Resolve the two forces into components along and at right 
angles to the rod. The former have no effect upon the motion 
of the rod : the latter are the only ones that have any effect 
upon its motion. The moment of the couple which they form 
is the product of Ad by AB, graphically represented by paral- 
lelogram AdBb ; and we can readily show that 

ADBC = AdBb. 

Hence follows the proposition. 

56. Effect of a Statical Couple on any Rigid Body. 

Refer the body (Fig. 32) to three rectan- 
gular axes, OX, OY, and OZ, assuming 
the origin at the centre of gravity of the 
body, and OZ as the axis about which 
the body is rotating. Let the mass of the 
particle P be AJ/, and its co-ordinates be 



Then will the force that would impart 




FIG. 32. 



to the mass AJf the angular velocity a in a unit of time be 



where r =. perpendicular from P on OZ, or 

r ^x 2 + y 2 . 

This force may be resolved into two, one parallel to O Y an<f 
the other to OX; the first component being ax&M, and the 
second a/y&M. 

Proceeding in the same way with each particle, and finding 
the resultant of each of these two sets of parallel forces, we 
shall obtain, finally, a single force parallel to OY and equal to 



and another parallel to OX, equal to 



56 APPLIED MECHANICS. 

But, since OZ passes through the centre of gravity of the body, 
we shall have 

= o and 2A M = o. 



Hence the resultant is in each case, not a single force, but a 
statical couple. Hence, to impart to a body a rotation about 
an axis passing through its centre of gravity requires the action 
of a statical couple ; and conversely, a statical couple so applied 
will cause such a rotation as that described. 

Further discussion of the motion of rigid bodies resulting 
from the action of statical couples is unnecessary for our pres- 
ent purpose, hence we shall pass to the deduction of the fol- 
lowing propositions, viz.: 

PROP. I. Two statical couples in the same plane balance 
each other when they have equal moments, and tend to pro- 
duce rotation in opposite directions. Let 
F 1 at a and F l at b represent one 
couple (left-handed in the figure), and 
let F t at d and F^ at e represent the 
other (right-handed in the figure), and let 

F\ab] = FJ&) ; 

then will these two couples balance each 
other. 

Proof. The resultant of F l at a and F^ 
FlG> 33> at d will be equal in amount, and directly 

opposed to the resultant of F l at b and F t at e and both 
will act along the diagonal fh of the parallelogram fchg. 
For we have (fg)(ab) = (fc)(de) 9 each being equal to the area 
of the parallelogram. 

. . -.&. 

fg " f< ^ ~f* ' 

hence follows the proposition. 

Hence follows that for a couple we may substitute another 
in the same plane, having the same moment, and tending to 
rotate the body in the same direction. 




COUPLES IN THE SAME OR PARALLEL PLANES. 57 




FIG. 33 (a). 



PROP. II. Two couples in parallel planes balance each 
other when their moments are equal, and the directions in 
which they tend to rotate the body are opposite. 

Let (Fig. 33 (a)) the planes of both couples, be perpendicular 
to OZ. Reduce them both so as to have their arms equal and 
transfer them, each in its own plane, 
till their arms are in the X plane. 
Let ab be the arm of one couple, 
and dc that of the other. Then will 
the two couples form an equilibrate 
system. For the resultant of the 
force at a and that at c acts at e, 
and is twice either one of its com- 
ponents, and hence is equal and di- 
rectly opposed to the resultant of the 
force at b and that at d. 

Hence we may generalize all our propositions in regard to 
the effect of statical couples and we may conclude that 

In order that two couples may have the same effect, it is 
necessary 

i. That they be in the same or parallel planes. 

2. That they have the same moment. 

3. That they tend to cause rotation in the same direction 
(i.e., both right-handed or both left-handed when looked at from 
the same side}. 

It also follows, that, for a given statical couple, we may sub- 
stitute another having the magnitudes of its forces different, 
provided only the moment of the couple remains the same. 

57. Composition of Couples in the Same or Parallel 
Planes. If the forces of the couples are not 
the same, reduce them to equivalent couples 
having the same force, transfer them to the 
same plane, and turn them so that their arms 
shall lie in the same straight line, as in Fig. 
34; the first couple consisting of the force F 
at A and F at B, and the second of F at B and F at C. 



L 



t 



FIG. 



APPLIED MECHANICS. 



The two equal and opposite forces counterbalance each other, 
and we have left a couple with force F and arm 

AC = AB + PC 
.*. Resultant moment = F. AC = F(AB) + F(BC). 

Hence : The moment of the couple which is the resultant of 
two or more couples in the same or parallel planes is equal t~ 
the algebraic sum of the moments of the component couples. 

EXAMPLES. 

i. Convert a couple whose force is 5 and arm 6 to an equivalent 
couple whose arm is 3. Find the resultant of this and another coupk 
in the same plane and sense whose force is 7 and arm 8 ; also find the 
force of the resultant couple when the arm is taken as 5. 



Solution. 



Moment of first couple = 5 x 6 = 30 

When arm is 3, force = ^- = 10 

Moment of second couple = 7 x 8 = 56 
Moment of resultant couple = 30 -f 56 = 86 
When arm is 5, force = - 8 ^ = 17 

1. Given the following couples in one plane : 



Force. 


Arm. 


12 


17 1 


3 


8 


5 


7 


6 


9 


12 


12 


10 


9 


14 


6 J 



Force. 

5 



Arm. 



Convert to equivalent 
couples having the < 
following : 



8 



20 



The first and the last three are right-handed ; the second, third, and 
fourth are left-handed. Find the moment of the resultant couple, and 
also its force when it has an arm n. 



COUPLES IN PLANES INCLINED TO EACH OTHER. 59 



58. Representation of a Couple by a Line. From the 
preceding we see that the effect of a couple remains the same 
as long as 

i . Its moment does not change. 

2. The direction of its axis (i.e., of the line drawn perpen- 
dicular to tJie plane of the couple} does not change. 

3. The direction in which it tends to make the body turn 
(right-handed or left-handed) remains the same. 

Hence a couple may be represented by drawing a line in 
the direction of its axis (perpendicular to its plane), and laying 
off on this line a distance containing as many units of length 
as there are units of moment in the couple, and indicating by a 
dot, an arrow-head, or some other means, in what direction one 
must look along the line in order that the rotation may appear 
right-handed. 

This line is called the Moment Axis of the couple. 

59. Composition of Couples situated in Planes inclined 
to Each Other. Suppose we have two couples situated 
neither in the same plane nor in parallel planes, and that we 
wish to find their resultant couple. We may proceed as fol- 
lows : Substitute for them equivalent couples with equal arms, 
then transfer them in their own plane respectively to such posi- 
tions that their arms shall 
coincide, and lie in the 
line of intersection of the 
two planes. 

This having been done, 
let OO, (Fig. 35) be the 
common arm, F and F 
the forces of one couple, 
F l and F t those of the 
other. The forces F and 
F, have for their resultant R, and F and F, have R,. 
Moreover, we may readily show that R and R, are equal and 




60 APPLIED MECHANICS. 

parallel, both being perpendicular to <9<9 2 . The resultant of 
the two couples is, therefore, a couple whose arm is OO^ and 
force R, the diagonal of the parallelogram on F and F lt so that 



R = \F 2 -h F* + 2FF l cos 0, 

where is the angle between the planes of the couples. Now, 
if we draw from O the line Oa perpendicular to OO I and to F, 
and hence perpendicular to the plane of the first couple, and if 
we draw in the same manner Ob perpendicular to the plane of 
the second couple, so that there shall be in Oa as many units 
of length as there are units of moment in the first couple, and 
in Ob as many units of length as there are units of moment in 
the second couple, we shall have 

i. The lines Oa and Ob are the moment axes of the two 
given couples respectively. 

2. The lines Oa and Ob lie in the same plane with F and 
F T , this plane being perpendicular to OO lt 

3. We have the proportion 

Oa-.0b = F. 00, -.F^OO^F: F,. 



4. If on Oa and Ob as sides we construct a parallelogram, 
it will be similar to the parallelogram on F and /v We shall 
have the proportion 

Oc : R = Oa : F = Ob\F*\ 

and since the sides of the two parallelograms are respectively 
perpendicular to each other, the diagonals are perpendicular to 
each other ; and since we have also 

Oc = R ' Oa and Oa = F. OO, .'. Oc = R . OO t , 
F 

it follows that Oc is perpendicular to the plane of the resultant 
couple, and contains as many units of length as there are units 
of moment in the moment of the resultant couple; in other 



COUPLE AND SINGLE FORCE IN THE SAME PLANE. 6l 

words, Oc will represent the moment axis of the resultant 
couple, and we shall have 



Oc = \Oa* -f Ob* + 2Oa . 
or, if we let 

Oa = Z, Ob = M, Oc = G, aOb = 0, 



G = VZ 2 + J/ 2 + 2ZJ/cos6>. 

This determines the moment of the resultant couple ; and, for 
the direction of its moment axis, we have 



and 



sin a Oc = sin 6 
G 



sin0. 



Hence we can compound and resolve couples just as we do 
forces, provided we represent the couples by their moment axes 

EXAMPLES. 

1. Given L = 43, M 15, 6 = 65; find resultant couple. 

2. Given Z = 40, M = 30, # = 30 ; find resultant couple. 

3. Given L i, M ' = 5, # = 45; find resultant couple. 

60. Resultant of a Couple and a Single Force in the 
Same Plane. Let M (Fig. 36 or 37) be the moment of the 





wF 



FIG. 36. FIG. 37. 

given couple, and let OF = F be the single force. For the 
given couple substitute an equivalent couple, one of whose 
forces is F at O, equal and directly opposed to the single 



62 



APPLIED MECHANICS, 



force F, these two counterbalancing each other, and leaving 
only the other force of the couple, which is equal and parallel 
to the original single force F, and acts along a line whose 



M 

distance from O is OA = . 

F 



Hence 



The resultant of a single force and a couple in the same plane 
is a force equal and parallel to the original force, having its 
line of direction at a perpendicular distance from the original 
force equal to the moment of the couple divided by the force. 

Fig. 36 shows the case when the couple is right-handed, and 
Fig. 37 when it is left-handed. 

61. Composition of Parallel Forces in General. In 
each case of composition of parallel forces ( 34, 35, and 36) 
it was stated that the method pursued was applicable to all 
cases except those where 

^F= o. 

We were obliged, at that time, to reserve this case, because we 
had not studied the action of a statical couple ; but now we will 
ad pt a method for the composition of parallel forces which will 
apply in all cases. 

(a) When all the forces are in one plane. Assume, as we did 
in 35, the axis OY to be parallel to 
the forces ; assume the forces and the 
co-ordinates of their lines of direction, 
as shown in the figure (Fig. 38). Now 
place at the origin O, along OY, two 
equal and opposite forces, each equal to 
x F, ; then these three forces, viz., F, at 
D, OA, and OB, produce the same effect 
as F, at D alone ; but F, at D and OR 
form a couple (left-handed in the figure) 
whose moment is F,*,. Hence the 
force Ft is equivalent to 



COMPOSITION OF PARALLEL FORCES. 



i. An equal and parallel force at the origin, and 

2. A statical couple whose moment is P\x^. 

Likewise the force F 2 is equivalent to (i) an equal and par- 
allel force at the origin, and (2) a couple whose moment is 
-F 2 x 2 , etc. 

Hence we shall have, if we proceed in the same way with 
all the forces, for resultant of the entire system a single force 

R = ^F along OY, 
and a single resultant couple 



(Observe that downward forces and left-handed couples are 
to be accounted negative.) 

Now, there may arise two cases. 
i. When ^F o, and 
2. When 2F><0. 

CASE I. When ^F = o, the resultant force along Y van- 
ishes, and the resultant of the entire system is 
a statical couple whose moment is 



CASE II. When %F > < o, we can reduce 
the resultant to a single force. 

Let (Fig. 39) OB represent the resultant 
force along OY, R = %F. With this, compound 
the couple whose moment is M 2<Fx, and 
we obtain as resultant ( 60) a single force 



FIG. 39. 



whose line of action is at a perpendicular distance from OY 
equal to 

AO = X r = 



6 4 



APPLIED MECHANICS. 



(b) When the forces are not confined to one plane. Assume, 
as before (Fig. 40), OZ parallel to the forces, and let F acting 

through A be one of the given 
forces, the co-ordinates of A be- 
ing x and y. Place at O two equal 
and opposite forces, each equal to 
F, and also at B two equal and 
opposite forces, each equal to F. 
These five forces produce the 
same effect as F alone at A, and 
they may be considered to con- 
sist of 
i. A single force F at the origin. 

2. A couple whose 'forces are F at B and F at O, and 
whose moment is Fx acting in the y plane. 

3. A couple whose forces are F at A and F at B, and 
whose moment is Fy acting in the x plane. Treating each of 
the forces in the same way, we shall have, in place of the entire 
system of parallel forces, the following forces and couples : 




FIG. 40. 



i. A single force R 
2. A couple My = 
3. A couple M x = 
Now, there may be 
two cases : 

i. When 2F >< O. 
2. When $F = o. 



along OZ. 
in the y plane. 
in the x plane. 



CASE I. When 
< o, we can reduce to a 
single resultant force 
having a fixed line of 
direction. Lay off (Fig. 
4 1 ) along OZ, OH $F. FIG. 4X . 

Combining this with the first of the above-stated couples, we 



R--SF 



COMPOSITION OF PARALLEL FOKCE.1. 



obtain a force R = 2<F at A, where OA = -' x r . Then 

2F 

combine with this resultant force R 2F at A, the second 
couple, and we shall have as single resultant of the entire 
system a single force 

R = 2F 

acting through B, where 



Hence the resultant is a force whose magnitude is 

R 2/7, 

the co-ordinates of its line of direction being 

CASE II. When S/ 7 o, there is no single resultant force ; 
but the system reduces to two couples, one in the x plane and 
one in the y plane, and these two can be reduced to one single 
resultant couple. (Observe that couples are to be accounted 
positive when, on being looked at by the observer from the posi- 
tive part of the axis towards 
the origin, they are t right- 
handed ; otherwise they are 
negative.) 

The moment axis of the 
couple in the x plane will 
be laid off on the axis OX 
from the origin towards the 
positive side if the moment 
is positive, or towards the 
negative side if it is nega- FIG. 42. 

tive, and likewise for the couple in the y plane. 




-~ x 



66 



APPLIED MECHANICS. 



Hence lay off (Fig. 42) OB M x , OA = M y) and by 
completing the rectangle we shall have OD as the moment 
axis of the resultant couple ; hence the resultant couple lies 
in a plane perpendicular to OD, and its moment bears to 
OD the same ratio as M x bears to OB. 

Hence we may write 

OD = M r = 



cosDOX = = cos0. 
M r 

If My had been negative, we should have OR as the moment 
axis of the resultant couple. 



EXAMPLES. 





p. 


X. 


y- 




P. 


X. 


>' 


I. 


5 


4 


3 


2. 


5 


-4 


3 




3 


2 


i 




2 


2 


i 




i 


3 


5 




-3 


3 


5 



Find the resultant in each example. 

62. Resultant of any System of Forces acting at Dif- 

ferent Points of a Rigid 
Body, all situated in One 
Plane. Let CF = F (Fig. 
43) be one of the given forces. 
Let all the forces be referred 
to a system of rectangular 
axes, as in the figure, and let 
a = angle made by F with 
FIG. 43. OX, etc. Let the co-ordi- 



M A 



nates of the point of application of F be AO = x, BO = 



SYSTEM OF FORCES ACTING ON RIGID BODY. 6? 

We first decompose CF = F into two components, parallel 
respectively to OX and O Y. These components are 

CD = Fcosa, CE = Fsina. 

Apply at O in the line O Y two equal and opposite forces, each 
equal to Fsin a, and at O in the line OX two equal and opposite 
forces, each equal to Fcosa. Since these four are mutually 
balanced, they do not alter the effect of the single force ; and 
hence we have, in place of Fat C, the six forces CD, OM, OK, 
CE, ON, OG. Of these six, CE and OG form a couple whose 
moment is 

(Fsma)x = Fxsina, 

CD and OK form a couple whose moment is 
(Fcosa)y = Fycosa. 

These two couples, being in the same plane, give as result- 
ant moment their algebraic sum, or 

F(y cos a x sin a) . 

We have, therefore, instead of the single force at C, the follow- 
ing: 

i. OM Fcos a along OX. 

2. ON = Fsin a along O Y. 

3. The couple M F(y cos a x sin a) in the given plane. 

Decompose in the same way each of the given forces ; and 
we have, on uniting the components along OX, those along OY, 
and the statical couples respectively, the following: 

i. A resultant force along OX, R x ^Fcos a. 

2. A resultant force along OY, R y ^Fsm a . 

3. A resultant couple in the plane, whose moment is 

M = %FLi> cos a x sin a}. 



68 



APPLIED MECHANICS. 



This entire system, on compounding the two forces at O t 
reduces to 



making with OX an angle a r , where 

^F cos a 



cos OT = 



R 



2. A resultant couple in the same plane, whose moment is 
M = *%F(y cos a x sin a) . 



Now compound this resultant force and couple, and we have, 
Y for final resultant, a single 

force equal and parallel to 
B E R, and acting along a line 
whose perpendicular dis- 
tance from O is equal to 

M 

R' 



G 

-\ 




,-- 


A* R 


C I 


^^ 




L 


E 


H 



FIG. 44 . 

Suppose (Fig. 44) the force 

OB = ^F cos a, 
614 = 
OR = 



The equation of this line 
may be found as follows : 



+ (S^ 1 sin a) 2 ; 



and let us suppose the resultant couple to be right-handed, and 
let 



then will the line ME parallel to OR be the line of direction 
of the single resultant force. 



CONDITIONS OF EQUILIBRIUM. 69 

Assuming the force R to act at any point C (x r , y r ) of this 
line, if we decompose it in the same way as we did the single 
forces previously, we obtain 

i. The force R cos a r = 2^ cos a along OX. 

2. The force R sina r = XFsina along OY. 

3. The couple R(y r cos a r x r sin a^). 

Hence we must have 

R(y r cos a r x r sin a r ) = ^F(y cos a x sin a) = J/~. 

Hence for the equation of the line of direction we have 

M 

y r cos a r x r sin a r = . ( i j 

R 

Another form for the same equation is 

cos a) _ Xr (2J?sma) = M. (2) 



63. Conditions of Equilibrium. If such a set of forces 
be in equilibrium, there must evidently be no tendency to h-an<r 
lation and none to rotation. Hence we must have 

R = o and M = o. 

Hence the conditions of equilibrium for any system of force? 
in a plane are three ; viz., 

2/7 cos a = o, 2/7 sin a = o, 2/7(j>cosa ^sina) = o. 

Another and a very convenient way to state the conditions of 
equilibrium for this case is as follows : 

If the forces be resolved into components along two direction? 
at right angles to each other, then the algebraic sum of the com- 
ponents along each of these directions must be zero, and th* 
algebraic sum of the moments of the forces about any axis 
pendicular to the plane of the forces must equal zero. 



APPLIED MECHANICS. 



EXAMPLES. 



i. Given 



2. Given 



p. 


X. 


y> 


5 


3 


2 


10 


i 


3 


-7 


4 


2 


P. 


X. 


* 


12 


27 


3 



-5 - 



54 



30 

45 



Find the resultant, and 
the equation of its 
line of direction. 

Find the resultant, and 
the equation of its 
line of direction. 



64. Resultant of any System of Forces not confined 

to One Plane Suppose we 

have a number of forces applied 
at different points of a rigid 
body, and acting in different 
directions, of which we wish to 
find the resultant. Refer them 
all to a system of three rect- 
x angular axes, OX, OY, OZ 
(Fig. 45). Let PR = F be 
one of the given forces. Re- 
solve it into three components, 
PK, PH, and PG, parallel 
Let 




FIG. 45- 

respectively to the three axes. 



RPK = a, RPH = 



RPG 



Let OA x, OB = y, OC z, be the co-ordinates of the 
point of application of the force F. Now introduce at B and 
also at O two forces, opposite in direction, and each equal to PK. 
We now have, instead of the force PK, the five forces PK, BM, 
BN, OS, and OT. The two forces PK and BN form a couple 
in the y plane, whose axis is a line parallel to the axis OY, and 
whose moment is (PK)(EB) (Fcos a )z = Fzcosa. The 



FORCES NOT CONFINED TO ONE PLANE. fl 

forces Mand OT form a couple in the z plane, whose moment 

is 

(BM)(OB} = -Jycosa. 

Now do the same for the other forces PH and PG, and we shall 
finally have, instead of the force PR, three forces, 

F cos a, F cos ft F cos y, 

acting at O in the directions OX, O V, and OZ respectively, 
together with six couples, two of which are in the x plane, two 
in the y plane, and two in the z plane. 

They thus form three couples, whose moments are as fol- 

lows : 

Around OX, F(y cos y z cos /?) ; 
Around OY, F(zcosa #cosy); 
Around OZ, F(x cos fty cos a) . 

Treat each of the given forces in the same way, and we shall 
have, in place of all the forces of the system, three forces, 

^F cos a along OX, 
^F cos J3 along OY, 
along OZ; 



and three couples, whose moments are as follows : 

Around OX, M x ^F(y cos y z cos ft) ; 
Around O Y, M y = ^F(z cos a x cos y) ; 
Around OZ, M z 2F(xcos(3 jycosa). 



The three forces give a resultant at O equal to 



R = V(cosa) 2 -f (XFcos/3) 2 4- -&F cosy) 2 , (i) 




a ( . 

cosa r = - - , cos ft- = - ~ S cosy r = - - *-. ( 2 ) 

. K 



APPLIED MECHANICS. 



For the three couples we have as resultant 



--* 



COS /a = 



M' 



COS v = 



M z 



(3) 

(4) 



A, p, and v being the angles made by the moment axis of the 
resultant couple with OX, O Y, and OZ respectively. 

Thus far we have reduced the whole system to a single result- 
ant force at the origin, and a couple. Sometimes we can reduce 

the system still farther, 
and sometimes not. The 
following investigation will 
show when we can do so. 
Let (Fig. 46) OP R be 
the resultant force, and 
OC =M the moment axis 
of the resultant couple. 
Denote the angle between 
them by 6 (a quantity thus 
far undetermined). Pro- 
ject OP = R on OC. Its 
projection will be OD = RcosO; then project, in its stead, the 
broken line OABP on OC. By the principles of projections, 
the projection of this broken line will equal OD. 

Now OA, AB, and BP are the co-ordinates of P, and make 
with OC the same angle as the axes OX, OY, and OZ ; i.e., 
A., //,, and v respectively : hence the length of the projection is 




FIG. 46. 



But 



Hence 



OA = 



OAcosX + 

AB = 



R COS = R COS Or COS A. 

COS0 = COS Or COS A. + COS fi r COS fJL 



BP = cosy r . 
R cos p r cos p, -f ^cosy r cosv 



+ cos y r cosv. (5) 



CONDITIONS OF EQUILIBRIUM. 73 

This enables us to find the angle between the resultant force 
and the moment axis of the resultant couple. 

The following cases may arise : 

i. When cos o, or 6 90, the force lies in the plane 
of the couple, and we can reduce to a single force acting at a 

distance from O equal to , and parallel to R at O. 

R 

2. When cos = I, or o, the moment axis of the 
couple coincides in direction with the force : hence the plane 
of the couple is perpendicular to the force, and no farther 
reduction is possible. 

3. When is neither o nor 90, we can resolve the couple 
M into two component couples, one of which, McosO, acts in a 
plane perpendicular to the direction of R, and the other, J/sin 0, 
acts in a plane containing R. The latter, on being combined 
with the force R at the origin, gives an equal and parallel force 
whose line of action is at a distance from that of R at O, equal 
to 

MsmO 
R 

4. When M = o, the resultant is a single force at O. 

5. When R o, the resultant is a couple. 

65. Conditions of Equilibrium. To produce equilibrium, 
we must have no tendency to translation and none to rotation. 
Hence we must have 

R = o and M = o. 
Hence we have, in general, six conditions of equilibrium ; viz., 



a = o, 2,J?cos/3 == o, ^F cos 7 == o. 
= o, My = o, M z = a 



74 APPLIED MECHANICS. 



EXAMPLES. 

1. Prove that, whenever three forces balance each other, they must 
lie in one plane. 

2. Show how to resolve a given force into two whose sum is given, 
the direction of one being also given. 

3. A straight rod of uniform section and material is suspended by two 
strings attached to its ends, the strings being of given length, and attached 
to the same fixed point : find the position of equilibrium of the rod. 

4. Two spheres are supported by strings attached to a given point, 
and rest against each other : find the tensions of the strings. 

5 . A straight rod of uniform section and material has its ends resting 
against two inclined planes at right angles to each other, the vertical 
plane which passes through the rod being at right angles to the line of 
intersection of the two planes : find the position of equilibrium of the 
rod, and the pressure on each plane, disregarding friction. 

6. A certain body weighs 8 Ibs. when placed in one pan of a false 
balance of equal arms, and 10 Ibs. in the other : find the true weight of 
the body. 

7. The points of attachment of the three legs of a three-legged table 
are the vertices of an isosceles right-angled triangle ; a weight of 100 Ibs. 
is supported at the middle of a line joining the vertex of one of the acute 
angles with the middle of the opposite side : find the pressure upon 
each leg. 

8. A heavy body rests upon an inclined plane without friction : find 
the horizontal force necessary to apply, to prevent it from falling. 

9. A rectangular picture is supported by a string passing over a 
smooth peg, the string being attached in the usual way at the sides, but 
one-fourth the distance from the top : find how many and what are the 
positions of equilibrium, assuming the absence of friction. 

16. Two equal and weightless rods are jointed together, and form a 
right angle ; they move freely about their common point : find the 
ratio of the weights that must be suspended from their extremities, that 
one of them may be inclined to the horizon at sixty degrees. 

ii. A weight of 100 Ibs. is suspended by two flexible strings, one 
of which is horizontal, and the other is inclined at an angle of thirty 
degrees to the vertical : find the tension in each string. 



D YNAMICS. DEFINITIONS. ?$ 



CHAPTER II. 

DYNAMICS. 

66. Definitions -- Dynamics is that part of mechanics 
which discusses the forces acting, when motion is the result. 

Velocity, in the case of uniform motion, is the space passed 
over by the moving body in a unit of time ; so that, if s repre- 
sent the space passed over in time t t and v represent the velocity, 
then 



Velocity, in variable motion, is the limit of the ratio of the 
space (AJ-) passed over in a short time (A/), to the time, as the 
latter approaches zero : hence 

r-* 

dt 

Acceleration is the limit of the ratio of the velocity ^A ; Im- 
parted to the moving body in a short time (A/), to the time, as 
the time approaches zero. Hence, if a represent the accelera- 
tion, 




* 



76 APPLIED MECHANICS. 

67. Uniform Motion In this case the acceleration is 

zero, and the velocity is constant ; and we have the equation 



s = vt. 



68. Uniformly Varying Motion. In this case the ac- 
celeration is constant : hence a is a constant in the equation 



and we obtain by one integration 

ds 
v = - = +,,. 

where c is an arbitrary constant : to determine it we observe, 
that, if v represent the value of v when / = o, we shall have 

v = o -f c 
.'. c = v 



and by another integration 

s = 

-.vJiera s is the space passed over in time // the arbitrary con- 
s f ant vanishing, because, when / = o, s is also zero. 

69. Measure of Force. It has already been seen, that, 
when a body is either at rest or moving uniformly in a straight 
line, there are either no forces acting upon it, or else the forces 
actr > upon it are balanced. If, on the other hand, the motion 
of -<>e body is rectilinear, but not uniform, the only unbalanced 
force acting is in the direction of the motion, and equal in mag- 
nitude to the momentum imparted in a unit of time in the direc- 
tion of the motion, or, in other words, to the limit of the ratio 
of the momentum imparted in a short time (A*), to the time, as 
the latter approaches zero. 



MECHANICAL WORK. UNIT OF WORK. // 

Thus, if F denote the force acting in the direction of the 
motion, m the mass, and a the acceleration, we shall have 

,., dv d 2 s (i) 

F = ma = m = m . v ' 

dt dt 2 

From (i) we derive 

mdv = Fdt; (2) 

and, if V Q be the velocity of the moving body at the time when 
/ = 4, and 2/ x its velocity when / = t lt we shall have 



Xvi r> 

mdv = I 
Jto 



Fdt 

JV Q J t Q 

or 

m(v, - v ) = J *Fdt; (3) 

or, in words, the momentum imparted to the body during the 
time / = (/, / ) by the force F, will be found by integrating 
the quantity Fdt between the limits / x and t Q . 

70. Mechanical Work. Whenever a force is applied to 
a moving body, the force is either used in overcoming resist- 
ances (i.e., opposing forces, such as gravity or friction), and 
leaving the body free to continue its original motion undis- 
turbed, or else it has its effect in altering the velocity of the 
body. In either case, the work done by the force is the prod- 
uct of the force, by the space passed through by the body *n 
the direction of the force. 

Unit qf Work. The unit of work is that work which is 
done when a unit of force acts through a unit of distance in 
the same direction as the force ; thus, if one pound and one 
foot are our units of force and length respectively, the unit of 
work will be one foot-pound. 

If a constant force act upon a moving body in the direction 
of its motion while the body moves through the space s, the 
work done by the force is 

Fs; 



APPLIED MECHANICS. 



and this, if the force is unresisted, is the energy, or capacity for 
performing work, which is imparted to the body upon which the 
force acts while it moves through the space s. 

Thus, if a lo-pound weight fall freely through a height of 
5 feet, the energy imparted to it by the force of gravity during 
this fall is 10 X 5 = 50 foot-pounds, and it would be necessary 
to do upon it 50 foot-pounds of work in order to destroy the 
velocity acquired by it during its fall. If, on the other hand, 
the force is a variable, the amount of work done in passing 
over any finite space in its own direction will be found by in- 
tegrating, between the proper limits, the expression 



The power which a machine exerts is the work which it 
performs in a unit of time. 

The unit of power commonly employed is the horse-power, 
which in English units is equal to 33000 foot-pounds per 
minute, or 550 foot-pounds per second. 

71. Energy. The energy of a body is its capacity for 
performing work. 

Kinetic or Actual Energy is the energy which a body pos- 
sesses in virtue of its velocity ; in other words, it is the work 
necessary to be done upon the body in order to destroy its 
velocity. This is equal to the work which would have to be 
done to bring the body from a state of rest to the velocity with 
which it is moving. Assume a body whose mass is m, and sup- 
pose that its velocity has been changed from V Q to v v Then if 
F be the force acting in the direction of the motion, we shall 
have, from equation (2), 69, that 

Fvdt = mvdv; (i) 

but 

vdt = ds 

/. Fds = mvdv. (2) 



ATWOOD'S MACHINE. 79 

Hence, by integration, 

I mvdv = / Fds 

*Jvo *J 

/. \m(v* - V 2 ) = fFds; (3) 

but fFds is the work that has been done on the body by the 
force, and the result of doing this work has been to increase 
its velocity from v to v t . It follows, that, in order to change 
the velocity from v to v u the amount of work necessary to per- 
form upon the body is 

*(*,* - *>o 2 ) = i (z> x * - *>o 2 ). (4) 

6 

If v = o, this expression becomes 

\mv*, or ^ (5) 

2g 

which is the expression for the kinetic energy of a body of mass 
m moving with a velocity v t . 

72. Atwood's Machine. A particular case of uniformly 
accelerated motion is to be found in Atwood's machine, in which 
a cord is passed over a pulley, and is loaded with unequal weights 
on the two sides. Were the weights equal, there would be no 
unbalanced force acting, and no motion would ensue ; but when 
they are unequal, we obtain as a result a uniformly accelerated 
motion (if we disregard the action of the pulley), because we 
have a constant force equal to the difference of the two weights 
acting on a mass whose weight is the sum of the two weights. 
Thus, if we have a lo-pound weight on one side and a 5-pound 
weight on the other, the unbalanced force acting is 

F = io- 5 = 5 Ibs. 




SO APPLIED MECHANICS. 



T O " i_ f 

The mass moved is M == - 3UL : hence the resulting ac- 


celeration is 



73. Normal and Tangential Components of the Forces 
acting on a Heavy Particle. If a body be in motion, either 
in a straight or in a curved line, and if at a certain instant all 
forces cease acting on it, the body will continue to move at a 
uniform rate in a straight line tangent to its path at that point 
where the body was situated when the forces ceased acting. 

If an unresisted force be applied in the direction of the 
body's motion, the motion will still take place in the same 
straight line; but the velocity will vary as long as the force 
acts, and, from what we have seen, the equation 

F=m* (i) 

dt 2 

will hold. 

If an unresisted force act in a direction inclined to the 
body's motion, it will cause the body to change its speed, and 
also its course, and hence to move in a curved line. Indeed, 
if a force acting on a body which is in motion be resolved into 
two components, one of which is tangent to its path and the 
other normal, the tangential component will cause the body to 
change its speed, and the normal component will cause it to 
change the direction of its motion. 

The measure of the tangential component is, as we have 
seen, 



and we will proceed to find an expression for the normal com- 
ponent otherwise known as the Deviating Force. For this 



CENTRIFUGAL FORCE. 8 1 

purpose we may substitute, for a small portion of the curve, a 
portion of the circle of curvature ; hence we will proceed to 
find an expression for the centrifugal force of a body which 
moves uniformly with a velocity v in a circle whose radius is r. 

CENTRIFUGAL FORCE. 

Let AC (Fig. 47) be the space described in the time A/. 

Then we have A B 

AC = 




The motion AC may be approximately consid- 
ered as the result of a uniform motion 

AB = z/A/ nearly, 
.and a uniformly accelerated motion PIG. 47 . 

BC = itf(A/) 2 = s, 
where a = acceleration due to centrifugal force. But 

(AB) 2 = BC . BD, 
or 

(vkty = %a(MY(2r + s) t 
where 

AO = OC = r 

/. v 2 = %a(2r + s) approximately 

2V 2 

.*. a = -- approximately. 

2r + s 

For its true value, pass to the limit where s = o. 

Hence we have, for the acceleration due to the centrifugal 
force, the expression 


r' 

Hence the centrifugal force is -equal to 



gr 



82 APPLIED MECHANICS. 



DEVIATING FORCE. 

If a body is moving in a curved path, whether circular or 
not, and the unbalanced force acting on it be resolved into tan- 
gential and normal components, the tangential component will 
be, as has already been seen, 



and the normal component will be 

mv 2 _ m/dsV 
r '- \dt)' 

where r is the radius of curvature of the path at the point in 
question. 

RESULTANT FORCE. 

Hence it follows that the entire unbalanced force acting on 
the body will be 



or 

F = m 



74. Components along Three Rectangular Axes of the 
Velocities of, and of the Forces acting on, a Moving 

Rociy. If we resolve the velocity into three components 

along OX, OY, and OZ, we shall have, for these components 
respectively, 

dx dy , dz 

- aDd ' 



this being evident from the fact that dx, dy, and dz are respec- 



COMPONENTS OF VELOCITIES AND FORCES. 83 

tively the projections of ds on the axes OX, OY, and OZ ; and, 
from the differential calculus, we have 



ds_ 
dt 

On the other hand, 



dx dy A dz 

*' it' and 7/ 



are not only the components of the velocity in the directions 

OX, OY, and OZ, but they are also the velocities of the body 
in these directions respectively. 

Now, the case of the accelerations is different ; for, while 



d 2 x d 2 y , d 2 z 
- 



are the accelerations in the directions OX, OY, and OZ respec- 
tively, they are not the components of the acceleration 



dt 2 
along the three axes. 

That they are the former is evident from the fact that , 

dt 

-f-, and are the velocities in the directions of the axes, and 
at at 

d 2 x d 2 v d 2 z 

, ~~, are their differential co-efficients, and hence repre- 

sent the accelerations along the three axes. But if we consider 
the components of the force acting on the body, we shall have 



84 APPLIED MECHANICS. 

for its components along OX t OY, and OZ, if a, ft, and y are 
the angles made by F with the axes respectively, 

Fcosa = m, F cos ft = m ^ Fcosy = m -. 
dt 2 dt 2 ^ 2 

.-. F 



and we found ( 73) for F, the value 



Hence, equating these values of F, and simplifying, we shall 
have the equation 



Hence it is plain that - , ^-, and - can only be the com- 
dt 2 df dt 2 

ponents of the actual acceleration 

when the last term f J vanishes, or when r = oo , i.e., when 



the motion is rectilinear. 

Moreover, we have the two expressions (i) and (2) for the 
force acting upon a moving body. 

The truth of the proposition just proved may also be seen 
from the following considerations : 

If a parallelopiped be constructed with the edges 

dx dy dz 



CENTRIFUGAL FORCE OF A SOLID BODY. 85 

the diagonal will be the actual velocity 

ds 
df 

and will, of course, coincide in direction with its path. 

On the other hand, if a parallelepiped be constructed with 

the edges 

d 2 x d 2 y d 2 z 
dt 2 ' dt 2 ' dt 2 ' 

its diagonal must coincide in direction with the force 



and can coincide in direction with the path, and hence with the 
actual acceleration 

d 2 s 

dt 2 ' 

only when the force is tangential to the path, and hence when 
the motion is rectilinear. 

75. Centrifugal Force of a Solid Body. When a solid 
body revolves in a circle, the resultant centrifugal force of the 
entire body acts in the direction of the perpendicular let fell 
from the centre of gravity of the body on the axis of rotation, 
and its magnitude is the same as if its entire weight were con- 
centrated at its centre of gravity. 

PROOF. Let (Fig, 48) the angular velocity = a, and the *eta' 
weight = W. Assume the axis of rotation perpendicular t 
the plane of the paper and passing through 
O ; assume, as axis of ;r, the perpendicular 
dropped from the centre of gravity upon 
the axis of rotation. The co-ordinates of 
the centre of gravity will then be (r , _^ ), 
and y will be equal to zero. 

FIG 8 

If, now, P be any particle of weight w, 
where r = perpendicular distance from P on axis of rotatsoo, 




86 APPLIED MECHANICS. 

and x OA, y = AP, we shall have for the centrifugal force 
of the particle at P 



w , 
-a. 2 r; 

g 



but if we resolve this into two components, parallel respectively 
to OX and OY, we shall have for these components 






and o.', = -wy, 

g sr g \g /r g ' 

and, for the resultant for the entire body we shall have, parallel 
to OX, 

(i) 



g 

and 

F y 2wy = Wy Q = o. (2) 

g g 

Hence the centrifugal force of the entire body is 

F-*-W*.; (3) 

ani if we let v = o,x = linear velocity of the centre of gravity, 
we have 

F- Wv * 

* ~~~ ) 

wnuh 13 the same as though the entire weight of the body 
;cic concentrated at its centre of gravity. 

EXAMPLES. 

H. A lo-pound weight is fastened by a rope 5 feet long to the 
centre, aroun 1 which it revolves at the rate of 200 turns per minute ; 
hrd the pull on the cord. 

2. A locomotive weighing 50000 Ibs., whose driving-wheels weigh 
toe Ibs., is running at 60 miles per hour, the diameter of the drivers 



UNIFORMLY VARYING RECTILINEAR MOTION. 8/ 

being 6 feet, and the distance from the centre of the wheel to the centre 
of gravity of the same being 2 inches (the drivers not being properly 
balanced) \ find the pressure of the locomotive on the track (a) when 
the centre of gravity is directly below the centre of the wheel, and (b) 
when it is directly above. 

3. Assume the same conditions, except- that the distance between 
centre of the wheel and its centre of gravity is 5 inches instead of 2. 

76. Uniformly Varying Rectilinear Motion. We have 

already found for this case ( 68) the equations 

- = a = a constant. 
(it* 



and we may write for the force acting, which is, of course, coin- 
cident in direction with the motion, 

F = m = ma = a constant. 

dr 

77. Motion of a Body acted on by the Force of Gravity 
only. A useful special case of uniformly varying motion is 
that of a body moving under the action of gravity only. 

The downward acceleration due to gravity is represented by 
g feet per second, the value of g varying at different points on 
the surface of the earth according to the following law : 

g = gi(i 0.00284 cos 2X)(i ^ feet per second, 

where 

g, = 32.1695 feet, 

A = latitude of the place, 

h = its elevation above mean sea-level in feet, 

R 20900000 feet. 



88 APPLIED MECHANICS. 

If, now, we represent by h the height fallen through by a 
descending body in time /, we shall have the equations, 

v. v + gt, 
h = v Q t + \gt*, 

where v is the initial downward velocity. 

If, on the other hand, we represent by v the initial upward 
velocity, and by h the height to which the body will rise in 
time / under the action of gravity only, we must write the equa- 
tions 



When v = o, the first set of equations gives 

v = gt y 
h = &/, 

which express the law of motion of a body starting from rest 
and subject to the action of gravity only. 

Eliminate / between these equations, and we shall have 



or 



h is called the height due to the velocity v, and represents the 
height through which a falling body must drop to acquire the 
velocity v ; and 

v = \2gh 



UNRESISTED PROJECTILE. 89 



is the velocity which a falling body will acquire in falling 
through the height h. Thus, if a body fall through a height of 
50 feet, it will, by that fall, acquire a velocity of about 



V 2 (3 2 i) (5) = V32i6.66 = 56.7 feet per second. 

Again : if a body has a velocity of 40 feet per second, we shall 
have 

v 2 1600 r , 

h = - - = 24.8 feet ; 
*g 64.3 

and we say that the body has a velocity due to the height 24.8 
feet, i. e., a velocity which it would acquire by falling through a 
height of 24.8 feet. 



EXAMPLES. 

1. A stone is dropped down a precipice, and is heard to strike the 
bottom in 4 seconds after it started : how high is the precipice ? 

2. How long will a stone, dropped down a precipice 500 feet high, 
take to reach the bottom ? 

3. What will be its velocity just before striking the ground? 

4. A body is thrown vertically upwards with a velocity of 100 feet 
per second ; to what height will it rise ? 

5. A body is thrown vertically upwards, and rises to a height of 50 
feet. With what velocity was it thrown, and how long was it in its 
ascent ? 

6. What will be its velocity in its ascent at a point 15 feet above 
the point from which it started, and what at the same point in its 
descent ? 

78. Unresisted Projectile. In the case of an unresisted 
projectile, we have a body on which is impressed a uniform 



APPLIED MECHANICS. 




motion in a certain direction (the direction of its initial motion), 
and which is acted on by the force of gravity only. 

Let OPC be 
the path (Fig. 49), 
OA the initial di- 
rection, and v the 
initial velocity, and 
the angle -4 CUT = 

K e. 

Then we shall 

FlG 49 have, for the hori- 

zontal and vertical 
components of the unbalanced force acting, when the projectile 
is at P (co-ordinates x and j), 

m = o along OX, and m = mg W along O Y. 
dP dt* 

Hence 

^ = ' ^ Tip = ~ g ' ^ 

Integrating, and observing, that, when t o, the horizontal 
and the vertical velocities were respectively z; cos and z> sin 0, 
we have 

dx n , , 

= V Q cos 0, (3) 

^ n t \ 

i- ***'-* W 

These equations could be derived directly by observing that 
the horizontal component of the initial velocity is V Q cos 0, and 
that this remains constant, as there is no unbalanced force act- 
ing in this direction, also that v sin 9 is the initial vertical 
velocity ; and, since the body is acted on by gravity only, this 
velocity will in time / be decreased by gt. 



UNRESISTED PROJECTILE. 91 

Integrating equations (3) and (4), and observing that for 
/ o, x and y are both zero, we obtain 

X = V Q COS O.t, (5) 

y = V Q sin O.t - \gt\ (6) 

Eliminate /, and we have 

^ = *tan0 -- $* - (7) 

2V * COS 2 

as the equation of the path, which is consequently a parabola. 

Equations (i), (2), (3), (4), (5), (6), and (7) enable us to solve 
any problem with reference to an unresisted projectile. 

Equation (7) may be written 



/ v 2 sin 2 0\ g / 

V ~~' ~~ ~ 2 Vo *ca*0 \ 



P sin0cos0 



which gives for the co-ordinates of the vertex 

_ v 2 sin 2 _ z/o 2 sin cos 

y\ ~~~ ) x\ - - 

2g g 



EXAMPLES. 

i. An unresisted projectile starts with a velocity of 100 feet pei* 
second at an upward angle of 30 to the horizon ; what will be its velocity 
when it has reached a point situated at a horizontal distance of icou teet 
from its starting-point, and how long will be required for it to 
that point? 

Solution. 

v = 100, = 30, v cos = 86.6, v sis as 50, 

g = 3 2 -i6. 
Equation (5) gives us 

1000 = 86.6 / 

.'. / = = 11.55 seconds. 

86,6 



9 2 



APPLIED MECHANICS. 



e> sin<9 - gt = 50 - 371.5 = -3 2I -5> 



v = V^(86.6) 2 + (3 2I -5) 2 = V75 + 103362 = 333. 

Hence the point in question will be reached in nj seconds after start- 
ing, and the velocity will then be 333 feet per second. 

2. An unresisted projectile is thrown upwards from the surface of 
the earth at angle of 39 to the horizontal : find the time when it will 
reach the earth, and the velocity it will have acquired when it reaches 
the earth, the velocity of throwing being 30 feet per second. 

3. A lo-pound weight is dropped from the window of a car when 
travelling over a bridge at a speed of 25 miles an hour. How long will 
it take to reach the ground 100 feet below the window, and what will be 
the kinetic energy when it reaches the ground ? 

4. With what horizontal velocity, and in what direction, must it be 
thrown, in order that it may strike the ground 50 feet forward of the 
point of starting? 

5. Suppose the same lo-pound weight to be thrown vertically up- 
wards from the car window with a velocity of 100 feet a minute, how 
long will it take to reach the ground, and at what point will it strike the 
ground ? 

79. Motion of a Body on an Inclined Plane without 

Friction. If a body move on 
an inclined plane along the line 
of steepest descent, subject to 
the action of gravity only, and 
if we resolve the force acting 
on it (i.e., its weight) into two 
components, along and perpen- 
dicular to the plane respec- 
tively, the latter component 
will be entirely balanced by 
the resistance of the plane, 

and the former will be the only unbalanced force acting on 

the body. 




MOTION OF A BODY ON AN INCLINED PLANE. 93 

Suppose a body whose weight is represented (Fig. 50) by 
HF = W to move along the inclined path AB under the action 
of gravity only. Let 9 be the inclination of AB to the horizon. 
Resolve W into two components, 




and HE = ^cos 9, 



respectively parallel and perpendicular to the plane. The 
former is the only unbalanced force acting on the body, and 
will cause it to move down the plane with a uniformly accel- 
erated motion ; the acceleration being 



(i) 



If the body is either at rest or moving downwards at the 
beginning, it will move downwards ; whereas, if it is first mov- 
ing upwards, it will gradually lose velocity, and move upwards 
more slowly, until ultimately its upward velocity will be de- 
stroyed, and it will begin moving downwards. 

The equations for uniformly varying motion are entirely 
applicable to these cases. Thus, suppose that the body has an 
initial downward velocity v ot this velocity will, at the end of the 
time /, become 

z> = ^ = z> + Crsintf)/ (2) 

at 

.-. s = v t -f k sin B . t*, (3) 

and, for the unbalanced force acting, we have 

F=ml = !(gsmO) = WsmO. (4) 

at 2 g 



94 APPLIED MECHANICS. 

If, on the other hand, the body's initial velocity is upward, 
and we denote this upward velocity by v of we shall have the 
equations 

v =|= v - (g*m$)t (5) 

s = vt - fesinfl ./ 2 (6) 

F= -WsinO. (7) 

Again, if the initial velocity is zero, equations (2) and (3) 
become 

(8) 



From these we obtain, for this case, 



2S 



do) 



and, substituting this value of / in (8), we have 



v = \2g(s sin 6), (n) 

or, if we let s sin 6 = h the vertical distance through which 
the body has fallen, we have 



v 2gh. (12) 

Hence, When a body, starting from rest, falls, under the 
action of gravity only, through a height h, the velocity acquired 
is \/2gh, whether the path be vertical or inclined. 



EXAMPLES. 



i. A body moves from the top to the bottom of a plane inclined 
to the horizon at 30, under the action of gravity only : find the time 
required for the descent, and the velocity at the foot of the plane. 



MOTION ALONG A CURVED LINE. 



95 




FIG. 51. 



2. In the right-angled triangle shown in the figure (Fig. 51), given 
AB = 10 feet, angle BAC = 30: find the time a A 
body would require, if acted on by gravity only, to fall 

from rest through each of the sides respectively, AB 
being vertical. 

3. Given inclination of plane to the horizon = 0, 
length of plane = /. compare the time of falling down 
the plane with the time of falling down the vertical. 

4. A loo-pound weight rests, without friction, on the 
plane of example 3. What horizontal force is required 
to keep it from sliding down the plane. 

5. Suppose 5 pounds horizontal force to be applied 

(a) so as to oppose the descent, () so as to aid the descent : find in 
each case how long it will take the weight to descend from the top to 
the bottom plane. 

80. Motion along a Curved Line under the Action of 
Gravity only. We shall consider two questions in this 
regard : (a) the velocity at any point of the curve (b) the time 
of descent through any part of the curve. 

(a) Velocity at any point. Let us suppose the body to have 

started from rest at A, and to have 
reached the point P in time /, 
where AB = x (Fig. 52). Then, 
since the curved line AP may be 
considered as the limit of a broken 
line running from A to P, and as 
it has already been seen that the 
velocity acquired by falling through 
c a certain height depends only upon 
the height, and not upon the incli- 
nation of the path, we shall have for a curved line also 




FIG. 52. 



where v is the velocity at P. 



APPLIED MECHANICS. 



(b) Time down a curve. Referring to the same figure, let / 
denote the time required to go from A to P, and &t the time to 
go from P to f, where PP' = AJ, and BB ! =. kx ; then, as we 
have seen that the velocity at P is \2gx, we shall have approx- 
imately for the space passed over in time A/, the equation 



or, passing to the limit, 



This equation gives 



tis 



ds 



or 



/ = c = r 

J^2gX J 



(2) 



v/here, of course, the proper limits of integration must be 
used. 

If / denote the time from A to P, we have 



= (""-*= 
J * ,,VV 



FK, 53 



EXAMPLE. 




A body acted on by gravity only is constrained to 
move in the arc of a circle from A to C (Fig. 53), radius 
10 feet. Find the time of describing the arc (quadrant) 
and the velocity acquired by the body when it reaches 



SIMPLE CIRCULAR PENDULUM. 



97 



8i. Simple Circular Pendulum. To find the time occu- 
pied in a vibration of a simple circu- ^c 
lar pendulum, we take D (Fig. 54) as 
origin, and DC as axis of x, and the 
axis of jj/at right angles to DC. Let 
AC /and BD = //, we shall have 
for the time of a single oscillation 
trom A to E 



/-, f 

J * = 





Now, from the equation of the circle AFDE, 

y 2 = 2/X X 2 , 



we have 



dy_ = I - x 
dx y 

ds I 



y s/2 ix - & 



Idx 



- x 2 )\_2g{h - *)] 



dx 



- x 2 V2/- 



or 




This can only be integrated approximately. 
Expanding f i J we obtain 



(-3T- 



7 + ~Ta 
4/ 32 / 2 



98 APPLIED MECHANICS. 

The greatest value of x is //; and if h is so small that we may 
omit , we shall have as our approximate result 

t = J-f / dx = \m vQ ~*^T\ k = nA o> 

* g J yhx x 2 V g( h ) o V ^ 

o 

If, however, the value of h as compared with / is too large 

to render it sufficiently accurate to omit , but so small that 

4/ 

we can safely omit the higher powers of ^, we shall have 

xdx 
h ' 4 / t 






h 4/[_2 
or 

' = V^ 1 + ^ (2) 

a nearer approximatioa 
The formula 



is the most used, and is more nearly correct, the smaller the 
value of h. 

EXAMPLES. 

i . Find the length of the simple circular pendulum which is to beat 
seconds at a place where g = 32^. 

Solution. 



SIMPLE CYCLOIDAL PENDULUM. 



99 




2. What is the time of vibration of a simple circular pendulum 5 
feet long? 

82. Simple Cycloidal Pendulum. The equation of the 
cycloid is 

x x 

y = # versin \- (2ax Jf 3 )^ 
a 

. dy_ \/ 2a ~ x 

dx V x 

ds_ = /2^\5 

dx \ x I 

Hence we shall have, for the time of a single oscillation, 

dx 



or 



This expression is independent of //, so that the time of vibra- 
tion is the same whether the arc be large or small. 

A body can be made to vibrate in a cycloidal arc by suspend- 
ing it by a flexible string between two cycloidal cheeks. This 
is shown from the fact that 
the evolute of the cycloid is 
another cycloid (Fig. 55). 

To prove this, we have, 
from the equation of the 
cycloid, 

y = a versin - -j- (2ax 



dy _ t / 
dx ~ V 



2a x ds 




a 



&.^_ 

<& *^ia - x 



I00 APPLIED MECHANICS. 

Hence the radius of curvature is 




and since we have for the evolute the relation 

ds' = dp, 
where ds f is the elementary arc of the evolute, 

f*x = za 

.-. /= I *; 

i/.*r^* 

and, observing that when x 2a p = o, we have 



If x l is the abscissa of the point of the evolute, 

- - x + d y = a - x 
ds 



and, transforming co-ordinates to B by putting x 2 . + 2a for 
we obtain 



which is the equation of another cycloid just like the first. 

The motion along a vertical cycloid may also be obtained by 
letting a body move along a groove in the form of a cycloid 
acted on by gravity alone ; and in this case the time of descent 
of the body to the lowest point is precisely the same at what- 
ever point of the curve the body is placed. 

83. Effect of Grade on the Tractive Force of a Rail- 
way Train. -Asa useful particular case of motion on an 
inclined plane, we have the case of a railroad train moving up 
or down a grade. It is necessary that a certain tractive force 



EFFECT OF GRADE ON TRACTIVE FORCE. IOI 

be exerted in order to overcome the resistances, and keep 
the train moving at a uniform rate along a level track. If, 
on the other hand, the track is not on a level, and if we 
resolve the weight of the train into components at right angles 
to and along the plane of the track, we shall have in the latter 
component a force which must be added to the tractive force 
above referred to when we wish to know the tractive force re- 
quired to carry it up grade, and must be subtracted when we 
wish to know the tractive force required to carry it down grade. 
The result of this subtraction may give, if the grade is suffi- 
ciently steep and the speed sufficiently slow, a negative quan- 
tity ; and in that case we must apply the brakes, instead of 
using steam, unless we wish the speed of the train to increase. 

EXAMPLES. 

i. A railroad train weighing 60000 Ibs., and running at 50 miles per 
hour, requires a tractive force of 618 Ibs. on a level ; what is the tractive 
force necessary when it is to ascend a grade of 50 feet per mile? What 
when it is to descend? Also what is the amount of work per minute 
in each case ? 

Solution. 

The resolution of the weight will give (Fig. 50, 7?^ tor the com- 
ponent along the plane, 

(60000)^ = 568.2 nearly. 
Hence 

Tractive force for a level = 618.0, 
Tractive force for ascent = 1186.2, 
Tractive force for descent 49.8. 

To ascertain the work done per minute in each case, we have 
(a) For a level track, 6l8 x 5 6 o x 528 = 2719200 foot-lbs. 

(l>) Up grade, 2719200 + 6ooo ^ x 5 = 5219200 foot-lbs. 
(c) Down grade, 2719200 - 6ooo X J x 5 = 219200 foot-lbs. 



102 



APPLIED MECHANICS, 



2. Suppose the tractive force required for each 2000 Ibs. of weight 
of train to be, on a level track, for velocities of 

5.0 miles per hour, 10.0 20.0 30.0 40.0 50.0 60 

6.1 Ibs., 6.6 8.3 ii. 2 15.3 20.6 27; 
find the tractive force required to carry the train of example i 

(a) Up an incline of 50 feet per mile at 30 miles per hour. 
(^) Down an incline of 50 feet per mile at 30 miles per hour. 
(<:) Down an incline of 10 feet per mile at 20 miles per hour. 
(//) What must be the incline down which the train must run to 
require no tractive force at 40 miles per hour? 

3. If in the first example the tractive force remains 618 Ibs. while 
the train is going down grade, what will be its velocity at the end of one 
minute, the grade being 10 feet per mile? 

84. Harmonic Motion If we imagine a body to be 

moving in a circle at a uniform rate (Fig. 56), and a second 

body to oscillate back and forth in 
the diameter AB, both starting 
from B, and 
if when the 
first body is 
* at C the other 
is directly un- 
der it at G, 
etc., then is 
the second 
body said to 




FIG. 56. 



move in harmonic motion. 

A practical case of this kind of mo- 
tion is the motion of a slotted cross-head 
of an engine, as shown in the figure 
ig- 57) i the crank moving at a 




form rate. In the case of the ordinary 
crank, and connecting-rod connecting 
the drive-wheel shaft of a stationary engine with the piston-rod, 



FIG. 57. 



HARMONIC MOTION. 1 03 

we have in the motion of the piston only an approximation to 
harmonic motion. We will proceed to determine the law of the 
force acting upon, and the velocity of, a body which is con- 
strained to move in harmonic motion. Let the body itself and 
the corresponding revolving body be supposed to start from 
B (Fig. 56), the latter revolving in left-handed rotation with an 
angular velocity a, and let the time taken by the former in 
reaching G be t: then will the angle BOC at; and we shall 
have, if s denote the space passed over by the body that moves 
with harmonic motion, 

s = BG OB - OCcosat, 
or, if 

r=O= OC t 

s = r rcosa/, (l) 

the velocity at the end of the time t will be 

V = = arsina/, (2) 

and the acceleration at the end of time / will be 

(3) 



Hence the force acting upon the body at that instant, in the 
direction of its motion, is 

F = m = ma 2 r cos at = ma 2 (OG). (4) 

dt* 



The force, therefore, varies directly as the distance of the body 
from the centre of its path. It is zero when the body is at the 



IO4 APPLIED MECHANICS. 

centre of its path, and greatest when it is at the ends of its 
travel, as its value is then 

W 
ma 2 r = o?r; 

S 

this being the same in amount as the centrifugal force of the 
revolving body, provided this latter have the same weight as the 
oscillating body. On the other hand, the velocity is greatest 

when at = - (i.e., at mid-stroke) ; and its value is then 

v = ar, 
this being also the velocity of the crank-pin at mid-stroke. 

EXAMPLE. 

Given that the reciprocating parts of an engine weigh 10000 Ibs., 
the length of crank being i foot, the crank making 60 revolutions per 
minute ; find the force required to make the cross-head follow the crank, 
(i) when the crank stands at 30 to the line of dead points, (2) when 
at 60, (3) when at the dead point. 

85. Work under Oblique Force. If the force act in 
any other direction than that of the motion, we must resolve it 
into two components, the component in the direction of the 
motion being the only one that does work. Thus if the force 
F is variable, and 6 equals the angle it makes with the direction 
of the motion, we shall have as our expression for the work 
done 

fFcosOds. 

Thus if a constant force of 100 Ibs. act upon a body in a direc- 
tion making an angle of 30 with the line of motion, then wil! 
the work done by the force during the time in which it moves 
through a distance of 10 feet be 

(100) (0.86603) (10) = 866 foot-lbs. 




ROTATION OF RIGID BODIES. 1 05 

86. Rotation of Rigid Bodies -- Suppose a rigid body 
(Fig. 58) to revolve about an axis perpendicular to the plane of 
the paper, and passing through O ; 
imagine a particle whose weight is 
w to be situated at a perpendicular 
distance OA = r from the axis of 
rotation, and let the angular accel- 
eration be a : let it now be required 
to find the moment of the force or 
forces required to impart this ac- 
celeration ; for we know that, if 

the axis of rotation pass through the centre of gravity of the 
body, the motion can be imparted only by a statical couple ; 
whereas if it do not pass through the centre of gravity, the 
motion can be imparted by a single force. 

We shall have, for the particle situated at A, 

Weight = w. 

Angular acceleration = a. 

Linear acceleration = o.r. 

Force required to impart this acceleration to this particle 

w 

a-r. 
g 

7ff 

Moment of this force about the axis = ar 2 . 

g 

Hence the moment of the force or forces required to impart 
to the entire body in a unit of time a rotation about the axis 
through O, with an angular velocity a, is 



8 8 S 



where / is used as a symbol to denote the limit of ^wr 2 , and is 
called the Moment of Inertia of the body about the axis through O. 



106 APPLIED MECHANICS. 

87. Angular Momentum. This quantity,, which ex- 

g 

presses the moment of the force or forces required to impart to 
the body the angular acceleration a about the axis in question 
is also called the Angular Momentum of the body when rotat- 
ing with the angular velocity about the given axis. 

88. Actual Energy of a Rotating Body. If it be re- 
quired to find the actual energy of the body when rotating 
with the angular velocity w, we have, for the actual energy of 
the particle at A, 



g 2 2g 

and for that of the entire body 

<u* w 2 / 

Iwr 2 = - . 

*g zg 

This is the amount of mechanical work which would have to be 
done to bring the body from a state of rest to the velocity w, or 
the total amount of work which the body could do in virtue 
of its velocity against any resistance tending to stop its 
rotation. 

89. Moment of Inertia. The term "moment of inertia" 
originated in a wrong conception of the properties of matter. 
The term has, however, been retained as a very convenient one, 
although the conceptions under which it originated have long 
ago vanished. The meaning of the term as at present used, in 
relation to a solid body, is as follows : 

The moment of inertia of a body about a given axis is the 
limit of the sum of the products of the weight of each of the ele- 
mentary particles that make tip the body, by the squares of their 
distances from the given axis. 

Thus, if w lt w 2 , w y etc., are the weights of the particles 
which are situated at distances r lf r r y etc., respectively from 



MOMENT OF INERTIA OF A PLANE SURFACE. IO/ 

the axis, the moment of inertia of the body about the given 

axis is 

/ = limit of 



90. Radius of Gyration. The radius of gyration of a 
body with respect to an axis is the perpendicular distance from 
the axis to that point at which, if the whole mass of the body 
were concentrated, the angular momentum, and hence the mo- 
ment of inertia, of the body, would remain the same as they are 
in the body itself. 

If p is the radius of gyration, the moment of inertia would 
be, when the mass is concentrated, 



hence we must have 
whence 



where W = entire weight of the body. 

91. Moment of Inertia of a Plane Surface The term , 

"moment of inertia," when applied to a plane figure, must, of 
course, be defined a little differently, as a plane surface has no 
weight ; but, inasmuch as the quantity to which that name is 
given is necessary for the solution of a great many questions. 

The moment of inertia of a plane surface about an axis, either 
in or not in the plane, is the limit of the sum of the products of 
the elementary areas into which the surface may be conceived to 
be divided, by the squares of their distances from the axis in 
question. 

In a similar way, for the radius of gyration p of a plane 
figure whose area is A, we have 



108 APPLIED MECHANICS. 

From this definition it will be evident, that, if the surface be 
referred to a pair of axes in its own plane, the moment of iner- 
tia of the surface about O Y will be 



(i) 

and the moment of inertia of the surface about OX will be 

J^fffdxdy. (2) 

The moment of inertia of the surface about an axis passing 
through the origin, and perpendicular to the plane XO Y, will be 

SS**dx*y, (3) 

where r=. distance from O to the point (x,y) ; hence r 2 =. x 2 -f 
y 2 , and the moment of inertia becomes 

ff(x 2 4- y 2 )dxdy = ffx 2 dxdy + ffy 2 dxdy = / + /. (4) 

This is called the "polar moment of inertia." If polar co-ordi- 
nates be used, this last becomes 

ffp 2 ( P dpdB) = ffptdpdO. (5) 

All these quantities are quantities that will arise in the discus- 
sion of stresses, and the letters /and./ are very commonly used 
to denote respectively 

ffx z dxdy and ffy*dxdy. 

Another quantity that occurs also, and which will be repre- 
sented by K, is 

ffxydxdy; (6) 

and this is called the moment of deviation. 



EXAMPLES OF MOMENTS OF INERl'IA. 



109 



EXAMPLES. 



The following examples will illustrate the mode of finding the 
moment of inertia : .x 

i. Find the moment of inertia of the rectangle 
ABCD about OY (Fig. 59). 



Solution. 

h 



FIG. 59. 



2. Find the moment of inertia of the entire circle (radius r) about 
the diameter OY (Fig. 60). 




FIG. 60. 



Solution. 



_ 
4 " " 64 



=2 



-xtf+ r * f Vr 

/ 4V 



3- Find the moment of inertia of the circular ring (outside radius r, 
inside radius r^ about OY (Fig. 61). 



Solution. 



"44 



64 



4. Find the moment of inertia of an ellipse 
(semi-axes a and b) about the minor axis OY. 




FIG. 61. 



no 



APPLIED MECHANICS. 



Solution. 



Equation of ellipse is -f- ^ 



7ta*b 



On the other hand, I x 

4 

92. Moments of Inertia of Plane Figures about Parallel 
Axes. 

PROPOSITION. The moment of inertia of a plane figure 
about an axis not passing through its centre of gravity is equal 
to its moment of inertia about a parallel axis passing through its 
centre of gravity increased by the product obtained by multiply- 
ing the area by the square of the distance between the two axes. 

PROOF. Let A B CD 
(Fig. 62) be the surface ; let 
0Fbe the axis not passing 
through the centre of grav- 
ity ; let P be an elementary 
area A^rAr, whose co-ordi- 




nates are OR x and RP 
y ; and let OO T a = a 
constant = distance be- 
tween the axes. 
Let O,R x, abscissa of P with reference to the axis 
passing through the centre of gravity, 



x = a -f- 
x 2 = x, 2 



2ax t 



Ay 



POLAR MOMENT OF INERTIA OF PLANE FIGURES. Ill 



Hence, summing, and passing to the limit, we have 

fftfdxdy = fjxfdxdy + zaffxjxdy + a*ffdxdy ; ( i ) 

but if we were seeking the abscissa of the centre of gravity 
when the surface is referred to Y^OY lt and if this abscissa be 
denoted by x m we should have 



_ 
= 



ffdxdy ' 

and, since X Q = o, /. ffx^dxdy = o ; hence, substituting this 
value in (l), we obtain 

ffx 2 dxdy = ffxfdxdy -f- a 2 ffdxdy. . (2) 

If, now, we call the moment of inertia about O Y, 7, that 
about O, Y lt / and let the area = A = ffdxdy, we shall have 

7=7, + a- A. (3) 

Q. E. D. 

93. Polar Moment of Inertia of Plane Figures. The 

moment of inertia of a plane 
figure about an axis perpen- 
dicular to the plane is equal 
to the sum of its moments 
of inertia about any pair of 
rectangular axes in its plane 
passing through the foot of 
the perpendicular. 

PROOF. Let BCD (Fig. 
63) be the surface, and P an ^ Y 
elementary area, and let 
OA x, AP = y, OP r; then the moment of inertia of 
the surface about OZ will be 

f ffdxdy = ff(x 2 +y*}dxdy = ffx*dxdy + f ffdxdy = / -f /. 
Q. E. D. 




FIG. 63. 



112 APPLIED MECHANICS. 

Hence follows, also, that the sum of the moments of inertia 
of a plane surface relatively .to a pair of rectangular axes in its 
own plane is isotropic ; i.e., the same as for any other pair of 
rectangular axes meeting at the same point, and lying in its 
plane. 



EXAMPLES. 



i. To find the moment of inertia of the rectangle (Fig. 59) about 
an axis through its centre perpendicular to the plane of the rectangle. 



Solution. 
Moment of inertia about YY , 

12 

Moment of inertia about an axis through its 



hence 



centre and perpendicular to YY = 

12 



Polar moment of inertia = 1 = (h 2 + 

12 12 12 



2. To find the moment of inertia of a circle about an axis through 
its centre and perpendicular to its pla'ne (Fig. 60). 



Solution. 

Moment of inertia about OY = , 

4 



hence 



Moment of inertia about OX = 

4 



-r, , , . Trr 4 TIT 4 TTf 

Polar moment of inertia = - f- = 

442 



3. To find the moment of inertia of an ellipse about an axis passing 
through its centre and perpendicular to its plane. 



MOMENTS OF INERTIA ABO^JT DIFFERENT AXES. 113 



Solution* 
From example 4, 91, we have 

/ -- 7ra ^ 3 
4 



.: Polar moment of inertia = (a 2 

4 



94. Moments of Inertia of Plane Figures about Different 
Axes compared. Given the surface KLM (Fig. 64), suppose 
we have already determined the quantities 

/ = ffx 2 dxdy, / = fffdxdy, K = ffxydxdy, 
it is required to determine, in terms of them, the quantities 



A 



the angles JTOFand X,OY, being both 
right angles, and YO Y, = a. 

We shall have, from the ordinary 
equations for the transformation of co- 
ordinates, to be found in any analytic 
geometry, the equations 

x t = x cos a -f- y sin a, 

y, = ycosa - *sina, FIG ^ 

x? = x 2 cos 2 a -f- y 2 sin 2 a -f- 2xy cos a sin a, 
jj 2 = ^ 2 sin 2 a -f- jy 2 cos 2 a 2^' cos a sin a, 
^jjj = ^y(cos 2 a sin 2 a) (x 2 y 2 ) cos a since. 




1 14 APPTIED MECHANICS. 



Hence 



= ffxfdxdy* = limit of Sxf&A 
= cos 2 a limit of 2x 2 &A + sin 2 a limit of 
2 cos a sin a limit 



2 (cos a sin a) ffxydxdy. 
J s = ffy l 2 dx l dy l = limit of lyf&A 

= (sin 2 a) limit of 2^A^ + (cos 2 a) limit of 
2 (cos a sin a) limit of 



2 (cos a sin a)ffxydxdy. 
K t = ffx l y 1 ^x j ^y l = limit of S^ij^iA^ 

= (cos 2 a sin 2 a) limit of 2<xy&A (cos a sin a) {limit of 

2x 2 &A - limit of ^y 2 ^A\ 
= (cos 2 a sin 2 a)ffxydxdy (cos a sin a) \ffx 2 dxdy 

fffdxdy}. 

Or, introducing the letters /, J t and TsT, we have 
7, = /cos 2 a + / sin 2 a -f 2^ cos a sin a, (i) 

y r = /sin 2 a + y COS 2 a 2 A" COS a sin a, (2) 

^ = (J /) cos a sin a + ^(cos 2 a sin 2 a). (3) 

The equations (i), (2), and (3) furnish the solution of the 
problem. 

95. Principal Moments of Inertia in a Plane. In every 
plane figure, a given point being assumed as origin, there is at 
least one pair of rectangular axes, about one of which the moment 
of inertia is a maximum, and a minimum about the other ; these 
moments of inertia being called principal moments of inertia, 
and the axes about which they are taken being called principal 
axes of inertia 



AXES OF SYMMETRY OF PLANE FIGURES. I 15 

PROOF. In order that / equation (i), 94, may be a maxi- 
mum or a minimum, we must have, as will be seen by differen- 
tiating its value, and putting the first differential co-efficient 
equal to zero, 

2/cos a sin a 4- 2/cos a sin a 4- 2^(cos 2 a sin 2 a) = o 
/. ^(cos 2 a sin 2 a) (/ /) cos a sin a = o (i) 

cos a sin a K iK , \ 

/. - = - .*. tan 2 a = - - -. (2) 

cos 2 a sin 2 a / J I J 

Hence, for the value of a given by (2), we have 7, a maximum 
or a minimum ; and as there are two values of 2a corresponding 
to the same value of tan 2a, and as these two values differ by 
1 80, the values of a will differ by 90, one corresponding to a 
maximum and the other to a minimum. 

Moreover, when the value of a is so chosen, we have 



as is proved by equation (i). Indeed, we might say that the 
condition for determining the principal axes of inertia is 

K, = o. 

96. Axes of Symmetry of Plane Figures. An axis 
which divides the figure symmetrically is always a principal 
axis. 

PROOF. Let us assume that the y axis divides the surface 
symmetrically ; then we shall have, with reference to this axis, 



K = 



And, since K is zero, the axis of y is one principal axis, and of 
course the axis of x is the other. The same method of reason- 
ing would show K = o if the x axis were the axis of symmetry. 



II 6 APPLIED MECHANICS. 

Hence, whenever a plane figure has an axis of symmetry, 
this axis is one of the principal axes, and the other is at 
right angles to it. Thus, for a rectangle, when the axis is to 
pass through its centre of gravity, the principal axes are par- 
allel to the sides respectively, the moment of inertia being- 
greatest about the shortest axis, and least about the longest. 
Thus in an ellipse the minor axis is the axis of maximum, 
and the major that of minimum, moment of inertia, etc. On 
the other hand, in a circle, or in a square, since the maximum 
and minimum are equal, it follows that the moments of inertia 
about all axes passing through the centre are the same. 

97. Conditions for Equal Values of Moment of In- 
ertia. When the moments of inertia of a plane figure about 
three different axes passing through the same point are the 
same, the moments of inertia about all axes passing through 
this point are the same. 

PROOF. Let / be the moment of inertia about O Y, 7 l 
about OY lt I 2 about OY 2 , and let 

YOY, = a, YOY 2 = ft, 
and let 

/, = /* = /. 

Then, from equation (i), 94, we have 

1=1 cos 2 a 4- J sin 2 a -f 2 K cos a sin a, 

/= /cos 2 /? +/sin 2 /3 + 2 A" cos/? sin/3. 
Hence 

(/ y)sin 2 a = 2 A' cos a sin a, (i) 

(7-/)sin 2 /3 == 2 K cos ft sin ft. (2) 

Hence 

(7-/)tana = 2 AT, (3) 

(7-/)tan0 = 2 AT. (4) 

And, since tan a is not equal to tan ft, we must have 
/ J o and K = o. 

Hence, since K o and / = J t we shall have, from eqja- 



MOMENTS OF INERTIA ABOUT PARALLEL AXES. I I/ 

tion (i), 94, for the moment of inertia /' about an axis, 
making any angle with O Y, 

I' = /cos 2 + /sin 2 + o = /. (5) 

Hence all the moments of inertia are equal. 

98. Components of Moments of Inertia of Solid 
Bodies. Refer the body to three rectangular axes, OX, OY, 
and OZ ; and let I x , I y , and I z represent its moment of inertia 
about each axis respectively. Then, if r denote the distance of 
any particle from OZ> we shall have 

I z = limit of ^wr 2 \ 
but 

r* = x 2 + y 2 

.'. I z = limit of So/C* 2 + y 2 } = limit of Saw 2 + limit of So/? 2 , (i) 
In the same way we have 

7* = limit of ^wy 2 + limit of ^wz 2 , (2) 

I y = limit of Sow 2 + limit of ^wz 2 . (3) 

99. Moments of Inertia of Solids around Parallel 
Axes. The moment of inertia of a solid body about an axis 
not passing through its centre of gravity is equal to its moment 
of inertia about a parallel axis passing through the centre of 
gravity, increased by the product of the entire weight of the 
body by the square of the distance between the two axes. 

PROOF. Refer the body to a system of three rectangular 
axes, OX, OY, and OZ, of which OZ is the one about which 
the moment of inertia is taken. Let the co-ordinates of the 
centre of gravity of the body with reference to these axes be 
(^oi Jo, #<>) Through the centre of gravity of the body draw a 
system of rectangular axes, parallel respectively to OX, OY, and 
OZ. Then we shall have for the co-ordinates of any point 

X = X o -\~ Xi, 

y = y* +y 

Z = Z + *,. 



APPLIED MECHANICS. 



Hence 



7 2 = limit of 2w(x 2 + y 2 ) = limit of *Zwx 2 4 limit of 
= limit of *2w(x 4- x,) 2 4- limit of %w(y + y t ) 2 
= x 2 limit of 2o> 4- 2 limit of 2w 4- 2x limit of 

4- limit of 



2 limit of 2o> 4- y 2 limit of 2w 
-f 2y limit of Soy^ -f limit of 



-f 



limit of 



= (* 2 -f 7o 2 ) 

-f limit of 
= r 2 W 4- // 4 2^ limit of 

But, since 6^ x is the centre of gravity, 

/. ^wx I = o and 
Hence 



4 



limit of 



limit of 



o. 



which proves the proposition. 

100. Examples of Moments of Inertia. 

i . To find the moment of inertia of a sphere whose radius is r and 
weight per unit of volume w, about the axis OZ drawn through its centre. 

Solution. 

Divide the sphere into thin slices (Fig. 65) by planes drawn perpen- 

dicular to OZ. Let the distance 
of the slice shown in the figure, 
above O be z, and its thickness dz : 
then will its radius be Vr 2 z 2 ; 
and we can readily see, from ex- 
ample 2, 93, that its moment of 
inertia about OZ will be 



dz. 




FIG. 65. 



Hence the moment of inertia 
of the entire sphere about OZ will 
be 



w 



- f V - 

2 J_ r V 



EXAMPLES OF MOMENTS OF INERTIA. 



which easily reduces to 



I z = 
15 



2. To find the moment of inertia of an ellipsoid (semi-axes a, b, c) 
about OZ (Fig. 66). 

SOLUTION. The equa- 
tion of the ellipsoid is 



Divide it into thin slices 
perpendicular to OZ, and 
let the slice shown in the 
figure be at a distance z 
from O. Then will this 
slice be elliptical, and its 
semi-axes will be 




FIG. 66. 



- V<T 2 - Z 2 



and 



C* 



and from example 3, 93, we readily obtain, for its moment of inertia 
about OZ, 



= 



Hence, for the moment of inertia of the ellipsoid about OZ, we 



have 



I 

U 



IS 

3. Find the moment of inertia of a right circular cylinder, length a, 
radius r, about its axis. 

Ans. 



120 APPLIED MECHANICS. 



4. Find the moment of inertia of the same about an axis perpen- 
dicular to, and bisecting its axis. Ans wwar* I a*\ 

4 V " 3/ 

5. Find the moment of inertia of an elliptic right cylinder, length 

2<r, transverse semi-axes a and b, about its longitudinal axis. 



Ans. ~(a 2 + b 2 }. 

6. Find the moment of inertia of the same about its transverse 
axis 2b. 



Ans. 



+ -) 

3/ 
7. Find the moment of inertia of a rectangular prism, sides 2a, 2b, 

2c, about central axis 2C. Ans. %wabc(a 2 -f b 2 }. 

101. Centre of Percussion. Suppose we have a body 
revolving, with an angular velocity a, about an axis perpendicu- 
lar to the plane of the paper, and 
passing through O. Join O with 
the centre of gravity, G, and take OG 
as axis of x ; the axis of y passing 
through (9, and lying in the plane of 
the paper. If, with a radius OA ;-, 
we describe an arc CA (Fig. 67), all 
particles situated in this arc have a 

linear velocity o.r. The force which would impart this velocity 
to any one of them, as that at A, in a unit of time, is 




g 

and this may be resolved into two, 

w , w 

ax and ay, 

S g 

respectively perpendicular and parallel to OG. The moment of 
this force about the axis is 



g 

hence the total moment of the forces which would impart to 



CENTRE OF PERCUSSION. \2l 



the body in a unit of time the angular velocity a, is, as has been 
shown already, 



g g 
The resultant of the forces acting on the body is 



g 

since, the centre of gravity being on OB, it follows that 
^wy = o ; and hence 

-*2wy o. 
g 

Hence the perpendicular distance from O to the line of direc- 
tion of the resultant force is measured along OG, and is 



g 7 f \ 



g 

and the point of application of the resultant force may be con- 
ceived to be at a point on OG at a distance / from O ; and this 
point of application of the resultant of the forces which pro- 
duce the rotation is called the Centre of Percussion. 

If p = radius of gyration about the axis through (9, and if 
x := distance from (9 to the centre of gravity, we have 

XoSw 

Hence 



or, in words, 

The radius of gyration is a mean proportional between the 
distance 1, and the distance x , betiveen the axis of oscillation and 
the centre of gravity. 

The centre of percussion with respect to a given axis of 
oscillation O has been defined as the point of application of the 



122 APPLIED MECHANICS. 

resultant of the forces which cause the body to rotate around the. 
point O. 

Another definition often given is, that it is the point at which) 
if a force be applied, there will be no shock on the axis of oscilla- 
tion ; and these two definitions are equivalent to each other. 

Let the particles of the body under consideration be con- 
ceived, for the sake of simplicity, to be distributed along a single 
line AB, and suppose a force F applied at D 
(Fig. 68). Conceive two equal and opposite 
forces, each equal to F, applied at C, the cen- 
tre of gravity of the body. 

\ Then these three forces are equivalent to 

a single force ^ applied at the centre of grav- 
ity C, which produces translation of the whole 
body ; and, secondly, a couple whose moment 
is F(CD), whose effect is to produce rotation 
FIG. 68. around an axis passing through the centre of 

gravity C. Under this condition of things it is evident that the 
centre of gravity C will have imparted to it in a unit of time a 

forward velocity equal to -, where M is the entire mass of the 

body ; the point D will have imparted to it a greater forward 
velocity ; while those points on the upper side of C will have 
imparted to them a less and less velocity as they recede from 
C, until, if the rod is sufficiently long, the particle at A will 
acquire a backward velocity. 

Hence there must be some point which for the instant in 
question is at rest; i.e., where the velocity due to rotation is just 
equal and opposite to that due to the translation, or about 
which, for the instant, the body is rotating : and if this point 
were fixed by a pivot, there would be no stress on the pivot 
caused by the force applied at D. 

An axis through this point is called the Instantaneous 
Axis. 



IMPACT OR COLLISION. 12$ 

102. Interchangeability of the Centre of Percussion 
and Axis of Oscillation. If we take, as axis of oscillation, a 
line perpendicular to the plane of the paper, and passing 
through D, then will O be the new centre of percussion. 

PROOF. We have seen ( 101) that 



where / = OD, X Q = OC, and p = radius of gyration about an 
axis through O perpendicular to the plane of the paper. 

Moreover, if /o represent the radius of gyration about an 
axis through C perpendicular to the plane of the paper, we shall 
have 

P* = p 2 + x<* 



XQ 

Now if D is taken as axis of oscillation, we shall have for the 
distance l t to the corresponding centre of percussion, 



CD I- Xo ' 

where p I = radius of gyration about the axis of oscillation 
through D. 

/ . Pi 2 po 2 + CD 2 p 2 r (j ~\ j 

I '-CD = -CD~ "D+ CJ >-**+ -*)-* 

Hence the new centre of percussion is at <9. Q. E. D. 

103. Impact or Collision. Impact or collision is a 
pressure of inappreciably short duration between two bodies. 

The direction of the force of impact is along the straight line 
drawn normal to the surfaces of the colliding bodies at their 
point of contact, and we may call this line the line of impact. 



124 APPLIED MECHANICS. 

The action that occurs in the case of collision may be de- 
scribed as follows : at first the bodies undergo compression ; 
the mutual pressure between them constantly increasing, until, 
when it has reached its maximum, the elasticity of the mate- 
rials begins to overpower the compressive force, and restore 
the bodies wholly or partially to their original shape and dimen- 
sions. 

Central impact occurs when the line joining the centres of 
gravity of the bodies coincides with the line of impact. 

Eccentric impact occurs when these lines do not coincide. 

Direct impact occurs when the line along which the relative 
motion of the bodies takes place, coincides with the line of 
impact. 

Oblique impact occurs when these lines do not coincide. 

CENTRAL IMPACT. 

104. Equality of Action and Re-action. One funda- 
mental principle that holds in all cases of central impact is the 
equality of action and re-action ; in other words, we must have, 
that, at every instant of the time during which the impact is 
taking place, the pressure that one body exerts upon the other 
is equal and opposite to that exerted by the second upon the 
first. 

The direct consequence of this principle is, that the algebraic 
sum of the momenta of the two bodies before impact remains 
unaltered by the impact, and hence that this sum is just the 
same at every instant of, and after, the impact. 

If we let 

m lt m 2 , be the respective masses, 
c lt c 2 , their respective velocities before impact, 

v u v 2 , their respective velocities after impact, 

i/, v" , their respective velocities at any given instant during 
the time while impact is taking place, 



CO-EFFICIEA r T OF RESTITUTION. 125 

then we must have the following two equations true ; viz., 
m 1 v l + m 2 v 2 = m l c l + m 2 c 2 , (i) 

mjf 4- m 2 v" = m l c l 4- m 2 c 2 . (2) 

105. Velocity at Time of Greatest Compression. At 

the instant when the compression is greatest i.e., at the 
instant when the elasticity of the bodies begins to overcome 
the deformation due to the impact, and to tend to restore them 
to their original forms the values of v' and v" must be equal 
to each other; in other words, the colliding bodies must be 
moving with a common velocity 

v = v' = v". (i) 

To determine this velocity, we have, from equation (2), 104, 
combined with (i), 

v = m ^ + m * c \ (2) 

m l 4- m 2 

106. Co-efficient of Restitution. In order to determine 
the values v lt v 2 , of the velocities after impact, we need two 
equations, and hence two conditions. One of them is fur- 
nished by equation (i), 104. The second depends upon the 
nature of the material of the colliding bodies, and we may dis- 
tinguish three cases : 

i. Inelastic Impact. In this case the velocity lost up to 
the time of greatest compression is not regained at all, and 
the velocity after impact is the common velocity ^ at the instant 
of greatest compression. In this case the whole of the work 
used up in compressing the bodies is lost, as none of it is 
restored by the elasticity of the material. 

2. Elastic Impact. In this case the velocity regained 
after the greatest compression, is equal and opposite to that 
lost up to the time of greatest compression ; therefore 

v z/j = c v v. (i) v 2 v v c 2 . (2) 



126 APPLIED MECHANICS. 

We may also define this case as that in which the work lost 
in compressing the bodies is entirely restored by the elasticity 
of the material, so that 



j . z2 , 

-- -- -- r 

2222 

Either condition will lead to the same result. 

3. Imperfectly Elastic Impact. In this case a part only 
of the velocity lost up to the time of greatest compression is 
regained after that time. 

If, when the two bodies are of the same material, we call e 
the co-efficient of restitution, then we shall so define it that 

v v l 



c* v v c 2 

or, in words, the co-efficient of restitution is the ratio of the 
velocity regained after compression to that lost previous to 
that time. 

In this case only a part of the work done in producing the 
compression is regained, hence there is loss of energy. Its 
amount will be determined later. 

Strictly speaking, all bodies belong to the third class ; the 
value of e being always a proper fraction, and never reaching 
unity, the value corresponding to perfect elasticity ; nor zero, 
the value corresponding to entire lack of elasticity. 

107. Inelastic Impact. In this case the velocity after 
impact is the common velocity at the time of greatest com- 
pression ; hence 

v = v, = v 2 (i) 



(2) 



And for the loss of energy due to impact we have 



m 2 c 2 , ^v* 

1 --- (M! -f- m 2 ) , 

2 2 



ELASTIC IMPACT. 12 J 



which, on substituting the value of v, reduces to 

,,"'?) (' - <> (3) 

2(m 1 +- m 2 ) 

1 08. Elastic Impact. In this case we have, of course, 
the condition, equation (i), 104, 



m\v-L + ^2^2 == m^c-i -f* 
and, for second equation, we may use equation (3), 106 ; viz., 

w^! 2 , m 2 v 2 2 _ mj? m^c^ 

2222 

Combining these two equations, we shall obtain 



m l 



We can obtain the same result without having to solve an 
equation of the second degree, by using instead the equations 
(i) and (2) of 106, together with (i) of 104; i.e., 

m l v I -f- m z v z = m 1 f l + m^\ 



or 

and ( 105) 



l ~\~ 



As the result of combining these equations, and eliminating 
v, we should obtain equations (i) and (2), as above, for the values 
of z\ and v 2 . In this case the energy lost by the collision is 
zero. 



128 APPLIED MECHANICS. 

109. Special Cases of Inelastic Impact. (a) Let the 
mass m 2 be at rest. Then c 2 o, 



v = m ' c ' 



< + * 

.' . Loss of energy = m * m * . ( 2 ) 

() Let w 2 be at rest, and let m 2 = oo ; i.e., let the mass ;;/ r 
strike against another which is at rest, and whose mass is in- 
finite. We have 

m 2 = oo , c 2 = o, 

- = o, (3) 



m 



W. ^ r Wi^ r 

Loss of energy = - - -- - = -L, (4) 



or the moving body is reduced to rest by the collision, and all 
its energy is expended in compression. 

(c) Let m l c l = m 2 c 2 ; i.e., let the two bodies move towards 
each other with equal momenta : 



o, (5) 



and the loss of energy = ^^ -f ^2!, (6) 

2 2 

the entire energy being lost. 

110. Special Cases of Elastic Impact. (a) Let the 
mass m 2 be at rest. Then c 2 =. o, 



EXAMPLES OF ELASTIC AND INELASTIC IMPACT. I2g 

(b) Let m 2 be at rest, and let m 2 oo . Then we have 

' 2 =0, 

^+ I 

m 2 
V 2 = o. (4) 

Hence the moving body retraces its path in the opposite direc- 
tion with the same velocity. 

(c) Let m^, = m 2 c 2 . Then our equations of condition 

become 

WiVi + m 2 v 2 = o, 

2222 

and from these we readily obtain 



i.e., both bodies return on their path with the same velocity 
with which they approached each other. 

in. Examples of Elastic and of Inelastic Impact. 

1. With what velocity must a body weighing 8 pounds strike one 
weighing 25 pounds in order to communicate to it a velocity of 2 feet 
per second, (a) when the bodies are perfectly elastic, (b) when wholly 
inelastic. 

2. Suppose sixteen impacts per minute take place between two bodies 
whose weights are respectively 1000 and 1200 pounds, their initial velo- 
cities being 5 and 2 feet per second respectively : find the loss of energy, 
the bodies being inelastic. 

112. Imperfect Elasticity. In this case we have the 

relations (see 106) 

V Vi _ 

v - c 2 



130 APPLIED MECHANICS. 

where 

v = m + * f * ; 

and we have also 

m^, -h m 2 v 2 = m^ -f m 2 c 2 . 

Determining from them the values of z/ x and v z , we obtain 
Vs = (i + e) - ect, (i) 

v 2 = (i + - ec v (2) 

or, by substituting for v its value, 



These may otherwise be put in the form 

-- -'-(+) - O, (5) 



Moreover, we have for the loss of energy due to impact 



E = (^ 2 - ^ 2 ) + ^(r 2 2 - v 2 

2 2 



or 



but, from (5) and (6) respectively, 

f s - Vl = ( T + <?) ^2(^*1 

Wj -f w 2 
^ - , a = _ (i + g),( 

W, -f- 



IMPERFECT ELASTICITY. 131 



2(m l 
/. E = . m 



But, from (i) and (2), 



or 



When ^ = i, or the elasticity is perfect, this loss of energy 
becomes zero. 

When e = o, or the bodies are totally inelastic, then the loss 
of energy becomes 



/ L ** \ v m f * V / 

*\rr*l ~T~ "*2/ 

as has been already shown in 107. 

An interesting fact in this connection is, that since (8) is 
the work expended in producing compression, and (7) is the 
work lost in all, therefore the work restored by the elasticity of 
the body is 



so that e 2 , or the square of the co-efficient of restitution, is the 
ratio of the work restored by the elasticity of the bodies, to 
the work expended in compressing the bodies up to the time 
of greatest compression. 



132 APPLIED MECHANICS. 

113. Special Cases. (a) Let m 2 be at rest, therefore 
2 = o. Then we shall have 

( _^(i + *) 1 = , ,*.-**. (I) 

( #*i + * 2 ) m t + m 2 



and for loss of energy 



When #z 2 = oo , and 2 = o, we have 

PI = ^i, (4) 

^ 2 = o, 

^ = (I _ , 2) ^!. (5) 

When m l c l = m 2 c 2 , then 



m,) ^ 



1 14. Values of e as Determined by Experiment. 
Since we have 



_ 
& 



IMPERFECT ELASTICITY. 133 

we shall have, when 

m 2 = oo and c 2 = o, 



m x 4- 
Hence 



Now, if we let a round ball fall vertically upon a horizontal 
slab from the height H y we shall have for the velocity of ap- 
proach 



and if we measure the height h to which it rises on its rebound, 
we shall have 



Hence 

In this way the value of e can be determined experimentally 
for different substances. 

Newton found for values of e: for glass, ||; for steel, |~ 
and Coriolis gives for ivory from 0.5 to 0.6. 

On the other hand, if we desired to adopt as our constant 
the ratio of the work restored, to the work spent in compres- 
sion, we should have for our constant ^ 2 , and hence the squares 
of the preceding numbers. 

EXAMPLES. 

i. If two trains of cars, weighing 120000 and 160000 Ibs., come 
into collision when they are moving in opposite directions with veloci- 
ties 20 and 15 feet per second respectively, what is the loss of mechan- 
ical effect expended in destroying the locomotives and cars ? 



134 APPLIED MECHANICS. 

2. Two perfectly inelastic balls approach each other with equal 
velocities, and are reduced to rest by the collision ; what must be the 
ratio of their weights ? 

3. Two steel balls, weighing 10 Ibs. each, are moving with velocities 
5 and 10 feet per second respectively, and in the same direction : find 
their velocities after impact, the fastest ball being in the rear, and over- 
taking the other ; also the loss of mechanical effect due to the impact, 
assuming e = 0.55. 

115. Oblique Impact. 

Let m lt m 2 , be the masses of the colliding bodies ; 
c lt c 2 , their respective velocities before impact ; 
a n a 2 , the angles made by c lt c 2 , with the line of centres ; 
v u v 2y the components of the velocities after impact ; 
, cos a,, c 2 cos a 2 , the components of c lt c 2 , along the line of 

centres ; 
c l sin a,, c 2 sin a 2 , the components of c u c 2t at right angles to 

the line of centres ; 
v the common component of the velocity at the instant of 

greatest compression along line of centres ; 
i/ t v", actual velocities after impact ; 
a', a", angles they make with line of centres ; 
vj, v", actual velocities when compression is greatest ; 
a/, a/', angles they make with line of centres. 

Then we shall have, by proceeding in the same way as was done 
m 112, 

V, = ^COSa, (l + e) ~ (VjCOSa, ^ 2 COSa 2 ), (l) 

#*i H- ^2 

V 2 = C 2 COS a 2 + (l 



OBLIQUE IMPACT. 135 





(3) 
(4) 

(5) 
(6) 

(7) 

(8) 
(9) 

(10) 


tf = VW + ^sin'a,, 


if' = y^ 2 2 + ^2 2 sin 2 a,, 
COS a' = ^, 

COS a" = J&, 
V 




Vc = Vz' 2 + <Tj 2 sin 2 a x , 


v/' = ^v 2 H~ ^sin 2 ^, 

00.*' -J, 



COS a/' = -^ 



/?* 



And for the energy lost in impact, we have 



, _ . 

2(m l -f- w 2 ) 

When the bodies are perfectly elastic, 

g = i, 
and equations (i), (2), and (12) become respectively 

Vi = ^ COS a, -- 2 - ( fi cos af _ ^ cos ^ 



C 2 COS a 2 H -- ?^ (f s COS a, ^ 2 COSa,), 

m l + m 2 



The rest remain the same in form. 

When the bodies are totally inelastic, 

* SB O, 



136 APPLIED MECHANICS. 

and equations (i), (2), and (12) become respectively 

#, = fj COS a, - - - (^TjCOStt! ^ COS 03), 

m^ + m z 

V 2 = c 2 cos a 2 H -- - - (Cj. cos a x <r 2 cos 03), 
/, + w 2 

, - r 2 cosa 2 ) 2 . 



2(m l 4- m 2 ) 
The rest remain the same in form. 

116. Impact of Revolving Bodies. Let the bodies A 
and B revolve about parallel axes, and impinge upon each other. 
Draw a common normal at the point of contact. This 
common normal will be the line of impact. 
Let c, angular velocity of A before impact, 
c 2 = angular velocity of B before impact, 
o> x = angular velocity of A after impact, 
<o 2 = angular velocity of B after impact, 
^ perpendicular from axis of A on line of impact, 
a z = perpendicular from axis of B on line of impact, 
7 X = moment of inertia of A about its axis, 
7 2 = moment of inertia of B about its axis. 

Then we shall have 

a l e l = Ci = linear velocity of A at point of contact before impact ; 

a 2 2 = c 2 = linear velocity of B at point of contact before impact ; 

a l (j} l = Vj. = linear velocity of A at point of contact after impact ; 

a 2 <a 2 v 2 = linear velocity of^ at point of contact after impact; 



2g 



// V 2 

_ / _!_ \JL = actual energy of A before impact : 

\*i 2 hg 

/ / \C 2 

= ( ^ ]- L - actual energy of B before impact ; 
\a 2 *)2& 

-^- = ( - ) = actual energy of A after impact : 

2g W/2- 

^- = ( - j- 2 - = actual energy of B after impact ; 
2 S \ a 2 2 / 2g 



IMPACT OF REVOLVING BODIES. 137 

Hence it follows that we have the case explained in 112 for 
imperfectly elastic impact, provided only we write 



- instead of m^g and instead of m 2 g. 
a, 2 a 2 2 



- instead 
a, 2 

Hence we shall have 

o>, = e, - *,(,, - <* 2 e 2 ) 8 f 2 r ,(l + ') (0 

/,#, -f/x 

W 2 = C 2 4- tfaOi*! ^2^2)7 - ' - (l + <?), (2) 

!&* + /I^2 2 

The case of perfect elasticity is obtained by making e = i. 
The case of total lack of elasticity is obtained by making 

f O. 

In the latter case the loss of energy is 

7 r / 2 x v 



as can be seen by substituting the proper values in equation (8), 
112. 



138 APPLIED MECHANICS. 



CHAPTER III. 
ROOF-TRUSSES. 

117. Definitions and Remarks. The term "truss" may 
be applied to any framed structure intended to support a load. 

In the case of any truss, the external loads may be applied 
only at the joints, or some of the truss members may support 
loads at points other than the joints. 

In the latter case those members are subjected, not merely 
to direct tension or compression, but also to a bending-action, 
the determination of which we shall defer until we have studied 
the mode of ascertaining the stresses in a loaded beam ; and 
we shall at present confine ourselves to the consideration of 
the direct stresses of tension and compression. 

For this purpose any loads applied between two adjacent 
joints must be resolved into two parallel components acting at 
those joints, and the truss is then to be considered as loaded at 
the joints. By this means we shall obtain the entire stresses in 
the members whenever the loads are concentrated at the joints; 
and, when certain members are loaded at other points, our re- 
sults will be the direct tensions and compressions of these mem- 
bers, leaving the stresses due to bending yet to be determined. 

A tie is a member suited to bear only tension. 

A strut is a member suited to bear compression. 

1 1 8. Frames of Two Bars. Frames of two bars may 
consist, (i) of two ties (Fig. 69), (2) of two struts (Fig. 70), 
(3) of a strut and a tie (Fig. 71). 



FRAMES OF TWO BARS. 



139 



CASE I. Two Ties (Fig. 69). Let the load be repre- 
sented graphically by CF = W. a 
Then if we resolve it into 
two components, CD and CE, 
acting along CB and CA re- 
spectively, CD will represent 
graphically the pull or tension 
in the tie CB, and CE that in 
the tie CA. 

The force acting on CB at 
B is equal and opposite to 




FIG. 69. 



CD, while that acting on CA at A is equal and opposite to CE. 
To compute these stresses analytically, we have 



CE = CF 



sin CFE 



= W 



sin 2 



sin CEF sin(Y + /,)' 



CD = CF 



sin CFD 
sin CDF 



= W 



sin/, 



sin(Y + /,) 

CASE II. Two Struts (Fig. 70). Let the load be repre- 
sented graphically by CF= W. 
Then will the components CD 
and CE represent the thrusts 
in the struts CB and CA re- 
spectively, and the re-actions 
of the supports at B and A 
will be equal and opposite to 
them. For analytical solution, 
we derive from the figure 




FIG. 70. 



CE = W 



smi. 



sin(* -f- 



CD = W 



sin* 



sin(i -f *,) 



CASE III. A Strut and a Tie (Fig. 71). Let the load be 
represented graphically by CF = W. Resolve it, as before, 
into components along the members of the truss. Then will 




140 APPLIED MECHANICS. 

CE represent the tension in the tie AC, and CD will represent 
the thrust in the strut BC ; and we may 
deduce the analytical formulae as before. 

1 19. Stability for Lateral Deviations. 
-In Case I, if the joint C be moved a little 
out of the plane of the paper, the load at 
C has such a direction that it will cause the 
truss to rotate around AB so as to return to 
its former position ; hence such a frame is 
stable as regards lateral deviations. 

In Case II the effect of the load, if C 
were moved a little out of the plane of the 
paper, would be to cause rotation in such a way as to overturn 
the truss ; hence such a frame is unstable as regards lateral 
deviations. 

In Case III the stability for lateral deviations will depend 
upon whether the load CF = W is parallel to AB, is directed 
away from it or towards it. If the first is the case (i.e., if A is 
the point of suspension of the tie), the frame is neutral, as the 
load has no effect, either to restore the truss to its former posi- 
tion, or to overturn it ; if the second is the case (i.e., if A t is 
the point of suspension of the tie), the truss is stable ; and, if 
the third is the case (i.e., if A is the point of suspension of the 
tie), it is unstable as regards lateral deviations. 

1 20. General Methods for Determining the Stresses in 
Trusses. In the determination of the stresses as above, it 
would have been sufficient to construct only the triangle CFD 
by laying off CF= W to scale, and then drawing CD parallel 
to CB, and FD parallel to CA, and the triangle CFD would have 
given us the complete solution of the problem. Moreover, the 
determination of the supporting forces of any truss, and of the 
stresses in the several members, is a question of equilibrium. 
Adopting the following as definitions, viz., 
External forces are the loads and supporting forces, 



TRIANGULAR FRAME. 14! 

Internal forces are the stresses in the members : 
we must have 

i. The external forces must form a balanced system; i.e., 
the supporting forces must balance the loads. 

2. The forces (external and internal) acting at each joint 
of the truss must form a balanced system ; i.e., the external 
forces (if any) at the joint must be balanced by the stresses in 
the members which meet at that joint. 

3. If any section be made, dividing the truss into two parts, 
the external forces which act upon that part which lies on one 
side of the section, must be balanced by the forces (internal) 
exerted by that part of the truss which lies on the other side 
of the section, upon the first part. 

The above three principles, the triangle, and polygon of 
forces, and the conditions of equilibrium for forces in a plane, 
enable us to determine the stresses in the different members 
of roof and bridge trusses. 

121. Triangular Frame. Given the triangular frame 
ABC (Fig. 72), and given the load W at C in magnitude and 
direction, given also the N 
direction of the support- 
ing force at B, to find the 
magnitude of this support- 
ing force, the magnitude 
and direction of the other 
supporting force, and the 
stresses in the members. 

SOLUTION. Join A \ b 

with D, the point of inter- FIG. 72. 

section of the line of direction of the load and the line BE. 
Then will DA be the direction of the other supporting force ; 
for the three external forces, in order to form a balanced sys- 
tem, must meet in a point, except when they are parallel. 
Then draw ab to scale, parallel to CD and equal to W. From 




142 



APPLIED MECHANICS. 



a draw ac parallel to BD, and from b draw be parallel to AD ; 
then will the triangle abca be the triangle of external forces, 
the sides ab, be, and ca, taken in order, representing respectively 
the load W, the supporting force at A, and the supporting force 
at A 

Then from a draw ad parallel to BC, and from c draw cd 
parallel to AB ; then will the triangle acd be the triangle of 
forces for the joint B, and the sides ca, ad, and dc, taken in 
order, will represent respectively the supporting force at B, the 
force exerted by the bar BC at the point B, and the force 
exerted by the bar AB at the point B. 

Since, therefore, the force ad exerted by the bar CB at B 
is directed away from the bar, it follows that CB is in compres- 
sion ; and, since the force dc exerted by the bar AB at B is 
directed towards the bar, it follows that AB is in tension. 

In the same way bdc is the triangle of forces for the point 
A ; the sides be, cd, and db representing respectively the sup- 

porting force at A, the force 
exerted by the bar AB at A, 
and the force exerted by the 
bar AC at A. 

The bar AB is again seen to 
be in tension, as the force cd 
exerted by the bar AB at A is 
directed towards the bar. 

So likewise the triangle abd 
is the triangle of forces for the 
point C. 

Fig. 73 shows the case when 
the supporting forces meet the load-line above, instead of 
below, the truss. 

122. Triangular Frame with Load and Supporting 
Forces Vertical. Fig. 74 shows the construction when the 
load and also the supporting forces are vertical. In this case 



\ 




FIG. 73. 



BOW'S NOTATION. 



143 




FIG. 74. 



the diagram becomes very much simplified, the triangle of 
external forces abd becom- 
ing a straight line. The 
diagram is otherwise con- 
structed just like the last 
one. 

123. Bow's Notation. 
The notation devised by 
Robert H. Bow very much 
simplifies the construction of the stress diagrams of roof- 
trusses. 

This notation is as follows : Let the radiating lines (Fig. 75) 
represent the lines of action of a system of forces in equilib- 
rium, and let the polygon abcdefa be the polygon representing 

these forces in magnitude 
and direction ; then denote 
the sides of the polygon 
in the ordinary way, by 
placing small letters at the 
vertices, but denote the 
radiating lines by capital 
letters placed in the angles. 
Thus the line AB is the 
line of direction of the 
force ab y etc. In applying the notation to roof-trusses, we letter 
the truss with capital letters in the spaces, and the stress dia- 
gram with small letters at the vertices. If, then, in drawing 
the polygon of equilibrium for any one joint of the truss, we 
take the forces always in the same order, proceeding always 
in right-handed or always in left-handed rotation, we shall be 
led to the simplest diagrams. Hereafter this notation will be 
used exclusively in determining the stresses in roof-trusses. 

124. Isosceles Triangular Frame: Concentrated Load 
(Fig. 76.) Let the load W act at the apex, the supporting 




FIG. 75. 



144 



APPLIED MECHANICS. 




FIG. 76. 



forces being vertical ; each will be equal to \ W : hence the 
polygon of external forces will be the triangle abc, the sides of 

which, ab, be, and ca, all lie in 
one straight line. Then begin 
at the left-hand support, and 
proceed again in right-handed 
rotation, and we have as the tri- 
angle of forces at this joint cad, 
the forces ca, ad, and dc, these 
being respectively the support- 
ing force, the stress in AD, and 
that in DC ; the directions of 
these forces being indicated by 
the order in which the letters follow each other : thus, ca is an 
upward force, ad is a downward force ; and, this being the 
force exerted by the bar AD at the left-hand support, we con- 
clude that the bar AD is in compression. Again : dc is 
directed towards the right, or towards the bar itself, and hence 
the bar DC is in tension. The triangle of forces for the other 
support is bed, and that for the apex abd. 

125. Isosceles Triangular Frame: Distributed Load. 
Let the load W be uniformly distributed over the two rafters 
AF and FB (Fig. 77) ; then will 
these two rafters be subjected to 
a direct stress, and also to a bend- 
ing action : and if we resolve the 
load on each rafter into two com- 
ponents at the ends of the rafter, 
then, considering these components 
as the loads at the joints, we shall 
determine correctly by our diagram the direct stresses in all 
the bars of the truss. 

The load distributed over AF is ; and of this, one-half is 




FIG. 77. 



POLYGONAL FRAME. 



the component at the support, and one-half at the apex, and 

similarly for the other rafter. This gives as our loads, at 

4 

each support, and-- at the apex. The polygon of external 
forces is eabcde, where the sides are as follows : 

W W , W , W , W 

ea = , ab , be , cd = , de = . 
42422 

Then, beginning at the left-hand support, we shall have for the 
polygon of forces the quadrilateral deafd, where de = = sup- 
porting force, ea = -- = downward load at support, af 

4 

stress in AF (compression), fd stress in FD (tension). The 
polygon for the apex is abf, and that for the right-hand support 
cdfbc. 

1 26. Polygonal Frame Given a polygonal frame (Fig. 

78) formed of bars jointed together at the vertices of the angles, 
and free to turn on these joints, 
it is evident, that, in order that 
the frame may retain its form, 
it is necessary that the direc- 
tions of, and the proportions 
between, the loads at the dif- 
ferent joints, should be speci- 
ally adapted to the given form : 
otherwise the frame will change 
its form. We will proceed to 
solve the following problem : 
Given the form of the frame, 
the magnitude of one load as AB, and the direction of all the 
external forces (loads and supporting forces) except one, we 
shall have" sufficient data to determine the magnitudes of all, 




\ 
\ 

\ 
\ 


F 




FIG. 78. 



146 



APPLIED MECHANICS. 



and the direction of the remaining external forces, and also the 
stresses in the bars 

Let the direction of all the loads be given, and also that of 
the supporting force EF, that of the supporting force AF being 
thus far unknown ; and let the magnitude of AB be given. 
Then, beginning at the joint ABG, we have for triangle of 
forces abg formed by drawing ab || and = AB, then drawing 
ga || AG, and bg || BG ; ga and bg both being thrusts. Then, 
passing to the joint BCG, we have the thrust in BG already 
determined, and it will in this case be represented by gb. If, 
now, we draw be || BC, and gc || GC, we shall have determined 
the load BC as be, and we shall have eg and gb as the thrusts 
in CG and GB respectively. Continuing in the same way, we 
obtain the triangles gcd, gde, and gfe, thus determining the 
magnitudes of the loads cd, de, and of the supporting force ef; 
and then the triangle gaf, formed by joining a and/, gives us af 
for the magnitude and direction of the left-hand support. The 
polygon abcdefa of external forces is called the Force Polygon, 
while the frame itself is called the Equilibrium Polygon. 

127. Polygonal Frame with Loads and Supporting 

Forces Vertical In this case (Fig. 79) we may give the 

form of the frame and the mag- 
nitude of one of the loads, to 
determine the other loads and 
the supporting forces, and also 
the stresses in the bars ; or we 
may give the form of the frame 
and the magnitude of the re- 
sultant of the loads, to find the 
loads and supporting forces. In 
the former case let the load AB 
be given. Then, proceeding in 
the same way as before, we find the diagram of Fig. 79 ; the 
polygon of external forces abcdefa falling all in one straight line. 




FIG. 79. 



FUNICULAR POLYGON. TRIANGULAR TRUSS. 



147 



If, on the other hand, the whole load ae be given, we observe 
that this is borne by the stresses in the extreme bars AG and 
GE ; hence, drawing ag || AG, and eg || EG, we find eg and ga 
as the stresses in EG and GA respectively. Then, proceeding 
to the joint ABG, we find, since ' ga is the force exerted by 
GA at this point, that, drawing gb || GB, we shall have ab as 
the part of the load acting at the joint ABG, etc. 

128. Funicular Polygon. If the frame of Fig. 79 be 
inverted, we shall have the 
case of Fig. 80, where all 
the bars, except FG, are sub- 
jected to tension; FG itself 
being subjected to compres- 
sion. The construction of the 
diagram of stresses being en- \ 
tirely similar to that already 
explained for Fig. 79, the ex- 
planation will not be repeated 
here. If the compression 
piece be omitted, the case 
becomes that of a chain hung 
at the upper joints (the supporting forces then becoming iden- 
tical with the tensions in the two extreme bars), the line gf 
would then be omitted from the diagram, and the polygon of 
external forces would become abcdega. 

129. Triangular Truss : Wind Pressure. Inasmuch as 
the pressure of the wind on a roof has been shown by experi- 
ment to be normal to the roof on the side from which it blows, 
we will next consider the case of a triangular truss with the 
load distributed over one rafter only, and normal to the rafter. 

There may be three cases : 

i. When there is a roller under one end, and the wind 
blows from the other side. 





FIG. 80. 



148 



APPLIED MECHANICS. 



2. When there is a roller under one end, and the wind 
tlows from the side of the roller. 

3. When there is no roller under either end. 

The last arrangement should always be avoided except in 
small and unimportant constructions ; for the presence of a 
roller under one end is necessary to allow the truss to change 
its length with the changes of temperature, and to prevent the 
stresses that would occur if it were confined. 



CASE I. Using Bow's notation, we have (Fig. 81) the 

whole load represented 
in the diagram by db. 
Its resultant acts at the 
middle of the rafter 
AE, whereas the sup- 
porting force at the 
right-hand end is (in 
consequence of the pres- 
ence of the roller) verti- 
cal. Hence, to find the 
line of action of the other 
supporting force, pro- 
duce the line of action 
of the load till it meets 
a vertical line drawn 

through the roller, and join their point of intersection with the 

support where there is no roller. We thus obtain CD as the 

line of action of the left-hand support. 

We can now determine the magnitude of the supporting 

forces be and cd by constructing the triangle bcdb of external 

forces. 

Now resolve the normal distributed force db into two single 

forces (equal to each other in this case), da and ab respectively, 

acting at the left-hand support and at the apex. 




FIG. 81. 



TRIANGULAR TRUSS: WIND PRESSURE. 



149 



Now proceed to the left-hand support. We find four forces 
in equilibrium, of which two are entirely known ; viz., cd and 
da: hence, constructing the quadrilateral cdaec, we have ae as 
the thrust in AE, and ec as the tension in EC. 

Next proceed to the apex, and construct the triangle of 
equilibrium abea, and we obtain be as the thrust in BE. 

The triangle bceb is then the triangle of equilibrium for the 
right-hand sup- 
port. 

CASE II. - 
In this case 
(Fig. 82) we fol- 
low the same 
method of pro- 
cedure, only the 
point of inter- 
section of the 
load and sup- 
porting forces 
is above, instead of below, the truss. The figure explains itself 

so fully that it is unnecessary to 

explain it here. 

CASE III. In this case the 
supports are capable of exerting 
resistance in any direction what- 
ever ; so that, if any circumstance 
should determine the direction 
of one of them, that of the other 
would be determined also. When there is no such circum- 
stance, it is customary to assume them parallel to the load 
(Fig. 83). Making this assumption, we begin by dividing the 
line db, which represents the load, into two parts, inversely 




FIG. 8 




FIG. 83. 



150 APPLIED MECHANICS. 

proportional to the two segments into which the line of action 
of the resultant of the load (the dotted line in the figure) 
divides the line EC. We thus obtain the supporting forces be 
and cd, and bcdb is the triangle of external forces. We then 
follow the same method as in the preceding cases. 

130. General Determination of the Stresses in Roof- 
Trusses. In order to compute the stresses in the different 
members of a roof-truss, it is necessary first to know the 
amount and distribution of the load. 

This consists generally of 

i. The weight of the truss itself. 

2. The weight of the purlins, jack-rafters, and superin- 
cumbent roofing, as the planks, slate, shingles, felt, etc. 

3. The weight of the snow. 

4. The weight of the ceiling of the room immediately 
below if this is hung from the truss, or the weight of the 
floor of the loft, and its load, if it be used as a room. 

5. The pressure of the wind ; and this may blow from 
either side. 

6. Any accidental load depending on the purposes for which 
the building is used. As an instance, we might have the case 
where a system of pulleys, by means of which heavy weights 
are lifted, is attached to the roof. 

In regard to the first two items, and the fourth, whenever 
the construction is of importance, the actual weights should 
be determined and used. In so doing, we can first make an 
approximate computation of the weight of the truss, and use it 
in the computation of the stresses ; the weights of the ceiling 
or of the floor below being accurately determined. After the 
stresses in the different members have been ascertained by the 
use of these loads, and the necessary dimensions of the mem- 
bers determined, we should compute the actual weight of the 
truss ; and if our approximate value is sufficiently different 
from the true value to warrant it, we should compute again 



STA'SSS IN ROOF-TRUSSES. 



the stresses. This second computation will, however, seldom be 
necessary. 

In making these computations, the weights of a cubic foot 
of the materials used will be needed ; and average values are 
given in the following table with sufficient accuracy for the 
purpose. 



WEIGHT OF SOME BUILDING MA- 
TERIALS PER CUBIC FOOT. 


Pounds. 


WEIGHT OF SLATING PER SQUARE 
FOOT. 
According to Trautwine. 


Pounds. 


TIMBER. 


41 


\ inch thick on laths . . . 
\ " " " i-inch boards . 


4-75 
6.75 


Hemlock 


2 C 


JL u jl 


7-30 


Maple 


^j 
41 


^ " " " laths . . . 


7-00 


Oak, live 


CQ 


& " " " i-inch boards, 


9-OO 


Oak white 


4Q 


& " " " 'i " " 


9-55 


Pine, white ** 


'Vy 
2 C to "3O 


i " " laths . . . 


9- 2 5 


Pine, yellow, Southern . . 
Spruce ... . . . . 


45 

2 C to 7O 


i " " " i-inch boards, 

JL << TJL 


11.25 
1 1. 80 


IRON. 




With slating-felt add . . . 
With |-mch mortar add . . 


ilb. 
3lbs. 




4 Co 






Iron, wrought ...... 


480 


NUMBER OF NAILS IN ONE POUND 


No 


Steel 


490 










3 -penny . 


4 CO 


OTHER SUBSTANCES. 


80 to 90 


4 " 
6 " 


340 
I CO 


Mortar, hardened .... 


IOT 


8 " 


IOO 


Snow, freshly fallen 


5 to 12 


10 " .... 


60 


Snow, compacted by rain . 


I C to CO 


12 " 


40 


Slate 


140 to 180 


20 " . 


2 c 











As to the weight of the snow upon the roof, Stoney recom- 
mends the use of 20 pounds per square foot in moderate 
climates ; and this would seem to the writer to be borne out by 
the experiments of Trautwine as recorded in his handbook, 



I5 2 APPLIED MECHANICS. 

although Trautwine himself considers 12 pounds per square 
foot as sufficient. 

131. Wind Pressure. While a great deal of work has 
been done to ascertain the direction and the greatest intensity 
of the pressure of the wind upon exposed surfaces, as those of 
roofs and bridges, nevertheless the amount of information on 
the subject is very small, inasmuch as but few experiments 
have been under the conditions of practice. Before giving a 
summary of what has been done the following statements will 
be made : 

i. The pressure of the wind upon a roof, or other surface, 
is assumed to be normal tc the surface upon which it blows ; 
and what little experimenting has been done upon the subject 
tends to confirm this view. 

2. Inasmuch as more attempts have been made to deter- 
mine experimentally the velocity of the wind than its pressure, 
hence there have been a good many experiments to determine 
the relation between the velocity and the pressure upon a sur- 
face to which the direction of the wind is normal. 

3. A few experimenters have tried to determine the rela- 
tion between the intensity of the pressure on a surface normal 
to the direction of the wind and one inclined to its direction. 

4. While the above have been the investigations most com- 
monly pursued, other subjects of experiment have been 

(a) The variation of pressure with density ; (b) with tem- 
perature ; (c) with humidity; (d) with the size of surface 
pressed upon ; (e) with the shape of surface pressed upon ; 
(/) whether the pressure corresponding to a certain velocity is 
the same whether the air moves against a body at rest, or 
whether the body moves in quiet air. 

By way of references to the literature of the subject may 
be given the following, as most of the work that has been 
done is included in them or in other references which they 
contain : 



WIND PRESSURE. 153 



i. Proceedings of the British Institution of Civil Engineers, vol. 
Ixix., year 1882, pages 80 to 218 inclusive. 

2. A. R. Wolff : Treatise on Windmills. 

3. C. Shaler Smith : Proceedings American Society of Civil En- 
gineers, vol. x., page 139. 

4. A. L. Rotch : Report of Work of the Blue Hill Meteorological 
Observatory, 1887. 

5. Engineering, Feb. 28th, 1890 : Experiments of Baker. 

6. Engineering, May 30, June 6, June 13, 1890: Experiments of 
O. T. Crosby. 

The first gives an account of a very full discussion of the 
subject, by a large number of Engineers. The second con- 
tains a recommendation that the temperature of the air be con- 
sidered in estimating the pressure. The fifth gives an account 
of Baker's experiments on wind pressure in connection with the 
building of the Forth Bridge. 

Before an account is given of the experimental work that 
has been done, the following statements will be made of what 
are some of the methods in most common use : 

1. A great many engineers very commonly call from 40 to 
55 pounds per square foot the maximum pressure on a vertical 
surface at right angles to the direction of the wind. One rather 
common practice, in the case of bridges, is to estimate 30 
pounds per square foot on the loaded, or 50 pounds per square 
foot on the unloaded structure. Nevertheless pressures of 80 
and 90 pounds per square foot have been registered and re- 
corded by the use of small pressure-plates, and by computation 
from anemometer records. 

2. By way of determining the intensity of the pressure on 
an inclined surface in terms of that on a surface normal to the 
direction of the wind, four methods more or less used will be 
enumerated here : 

(a) Duchemin's formula, which Professor W. C. Unwin 
recommends, is as follows, viz. : 



154 



APPLIED MECHANICS. 



sin 6 



-*\ + sin'0' 

where / = intensity of normal pressure on roof, /, = intensity 
of piessure on a plane normal to the direction of the wind. 
(b) Hutton's formula, 

p=p l (sin 0)*.*4 &-*. 

Unwin claims that this and Duchemin's formula give nearly 
the same results for all angles of inclination greater than 15. 

The following table gives the results obtained by the use of 
each, on the assumption that p l = 40 : 



e 


Duchemin. 


Hutton. 


9 


Duchemin. 


Hutton. 


5 


6.89 


5-10 


50 


38.64 


38.10 


10 


13-59 


9.60 


55 


39-21 


39-40 


15 


19.32 


14.20 


60 


39-74 


40.00 


20 


24.24 


18.40 


65 


39-82 


40.00 


25 


28.77 


22.6O 


70 


39-91 


40.00 


30 


32.00 


26.50 


75 


39 -9 6 


40.00 


35 


34-52 


30.10 


80 


40.00 


40.00 


40 


36.40 


33-30 


85 


40.00 


40.00 


45 


37-73 


36.00 


90 


40.00 


40.00 



(c) A formula very commonly favored, but which does not 
agree with any experiments that have been made, is 



sn 



6. 



It gives much lower results, as a rule, than either of the others, 
but it is favored by many because, if we assume the wind to 
blow in parallel lines till it strikes the surface, and then to get 
suddenly out of the way, forming no eddies on the back side 
of the surface and meeting no lateral resistance on the front 



WIND PRESSURE. 155 



side, all of which are conditions that do not exist, we could 
then deduce it as follows: 

Assume a unit surface making an angle 6 with the direction 
of the wind, the total pressure on this surface in the direction 
of the wind would be^j sin #; and by resolving this into nor- 
mal and tangential components we should have, for the former, 



(d] Another rule which is sometimes used, but which has 
nothing to recommend it, is to consider the normal intensity 
of the wind pressure per square foot of roof surface as equal to 
the number of degrees of inclination of the roof to the hori- 
zontal. The wind pressure allowed for by this rule is very 
excessive, as it would be 90 pounds per square foot for a ver- 
tical surface. 

Taking up, now, the experimental work that has been done, 
we will begin with the attempts to determine velocities and 
pressures, and the relation between them. 

i. In regard to velocities, these are determined by using 
some kind of an anemometer, and in all these cases there are 
several difficulties and sources of error, as follows : 

(a) In many cases the anemometers have not even been 
graduated experimentally, but it has been assumed outright 
that the velocity of the air is just three times the linear velocity 
of the cups of a cup anemometer. 

(b) When they have been graduated, it has generally been 
done by attaching them to the end of the arm of a whirling 
machine, which, when the arm is long, and the velocity moder- 
ate, will do very well, but is the more inaccurate the shorter 
the arm and the higher the velocity of motion. 

(c) The wind always comes in gusts, and hence the ane- 
mometer does not register the average velocity of the wind at 
any one moment, but that of the particular portion that comes 



156 APPLIED MECHANICS. 

in contact with it, and this is always a small portion, on ac- 
count of the small size of the anemometer. 

(d) In order to get an indication which is not affected by 
the cross-currents reflected from the surrounding buildings and 
chimneys, it is necessary to put the anemometer very high up, 
and then, of course, we obtain the indications corresponding 
to that height, which is greater than that of the buildings, and 
it is well known that the velocity of the wind increases very 
considerably with the height. 

Next, as to the direct determination of pressure, this has 
usually been done by means of some kind of pressure-plate, 
either round or square, but of small size, thus allowing the 
eddies formed on the back side of the plate to have a con- 
siderable effect. The results obtained by the use of different 
sizes and different shapes of .plates have therefore differed very 
considerably ; and while some have claimed that the pressure 
per square foot increases with the size of the surface pressed 
upon, it has been very thoroughly proved by the more modern 
investigations that the opposite is true, and that the pressure 
decreases with the size. 

While the records from small pressure-plates have fre- 
quently shown very high pressures per square foot, as 80, 90, 
or even over 100 pounds per square foot, it has become very 
generally recognized by engineers that by far the greater part 
of existing buildings and bridges would be overturned by winds 
of such force, or anywhere near such force, and it has not been 
customary among them to make use of such high figures for 
wind pressure on bridges and roofs in computing the stability 
of structures. While some of the figures in general use have 
already been given, nevertheless the tendency of modern inves- 
tigation seems to be to obtain rather lower figures. In this con- 
nection it is well to refer to the work done by Baker in connec- 
tion with the construction of the Forth Bridge. The following 
description is taken from " Engineering" of Feb. 28th, 1890: 



WIND PRESSURE. 



157 



" The wind pressure to be provided for in the calcu- 
lations for bridges in exposed positions is 56 Ibs. per square 
foot, according to the Board of Trade regulations, and this 
twice over the whole area of the girder surface exposed, the 
resistance to such pressure to be by dead-weight in the struc- 
ture alone. 

44 The most violent gales which have occurred during the 
construction of the Forth Bridge are given, with the pressures 
recorded on the wind gauges, in the annexed table : 



Year. 


Month 
and 
Day. 


Pressure in pounds per square foot. 


Direction 
of 
Wind. 


Revolving 
Gauge. 


Small 
fixed 
Gauge. 


Large 
fixed 
Gauge. 


In centre 
of large 
Gauge. 


Right- 
hand top 
of large 
Gauge. 


1883 


Dec. ii, 


33 


39 


22 






s. w.* 


1884 


Jan. 26, 


65 


4i 


35 






s. w.* 


1884 


Oct. 27, 


29 


23 


18 






s. w. 


1884 


Oct. 28, 


26 


29 


!9 






s. w. 


1885 


Mar. 20, 


30 


25 


17 






w. 


1885 


Dec. 4, 


25 


27 


19 






w. 


1886 


Mar. 31, 


26 


3i 


J 9 






s. w. 


1887 


Feb. 4, 


26 


4i 


15 






s. w. 


1888 


Jan. 5, 


27 


16 


7 






S. E. 


[888 


Nov. 17, 


35 


41 


27 






w. 


1889 


" 2, 


27 


34 


12 






s. w. 


890 


Jan. 19, 


27 


28 


16 






s. w. 


1890 


" 21, 


26 


38 


15 






w. 


1890 


" 22, 


27 


24 


18 


231 


22 


S. W. by W. 



* These data are unreliable, owing to faulty registration by the indicator-needle, as will 
b' presently explained. They were altered after this date. The barometer fell to 27.5 inches 
ot, <hat occasion, over three quarters of an inch within an hour. 



158 APPLIED MECHANICS. 

"The pressure-gauges, which were put up in the summer 
of 1882 on the top of the old castle of Inchgarvie, and from 
which daily records have been taken throughout, were of very 
simple construction. The maximum pressures only were taken. 
The most unfavorable direction from which the wind pressure 
can strike the bridge is nearly due east and west, and two out 
of the three gauges were fixed to face these directions, while 
a third was so arranged as to register for any direction of 
wind. 

"The principal gauge is a large board 20 feet long by 15 
feet high, or 300 square feet area, set vertically with its faces east 
and west. The weight of this board is carried by two rods sus- 
pended from a framework surrounding the board, and so ar- 
ranged as to offer as little resistance as possible to the passage 
of the wind, in order not to create eddies near the edge of the 
board. In the horizontal central axis of the board there are 
fixed two pins which fit into the lower eyes of the suspension- 
rods, the object being to balance the board as nearly as pos- 
sible. Each of the four corners of the board is held between 
two spiral springs, all carefully and easily adjusted so that any 
pressure exerted on either face will push it evenly in the op- 
posite direction, but upon such pressure being removed the 
compressed springs will force the board back to its normal 
position. To the four corners four irons are attached, uniting 
in a pyramidal formation in one point, whence a single wire 
passes over a pulley to the registering apparatus below. In 
order to ascertain to some extent how far great gusts of wind 
are quite local in their action, and exert great pressure only 
upon a very limited area, two circular spaces, one in the exact 
centre and one in the right-hand top corner, about 18 inches 
in diameter, were cut out of the board and circular plates in- 
serted, which could register independently the force of the 
wind upon them. 

" By the side of the large square board, at a distance of 



WIND PRESSURE. 159 



about 8 feet, another gauge, a circular plate of 1.5 square feet 
area, facing east and west, was fixed up with separate regis- 
tration. This was intended as a check upon the indications 
given by the large board. 

"Another gauge of the same dimensions as the last, but 
with the disc attached to the short arm of a double vane, so 
that it would face the wind from whatever direction it might 
come, was set up. 

" On one occasion the small fixed board appeared to regis- 
ter 65 pounds to the square foot a registration which caused 
no little alarm and anxiety. Mr. Baker found, upon inves- 
tigation, that the registering apparatus was not in good order, 
and after adjusting it the highest pressure recorded was 41 
pounds. 

" In order to determine the effect of the wind upon surfaces 
like that of the exposed surface of the bridge, he devised an 
apparatus which consisted of a light wooden rod suspended in 
the middle, so as to balance correctly, by a string from the 
ceiling. At one end was attached a cardboard model of the 
surface, the resistance of which was to be tested, and at the 
opposite end was placed a sheet of cardboard facing the same 
way as the model, so arranged that by means of another and 
adjustable sheet, which would slide in and out of the first, 
the surface at that end could be increased or decreased at 
the will of the operator. The mode of working is for a 
person to pull it from its perpendicular position towards 
himself, and then gently release it, being careful to allow 
both ends to go together. If this is properly done, it is evi- 
dent that the rod will in swinging retain a position parallel 
to its original position, supposing that the model at one 
end and the cardboard frame at the other are balanced as 
to weight, and that the two surfaces exposed to the air 
pressure coming against it in swinging are exactly alike. 
Should one area be greater than the other, the model or card- 



160 APPLIED MECHANICS. 



board sheet, whichever it may be, will b~ lagging behind, and 
twist t^e string." 

The experiments carried on in various ways by different 
people and at different times are generally in agreement with 
each other and with the results of more elaborate processes. 
The information specially desired was in regard to the wind 
pressure upon surfaces more or less sheltered by those imme- 
diately in front of them. In this regard Mr. Baker satisfied 
himself that, while the results differed very considerably ac- 
cording to the distance apart of the surfaces, in no case was 
the area affected by the wind, in any girder which had two or 
more surfaces exposed, more than 1.8 times the area of the 
surface directly fronting the wind, and, as the calculations had 
been made for twice this surface, the stresses which the struc- 
ture will receive from this cause will be less than those pro- 
vided for. 

Next, as to the relation between velocity and pressure, a 
great many formulae have been devised, to agree with the 
results of different experimenters. Most all of them make the 
pressure proportional to the square of the velocity; while 
some add a term proportional to the velocity itself, and when 
higher velocities are reached, as those usual in gunnery, terms 
have been introduced with powers of the velocity higher than 
the second. It is hardly worth while to consider these dif- 
ferent formulae, as it is rather the pressure than the velocity 
that the engineer is interested in, and correct information in 
this regard is to be obtained rather from pressure-boards than 
from anemometers. Nevertheless, it may be stated that one 
of the most usual formulae is that of Smeaton, and is 



200 

where P= pressure in pounds per square foot, and V '= velocity 



WIND PRESSURE. l6l 



in miles per hour. This formula agrees very well with a num- 
ber of experiments that have been made where anemometers 
have been used to determine the velocity, and small pressure- 
plates (say one square foot) to determine the pressure ; thus 
this formula satisfies very well the experiments made at the 
Blue Hill Meteorological Observatory, near Boston, Mass., 
U. S. A. 

It was originally deduced from some very old experiments 
of Rouse ; and it agrees with a good many, but disagrees with 
other experiments. It is probably the formula that has been 
more quoted than any other. 

A little ought also to be said in regard to the pressure of 
the wind on very high structures, as on the piers of high via- 
ducts and on tall chimneys. In this regard it is to be ob- 
served : 

i. The pressure, as well as the velocity of the wind, be- 
comes greater the higher up from the ground the surface ex- 
posed is situated. 

2. From calculations on chimneys that have stood for a 
long time, Rankine deduced, as the greatest average wind 
pressure that could be realized in the case of tall chimneys, 55 
pounds per square foot. 

3. In making the piers of high viaducts, it would seem 
desirable not to make them solid, but to use only four up- 
rights at the corners connected by lattice work, in order to- 
expose a smaller surface to the wind. Nevertheless, as was ex- 
plained, it will not do to separate the structure into its com- 
ponent parts, and to estimate the pressure on each part 
separately and then add the results together to get the total 
effect ; but we really need some such experiments as those of 
Baker. 

4. Some old experiments of Borda bear out the common 
practice of assuming the wind pressure on the surface of a cir- 



1 62 APPLIED MECHANICS 

cular cylinder one half that which would exist on its projection 
on a plane normal to the direction of the wind, 

There remains now only to refer to a serial article by O. T. 
Crosby, in " Engineering" of May 30, June 6th, and June I3th, 
1890, containining some experiments made by him on wind 
pressure near Baltimore, Md. The first two numbers contain 
rather a summary of what has been done by others, and it is 
in the copy of June I3th that is to be found the account of his 
own work, which was done in order to determine the resistances 
of the air to fast-moving trains. 

He used a whirling arrangement turning about a vertical 
axis, to the end of which was attached a car, the circumference 
through which the car moved being 36 feet. 

In order to determine whether the circular motion produced 
any disturbing effect, he ran a car having a cross-section of 5.1 
square feet on a circular track about two miles in circumference, 
the speed of the car being about 50 miles per hour, and the 
results obtained in this way agreed very nearly with those ob- 
tained from his whirling table. The special peculiarity of his 
results is that he obtained, by plotting them, the law that the 
pressure varies directly as the first power of the velocity, and 
not as the square or some higher power; also, his pressures, 
after the velocity had passed 25 or 30 miles per hour, are 
much lower than those given by Smeaton and others, the pres- 
sure on a normal plane surface moving at 115 miles per hour 
being about 27 pounds per square foot. 

The cars used were generally about 3 feet long without the 
front. The fronts attached were: i. Normal plane surface; 
2. Wedge, base i, height I ; 3. Pyramid, base I, height 2; 4. 
Wedge and cyma, base I, height 2; 5. Parabolic wedge, 
base i, height 2. 

His experiments covered a range of velocities from 30 to 
130 miles per hour. 



DISTRIBUTION OF THE LOADS. 163 

The law of the first powers of the velocities seems peculiar, 
and certainly ought not to be accepted without further cor- 
roborative evidence ; but the low values of the pressures agree 
with Baker's results and with the tendency of the more modern 
investigations. 

132. Approximate Estimation of the Load. In all 
important constructions, the estimates of the loads should be 
made as above described. For smaller constructions, and for 
the purposes of a preliminary computation in all cases, we only 
estimate the roof-weight roughly ; and some authors even as- 
sume the wind pressure as a vertical force. 

Trautwine recommends the use of the following figures for 
the total load per square foot, including wind and snow, when 
the span is, 75 feet or less : 

Roof covered with corrugated iron, unbearded ... 28 Ibs. 

Roof plastered below the rafters 38 " 

Roof, corrugated iron on boards 31 " 

Roof plastered below the rafters 41 " 

Roof, slate, unboarded or on laths 33 " 

Roof, slate, on boards ij inches thick 35 " 

Roof, slate, if plastered below the rafters 46 " 

Roof, shingles on laths 30 " 

Roof plastered below rafters or below tie-beam . . . 40 " 
From 75 to 100 feet, add 4 Ibs. to each. 

133. Distribution of the Loads. The methods for de- 
termining the stresses, which will be used here, give correct 
results only when the loads are concentrated at joints, and are 
not distributed over any members of the truss. 

In. constructions of importance, this concentration of the 
loads at the joints should always be effected if possible ; 
because, when this is the case, the stresses in the members 
of the truss can be, if properly fitted, obliged to resist only 
stresses of direct tension, or of direct compression ; but, when 
there is a load distributed over any member of the truss, that 
member, in addition to the direct compression or direct tension, 
is subjected to a bending-stress- The effect of this bending 



164 



APPLIED MECHANICS. 



cannot be discussed until we have studied the theory of beams. 
Nevertheless, it is a fact that we have no experimental knowl- 
edge of the behavior of a piece under combined compression 
and bending ; and we know that when certain pieces are to 
resist bending, in addition to tension, they must be made much 
larger in proportion than is necessary when they bear tension 
only. 




FIG. 84. 

The manner in which this concentration of the loads is 
effected, is shown in Fig. 84, which is intended to represent one 
of a series of trusses that supports a roof, the rafters being the 
two lower ones in the figure. Resting on two consecutive 
trusses, and extending from one to the other, are beams called 
purlins, which should be placed only above the joints of the truss, 
and which are shown in cross-section in the figure. On these 
purlins are supported the jack-rafters parallel to the rafters, and 
at sufficiently frequent intervals to support suitably the plank 
and superincumbent roofing-materials. 

By this means we secure that the entire bending-stress comes 
upon the jack-rafters and purlins, and that the rafters proper 
are subjected only to a direct compression. When, however, 
the load is distributed, i.e., when the roofing rests directly on the 
rafters, or when the purlins are placed at points other than the 
joints, the bending-stress should be taken into account; and 
while the methods to be developed here will give the stresses 



DIRECT DETERMINATION OF THE STRESSES. 165 

in all the members that are not subjected to bending, the bend- 
ing-stress may be largely in excess of the direct stress in those 
pieces that are subjected to bending. How to take this into 
account will be explained later. 

Another important item to consider is, that, in constructions 
of importance, a roller should be placed under one end of the 
truss to allow it to move with the change of temperature ; as 
otherwise some of the members will be either bent, or at least 
subjected to initial stresses. The presence of a roller obliges 
the supporting force at that point to be vertical, whether the 
load be vertical or inclined. 

It is customary, and does not entail any appreciable error, 
to consider the weight of the truss itself, as well as that of the 
superincumbent load, as concentrated at the upper joints ; i.e., 
those on the rafters. 

In the case of a ceiling on the room below, or of a loft 
whose floor rests on the lower joints, we must, of course, ac- 
count the proper load as resting on the lower joints. 

134. Direct Determination of the Stresses. This, as 
we have seen, is merely a question of equilibrium of forces in 
a plane, where certain forces acting are known, and others are 
to be determined. 

As to the methods of solution, we might adopt 

i. A graphical solution, laying off the loads to scale, and 
constructing the diagram by the use of the propositions of 
the polygon, and the triangle of forces, and then determining the 
results by measuring the lines representing the stresses to 
the same scale. 

2. An analytical solution, imposing the analytical conditions 
of equilibrium, as given under the " Composition of Forces," 
between the known and unknown forces. 

3. A third method is to construct the diagram as in the 
graphical solution, but then, instead of determining the results 
by measuring the resulting lines to scale, to compute the un- 



1 66 APPLIED MECHANICS. 

known from the known lines of the diagram by the ordinary 
methods of trigonometry. 

.The first, or purely graphical, method, is one which has 
received a very large amount of attention of late years, and 
in which a great deal of progress has been made. It is, doubt- 
less, very convenient for a skilled draughtsman, and especially 
convenient for one who, though skilled in draughting, is not 
free with trigonometric work ; but it seems to me, that, when 
the results are determined by computation from the diagram, 
there is less chance of a slight error in some unfavorable tri- 
angle vitiating all the results. I am therefore disposed to 
recommend for roof-trusses the third method. 

In the case of bridge-trusses, on the other hand, I believe 
the graphical not to be as convenient as a purely analytic 
method. 

135. Roof-Trusses. In what follows, the graphical solu- 
tions will be explained with very little reference to the trigono- 
metric work, as that varies in each special case, and to one who 
has a reasonable familiarity with the solution of plane triangles, 
it will present no difficulty ; whereas to deduce the formulae 
for each case would complicate matters very much. Proceed- 
ing to special examples, let us take, first, the truss shown in 
Fig. 85, and let the vertical load upon it be W uniformly dis- 
tributed over the top of the roof, the purlins being at the joints 
on the rafters. 

The loads at the several joints will then be as follows, viz. 
(Fig. 85*), 

ab = kl = ~, be = cd = de = ef = fg = gh = hk = ~. 
16 8 

Then the supporting forces will be 

lm = ma = . 

2 

We thus have, as polygon of external forces, abcdefghklma. 



ROOF-TRUSSES. 



I6 7 



Now proceed to either support, say, the left-hand one ; and 
there we have the two forces ab and ma known, while by and 
ym are unknown. We then construct 
the quadrilateral maby in the figure, and 
thus determine by and ym. As to whether 





FIG. 850. 




FIG. 8s<5. 



dbcde 



FIG. 85. 

these represent thrust or tension, 
we need only remember that they 
are the forces exerted by the re- 
spective bars at the joints : and, since by is directed away from 
the bar BY, this bar is in compression; whereas, ym being 
directed towards the bar YM, that bar is in tension. 




l68 APPLIED MECHANICS. 

Having determined these two stresses, we next proceed to 
another joint, where we have only two unknown forces. Take 
the joint at which the load be acts, and we have as known 
quantities the load be, and also the force exerted by the bar 
YB, which is in compression. This force is now directed away 
from the bar, and hence is represented by yb. The unknown 
forces are the stresses in CX and XY. Hence we construct 
the quadrilateral cxybc ; and we thus determine the stresses in 
CX and XY as ex and xy t both being thrusts. 

Next proceed to the joint YXW, and construct the quadri- 
lateral myxwm, and thus determine the tension xw and the 
tension wm. 

Next proceed to the joint where cd acts, and so on. We 
thus obtain the diagram (Fig. 85*2) giving all the stresses. 

The truss in the figure was constructed with an angle of 30* 
at the base, and hence gives special values in accordance with 
that angle. 

In order to show how we may compute the stresses from the 
diagram, the computation will be given. 

From triangle bmy, we have bm = - W 

10 



ym = -Wcot 30 = 

16 16 



by = --^cosec 30 = w = ky. 
1 6 8 



From the triangle umc, we have cm W, 

16 



um = w 
16 



ROOF-TRUSSES. 



169 



yx yw sec 30 



- 
16 



= ( ^ 

\i6 



= = xv = vt, 



i6 



256 256 



, 
8 



ex = wm sec 



256 256 



vd = urn sec 30 = W\ -4 

\ 16 / y/ 8 



4 ' 



Hence we shall have for the stresses, 



RAFTERS 


(compression) . 


VERTICALS 


(tension). 


by = kn 
ex = ho 


= \W. 


xw = op 


W 
16* 


dv = gq 
ct =fs 


= \Z. 


vu = qr 


: T' 


HORIZONTAL TIES (tension). 

_4/7 


ts = 1 07 
8 

DIAGONAL BRACES (compression). 


my = mn 


16 


xy = 


- if 

8 ' 


mw = mp 


= g .- 


o/z; = qp 


~"i6 


mu = mr 


16 


fu = sr 


** 8 



1/0 APPLIED MECHANICS. 

Next, as to the stresses due to wind pressure, we will sup- 
pose that there is a roller under the left-hand end of the truss, 
and none under the right-hand end ; and we will proceed to 
determine the stresses due to wind pressure. 

First, suppose the wind to blow from the left-hand side of 
rhe truss, and let the total wind pressure be (Fig. 8$b) af= W^. 
The resultant, of course, acts along the dotted line drawn per- 
pendicular to the left-hand rafter at its middle point, as shown 
in Fig. 85. 

The left-hand supporting force will be vertical : hence, pro- 
ducing the above-described dotted line, and a vertical through 
the roller to their intersection, and joining this point with the 
right-hand end of the truss, we have the direction of the right- 
hand supporting force. In this case, since the angle of the 
truss is 30, the line of action of the right-hand supporting 
force coincides in direction with the right-hand rafter. We 
now construct the triangle of external forces afm y and we 
obtain the supporting forces fm and ma. We then have, as 
the loads at the joints,' 

ab - = ef, 



be = - = cd = de. 

4 

Then proceed as before to the left-hand joint ; and we find that 
two of the four forces acting there are known, viz., ma and ab, 
and two are unknown, viz., the stresses in .Z? Fand YM. Then 
construct the quadrilateral mabym, and we have the stresses by 
and ym ; the first being compression and the second tension, 
as shown by reasoning similar to that previously adopted. 

Then pass to the next joint on the rafter, and construct the 
quadrilateral ybcxy, where yb and be are already known, and we 
obtain ex and xy ; and so proceed as before from joint to joint, 



ROOF-TRUSSES WITH LOADS AT LOWER JOINTS. I/T 

remembering, that, in order to be able to construct the polygon 
of forces in each case, it is necessary that only two of the forces 
acting should be unknown. 

When the wind blows from the other side, we shall obtain 
the diagram shown in Fig. 85^. 

After having determined the stresses from the vertical load 
diagram and those from the two wind diagrams, we should, in 
order to obtain the greatest stress that can come on any one 
member of the truss, add to the stress due to the vertical load 
the greater of the stresses due to the wind pressure. 

136. Roof-Truss with Loads at Lower Joints. In 
Fig. 86 is drawn a stress diagram 
for the truss shown in Fig. 84 on 
the supposition that there is also X. 
a load on the lower joints. In 
this case let W be the whole load 
of the truss, except the ceiling, 
^ the weight of the ceiling 



and 

below ; the latter being supported 

a,t the lower joints and on the 

two extreme vertical suspension FlG - 86 - 

rods. Then will the loads at the joints be as follows; viz., 




ab = 

be = \( 
cd = \W 
mn 



= rq 



= kl, 

= gh = de = jfe 

= on = qp = op. 



Observe that there is no joint at the lower end of either of the 
end suspension rods, but that whatever load is supported by 
these is hung directly from the upper joints, where be and hk act 
We have also for each of the supporting forces Im and ra 



1/2 APPLIED MECHANICS. 

Hence we have, for the polygon of external forces, 
abcdefghklm nopqra, 

which is all in one straight line, and which laps over on 
itself. 

In constructing the diagram, we then proceed in the same 
way as heretofore. 

137. General Remarks. As to the course to be pursued 
in general, we may lay down the following directions : 

I . Determine all the external forces ; in other words, the loads 
being known, determine the supporting forces. 

2. Construct the polygon of forces for each joint of the truss, 
beginning at some joint where only two of the forces acting at 
that joint are unknown. This is usually the case at the support. 
Then proceed from joint to joint, bearing in mind that we can 
only determine the polygon of forces when the magnitudes of 
all but two sides are known. 

3. Adopt a certain direction of rotation, and adhere to it 
throughout; i.e., if we proceed in right-handed rotation at one 
joint, we must do the same at all, and we shall thus obtain neat 
and convenient figures. 

4. Observe that the stresses obtained are the forces exerted 
by the bars under consideration, and that these are thrusts when 
they act away from the bars, and tensions when they are directed 
towards the bars. 

We will next take some examples of roof-trusses, and con- 
struct the diagrams of some of them, calling attention only to 
special peculiarities in those cases where they exist. 

It will be assumed that the student can make the trigono- 
metric computations from the diagram. 

The scale of load and wind diagram will not always be the 
same ; and the stress diagrams will in general be smaller than 
is advisable in using them, and very much too small if the 



ROOF-TRUSSES WITH LOADS AT LOWER JOINTS. 173 

results were to be obtained by a purely graphical process with- 
out any computation. 

The loads will in all cases be assumed to be distributed 
uniformly over the jack-rafters, or, in other words, concen- 
trated at the joints. 

Those cases where no stress diagram is drawn may be con- 
sidered as problems to be solved. 




FIG. 87. 



FIG. 870. 




FIG. 87*. 



174 



APPLIED MECHANICS. 




PIG. 88. 



FIG. 883. 



abcdl 



FIG. 



R O OF- TR USSES WITH LOADS A % L O WER JOINTS. I 7 5 




FIG. 89. 



FIG. 8ga. 




FIG. 90. 



FIG. 




a 
b 

c A 




X 



FIG. 92. 



FIG. 92*. 



FIG. 93. 



FIG. 93 a. 



APPLIED MECHANICS. 



138. Hammer- Beam Truss (Fig. 94). This form of 
truss is frequently used in constructions where architectural 
effect is the principal consideration rather than strength. It 
is not an advantageous form from the point of view of strength, 




FIG. 



FIG. 94. 





FIG. 944. 




FIG. 94,:. 



for the absence of a tie-rod joining the two lower joints causes 
a tendency to spread out at the base, which tendency is usually 
counteracted by *the horizontal thrust furnished by the but 
tresses against which it is supported. 



HAMMER-BEAM TRUSS. 177 

When such a thrust is furnished (or were there a tie-rod), 
and the load is symmetrical and vertical, the bars are not all 
needed, and some of them are left without any stress. In 
the case in hand, it will be found that UV, VM, MQ, and QR 
are not needed. We must also observe that the effect of the 
curved members MY, MV, MQ, and MAT on the other parts of 
the truss is just the same as though they were straight, as 
shown in the dotted lines. The curved piece, of course, has to 
be subjected to a bending-stress in order to resist the stress 
acting upon it. If, as is generally the case, the abutments are 
capable of furnishing all the horizontal thrust needed, it will 
first be necessary to ascertain how much they will be called 
upon to furnish. To do this, observe that we have really a truss 
similar to that shown in Fig. 92, supported on two inclined 
framed struts, of which the lines of resistance are the dotted 
lines (Fig. 94) I 4 and 7 8, and that, under a symmetrical load, 
this polygonal frame will be in equilibrium, and, moreover, the 
curved pieces MV and MQ will be without stress, these only- 
being of use to resist unsymmetrical loads, as the snow or 
wind. 

Let the whole load, concentrated by means of the purlins 
at the joints of the rafters, be W. Then will the truss 467 have 

W 

to bear \ W, and this will give to be supported at each of 

4 
the points 4 and 7. Moreover, on the space 2 4 is distributed 

, which has, as far as overturning the strut is concerned, the 
4 

W W 

same effect as at 2, and at 4. Hence the load to be sup- 
8 8 

ported at 4 by the inclined strut is a vertical load equal to 

(i + J) w 1 w - We ma y then find the force that must be 

furnished by the abutment, or by the tie-rod, in either of the 
two following ways : 



178 APPLIED MECHANICS 

i. By constructing the triangle ySe (Fig. 94*2), with 78 = 
| W, ye || 14, and eS parallel to the horizontal thrust of the abut- 
ment ; then will y& be the triangle of forces at I, and eS will be 
the thrust at i. 

2. Multiply f W by the perpendicular distance from 4 to 
i 2, and divide by the height of 4 above I 8 for the thrust of the 
abutment ; in other words, take moments about the point i. 

Now, to construct the diagram of stresses, let, in Fig. 94^, 
the loads be 

ab, be, cd, </<?, ef,fg, gh, hk, and kl t 
and let 

lz = za = \W 

be the vertical component of the supporting force ; let zm be 
the thrust of the abutment : then will Im and ma be the real 
supporting forces ; and we shall have, for polygon of external 
forces, 

abcdefghklma. 

Then, proceeding to the joint i, we obtain, for polygon of forces, 

maym ; 

and, proceeding from joint to joint, we obtain the stresses in all 
the members of the truss, as shown in Fig. 94^. 

It will be noticed that UV and RQ are also free from 
stress. 

If we had no horizontal thrust from the abutment, and the 
supporting forces were vertical, the members MV and MQ 
would be called into action, and J/Fand MN would be inactive. 
To exhibit this case, I have drawn diagram 94/7, which shows 
the stresses that would then be developed. A Fand NL would 
become merely part of the supports. 

In this latter case the stresses are generally much greater 
than in the former, and a stress is developed in UV. 



SCISSOR-BEAM TRUSS. 



139. Hammer-Beam Truss: Wind Pressure. Fig. 95 
shows the stress diagram of the hammer-beam truss for wind 
pressure when there is no roller under either end, and when 
the wind blows from the left. A similar diagram would give the 
stresses when it blows from the right. 




FIG. 95. 



FIG. 



The cases when there is a roller are not drawn : the student 
may construct them for himself. 

140. Scissor-Beam Truss. We have already discussed 
two forms of scissor-beam truss 
in Figs. 90 and 91. These 
trusses having the right number 
of parts, their diagrams present 
no difficulty. Another form of A 
the scissor-beam truss is shown 
in Fig. 96, and its diagram pre- 
sents no difficulty. 

The only peculiarity to be noticed is, that, after having coa 
structed the polygon of external forces, 

abcdefma, 

we cannot proceed to construct the polygon of equilibrium for 
one of the supports, because there are three unknown forces 




FlG - 



FlG - ^ 



1 8o 



APPLIED MECHANICS. 



there. We therefore begin at the apex CD, and construct the 
triangle of forces cdl for this point ; then proceed to joint CB, 
and construct the quadrilateral 

bclkb; 
then proceed to the left-hand support, and obtain 

mabkgm ; 
and so continue. 

141. Scissor-Beam Truss -without Horizontal Tie. 

Very often the scissor-beam truss is constructed without any 
horizontal tie, in which case it has the appearance of Fig. 97, 
where there is sometimes a pin at GKLH and sometimes not. 





FIG. gja. 




FIG. 97. 



FIG. 97<5. 



FIG. 97 c. 



In this case, if the abutments are capable of furnishing hori- 
zontal thrust to take the place of the horizontal tie of Fig. 96, 
we are reduced back to that case. If the abutments are not 
capable of furnishing horizontal thrust, we are then relying on 
the stiffness of the rafters to prevent the deformation of the 
truss ; for, were the points BC and DE really joints, with pins, 
the deformation would take place, as shown in Fig. 97^ or Fig. 
97^, according as the two inclined ties were each made in one 
piece or in two (i.e., according as they are not pinned together 
at KH, or as they are pinned). This necessity of depending 
on the stiffness of the rafters, and the liability to deformation 
if they had joints at their middle points, become apparent as 
soon as we attempt to draw the diagram. Such an attempt is 



SCISSOR-BEAM TRUSS WITHOUT HORIZONTAL TIE. l8l 

made in Fig. 97^, where abcdefga is the polygon of external 
forces, gabkg the polygon of stresses for the left-hand support, 
kbclk that for joint BC. Then, on proceeding to draw the tri- 
angle of stresses for the vertex, we find that the line joining d 
and / is not parallel to DL, and hence that the truss is not 
stable. We ought, however, in this latter case, when the sup- 
porting forces are vertical, and when we rely upon the stiffness 
of the rafters to prevent deformation, to be able to determine 
the direct stresses in the bars ; and for this we will employ an 
analytical instead of a graphical method, as being the most con- 
venient in this case. 

Let us assume that there is no pin at the intersection of the 
two ties, and that the two rafters are inclined at an angle of 45 
to the horizon. 

We then have, if W = the entire load, and a = angle 
between BK and KG, 

w w 

ab = cf = , be ^ cd = de = , 
8 4 

T 2 

tana = 4, sin a = , cos a = , 

Vs ^s 

Let x be the stress in each tie, and let y = cl dl = thrust 
in each upper half of the rafters. 

Then we must observe that the rafter has, in addition to its 
direct stresses, a tendency to bend, due to a normal load at the 
middle, this normal load being equal to the sum of the normal 
components of be and of x> when these are resolved along and 
normal to the rafter. Hence 



normal load = x cos a -\ sin 45 

4 



This, resolved into components acting at each end of the rafter r 
gives a normal downward force at each end equal to 

-f- 



I 82 APPLIED MECHANICS. 

Hence, resolving all the forces acting at the left-hand support 
into components along and at right angles to the rafter, and 
imposing the condition of equilibrium that the algebraic sum 
of their normal components shall equal zero, we have, if we call 
upward forces positive, 

-f JFsin45 (%xcosa + ^-fFsin45) #sina = o; (i) 
but, since 



we have from (i) 

W 
2#sina = sin 45 

4 

W o 

/. jfsma = sin 45 
8 



( 

Then, proceeding to the apex of the roof, we have that the load 

, W 
cd = 

4 
gives, when resolved along the two rafters, a stress in each 

equal to 



4 

Hence the load to be supported in a direction normal to the 
rafter at the apex is 

sin 45 -f- (^ cos a -\ -- sin 45). 
4 8 

Hence, substituting for x its value, we have 

y = cl=dl= 5Tsin 4 5. (3) 

Then, proceeding to the left-hand support, and equating to zero 
the algebraic sum of the components along the rafter, we have 

bk = (ga 0)cos45 ~f~ -^cosa 

-f JWsii^ = f ^5^45. (4) 



SCISSOR-BEAM TRUSS WITHOUT HORIZONTAL TIE. 183 



We have thus determined in (2), (3), and (4) the values of x y y, 
and bk eh. 

By way of verification, proceed to the middle of the left- 
hand rafter, and we find the algebraic sum of the components 
of be and x along the rafter to be 



and this is the difference between bk and cl, as it should be. 

We have thus obtained the direct stresses ; and we have, in 
addition, that the rafter itself is also subjected to a bending- 
moment from a normal load at the centre, this load being equal 
to 

xcosa H -- sin 45 = sin 45. 
4 2 

How to take this into account will be explained under the 
" Theory of Beams." 

142. Examples. The following figures of roof -trusses 
may be considered as a set of examples, for which the stress 
diagrams are to be worked out. 

Observe, that, wherever there is a joint, the truss is to be 
supposed perfectly flexible, i.e., free to turn around a pin. 




FIG. 98. 



FIG. 99- 



FIG. too. 



FIG. 101. 




FIG. 102. 



FIG. 103. 



FIG. 104. 



FIG. 






FIG. 106. 



FIG. 107. 



FIG. 108 



1 84 APPLIED MECHANICS. 



CHAPTER IV. 
BRIDGE-TRUSSES. 

143. Method of Sections. It is perfectly possible to 
determine the stresses in the members of a bridge-truss 
graphically, or by any methods that are used for roof-trusses. 

In this work an analytical method will be used ; i.e., a method 
of sections. This method involves the use of the analytical con- 
ditions of equilibrium for forces in a plane explained in 63. 
These are as follows ; viz., 

If a set of forces in a plane, which are in equilibrium, be 
resolved into components in two directions at right angles to 
each other, then 

i. The algebraic sum of the components in one of these 
directions must be zero. 

2. The algebraic sum of the components in the other of 
these directions must be zero. 

3. The algebraic sum of the moments of the forces about 
any axis perpendicular to the plane of the forces must be zero. 

Assume, now, a bridge-truss (Figs. 109, no, in, 112, pages 
186 and 187) loaded at a part or all of the joints. Conceive a 
vertical section ab cutting the horizontal members 6-8 and 7~9 
and the diagonal 7-8, and dividing the truss into two parts. 
Then the forces acting on either part must be in equilibrium, 
in other words, the external forces, loads, and supporting forces, 
acting on one part, must be balanced by the stresses in the 
members cut by the section ; i.e., by the forces exerted by the 
other part of the truss on the part under consideration. Hence 
we must have the three following conditions ; viz., - 



SHEARING-FORCE AND BENDING-MOMENT. .185 

i. The algebraic sum of the vertical components of the 
above-mentioned forces must be zero, 

2. The algebraic sum of the horizontal components of these 
forces must be zero. 

3. The algebraic sum of the moments of these forces about 
any axis perpendicular to the plane of the truss must be zero. 

144. Shearing-Force and Bending-Moment. Assum- 
ing all the loads and supporting forces to be vertical, we shall 
have the following as definitions. 

The Shearing-Force at any section is the force with which 
the part of the girder on one side of the section tends to slide 
by the part on the other side. 

In a girder free at one end, it is equal to the sum of the 
loads between the section and the free end. 

In a girder supported at both ends, it is equal in magnitude 
to the difference between the supporting force at either end, 
and the sum of the loads between the section and that support- 
ing force. 

The Bending-Moment at any section is the resultant moment 
of the external forces acting on the part of the girder to one side 
of the section, tending to rotate that part of the girder around 
a horizontal axis lying in the plane of the section. 

In a girder free at one end, it is equal to the sum of the 
moments of the loads between the section and the free end, 
about a horizontal axis in the section. 

In a girder supported at both ends, it is the difference be- 
tween the moment of either supporting force, and the sum of 
the moments of the loads between the section and that sup- 
port ; all the moments being taken about a horizontal axis in 
the section. 

145. Use of Shearing-Force and Bending-Moment. 
The three conditions stated in 143 may be expressed as fol- 
lows : 

i. The algebraic sum of the horizontal components of the 
stresses in the members cut by the section must be zero. 



1 86 



APPLIED MECHANICS. 



2. The algebraic sum of the vertical components of the 
stresses in the members cut by the section must balance the 
shearing-force. 

3. The algebraic sum of the moments of the stresses in 
the members cut by the section, about any axis perpendicular to 
the plane of the truss, and lying in the plane of the section, 
must balance the bending-moment at the section. 

As the conditions of equilibrium are three in number, they 
will enable us to determine the stresses in the members, pro- 
vided the section does not cut more than three ; and this 
determination will require the solution of three simultaneous 
equations of the first degree with three unknown quantities 
(the stresses in the three members). 

By a little care, however, in choosing the section, we can 
very much simplify the operations, and reduce our work to the 
solution of one equation with only one unknown quantity ; the 
proper choice of the section taking the place of the elimination. 

146. Examples of Bridge-Trusses. Figs. 109-1 12 rep- 
resent two common kinds of bridge-trusses : in the first two 

the braces are all 

i 3 5 _7]ajk.n..i3 45 17 19 21 23 25 27 29 diagonal, in the 

last two they are 
partly vertical and 
partly diagonal. 

The first two are called Warren girders, or half-lattice girders ; 
since there is only one system of bracing, 
as in the figures. When, on the other 
hand, there are more than one system, so 
that the diagonals cross each other, they 
are called lattice girders. 

147. General Outline of the Steps 
to be taken in determining the Stresses 
in a Bridge-Truss under a Fixed Load. 

i. If the truss is supported at both ends, find the sup- 
porting forces. 



VVV\/K/V\/\/\A/\/\/\A/ 

2 4 6 b\ 8 *| 10 12 14 16 18 20 22 24 2628 
FIG. 109. 



1357 


a 91 a- 11 13 


vw 


2157 


2466 


8 j 10 13 



FIG. no. 



DETERMINING THE STRESSES IN A BRIDGE-TRUSS. 1 87 



2. Assume, in all cases, a section, in such a manner as not 
to cut more than three members if possible, or, rather, three 
of those that 

1 13 15 17 19 21 23 25 27 28 

XlXIXbd/l/l/1// 



brought 



7 a 



\ 



10 12 14 16 18 20 22 24 26 



FlG - 



2 4 6 


R 


10 12 14 


/\/]/\, 


/ 


/MX 


1357 


9 


11 13 



FIG. 



are 

into action 
by the loads 
on the truss ; 
and it will 

save labor if we assume the section so as to cut two of the 

three very near their point of inter- 
section. 

3. Find the shearing-force at the 
section. 

4. Find the bending-moment at 
the section. 

5. Impose the analytical conditions of equilibrium on all 
the forces acting on the part of the girder to one side of the 
section, the part between the section and the free end when 
the girder is free at one end, or either part when it is supported 
at both ends. 

In the cases shown in Figs. 109 and no, we may describe 
the process as follows ; viz., 

(a) Find the stress in the diagonal from the fact, that (since 
the stress in the diagonal is the only one that has a vertical 
component at the section) the vertical component of the stress 
in the diagonal must balance the shearing-force. 

(b) Take moments about the point of intersection of the 
diagonal and horizontal chord near which the section is taken ; 
then the stresses in those members will have no moment, so 
that the moment of the stress in the other horizontal must 
balance the bending-moment at the section. Hence the stress 
in the horizontal will be found by dividing the bending-moment 
at the section by the height of the girder. 

The above will be best illustrated by some examples. 



I 88 APPLIED MECHANICS. 

EXAMPLE I. Given the semi-girder shown in Fig. no, 
loaded at joint 13 with 4000 pounds, and at each of the joints 
l > 3> 5> 7> 9 an d ii with 8000 pounds. Suppose the length of 
each chord and each diagonal to be 5 feet. Required the stress 
in each member. 

Solution. For the purpose of explaining the method of 
procedure, we will suppose that we desire to find first the 
stresses in 8-10 and 9-10. 

Assume a vertical section very near the joint 9, but to the 
right of it, so that it shall cut both 8-10 and 9-10. 

If, now, the truss were actually separated into two parts at 
this section, the right-hand part would, in consequence of the 
loads acting on it, separate from the other part. This tendency 
to separate is counteracted by the following three forces : 

i. The pull exerted by the part <$-x of the bar 9-11 on the 
part x-\\ of the same bar. 

2. The thrust exerted by the part 8-2 of the bar 8-10 on 
the part ^-10 of the same bar. 

3. The pull exerted by the part 9-7 of the bar 9-10 on the 
part y-io of the same bar. 

The shearing-force at this section is 

8000 -f- 4000 = 12000 Ibs., 

and this is equal to the vertical component of the stress in the 
diagonal. Hence 

T 2OOO 

Stress in 9-10 = = 12000(1.1547) = 13856 Ibs. 

This stress is a pull, as may be seen from the fact, that, in 
order to prevent the part of the girder to the right of the 
section from sliding downwards under the action of the load, 
the part 9-7 of the diagonal 9-10 must pull the part y-io of 
the same diagonal. 

Next take moments about 9 : and, since the moment of the 
stresses in 9-1 1 and 9-10 about 9 is zero, we must have that the 
moment of the stress in 8-10; i.e., the product of this stress 
by the height of the girder, must equal the bending-moment. 



DETERMINING THE STRESSES 2N A BRIDGE-TRUSS. 189 



The bending-moment about 9 is 

8000 x 5 4- 4000 x 10 = 80000 foot-lbs. 
80000 



Hence 



Stress in 8-10 



4-33 



80000(0.23094) = 18475 



Proceed in a similar way for all the other members. The 
work may be arranged as in the following table ; the diagonal 
stresses being deduced from the shearing-forces by multiplying 
by 1.1547, and the chord stresses from the bending-moments 
by multiplying by 0.23094. 



2_ 




Stresses in Diagonals cut 




Stresses in Chords opposite the 


JJ 


Shearing- 


by Section, in Ibs. 


Bending- 


respective Joints. 


c .S> 


Force 




Moment, in 




O * 


in Ibs. 






foot-lbs. 






* J 




Tension. 


Compression. 




Tension. 


Compression. 


I 


44OOO 


50806 




72OOOO 




166277 


2 


44OOO 




50806 


6lOOOO 


140873 




3 


36000 


4^69 




500000 




11547 


4 


36OOO 




41569 


4IOOOO 


94685 




5 


28000 


32331 


\ 320000 




73901 


6 


28OOO 




32331 \ 250000 


57735 




7 


2OOOO 


23094 




' ISOOOO 




41569 


8 


20000 




23094 


I3OOOO 


30022 




9 i 2000 


13856 




80000 




18475 


10 


I2OOO 




13856 


5OOOO 


,"547 




II 


4OOO 


4619 




2OOOO 




4618 


12 


4OOO 




4619 


IOOOO 


2309 





EXAMPLE II. Given the truss (Fig. 109) loaded at each oi 
the lower joints with 10000 Ibs. : find the stresses in the members. 
The length of chord is equal to the length of diagonal = 10 ft. 

Throughout this chapter, tensions will be written with the 
minus, and compressions with the plus sign. 

Solution. Total load = 14(10000) = 140000 Ibs. 

Each supporting force = 70000 " 
The entire work is shown in the following tables: 



i go 



APPLIED MECHANICS. 



CO ^O O\ 



II II 



o o o 

to >-o to 

*t VO 00 



CO * CO rt CO ^ CO 



x x 



to o ^o O to O 

H M ** M <-> 

o 4- + + + + + 

to o to o to O 



I I I I I I I I I I I I I 
O to o to O 

X X X X X X X 



10 O to 



>-o O to O 
N co co Tj- 



X X X X X X 



888 

N CO CO 



1 1 1 1 1 



CO. CO N M >-> *- 

II II II II II II II II 

I I I I I I I I 



tt 

N 



ONOO t>sVO torfcoN "-I O ONOO txO 
MNNNNMNNMNH-,H4 



i- M CO 



00 ON O NH N CO 



DETERMINING THE STRESSES IN A BRIDGE-TRUSS, 



Numbers of Diagonals. 


Stresses 


in Diagonals, in Ibs. 


I- 2 


28-29 


70000 X 


I-I547 = 


80829 


2- 3 


27-28 


+ 60000 X 


I.T547 = 


+ 69282 


3- 4 


26-27 


60000 X 


I.I547 = 


69282 


4- 5 


25-26 


+ 50000 X 


I.I547 = 


+ 57735 


5-6 


24-25 


50000 X 


LI547 = 


-57735 


6- 7 


23-24 


+40000 X 


I.I547 = 


+46188 


7- 8 


22-23 


40000 X 


LI547 = 


-46188 


8- 9 


21-22 


+ 30000 X 


LI547 = 


+ 34641 


9-10 


2O-2I 


30000 X 


LI547 = 


34641 


IO-II 


I9-2O 


+ 20000 X 


LI547 = 


+ 23094 


11-12 


18-19 


20000 X 


I.I547 = 


-23094 


12-13 


I7-I8 


+ IOOOO X 


LI547 = 


+ II547 


I3-H 


16-17 


i oooo x 


I.I547 = 


-H547 


14-15 


I5-I6 


+ 








LOWER CHORDS. 



IM umbers of Chords. 


Stresses in Chords, in Ibs. 


2- 4 


26-28 


65OOOO 


X 0.11547 = 


- 755 6 


4- 6 


24-26 


I2OOOOO 


X 0.11547 = 


138564 


6- 8 


22-24 


I65OOOO 


X 0.11547 = 


190526 


8-10 


20-22 


2OOOOOO 


X 0.11547 = 


-230940 


10-12 


I 8-20 


225OOOO 


X 0.11547 = 


259808 


12-14 


16-18 


245OOOO 


X 0.11547 = 


-277128 


I4-l6 




245OOOO 


X 0.11547 = 


282902 



1 9 2 



APPLIED MECHANICS. 



UPPER CHORDS. 



Numbers of Chords. 


Stresses in Chords, in Ibs. 


'- 3 


27-29 


350000 


X 0.11547 = 


+ 40415 


3- 5 


25-27 


950000 


x 0.11547 = 


+ 109697 


5- 7 


23-25 


I45OOOO 


X 0.11547 = 


+ 167432 


7- 9 


21-23 


1850000 


X 0.11547 = 


+ 213620 


9-1 1 


19-21 


2I5OOOO 


X 0.11547 = 


+248261 


ii 13 


17-19 


2350000 


X 0.11547 = 


+ 267355 


'j -'5 


i5-i7 


245OOOO 


X 0.11547 = 


+ 282902 



EXAMPLE III. Given the same truss as in Example II., 
loaded at 2, 4, 6, 8, 10, and 12 with 10000 Ibs. at each point, 
the remaining lower joints being loaded with 50000 Ibs. at each 
joint : find the stresses in the members. 

EXAMPLE IV. Given a semi-girder, free at one end (Fig. 
112), loaded at 2, 4, and 6 with 10000 Ibs., and at 8, 10, and 12 
with 5000 Ibs. : find the stresses in the members. 

TRAVELLING-LOAD. 

148. Half-Lattice Girder: Travelling-Load. When a 
girder is used for a bridge, it is not subjected all the time to 
the same set of loads. 

The load in this case consists of two parts, one, the dead 
load, including the bridge weight, together with any permanent 
load that may rest upon the bridge ; and the other, the moving 
or variable load, also called the travelling-load, such as the 
weight of the whole or part of a railroad train if it is a railroad 
bridge, or the weight of the passing teams, etc., if it is a common- 
road bridge. Hence it is necessary that we should be able to 
determine the amount and distribution of the loads upon the 
bridge which will produce the greatest tension or the greatest 



GREATEST DIAGONAL STRESSES IN GIRDER. 193 

compression in every member, and the consequent stress pro- 
duced. 

149. Greatest Stresses in Semi-Girder. Wherever the 
section be assumed in a semi-girder, it is evident that any- load 
placed on the truss at any point between the section and the 
free end increases both the shearing-force and the bending- 
momerit at that section, and that any load placed between the 
section and the fixed end has no effect whatever on either 
the shearing-force or the bending-moment at that section. 

Hence every member of a semi-girder will have a greater 
stress upon it when the entire load is on, than with any partial 
load. 

150. Greatest Chord Stresses in Girder supported at 
Both Ends. Every load which is placed upon the truss, no 
matter where it is placed, will produce at any section whatever a 
bending-moment tending to turn the two parts of the truss on 
the two sides of the section upwards from the supports ; i.e., so 
as to render the truss concave upwards. 

Hence every load that is placed upon the truss causes com- 
pression in every horizontal upper chord, and tension in every 
horizontal lower chord. Hence, in order to obtain the greatest 
chord stresses, we assume the whole of the moving load to be 
upon the bridge. 

151. Greatest Diagonal Stresses in Girder supported 
at Both Ends. To determine the distribution of the load 
that will produce the greatest stress of a certain kind (tension 
or compression) in any given diagonal, let us suppose the diag- 
onal in question to be 7-8 (Fig. 109), through which we take 
our section ab. Now it is evident that any load placed on the 
truss between ab and the left-hand (nearer) support will cause a 
shearing-force at that section which will tend to slide the part 
of the girder to the left of the section downwards with refer- 
ence to -the other part, and hence will cause a compressive 
stress in 7-8 ; while any load between the section and the right- 



194 APPLIED MECHANICS. 

hand (farther) support will cause a shearing-force of the oppo- 
site kind, and hence a tension in the bar 7-8. 

Now, the bridge weight itself brings an equal load upon each 
joint ; hence, when the bridge weight is the only load upon the 
truss, the bar 7-8 is in tension. 

Hence, any load placed upon the truss between the section 
and the farther support tends to increase the shearing-force at 
that section due to the dead load (provided this is equally dis- 
tributed among the joints) ; whereas any load placed between 
the section and the nearer support tends to decrease the shear- 
ing-force at the section due to the dead load, or to produce a 
shearing-force of the opposite kind to that produced by the dead 
load at that section. 

Hence, if we assume the dead load to be equally distributed 
among the joints, we shall have the two following propositions 
true : 

(a) In order to determine the greatest stress in any diagonal 
which is of the same kind as that produced by the dead load, 
we must assume the moving load to cover all the panel points 
between the section and the farther abutment, and no other 
panel points. 

(b) In order to determine the greatest stress in any diagonal 
of the opposite kind to that produced by the dead load, we must 
assume the moving load to cover all the panel points between 
the section and the nearer abutment, and no others. 

This will be made clear by an example. 

EXAMPLE I. Given the truss shown in Fig. 113. Length 

of chord = length of diagonal = 
A A A g !Lu 10 feet. Dead load = 8000 Ibs. 



Y 4 Y Y Y Y Y Ypl applied at each upper panel point. 
FIG Moving load = 30000 Ibs. applied 

at each upper panel point. Find 
the greatest stresses in the members. 



EXAMPLE OF BRIDGE-TRUSS, 



195 



Solution, (a) Chord Stresses. Assume the whole load to 
be upon the bridge : 
this will give 38000 
each 



(1) + 76788 (3) 4-. 20R423 (5) -f- 296181 (7; + 340059 (9) 




(2) -153575 (4) -263272(6) -329090 (8)- 3510251 



Ibs. at eacn upper 
panel point ; i.e., omit- 
ting I and 17, where 
the load acts directly 
on the support, and 
not on the truss. FlG ' II4- 

Hence, considering the bridge so loaded, we shall have the fol- 
lowing results for the chord stresses : 

Each supporting force = sSooof-J 133000. 



Section at 


Bending-Moment, in foot-lbs. 


2 16 


133000 x 5 




= 665000 


3 15 


133000 X IO 




= 1330000 


4 14 


133000 X 15 


38000 x 5 


= 1805000 


5 13 


133000 X 2O 


38000 X 10 


= 2280000 


6 12 


133000 X 25 


- 3 8ooo( 5 -f- 15) 


= 2565000 


7 ii 


133000 X 30 


- 38000(10 + 20) 


= 2850000 


8 10 


133000 x 35 


38ooo( 5 + 15 -{- 


25) = 2945000 


9 


133000 X 40 


38000(10 4- 20 -j- 


30) = 3040000 



Numbers of Chords. 


Stresses in Upper 


Chords. 


i-3 


I5-I7 


665000 X 0.11547 


= + 76788 


3-5 


I3-I5 


1805000 X 0.11547 


= +208423 


5-7 


11-13 


2565000 X 0.11547 


= +296181 


7-9 


9-1 1 


2945000 X 0.11547 


= +340059 



APPLIED MECHANICS. 



Numbers of Chords. 


Stresses 


in Lower Chords. 


2- 4 


14-16 


1330000 X 


O.II547 = 


-153575 


4- 6 


12-14 


2280000 X 


O.II547 = 


-263272 


6- 8 


IO-I2 


2850000 X 


O.II547 = 


329090 


8-10 




3040000 X 


O.II547 = 


-351029 



Next, as to the diagonals, take, for instance, the diagonal 
7-8. When the dead load alone is on the bridge, the diagonal 
7-8 is in tension. From the preceding, we see that the greatest 
tension is produced in this bar when the moving load is on the 
points 9, n, 13, and 15, and the dead load only on the points 3, 
5, 7. Now, a load of 38000 Ibs. at 13, for instance, causes a 

shearing-force of (38000) = 9500 Ibs. at any section to the 

10 

left of 13; and this shearing-force tends to cause the part to 
the left of the section to slide upwards, and that to the right 
downwards. 

On the other hand, with the same load at the same place, 

there is produced a shearing-force of (38000) = 28500 Ibs. 

16 

at any section to the right of 13 ; and this shearing-force tends 
to cause the part to the left to slide downwards, and that to the 
right upwards. Paying attention to this fact, we shall have, 
when the loads are distributed as above described, a shearing- 
force at the bar 7-8 causing tension in this bar ; the magnitude 
of this shearing-force being 



6 + 8) _ 



i6 16 

Hence, we may arrange the work as follows : 



6 ) = 41500. 



GREATEST DIAGONAL STRESSES IN GIRDER. 



197 







Greatest 






Stresses in 


Numbers of 


Greatest Shearing-Forces producing Stresses of Same Kind as 


Diagonals of 


Diagonals. 


Dead Load. 


as those due 






to Dead 






Load. 


1-2 


17-16 


3^ (2 + 4+6+8+IO+I2+I4) = 133000 


-'53575 


2-3 


16-15 


^(2+4+6+8+10+12+14) = I33 ooo 


+ '53575 


3-4 


'5-H 


3 ^(2+ 4 +6+8+io+ I2 )-~(2) = 98750 


114027 


4-5 


14-13 


^(2+4+6+8+10+12) -^(2) = 98750 


+ i 14027 


5-6 


13-12 


^(2+4+6+8+10) -^(2+4) = 68250 


- 78808 


6-7 


I 2-1 1 


^(2+4+6+8 + 10) -^(2+4) - 68250 


+ 78808 


7-8 


II-IO 


^(2+4+6+8) -^(2+4+6) = 41500 


- 47920 


8-9 


io- 9 


^(2+4+6+8) - ^(2+4+6) - 41500 


+ 479 20 







Greatest 






Stresses in 


Numbers of 


Greatest Shearing-Forces producing Stresses of Kind Opposite 


Diagonals ^pf 


Diagonals. 


from Dead Load. 


Kind Oppo- 






site from 






Dead Load. 


8-9 


io- 9 


^(2+ 4 +6) - ^(2+4+6+8) - 18500 


-21362 


7-8 


II-IO 


^(2+4+6) - ?^( 2 + 4 +6+8) - 18500 


+ 21362 



The diagonals 7-8, 8-9, 9-10, and 10-11 are the only ones 
that, under any circumstances, can have a stress of the kind 
opposite to that to which they are subjected under the dead 
load alone. 



I9 8 APPLIED MECHANICS. 

Fig. 114 exhibits the manner of writing the stresses on the 
diagram. 

152. General Application of this Method. It is plain 
that the method used above will apply to any single system of 
bridge-truss with horizontal chords and diagonal bracing, what- 
ever be the inclination of the braces. 

When seeking the stress in a diagonal, the section must be 
so taken as to cut that diagonal ; and, as far as this stress alone 
is concerned, it may be equally well taken at any point, as well 
as near a joint, provided only it cuts that diagonal which is in 
action under the load that produces the greatest stress in this 
one, and no other. 

On the other hand, when we seek the stress in a horizontal 
chord, the section might very properly be taken through the 
joint opposite that chord. 

Taking it very near the joint, only serves to make one sec- 
tion answer both purposes simultaneously. 

153. Bridge-Trusses with Vertical and Diagonal Bra- 
cing. When, as in Figs, in and 112, there are both vertical 
and diagonal braces, and also horizontal chords, we may deter- 
mine the stresses in the diagonals and in the chords just as 
before ; only we must take the section just to one side of a joint, 
and never through the joint. 

As to the verticals, in order to determine the stress in any 
vertical, we must impose the conditions of equilibrium between 
the vertical components of the forces acting at one end of that 
vertical: thus, if the loads are at the upper joints in Fig. in, 
then the stress in vertical 3-2 must be equal and opposite to 
the vertical component of the stress in diagonal 1-2, as these 
stresses are the only vertical forces acting at joint 2. 

Vertical 5-4 has for its stress the vertical component of the 
stress in 3-4, etc. Thus 

Stress in 3-2 = shearing-force in panel 1-3, 
Stress in 5-4 = shearing-force in panel 3-5, etc.- 



TRUSSES WITH VERTICAL AND DIAGONAL BRACING. 199 

On the other hand, if the loads be applied at the lower 
joints, then 

Stress in 3-2 = shearing-force in panel 3-5, 
Stress in 5-4 = shearing-force in panel 5-7, etc. 

EXAMPLE. Given the truss shown in Fig. in. Given 
panel length = height of truss 10 feet, dead load per panel 
point = 12000 Ibs., moving load per panel point = 23000 Ibs. ; 
load applied at upper joints. 

Solution, (a) Chord Stresses. Assume the entire load on 
the bridge, i.e., 35000 Ibs. per panel point. Hence 

Total load on truss =13 (35000) = 455000 Ibs., 
Each supporting force = 227500 Ibs. 



Joint near 
which 
Section is 
taken. 


Bending-Moment at the Section very near the Joint, on 


Either Side of the Joint. 


I 28 







3 27 


227500 x 10 


= 2275000 


5 25 


227500 X 20 35000 X 10 


= 4200000 


7 23 


227500 X 30 35000(10 + 20) 


= 5775000 


9 21 


227500 X 40 35000 ( 10 + 20 + 30) 


7000000 


ir 19 


227500 X 50 35000 (10 + 20 + 30 + 40) 


= 7875000 


13 17 


227500 X 60 35000(10 + 20 -h 30 + 40 + 


50) = 8400000 


IS 


227500 X 70 35000 (10 + 20 + 30 + 40 + 


50 -f- 60) = 8575000 



To find any chord stress, divide the bending-moment at a 
section cutting the chord, and passing close to the opposite 
joint, by the height of the girder, which in this case is 10. 
Hence we have for the chord stresses (denoting, as before, com- 
pression by +, and tension by ) : 



2OO 



APPLIED MECHANICS. 



Stresses in Upper Chords. 


Stresses in Lower Chords. 


i- 3 


27-28 


+ 227500 


2- 4 


24-26 


227500 


3- 5 


25-27 


4-420000 


4- 6 


22-24 


420000 


5- 7 


2 3- 2 5 


+ 5775 


6- 8 


20-22 


-5775 


7- 9 


21-23 


+ 700000 


8-10 


18-20 


700000 


91 1 


19-21 


+ 787500 


IO-I2 


1 6-1 8 


-787500 


11-13 


17-19 


+ 840000 


12-14 


14-16 


840000 


i3-!5 


iS- 1 ? 


+ 8575 









Diagonals. It is evident, that, for the diagonals, the same 
rule holds as in the case of the Warren girder : i.e., the greatest 
stress of the same kind as that produced by the dead load 
occurs when the moving load is on all the joints between the 
diagonal in question and the farther abutment ; whereas the 
greatest stress of the opposite kind occurs when the moving 
load covers all the joints between the diagonal in question and 
the nearer abutment. 

The work of determining the greatest shearing-forces may 
be arranged as in tables on p. 191. 

Counterbraces. If the truss were constructed with those 
diagonals only that slope downwards towards the centre, and 
which may be called the main braces, the diagonals 1 1-12, 
13-14, 14-17, and 16-19 would sometimes be called upon to 
bear a thrust, and the verticals 12-13 and 17-16 a pull : this 
would necessitate making these diagonals sufficiently strong 
to resist the greatest thrust to which they are liable, and fixing 
the verticals in such a way as to enable them to bear a pull. 

In order to avoid this, the diagonals 10-13, 12-15, I 5~ I 6, 
and 17-18 are inserted, which are called counterbraces, and 
which come into action only when the corresponding main 



TRUSSES WITH VERTICAL AND DIAGONAL BRACING. 2OI 



braces would otherwise be subjected to thrust. They also 
prevent any tension in the verticals. 



Diagonals. 


Greatest Shearing- Forces of the Same Kind as those produced by 
Dead Load. 


I- 2 


28-26 


^ I+2 + 3 + ... +I3 ) 


= 227500 


3- 4 


27-24 


3J^ (l + 2 + 3+ ... + I2) _I5^( l) 


= I94H3 


5-6 


25-22 


3 -^P(l + 2 + 3+ ... + II)- '-^(1 + 2) 


= 162429 


7-8 


23-20 


^(i + 2+3+ . . . + 10) - x -^?(i + 2+3) 

J-4 J-4 


= 132357 


9-10 


2I-I8 


^(1 + 2+3+ ...+ 9)-^p(i + 2+.. 


+ 4) = 103929 


11-12 


19-16 


33222(1 + 2+3+...+ 8)-^(i + 2+.. 


+ 5)= 77H3 


I3T4 


17-14 


3^ (l + 2 + 3 +...+ 7) _^ I + 2+ .. 


+ 6)= 52000 



Diagonals. 


Greatest Shearing-Forces of the Opposite Kind to those produced by 
Dead Load. 


3-14 


17-14 


^ (I+2+3+ ... + 6) _^ I+2+ ... +7) 


= 28500 


11-12 


19-16 


22f(i + 2+ ... + 5 )_H5p ( r + 2 + ... + 8) 


= 6643 


9-10 


2I-I8 


^(, + 2+ ... + 4) -^ (l + 2 + ... + 9) 


= ~i357i 



The main braces and counterbraces of a panel are never in 
action simultaneously. Hence we have, for the greatest stresses 
in the diagonals, the following results, obtained by multiplying 

the corresponding shearing-forces by - 1.414. 

cos 45 



2O2 



APPLIED MECHANICS. 



In the following I have used this number to three decimal 
places, as being sufficiently accurate for practical purposes. 



Stresses in Main Braces. 


Stresses in Counterbraces. 


I- 2 


28-26 


-321685 


15-12 


15-16 


40299 


3- 4 


27-24 


-274518 


I3-IO 


I7-I8 


- 9393 


5-6 


25-22 


-229675 








7- 8 


23-20 


-187153 








9-10 


21-18 


146956 








11-12 


19-16 


109080 








i3~ I 4 


17-14 


- 735 28 









Vertical Posts. Since the loads are applied at the upper 
joints, the conditions of equilibrium at the lower joints require 
that the thrust in any vertical post shall be equal to the vertical 
component of the tension in that diagonal which, being in action 
at the time, meets it at its lower end. 

Hence it is equal to the shearing-force in that panel where 
the acting diagonal meets it at its lower end. 

We therefore have, for the posts, the following as the greatest 
thrusts : 

STRESSES IN VERTICALS. 



3- 2 


27-26 


+ 2275OO 


5- 4 


25-24 


+ I94M3 


7- 6 


23-22 


+ 162429 


9- 8 


2I-2O 


+ 132357 


II-IO 


I9-I8 


+ 103929 


13-12 


I7-l6 


+ 77143 


i5- J 4 


+ 52000 



CONCENTRATING THE LOAD AT THE JOINTS. 



203 



X 



X 



X 



FIG. 





Fig. 115 shows the stresses marked on the diagram. 

154. Manner of Concentrating the Load at the Joints. 

In using the methods given above, we are 

assuming that all the loads are concentrated 

at the joints, and that none are distributed 

over any of the pieces. As far as the mov- 
ing load is concerned, and also all of the 

dead load except the weight of the truss 

itself, this always is, or ought to be, effected ; 

and it is accomplished in a manner similar 

to that adopted in the case of roof-trusses. 

This method is shown in the figure (Fig. 

1 1 6); floor-beams being laid across from 
girder to girder at the joints, 
on top of which are laid longi- 
tudinal beams, and on these 
the sleepers if it is a railroad 
bridge, or the floor if it is a 
road bridge. The weight of 
the truss itself is so small a 
part of what the bridge is 
called upon to bear, that it 
can, without appreciable error, 
be considered as concentrated 
at the joints either of the up- 
per chord, of the lower chord, 
or of both, according to the 
manner in which the rest of 
the load is distributed. 

155. Closer Approxima- 
tion to Actual Shearing- 
Force. In our computations 
of greatest shearing-force, we 








FIG. 115. 



make an approximation which is generally considered to be 



APPLIED MECHANICS. 



sufficiently close, and which is always on the safe side. To 
illustrate it, take the case of panel 3-5 of the last example. 
In determining its greatest shearing-force, we considered a load 
of 35000 Ibs. per panel point to rest on all the joints from the 
right-hand support to joint 5, inclusive, and the dead load to 
rest on all the other joints of the~truss. Now, it is impossible, 
if the load is distributed uniformly on the floor of the bridge, 
to have a load of 35000 Ibs. on 5 and 12000 on 3 simultaneously ; 
for, if the moving load extended on the bridge floor only up to 
5, the load on 5 would be only 12000 + ^-(23000) = 23500 Ibs., 
and that on 3 would then be 12000 Ibs. If, on the other hand, 
the moving load extends beyond 5 at all, as it must if the load 
on 5 is to be greater than 23500 Ibs., then part of it will rest 
on 3, and the load on 3 will then be greater than 12000 Ibs. ; 
for whatever load there is between 3 and 5 is supported at 
3 and 5. 

Moreover, we know that the effect of increasing the load on 
5 is to increase the shearing-force, provided we do not at the 
same time increase that on 3 so much as to destroy the effect 
of increasing that on 5. 

Hence, there must be some point between 3 and 5 to which 
the moving load must extend in order to render the shearing- 
force in panel 3-5 a maximum. 

Let the distance of this point from 5 be^r; then, if we let 



w 






= moving load per foot of length, 

Moving load on panel = wx, 

Part supported at 3 = -- , 

20 



Part supported at 5 = wx -. 

20 

Hence, portion of shearing-force due to the moving load on 
panel 3-5 equals 



CONCENTRATING THE LOAD AT THE JOINTS. 



I2/ WX 2 \ I WX 2 W I I*X 2 \ 

( WX -- ) --- = ( I2X -- - ). 

i4\ 20 / 14 20 i4\ 20 / 

This becomes a maximum when its first differential co-efficient 
becomes zero, i.e., when 



therefore 



12 - x = o 



X = 9.2 3 . 



Hence, when the moving load extends to a distance of 9.23 feet 
from 5, then the shearing-force in panel 3-5, and hence the 
stress in diagonal 3-4, is a maximum. 



Panels. 


Portion of Shearing-Force 
due to Moving Load on 
Panel. 


Value 
of x t in 
feet. 


Portion of Load 
at Joints named 
below. 


Portion of Load 
at Joints named 
below. 


i- 3 


27-28 


iv ( I 3 *A 


IO.OO 





11500 


3 


11500 


i4\ J 20 y 


3- 5 

5- 7 


25-27 
23-25 


I4\ 20 / 


9-23 
8.46 


3 

5 


9797 
8230 


5 
7 


11432 
11227 


I4\ 20 / 


7- 9 


21-23 


I4\ 20 / 


7.69 


7 


6801 


9 


10886 


9-1 1 


19-21 


H( 9X ~ ^f?) 


6.92 


9 


5507 


n 


10409 


11-13 
13-15 


17-19 
15-17 


14 \ 20 / 

u<l ^ I3^" 2 \ 


6.15 
5.38 


ii 
13 


4350 
3329 


13 
15 


9795 
9045 1 


I4\ '' 20 / 



To show how the adoption of this method would affect the 
resulting stresses in the diagonals and verticals, I have given 
the work above, and shown the difference between these and 



206 



APPLIED MECHANICS. 



the former results. In this table x = distance covered by load 
from end of panel nearest the centre. 



Panels. 



Greatest Shearing-Force of Same Kind as that due to Dead Load. 



3- 5 
5- 7 
7- 9 



11-13 



27-28 
25-27 
23-25 

2I-2 3 

19-21 
17-19 
15-17 



3500Q, 
14 l 



227500 

= 193385 

= 161038 



-101654 



(i+...+S)= 49345 



Hence, for stresses in main braces, we have 



Diagonals. 


Stresses. 


I- 2 


28-26 


-321685 


3- 4 


27-24 


-273446 


5- 6 


25-22 


227708 


7- 8 


23-20 


-184472 


9-10 


21-18 


143739 


11-12 


19-16 


105507 


i3- J 4 


17-14 


69774 



Moreover, for the shearing-forces of opposite kind from 



CONCENTRATING THE LOAD AT THE JOINTS. 2O/ 



those due to dead load, we have, if x = distance from end of 
panel nearest support which is covered by moving load, 



Panels. 


Portion of Shear due 
to Moving Load on Panel. 


Value 
otx. 


Portion of Load 
at Joints named 
below. 


Portion of Load 
at Joints named 
below. 




17-15 


/6* - ^} 


4.62 


15 


2455 


13 


8171 






I4\ 20 / 












11-13 


19-17 


E( SX - l^f] 


3.84 


13 


1695 


II 


7137 






I4\ 20 / 













Panels. 


Greatest Shearing-Forces of Opposite Kind from those due to Dead 


Load. 


13-5 


17-15 


S22(i+...+ 


5) + l(3:6 7I )-f 4 , I4455 ,-" i 7 (l+ ... + 6) = 


25846 


M-.3 


1^-17 


35000 <i -j- + 


4)+ l (3o637) _ f4(l3695) _^ (l+ ... +7) = 


4116 



Hence we have the following as the stresses in the counter- 
braces : 



Counter-Braces. 


Stresses. 


15-12 
13-10 


15-16 
I7-l8 


- 36546 

5820 



And, for the verticals, we have the new, instead of the old, 
shearing-forces. 



208 



APPLIED MECHANICS. 



The following table compares the results : 



Diagonals. 


Stress, Ordinary 
Method. 


Stress, New Method. 


Difference. 


I- 2 


28-26 


-321685 


-321685 




3- 4 


27-24 


-274518 


-273446 


1072 


5- 6 


25-22 


-229675 


-227708 


1967 


7- 8 


23-20 


-187153 


-184472 


268l 


9-10 


21-18 


-146956 


- J 43739 


3217 


11-12 


19-16 


109080 


105507 


3573 


i3- J 4 


17-14 


- 73528 


- 69774 


3754 


1512 


15-16 


- 40299 


36546 


3753 


13-10 


17-18 


- 9393 


5820 


3573 



Verticals. 


Stress, Ordinary 
Method. 


Stress, New Method. 


Difference. 


3- 2 


27-26 


-f227500 


+ 227500 


O 


5- 4 


25-24 


+ I94H3 


+ 193385 


758 


7- 6 


23-22 


4-162429 


+ 161038 


I39 1 


9- 8 


21-20 


+ T 3 2 357 


+ 130461 


1896 


II-IO 


I9-I8 


4-103929 


+ 101654 


2275 


13-12 


I7-l6 


+ 77H3 


+ 74616 


2527 


i5- J 4 


4-' 28500 


+ 49345 


2655 



156. Compound Bridge-Trusses The trusses already 

discussed have contained but a single system of latticing, or 



COMPO UND BRID GE- TR USSES. 



209 



at least only one system that comes in play at one time ; so that 
a vertical section never cuts more than three bars that are in 
action simultaneously, the main brace having no stress upon it 
when the counterbrace is in action, and vice versa. 

We may, however, have bridge-trusses with more than- one 
system of lattices ; and, in determining the stresses in their 
members, we must resolve them into their component systems, 
and determine the greatest stress in each system separately, 
and then, for bars which are common to the two systems, add 
together the stresses brought about by each. 

In some cases, the design is such that it is possible to 
resolve the truss into systems in more than one way, and then 
there arises an uncertainty as to which course the stresses will 
actually pursue. 

In such cases, the only safe way is to determine the greatest 
stress in each piece with every possible mode of resolution of 
the systems, and then to design each piece in such a way as to 
be able to resist that stress. 

Generally, however, such ambiguity is an indication of a 
waste of material ; as it is most economical to put in the bridge 
only those pieces that are absolutely necessary to bear the 
stresses, as other pieces only add so much weight to the struc- 
ture, and are useless to bear the load. 

The mode of proceeding can be best explained by some 
examples. 

EXAMPLE I. Given the lattice-girder shown in Fig. 117, 
loaded at the lower panel points 

1 TA i i i 11 1 3 5 7 9 11 13 13 17 19 21 23 

only. Dead load = 7200 Ibs. 

per panel point, moving load 

18000 Ibs per panel point; 

let the entire length of bridge FlG ' " 7 * 

be 60 feet ; let the angle made by braces with horizontal 

= 60. 




210 



APPLIED MECHANICS. 







+75600 
FIG. ujc. 




19 23 



10 14 18 22 



This truss evidently consists of the two single trusses shown 
in Figs. \\ja 
njb; 



and njb ; and 
we can compute 
the greatest 
stress of each 

kind in each member of these trusses, and thus 
obtain at once 
all the diag- 
onal stresses, 
and then, by E3 Ec4 

.... FIG. 117*. 

addition, the 
greatest chord stresses. 

Thus the stress in 1-3 (Fig. 117) is the 
same as the stress in 1-5 (Fig. I ija). 

The stress in 3-5 = stress in 1-5 (Fig. 
1170) + stress in 3-7 (Fig. 117^). 

The stress in 5-7 = stress in 5-9 (Fig. 
117*7) -f stress in 3-7 (Fig. 117^). 

The results are given on the diagram (Fig. 
117^); the work being left for the student, as 
it is similar to that done heretofore. 

EXAMPLE II. Given the lattice-girder 
shown in Fig. 1 1 8. Given, as before, Dead 
load = 7200 Ibs. per panel point, moving load 
= 18000 Ibs. per panel point, entire length of 
bridge = 25 feet ; load applied at lower panel 
points. 



Solution. In this case, there are two possible modes of 
resolving it into systems. The first is shown in Figs. uSa and 
n%b : and this is necessarily the mode of division that must 
hold whenever the load is unevenly distributed, or when the 



COMPO UND BRID GE- TR USSES. 



211 



travelling-load covers only a part of the bridge ; for a single 
load at 6 is necessarily put in communication with the support 
at 2 by means of the diagonals 6-3 and 3-2, and with the sup- 
port at 12 by means of the diagonals 6-7, 7-10, lo-n, and the 
vertical 11-12, and can cause no stress in the other diagonals 



1 3 5 7 9 11 




7 11 



6 10 12 

FIG. ii&r. 



24 8 12 

FIG. i i 83. 



5 7 11 



Z37I 



10 12 



FIG. II&T. 



5 7 




When, however, the whole travelling-load is on the bridge, 
it is perfectly possible to divide it into the two trusses shown 
in Figs. II&T and n&/, the diagonals 4-5, 7-10, 6-7, and 5-8 
having no stress upon them. 

When the load is unevenly distributed, we have certainly 
the first method of division ; and when evenly, we are not sure 
which will hold. 

Hence we must compute the greatest stresses with each 
mode of division, and use for each member the greatest ; for 
thus only shall we be sure that the truss is made strong 
enough. 

We shall thus have the following results : 



212 



APPLIED MECHANICS. 



FIRST MODE OF DIVISION (FIGS. n8 AND 



Diagonals. 


Greatest Shearing-Force 
of One Kind. 


Greatest Shearing-Force 
of Opposite Kind. 


Corresponding 
Stresses. 


Fig. 
n8a. 


Fig. 
118*. 


2- 3 


12-9 


~~~(3 + J ) = 20160 


O 


+23279 





3-6 


9-3 


~(3+.i) = 20160 





-23279 


+ o 


6- 7 


8-5 


25200 7200, 
z (2) = 2160 


25200., . 7200 ..... 
(2) = 0040 


+ 2494 


9976 


7-10 
10-11 


5-4 
4-1 


25200 7200, . 
-- - (2) = 2160 

= 


25200, , nroin 


- 2494 




+ 9976 
34918 


. ( 2 + 4; 3024 



Chords. 
Supporting force at 2 (Fig. uSa) or 12 (Fig. 

= '-*? (3 + 
Supporting force at 12 (Fig. u8a) or 2 (Fig. 



= 20160, 



Section. 




Chords. 


Maxi- 












i mum 




Com- 






Bending-Moment. 




i Stresses 
in 


Chords. 


ponents 
of 


Greatest 
Resultant 


s 


S 





si 


i "2 






00 


H ! Separate 


Stresses. 


Stresses. 


M 


bib 




bi) 


bi Trusses. 




j 













i 






1 

i 3 


9 


20i6oX 5 = 100800 2- 6 


8-12 


11639 


1-3 


9-1 1 i o-f- 1-5 


+ 17459 


6 


8 


20160X10=201600! 3- 7 


5- 9+23279 


3-5 


7- 9> 7+1-5 


+40738 : 


7 


5 


20160X15 25200 


















X5 = i?6400 


6-10 


4- 8 20369 


5-7 


3- 7+5-9 


+46558 


10 


4 


30240X 5 = 151200 


7-1 1 


I- 5 +17459 2 -4 


10-12 2- 6+2-4 


-11639 








10-12 


2- 4 


O 


4-6 


8-102- 6+4-8 


32008 ! 














6-8 




6-10+4-8 


-40738 ! 



COMPOUND BRIDGE-TRUSSES. 



2I 3 



SECOND METHOD OF DIVISION (FIGS. nSc AND 
Diagonals (Fig. n8<r). 



Diagonals. 


Maximum 
Shear. 


Corresponding 
Stresses. 


1-4 


10-11 


252OO 


29098 


4-5 


7-10 





O 



Fig. u&/. 



Diagonals. 


Maximum 
Shear. 


Corresponding 
Stresses. 


2-3 


9-12 


25200 


+ 29098 


3-6 


8-9 


25200 


29098 


6-7 


5-8 


O 


O 



Chords. 

Each supporting force in either figure = 25200. 
Fig. n8c. 

Bending-moment anywhere between 4 and 10 = (25200) (5) = 126000; 

/. Stress in i-n = +14549, 
.*. Stress in 4-10 = 14549. 

Fig. n8d. 

Bending-moment at 3 or 9 = 126000, 

Bending-moment anywhere between 6 and 8 = 252000; 

/. Stress in 3-9 = 4-29098, 
Stress in 2-6 or 8-12 = 14549, 
Stress in 6-8 = 29098. 



214 



APPLIED MECHANICS. 



Hence we have for chord stresses, with this second divis- 
ion, 



Chords. 




Stresses. 


i-3 


9-1 1 


I-II -|- 


+ 14549 


3-5 


7- 9 


i-n + 3-9 


+ 43 6 47 


5-7 


- . 


i-n + 3-9 


+ 43647 


2-4 


IO-I2 


-f- 2-6 


-14549 


4-6 


8-10 


410 4- 2-6 


29098 


6-8 





4-10 + 6-8 


43 6 47 



Hence, selecting for each bar the greatest, we shall have, as 
the stresses which the truss must be able to resist, 



1-4 


IO-II 


+ o 


-34918 


i-3 


9-1 1 


+ 17459 


2 -3 


12-9 


+ 29098 





3-5 


7- 9 


+43647 


3-6 


9-8 


+ o 


29098 


5-7 


- 


+46558 


4-5 


10- 7 


+ 9976 


2494 


2-4 


IO-I2 


-14549 


5-8 


7- 6 


+ 2494 


- 9976 


4-6 


8-10 


32008 










6-8 




43647 



These results are recorded in Fiff. uSe. 



(1)+17459 (3)+ 43647(5)+ 46558(7)+ 43647(9^17459(11) 




)- 32008 (6) - 43647 (8)-32008(10)-14549(12) 
FIG. ii&?. 



157. Other Trusses. In Figs. 119, 120, and 121, we 
have examples of the double-panel system with the load placed 



OTHER TRUSSES. 



21 5 



at the lower panel points only. When, as in 119 and 120, the 
number of panels is odd, the same ambiguity arises as took place 
in Fig. 118. When, on the other hand, the number of panels 
is even, as shown in Fig. 121, there is only one mode of division 
into systems possible. The diagrams speak for themselves, and 
need no explanation. 



24 6 8 10 12 14 16 18 20 22 24 26 28 30 



1 8 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 34 




24 8 12 16 20 24 






13 17 

FIG. 



25 29 33 34 



22 26 30 32 




7 11 15 19 23 27 31 38 34 

FlG. no*. 



2 4 



22 26 30 32 



5 9 13 17 19 23 27 31 33 31 

FIG. ngc. 



2 6 10 



24 28 32 



1 11 15 21 26 2 3S 14 

Fro. norf. 



216 



APPLIED MECHANICS. 



246 8 10 12 14 16 18 20 22 24 26 




2 4 



12 16 



24 28 32 






11 15 19 23 27 

FIG. izoa. 



35 



6 10 14 18 22 26 30 S4 36 






159 



13 17 21 25 

FIG. i2o. 



33 35 



6 10 14 



24 28 32 36 




2 4 



26 30 34 36 




13 7 11 " 15 21 25 29 33 35 

FIG. izod. 



2 4 6 8 10 12 14 16 18 



1 3 5 7 9 11 13 15 17 19 20 




FINK'S TRUSS. 



217 



2 6 10 14 18 







11 15 

FIG. i2i. 



12 16 18 



V 

/ \ 




5 9 13 17 

FIG. i2i3. 




The trusses given above may be considered as examples, to 
be solved by the student by assuming the dead and the moving 
load per panel point respectively. 

158. Fink's Truss. The description of this truss will 
be evident from the figure. There is, first, the primary truss 
1-8-16; then on each side 
of 9-8 (the middle post of 
this truss) is a secondary 
truss (1-4-9 on tne left, 




and 9-12-16 on the right). 

Each of these secondary 
trusses contains a pair of smaller secondary trusses, and the 
division might be continued if the segments into which the 
upper chord is thus divided were too long. 

Of the inclined ties, there is none in which any load tends 
to produce compression ; in other words, every load either in- 
creases the tension in the tie, or else does not affect it. Hence 



218 



APPLIED MECHANICS. 



the greatest stresses in all the members will be attained when 
the entire travelling-load is on the truss, and we need only con- 
sider that case. 

The determination of the stress in any one member can 
readily be obtained by determining, by means of the triangle 
of forces, the stress in that member due to the presence of 
the total load per panel point, at each point, and then adding the 
results. This will be illustrated by a few diagonals. 

Let angle 8-1-9 *'> 
Let angle 4-1-5 = i n 
Let angle 2-1-3 ** \ 

we shall have, if w -{- w^ entire load per panel point, 



Designation 
of Ties. 


EFFECT OF LOADS AT 


Resultant 
Tensions. 


3 


5 


7 


9 


11 


13 


15 


1-2 

2-5 
5-6 
6-9 
1-4 
4-9 
1-8 


W + Wi 


O 

o 



o 

w + w l 


O 



W + Wi 


o 




o 



o 

w + Wi 


o 



o 




o 
3 w-fw/t 




o 



o 



o 

W + Wi 




O 




o 

W + Wi 

8 sin i 


W + Wi 


2 sin i 2 
w + w l 


2 sin z' 2 

W + Wj 


2 sin z' 2 



o 

w -f -Wi 


2 sin z' 2 
w + w t 


2 sin z' 2 

W + Wi 


2 sin /' 2 
w + w x 


2 sin / 2 

W + Wi 


2 sin /' 2 

W + Wi 


4 sin * x 

W -\~ Wi 


2 sin *! 

7f -}- W x 


4 sin z'i 

W + Wi 


sin /! 
ze/ + Wi 


4 sin /i 

W + Wi 


2 sin z'i 

W + Wi 


4 sin /'x 

SW+Wi 


sin /j , 

2(/+W,) 


8 sin * 


4 sin / 


8 sin z 


2 sin 2 


8 sin* 


4 sin / 


sin 2 i 

i 



The stresses in all the other members may be found in a 
similar manner. 



GENERAL REMARKS. 



2I 9 




159. Bollman's Truss. The description of this truss is 
made sufficiently clear by the figure. The upper chord is made 
in separate pieces ; and 

1 3 5 7 9 11 12 

the short diagonals 2-5, 
3-4, 4-7, 5-6, 7-8, 6-9, 
8-1 1, and 9-10 are only 
needed to prevent a 
bending of the upper 
chord at the joints. FlG - I24> 

When this is their only object, the stress upon them cannot be 
calculated : indeed, it is zero until bending takes place ; and 
then it is the less, the less the bending. Hence, in this case, 
the stress is wholly taken up by the principal ties ; and these 
have their greatest stress when the whole load is on the bridge. 
The computation of the stresses is made in a similar man- 
ner to that used in the Fink. 




1 60. General Remarks. The methods already explained 
are intended to enable the student to solve any case of a bridge- 
truss where there is no ambiguity as to the course pursued by 
the stresses. 

In cases where a large number of trusses of one given type 
are to be computed, it would, as a rule, be a saving of labor to 
determine formulae for the stresses in the members, and then 
substitute in these formulae. 

Such formulae may be deduced by using letters to denote 
the load and dimensions, instead of inserting directly their 
numerical values ; and then, having deduced the formulae for 
the type of truss, we can apply it to any case by merely sub- 
stituting for the letters their numerical values corresponding 
to that case. 

Such sets of formulae would apply merely to specific styles 
of trusses, and any variation in these styles would require the 
formulae to be changed. 



220 APPLIED MECHANICS. 

In order to show how such formulas are deduced, a few will 
be deduced for such a bridge as is shown in Fig. 1 1 1. 

Let the load be applied at the upper panel points only ; let 
dead load per panel point = w, moving load per panel point 
= w,. Let the whole number of panels be N, N being an even 
number. Let the length of one panel = height of truss = /. 
Then length of entire span = Nl. 

Consider the (n + i) th panel from the middle. 

The stress in the main tie is greatest when the moving load 
is on all the panel points from the farther abutment up to the 
panel in question, (n + th - 

Hence, for the n tb panel from the middle, the greatest shear' 
ing-force that causes tension in the main tie is equal to 

w-\-w 1 



Hence stress in main tie 

N 



For the counterbrace, we should obtain, in a similar way, the 
formula 



_N 

H T \ ^~ 1 I \ fl I 

2N LA 2 / 2 



] - wN(n + ,) } , 



which represents tension when it is positive. Proceed in a 
similar way for the other members. 

When there is more than one system, we must divide the 
truss into its component systems; and when there is ambiguity, 
we must use, in determining the dimensions of each member, 
the greatest stress that can possibly come upon it. 



CENTRE OF GRAVITY. 221 



CHAPTER V. 

CENTRE OF GRAVITY. 

161. The centre of gravity of a body or system of bodies, is 
that point through which the resultant of the system of parallel 
forces that constitutes the weight of the body or system of 
bodies always passes, whatever be the position in which the 
body is placed with reference to the direction of the forces. 

162. Centre of Gravity of a System of Bodies. If 
we have a system of bodies whose weights are W iy W 2y W y etc., 
the co-ordinates of their individual centres of gravity being 
fo, y *i), (* y **}> (* y Iv * 3 ) etc., respectively, and if we 
denote by x m y , z , the co-ordinates of the centre of gravity of 
the system, we should obtain, just as in the determination of the 
centre of any system of parallel forces, 

i. By turning all the forces parallel to OZ> and taking 
moments about OY, 



(W, + W 2 + W 3 + etc.)* = W,x, + W 2 x 2 + W t x 3 + etc., 
or 



and, taking moments about OX, 

etc., 



or 



222 



APPLIED MECHANICS. 



2. By turning all the forces parallel to OX, and taking 
moments about OY, 



(W* + W, + W z + etc.K = W& + W 2 z 2 + W z z z + etc., 



or 



Hence we have, for the co-ordinates of the centre of gravity 
of the system, 



EXAMPLES. 

I. Suppose a rectangular, homogeneous plate of brass (Fig. 125), 

where AD = 1 2 inches, AB = 5 inches, 
and whose weight is 2 Ibs., to have 
weights attached at the points A, B, C, 
and D respectively, equal to 8, 6, 5, and 
x $ Ibs. ; find the centre of gravity of the 
system. 



4- 



Solution. 

Assume the origin of co-ordinates at 
the centre of the rectangle, and we have 

W, = 2, W 2 = 8, W, = 6, W 4 = 5, W s = 3, 
*, =o, x 2 = 6, x z = 6, * 4 = -6, ^ s = -6, 
Ji =o, ^ 2 = f, J 3 = -f, y 4 = -|, j s = f ; 

= o -f 48 -f 36 30.0 18.0 = 36, 
= o -f- 20 15 12.5 4- 7.5 = o, 
= 2 -f- 8 -f 6 + 5.0 4- 3.0 = 24; 

_ 3^> _ ^ 

= 2 4 = = 2 4 = 

Hence the centre of gravity is situated at a point E on the line OX, 
where OE = 1.5. 



CENTRE OF GRAVITY OF . HOMOGENEOUS BODIES. 22$ 

2. Given a uniform circular plate of radius 8, and weight 3 Ibs. 
(Fig. 126). At the points A, B, C, and D, 
weights are attached equal to 10, 15, 25, and 23 
Ibs. respectively, also given AB = 45, BC = 
105'', CD = 120 ; find the centre of gravity of 
the system. 



163. Centre of Gravity of Homogeneous Bodies. For 
the case of a single homogeneous body, the formulae have been 
already deduced in 44. They are 




fxdV 

~ JdV 



and for the weight of the body, 

W = wfdV, 

where x& y , z m are the co-ordinates of the centre of gravity of 
the body, W its weight, and w its weight per unit of volume. 

From these formulae we can readily deduce those for any 
special cases ; thus, 

(a) For a volume referred to rectangular co-ordinate axes, 
d V dxdydz. 

x _ = fffxdxdydz _ = fffydxdydz = fffzdxdydz 

SSfdxdydz' y " Sffdxdydz* * == fffdxdydz 



(b) For a flat plate of uniform thickness, t, the centre of grav- 
ity is in the middle layer; hence only two co-ordinates are 
required to determine it. If it be referred to a system of rect- 
angular axes in the middle plane, dV '= tdxdy, 

_ ffxdxdy _ ffydxdy 



224 APPLIED MECHANICS. 

The centre of gravity of such a thin plate is also called the 
centre of gravity of the plane area that constitutes the middle 
plane section ; hence 

(c) For a plane area referred to rectangular co-ordinate axes 
in its own plane, 

Sfxdxdy ffydxdy 



(d) For a slender rod of uniform sectional area, a, if x, y, z, 
be the co-ordinates of points on the axis (straight or curved) of 
the rod, we shall have dV ads a^(dx) 2 + (dy)* + (dzf> 



(I)* 



fyds 

= l- = 

fds 



A* 



(e) For a slender rod whose axis lies wholly in one plane, 
the centre of gravity lies, of course, in the same plane ; and if 
our co-ordinate axes be taken in this plane, we shall have z = c 

-=- =r o, and also Z Q = o. Hence we need only two co 
ax 



CENTRE OF GRAVITY OF HOMOGENEOUS BODIES. 22$ 

ordinates to determine the centre of gravity, hence dV =. ads 
m fxds _ J 



AMI) 



+ }<tx 



* 



4 \dx 



(/) For a line, straight or curved, which lies entirely in one 
plane, we shall have, again, 



+ 



Sds 



fds 

' ^ } doc 



Whenever the body of which we wish to determine the 
centre of gravity is made up of simple figures, of which we 
already know the positions of the centres of gravity, the method 
explained in 162 should be used, and not the formulae that 
involve integration ; i.e., taking moments about any given line 
will give us the perpendicular distance of the centre of gravity 
from that line. 

In the case of the determination of the strength and stiff- 
ness of beams, it is necessary to know the distance of a hori- 
zontal line passing through the centre of gravity of the section, 



226 APPLIED MECHANICS. 

from the top or the bottom of the section ; but it is of no prac- 
tical importance to know the position of the centre of gravity 
on this line. In most of the examples that follow, therefore, 
the results given are these distances. These examples should 
be worked out by the student. 

In the case of wrought-iron beams of various sections, on 
account of the thinness of the iron, a sufficiently close approxi- 
mation is often obtained by considering the cross-section as 
composed of its central lines ; the area of any given portion 
being found by multiplying the thickness of the iron by the 
corresponding length of line, the several areas being assumed 
to be concentrated in single lines. 

EXAMPLES. 

i. Straight Line AB (Fig* 127). The centre of gravity is evidently 
at the middle of the line, as this is a point of 



FIG. 127. 



- B symmetry. 



2. Combination of Two Straight Lines. The centre of gravity in 
each case lies on the line OO I} Figs. 128, 129, 130, and 131. 

(a) Angle- Iron of Unequal Arms (Fig. 128). Length AB = b, 
length BC = h, area AB = A, area 

BC = B; 

q _^^ E;\ p, 

/. BE = DE = % . A D"C" 

\Jb 2 -f- h 2 FIG. 128. 

() Angle-Iron of Equal Arms (Fig. 129). Length AB = BC 
B = b; 

\^ 

b b 




FIG. 129. 



CENTRE OF GRAVITY OF HOMOGENEOUS BODIES. 22 / 



(c) Cross of Equal Arms (Fig. 130). AB = OO t = h; 



:. AC = BC 



c p, 



B 

FIG 130. 

(d) T-Iron (Fig. 131). Area AB = A, area CE = B, length 
A E B CE = h; 

Bh 



Fi?. 131. 2 ^ -}- &) 

3. Combination of Three Lines. OO l = line passing through the 
centre of gravity. 

(a) Thin Isosceles Triangular Cell (Fig. 132). Length 
BC = a, length AC = b, area AB = BC A D c 
= ^ area ^4C = B; o XZ "7 



/. DB 




B 

FIG. 132. 



-(- B 



BE - 



Same in Different Position (Fig. 133). 



BD = DC = - 



FIG. i J3 . 



228 



APPLIED MECHANICS. 



(c) Channel-Iron (Fig. 134). Area of flanges = A, area of web 
= B, depth of flanges + J thickness of 



Ah 



FIG. 134. 



(d} H-Beam (Fig. 135). Area of upper flange = A I9 area of 
lower flange = A 2 , area of web = B, height = h. 

4 2 + B 



_ 
= 



k 



C B 



EOF 

FIG. 135. 



2 A s + A 2 + B 

4. Combination of Four Lines. OO^ = line passing through the 
centre of gravity. 

(a) Thin Rectangular Cell (Fig. 136). Length AB = h; 




/. AE = BE = - 

2 



FIG. 136. 

() Thin Square Cell (Fig. 137). 



Z? 77 /~* Z7 

yj y^ ^^ OA> * 

2 



= BC = 




-O, 



FIG. 137. 

5. Circular Arcs. 

(a) Circular Arc AB (Fig. 138). Angle A OB = 0,, radius = r, 



Use formula 




fyds 



FIG. i 3 i. 



" fds ' - fds ' 

but use polar co-ordinates, where 

ds = rdQ, sf = rcosO, y = rsinO, 



CENTRE OF GRAVITY OF HOMOGENEOUS BODIES. 229 



r 2 cos OdO 



2 f 



i 



Oi 

si 



sin OdB 



(i cos0,) 



Circular Arc AC (same figure). 
r sin #! 



, y<> = o. 



(c) Quarter-Arc of Circle AB, Radius r (Fig. 139). 
r 2 I 2 cos OdB 

Jo 



2r 



Semi-circumference ABC (same figure). 



FIG- 13* 



6. Combination of Circles and Straight Lines. 

Barlow Rail (Fig. 140). Two quadrants, radius r, and web, 
c , whose area = ^- the united area of the quadrants. 
Let united area of quadrants = A, area of web 
; let 



AI IB 

FIG. 140. 



230 



APPLIED MECHANICS. 



7. Areas. 

(a) T-Section (Fig. 141). Let length AB = B, EF = b, entire 
B height = H, GE = h. Let distance of centre 

x.*mmmmmm^i o f gravity below AB = x t ; therefore, taking 
moments about AB as an axis, 
-h(B-t)\ 

-$ 




- k(S - 



whence we can readily derive x t . 



(b) I-Section (Fig. 142). Let AB = B, GH = b, MN = b n 
entire height = H, BC = H h, EH = h t ; and let x t = distance 
of centre of gravity below AB. A 

Hence, taking moments about AB, we have 



Xl \B(H - h) 
B 



- h,) 




whence we can deduce x t 



(c} Triangle (Fig. 143). If we consider the triangle OBC as 
composed of an indefinite number of narrow strips parallel to the side 
CB, of which FLHK is one, the centre of gravity 
of each one of these strips will be on the line OD 
drawn from O to the middle point of the side 
CB ; hence the centre ^f gravity of the entire tri- 
c angle must be on the line OD. For a similar rea- 
son, it must be on the median line CE ; hence the 
centre of gravity mhst be at the intersection of the median lines, and 
hence 

BC . ODsv^ODC 




o 

FIG. 143. 



X Q = 



= ^OD. Moreover, area = 



CENTRE OF GRAVITY OF HOMOGENEOUS BODIES. 



(d) Trapezoid (Fig. 144). 



First Solution. Bisect AB in 6>, and CE in >; let g^ be the 



centre of gravity of CEJB, and g t that of ABC. 
Then will 6 1 , the centre of gravity of the trape- 
zoid, be on the line g^^ and 



Gg, -. 




Gg, CEB* 



But it must be on the line OD; hence it is at their intersectioa 
From the similarity of GG l g l GG^g^ we have 



GG, 

GG ~ 



ABC 



BEC ~ CE 



B m 
b ; 



GG, 



andsince 



OD 



Second Solution. Fig. 144 (a). Let O be the 
point of intersection of the non-parallel sides AC 
and BE. Let OF = x lt OD * OG = x n . Take 
moments about an axis through O, and perpen- 
dicular to OF) and we readily obtain 




Fie. 



232 APPLIED MECHANICS. 

(e) Parabolic Half-Segment OAB (Fig. 145). Let OA ~ x,, 

AB = jh ; let x ,y , be the co-ordinates of the centre of gravity ; let 

the equation of the parabola be y 2 = 

Y 



'.r, /-' 2 *M f Xl 3 

Jo xdocd y 2a J ** d 



X = 









/*' rya 

t/o t/o ' 



r<^ /.r, ^ 3 ^ 

t/o y ~~ s i ~ 3v i MI 



Area 



(/) Parabolic Spandril OBC (Fig. 145). Let x ,y , be co-ordi- 
nates of centre of gravity of the spandril. 



xdxdy 



x^ (*y\ 
I 

_ l_?y 
/*Jr, /*y t 

/ / L 

1/0 ^ 



_ 

> /*, /^ 

/ / 

1/0 ^ 2 * 
Area = ^j, - 



CENTRE OF GRA VI TY OF HOMOGENEOUS BODIES. 233 



(g) Circular Sector OAC (Fig. 146). Let OA = r, AOX = 0,, 
be the co-ordinates of the centre of gravity : 



. . ^o^O, 



Xo = 



/r r*V r"* x* /rcos0j /*jr tan 0, 

/ *<**#+ / / JCflkfljK 

.ooggy-V^rra y ^-rtanfl, ^ 



Area 



c 

FIG. 146. 



Second Solution. 

Consider the sector to be made up of an indefinite number of 
narrow rings ; let p be the variable radius, and dp the thickness : 

Elementary area = 2pft l dp, 

and centre of gravity of this elementary area is on OX, at a distance 
from O equal to p ^ 1 [see Example 5 ()] ; 



X = 



(ti) Circular Half-Segment ABX (Fig. 146), 



f" f Q " xdxdy f" xVr* - x*dx 

Sector minus triangle %r*S l |r 2 sin 0, cos 0, 



, sn <, cos 



r r^ 

^rcosgyo '"""' = . 4sin B ig 1 -sin a 0.cosg 1 
, sin 0, cos 0,) ~~ 0, sin 0, cos 0, 



234 APPLIED MECHANICS. 

164. Pappus's Theorems. The following two theorems 
serve often to simplify the determination of the centres of 
gravity of lines and areas. They are as follows : 

THEOREM I. If a plane curve lies wholly on one side of a 
straight line in its own plane, and, revolving about that line, 
generates thereby a surface of revolution, the area of the sur- 
face is equal to the product of the length of the revolving line, 
and of the path described by its centre of gravity. 

Proof. Let the curve lie in the xy plane, and let the axis 
of y be the line about which it revolves. We have, from what 

fxds 
precedes, 163 (e\ X Q =- -' 



.*. x fds = fxds, 

where x equals the perpendicular distance of the centre of 
gravity of the curve from O Y, ds = elementary arc, 

2irx fds = f(2irx)ds; 
or, reversing the equation, 

f(2irx)ds = 



But f(2irx)ds = surface described in one revolution, while s =. 
length of arc, and 2irx Q = path described by the centre of 
gravity in one revolution. Hence follows the proposition. 

THEOREM II. If a plane area lying wholly on the same 
side of a straight line in its own plane revolves about that line, 
and thereby generates a solid of revolution, the volume of the 
solid thus generated is equal to the product of the revolving 
area, and of the path described by the centre of gravity of the 
plane area during the revolution, 



PAPPUS'S THEOREMS. 235 

Proof. Let the area lie in the xy plane, and let the axis 
OY be the axis of revolution. We then have, from what has 
preceded, if x = perpendicular distance of the centre of gravity 
of the plane area from OY t the equation, 163 (b), 

Sfxdxdy 
*- ffdxdy' 
Hence 

Xo ffdxdy = ffxdxdy; 

/. (2irx ) ffdxdy 
or 

ff(2trx)dxdy 



But ff(2irx)dxdy = volume described in one revolution, and 
2iex Q = path described by the centre of gravity in one revolu- 
tion. Hence follows the proposition. 

The same propositions hold true for any part of a revolution, 
as well as for an entire revolution, since we might have multi- 
plied through by the circular measure 6, instead of by 2ir. 

It is evident that the first of these two theorems may be 
used to determine the centre of gravity of a line, when the 
length of the line, and the surface described by revolving it 
about the axis, are known ; and so also that the second theorem 
may be used to determine the centre of gravity of a plane area 
whenever the area is known, and also the volume described by 
revolving it around the axis. 

EXAMPLES. 

i. Circular Arc AC (Fig. 138). Length of arc = s = 2rO, sur- 
face of zone described by revolving it about O Y = circumference of a 
great circle multiplied by the altitude = (ztrr) (2rsmO l ); 



x l = rsinfl, 



sm0 z 
r- 



236 APPLIED MECHANICS. 

2. Semicircular Arc (Fig. 139). Length of arc = nr, surface of 
sphere described = 4?rr 2 ; 

2r 
.'. 2Trx (Trr) = 4?rr 2 .*. x = 

7T 

3. Trapezoid (Fig. 147). Let AD = b, BC b ; let it revolve 
around AD : it generates two cones and a cylinder. 

AD + BC 
Y Area of trapezoid = - BG, 

B Volume = ~ -(AG + HD) + 7r(G) 2 . BC 



\-HD+ 3 BC) 
FlG - J 47. = ^ ^(^Z) + BC + ^C) 



GBI BC \ GBI 

= KL 



4. Circular Sector AGO (Fig. 146). Area of sector = r*0 lt 
volume described = -Jr (surface of zone) = \r(2-rrr} (?r sin 0,) = 
sin 0! 



165. Centre of Gravity of Solid Bodies. The general 
formulae furnish, in most cases, a very complicated solution, and 
hence we generally have recourse to some simpler method. A 
few examples will be given in this and the next section. 



CENTRE OF GRAVITY OF SYMMETRICAL BODIES. 237 



Tetrahedron ABCD (Fig. 148). The plane ABE, containing the 
edge AB and the middle point E of the edge CD, bisects all lines 
drawn parallel to CD, and terminating in the faces A 

ABD and ABC : hence a similar reasoning to that 
used in the case of the triangle will show that the cen- 
tre of gravity of the pyramid must be in the plane 
ABE ; in the same way it may be shown that it must 
lie in the plane ACF. Hence it must lie in their 
intersection, or in the line AG joining the vertex A 
with the centre of gravity (intersection of the medians) 
of the opposite face. 




FIG. 148. 
In the same way it can be shown that the centre 



of gravity of the triangular pyramid must lie in the line drawn from 
the vertex B to the centre of gravity of the face A CD. Hence the 
centre, of gravity of the tetrahedron will be found on the line AG at 
a distance from G equal to \A G. 

1 66. Centre of Gravity of Bodies which are Symmet- 
rical with Respect to an Axis. Such solids may be gener- 
ated by the motion of a plane figure, as ABCD 
(Fig. 149), of variable dimensions, and of any 
form whose centre G remains upon the axis 
OX ; its plane being always perpendicular to 
OX, and its variable area X being a function 
of x, its distance from the origin. 

Here the centre of gravity will evidently 
FIG. i 49 . jj e on tne ax j s QX^ an d the elementary vol- 

ume will be the volume of a thin plate whose area is X and 
thickness A;r ; hence the elementary volume will be 

Take moments about OY, and we shall have 




or 



x fXdx = fXxdx and Volume = fXdx, 
fXxdx 

** = 7xJ*" F =/^ 



APPLIED MECHANICS. 



EXAMPLES. 

x 2 y 2 z 2 

I. Ellipsoid -f ^- + = i (Fig. 150). Find centre of gravity 
a D c 

of the half to the right of the x plane. Let OK 
= x. Now if, in the equation of the ellipsoid, 

v X 2 Z 2 

we make y = o, we have H = I ; 




where z = 

Make z = o in the equation of the ellipsoid, and + ij = I > 



where ^ = 



.-. EK 



are the semi-axes of the variable ellipse EGFH, which, by moving along 
OX, generates the ellipsoid. Hence 



hence 



irbc 
Area EGFH = Tr(EK . GK) = (a* - x 2 ) = X; 



Elementary volume = (a 2 



irbc (* a . ( a 2 x 2 x 4 ) 

I (a 2 x x*)dx < > 

a 2 Jo _ ( 2 4 ). 

trbc 



^ J ( x*} a 

a 2 -x*)dx \a 2 x--\ 

3)0 

V = - - I a (a 2 - x 2 )dx = \irabc. 

d 2 t/0 

a. Hemisphere. Make a = b = c, and x = f a, V 



CENTRE OF GRAVITY OF SYMMETRICAL BODIES. 239 

If the section X were oblique to OX t making an angle 0. 
with it, the elementary volume would not be Xdx, but Xdx sin 0, 

and we should have 



3. Oblique Cone (Fig. 151). Let OA = h; let area of base be 
and let the angle made by OX with the base be 6; 

X x> A 






FIG. 151. 



r h 

sin0 / ^^ 

** o 



4. Truncated Cone (Fig. 151). Let height of entire cone be 
h = OA ; let height of portion cut off be h l ; 

AT* h*- h* 

I x*dx 

4 ,/^-A 4 



^TTSi 



240 APPLIED MECHANICS. 



CHAPTER VI. 

STRENGTH OF MATERIALS. 

167. Stress, Strain, and Modulus of Elasticity When 

a body is subjected to the action of external forces, if we 
imagine a plane section dividing the body into two parts, the 
force with which one part of the body acts upon the other 
at this plane is called the stress on the plane ; it may be a 
tensile, a compressive, or a shearing stress, or it may be a com- 
bination of either of the two first with the last. In order to 
know the stress completely, we must know its distribution and 
its direction at each point of the plane. If we consider a small 
area lying in this plane, including the point O, and represent 
the stress on this area by /, whereas the area itself is repre- 
sented by a, then will the limit of <- as a approaches zero be the 

a 

intensity of the stress on the plane under consideration at the 
point 0. 

When a body is subjected to the action of external forces, 
and, in consequence of this, undergoes a change of form, it 
will be found that lines drawn within the body are changed, by 
the action of these external forces, in length, in direction, or 
in both ; and the entire change of form of the body may be 
correctly described by describing a sufficient number of these 
changes. 

If we join two points, A and B, of a body before the 
external forces are applied, and find, that, after the application 
of the external forces, the line joining the same two points of 
the body has undergone a change of length &(AB), then is the 



STRESS, STRAIN, AND MODULUS OF ELASTICITY. 241 

limit of the ratio ' as AB approaches zero called the 



strain of the body at the point A in the direction AB. 

If AB 4- &(AB) > AB, the strain is one of tension. 

If AB + A (^4-5) < ^4-#, the strain is one of compression. 



Suppose a straight rod of uniform section A to be subjected 
to a pull P in the direction of its length, and that this pull is 
uniformly distributed over the cross-section : then will the in- 
tensity of the stress on the cross-section be 



If P be measured in pounds, and A in square inches, then will 
/ be measured in pounds per square inch. 

If the length of the rod before the load is applied be /, 
and its length after the load is applied be I ~\- e, then is e the 
elongation of the rod ; and if this elongation is uniform through- 

x> 

out the length of the rod, then is - the elongation of the rod 

per unit of length, or the strain. 

Hence, if a represent the strain due to the stress / per 
unit of area, we shall have 



The Modulus of Elasticity is commonly defined as the ratio 
of the stress per unit of area to the strain, or 

*-*-; 

a 

and this is expressed in units of weight per unit of area, as in 
pounds per square inch. 

This definition is true, however, only for stresses for which 
Hooke's law " The stress is proportional to the strain " holds. 



242 APPLIED MECHANICS. 

For greater stresses the permanent set must first be deducted 
from the strain, and the remainder be used as divisor. 

The limit of elasticity of any material is the stress above 
which the stresses are no longer proportional to the strains. 

The modulus of elasticity was formerly defined as the 
weight that would stretch a rod one square inch in section to 
double its length, if Hooke's law held up to that point, and 
the rod did not break. 

EXAMPLES. 

1. A wrought-iron rod 10 feet long and i inch in diameter is loaded 
in the direction of its length with 8000 Ibs. ; find (i) the intensity of 
the stress, (2) the elongation of the rod ; assuming the modulus of the 
iron to be 28000000 Ibs. per square inch. 

2. What would be the elongation of a similar rod of cast-iron 
under the same load, assuming the modulus of elasticity of cast-iron to 
be 1 7000000 Ibs. per square inch ? 

3. Given a steel bar, area of section being 4 square inches, the 
length of a certain portion under a load of 25000 Ibs. being 10 feet, 
and its length under a load of 100000 Ibs. being 10' o".o75 ; find the 
modulus of elasticity of the material. 

4. What load will be required to stretch the rod in the first example 
Y 1 ^ inch ? 

1 68. Resistance to Stretching and Tearing. The most- 
used criterion of safety against injury for a loaded piece is, 
that the greatest intensity of the stress to which any part of it 
is subjected shall nowhere exceed a certain fixed amount, called 
the working-strength of the material ; this working-strength 
being a certain fraction of the breaking-strength determined 
by practical considerations. 

The more correct but less used criterion is, that the great- 
est strain in any part of the structure shall nowhere exceed 
the working-strain ; the greatest allowable amount of strain 
being a fixed quantity determined by practical considerations. 



RESISTANCE TO STRETCHING AND TEARING. 243 

This is equivalent to limiting the allowable elongation or 
compression to a certain fraction of its length, or the deflection 
of a beam to a certain fraction of the span. 

If the stress on a plane surface be uniformly distributed, 
its resultant will evidently act at the centre of gravity of the 
surface, as has been already shown in 42 to be the case with 
any uniformly distributed force. 

If a straight rod of uniform section and material be sub- 
jected to a pull in the direction of its length, and if the result- 
ant of the pull acts along a line passing through the centres 
of gravity of the sections of the rod, it is assumed in practice 
that the stress is uniformly distributed throughout the rod, and 
hence that for any section we shall obtain the stress per square 
inch by dividing the total pull by the number of square inches 
in the section. 

If, on the other hand, the resultant of the pull does not act 
through the centres of gravity of the sections, the pull is not 
uniformly distributed ; and while 



will express the mean stress per square inch, the actual inten- 
sity of the stress will vary at different points of the section, 

p 
being greater than at some points and less at others. How 

A 

to determine its greatest intensity in such cases will be shown 
later. 

With good workmanship and well-fitting joints, the first 
case, or that of a uniformly distributed stress, can be practi- 
cally realized ; but with ill-fitting joints or poor workmanship, 
or with a material that is not homogeneous, the resultant of 
the pull is liable to be thrown to one side of the line passing 
through the centres of gravity of the sections, and thus there 



244 APPLIED MECHANICS. 

is set up a bending-action in addition to the direct tension, and 
therefore an unevenly distributed stress. 

It is of the greatest importance in practice to take cogni- 
zance of any such irregularities, and determine the greatest 
intensity of the stress to which the piece is subjected : though 
it is too often taken account of merely by means of a factor of 
safety ; in other words, by guess. 

Leaving, then, this latter case until we have studied the 
stresses due to bending, we will confine ourselves to the case 
of the uniformly distributed stress. 

If the total pull on the rod in the direction of its length 
be P, and the area of its cross-section A, we shall have, for the 
intensity of the pull, 

P 



On the other hand, if the working-strength of the material 
per unit of area be /, we shall have, for the greatest admissible 

load to be applied, 

P = fA. 

If / be the working-strength of the material per square 
inch, and E the modulus of elasticity, then is the greatest 
admissible strain equal to 



Thus, assuming 12000 Ibs. per square inch as the working 
tensile strength of wrought-iron, and 28000000 Ibs. per square 
inch as its modulus of elasticity, its working-strain would be 



1 2000 



28000000 7000 

Hence the greatest safe elongation of the bar would be 
of its length. Hence a rod 10 feet long could safely be 
stretched ^ of a foot = 0.05 14". 



VALUES OF BREAKING AND WORKING STRENGTH. 245 

169. Approximate Values of Breaking Strength, and 
of Modulus of Elasticity. In a later part of this book the 
attempt will be made to give an account of the experiments 
that have been made to determine the strength and elas- 
ticity of the materials ordinarily used in construction, in such 
a way as to enable the student to decide for himself, in any- 
special case, upon the proper values of the constants that he 
ought to use. 

For the present, however, the following will be given as a 
rough approximation to some of these quantities, which we may 
make use of in our work until we reach the above-mentioned 
account. 

(a) Cast-Iron. 

Breaking tensile strength per square inch, of common quali- 
ties, 14000 to 20000 Ibs. ; of gun iron, 30000 to 33000 Ibs. 

Modulus of elasticity for tension and for compression, about 
17000000 Ibs. per square inch. 

(b) Wrought-Iron. 

Breaking tensile strength per square inch, from 40000 to* 
60000 Ibs. 

Modulus of elasticity for tension and for compression, about 
28000000. 

(c) Mild Steel. 

Breaking tensile strength per square inch, 55000 to 70000 
Ibs. 

Modulus of elasticity for tension and for compression, from 
28000000 to 30000000 Ibs. per square inch. 

(d) Wood. 

Breaking compressive strength per square inch : 

Oak, green 3000 Ibs. 

Oak, dry 3000 to 6000 Ibs. 

Yellow pine, green 3000 to 4000 Ibs. 

Yellow pine, dry 4000 to 7000 Ibs. 



246 APPLIED MECHANICS. 

Modulus of elasticity for compression (average values) : 

Oak 1300000 Ibs. per square inch. 

Yellow pine 1600000 Ibs. per square inch. 

170. Sudden Application of the Load. If a wrought- 
iron rod 10 feet long and I square inch in section be loaded 
with 12000 pounds in the direction of its length, and if the 
modulus of elasticity of the iron be 28000000, it will stretch 
0.05 14" provided the load be gradually applied : thus, the rod 
begins to stretch as soon as a small load is applied ; and, as the 
load gradually increases, the stretch increases, until it reaches 
0.05 14". 

If, on the other hand, the load of 12000 Ibs. be suddenly 
applied (i.e., put on all at once) without being allowed to fall 
through any height beforehand, it would cause a greater stretch 
at first, the rod undergoing a series of oscillations, finally 
settling down to an elongation of 0.05 14". 

To ascertain what suddenly applied load will produce at 
most the elongation 0.05 14", observe, that, in the case of the 
gradually applied load, we have a load gradually increasing from 

o to 12000 Ibs. 

Its mean value is, therefore, ^(12000) = 6000 Ibs. ; and this 
force descends through a distance of 

0.05 14". 

Hence the amount of mechanical work done on the rod by the 
gradually applied load in producing this elongation is 

(6000) (0.0514) = 308.4 inch-lbs. 

Hence, if we are to perform upon the rod 308.4 inch-lbs. of 
work with a constant force, and if the stretch is to be 0.05 14", 
the magnitude of the force must be 

308.4 



0-0514" 



= 6000 Ibs. 



RESILIENCE OF A TENSION-BAR. 247 

Hence a suddenly applied load will produce double the strain 
that would be produced by the same load gradually applied ; 
and, moreover, a suddenly applied load should be only half as 
great as one gradually applied if it is to produce the same 
strain. 

171. Resilience of a Tension-Bar. The resilience of a 
tension-rod is the mechanical work done in stretching it to the 
same amount that it would stretch under the greatest allowable 
gradually applied load, and is found by multiplying the greatest 
allowable load by half the corresponding elongation. 

Thus, suppose a load of 100 Ibs. to be dropped upon the 
rod described above in such a way as to cause an elongation not 
greater than 0.05 14", it would be necessary to drop it from a 
height not greater than 3.08". 

EXAMPLES. 

1 . A wrought-iron rod is 1 2 feet long and i inch in diameter, and 
is loaded in the direction of its length; the working-strength of the 
iron being 12000 Ibs. per square inch, and the modulus of elasticity 
28000000 Ibs. per square inch. 

Find the working-strain. 
Find the working-load. 
Find the working-elongation. 
Find the working-resilience. 

From what height can a 5o-pound weight be dropped so as to produce 
tension, without stretching it more than the working- elongation? 

2. Do the same fora cast-iron rod, where the working-strength is 
5000 pounds per square inch, and the modulus of elasticity 17000000; 
the dimensions of the rod being the same. 

172. Results of Wohler's Experiments on Tensile 
Strength. According to the experiments of Wohler, of which 
an account will be given later, the breaking-strength of a piece 



248 APPLIED MECHANICS. 

depends, not only on whether the load is gradually or suddenly 
applied, but also on the extreme variations of load that the 
piece is called upon to undergo, and the number of changes to 
which it is to be submitted during its life. 

For a piece which is always in tension, he determines the 
following two constants ; viz., /, the carrying-strength per square 
inch, or the greatest quiescent stress . that the piece will bear, 
and u, the primitive safe strength, or the greatest stress per 
square inch of which the piece will bear an indefinite number 
of repetitions, the stress being entirely removed in the inter- 
vals. 

This primitive safe strength, u y is used as the breaking- 
strength when the stress varies from o to u every time. Then, 
by means of Launhardt's formula, we are able to determine the 
ultimate strength per square inch for any different limits of 
stress, as for a piece that is to be alternately subjected to 80000 
and 6000 pounds. 

Thus, for Phoenix Company's axle iron, Wohler finds 

/ = 3290 kil. per sq. cent. = 46800 Ibs. per sq. in., 
u 2100 kil. per sq. cent. = 30000 Ibs. per sq. in. 

Launhardt's formula for the ultimate strength per unit of area 
is 

t u least stress ) 



a = u{\ + 



u greatest stress)' 



Hence, with these values of t and u y we should have, for the 
ultimate strength per square inch, 

( i least stress ) 

a 2100% i -h - ->kil. per sq. cent., 

( 2 greatest stress) 

or 

ii least stress ) 
i -f- - -\ Ibs. per sq. in. 

2 greatest stress) 




WOH LEWS EXPERIMENTS ON TENSILE STRENGTH. 249 



Thus, if least stress = 6000, and greatest = 80000, we should 

have 

a = 30000$ i -f . -fo\ = 30000^1 + isul = 3 II2 5> 

if least stress = 60000, and greatest = 80000, 

a = 30000 | i -f i . f I = 30000^1 + fj = 41250; 

if least stress = greatest stress = 80000, 

a 30000^1 -j- \\ = 45000 = carrying-strength. 

Hence, instead of using, as breaking-strength per square inch 
in all cases, 45000, we should use a set of values varying from 
45000 down to 30000, according to the variation of stress which 
the piece is to undergo. 

For working-strength, Weyrauch divides this by 3 : thus 
obtaining, for working-strength per square inch, 

( i least stress ) 

b 10000 < i H [ Ibs. per sq. in. ; 

( 2 greatest stress ) 

lor Krupp's cast-steel, notwithstanding the fact that Wohler 

finds 

/ = 7340 kil. per sq. cent. = 104400 Ibs. per sq. in., 
u = 33 kil. per sq. cent. = 46900 Ibs. per sq. in., 

Weyrauch recommends 

( q least stress ) 

a 3300 \ i + - -Vkil. per sq. cent., 

( ii greatest stress) 

( o least stress ) 

a = 46000 < i 4- >lbs. per sq. in., 

( ii greatest stress) 



= I5633J 



o least stress ) 

i 4- - ; - - > Ibs. per sq. in. 

1 1 greatest stress j 



EXAMPLES. 

Find the breaking-strength per square inch for a wrought-iron tension 
rod. 

1. Extreme loads are 75000 and 6000 Ibs. 

2. Extreme loads are 120000 and 100000 Ibs. 

3. Extreme loads are 300000 and 10000 Ibs. 
Find the safe section for the rod in each case. 



250 APPLIED MECHANICS. 

173. Suspension-Rod of Uniform Strength. In the 
case of a long suspension-rod, the weight of the rod itself some- 
times becomes an important item. The upper section must, of 
course, be large enough to bear the weight that is hung from 
the rod plus the weight of the rod itself ; but it is sometimes 
desirable to diminish the sections as they descend. This is often 
accomplished in mines by making the rod in sections, each section 
being calculated to bear the weight below it plus its own weight. 
Were the sections gradually diminished, so that each section 
would be just large enough to support the weight below it, we 
should, of course, have a curvilinear form ; and the equation of 
this curve could be found as follows, or, rather, the area of any 
section at a distance from the bottom of the rod. 
Let W = weight hung at O (Fig. 152), 

Let w = weight per unit of volume of 

the rod, 

Let x = distance AO, 

Let 5 = area of section A, 

Let x + dx = distance BO, 



Let 5 + dS = area of section at B t 



Let f = working-strength of the mate- 

rial per square inch. 

i. The section at O must be just large enough 
to sustain the load W; 
*ff ' W 

FIG. 152. * S Q = f- 

2. The area in dS must be just enough to sustain the 
weight of the portion of the rod between A and B. 
The weight of this portion is wSdx ; 

_ wSdx 
.'. <>= 

dS w iv 

.*. -= = -fdx /. log, S = -7X H- a constant 

> / / 



CYLINDERS SUBJECTED TO INTERNAL PRESSURE. .2$ I 



W 

When x = o, 5 = -^ ; 

W IW\ w 

.-. log* Y = the constant .% log,.S log,( -y- ) = -^ 



This gives us the means of determining the area at any dis- 
tance x from O. 

EXAMPLES. 

1. A wrought-iron tension-rod 200 feet long is to sustain a load of 
2000 Ibs. with a factor of safety of 4, and is to be made in 4 sections, 
each 50 feet long; find the diameter of each section, the weight of the 
wrought iron being 480 Ibs. per cubic foot. 

2. Find the diameter needed if the rod were made of uniform 
section, also the weight of the extra iron necessary to use in this case. 

3. Find the equation of the longitudinal section of the rod, assum- 
ing a square cross-section, if it were one of uniform strength, instead of 
being made in 4 sections. 

174. Thin Hollow Cylinders subjected to an Internal 
Normal Pressure. Let/ denote the uniform intensity of the 
pressure exerted by a fluid which is confined within a hollow 
cylinder of radius r and of thickness / (Fig, 153), 
the thickness being small compared with the radius. 
Let us consider a unit of length of the cylinder, and c ( 
let us also consider the forces acting on the upper 
half-ring CED. PIG. 153. 

The total upward force acting on this half-ring, in conse- 
quence of the internal normal pressure, will be the same as 
that acting on a section of the cylinder made by a plane pass- 
ing through its axis, and the diameter CD. The area of this 




252 APPLIED MECHANICS. 

section will be 2r X I = 2r : hence the total upward force will be 
2r X / = 2pr; and the tendency of this upward force is to cause 
the cylinder to give way at A and B, the upper part separating 
from the lower. 

This tendency is resisted by the tension in the metal at the 
sections AC and BD ; hence at each of these sections, there has 
to be resisted a tensile stress equal to \(2pr) pr. This stress 
is really not distributed uniformly throughout the cross-section 
of the metal ; but, inasmuch as the metal is thin, no serious 
error will be made if it be accounted as distributed uniformly. 
The area of each section, however, is t X i = / / therefore, if 
T denote the intensity of the tension in the metal in a tangential 
direction (i.e., the intensity of the hoop tension), we shall have 



Hence, to insure safety, T must not be greater than f, the 
working-strength of the material for tension ; hence, putting 



f-pr 
/- / , 

we shall have 

/ = 7 

as the proper thickness, when/ = normal pressure per square 
inch, and radius = r. 

The above are the formulas in common use for the deter- 
mination of the thickness of the shell of a steam-boiler ; for in 
that case the steam-pressure is so great that the tension 
induced by any shocks that are likely to occur, or by the weight 
of the boiler, is very small in comparison with that induced 
by the steam-pressure. On the other hand, in the case of an 
ordinary water-pipe, the reverse is the case. 



RESISTANCE TO DIRECT COMPRESSION. 



To provide for this case, Weisbach directs us to add to the 
thickness we should obtain by the above formulae, a constant 
minimum thickness. 

The following -are his formulae, d being the diameter in 
inches, / the internal normal pressure in atmospheres, and / 
the thickness in inches. For tubes made of 

Sheet-iron ......... /* = 0.00086 pd -f 0.12 

Cast-iron ......... t = 0.00238^ -f- 0.34 

Copper ....... . . . / = o.ooi48/^/ + 0.16 

Lead .......... /= 0.00507^+ 0.21 

Zinc ........... / = 0.00242 pd -+- 0.16 

Wood .......... / = 0.03230^ -f- 1.07 

Natural stone ........ /= 0.03690/^4- 1.18 

Artificial stone ....... t = 0.05380/^4- 1.58 

175. Resistance to Direct Compression. When a piece 
is subjected to compression, the distribution of the compressive 
stress on any cross-section depends, first, upon whether the 
resultant of the pressure acts along the line containing the cen- 
tres of gravity of the sections, and, secondly, upon the dimen- 
sions of the piece ; thus determining whether it will bend or 
not. 

In the case of an eccentric load, or of a piece of such length 
that it yields by bending, the stress is not uniformly distributed ; 
and, in order to proportion the piece, we must determine the 
greatest intensity of the stress upon it, and so proportion it 
that this shall be kept within the working-strength of the ma- 
terial for compression. 

Either of these cases is not a case of direct compres- 
sion. 

In the case of direct compression (i.e., where the stress over 
each section is uniformly distributed), the intensity of the stress 
is found by dividing the total compression by the area of the 



254 APPLIED MECHANICS. 

section ; so that, if P be the total compression, and A the area 
of the section, and / the intensity of the compressive stress, 



On the other hand, if f is the compressive working-strength oi 
the material per square inch, and A the area of the section in 
square inches, then the greatest allowable load on the piece 
subjected to compression is 



The same remarks as were made in regard to a suddenly 
applied load and resilience, in the case of direct tension, apply 
in the case of direct compression. 

176. Results of Wohler's Experiments on Compressive 
Strength. Wohler also made experiments in regard to pieces 
subjected to alternate tension and compression, taking, in the 
experiments themselves, the case where the metal is subjected 
to alternate tensions and compressions of equal amount. 

The greatest stress of which the piece would bear an indefi- 
nite number of changes under these conditions, is called the 
vibration safe strength, and is denoted by s. 

Weyrauch deduces a formula similar to that of Launhardt 
for the greatest allowable stress per unit of area on the piece 
when it is subjected to alternate tensions and compressions of 
different amounts. 

Thus, for Phoenix Company's axle iron, Wohler deduces 

/ = 3290 kil. per sq. cent. = 46800 Ibs. per sq. in., 
u = 2100 kil. per sq. cent. = 30000 Ibs. per sq. in., 
s = 1 1 70 kil. per sq. cent. = 1 6600 Ibs. per sq. in. 



EXPERIMENTS ON COMPRESSIVE STRENGTH. 2$$ 

Weyrauch's formula for the ultimate strength per unit of 
area is 

{u s least maximum stress | 
u greatest maximum stress \ ' 

and, with these values of u and s, it gives 



least maximum stress 
a = 2100 



!i least maximum stress | 
1 ~ 2 greatest maximum stress j ' per S( ** cent * 

11 least maximum stress ) 
i - = - / Ibs. per sq. in. 

2 greatest maximum stress ) 



With a factor of safety of 3, we should have, for the greatest 
admissible stress per square inch, 

( i least maximum stress ) 
b = i oooo < i : Jibs. 

( 2 greatest maximum stress) 

For Krupp's cast-steel, 

/ = 7340 kil. per sq. cent. = 104400 Ibs. per sq. in., 

u = 3300 kil. per sq. cent. = 46900 Ibs. per sq. in. approximately, 

s = 2050 kil. per sq. cent. = 29150 Ibs. per sq. in. approximately. 

We have, therefore, for the breaking-strength per unit of 
area, according to Weyrauch's formula, 

least maximum stress 
a 



or 

a 



!c least maximum stress ) 
- il greatest maximum stress } kiL per Sq ' Cent ' 

( c least maximum stress ) 

- 4 6 9 oo| , - fi greatestmaximumstress [lbs. per sq. .; 



256 APPLIED MECHANICS. 

and, using a factor of safety of 3, we have, for the greatest admis- 
sible stress per square inch, 

!r least maximum stress ) , 
i f: - : - > Ibs. per sq. in. 

1 1 greatest maximum stress j 



b 15630 



The principles respecting an eccentric compressive load, and 
those respecting the giving-way of long columns so far as they 
are known, can only be treated after we have studied the resist- 
ance of beams to bending; hence this subject will be deferred 
until that time. 

EXAMPLES. 

Find the proper working and breaking strength per square inch to 
be used for a wrought- iron rod, the extreme stresses being 

1. 80000 Ibs. tension and 6000 Ibs. compression. 

2. i ooooo Ibs. tension and 100000 Ibs. compression. 

3. 70000 Ibs. tension and 60000 Ibs. compression. 
Do the same for a steel rod. 

177. Resistance to Shearing. One of the principal cases 
where the resistance to shearing comes into practical use is 
that where the members of a structure, which are themselves 
subjected to direct tension or compression or bending, are united 
by such pieces as bolts, rivets, pins, or keys, which are sub- 
jected to shearing. Sometimes the shearing is combined with 
tension or with bending ; and whenever this is the case, it is 
necessary to take account of this fact in designing the pieces. 
It is important that the pins, keys, etc., should be equally 
strong with the pieces they connect. 

Probably one of the most important modes of connection is 
by means of rivets. In order that there may be only a shearing 
action, with but little bending of the rivets, the latter must 
fit very tightly. The manner in which the riveting is done will 
necessarily affect very essentially the strength of the joints; 



RESISTANCE TO SHEARING. 



257 



hence the only way to discuss fully the strength of riveted 
joints is to take into account the manner of effecting the rivet- 
ing, and hence the results of experiments. These will be 
spoken of later ; but the ordinary theories by which the strength 
and proportions of .some of the simplest forms of riveted joints 
are determined will be given, which theories are necessary also 
in discussing the results of experiments thereon. 

The principle on which the theory is based, in these simple 
cases, is that of making the resistance of the joint to yielding 
equal in the first three, and also in either the fourth or the 
fifth of the ways in which it is possible for it to yield, as 
enumerated on pages 258 and 259. 



A single-riveted lap-joint is one 
with a single row of rivets, as 
shown in Fig. 154. 



A single-riveted butt-joint with 
covering plate is shown in 

T C C 



one 
Fig. 155 



A single-riveted butt-joint with 
two covering, plates is shown in 
Fig. 156. 



FIG. 154. 






FIG. 155. 




FIG. 156. 



25 8 



APPLIED MECHANICS. 



FIG. 157. 



FIG. 158. 



A double-riveted lap-joint with 
the rivets staggered is shown in 
Fig. 157; one with chain riveting, 
in Fig. 158. 



Taking the case of the single-riveted lap-joint shown in Fig. 
1 54, it may yield in one of five ways : 



i. By the crushing of the plate 
in front of the rivet (Fig. 159). 



FIG. 159. 



FIG. 160. 



2. By the shearing of tne nvet 
(Fig. 160). 



RESISTANCE TO SHEARING. 



259 











3. By the tearing of the plate 
between the rivet-holes (Fig. 161). 


1 







FIG. 161. 



4. By the rivet breaking 
through the plate (Fig. 162). 

5. By the rivet shearing out 
the plate in front of it. 



Let us call 

d the diameter of a rivet. 

/ the pitch of the rivets ; i.e., FlG - l62 - 

their distance apart from centre to centre. 
t the thickness of the plate. 
/ the lap of the plate ; i e., the distance from the centre 

of a rivet-hole to the outer edge of the plate. 



f t the ultimate tensile strength of the iron. 
/, the ultimate shearing-strength of the rivet-iron. 
f s > the ultimate shearing-strength of the plate. 
f c the ultimate crushing-strength of the iron. 
We shall then have 

i. Resistance of plate in front of rivet to crushing =r f c td. 

2. Resistance of one rivet to shearing = //- Y 

3. Resistance of plate between two rivet-holes to tearing 
= /'(/ - d). 

4. Resistance of plate to being broken through = a~ , 

d 



where a is a constant depending on the material, 
taken as empirical for the present. 

A reasonable value of this constant is /". 



This may be 



260< APPLIED MECHANICS. 

5. Resistance of plate in front of the rivet to shearing 
= 2/,7/. 

Assuming that we know the thickness of the plate to start 
with, we obtain, by equating the first two resistances, 



which determines the diameter of the rivet. 
Equating 3 and 2, we obtain 



which gives the pitch of the rivets in terms of the diameter of 
the rivet, and the thickness of the plate. 
Equating, next, 4 and i, we have 



which gives the lap of the plate needed in order that it may not 
break through. 

By equating 5 and i, we find the lap needed that it may 
not shear out in front of the rivet. 

A similar method of reasoning would enable us to determine 
the corresponding quantities in the cases of double-riveted 
joints, etc. 

There are a number of practical considerations which 
modify more or less the results of such calculations, and which 
can only be determined experimentally. A fuller account of 
this subject from an experimental point of view will be given 
later. 

178. Intensity of Stress. Whenever the stress over a 
plane area is uniformly distributed, we obtain its intensity at 
each point by dividing the total stress by the area over which 
it acts, thus obtaining the amount per unit of area. When, how- 
ever, the stress is not uniformly distributed, or when its inten- 



INTENSITY OF STRESS. 



26l 



sity varies at different points, we must adopt a somewhat differ- 
ent definition of its intensity at a given point. In that case, if 
we assume a small area containing that point, and divide the 
stress which acts on that area by the area, we shall have, in the 
quotient, an approximation to the intensity required, which will 
approach nearer and nearer to the true value of the intensity at 
that point, the smaller the area is taken. 

Hence the intensity of a variable stress at a given point is, -- 

The limit of the ratio of the stress acting on a small area 
containing that point, to the area, as the latter grows smaller and 
smaller. 

By dividing the total stress acting on a certain area by the 
entire area, we obtain the mean intensity of the stress for the 
entire area. 

179. Graphical Representation of Stress A conven- 
ient mode of representing stress 
graphically is the following: 

Let AB (Fig. 163) be the plane 
surface upon which the stress acts ; 
let the axes OX and OY be taken 
in this plane, the axis OZ being at 
right angles to the plane. 

Conceive a portion of a cylinder 
whose elements are all parallel to 
OZ, bounded at one end by the 
given plane surface, and at the 
other by a surface whose ordinate 
many units of length as there are units of force in the intensity 
of the stress at that point of the given plane surface where the 
ordinate cuts it. 

The volume of such a figure will evidently be 

V = ffzdxdy = ffpdxdy, 
where z / = intensity of the stress at the given point. 




FIG. 163. 

at any point contains as 



262 



APPLIED MECHANICS. 



Hence the volume of the cylindrical figure will contain as 
many units of volume as the total stress contains units of 
force ; or, in other words, the total stress will be correctly repre- 
sented by the volume of the body. 

If the stress on the plane 
figure is partly tension and 
partly compression, the sur- 
face whose ordinates repre- 
sent the intensity of the 
stress will lie partly on one 
side of the given plane sur- 
face and partly on the other ; 
this surface and the plane 
surface on which the stress 
acts, cutting each other in 
some line, straight or curved, 
as shown in Fig. 164. In that 




FlG - 



case, the magnitude of the resultant stress P V 

will be equal to the difference of the wedge-shaped volumes 

shown in the figure. 

It will be observed that the above method of representing 
stress graphically represents, i, the intensity at each point of 
the surface to which it is applied ; and, 2, the total amount 
of the stress on the surface. It does not, however, represent 
its direction, except in the case when the stress is normal to 
the surface on which it acts. 

In this latter case, however, this is a complete representa- 
tion of the stress. 

The two most common cases of stress are, i, uniform stress, 
and, 2, uniformly varying stress. These two cases are repre- 
sented respectively in Figs. 165 and 166; the direction also 
being correctly represented when, as is most frequently the 
case, the stress is normal to the surface of action. In Fig. 
165, AB is supposed to be the surface on which the stress 



GRAPHICAL REPRESENTATION OF STRESS. 



263 




acts ; the stress is supposed to be uniform, and normal to the 

surface on which it acts ; the bound- 
ing surface in this case becomes a 

plane parallel to AB ; the intensity 

of the stress at any point, as P, will 

be represented by PQ; while the 

whole cylinder will contain as many 

units of volume as there are units of 

force in the whole stress. 

Fig. i(56 represents a uniformly 

varying stress. Here, again, AB is 

the surface of action, and the stress 

is supposed to vary at a uniform rate FlG> l65 ' 

from the axis O Y. The upper bounding surface of the cylin- 
drical figure which represents the stress 
becomes a plane inclined to the XOY 
plane, and containing the axis O Y. 

In this case, if a represent the in- 
tensity of the stress at a unit's distance 
from O Y, the stress at a distance x from 
OY will be/ = ax, and the total amount 
of the stress will be 




FIG. 166. 



P = ffpdxdy = affxdxdy. 



When a stress is oblique to the surface of action, it may be 
represented correctly in all particulars, except in direction, in 
the above-stated way. 

1 80. Centre of Stress. The centre of stress, or the 
point of the surface at which the resultant of the stress acts, 
often becomes a matter of practical importance. If, for con- 
venience, we employ a system of rectangular co-ordinate axes, 
of which the axes OX and OY are taken in the plane of the 
surface on which the stress acts, and if we let p = $(x, y) be 



264 APPLIED MECHANICS. 

the intensity of the stress at the point (x, y), we shall have, 
for the co-ordinates of the centre of stress, 

ffxpdxdy J'Sypdxdy 

: 



(see 42), where the denominator, or ffpdxdy, represents the 
total amount of the stress. 

When the stress is positive and negative at different parts 
of the surface, as in Fig. 164, the case may arise when the posi- 
tive and negative parts balance each other, and hence the 
stress on the surface constitutes a statical couple. In that case 

Sfpdxdy = o. 

181. Uniform Stress. In the case of uniform stress, we 
have 

i. The intensity of the stress is constant, or / = a con- 
stant. 

2. The volume which represents it graphically becomes a 
cylinder with parallel and equal bases, as in Fig. 165. 

3. The centre of stress is at the centre of gravity of the 
surface of action ; for the formulae become, when / is constant, 

_ pffxdxdy _ ffxdxdy _ 
Xl ~~~ pffdxdy ~~~' ~~ 

pffydxdy 
= 



pffdxdy ~' ffdxdy ' 

"where x , y , are the co-ordinates of the centre of gravity of the 
surface. 

Examples of uniform stress have already been given in the 
cases of direct tension, direct compression, and, in the case of 
riveted joints, for the shearing-force on the rivet. 



UNIFORMLY VARYING STRESS. 26$ 

182. Uniformly Varying Stress. Uniformly varying 
stress has already been denned as a stress whose intensity varies 
uniformly from a given line in its own plane ; and this line will 
be called the Neutral Axis. Thus, if the plane be taken as the 
XOY plane (Fig. 166), and the given line be taken as OY, we 
shall have, if a denotes the intensity of the stress at a unit's 
distance from OY, and x the distance of any special point from 
O Y, that the intensity of the stress at the point will be 

p = ax. 

The total amount of the stress will be 

P= affxdxdy. 

The total moment of the stress about O Y will be found by 
multiplying each elementary stress by its leverage. This lever- 
age is, in the case of normal stress, x ; hence in that case the 
moment of any single elementary force will be 



and the total moment of the stress will be 

M - affx^dxdy = al. 

In the case of oblique stres x s, this result has to be modified, 
as the leverage is no longer x. Confining ourselves to stress 
normal to the plane of action, we have, for the co-ordinates of 
the centre of stress, 

_ ffpxdxdy _ affx*dxdy _ ffx^dxdy _ ffx*dxdy_ I 
ffpdxdy ~~ P = ffxdxdy '' x Q A ~ x A 

_ ffpydxdy _ affxydxdy _ ffxydxdy _ ffxydxdy 
~~ ffpdxdy ~~ P ~~ ffxdxdy = x*A 

since 

P = affxdxdy = aXoA, 

where x m y m are the co-ordinates of the centre of gravity, and 
A is the area of the surface of action. 



266 APPLIED MECHANICS. 



183. Case of a Uniformly Varying Stress which 
amounts to a Statical Couple. Whenever P = o, we have 

affxdxdy = o /. ffxdxdy = o .'. x^A = o .*. x = o. 

In this case, therefore, we have 

i. There is no resultant stress, and hence the whole stress 
amounts to a statical couple. 

2. Since X Q = o, the centre of gravity of the surface of 
action is on the axis OY, which is the neutral axis. 

Hence follows the proposition : 

When a uniformly varying stress amounts to a statical couple, 
the neutral axis contains (passes through) the centre of gravity 
of the surface of action. 

In this case there is no .single resultant of the stress ; but 
the moment of the couple will be, as has been already shown, 

M = affx 2 dxdy. 

184. Example of Uniformly Varying Stress. One of 
the most common examples of uniformly varying stress is that 
of the pressure of water upon the sides of the vessel contain- 
ing it. 

Thus, let Fig. 167 represent the vertical cross-section of a 
reservoir wall, the water pressing against the 
vertical face AB. It is a fact established by 
experiment, that the intensity of the pressure 
of any body of water at any point is propor- 
tional to the depth of the point below the 
free upper level of the water, and normal to 
the surface pressed upon. Hence, if we sup- 
pose the free upper level of the water to be 
even with the top of the wall, the intensity 
of the pressure there will be zero ; and if we represent by CB 
the intensity of the pressure at the bottom, then, joining^ and 




STRESSES IN BEAMS UNDER TRANSVERSE LOAD. 267 



C we shall have the intensity of the pressure at any point, as 
D t represented by ED, where 

ED : CB = AD : AB. 

Here, then, we have a case of uniformly varying stress nor- 
mal to the surface on which it acts. 

185. Fundamental Principles of the Common Theory 
of the Stresses in Beams under 
a Transverse Load. Fig. 168 
shows a beam fixed at one end and 
loaded at the other, while Fig. 169 
shows a beam supported at the 
ends and loaded at the middle. 
Let, in each case, the plane of the 
paper contain a vertical longi- 
tudinal section of the beam. In 

Fig. 1 68, 
it is evi- 
dent that 
the upper 

fibres are lengthened, while the lower 
ones are shortened, and vice versa in 
Fig. 169. In either case, there is, 
somewhere between the upper and 
lower fibres, a fibre which is neither 
elongated nor com- 
pressed. 

Let CN repre- 
sent that fibre, Fig. 
.168, and CP, Fig. 
169. This line may 
be called the neutral 
FIG. i6g. line of the longitu- 

dinal section ; and, if a section be made at any point at right 






268 APPLIED MECHANICS. 

angles to this line, the horizontal line which lies in the cross- 
section, and cuts the neutral lines of all the longitudinal sec- 
tions, or, in other words, the locus of the points where the 
neutral lines of the longitudinal sections cut the cross-section, 
is called the Neutral Axis of the cross-section. In the ordinary 
theory of the stresses in beams, a number of assumptions are 
made, which will now be enumerated. 



ASSUMPTIONS MADE IN THE COMMON THEORY OF BEAMS. 

ASSUMPTION No. I. If, when a beam is not loaded, a 
plane cross-section be made, this cross-section will still be a 
plane after the load is put on, and bending takes place. From 
this assumption, we deduce, as a consequence, that, if a certain 
cross-section be assumed, the elongation or shortening per unit 
of length of any fibre at the point where it cuts this cross-sec- 
tion, is proportional to the distance of the fibre from the neutral 
axis of the cross-section. 

Proof. Imagine two originally parallel cross-sections so 
near to each other that the curve in which that part of the 
neutral line between them bends may, without appreciable error, 
be accounted circular. Let ED and GH (Fig. 168 or Fig. 169) 
be the lines in which these cross-sections cut the plane of the 
paper, and let O be the point of intersection of the lines ED 
and GH. Let OF = r, FL = y, FK = /, LM = / + a/, in 
which a is the strain or elongation per unit of length of a fibre 
at a distance y from the neutral line, y being a variable ; then, 
because FK and LM are concentric arcs subtending the same 
angle at the centre, we shall have the proportion 

r + y I -\- ol y 

^ = -y- or i + a = i + 

y 

.'.a = - Or a = 

r 



ASSUMPTIONS IN THE COMMON THEORY OF BEAMS. 269 



but as y varies for different points in any given cross-section, 
while r remains the same for the same section, it follows, that, 
if a certain cross-section be assumed, the strain of any fibre at 
the point where it cuts this cross-section is proportional directly 
to the distance of this fibre from the neutral axis of the cross- 
section. 

ASSUMPTION No. 2. This assumption is that commonly 
known as Hooke s Law. It is as follows : " Ut tensio sic vis ; " 
i.e., The stress is proportional to the strain, or to the elonga- 
tion or compression per unit of length. As to the evidence in 
favor of this law, experiment shows, that, as long as the mate- 
rial is not strained beyond safe limits, this law holds. Hence, 
making these two assumptions, we shall have : At a given 
cross-section of a loaded beam, the direct stress on any fibre 
varies directly as the distance of the fibre from the neutral axis. 
Hence it is a uniformly varying stress, and we may repre- 
sent it graphically as follows : Let 
ABCD, Fig. 170, be the cross-sec- 
tion of a beam, and KL the neutral 
axis. Assume this for axis OY, and 
draw the other two axes, as in the 
figure. If, now, EA be drawn to 
represent the intensity of the direct 
(normal) stress at A, then will the 
pair of wedges AEFBKL and 
DCHGKL represent the stress graphically, since it is uni- 
formly varying. 

POSITION OF NEUTRAL AXIS. 

ASSUMPTION No. 3. It will next be shown that, on the 
two assumptions made above, and from the further assumption 
that the deformation of each fibre of the beam parallel to its* 
longitudinal axis is due to the forces acting on its ends 




FIG. 170. 



2~0 APPLIED MECHANICS. 

and that it suffers no traction from neighboring fibres, it fol- 
lows that the neutral axis must pass through the centre of 
gravity of the cross-section. 



D 


N 




1 1 


|I C E 


1 A A iff * ""vi B ' 



FIG. 171. FIG. 172. 

Since the curvatures in Figs. 168 and 169 are exaggerated 
in order to render them visible, Figs. 171 and 172 have been 
drawn. If, now, we assume a section DE, such that AD = x 
(Fig. 171) and NE = x (Fig. 172), and consider all the forces 
acting on that part of the beam which lies to the right of DE 
(i.e., both the external forces and the stresses which the other 
parts of the beam exert on this part), we must find them in 
equilibrium. The external forces are, in Fig. 172, 

i. The loads acting between B and E ; in this case there 
are none. 

2. The supporting force at B ; in this case it is equal to 

W 

, and acts vertically upwards. 

In Fig. 171 they are, 

The loads between D and N ' ; in this case there is only the 
one, W at N. 

The internal forces are merely the stresses exerted by the 
other parts of the beam on this part : they are, 

i. The resistance to shearing at the section, which is a 
vertical stress. 

2. The direct stresses, which are horizontal. 

Now, since the part of the beam to the right of DE is at 
rest, the forces acting on it must be in equilibrium ; and, since 



POSITION OF NEUTRAL AXIS. 2 7 l 

they are all parallel to the plane of the paper, we must have 
the three following conditions ; viz., 

i. The algebraic sum of the vertical forces must be zero. 

2. The algebraic sum of the horizontal forces must be zero. 

3. The algebraic sum of the moments of the forces about 
any axis perpendicular to the plane of the paper must be 
zero. 

But, on the above assumptions, the only horizontal forces 
are the direct stresses : hence the algebraic sum of these direct 
stresses must be zero ; or, in other words, the direct stresses 
must be equivalent to a statical couple. 

Now, it has already been shown, that, whenever a uniformly 
varying stress amounts to a statical couple, the neutral axis 
must pass through the centre of gravity of the surface acted 
upon. Hence in a loaded beam, if the three preceding assump- 
tions be made, it follows that the neutral axis of any cross- 
section must contain the centre of gravity of that section. 

By way of experimental proof of this conclusion, Barlow 
has shown by experiment, that, in a cast-iron beam of rectangu- 
lar section, the neutral axis does pass through the centre of 
gravity of the section. 

RESUME. 

The conclusions arrived at from the foregoing are as fol- 
lows : 

i. That at any section of a loaded beam, if a horizontal 
line be drawn through the centre of gravity of the section, 
then the fibres lying along this line will be subjected neither 
to tension nor to compression ; in other words, this line will be 
the neutral axis of the section. 

2. The fibres on one side of this line will be subjected to 
tension, those on the other side being subjected to compres- 
sion ; the tension or compression of any one fibre being proper 
tionai to its distance from the neutral axis. 



2/2 APPLIED MECHANICS. 

The first of the three assumptions of the common theory 
was not accepted by St. Venant, who developed by means of 
the methods of the Theory of Elasticity a theory of beams 
based upon the second and third assumptions only. A study 
of. St. Venant's theory involves, however, far more complica- 
tion, and requires a good previous knowledge of the Theory of 
Elasticity. Moreover the results of the two theories as far as 
the determination of the outside fibre-stresses and of the de- 
flections are practically in agreement, while, on the other hand, 
the intensities of the shearing-forces as computed by the two 
theories are not in agreement. 

The St. Venant theory may be found in several treatises 
upon the Theory of Elasticity. 

1 86. Shearing-Force and Bending-Moment. In deter- 
mining the strength of a beam, or the proper dimensions of a 
beam to bear a certain load, when we assume the neutral axis 
to pass through the centre of gravity of the cross-section, we 
have imposed the second of the three last-mentioned conditions 
of equilibrium. The remaining two conditions may otherwise 
be stated as follows : 

i. The total force tending to cause that part of the beam 
that lies to one side of the section to slide by the other part, 
must be balanced by the resistance of the beam to shearing at 
the section. 

2. The resultant moment of the external forces acting on 
that part of the beam that lies to one side of the section, about 
a horizontal axis in the plane of the section, must be balanced 
by the moment of the couple formed by the resisting stresses. 

The shearing-force at any section is the force with which the 
part of the beam on one side of the section tends to slide by the 
part on the other side. In a beam free at one end, it is equal to 
the sum of the loads between the section and the free end. In 
a beam supported at both ends, it is equal in magnitude to the 
difference between the supporting force at either end, 'and 
the sum of the loads between the section and that support. 



SHEARING-FORCE AND BEND ING-MOMENT. 2? 3 

The bending-moment at any section is the resultant moment 
of the external forces acting on the part of the beam to one 
side of the section, these moments being taken about a hori- 
zontal axis in the section. 

In a beam free at one end, it is equal to the sum of the 
moments of the loads between the section and the free end, 
about a horizontal axis in the section. 

In a beam supported at both ends, it is the difference be- 
tween the moment of either supporting force, and the sum of 
the moments of the loads between the section and that sup- 
port ; all the moments being taken about a horizontal axis in 
the section. 

Hence the two conditions of equilibrium may be more 
briefly stated as follows : 

i. The shearing-force at the section must be balanced 
by the resistance opposed by the beam to shearing at the 
section. 

2. The bending-moment at the section must be balanced 
by the moment of the couple formed by the resisting stresses. 

It is necessary, therefore, in determining the strength of a 
beam, to be able to determine the shearing-force and bending- 
moment at any point, and also the greatest shearing-force and 
the greatest bending-moment, whatever be the loads. 

A table of these values for a number of ordinary cases will 
now be given ; but I should recommend that the table be merely 
considered as a set of examples, and that the rules already 
given for finding them be followed in each individual case. 

Let, in each case, the length of the beam be /, and the 
total load W. When the beam is fixed at one end and free at 
the other, let the origin be taken at the fixed end ; when it is 
supported at both ends, let it be taken directly over one support. 
Let x be the distance of any section from the origin. Then we 
shall have the results given in the following table : 



274 



APPLIED MECHANICS. 



At Dista 
from O 






ifeh. 



ft 



Distance 
m Origin. 



' 




" 

pq 



Si 



I 



T? & 

g '-3 

>, 



IS 



11 



4'g 
! 
It 



W iJ 



.g 



r- 4> 

u 



PQ 



MOMENTS OF INERTIA OF SECTIONS. 2/5 

In a beam fixed at one end and free at the other, the great- 
est shearing-force, and also the greatest bending-moment, are at 
the fixed end. In a beam supported at both ends, and loaded 
at the middle, or with a uniformly distributed load, the greatest 
shearing-force is at either support, the greatest bending-moment 
being at the middle. In the last case (i.e., that of a beam sup- 
ported at the ends,- and having a single load not at the middle),. 
the greatest bending-moment is at the load ; the greatest shear- 
ing-force being at that support where the supporting force is 
greatest. 

187. Moments of Inertia of Sections. In the usual 
methods of determining the strength of a beam or column, it 
is necessary to know, i, the distance from the neutral axis of 
the section to the most strained fibres ; 2, the moment of in- 
ertia of the section about the neutral axis. The manner of 
finding the moments of inertia has been explained in Chap. II. 

In the following table are given the areas of a large number 
of sections, and also their moments of inertia about the neutral 
axis, which is the axis YY in each case. These results should 
be deduced by the student. 



276 



APPLIED MECHANICS. 



Distance of YY from 
Extreme Fibres. 



S. 
HP 



| 1! II 




MOMENTS OF INERTIA OF SECTIONS. 



277 



fa ^r 

S- SS 



5ia 5lx 



13 
w rt 



-o 



CQ 






278 



APPLIED MECHANICS. 



o a 

1 

rt X 
w W 
Q 







vO 



f 



I I 



-S 

-8 S 



3 ^ 

^H ^ 






> MH 

55 

"5 



g -s 

^ bfl tfl ^ bo 



!* 






MOMENTS OF INERTIA OF SECTIONS. 



'279 



+ + 






4- 

^ IN 



*?! 



II- ^ I 

H ^ 



c/T > 

| -S'S 

4 s,-^ 

's ^ s 

N 2 

" s * 

* K 



if fi ii + 



S o 



< H 



o 

'^3 rt oS rt 

Vj (U (L) 

o i- >_ i- 




JA, 





^ U 



280 



APPLIED MECHANICS. 



o g 
4> c 



SJIN 

u 



II 



** rj- 

fp 



1 



8" 



V V. 
II II 






i2 
x a 
J5 

131 

' 



11 



I 

'~ rt 



i 










MOMENTS OF INERTIA OF SECTIONS. 



28l 



1 -^1 


^ 8 




8 u 


3 1 




V* 


- 1 . 





- 1* 


* 




^ "5 <u 


i; 




-G "-Q 


rt K 




~^ 


s 




w 




f 1 


T5 yj 

<U G nj 







w ^J o 




o ^ 


III 




C 

s 


!*l 

rt bJO c/3 


-' 


I 1 ^ fi 


c ^ 




'1 ^ 


j^ o 




+ I' 


(J^ ^.| ^J 






1|l 

8 S -g 




1 




" 


<U i^ O 




N 


sli 


^ 


^ ^ ^ I, 


O ^^"^ CT3 




||) g|) | | ||||^ 


c/3 '53 ^ 




^J ^J ^^ J *5"^ s ^ 


n G 3 


g 


^v ^^ ^ w to 2 ^ "^ 


-2 ""* 


'ft 


oj^i 4)^i<u t>rt ^-t-> ^3'^g 


<U O G 

-G o,^ 

4-> 


1 


~ .2 " '?:! '-= ^ ^ | ^ f 

X rt<U ^rtS ^"3^ c3 2i "P 

to <u w . Mo'' tt '3.2'i'aow 
"Sec 'Sex 'J3-Qo o'xuc^w 

j-jrtO _j,j rt *- j-.rt*. ^rtortofi 


"rt *hO ^ 




C/2 C/2 C/2 U 


>> G 




O "-> N ro 


bJO C JU 




2" 2" 2 s 2" 


^ O C4 




jS>- 


< crj *"O 




o>- \ ^ 


'o ^ c 


w 


<---i \ m ? ^' 


s - r >. 

<U O 00 ^ 


1 


V i x fe % x 


.X3 J3 .t3 

;t-|| 




J- L-N "FT 


^ "5 ^o 




>fo 



282 



APPLIED MECHANICS. 



.5 + 



C <L> T3 
rt X! fi 

it!! 

^ P S <u 






. M_ 



S 

M U) rt e 

3 c a S 

.g rt T3 

^1 ^ 

oq .5P - *- 



u ,e <u 

< bfl H 

O 13 C 

1-. O nj 



. 





.22 - 






MOMENTS OF INERTIA OF SECTIONS. 



28 3 



2 a 

aj **-i w 
O V 



k|M 



CT 1 



Cfl M 

O 

CJ 



Sh 
II 
fe 



! 

ii 

I 



5 



st: 



VM " 

O fi 

si S 



3 ^ 




284 



APPLIED MECHANICS. 



o o D 

u H* a 

u <u 

a is 

rt x 

S5 W 
Q 









+ 
^ 



I 






. 



^ k 
w S 

"i '-3 



1 < 1 



3 C 42 

- 




MOMENTS OF INERTIA OF SECTIONS. 



285 



.< a 

K JO 



a 

rl 



IN s | N s I N <i I ( 
II II II H 



ripti 



-1 1 -1 

z I Egg 

< U J < U 



in 



Oq 



?< 




Q CQ 



> < 



: i 



286 



APPLIED MECHANICS. 



188. Cross-Sections of Phoenix Columns considered 
as made of Lines. It is to be observed that the moments 
of inertia are the same for all axes passing through the centre. 
Thickness /, radius of round ones = r, area of each flange 
= a, length of each flange = /. 



Figure. 



Description. 



A. 



Y 2 II 



Y2I2 




Four flanges 



2-nrt + 



20. 



Eight flanges 



2irrt + 8a 



(-0 



Square, four flanges, 
r radius of cir- 
cumscribed circle 



Six flanges 



6a 



REPRESENTATION OF B ENDING-MOMENTS. 



287 



189. Graphical Representation of Bending-Moments. 

The bending-moment at each point of a loaded beam may be 
represented graphically by lines laid off to scale, as will be 
shown by examples. 

I. Suppose we have the cantilever shown in Fig. 215, 
loaded at D with a load W ': then 
will the bending-moment at any 
section, as at F t be obtained by 
multiplying W by FD ; that at AC 
being W X (AB). If, now, we lay 
off CE to scale to represent this, 
i.e., having as many units of length 




FIG. 215. 



as there are units of moment in the product W X (AB), and 
join E with D, then will the ordinate FG of any point, as G, 
represent (to the same scale) the bending-moment at a section 
through F. 

II. If we have a uniformly distributed load, we should have, 
for the line corresponding to CE in Fig. 215, a curve. This is 

shown in Fig. 216, where we have the 
uniformly distributed load EIGF. If 
we take the origin at D, as before, 
we have, for the bending-moment, at a 
distance ^ from the origin, as has been 

W 

shown, -(/ xY ; and by giving x dif- 
ferent values, and laying off the corresponding value of the 
bending-moment, we obtain the curve CA, any ordinate of 
which will represent the bending-moment at the corresponding 
point of the beam. 

When we have more than one load on a beam, we must draw 
the curve of bending-moments for each load separately, and 
then find the actual bending-moment at any point of the beam 




FIG. 216. 



283 



APPLIED MECHANICS. 



by taking the sum of the ordinates (drawn from that point) of 
each of these separate curves or straight lines. If we then 
draw a new curve, whose ordinates are these sums, we shall have 
the actual curve of bending-moments for the beam as loaded. 
Some examples will now be given, which will explain them- 
selves. 

III. Fig. 217 shows a cantilever with three concentrated 
loads. The line of bending-moments 
for the load at C is CE, that for the 
load at O is OF, and for the load at P 
is PG. They are combined above the 
beam by laying off AH DE, HK = 
DF, and KL = DG, and thus obtaining 
the broken line LMNB, which is the 
line of bending-moments of the beam 
loaded with all three loads. 



FIG. 217. 

IV. Fig. 218 shows the case of a beam supported at both 
ends, and loaded at a single . point 

D; ALB is the line of bending- 
moments when the weight of the 
beam is disregarded, so that xy = 
bending-moment at x. FIG. 218 

V. Fig. 219 shows the case of a beam supported at the ends, 

and loaded with three concentrated 
loads at the points B, C, and D re- 
spectively; the lines of bending-mo- 
ments for each individual load being 
respectively AFE, AGE, and AHE, 
FIG. 219. and the actual line of bending-mo- 

ments being AKLME. 






REPRESENTATION OF BEND ING-MOMENTS. 



289 



VI. Fig. 220 shows the case of a beam supported at the 
ends, and loaded with a uniformly dis- A E F B 

tributed load ; the line of bending- 
moments being a curve, ACDB, as 



shown in the figure. FlG . 220> 



VII. In Fig. 221 we have the case of a beam, over a part of 
which, viz., EF, there is a distributed load ; the rest of the 

beam being unloaded. The line of 
bending-moments is curvilinear be- 
tween E and F, and straight outside 



Xjs^/tt y^ of these limits. It isAGSHB; and, 
when the curve is plotted, we can 



N/ 

find the greatest bending-moment 

graphically by finding its greatest ordinate. We can also 
determine it analytically by first determining the bending- 
moment at a distance x from the origin, and on the side 
towards the resultant of the load, and then differentiating. 
This process is shown in the following: 
Let A (Fig. 222) be the point where 
the resultant of the load acts, and O the 



middle of the beam, and let w be the c OA B 

load per unit of length ; let OA = a, AB = 

AC = b, and ED = 2^, so that the whole load = 2wb : there- 

a A- c wb(a 4- c] 
fore supporting force at D = 2ivb = - -. 

If we take a section at a distance x from O to the right, we 
shall have, for the bending-moment at that section, 

wb(a -+ c) w 

(c x) (a -f b x) 2 = a maximum. 

Differentiate, and we have 

wb(a -\- c\ a(c &) 

I tioi ( /T j h */\ /~\ * 

c T V -r ; ... - c 



290 



APPLIED MECHANICS. 



hence the greatest bending-moment will be 

^ \ C ) 2 \ C ) 



7 ( a 



ac 










VIII. In Figs. 223 and 224 we have the case of a beam 
supported at 
the ends, and 

1 J J <.!_ 

loaded with a 
uniformly dis- 
tributed load, 
and also with 
a c o n c e n- 
trated load. 
In the first 





Fi G .2 24 . 



FlG - 22 3- 



figure, the greatest bending-moment 



is at/?, and in the second at C. 



IX. In Fig. 225 we have a beam supported at A and B, and 
loaded at C and D with equal 
weights; the lengths of AC and 
BD being equal. We have, con- 
sequently, between A and , a 
uniform bending-moment ; while 
on the left of A and on the right 




The line of bending- 



FlG - 225 ' 

of B we have a varying bending-moment. 
moments is, in this case, CabD. 

We may, in a similar way, derive curves of bending-moment 
for all cases of loading and supporting beams. 



AT DIFFERENT PARTS OF A BEAM. 2QT 



190. Mode of Procedure for Ascertaining the Stresses^ 
at Different Parts of a Beam when the Loads and the Di- 
mensions are given, and when no Fibre at the Cross- 
section under Consideration is Strained beyond the 
Elastic Limit. When the dimensions of a beam, the 
load and its distribution, and the manner of supporting are 
given, and it is desired to find the actual intensity of the stress 
on any particular fibre at any given cross-section, we must pro- 
ceed as follows :-- 

i. Find the actual bending-moment (M) at that cross-sec- 
tion. 

2. Find the moment of inertia (/) of the section about its 
neutral axis. 

3. Observe, that, from what has already been shown, the 
moment of the couple formed by the tensions and compressions 
is al, where a = intensity of stress of a fibre whose distance 
from the neutral axis is unity, and that this moment must equal 
the bending-moment at the section in order to secure equilib- 
rium. Hence we must have 



Moreover, if / denote the (unknown) intensity of the stress 
of the fibre where the* stress is desired, and if y denote the 
distance of this fibre from the neutral axis, we shall have 



from which equation we can determine/. 

EXAMPLES. 

i. Given a beam 18 feet span, supported at both ends, and loaded 
uniformly (its own weight included) with 1000 Ibs. per foot of length. 
The cross- section is a T, where area of flange = 3 square inches, 
area of web = 4 square inches, height = 10 inches. Find (a) the 



292 APPLIED MECHANICS. 



bending-moment at 3 feet from one end ; (b) the greatest bending- 
moment; (c) the greatest intensity of the tension at each of the 
above sections ; (d) the greatest intensity of the compression at each 
>of these sections. 

2. Given an I-beam with equal flanges, area of each flange = 3 
square inches, area of web = 3 square inches, height = 10 inches; the 
beam is 1 2 feet long, supported at the ends, and loaded uniformly (its 
own weight included) with a load of 2000 Ibs. per foot of length. Find 
J(a) the bending-moment at a section one foot from the end ; (<) the 
greatest bending-moment ; (<r) the greatest intensity of the stress at 
each of the above cross-sections. 



191. Mode of Procedure for Ascertaining the Dimen- 
sions of a Beam to bear a Certain Load, or the Load that 
a Beam of Given Dimensions and Material is Capable of 
Bearing. If we wish to determine the* proper dimensions 
of the beam when the load and its distribution, as well as the 
manner of supporting, are given, so that it shall nowhere be 
strained beyond safe limits, or if we wish to determine the 
greatest load consistent with safety when the other quantities 
are given, we must impose the condition that the greatest 
intensity of the tension to which any fibre is subjected shall 
not exceed the safe working-strength for tension of the mate- 
rial of which the beam is made, and the greatest intensity of 
the compression to which any fibre is subjected shall not exceed 
the safe working-strength of the material (or compression. 

Thus, we must in this case first determine where is the 
section of greatest bending-moment (this determination some- 
limes involves the use of the Differential Calculus). 

Next we must determine the magnitude of the greatest 
bending-moment, absolutely if the load and length of the beam 
are given (if not, in terms of these quantities), and then equate 
this to the moment of the resisting couple. 

Thus, if MQ is the greatest bending-moment, when the loads 
are such that no fibre is strained beyond the elastic limit, 7 the 



WORKING-STRENGTH. . 2$$ 

moment of inertia of that section where this greatest bending-moment 
acts, and if } t = greatest tensile fibre stress per square inch, f c = 
greatest compressive fibre strength per square inch, y t = distance 
of most stretched fibre from the neutral axis, and y c = distance 

of most compressed fibre from the neutral axis, then will be 

yt 

the greatest tension per square inch, at a unit's distance from the 
neutral axis, and the greatest compression per square inch, at a 

unit's distance from the neutral axis. 

Moreover, in this case, these two ratios are equal, and hence 

l/f f* T ? c T 
MQ = I = I. 

yt y c 

SAFE OR WORKING-LOAD. 

If // = safe working-strength per square inch for tension, 
/ c '=safe working-strength per square inch for compression, and 
M Q = greatest safe working bending-moment, then the ratios, 

and , are not equal. 

yt yc 

f r f ' f ' 

Hence, when is less than we have M o' = /, and when. 

yt y c yt 

It' ]c fc 

is greater than we have MQ =I. 

yt y c y c 

BREAKING-LOAD AND MODULUS OF RUPTURE. 

If M is the greatest bending-moment when the beam is 
subjected to its breaking-load, the formulae given above do not 
apply, inasmuch as a portion of the fibres are strained beyond 
the elastic limit, and Hooke's law no longer holds, since, after 
the elastic limit is passed, the ratio of stress to strain decreases 
when the stress increases. 

Indeed, the stresses in the different fibres are no longer pro- 




294 APPLIED MECHANICS. 

portional to the distances of those fibres from the neutral axis. 
A graphical representation of the stress at different points of any 
given section AB would be of the character shown in the figure, 
o , the form of the curve CDE varying with the shape 
of the cross-section. 

Nevertheless, it is customary to compute the 
breaking-strength of a beam by means of the 

My 
formula /= = , where y is taken as the distance from the neutral 

axis to that outer fibre which gives way first, i.e., to the most 
stretched fibre if the beam breaks by tension, or to the most com- 
pressed fibre, if it breaks by compression. The quantity /, which 
may thus be computed from the formula 

My 



is defined as the Modulus oj Rupture. 

Inasmuch as this formula would give the outside fibre stress, 
if the stress were uniformly varying, it follows that, in the case of 
materials for which the tensile is less than the compressive strength, 
the modulus of rupture is greater than the tensile strength, while 
in that of materials for which the compressive is less than 
the tensile strength the modulus of rupture is greater than the 
compressive strength. 

For experimental work bearing upon this matter, see an 
article by Prof. J. Sondericker, in the Technology Quarterly for 
October, 1888. 

WORKING-STRENGTH. 

The working-strength per square inch of a material for trans- 
verse strength is the greatest stress per square inch to which it 
is safe to subject the most strained fibre of the beam. It is usually 
obtained by dividing the modulus of rupture by some factor of 
safety, as 3 or 4. 



WORKING-STRENGTH. 2$$ 

192. EXAMPLES. 

i. Given a beam (Fig. 226) supported at both ends, and loaded, 
i, with w pounds per unit of length uniformly, and 2, with a single 
load Wat a. distance a from the left-hand support: find the position 
of the section of greatest bending-moment, and the value of the greatest 
bending-moment. 
o A a Solution. 



1 



(i) Left-hand supporti ng- force = - -\ . 

Right-hand supporting-force = _ _j_ t 



(2) Assume a section at a distance x from the left-hand support 
(this support being the origin), and the bending-moment at that sec- 
tion is, 

(wl W(l - a)\ wx* 
when x < a, -\ 



2 ) 2 

and when x > a, 

wx* 



To find the value of x for the section of greatest bending-moment, 
differentiate each, and put the first differential co-efficient = zero. 
We shall thus have, in the first case, 

W l W(l -a) I W(l - a) 

-- 1 -- ~ - - w# = o, or x = - H -- - -. - : 
2 / 2 wl 

and in the second case, 

wl W(l -a) I W(l -a) W 

- + , - - -wx- W** o, or x = - + v . - -- . 

2 / 2 Wl W 

Now, whenever the first is < #, or the second is > a, we shall have 
in that one the value of x corresponding to the section of greatest 
bending-moment. But if the first is > a, and the second < 0, then the 
greatest bending-moment is at the concentrated load. 

These conclusions will be evident on drawing a diagram representing 
the bending-moments graphically, as in Figs. 223 and 224; and the 
greatest bending-moment may then be found by substituting, in the cor- 
responding expression for the bending-moment, the deduced value of x. 



296 APPLIED MECHANICS. 

2. Given an I-beam, 10 feet long, supported at both ends, and 
loaded, at a distance 2 feet to the left of the middle, with 20000 pounds. 
Find the bending-moment at the middle, the greatest bending-moment, 
also the greatest intensity of the tension, and that of the compression at 
each of these sections. 

Given Area of upper flange = 8 sq. in. 

Area of lower flange = 5 sq. in. 

Area of web = 7 sq. in. 
Total depth = 14 in. 

193. Beams of Uniform Strength. Abeam of uniform 
strength (technically so called) is one in which the dimensions 
of the cross-section are varied in such a manner, that, at each 
cross-section, the greatest intensity of the tension shall be 
the same, and so also the greatest intensity of the com- 
pression. 

-Such beams are very rarely used ; and, as the cross-section 
varies at different points, it would be decidedly bad engineering 
to make them of wood, for it would be necessary to cut the 
wood across the grain, and this would develop a tendency to 
split. 

In making them of iron, also, the saving of iron would gen- 
erally be more than offset by the extra cost of rolling such a 
beam. Nevertheless, we will discuss the form of such beams in 
the case wlien the section is rectangular. 

In all cases we have the general equation 



y 

applying at each cross-section, where M = bending-moment 
''section at distance x from origin), / = moment of inertia of 
same section, ' y = distance from neutral axis to most strained 
fibre, and p intensity of stress on most strained fibre ; the 
condition for this case being that / is a constant for all values 
of x (i.e., for all positions of the section), while M, /, and y 
are functions of x. 



BEAMS OF UNIFORM STRENGTH. 297 

As we are limiting ourselves to rectangular sections, if we 
let b = breadth and h = depth of rectangle (one or both vary- 
ing with x\ we shall have 



as the condition for such a beam, with/ a constant for all values 
of x, when the same load remains on the beam. 

We must, therefore, have bk 2 proportional to M. Hence, 
assuming the origin as before, 

i. Fixed at one end, load at the other, bh* =(-) W(l *). 



2. Fixed at one end, uniformly loaded, bh 2 = ( - ) (/ x) 2 . 

\ 21' 



, 

3. Supported at ends, loaded at I 2 \/ 2 

middle - | 

2 p 2 

4. Supported at ends, uniformly loaded, bh 2 = ( --- }(lx x 2 ). 

\j> 2l ' 

Now, this variation of section may be accomplished in one 
of two ways: ist, by making h constant, and letting b vary; 
and 2d, by making b constant, and letting h vary. Thus, in 
the first case above mentioned, if h is constant, we have, for the 
plan of the beam, 



and if one side be taken parallel to the axis of the beam, this 
will be the equation of the other side ; and, as this is the equa- 
tion of a straight line, the plan will be a triangle. 



APPLIED MECHANICS. 



If, on the other hand, b be constant, and h vary, we shall 
nave, for the vertical longitudinal section of the beam, 



and, if one side be taken as a straight line in the direction of 
the axis, the other will be a parabola. 

A similar reasoning will give the plan or elevation respect- 
ively in each case ; and these can be readily plotted from their 
equations. 

CROSS-SECTION OF EQUAL STRENGTH. 

A cross-section of equal strength (technically so called) is 

one so proportioned that the greatest intensity of the tension 

shall bear the same ratio to the breaking tensile strength of the 

material as the greatest intensity of the compression bears to 

the breaking compressive strength of the material. This is 

accomplished, as will be shown directly, by so arranging the 

form and dimensions of the section that the distance of the 

neutral axis from the most stretched fibre shall bear to its 

distance from the most compressed fibre the same ratio that 

the tensile bears to the compressive strength of the material. 

Let f c breaking-strength per square inch for compression, 

f t = breaking-strength per square inch for tension, 

y c = distance of neutral axis from most compressed 

fibre, 

y t = distance of neutral axis from most stretched fibre. 

If p c = actual greatest intensity of compression, and p t = 

actual greatest intensity of tension, then, for a cross-section 

of equal strength, we must have, according to the definition, 

<=; but we have = = intensity of stress at a unit's 

pt ft yc yt 



CROSS-SECTION OF EQUAL STRENGTH. 2Q9 

distance from the neutral axis. Hence, combining these two, 
we obtain 

y - = 7 

y t ft 

EXAMPLE. 

Suppose we have^ = 80000 Ibs. per square inch, and/ = 20000 
Ibs. per square inch. : find the proper proportion between the flange A t 
and the web A 2 of a T-section whose depth is h. 

194. Deflection of Beams. We have already seen ( 185), 
that, in the case of a beam which is bent by a transverse load, 
we have 

-'oo, 



a 

r 



where (having assumed a certain cross-section whose distance 
from the origin is x) a = the strain of a fibre whose distance 
from the neutral axis is y, and r = radius of curvature of 
the neutral lamina at the section in question. Hence follows the 
equation 



but from the definition of E t the modulus of elasticity, we shall 
have 

V* 

where / = intensity of the stress, at a distance y from the 
neutral axis. 

Hence it follows, assuming Hooke's law, that 

r Ey E y 
We have already seen, that, disregarding signs, M = - / 



3^0 APPLIED MECHANICS. 

(making, of course, the two assumptions already spoken of 
when this formula was deduced), where M = bending-moment 
at, and / = moment of inertia of, the section in question ; i.e., 
of that section whose distance from the origin is x. This gives 

- = , if, denoting tension by the + sig n > and taking y 

positive upwards, we call M positive when it tends to cause 
tension on the lower, and compression on the upper, side; these 
being the conventions in regard to signs which we shall adopt 
in future. Hence, by substitution, we have 

1 - p - M (i\ 

~r~Ey- El 

Now, if we assume the axis of x coincident with the neutral 
line of the central longitudinal section of the beam, and the 
axis of v at right angles to this, and v positive upwards, no 
matter where the origin is taken, we shall always have, as is 
shown in the Differential Calculus, 



' (+(1)7 

Hence equation (i) becomes 



(*) 



M and / being functions of x : and, when we can integrate 
this equation, we can obtain v in terms of x, thus having the 
equation of the elastic curve of the neutral line ; and, by com- 
puting the value of v corresponding to any assumed value of x, 
we can obtain the deflection at that point of the beam. 



FORMULA FOR SLOPE AND DEFLECTION. 3OI 

The above equation (2) is, as a rule, too complicated to be 
integrated, except by approximation ; and the approximation 
usually made is the following : 

Since in a beam not too heavily loaded, the slope, and con- 
sequently the tangent of the slope (or angle the neutral line 
makes with the horizontal at any point), is necessarily small, it 

follows that is very small, and hence (--) is also very small, 
dx \dxl 

and i + ( ) is nearly equal to unity. Making this substitu- 
tion, we obtain, in place of equation (2), 



-. 

d* ~ EI* 

and this is the equation with which we always start in com- 
puting the slope and deflection of a loaded beam, or in finding 
the equation of the elastic line. 

By one integration (suitably determining the arbitrary con- 

stant) we obtain the slope whose tangent is , and by a second 

dx 

integration we obtain the deflection v at a distance x from the 
origin ; and thus, by substituting any desired value for x, we 
can obtain the deflection at any point. 

195. Ordinary Formulae for Slope and Deflection. 
We may therefore write, if i is the circular measure of the 

slope at a distance x from the origin, since i = tan i = -j- 

dx 

nearly, 

<v = M_ 

dx 2 ~~~ Ef 



f 



M 



3O2 APPLIED MECHANICS. 

In these equations, of course, E is taken as a constant, M 
must ALWAYS be expressed in terms of x, and so also must / 
whenever the section varies at different points. When, how- 
ever, the section is uniform, / is constant, and the formulae 
reduce to 



"Jiff- 



196. Special Cases -- i. Let us take a cantilever loaded 
with a single load at the free end. Assume the origin, as 
before, at the fixed end, and let the beam be one of uniform 
section. We then have M = W(l x\ 



*'~<> -*< ?- 

To determine c, observe that when x = o, i = o ; 



c = o 



is the slope at a distance x from the origin. 
The deflection at the same point will be 

, . f,** = - y& = -^- 2 - ^w 

J EIJ \ 2 ) EI\ 2 6 ) 

but when x = o, v = o .*. ^ = o /. the deflection at 
a distance x from the origin will be 



The equations (i) and (2) give us the means of finding the 
slope and deflection at any point of the beam. 

To find the greatest slope and deflection, we have that both 
expressions are greatest when x = I. Hence, if * and V Q rep- 
resent the greatest slope and deflection respectively, 

Wl 2 



SPECIAL CASES. 33 



2. Next take the case of a beam supported at both ends 
and loaded uniformly, the load per unit of length being w. 
Assume the origin at the left-hand end ; then 



wl wx 2 w 
M x --- = Ux x 2 ) and W= wl 

2 2 2 V 



w 



/w /lx 2 x*\ 
(lx - x ^ x = _(___) + ,. 



2EI ^ 

To determine c, we have that when x = -, then i = o; 

W // 3 / 3 \ 2// 3 

V 8 " V + 
- } - W ^ W (6/x 2 - /3^ 

f ff '/ ' V -5 "?/ r> A 7? T <y A Ji ' 7 ^ ^ ' V / 



/w /* 
to = ^7j (6 ^ " ^ ~ /3) ^ 



24^7 

But when x = o, v = o ; 



/. r = o 



For the greatest slope, we have ;tr = o, or x = I; 



24^*7 24^7 



For the greatest deflection, x = - ; 



w 5/ 4 5 a// 4 

384^7 



304 APPLIED MECHANICS. 

3. Take the case of a beam supported at both ends, and 
loaded at the middle with a load W. 

Assume, as before, the origin at the left-hand support. 
Then we shall have 

W I W I 

M = x, x < -, and M (/ x} when x > 

Therefore, for the slope up to the middle, we have 

w r w x 2 

i = ^r, I xdx = =- T h c. 

2EIJ 2EI 2 

When x ~ , then i = o ; 

wr 



W . 



and 



w /v , r\. w (x> rx\ 

v - I \x lax = I 

AJcl'J \ 4/ 4.fi/\3 4/ 

But when x o, v = o ; 



c = o. 



i 4 

The slope is greatest when x o ; 

.'. z' = 



The deflection is greatest when x = -; 



4. In the following table 7 denotes the moment of inertia 
of the largest section : 



SPECIAL CASES. 



305 



Uniform Cross-Section. 


Greatest Slope. 


Greatest 
Deflection. 


Fixed at one end, loaded at the other 
Fixed at one end, loaded uniformly . . 
Supported at ends, load at middle . . . 
Supported at ends, uniformly loaded . . 


i Wl 2 
2 7?7 
i Wl 2 


i Wl* 
ZEI 

i Wl* 


*>EJ 
i Wl 2 


*EI 
i Wl* 


'6 El 
i Wl* 
^EI 


48^7 
5 #73 
384^/ 


Uniform Strength and Uniform Depth, 
Rectangular Section. 






Fixed at one end, load at the other . . 
Fixed at one end, uniformly loaded . . 
Supported at both ends, load at middle . 
Supported at both ends, uniformly loaded, 


Wl 2 
~EI 
i Wl 2 
2 El 
i Wl* 
*~EI 
i Wl 2 


i Wl* 
~*~EI 
iWZ* 

4^7 
i Wl* 
V~EI 

i m* 


& EI 


64^7 


Uniform Strength and Uniform Breadth, 
Rectangular Section. 






Fixed at one end, loaded at the other, 
Supported at both ends, load at middle, 
Supported at both ends, uniformly loaded 


wr 

2 ~EI 

i wr 


2 wr 

3 EI 

i wr 


4 El 

o wr 


24 EI 
0018 Wr 


0-098 EI 


0.010 EJ 



306 APPLIED MECHANICS. 

197. Deflection with Uniform Bending-Moment If 

the bending-moment is uniform, then M is constant ; and, if / 
is also constant, we have 

_ ^L f - Mx 

but when x = -, then i = o; 

Ml 



2EI 



Ml l\ dv 

" t = T^rl x -- ) = -r 
EI\ 2] dx 



lx 



the constant disappearing because v = o when x = o. 

Hence, for a beam where the bending-moment is uniform, 
we have 

_ J ^\ M Y* 2 ^ 

and for greatest slope and deflection, we have 

-Ml Mil* / 2 \ i Ml 2 

1 .71 

^O * ~~ r\ T * V<~\ -r- 



198. Resilience of a Beam. 7^ resilience of a beam 
is the mechanical work performed in deflecting it to the amount 
it would deflect under its greatest allowable gradually applied 
load. In the case of a concentrated load, if W is the greatest 
allowable gradually applied load, and v l the corresponding 
deflection at the point of application of the load, then will the 

W 

mean value of the load that produces this deflection be 

W 

and the resilience of the beam will be z/,. 

2 



SLOPE AND DEFLECTION OF A BEAM. 3O/ 



i99- Slope and Deflection of a Beam with a Con- 
centrated Load not at the 

Middle. Take, as the next A a 

case, a beam (Fig. 228). Let < a 

the load at A be W, and dis- 
tance OA = a y and let a > -. 

2 FIG. 228. 

W(l - a) 
x < a M = ^ '- x, 



^f 



- a) 

- x<a ' = 



When x = o, * = 4 = undetermined slope at 

= . = W(l - a 

and 



When x =. o, v = o ; 



To determine r, observe that when ^r ^, this value of i 
and that deduced from (i) must be identical. 

Waf _^!\, W(l - a)a* . Wa* . 

TEI\ a 2 ) H 2/^7 2^7 



308 APPLIED MECHANICS. 



Wat , x*\ Wa* , . , 



2 / 2EI 

or 



and 

v = Jlx -**- ld)dx 



To determine c, observe that when x = a, this value of v 
and (2) must be identical ; 

Wa f 3 \ . W(l a) 



/ a 

( 

\ i 



4 4- tf 4 ) = 



61EI 6EI 
+ ^ 2 ) H- ^- (4) 
To determine 4> we have that when x /, v = o ; 



Substituting this value of i in the equations (i), (2), (3), and 
(4), we obtain for 



SLOPE AND DEFLECTION OF A BEAM. 309 



Wa , r Wa 



(4) v = 



To find the greatest deflection, differentiate (2), and place 
the first differential co-efficient equal to zero : or, which is the 
same thing, place i = o in (i), and find the value of x ; then 
substitute this value in (2), and we shall have the greatest 
deflection. 

We thus obtain 

(/-*)*> = ^( 3 / -*/-*>) .-. = "-(* ~ & + a -\ 
3 3\ ' ' / 

or 



' *=' ' a ~'> 



and the greatest deflection becomes 

Wa(l - a)(2l - a) 



_ 



2OO. EXAMPLES. 

1. In example i, p. 294, find the greatest deflection of the beam 
when it is loaded with \ of its breaking-load, assuming E = 1200000. 

2. In the same case, find what load will cause it to deflect ^J^ of its 
span. 

3. What will be the stress at the most strained fibre when this occurs- 

4. In example 3, p. 294, find the load the beam will bear without 
deflecting more than ^J^ of its span, assuming E = 24000000. 

5. Find the stress at the most strained fibre when this occurs. 

6. In example 6, p. 295, find the greatest deflection under a load 
J the breaking-load. 



3io 



APPLIED MECHANICS. 



2Oi. Deflection and Slope under Working-Load. If 

we take the four cases of deflection given in the first part of 

the table on p. 305, and calling/ the working strength of the 

material, and y the distance of the most strained fibre from 
the neutral axis, and if we make the applied load the working- 
load, we shall have respectively 

! m =*L W=^ 

y ty 

Wl // 2/7 

2 . = /. W 

2 y ly 

m = fj 

4 '" y 
wi fl 



4- T -T 



*y 

: W= ^ 



And the values of slope and deflection will become respectively, 



Slope. 



Deflection. 



Slope. 



Deflection. 



/ 



2. 



From these values, and those given on p. 305, we derive the 
following two propositions : 

i. If we have a series of beams differing only in length; 
and we apply the same load in the same manner to each, their 
greatest slopes will vary as the squares of their lengths, and 
their greatest deflections as the cubes of their lengths. 



SLOPE AND DEFLECTION OF RECTANGULAR BEAMS. 31! 



2. If, however, we load the same beams, not with the same 
load, but each one with its working-load, as determined by 
allowing a given greatest fibre stress, then will their greatest 
slopes vary as the lengths, and their greatest deflections as the 
squares of their lengths. 

202. Slope and Deflection of Rectangular Beams 

bfc h 

If the beams are rectangular, so that / = and y = -, the 

values of slope and deflection above referred to become further 
simplified, and we have the following tables : 





Given Load W. 


Working-Load. 
Greatest Fibre Stress =/. * 


Slope. 


Deflection. 


Slope. 


Deflection. 


1. 

2. 

3. 
4- 


6W1 2 


4 #73 


I 


2 / 2 


Ebfc 
2W* 


Ebte 

3 Wl* 


Eh 

2_fl 


3 Eh 

i 


Ebte 

3 Wl* 
46fc 
i Wl* 


*Ebh* 
i Wl* 


3 Eh 
i fl 


2 Eh 
i/' 


lEbfc 

5 wr* 


*Eh 

2ft 

1 Eh 


(>Eh 

sfi* 

^Eh 


* Ebfc 


32 Ebh* 



So that, in the case of rectangular beams similarly loaded and 
supported, we may say that 

Under a given load W, the slopes vary as the squares of 
the lengths, and inversely as the breadths and the cubes of the 
depths ; while the deflections vary as the cubes of the lengths, 
and inversely as the breadths and the cubes of the depths. 



312 APPLIED MECHANICS. 

On the other hand, under their working-loads, the slopes vary 
directly as the lengths, and inversely as the depths ; while the 
deflections vary as the squares of the lengths, and inversely as 
the depths. 

203. Beams Fixed at the Ends. The only cases which 
we shall discuss here are the two following ; viz., 

i. Uniform section loaded at the middle. 

2. Uniform section, load uniformly distributed. 

CASE I. Uniform Section loaded at the Middle. The 
fixing at the ends may be effected by building the beam for 

some distance into the wall, as 
shown in Fig. 229. The same 
result, as far as the effect on 
|w the beam is concerned, might 

be effected as follows : Hav- 
ing merely supported it, and 

placed upon it the loads it has to bear, load the ends outside 
of the supports just enough to make the tangents at the sup- 
ports horizontal. 

These loads on the ends would, if the other load was re- 
moved, cause the beam to be convex upwards : and, moreover, 
the bending-moment due to this load would be of the same 
amount at all points between the supports ; i.e., a uniform 
bending-moment. Moreover, since the effect of the central 
load and the loads on the ends is to make the tangents over 
the supports horizontal, it follows that the upward slope at the 
support due to the uniform bending-moment above described 
must be just equal in amount to the downward slope due to the 
load at the middle, which occurs when the beam is only sup- 
ported. 

Hence the proper method of proceeding is as follows : 

i. Calculate the slope at the support as though the beam 
were supported, and not fixed, at the ends ; and we shall 
if we represent this slope by i u the equation 



BEAMS FIXED AT THE ENDS. 313 



w* 



2. Determine the uniform bending-moment which would 
produce this slope. 

To do this, we have, if we represent this uniform bending- 
moment by M lf that the slope which it would produce would be 



and, since this is equal to * we shall have the equation 

_^/_ ;w_. M 

TEI -*"- V3) 



.: Jf, = ~ (4) 

This is the actual bending-moment at either fixed end ; and the 
bending-moment at any special section at a distance x from 
the origin will be 



where M is the bending-moment we should have at that sec- 
tion if the beam were merely supported, and not fixed. Hence, 
when it is fixed at the ends, we shall have, for the bending- 
moment at a distance x from O, where O is at the left-hand 
support, 

W W, 
M=oc-l. (5) 

When x = -, we obtain, as bending-moment at the middle, 

*-?; (6) 

o 

and, since M l = M , it follows that the greatest bending- 
moment is 

W 

8 ; 



3H APPLIED MECHANICS. 

this being the magnitude of the bending-moment at the middle 
and also at the support. 

POINTS OF INFLECTION. 

The value of M becomes zero when 

x = - and when x = ; 
4 4' 

hence it follows that at these points the beam is not bent, and 
that we thus have two points of inflection half-way between the 
middle and the supports. 

SLOPE AND DEFLECTION UNDER A GIVEN LOAD. 

We shall have, as before, 

W& Wlx . 



/M , 
EI dX = 



and since, when x = o, i = o, 
.. c ' o 

""" * = ~dx = 



W I 2X l IX 2 

v I 



3 



2 



the constant vanishing because v = o when x = o. The slope 

becomes greatest when x = -, and the deflection when x = -. 

4 
Hence for greatest slope and deflection, we have 

Wl 2 f . 

64^7' 



BEAMS FIXED AT THE ENDS. 315 



SLOPE AND DEFLECTION UNDER THE WORKING-LOAD. 

If f represent the working-strength of the material per 
square inch, and if W represent the centre working-load, we 

shall have 

fiP7 = /7 

8 " y 



CASE II. Uniform Section, Load uniformly Distributed. 
Pursuing a method entirely similar to that adopted in the former 
case, we have 

i. Slope at end, on the supposition of supported ends, is 

Wl* 



2 4 ^/ 
2. Slope at end under uniform bending-moment M t is 

() 



Hence, since their sum equals zero, 

Wl 

12 ' 

which is the bending-moment over either support. 
The bending-moment at distance x from one end is 

W Wl 
M - \lx x 2 ) . ^4) 

2/ 12 

Wl 
This is greatest when x = o, and is then . Hence great- 

1 2 

est bending-moment is, in magnitude, 

(5) 

12 



316 APPLIED MECHANICS. 



POINTS OF INFLECTION. 

M becomes zero when x = - =. (6) 

2 2 ^3 

Hence the two points of inflection are situated at a distance 
/ 

on either side of the middle. 



SLOPE AND DEFLECTION. 

, M 7 

t s= I -~dx = 



the constant vanishing because z = o when x = Oi 
W 



v = 



the constant vanishing because ^ = o when .r = o. Hence for 
greatest slope and deflection we have, t is greatest when x = 



-f i zt -y=\ 



and z; is greatest when * = - ; 



SLOPE AND DEFLECTION UNDER WORKING-LOAD. 

For working-load we have 

Wl fl 

77 = 7 



B ENDING-MOMENT AND SHEARING-FORCE. 



3'7 



EXAMPLES. 

1. Given a 4-inch by 12 -inch yellow-pine beam, span 20 feet, fixed 
at the ends ; find its safe centre load, its safe uniformly distributed load, 
and its deflection under each load. Assume a modulus of rupture 5000 
Ibs. per square inch, and factor of safety 4. Modulus of elasticity, 
1200000. 

2. Find the depth necessary that a 4-inch wide yellow-pine beam, 20 
feet span, fixed at the ends, may not deflect more than one four-hun- 
dredth of the span under a load of 5000 Ibs. centre load. 

204. Variation of Bending-Moment with Shearing- 
Force. If, in any loaded beam whatever, M represent the 
b ending-moment, and F the shearing-force at a distance x from 
the origin, then will 

*-<* > 

Proof (a). In the case of a cantilever (Fig. 230), assume 
the origin at the fixed end ; then, if M 
represent the bending-moment at a 
distance x from the origin, and M '+ ^M 
that at a distance x + tx from the 
origin, we shall have the following 
equations : 

x = l 

M= S W(a- x), 

X = X 

x = l 

M + AJ/ = -2 W(a- x A#) nearly. 

X = X 

a being the co-ordinate of the point of application of W, 

x = l 

AJ/= AjeS W nearly 



====- = 2 W: 



3 I 8 APPLIED MECHANICS. 



and, if we pass to the limit, and observe that 



we shall obtain 



(b) In the case of a beam supported at the ends (Fig. 231), 
, A.,*. assume the origin at the left-hand 

/|^ I ^j,j 7\ end, and let the left-hand support- 
ing-force be S ; then, if a represent 
FIG. 231. the distance from the origin to the 

point of application of W, we shall have the equations 



M = Sx - 2 W(x - a), 

M 4- bM = S(* + A#) - S ^(tf d5 4- A*) nearly. 
Hence, by subtraction, 

X- X 

= S . tec 2 WA* nearly 



JT = o 



= o 2 ^nearly; 

mt-' x = o 

and, if we pass to the limit, and observe that 
p ^ g 2 iff 

Jf = O 

we shall obtain 
as before. 



LONGITUDINAL SHEARING OF BEAMS. 319 

205. Longitudinal Shearing of Beams. The resistance 
of a beam to longitudinal shearing sometimes becomes a mat- 
ter of importance, especially in timber, where the resistance to 
shearing along the grain is very small. We will therefore pro- 
ceed to ascertain how to compute the intensity of the longi- 
tudinal shear at any point of the beam, under any given load ; 
as this should not be allowed to exceed a certain safe limit, to 
be determined experimentally. Assume a A 



section AC (Fig. 232) at a distance x from V 






the origin, and let the bending-moment at 
that section be M. Let the section BD be 

at a distance x + &>* from the origin, and 

let the bending-moment at that section be FIG. 232 . 

M + kM. 

Let y be the distance of the outside fibre from the neutral 
axis ; and let ca y^ be the distance of a, the point at which 
the shearing-force is required, from the neutral axis. 

Consider the forces acting on the portion ABba, and we 
shall have 




i. Intensity of direct stress at A = -j^. 

2. Intensity of direct stress at a unit's distance from neu- 

M 
tral axis = -j. 

My 
3. Intensity of direct stress at ^, where ce = y, is =-. 

(M + 
So, likewise, intensity of direct stress at / is 

Therefore, if z represent the width of the beam at the point 
?, we shall have 

M (*y 

Total stress on face Aa = -j- I yzdy, 

1 Jyi 
M + ^M r?o 

Total stress on face Bb = - j - I yzdy ; 

l J yi 



320 APPLIED MECHANICS. 



^.rr bM C y 

,*. Difference I yzdy : 
/ J yi 



and this is the total horizontal force tending to slide the piece 
AabB'on the face ab. 

Area of face ab, if #, is its width, is 



therefore intensity of shear at a is approximately 

&M C? 

-r I yzJy 



or exactly (by passing to the limit) 

/dM\ 

dM 
And, observing that F = -T-, this intensity reduces to 



(0 

We may reduce this expression to another form by observ- 
ing, that, if y z represent the distance from c to the centre of 
gravity of area Aa, and A represent its area, we have 

/Vo 

J yzdy=y 2 A; 

therefore intensity of shear (at distance j/j from neutral axis) at 
point a = 

( -M } - (a) 

This may be expressed as follows : 



LONGITUDINAL SHEARING OF BEAMS. 321 

Divide the shearing-force at the section of the beam under 
consideration, by the product of the moment of inertia of the 
section and its width at the point where the intensity of the 
shearing-force is desired, and multiply the quotient by the statical 
moment of the portion of the cross-section between the point in 
question and the outer fibre ; this moment being taken about the 
neutral axis. The result is the required intensity of shear. 

The last factor is evidently greatest at the neutral axis ; 
hence the intensity of the shearing-force is greatest at the 
neutral axis. 

LONGITUDINAL SHEARING OF RECTANGULAR BEAMS. 

For rectangular beams, we have 

th* 

/=-, *, = *. 

Hence formula (2) becomes 



^)- (3) 

For the intensity at the neutral axis, we shall have, therefore, 

I2F /h bh\ 3 F 
b*h?> \4 2 / 2 bh? 

since for the neutral axis we have 

h bh 

v a = - and A = . 
4 2 

EXAMPLES. 

i . What is the intensity of the tendency to shear at the neutral axis 
of a rectangular 4-inch by 1 2-inch beam, of 14 feet span, loaded at the 
middle with 5000 (bs. 



3 22 



APPLIED MECHANICS. 



2. What is that of the same beam at the neutral axis of the cross- 
section at the support, when the beam has a uniformly distributed load 
of T 2000 Ibs. 

3. What is that of a 9-inch by 14-inch beam, 20 feet span, loaded 
with 15000 Ibs. at the middle. 



206. Strength of Hooks. The following is the method 
to be pursued in determining the stresses in a 
hook due to a given load ; or, vice versa, the 
proper dimensions to use for a given load. 

Suppose (Fig. 233) a load hung at E; the 
load being P, and the distances 
AB n\ 




OF=y y 

O being the centre of gravity of this 
section, conceive two equal and opposite 
forces, each equal and parallel to P, acting 
at O. 

Let A = area of section, and let 7 = its 
moment of inertia about CD (BCDF represents the section 
revolved into the plane of the paper) ; then 

i . The downward force at O causes a uniformly distributed 
stress over the section, whose intensity is 



2. The downward force at E and the upward force at O 

constitute a couple, whose moment is 



and this is resisted, just as the bending-moment in a beam, by 
a uniformly varying stress, producing tension on the left, and 
compression on the right, of CD. 



COLUMNS. 323 



If we call p^ the greatest intensity of the tension due to 
this bending-moment, viz., that at B, we have 



and if / 3 denote the greatest intensity of the compression due 
to the bending moment, viz., that at F, we have 



therefore the actual greatest intensity of the tension is 



and this must be kept within the working strength if the load 
is to be a safe one ; and so also the actual greatest intensity 
of the compression, viz., that at F, is, when/, >/,, 

,-, _, -A*+*)y, ^ 
A -A A- 7 ^, 

which must be kept within the working strength for compression. 

207. Strength of Columns. The formulae most commonly 
employed for the breaking-strength of columns subjected to a load 
whose resultant acts along the axis have been, until recently, 
the Gordon formulae with Rankine's modifications, the so-called 
Euler formulae, and the avowedly empirical formulae of Hodg- 
kinson. These formulae do not give results which agree with 
those obtained from tests made upon such full-size columns as 
are used in practice. 

The deductions of the first two are not logical, certain assump- 
tions being made which are not borne out by the facts. 

When a column is subjected to a load which strains any fibre 
beyond the elastic limit, the stresses are not proportional to the 
strains, and hence there can be no rational formula for the break- 
ing-load. 

Hence, all formulae for the breaking-load are, of necessity 



3 2 4 APPLIED MECHANICS. 

empirical, and depend for their accuracy upon their agreement 
with the results of experiments upon the breaking-strength of 
such full-size columns as are used in practice. 

Nevertheless, the ordinary so-called deductions of the Gor- 
don, and of the so-called Euler formula? will be given first. 

208. Gordon's Formulae for Columns. (a) Column fixed in 
fc Direction at Both Ends. Let CAD be the central axis of the 
column, P the breaking-load, and v the greatest deflection, AB. 
Conceive at A two equal and opposite forces, each equal to P; 
then 

i. The downward force at A causes a uniformly distributed 
stress over the section, of intensity, 

^P_ 

[ D 2. The downward force at C and the upward force at A 
Fio.234. constitute a bending couple whose moment is 

M=Pv. 
If p 2 = the greatest intensity of the compression due to this bending, 



where 7= distance from the neutral axis to the most strained fibre of 
the section at A. Then will the greatest intensity of stress at A be 



and, since P is the breaking-load, p must be equal to the breaking- 
strength for compression per square inch=/'. 

(i) 

where p= smallest radius of gyration of section at A. 

Thus far the reasoning appears sound; but in the next step it is 
assumed that 



GORDON'S FORMULA FOR COLUMNS. 



325 



where c is a constant to be determined by experiment. Hence, sub- 
stituting this, and solving for P, 

P = fA , 9 , (2) 



which is the formula for a column fixed in direction at both ends. 

(b) Column hinged at the Ends. It is assumed that the points of 
inflection are half-way between the middle and the ends, and -jr~ 
hence that, by taking the middle half, we have the case of bending 
of a column hinged at the ends (Fig. 235). Hence, to obtain 
the formula suitable for this case, substitute, in (2), 2/ for /, and 

we obtain 

* 

FIG. 235. 

" M <3) 



(c) Column fixed at One End and hinged at the Other (Fig. 236). 
~~r In this case we should, in accordance with these assumptions, 
take J of the column fixed in direction at both ends; hence, to 
obtain the formula for this case, substitute, in (2), J/ for /, and 

we thus obtain 

j- 1 

(4) 



i6/ 2 ' 
gcp 2 



FIG. 236. 



>. 236. 

Rankine gives, for values of / and c, the following, based upon 
Hodgkinson's experiments: 





f 
(Ibs. per sq. in.). 


c. 


Wrought-iron 


36000 


36000 


Cast-iron 


80000 




Dry timber . 


72OO 


3OOO 









326 



APPLIED MECHANICS. 



2o8a. So-called Euler Formulae for the Strength of 
Columns. 

(a) Column fixed in Direction at One End only, which bends, as 
shown in the Figure, 

i. Calculate the breaking-load on the assumption that the column 
will give way by direct compression. This will be 

PI=/A, CO 

where /"= crushing-strength per square inch, and A = area of cross- 
section in square inches. 

2. Calculate the load that would break the column if it were to 
give way by bending, by means of the following formula : 

f, = 

where E= modulus of elasticity of the material, 7= smallest moment 
of inertia of the cross-section, and /= length of column. 

Then will the actual breaking-strength, according to Euler, be the 
smaller of these two results. 

To deduce the latter formula, assume the origin at the 
upper end, and take x vertical and y horizontal. 

Let p= radius of curvature at point (x, y), and let 
3/=bending-moment at the same point. 

Then we have, with compression plus and tension 
minus, 




PIG. as?. 



M_ 
El 



Py 

El' 



(3) 



But 



d*y P 

"dx* = "El*' 



dy d*y , P f dy, 
JL . -J-dx I y-~dx 
dx dx* EIJ J dx 



dx 



^\ = --^y 2 4- c; 



and, since for y 



Er 

~dx 



EULER FORMULA FOR STRENGTH OF COLUMNS. 327 



dy 

J 



y 

' Sm - a = 
And since, when x=o, y=> " c=o, we have 



When y=a, x=l\ hence, substituting in (5), and solving for P, 

>-(=) 

(b) Column hinged at Both Ends (Fig. 235). 
i. For the crushing-load, 

P~/A. 
2. For the breaking-load by bending, put 1/2 for I in (6) ; hence 

(7) 

(c) Column fixed in Direction at One End, and hinged at the other 
<Fig. 236). 

i. For the crushing-load, 

2. For the breaking-load by bending, put //3 for / in (6) ; hence 



((/) Column fixed in Direction at Both Ends (Fig. 234). 
i. For the crushing-load, 

P.-/4. 
2. For the breaking-load by bending, 



:his being obtained from (2) or (6) by substituting 1/4 for /. 



328 APPLIED MECHANICS. 

(e) In order to ascertain the length wnere incipient flexure occurs, 
according to this theory we should place the two results equal to each 
other, and from the resulting equation determine /. We should thus 
obtain, for the three cases respectively, 

<) /- 4/, (10) 



(r) l= 

Hence all columns whose length is less than that given in these 
formulae will, according to Euler, give way by direct crushing; and 
those of greater length, by bending only. 

209. Hodgkinson's Rules for the Strength of Columns. 
Eaton Hodgkinson made a large number of tests of small columns, 
especially of cast-iron, and deduced from these tests certain empirical 
formulas. The strength of pillars of the ordinary sizes used in practice 
has been computed by means of Hodgkinson's formulae, and tabulated 
by Mr. James B. Francis; and we find in his book the following rules 
for the strength of solid cylindrical pillars of cast-iron, with the ends 
flat, i.e., "finished in planes perpendicular to the axis, the weight 
being uniformly distributed on these planes": 

For pillars whose length exceeds thirty times their diameter, 

^=99318^, (.) 

where D= diameter in inches, /= length in feet, W= breaking- weight 
in Ibs. 

If, on the other hand, the length does not exceed thirty times the 
diameter, he gives, for the breaking-weight, the following formula: 



where W= breaking- weight that would be derived from the preceding 
formula, W'= actual breaking- weight, 



BREAKING-LOAD OF FULL-SIZE COLUMNS 329 

For hollow cast-iron pillars, if D= external diameter in inches, d= 
internal diameter in inches, we should have, in place of (i), 

7*. 5S _^ ( ^ 



/'? 

and in place of (3), 

c = i 0080 i 

4 

For very long wrought-iron pillars, Hodgkinson found the strength 
to be 1.745 times that of a cast-iron pillar of the same dimensions; but, 
for very short pillars, he found the strength of the wrought-iron pillar 
very much less than that of the cast-iron one of the same dimensions. 
With a length of 30 diameters and flat ends, the wrought-ir on exceeded 
the cast-iron by about 10 per cent. 

210. Breaking-load of Full-size Columns. The tests 
made upon full-size columns are not as many as would be desir- 
able. The details will be given in Chapter VII, but a few of the 
empirical formulae which represent their results will be given here. 

If P = breaking-load, A= area of smallest section, / = length 
of column, p = least radius of gyration of section, and } e = crushing- 
strength of the material per unit of area, it will be found that for 
values of I/ p less than a certain amount, the column remains 
straight, and the breaking-load may be computed by means 01 
the formula P = f c A . 

For greater values of l/p, the breaking-load is smaller than that 
given by this formula, and may be computed by mean;.; of the 
formula P = fA, 

by using for / a value smaller than f c , this value varying with the 
value of l/p t and being determined empirically from the results of 
tests of full-size columns. 

(a) In the case of cast-iron columns no tests have been made 
of full-size columns of the second class, while those made upon 
the first class indicate that the value of } c suitable for use in 
practice is from 25,000 to 30,000 Ibs. per square inch. 

(b) In the case of wrought-iron columns, the tests of the first 
class indicate that the value of } c suitable for use in practice is 
from 30,000 to 35,000 Ibs. per square inch. 



330 APPLIED MECHANICS 

(c) In the case of wrought-iron columns of the second class, 
the formula of Mr. C. L. Strobel for bridge columns with either 
flat or pin ends, when l/p> 90, is 

p A * 

= 46000-125-. 

A p 

On the other hand, those recommended by Prof. J. Sonde- 
ricker, of which the first was devised by Mr. Theodore Cooper, 
are as follows: 

(a) For Phoenix columns with flat ends l/p > So, 
P_ 36000 
A 



18000 

For lattice columns with pin-ends and l/p>6o, 
P_ = 340QO 
A 



12000 

(7-) For solid web, square, or box columns with flat ends, and 
l/P>8o, 

P 33000 

Z = ~(//fl-8o)2' 

10000 

($) For solid web, square, or box columns with pin-ends, and 
l/P>6o, 

P _ 31000 

A (l/p -60)* 



6000 

The number of tests that have been made upon full-size steel 
columns is very small, hence no formulae will be given here, but 
the subject will be discussed in Chapter VII. The number of 
tests that have been made upon full-size timber columns is con- 
siderable, but this subject will also be discussed in Chapter VII. 

211. Columns subjected to Loads which do not Strain 
any Fibre beyond the Elastic Limit. Under this head will 
be discussed, first, the mode of determining the greatest fibre 



THEORY OF COLUMNS. 331 

stress in a straight column subjected to an eccentric load, and, 
secondly, the general theory of columns. 

(a) Straight column, under eccentric load. Let O' be the 
centre of gravity of the lower section, and let A'O' = x , where 
A' is the point of application of the resultant of the 
eccentric load. Conceive two equal and opposite 
forces at O', each equal and parallel to P. Then we 
have: 

i. Downward force along OO r causes uniform 

P 

stress of intensity p\ = -r . 

2. The other two form a couple whose moment 
is Px , and the greatest intensity of the stress due to 

this couple is p2= - r > where a = O'B'. Hence, 
FIG. 238. 1 

the greatest intensity of the stress is 
P Px a 



and this should be kept within the limits of the working-strength. 

(b) Theory of columns. The theory of columns is that of 
the Inflectional Elastica, and is explained in several treatises, 
among which is that of A. E. H. Love on the Theory of Elasticity. 
It is as follows: 

Let the curve OP be an elastic line, on which O is a point of 
inflection. It follows that there is no bending-moment at this 
point, and hence we may assume 
that at O a single force R acts. Take Y| 

the origin at O, and axis of X along the 
line of action of the force R. Let 
E\ = modulus of elasticity of the 
material, 7 = moment of inertia of 
section about an axis through its centre of gravity, and perpen- 




33 2 APPLIED MECHANICS. 

dicular to the plane of the curve, (> = angle between OX and the 
tangent at any point P whose coordinates are x and y, a = value 
of <j) at point O, r= radius of curvature of the curve at P, s 
length of arc OP, / = length of one bay, i.e., measured from O to 

the next point of inflection, 0=T> -4= area of section, p = -*rr> 



R 

= ' 



Then we have for any such elastic line, when compressions are 
plus and tensions minus, 

i M_ 
p ET 

Moreover, since = 3 and M = Ry, we have, for a column 
p ds 

d6 R 
of the same cross-section throughout its length, ~y~ = ~~7>'> 

ID 

where the quantity j^j is a constant. 
By differentiation we obtain 

R dy R 



Integrating, and observing that at O, ~^J =O an ^ <i> =a > 



obtain 



The integration of this equation requires the use of elliptic 
integrals, hence only the results will be given here. 



THEORY OF COLUMNS. 



332* 



They are : 



(2) 



and 



(3) 



(4) 



where E denotes the elliptic integral of the second kind, and K 
the complete elliptic integral of the first kind. 

Moreover, for the determination of the load R, we obtain 
from equation (4) 



K=- 



and hence 



4K2 



(6) 



From these equations, we can, by using a table of elliptic 
functions, deduce the following results for the coordinates of points 
on the inflectional elastica, for various values of a : 



a 


5 

T 


X 

/ 


y 
I 


10 


o.oo 


o . oooo 


. OOOO 




0.25 

0.50 


0.2476 

o . 4962 


o .0392 
-554 


20 


o.oo 


o . oooo 


. OOOO 




0.25 
o. 50 


0.2376 

o . 4849 


0.0773 

o. 1079 


30 


o.oo 


. OOOO 


. OOOO 




0.25 

0.50 


0.2224 
o . 4662 


0.1135 

o. 1620 



Moreover, these results agree with those which we obtain by 



APPLIED MECHANICS. 



experiment, and thus we can, by making use of our calculations, 
compute the load required to produce a given elastica, determined 
by the slope at the points of inflection, which, in the case of pin- 
ended columns, are at the ends, and, in the case of columns fixed 
in direction at the ends, are half-way between -the middle and the 
ends. 

All this can be done, and can be verified by experiment, 
provided that the load is not so great that any fibre is strained 
beyond the elastic limit of the material, and provided the value of 
l/p is not so small that the curvilinear form is unstable. 

For smaller values of l/p the only stable form is a straight line, 
and the column does not bend. 

To ascertain the least value of l/p for which a curved form is 
stable, observe that K cannot be less than 71/2, and since this cor- 
responds to one bay, and hence to the case of a pin-ended column, 
we have in that case, by substituting n/2 for K in equation (6) , 

7T 2 



n 

and, since I=AfP and -7=<7, 



we have for the line of demarcation between the straight and 
curved form in a pin-ended column 



-; (7) 

and for that in the case of a column fixed in direction at the ends 



As an example, if a= 10,000 and 1=30,000,000 we should 
find that a pin-ended column would not bend unless l/p were 
greater than 172, and that a column fixed in direction at the ends 




STRENGTH OF SHAFTING. 333 

would not bend unless l/p were greater than 344. Columns with 
smaller values of l/p would remain straight when the resultant of 
the load acts along the axis, and no fibre is strained beyond the 
elastic limit. 

212. Strength of Shafting. The usual criterion for the 
strength of shafting is, that it shall be sufficiently strong to 
resist the twisting to which it is exposed in the transmission of 
power. 

Proceeding- in this way, let EF (Fig. 239) be a shaft, AB the 
driving, and CD the following, pulley. 
Then, if two cross-sections be taken 
between these two pulleys, the por- 
tion of the shaft between these two 
cross-sections will, during the trans- 
mission of power, be in a twisted con- F 

riG. 239. 

dition ; and if, when the shaft is at 

rest, a pair of vertical parallel diameters be drawn in these sec- 
tions, they will, after it is set in motion, no longer be parallel, 
but will be inclined to each other at an angle depending upon 
the power applied. Let GH be a section at a distance x from 
O, and let KI be another section at a distance x -f- dx from O. 
Then, if di represent the angle at which the originally parallel 
diameters of these sections diverge from each other, and if r = 
the radius of the shaft, we shall have, for the length of an arc 
passed over by a point on the outside, 

rdi; 

and for the length of an arc that would be passed over if the 
sections were a unit's distance apart, instead of dx apart, 

rdi _ di 
dx dx 

This is called the strain of the outer fibres of the shaft, as it 
is the distortion per unit of length of the shaft. 



334 APPLIED MECHANICS. 

In all cases where the shaft is homogeneous and symmet- 
rical, if i is the angle of divergence of two originally parallel 
diameters whose distance apart is x, we shall have the strain, 

di i 
v = r = r-. 

dx x 

This also is the tangent of the angle of the helix. 

A fibre whose distance from the axis of the shaft is unity, 
will have, for its strain, 

dt_ = / 

dx x 

A fibre whose distance from the axis of the shaft is p, will have, 
for its strain, 

di i 

v = p - = p-. 

dx x 

Fixing, now, our attention upon one cross-section, GH, we have 
that the strain of a fibre at a distance p from the axis (p varying, 
and being the radius of any point whatever) is 



where - is a constant for all points of this cross-section. 

X 

Hence, assuming Hooke's law, " Ut tensio sic vis" we shall 
have, if C represent the shearing modulus of elasticity, that the 
stress of a fibre whose distance from the axis is p, is 



which quantity is proportional to p, or varies uniformly from the 
centre of the shaft. 

The intensity at a unit's distance from the axis is 



0- 



STRENGTH OF SHAFTING. 335 

and if we represent this by a, we shall have for that at a dis- 
tance p from the axis, 




Hence we shall have (Fig, 240), that, on a small 
area, V J 

dA = dp( P dB) _ pdpdO, ^^ 

the stress will be 

pdA = apdA = ap 2 dpd9. 

The moment of this stress about the axis of the shaft is 

ppdA = ap z dA = ap^dpdO, 

and the entire moment of the stress at a cross-section is 
afp*dA = affpidpdO = al, 

where / = fp 2 dA is the moment of inertia of the section about 
the axis of the shaft. 

This moment of the stress is evidently caused by, and hence 
must be balanced by, the twisting-moment due to the pull of the 
belt. Hence, if M represent the greatest allowable twisting- 
moment, and a the greatest allowable intensity of the stress at 
a unit's distance from the axis, we shall have 

M = al = - /. 

P 

If / is the safe working shearing-strength of the material 
per square inch, we shall have / as the greatest safe stress per 
square inch at the outside fibre, and hence 

M=- I 
r 

will be the greatest allowable twisting-moment. 



33^ APPLIED MECHANICS. 

For a circle, radius r t 

2 " ~ * ~~2~ ~ J ~i6~* 
For a hollow circle, outside radius r v inside radius r M 



Moreover, if the dimensions of a shaft are given, and the 
actual twisting-moment to which it is subjected, the stress at a 
fibre at a distance p from the axis will be found by means of the 
formula 



The more usual data are the horse-power transmitted and 
the speed, rather than the twisting-moment. 

If we let P = force applied in pounds and R = its leverage 
in inches, as, for instance, when P = difference of tensions of 
belt, and R = radius of pulley, we have 



and if HP number of horses-power transmitted, and 
N = number of turns per minute, then 



TT T) _ \ *' *_ /_ . 

12 X 33000 ' 
12 X -l^OOoIfP 

^r^r. ^ Jyi 

271 N 

EXAMPLE. 

Given working-strength for shearing of wrought-iron as 10000 
Ibs. per square inch ; find proper diameter of shaft to transmit 
2o-horse power, making 100 turns per minute. 



TRANSVERSE DEFLECTION OF SHAFT. 337 



Mp 
Angle of Torsion. From the formula, page 336, p~ =- % 



combined with 



we have 



= ap = Cp-, 

oc 



. _ MX 
" ~' 



which gives the circular measure of the angle of divergence of 
two originally parallel diameters whose distance apart is x ; the 
twisting-moment being M, and the modulus of shearing elas- 
ticity of the material, C. 

EXAMPLES. 

1. Find the angle of twist of the shaft given in example i, 212, 
when the length is 10 feet, and C = 8500000. 

2. What must be the diameter of a shaft to carry 80 horses-power, 
with a speed of 300 revolutions per minute, and factor of safety 6, break- 
ing shearing-strength of the iron per square inch being 50000 Ibs. 

213. Transverse Deflection of Shafts. In determining 
the proper diameter of shaft to be used in any given case, we 
ought not merely to consider the -resistance to twisting, but 
also the deflection under the transverse load of the belt-pulls, 
weights of pulleys, etc. This deflection should not be allowed 
to exceed y^- of an inch per foot of length. Hence the de- 
flection should be determined in each case. 

The formulae for computing this deflection will not be given 
here, as the methods to be pursued are just the same as in the 
case of a beam, and can be obtained from the discussions on 
that subject. 



APPLIED MECHANICS. 



214. Combined Twisting and Bending. The most com- 
mon case of a shaft is for it to be subjected to combined twisting 
and bending. The discussion of this case involves the theory 
of elasticity, and will not be treated here ; but the formulae com- 
monly given will be stated, without attempt to prove them until 
a later period. These formulae are as follows : 
Let M l = greatest bending-moment, 

M 2 = greatest twisting-moment, 

r = external radius of shaft, 

/ = moment of inertia of section about a diameter, 

TTf 4 

for a solid shaft / = , 
4 

f = working-strength of the material = greatest al- 

lowable stress at outside fibre ; 
then 

i. According to Grashof, 



/= LjfJ/i + fVJ/x 2 + M*\. (i) 

2. According to Rankine, 



/ = j M, + \!M* + M; j . (2) 

215. Springs. The object of this discussion is to enable us 
to answer the following three questions : (a) Given a spring, 
to determine the load that.it can bear without producing in the 
metal a maximum fibre stress greater than a given amount. 
(&) Given a spring, to determine its displacement (elongation, 
compression, or deflection) under any given load, (c) Given a 
load P and a displacement & t ; a spring is to be made of a 
given material such that the load P shall produce the displace- 
ment 6 I , and that the metal shall not, in that case, be subjected 
to more than a given maximum fibre stress. Determine the 
proper dimensions of the spring. 



SPRINGS. 339 



There are practically only two cases to be considered as far 
as the manner of resisting the load is concerned. In the first, 
the spring is subjected to transverse stress, and is to be calcu- 
lated by the ordinary rules for beams. In the second, the 
spring is subjected to torsion, and the ordinary rules for re- 
sistance to torsion apply. It is true that in most cases where 
the spring is subjected to torsion there is also a small amount 
of transverse stress in addition to the torsion ; but in a well- 
made spring this transverse stress is of very small amount, and 
we may neglect it without much error. 

We will begin with those cases where the spring is subjected 
to torsion, and for all cases we shall adopt the following nota 
tion : 

P = load on spring producing maximum fibre stress/; 

f = greatest allowable maximum fibre stress for shearing ; 

C = shearing modulus of elasticity ; 

x = length of wire forming the spring ; 
M l = greatest twisting moment under load P\ 

L = any load less than the limit of elasticity ; 
M = twisting moment under this load ; 

p = maximum fibre stress under load L ; 

p distance from axis of wire to most strained fibre ; 

/ = moment of inertia of section about axis of wire ; 

z'j = angle of twist of wire under load P- 
i = angle of twist of wire under load L ; 

V = volume of spring ; 

#j = displacement of point where load is applied when load 
isP; 

d displacement of point where load is applied when load 
isZ. 

Then from pages 335 and 337 we obtain the following four 
formulae : 

*=/, (i) 



34O APPLIED MECHANICS. 



MX 
'=C7' 



(3) 



These four formulae will enable us to solve all the cases of 
springs subjected to torsion only. Moreover, in the cases 
which we shall discuss under this head, the wire will have either 
a circular or a rectangular section : in the former case we will 
denote its diameter by d, and we shall then have 

net* d 

/= -- and p = ; 
32 2 

while in the latter case we will denote the two dimensions of 
the rectangle by b and h, respectively, and we shall then have 



We will now proceed to determine the values of P, #, S l , and 
V in each of the following four cases, all of which are cases of 
torsion : 

CASE i. Simple round torsion wire. Let AB, the leverage 
of the load about the axis, be R ; then we shall have 

M = LR, M, = PR ; 
and we readily obtain from the formulae (i), (2), (3), and (4) 



^\ 



,f C- <" 



(7) 



SPRINGS. 



341 



and from these we readily obtain 



(8) 



CASE 2.' Simple rectangular torsion wire. In this case we 
readily obtain 



(9) 



, D . , . 

= =ri ' (IC 




>=*'> = "' 



CASES 3 and 4. Helical springs made of round and of rec- 
tangular wire respectively. A helical spring may be used either 
in tension or in compression. In either case it is important 
that the ends should be so guided that the pair of equal and 
opposite forces acting at the ends of the spring should act ex- 
actly along the axis of the spring. 

This is of especial importance when the spring is used for 
making accurate measurements of forces, as in the steam-en- 
gine indicator, in spring balances, etc. 

Moreover, it is generally safer, as far as accuracy is con- 
cerned, to use a helical spring in tension rather than in com- 
pression, as it is easier to make sure that the forces act along 



34 2 APPLIED MECHANICS. 

the axis in the case of tension than in the case of compres- 
sion. 

Whichever way the spring is used, however, provided only 
the two opposing forces act along the axis of the spring, the 
resistance to which the spring is subjected is mainly torsion, 
inasmuch as the amount of bending is very slight. 

This bending, however, we will neglect, and will compute 
the spring as a case of pure torsion, the same notation being 
used as before, except that we will now denote by R the radius 





of the spring, and we shall have 

M = LR, M, = 

and now formulae (5), (6), (7), and (8) become applicable to a 
spring made of round wire, and formulae (9) and (10), (n) and 
(12), to one made of rectangular wire. 

We must bear in mind, however, that x denotes the length 
of the wire composing the spring, and not the length of the 
spring, d and d l now denote the elongations or compressions 
of the spring. 

GENERAL REMARKS. 

By comparing equations (8) and (12), it will be seen 
that if a spring is required for a given service, its volume 
and hence its weight must be 50 per cent greater if made 
of rectangular than if made of round wire. Again, it is 
evident that when the kind of spring required is given. 



SPRINGS. 343 



and the values of C and f for the material of which it is to 
be made are known, the volume and hence the weight of 
the spring depends only on the product Pd lt and that as soon 
as P and d\ are given, the weight of the spring is fixed inde- 
pendently of its special dimensions. If, however, we fix any 
one dimension arbitrarily, the others must be so fixed as to 
satisfy the equations already given. Next, as to the values to 
be used for /and C, these will depend upon the nature of the 
special material of which the spring is made, and these can 
only be determined by experiment. Confining ourselves now 
to the case of steel springs, it is plain that /and 7 should be 
values corresponding to tempered steel. 

As an example, suppose we require the weight of a helical 
spring, which is to bear a safe load of 10000 Ibs. with a deflec- 
tion of one inch, assuming C= 12600000 and/= 80000 Ibs. 
per sq. in., and as the weight of the steel 0.28 Ib. per cubic 
inch. 

From formula (8) we obtain 

_, 2 X 12600000 X 10000 X i 

=39.4cu.m. 



Hence the weight of the spring must be (39.4) (0.28) = II Ibs. 
We may use either a single spring weighing 1 1 Ibs., or else 
two or more springs either side by side or in a nest, whose com- 
bined weight is 1 1 Ibs. Of course in the latter case they must 
all deflect the same amount under the portion of the load 
which each one is expected to bear, and this fact must be 
taken into account in proportioning the separate springs that 
compose the nest. 

FLAT SPRINGS. 

Let P, L, V, d, and d^ have the same meanings as before, 
and let 



344 



APPLIED MECHANICS, 



f= greatest allowable fibre stress for tension or compres- 

sion : 

R = modulus of elasticity for tension or compression ; 
/= length of spring; 

M l = maximum bending-moment under load P ; 
M= maximum bending-moment under load L. 

Moreover, the sections to be considered are all rectangular, 
and we will let b = breadth and h = depth at the section 
where the greatest bending-moment acts, the depth being 
measured parallel to the load. 

Then if / denote the moment of inertia of the section of 
greatest bending-moment about its neutral axis, we shall have 

f= M 

12 

We will now consider six cases of flat springs, and will de- 
termine P, tf, tf z , and V for each case, and for this purpose we 
only need to apply the ordinary rules for the strength and de- 
flection of beams. 

CASE i. Simple rectangular spring, fixed at one end and 
loaded at the other. 



I 3 L 
i*> 

I* f 



(24) 




E 
E 






(26) 



SPRINGS. 



345 



CASE 2. Spring- of uniform depth and uniform strength, tri- 
in plan , fixed at one end and loaded at the other. 



(27) 
(28) 

(29) 




(30) 



CASE 3. Spring of uniform breadth and uniform strength, 
parabolic in elevation, fixed at one end and loaded at the 
other. 



(31) 



(33) 



(34) 



CASE 4. Compound wagon spring, made up of n simple rec- 
tangular springs laid one above the other, fixed at one end and 
loaded at the other. 



346 



APPLIED MECHANICS. 



Let the breadth be b, and the depth of each separate layer 


be h. Then 








'- 


n bh* 
6 / /' 


(35) 


i 


N^ 


6 = 


4 /* L 

nbh* E* 


(30 


i 


=3=^ 




/'/ 


(37) 


i 




\ 


i 


y^ ^fi 1 * 





(38) 



CASE 5. Compound spring composed of n triangular springs 
laid one above the other, fixed at one end and loaded at the 
other. 



*-\r=r ^ 

6 = -nW L E> < 4 > 

*=ji ; (4I) 




CASE 6. This case differs from the last in that in order to 
economize material we superpose springs of different lengths, 



SPRINGS. 



347 



and make them of such a shape that by the action of a single 
force at the free end they are bent in arcs of circles of nearly 
or exactly the same radius. 
The force P bends the 
lowest triangular piece AA 



in 



the arc of a circle. 



length of this piece is -. 




In order that the re- 
maining parallelopipedical 
portion may bend into an 
arc of the same circle it is 
necessary that it should have 

acting on it a uniform bending-moment throughout, and this 
is attained if it exerts a pressure at A l upon the succeeding 
spring equal to the force P, and following this out we should 
find that the entire spring would bend in an arc of a circle. 

The values of P, 6, d z , and Fare correctly expressed for 
this case by (39), (40), (41), and (42). 

For any flat springs which are supported at the ends and 
loaded at the middle, or where two springs are fastened to- 
gether, it is easy to compute, by means of the formulae already 
developed, by making the necessary alterations, the quantities 
P. 3 d z , and V, and this will be left to the student. 

COILED SPRINGS SUBJECTED TO TRANSVERSE STRESS. 

Three cases of coiled springs will now be given as shown 
in the figures, and the values of P, 3, d lt and Fwill be deter- 
mined for each. 

In each of these cases let R be the leverage, of the load, 
and let GO = angle turned through under the load. Then we 
may observe that all the three cases are cases of beams sub- 
jected to a uniform bending-moment throughout their length, 
this bending-moment being LR for load L and PR for load P. 



348 



APPLIED MECHANICS. 



CASES I and 2. Coiled spring, rectangular in section. 

f b>i i \ 

^ = i/-^> (43) 

UP L , , 

(44) 

(45) 



f Rl 




CASE 3. Coiled spring, cir- 
cular in section. 

f = -^f^> (47) 

64 l^_L . . 

~* j* z^> \4w 

(49) 



E 



TIME OF OSCILLATION OF A SPRING. 



(46) 




(So) 



Since in any spring the load producing any displacement 
is proportional to the displacement, it follows that when a 
spring oscillates, its motion is harmonious. 



SPRINGS. 349 



Suppose the load on the spring to be P t and hence its nor- 
mal displacement to be <S\. Now let the extreme displacements 
on the two sides of #, be # , and the force producing it />, so that 
the actual displacement varies from # x -f- tf to <$, <? , and the 
force acting varies from P -\- p to P p. 

Now. from the properties of the spring we must have 

=; /.*. = *.. (so 

Moreover, in the case of harmonic motion the maximum 



value of the force acting is - (see p. 104). But the load 

o 

oscillating is P instead of W, and the extreme displacement is 
6 e instead of r. 
Hence we have 



(52) 

S d 



(S3) 
Hence the time of a double oscillation 

(54) 

g 



35 APPLIED MECHANICS. 



CHAPTER VII. 

STRENGTH OF MATERIALS AS DETERMINED BY 
EXPERIMENT. 

216. Whatever computations are made to determine the 
form and dimensions of pieces that are to resist stress and 
strain should be based upon experiments made upon the mate- 
rials themselves. 

The most valuable experiments are those made upon pieces 
of the same quality, size, and form as those to which the results 
are to be applied, and under conditions entirely similar to those 
to which the pieces are subjected in actual practice. 

From such experiments the engineer can learn upon what he 
can rely in designing any structure or machine, and this class of 
tests must be the final arbiter in deciding upon the quality of 
material best suited for a given service. An attempt will be made 
in this chapter to give an account of the most important results 
of experiments on the strength of materials, and to explain the 
modes of using the results. 

While the importance of making tests upon full-size pieces, 
and of introducing into the experiments the conditions of 
practice, is pretty generally recognized to-day, nevertheless 
there are some who have not yet learned to recognize the fact 
that attempts to infer the behavior of full-size pieces under 
practical conditions from the results of tests on small models, 
made under conditions which are, as a rule, necessarily, quite 
different from those of practice, are very liable to lead to con- 
clusions that are entirely erroneous. 



GENERAL REMARKS. 351 

Such a proceeding is in direct violation of a principle that 
the physicist is careful to observe throughout his work, viz.: 
not to apply the results to cases where the conditions are essentially 
different from those of the experiments. 

When the quality of the material suited for a given 
service is known, tests of the material furnished must be 
made to determine its quality. Such tests, made upon 
small samples, should be of such a kind that there may 
be a clear understanding, as to the quality desired, between 
the maker of the specifications and the producer. Whenever 
possible, standard forms of specimens and standard methods 
of tests should be used. 

The determination of standards is occupying the at- 
tention of the Int. Assoc. for Testing Materials, the British 
Standards Committee, the Am. Soc. for Testing Materials, 
and others. 

To ascertain the quality of the material tensile tests are most 
frequently employed, their objects being to determine the tensile 
strength per square inch, the limit of elasticity, the yield-point, 
the ultimate contraction of area per cent, the ultimate elongation 
per cent in a certain gauged length, and sometimes the modulus 
of elasticity. 

While the standard forms and dimensions will be given later, 
the following general classification of the forms in use will be 
given here, viz. 

i. The specimen may be provided with a shoulder at each 
end, having a larger sectional area than the main body of the 
specimen, the section of this being uniform throughout as shown 
in Fig. a, the latter being of so great a length in proportion to the 

diameter that the stretch of i i 

i i i i i i i i i 

the specimen is not essentially I 

different from what it would FIG. a. 

be if the section were uniform throughout. The shoulders are, 



35 2 APPLIED MECHANICS. 

of course, the portions of the specimen where the holders (or 
clamps) of the testing-machine are attached. 

2. In the case of a round specimen of that kind there may 
be a screw-thread on the shoulders as shown in Fig. b. 

In the case of a brittle material, as 
1^ JJJII cast-iron or hard steel, it is desirable to 

use a holder with a ball-joint, and to 
screw the specimen into the holder. 

3. The specimen may be provided with a shoulder at each 
end, the main body of the specimen being, however, so short 
in proportion to the diameter that the stretch is essentially 
modified. Such a form is shown in Fig. c. 



FIG. c. FIG. d. 

4. The specimen may be a grooved specimen as shown in 
Fig. d, where the length of the smallest section is zero. 

5. The section of the specimen may be uniform through- 
out, the length between the holders being so great in propor- 
tion to the diameter that the stretching of the fibres is not 
interfered with. This form of specimen is shown in Fig. e. 

Assume a specimen of duc- 

I I 'tile material, as mild steel or 

wrought-iron, of the ist or the 
FlG - ' 5th shape, subjected to stress 

in the testing-machine, or else by direct weight, and suppose 
that we mark off upon the main body, i.e., the parallel section 
of the specimen, a gauged length of 8 or 10 inches (preferably 
8 inches), and measure, by means of some form of extensom- 
eter, the elongations in the gauged length, corresponding to 
the stresses applied ; then plot a stress-strain diagram as shown 
in Fig. /, having stresses per square inch for abscissae, and the 
corresponding strains for ordinates. 



GENERAL REMARKS. 



353 



CO 

111 

5- 

2 0" 




































</ 


































f ^-- 





'B 




"eo -too 

Z .m 

I 

O .002 
h 
u o 





























-- 


**-*" 


























.--* 


^ 


-""" 


























^~~- 


^-- 


~^~ 


























^ - 


-- 


r^" 


-"" 
































OC V 2000 6000 KM 
CO 


XX) 14( 


XX) 18 
LOAD 


XX) 22 
PER SQ. 


K)0 20000 30< 
IN, 


XX) 31000 3800C 



FIG. /. 

We shall find that the strains begin by being proportional 
to the stresses, but when a certain stress is reached, called the 
" limit of elasticity " or " elastic limit," shown at A, the strains 
increase more rapidly than the stresses, but the rate of increase 
in the ratio of the strain to the stress is not large until a stress 
is reached called the " yield-point " or "stretch-limit," shown 
at B, which is usually a little larger than the elastic limit ; and 
then the rate of increase of the ratio of strain to stress becomes 
much larger. 

Observe, also, that if a small load be applied to the piece 
under test, and then removed, the deformation or distortion 
caused by the application of the load apparently vanishes, and 
the piece resumes its original form and dimensions on the 
removal of the load ; in other words, no permanent set takes 
place. When the load, however, is increased beyond a certain 
point, the piece under test does not return entirely to its 
original dimensions on the removal of the load, but retains a 
certain permanent set. While permanent set that is easily 
determined begins at or near the elastic limit, and while the 
permanent sets corresponding to stresses greater than the 
elastic limit are much greater than the corresponding recoils, 
and hence form the greater part of the strains corresponding 
to such stresses, nevertheless experiments show that even a 
very small load will often produce a permanent set, and that 
the apparent return of the piece to its original dimensions is, 



354 APPLIED MECHANICS. 

in a number of cases, only due to the want of delicacy in the 
measuring-instruments at our command. 

I- After the elastic limit and the yield-point have been passed, 
the ratio of the strain to the stress is much greater than before, 
the stretch becomes local, with a local contraction of area, this 
being due to the plasticity of the metal. 

Finally, when the maximum stress is applied, or, in other 
words, the breaking-stress, the behavior is apparently some- 
what different when the piece is subjected to dead weight from 
what it is when in a testing-machine. In the former case, 
when the maximum load is reached, the specimen continues to 
stretch rapidly, without increase in the load, until the specimen 
breaks. 

In the case of the testing-machine, however, the application 
of the maximum load causes, of course, the specimen to 
stretch, but this stretch naturally reduces the load applied, and 
the actual load under which the specimen separates into two 
parts is less, and often very considerably less, than the maxi- 
mum or breaking stress. 

Observe that the terms " breaking-load " and " breaking- 
stress " are always used to mean the " maximum load " and 
"maximum stress " respectively, and are never used to denote 
the load or the stress under which the specimen separates into 
two parts when the latter differs from the former. 

If the stretch of the specimen, as described above, is in any 
way interfered with, the behavior of the specimen will not be 
a proper criterion of the properties of the material ; the per- 
centage contraction of area at fracture will vary with the 
amount of interference with the stretch, and hence with the 
proportions of the specimen ; and the maximum or breaking 
strength will be greater than the real maximum or breaking 
strength per square inch of the material. Hence it follows 
that the 3d and 4th forms of specimen do not indicate cor- 
rectly the quality of the material, furnishing, as they do, 
erroneous values for both breaking-strength and ductility. 



CAST-IRON. 355 



The quantities sought in such tests as those described 
above (with specimens of the 1st, 2d or 5th forms) are, as 
already stated : 

i. The breaking-strength per square inch of the material; 

2. The limit of elasticity of the material ; 

3. The yield-point or stretch-limit of the material; 

4. The ultimate contraction of area per cent : 

5. ^he ultimate elongation per cent in a given gauged 
length ; 

6. The modulus of elasticity. 

The first gives, of course, the tensile str :h of the ma- 
terial ; the second and third ought both to be determined, but 
many content themselves with the third alone, since it is much 
easier to obtain. While they are commonly not far apart, it 
is a fact that certain kinds of stress to which the piece may be 
subjected may cause them to become very different from each 
other. The fourth and fifth are the usual ways of measuring 
the ductility of the metal ; and while the fourth is the most 
definite, the fifth is very much employed, and finds favor with 
most iron and steel manufacturers. The sixth is not often 
determined for commercial work, but it is one of the important 
properties of the metal. 

Of these six properties the two most universally insisted 
upon in specifications for material to be used in the construc- 
tion of structures or of machines are ductility, which is 
universally recognized as an all-important matter, and a suit- 
able breaking-strength per square inch, both a lower and an 
upper limit being generally prescribed for this last. 

On the other hand, although cast-iron and hard steel are 
brittle metals when compared with wrought-iron and mild 
steel, nevertheless it is true that the third and fourth forms 
of specimen will show too high results for tensile strength even 
in these materials on account of the interference with the 
stretch of the metal. 



APPLIED MECHANICS. 



217. Cast- Iron. Cast-iron is a combination of iron with 
carbon, the most usual quantity being from 3 to 4 per cent. The 
large amount of carbon which it contains is its distinguishing 
feature, and determines its behavior in most respects. Besides 
carbon, cast-iron contains such substances as silicon, phosphorus, 
sulphur, manganese, and others. A considerable amount (more 
than 1.37 per cent as stated by Prof. Howe) of silicon forces 
carbon out of v combination and into the graphitic form, thus 
lowering the strength. 

Pig-Iron is the result of the first smelting, being obtained 
directly from the blast-furnace. The ore and fuel (usually 
coke, though anthracite coal is used to some extent, and some- 
times charcoal) are put into the furnace, together with a flux, 
which is usually limestone, in suitable proportions. The mass 
is brought to a high heat, a strong blast of heated air being intro- 
duced. The mass is thus melted, the fluid metal settling to 
the bottom, while slag, which is the result of the combination 
of the flux with impurities of the ore and fuel, rises to the top. 
The iron is drawn off in the liquid state and run into moulds, 
the result being pig-iron. 

The metal usually contains from 3 to 4 per cent of carbon, 
a part being chemically combined with the iron, and a part in 
the form of graphite. The larger the proportion of combined 
carbon, the whiter the fracture, and the harder and more brittle 
the product, while the larger the proportion of graphite, the darker 
the fracture, and the softer and less brittle the product. That 
which has most of its carbon in combination is called white iron, 
while that which contains a large proportion of graphite is called 
gray cast-iron. 

Pig-iron also contains silicon, sulphur, phosphorus, etc. 
The quantity of the first two can, to a certain extent, be controlled 
in the furnace, but not that of the last, so that if low phosphorus is 
desired, the ore and the fuel used must both be low in phosphorus. 

Gray cast-iron has been, and is sometimes classified in various 



CAST-IRON. 357 



ways, according to the proportions of the combined carbon, and 
of the graphite, but the most modern practice is to sell, buy, and 
specify the iron by means of its chemical composition, and not by 
brands. 

That which contains the largest amount of carbon in mechan- 
ical mixture is, as a rule, soft and fusible, and hence suitable for 
making castings where precision of form is the chief desidera- 
tum, as its fusibility causes it to fill the mould well. For general 
use in construction, where strength and toughness are all-import- 
ant considerations, those grades are required which are neither 
extremely soft nor extremely hard. 

As to the adaptability of cast-iron to construction, it presents 
certain advantages and certain disadvantages. It is the cheapest 
form of iron. It is easy to give it any desired form. It resists 
oxidation better than either wrought-iron or steel. Its com- 
pressive strength is comparatively high when the castings are 
small and perfect. On the other hand, its tensile strength is 
much less than that of wrought-iron, or that of steel, averaging 
in common varieties from 16000 or 17000 to about 26000 
pounds per square inch. It cannot be riveted or welded. It 
is a brittle and not a ductile material, it does not give much 
warning before fracture, and, while the stretch under any 
given load per square inch is decidedly larger than that of 
wrought-iron or steel, its total stretch before fracture is small 
when compared with wrought iron and steel. One of the dif- 
ficulties in the use of cast -iron in construction is its liability to 
initial strains from inequality in cooling. Thus if one part of 
the casting is very thin and another very thick, the thin part 
cools first, and the other parts, in cooling afterwards, cause 
stresses in the thin part. 

The fracture of good cast-iron should be of a bluish-gray 
color and close-grained texture. 

At one time cast-iron was extensively used for all sorts 
of structural work, but it was soon superseded by wrought-iron, 
and later by steel. 

Thus it is no longer used in bridgework, nor for floor- 



APPLIED MECHANICS. 



beams of a building, though it is still used to a considerable 
extent for the columns of buildings; and for this purpose it 
has in its favor the fact that it resists the action of a fire better 
than wrought iron or steel. Thus, in the present day, when 
the steel skeleton construction of buildings is so extensively 
employed, it is very necaesary to protect the steel beams and 
columns by covering them with some non-conducting material, 
as, otherwise, they would be liable to collapse in case of fire. 

It is used in cases where the form of the piece is of more 
importance than strength, and also where, on account of its 
form, it would be difficult or expensive to forge ; thus hangers, 
pulleys, gear-wheels, and various other parts of machinery of a 
similar character are usually made of cast-iron, as well as a 
great many other pieces used in construction. It is also used 
where mass and hence weight is an important consideration, 
as in the bed-plates and the frames of machines, etc. 

Malleable Iron. When a casting, in which toughness is 
required is to be made of a rather intricate form, it is frequently 
the .custom to malleableize the cast-iron, i.e., to remove a part 
of its carbon, and the result is provided the casting is small 
a product that can be hammered into any desired shape wher* 
c old, but is brittle when hot. 

A list of references to some of the principal experimental 
works on the strength and elasticity of cast iron will be given. 

i. Eaton Hodgkinson : (a) Report of the Commissioners on the 
Application of Iron to Railway Structures. 

(b) London Philosophical Transactions. 1840. 

(c) Experimental Researches on the Strength and other Prop- 
erties of Cast-Iron. 1846. 

2. W. H. Barlow : Barlow's Strength of Materials. 

3. Sir William Fairbairn : On the Application of Cast and Wrought 

Iron to Building Purposes. 
4. Major Wade (U.S.A.) : Report of the Ordnance Department 

on the Experiments on Metals for Cannon. 1856. 
5. Capt. T. J. Rodman : Experiments on Metals for Cannon. 
6. Col. Rosset: Resistenzadei Principal! Metalli daBocchidi Fuoco. 



TENSILE STRENGTH OF CAST-IRON. 359 

7. Tests of Metals made on the Government Testing Machine at 
Watertown Arsenal, 1887, 1888, 1889, 1890, 1891, 1892, 
1893, l8 94 1896, 1897, 1898. 

8. Transactions Am. Soc. Mechl. Engrs. for 1889, p. 187 et seq. 

9. W. J. Keep: (a) Transverse Strength of Cast-iron. Trans. Am, 
Soc. Mechl. Engrs., 1893. 

(b) Relative Tests of Cast-iron. Trans. Am. Soc. 

Mechl. Engrs., 1895. 

(c) Transverse Strength of Cast-iron. Trans. Am. 

Soc. Mechl. Engrs., 1895. 

(d) Keep's Cooling Curves. Trans. Am. Soc. 

Mechl. Engrs., 1895. 

(e) Strength of Cast-iron. Trans. Am. Soc. 

Mechl. Engrs., 1896, 
10. Bauschinger: Mittheilungenausdem Mech. Tech. Lab. Miinchen. 

Heft 12, 1885; Heft 15, 1887; Heft 27, 1902; Heft 28. 1902. 
ii. Tetmajer; Mittheilungen der Materialpriifungsanstalt Zurich. 

Heft 3, 1886; Heft 4, 1890; Hefte 5 and 9, 1896. 
12. Technology Quarterly. October 1888, page 12 et seq. 
13. Technology Quarterly. Vol. 7, No. 2; VoL 10. No. 3. 
14. Transactions of the American Foundrymen's Association. 
15. Transactions of the American Society for Testing Materials. 

218. Tensile Strength of Cast-iron. As the use of 
cast-iron to resist tension has been almost entirely superseded by 
that of wrought-iron and steel, results of tests of full-size pieces 
of cast-iron in tension are not available. Tensile tests, however, 
have been extensively employed to determine the quality ; especially 
so when cast-iron cannon were in use; and tensile tests of cast- 
iron are still made, to a certain extent, for the determination 
of quality. For such tests standard specimens should be used, 
and attempts are being made to reduce their number. 

As the strength that should be attained in such specimens 
will become evident from the Standard Specifications of the 
Am. Soc. for Testing Materials, on page 385 et seq., only a few 
tensile tests will be quoted here, and those, for the purpose of 



360 



APPLIED MECHANICS. 



acquainting the reader with the results of some tensile tests of 
cast-iron. 

About 1840 Eaton Hodgkinson made a few experiments to 
determine the laws of extension of cast-iron, and for this purpose 
used rods lofeet long and i square inch in section. The tables 
of average results are given below. 

These tables show that the ratio of the stress to the strain of 
cast-iron varies with the load, growing gradually smaller as the 
load increases, that with moderate loads the ratio of stress to 

RESULTS OF NINE TENSILE TESTS. RESULTS OF EIGHT COMPRESSIVE TESTS. 



Weights 


Strains in 


Ratio of 




Weights 


Strains in 


Ratio of 


Laid on 


Fractions of 


Stress to 




Laid on 


Fractions of 


Stress to 


in Pounds. 


the Length. 


Total Strain. 




in Pounds. 


the Length. 


Total Strain. 


1053.77 


O.OOCO7 


14050320 




2064.75 


ocoo. i 6 


13214400 


1580.65 


o . ooo 1 1 


13815720 




4129.49 


0.00032 


12778200 


2107.54 


o . ooo i 6 


13597080 




6194.24 


0.00050 


12434040 


3161.31 


0.00024 


13218000 




8258.98 


0.00066 


12578760 


4215.08 


0.00033 


12936360 




10323.73 


o . 00083 


12458280 


5268.85 


o . 00042 


12645240 




12388.48 


O.OOIOO 


12357600 


6322.62 


0.00051 


12377040 




14453.22 


0.00188 


12245880 


7376.39 


0.00061 


12059520 




16517.97 


0.00136 


12132240 


8430.16 


0.00072 


11776680 




18582.71 


0.00154 


12050400 


9483 . 94 


0.00083 


11437920 




20647.46 


0.00172 


12013680 


I0537-7I 


0.00095 


11314440 




24776.95 


0.00208 


11911560 


11591.48 


0.00107 


10841640 




28906.45 


0.00247 


11679720 


12645.25 


O.OOI2I 


10479480 




33030 . 80 


0.00295 


11215560 


13699.83 


0.00139 


9855960 










14793.10 


O.OOI55 


9549120 











strain for tension of cast-iron does not differ materially from 
that for compression, and that the difference increases as the 
load becomes greater. The agreement is even closer in the 
case of wrought-iron and steel. 

The gradual decrease of the ratio of stress to strain with the 
increase of load shows that Hooke's law, " Ut tensio sic vis" 
(the stress is proportional to the strain), does not hold true in 



RESULTS OF TESTS. 361 

cast-iron. Hence, strictly speaking, cast-iron has no elastic 
limit and no modulus of elasticity, nevertheless we are accustomed 
to call the ratio of the stress to the strain under moderate loads 
the modulus of elasticity of the cast-iron. 

In making specifications intended to secure a good quality 
of 'cast-iron it is very common to call for a transverse test. 
Indeed the resolutions of the international conferences relative 
to uniform methods of testing recommend, in the case of cast- 
iron: 

(a) Test-pieces to be of the shape of prismatic bars no cm. 
standard length (43") and to have a section of 3 cm. square 
(i".i8), one having an addition on one end, from which cubes 
can be cut for compression tests. 

(b) Three such specimens to be tested for transverse strength. 

(c) The tensile strength to be determined from turned test- 
pieces 20 mm. (o".785) diameter and 200 mm. (7". 85) long, cut 
from the two ends of the test-pieces broken by flexure. 

(d) The compressive strength to be determined from cubes 
3 cm. (i".i8) on a side cut from the first specimens, pressure 
to be applied in the direction of the axis of the original bar. 

These requirements, while calling for transverse tests, call 
also for tensile and compressive tests. 

T .* Siandard Specifications of the Am. Soc. for Testing 
Mater.;. Js w.ll be found on page 385 et seq. 

Inasmuch as the tensile strength has been, and is also made 
the basis of specifications for cast-iron, it is important to con- 
sider what should be attained in this regard. 

For this purpose a few tables of comparatively modern tests 
will be given here, and it will be seen that in the ordinary 
varieties of cast-iron it is easy to secure tensile strengths 
from 16,000 to 25,000 pounds per square inch, and that 
more can be secured by taking proper precautions in the 
manufacture. 

Indeed cast-iron which, when tested in the form of a 
grooved specimen, shows a tensile strength of at least 30,000 



5 62 



APPLIED MECHANICS. 



pounds per square inch is called gun-iron, this having been a 
requirement of the United States Government, in the days of 
cast-iron cannon, for all cast-iron that was to be used in their 
manufacture. 

The following table is taken from a paper on the Strength 
of Cast-Iron, by Mr. W. J. Keep, published in the Transactions 
of the American Society of Mechanical Engineers for 1896, 
and it gives the averages of the tensile strengths of the fifteen 
different series of tests recorded in the paper. This table is 
given here merely as an example of the results that can be 
obtained by tension tests upon usual varieties of cast-iron. 
The table is as follows : 



AVERAGES OF TENSION TESTS OF ROUND BARS. 





Area of Section, 


Area of Section 




Area of Section 


Area of Section, 




0.375 Sq. In. 


1.12 Sq In. 




o 375 Sq. In. 


1. 12 Sq. In. 


. 






No. of 








Breaking Load 
per Sq. Inch. 


Breaking Load 
per Sq. Inch 


Series. 


Breaking Load 
per Sq Inch. 


Breaking Load 
per Sq. Inch. 


J 


2OOOO 


1 57OO 






14800 


2 


20580 


22500 


IO 








oeo^O 


2O4CO 


1 1 




I7OOO 


4 


21850 


19350 


12 


17700 


17500 


5 


22425 


19750 


13 


I4OOO 


2I3OO 


6 


25550 


17200 


H 


24400 


2O3OO 


7 


18950 


17700 


15 


23525 


20500 


8 


17700 


15350 









The following table of results of tension tests of ordinary 
cast-iron from another source will also be given for the same 
purpose as Mr. Keep's results : 



CAST-IRON. 



363 



CAST-IRON TENSION. 





a 

.2 


|.S 






c 

.2 


O.S 






u 


1-3 "" 






u 


J <" 




Dimensions. 


*3 


ll 


Modulus of 
Elasticity. 


Dimensions. 


c 


S v 
3 a 
8 en 


Modulus of 
Elasticity. 














.2 .Q 






f" 









Ir 


3d 




.03 X .04 


.06 


19340 


14857000 


.00 X .00 


.00 


17100 


13333000 


.03 X .02 


.05 


23910 


15481000 


.00 X .02 


.02 


19068 


13680000 


.00 X 98 


.98 


21180 


15238000 


.00 X .00 


.00 


1 8000 


13333000 


.00 X 97 


97 


23227 


15881000 


.00 X .02 


.02 


19299 


12057000 


.ot X .06 


.08 


19830 


14539000 


.06 X .98 


o? 


17488 


13249000 


. X .03 


03 


20413 


17632000 


.00 X .98 


.98 


19500 


13250000 


.93 X .00 


93 


16774 


14337000 


.02 X .02 


03 


20747 


14543000 


.00 X -oo 


.00 


18600 


15383000 


.03 X .03 


.06 


18620 


13434000 


.00 X .00 


.00 


18000 


16666000 


.00 X .00 


.00 


18910 


13043000 


.00 X .00 


.00 


19400 


17911000 










.00 X .00 


.00 


20950 


15789000 


.00 X .00 


.00 


19900 


15000000 


.00 X .00 


.00 


22900 


15000000 


.00 X .02 


.02 


19594 


13373000 


.00 X .00 


.00 


22400 


15564000 


.01 x .03 


.04 


16341 


13108000 


.00 X .00 


.00 


21300 


15384000 


.01 X .03 


.04 


'3844 


13640000 


.00 X .02 


.02 


19692 


i 5966000 


.02 X .08 


.01 


13798 


11840000 


.ot X .03 


.05 


21005 


15075000 


.00 X .C2 


.02 


17647 


12787000 


08 < 2; 


33 


20600 


11900000 


.03 X .03 


.06 


14025 


12^68000 


.05 X .03 


03 


17067 


12676000 


.04 X .02 


.06 


15083 


13466000 


00 X .03 


.03 


19900 


12929000 


.02 X .04 


06 


16874 


9751900 


.08 X .03 


.02 


16404 


12577000 


.00 X .00 


.00 


aoooo 


13043000 


i .co X .02 


.02 


16450 


12570000 











Colonel Rosset, of the Arsenal at Turin, made a series of 
experiments upon the influence of the shape of the specimen 
upon the tensile strength. For this purpose he used specimens 
with shoulders ; and, among other tests, he compared the 
strength of the same iron by using specimens the lengths of 
whose smallest parts were respectively i metre, 30 millimetres, 
and o millimetres, with the following results : 



Length of Specimen. 


Tensile Strength, in Ibs., per Square Inch. 


ist Cannon. 


2d Cannon. 


3d Cannon. 


i metre . . 
30 millimetres . 
o millimetres . 


31291 
3 2 57I 
33993 


25601 

345 62 
36411 


28019 
30011 
30011 



APPLIED MECHANICS. 



It will thus be seen that, before we can decide upon the 
quality of cast-iron as affected by the tensile strength, it is 
necessary to know the length of that part of the specimen 
which has the smallest area. Colonel Rosset's tests of cast 
iron were almost entirely confined to high-grade irons, suitable 
to use in cannons. 

He deduced, for mean value of the modulus of elasticity of 
the specimens i metre in length, 20419658 Ibs. per square inch : 
this, of course, is a modulus only adapted to these high grades, 
and is not applicable to common cast-iron. 

219. Cast-Iron Columns. In consequence of the high 
compressive strength shown by cast-iron when tested in small 
pieces, and in pieces free from imperfections, it was once 
considered a very suitable material for all kinds of columns. 
Nevertheless, its use for the compression members of bridge 
and roof trusses has been abandoned; cast-iron having been 
displaced first by wrought-iron and subsequently by steel, 
which is the substance now in use for these purposes. 

The principal reasons for the change are the lack of 
ductility, and the consequent brittleness of cast-iron, that it 
cannot be riveted, and that if it breaks it cannot be eas'ly 
repaired. Cast-iron is, however, used to a very considerable 
extent for the columns of buildings. 

The Gordon, the so-called Euler, and the Hodgkinson 
formulae for the breaking-strength of cast-iron columns, have 
all been given in paragraphs 208, 2080, and 209. They are, 
however, all based upon tests made upon very small columns, 
and do not give results agreeing with the tests of such full-size 
columns as are used in practice. We will next consider, 
therefore, the tests that have been made upon full-size cast-iron 
columns, and the conclusions that are warranted in the light of 
these tests. 

Two sets of tests of cast-iron mill columns have been made 
on the Government testing-machine at Watertown Arsenal; an 
account of these sets of tests is published in their reports of 
1887 and of 1888. 



CAST-IRON COLUMNS. 365 



The first lot consisted of eleven old cast-iron columns, which 
had been removed from the Pacific Mills at Lawrence, Mass., 
during repairs and alterations. 

The second lot consisted of five new cast-iron columns cast 
along with a lot that was to be used in a new mill. 

Of these five, the strength of two was greater than the 
capacity of the testing-machine, hence only three were broken ; 
while in the case of the other two the test was discontinued 
when a load of 800000 Ibs. was reached. All the columns con- 
tained a good deal of spongy metal, which of course rendered 
their strength less than it would otherwise have been ; never- 
theless, inasmuch as this is just what is met with in building, 
it is believed that these tests furnish reliable information as to 
what we should expect in practice, and that this information 
is much more reliable than any that can be derived from test- 
ing small columns. 

In all the tests the compressions were measured under a 
large number of loads less than the ultimate strength ; but in- 
asmuch as it is not possible, in the case of cast-iron, to fix any 
limits within which the stress is proportional to the strain, no 
attempt will here be made to compute the modulus of elas- 
ticity. Hence there will be given here a table showing the 
dimensions of the columns tested, their ultimate strengths, 
and, in those cases where they were measured, the horizontal 
and vertical components of their deflections, measured at the 
time when their ultimate strengths were reached, as the Govern- 
ment machine is a horizontal machine. A glance at the table 
will make it evident that we Cannot, in the case of such columns, 
rely upon a crushing strength any greater than 25000 or 30000 
Ibs. per square inch of area of section. Hence it would seem 
to the writer that, in order to proportion a cast-iron column 
to bear a certain load in a building, we should determine the 
outside diameter in such a way as to avoid an excessive ratio 
of length to diameter ; if this ratio is not much in excess 



APPLIED MECHANICS. 



of twenty, the extra stress produced by any eccentricity of the 
load due to the deflection of the column will be very slight. 
At the same time see that the thickness of metal is sufficient 
to insure a good sound casting. 

Now, having figured the column in this way, compute the 
outside fibre stress (using the method of 207) that would 
occur with the loading of the floors assumed to be such as to 
give as great an eccentricity as it is possible to bring upon the 
column. If this distribution of the load is one that is likely 
to occur, then the maximum fibre stress in the column due to 
it ought not to be greatly in excess of 5000 Ibs. per square 
inch ; but if it is one which there is scarcely a chance of realiz- 
ing, then the maximum fibre stress under it might be allowed 
to reach 10000 Ibs. per square inch. If by adopting the di~ 
mensions already chosen these results can be obtained, we may 
adopt them ; but if it is necessary to increase the sectional 
area in order to accomplish them, we should increase it. 

Another matter that should be referred to here is the fact 
that a long cap on a column is more conducive to the produc- 
tion of an eccentric loading than a short one ; hence, that a 
long cap is a source of weakness in a column. 

Other sources of weakness in cast-iron columns are spongy 
places in the casting (which correspond in a certain way with 
knots in wood), and also an inequality in the thickness of the 
two sides of the column, the result of this being the same as 
that of eccentric loading; and it is especially liable to occur in 
consequence of the fact that it is the common practice to cast 
columns on their side, and not on end. The engineer should, 
however, inspect all columns to be used in a building, and reject 
any that have the thickness of the shell differing in different 
parts by more than a very small amount. 

A series of tests of full-size cast-iron columns was made by 
the Department of Buildings of New York City, under the 
direction of Mr. W. W. Ewing, in December, 1897, upon the 



CAST-IKON COLUMNS. 



367 



Remarks. 


rt T3 rt rt ^aJ 
w nj w T3 W 














iple flexure. Post con- 
t middle before testing, 
iple flexure. 


C 

U 
V 

en 


- ^ H 2 |i o ,_, ^ ** 


j 


ID 


U 


aanx; 


e flexure 


t 

3 
X 


U,^ *u* *>1'X X 


3 


3 
X 


3 
X 


c g2 c 1 JfS S 











1C 


oS^o-S "S-e-c 


cx 


CX 


CX 


ex 


a. 


a. 


.0^ -5 . w . _^tc ^ ^ 

" S "3 T3 - - 


1 


-o 


a 


3 

'3 


j5' 

3 

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js 
'3 


*o 2f 'O 
<u <u 

(3 (3 


i 

'rt 


rt rt rt rt rt 


'rt 


"rt 


'1 














lid 

3 "5 -2 









o 





0^ 





CO rfr 

C^ HH 

M M 


en 

O 


t i 

a s > 

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6 




m en 

6 6 


m 

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'S a s 

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co en en o O 
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R 2 

m r^ 


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lip 

p w 


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M en O M o 
in in ^- en en 


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CO 

00 


m 

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8 

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m rt 


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en 


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vn co m en co 

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CO 

o 


8 




Cl 


8 
-f 




4- 

-t 


r^ 


Tf 

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t*** QO 


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f|f s 

5 tJ < cr 

i^ 

.be <i 

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in O 
rt rt rt *^ ^ 


eg 


. 


ST 


CN 


o 

m 


ft 


IO CO 


M 

en 


* 2 S " " 


" 


co 


n 




in 


O 


^ co en r^ en 
r*^ t^ \o ^ ^ 


5 


in 


O 

-1- 


CM 


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1 


11 
U 


O M N en *f 


in 

8: 





r-. 
$ 


1 


1 


1 


sutunjoD 

M3JSI 


sutunjoo 


PIO 



3 68 



APPLIED MECHANICS. 



i 

* 



513 



Mb! 

--tMAJsJi. 



S-'OI- 




Sg-'6-OI 




CAST-IRON COLUMNS. 



369 




370 



APPLIED MEGHAN rCS. 



hydraulic press of the Phoenix Bridge Works. This press 
weighs the load on the specimen plus the friction of the piston, 
the latter being, of course, a variable quantity. Nevertheless 
great pains were taken to determine this friction, and hence 
the results are doubtless substantially correct. 

The results are, it will be seen, similar to those obtained in 
the Watertown tests. The table of results is given below, 
and no farther comments are needed. Subsequently tests 
were made to determine the strength of the brackets. For 
this, however, the reader is referred to the Report itself, or to 
Engineering News of January 2O, 1898, and for further details 
of the tests of the columns, to the Report itself, or to Engi- 
neering News of January 13, 1898. 



Column 
Number. 


Length, 
Inches. 


Outside 
Diameter, 
Inches. 


Average 
Thick- 
ness, 
Inches. 


Breaking 
Load, 
Lbs. 


Average 
Area Sec- 
tion, 
Sq. In. 


Inches 


/ 
P 


Break- 
ing 
Load 
per sq. 
in., IDS. 


I 


190.25 


15 




1356000 


43.98 


4.96 


.38.36 


30832 


II 


190.25 


15 


^ 


1330000 


49-03 


4.92 


38.67 


27126 


B* 


190.25 


15 


\ 


1198000 


49-03 


4.92 


38.67 


24434 


B< 


190.25 


15* 


\ 


1246000 


49.48 


4.98 


38.20 


25l8l 


5 


190.25 


15 


H 


1632000 


50.91 


4.91 


38.75 


3 20 57 










over 








over 


6 


I90.2 


15 


'A 


2082000 


51.52 


4,90 


38.73 

** ' +* 


40411 


XVI 


1 60 


81 to 7 f 


i 


651000 


21.99 


2.50 


64.00 


29604 


XVII 


1 60 


8 


'& 


645600 


22.87 


2.48 


64.52 


28229 


7 


t20 


6 T V 


*& 


455200 


17.64 


1.78 


67.41 


25805 


8 


1 2O 


6ft 


i*V 


474100 


17-37 


1.8o 


66.67 


27236 



CAST-IRON COLUMNS. 



50000 
40000 
30000 
20000 



70 80 



90 100 110 120 



Abscissge= length di- 
vided by radius of 
gyration of small- 
est section. 

Ordinates:= breaking 
strengths per 
square inch of 
smallest section. 



130 no 150 



CAST-IRON COLUMNS. 



371 



The cut on page 370 shows a graphical representation of 
the preceding tests of full-size cast-iron columns. 

In Heft VIII (1896) of the Mitt. d. Materialpriifungsanstalt 
in Zurich is an account of 296 cast-iron struts tested by Prof. 
Tetmajer; 46 being 3 cm. (i". 1 8) square will not be men- 
tioned farther. The other 250 were hollow circular, the inside 
diameters being 10 cm. (3". 94), 12 cm. (4". 72), or 15 cm. 
(5".9i); the thicknesses being i cm. (o".39) oro.8 cm. (0^.31). 
The lengths varied from 4 m. (13'. 12) to 20 cm. (?".g). They 
are not the most usual thicknesses of columns for buildings, 
though used to a considerable extent. They might be called 
cast-iron pipe columns. The following table contains all those 
250 cm. (8'.2) long and over, and i cm. thick, and one set of 
those 0.8 cm. thick. This will exhibit the character of the results 

for such columns of usual lengths. In computing the actual 



Thickness o". 39. 


Thickness o".3i. 






Outside 




Ultimate 






Outside 




Ultimate 


No. of 
Test. 


Length, 
Feet. 


Diame- 
ter, 


I 

p 


Strength, 
Pounds 


No. of 
Test. 


Length, 
Feet. 


Diame- 
ter, 


/ 
p 


Strength, 
Pounds 






Inches. 




per sq. in. 






Inches, 




per sq. in. 


55 
56 


9.84 
9.84 


S3 


7 7?: 7 


18481 
20761 


207 
208 


13.12 
13.12 


4.62 
.61 


103.9 
103.9 


11518 

11660 


57 


8.20 


4.76 


63.4 


28156 


209 


n. 4 8 


58 


91.1 


16922 


58 


8.20 


4.78 


639 


2986-2 


210 


n. 4 8 


59 


91.9 


-0577 


69 
70 
7* 


9.84 

9.84 

8.20 


5-63 

5.6z 
5.65 


64.4 
64.4 
53-3 


24174 
32564 
36546 


211 
212 
213 


9.84 
9.84 

8.20 


56 
.60 
.56 


78.9 
77-7 
65-4 


194.12 
19482 
3*843 


72 


8.20 


5.63 


53-4 


47353 


3I 4 


8.20 


4.61 


64.7 


33 '33 


86 


9.84 


6.69 


53-2 


32564 


225 


13.12 


5-41 


87.8 


15216 


87 


9.84 


6.67 


53-3 


34270 


226 


13-12 


5-43 


87.5 


17623 


88 


8.20 


6-73 


43-9 


44224 


227 


11.48 


5-41 


76.7 


22326 


89 


8.20 


6.69 


44.1 


46642 


228 
229 


11.48 
9.84 


5-39 
5-39 


77-3 
66.4 


2I3I 

23748 




230 


9.84 


5-41 


66.1 


23463 




23 1 


8.20 


5-41 


54- 8 


38110 




2 3 2 


8.20 


5-41 


54-5 


36688 




243 


13.12 


6.56 


71.8 


22041 




244 


13.12 


6-54 


71.9 


24885 




245 


11.48 


6.56 


62.6 


27729 




2 4 6 


11.48 


6.56 


62.5 


28156 




247 


9.84 


6-53 


S3-9 


355 20 




2 4 8 


9.84 


6-54 


53-9 


31853 




249 


8.20 


6.56 


44-8 


4*949 




2 5 


8.20 


6.56 


44.8 


453^2 



372 APPLIED MECHANICS. 

length of the strut has been used, whereas Tetmajer adds to 
this 9 ".84, the thickness of the platforms of the machines, as 
they bore o;i knife-edges. 

Prof. Bauschinger of Munich made two series of tests of 
full-size cast- and of wrought-iron columns to determine the 
effect of heating them red-hot and sprinkling them with water 
while under load. They were loaded in his testing-machine 
with their estimated safe load as calculated from the formulas. 

For cast-iron, 



19912^4 

./" 

For wrought-iron, 



i -|- 0.0006-5 



11378^4 
*' r , 

i -f- 0.00009 2 

where P = safe load (factor of safety five), A = area of section, 
/ = length, p least radius of gyration, pounds and inches 
being the units. 

A fire was made in a U-shaped receptacle under the post, 
so arranged that the flames enveloped the post. The tem- 
perature was determined from time to time by means of alloys 
of different melting-points ; and the horizontal and vertical 
components of the deflections were read off on a dial as indi- 
cated by a hand attached to the post by a long wire. The 
post was also examined for cracks or fractures. 

In the 1884 series he tested six cast-iron posts of various 
styles, and three wrought-iron posts, one of them being made 
of channel-irons and plates put together with screw-bolts, one 
of I irons and plates also put together with screw-bolts, and 
one hollow circular. 

The details of the tests will not be given here, but only 
Bauschinger's conclusions. He said : 

That wrought-iron columns, even under the most favorable 



CAST- 1 RON- COLUMNS. 373 

adjustment of their ends and of the manner of loading, bend 
so much that they cannot hold their load, sometimes with a 
temperature less than 600 Centigrade, and always when they 
are at a red heat ; and this bending is accelerated by sprink- 
ling on the opposite side, even when only the ends of the post 
are sprinkled. 

. That under similar circumstances cast-iron posts bend, and 
this bending is increased by sprinkling ; but it does not exceed 
certain limits, even when the post is red for its entire length 
and the stream of water is directed against the middle, and the 
post does not cease to bear its load even when cracks are de- 
veloped by the sprinkling. Only when both ends of a cast-iron 
post are free to change their directions does sprinkling them 
at the middle of the opposite side when they are red make 
them break, but such an unfavorable case of fastening the ends 
hardly ever occurs in practice. 

That the cracks in the columns tested occurred in the 
smooth parts, and not at corners or projections. 

That the result of these tests warns us to be much more 
prudent in regard to the use of wrought-iron in building. If 
posts which are subjected to a longitudinal pressure bend so 
badly when subjected to heat on one side that they lose the 
power of bearing their load, how much more must this be the 
case with wrought-iron beams ; and he urges the importance of 
making more experiments. 

In Heft XV of the Mittheilungen he says that the results 
were criticised in two ways, viz. : Moller claiming that he 
should have used different constants, and Gerber that the 
wrought-iron posts were not properly made. 

Bauscliinger therefore concluded to make a new set of tests, 
and for this purpose he had made two cast-iron and five 
wrought-iron columns the former being carefully cast, but on 
the side, while the wrought-iron ones were made by a bridge 
company of very good reputation, and four of them were 
similar to those made at the time for a new warehouse in 
Hamburg. 



374 APPLIED MECHANICS. 



The tests were made just as before, and the following are 
his conclusions : 

That when wrought-iron posts are as well constructed as 
the two referred to, they resist fire and sprinkling tolerably 
well, though not as well as cast-iron ; but that posts con- 
structed like the other three, even with the fire alone, and 
before the sprinkling begins, get so bent that they can no 
longer hold their load. Good construction requires that the 
rows of rivets shall extend through the entire length of the 
post, and the rivets should be quite near each other ; but the 
tests are not extensive enough to show what are the necessary 
requirements to make wrought-iron posts able to stand fire and 
sprinkling ; in order to know this more experiments are needed. 

In Dingler's Polytechnisches Journal for 1889, page 259 et 
seq., is an article by Professor A. Martens, of Berlin, uprn 
the behavior of cast- and wrought-iron in fires, considering 
especially the burning of a large warehouse in Berlin, and 
advocating the protection of iron-work by covering it with 
cement. He says that there are two series of tests upon 
this subject, one of which is the tests of Bauschinger already 
explained, and the other a set of tests made by Moller and 
Luhmann. 

No detailed account of these tests will be given here, but 
only Holler's conclusions, as stated by Prof. Martens, which 
are as follows: 

i. With ten cast-iron posts he could not get any cracks 
by sprinkling at a red heat; but it is to be noted that his were 
new posts, while those used in Bauschinger's first series were 
old ones, and that those in Bauschinger's second series, which 
were new and very carefully cast, did not show cracks either. 

2. He claims that while the cracks would allow the post 
still to bear a centre load, it could not bear an eccentric load 
or a transverse load. 

3. "He claims that the load on a cast-iron post should be 
limited to one which shall not produce sufficient bending to 
bring about a tensile stress anywhere when the post is bent by 
the heat and sprinkling. 



TRANSVERSE STRENGTH OF CAST-IRON. 375 

4. He claims that in either cast- or wrought-iron posts, if 
the ends are not fixed, the ratio of length to diameter should 
not exceed I o, whereas if they are it should not exceed 17; 
also, that there is no such thing as absolute safety from fire 
with iron. 

5. A covering of cement delays the action of the fire, and 
that therefore such a covering is a protection to the post 
against excessive one-sided heating and cooling. 

6. Cast-iron is more likely to have at any one section a 
collection of hidden flaws than wrought-iron. 

220. Transverse Strength of Cast-iron. At one time 
cast-iron was very largely used for beams and girders in build- 
ings to support a transverse load. Its use for this purpose has 
now been almost entirely abandoned, as it has been superseded 
by wrought-iron and steel. 

A great many experiments have been made on the trans- 
verse strength of cast-iron ; the specimens used in some cases 
being small, and in others large. The records of a great many 
experiments of this kind are to be found in the first four books 
of the list already enumerated in 217. The details of these 
tests will not be considered here, but an outline will be given 
of some of the main difficulties that arise in applying the results 
and in using the beams. 

Cast-iron is treacherous and liable to hidden flaws ; it is 
brittle. It is also a fact that in casting any piece where the 
thickness varies in different parts, the unequal cooling is liable 
to establish initial strains in the metal, and that therefore 
those parts where such strains have been established have 
their breaking-strength diminished in proportion to the amount 
of these strains. 

In the case of cast-iron also, the ratio of the stress to the 
strain is not constant, even with small loads, and is far from 
constant with larger loads ; also, inasmuch as the compressive 
strength is far greater than the tensile, it follows that, in a 
transversely loaded beam which is symmetrical above and be- 
low the middle, the fibres subjected to tension approach their 




37 6 APPLIED MECHANICS. 

full tensile strength long before those subjected to compression 
are anywhere near their compressive strength. The result of 
all this is, that if a cast-iron beam be broken transversely, and 
the modulus of rupture be computed by using the ordinary 
formula, 

f-My 
7 " I ' 

we shall find, as a rule, a very considerable disagreement be- 
tween the modulus of rupture so calculated and either the 
tensile or compressive strength of the same iron. Indeed, 
Rankine used to give, as the modulus of rupture for rectangu- 
lar cast-iron beams, 40000 Ibs. per square inch, and for open- 
work beams 17000 Ibs. per square inch, which latter is about 
the tensile strength of fairly good common cast-iron. 

A great deal has been said and written, and a good many 
experiments have been made, to explain this seeming disagree- 
ment between the modulus of rupture as thus computed, and 
the tensile strength of the iron. Barlow proposed a theory 
based upon the assumption of the existence of certain stresses 
in addition to those taken account of in the ordinary theory of 
beams, but his theory has no evidence in its favor. 

Rankine claimed that the fact that the outer skin is harder 
than the rest of the metal would serve to explain matters, but 
this would not explain the fact that the discrepancy exists in 
the case of planed specimens also. 

Neither Barlow nor Rankine seems to have attempted to 
find the explanation in the fact that the formula 



assumes the proportionality of the stress to the strain, and 
hence that is less and less applicable the greater the load, and 
hence the nearer the load is to the breaking load. An article 
by Mr. Sondericker in the Technology Quarterly of October, 



TRANSVERSE STRENGTH OF CAST-IRON. 377 

1888, gives an account of some experiments made by him to 
test the theory that " the direct stress, tension, or compression, 
at any point of a given cross-section of a beam, is the same 
function of the accompanying strain, as in the case of the cor- 
responding stress when uniformly distributed," and the results 
bear out the theory very well ; hence it follows that, if we use 
the common theory of beams, determining the stresses as such 
multiples of the strains as they show themselves to be in direct 
tensile and compressive tests, the discrepancies largely vanish, 
and those that are left can probably be accounted for by initial 
stresses due to unequal rate of cooling, and by the skin, or 
by lack of homogeneity. In the same article he quotes the 
results of other tests bearing more or less on the matter, and 
there will be quoted here the table on page 378. 
If, therefore, we wish to make use of the formula 



y 

in calculating the strength of cast-iron beams, we cannot use 
one fixed value of f for all beams made of one given quality 
of cast-iron, but we shall have to use a very varying modulus 
of rupture, varying especially with the form, and also with the 
size of the beam under consideration. Now, in order to do 
this, and obtain reasonably correct results, we need, wherever 
possible, to use values of f that have been deduced from ex- 
periments upon pieces like those which we are to use in prac- 
tice, and under, as nearly as possible, like conditions. 

There are not very many records of such experiments avail- 
able, and, in cases where we cannot obtain them, it will prob- 
ably be best to use a value of f no greatei than the tensile 
strength for complicated forms, and forms having thin webs. 
For pieces of rectangular or circular section we might probably 
use, for good fair cast-iron, 25000 to 30000 Ibs. per square 
inch. 

A few tests of the character referred to have been made in 
the engineering laboratories of the Massachusetts Institute of 



3 ;8 



APPLIED MECHANICS. 








Modulus oil 






Form of 
Beam Sec- 


Tensile 
Strength, 


Rupture 

, My 
J ~7~< 


Ratio. 


Condition 
of 


Experimenter. 


tion, 


\bs. per Sq. 
In. 


/ 
Ibs. perSq. 




Specimen. 








In. 










19850 


41320 


2.08 


Turned 


C. Bach.* 


x?H^, 


16070 


35500 


2.21 


Turned 


Considere.f 




34420 


63330 


1.84 


Turned 


Considere. 


\Hx 


24770 


54390 


2.19 


Turned 


Robinson and Segundo. J 




25040 


46280 


1.8 5 


Rough 


Robinson and Segundo. 




Mean. 


2.03 








16070 


29250 


1.82 


Planed 


Considere. 


ijlflp 


36270 


58760 


1.62 


Planed 


Considfere. 




19090 


33740 


1-77 


Planed 


C. Bach. 




Mean. 


1.74 








19470 


34000 


1-75 


Planed 


C. Bach. 


in 


31430 


49030 


1.56 


Planed 


Considere. 


Hi 


19880 


33860 


1.70 


Planed 


Sondericker. 




24770 


42340 


1.71 


Planed 


Robinson and Segundo. 


g^^ 


25040 


42IIO 


1.68 


Rough 


Robinson and Segundo. 




Mean. 


1.68 








19470 


28150 


1-45 


Planed 


C. Bach. 




16070 


225OO 


1.40 


Planed 


Considere. 




31860 


36640 


I-I5 


Planed 


Considere. 




25040 


3I3IO 


1.25 


Rough 


Robinson and Segundo, 




Mean. 


I-3I 








16070 


23780 


1.48 


Planed 


Considere. 


n 


31290 


34730 


I. II 


Planed 


Considere. 


n 


18050 


24550 


1.36 


Planed 


Sondericker. 


^^ 


22470 


26150 


1.16 


Rough 


Burgess and Viel6. 




Mean. 


1.28 







See Zeitschrift des Vereines Deutscher Ingenieure, Mar. 3d and loth, 1888. 

t See Annales des Fonts et Chausse"es, 1885. 

t See Proceedings Institute of Civil Engineers, Vol 86. 

5 Sec Proceedings Am. Soc. Mecbl. Engrs. 1889, pp. 187 et seq. 



TRANSVERSE STRENGTH OF CAST-IRON. 



379 



Technology, and a brief statement of" them will be given here. 
The first that will be referred to here is a series of experiments 
made by two students of the Institute, an account of which is 
given in the Proceedings of the American Society of Mechani- 
cal Engineers for 1889, pp. 187 et seq. 

The object of this investigation was to determine the trans- 
verse strength of cast-iron in the form of window lintels, and 
also the deflections under moderate loads, and from the latter 
to deduce the modulus of elasticity of the cast-iron, and to 
compare it with the modulus of elasticity of the same iron, as 
determined from tensile experiments ; also the tensile strength 
and limit of elasticity of specimens taken from different parts 
of che lintel were determined. 

The iron used was of two qualities, marked P and 5 respec- 
tively. 

The tensile specimens were cast at the same time, and from 
the same run as the lintels. 

Besides this, one of each kind of window lintels was cut up 
into tensile specimens, and the specimens were so marked as to 
show from what part of the lintel they were cut. 

The tables of tests will now be given, and the following ex- 
planation of the symbolism employed. 

P and S are used, as already stated, to denote the quality 
of the iron. 

A and B are used to denote, respectively, that the specimen 
was unplaned or planed. 

I, 2, 3, etc., denote the number of the test made on that 
particular kind and condition. 



380 



APPLIED MECHANICS. 



I., II., III., denote that the piece has been taken from a 
lintel, and also from what part, as will easily be seen by the 
sketch on page 379. 

Thus P. B. 3 would signify that the specimen was of quality 
P 9 had been planed, and was the third test of this class. 

On the other hand, P. B. 3 II., would signify in addition 
that it had been taken from a lintel, and was a piece of one of 
the strips marked II. in the sketch. 

The following is a summary of the breaking-weights per 
square inch of the specimens not cut from the lintels : 

P. A. i 23757 S. A. i 24204 

P. A. 2 21423 S. A. 2 25258 

P. A. 3 18938 S. A. 3 24706 

P. A. 4...., 21409 

3)74168 



24723 




21382 

P. B. i 21756 

P. B.3 25207 

2)46963 



S. B. i 
S. B, 2 



29574 
23201 



2)52775 



23482 26388 

The following are the breaking-weights per square inch of 
the specimens cut from the iinteis : 



P B. 



' 5 


I 


19651 




6 


I 


20715 




9 


I 


21076 




10 


I 


21483 




4 


II 


19016 




7 


II 


19376 




ii 


II 


22146 




12 


II 


20552 




2 


III 


10594 


(Broke at 
a flaw.) 


13 


III 


16141 




I 8 


IV 


10616 





S. B. 



6 


I 


29124 


7 


I 


28372 


8 


I 


25425 


3 


II 


24704 


4 


II 


29414 


5 


II 


23610 


9 


III 


27523 


10 


III 


18301 


4 


IV 


19616 



TKAffSVERSE STRENGTH OF CAST-IRON. 



381 



All the window lintels tested were of the form shown in 
the figure, and all were supported at the ends and loaded at 
the middle, the span in every case being 52". From the cut 
it will be seen that the web varied in height, being 4 inches 
high above the flange in the centre, and decreasing to 2.5 inches 
at the ends over the supports. 

The following are the results of the separate tests, where 
tensile modulus of rupture means the outside fibre stress per 
square inch on the tension side, and compressive modulus of 
rupture that on the compression side, both being calculated 
from the actual breaking load by the formula 



f- 
J ~ ' ' 



Mark on Lintel. 


Breaking-Load, 
Lbs. 


Tensile Modulus of 
Rupture, 
Ibs. per Sq. In. 


Compressive Modulus 
of Rupture, 
Ibs. per Sq. In. 


P. I 


27220 


26648 


81578 


P. 2 


30520 


29879 


91467 


P. 3 


27200 


26659 


81608 


S. i 


26750 


26198 


80164 


S. 2 


19850 


19433 


59490 


S. 3 


28670 


28068 


85924 


S. 4 


25120 


24592 


75285 



The second series of experiments was made by two other 
students, and an account of the work is given in the same 
article as the former one. 

The object was to determine the constants suitable to use 
in the formulae for determining the strength of the arms of 
cast-iron pulleys ; and also, incidentally, to determine the hold- 
ing power of keys and set-screws. 

Some old pulleys with curved arms, which had been in use 
at the shops, were employed for these tests. They were all 



382 APPLIED MECHANICS. 

about fifteen inches in diameter, and were bored for a shaft 
I T ^ inches in diameter. 

Inasmuch as this size of shaft would not bear the strain 
necessary to break the arms, the hubs were bored out to a 
diameter of i-j-J- inches diameter, and key-seated for a key one- 
half an inch square. 

In order to strengthen the hubs sufficiently, two wrought- 
iron rings were shrunk on them, so as to make it a test of the 
arms and not of the hub. 

The pulley under test is keyed to a shaft which, in its turn, 
is keyed to a pair of castings supported by two wrought-iron I- 
beams, resting upon a pair of jack-screws, by means of which 
the load is applied. A wire rope is wound around the rim of 
the pulley, and leaves it in a tangential direction vertically. 
This rope is connected with the weighing lever of the machine, 
and weighs the load applied. 

In a number of the experiments one arm gave way first, 
and then the unsupported part of the rim broke. 

The breaking-load of the separate pulleys was, of course, 
determined, and then it was sought to compute from this the 
value of f from the formula 



which is the one most commonly given for the strength of 
pulley arms, and which is based upon several erroneous assump- 
tions, one of which is that the bending-moment is equally 
divided among the several arms. In this formula 

/= moment of inertia of section, 

n = number of arms, 

y = half depth of each arm = distance from neutral axis to 
outside fibre, 

x = length of each arm in a radial direction, 

P = breaking-load determined by experiment. 



TRANSVERSE STRENGTH OF CAST-IRON. 



383 



The results are given in the following table, the units being 
inches and pounds : 



the 
the 



o o 

H 

rt cl 

II 



tly inc 
broke. 



H 

!i 
ii 



sad subsequently 
when the rim brok 



.sg ~| 

S "5 - 



II S*53 

j- -S:r c 3 
13 33 la 



5*1 



8JS 



u i 



!! iflSrf" 2 

S^ 13 

s2 is^-sl. 

O H 






C- J3- 

O H 



rt 
| 

1 I 



=3 s- 



U^3 

O 









I 



.2 "u 





Q S 



XXX 



P 
x 



L-^C 

X 



X 

< 



t-pj t-bj 
Mps otjeo 

X X 



X 

-p 



* 

X 



N|m 
X X 



stnay jo aaqtnnj^ | 



souy 



a 



" 



qn H 



co co co 



jo ssaujpiqi, 



-C -C 



-^ M H W H 
co co CO 



Xannj jo -oreia 



IT rt 2" "? iT ? 



APPLIED MECHANICS. 



In the cases of numbers 5, 7, 8, 9, and 10 some of the arms 
were not broken, the rims were now broken off, and the re- 
maining arms were tested separately, the pull being exerted by 
a yoke hung over the end of the arm, the lower end being at- 
tached to the link of the machine. 

The arms were always placed so that the direction of the 
pull was tangent to the curve of the rim at the end of the arm. 
The actual modulus of rupture was then determined by calcula- 
tion from the experimental results, and is recorded in the 
following table, the units being inches and pounds : 



Number of 
Arms. 


Dimensions of Sec- 
tion at Fracture : 
all elliptical. 


Bend of Arm with 
or against Load. 


Modulus of 
Rupture. 


AverageModulus of 
Rupture for each 
Pulley. 


5 i 


i* X iJ 


against 


4539 6 


45396 


7 i 


i* X* 


against 


36802 




7 2 


'if Xf 


against 


39537 




7 3 


itfxi 


with 


46407 


40915 


8 i 


itt x H 


against 


35503 




8 2 


iH x|| 


against 


36091 




8-3 


iX 


with 


39939 




8-4 


iHxtt 


with 


42469 


38500 


9 1 


i*Xf 


against 


41899 




9 2 


'*XH 


against 


44148 




9 3 


i*xf 


with 


55442 


47163 


10 I 


if XH 


against 


54743 




10 2 


IXH 


against 


5943 




io 3 


ix 


against 


38605 




10 4 


if x ft 


with 


55 2 29 


49880 

-^^ 



STANDARD SPECIFICATIONS FOR CAST-IRON. 385 

STANDARD SPECIFICATIONS FOR CAST-IRON, OF THE AMERICAN 
SOCIETY FOR TESTING MATERIALS. 

The standard specifications for cast-iron, of the American 
Society for Testing Materials, contain specifications for i 
Foundry Pig-iron, 2 Gray Iron Castings, 3 Malleable Iron 
Castings, 4 Locomotive Cylinders, 5 Cast-iron Pipe and Special 
Castings, 6 Cast-iron Car-wheels. Of these, i, 2, and 4 will 
be quoted in full, and extracts will be given from 5. For the 
remainder see the proceedings of the Society. 

AMERICAN SOCIETY FOR TESTING MATERIALS. 
SPECIFICATIONS FOR FOUNDRY PIG-IRON. 

ANALYSIS. 
It is recommended that all purchases be made by analysis. 

SAMPLING. 

In all contracts where pig-iron is sold by chemical analysis, each 
car load, or its equivalent, shall be considered as a unit. At least one 
pig shall be selected at random from each four tons of every car load, 
and so as to fairly represent it. 

Drillings shall be taken so as to fairly represent the fracture-surface 
of each pig, and the sample analysed shall consist of an equal quantity 
of drillings from each pig, well mixed .and ground before analysis. 

In case of disagreement between buyer and seller, an independent 
analyst, to be mutually agreed upon, shall be engaged to sample and 
analyze the iron. In this event one pig shall be taken to represent 
every two tons. 

The cost of this sampling and analysis shall be borne by the buyer 
if the shipment is proved up to specifications, and by the seller if other- 
wise. 

ALLOWANCES AND PENALTIES. 

In all contracts, in the absence of a definite understanding to the 
contrary, a variation of 10 per cent in silicon, either way, and of o.oi 
sulphur, above the standard, is allowed. 

A deficiency of over 10 per cent and up to 20 per cent, in the silicon, 
subjects the shipment to a penalty of 4 per cent of the contract price. 



386 



APPLIED MECHANICS. 



BASE ANALYSIS OF GRADES. 

In the absence of specifications, the following numbers, known to 
the trade, shall represent the appended analyses for standard grades 
of foundry pig-irons, irrespective of fracture, and subject to allowances 
and penalty as above: 



Grade. 


Per Cent 

Silicon. 


Per Cent 
Sulphur 
(Volumetric). 


Per Cent 

Sulphur 
(Gravimetric). 


No i . . . 


2-75 


0-035 


0.045 


N >. 2 . . . 


2.25 


0.045 


0-055 


No 3 ... 


i-75 


0-55 


0.065 


No 4 ... 


1.25 


0.065 


0.075 



PROPOSED SPECIFICATIONS FOR GRAY IRON CASTINGS. 
PROCESS OF MANUFACTURE. 

Unless furnace iron is specified, all gray castings are understood to 
be made by the cupola process. 

CHEMICAL PROPERTIES. 
The sulphur contents to be as follows: 

Light castings not over o . 08 per cent. 

Medium castings . . . . " " o.io " " 
Heavy castings " " 0.12 " " 

DEFINITION. 

In dividing castings into light, medium, and heavy classes, the 
following standards have been adopted : 

Castings having any section less than \ of an inch thick shall be 
known as light castings. 

Castings in which no section is less than 2 ins. thick shall be known 
as heavy castings. 

Medium castings are those not included in the above definitions. 
PHYSICAL PROPERTIES. 

Transverse Test. The minimum breaking-strength of the "Arbi- 
tration Bar " under transverse load shall not be under: 

Light castings 2500 Ibs. 

Medium castings 2900 " 

Heavy castings 3300 " 



STANDARD SPECIFICATIONS t-'OR CAST-IRON. 



387 



In no case shall the deflection be under .10 of an inch. 

Tensile Test. Where specified, this shall not run less than: 

Light castings 18000 Ibs. per square inch. 

Medium castings .... 21000 " " " " 
Heavy castings 24000 " ' ' ' ' " 

THE " ARBITRATION BAR" AND METHODS OF TESTING. 

The quality of the iron going into castings under specification 
shall be determined by means of the " Arbitration Bar." This is 
a bar ij ins. in diameter and 15 ins. long. It shall be prepared as 
stated further on and tested transversely. The tensile test is not 
recommended, but in case it is called for, the bar as shown in Fig. i, 
and turned up from any of the broken pieces of the transverse test, 
shall be used. The expense of the tensile test shall fall on the purchaser. 

Two sets of two bars shall be cast from each heat, one set from the 
first and the other set from the last iron going into the castings. Where 



i 





:r 



the heat exceeds twenty tons, an additional set of two bars shall be 
cast for each twenty tons or fraction thereof above this amount. In 
case of a change of mixture during the heat, one set of two bars shall 
also be cast for every mixture other than the regular one. Each set 
of two bars is to go into a single mold. The bars shall not be rumbled 
or otherwise treated, being simply brushed off before testing. 



$88 APPLIED MECHANICS. 

The transverse test shall be made on all the bars cast, with supports 
12 ins. apart, load applied at the middle, and the deflection at rupture 
noted. One bar of every two of each set made must fulfill the re- 
quirements to permit acceptance of the castings represented. 

The mold for the bars is shown in Fig. 2 (not shown here). The 
bottom of the bar is iV of an inch smaller in diameter than the top, 
to allow for draft and for the strain of pouring. The pattern shall not 
be rapped before withdrawing. The flask is to be rammed up with 
green molding-sand, a little damper than usual, well mixed and put 
through a No. 8 sieve, with a mixture of one to twelve bituminous 
facing. The mold shall be rammed evenly and fairly hard, thoroughly 
dried and not cast until it is cold. The test-bar shall not be removed 
from the mold until cold enough to be handled. 

SPEED OF TESTING. 

The rate of application of the load shall be thirty seconds for a 
deflection of .10 of an inch. 

* 
SAMPLES FOR CHEMICAL ANALYSIS. 

Borings from the broken pieces of the " Arbitration Bar " shall 
be used for the sulphur determinations. One determination for each 
mold made shall be required. In case of dispute, the standards of 
the American Foundrymen's Association shall be used for comparison. 

FINISH. 

Castings shall be true to pattern, free from cracks, flaws, and ex- 
cessive shrinkage. In other respects they shall conform to whatever 
points may be specially agreed upon. 

INSPECTION. 

The inspector shall have reasonable facilities afforded him by the 
manufacturer to satisfy him that the finished material is furnished in 
accordance with these specifications. All tests and inspections shall, 
as far as possible, be made at the place of manufacture prior to ship- 
ment. 



STANDARD SPECIFICATIONS FOR CAST-IRON. 389 

SPECIFICATIONS FOR LOCOMOTIVE CYLINDERS. 

PROCESS OF MANUFACTURE. 

Locomotive cylinders shall be made from good quality of close- 
grained gray iron cast in a dry sand mold. 

CHEMICAL PROPERTIES. 

Drillings taken from test-pieces cast as hereafter mentioned shall 
conform to the following limits in chemical composition : 

Silicon from 1.25 to i . 75 per cent 

Phosphorus not over .9 ' ' " 

Sulphur " " .10 " " 

PHYSICAL PROPERTIES. 

The minimum physical qualities for cylinder iron shall be as 
follows : 

The ''Arbitration Test-Bar," ij ins. in diameter, with supports 
12 ins. apart shall have a transverse strength not less than 30x50 Ibs., 
centrally appliedj and a deflection not less than o.io of an inch. 

TEST-PIECES AND METHOD OF TESTING. 

The standard test shall be ij ins. in diameter, about 14 ins. long, 
cast on end in dry sand. The drillings for analysis shall be taken 
from this test-piece, but in case of rejection of the manufacturer shall 
have option of analyzing drillings from the bore of the cylinder, upon 
which analysis the acceptance or rejection of the cylinder shall be 
based. 

One test-piece for each cylinder shall be required. 

CHARACTER OF CASTINGS. 

Castings shall be smooth, well cleaned, free from blow-holes, shrink- 
age cracks, or other defects, and must finish to blue-print size. 

Each cylinder shall have cast on each side of saddle manufacturer's 
mark, serial number, date made, and mark showing order number. 

INSPECTOR. 

The inspector representing the purchaser shall have all reasonable 
facilities afforded to him by the manufacturer to satisfy himself that the 
finished material is furnished in accordance with these specifications. 
All tests and inspections shall be made at the place of the manufacturer. 



39 APPLIED MECHANICS. 

CAST-IRON PIPE AND SPECIAL CASTINGS. 

This specification is divided into the following sections, viz.: i 
Description of Pipes, 2 Allowable Variation in Diameter of Pipes and 
Sockets, 3' Allowable Variation in Thickness, 4 Defective Spigots may 
be Cut, 5 Special Castings, 6 Marking, 7 Allowable Percentage of 
Variation in Weight, 8 Quality of Iron, 9 Tests of Material, 10 Cast- 
ing of Pipes, 11 Quality of Castings, 12- Cleaning and Inspection, 13 
Coating, 14 Hydrostatic Test, 15 Weighing, 16 Contractor to Furnish 
Men and Materials, 17 Power of Engineer to Inspect, 18 Inspector 
to Report, 19 Castings to be Delivered Sound and Perfect, 20 Defi- 
nition of the Word Engineer. 

Of these, only sections 8 and 9 will be quoted here, as follows: 

QUALITY OF IRON. 

SECTION 8. All pipes and special castings shall be made of cast- 
iron of good quality, and of such character as shall make the metal 
of the castings strong, tough, and of even grain, and soft enough to 
satisfactorily admit of drilling and cutting. The metal shall be made 
without any admixture of cinder-iron or other inferior metal, and shall 
be remelted in a cupola or air furnace. 

TESTS OF MATERIAL. 

SECTION 9. Specimen bars of the metal used, each being 26 inches 
long by 2 inches wide and i inch thick, shall be made without charge 
as often as the engineer may direct, and, in default of definite instruc- 
tions, the contractor shall make and test at least one bar from each heat 
or run of metal. The bars, when placed flatwise upon supports 24 
inches apart and loaded in the centre, shall for pipes 12 inches or less 
in diameter support a load of 1900 pounds and show a deflection of 
not less than .30 of an inch before breaking, and for pipes of sizes larger 
than 12 inches shall support a load of 2000 pounds and show a deflection 
of not less than .32 of an inch. The contractor shall have the right to 
make and break three bars from each heat or run of metal, and the test 
shall be based upon the average results of the three bars. Should 
the dimensions of the bars differ from those above given, a proper 
allowance therefor shall be made in the results of the tests. 



WROUGHT-IRON, 39! 



221. Wrou gilt-Iron. Wrought -iron is obtained by melt- 
ing pig-iron in contact with iron ore, oxidizing, and burning out, 
as far as may be, the carbon, the phosphorus, and the silicon. 
In many cases, however, the charge consists largely of wrought- 
iron or. steel scrap, and cast-iron borings. 

The process is commonly carried on in a puddling furnace, 
where an oxidizing flame is passed over the melted pig-iron. 

As the heat is not sufficiently intense to melt the wrought- 
iron produced, the metal is left in a plastic condition, full of 
bubbles and holes, which contain considerable slag. It is then 
squeezed, and rolled or hammered, to eliminate, as far as possible, 
the slag, and to weld the iron into a solid mass. 

The result of this first rolling is known as muck-bar, and must 
be "piled," heated, and rolled or hammered at least once more 
before it is suitable for use in construction. 

In making the piles, while muck-bar is sometimes used 
exclusively, a considerable part, and often the greater part, is 
made of scrap. 

Wrought-iron is thus, throughout its manufacture, a series 
of welds. Moreover, wherever slag is present, these welds cannot 
be perfect. It is also subject to the impurities of the cast-iron 
from which it is made. Thus, the presence of sulphur makes 
it red-short, or brittle when hot; and the presence of phosphorus 
makes it cold-short, or brittle when cold. 

It cannot, like cast-iron, be melted and run into moulds; 
but it can be easily welded by the ordinary methods 

Wrought-iron is much more capable of bearing a tensile or 
transverse stress than cast-iron: it is tougher, it stretches more, 
and gives more warning before fracture. At one time cast-iron 
was the principal structural material, but it was soon displaced 
by wrought-iron, which became the principal metal used in 
construction, but now, since the modern methods of steel-making 
supply a more homogeneous product at a cheaper price, wrought- 
'iron has been superseded by mild steel in most pieces used in 
construction. 



39 2 APPLIED MECHANICS. 

Wrought-iron is also expected to withstand a great many 
trials that would seriously injure cast-iron: thus, two pieces 
of wrought-iron are generally united together by riveting; the 
holes for the rivets have to be punched or drilled, and then the 
rivets have to be hammered; the entire process tending to injure 
the iron. Wrought-iron has to withstand flanging, and is liable 
to severe shocks when in use; as, for instance, those that occur 
from the changes of temperature in the different parts of a steam- 
boiler. 

The following references to a large number of tests of wrought- 
iron will be given : 

i. Eaton Hodgkinson: (a) Report of Commissioners on the Applica- 

tion of Iron to Railway Structures. 
(b) London Philosophical Transactions. 1840. 
2. William H. Barlow: Barlow's Strength of Materials. 
3. Sir William Fairbairn: On the Application of Cast and Wrought 

Iron to Building Purposes. 
4. Franklin Institute Committee: Report of the Committee of the 

Franklin Institute. In the Franklin Institute Journal of 



5. L. A. Beardslee, Commander U.S.N. : Experiments on the Strength 
of Wrought-iron and of Chain Cables. Revised and enlarged 
by William Kent, M.E., or Executive Document 98, 45th 
Congress, as stated below. 

6. David Kirkaldy: Experiments on Wrought-iron and Steel. 

7. G. Bouscaren: Report on the Progress of Work on the Cincinnati 
Southern Railway, by Thomas D. Lovett. Nov. i, 1875. 

8. Tests of Metals made at Watertown Arsenal. Of these the first 
two volumes were published before 1881, and since that 
time one volume has been published every year. Nearly all 
of them contain tests of wrought-iron and a great many of 
them contain tests of full-size pieces of wrought-iron. 

9. A. Wohler: (a) Die Festigkeits versuche mit Eisen und Stahl. 

(b) Strength and Determination of the Dimensions of Structures 



TENSILE STRENGTH OF WROUGHT-IRON. 393 

of Iron and Steel, by Dr. Phil. Jacob J. Weyrauch. Translated 

by Professor Dubois. 
10. Technology Quarterly, Vol. VII. No. 2, Vol. VIII. No. 3, Vol. 

IX. Nos. 2 and 3, and Vol. X. No. 4. 
11. Mitt, der Materialpriifungsaustalt in Zurich. 
12. Mitt, aus dem Mech. Tech. Lab. in Berlin. 
13. Mitt, aus dem Mech. Tech. Lab. in Miinchen. 

222. Tensile Strength of Wrought-iron. About the 
year 1840 was published the report of the Commission appointed 
by the British Government to investigate the application of iron 
to railway structures. While a number of tests of iron had been 
previously made, this work may properly be regarded as having 
been the first investigation of the kind that was at all thorough. 
At that time cast-iron was the metal most used in construction, 
and hence the greater part of the work of the Commission was 
devoted to a study of that metal. They made, however, a number 
of tests of wrought-iron, which, though they were of the greatest 
value at the time, and still have some value, will not be quoted 
here. 

At about that time the use of wrought-iron began to increase 
at a rapid rate, the necessary appliances were introduced to roll 
it into I beams, channel-irons, angle-irons, and other shapes, 
and it began to displace cast-iron for one after another purpose 
until it came to be the metal most extensively used in construction, 
both in the case of structures and machines. 

At first the chief desideratum was assumed to be that it 
should have a high tensile strength, and scarcely any attention 
was paid to its ductility. 

About 1865, however, engineers began to realize that duc- 
tility is an all-important property of a metal to be used in 
construction, and that this is not necessarily and not generally 
obtainable with a very high tensile strength. The most 



394 APPLIED MECHANICS. 

prominent advocate, at that time, of the importance of duc- 
tility was David Kirkaldy, who published a book, entitled 
" Experiments on Wrought Iron and Steel," containing the 
results of his tests down to 1866. 

In the early part of his book will be found a summary of 
what had been done by earlier experimenters in this line. 

Kirkaldy tested a large number of English irons, determin- 
ing both their breaking-strengths and their ductility. 

In the light of the results obtained by him, he proceeded 
to draw up his famous sixty-six conclusions. 

These sixty-six conclusions will not be quoted here, but 
the following statement will be made regarding the main 
results of his work : 

i. He proved that the results obtained by testing grooved 
specimens (or specimens of such form as to interfere with the 
flow of the metal while under test) did not indicate correctly 
the quality of the metal, but that such specimens should be 
used as did not interfere with the flow of the metal.. 

2. He advocated, with all the earnestness of which he was 
capable, the conclusion that it was of the greatest importance 
that all wrought-iron used in construction should have a good 
ductility, and, in his tests, he adopted five different methods 
of measuring ductility. 

These methods are : i. Contraction of area at fracture per 
cent ; 2. Ultimate elongation per cent ; 3. Breaking-strength 
per square inch of fractured area ; 4. Contraction of stretched 
area per cent, i.e., the contraction of area attained when the 
maximum load is first reached; 5. Breaking-weight per square 
inch of stretched area. Of these only two are used at the present 
time, the first and second, and they serve as measures of 
ductility. These two are the principal conclusions from Kir- 
kaldy's tests, though he cites a great many more, one of the 
principal of them being his conclusion regarding so-called cold 
crystallization, which will be mentioned later. 



SPECIFICATIONS FOR WROUGHT-IRON. 



395 



Tests of the tensile strength of wrought-iron may be divided 
into two classes: i those made mainly for the purpose of deter- 
mining the quality of the material, and 2 those made upon such 
full-size pieces as are used in practice to resist tension. 

The tests of the first class are made upon small specimens, 
and, in order that the results may be comparable, the use of 
standard forms and dimensions is, generally, a desideratum. 
The specifications for wrought-iron of the American Society for 
Testing Materials will be given first, as they refer to the kind 
of wrought-iron that is in most common use, and then some 
other tensile tests of various kinds of wrought-iron in small pieces 
will be given. Subsequently tests of wrought-iron eye-bars will 
be quoted. 

AMERICAN SOCIETY FOR TESTING MATERIALS. 
SPECIFICATIONS FOR WROUGHT-IRON. 

PROCESS OF MANUFACTURE. 

1. Wrought-iron shall be made by the puddling process or rolled 
from fagots or piles made from wrought-iron scrap, alone or with 
muck-bar added. 

PHYSICAL PROPERTIES. 

2. The minimum physical qualities required in the four classes of 
wrought-iron shall be as follows : 





Stay-bolt 
Iron. 


Merchant 
Iron. 
Grade "A." 


Merchant 
Iron. 
Grade "B." 


Merchant 
Iron, 
Grade "C." 


Tensile strength, pounds 










per square inch . 


46000 


50000 


48000 


48000 


Yield-point, pounds per 










square inch 


25000 


25000 


25000 


25000 


Elongation, per cent in 8 










inches 


28 


25 


20 


2O 



3. In sections weighing less than 0.654 pound per lineal foot, the 
percentage of elongation required in the four classes specified in para- 



39^ APPLIED MECHANICS. 

graph No. 2 shall be 12 per cent., 15 per cent., 18 per cent., and 
21 per cent., respectively. 

4. The four classes of iron when nicked and tested as described in 
paragraph No. 9 shall show the following fracture : 

(a) Stay-bolt iron, a long, clean, silky fibre, free from slag or dirt 
and wholly fibrous, being practically free from crystalline spots. 

(b) Merchant iron, Grade "A," a long, clean, silky fibre, free from 
slag or dirt or any course crystalline spots. A few fine crystalline 
spots may be tolerated, provided they do not in the aggregate exceed 
10 per cent of the sectional area of the bar. 

(c) Merchant iron, Grade "B," a generally fibrous fracture, free 
from coarse crystalline spots. Not over 10 per cent of the fractured 
surface shall be granular. 

(d) Merchant iron, Grade "C," a generally fibrous fracture, free 
from coarse crystalline spots. Not over 15 per cent of the fractured 
surface shall be granular. 

5. The four classes of iron, when tested as described in paragraph 
No. 10, shall conform to the following bending tests: 

(e) Stay-bolt iron, a piece of stay-bolt iron about 24 inches long, 
shall bend in the middle through 180 flat on itself, and then bend in 
the middle through 180 flat on itself in a plane at a right angle to 
the former direction without a fracture on outside of the bent 
portions. Another specimen with a thread cut over the entire length 
shall stand this double bending without showing deep cracks in the 
threads. 

(/) Merchant iron, Grade "A," shall bend cold 180 flat on itself, 
without fracture on outside of the bent portion. 

(g) Merchant iron, Grade "B," shall bend cold 180 around a 
diameter equal to the thickness of the tested specimen, without fracture 
on outside of bent portion. 

(h) Merchant iron, Grade "C," shall bend cold 180 around a 
diameter equal to twice the thickness of the specimen tested, without 
fracture on outside of the bent portion. 

6. The four classes of iron when tested as described in paragraph 
No. n, shall conform to the following hot bending tests: 

(i) Stay-olt iron, shall bend through 180 flat on itself, without 



SPECIFICATIONS FOR WROUGHT-IRON. 397 

showing cracks or flaws. A similar specimen heated to a yellow heat 
and suddenly quenched in water between 80 and 90 F. shall bend, 
without hammering on the bend, 180 flat on itself, without showing 
cracks or flaws. 

(/) Merchant iron, Grade "A," shall bend through 180 flat on 
itself, without showing cracks or flaws. A similar specimen heated 
to a yellow heat and suddenly quenched in water between 80 and 
90 F. shall bend, without hammering on the bend, 180 flat on itself, 
without showing cracks or flaws. A similar specimen heated to a bright- 
red heat shall be split at the end and each part bent back through an 
angle of 180. It will also be punched and expanded by drifts until 
a round hole is formed whose diameter is not less than nine-tenths of 
the diameter of the rod or width of the bar. Any extension of the 
original split or indications of fracture, cracks, or flaws developed by 
the above tests will be sufficient cause for the rejection of the lot rep- 
resented by that rod or bar. 

(k) Merchant iron, Grade "B," shall bend through 180 flat on 
itself, without showing cracks or flaws. 

(/) Merchant iron, Grade "C," shall bend sharply to a right angle, 
without showing cracks or flaws. 

7. Stay-bolt iron shall permit of the cutting of a clean sharp thread 
and be rolled true to gauges desired, so as not to jam in the threading 
dies. 

TEST PIECES AND METHODS OF TESTING. 

8. Whenever possible, iron shall be tested in full size as rolled, to 
determine the physical qualities specified in paragraphs Nos. 2 and 3, 
the elongation being measured on an eight inch (8") gauged length. 
In flats and shapes too large to test as rolled, the standard test specimen 
shall be one and one-half inches (ii") wide and eight inches (8") 
gauged length. 

In large rounds, the standard test specimen of two inches (2") 
gauged length shall be used; the center of this specimen shall be half- 
way between the center and outside of the round. Sketches of these 
two standard test specimens are as follows: 



39* APPLIED MECHANICS. 



. 4*: --- \ 




I " 



jt -18-about \ 

PIECE TO BE OF SAME THICKNESS AS T&E PLATE. 



9. Nicking tests shall be made on specimens cut from the iron as 
rolled. The specimen shall be slightly and evenly nicked on one side 
and bent back at this point through an angle of 180 by a succession of 
light blows. 

10. Cold bending tests shall be .made on specimens cut from the 
bar as rolled. The specimen shall be bent through an angle of 180 
by pressure or by a succession of light blows. 

11. Hot bending tests shall be made on specimens cut from the 
bar as rolled. The specimens, heated to a bright red heat, shall be 
bent through an angle of 180 by pressure or by a succession of light 
blows and without hammering directly on the bend. 

If desired, a similar bar of any of the four classes of iron shall be 
worked and welded in the ordinary manner without showing signs of 
red shortness. 

12. The yield -point specified in paragraph No. 2 shall be deter- 
mined by the careful observation of the drop of the beam or halt in 
the gauge of the testing-machine. 



TESTS OF COMMANDER BEARDSLEE. 399 

FINISH. 

13. All wrought-iron must be practically straight, smooth, free 
from cinder spots or injurious flaws, buckles, blisters or cracks. 

In round iron, sizes must conform to the Standard Limit gauge 
as adopted by the Master Car Builders' Association in November, 

1883. 

INSPECTION. 

14. Inspectors representing the purchasers shall have all reason- 
able facilities afforded them by the manufacturer to satisfy them that 
the finished material is furnished in accordance with these specifications. 
All tests and inspections shall be made at the place of manufacture 
prior to shipment. 



TESTS OF COMMANDER BEARDSLEE. 

One of the most valuable sets of tests of wrought-iron is that 
obtained by committees D, H, and M of the Board appointed 
by the United States Government to test iron and steel; the 
special duties of these committees being to test such iron as would 
be used in chain-cable, and the chain-cable itself. The chairman 
of these three committees, which were consolidated into one, was 
Commander L .A. Beardslee of the United States Navy. The 
full account of the tests is to be found in Executive Document 
98, 45th Congress, second session; and an abridged account of 
them was published by William Kent, as has been already 
mentioned. 

The samples of bar-iron tested were round, and varied from 
one inch to four inches in diameter. 



AOO APPLIED MECHANICS. 

Certain conclusions which they reached refer to all kinds- 
of wrought-iron, and will be given here before giving a table of 
the results of the tests. 

i. Kirkaldy considers the breaking-strength per square 
inch of fractured area as the main criterion by which to deter- 
mine the merits of a piece of iron or steel. Commander 
Beardslee, on the other hand, thinks that a better criterion is 
what he calls the "tensile limit;" i.e., the maximum load the 
piece sustains divided by the area of the smallest section when 
that load is on, i.e., just before the load ceases to increase in 
the testing-machine. 

2. Kirkaldy had already called attention to the fact that 
the tensile strength of a specimen is very much affected by its 
shape, and that, in a specimen where the shape is such that 
the length of that part which has the smallest cross-section is 
practically zero (as is the case when a groove is cut around 
the specimen), the breaking-strength is greater than it is when 
this portion is long ; the excess being in some cases as much 
as 33 per cent. 

Commander Beardslee undertook, by actually testing speci- 
mens whose smallest areas varied in length, to determine what 
must be the least length of that part of the specimen whose 
cross-section area is smallest, in order that the tensile strength 
may not be greater than with a long specimen. The conclusion 
reached was, that no test-piece should be less than one-half inch 
in diameter, and that the length should never be less than four 
diameters ; while a length of five or six diameters is necessary 
with soft and ductile metal in order to insure correct results. 
The following results of testing steel are given in Mr. Kent's 
book, as confirming the same rule in the case of steel. The 
tests were made upon Bessemer steel by Col. Wilmot at the 
Woolwich arsenal. 



TESTS OF COMMANDER BEARDSLEE. 



4OI 







Tensile Strength. 


Pounds per 
Square Inch. 






Highest 
Lowest 


162974 






Average . . . . . 
Highest 
T owpst 


153677 
123165 






Average 


10 3 2 55 
114460 



3. Commander Beardslee also noticed that rods of certain 
diameters of the same kind of iron bore less in proportion than 
rods of other diameters ; and, after searching carefully for the 
reason, he found it to lie in the proportion between the diam- 
eter of the rod and the size of the pile from which it is 
rolled. The following examples are given : 

ij-in. diameter, 6.62% of pile, 56543 Ibs. per sq. in. tensile strength. 



If 
I* 
If 
'I 



8.i8% 


tt 


56478 ' 


9.90% 


" 


54277 " 


11.78% 


tt 


5355 " 


7.68% 


" 


56344 " 


8.90% 


tt 


55018 " 


10.22% 


(i 


54034 " 


II-63% 


(i 


51848 " 



He therefore claims, that, in any set of tests of round iron, 
it is necessary to give the diameter of the rod tested, and not 
merely the breaking-strength per square inch. 

4. He gives evidence to show, that if a bar is under-heated, 
it will have an unduly high tenacity and elastic limit ; and that 
if it is over-heated, the reverse will be the case. 



402 



APPLIED MECHANICS 



5. The discovery was made independently by Commander 
Beardslee and Professor Thurston, that wrought-iron, after 
having been subjected to its ultimate tensile strength without 
breaking it, would, if relieved of its load and allowed to rest, 
have its breaking-strength and its limit of elasticity increased. 

His experiments show that the increase is in irons of a 
fibrous and ductile nature, rather than in brittle and steely 
ones ; hence the latter class would be but little benefited by 
the action of this law. 

The most characteristic table regarding this matter is the 
following : 

EFFECT OF EIGHTEEN HOURS' REST ON IRONS OF WIDELY DIFFER 
ENT CHARACTERS. 







I 




Ultimate Strength 


i 
i 




per Square Inch. 


i 






"O 1 




First 


Second 


JxGXEl&rJCS* 




Strain. 


Strain. 




Boiler iron . . . 


48600 


56500 


Not broken. 


tt tt 


49800 


57000 


Broken \ 


tt (t 


49800 


58000 


Broken 1 Average gain, 


H ft 


48100 


54400 


Broken f 15.8%. 


(I 11 


48150 


5555 


Broken J 


Contract chain iron, 


50200 


54000 


Broken *j 


(t ii 


50250 


53200 


Not broken 1 Average 


d ti tt 


50700 


553oo 


Not broken j* gain, 


n t( ft 


49600 


52900 


Not broken j 6.4%. 


ft tt a 


51200 


52800 


Not broken ) 


Iron K . . . . 
ft ft 


58800 64500 

59000 ! 65800 


Broken ^ 
Broken I Avera g e ^ 


ft tt 


56400 j 60600 


Broken J 9-4%- 






j 



CHAIN CABLE. 403 



233. Chain Cable. The most thorough set of tests of the 
strength of chain cable is that made by Commander Beardslee 
for the United-States government, an account of which may be 
found either in the report already referred to, or in the abridg- 
ment by William Kent. 

In this report are to be found a number of conclusions, 
some of which are as follows : 

i. That cables made of studded links (i.e., links with a 
cast-iron stud, to keep the sides apart) are weaker than open- 
link cables. 

2. That the welding of the links is a source of weakness ; 
the amount of loss of strength from this cause being a very 
uncertain quantity, depending partly on the suitability of the 
iron for welding, and partly on the skill of the chain-welder. 

3. That an iron which has a high tensile strength does not 
necessarily make a good iron for cables. Of the irons tested, 
those that made the strongest cables were irons with about 
51000 Ibs. tensile strength. 

4. The greatest strength possible to realize in a cable per 
square inch of the bar from which it is made being 200 per 
cent of that of the bar-iron from which it was made, the cables 
tested varied from 155 to 185 per cent of that of the bar- 
iron. 

5. The Admiralty rule for proving chain cables, by which 
they are subjected to a load in excess of their elastic limit, 
is objected to, as liable to injure the cable : and the report 
suggests, in its place, a lower set of proving-strengths, as given 
in the following table ; the Admiralty proving-strengths being 
ilso given in the table. 

In these recommendations, account is taken of the different 
proportion of strength of different size bars as they come from 
th: rolls, also no proving-stress is recommended greater than 
50 per cent of the strength of the weakest link, and 45.5 per 
cent rf the strongest ; v/hereas in the Admiralty tests, 66.2 



404 



APPLIED MECHANICS. 



per cent of the strength of the weakest, and 60.3 per cent of 
the strongest, is sometimes used. 

For the details of this investigation, see the report, Execu- 
tive Document No. 98, 45th Congress, second session, or the 
abridgment already referred to. 



Diameter of 
Iron, 
in inches. 


Recommended 
Proving-Strains. 


Admiralty 
Proving-Strains. 


Diameter of 
Iron, 
in inches. 


Recommended 
Proving-Strains. 


Admiralty 
Proving-Strains. 


2 


121737 


161280 


I* 


66138 


83317 


lit 


114806 


I5I357 


If 


60920 


76230 


I 8 


108058 


I4I75 


i-iV 


55903 


69457 


lit 


101499 


I3 2 457 


li 


5I08 4 


63000 


If 


95128 


123480 


IA 


46468 


56857 


Itt 


88947 


114817 


It 


42053 


5 I0 3 


If 


82956 


106470 


nV 


37820 


455*7 


I* 


77159 


98437 


i 


33 8 40 


40320 


it 


7I55 


90720 









While steel long ago displaced wrought-iron for boiler-plate, 
and while steel I beams, channel-bars, angle-irons, and other 
shapes, as well as eye-bars, have, of late years, displaced 
wrought-iron to a very great extent, nevertheless wrought-iron 
is still very extensively used, and for a great variety of struc- 
tural purposes. 

For wrought-iron to be used in construction, ductility, 
homogeneity, and often weldability are the great desiderata, 
together with as large a tensile strength as is consistent with 
these. As to the requirements made by different engineers for 
wrought-iron for structural purposes, the minimum tensile 
strength called for varies from about 46000 to about 50000 
pounds per square inch, with ultimate elongations varying from 
15$ to 30$ in 8 inches, according to the purpose for which it is 
wanted. It is also very common, when good iron is wanted, 



CHAIN CABLE. 



405 



to insist that it shall not be made of scrap. The following 
tables of tensile tests of wrought iron of various kinds will 
show what results can be obtained. 



Norway Iron. 


Burden's Best. 




od 


c 


c 






e"a 


d 


a 






o"". 


-.'". 


4> 






o"" 


4-r"! 


-8 




u . 


1& 


o a 


1 5 


If 


it 


i 


1? 

o a 


o u 
c w 
.2 
5 jf 


" >> 
& G 


11 


Is' 


u 


|| 


"5 ! 


i! 


u 


11 




Is 


5 


s 







s 


5 


s 


5 


OS 


s 


75 


48390 


23620 


62.6 


30090000 


.76 


53566 


27554 


57-6 


29175000 


75 


46340 


21160 


62.7 


30780000 


75 


50023 


26030 


49-8 


30643000 


75 


48280 


28030 


62.6 


29020000 


76 


47724 


25350 


47-6 


30310000 


77 


45160 


20400 


68.8 


27388000 


77 


46772 


24700 


45-2 


28347000 


75 


46063 


19240 


68.6 


27666000 


77 


46600 


22550 


46.2 


29528000 


77 


44490 


20510 


67-5 


28452000 


77 


47395 


22550 


46.2 


28347000 


74 


43233 


22079 


70.5 


29026000 


77 


47963 


22695 


48.6 


29475000 


75 


43470 


19400 


75-5 


26700000 


77 


47860 


26948 


46.4 


26948000 


73 


38950 


22030 


72.3 


30140000 


77 


475CQ 


26927 


42-3 


28435000 


74 


43240 


21970 


75-2 


27726000 


76 


47610 


23036 


53- l 


29551000 


74 


44564 


21970 


73.8 


28663000 


77 


49238 


22725 


49-2 


27470000 


74 


43860 


19658 


75-o 


18000000 


76 


50037 


27700 


53-6 


29251000 


i .00 


41620 


15560 


7-3 


27295000 


76 


48538 


27224 


48.8 


29355000 


75 


42215 




68.6 


29292000 


76 


50060 


23201 


S3- 


31028000 


75 


42033 


19239 


62.4 


29729000 


.76 


49143 


23240 


5-4 


30438000 


.76 


4'574 


14328 


69-5 


27450000 


.76 


48655 


23414 


49.6 


30062000 


.76 
75 


41574 
426^6 


16531 
19240 


68.7 
59- 


29098000 
31785000 


76 
76 


47220 
47090 


22880 
23020 


53-4 

54-i 


29969000 
33657000 


75 


41875 


16978 


70.1 


30487000 


.76 


49690 


27480 


53-3 


29614000 


75 


43396 


19112 


59-3 


28000000 


.76 


47430 


22950 


51-8 


29443000 


74 


39210 


15216 


73' 2 


30294000 


76 


4795 


23000 


57-o 


29504000 


74 




12603 


70.5 


28810000 


.76 




22892 


45-8 


28779000 


74 


39896 


15187 


69.7 


31153000 


77 


49411 


18420 


46.6 


30112000 


74 


39156 


16123 


76.4 


29807000 


.76 


49660 


23186 


51-3 


30160000 


74 


41030 


17490 


69.8 


29310000 


76 


48055 


20940 


56.7 


28809000 


75 


41180 


18000 


72.5 


31073000 


77 


49026 


22578 


40.5 


27292000 


74 


42320 


19660 


68.0 


30834000 


76 


47220 


23060 


51-2 


33710000 


74 


43913 


198:53 


69.8 


26970000 


.76 


5 OI 49 


20940 


41.5 


27450000 


74 


42102 


191.81 


78.3 


29127000 


75 


48553 


23767 


74-3 


31124000 


74 


39698 


17638 


70-5 


30023000 




49350 


21503 


66.5 


31793000 


73 


43187 


17846 


68.6 


28553000 


.76 


50083 


20940 




29097000 


73 


40669 


17820 


73-8 


30159000 


76 


47019 


23140 


5 1 -4 


29978000 


73 


39348 


16593 


69-3 


29518000 


76 


47504 


20942 


53-2 


28527000 


73 


39671 


12987 


78.1 


28861000 


76 


47747 


20942 


49-5 


29874000 


75 


39951 


16886 


77.2 


30020000 


.76 


50927 


23453 


46.9 


28350000 


74 


41093 


16400 


74- 1 


28634000 


75 


51269 


21182 


51 -9 


32551000 


74 


40192 


14053 


76.3 


28627000 


75 


50930 


23770 


53-7 


29293000 


73 


44470 


16844 


73-4 


31114000 


76 


50083 


23146 


45 - 


29097000 


74 


41940 


17523 


78.5 


29373000 


75 


48168 


23767 


55-6 


29879000 


74 


42531 


16449 


7.S 


30410000 


76 


49500 


26500 


55-o 


3 i 600000 




75 


48400 


27200 


46.2 


28700000 




75 


47600 


27200 


50.1 


29300000 




77 


47200 


23600 


56.1 


27800000 




.76 


46700 


24200 


41.8 


29700000 




77 . 


45600 


23600 


54-5 


28200000 



406 



APPLIED MECHANICS. 



Refined Iron. 


Wrought-iron Wire. 




|a 


a 


c 
u 








"rt'.S 


,H 


C 
V 


% 




J a- 


f sr 


o 


"- 1 x 






Jj cr 


'iif 


c <> 
"" u 


S <n 


fc 


!* 




8. 


o ^ 

01 JO 


Kind of Wire. 


feg 


i 




.2 s - 


^~S 


fjj 


S ^ 


o a, 


2 


f i 




' i> 
By 


8 ^ 


.H ^~ 


is 


"5.ti 3 


a 


5 


|| 


"S-< 


"83 




E9 


g.S 


1 j 


"^ <5 




5 


5"" 


5 


& 






5"* 





w~~ 


(S 


s* 1 




56270 


28293 


33 *9 


28618000 


Annealed wire 






^3800 


g, j 




77 


53450 


28990 


22.0 


26997000 


Annealed wire 


ii3 


61500 


4jJCH_XJ 


03. i 
75- 





.76 


55880 


29758 


33-5 


27711000 


Annealed wire 


135 


61100 


39800 


49-4 


23000000 


77 


53850 


29370 


33-3 


28718000 


Annealed wire 


.136 


59500 


39200 


71.2 


25500000 


77 


52770 


33722 


14.8 


27355000 


Annealed wire 




45100 


35800 


76.8 


23500000 


74 


52770 


28829 


33-5 


29273000 


Annealed wire 


136 


59800 


34000 


72.7 




77 


51320 


29294 


22 6 


28082000 


Annealed wire 


J 35 


62400 








77 
74 



53778 , 
4888* 


27138 
28822 


25-4 
13 8 


28659000 
28137000 


Common wire 
Common wire 


. no 
.109 


90900 

103000 


64000 


.1 


30300000 
27500000 


75 


49240 


28190 


14.0 


27520000 


Common wire 


. no 


104000 


60000 


51.0 


22900000 


75 


50190 


30590 


17.8 


26237000 


Common wire 


"3 


93700 


68200 


60.5 


25200000 


77 
75 


51460 

47495 


29256 
30387 


22.4 

12.2 


25680000 
27613000 


Common wire 
Common wire 


.080 
.080 


113000 
113000 


45800 
56700 


41.9 
51.0 


27000000 
26500000 




48352 


30574 


17.3 


27177000 


Common wire 


.079 


IT2OOO 


54300 


53-3 


26600000 


.76 


47I5I 


25982 


75-4 


21628000 


Common wire 


079 


120000 


73600 


28.1 


26100000 


77 


5035 1 


35720 


25-3 


27477000 


Common wire 


.079 


IO9OOO 


54300 


40.4 


26400000 


75 


48202 


28521 


14.7 


27888000 


Common wire 


.080 


98300 




43-8 


27100000 


75 


50703 


30558 


13.0 


23713000 


Annealed wire 


.081 


99600 




61 .9 


26600000 


75 


49223 


30517 


J5-2 


27126000 


Annealed wire 


.082 


93500 


50400 


64-3 




75 


49120 


29000 


17.8 


28290000 


Annealed wire 


.082 


86300 


50400 


68.5 


27100000 


75 


47060 


31700 


j c 4 




Annealed wire 


.082 


89900 


54OOO 


ej 7 


24900000 


:3 


47830 
51300 


29400 
26000 


17.8 

29.1 


29290000 
30100000 


Annealed wire 
Annealed wire 


.082 
.082 


97100 
93500 


57600 
39600 


55-0 


26100000 


76 


52400 


35000 


29.1 


25400000 


Annealed wire 


.082 


71000 


50400 


67 .'i 


27000000 


:8 


53400 

52IOO 


29000 
26000 


24.9 
29. i 


28200000 


Common wire 
Annealed wire 


.167 
.081 


57200 


45100 


65.6 
60.4 







CJTOO 


29000 


24.9 




Annealed wire 


.082 


959 


553 






76 


04- l *" K -' 

51500 


26500 


24.6 


33100000 


Common wire 


.163 


935OO 
67400 


40100 


56-9 





76 


52500 


242OO 


22 3 




Common wire 




6l5OO 


.___- 


52.8 




77 


77300 


34400 


26.5 


26800000 


Piano wire, 


3 










75 
75 


53100 
52900 


31700 


24.9 
27.2 


27200000 
26100000 


No. 13 
Piano wire, 


031 


345000 






29500000 






76 


51600 


28700 


22 . 3 


26000000 


No, 23 


.048 


ofi^enn 






ononoooo 


4-ooX 












j " 


1. 01 


40700 




14.4 







76 


53100 


28700 


24.6 







.76 


52200 


33100 


26.9 


31000000 




75 


50100 


31700 


22.6 


28700000 




76 


49400 


26500 


26.9 






.02 


50300 


31800 


16.9 


27200000 




.01 


47000 


32500 


20.6 


27700000 




.01 


50400 


30000 


25.8 


28300000 




.02 


49600 


31800 


23-9 


26500000 




.OI 


50200 


30000 


32-5 


26800000 




.02 


50500 


29400 


30.6 


26200000 




.01 


51400 


30000 


29.2 


28300000 




.02 


50400 





20.4 


28200000 




.02 


50200 


31800 


I 5- 1 


27200000 




.01 


48100 


30000 


32-5 


27700000 




.OI 


50600 


30000 


27-5 


25800000 




77 


48600 


25800 


52.6 


28000000 




74 


53900 


27900 


38.6 


29700000 





74 


54000 


25600 


18.0 


29700000 




76 






AI 8 






.70 

.76 


53500 


30900 


41.0 

33-4 


27600000 





TENSILE TESTS OF WROUGHT IRON. 



407 



In Heft IV (1890) of the Mitt. d. Materialpriifungsanstalt 
in Zurich is an account of a set of tensile tests of wrought-iron 
and mild-steel angles, tees, and channels. The following is a 
summary of his results for wrought-iron shapes : 



ANGLE-IRONS. 









a' 
















rt 






g 








u 


J w"o 


- v-"5 


*j' u> S 


SU 


Modulus of 


5 


Dimensions, 
Inches. 


V 

a 


I|s 

X g 3 


*ls 


(2-S jj 

*O 3 2 


Ifc 
1* 


Elasticity, 
Pounds per 
Square Inch. 


3 




& 




J o a" 


13 o o* 
j^CXyj 


T3 u 








Lbs. 












2 


2.76 X 2.76 X 0.31 


l6 -53 


49910 


25020 


37680 


9-5 


28824000 


4 


2.76 X 2.76 X 0.51 


27.62 


49060 


20190 


32560 


II. 7 


28070000 


6 


3-54 X 3-54 X 0.35 


26.21 


50620 


253 10 




15.8 


28269000 


8 


3- l 5 X 3-54 X 0.55 


38-51 


5II90 


25310 


32000 


l6. 4 


27786000 


1O 


4.13 X 4'*3 X 0.47 


35.48 


49200 


28010 


33*30 


12. O 


28537000 


12 


4.13 X4-13 Xo.6 7 


55-24 


46070 


22750 


32280 


10.2 


28554000 


M 


5.12 X 5.12 X 0.67 


62.09 


47780 


22610 


3043 


12. 


27985000 


16 


5.12 X 5-12 X 0.87 


102.61 


48490 




31140 


12.3 





















TEE-IRONS. 



3 


3". 60 X 3" -35 


22.88 


5 2 470 


25880 


3768o 


14.2 


27615000 


4 




" 


49200 


23610 


34700 


15-5 


27672000 


7 


3". 94 X 3"-94 


31.75 


51760 


21610 


38820 


11.7 


27857000 


8 






54040 


18630 


354*0 


21.3 


27743000 


9 

10 


5". 90 X S'^94 


46.^91 


53 6l o 
52900 


23600 
22890 


36970 
3598o 


19.0 
14-5 


27402000 
28255000 



CHANNEL-IRONS. 



I 


4.13 X 2.56 


28.43 


50630 


23329 


35120 


15-9 


27544000 


a 




44 


49200 


24170 


33700 


12.7 


27885000 


4 


4.13 X 2.64 


31.65 


543 2 o 


23040 


36690 


20.6 


27772000 


5 


6.93 X 2.83 


48.89 


51760 


24460 


35550 


19.9 


27658000 


6 


" " 


" 


54610 


23320 


34270 


17-5 


27999000 


7 
8 


6.93 X 2.^91 


S4 ti 43 


51900 
52900 


19620 
24320 


35690 
30860 


20.4 

14.5 


27487000 
29663000 


9 


8.46 X 3-35 


85.68 


52050 


22330 


3456o 


20.9 


27701000 


10 


4k 4*T 


'* 


5347 


24170 


36260 


17.0 


28710000 


la 


8.46 X 3-50 


92.33 


52760 


32040 


34840 


ix. 9 


28568000 



408 



APPLIED MECHANICS. 



TENSILE TESTS MADE SUBSEQUENTLY AT THE WATERTOWN 

ARSENAL. 

Here will next be given, in tabulated form, the results of a 
number of tensile tests made on the government machine at the 
Watertown Arsenal. 

The following tables of results on rolled bars, from the Elmira 
Rolling-Mill Company (mark L) and from the Passaic Rolling- 
Mills (mark S), are given in Executive Document 12, ^.Jth Con- 
gress, 1st session, and in Executive Document /, <ffth Congress, 
2d session. 

SINGLE REFINED BARS. 



1 

c 



^ 

rt 
% 


Sectional Area, in 
square inches. 


. U 

id 

J 8, . 
2 rf-5 
a a 

w 


Ultimate Strength, 
in Ibs., per 
Square Inch. 


$> 

c 

'*? 

S-5 
I' S 


Contraction of 
Area, %. 


Appearance of 
Fracture. 


Modulus of Elas- 
ticity at Load of 
20000 Lbs. per 
Square Inch. 




1^ 
ta 


t* 

II 


L i 


3.06 


28500 


52710 


18.4 


33-3 


95 


5 


26981450 




L 2 


3 06 


29500 


53630 


16.4 


36.0 


92 


8 


27826036 




L 3 


3.06 


29000 


52090 


21.4 


34.6 


95 


5 


28419182 




L 4 


3.06 


29000 


5 '440 


15.0 


20.3 


90 


IO 


30888030 




L 5 


6.46 


27500 


505 00 


H-5 


27.6 


95 


5 


27826036 




L 6 


6.40 


27500 


50530 


17-3 


22.3 


70 


30 


27118644 




L 7 


6-39 


27000 


50200 


18.0 


22.5 


95 


5 


27444253 




L 8 


3- 2 4 


- 


51667 


22.0 


36.0 


70 


3 


28318584 


Round. 


L 9 


3.20 


- 


50844 


I6. 3 


22.0 


'5 


85 


27972027 


u 


L 10 


3.20 


- 


53062 


2I.O 


4O.O 


95 


5 


28119507 





S ii 


3.08 


28500 


48640 


'3-3 


24-3 


100 


Slightly 


27586206 




S 12 


3.08 


28000 


50390 


16.9 


35-i 


100 


o 


27586206 




s 13 


3-05 


28500 


47050 


9.0 


22.0 


IOO 


o 


27874564 




S i S 


6.40 


26000 


49700 


17.1 


19.2 


85 


15 


29906542 




S 16 


6.40 


24000 


49280 


15-7 


177 


85 


!5 


26490066 




S 17 


6.41 


24500 


48740 


14-3 


17.3 


80 


20 


28119507 




S 18 


3-i7 


24600 


49680 


19-5 


32.0 


IOO 


Slightly 


27972027 


Round. 


S 19 


3-17 


25870 


49338 


18.3 


38.0 


IOO 





29357798 


" 
















Cinder 






S 20 


3-i7 


24920 


48864 


18.4 


37-o 


IOO 


at centre 


27729636 





DOUBLE REFINED BARS. 



409 



DOUBLE REFINED BARS. 



Mark on Bar. 


Sectional Area, in 
square inches. 


S 

~- H, 

! j? 

3 I . 

u - -c 

llJ 

H 


Ultimate Strength, 
in Ibs., per 
Square Inch. 


S 

g 

g^ 
11 

I -s 
w 


Contraction of 
Area, %. 


Appearance of 
Fracture. 


Modulus of Elas- 
ticity at Load of 
20000 Ibs. per 
Square Inch. 




1 ^ 
fa 


si ^ 

\ i 

u - 


L 20 1 3.06 


29000 


5356o 15-3 


37-9 


100 


o 


2 7633 8 5 l 




L 2O2 


3-03 


30000 


52650 16.2 


20.6 


85 


15 


34042553 




L 203 


3.06 


32500 


53500 1 16.5 


27-5 


TOO 





28169014 




L 204 


3.06 


32500 


54480 15.4 


24.8 


TOO 


o 


29090909 




L 205 


6-33 


27000 


51230 17.8 


24.2 


80 


20 


28119507 




L 206 


6-34 


27500 


50500 17.6 


21. 1 


100 


Slightly 


29629629 




L 207 


6-34 


27000 


51030 ; 21.4 


31-9 


100 


o 


27826086 
















C^ up- 






L 208 

i 


3.20 





50156 '22.7 


43 


IOO 


shaped 


28021015 


Round. 


L 209 


3.20 


- 


49937 22.6 


45-o 


TOO 


" 


28622540 


" 


L 210 


3.20 


- 


50188 19.9 


43-o 


IOO 


" 


28985507 


" 


S 211 


3-05 


29500 


5II50 J22.0 


3i-5 


IOO 


o 


32989690 




S 212 


3-5 


28500 


5IIIO 22.0 


36.1 


IOO 


o 


25559105 




S 213 


3-" 


29500 


51860 22-5 


39-2 


IOO 


o 


26446280 




S 215 


6.31 


27500 


50980 


19.1 


23.6 


95 


5 


29357798 




S 216 


6.38 


27OOO 


50770 


20.7 


29.6 


IOO 





28268551 




S 217 


6-33 


27000 


51340 


19-3 


35-2 


IOO 


o 


28070175 




S 218 


3-i7 


24610 


50631 


20.4 


41.0 


IOO 


o 


28622540 


Round. 
















Cup- 






S 219 


3-i7 





50915 


25-5 


44.0 


IOO 


shaped 


28268551 


" 


S 220 


3-^7 





50205 


23-7 


44.0 


IOO 




28070175 





The moduli of elasticity had not been computed in the 
report, but have been computed in these tables from the elon- 
gations under a load of 20000 Ibs. per square inch in each case, 
as recorded in the details of the tests. 

In these reports are also to be found tensile tests of iron 
from other companies, as the Detroit Bridge Company, the 
Phoenix Company, the Pencoyd Company, etc. Some of these 



4io 



APPLIED MECHANICS. 



tests were made to determine the effect of 


rest upon the bar 


after it had been strained to its ultimate 


strength, also to 


determine the strength after 


annealing. The following table 


shows these latter results : 










o-o 




5 




8| . 




a 




v c-2 




& 




li! 






c a c c 

rt rt rt rt 


1 


S S 


o 


far hfl ** M M 


hi) 


& fci hli 


u 


5^ . (c" g.^ vC i5C 


* w 


iJC "*C *C <3 


fits 


3 R>* 3 8. J 2 ^ v 


&*!,: 


8* vg^ &* g 


p 


en w ^ ^S CB' "> g" 

gjftif E&8 2'2o 

0.0*5 0^2*3 iJ 2^ ^ 3J3 1 ; 


- 10 wwrt i 

!i III il 


& ,?3. . & g 

2 2 S" 1 I S" ? 

ja"3 ja g ^ .0 3 a 




o 


fc Q 


___ E E ' 


o -d 'uopoas 


. , 




. 










B9JV JO 


M 1 VO CO M * tx 


1 ^ S S co 


* * " " 


3 'd 'qiSuaq 
\vu\BuQ cuoaj 
uoiiBxfuo^g 


M-^-Min^-O coco 

MM MM HM H M 


ro vo co N sn 


C\ O^ VO H M 

M H M C1 ti 


d 5 ti 


S> 8 2 vg 8>S\ 8 (8 

MO HONCOCO OOv 


8 5- ft & 5- vS 

WVO VO H IN 10 


R. 8 8 v8 S 

O S> ij- in co o 


O (J "" S 5 ri 

** & 


n vo m vo invo in n 


co * in vo * in 


vo in vo n vo vo 


a .H -- v c 

{/) tC w Q P * H ~ I 


S O Q O Q 
o o ^ o o o 

o I >n * I m m 


J. 8 8 | 


O Q O 

S o o 

0> Jo f in oo | 


o s^lcfl 


I? ^ S5 -S ST ^ 


in N 


\ft in co in 


jlta 


S 1 . ff i 'Si ^ i 


^0 vg ^ =0^ 


CO vo tv 

rx i I o vo vo 


w? - "^cJf 


to ro ro vo 


in co ei N co 


M ro N 


*| ^g d 


181*81 


ET S S 8.8 


jn w M ^t* w 


u c c"" 1 
H j I" 1 


M M M 




" 


K * 5. 

T3 C 


S , o" . S 1 ! 5* 


R ? <? <2 ? 


<^ ff , S 1 * o? 




m ^_ fr> _^ ^ 


* ro M ro 


M M ro d M 


Condition of Bar 
when Tested. 


:p 11 :, : i 

11 - a l -I 2 

|| 'l| "S ' 

llgl^ll? 1 s 
Hitiln * s 

O&< OPS Ooi O OS 


og ' ' -a" S ' ' 

& ; IJ' ; 

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g-a . . fc s-o . . 

si . _ r-3-_ 
il 4 nl 4 

o.h .SP w c.h .5P 

s rtrt o E rtrt o 

/ 


1 !!!i i i 

= !ls s s 

s 2~s - s -5 

73 *?, T) 1- c -0 -0 

8 818558 S 

4) OJCJDrt'CvD OJ 

pi D: as o oi rt 


i g; i I 


s R ** R 


oo oo' P. oo oo" 
co co s co co 



DOUBLE REFINED BARS. 



411 



Some tests were made to determine the values of the 
modulus of elasticity of the same iron for tension and for 
compression ; and these were found experimentally to be 
almost identical, as was to be expected. For these tests the 
student is referred to the reports themselves ; and only cer- 
tain tests on eye-bars of the Phcenix Company will be 
appended here. 



Arsenal 
Number. 


Outside 
Length, 
Inches. 


Gauged 
Length, 
Inches. 


Sectional 
Area, 
Sq. In. 


Ultimate 
Strength, 
Pounds 
per sq. in. 


Contraction 
of Area at 
Fracture, 
per cent. 


511 


67.75 


50 


1.478 


40600 


16.8 


513 


67.80 


50 


1.940 


39480 


13-9 


518 


96.05 


75 


5-103 


46720 


8.1 



Quite a number of tests of the iron of different American 
companies are to be found in the "Report on the Progress of 
Work on the Cincinnati Southern Railway," by Thomas D. 
Lovett, Nov. i, 1875. 

For these the student is referred to the report named. 



WROUGHT-IRON PLATE. 



The following table contains some tests of wrought-iron 
plate and bars made on the Government testing-machine at 
Watertown in 1883 and 1884 for the Supervising Architect at 
Washington, D.C. 



412 



APPLIED MECHANICS. 








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Strengt 



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p 



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NONONONONONONONONONONONONONONOVONONONONONONONONOVONO NONONONO 

WROUGHT-IRON AND STEEL EYE-BARS. 

In the report of the Government tests for 1886 is given the 
following table of tensile tests of wrought-iron eye-bars. The 
wrought-iron ones were furnished by the General Manager of 
the Boston and Maine Railroad, and the steel ones by the 
Chief Engineer for the American Committee of the Statue of 
Liberty. 



CO M PRESS IV E STRENGTH OF WROUGHT-IRON. 413 



WROUGHT- IRON EYE-BARS. 



Dimensions. 




K 


Elongation. 


1 


O 

"*"* 0) 


01 in . 


Fracture. 


|-s , 






*| 


bcx 




i ( 


*o 


J3S 
l 


O C oj 






$ V 0> 






. HH 


SHH 


T) 


u.S 


o 


^co 


^.^ 






fill 





1 


"!! 


W g 

r=3 


if 


t^s 


"o 

1 


W3 4; 


ia? 


G 
_0 


Appearance. 


c2 


8 


H 


He/2 


cw 


Ojj 


Isl 




-d^^c 


'x-2 


1 




J 


5 


H 


2 


H 


c 


J 




S 


S 


3 




Ins. 


Ins. 


Ins. 


Lbs. 


Lbs. 


% 


% 


% 


Lbs. 


Lbs. 






238.55l5.oo 


1.14 


22456 


45105 


II .7 


ii .6 


31 .2 


28037000 


52763 


Stem 


Fibrous, traces 
























of granulation. 


238.60 


5.00 


1.15 


22610 


44540 


9.4 


9.4 


29.6 


28125000 


50588 


" 


Fibrous, 70% 
























Granular, 30% 


238.57 


4-99 


1.14 


21790 


43320 


7-8 


3.0 


26.4 


27950000 


48492 


1 ' 


Fibrous, 70% 
























Granular, 30% 


238.64 


5.00 


1. 16 


22410 


39550 


5 i 


4-8 


9.8 


27355000 


54013 


1 ' 


Fibrous, 70% 
























Granular, 30% 


238.62 


6.05 


i .44 


19750 


43260 


12.05 


I 2 .06 


24.8 


28800000 


43166 


1 ' 


Granular, 80% 
























Fibrous, 20% 


238.62 


6.05 


i .44 


22730 


42020 


6.5 


6.6 


19. 2 


28301000 


41929 


" 


Granular 5% at 
























one edge, fi- 
























brous for bal- 
























ance of fracture. 



The gauged length of the bars was 180 inches. The moduli 
of elasticity computed between 5000 and 10,000 pounds per 
square inch. 

COMPRESSIVE STRENGTH OF WROUGHT-IRON. 

In regard to the compressive strength of wrought-iron, we 
may wish to study it with reference to 

1. The strength of wrought-iron columns; 

2. The strength of wrought-iron beams ; 

3. The effects of a crushing force upon small pieces not 
laterally supported ; 

4. The effects of a crushing force upon small pieces laterally 
supported. 

i. In this case it may be said that, by reference to the 
tests of wrought-iron bridge columns, the compressive strength 
per square inch of wrought-iron in masses of such sizes is given 
by the tests of the shorter lengths of such columns, i.e., by 
those that are short enough not to acquire, when the maximum 
load is just reached, a deflection sufficient to throw any appreci- 
ably greater stress per square inch on any part of the column in 
consequence of the eccentricity of the load due to the deflec- 



4H APPLIED MECHANICS. 

tion. The results thus obtained are naturally lower than we 
should expect to obtain in smaller masses. 

2. In this case the evidence that there is goes to show that 
the compressive strength is the same as in the case of i, and 
hence that it is less than the tensile strength. Indeed, the results 
of tests of full-size beams show a modulus of rupture greater 
than the compressive strength, less than the tensile strength in I 
sections, and greater in circular sections; all this being what 
would naturally be expected. 

3. If a small cylinder of ductile wrought-iron is tested with- 
out lateral . support, and with flat ends, the friction of the ends 
against the platforms of the testing-machine comes in to interfere 
with the flow of the metal; and if, besides this, the ratio of length 
to diameter is so small as to prevent buckling, then the specimen 
will gradually flatten out, and it becomes impossible to find any 
maximum load, because the area of the central part is constantly 
increasing. 

4. In this case the crushing strength per square inch that 
causes continuous flow, and also the maximum strength per 
square inch, is greater than that where the specimen has no 
lateral support. Hence follows, that in the case of wrought-iron 
rivets it is .entirely safe to allow a bearing pressure in the neighbor- 
hood of 90,000 or 100,000 pounds per square inch, according to 
circumstances. 

223. Wrought-iron Columns. Until after about 1880 
there was but little experimental knowledge on this subject beyond 
the experiments of Hodgkinson, which have furnished the con- 
stants for Hodgkinson's, and also for Gordon's formula, as already 
given in 208 and 209. 

These formula have been in very general use, and it is only 
of late years that we have been able to test their accuracy by 
tests on full-size wrought-iron columns. The disagreement 
of the formulae already referred to, with the results of the tests, 
has led to the proposal of a large number of similar formulae, 



WRO UGHT-IRON COL UMNS. 4 1 5 

each having its constants determined to suit a certain definite 
set of tests, and hence all these formulae thus proposed, which 
are, of course, empirical, and can only be applied with safety 
within the range of the cases experimented upon. 

A few of these will now be enumerated; and then will follow 
tables of the actual tests, which furnish the best means of deter- 
mining the strength of these columns ; and it would appear that 
it is these tables themselves which the engineer would wish to use 
in designing any structure. 

On the 1 5th of June, 1881, Mr. Clark, of the firm of Clark, 
Reeves & Co., presented to the American Society of Civil Engineers 
a report of a number of tests on full-size Phoenix columns, made 
for them at the Watertown Arsenal, together with a comparison 
of the actual breaking-weights with those which would have 
been obtained by using the common form of Gordon's formula 
for wrought-iron. The table is shown on page 416. 

The very considerable disagreement between the breaking- 
loads as calculated by Gordon's formula, and the actual break- 
ing-loads, led a number of people to propose empirical formulae 
of one form or another which should represent this set of 
tests, and also others which should represent some other tests 
on full-size bridge columns, which had been previously made 
in other places. 

Of these I shall only give those proposed by Mr. Theodore 
Cooper, which are as follows : 

p f 

For square-ended columns . . . -j = - -n - ry 

' + -*>) 



18000 



P f 

For phi-ended columns . . . . -j = 



7.8000 



416 



APPLIED MECHANICS. 



frlil 

- 



vO ONfOONi^toior^>- O vO VO r^ M -TJ- N t 
Tj-vo-. rorfLOO\O O^OvOOO -^-M N ro 
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1 1 1 1 



tf r^ r 



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w 1-1 M O ""If 

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^ -O ^ CO ^OSO 10 

ro ro to ro to to to 



10 II 






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^" fO O\ M i i ^ i M O i i 

N N S M 1 1 i-i 1 w M 1 



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l NMMM>-iM' l -i l i-it-c l 'O O O 1-1 p 

dddddd d do d dddd 



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WROUGHT-IRON COLUMNS. 



417 




418 



APPLLED MECHANICS. 



And he gives, for the values off, 

For Phoenix columns /*= 36000; 

" American Company's columns . . . ./"= 30000; 
" box and open columns /= 31000. 

He deduces these values of f from some tests made in 1875 
by Mr. Bouscaren, combined with those, already referred to, 
made at the Watertown Arsenal. The box and open columns 
were made of channel-bars and latticing. The tables or dia- 
grams presented to justify the formulae proposed can be found 
in the "Transactions of the American Society of Civil Engineers " 
for 1882. 

Besides the above there will be given here tables of three 
sets of tests of full-size wrought-iron columns, viz. : 

i. The series made at Watertown Arsenal, this being the 
most complete set of tests of full-size wrought-iron columns in 
existence. 

2. A series of tests of Z-bar columns made by Mr. C, L. 
Strobel. 

3. A few tests made at the Mass. Institute of Technology. 
Reference will also be made to the tests of Mr. G. Bouscaren, 
and to those made by Prof. Tetmajer, at the Materialprufungs- 
anstalt in Zurich. 

Graphical rep esentations, however, will first be given of the 




10000 



results of those tested at Watertown Arsenal, with the correspond- 
ing curves, representing (a) the formulae of Prof. Sondericker (see 



WROUGHT-IRON COLUMNS. 419 

page 417), and (b) that of Mr. Strobel (see page 418). These 
diagrams will be preceded by the corresponding formulae. 

A perusal of them will show that, for values of less than 

P 

a certain quantity, which Mr. Strobel assumes as 90, arid Prof. 
Sondericker as 80 for flat-ended, and 60 for pin-ended columns; 

p 

the value of (i.e., the breaking-load divided by the area) is 
A 

I P 

constant. For greater values of the value of decreases, and 

P A 

for this portion of the curve, Prof. Sondericker's formulae are as 

follows : 

For flat-ended Phcenix columns he recommends Cooper's 
formula. 

For lattice columns with pin-ends, reported in Exec. Doc. 
12, 47th Congress, ist session, and Exec. Doc. 5, 48th Congress, 
ist session, he recommends the formula 

P_ = 34000 

A = 

60 



12000 

For the box and solid web columns reported in Exec. Doc. 5, 
48th Congress, ist session, and Exec. Doc. 35, 49th Congress, 
ist session, taken together with Bouscaren's results on box and 
on American Bridge Company's columns, he recommends 

P 33000 

For flat-ends -r= - 

A. 



/ \ 2 ' 

-80 
p / 



10000 



p 31000 

For pin-ends .......... -;-= 



6000 



4 2 APPLIED MECHANICS. 

In these formulae P = breaking-load in pounds, A= sectional 
area in square inches, / = length in inches, and p = least radius 
of gyration of section in inches. 

Moreover, the numerator in each of these formulae is the 

*P / 

value of -j- corresponding to the case when is less than 80 in 
A p 

flat-ended, and less than 60 in pin-ended columns. 

Instead of the above Mr. Strobel recommends for value of 

P I 

-r when - is less than 90, 35000 pounds per square inch, and, 
A P 

for values of greater than 90, the formula 



r =46ooo-i2 5 -. 

Moreover, if P'=safe load, in pounds, he recommends 

/ P' 

(a)For-<90, -^- = 8000; 
p A. 

I P' I 

(b) For > 90, r = 1 0600 30. 
p A p 

While Gordon's formula, or a modification of it, is still in 
use in many bridge specifications, quite a number of them have 
substituted the Strobel formula, or a modification of it. 

WROUGHT-IRON COLUMNS SUBJECTED TO ECCENTRIC LOAD. 

All the formulae given thus far for the breaking or for the 
safe load on wrought-iron columns are only applicable when 
the load is so applied that its resultant acts along the axis of the 
column, and either the diagrams on pages 417 and 418, or the 
corresponding formulae, give us the breaking-strength per square 
inch, i.e., the number of pounds per square inch which, multiplied 



WROUGHT-IRON COLUMNS. 421 



by the area in square inches, gives the breaking-load of the column; 
the safe load per square inch being obtained by dividing the 
breaking-load per square inch by a suitable factor of safety. On 
the other hand, whenever the resultant of the load on the column 
does not act along the axis of the column, we must determine the 
fibre-stress due to the direct load, and to this add the greatest 
fibre stress due to the bending-moment, the sum of the two being 
the actual greatest fibre stress, and the column must be so pro- 
portioned that this greatest fibre stress shall not exceed the safe 
strength per square inch, as determined by dividing the breaking- 
strength per square inch by the proper factor of safety; and this 
proceeding should be followed whatever be the cause of the 
eccentric load whether it be due to the beams supported by the 
column on one side being more heavily loaded than those on the 
other, whether it be due to the load transmitted from the columns 
above being eccentric, whether it be due to the mode of connection 
of the column to the other parts of the structure, whether it be 
due to poor fitting, or to any other cause. 

TESTS OF FULL-SIZE WROUGHT-IRON COLUMNS. 

The tests made at the Watertown Arsenal will next be given, 
together with cuts showing the form of the columns ; these being 
taken from the Tests of Metals for 1881, 1882, 1883, 1884, and 
1885. 

The following tables are taken from the volume for 1881.: 



422 



APPLIED MECHANICS. 



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WROUGHT-IRON COLUMNS. 



423 



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424 



APPLIED MECHANICS. 



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WROUGHT-IRON COLUMNS. 



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APPLIED MECHANICS. 






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c^^ 


" ~ 




-C 


.5 











J 


N 


o g" 


-s 





c 

ID 

.s 


a 


o 


"rt 


V 


C 





^^ 


& 


it- 



WR UGHT-IR ON COL UMNS. 



427 




428 



APPLIED MECHANICS. 




WROUGHT-IRON COLUMNS. 



429 



The next table taken from the volume for 1882 men- 
tioned above contains the results of some compressive tests 
of wrought-iron I-beams placed in the machine with the ends 
vertical and tested with flat-ends ; also of some tensile speci- 
mens cut off from two of them. 

TESTS OF I-BEAMS BY COMPRESSION. 







Width 
of 


Thick- 


Total 


*J 

C 

tx 


Sectional. 


Ultimate Strength. 




Length. 


Flange. 


Web. 


Depth. 


(U 


Area. 


Actual. 


PerSq. In 




In. 


In. 


In. 


In. 


L s. 


Sq. In. 


Lbs. 


Lbs. 


I 


57.06 


5-45 


0.64 


9.00 


228 


14.40 


545 TOO 


37854 


2 


155-45 


4.40 


0.40 


10.52 


443 


IO.26 


207000 


20170 


3 


191.90 


3-56 


0.40 


9.08 


365 


6.85 


85380 


12460 


4 


191.90 


3-59 


0-43 


9.09 


38i 


7-15 


85200 


11916 


5 


119-85 


2.98 


0.28 


6. ii 


139 


4.18 


IOI200 


24210 


6 


i8o.33 


3.60 


0.42 


6.96 


303 


6.05 


84650 


13990 


7 


192.04 


3.58 


0-45 


7-94 


355 


6.65 


83400 


12540 


8 


192.90 


3.60 


0.44 


7.98 


353 


6-59 


92300 


I40IO 


9 


215-88 


4.28 


0.40 


10.52 


561 


9-30 


I49OOO 


I6O2O 


10 


264.08 


4-49 


0.48 


10-53 


747 


10.19 


II3IOO 


IIIOO 


ii 


264.08 


4-43 


0.50 


10.51 


767 


10.46 


107800 


10306 


12 


264.00 


4.90 


0.53 


I5-I5 


1085 


14.80 


184700 


I24OO 


13 


263.95 


4.84 


0-53 


14.74 


1081 


14.74 


I87OOO 


12686 



TESTS OF SPECIMENS FROM NOS. I AND 2 BY TENSION. 





Cut from 
Flange 
or Web. 


Width. 
In. 


Depth. 
In. 


Sectional 
Area. 

Sq. In. 


Ultimate Strength. 


Contrac- 
tion of 
Area. 

Per Cent. 


Actual. 
Lbs. 


Per Sq. In. 
Lbs. 


f 


Web. 


3-00 


0.65 


1-95 


103300 


52970 


IO 




Web. 


3-00 


0.50 


I-5I 


65400 


43340 


3-9 


1 


Flange. 


4.00 


0.75 


3-01 


146400 


48640 


19.6 


1 


Flange. 


4.00 


0.76 


3-02 


I47IOO 


48640 


15-9 


f 


Flange. 


3.00 


0.51 


1-53 


55400 


36210 


n. I 


I 


Web. 


3.00 


0.40 


I.I 9 


52900 


44640 


16.5 



43 APPLIED MECHANICS. 

Next will be given the set of tests which is reported in the 
volumes for 1883 and 1884. 

The following is quoted from the first of the two : 



" COMPRESSION TESTS OF WROUGHT-IRON COLUMNS, LATTICED, BOX, 

AND SOLID WEB. 

" This series of tests comprises seventy-four columns, forty 
of the number having been tested, the results of which are 
herewith presented. 

"The columns were made by the Detroit Bridge and Iron 
Company. 

" The styles of posts represented are those composed of 

" Channel-bars with solid webs ; 

" Channel-bars and plates ; 

" Plates and angles ; 

" Channel-bars latticed, with straight and swelled sides ; 

" Channel-bars, latticed on one side, and with continuous 
plate on one side. 

" All the posts were tested with 3^-inch pins placed in the 
centre of gravity of cross-section ; except two posts of set N y 
which had the pins in the centre of gravity of the channel- 
bars. 

" This gave an eccentric loading for these columns, on ac- 
count of the continuous plate on one side of the channel- 
bars. 

" The pins were used in a vertical position, unless other- 
wise stated in the details of the tests. 

" In the testing-machine the posts occupied a horizontal 
position. 

" They were counterweighted at the middle. 

" Cast-iron bolsters for pin-seats were used between the ends 



WROUGHT-IRON COLUMNS. 43! 

of the columns and the flat compression platforms of the test- 
ing-machine. 

" The sectional areas were obtained from the weights of the 
channel-bars, angles, and plates, which were weighed before 
any holes were punched, calling the sectional area, in square 
inches, one-tenth the weight in pounds per yard of the iron. 

" Compressions and sets were measured within the gauged 
length by a screw micrometer. 

" The gauged length covered the middle portion of the 
post, and was taken along the centre line of the upper chan- 
nel-bar or plate, always using a length shorter than the dis- 
tance between the eye-plates, to obtain gaugings unaffected by 
the concentration of the load at those points. 

" The deflections were measured at the middle of the post. 
The pointer, moving over the face of a dial, indicated the 
amount and direction of the deflection. 

" Loads were gradually applied, measuring the compressions 
and deflections after each increment ; returning at intervals to 
the initial load to determine the sets. 

" The maximum load the column was capable of sustaining 
was recorded as the ultimate strength, although, previous to 
reaching this load, considerable distortion may have been pro- 
duced. 

" Observations were made on the behavior of the posts 
after passing the maximum load, while the pressure was fall- 
ing, showing, in some cases, a tendency to deflect with a sudden 
spring, accompanied by serious loss of strength. 

" The slips of the eye-plates along the continuous plates 
and channel-bars during the tests were measured for certain 
posts in sets F, G, H, and /. The measurements of slip were 
taken in a length of 10 inches or 20 inches, one end of the 
micrometer being secured to the eye-plate, and one end to the 
channel-bar. The readings include both the compression 
movement of the material and the slip of the plates. 



43 2 APPLIED MECHANICS. 

" Columns H, 7, Z, and J/ were provided with pin-holes for 
placing the pins either parallel or perpendicular to the webs of 
the channel-bars. 

" After the ultimate strength had been determined with the 
pins in their first position, a supplementary test was made, if 
the condition of the column justified it, with the pins at right 
angles to their former position ; thus changing the moment of 
inertia of the cross-section, taken about the pin as an axis. 

" The experiments with columns N show how much strength 
is saved by employing pins in the centre of gravity of the cross- 
section. Where such was not the case, the columns showed 
the effect of the eccentric loading by deflections perpendicular 
to the axis of the pins, from the initial loads, which resulted in 
their early failure." 



WKO UGHT-1RON COL UMNS. 



433 



TABULATION OF EXPERIMENTS ON WROUGHT-IRON COLUMNS 
WITH 3J-INCH PIN-ENDS. 









Ultimate 








Length, 




Strength. 








Centre 


Sec- 






No. of 
Test. 


Style of Column. 


to 
Centre 


tional 
Area. 


Total, 


Lbs. 


Manner of Failure. 






of Pins. 




Lbs. 


per 








In. 


Sq. In. 




Sq. In. 






Set! 


A. 












752 


u 


_J 


126.20 


9-831 


297100 


30220 


Deflected perpendicular 
to axis of pins. 


757 


*- " 




120.07 


10.199 


320000 


31380 


Sheared rivets in eye- 




if 


' t; 1 










plates. 


755 


X> 


10 


180.00 


9-977 


251000 


25160 


Deflected perpendicular 
to axis of pins. 


756 


r 


~~n~\ 


180.00 


9-977 


210000 


21050 


Do. do. 


753 


*~~z 


6 r> 


240.00 


9-732 


188600 


19380 


Do. do. 


754 


i 




240.10 


9.762 


158300 


16220 


Do. do. 




i 














SetlD, 












1642 








240.00 
240.00 


16.077 
16.281 


425000 
367000 


26430 
22540 


Deflected perpendicular 
to axis of pins. 
Do. do. 


~] 


r =! 


1646 


n? 




320.00 


16.179 


3I8800 


19700 


Do. do. 


1647 


\ 




320.10 


16.141 


283600 




Do. do. 




Hj, ' 
8 > 
















. Set 


a 












,653 




*/ 8 


I 


320.00 


17.898 


474000 


26480 


Deflected perpendicular 
to axis of pins. 


1654 






i 


320.00 


19.417 


49IOOO 


25290 


Do. do. 






Le^ 


1 1 














Setjp. 












,645 




e s' 


T r 


319.95 


16.168 


453000 


28020 


Deflected parallel to axis 

of pins. 


1*50 




"*'" 


IL 


320.00 


16.267 


454000 


27910 


Deflected perpendicular 
to axis of pins. 


? 





434 



A PPLIE /) ME CHA A ICS. 



TABULATION OF EXPERIMENTS ON WR OUGHT-IRON COLUMNS 
WITH 3^-INCH PIN-ENDS. 







Length, 


Ultimate 
Strength. 








Centre 


Sec- 






No. of 
Test. 


Style of Column. 


to 
Centre 


tional 
Area. 


T/-\t<i 1 


Lbs. 


Manner of Failure. 






of Pins. 




i otai. 
Lbs. 


per 








In. 


Sq. In. 




Sq. In. 








Set- G. 












1651 






320.00 


20.954 


540000 


25770 


Deflected m diagonal 
direction. 


1652 




l^6 T 


320.10 


20.613 


535000 


25950 


Sheared rivets in eye- 







' 2* ' \ 










plates. 


746 


SetiH. 


159-20 


7.628 


258700 


339X0 


Deflected perpendicular 
to axis of pins. 


747 


?^i- 


f Tt 


159-27 


8.056 


294700 


36580 


Do. do. 


748 




^- 8 *44 


239.60 


7.621 


260000 


34120 | Do. do. 


749 




i 


239.60 


7.621 


254600 j 33410 Deflected in diagonal 














direction. 


1648 


ffi 


/ * 






x- 


31610 1 Deflected narallel to ?xis 






319.90 


7-7 


243000 




of pins. 


1649 




3^9-85 


7 '673 


229200 


29870 


Deflected in diagonal 
direction. 


740 


I 


i59-9o 


7-645 


262500 


34340 


Deflected perpendicular 




Set I. 










to axis of pins. 


74 l 


(swelled.) 


i59-9o 


7.624 


255650 


33530 


Do. do. 


739 


~~ 


i |~~t 


239.70 


7-5 1 ? 


251000 


33390 


Deflected parallel to axis 






1 








of pins. 


75 




( | 3 '$ "U 


239.70 


7 -S3 1 


259000 


34390 


Deflected perpendicular 






. 










to axis oi pins. 


1643 


_ 


1 _l 


319.80 


7.691 


237200 


30840 


Deflected parallel to axis 

rf rin<* 


1644 


1 


319.92 


7.702 


237000 


30770 Deflected in diagonal 
direction. 




< 













1640 




J-K 


199.84 


"944, 


403000 


33740 


Deflected perpendicular 






~i p* 










to axis of pins. 


1641 
1634 




4- - 


200.00 
300.00 


12.302 
12.148 


426500 
408000 


34670 
33630 


Deflected in diagonal 
direction. 
Deflected perpendicular 






7 










to axis of pins. 


1635 




_i 


3OO.OO 


12.175 


395000 


32440 


Do. do. 






r 











WROUGHT-1RON COLUMNS. 



435 



TABULATION OF EXPERIMENTS ON WROUGHT-IRON COLUMNS 
WITH 3J-INCH PIN-ENDS. 









Length, 
Centre 


Sec- 


Ultimate 
Strength. 


- 


No. of 
Test. 


Style of 


Column. 


to 
Centre 


tional 
Area. 


Total, 


Lbs. 


Manner of Failure. 








of Pins. 




Lbs. 


per 










In. 


Sq. In. 




Sq. In. 






Sejt M. 














(swelled.) 












1638 






1 


199-25 


12.366 


385000 


330 


Deflected perpendicular 
to axis of pins. 


1639 




n 


*"o 


199.50 


12.659 


405000 


31990 


Do. do. 


1636 






I 


300.20 


11.920 


391400 


32830 


Deflected in diagonal 
direction. 


l6 37 




. 


J 


300.15 


11.932 


390700 


32740 


Do. do. 


1630 


1 

*rf 

ik s! s_l 




300.00 


17.622 


461500 


26190 


Deflected perpendicular 
to axis of pins. 


itS 





1631 


1 






300.00 


17.231 


485000 


28150 


Do. do. 


1632 


|p-10 


I 


.2 

4 


300.00 


17-57 


306000 


17420 


Do. do. 






^| 1 




*^o 












1633 


jr^-f 






300.00 


17.721 


307000 


17270 


Do. do. 




K 
















I 1 " 













The remainder of the tests of this series of seventy-four 
columns is reported in the volume for 1884. 

The only portion of the description that it is worth while 
to quote is the following, as the tests were made in a similar 
way to what has been already described : 

" Sixteen posts were tested with flat ends ; eighteen were 
tested with 3^-inch pin-ends. 



436 



APPLIED MECHANICS. 



" The pins were placed in the centre of gravity of cross- 
section, except two posts of set K, which had the pins in the 
centre of gravity of the channel-bars, giving an eccentric bear- 
ing to these columns, on account of the continuous plate on 
one side of the channel-bars." 



TABULATION OF EXPERIMENTS ON WROUGI IT-IRON COLUMNS 
WITH FLAT ENDS. 











Ultimate 








Total 


Sec- 


Strength. 




No. of 
Test. 


Style of Column. 


Length. 


tional 
Area. 


Total, 


Per 


Number of Failure. 












Sq. In., 








Ft. In. 


Sq. In. 




Lbs. 




377 
378 


SetB. 


1 


*" 


10 7.90 
10 7.90 


12.08 
ii. ii 


383200 
372900 


31722 
33564 


Buckling-plate D be- 
tween the riveting. 
Buckling-plates. 


"4 






I 














379 


SetE. 


- 8 4 


^ 


13 i i. 80 
13 11.80 


17.01 
17.80 


633600 


34950 
35595 


Buckling - plates be' 
tween the riveting. 
Triple flexure. 


346 








13 11.9 


15.74 


517000 


32846 


Buckling-plates. 





v^" "T" 




347 




1 


, f 


13 11.65 


15.84 


555200 


35050 


Do. do. 




Set F. 


.a//' ^J 


7j6 












342 




< 7^ 




20 7 . 63 


15.68 


517500 


33003 


Deflecting upward. 


344 


w 


20 7 . 80 


15-56 


536900 


3455 


Buckling-plates. 


348 




T 3 "-75 


21 .02 


708000 


33682 


Buckling-plates. 




ll ^ ^ 


1 












349 


; i" ? | 


1 


X 3 "-75 


21 .46 


709500 


33061 


Triple flexure. 


343 


SetG. jfc-Ka" 
_>- 6.90% 


it> 


20 7.60 

20 7.63 


21.20 
21.49 


700000 
729450 


330'9 
33943 


Deflecting upward. 
Deflecting downward. 




^ 8 " 



WRO UC,H 7 '-IRON COL UMNS. 



437 



TABULATION OF EXPERIMENTS ON WROUGHT-IRON COLUMNS 
WITH FLAT ENDS. 







Ultimate 








Total 


Sec- 


Strength. 




No. of 
Test. 


Style of Column. 


Length. 


tional 
Area. 




Manner of Failure. 


Total, 


Per Sq. 






Ft. In. 


Sq. In. 


Lbs. 


In., Ibs. 




339 




20 7-94 


12.64 


412900 


32666 


Deflecting upward. 




" 7* 1 T 

*** H 




SetK. 


T 












340 


er= 


=LI 


20 7.94 


12.74 


431400 


33862 


Do. do. 




Latticed 
















%,'plate. 71 












337 
338 


SetN. 


-"I! 


25 7-75 
25 7-88 


16.99 
17.40 


582400 
580000 


34279 

33333 


Deflecting downward 
and sideways. 
Deflecting diagonally 
channel B and lattic- 
ing on the concave 




Latticed 










side. 



TABULATION OF EXPERIMENTS ON WROUGHT-IRON COLUMNS 
WITH 3^-INCH PIN-ENDS. 







Length, 


Ultimate 










Centre Sec- 


Strength. 




No. of 
Test. 


Style of Column. 


to i tional 

Centre Area, 
of Pins. 




Manner o( Failure. 


Total, Per Sq. 










Ft. In. 


Sq. In. 


Lbs. In., Ibs. 

l 




368 




1 






15 o.i 


11.42 


379200 33205 


Hor. deflection perpen- 
dic. to plane of pins. 


356 


Set B. 


jji 


if 


20 0.0 


11.42 
ii .42 


342000 


29947 


Do. do. 


357 






20 0.0 


11.31 


330100 


29186 


Do. do. 

% 


< 




371 








9 ir -9 


9.14 


286100 


31302 


Buckling - plates be- 
tween rivets. 


372 


J 






IO O.O 


10.07 


319200 


31698 


Do. do. 


370 


kf 


i 


4 


a 


15 o.o 


9.21 


291500 


31650 I Hor. deflec. and buck- 


369 


Set C. 




J (. 
o. 


4 


L_ 


15 o.o 


9-44 


290000 


30720 


ling between rivets. 
Do. do. 


354 


'* 


20 o.o 


9.24 


267500 


28950 


Triple flexure. 


365 


20 o.o 


9-36 


279700 


29879 


Hor. deflection. 



438 



APPLIED MECHANICS. 



TABULATION OF EXPERIMENTS ON WROUGHT-IRON COLUMNS 
WITH 3^-INCH PIN-ENDS. 





Style of Column. 






Ultimate 






Length, 
Centre 


Sec- 


Strength. 




No. of 
Test. 


to 
Centre 


tional 
Area. 


Total, 


Per 


Manner of Failure. 




of Pins. 






Sq. In., 






Ft. In. 


Sq. In. 




Lbs. 






I l 


_l 












360 
361 


<-^t> 

SetD. 

l| 


<* 


*3 4- I 3 
13 4.00 


15-34 
15.40 


475000 
485000 


30965 
3*494 


Deflecting upward in 
plane of pins. 
Hor. deflection perpen- 
dicular to plane of 
pins. 


358 


Sot E. 


, 




h 


20 o.o 


17.77 


570000 


32077 


Hor. deflection perpen- 
dicular to plane of 


." 


"jj 






1 


l 












pins. 


359 




- s"\ 


4* 




20 o.o 


17.22 


5554oo 


32253 


Do. do. 


350 




^ 


C 




20 0.25 


12.48 


202700 


16242 


Hor. deflection, con- 






C0| 


o- 












cave on lattice side. 


351 


SetK. 


!l 


IJ 

00 


1 


20 0.00 


10.84 


208200 


10207 


Do. do. 


352 


i 


"\ 


Lt 


5 


2O O.25 


12.65 


350000 


27668 


Do. do. 


353 




51 


E 




2O O.25 


12.7 


390400 


30596 


Hor. deflection perpen- 
. dicular to plane of 




-I- 












pins, convex on lat- 
















tice side. 



Besides the above, there are four tests of lattice columns 
reported in Exec. Doc. 36, 49th Congress, 1st session, but as 
these columns were rather poorly constructed and form rather 
special cases they will not be quoted here. 

In determining the strength of a bridge column made of 
channel-bars and latticing, these results of tests on full-size 
columns furnish us the best data upon which to base our con- 
clusions. 



WROUGHT-IRON COLUMNS. 



439 



In the Trans. Am. Soc. C. E. for April, 1888, Mr. C. L. Stro- 
bel gives an account of his tests on wrought-iron Z-bar columns, 
from which the following is condensed, viz.: The Z-irons used 
in making the columns were 2^X3X2^ inches in size, and & 
inch thick. 

Two columns were about n ft. long, two 15 ft., two 19 ft., 
three 22 ft., three 25 ft., and three 28 ft., a total of fifteen columns. 
The table of results follows: 











Ultimate 






Ultimate 




Strength, 


Length, 


Sectional 
Area, 


Strength 
by Tests 


/ 

Q 


Strobel's 






per Sq. In. 




Formula 










per Sq. In. 


Inches. 


Sq. Ins. 


Lbs. 




Lbs. 


13'i 


9-435 


36800 


64 





131* 




34600 


64 





180 


9.480 


34600 


88 


35000 


180 


9.280 


36600 


88 


35000 


228f 


9.241 


33800 


112 


32200 


228f 


10. 104 


33700 


112 


32200 


264 


9.286 


30700 


129 


29900 


264' 


9.286 


29500 


129 


29900 


264 


9.286 


30700 


129 


29900 


300 


9.156 


28100 


I 4 6 


27750 


3 00 


9-45 6 


28000 


I 4 6 


27750 


300 


9.516 


28400 


I 4 6 


27750 


336 


9-375 


27700 


164 


25500 


336 


9-643 


28000 


l6 4 


25500 


336 


9-375 


27600 


164 


25500 



The following table shows the results of compression tests 
made in the engineering laboratories of the Massachusetts In- 
stitute of Technology upon some wrought-iron pipe columns. 
They were tested with the ordinary cast-iron flange-coupling 
screwed on to the ends, bearing against the platforms of the 
testing- machine, which were adjustable, inasmuch as they were 
provided with spherical joints. 



440 



APPLIED MECHANICS. 



The tests of full-size wrought-iron columns made by Mr. G. 
Bouscaren, are given in the Report of the Progress of Work on 
the Cincinnati Southern Railway, by Thos. D. Lovett, Nov. i, 

1875- 

In Heft IV (1890) of the Mittheilungen der Materialprii- 
fungsanstalt in Zurich is given an account of a large number 
of tests of wrought-iron and steel columns of the following 
forms, viz.: i. Angle-irons; 2. Tee iron; 3. Channel- bars ; 
4*. Two angle-irons riveted together ; 5. Four angle-irons 
riveted together; 6. Two channel-bars riveted together; 
7. Two tee irons riveted together ; also quite a number of tests 
of columns of some of these forms subjected to eccentric 
loads, the eccentricity of the load being, in some cases, as much 
as 8 cm. (3". 15). The columns tested were of a variety of 
lengths, the longest ones being 560 cm. (18.37) ^ eet l n g- 

In Heft VIII (1896) of the same Mittheilungen is an ac- 
count of another set of tests of columns of the above-described 
forms. The results of these valuable tests will not be quoted 
here, but for them the reader is referred to the Mittheilungen 
themselves. 



"o 


u 

V 






V 

I 


_H 


2 


i 


a 
a 




5 c 


ominal Size 
Pipe. 


iside Diame 


utside Diam 
eter. 


iameter of 
Flanges. 


li 

"0 


auge Lengtl 


aximum Lo; 


rea of Cross 
section. 


'aximum Lo 
per Sq. In. 


ompression, 
Modulus of 
Elasticity. 


o o 2 

u c'o 


* 


M 





C 




O 


s 


< 


2 


U 


^ TQ. 


In. 


In. 


In. 


In. 


In. 


In. 


Lbs. 


Sq. In. 


Lbs. 






2 


2.06 


2-37 


7 




I 


51 


30000 


i. 08 


27800 


24300000 


88.8 


2 


2.04 


2-39 


7 


69 


1 


51 


29800 


1 .22 


24500 


222OOOOO 


89.1 


2 5 


2.50 


2.89 


8 


93 




86 


34500 


1.65 


20900 


25200000 


98.1 


2 i 


2.48 


2.88 


8 


93 : 




86 


37000 


1.68 


22000 


259OOOOO 


98.4 


3 


3.06 


3-44 


8f 


93 


| 


86 


45500 


1.94 


23500 


27700000 


81.4 


3 




3.48 


8| 


93' 




86 


51000 


2.01 


25300 


25IOOOOO 


80.5 




3 '.60 


4.00 




105! 




100.5 


55000 


2-39 


23000 


25200000 


78.2 


32 


3-59 


3-99 


9i 


1:051 




100.5 


65000 


2-39 


27200 


246OOOOO 


78.5 


4 


4.07 


4'53 


9fr 




100.5 


80000 


3-" 


25700 


258OOOOO 


77-1 


4 


4.09 


4-50 




"7ft 


100.5 


69000 


2.76 


25000 


24900000 


77-3 



224. Transverse Strength of Wrought-iron. 

Wrought-iron owes its extensive introduction into con- 
struction as much or more to the efforts of Sir William 
Fairbairn than to anyone else; and while he was furnishing 



TRANSVERSE STRENGTH OF WROUGHT-IRON. 441 

the means to Eaton Hodgkinson to make extensive experiments 
on cast-iron columns, and while he made experiments himself 
on cast iron beams, which were in use at that time, he also 
carried on a large number of tests on beams built of wrought- 
iron, more especially those of tubular form, and those having 
an I or a T section, and made of pieces riveted together. In 
his book on the " Application of Cast and Wrought Iron to 
Building Purposes " he gives an account of a large number 
of these experiments, including those made for the purpose of 
designing the Britannia and Conway tubular bridges, a fuller 
account of which will be found in his book entitled " An Ac- 
count of the Construction of the Britannia and Conway Tubular 
Bridges." In the first-named treatise he urges very strongly 
the use of wrought-iron, instead of cast-iron, to bear a trans- 
verse load. 

Fairbairn tested a number of wrought-iron built-up beams, 
but they were of small dimensions and are hardly comparable 
with those used in practice. 

In the light of the tests made upon wrought-iron columns, 
it is evident that the compressive strength of wrought-iron is 
less than the tensile strength. Hence we should naturally ex- 
pect that the modulus of rupture would be, in all cases, greater 
than the compressive strength, and that it might or might not 
be greater than the tensile strength of the iron. Of course 
the modulus of rupture varies very much with the shape of the 
cross-section, for the same reasons as were explained in the 

paragraph 191, i.e., that the formula M = f assumes Hooke's 

law, "the stress is proportional to the strain," to hold, and that 
this is not true near the breaking-point. 

The value of the modulus of rupture is also influenced 
by the reduction in the rolls, and hence somewhat by the size 
of the beam. 

Small round or rectangular bars tested for transverse 
strength show a modulus of rupture very much in excess of 
the compressive strength per square inch of the iron, and ex- 
ceeding very considerably even the tensile strength. 

While a great many tests of such specimens have been 



44 2 APPLIED MECHANICS. 

made, none will be quoted here, but the last five tests of the 
table on page 542 show that for a wrought-iron having a ten- 
sile strength per square inch from 58700 to 60250 pounds, mo- 
duli of rupture were obtained from Soooo to 90000 pounds, 
as, the number of turns of these rotating shafts being com- 
paratively small, the breaking-loads were not far below the 
quiescent breaking loads. On the other hand the moduli of 
rupture of I beams and other shapes used in building have 
very much lower values, but for these, tests will be cited. 

As to experiments on large beams, we have : 

i. Some tests made by Mr. William Sooy Smith and jy 
Col. Laidley at the Watertovvn Arsenal. 

2. Some tests made in Holland on iron and steel beams, 
an account of which is given in the Proceedings of the Brit- 
ish Institute of Civil Engineers for 1886, vol. Ixxxiv. p. 412 et 
seq. 

3. Some tests made in the laboratory of Applied Mechan- 
ics of the Massachusetts Institute of Technology, on iron and 
steel I beams. 

4. Tests made by the different iron companies upon beams 
of their own manufacture, and recorded in their respective 
hand-books. 

Mr. Smith's tests are recorded in Executive Document 23, 
46th Congress, second session. 

5. In Heft IV (1890) of the Mittheilungen der Material- 
priifungsanstalt in Zurich will be found accounts of tests 
made by Prof. Tetmajer upon the transverse strength of 
I beams, of deck-beams, and of plate girders. 

The results of these tests will be given in the table on top 
of page 443- 

Specimens cut from the flanges, and also from the webs of 
the last seven of these beams, were tested for tension. In the 
case of those cut from the flanges, the tensile strength varied 



TRANSVERSE STRENGTH OF WROUGHT-IRON. 443 



Depth. 
(Inches.) 


Moment of 
Inertia. 
(Inches)*. 


Span. 
(Inches.) 


Modulus of 
Rupture. 
^Lbs. per Sq. In.) 


Modulus of 
Elasticity. 
(Lbs. per Sq. In.) 


7.87 
7.87 
3-93 
5-9i 
7.87 
9-45 
n.8i 


52.04 
52.04 

4.13 
17-85 
5 T -95 
103. .-.4 


62.96 
62.96 
3i-44 
47.28 
62.96 
75.60 
94.48 


51190 

56453 
62852 

56453 
53894 
51619 

CT t8^? 


27501500 
28937700 
28767101 
28212500 
28226700 
273735 o 










4 34 























from 50200 in the 1 5". 75 beam to 57300 pounds per square inch 
in the 3^.93 beam. On the other hand, in the case of the 
specimens cut from the web, the tensile strengths varied from 
44900 in the I i^.Si beam to 54400 pounds per square inch in the 
3"-93 beam, the contraction of area per cent varying from 23.6 
to 32.1 per cent in the flanges, and from 12.5 to 15.9 per cent 
in the web. 

The results obtained with the deck-beams are as follows : 



Depth. 
(Inches.) 


Moment of 
Inertia. 
(Inches) 4 . 


Span. 
(Inches.) 


Modulus of 
Rupture. 
(Lbs. per Sq. In.) 


Modulus of 
Elasticity. 
(Lbs. per Sq. In.) 


4-93 


19.88 


70.86 


56170 


25112500 


4.26 


9-38 


59.06 


48920 


25823500 


3-52 


5-33 


47.24 


553 20 


25596000 


3.48 


4.71 


39-37 


54180 


26804700 


2.36 


1.30 


31.50 


52760 


24202400 


'93 


0.60 


23.62 


58160 





Tensile tests of specimens cut from these deck-beams 
showed tensile strengths of from 47540 in the i".93 beam to 
54750 pounds per square inch in the 2". 36 beam, and contrac- 
tions of area of from 14.1 per cent to 18.4 per cent. 

The results obtained with the plate girders are as follows, 
viz. : 



Depth of 
Web. 
(Inches.) 


Modulus of 
Rupture. 
(Lbs. per Sq. 
In.) 


Modulus of 
Elasticity. 
(Lbs. per Sq. 
In.) 


Depth of 
Web. 
(Inches.) 


Modulus of 
Rupture. 
(Lbs. per Sq. 
In.) 


Modulus of 
Elasticity. 
(Lbs. per Sq. 
In.) 


iS-75 
iS-75 

19.69 
19.69 


51480 
53180 
SH? 6 
52610 


26449200 
25539100 
24813900 
25605500 


23.62 
23.62 
27.56 
27-56 


52760 
48490 
47780 
46500 


26321200 

26548700 
25667100 
26776300 



444 



APPLIED MECHANICS. 



The tensile strength of the material of the webs varied from 
29860 to 41240 pounds per square inch, while the contraction of 
area was only 0.4 per cent. The tensile strength of the material 
of the flange-plates was 51050 pounds per square inch, with a 
contraction of area of 17 percent. The tensile strength of the 
angle-irons was 46357 pounds per square inch, with a contraction 
of area of 14 per cent. 

The following table gives the results that have been obtained 
in the tests that have been made upon wrought-iron I beams in 
the laboratory of Applied Mechanics of the Massachusetts Insti- 
tute of Technology. This table will give a fair idea of the strength 
and elasticity of such beams. 

TESTS OP WROUGHT-IRON BEAMS MADE IN THE LABORATORY OF APPLIED 
MECHANICS OF THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY, 
ALL LOADED AT THE CENTER. 



No. 
of 
Test. 


Ins. 


Moment 
of 
Inertia. 


Span. 
Ft. Ins. 


Break- 
ing Load. 

Lbs. 


Moduli 
of 
Rupture. 
Lbs. per 
Sq. In. 


Moduli of 
Elasticity. 
Lbs. per 
Sq. In. 


Remarks. 


121 


6 


24.41 


12 


9500 


42386 


26679000 


From Phoenix Co. 


124 


7 


43-5 


14 o 


14100 


48082 


28457000 


< a 


126 


5 


12.47 


12 


6450 


46624 


29549000 


< t t 1 ! 


209 


7 


44-93 


13 8 


1 2 2OO 


39670 


31057000 


t 1 I 


211 


8 


67.32 


13 8 


I7OOO 


42000 


28532000 


< ( 4 


215 


9 


no. 78 


14 8 


23000 


41680 


27165000 


' . 1 ' ' 


227 


8 


61 .20 


13 6 


18300 


49640 


27397000 


From Belgium. 


230 


9 


86.41 


13 8 


21300 


45850 


27365000 


< * < > 


235 


9 


40.91 


13 8 


13800 


49140 


27923000 


E < 


253 


7 


43-05 


14 8 


II3I9 


40660 


28045000 


From Phoenix Co. 


256 


8 


66.56 


14 8 


14547 


38460 


28187000 


i 


263 


9 


108.67 


14 8 


19694 


36160 


27050000 





291 


7 


45-96 


14 6 


IO7OO 


36340 


26790000 


i < 


292 


8 


66.39 


14 6 


14300 


38200 


27380000 





294 


9 


92.89 


14 6 


I92OO 


41470 


27050000 


i '- 


338 


, 6 


25.92 


i4 7 


72OO 


37800 


27860000 


N. T. Steel & Iron Co. 


34i 


7 


46.73 


12 II 


13600 


40600 


27410000 


I It H 


345 


8 


7I-25 


14 7 


15400 


38400 


26940000 


i ,: T, , t ,, 


379 


7 


48.84 


12 IO 


15500 


44300 


26170000 





STEEL. 445 



225. Steel. While steel is a malleable compound of iron, 
with less than 2 per cent of carbon and with other substances, 
the definition recommended by an international committee of 
metallurgists in 1876, and used to some extent in German and 
Scandanavian countries, is different from that in general use in 
English-speaking countries, and in France. 

The definition recommended by the international committee 
may be found in the Trans. Am. Inst. Min. Engrs. for October, 
1876, and is in the following: 

i. That all malleable compounds of iron with its ordinary 
ingredients, which are aggregated from pasty masses, or from 
piles, or from any form of iron not in a fluid state, and which 
will not sensibly harden and temper, and which generally 
resemble what is called "wrought-iron," shall be called weld-iron 
(Schweisseisen). 

2. That such compounds, when they will, from any cause, 
harden and temper, and which resemble what is now called 
puddled steel, shall be called weld-steel (Schweissstahl). 

3. That all compounds of iron, with its ordinary ingredients 
which have been cast from a fluid state into malleable masses, 
and which will not sensibly harden by being quenched in water 
while at a red heat, shall be called ingot-iron (Flusseisen). 

4. That all such compounds, when they will, from any cause, 
so harden, shall be called ingot-steel (Flussstahl). 

On the other hand, in English-speaking countries, those 
compounds which have been aggregated from a pasty mass, 
usually in the puddling -furnace, and which contain slag, are 
generally called wrought-iron, while those which have been cast 
from a molten state into a malleable mass are generally called steel. 

While this classification is not perfect, it states the most 
common practice in a general way. Exceptions, two of which 
are that it does not include the cases of cementation steel and 
of puddled steel, will not be discussed here. 

In view of the above, it will be plain that what is commonly 



44 APPLIED MECHANICS. 

called mild steel in America, would be called ingot-iron under 
the definition of the international committee. Steel is usually 
made by one of three processes, viz. : the crucible process, the 
Bessemer process, or the open-hearth process. 

While other processes, as the cementation process and others, 
are sometimes used, the three enumerated above are in most 
common use at the present time. 

Crucible Steel. This is very commonly made by re-melting 
blister-steel in crucibles; the blister-steel being made by the 
cementation process, in which bars of very pure wrought-iron, 
especially low in phosphorus, are heated in contact with charcoal 
until they have absorbed the necessary amount of qarbon. 

A cheaper process, and one much used at the present day, 
is to melt a mixture of charcoal and crude bar-;ron in a 
crucible. 

Crucible steel, which is always high-carbon steel, is used for 
the finest cutlery, tools, etc., and "wherever a very pure and 
homogeneous quality of steel is required. 

Bessemer Steel. In the Bessemer process, a blast of air is 
blown into melted cast-iron, removing the greater part of its 
carbon and burning out more or less of the other ingredients. 
The process is conducted in a converter, which is usually so 
arranged that, when the operation is complete, it can be rotated 
around a horizontal axis to such an extent that the tuyeres are 
above the surface of the molten steel, and the blast is shut off. 

In the acid Bessemer process, the lining of the converter is 
made of some silicious substance, the burning of silicon being 
relied upon to develop a sufficiently high temperature to keep 
the metal fluid. 

In the basic Bessemer process, the lining of the converter is 
of such a nature as to resist the action of basic slags. It is usually 
made of dolomite, or of some kind of limestone. Burned lime is 
added to the charge to seize the silicon and phosphorus, the latter 
serving to develop a sufficiently high temperature. 



STEEL. 447 



In the latter part of the operation, the phosphorus is largely 
burned out, whereas in the acid process, in order to produce a 
steel that is low in phosphorus, it is necessary to use a pig-iron 
that is low in phosphorus. 

Open-hearth Steel. In the open-hearth process, a charge of 
pig-iron and scrap is placed on the bed of a regenerative furnace, 
and exposed to the action of the flame, and is thus converted into 
steel. 

In the acid open-hearth process, the lining of the furnace is 
of a silicious nature, and is covered with sand, while in the basic 
it is usually of dolomite, or of some kind of limestone. 

Bessemer and open-hearth steel contain more impurities 
than crucible steel, but they are very much cheaper, and are 
just as suitable for many purposes. It is only in consequence 
of their introduction that steel can be extensively used on the 
large scale, as crucible steel would be too expensive for many 
purposes. 

Steel, unlike wrought-iron, is fusible; unlike cast-iron, it can 
be forged; and, with the exception of the harder grades, it can 
be welded by heating and hammering, the welding of high -carbon 
steel in large masses being a very uncertain operation, though 
small masses can be welded by taking proper care. 

The special characteristic, however, is, that, with the exception 
of the milder grades, when raised to a red heat and suddenly 
cooled, it becomes hard and brittle, and that, by subsequent 
heating and cooling, the hardness may be reduced to any desired 
degree. The first process is called hardening and the second 
tempering. 

The principal element in the steels that are ordinarily used is 
carbon; nevertheless, both Bessemer and open-hearth steel con- 
tain also silicon, manganese, sulphur, phosphorus, etc., which 
have more or less effect upon the resisting properties of the metal. 
Sulphur, silicon, and phosphorus usually come from the ore, 
the fuel, and the flux, while manganese, which is added, operates, 



44-8 APPLIED MECHANICS. 

among other things, to render the steel ductile while hot, and 
therefore workable, and to absorb oxygen from the melted mass. 

Sulphur is injurious by causing brittleness when hot, and 
phosphorus by causing brittleness when cold. Phosphorus is 
the most harmful ingredient in steel, so that when steel is to be 
used for structural purposes, it is important to have as little 
phosphorus as possible, and any excess of phosphorus is not to 
be tolerated. 

The injury done to steel plates by punching is greater than 
that done to iron plates: this injury can, however, be removed 
by annealing. Steel requires greater care in working it than 
iron, whether in punching, flanging, riveting, or other methods 
of working; otherwise it may, if overheated, burn, or receive 
other injury from careless workmanship. 

The chemical composition of steel is one important element 
in its resisting properties; but, on the other hand, the mode of 
working also has a great influence on the quality. 

The introduction of the Bessemer process was quickly fol- 
lowed by the general use of steel rails, and later, as this and the 
other processes for making steel for structural purposes have 
been developed, there has been a constant increase in the pur- 
poses for which steel has been used. 

One of the earlier applications was to the construction 
of steam-boilers, steel boiler-plate displacing almost entirely 
wrought-iron boiler-plate. Of late years the development of 
the steel manufacture has so perfected, and at the same time 
cheapened, structural steel that it is now used in most cases 
where wrought-iron was formerly employed. Thus the eye-bars 
and the struts of bridges are almost exclusively made of steel, 
also such shapes as angle-irons, channel-bars, Z bars, tee iron, 
I beams, etc., are almost exclusively made of steel, and while 
steel has long been used for many parts of machinery, never- 
theless it is now generally used in many cases where a con- 
siderable fear of it formerly existed, as in main rods, parallel 



^ TEEL. 449 



rods, and crank-pins, and in a large number of parts of machinery 
subjected to more or less vibration. On the other hand, the 
steel used for tools is, of course, high-carbon steel. 

Tools are almost always made of crucible steel, and they 
have of course a high percentage of carbon, a high tensile strength, 
and especially should they be capable of being well hardened 
and taking a good temper. 

The usual steel of commerce may be called carbon steel, 
because, although it always contains small percentages of other 
ingredients, nevertheless carbon is the ingredient that princi- 
pally determines its properties. When iron or steel is alloyed 
with large percentages of certain substances, the resulting al- 
loys enjoy certain special properties, and these alloys still bear 
the name of steel. Two of the most prominent of these are 
manganese steel and nickel steel. 

Regarding the first it may be said that although carbon steel 
becomes practically useless when the manganese reaches about 
ij per cent, nevertheless with manganese exceeding about 7 per 
cent we obtain manganese steel which is so hard that it is exceed- 
ingly difficult to machine it. 

The alloy that has come into most prominent notice recently 
is nickel steel, which consists most commonly of a carbon steel 
with from 0.2 to 0.4 per cent of carbon and with from 3 to 5 
per cent of nickel. With this amount of nickel the tensile strength 
is very much increased, but more especially is the limit of elasticity 
increased by a very large amount; and while the contraction of 
area at fracture and the ultimate elongation per cent are a little 
less than that of carbon steel with the same percentage of carbon, 
they are not less than those of carbon steel of the same tensile 
strength. 

It is used for armor-plates, for which it is specially suitable 
on account of the fact that the nickel renders the steel more 
sensitive to hardening. It is finding, also, a great many other 
uses to which it is specially adapted by its peculiar properties. 



450 APPLIED MECHANICS. 

It has been used for bicycle-spokes, for shafts for ocean steam- 
ships, for piston-rods, and for various other purposes. Among 
the many examples given by Mr. D. H. Browne in a paper before 
the American Institute of Mining Engineers is a case where the 
presence of 3.5 per cent nickel increased the ultimate strength 
of 0.2 per cent carbon steel from 55000 to 85000, and the elastic 
limit from 28000 to 48000 pounds per square inch, while the 
contraction of area at fracture was only decreased from 60 per 
cent to 55 per cent. 

The quality of steel to be used for different purposes differs, 
and while the specifications for any one purpose, made by different 
engineers, and by different engineering societies, often differ, the 
work of the American Society for Testing Materials is tending 
to harmonize them as far as possible. The result of their efforts 
is shown in the following set of specifications. 

AMERICAN SOCIETY FOR TESTING MATERIALS. 
SPECIFICATIONS FOR STEEL. 

STEEL CASTINGS. 

Adopted 1901. Modified 1905. 

PROCESS OF MANUFACTURE. 

1. Steel for castings may be made by the open-hearth, crucible, 
or Bessemer process. Castings to be annealed unless otherwise 
specified. 

CHEMICAL PROPERTIES. 

2. Ordinary castings, those in which no physical requirements are 
Ordinary specified, shall not contain over 0.40 per cent of carbon, 

Castings. nor ove] . Q Q g p er cent Q f ph os p norus> 

3. Castings which are subjected to physical test shall not contain 
Tested oy er 0.05 per cent of phosphorus, nor over 0.05 per cent 

Castings. of sulphun 

PHYSICAL PROPERTIES. 

4. Tested castings shall be of three classes: "hard," " medium," 
Tensile & n d "soft." The minimum physical qualities required in 
Tests * each class shall be as follows : 



STEEL CASTINGS. 



451 





Hard 


Medium 


Soft 




Castings. 


Castings. 


Castings. 


Tensile strength, pounds per square inch . 


85000 


70000 


60000 


Yield-point, pounds per square inch . 


38250 


3I5 00 


27000 


Elongation, per cent in 2 inches .... 


15 


18 


22 


Contraction of area, per cent 


20 


25 


3 



5. A test to destruction may be substituted for the tensile test in 
the case of small or unimportant castings by selecting 

three castings from a lot. This test shall show the material 
to be ductile and free from injurious defects and suitable for the pur- 
poses intended. A lot shall consist of all castings from the same melt 
or blow, annealed in the same furnace charge. 

6. Large castings are to be suspended and hammered all over. 
No cracks,- flaws, defects, nor weakness shall appear after Percussive 
such treatment. Test - 

7. A specimen one inch by one-half inch (i"Xi") shall bend cold 
around a diameter of one inch (i") without fracture on Bending 
outside of bent portion, through an angle of 120 for "soft" Test ' 
castings and of 90 for "medium " castings. 

TEST PIECES AND METHODS OF TESTING. 

8. The standard turned test specimen one-half inch (J") diameter 
and two inch (2") gauged length shall be used to determine Test Speci- 
the physical properties specified in paragraph No. 4. It Tensiie"Test. 
is shown in Fig. i. (See page 398.) 

9. The number of standard test specimens shall depend upon the 
character and importance of the castings. A test piece 

shall be cut cold from a coupon to be moulded and cast 
on some portion of one or more castings from each melt JSens? Spec " 
or blow or from the sink-heads (in case heads of sufficient 
size are used). The coupon or sink-head must receive the same treat- 
ment as the casting or castings before the specimen is cut out, and 
before the coupon or sink-head is removed from the casting. 

10. One specimen for bending test one inch by one-half inch (i"X i") 
shall be cut cold from the coupon or sink-head of the cast- jest specimen 
ing or dastings as specified in paragraph No. 9. The for Bending - 
bending test may be made by pressure or by blows. 



452 



APPLIED MECHANICS. 



n. The yield-point specified in paragraph No. 4 shall be deter- 
mined by the careful observation of the drop of the beam 

Yield-point. . J . 

or halt in the gauge of the testing-machine. 

12. Turnings from tensile specimen, drillings from the bending 

specimen, or drillings from the small test ingot, if preferred 
chemical by the inspector, shall be used to determine whether or 

not the steel is within the limits in phosphorus and sulphur 
specified in paragraphs Nos. 2 and 3. 

FINISH. 

13. Castings shall be true to pattern, free from blemishes, flaws, 
or shrinkage cracks. Bearing-surfaces shall be solid, and no porosity 
shall be allowed in positions where the resistance and value of the 
casting for the purpose intended will be seriously affected thereby. 

INSPECTION. 

14. The inspector, representing the purchaser, shall have all 
reasonable facilities afforded to him by the manufacturer to satisfy 
him that the finished material is furnished in accordance with these 
specifications. All tests and inspections shall be made at the place of 
manufacture, prior to shipment. 

STEEL FORCINGS. 

Adopted 1901. Modified 1905. 

PROCESS OF MANUFACTURE. 

1. Steel for forgings may be made by the open-hearth, crucible, or 
Bessemer process. 

CHEMICAL PROPERTIES. 

2. There shall be four classes of steel forgings which shall conform 
to the following limits in chemical composition: 





Forgings 
of Soft or 
Low-car- 
bon Steel. 


Forgings 
of Carbon 
Steel not 
Annealed. 


Forgings of 
Carbon Steel 
Oil-tempered 
or Annealed. 


Loco- 
motive 
Forg- 
ings. 


Forgings of 
Nickel Steel, 
Oil-tempered 
or Annealed. 


Phosphorus shall not exceed 
Sulphur 
Manganese " " " 
Nickel 


Per Cent. 

0. 10 
0. 10 


Per Cent. 
0.06 
0.06 


Per Cent. 

0.04 
0.04 


Per Ct, 

0.05 
0.05 
0.60 


Per Cent. 
0.04 
0.04 

3 . o to 4 . o 



Tensile 
Tests. 



PHYSICAL PROPERTIES. 

3. The minimum physical qualities required of the 
different-sized forgings of each class shall be as follows: 



STEEL FORCINGS. 



453 



Tensile 
Strength. 
Lbs. per 
Sq. In. 


Yield- 
point. 
Lbs. per 
Sq. In. 


Elonga- 
tion in 2 
Inches. 
Per Cent. 


Contrac- 
tion of 
Area. 
Per Cent. 












SOFT STEEL OR LOW-CARBON STEEL. 


58000 


29000 


28 


35 


For solid or hollow f orgings, no diameter 




- 






or thickness of section to exceed 10". 










CARBON STEEL NOT ANNEALED. 


75000 


375 


18 


30 


For solid or hollow forgings, no diameter 










or thickness of section to exceed 10". 




Elastic 










Limit. 






CARBON STEEL ANNEALED. 


80000 


40000 


22 


35 


For solid or hollow forgings, no diameter 










or thickness of section to exceed 10". 


75000 


375 


23 


35 


For solid forgings, no diameter to exceed 










20" or thickness of section 15". 


7OOOO 


35000 


24 


3 


For solid forgings, over 20" diameter. 










CARBON STEEL OIL-TEMPERED. 


90000 


55000 


2O 


45 


For solid or hollow forgings, no diameter 










or thickness of section to exceed 3". 


85000 


50000 


22 


45 


For solid forgings of rectangular sections 










not exceeding 6" in thickness or hol- 










low forgings, the walls of which do not 










exceed 6" in thickness. 


80000 


45000 


23 


40 


For solid forgings of rectangular sections 










not exceeding 10" in thickness or hol- 










low forgings, the walls of which do not 
exceed 10" in thickness. 


80000 


40000 


20 


25 


LOCOMOTIVE FORCINGS. 










NICKEL STEEL ANNEALED. 


80000 


50000 


2 5 


45 


For solid or hollow forgings, no diameter 










or thickness of section to exceed 10". 


8OOOO 


45000 


25 


45 


For solid forgings, no diameter to exceed 
20" or thickness of section 15". 


80000 


45000 


24 


40 


For solid forgings, over 20" diameter. 










NICKEL STEEL, OIL-TEMPERED. 


95000 


65000 


21 


50 


For solid or hollow forgings, no diameter 










or thickness of section to exceed 3". 


9OOOO 


60000 


22 


50 


For solid forgings of rectangular sections 










not exceeding 6" in thickness or hol- 










low forgings, the walls of which do not 










exceed 6" in thickness. 


85000 


55ooo 


24 


45 


For solid forgings of rectangular sections 










not exceeding 10" in thickness or hol- 










low forgings, the walls of which do not 










exceed 10" in thickness. 



454 APPLIED MECHANICS. 

4. A specimen one inch by one-half inch (i"Xj") shall bend 

cold 180 without fracture on outside of the bent portion, 
Test. as follows: 

Around a diameter of J n ', for forgings of soft steel. 

Around a diameter of ij", for forgings of carbon steel not annealed. 

Around a diameter of ij", for forgings of carbon steel annealed, if 
20" in diameter or over. 

Around a diameter of i", for forgings of carbon steel annealed, 
if under 20" diameter. 

Around a diameter of i", for forgings of carbon steel, oil tempered. 

Around a diameter of J", for forgings of nickel steel annealed. 

Around a diameter of i", for forgings of nickel steel, oil tempered, 

For locomotive forgjngs no bending tests will be required. 

TEST PIECES AND METHODS OF TESTING. 

5. The standard turned test specimen, one-half inch (J") diameter 
Test sped- and two (2") gauged length, shall be used to determine 
site Test. the physical properties specified in paragraph No. 3. 

It is shown in Fig. i. (See page 398.) 

6. The number and location of test specimens to be taken from 

a melt, blow, or a forging, shall depend upon its character 
LoSx>n a of d and importance, and must therefore be regulated by 
hSensf pec " individual cases. The test specimens shall be cut cold 

from the forging or full-sized prolongation of same parallel 
to the axis of the forging and half-way between the centre and outside, 
the specimens to be longitudinal; i.e., the length of the specimen to 
correspond with the direction in which the metal is most drawn out 
or worked. When forgings have large ends or collars, the test specimens 
shall be taken from a prolongation of the same diameter or section as 
that of the forging back of the large end or collar. In the case of 
hollow shafting, either forged or bored, the specimen shall be taken within 
the finished section prolonged, half-way between the inner and outer 
surface of the wall of the forging. 

7. The specimen for bending test one inch by one-half inch. 
Test Specimen ( I// Xj // ) shall be cut as specified in paragraph No. 6. 
for Bending. The bending test may be made by pressure or by blows. 



OPEN-HEARTH BOILER PLATE AND RIVET STEEL. 455 

8. The yield-point specified in paragraph No. 3 shall be determined 
by the careful observation of the drop of the beam, or 

halt in the gauge of the testing machine. Yield-point 

9. The elastic limit specified in paragraph No. 3 shall be determined 
by means of an extensometer, which is to be attached to Elastic 

the test specimen in such manner as to show the change Limit. 

in rate of extension under uniform rate of loading, and will be taken 

at that point where the proportionality changes. 

10. Turnings from the tensile specimen or drillings from the bend- 
ing specimen or drillings from the small test ingot, if pre- 
ferred by the inspector, shall be used to determine whether 

or not the steel is within the limits in chemical composition na ysis * 
specified in paragraph No. 2. 

FINISH. 

11. Forgings shall be free from cracks, flaws, seams, or other 
injurious imperfections, and shall conform to the dimensions shown 
on drawings furnished by the purchaser, and be made and finished in 
a workmanlike manner. 

INSPECTION. 

12. The inspector, representing the purchaser, sha.ll have all reason- 
able facilities afforded him by the manufacturer to satisfy him that 
the finished material is furnished in accordance with these specifications. 
All tests and inspections shall be made at the place of manufacture, 
prior to shipment. 

OPEN-HEARTH BOILER PLATE AND RIVET STEEL. 

Adopted 1901. 

PROCESS OF MANUFACTURE. 

1. Steel shall be made by the open-hearth process. 

CHEMICAL PROPERTIES. 

2. There shall be three classes of open-hearth boiler plate and 
rivet steel; namely, flange, or boiler steel, fire-box steel, and extra- 
soft steel, which shall conform to the following limits in chemical 
composition: 



APPLIED MECHANICS. 





Flange or 
Boiler Steel. 
Per Cent. 


Fire-box 
Steel. 
Per Cent. 


Extra Soft 
Steel. 
Per Cent. 


Phosphorus shall not exceed . 

Sulphur " " " . . 
Manganese 


("Acid 0.06 
\ Basic o . 04 
0.05 
o 30 to o 60 


Acid o . 04 
Basic 0.03 
0.04 
o 30 to o co 


Acid o . 04 
Basic 0.04 
0.04 











Boiler-rivet 
Steel. 



3. Steel for boiler rivets shall be of the extra-soft 
class as specified in paragraphs Nos. 2 and 4. 



PHYSICAL PROPERTIES. 
4. The three classes of open-hearth boiler plate and rivet steel- 



Tensile 

Tests. 

ities : 



namely, flange or boiler steel, fire-box steel, and extra- 
soft steel shall conform to the following physical qual- 





Flange or 
Boiler Steel. 


Fire-box 
Steel. 


Extra Soft 
Steel. 


Tensile strength, pounds 
per square inch . 
Yield-point, in pounds per 
square inch, shall not be 
less than 
Elongation, per cent in 8 
inches shall not be less 
than 


55000 to 65000 

JT.S. 

2C 


52000 to 62000 

IT. s. 

26 


45000 to 55000 
JT.S. 

28 











5. For material less than five-sixteenths inch (&") and more than 

three-fourths inch ( j") in thickness the following modifica- 

inEio'n*ation t * ons s ^ a ^ ^ e ma( ^ e m t ^ le requirements for elongation: 
for Thin and (#) For each increase of one-eighth inch (") in thick- 
ness above three-fourths inch (f") a deduction of one 
per cent (i%) shall be made from the specified elongation. 

(b). For each decrease of one-sixteenth inch (&") in thickness 
below five-sixteenths inch (&") a deduction of two and one-half per 
cent (2^%) shall be made from the specified elongation. 

6. The three classes of open-hearth boiler plate and rivet steel 
B nd'n shall conform to the following bending tests; and for this 
Tests. purpose the test specimen shall be one and one-half inches 
(ii") wide, if possible, and for all material three-fourths inch (}") or 
less in thickness the test specimen shall be of the same thickness as that 



OPEN-HEARTH BOILER PLATE AND RIVET STEEL. 

of the finished material from which it is cut, but for material more than 
three-fourths inch }") thick the bending-test specimen may be one- 
half inch (J") thick: 

Rivet rounds shall be tested of full size as rolled. 

(c). Test specimens cut from the rolled material, as specified above, 
shall be subjected to a cold bending test, and also to a quenched bending 
test. The cold bending test shall be made on the material in the con- 
dition in which it is to be used, and prior to the quenched bending test 
the specimen shall be heated to a light cherry-red, as seen in the dark, 
and quenched in water the temperature of which is between 80 and 
90 Fahrenheit. 

(d). Flange or boiler steel, fire-box steel, and rivet steel, both before 
and after quenching, shall bend cold one hundred and eighty degrees 
(180) flat on itself without fracture on the outside of the bent portion. 

7. For fire-box steel a sample taken from a broken tensile-test 
specimen shall not show any single seam or cavity more Homogeneity 
than one-fourth inch (J") long in either of the three fractures Tests - 
obtained on the test for homogeneity as described below in paragraph 12. 

TEST PIECES AND METHODS OF TESTING. 

8. The standard specimen of eight inch (8") gauged length shall be 
used to determine the physical properties specified in 
paragraphs Nos. 4 and <. The standard shape of the men for 

. , , ,, , , T^- Tensile Test. 

test specimen for sheared plates shall be as shown in rig. 
2. (See page 398.) 

For other material the test specimen may be the same as for 
sheared plates, or it may be planed or turned parallel throughout its 
entire length ; and in all cases, where possible, two opposite sides of the 
test specimens shall be the rolled surfaces. Rivet rounds and small 
rolled bars shall be tested of full size as rolled. 

9. One tensile-test specimen will be furnished from each plate as 
it is rolled, and two tensile-test specimens will be furnished 

from each melt of rivet rounds. In case any one of these 
develops flaws or breaks outside of the middle third of its 
gauged length, it may be discarded and another test specimen sub- 
stituted therefor. 



APPLIED MECHANICS. 



10. For material three-fourths inch (-") or less in thickness the 

bending-test specimen shall have the natural rolled surface 
mens S ?or CI ~ on two opposite sides. The bending-test specimens cut 

from plates shall be one and one-half inches (ij") wide, 
and for material more than three-fourths inch (") thick the bending- 
test specimens may be one-half inch (J") thick. The sheared edges 
of bending-test specimens may be milled or planed. The bending- 
test specimens for rivet rounds shall be of full size as rolled. The 
bending test may be made by pressure or by blows. 

11. One cold-bending specimen and one quenched-bending specimen 

will be furnished from each plate as it is rolled. Two 
Bending cold-bending specimens and two quenched-bending speci- 

mens will be furnished from each melt of rivet rounds. 
The homogeneity test for fire-box steel shall be made on one of the 
broken tensile-test specimens. 

12. The homogeneity test for fire-box steel is made as follows: A 

portion of the broken tensile-test specimen is either nicked 
Homogeneity with a chisel or grooved on a machine, transversely about 
Phi-box a sixteenth of an inch (;&") deep, in three places about 

two inches (2") apart. The first groove should be made 
on one side, two inches (2") from the square end of the specimen; 
the second, two inches (2") from it on the opposite side; and the third, 
two inches (2") from the last, and on the opposite side from it. The 
test specimen is then put in a vise, with the first groove about a quarter 
of an inch (J") above the jaws, care being taken to hold it firmly. The 
projecting end of the test specimen is then broken off by means of a 
hammer, a number of light blows being used, and the bending being 
away from the groove. The specimen is broken at the other two 
grooves in the same way. The object of this treatment is to open 
and render visible to the eye any seams due to failure to weld up, or to 
foreign interposed matter, or cavities due to gas bubbles in the ingot. 
After rupture, one side of each fracture is examined, a pocket lens 
being used, if necessary, and the length of the seams and cavities is 
determined. 

13. For the purposes of this specification the yield -point shall be 

determined by the careful observation of the 'drop of the 
Yield-point. ^ eam or na j t m tne g au g e of the testing machine. 



OPEN-HEARTH BOILER PLATE AND RIVET STEEL. 459 

14. In order to determine if the material conforms to the chemical 
limitations prescribed in paragraph 2 herein, analysis 

shall be made of drillings taken from a small test ingot. aiemicai r 
An additional check analysis may be made from a tensile Anal * sis - 
specimen of each melt used on an order, other than in locomotive 
fire-box steel. In the case of locomotive fire-box steel a check analysis 
may be made from the tensile specimen from each plate as rolled. 

VARIATION IN WEIGHT. 

15. The variation in cross section or weight of more than 2j per 
cent from that specified will be sufficient cause for rejection, except in 
the case of sheared plates, which will be covered by the following per- 
missible variations: 

(e) Plates 12 J pounds per square foot for heavier, up to 100 inches 
wide when ordered to weight, shall not average more than 2^ per cent 
variation above or 2\ per cent below the theoretical weight. When 
100 inches wide and over, 5 per cent above or 5 per cent below the 
theoretical weight. 

(/) Plates under 12 J pounds per square foot, when ordered to 
weight, shall not average a greater variation than the following: 

Up to 75 inches wide, 2 J per cent above or 2 J per cent below the theo- 
retical weight. Seventy -five inches wide up to 100 inches wide, 5 per cent 
above or 3 per cent below the theoretical weight. When 100 inches 
wide and over, 10 per cent above or 3 per cent below the theoretical 
weight. 

(g) For all plates ordered to gauge there will be permitted an average 
excess of weight over that corresponding to the dimensions on the order 
equal. in amount to that specified in the following table: 

TABLE OF ALLOWANCES FOR OVERWEIGHT FOR RECTANGULAR PLATES 
WHEN ORDERED TO GAUGE. 

Plates will be considered up to gauge if measuring not over -^ 
inch less than the ordered gauge. 

The weight of one cubic inch of rolled steel is assumed to be 0.2833 
pound. 



460 



APPLIED MECHANICS. 



PLATES \ INCH AND OVER IN THICKNESS. 





Width of Plate. 


Thickness of 








Plate. 
Inch. 


Up to 75 
Inches. 


75 to 100 
Inches. 


Over 100 
Inches. 




Per Cent. 


Per Cent. 


Per Cent. 


i 


IO 


14 


18 


A 


8 


12 


16 


1 


7 


IO 


13 


tk 


6 


8 


10 


1 


5 


7 


9 


A 


4* 


6J 


8i 


^ 


4 


6 


8 


Overf 


3* 


5 


6* 



PLATES UNDER INCH IN THICKNESS. 



Thickness of 
Plate. 
Inch. 


Width of Plate. 


Up to 50 
Inches. 
Per Cent. 


50 Inches 
and Above. 
Per Cent. 


up to ^j 

5 ' i t t 3 
32 16 
3 < ( "1 
16 1 


10 

7 


Si 

IO 



FINISH. 

1 6. All finished material shall be free from injurious surface defects 
and laminations, and must have a workmanlike finish. 

BRANDING. 

17. Every finished piece of steel shall be stamped with the melt 
number, and each plate and the coupon or test specimen cut from it 
shall be stamped with a separate identifying mark or number. Rivet 
steel may be shipped in bundles securely wired together with the melt 
number on a metal tag attached. 

INSPECTION. 

1 8. The inspector, representing the purchaser, shall have all reason- 
able facilities afforded to him by the manufacturer to satisfy him that 
the finished material is furnished in accordance with these specifica- 
tions. All tests and inspections shall be made at the place of manu- 
facture, prior to shipment. 



STRUCTURAL STEEL FOR BUILDINGS. 



461 



STRUCTURAL STEEL FOR BUILDINGS. 

Adopted 1901. 

PROCESS OF MANUFACTURE. 

1. Steel may be made by either the open-hearth or Bessemer process. 

CHEMICAL PROPERTIES. 

2. Each of the two classes of structural steel for buildings shall 
not contain more than o. 10 per cent of phosphorus. 

PHYSICAL PROPERTIES. 

3. There shall be two classes of structural steel for buildings, 
namely, rivet steel and medium steel, which shall con- Classes, 
form to the following physical qualities: 

4. Tensile Tests. 





Rivet Steel. 


Medium Steel. 


Tensile strength, pounds per square inch . 
Yield-point, in pounds per square inch, shall 
not be less than 


50000 to 60000 

*T. S. 


60000 to 70000 
*T. S. 


Elongation, per cent in 8 inches shall not be 
less than 


26 


22 



5. For material less than five-sixteenths inch (h") and more than 
three-fourths inch (}") in thickness the following modifica- 
tions shall be made in the requirements for elongation: Modifications 

(a) For each increase of one-eighth inch (") in thick- f^* and" 
ness above three-fourths inch (f") a deduction of one Thick Material - 
per cent (i%) shall be made from the specified elongation. 

(b) For each decrease of one-sixteenth inch (T&") in thickness 
below five-sixteenths inch (&") a deduction of two and one-half per 
cent (2^%) shall be made from the specified elongation. 

(c) For pins the required elongation shall be five per cent (5%) 
less than that specified in paragraph No. 4, as determined on a test 
specimen the centre of which shall be one inch (i") from the surface. 

6. The two classes of structural steel for buildings shall conform 
to the following bending tests ; and for this purpose the test Bending> 
specimen shall be one and one-half inches (ij") wide, if Tests - 
possible, and for all material three-fourths ( J") or less in thickness the test 
specimen shall be of the same thickness as that of the finished material 



462 APPLIED MECHANICS. 

from which it is cut, but for material more than three-fourths inch (f ") 
thick the bending-test specimen may be one-half inch (J") thick. 
Rivet rounds shall be tested of full size as rolled. 

(d) Rivet steel shall bend cold 180 flat on itself without fracture 
on the outside of the bent portion. 

(e) Medium steel shall bend cold 180 around a diameter equal 
to the thickness of the specimen tested, without fracture on the outside 
of the bent portion. 

TEST PIECES AND METHODS OF TESTING. 

7. The standard test specimen of eight-inch (8") gauged length 

shall be used to determine the physical properties specified 
men for Ten- in paragraphs Nos. 4 and 5. The standard shape of the 

test specimen for sheared plates shall be as shown by 
Fig. 2. (See page 398.) For other material the test specimen may be 
the same as for sheared plates or it may be planed or turned parallel 
throughout its entire length and, in all cases where possible, two oppo- 
site sides of the test specimen shall be the rolled surfaces. Rivet 
rounds and small rolled bars shall be tested of full size as rolled. 

8. One tensile-test specimen shall be taken from the finished 
Number of material of each melt or blow ; but in case this develops 
Tensile Tests. fl aws> or breaks outside of the middle third of its gauged 
length, it may be discarded and another test specimen substituted 
therefor. 

9. One test specimen for bending shall be taken from the finished 

material of each melt or blow as it comes from the rolls, and 



Test 

men for for material three-fourths inch (f ") and less in thickness 
this specimen shall have the natural rolled surface on two 
opposite sides. The bending-test specimen shall be one and one- 
half inches (ij") wide, if possible; and for material more than three- 
fourths inch (I") thick the bending-test specimen may be one-half 
inch (I") thick. The sheared edges of bending-test specimens may 
be milled or planed. 

Rivet rounds shall be tested of full size as rolled. 

(/) The bending test may be made by pressure or by blows. 

10. Material which is to be used without annealing or further 
Annealed treatment shall be tested for tensile strength in the con- 
dition in which it comes from the rolls. Where it is 



STRUCTURAL STEEL FOR BUILDINGS. 463 

impracticable to secure a test specimen from material which has 
been annealed or otherwise treated, a full-sized section of tensile- 
test specimen length shall be similarly treated before cutting the tensile- 
test specimen therefrom. 

11. For the purposes of this specification the yield-point shall be 
determined by the careful observaton of the drop of the Yield-point. 
beam or halt in the gauge of the testing machine. 

12. In order to determine if the material conforms 

(o the chemical limitations prescribed in paragraph No. 2 ch?micai r 
herein, analysis shall be made of drillings taken from a AnJ 
small test ingot. 

VARIATION IN WEIGHT. 

13. The variation in cross section or weight of more than 2\ per 
cent from that specified will be sufficient cause for rejection, except in 
the case of sheared plates, which will be covered by the following per- 
missible variations : 

(g) Plates 12^ pounds per square foot or heavier, up to 100 inches 
wide, when ordered to weight, shall not average more than 2j per 
cent variation above or 2\ per cent below the theoretical weight. 
When 100 inches wide, and over 5 per cent above or 5 per cent below 
the theoretical weight. 

(h) Plates under 12 J pounds per square foot, when ordered to 
weight, shall not average a greater variation than the following : 

Up to 75 inches wide, 2^ per cent above or 2\ per cent below the 
theoretical weight. Seventy-five inches wide up to 100 inches wide, 5 
per cent above or 3 per cent below the theoretical weight. When 100 
inches wide and over, 10 per cent above or 3 per cent below the 
theoretical weight. , 

(i) For all plates ordered to gauge, there will be permitted an 
average excess of weight over that corresponding to the dimensions 
on the order equal in amount to that specified in the following table : 
TABLE OF ALLOWANCES FOR OVERWEIGHT FOR RECTANGULAR PLATES 
WHEN ORDERED TO GAUGE. 

Plates will be considered up to gauge if measuring not over T ^ 
inch less than the ordered gauge. 

The weight of i cubic inch of rolled steel is assumed to be 0.2833 
pound. 



464 



APPLIED MECHANICS. 



PLATES J INCH AND OVER IN THICKNESS. 





Width of Plate. 






Plate. 








Inch. 


Up to 75 


75 to 100 


Over 100 




Inches. 
Per Cent. 


Inches. 
Per Cent. 


Inches. 
Per Cent. 


t 


10 

8 


14 

12 


18 
16 


I 


7 


10 


*3 


Tff 


6 


8 


10 


I 


5 


7 


Q 


TS 


4i 


6| 


8* 





4 


6 


8 


Over 


3* 


5 


6* 



PLATES UNDER \ INCH IN THICKNESS. 





Width of Plate. 






Plate. 






Inch. 


Up to 50 
Inches. 


50 Inches 
and Above. 




Per Cent. 


Per Cent. 


\ up to & 


10 


15 


& " " & 


8J 


za| 


A " " i 


7 


10 



FINISH. 

14. Finished material must be free from injurious seams, flaws, or 
cracks, and have a workmanlike finish. 

BRANDING. 

15. Every finished piece of steel shall be stamped with the melt or 
blow number, except that small pieces may be shipped in bundles 
securely wired together with the melt or blow number on a metal tag 
attached. 

INSPECTION. 

1 6. The inspector, representing the purchaser, shall have all reason- 
able facilities accorded to him by the manufacturer to satisfy him that 
the finished material is furnished in accordance with these specifications. 
All tests and inspections shall be made at the place of manufacture, 
prior to shipment. 



SPECIFICATIONS FOR STEEL FOR BRIDGES. 



465 



STRUCTURAL STEEL FOR BRIDGES. 

Adopted 1905. 

1. Steel shall be made by the open-hearth process. Manufacture. 

2. The chemical and physical properties shall conform chemical and 



to the following limits: 






Physical 
Properties. 


Elements Considered. 


Structural Steel. 


Rivet Steel. 


Steel Castings. 


_ f Basic . 


o 04 per cent 


o 04 per cent 




Phosphorus Max. | Acid 
Sulphur Max 


0.08 
0.05 " 


0.04 " 
0.04 " 


0.08 
0.05 " 


Lit tensile strength 


Desired 


D esired 


Not less than 


Pounds per sq in 


60,000 


50,000 


6c ooo 


Elong.: Min. per cent, in 8 in. 
(Fig. i) 


f i, =500,000* 


1,^00,000 




Uong.: Min. per cent, in 2 in. 

(Fig 2) . 


\ Ult. tens. str. 

22 


Ult. tens. str. 


18 


Character of fracture 


Silky 


Silky 




Cold bend without fracture 


1 80 flat f 


1 80 flat % 


granular. 
00 d 3t 











Retests. 



* See par. n. \ See par. 12, 13 and 14. \ See par. 15. 

The yield-point, as indicated by the drop of beam, shall be recorded 
in the test reports. 

3. If the ultimate strength varies more than 4,000 Ibs. from that 
desired, a retest may be made, at the discretion of the inspec- 
tor, on the same gauge, which, to be acceptable, shall be 

within 5,000 Ibs. of the desired ultimate. 

4. Chemical determinations of the percentages of carbon, phos- 
phorus, sulphur, and manganese shall be made by the Chemica , r^. 
manufacturer from a test ingot taken at the time of the terminations. 
pouring of each melt of steel and a correct copy of such analysis shall 
be furnished to the engineer or his inspector. Check analyses shall be 
made from finished material, if called for by the purchaser, in which 
case an excess of 25 per cent above the required limits will be allowed. 

5. Specimens for tensile and bending tests for plates, shapes, and 
bars shall be made by cutting coupons from the finished pjates shapes 
product, which shall have both faces rolled and both edges and Bar *- 
milled to the form shown by Fig. 2, page 398; or with both edges 



466 APPLIED MECHANICS. 

parallel; or they may be turned to a diameter of f inch for a length 

of at least 9 inches, with enlarged ends. 

Rivets. 6. Rivet rods shall be tested as rolled. 

7. Specimens shall be cut from the finished rolled or forged bar in 
Pins and such manner that the centre of the specimen shall be i 
Rollers. i ncn f rom the surface of the bar. The specimen for tensile 
test shall be turned to the form shown by Fig. i, page 398. The 
specimen for bending test shall be i inch by J inch in section. 

8. The number of tests will depend on the character and import- 
steel Cast- ance of the castings. Specimens shall be cut cold from 
ings. coupons moulded and cast on some portion of one or more 
castings from each melt or from the sink-heads, if the heads are of 
sufficient size. The coupon or sink-head, so used, shall be annealed 
with the casting before it is cut off. Test specimens to be" of the form 
prescribed for pins and rollers. 

9. Material which is to be used without annealing or further treat- 
Conditions m ent shall be tested in the condition in which it comes 
for Tests. from the rolls. When material is to be annealed or other- 
wise treated before use, the specimens for tensile tests, representing 
such material, shall be cut from properly annealed or similarly treated 
short lengths of the full section of the bar. 

10. At least one tensile and one bending test shall be made from 
Number of each melt of steel as rolled. In case steel differing f inch 
Tests. an d more in thickness is rolled from one melt, a test shall be 
made from the thickest and thinest material rolled. 

11. For material less than 5-16 inch and more than f inch in 

thickness the following modifications will be allowed in the 

Elongation. . r , . 

requirements for elongation: 

(a) For each 1-16 inch in thickness below 5-16 inch, a deduction 

of 2^ will be allowed from the specified percentage. 

(b) For each J inch in thickness above } inch, a deduction of i 

will be allowed from the specified percentage. 

12. Bending tests may be made by pressure or' by blows. Plates, 
Bendin shapes, and bars less than i inch thick shall bend as called 
Tests. f or m paragraph 2. 

13. Full-sized material for eye-bars and other steel i inch thick 



SPECIFICATIONS FOR STEEL FOR BRIDGES. 467 

and over, tested as rolled, shall bend cold 180 around F d 
a pin the diameter of which is equal to twice the thickness Bends 
of the bar, without a fracture on the outside of bend. 

14. Angles f inch and less in thickness shall open flat, and angles 
J inch and less in -thickness shall bend shut, cold, under Testson 
blows of a hammer, without sign of fracture. This test Angles, 
will be made only when required by the inspector. 

15. Rivet steel, when nicked and bent around a bar of the same 
diameter as the rivet rod, shall give a gradual break and a Tests on 
fine, silky, uniform fracture. Rivet stee! - 

1 6. Finished material shall be free from injurious seams, flaws, 
cracks, defective edges, or other defects, and have a smooth 
uniform, workmanlike finish. Plates 36 inches in width IS * 
and under shall have rolled edges. 

17. Every finished piece of steel shall have the melt number and 
the name of the manufacturer stamped or rolled upon it. Markin 
Steel for pins and rollers shall be stamped on the end. 

Rivet and lattice steel and other small parts may be bundled with the 
above marks on an attached metal tag. 

1 8. Material which, subsequent to the above tests at the mills and 
its acceptance there, develops weak spots, brittleness, 

cracks or other imperfections, or is found to have injurious ejecl 
defects, will be rejected at the shop and shall be replaced by the manu- 
facturer at his own cost. 

19. A variation in cross-section or weight of each piece of steel 
of more than 2\ per cent from that specified will be suffi- p ei . miss j b | e 
cient cause for rejection, except in case of sheared plates, Variations, 
which will be covered by the following permissible variations, which 
are to apply to single plates. 

WHEN ORDERED TO WEIGHT. 

20. Plates 12^ pounds per square foot or heavier: Variations? 

(a) Up to 100 inches wide, 2\ per cent above or below the pre- 

scribed weight. 

(b) One hundred inches wide and over, 5 per cent above or below. 

21. Plates under i2\ pounds per square foot: 

(a) Up to 75 inches wide, 2j per cent above or below. 



468 



APPLIED MECHANICS. 



(b) Seventy-five inches and up to 100 inches wide, 5 per cent above 

or 3 per cent below. 

(c) One hundred inches wide and over, 10 per cent above or 3 

per cent below. 



Permissible 
Variations. 



WHEN ORDERED TO GAUGE. 

22. Plates will be accepted if they measure not more 
than o.oi inch below the ordered thickness. 

23. An excess over the nominal weight corresponding to the dimen- 
sions on the order, will be allowed for each plate, if not more than that 
shown in the following tables, one cubic inch of rolled steel being 
assumed to weigh 0.2833 pound. 

24. Plates i inch and over in thickness. 



Thickness 
Ordered. 


Nominal 
Weights. 


Width of Plate. 


Up to 75". 


75" and up 
to too". 


ioo"and up 
to 115'. 


Over 1 1 5". 


14 in 
5-16 
3-8 
7-16 

1-2 
9~l6 

5-8 
Over 5-8 ' 


ch. 


10.20 It 

12.75 

15-30 
17-85 

20.40 

22.95 

25-50 


)S. 


10 p 
8 

6 

& 

4 
3* 


er ce 


nt. 


14 p 
12 
IO 

8 

i j 

5 


er ce 


nt. 


18 p 
i6 

13 

IO 

!> 

6* 


er ce 


nt. 


17 per cent. 
13 ' 

12 ' 

ii ' 

10 ' 

9 ' 





25. Plates under J inch in thickness. 



Thickness Ordered. 


Nominal Weights. 
Pounds per 
Square Feet. 


Width of Plate. 


Up to 50". 


50" and up to 

70". 


Over 70", 


1-8" up to 5-32" 
5-32" " 3-i6" 
3-16'" " 1-4" 


5.10 to 6.37 
6-37 " 7-65 

7.65 " IO.2O 


10 per cent. 
SJ " 

7 " " 


15 per cent. 

12* " " 
10 " " 


20 per cent. 
17 " " 

15 " " 



26. The purchaser shall be furnished complete copies of mill orders 
ins ction an d no material shall be rolled, nor work done, before the 
and Testing, purchaser has been notified where the orders have been 
placed, so that he may arrange for the inspection. 



STRUCTURAL STEEL FOR SHIPS. 



469 



27. The manufacturer shall furnish all facilities for inspecting and 
testing the weight and quality of all material at the mill where it is 
manufactured. He shall furnish a suitable testing machine for testing 
the specimens, as well as prepare the pieces for the machine, free of cost. 

28. When an inspector is furnished by the purchaser to inspect 
material at the mills, he shall have full access, at all times, to all parts 
of mills where material to be inspected by him is being manufactured. 

STRUCTURAL STEEL FOR SHIPS. 

Adopted 1901 for bridges and ships. Restricted to ships, 1905. 

PROCESS OF MANUFACTURE. 

1. Steel shall be made by the open-hearth process. 

CHEMICAL PROPERTIES. 

2. Each of the three classes of structural steel for ships shall con- 
form to the following limits in chemical composition : 





Steel Made by 
the Acid 
Process. 
Per Cent. 


Steel Made by 
the Basic 
Process. 
Per Cent. 


Phosphorus shall not exceed . 
Sulphur 


O.o8 
0.06 


O.o6 
0.06 



PHYSICAL PROPERTIES. 

3. There shall be three classes of structural steel for ships, 
namely, rivet steel, soft steel, and medium steel, which 

shall conform to the following physical qualities: 

4. Tensile Tests. 



Classes. 





Rivet Steel. 


Soft Steel. 


Medium Steel. 


Tensile strength, pounds 








per square inch . 


50000 to 60000 


52000 to 62000 


60000 to 70000 


Yield -point, in pounds per 








square inch shall not be 








less than 


IT.S. 


JT. S. 


JT.S. 


Elongation, per cent in 8 








inches shall not be less 








than 


26 


25 


22 



4/0 APPLIED MECHANICS. 



5. For material less than five-sixteenths inch (&") and more than 

three-fourths inch (}") in thickness the following modifi- 
Modifications cations shall be made in the requirements for elongation : 
for Thin and (#) For each increase of one-eighth inch ( J") in thickness 

'above three-fourths inch (}") a deduction of one per cent 
(i%) shall be made from the specified elongation. 

(b) For each decrease of one-sixteenth inch (TS") in -thickness 
below five-sixteenths inch (&") a deduction of two and one-half per 
cent (2 J%) shall be made from the specified elongation. 

(c) For pins made from any of the three classes of steel the required 
elongation shall be five per cent (5%) less than that specified in para- 
gaph No. 4, as determined on a test specimen, the center of which shall 
be one inch (i") from the surface. 

6. Eye-bars shall be of medium steel. Full-sized tests shall show 
Tensile Tests I2 i P er cent elongation in fifteen feet of the body of the 
of Eye-bars, eye-bar, and the tensile strength shall not be less than 
55,000 pounds per square inch. Eye-bars shall be required to break 
in the body; but, should an eye-bar break in the head, and show twelve 
and one-half per cent (12^%) elongation in fifteen feet and the tensile 
strength specified, it shall not be cause for rejection, provided that not 
more than one-third (J) of the total number of eye-bars tested break 
in the head. 

7. The three classes of structural steel for ships shall conform 
Bendin * ^ e fM wm g bending tests; and for this purpose 
Tests. the test specimen shall be one and one-half inches wide, 
if possible, and for all material three-fourths inch (f ") or less in thick- 
ness the test specimen shall be of the same thickness as that of the 
finished material from which it is cut, but for material more than 
three-fourths inch (f") thick the bending-test specimen may be one- 
half inch (") thick. 

Rivet rounds shall be tested of full size as rolled. 

(d) Rivet steel shall bend cold 180 flat on itself without fracture 
on the outside of the bent portion. 

(e) Soft steel shall bend cold 180 flat on itself without fracture on 
the outside of the bent portion. 

(/) Medium steel shall bend cold 180 around a diameter equal to 
the thickness of the specimen tested, without fracture on the outside of 
the bent portion. 



STRUCTURAL STEEL FOR SHIPS. 47 1 

TEST PIECES AND METHODS OF TESTING. 

8. The standard test specimen of eight inch (8") gauged length shall 
be used to determine the physical properties specified in 



paragraphs Nos. 4 and 5. The standard shape of the test men 
specimen for sheared plates shall be as shown by Fig. 2, 
page 398. For other material the test specimen may be the same as 
for sheared plates, or it may be planed or turned parallel throughout 
its entire length; and, in all cases where possible, two opposite sides 
of the test specimens shall be the rolled surfaces. Rivet rounds and 
small rolled bars shall be tested of full size as rolled. 

9. One tensile-test specimen shall be taken from the finished material 
of each melt; but in case this develops flaws, or breaks Numb ^ 
outside of the middle third of its gauged length, it may Tensile Tests. 
be discarded, and another test specimen substituted therefor. 

10. One test specimen for bending shall be taken from the finished 
material of each melt as it comes from the rolls, and for 
material three-fourths inch (f ") and less in thickness nSns for CI " 
this specimen shall have the natural rolled surface on Bendmg - 
two opposite sides. The bending-test specimen shall be one and one 
half inches (i J") wide, if possible, and for material more than three- 
fourths inch (I") thick the bending-test specimen may be one-half 
inch (J") thick. The sheared edges of bending-test specimens may 
be milled or planed. 

(g) The bending test may be made by pressure or by blows. 

11. Material which is to be used without annealing or further 
treatment shall be tested for tensile strength in the con- 

dition in which it comes from the rolls. Where it is imprac- Test Speci- 
ticable to secure a test specimen from material which 
has been annealed or otherwise treated, a full-sized section of tensile 
test, specimen length, shall be similarly treated before cutting the 
tensile-test specimen therefrom. 

12. For the purpose of this specification the yield-point shall be 
determined by the careful observation of the drop of the 

beam or halt in the gauge of the testing machine. 

13. In order to determine if the material conforms to 

the chemical limitations prescribed in paragraph No. 2 Ch?mica? r 
herein, analysis shall be made of drillings taken from a 
small test ingot. 



472 



APPLIED MECHANICS. 



VARIATION IN WEIGHT. 

14. The variation in cross section or weight of more than 2j per 
cent from that specified will be sufficient cause for rejection, except in 
the case of sheared plates, which will be covered by the following per- 
missible variations: 

(h) Plates i2j pounds per square foot or heavier, up to 100 inches 
wide, when ordered to weight, shall not average more than 2 J per cent 
variation above or 2^ per cent below the theoretical weight. When 
100 inches wide and over, 5 per cent above or 5 per cent below the 
theoretical weight. 

(i) Plates under 12^ pounds per square foot, when ordered to weight, 
shall not average a greater variation than the following: 

Up to 75 inches wide, 2j per cent above or 2\ per cent below the 
theoretical weight. 75 inches wide up to 100 inches wide, 5 per cent 
above or 3 per cent below the theoretical weight. When 100 inches wide 
and over, 10 per cent above or 3 per cent below the theoretical weight. 

(j) For all plates ordered to gauge there will be permitted an average 
excess of weight over that corresponding to the dimensions on the order 
equal in amount to that specified in the following table : 

TABLE OF ALLOWANCES FOR OVERWEIGHT FOR RECTANGULAR PLATES 

WHEN ORDERED TO GAUGE. 

Plates will be considered up to gauge if measuring not over T ^- 
inch less than the ordered gauge. 

The weight of i cubic inch of rolled steel is assumed to be 0.2833 

pound. 

PLATE INCH AND OVER IN THICKNESS. 





Width of Plate. 


Plate. 
Inch. 


Up to 75 
Inches. 


75 to TOO 
Inches. 


Over 100 
Inches. 




Per Cent. 


Per Cent. 


Per Cent. 





10 


14 


18 


& 


8 


12 


16 


| 


7 


10 


13 


A 


6 


8 


IO 


i 


5 


7 


Q 


& 


4* 


6* 


8i 


f 


4 


6 


8 


Overf 


3i 


5 


6* 



STEEL AXLES. 



473 



PLATES UNDER \ INCH IN THICKNESS. 



Thickness of 
Plate. 
Inch. 


Width of Plate. 


Up to 50 
Inches. 
Per Cent. 


50 Inches 
and Above. 
Per Cent. 


\ up to & 

A " "A 
& " "i* 


10 

8* 

7 


11, 

10 



FINISH. 

15. Finished material must be free from injurious seams, flaws, or 
cracks, and have a workmanlike finish. 

BRANDING. 

1 6. Every finished piece of steel shall be stamped with the melt 
number, and steel for pins shall have the melt number stamped on the 
ends. Rivets and lacing steel, and small pieces for pin plates and 
stiffeners, may be shipped in bundles, securely wired together, with the 
melt number on a metal tag attached. 

INSPECTION. 

17. The inspector, representing the purchaser, shall have all reason- 
able facilities afforded to him by the manufacturer to satisfy him that 
the finished material is furnished in accordance with these specifica- 
tions. All tests and inspections shall be made at the place of manu- 
facture, prior to shipment. 

STEEL AXLES. 

Adopted 1901. Modified 1905. 

PROCESS OF MANUFACTURE. 

1. Steel for axles shall be made by the open-hearth process. 

CHEMICAL PROPERTIES. 

2. There shall be three classes of steel axles, which shall conform 
to the following limits in chemical composition: 



474 



APPLIED MECHANICS. 





Car and 
Tender-truck 
Axles. 

Per Cent. 


Driving and 
Engine -truck 
Axles. 
(Carbon Steel.) 
Per Cent. 


Driving-wheel 
Axles. 
(Nickel-steel.) 

Per Cent. 


Phosphorus shall not exceed 


0.06 


O.o6 


O O4 


Sulphur " " " 


o 06 


c 06 


o 04 


Manganese " " " 
Nickel 




0.60 


3O to 4 O 











PHYSICAL PROPERTIES. 

3. For car and tender- truck axles, no tensile test shall 
be required. 

4. The minimum physical qualities required in the two classes of 
driving-wheel axles shall be as follows : 



Tensile Tests. 





Driving and 
Engine-truck 
Axles. 
(Carbon Steel.) 


Driving and 
Engine-truck 
Axles. 
(Nickel steel.) 


Tensile strength pounds per square inch .... 


80,000 
40,000 

20 

25 


80,000 
50,000 
25 
45 


Yield-point pounds per square inch 


Elongation per cent in two inches 


Contraction of area per cent . 





5. One axle selected from each melt, when tested by the drop test 
described in paragraph No. 9, shall stand the number of 
blows at the height specified in the following table without 
rupture and without exceeding, as the result of the first blow, the deflec- 
tion given. Any melt failing to meet these requirements will be rejected. 



Diameter of 
Axle at Center. 
Inches. 


Number of 
Blows. 


Height of 
Drop. 
Feet 


Deflection. 
Inches. 


4i 


5 


24 


81 


4 


5 


26 


8i 


4^T5 


5 


28} 


81 


4i 


5 


31 


, 8 


4f 


5 


34 


8 


5l 


5 


43 


7, 


5i 


7 


43 


Si . 



6. Carbon-steel and nickel-steel driving-wheel axis shall not 
subject to the above drop test. 



be 



STEEL AXLES. 475 



TEST PIECES AND METHODS OF TESTING. 



7. The standard test specimen one-half inch (J") diameter and 
two inch (2") gauged length shall be used to determine 

the physical properties specified in paragraph No. 4. It men fo?Ten- 

, -,-,. /c , \ sile Tests. 

is shown in rig. i. (See p. 398.) 

8. For driving and engine-truck axles one longitudinal test specimen 
shall be cut from one axle of each melt. The center of Numberand 
this test specimen shall be half-way between the center T^/e of ci _ 
and outside of the axle. mens - 

9. The points of supports on which the axle rests during tests must 
be three feet apart from center to center; the tup must DropTest 
weigh 1,640 pounds; the anvil, which is supported on Described. 
springs, must weigh 17,500 pounds; it must be free to move in a ver- 
tical direction; the springs upon which it rests must be twelve in number, 
of the kind described on drawing; and the radius of supports and of 
the striking face on the tup in the direction of the axis of the axle must 
be five (5) inches. When an axle is tested, it must be so placed in the 
machine that the tup will strike it midway between the ends; and it 
must be turned over after the first and third blows, and, when required, 
after the fifth blow. To measure the deflection after the first blow, 
prepare a straight edge as long as the axle, by reinforcing it on one side, 
equally at each end, so that, when it is laid on the axle, the reinforced 
parts will rest on the collars or ends of the axle, and the balance of the 
straight edge not touch the axle at any place. Next place the axle in 
position for test, lay the straight edge on it, and measure the distance 
from the straight edge to the axle at the middle point of the latter. 
Then, after the first blow, place the straight edge on the now bent axle 
in the same manner as before, and measure the distance from it to that 
side of the axle next to the straight edge at the point farthest away from 
the latter. The difference between the two measurements is the de- 
flection. The report of the drop test shall state the atmospheric tem- 
perature at the time the tests were made. 

10. The yield-point specified in paragraph No. 4 shall be determined 
by the careful observation of the drop of the beam or halt 

. J . Yield-point. 

in the gauge of the testing machine. 



4/6 



APPLIED MECHANICS. 



11. Turnings from the tensile-test specimen of driving and engine- 

truck axles, or drillings taken midway between the center 
Sample for and outside of car, engine, and tender-truck axles, or 
Analysis! drillings from the small test ingot, if preferred by the 

inspector, shall be used to determine whether the melt is 
within the limits of chemical composition specified in paragraph No. 2. 

FINISH. 

12. Axles shall conform in sizes, shapes, and limiting weights to the 
requirements given on the order or print sent with it. They shall be 
made and finished in a workmanlike manner, and shall be free from 
all injurious cracks, seams, or flaws. In centering, sixty- (60) degree 
centers must be used, with clearance given at the point to avoid dulling 
the shop lathe centers. 

BRANDING. 

13. Each axle shall be legibly stamped with the melt number and 
initials of the maker at the places marked on the print or indicated by 
the inspector. 

INSPECTION. 

14. The inspector, representing the purchaser, shall have all reason- 
able facilities afforded to him by the manufacturer to satisfy him that 
the finished material is furnished in accordance with these specifications. 
All tests and inspections shall be made at the place of manufacture, 
prior to shipment. 

STEEL TIRES. 
Adopted 1901. 

PROCESS OF MANUFACTURE. 

1. Steel for tires may be made by either the open-hearth or crucible 

process. 

CHEMICAL PROPERTIES. 

2. There will be three classes of steel tires which shall conform 
to the following limits in chemical composition: 





Passenger 
Engines. 

Per Cent. 


Freight-engine 
and 
Car-wheels. 
Per Cent. 


Switching- 
engines. 

Per Cent. 


Manganese shall not exceed 
Silicon shall not be less than 


0.80 
o. 20 


0.8o 
O.2O 


0.8o 

O.2O 


Phosphorus shall not exceed 


0.05 

O Os 


O.O5 
O O? 


O.O5 

O O ^ 











STEEL TIRES. 



477 



PHYSICAL PROPERTIES. 

3. The minimum physical qualities required in each of 
the three classes of steel tires shall be as follows : 



Tensile Tests. 





Passenger- 
engines. 


Freight- 
engine and 
Car-wheels. 


Switching- 
engines. 


Tensile strength, pounds per square inch. 
Elongation, per cent in two inches .... 


100,000 

12 


110,000 
IO 


120,000 

g 











Drop Test. 



4. In the event of the contract calling for a drop test, a test tire 
from each melt will be furnished at the purchaser's expense, 
provided it meets the requirements. This test tire shall 

stand the drop test described in paragraph No. 7, without breaking or 
cracking, and shall show a minimum deflection equal to D 2 -r- 
(4oT 2 +2D), the letter "D" being internal diameter and the letter 
"T " thickness of tire at center of tread. 

TEST PIECES AND METHODS OF TESTING. 

5. The standard turned test specimen, one-half inch (J") diameter 
and two inch (2") gauged length, shall be used to determine Test Speci- 
the physical properties specified in paragraph No. 3. It {ensile Tests. 
is shown in Fig. i. (See p. 398.) 

6. When the drop test is specified, this test specimen shall be cut cold 
from the tested tire at the point least affected by the drop Location of 
test. If the diameter of the tire is such that 'the whole "SSSf Speci " 
circumference of the tire is seriously affected by the drop test, or if no 
drop test is required, the test specimen shall be forged from a test ingot 
cast when pouring the melt, the test ingot receiving, as nearly as pos- 
sible, the same proportion of reduction as the ingots from which the 
tires are made. 

7. The test tire shall be placed vertically under the drop in a run- 
ning position on solid foundation of at least ten tons in Dro Test 
weight and subjected to successive blows from a tup weigh- Described, 
ing 2,240 pounds, falling from increasing heights until the required 
deflection is obtained. 

8. Turnings from the tensile specimen, or drillings from the small 
test ingot, or turnings from the tire, if preferred by the 
inspector, shall be used to determine whether the melt is chemical* 
within the limits of chemical composition specified in ' 
paragraph No. 2. 



478 



APPLIED MECHANICS. 



FINISH. 

9. All tires shall be free from cracks, flaws, or other injurious im- 
perfections, and shall conform to dimensions shown on drawings fur- 
nished t>y the purchaser. 

BRANDING. 

10. Tires shall be stamped with the maker's brand and number in 
such a manner that each individual tire may be identified. 

INSPECTION. 

11. The inspector representing the purchaser shall have all reason- 
able facilities afforded to him by the manufacturer to satisfy him that 
the finished material is furnished in accordance with these specifications. 
All tests and inspections shall be made at the place of manufacture, 
prior to shipment. 

STEEL RAILS. 

Adopted 1901. 

PROCESS OF MANUFACTURE. 

1. (a) Steel may be made by the Bessemer or open-hearth process. 

(b) The entire process of manufacture and testing shall be in accord- 
ance with the best standard current practice, and special care shall be 
taken to conform to the following instructions : 

(c) Ingots shall be kept in a vertical position in pit heating furnaces. 

(d) No bled ingots shall be used. 

(e) Sufficient material shall be discarded from the top of the ingots 
to insure sound rails. 

CHEMICAL PROPERTIES. 

2. Rails of the various weights per yard specified below shall con- 
form to the following limits in chemical composition: 





50 to 59 + 
Pounds. 
Per Cent. 


60 to 69 + 
Pounds. 
Per Cent. 


70 to 79 + 
Pounds. 
Per Cent. 


80 to 89 + 
Pounds. 
Per Cent. 


90 to 100 
Pounds. 
Per Cent. 


Carbon .... 


o. 3S 0.45 


. 38-0 . 48 


o . 400 . so 


0.430 S3 


O 4S O SS 


Phosphorus shall not 
exceed 


O. IO 


O. IO 


O. IO 


O IO 


'IO u Oo 
O IO 


Silicon shall not ex- 
ceed 


o 20 


o 20 


o 20 


o 20 




M^anganese 


o 701 oo 


o . 701 . oo 


O 7S I OS 


o 80 i 10 


o 80 i 10 















STEEL RAILS. 



479 



PHYSICAL PROPERTIES. 
3. One drop test shall be made on a piece of rail not more than six 



feet long, selected from every fifth blow of steel. The rail 
shall be placed head upwards on the supports, and the 
various sections shall be subjected to the following impact tests: 



Drop Test. 



Weight of Rail. 
Pounds per Yard. 


Height of 
Drop. 
Feet. 




45 to and including 


55"" 


15 


More than 


55 " 


65.... 


16 


< < < < 


55 


75- - 


17 


< 


75 " 


85.... 


18 




85 " 


100. . . . 


J 9 



If any rail break when subjected to the drop test, two additional tests 
will be made of other rails from the same blow of steel, and, if either of 
these latter tests fail, all the rails of the blow which they represent will be 
rejected ; but, if both of these additional test pieces meet the require- 
ments, all the rails of the blow which they represent will be accepted. 
If the rails from the tested blow shall be rejected for failure to meet the 
requirements of the drop test, as above specified, two other rails will 
be subjected to the same tests, one from the blow next preceding, and 
one from the blow next succeeding the rejected blow. In case the 
first test taken from the preceding or succeeding blow shall fail, two 
additional tests shall be taken from the same blow of steel, the accept- 
ance or rejection of which shall also be determined as specified above; 
and, if the rails of the preceding or succeeding blow shall be rejected, 
similar tests may be taken from the previous or following blows, as the 
case may be, until the entire group of five blows is tested, if necessary. 

The acceptance or rejection of all the rails from any blow will 
depend upon the result of the tests thereof. 

TEST PIECES AND METHODS OF TESTING. 

4. The drop-test machine shall have a tup of two thousand (2,000) 
pounds weight, the striking face of which shall have a Drop _ testing 
radius of not more than five inches (5"), and the test rail Machine. 
shall be placed head upwards on solid supports three test (3') apart. 
The anvil-block shall weigh at least twenty thousand (20,000) pounds, 
and the supports shall be a part of, or firmly secured to, the anvil. 



480 APPLIED MECHANICS. 

The report of the drop test shall state the atmospheric temperature at 
the time the tests were made. 

5. The manufacturer shall furnish the inspector daily with carbon 

determinations of each blow, and a complete chemical 
ChTm!ca? r ^analysis every twenty-four hours, representing the average 
Analysis. of the Qther elements conta ined in the steel. These analy- 
ses shall be made on drillings taken from a small test ingot. 

FINISH. 

6. Unless otherwise specified, the section of rail shall be the Amer- 

ican Standard, recommended by the American Society 
of Civil Engineers, and shall conform, as accurately as 
possible, to the templet furnished by the railroad company, consistent 
with paragraph No. 7, relative to specified weight. A variation in 
height of one-sixty-fourth of an inch ( T V) less and one-thirty-second 
of an inch (fa") greater than the specified height will be permitted. A 
perfect fit of the splice-bars, however, shall be maintained at all times. 

7. The weight of the rails shall be maintained as nearly as possible, 

after complying with paragraph No. 6, to that specified in 
contract. A variation of one-half of one per cent (4%) for 

an entire order will be allowed. Rails shall be accepted and paid for 

according to actual weights. 

8. The standard length of rails shall be thirty feet (30'). Ten pe 

cent (10%) of the entire order will be accepted in shorter 
lengths, varying by even feet down to twenty-four feet 

(24'). A variation of one-fourth of an inch (J") in length from that 

specified will be allowed. 

9. Circular holes for splice-bars shall be drilled in accordance with 

the specifications of the purchaser. The holes shall ac- 
curately conform to the drawing and dimensions furnished 
in every respect, and must be free from burrs. 

10. Rails shall be straightened while cold, smooth on head, sawed 

square at ends, and prior to shipment shall have the burr 
occasioned by the saw-cutting removed, and the ends made 

clean. No. i rails shall be free from injurious defects and flaws of all 

kinds. 

BRANDING. 

11. The name of the maker, the month and year of manufacture, 



STEEL SPLICE-BARS. 481 

shall be rolled in raised letters on the side of the web, and the number 
of the blow shall be stamped on each rail. 

INSPECTION. 

12. The inspector, representing the purchaser, shall have all reason- 
able facilities afforded to him by the manufacturer to satisfy him that 
the finished material is furnished in accordance with these specifications. 
All tests and inspections shall be made at the place of manufacture, 
prior to shipment. 

No. 2 RAILS. 

13. Rails that possess any injurious physical defects, or which 
for any other cause are not suitable for first quality, or No. i rails, 
shall be considered as No. 2 rails, provided, however, that rails which 
contain any physical defects which seriously impair their strength shall 
be rejected. The ends of all No. 2 rails shall be painted in order to 
distinguish them. 

STEEL SPLICE-BARS. 

Adopted 1901. 

PROCESS OF MANUFACTURE. 

1. Steel for splice-bars may be made by the Bessemer, or open- 
hearth process. 

CHEMICAL PROPERTIES. 

2. Steel for splice-bars shall conform to the following limits in 
chemical composition: 

Per Cent. 

Carbon shall not exceed - I 5 

Phosphorus shall not exceed o . 10 

Manganese o . 30-0 . 60 

PHYSICAL PROPERTIES. 
*. Splice-bar steel shall conform to the following physi- 

. ... Tensile Tests. 

cal qualities: 

Tensile strength, pounds per square inch 54,ooo to 64,000 

Yield -point, pounds per square inch ._.... 32,000 

Elongation, per cent in eight inches shall not be 

less than. 25 



482 APPLIED MECHANICS. 

4. (a) A test specimen cut from the head of the splice-bar shall 
bend 180 flat on itself without fracture on the outside of 

Bending 

Tests. the bent portion. 

(b) If preferred, the bending tests may be made on an unpunched 
splice-bar, which, if necessary, shall be first flattened, and shall then be 
bent 1 80 flat on itself without fracture on the outside of the bent por- 
tion. 

TEST PIECES AND METHODS OF TESTING. 

Test Sped- 5- A test specimen of eight inch (8") gauged length, cut 

Te e nsi f ie r Tests. fr m the head of the splice-bar, shall be used to determine 
the physical properties specified in paragraph No. 3. 

6. One tensile-test specimen shall be taken from the rolled splice- 

bars of each blow or melt ; but in case this develops flaws, 
Number of or breaks outside of the middle third of its gauged length, 
Tensile ests. ^ mav ^ discarded, and another test specimen substituted 

therefor. 

7. One test specimen cut from the head of the splice-bar shall be 

taken from a rolled bar of each blow or melt, or, if preferred, 
nfen foi^ 1 " the bending test may be made on an unpunched splice-bar 
Bending. which, if necessary, shall be flattened before testing. The 
bending test may be made by pressure or by blows. 

8. For the purposes of this specification the yield-point shall be de- 

termined by the careful observation of the drop of the beam 
Yield-point. or nalt j n the g auge o f tne testing machine. 

9. In order to determine if the material conforms to the 
Sample for chemical limitations prescribed in paragraph No. 2 herein, 
Analysis! analysis shall be made of drillings taken from a small test 
ingot. 

FINISH. 

10. All splice-bars shall be smoothly rolled and true to templet. 
The bars shall be sheared accurately to length and free from fins and 
cracks, and shall perfectly fit the rails for which they are intended. The 
punching and notching shall accurately conform in every respect to the 
drawing and dimensions furnished. A variation in weight of more 
than 2 J per cent from that specified will be sufficient cause for rejection. 



STRENGTH OF STEEL. 483 

BRANDING. 

11. The name of the maker and the year of manufacture shall be 
rolled in raised letters on the side of the splice-bar. 

INSPECTION. 

12. The inspector, representing the purchaser, shall have all reason- 
able facilities afforded to him by the manufacturer, to satisfy him that 
the finished material is furnished in accordance with these specifications. 
All tests and inspections shall be made at the place of manufacture, 
prior to shipment. 

226. Strength of Steel. The literature upon steel is 
exceedingly voluminous, and many books and articles written 
upon the metallurgy of steel, such as "Metallurgy of Steel," by 
Henry M. Howe, and "The Manufacture and Properties of Iron 
and Steel," by H. H. Campbell, contain a great many tests, which 
have, as a rule, to do with its properties and the effects of different 
compositions and treatments. They do not often contain, how- 
ever, tests upon full-size pieces, such as columns for bridges or 
buildings, beams, large riveted joints, full-size parts of machinery, 
etc. The greater part of this latter class of tests are to be found in 
the reports of the various testing laboratories, such as those "of 
the laboratories at Munich, at Berlin, and at Zurich in Europe, 
and the Watertown Arsenal reports and the Technology Quarterly 
in America; and also in various articles in the Proceedings of 
the various Engineering Societies in Europe and America. A 
number of these have already been mentioned among the refer- 
ences to tests of wrought-iron, and the greater part of them contain 
also experiments on steel. 

References to such full-size tests of steel as are quoted here 
will be given in connection with the tests themselves. 

A detailed study of the effect of the different ingredients 
and combinations of ingredients, upon the strength, elasticity, 
and ductility of steel, is a very complicated matter; it belongs 
to the study of Metallurgy and is beyond the scope of this 
work. Nevertheless, the engineer needs, of course, some general 



484 APPLIED MECHANICS. 



knowledge of these matters, and especially of the effect, within 
certain limits, of different percentages of carbon. 

This subject has been dealt with by Mr. Wm. R. Webster 
in the Trans. Am. Inst. Mining Engineers, of October, 1892, 
August, 1893, and October, 1898, and in the Journal of the Iron 
and Steel Institute, No. i, 1894; also by Mr. A. C. Cunningham 
in the Trans. Am. Soc. Civil Engineers of December, 1897; and 
by Mr. H. H. Campbell, in his book, " Metallurgy of Iron and 
Steel." Of course none of them claims anything more than 
approximation for their various rules and formulae, and then only 
in the case of what they call normal steel, i.e., such steel as is 
most frequently manufactured by the mills. 

Mr. Webster made an investigation of the effects of carbon, 
phosphorus, manganese, and sulp'hur upon the tensile strength 
of the steel. He gives a set of tables from which to determine, 
approximately, the tensile strength of normal steel, of a given 
chemical composition. His investigations were principally made 
upon basic Bessemer, and basic open-hearth steel. 

Mr. Campbell gives a formula for the tensile strength of acid, 
and another for the tensile strength of basic steel, and states that 
they represent the facts with a good degree of accuracy. His 
formulae are as follows : 

For acid steel, 

38600 + 1 2iC + 89? + R = ultimate strength; 
For basic steel, 

= ultimate strength; 



where C indicates carbon, P phosphorus, and Mn manganese, 
in units of o.ooi per cent, and R depends upon the finishing 
temperature, and may be plus or minus. 

Mr Cunningham gives the following rule: To find the approx- 
imate tensile strength of structural steel; to a base of 40000 add 
1000 pounds for every o.oi per cent of carbon, and 1000 pounds for 



STRENGTH OF STEEL. 



485 



every o.oi per cent of phosphorus, neglecting all other elements 
in normal steel. 

In this connection a set of tests will be quoted which were 
made on the government testing-machine at Watertown Arsenal, 
upon specimens of steel containing different percentages of carbon, 
the tests themselves forming a portion of a series denominated 
in the government report as the "Temperature Series." The 
account of the tests to be quoted is to be found in their report 
for 1887. 

Ten grades of open-hearth steel are here represented, in which 
the carbon ranges from 0.09 to 0.97 per cent, varying by tenths 
of a per cent as nearly as was practicable to obtain the steel. 

The other elements do not follow any regular succession. 

TENSILE TESTS OF STEEL BARS TEMPERATURE SERIES. 
Tests at Atmospheric Temperature. 







c 
i) 






a 


I 




$ 


Is 


a 


ss 


3.S 


M 


n 


1 

1 
d 


Carbon, Per Cent 


Manganese, Per C 


Silicon, Per Cent. 


Diameter, Inches. 


Sectional Area, Sq 


i. 


Length of Rest, 
Months. 


|a 
ojf 

li 


fe 

C/3 . 

&~4 
| W 


Elongation in 30 
Per Cent. 


Contraction of A 
at Fracture, 
Cent. 


Mechanical Work 
Elastic Limit, 
Inch-Lbs. 


Mechanical Work 
Tensile Streng 
in Inch-Lbs. 


Pounds per Sq. In. 
Ruptured Sectio 


753 


0.09 


O.II 




1.009 


0.80 


21000 


3 


30000 


52475 


23-6 


63-5 


15-85 


9808.36 


106434 


754 


0.20 


0.45 




1.009 


0.80 


25000 


3 


395oo 


68375 


21.2 


49.1 


26.40 


10651.90 


"3704 


755 


0.31 


0.57 




0.798 


0.50 


25000 


6 


46500 


80600 


18.0 


43-5 


37-27 


10660.77 


126640 


7560.37 


0.70 




0.798 


0.50 


25000 


6 


5OOOO 


85160 


17-5 


45-3 


42.50 


10935-48 


134600 


| 


0.58 


0.02 


0.798 


0.50 


30000 


6 


58000 


98760 


14.9 


41.6 


58.00 


11380.62 


152380 


758 


0-57 


o-93 


0.07 


0.798 


0.50 


30000 


6 


55000 


117440 


10. I 


14.0 


52-43 


11169.34 


134880 


759 


0.71 


0.58 


0.08 


0-757 


0.45 


35000 


12 


57000 


116000 


8.8 


26.2 


56.53 


9231.21 


151510 


760 


0.81 


0.56 


0.17 


0.798 


0.50 


40000 


12 


70000 


149600 


5-o 


5-4 


84-35 


7872.20 


158140 


761 


0.89 


o-57 


0.19 


0-757 


0-45 


45000 


12 


75000 


141290 


4-3 


4-4 


95-00 


6418.53 147860 


762 


o-97 


0.80 


0.28 


0-757 


0-45 


50000 


12 


79000 


'52550 


4-3 


5-8 


108.62 


7550-23 161910 



486 



APPLIED MECHANICS. 



The following tables include sets of miscellaneous tests of 
various kinds of steel. 



Bessemer Steel. 


Open-hearth Steel. 


Q u - 

' W 


| ^ 


i. 


| ,1 


O ^v 

S! 


5 . 
c 


S fc 

3 Q.^, 


t* 


la? 


ojj 
w"G 


If 


1-sjjS 


'S'- in C ' 


3 o 


si 


% 


S-6 ,.= 


^ *^ C 


Ps 


2 '^ 


*j B 

u s 


!-ssT 


$3* 


|*t 


o 
W 


!l 


l^esr 


J^sr 




Is 


.7426 


70983 


40397 


54-7 


29139000 


.7600 


64169 


47395 


56.7 


29392600 


.7481 


57700 


33000 


53-5 


28885000 


.7500 


63083 


44137 


64.0 


30179000 


7463 


58408 


28575 


58-9 


32799000 


.7600 


64477 


47394 


60. i 


30780000 


.7285 


62761 


34787 


65-2 


32135000 


.7700 


62449 


46171 


63.5 


30481000 


.7476 


50505 


19364 


72-5 


29479000 


.7700 


62556 


46171 


64.3 


29073000 


7442 


51230 


r 9550 


69-7 


30653000 


.7700 


62857 


46171 


59-5 


29073000 


75 


51110 


21503 


7 1 '5 


28457000 


.7600 


643 '5 


45189 


64.1 


29843000 


75 


51518 


21503 


50.1 


27665000 


.7650 


63527 


44600 


64.6 


29008000 


.7300 


73865 


46584 


56.8 


29600000 


.7600 


64830 


42984 


61.8 


28527000 


.7500 
.7400 
.7400 


50294 
97655 
87086 


26029 
54641 
47666 


27.0 
44.8 
46.8 


18055000 
30539000 
30090000 


755 
.7600 

7575 


65020 
65140 
65240 


45790 
45*90 
43270 


57-9 
64.9 
62.3 


29338000 
31288000 

30040000 


.7600 


87508 
65235 


50673 
49598 


48.0 
62.5 


30057000 
30310000 


7575 
.7600 


65125 
64500 


45487 
40780 


59 9 
61.9 


29912000 
30060000 


7350 


87014 


50673 


45-o 


30058000 


7550 


65089 


41320 


61.3 


291340:0 


.7400 


87356 


49991 


38.6 


30090000 


.87 


43300 


21900 


75-6 


28500000 


75 
.7420 


86720 


48665 
49650 


46.2 
47-2 


28868000 
29887000 


73 
72 


44900 
46300 


21500 

22IOO 


75-7 
73-6 


29900000 
29900000 


.7691 


60465 


35526 


61.5 


29149000 


.60 


46800 


24800 


73 '3 


30800000 


7730 


66077 


35 J 59 


61.5 


30244000 


.60 


46700 


24800 


75-0 


30000000 


.7690 


66745 


35526 


62.9 


30075000 




.7690 


66445 


39832 


62.5 


30560000 




.7690 


66142 


35526 


60.7 


27864000 




.7680 


66530 


356i8 


60.9 


29225000 




.7690 


67068 




61.5 


30075000 





Machine-steel. 


Boiler-plate. 


<u 
ffjj 

Si 


W 


i_ 

y *j . c 


C -! 

_o rt - c 
3 <3 u 


"o'S* 
II 


Section. 


. C 

"2 >'" 


0i .!U 


!< 


*o 

1-S 


I 5 


|3dSf 


Jsdsr 


1 


|s 




esr 


|;3d5f 




| 


.7608 


91795 


62693 


53-3 


29316000 


379 i-458 


5945 


31670 


47-3 


20459000 


.7629 


96256 


66723 


44-2 


29391000 


.384 1-48 


58770 


39590 


56.3 


30270000 


7633 


96767 


65561 


44.1 


29586000 


.365 1.65 




32380 


45-6 


29305000 


7520 


92087 


59665 


40-4 


30482000 


.369 1.49 


61657 


37284 




30135000 


7593 


92091 


62940 




30848000 


.398 1.511 


5437 


30760 


58.0 


28608000 


7598 


92191 


58445 


50-4 


28968000 


.376 1.496 




32889 


57-1 


28511000 


7625 


86941 


62413 


48.7 


28340000 


.4095x1.3647 


5493 


33" 




29826000 


.7623 


9575 


62445 


46.2 


30604000 


375 Xi. 494 


55173 


29451 


58.5 


28849000 


.7620 
.7620 


96045 
91220 


62495 
62495 


44.6 
51-6 


28802000 
32706000 


.3737x1.4974 
.475 x 1.0295 


54220 
47954 


31270 


g; 


27490000 


.7560 


96684 


63490 


42.8 


29400000 


4702X 1.0064 


51035 




67.5 





.7634 


96567 


66638 


43 -3 


30518000 


.4292x1.0235 


5556o 





64.9 





.7609 


92804 


60755 


51-3 


28884000 


.4258x1.0123 


54984 





67.7 




7597 


86678 


58460 


50-2 


29867000 


.4225 x 1.0025 


60680 





56.3 




.7580 


96119 


54292 


41.8 


29818000 


.4125 x 1.0025 


61500 





55-3 




. 7600 


86741 


58415 


45-7 


28738000 


40X 1.02 


5719 


26552 


58.5 


25800000 


75 T 3 


99635 


62032 


27.4 


21813000 


. 50 x i .02 


60352 


28431 


58.8 


36199000 


7613 


106980 


60413 


48.4 


29291000 


.49 X 1.02 


63825 


31010 


52.0 


25273000 


759 


96142 


54149 


45-6 


26643000 


.49 X 1.02 


60024 


29012 


58.8 


26677000 


.7699 


94513 


5907 1 


46.6 


28148000 


.50 X 1.02 


59803 


29012 


50-5 


30012000 


.7622 


9H35 


64654 


54-2 


27164000 


. 49 x i . 02 


61024 


29012 


58.8 


30012000 


.7613 


86775 


60413 


47-8 


29291000 


5GX 1.02 


60393 


29412 


60.2 


29412000 


7567 


9.6249 


61150 


45-2 


28776000 


49X 1.02 


63625 


30012 


46.1 


31866000 


7579 


95303 


67606 


44.9 


30005000 


39X 1.28 


50480 


26041 


62.5 


32051000 


754 




54870 


40.8 


29416000 


. 3 8x 1.27 


53543 


29009 


59-8 


26168000 


7554 


84990 


45752 


55-5 


29751000 


.41x1.27 


58144 


27846 


50.1 


35455000 



TENSILE STRENGTH OF STEEL. 



487 



BESSEMER STEEL WIRE. 


Diameter 
of Cross- 
section. 
(Inches.) 


Elastic 
Limit. 
(Lbs. per 
sq. in.) 


Maximum 
Load. 
(Pounds.) 


Maximum 
Load. 
(Lbs. per 
sq. in.) 


Reduction 
of Area. 
(Per cent.) 


Modulus of 
Elasticity. 
(Lbs, per 
sq. in. 


. 1290 




1013 


77500 


64.4 


30OCOOOO 


.1280 


66100 


1021 


79400 


55-9 


30400000 


.1288 


68300 


1OIO 


77500 


61.4 


30900000 


.I2yr 


67200 


970 


74100 


63-5 


30000000 


.1283 


66500 


996 


77000 


57-1 


28500000 


.1283 


69600 


IO2I 


79000 


57 - 1 


30000000 


.1289 


69700 


I00 5 


77000 


60.5 


3 i 200000 


.1281 


71400 


I43 


80900 


63-9 


29200000 


.128, 


65800 


1004 


77700 


62.6 


30700000 


.1286 


68500 


IOOO 


77000 


63.2 


31000000 


BESSEMER SPRING-STEEL WIRE. 


.0911 


76000 


9^0 


146000 


34-6 


24500000 


.0910 


69900 


974 


149000 


51 .6 


25900000 


.0905 


79600 


969 


150000 


42.0 


23000000 


.0911 


72900 


95 


146000 


37-5 


24200000 


.0905 




93i 


143000 


39-6 


25400000 



TESTS OF STEEL EYE-BARS. 

Tests of Steel Eye-bars made on the Government Machine. 
In the Tests of Metals at Watertown Arsenal for 1883 is 
the record of the tests of six eye-bars of steel, presented by 
the president of the Keystone Bridge Company. 

The following is an extract from the report in regard to 
these eye-bars : 

" The eye-bars were made of Pernot open-hearth steel, fur- 
nished by the Cambria Iron Company of Johnstown, Penn. 

"The furnace charges, about 15 tons each of cast-iron, 
magnetic ore, spiegeleisen, and rail-ends, preheated in an aux- 
iliary furnace, required six and one-half hours for conversion. 

" All these bars were rolled from the same ingot. 

" Samples were tested at the steel-works taken from a test 
ingot about one inch square, from which were rolled |-inch 
round specimens. 



488 



APPLIED MECHANICS. 



"The annealed specimen was buried in hot ashes while still 
red-hot, and allowed to cool with them. 

" The following results were obtained by tensile tests : 





Elastic 
Limit, in 
Ibs., per 
Sq. In. 


Ultimate 
Strength, in 
Ibs., per 
Sq. In. 


Contrac- 
tion of 
Area. 


Modulus 
of 
Elasticity. 


Carbon. 


f-inch round rolled bar . 


48040 


73 I 5 


% 

45-7 


28210000 


%- 

0.27 


f-inch round rolled and 












annealed bar .... 


422IO 


69470 


54-2 


292IOOOO 


0.27 



" The billets measured 7 inches by 8 inches, and were 
bloomed down from 14-inch square ingot. 

" They were rolled down to bar-section in grooved rolls at 
the Union Iron Mills, Pittsburgh. 

" The reduction in the roughing-rolls was from 7 inches by 
8 inches to 6J inches by 4 inches ; and in the finishing-rolls, to 
6^ inches by I inch. 

"The eye-bar heads were made by the Keystone Bridge 
Company, Pittsburgh, by upsetting and hammering, proceeding 
as follows : 

"The bar is heated bright red for a length of (approxi- 
mately) 27 inches, and upset in a hydraulic machine ; after 
which the bar is reheated, and drawn down to the required 
thickness, and given its proper form in a hammer-die. 

"The bars are next annealed, which is done in a gas-furnace 
longer than the bars. They are placed on edge on a car in the 
annealing-furnace, separated one from another to allow free 
circulation of the heated gases. They are heated to a red heat, 
when the fires are drawn, and the furnace allowed to cool. 
Three or four days, according to conditions, are required before 
the bars are withdrawn. 



TENSILE STRENGTH OF STEEL. 489 

" The pin-holes are then bored. 

"The analyses of the heads before annealing were: 

" Carbon, by color 0.270 per cent 

Silicon 0.036 " 

Sulphur -75 " 

Phosphorus 0.090 " 

Manganese 0.380 " 

Copper Trace. 

" The bars were tested in a horizontal position, secured at 
the ends, which were vertical. 

" To prevent sagging of the stem, a counterweight was used 
at the middle of the bar. 

" Before placing in the testing-machine, the stem from neck 
to neck was laid off into lo-inch sections, to determine the 
uniformity of the stretch after the bar had been fractured. 

"A number of intermediate lo-inch sections were used as 
the gauged length, obtaining micrometer measurements of 
elongation, and the elastic limit for that part of the stem -which 
was not acted upon during the formation of the heads. Elon- 
gations were also measured from centre to centre of pins, taken 
with an ordinary graduated steel scale. 

" The moduli of elasticity were computed from elongations 
taken between loads of 10000 and 30000 Ibs. per square inch, 
deducting the permanent sets. 

"The behavior of bars Nos. 4582 and 4583, after having 
been strained beyond the elastic limit, is shown by elongations 
of the gauged length measured after loads of 40000 and 50000 
Ibs. per square inch had been applied ; and with bar No. 4583, 
after its first fracture under 64000 Ibs. per square inch, a rest 
of five days intervening between the time of fracture and the 
time of measuring the elongations. 

"Considering the behavior between loads of roooo and 
30000 Ibs. per square inch, we observe the elongations for the 



49 APPLIED MECHANICS. 

primitive readings are nearly in exact proportion to the incre- 
ments of load. 

" Loads were increased to 40000 Ibs. per square inch, passing 
the elastic limit at about 37000 Ibs. per square inch ; the respec- 
tive permanent stretch of the bars being 1.31 and 1.26 per cent. 

" Elongations were then immediately redetermined, which 
show a reduction in the modulus of elasticity, as we advanced 
with each increment, of 5000 Ibs. per square inch. 

" Corresponding measurements after the bars had been 
loaded with 50000 Ibs. per square inch reach the same kind of 
results. 

"The first fracture of bar No. 4583, under 64000 Ibs. per 
square inch, occurred at the neck, leaving sufficient length to 
grasp in the hydraulic jaws of the testing-machine, and con- 
tinue observations on the original gauged length. This was 
done after the fractured bar had rested five days. 

" The elongations now show the modulus of elasticity con- 
stant or nearly so, the only difference in measurements being 
in the last figures, up to 50000 Ibs. The readings were then 
immediately repeated, and the same uniformity of elongations 
obtained. 

" An illustration of the serious influence of defective metal 
in the heads is found in the first fracture of bar No. 4583. 

" There was about 27 per cent excess of metal along the 
line of fracture over the section of the stem." 



TENSILE STRENGTH OF STEEL. 



49 I 













z 


s * 





2 


W* W^ 

^* ft 


S 


!?*(. 


8. 8. 





w 




g S 


3- fr 


* f* 


r* 


J* 


1 1 


r* ^ 


=r 5 ,S 


W N 


<s? 


3 




f S 


8 S, 


8 8 


8 





8- 


- r 


Gauged Length, in inches. 


C 1" 


i 


C 


1 1 


ON 00 


Width, in inches. 


p p 

vO <O 

V| V| 


p 


p 


1 1 


P P 


N! 

j< . w PT 


W W 


OJ 


(A) 




w w 


^w 5V_w 


O> M 


CO 





1 1 


oJ ^ 


o C "ft -; ^ 








8 







s j" -" s 1 


If 


i 


1 


"3 ^ 



<8 o* 


f ! 


[fvril 


OJ Ov 


M 

M 

8 


M 
10 

^ 


OJ 


1 1 


In Gauged W 
Length. J 


<2 < 


M 


M 




1 ^* 


Centre to Cen- S 5- 


in M 


w 







' vb 


tre of Pins. ' ? 


* -f" 


% 


S 


00 ON 


W 


Contraction of Area, per 


* *" 


w 


M 


*. <J\ 


^ 


cent. 


NO ON 


f 


S 


1 I 


it 


W 


ff 3 


f 


s 


1 1 


2 


Maximum Compression 
on Pin-Holes, in Ibs., 


8 s 8 s 


6 


o 




o 8 


per Square Inch. 








W 


ed w 




s s 


8 


5 


* 1 
n 


1 1 


*J* 


s s 


S 


S 


R 5' 


S 5' 


3 I- 


5 5 


a 


8 


& 8 


s 


H 










S- 




3 ?. 





fi 


^^ ^ 


O Jrt 


^ 


8 Z 


p 


JC 

< 


1 8 


i I s 


2 


II 


5r 
vj 


1 

w 


ff. < 
P I f 


c rj 

? g. 


i 


I ? 


t> 

i 


1 


S s 

o- 


c 


3, 


i 


^ 


$ 


2, ? 


55 


3 1 


o 


3 




W B 




O 














? 







? P 




p 



492 



APPLIED MECHANICS. 



ELONGATIONS OF No. 4582 FOR EACH INCREMENT OF 5000 LBS. PER 

SQUARE INCH. 



Loads, in Ibs., 
per 
Square Inch. 


Elongations. 


Primitive Load- 
ing. 


After Load of 
40000 Ibs. per 
Square Inch. 


After Load of 
50000 Ibs. per 
Square Inch. 


IOOOO 


_ 


. 


_ 


15000 
20000 
25000 


0.0274 
0.0269 
0.0269 


0.0300 
0.0305 
0.0320 


0.03II 
0.0322 
0.0337 


30000 


0.0269 


0.0330 


0.0341 



ELONGATIONS OF No. 4583 FOR EACH INCREMENT OF 5000 LBS. PER 

SQUARE INCH. 



Loads, in 


Elongations. 


Elongations after 64000 Ibs. per 
Square Inch. 


Ibs., per 












Square Inch. 


Primitive Load- 


After 40000 Ibs. 


After 50000 Ibs. 


First 


Second 




ing. 


per 
Square Inch. 


per 
Square Inch. 


Reading. 


Reading. 


IOOOO 


_ 


. 


. 


. 


. 


15000 


0.0272 


0.0291 


0.0302 


0.0311 


0.0310 


2OOOO 


0.0272 


0.0305 


0.0315 


0.0308 


0.0310 


25OOO 


0.0268 


0.0314 


0.0325 


0.0311 


0.0310 


3OOOO 


0.0267 


0.0326 


0.0340 


0.0312 


0.0310 


35000 


- 


- 


- 


0.0311 


- 


40000 


- 


- 


- 


0.0312 


- 


45000 


- 


- 


- 


0.0310 


- 


50000 


" 


~ 


" 


0.0315 





In the Tests of Metals for 1886 is given the following table 
of tensile tests of steel eye-bars, furnished by the Chief Engineer 
of the Statue of Liberty. 



TENSILE STRENGTH OF STEEL. 



493 



Dimensions. 


ft 


ft 


Elongation. 


i 


. 


*l- 


Fractxire. 


o , 








f< 




. 




+3 o 


Q, rj O 












|| 






V 

M 


*o 

c 


J~ 


cS(^ 






























-S 

bo "0 

F 


Width. 


Thickness 


Elastic 
Square 


Tensile S1 
Square 


it 
fj 


Center to 
of Pin-1 


Contracti 


Modulus c 
per Sqx 


Maximun 
sion on 
per Sqx 


Location. 


Appearance. 


Ins. 


Ins. 


Ins. 


Lbs. 


Lbs. 


% 


% 


% 


Lbs. 


Lbs. 






308.00 


5- 16 


I .02 


34610 


64870 


7-4 


7-3 




31400000 


74173 






308.00 


5-14 


I .02 


34730 


69330 


10.4 


10.3 




29279000 


84093 






308.00 


5-15 


I .02 


37330 


70286' 11.7 


ii. 5 




29017000 


80043 






308. 10 


5-14 


I .02 


35000 


70229 


ii. 6 


ii-4 


13-4 




79826 


Stem 


Granular, radi- 
























ating from a 
























button of 
























hard metal. 


308.00 


5-13 


I .02 


35950 


71680 


ii. 8 


ii. 5 






81323 






307.95 


5-iS 


I .02 


35000 


70895 


12. I 


ii. 8 




30162000 


80737 







The gauged length of the bars was 260 inches. The moduli 
of elasticity computed between 25000 and 30000 pounds per 
square inch. 

In connexion with the work upon the bridge over the Missis- 
sippi at Memphis, Mr. Geo. S. Morison, the Chief Engineer, 
had 56 full-size stee eye-bars tested. The results are given in 
his Report, dated March, 1894, and furnish valuable information 
regarding the behavior of the steel, and the design, and con- 
struction of the bars. Only the following table (see page 494) 
will be given here, containing a portion of the results of the tests 
upon 31 of the bars, all made of basic open-hearth steel, and 
all of which broke in the body. 

This table will aid the reader in comparing the tensile strength 
and the limit of elasticity of full-size steel eye-bars, with those 
obtained from the tests of small samples of the steel. 

In Engineering News of Feb. 2, 1905, 'is an article containing 
a comparison of full-size and specimen tests of eleven steel eye- 
bars, made at the Phoenix Iron Co. Each of these bars was 15 
inches wide; two of them were ij inches thick; one was i& 
inches thick, six were 2 inches thick, and two were 2& inches 
thick. The specimen tests gave tensile strengths varying from 
60310 to 67000 pounds per square inch, and limits of elasticity 
varying from 31550 to 41760 pounds per square inch. 



494 



APPLIED MECHANICS. 



FULL-SIZE EYE-BARS. 



SAMPLE BARS FROM SAME MELT. 







u 


u 


J. 




i 


Jl 


Ii 


. 


jd 


a 
44 


o 


Js^ 


ff 







c O 


W & 


*& 


P 


'1 


JT 


3 




Ins. 


Ins. 


Ins. 


Lbs. 


Lbs. 


10.07 


1.50 


160.63 


35 ICO 


67490 


9-95 


1-73 


358.93 


3768o 


70160 


9.98 


i-75 


361-23 


39700 


65500 


10.05 


1.50 


162.38 


33140 


65060 


6.08 




291.26 


29690 


56700 


10.07 


iley 


287.37 


32860 


65600 


9.92 




284.28 


31110 


61060 


9-94 


0-99 


287.88 


3399 


63220 


10.05 


2.20 


222.88 


2933 


63100 


10. 12 


1.86 


464.03 


31970 


53860 


7-12 


1.17 


314.04 


30270 


51500 


IO.07 


2.20 


338.73 


28080 


55160 


10.03 


I.8l 


25L58 


29670 


62140 


9-97 


i-37 


250.28 


32700 


65400 


7.02 




385.73 


28980 


52010 


7.01 


i! 2 6 


385.78 


28410 


54740 


9-99 


1.62 


249.98 


30500 


58870 


9.96 


2.05 


341.28 


3336o 


7355 


10.13 


1.30 


249.48 


32520 


60710 


9.98 


1.81 


284.82 


28000 


58720 


10. 15 


1-83 


221 .98 


32290 


62270 


10.04 


o-99 


361.68 


29970 


58680 


7.01 


1.27 


258.68 


28640 


56830 


7.98 


i .20 


254.63 


31930 


63870 


8.03 


2.32 


338.58 


32840 


62400 


7.00 


1.18 


258.68 


27870 


53520 


9.09 


1.25 


206.58 


3259 


574io 


8. ii 


1.79 


279.98 


28940 


58010 


7.00 


i .00 


289.23 


31380 


59850 



t 




3d 


i-5 c 




I* 


-"+J 


e~ 




P *-* 


"3 
| 


Is 


So* 


If 


fs 


1 


W 


Js 

w 


1 


O, 


Sq. In. 




Lbs. 


Lbs. 




9500 


27-5 


41580 


73050 


.027 


.9918 


24.4 


42650 


75620 


015 


.9520 


28.8 


40280 


70280 


.062 


.9500 


27-5 


41580 


73050 


.027 


9756 


28.1 


40490 


69700 


.026 


1.1590 


20. o 


43750 


75000 


.021 


1.0140 


28.8 


42210 


69730 


.046 


.9868 


28.1 


40230 


69720 


.025 


9635 


28.8 


38090 


71300 


.017 


I .O2OI 


27.0 


40200 


71860 


.017 


I .0180 


28.8 


33400 


57170 


.014 


. I22O 


24.2 


38320 


70220 


.023 


.O20O 


26.3 


40200 


71080 


.028 


.0670 


25.0 


3936o 


69360 


.041 


. I70O 


31-3 


34190 


58460 


039 


.0170 


28.1 


41400 


67840 


.OIO 


9338 


25.0 


40910 


70360 


.014 


.9700 


25-5 


40410 


69900 


.063 


954 


27.0 


40400 


70490 


.023 


5557 


29-5 


40000 


66800 


.008 


.9746 


21.3 


40530 


72240 


.056 


.1720 


27.0 


40610 


70480 


.O6O 


.0200 


28.! 


4079 


68730 


.030 


.0100 


21.9 


40900 


69800 


.024 


.0620 


23.1 


41710 


71000 


.066 


.0560 




32480 


58050 


.027 


9734 


28^7 


38110 


60920 


.014 


.114 


23.0 


40480 


66880 


.030 


.020 


28.1 


40790 


68730 


.030 



The decrease of tensile strength in the full-size eye-bars 
varied from 6.3 per cent to 11.9 per cent, while the decrease in 
elastic limit varied from 8.3 per cent to 17 per cent. 

STEEL COLUMNS. 

In the Trans. Am. Soc. C. E., of June, 1889, will be found a 
paper by Mr. J. G. Dagron, giving an account of a set of tests of 
eight full-size Bessemer-steel bridge columns, made for the Sus- 



STEEL COLUMNS. 



495 



quehanna River Bridge of the Baltimore and Ohio R.R. The steel 
varied in tensile strength from 83680 to 84440 pounds per square 
inch, in elastic limit from 51190 to 53890 pounds per square inch, 
in elongation in 8 inches from 18.75 P er cent to 2O -75 P er cent, 
and in contraction of area from 30.55 per cent to 39.7 per cent. 
The columns were made by the Keystone Bridge Company and 
tested in their hydraulic press, with the columns in a horizontal 
position, and with the pins horizontal. 

The results obtained are given by the accompanying table : 



No. of 
Column. 


Depth. 
Inches. 


Sectional 
Area. 
Sq. Ins. 


Length 
Center to 
Center 
Pin-holes. 


Ratio of 
Length to 
Radius of 
Gyration. 


Square of 
Radius of 
Gyration. 


Ultimate 
Strength, 
in Lbs. 
per 
Sq. In. 


Modulus of 
Elasticity. 
Lbs. 
per Sq. In. 


I 


8 


8.24 


i6'o' 


42.05 


20.86 


41020 


27705000 


2 


8 


8.24 


i6'o' 


42.05 


20.86 


41650 


27705000 


3 


8 


8.24 


20' 0' 


5 2 -5 6 4 


20.86 


39440 


26113000 


4 


8 


8.24 


20' 


52.5 6 4 


20.86 


41050 


25816000 


5 


8 


8.24 


2 4 '0' 


63.075 


20.86 


40230 


29504000 


6 


8 


8.24 


24'0' 


63-075 


20.86 


40070 


28398000 


7 


9 


13-23 


25'7-T 


5 s -795 


27-34 


35570 


26557000 


8 


9 


13-23 


25-7-i" 


58.795 


27.34 


38810 


29478000 



The columns failed as follows : 



i. 



No. 



No. 



No. 3. 
No. 4 

No. 5 



Failed by bending downwards at rivet in latticing, i foot 

loj inches from the center, buckling flange angles and 

web-plate. 
2. Failed by bending upwards at rivet in latticing at center, 

buckling flange angles and web-plate. One angle 

was fractured at point of buckling, and also at the two 

adjacent rivets in latticing 
Failed by bending upwards between latticing, 3 feet from 

center, buckling flange angles and web-plate. 
Failed by bending upwards between latticing, 4 inches from. 

center, buckling flange angles and web-plate. 
Failed by bending upwards between latticing, 9^ inches 

from center, buckling flange angles and web-plate. 



49<5 



APPLIED MECHANICS. 



No. 6. Failed by bending upwards between latticing, i foot 

5! inches from center, buckling flange angles and web 

plate. 
No. 7. Failed by bending upwards at rivet in latticing, 3 inches 

from center, buckling flange angles and web-plate. 
No. 8. Failed by bending upwards at rivet in latticing, i foot 

from center, buckling flange angles and web-plate. 

In every case, after test, the rivets of each column were found 
by hammer test to be perfectly right. 

The following table gives the results of a set of tests by direct 
compression, of eight connecting-rods specially made for these 
tests, by the Baldwin Locomotive Works, and tested in the Labora- 
tory of Applied Mechanics of the Mass. Institute of Technology. 













Breaking- 








Area. 


Tensile Properties of the Steel. 


strength per Sq. 




o *. 








In. of the Rod. 














,3 


oo 












!3 


J_ 


g 


g 


'(3 


bo 





g 


Modulus 


d 
o 


c 




-i< 


P 


"o 


"o 




Id- 
*w 


ll 

SO" 


J 

'li 


of 
Elasticity 
per 


1 


o 


1 


f<3 




1 


re 


J"^ % 
ft 


|& 




C 


Sq. In. 


1 


1 




M 




M 







H 


w 


O 




M 






Ins. 




Sq.In. 


Sq.In. 


Lbs. 


Lbs. 


Pr.Ct. 


Pr. Ct. 


Lbs. 


.Lbs. 


Lbs. 


A 


89-38 


100.5 


7.19 


7 .6o 


57730 


80280 


25-8 


30.9 


28000000 


38700 


36700 


B 


98-38 


109.4 


7.19 


7.78 


45 6 5o 


78830 


20.8 


34.1 


28300000 


40600 


37500 


C 


107.38 


118.5 


6-73 


7.21 


43900 


77840 


20.4 


42.5 


30000000 


39300 


36700 


D 


in-75 


125 .0 


7.27 


7.78 


4.7560 


79270 


22.3 


43-2 


28500000 


36100 


33700 


E 


116.25 


130.0 


7.38 


7.96 


45820 


79250 






30500000 


39300 


36400 


F 


120.63 


134.8 


7.21 


7-55 


49440 


81660 


24.1 


39-9 


28800000 


39300 


37500 


G 


125-13 


139-7 


7.06 




3959 


79690 


24.4 


45-5 


30300000 


38000 


35000 


H 


134-13 


149.4 


7.28 


7*78 


39470 


78650 


21.0 


28.3 


30800000 


37400 


35oo 



TRANSVERSE STRENGTH OF STEEL. 



The following table gives the results of tests of a number of 
steel I beams, made in the Laboratory of Applied Mechanics 
of the Mass. Institute of Technology. 



^S o E V B E ^ 

UNIVERSITY 



OF 



TRANSVERSE S: 



T GTH OF STEEL. 



497 













Mo- 


Modulus 








Mo- 




Break- 


dulus of 


of 




No. of 
Test. 


Depth. 
Inches. 


ment of 
Inertia. 


Span. 
Feet and 


ing 
Load. 


Rup- 
ture 


Elasticity 
per 


Remarks. 






Ins. 


Inches. 


Lbs. 


per 

Sq. In. 


Sq. In. 
Lbs. 














Lbs. 






290 


7 


38.00 


14' 6" 


10500 


42874 


29030000 


From Phoenix Co. 


2 93 


8 


57-11 


14' 6" 


14200 


44270 


29410000 


t t 1 1 it 


2 95 


9 


81.34 


14' 6" 


16700 


40200 


29890000 


n 1 1 it 


337 


6 


24.86 


14' 7" 


8200 


44900 


28170000 


N. J. Iron & Steel Co. 


340 


7 


39.63 


12' II" 


I2OOO 


42100 


27480000 


< < if < < it 


343 


8 


5 J -67 


14' 7" 


14900 


46400 


29040000 


11 I ( < C (I 


63ia 


10 


i34.oo 


14' o" 


24200 


3 8 500 


28400000 


Carnegie Steel Co. 


638 


10 


134.00 


14' o" 


25100 


395 


29300000 


( ( i i t ( 


674 


IO 


129.00 


14' o" 


249OO 


41300 


27450000 


1C C C t I 


675 


IO 


131 .20 


14' o" 


25600 


41700 


27850000 


C ( t ( t I 



In Heft IV of the Mitth. der Materialprtifungsanstalt in 
Zurich are given the following results of tests of the transverse 
strength of ten steel plate girders : 



Depth of 
Web. 
(Inches.) 


Span. 
(Inches.) 


Modulus of 
Rupture. 
(Lbs. per 
sq. in.) 


Modulus of 
Elasticity. 
(Lbs. persq.in.) 


19.76 


177.17 


53325 


29193660 


19.76 


I77.I7 


55316 


27430380 


15-75 


I4L73 


55174 


26662500 


15-75 


I4L73 


55316 


28738620 


19.69 


177.17 


53325 


29193560 


19.69 


177.17 


55316 


27430380 


23.62 


2I2.6O 


57591 


28795500 


23.62 


2I2.6O 


52472 


28155600 


27.56 


248 . 03 


54320 


27529920 


27.56 


248.03 


53041 


28752840 



498 APPLIED MECHANICS. 



COLD CRYSTALLIZATION OF IRON AND STEEL. 

The question of cold crystallization of wrought-iron and 
steel is one that has been agitated from the earliest times, and, 
although Kirkaldy tried to dispose of it finally by offering evi- 
dence showing that it does not exist, nevertheless we find the 
same old question cropping out every little while, and although 
the bulk of the evidence is admitted to be against it, and, as it 
seems to the writer, there is no evidence in its favor, we find 
every now and then some one who thinks that certain observed 
phenomena can be explained in no other way. 

The most usual phenomenon which cold crystallization is 
called upon to explain is the crystalline appearance of the 
fracture of some piece of wrought-iron or steel that has been 
in service for a long time, and which has, as a rule, been sub- 
jected to more or less jars or shocks. The cases most fre- 
quently cited are those of axles of some sort which have been 
broken, and, in the case of which, the fracture has had a crys- 
talline appearance, and where samples cut from the other parts 
of the axle and tested have shown a fibrous fracture. The 
assumption has therefore been made that the iron was origi- 
nally fibrous, and that crystallization has been caused by the 
shocks or the jarring to which it has been subjected in the 
natural service for which it was intended. 

Kirkaldy showed (see his sixty-six conclusions) that when 
fibrous iron was broken suddenly, or when the form of the 
piece was such as not to offer any opportunity for the fibres to 
stretch, the fibres always broke off shorthand the fracture was 
at right angles to their length, and hence followed the crystal- 
line appearance ; whereas if the breaking was gradual, and the 
fibres had a chance to stretch, they produced a fibrous appear- 
ance : in short, he claimed that the difference between the crys- 
talline or the fibrous appearance of the fracture was only a 



COLD CRYSTALLIZATION OF IRON AND STEEL. 499 

difference of appearance, and not a change of internal structure 
from fibrous to crystalline. 

The facts that Kirkaldy showed in this regard are generally 
acknowledged to-day, and doubtless answer by far the greater 
part of those who claimed cold crystallization at the time that 
he wrote, and also a great many of those who claim its exist- 
ence to-day. 

But it is easy, if suitable means be taken, to distinguish 
cases of crystalline appearance of fracture from cases where 
there are actual crystals in the piece ; and it is rather about 
those cases where the iron near the fracture actually contains 
distinct crystals that what discussion there is to-day that is 
worth considering takes place. 

The number of such cases is, of course, small, but every 
once in a while some one is cited, and the claim is put forward 
that the iron was originally fibrous, and that these crystals 
must therefore have been produced without heating the iron 
to a temperature where chemical change is known to occur. 

Inasmuch as the one who claims the existence of cold crys- 
tallization is announcing a theory which is manifestly opposed 
to the well-known chemical law that crystallization requires 
freedom of molecular motion, and hence can only take place 
from solution, fusion, or sublimation, it follows that the burden 
of proof rests with him, and before he can substantiate his 
theory in any single case he must prove beyond the possibility 
of doubt, i, that the iron or steel was originally fibrous, i.e., 
not only that fibrous iron was used in manufacturing the pieces, 
but also that it had not been overheated during its manufac- 
ture, and, 2, that it has never been overheated during its period 
of service. 

It is because the writer is not aware of any case where these 
two circumstances have been proved to hold that he says that 
he knows of no evidence for cold crystallization. In this con- 
nection it is not worth while to quote very much of the exten- 



500 APPLIED MECHANICS. 

sive literature on the subject ; hence only a little of the most 
modern evidence will be given here. 

On page 1007 et seq. of the report of tests on the govern- 
ment testing-machine at Watertown Arsenal for 1885 is given 
an account of a portion of a series of tests upon wrought-iron 
railway axles, and the following is quoted from that report : 

" This series of axle tests, begun September, 1883, is carried 
on for the purpose of determining whether a change in struc- 
ture takes place in a metal originally ductile and fibrous to a 
brittle, granular, or crystalline state, resulting from exposure 
to such conditions as are met with in the ordinary service of a 
railway axle. 

" Twelve axles were forged from one lot of double-rolled 
muck-bars, and in their manufacture were practically treated 
alike. Each axle was made from a pile composed of nine bars, 
each 6 in. wide, f in. thick, and 3 ft. 3 in. long, and was finished 
in four heats, two heats for each end. 

" The forging was done by the Boston Forge Company in 
their improved hammer dies, which finish the axle very nearly 
to its final dimensions. 

"Two axles were taken for immediate test, to show the 
-quality of the finished metal before it had performed any rail- 
way service, and serve as standards to compare with the 
remaining ten axles, to be tested after they had been in 
use. 

" The axles are in use in the tender-trucks of express loco- 
motives of the Boston and Albany Railroad. Mr. A. B. Under- 
hill, superintendent of motive-power, contributes the axles and 
furnishes the record of their mileage." 

The results of some measurements of deflection are given 
concerning one of the axles in tender 134, after it had run 
95000 miles ; and then follows : 

" Regarding the axle for the time being as cylindrical, 3.96 



COLD CRYSTALLIZATION OF IRON AND STEEL. 50 1 



inches diameter, the modulus of elasticity by computation will 
be 28541000 pounds. 

" Applying this modulus to the deflections observed in rear 
axle of the rear trucks of tender No. 150, the maximum fibre 
strain is found to be 9935 pounds per square inch when the 
tender was partially loaded, and 14900 pounds per square inch 
when fully loaded. 

" Taken together, the tensile and compressive stresses, 
which are equal, amount to 19870 and 29800 pounds per square 
inch respectively, as the range of stresses over which the metal 
works. 

" This definition of the limits of stresses must be regarded 
as approximate. There are influences which tend to increase 
the maximum fibre strain, such as unevenness of the track, the 
side thrust of the wheel-flanges against the rails. On the other 
hand, the inertia of the axle, particularly under high rates of 
speed, would exert a restraining influence on the total deflec- 
tion. 

" Nine tensile specimens were taken from each axle ; three 
from each end, including the section of axle between the box 
and wheel bearings, and three from the middle of its length. 
They are marked M.B., with the number of the axle ; also a 
sub-number and letter to indicate from what part of the axle 
each was taken. 

" The tensile test-pieces showed fibrous metal, and generally 
free from granulation. 

" The muck-bar had a higher elastic limit and lower tensile 
strength, and less elongation than the axles. The moduli of 
elasticity of the two are almost identical. 

" Between loads of 15000 and 25000 pounds per square inch 
the muck-bar had a modulus of elasticity of 29400000 pounds, 
the axles (average of all specimens) between 5000 and 20000 
pounds per square inch was 29367000 pounds. Individually 



502 APPLIED MECH AXILS. 

the axles showed the modulus of elasticity to be substantially 
the same in each." 

Two specimens were subjected to their maximum load and 
removed from the testing-machine before breaking in order to 
see whether the straining followed by rest will cause any 
change. 

" It does not appear from these tests that 95000 miles 
run has produced any effect on the quality of the metal." 

On page 1619 et seq. of the Report for 1886 is given an 
account of the tests made on some more of these axles which 
had run 163138 miles, and the following is quoted from that 
account : 

" Specimens from muck-bar axle No. 4 after the axle had 
run 163138 miles. 

" Comparing these results with earlier tests of this series, the 
tensile strength of the metal in this axle is lower, and the 
modulus of elasticity less than shown by the preceding axles. 

" The variations in strength, elasticity, and ductility are no 
greater, however, than those met in different specimens of new 
iron of nominally the same grade, and while apparently there 
is a deterioration in quality, it needs confirmation of a more 
decisive nature from the remaining axles before attributing 
this result to the influence of the work done in service." 

Another set of tests made at Watertown Arsenal is to be 
found on page 1044 et seq. of the Report for 1885. There were 
tested - 

i. Two side-rods of a passenger locomotive which had been 
in service about twelve years. 

2. One side-rod of a passenger engine which had been run 
twenty-eight years and eight months. 

3. One main-rod which had been run thirty-two years and 
eight months in freight and five years in passenger service. 

In none of these tests were there any evidences of crystal- 
lization, as the metal was in all cases fibrous when fractured. 



COLD CRYSTALLIZATION OF IRON AND STEEL. 503 

In the report is said : 

" There are no data at command telling what the original 
qualities of the metal of these bars were : it is sufficient, how- 
ever, to find toughness and a fibrous appearance in the iron to 
prove that brittleness or crystallization has not resulted from 
long exposure to the stresses and vibrations these bars have 
sustained." 

The only other evidence that will be referred to is the paper 
of Mr. A. F. Hill upon the " Crystallization of Iron and Steel," 
contained in the Proceedings of the Society of Arts of the 
Massachusetts Institute of Technology for 1882-83. In this 
article Mr. Hill covers the ground very fully, and distinctly 
asserts that 

" The fact is that there is at present not a single well- 
authenticated instance of iron or steel ever having become 
crystallized from use under temperatures below 900 F." 

He claims to have investigated a great many cases where 
cold crystallization has been claimed, and to have found, in 
every case where crystals existed, that at some period of its 
manufacture or working the metal was overheated. He 
says : 

" That the crystalline appearance of a fracture is not neces- 
sarily an indication of the presence of genuine crystals is proven 
by the well-known fact that a skilful blacksmith can fracture 
fine fibrous iron or steel in such a manner as to let it appear 
either fibrous and silky, or coarse and crystalline, according to 
his method of breaking the bar. On the other hand, where 
there is genuine crystallization, no skill of manipulation will 
avail to hide that fact in the fracture. The most striking 
illustrations of this that have come under my notice are the 
fractures of the beam-strap of the Kaaterskill, and of the 
connecting-rod of the chain-cable testing-machine at the Wash- 
ington Navy Yard. The photographs of both fractures are 
submitted to you, and the similarity of their appearance is 



5O4 APPLIED MECHANICS. 

most singular. Yet what a difference in the development of 
the longitudinal sections by acid treatment, which are also 
presented to you. 

" In the Kaaterskill accident the fractures of both the 
upper and lower arms of the strap were found to be short and 
square. The appearance of the fractured faces showed no 
trace of fibre, and was altogether granular. Yet the longitudi- 
nal section, taken immediately through the break, and devel- 
oped by acid treatment, shows the presence of but few and 
small crystals, and the generally fibrous character of the iron 
used in the strap. 

" In the connecting-rod of the chain-cable testing-machine 
we find the crystalline appearance of the fracture less, if any- 
thing, than that of the beam-strap, while the development of 
the longitudinal section by acid treatment reveals most beauti- 
fully, in this case, the thoroughly crystalline character of the 
metal. As is well known, this rod, after many years of service, 
finally broke under a comparatively light strain, and having all 
along been supposed to have been carefully made, and from 
well-selected scrap, its intensely crystalline structure, as re- 
vealed by the fractures, has done service for quite a number of 
years as piece de resistance in all the ' cold-crystallization ' 
arguments which have been served up in that time." 

He then goes on to say that he cut the rod in a longitudinal 
direction, and treated the section with acid ; that some of the 
crystals shown are so large as to be discernible with the naked 
eye ; that the treated section furnished incontrovertible evi- 
dence that 'the rod, aside from the fact of being badly dimen- 
sioned anyhow, was made of poor material, badly heated, and 
msufrlciently hammered, all records, suppositions, ^and asser- 
tfons to the contrary notwithstanding ; that there are a large 
number of crystals composed of a substance, presumably a 
ferro-carbide, which is not soluble in nitric acid, and is found 
in steel only ; that the deduction from the large amount of this 



COLD CRYSTALLIZATION OF IRON AND STEEL. 505 

substance is that the pile was formed of rather poorly selected 
scrap, with steel scrap mixed in ; that evidences of bad heating 
are abundant throughout ; and that the strongest evidence 
against the presumption that these crystals were formed during 
the service of the rod, or while the metal was cold, is found in 
the groupings of the crystals during their formation, as shown 
in the tracing developed by the acid ; that they are not of the 
same chemical composition, the lighter parts containing much 
more carbon than the darker ones ; it is therefore pretty evi- 
dent that with the grouping of the crystals a segregation of 
like chemical compounds took place, and this of course would 
have been impossible in the solid state. He then cites an 
experiment he made, in which he took a slab of best selected 
scrap weighing about 200 pounds and forged it down to a 
3-inch by 3-inch square bar, one-half being properly forged, 
and the other half being exposed to a sharp flame bringing it 
quickly to a running heat, keeping it at this heat some time, 
and then hammering lightly and then treating it a second time 
in a similar manner ; the result being, that while no difference 
was discernible in the appearance of the two portions, when 
cut and treated with acid the portion that was properly made 
showed itself to be a fair representative of the best quality of 
iron, while in the other portion the crystallization was strongly 
marked, the majority of the crystals being large and well 
developed. 

He also says : 

" The fact is, all hammered iron or steel is more or less 
crystalline, the lesser or greater degree of crystallization de- 
pending altogether upon the greater or lesser skill employed 
in working the metal, and also largely upon the size of the 
forging. Crystallization tends to lower very sensibly the elastic 
limit of iron and steel, and therefore hastens the deterioration 
of the metal under strain. It is for this reason that large a:id 
heavy forgings ought to be, and measurably are, excluded as 



5O6 APPLIED MECHANICS. 

much as possible from permanent structures. In machine con- 
struction we cannot do without them, and must therefore 
accept the necessity of replacing more or less frequently the 
parts doing the heaviest work." 

The evidence given above seems to the writer to be suffi- 
cient, and to warrant the conclusions stated on pages 475, 476. 

EFFECT OF TEMPERATURE UPON THE RESISTING PROPER- 
TIES OF IRON AND STEEL. 

Much the best and most systematic work upon this subject 
has been done at the Watertown Arsenal, and an account of it 
is to be found in " Notes on the Construction of Ordnance, 
No. 50," published by the Ordnance Department at Washing- 
ton, D. C, U.S.A. 

Other references are the following: 

Sir William Fairbairn: Useful Information for Engineers. 
Committee of Franklin Institute: Franklin Institute Journal. 
Knutt Styffe and Christer P. Sandberg: Iron and Steel. 
Kollman: Engineering, July 30, 1880. 
Massachusetts R. R. Commissioners' Report of 1874. 
Bauschinger: Mittheilungen, Heft 13, year 1886. 

A summary of the Watertown tests, largely quoted from 
the above-mentioned report, will be given here, and then a few 
remarks will suffice for the others. 

The subjects upon which experiments were made at Water 
town were the effect of temperatures upon 

i. The coefficient of expansion. 

2. The modulus of elasticity. 

3. The tensile strength. 

4. The elastic limit. 

5. The stress per square inch of ruptured section- 

6. The percentage contraction of area. 

7. The rate of flow under stress. 

8. The specific gravity. 



EPFECT OF TEMPERATURE ON TRON AND STEEL. S7 



9. The strength when strained hot and subsequently rup- 
tured cold. 

10. The color after cooling. 
11. Riveted joints. 

1. THE COEFFICIENTS OF EXPANSION. 

These were determined from direct measurements upon the 
experimental bars, first measuring their lengths on sections 
35 inches long, while the bars were immersed in a cold bath of 
ice-water, and again measuring the same sections after a period 
of immersion in a bath of hot oil. 

The range of temperature employed was about 210 degrees 
Fahr., as shown by mercurial thermometers. 

Observations were repeated, and again after the steel bars 
had been heated and quenched in water and in oil. 

The average values are exhibited in the following : 

TABLE I. 
First Series of Bars. 



Metal. 


Chemical Composition. 


Coefficients of Expansion 
per Degree Fahr., per 
Unit of Length. 


C. 


Mn. 


Si. 


Wrought-iron. 








.0000067302 


Steel. 


.09 


. II 




.0000067561 


" 


.20 


45 




.0000066259 


" 


31 


57 




.0000065149 




37 


.70 




.0000066597 


4 ' 


.51 


58 


.02 


.OOOOO662O2 


tt 


57 


93 


.07 


.0000063891 


" 


71 


58 


.08 


.0000064716 


" 


.81 


56 


17 


.0000062167 


11 


.89 


57 


.19 


.0000062335 


* * 


97 


.80 


.28 


.0000061700 


Cast- (gun) iron. 








.0000059261 


Drawn copper. 








0000091286 



APPLIED MECHANICS 



Subsequent determinations of the coefficient of expansion 
of a second series of steel bars gave 



TABLE II. 







Jnemical L 


omposition 






Coefficients of Expansion 


c. 


Mn. 


Si. 


S. 


P. 


Cu. 


per Degree Fahr., per 
Unit of Length. 


17 


I.I3 


.023 


.122 


.079 


.04 


.0000067886 


.20 


.69 


037 


13 


.078 


.26 


.0000068567 


.21 


.26 


.08 


.14 


-.059 


.00 


.0000067623 


.26 


.07 


.11 


.096 


.08 


047 s 


.0000067476 


.26 


.26 


.07 


.112 


.06 


.038 


.0000067102 


.26 


.28 


.07 


."5 


.062 


035 


.0000067175 


.28 


23 


.09 


.168 


.09 


.178 


.0000067794 


43 


97 


05 


.08 


.096 


.024 


.0000066124 


43 


i. 08 


037 


.08 


.114 


233 


.0000066377 


53 


75 


.10 


.078 


.087 


.174 


.0000064181 


55 


1.02 


.05 


.078 


.12 


15 


.OOOOo66l22 


.72 


.70 


.18 


.07 


13 


23 


.0000064330 


.72 


.76 


.20 


.056 


.086 


.186 


. 0000063080 


79 


.86 


.21 


.084 


093 


.096 


.0000063562 


.07 


.07 


13 


.01 


.018 


.006 


.0000061528 


.08 


.12 


.I 9 


.Oil 


.02 


trace 


.0000061702 


.12 


.10 


.09 


.013 


.018 


trace 


.0000060716 


.14 


.IO 


15 


trace 


.018 


trace 


.0000062589 


17 


.10 


.10 


trace 


.018 


o 


.0000061332 


31 


13 


.19 


.Oil 


.026 


trace 


.0000061478 



Ten bars of the first series were now heated a bright cherry- 
red and quenched in oil at 80 Fahr., the hot bars successively 
raising the temperature of the oil to about 240 Fahr., the bath 
being cooled between each immersion. 

The behavior of the bars under rising temperature, when 
examined for coefficients of expansion, seemed somewhat 
erratic, the highest temperature reached being 235 ; but this 
behavior was subsequently explained by the permanent changes 
in length found when the bars were returned to the cold bath. 



EFFECT OF TEMPERATURE ON IRON AND STEEL. $09 

Generally the bars were found permanently shortened at the 
close of these observations. 

The bars were again heated bright cherry-red and quenched 
in water at 50 to 55 Fahr., the water being raised by the 
quenching to 1 10 to 125 Fahr. 

After resting 72 hours, measurements were taken in the cold 
bath, followed by a rest of 18 hours, when they were heated 
and measured in the hot bath, after which they were measured 
in the cold bath ; the maximum temperature reached with the 
hot bath being 233. 7 Fahr., erratic behavior occurring still. 

They were next heated in an oil bath at 300 Fahr., and 
kept at this temperature 6 hours, then cooled in the bath ; 15 
hours later they were heated to 243 Fahr., and again measured 
hot, and then cold. These downward readings showed the 
quenched in water bars to have their coefficients elevated 
above the normal, as shown in the following table, these 
being the same steel bars as in Table I, and in the sarr.e 
order : 

TABLE III. 



Coefficients of Expansion 


Apparent Shortening of Bars 
Due to Six Hours at 300 


per Degree per Unit of 
Length. 


Fahr., and the following 
Immersion in the Hot Bath. 


.0000067641 


.0006 


.0000066622 


.0002 


.0000066985 


.OOl6 


.0000067377 


.OO23 


.0000069776 


.0004 


.0000067041 


.0082 


.0000066939 


.0064 


.0000068790 


.0054 


.0000072906 


.0055 


.0000071578 


.0048 



510 APPLIED MECHANICS. 

Finally the bars were annealed by heating bright red and 
cooling in pine shavings, the effect of which was to approxi- 
mately restore the rate of expansion to the normal, as shown 
by Table I for these ranges of temperature. 

2. MODULUS OF ELASTICITY. 

These were obtained with the first series of bars at atmos^ 
pheric temperatures, and at higher temperatures, up to 495 
Fahr. 

There occurred invariably a decrease in the modulus of 
elasticity with an increase in temperature, and, in the case of 
the specimens tested, the low carbon steels showed a greater 
reduction in the modulus than the high carbon steels, the 
first specimen having a modulus of elasticity at the minimum 
temperature 30612000, and at the maximum 27419000, while 
the last specimen had at the minimum temperature 29126000, 
and at the maximum 27778000. 

3. TENSILE STRENGTH. 

The tests were made upon the first series of steel bars, 
wrought-irons marked A and B, a muck-bar railway axle, and 
cast-iron specimens from a slab of gun-iron. 

The specimens were o".798 diameter, and 5" length of stem, 
having threaded ends \ fl '.25 diameter. 

Wrought-iron A was selected because it was found very hot 
short at a welding temperature. It had been strained with a 
tensile stress of 42320 pounds per square inch seven years 
previous to being cut up into specimens for the hot tests. 

The specimens while under test were confined within a 
sheet-iron muffle, through the ends of which passed auxiliary 
bars screwed to the specimens, the auxiliary bars being secured 
to the testing-machine. 



EFFEC7" OF TEMPERATURE ON IRON AND STEEL. $11 

The heating was done by means of gas-burners arranged 
below the specimen and within the muffle. 

The temperature of the test-bar was estimated from the 
expansion of the metal, observed on a specimen length of six 
inches, using the coefficients which were determined at lower 
temperatures, as hereinbefore stated, assuming there was a 
uniform rate of expansion. 

Access to the specimen for the purpose of measuring the 
expansion was had through holes in the top of the muffle. 
The temperature was regulated by varying the number of gas- 
burners in use, the pressure of the gas, and also by means of 
diaphragms placed within the muffle for diffusing the heat. 

The approximate elongations under different stresses were 
determined during the continuance of a test from measurements 
made on the hydraulic holders of the testing-machine, at a 
convenient distance from the hot muffle, correcting these 
measurements from data obtained by simultaneous micrometer 
readings made on the specimen and the hydraulic holders at 
atmospheric temperatures. 

While it does not seem expedient in one series of tests to 
obtain complete results upon the tensile properties at high 
temperatures, yet, incidentally, much additional valuable infor- 
mation may be obtained while giving prominence to one or 
more features. 

From these elongations the elastic limits were established 
where the elongations increased rapidly under equal incre- 
ments of load. Proceeding with the test until the maximum 
stress was reached, recorded as the tensile strength, observing 
the elongation at the time, then, when practicable, noting 
the stress at the time of rupture." 

For the detailed tables of tests the student is referred to 
the " Notes on the Construction of Ordnance." 

The elastic limits and tensile strengths are computed in 
pounds per square inch, both on original sectional areas of the 



APPLIED MECHANICS. 



specimens and on the minimum or reduced sections, as meas- 
ured at the close of the hot tests. 

From the results it appears that the tensile strength of the 
steel bars diminishes as the temperature increases from zero 
Fahr., until a minimum is reached between 200 and 300 Fahr., 
the milder steels appearing to reach the place of minimum 
strength at lower temperatures than the higher carbon bars. 

From the temperature of this first minimum strength the 
bars display greater tenacity with increase of temperature, until 
the maximum is reached between the temperatures of about 
400 to 650 Fahr. 

The higher carbon steels reach the temperature of maximum 
strength abruptly, and retain the highest strength over a lim- 
ited range of temperature. The mild steels retain the increased 
tenacity over a wider range of temperature. 

From the temperature of maximum strength the tenacity 
diminishes rapidly with the high carbon bars, somewhat less 
so with mild steels, until the highest temperatures are reached, 
covered by these experiments. 

The greatest loss observed in passing from 70 Fahr. to the 
temperature of first minimum strength was 6.5 per cent at 
295 Fahr. 

The greatest gain over the strength of the metal at 70 was 
25.8 per cent at 460 Fahr. 

The several grades of metal approached each other in 
tenacity as the higher temperatures were reached. Thus steels 
differing in tensile strength nearly 90000 pounds per square 
inch at 70, when heated to 1600 Fahr. appear to differ only 
about loooo pounds per square inch. 

The rate of speed of testing which may modify somewhat 
the results with ductile material at atmospheric temperatures 
has a very decided influence on the apparent tenacity at high 
temperatures. 

A grade of metal which, at low temperatures, had little 



EFFECT OF TEMPERATURE ON IRON AND STEEL. $13 

ductility, displayed the same strength whether rapidly or slowly 
fractured from the temperature of the testing-room up to 600 
Fahr. ; above this temperature the apparent strength of the 
rapidly fractured specimens largely exceeded the others. 

At 1410 Fahr. the slowly fractured bar showed 33240 
pounds per square inch tensile strength, while a bar tested in 
two seconds showed 63000 pounds per square inch. 

Cast-iron appeared to maintain its strength with a tendency 
to increase until 900 Fahr. is reached, beyond which the 
strength diminishes. Under the higher temperatures it devel- 
oped numerous cracks on the surface of the specimens preced- 
ing complete rupture. 

4. ELASTIC LIMIT. 

The report says of this that it appears to diminish with in- 
crease of temperature. Owing to a period of rapid yielding with- 
out increase of stress, or even under reduced stress, the elastic 
limit is well defined at moderate temperatures with most of the 
steels. 

Mild steel shows this yielding point up to the vicinity of 
500; in hard steels, if present, it appears at lower temperatures. 

The gradual change in the rate of elongation at other times 
often leaves the definition of the elastic limit vague and doubt- 
ful, especially so at high temperatures. The exclusion of de- 
terminable sets would in most cases place the elastic limit below 
the values herein given. 

In approaching temperatures at which the tensile proper- 
ties are almost eliminated exact determinations are correspond- 
ingly difficult, the tendency being to appear to reach too high 
values. 

5. STRESS ON THE RUPTURED SECTION. 

This, generally, follows with and resembles the curve of 
tensile strength. 



514 APPLIED MECHANICS. 

Specimens of large contraction of area, tested at high 
temperature, have given evidence on the fractured ends of 
having separated at the centre of the bar before the outside 
metal parted. 

Elongation under Stress^ 

Although the metal is capable of being worked under the 
hammer at high temperatures, it does not then possess sufficient 
strength within itself to develop much elongation, general 
elongation being greatest at lower temperatures. 

Greater rigidity exists under certain stresses at intermedi- 
ate temperatures than at either higher or lower temperatures. 

Thus one of the specimens tested at 569 Fahr. showed less 
elongation under stresses above 50000 pounds per square inch 
than the bars strained at higher or lower temperatures. 

Two other specimens showed a similar behavior at 315 and 
387 respectively, and likewise other specimens. 

In bars tested at about 200 to 400 Fahr. there are dis- 
played alternate periods of rigidity and relaxation under in- 
creasing stresses, resembling the yielding described as occur- 
ring with some bars immediately after passing the elastic 
limit. 

The repetition of these intervals of rigidity and relaxation 
is suggestive of some remarkable change taking place within 
the metal in this zone of temperature. 

6. PERCENTAGE CONTRACTION OF AREA. 

This varies with the temperature of the bar ; it is somewhat 
less in mild and medium hard steels at 400 to 600 than at 
atmospheric temperatures. 

Above 500 or 600 the contraction increases with the 
temperature of the metal ; with three exceptions, which showed 
diminished contraction at 1100 Fahr., until at the highest 
temperatures some of them were drawn down almost to points. 



EFFECT OF TEMPERATURE ON IRON AND STEEL. 



7. RATE OF FLOW UNDER STRESS. 

The full effect of a load superior to the elastic limit is not 
immediately felt in the elongation of a ductile metal, and the 
same is true at higher temperatures. 

The flow caused by a stress not largely in excess of the 
elastic limit has a retarding rate of speed, and eventually ceases 
altogether ; whereas under a high stress the rate of flow may 
accelerate, and end in rupture of the metal. 

Hence the apparent tensile strength maybe modified within 
limits by the time employed in producing fracture. 

8. SPECIFIC GRAVITY. 

In general, the specific gravity is materially diminished in 
the vicinity of the fractured ends of tensile specimens, and this 
diminution takes place in the different grades of steel, in bars 
ruptured under different conditions of temperature, stress, and 
contraction of area. 



9 BARS STRAINED HOT, AND SUBSEQUENTLY RUPTURED COLD. 

The effect of straining hot on the subsequent strength cold 
appears to depend upon the magnitude of the straining force 
and the temperature in the first instance. 

There is a zone of temperature in which the effect of hot 
straining elevates the elastic limit above the applied stress, and 
above the primitive value, and if the straining force approaches 
the tensile strength, there is also a material elevation of that 
value when ruptured cold. These effects have been observed 
within the limits of about 335 and 740 Fahr. 

After exposure to higher temperatures there occurs a 
gradual loss in both the elastic limit and tensile strength, and 
generally a noticeable increase in the contraction of area. 



5l6 APPLIED MECHANICS. 



This was not sensibly changed by temperatures below 200. 
After 300 the metal was light straw-colored : after 400, deep 
straw ; from 500 to 600, purple, bronze-colored, and blue ; 
after 700, dark blue and blue black. 

After 800 some specimens still remained dark blue. After 
heating above about 800 the final color affords less satisfactory 
means of approximately judging of the temperature, the color 
remaining a blue black, and darker when a thick magnetic 
oxide is formed. 

At about 1100 the surface oxide reaches a tangible thick- 
ness, a heavy scale of o".ooi to o".oo2 thickness forming as 
higher temperatures are reached. The red oxide appears at 
about 1500. 

11. IN THE TESTS OF RIVETED JOINTS 

of steel boiler-plates at temperatures ranging from 70 to about 
700 Fahr. the indications of the tensile tests of plain bars were 
corroborated. 

Joints at 200 Fahr. showed less strength than when cold ; 
at 250 and higher temperatures the strength exceeded the 
cold joints ; and when overstrained at 400 and 500 there was 
found, upon completing the test cold, an increase in strength. 

Rivets which sheared cold at 40000 to 41000 pounds per 
square inch, at 300 Fahr. sheared at 46000 pounds per square 
inch ; and at 600 Fahr., the highest temperature at which the 
joints failed in this manner, the shearing-strength was 42130 
pounds per square inch. 

In addition to the work at Watertown which has just been 
detailed two other matters will be referred to here. 



EFFECT OF TEMPERATURE ON IRON AND STEEL. $1? 

1. It is well known that wrought-iron and steel are very 
brittle at a straw heat and a pale blue, as shown by the fact that 
when the attempt is made to bend a specimen at these tempera- 
tures it results in cracking it some time before a complete bend- 
ing can be effected, even in the case of metal which is so ductile 
that it can be bent double cold, red hot, or at a flanging heat, 
without showing any signs of cracking. 

2. Bauschinger defines the elastic limit as the load at which 
the stress is no longer proportional to the strain ; whereas he 
calls stretch-limit (Streckgrenze) the load at which the strain 
diagram makes a sudden change in its direction ; i.e., where 
instead of showing a gradually increasing ratio of strain to stress 
it shows a sudden and rapid increase. 

From his experiments (see Heft 13 of the Mittheilungen, 
year 1886) he draws the following conclusions : 

(a) That the effect of heating and subsequent cooling in 
lowering both the elastic and the stretch limits in mild steel 
begins at about 660 Fahr. when the cooling is sudden, and at 
about 840 Fahr. when it is slow, and for wrought-iron at about 
750 with either rapid or slow cooling. 

(b) That the operation of heating above those temperatures, 
and of subsequent slow or quick cooling, is that both the elastic 
and the stretch limit are lowered, and the more so the greater 
the heating ; also, that this effect is greater on the elastic than 
on the stretch limit. 

(c) Quick cooling after heating higher than the above-stated 
temperatures lowers the elastic and the stretch limit, especially 
the first, much more than slow cooling, dropping the elastic 
limit almost immediately at a heat of about 930 and certainly 
at a red heat to nothing or nearly nothing in wrought-iron, and 
in both mild and hard steel, while slow cooling cannot bring 
about such a great drop of the elastic limit, even from more 
than a red heat. 



APPLIED MECHANICS. 



Effect of Cold-Rolling on Iron and Steel. It has already 
been stated, p. 410, that it was discovered independently by 
Commander Beardslee and Professor Thurston, that if a load 
were gradually applied to a piece of iron or steel which exceeded 
its elastic limit, and the piece then allowed to rest, the elastic 
limit and the ultimate strength would thus be increased. This 
may be accomplished with soft iron and steel by cold-rolling or 
cold-drawing, but cannot be taken advantage of in hard iron 
or steel. 

Professor Thurston, who has investigated this matter at 
great length, and made a large number of tests on the subject, 
gives the following as the results of cold-rolling: 



Increase in 


Per Cent. 


Tenacity 


2C to 4.0 


Transverse stress .... 


CQ to 80 


Elastic limit (tension, torsion, and transverse), 


80 to 125 
300 to 400 


Elastic resilience (transverse) 


I CO to A.2 C 







He also says, in regard to the modulus of elasticity, 

" Collating the results of several hundred tests, the author 
[Professor Thurston] found that the modulus of elasticity rose, 
in cold-rolling, from about 25000000 Ibs. per square inch to 
26000000, the tenacity .from 52000 Ibs. to nearly 70000, the 
elastic limit from 30000 Ibs. to nearly 60000 Ibs. ; and the ex- 
tension was reduced from 25 to ioj per cent. 

" Transverse loads gave a reduction of the modulus of elas- 
ticity to the extent of about 1000000 Ibs. per square inch, an 
increase in the modulus of rupture from 73600 to 133600, and 
reduction of deflection at maximum load of about 25 per cent. 
The resistance of the elastic limit was doubled, and occurred 
at a much greater deflection than with untreated iron." 

On the other hand, the two steel eye-bars referred to on 



FACTOR OF SAFETY. 



519 



p. 472 show a decrease of modulus of elasticity with increasing 
overstrain. 

Whitworth's Compressed Steel. Sir Joseph Whitworth pro- 
duces steel of great strength by applying to the molten metal, 
directly after it leaves the furnace, a pressure of about 14000 
Ibs. per square inch; this being sufficient to reduce the length 
of an eight-foot column by one foot. He claims, according to 
D. K. Clark, to be able to obtain with certainty a strength of 
40 English tons with 30 per cent ductility, and mild steel of a 
strength of 30 English tons with 33 or 34 per cent ductility. 

The following tests were made on the Watertown machine, 
upon some specimens of Whitworth steel taken from a section 
of a jacket which was shrunk upon a wrought-iron tube, and 
removed from shrinkage by the application of high furnace heat : 

TENSILE TESTS. 



Diameter, 
Inches. 


Tensile 
Strength, 
Ibs. per Sq. In. 


Elastic Limit, 
Ibs. per Sq. In. 


Contraction 
of Area, 
per cent, 


o 564 


103960 


55000 


41.9 


0.564 


90040 


48000 


47.2 


0.564 


104200 


57000 


24.6 


0.564 


IOOI20 


57000 


44.6 


0.564 


93040 


53000 


39-2 


0.564 


104160 


60000 


24.6 


0.564 


93160 


47000 


39-2 



COMPRESSIVE TESTS. 



Length, 
Inches. 


Diameter, 
Inches. 


Compressive 
Strength, 
Ibs. per Sq. In. 


Elastic Limit, 
Ibs. per Sq. In. 


5 


0.798 


IO2IOO 


61000 


5 


0.798 


89000 


57000 


3-94 


0.798 


IOI6OO 


53000 


3-94 


0.798 


IOI6OO 


54000 



227. Factor of Safety In order to determine the 

proper dimensions of any loaded piece, it becomes necessary 



52O APPLIED MECHANICS. 

to fix, in some way, upon the greatest allowable stress per 
square inch to which the piece shall be subjected. 

The most common practice has been to make this some 
fraction of the breaking-strength of the material per square 
inch. 

As to how great this factor should be, depends upon 

i. The use to which the piece is to be subjected ; 

2. The liability to variation in the quality of the material ; 

3. The question whether we are considering, as the load 
upon the piece, the average load, or the greatest load that can 
by any possibility come upon it ; 

4. The question as to whether the structure is a temporary 
or a permanent one; 

5. The amount of injury that would be done by breakage 
of the piece ; 
and other considerations. 

The factors most commonly recommended are, 3 for a dead 
or quiescent load, and 6 for a live or moving load. 

A common American and English practice for iron bridges 
is to use a factor of safety of 4 for both dead and moving load. 
In machinery a factor as large as 6 is desirable when there is 
no liability to shocks ; and when there is, a larger factor should 
be used. 

A method sometimes followed for tension and compression 
pieces is, to prescribe that the stretch under the given load 
should not exceed a certain fixed fraction of the length. This 
requires a knowledge of the modulus of elasticity of the mate- 
rial. 

In the case of a piece subjected to a transverse load, it is 
the most common custom to determine its dimensions in accord- 
ance with the principle of providing sufficient strength ; and 
for this purpose a certain fraction (as one-fourth) of the mod- 
ulus of rupture is prescribed as the greatest allowable safe 
stress per square inch at the outside fibre. Thus, for wrought- 
iron from 10000 to 12000 Ibs. per square inch is often adopted 



REPEATED STRESSES. $21 

as the greatest allowable stress at the outside fibre, this being 
about one-fourth of the modulus of rupture. 

The other method for dimensioning a beam is, to prescribe 
its stiffness ; i.e., that it shall not deflect under its load more 
than a certain fraction of the span. This fraction is taken as 

rb- to 7TT<7- 

This latter method depends upon the modulus of elasticity 

of the beam ; and while it is the most advisable method to 
follow, and as a rule would be safer than the other method, 
nevertheless, in the case of very stiff and brittle material it 
might be dangerous ; hence we ought to know also the break- 
ing-weight and the limit of elasticity of the beam we are to use, 
and not allow it to approach either of these. This precaution 
will be especially important to observe in the case of steel 
beams, which are only now being introduced. 

On the other hand, in moving machinery a factor of safety 
of six is usually required when there is no unusual exposure to 
shocks, as in smooth-running shafting, etc. ; and when there 
are irregular shocks liable to come upon the piece, a greater 
factor is used. 

WOHLER'S RESULTS. 

228. Repeated Stresses. The extensive experiments of 
Wohler for the Prussian government, which were subsequently 
carried on by his successor, Spangenberg, were made to deter- 
mine the effect of oft-repeated stresses, and of changes of 
stress, upon wrought-iron and steel. 

In the ordinary American and English practice, it is cus- 
tomary, in determining the dimensions of a piece, as of a bridge 
member, to ascertain the greatest load which the piece can 
ever be called upon to bear, and to fix the size of the piece in 
accordance with this greatest load. 

Wohler called attention to the fact that the load that would 
break a piece depends upon both the greatest and least load 
that it would ever be called upon to bear. Thus, a tension-rod 



522 APPLIED MECHANICS. 

which is subjected to alternate changes of load extending from 
20000 to 80000 Ibs. would require a greater area for safety than 
one which was subjected to loads varying only between the 
limits of 60000 and 80000 Ibs. ; and this would require more 
area than one which was subjected to a steady load of 80000 
Ibs. 

Wohler expresses this law as follows, in his " Festigkeits 
versuche mit Eisen und Stahl." 

"The law discovered by me, whose universal application 
for iron and steel has been proved by these experiments, is as 
follows : The fracture of the material can be effected by 
variations of stress repeated a great number of times, of 
which none reaches the breaking-limit. The differences of 
the stresses which limit the variations of stress determine the 
breaking-strength. The absolute magnitude of the limiting 
stresses is only so far of influence as, with an increasing stress, 
the differences which bring about fracture grow less. 

" For cases where the fibre passes from tension to compres- 
sion and vice versa, we consider tensile strength as positive 
and compressive strength as negative ; so that in this case the 
difference of the extreme fibre stresses is equal to the greatest 
tension plus the greatest compression." 

Besides the ordinary tests of tensile, compressive, shearing, 
and torsional strength, he made his experiments mainly on the 
following two cases : 

i. Repeated tensile strength; the load being applied and 
wholly removed successively, and the number of repetitions 
required for fracture counted. 

2. Alternate tension and compression of equal amounts 
successively applied, the number of repetitions required for 
fracture being counted. 

In making these two sets of tests, he made the first set in 
two ways : 

(a) By applying direct tension. 



LAUNHARDT'S FORMULA. 523 

(b) By applying a transverse load, and determining the 
greatest fibre stress. 

The second set of tests was made by loading at one end a 
piece of shaft fixed in direction at the other, and then causing 
it to revolve rapidly, each fibre passing alternately from tension 
to an equal compression, and vice versa. 

He also tried a few experiments where the lower limit of 
stress was neither zero nor equal to the upper limit, with a 
minus sign, also some experiments on torsion, on shearing, 
and on repeated torsion. 

When Wohler had made his experiments, and published his 
results, there were a number of attempts made by different 
persons to deduce formulae which should depend upon these 
experiments for their constants, and which should serve to deter- 
mine the breaking-strength for any given variation of stresses. 

Only two of these formulae will be given here, viz. : 

i That of Launhardt for one kind of stress, 

2 That of Weyrauch for alternate tension and compression. 

LAUNHARDT'S FORMULA. 

The constants used in this formula are : 

i. /, the carrying-strength (Tragfestigkeit) of the material 
per unit of area, which is the same as the tensile strength as 
determined by the ordinary tensile testing-machine. 

2. u, the primitive breaking-strength (Ursprungsfestigkeit), 
i.e., the greatest stress per unit of area of which the piece can bear, 
without breaking, an unlimited number of repetitions, the load 
being entirely removed between times. These two quantities 
have been determined experimentally by Wohler; and it is the 
object of Launhardt's formula to deduce, in terms of /, u, and the 
ratio between the greatest and least loads to which the piece is 
ever subjected, the value a of the breaking-strength per unit of 
area when these loads are applied. 



524 APPLIED MECHANICS. 



Let the greatest stress per unit area be a. 
the least stress per unit area be c. 

Plot the values of - as abscissae, and those of a as ordinates, 
making OA = u (since when - = o, a = u), OC=i, and CB = t 

(since when -=i, a = /). Then will any curve 

, E B which passes through the points A and B have 
\\^~\~\ ^ or * ts or( ^ mates values of a that will satisfy the 



conditions that when c = o, a u, and when c = /, 
a = t. By assuming for this curve, the straight 



line AB we obtain DE = AO + FE = AO + (BG)^ , and hence 

a=w + (/-w)~, (i) 

which is Launhardt's formula. 

Moreover, if we denote by max L the greatest load on the en- 
tire piece, and by min L the least, we shall have 

c_ min L 
a max L' 
Hence 

min L 



- r, 

max L 



(2) 



this being in such a form as can be used. Or we may write it 
thus: 

!/ u min L 
i+ j 

this being the more common form. 

The values of the constants as determined by Wohler's experi- 
ments, and the resulting form of the formula for Phcenix axle- iron 
and for Krupp cast-steel, have already been given in 172. 



WEYRAUCH'S FORMULA. $2$ 

In the same paragraph are given the corresponding values of 
by the safe working-strength, the factor of safety being three. 

WEYRAUCH'S FORMULA FOR ALTERNATE TENSION AND 

COMPRESSION. 

The constants used in this formula are : 

i. u, the primitive breaking-strength, which has been already 
defined. 

2. s, the vibration breaking-strength (Schwingungsfestigkeit) 
i.e., the greatest stress per unit of area, of which the piece can 
bear, without breaking, an unlimited number of applications, 
when subjected alternately to a tensile, and to a compressive 
stress of the same magnitude. 

He lets a = greatest stress per unit of area, c= greatest stress 
of the opposite kind per unit of area. If a is tension, c is com- 
pression, and vice versa. 

Plot the values of - as abscissae, and those of a as ordinates, 
making OA=u (since when - = i, a=w), OC = i, and CB=s 

(since when = i , a =s) . Then will any curve 

which passes through the points A and B 
have for its ordinates values of a that will 
satisfy the conditions that when c=o, a=u, 
and when c=s, a=s. 

By assuming for this curve the straight line AB we obtain 

t and hence 




a=u-(u-s)-, (4) 

which is the Weyrauch formula. 



526 APPLIED MECHANICS. 

Moreover, if we write 

c max U 
a max L ' 

where max L= greatest load on the piece, and max Z/= greatest 
load of opposite kind, so that, if L is tension, L' shall be com- 
pression, and vice versa, we shall have 

.max L' 



this being in a form suitable to use, the more common form being 

(u s max L' } 
i - --- r \ - (6) 

u max L J 

The values of the constants as determined from Wohler's 
experiments, and the resulting form of the formulae for Phoenix 
axle-iron and for Krupp cast-steel, are given in 176. 

GENERAL REMARKS. 

In each case the value of a given by the formula (3) or (6) 
is the breaking-strength per unit of area. 

If either of these values of a be divided by 3, we have, accord- 
ing to Weyrauch, the safe working-strength. 

WOHLER'S EXPERIMENTAL RESULTS. 

Wohler himself made his tests upon the extremes of fibre 
stresses of which a piece could bear, without breaking, an 
unlimited number of applications. He gives, as a summary of 
these results, the following: 

In iron, 

Between +16000 Ibs. per sq. in. and 16000 Ibs. per sq. in. 
+ 30000 " " " o " " 

+44000 " " " +24000 " " 

In axle-steel, 

Between +28000 Ibs. per sq. in. and 28000 Ibs. per sq. in. 
" +48000 " " " o " " 

" +80000 " " " +35000 " " 



WOHLER'S EXPERIMENTAL RESULTS. 527 

In untempered spring steel, 

Between +50000 Ibs. per sq. in. and o Ibs. per sq. in. 
-f 70000 " " " +25000 " " 

+ 80000 " " " +40000 " " 

+ 90000 " " " +60000 " " 

For shearing in axle-steel, 

Between +22000 Ibs. per sq. in. and 22000 Ibs. per sq. in. 
+ 38000 " " o " 

This table would justify the use, in Launhardt's and Wey- 
rauch's formulae, of the following values of u and s ; viz., 
In iron, 

u = 30000 Ibs. per sq. in., 

s = 16000 Ibs. per sq. in. 

In axle steel, 

u = 48000 Ibs. per sq. in., 
s = 28000 Ibs. per sq. in. 

In untempered spring steel, 

u = 50000 Ibs. per sq. in. 

And it would require, that if, with these values of u, and the 
values of / given in 172 and 176, we put 

c 24000 

in Launhardt's formula for iron, we ought to obtain approxi- 
mately 

a = 44000 ; 

and if we put c = 35000 in that for steel, we should obtain 
approximately 

a = 80000. 



528 APPLIED MECHANICS. 



FACTOR OF SAFETY. 

We have seen that Weyrauch recommends, to use with 
Wohler's results, a factor of safety of three for ordinary bridge 
work and similar constructions. 

Wohler himself, however, in his " Festigkeits versuche mit 
Eisen und Stahl," says, 

i. That we must guard against any danger of putting on 
the piece a load greater than it is calculated to resist, by assum- 
ing as its greatest stress the actually greatest load that can 
ever come upon the piece ; and 

2. This being done, that the only thing to be provided for 
is the lack of homogeneity in the material. 

3. That any material which requires a factor of safety 
greater than two is unfit for use. This advice would hardly be 
accepted by engineers, however. 

He also claims that the reason why it is safe to load car- 
springs so much above their limit of elasticity, and so near 
their breaking-load, is, that the variation of stress to which they 
are subjected is very inconsiderable compared with the greatest 
stress to which they are subjected. 

GENERAL REMARKS. 

It is to be observed, 

i. 'The tests were all made on a good quality of iron and 
of steel, consequently on materials that have a good degree of 
homogeneity. 

2. The specimens were all small, and the repetitions of load 
succeeded each other very rapidly, no time being given for the 
material to rest between them. 

3. No observations were made on the behavior of the piece 
during the experiment before fracture. 



SHEARING-STRENGTH OF IRON AND STEEL. 529 

4. As long as we are dealing only with tension, we can say 
without error that 

c_ _ min L t 
a max L ' 

but as soon as both stresses or either become compression, if 
the piece is long compared with its diameter, we cannot assert 
with accuracy the above relation, nor that 

c max U 
= 

a maxZ 

and hence results based on these assumptions must be to a 
certain extent erroneous. 

5. When a piece is subjected to alternate tension and com- 
pression, it must be calculated so as to bear either : thus, if 
sufficient area is given it to enable it to bear the tension, it may 
not be able to bear the compression unless the metal is 'so dis- 
tributed as to enable it to withstand the bending that results 
from its action as a column. 

While Wohler's tests were mostly confined to ascertaining 
breaking-strengths, the later experimenters upon this subject, 
especially Prof. Bauschinger at Munich, Mr. Howard at the 
Watertown Arsenal, and Prof. Sondericker at the Mass. Institute 
of Technology, have all undertaken to study the elastic change.3 
developed in the material by repeated stresses, and also, to some 
extent, the effect upon resistance to repeated stress, of flaws, of 
indentations, and of sudden changes of section, including sharp 
corners. 

They all agree in the conclusion that flaws and indentations 
(even though very slight) and sharp corners, including keyways, 
reduce the resistance to repeated stress very considerably. 

A brief account will be given of some of their principal con- 
clusions. 



53 APPLIED MECHANICS. 



BAUSCHINGER'S TESTS ON REPEATED STRESSES. 

Bauschinger's tests upon repeated stress include work upon 
the properties of metals at or near the elastic limit. Of the 
properties which he enumerates, the following will be quoted 
here: 

(a) The sets within the elastic limit are very small, and in- 
crease proportionally to the load, while above that point they 
increase much more rapidly. 

(b) With repeated loading, inside of the elastic limit, dropping 
to zero between times, we find each time the same total 
elongations. 

(c) While within the elastic limit the elongations remain 
constant as long as the load is constant; with a load above the 
elastic limit the final elongations under that load are only reached 
after a considerable length of time. 

(d) If by subjecting a rod to changing stresses between an 
upper and lower limit, of which at least the upper is above the 
original elastic limit, the latter were either unchanged or lowered, 
or i f , in the case of its being raised, it were to remain below the 
upper limit, then the repetition of such stresses must finally end 
in rupture, for each new application of the stress increases the 
strain; but if both limits of the changing stress are and 
remain below the elastic limit, the repetition will not cause 
breakage. 

(e) Bauschinger says that by overstraining, the stretch limit 
is always raised up to the load with which the stretching was 
done; but in the time of rest following the unloading the stretch 
limit rises farther, so that it becomes greater than the max- 
imum load with which the piece was stretched, and this rising 
continues for days, months, and years; but, on the other hand, 
that the elastic limit is lowered by the overstraining, often to 
zero; and that a subsequent rest gradually raises it until it 
reaches, after several days, the load applied, and in time 



EXPERIMENTS WITH A REPEATED TENSION MACHINE. 531 

rises above this ; that, as a rule, the modulus of elasticity 
is also lowered under the same circumstances, and is also 
restored by rest, and rises after several years above its 
original magnitude. 

(/) By a tensile load above the elastic limit the elastic 
limit for compression is lowered, and vice versa for a compres- 
sive load ; and a comparatively small excess over the elastic 
limit for one kind of load may lower that for the opposite 
kind down to zero at once. Moreover, an elastic limit which 
has been lowered in this way is not materially restored by a 
period of rest at any rate, of three or four days. 

(<") With gradually increasing stresses, changing from 
tension to compression, and vice versa, the first lowering of 
the elastic limit occurs when the stresses exceed the original 
elastic limit. 

(//) If the elastic limit for tension or compression has been 
lowered by an excessive load of the opposite kind, i.e., one ex- 
ceeding the original elastic limit, then, by gradually increasing 
stresses, changing between tension and compression, it can 
again be raised, but only up to a limit which lies considerably 
below the original elastic limit. 

EXPERIMENTS WITH A REPEATED TENSION MACHINE. 

Bauschinger states that in 1881 he acquired a machine 
similar to that used by Wohler for repeated application of a 
tensile stress. 

The plan of the experiments which he made with it, and 
which are detailed in the I3th Heft of the Mittheilungen, is as 
follows : 

From a large piece of the material there were cut at least 
four, and sometimes more, test-pieces for the Wohler machine. 
One of them was tested in the Werder machine to determine 
its limit of elasticity and its tensile strength ; the others were 



532 APPLIED MECHANICS. 

tested in the Wohler machine, so arranged that the upper limit 
of the repeated stress should be, for the first specimen, near 
the elastic limit ; for the second, somewhat higher, etc., the 
lower limit being in all cases zero. 

From time to time the test-pieces, after they had been sub- 
jected to some hundred thousands, or some millions, of repeti- 
tions, were taken from the Wohler machine and had their limits 
of elasticity determined in the Werder machine. 

The tables of the tests are to be found in the Mittheilungen, 
and from them Bauschinger draws the following conclusions : 

i. With repeated tensile stresses, whose lower limit was 
zero, and whose upper limit was near the original elastic limit, 
breakage did not occur with from 5 to 16 millions of repeti- 
tions. 

Bauschinger says that in applying this law to practical cases 
we must bear in mind two things : (a) that it does not apply 
when there are flaws, as several specimens which contained flaws, 
many of them so small as to be hardly discoverable, broke with 
a much smaller number of repetitions ; (b) another caution is 
that we should make sure that we know what is really the origi- 
nal elastic limit, as this varies very much with the previous 
treatment of the piece, especially the treatment it received 
during its manufacture, and it may be very small, or it may be 
very near the breaking-strength. 

2. With oft-repeated stresses, varying between zero and an 
upper stress, which is in the neighborhood of or above the 
original elastic limit, the latter is raised even above, often far 
above, the upper limit of stresses, and the higher the greater 
the number of repetitions, without, however, its being able to 
exceed a known limiting value. 

3. Repeated stresses between zero and an upper limit, 
which is below the limiting value of stress which it is possible 
for the elastic limit to reach, do not cause rupture ; but if the 
upper limit lies above this limiting value, breakage must occur 
.after a limited number of repetitions. 



EXPERIMENTS WITH A REPEATED TENSION-MACHINE. 533 

4. The tensile strength is not diminished with a million 
repetitions, but rather increased, when the test-piece after hav- 
ing been subjected to repeated stresses is broken with a steady 
load. 

5. He discusses here the probability of the time of forma- 
tion of what he considers to be a change in the structure of 
the metal at the place of the fracture. 

Besides the above will be given the numerical values which 
Bauschinger obtained for carrying strength and for primitive 
safe strength as average values. 

i. For wrought-iron plates : 

/ = 49500 Ibs. per sq. in. 
u 28450 " " " " 

2. For mild-steel plates (Bessemer) : 

t = 62010 Ibs. per sq. in. 

= 34140 " " " " 

> i 

3. For bar wrought-iron, 80 mm. by 10 mm. : 

t = 57600 Ibs. per sq. in. 
u= 31290 " " " " 

4. For bar wrought-iron, 40 mm. by 10 mm. : 

/ == 57180 Ibs. per sq. in. 
u = 34140 " " " " 

5. For Thomas-steel axle : 

t = 87050 Ibs. per sq. in. 
u = 42670 " " " " 

6. For Thomas-steel rails : 

/ = 84490 Ibs. per sq. in. 
u = 39820 " " " " 



534 APPLIED MECHANICS. 

7. For Thomas-steel boiler-plate : 

=57600 Ibs. per sq. in. 
u = 34140 " " " " 

For Thomas-steel axle, and Thomas-steel rails, Bauschinger's 
obtained for the vibration breaking-strength the same values as 
those for primitive breaking-strength. His experiments on the 
other five niaterials, however, give lower values for 5 than for 
u. These values will not be quoted here, however, because they 
were obtained from experiments upon rotating bars of rectangular 
section transversely loaded. 

EXPERIMENTS UPON ROTATING SHAFTING SUBJECTED TO TRANS- 
VERSE LOADS, BY PROF. SONDERICKER. 

Accounts of these tests are to be found in the Technology 
Quarterly of April, 1892, and of March, 1899. In every case the 
(transverse) loads were so applied, that a certain portion, greater 
than ten inches in length, was subjected to a uniform bending- 
moment. At various times, the shaft was stopped, the load was 
removed, then replaced, and again removed, and measurements 
made of the strains and sets. The diameter of the shaft was, 
in every case, approximately one inch. Some extracts from the 
paper of March, 1899, will be given. The investigations were 
conducted along two lines. 

i. The determination of elastic changes, resulting from 
the repeated stresses, and the influence of such changes in pro- 
ducing fracture. 

2. The influence of form, flaws, and local conditions generally 
in causing fracture. 

Accurate measurements of the elastic strains, and sets were 
made at intervals during each test. Characteristic curves of set 
indicate the general character of the 
changes which occurred in the set, the 
abscissa? being the number of revo- 
lutions, and the ordinates the amount 
of the set, a is the characteristic curve 





EXPERIMENTS WITH A REPEATED TENSION MACHINE. 535 



for wrought-iron, and also occurred in one kind of soft steel. No 
change is produced until the elastic limit is reached, and then 
the change consists in a decrease of set. b is the characteristic 
curve for all the steels tested with the single exception mentioned. 
It is the reverse of the preceding, beginning commonly below the 
elastic limit, and consisting of an increase of set; rapid at first, 
but finally ceasing. Under heavy loads, the increase of set L 
very rapid, and ceases comparatively quickly. Accompanying 
the change of set there is a change in the elastic strain in the 
same direction but much smaller in amount. From the fact that 
these changes finally cease, we conclude that, if of sufficiently 
small magnitude, they do not necessarily result in fracture. 

The table on page 536 gives a number of his results. 

Regarding these results he says : 

i. In several cases, changes would have been detected under 
smaller stresses had observations been taken. 

2. Changes of set may be expected to begin at stresses vary- 
ing from J to J of the tensile strength. 

3. The set does not appear to have a notable influence in 
causing fracture until it reaches o".ooi or o".oo2 in a length of 
ten inches. 

4. The effect of rest is to decrease the amount of set. In 
most cases, however, the set lost is soon regained, when the bar is 
again subjected to repeated stress, especially in the case of the 
harder steels. 

Prof. Sondericker also cites a few experiments to determine 
the loss of strength due to indentations, grooves, and key ways. 
In one case, the result of cutting a groove around the steel shaft 
about o".cx>3 deep was a loss of strength of about 40 per cent, 
while similar results were obtained with indentations, and with 
square shoulders. He also cites the case of two pieces of steel 
shafting united by a coupling, where the result of cutting the 
necessary keyways in the shafts caused, apparently, a loss of 
about 50 per cent. 



536 



APPLIED MECHANICS. 





Tensile Prop- 




i 








! erties of the 




Revolu- 








Metal. 




tions at 




Maxi- 




X 


Material. 


Elastic 
Limit 


Tensile 
St'gth 


Stress 
per 
Sq. In. 


which 
Change 
was First 
Ob- 


Revolu- 
tions. 


mum 
Observed 
Sets. 


Remarks. 


rt 

B 




per 
Sq. In. 
Lbs. 


per 
Sq. In. 
Lbs. 


Lbs. 


served. 




Inches. 




D 


Wt.-Iron 


15700 


45080 


30000 


42300 


86400 


< .01200 


Broke at one end at 


















shoulder, and at 


















other where arm 


















was attached. 


40 


1 


24000 


50700 


24000 


1500000 


1500000 














26000 




2427000 




Broke. 


i 


' 


25900 


51390 


26000 


2214000 


2285000 


.00026 


Broke near center. 


2 


" 


25900 


5139 


32000 


486000 


486000 


.00136 


Broke at mark burn- 








. 










ed by electric cur- 


















rent. 


3 


1 ' 


23400 


50510 


24000 


6593000 


6593000 


.00037 












24000 




4059000 


. OOOI I 






. 






25000 




8962000 


.00016 












26000 




3932000 


.00022 












27000 




8155000 


.00037 












28000 




589000 


.00038 


Broke at shoulder. 


4 
33 


Steel 


23400 
24800 


50510 
47400 


28000 
32000 


2506000 
85900 


2506000 
89750 


.00042 

.00771 


Broke at center. 
Broke outside of arm 


















near bearing; color 


















blue black. 


34 





24800 47400 


32000 


103500 


116600 


.00832 


Do. 


21 


' ' 


30400 62590 


32000 


4395000 


4395000 


. OOO29 












34000 




8339000 


.00032 












36000 




4627000 


.00041 












36000 




1428000 


.OO008 


After resting unload- 


















ed 1 8 days. 










38000 




3769000 


.00023 












40000 




4523000 


.0005 4 












42000 




505000 


.00072 


Broke near shoulder. 


54 





42000 


63130 


45000 


163000 


163000 


.00312 


Broke at shoulder. 


50 


4 


23200 


73760 


30000 


339000 


339000 


.OOIOO 












35000 




16400 


.00282 


Not broken. 


25 


' ' 


38300 


78010 


40000 


5031000 


5031000 


.00028 












42000 




2483000 


.00046 


Broke at shoulder. 


26 


" 


38300 


78010 


40000 


20838000 


20838000 


.00037 












42000 




3311000 


.00044 


Not broken. 


18 


" 


50000 


81010 


36000 


6463000 


6982000 


.00052 


Broke where arm was 


















attached. 


iQ 


1 


50000 


81010 


36000 


7252000 


7686000 


.00069 


Broke at shoulder. 


20 


' 


50000 


81010 


34000 


2I22IOOO 


21 22IOOO 


.00028 












36000 




13577000 


.00067 












38000 




2263OOO 


.00113 












38000 




9237000 


.00116 


After resting 6 mos. 


















unloaded. 










40000 




932000 


.00177 


Broke at shoulder. 


S3 





58000 


96580 


50000 


24000 


146500 


.00249 


Broke at shoulder. 


58 




58000 


96580 


45000 
50000 


50100 


50100 
156900 


.00020 
.00289 


Broke near middle. 


29 


" 


54000 


104480 


40000 


5257000 


5257000 


.00046 












42000 




7125000 


.00067 












44000 




4626000 


.00100 












46000 




6/60000 


.00145 












48000 




4965000 


.00196 












50000 
50000 




I 7OOOO 
1000 


.00197 
.00203 


After 24 days rest un- 


















loaded ; not broken. 


55 


" 


50000 


104830 


35000 


276900 


276900 


.00060 












40000 




237900 


.00274 












50000 




22530 


.00615 


Broke near shoulder; 


















color dark straw. 


57 




50000 


104830 


60000 


14300 


14900 


.00768 


Broke near shoulder; 
color dark blue. 



EXPERIMENTS WITH A REPEATED TENSION MACHINE. 537 

TESTS OF ROTATING SHAFTING UNDER TRANSVERSE LOAD, BY 
MR. HOWARD AT THE WATERTOWN ARSENAL. 

A large number of tests of this character have been made at 
the Watertown Arsenal. A few extracts will be given from the 
remarks of Mr. Howard upon the subject, which may be found 
in the Technology Quarterly of March, 1899, as follows: 

"In the Watertown tests, two principal objects have been 
in view, namely, to ascertain the total number of repetitions of 
stresses necessary to cause rupture, and to observe through what 
phases the physical properties of the metal pass prior to the 
limit of ultimate endurance. The Watertown tests have included 
cast-iron, wrought-iron, hot and cold rolled metal, and steels 
ranging in carbon from o.i per cent to i.i per cent, also milled 
steels. The fibre-stresses have ranged from 10000 pounds 
per square inch on the cast-iron bars up to 60000 pounds pel- 
square inch on the higher tensile-strength steel bars. 

The speed of rotation was from 400 per minute up to 2200 
per minute, in different experiments. Observations were made 
on the deflection of the shafts, and on the sets developed. It 
was early observed that intervals of rest were followed by tem- 
porary reduction in the magnitude of the sets. In the Report 
of Tests of Metals of 1888, he says the deflections tend to 
diminish under high speeds of rotation, when the loads exceed 
the elastic limit of the metal, and tend to cause permanent sets; 
but, on the other hand, when the elastic limit is not passed, the 
deflections are the same within the range of speeds yet experi- 
mented upon. 

Efforts were inaugurated at this time to ascertain the effect 
of repeated alternate stresses on the tensile properties of the 
metal, and it appeared that such treatment tended to raise the 
tensile strength of the metal before rupture ensued. 

Concerning the limit of indefinite endurance to repeated 
stress we know but very little. In most experiments rupture 
occurs after a few thousand repetitions, so high have been the 



533 



APPLIED MECHANICS. 



applied stresses. Examples are not uncommon in railway prac- 
tice of axles having made 200000000 rotations. In order to 
establish a practical limit of endurance, indefinite endurance, 
if we choose to call it so, our experimental stresses will need to 
be somewhat lowered, or new grades of metal found. 

The following table which accompanied the Watertown 
Arsenal Exhibit at the Louisiana Purchase Exposition gives a 
summary of some of the repeated stress tests upon three different 
grades of steel: 

STEEL BARS. 
Tensile Tests and Repeated Stress Tests on Different Carbon Steels. 





Tensile Tests. 


Repeated Stress Tests. 












Mechan- 


Maxi- 








Elastic 


Ten- 


Elon- 


Con- 


ical 


mum 




Mechan- 


Description. 


Limit 
per 


sile 
St'gth 


gation 
in 4 


trac- 
tion of 


Work at 
Rupture 


Fiber 
Stress 


Number 
of Rota- 


ical Work 
at Rup- 




Sq. In. 
Lbs. 


per 


Ins. 
Per ct. 


Area 
Per ct. 


per 
Cu.In. 


per 
Sq. In. 


tions at 
Rupture. 


ture per 
Cu. In. 






Lbs. 






Ft.-lbs. 


Lbs. 




Ft.-lbs. 














(60000 


6470 


32835 














50000 


17790 


62635 


0.17 Carbon steel . 


51000 


68000 


33-5 


51-9 


982 


45000 
j 40000 


70400 
293500 


201960 
665290 














1 35000 


5757920 


9992390 














30000 


*236ooooo 


*295ooooo 














60000 


12490 


63387 














50000 


93160 


328000 


0.5 5 Carbon steel. 


57ooo 


106100 


16.2 


18.7 


1.047 


45000 
40000 


166240 
455350 


476900 
1032130 














35000 


9007 20 


1563125 














30000 


*i 9870000 


*24838ooo 














60000 


37250 


189044 














55000 


93790 


399780 


0.82 Carbon steel. 


63000 


142250 


8.5 


6.5 


888 


50000 
45000 


213150 
605460 


750465 
1736910 














40000 


*i756oooo *409730oo 














35000 


*I9220000 


*33635ooo 



















* Not ruptured. 
GENERAL REMARKS. 

That the amount of detailed information regarding repeated 
stresses is small compared with what is needed will be evident 
when we consider the number of cases in which metal is subjected 
to such stresses in practice, among which are shafting, connecting- 
rods, parallel rods, propeller-shafts, crank-shafts, railway axles, 
rails, riveted and other bridge members, etc. In the case of 



TORSIONAL STRENGTH OF WROUGHT IRON AND STE^L. 539 



some of them, notably, railway axles, attempts have been made 
to base specifications for the material upon such tests as have 
become available upon repeated stresses. 

229. Shearing-strength of Iron and Steel. Some of 
the most common cases where the shearing resistance of iron 
and steel is brought into play are : 

i. In the case of a torsional stress, as in shafting. 

2. In the case of pins, as in bridge-pins, crank-pins, etc. 

3. In the case of riveted joints. 

The so-called apparent outside fibre-stress at fracture, as 
determined from experiments on torsional strength, is found 
to be not far from the tensile strength of the metal, and is, of 
course, greater than the shearing-strength, for the same reasons as 
render the modulus of rupture greater than the actual outside 
fibre-stress at fracture in transverse tests. 

Moreover, the shearing strength of wrought-iron rivets is 
shown by experiment to be about f the tensile strength of the 
rivet metal. 

In regard to cast-iron, Bindon Stoney found the shearing and 
tensile strength about equal. 

The cases where shearing comes in play in wrought-iron and 
steel will therefore be treated separately. 

230. Torsional Strength of Wrought-iron and Steel. 
The method formerly followed, and in use by some at the present 
day, was to compute the strength of a shaft from the twisting- 
moment only, neglecting the bending, but varying the working- 
strength per square inch to be used according to the character 
of the service. It is generally the fact, however, that when 
shafting is running the pulls of the belts create a bending back- 
wards and forwards, bringing the same fibre alternately into 
tension and compression; and this is combined with the shearing- 
stresses developed due to the twisting-moment alone. At the 
two extremes of these general cases are : 

i. The case when the portion of a shaft between two hangers 



540 APPLIED MECHANICS. 

has no pulleys upon it, and when the pulls on the neighboring 
spans are not so great as to deflect this span appreciably. That 
is a case of pure torsion: and if the shaft is running smoothly, 
with no jars or shocks, and no liability to have a greater load 
thrown upon it temporarily, we may compute it by the usual 
torsion formula, given in 212; using for breaking-strength of 
wrought-iron and steel the so-called apparent outside fibre-stress 
at fracture as determined from torsional tests, and a factor of 
safety six, and such a proceeding will probably give us a reasonable 
degree of safety. 

2. The case when, pulleys being placed otherwise than near 
the hangers, the belt-pulls are so great that the torsion becomes 
insignificant compared with the bending, and then it would be 
proper to compute our shaft so as not to deflect more than y^-g-g- 
of its span under the load, or better, not more than yeV o : f 
course we- should compute also the breaking transverse load, and 
see that we have a good margin of safety. 

In other cases, the methods pursued, the first two of which 
are incorrect, have been 

i. By using the ordinary torsion formula combined with a 
large factor of safety. 

2. By computing the shaft also for deflection, and providing 
that its deflection shall not exceed rsV<r or Trinr f i ts span. 

This, however, neglects the torsion, and also the rapid change 
of stress upon each fibre from tension to compression. 

3. By using the formula of Grashof or of Rankine for com : 
bined bending and twisting, with the constants that have been 
derived from experiments on simple tension or simple torsion. 

The results given on pages 544 and 545 are from pieces of 
shafting of considerable length. As has been stated, the so-called 
"apparent outside fibre-stress at fracture" appears to be not very 
far from the tensile strength of the material, and the torsional 
modulus of elasticity appears to be from three-eighths to two- 
fifths of the tensile modulus of elasticity. 



TORSIONAL STRENGTH OF WROUGHT-IRON AND STEEL. 541 

Under certain circumstances the bending may have the 
greatest influence, while the twisting may be predominant in 
others, or their influence may be equally divided. Which of these 
is the case will depend upon the location of the hangers and of 
the pulleys, the width of the belts, etc., etc. 

As to the formulae which take into account both twisting and 
bending, there are two, both of which are based upon the theory 
of elasticity. The first, which is the most correct from a theo- 
retical point of view, is that given by Grashof and other writers 
on the theory of elasticity, and is 




where Mi='greatest bending-moment ; 
M% = greatest twisting-moment ; 
r = external radius of shaft; 

/ = moment of inertia of section about a diameter; 
/ = greatest allowable stress at outside fibre; 
w = a constant depending on the nature of the material. 
In the case of iron or steel the value of m is often taken as 4, 
though it is, in most cases, nearer 3. When m = 4 we have 



f __r 

'-/ 

The other formula, which is also based upon the theory of 
elasticity, but which is not as correct, is that given by Rankine, 
and is 



With a view to determine the behavior of shafting under a 
combination of twisting and bending, suitable machinery was 
erected in the engineering laboratories of the Mass. Institute of 
Technology, and a number of tests were made. 



542 



APPLIED MECHANICS. 



The principal points of the method of procedure are the 
following, viz.: 

i st. The shaft under test is in motion, and is actually driving 
an amount of power which is weighed on a Prony brake. 

2d. A tr nsverse load is applied which may be varied at the 
option of the experimenter, and which is weighed on a platform 
scale. 

3d. The proportion between the torsional and transverse 
loads may be adjusted to correspond with the proportion be- 
tween the power transmitted and the belt-pull sustained by a 
shaft in actual use. 

4th. Tests are made not only of breaking-strength, but also 
angle of twist and deflection under moderate loads are measured. 

The following table will give the results oT the tests on 
iron shafts, and they will then be discussed : 





Time 






ji/i, 


*., 


A, 


/a, 








No. 


of 


Total 


H. P. 


max. 


max. 


max. 


max. 




j- 




of 
Test. 


run- 
ning, 
min- 


revolu- 
tions. 


trans- 
mitted. 


bending 
moment. 


twisting 
moment. 


bend, 
fibre 


twist, 
fibre 


Grashof. 


Ran- 
kine. 


Diam. 
ins. 




utes. 






In.-lbs. 


In.-lbs. 


stress. 


stress. 








8 


37-5 


7040 


11.717 


11514.1 


3926.4 


60024 


10234 


62162 


6i755 


"25 


9 


200 


38839 


8.181 


10507.8 


2656.8 


54777 


6925 


55876 


55671 


".25 


10 


l62 


31641 


5.291 


9891.0 


1714.6 


5*562 


4469 


52062 


5*976 


".25 


ii 


553 


108002 


4-331 


9241.7 


1399.2 


48179 


3647 


48539 


48769 


"25 


12 


408 


80694 


6.276 


9241.7 


2027.6 


48179 


5287. 


48911 


48769 


".25 


13 


98 


19333 


6.342 


8917.1 


2028.2 


46485 


5287 


47245 


47105 


"25 


14 


423 


82741 


6.283 


8917.1 


2029.7 


46485 


5290 


47246 


47106 


"25 


15 


565 


108739 


6.192 


8592-5 


2031.6 


44793 


5295 


45582 


45436 


"25 


10 


443 


88208 


6.338 


8267.8 


2026.8 


43100 


5283 


439H 


437*3 


".25 


17 


95i 


185233 


6.283 


3781-5 


2029.7 


38503 


10333 


41768 


41117 


" 








14 874 


8218 




84185 




0X028 


68 


It 


2O 






*T"JT 

7, 562 


7976 


2394 


82112 


I2l88 




8 o i 





21 






9.972 


/y/^ 
8917 


3232 


90793 


l6454 




03716 





22 






T C T CQ 


8017 






2468l 


O86l2 













*-> A jy 
2 . 955 


y L / 

7652 


945 


77OI7 


48ll 




82 


II 














//y 1 j 


T- OI> L 









TOR SIGNAL STRENGTH OF WROUGHT-IRON, ETC. 543 

In 19 to 23 inclusive the number of revolutions was small and 
the outside fibre stress at fracture was correspondingly large. 

Two specimens of the \' '.25 shafting and two of the i" 
were tested for tension, the results being as follows : 

Breaking-strength, per sq. in. 

,, ,. ( No. i . 46800 

,".35 diameter ] NQ 2 ...... 

Average ... 




Average .... 60250 
As to conclusions : 

1st. It is plain from these results that a shaft whose size is 
determined by means of the results of a quick test would be 
too weak, and that our constants should be obtained from tests 
which last for a considerable length of time. 

2d. A perusal of the tables will show that the results ob- 
tained apply more to the bending than to the twisting of a 
shaft, as the transverse load used in these tests was so large 
compared with the twist as to exert the controlling influence. 
This will be plain by a comparison of the values of f lt f t , 
and/. 

3d. Nevertheless, the bending-moments actually used were 
generally less than such as might easily be realized in practice 
with the twisting-moments used. 

4th. It seems fair to conclude that, in the greater part of 
cases where shafting is used to transmit power, as in line-shaft- 
ing or in most cases of head shafting, the breaking is even 'more 
liable to occur from bending back and forth than from twist- 
ing, and hence 'that in no such case ought we to omit to 
make a computation for the bending of the shaft as well as the 
twist. 

5th. As to the precise value of the greatest allowable out- 
side fibre stress to be used in the Grashof formula, it is plain 



544 



APPLIED MECHANICS. 



that it is not correct to use a value as great as the tensile 
strength of the iron, and while the tests show that this figure 
should not for common iron exceed 40000 Ibs. per square inch, 
it is probable that tests where a longer time is allowed for 
fracture will show a smaller result yet. 

TORSIONAL TESTS OF WROUGHT-IRON. 



Norway Iron. 


Burden's Best. 


x 





C " 


-*- 


^ c 


(0 O 


J= <' 


--. 


be 

C A 


*-> 
M rt 





01 '"' 


V 


OJ _C 


(fl C 


is, 




3 C 
"3 .'". 


ir. o 

8* 


! 




g& 


."S -E 


2 -E 


Diameter. (In< 


Distance betw 
Grips. (Incl 


Maximum Twi 
Moment. (I 
Lbs.) 


Number of Tu 
between Gri 
Fracture. 


sarent Out 
ibre Stress, 
-bs. per sq. 


a >,cr 
o.ti tn 


Diameter of C 
section. (In< 


| 

u 

p 


MaximumTwi 
Moment. (] 
Lbs.) 


Number of Tu 
between Gri 
Fracture. 


Apparent Out 
Fibre Stress 
(Lbs. per sq. 


Shearing Modi 
of Elasticity 
(Lbs. per sq 


au^ 


j= ^ 
1/3 


.00 


70.40 


72360 


16.50 


46065 


11406000 


.01 


63.8 


85050 


9-50 


533oo 


11300000 


.02 


72.00 


74970 


16.00 


46600 


13215000 


.01 


59-0 


86400 


8.63 


54200 


11500000 


3 


7 1 -3 


72000 


14.00 


43757 


12902000 


.01 


53-o 


84510 


6.87 


53000 


11200000 


02 


70.40 


74520 


17.00 


46321 


12247000 


.00 


58-8 


87480 


8. 4 o 


557oo 


II600000 


.02 


69.80 


72000 


14-25 


45837 


12738000 


.00 


65-5 


85410 


8.52 


544oo 


11600000 


03 


70.30 


74880 


15-50 


456Qa 


11361000 


.01 


60.2 


8559 


8.82 


537 


1 1200000 


03 




74880 


20.00 


44658 


11957000 


.02 


58.5 


85140 


8.05 


52600 


II3OOOOO 


03 


70.20 


7956o 


16.00 


48437 


11554000 


.OO 


57-0 


82650 


7-31 


52600 


II500000 


03 


84 


74880 


15-5 


45590 


11900000 


.02 


57-8 


86580 


8.54 


535oo 


II2OOOOO 


54 


54 


35100 


12. 


48950 


9840000 


.02 


59-5 


86040 


8.61 


53200 


II200000 


52 


49 


34200 


11.25 


49600 


11410000 


.02 


60.0 


87840 


8-93 


543 


11300000 


53 


53 


33840 


8.50 


48120 


Il6oOOOO 


.OX 


60.0 


88200 


8.48 


544oo 


I 1200000 






3-384.0 


II . IO 


48110 




.OI 


53- 3 


87480 


7.85 




1 I 300000 


53 


49 


6 JT- U 

34920 


14.56 


49650 


11840000 


.OI 


59-5 


83970 


8.01 


52700 


11400000 


52 


53 


34200 


8.98 


49600 


12480000 


.01 


59-5 


84780 


8.32 


53200 


II200000 


25 


70 


111960 


5-8o 


50060 


11830000 


03 


61.0 


83520 


8.98 


50900 


I I IOOOOO 


27 


75 


106920 


12-30 


46600 


10900000 


.00 


63.0 


84050 


9.24 


535o 


11700000) 


25 


70 


108360 


10.90 


48500 


IlSooooo 


.02 


60.3 


85950 


7-94 


53*00 


1I2OOOOO 


25 


76 


109800 


9.90 


49100 


11700000 


.OI 


60.0 


84600 


8.62 


53100 


11400000 


23 


70 


113670 


11.00 


52200 


12000000 


.02 


61.0 


83520 


8.50 


51600 


IIOOOOOO 


.26 


7 


107640 


IO.OO 


47500 


II400000 


.OO 


60.5 


86040 


8.80 


54000 


11600000! 


Refined Iron. 


.01 
2.01 


3o!8 


85680 
87480 


siii 


53700 
54900 


11300000 
11500000 


.03 


72.10 


84240 


4.00 


51285 


I247IOOO 


2.01 


47-8 


85860 


7.24 


539o 


11300000 


.03 


71 .20 


66960 




40646 


12576000 


2.OI 


59-4 


85050 


8.92 


533 


11500000 


03 


70.50 


70200 


3-50 


41743 


II372000 


2.01 


60.9 


86400 


8.75 


54200 


11300900 




72.10 


72000 


2.30 


43834 


10960000 


2.00 


59-5 


86400 


9.08 


55oo 


11700000 


03 


71.80 


61920 


2.80 


36820 


11393000 


2.01 


58.5 


85650 


8.30 


537 


11500000 


03 


7!-3 


68760 


2.50 


40887 


H36OOOO 


2.00 


58.5 


84870 


7-87 


54200 


11800000 




69.30 


78120 


2.80 


46453 


I287IOOO 


2.OI 


58.3 


87300 


9-45 


54800 


11500000 


30 




22320 


14.60 


52127 


II436000 


2.01 


58-4 


86490 


8-73 


54200 


11900000 


50 


7I-25 


36360 


10.30 


54867 


II482000 


2.OI 


58.4 


87120 


8.09 


54600 


11600000 


33* 


71-75 


45360 


6.70 


50852 


12359000 


2.00 


59-8 


84870 


8.80 


54000 


11700000 


So 


79-5 


32760 


14.60 


49435 


IO7IOOO 














2 7 


62 


23800 


4.71 


53920 


2720000 














.26 


63 


22950 


5.60 


54250 


223OOOO 














.29 


64 


24600 


5-24 


52840 


2510000 














25 


60 


16640 


3-70 


52150 


2190000 














2 5 


61 


2159 


4.40 


5437 


2510000 














27 


66 


23750 


5-oo 


53890 


2840000 














77 


7 1 


38970 


2. IO 


35790 


I2OOOOO 














75 


61.1 


56150 


7-1 


534oo 


1200000 














75 


64.0 


55350 


8.7 


52600 


12OOOOO 














7 2 


64-5 


45090 


5-4 


44900 


1800000 














75 


61.0 


53360 


6. i 


50700 


II500000 














- /4 


63.0 


557* 


1.49 


52900 


IT3OOOOO 















The above tables show the results of tests made in the engi- 
.leering laboratories of the Massachusetts Institute of Technol- 



TO RSI ON A L S TRENG TH OF IV RO UGHT-IRON AND S TEEL. 545 

ogy upon the torsional strength of various kinds of wrought-iron. 
The figures in the column headed " Apparent outside fibre-stress " 

Mr 

are obtained from the formula / = -j- 9 where M = maximum 

twisting-moment, r= outside radius of shaft, and / = polar moment 
of inertia of section. Of course it is not the outside fibre -stress. 



TORSIONAL TESTS OF BESSEMER STEEL. 





1 
O 


ri 

J8 

o 

c 


1 


+i (-1 


S 

M 


|jti 


& 


*| 


0) 


a 


1 

1 


jj 


f +>" 


d 
IjJ 


iF! 


I 5 


f: 


S 


1 


bo 
J 


S 


CJ 


^1 


23 rt 


* 


s 


1u 


11 

rt o 


'S 


.3 

+3 


<u- c 

'O W5 


Sf" 


**"* S 


g|S 


^ 1 8 


ri 


!> 


3 


J 


^S ^ 


^ G 


CX^^g 


SM ^ 


E & p 


S 


3 


S 




o 


1 


< w 


S 


I*" 


I 7C 


60 .00 








66960 


63632 


12418000 


ii .88 


- 1 /o 


en 7r 








30600 


"O w 3'* 

7^031 


I I243OOO 




I - "3Q 


oy i j 

fQ CQ 








oy 
3762O 


/O^v)- 1 - 
712 CO 


A i^ if ^^WV-* 

12 CQ4OOO 


I C .OO 


2 .OO 


oy o w 

56 .00 








o / 
101520 


/ o w 

64630 


A^ ^yifwwvy 

1 1 82 OOOO 


7 ^5 


2 .OO 


55-Qo 


30 








100260 


63830 


10320000 


7.87 


2 O2 


c6 . oo 


?lS 






I I 1960 


*"o^S?r> 


I IQQOOOO 






















T &C 


A A Ort 


2 4 






81240 


A - 2 -/-\ 


I O2 COOOO 


* R^7 




















jo OI 


55 .00 


36 






112 COO 


'roT) i n 


I34IOOOO 


<; 76 


1 * ' w 

2 O2 


190 . oo 


o 
144, 






oy 

I 34IOO 


82860 


I l83OOOO 


D * / 

6 77 




96.00 


A if if 

75 


20160 


30400 


'O ' 

56000 


84500 


12200000 


u / / 
16.30 


5 


94-00 


75 


19800 


29900 


5328o 


80400 


I220OOOO 


1-570 


52 


93.00 


75 


21600 


31300 


53280 


77300 


IO70000O 


16.10 


53 


94.00 


75 


21600 


30800 


52560 


7470 


I09OOOOO 


15.80 


49 


60.00 


40 







43920 


66280 


IlSoOOOO 


8.60 


5 


58.00 


40 


14400 


21700 


44640 


67400 


H7OOOOO 


13.30 


5 


57 -5P 


40 


17000 


25600 


44820 


67600 


II900000 


ii .50 


5 


57.60 


40 


17000 


25600 


45810 


69100 


II90OOOO 


9.90 


5 


59-oo 


40 


16000 


24100 


4545 


68600 


Il6oOOOO 


10. 80 


5 


58.80 


40 


18000 


27200 


44460 


67100 


II700000 


10.50 


5 


59.10 


40 


l6j2OO 


24400 


45000 


67900 


II500000 


13.20 


5 


58.20 


40 


18000 


27200 


' 44920 


67800 


I I 7OOOOO 


10.80 


5 


58.00 


40 


18000 


27200 


45540 


68700 


I I 7OOOOO 


11.40 



546 APPLIED MECHANICS, 

232. Riveted Joints. The most common way of uniting 
plates of wrought-iron or steel is by means of rivets. It is, 
therefore, a matter of importance to know the strength of such 
joints, and also the proportions which will render their efficie,n- 
cies greatest; i.e., that will bring their strength as near as 
possible to the strength of the solid plate. 

In 177 was explained the mode of proportioning riveted 
joints usually taught, based upon the principle of making all 
the resistances to giving way equal, and assuming, as the modes 
of giving way, those there enumerated. This theory does not, 
however, represent the facts of the case, as 

i. The stresses which resist the giving- way are of a more 
complex nature than those there assumed, so that the efficiency 
of a joint constructed in the way described above may not be 
as great as that of one differently constructed ; 

2. The effects of punching, drilling, and riveting, come in 
to modify further the action ; and 

3. The purposes for which the joint is to be used, often fix 
some of the dimensions within narrow limits beforehand. 

In order to know, therefore, the efficiency of any one kind 
of joint, we must have recourse to experiment. And here again 
we must not expect to draw correct conclusions from experi- 
ments made upon narrow strips of plate riveted together with 
one or two rivets ; but we need experiments upon joints in wide 
plates containing a sufficiently long line of rivets to bring into 
play all the forces that we have in the actual joint. The greater 
part of the experiments thus far made have been made upon 
narrow strips, with but few rivets. The number of tests of the 
other class is not large, and of those that have been made, the 
greater part merely furnish us information as to the behavior 
of the particular form of joint tested, and do not teach us how 
to proportion the best or strongest joint in any given plates, as 
no complete and systematic series of tests has thus far been 
carried out, though such a series has been begun on the govern- 
ment testing-machine at the Watertown Arsenal. 



RIVETED JOINTS. 



The only tests to which it seems to the writer worth while 
to make reference here are : 

i. A portion of those made by a committee of the British 
Institution of Mechanical Engineers, inasmuch as, although a 
very large part were made upon narrow strips with but few 
rivets, nevertheless a portion were made upon wide strips. 

2. The tests on riveted joints that have been made on the' 
government testing-machine at Watertown Arsenal. 

i. The account of this series is to be found at intervals 
from 1880 to 1885 inclusive, with one supplementary set in 1888, 
in the proceedings of the British Institution of Mechanical 
Engineers; but as all except the supplementary set has also 
been published in London Engineering, these latter references 
will be given here as follows : 

Engineering for 1880, vol. 29, pages no, 128, 148, 254, 300, 350. 

" 1881, vol. 31, " 427, 436, 458, 508, 588. 

" 1885, vol. 39, " 524. 

" 1885, vol. 40, " 19,43. 
Also, Proc. Brit. Inst. Mechl. Engrs., Oct. 1888. 

2. The second series, referred to above, or those made on 
the government testing-machine at Watertown Arsenal, are to 
be found in their reports of the following years, viz., 1882, 
1883, 1885, 1886, 1887, and 1895. 

3. Report of tests of structural material made at the 
Watertown Arsenal, Mass., June, 1891. 

While it is from tests upon long joints that we can derive 
correct and reliable information to use in practice, and hence 
while the experiments already made give us a considerable 
amount of information, nevertheless as the tests have not yet 
been carried far enough to furnish all the information we need, 
and to settle cases that we are liable to be called upon to 
decide, therefore, before quoting the above experiments, a few 
of the rules and proportions more or less used at the present 



54$ APPLIED MECHANICS. 

time, and the modes of determining them, will be first ex- 
plained. 

In this regard we must observe that practical considerations 
render it necessary to make the proportions different when the 
joint is in the shell of a steam-boiler, from the case when it is 
in a girder or other part of a structure. 

In the case of boiler-work, the joint must be steam-tight, and 
hence the pitch of the rivets must be small enough to render 
it so : whereas in girder-work this requirement does not exist ; 
and hence the pitch can, as far as this requirement goes, be 
made greater. 

It is probable, that, with good workmanship, we might be able 
to secure a steam-tight joint with considerably greater pitches 
than those commonly used in boiler-work ; and now and then 
some boiler-maker is bold enough to attempt it. 

Some years ago punching was the most common practice , 
but now drilling has displaced punching to such an extent that 
all the better class of boiler-work is now drilled, and drilling is 
also used to a very considerable extent in girder-work. When 
drilling is used, the plates, etc., to be united should be clamped 
together and the holes drilled through them all together. In 
this regard it should be said : 

i. When the holes are drilled, and hence no injury is done 
to the metal between the rivet-holes, this portion of the plate 
comes to have the properties of a grooved specimen, and hence 
has a greater tensile strength per square inch than a straight 
specimen of the same plate, as the metal around the holes has 
not a chance to stretch. This excess tenacity may amount 
to as much as 25 per cent in some cases, though it is usually 
nearer 10 or 12 per cent, depending not only on the nature of 
the material, but also on the proportions. 

2. When the holes are punched, we have, again, a grooved 
specimen, but the punching injures the metal around the hole, 
and this injury is greater the less the ductility of the metal : 
thus, much less injury is done by the punch to soft-steel plates 



RIVETED JOINTS. 549 



than to wrought-iron ones, and less to thin than to thick plates. 
This injury may reach as much as 35 per cent, or it may be 
very small. Besides this, in punching there is liability of crack- 
ing the plate, and of not having the holes in the two plates that 
are to be united come exactly opposite each other. A number of 
tests on the tenacity of punched and drilled plates of wrought- 
iron, and of mild steel, made on the government testing-machine 
at Watertown Arsenal, are given on page 564 tt seq. 

The hardening of the metal by punching also decreases the 
ductility of the piece. 

The injury done by punching may be almost entirely re- 
moved in either of the following ways : 

i. By annealing the plate. 

2. By reaming out the injured portion of the metal around 
the hole ; i.e., by punching the hole a little smaller than is de- 
sired, and then reaming it out to the required size. 

There is a certain friction developed by the contraction of 
the rivets in cooling, tending to resist the giving way of the 
joint ; and some have advocated the determination of the safe 
load upon a riveted joint on the basis of the friction developed, 
instead of on the basis of strength notably M. Dupuy in the 
Annales des Fonts et Chausees for January, 1895 ; but this 
seems to the author an erroneous and unsafe method of pro- 
ceeding: i, because tests show that slipping occurs at all 
loads, beginning at loads much smaller than the safe loads on 
the joint ; 2, because all friction disappears before the break- 
ing load is reached. 

Hence it is safer to disregard friction in designing a tensile 
riveted joint. 

The shearing-strength of the rivets would appear to be 
about two thirds the tensile strength of the rivet metal. 

Before proceeding to give an account of Kennedy's tests, 
and of those made at the Watertown Arsenal, which form the 
principal basis for determining the constants, i.e., the tearing- 
strength of the plate, the shearing-strength of the rivet iron, 



550 APPLIED MECHANICS. 

and the ultimate compression on the bearing surface, it will be 
best to outline the proper method of designing a riveted joint, 
and for this purpose a discussion of a few cases of tensile riveted 
joints, as given by Prof. Peter Schwamb, will be given by way 
of illustration. 

The letters used will be as follows, viz. : 

d = diameter of driven 'rivet in inches ; 

t = thickness of plate in inches ; 

/! thickness of one cover-plate in inches ; 

f s = shearing-strength of rivet per square inch ; 

f t tearing-strength of plate per square inch ; 

f c = crushing-strength of rivet or plate per square inch; 

/ pitch of rivets in inches ; 

p d = diagonal pitch in inches ; 
/ = lap in inches. 

In every case of a tension-joint we begin by selecting a 
repeating section and noting all the ways in which it may fail. 
It would seem natural, then, to determine the diameter of the 
rivet to be used by equating the resistance to shearing and 
the resistance to crushing, and in some cases it is desirable to 
adopt the resulting diameter of rivet ; but there are also many 
cases where there is good reason for adopting either a larger 
or a smaller rivet, and others where there is good reason for 
determining the trial diameter in some other way. 

Thus we may find that the rivet which presents equal re- 
sistance to shearing and crushing may be too large to be suc- 
cessfully worked, or it may require a pitch too large for the 
purposes for which the joint is to be used; or, on the other 
hand, it may be so small that it would lead to a pitch too 
small to be practicable ; or it might, in a complicated joint, 
where there are a good many ways of possible failing, lead to 
a low efficiency. In all cases, a commercial diameter must be 
selected. 



SINGLE-RIVETED LAP-JOINT. 55 1 

Single-riveted Lap-joint. Repeating section containing 
one rivet may fail by 

1, shearing one rivet. Resistance =^ t 

4 

2, tearing the plate. Resistance =f t (pd)t. 

3, compression. Resistance f c td. 

Equating i and 3 gives d = y (i) 

71 ft 

A larger rivet will crush, a smaller one will shear. 
The diameter given by (i) will frequently be found to be 
larger than can be successfully worked. 

Equating 2 and 3 gives p d(\ + y ). (2) 

Equating i and 2 gives p = d (i + ^ }. (3) 

4* ft/ 

If the value of d given in (i) is used, then (2) and (3) give 
the sameVesult. If, however, a different value of d is used, 
then the pitch should be determined by (2) for a larger and 
by ($) for a smaller rivet. 

It may be well to note that whenever compression fixes 
the pitch, the computed efficiency 

P~d_ f< 

P -f t +f. 

is independent of the diameter of the rivet, and that this is 
the maximum efficiency obtainable with this style of joint. 

SINGLE-RIVETED DOUBLE-SHEAR BUTT-JOINT. 

The combined thickness of the two cover-plates should 
always be greater than /, and, this being the case, we proceed 
as follows : 



55 2 APPLIED MECHANICS. 

Repeating section containing one rivet may fail by 
i , shearing one rivet in two places. Resistance = /, . 

2, tearing the plate. Resistance = f t (p d)t. 

3, compression Resistance =f e td. 

Equating i and 3 gives d= j (4) 

71 ft 

A larger rivet will crush, a smaller one will shear. 

The diameter given by (4) is just one half that given by (i), 
f nd will frequently be found to lead to a pitch too small to use in 
practice. In such cases we should use a larger rivet. 

Equating 2 and 3 gives p=d[i + j). (5) 

v It' 

Equating i and 2 gives p=d(i+~j}. (6) 

If the value of d given by (4) be used, then (5) and (6) give 
the same result. If, however, a different value of d be used, then 
the pitch should be determined by (5) for a larger and by (6) for 
a smaller rivet. 

For the diagonal pitch, in the case of staggered riveting, we 
should have, at least, according to Kennedy's sixth conclusion 
(see page 566) 2(p d d)=^(pd) and hence p d = $p + $d. 

DOUBLE-RIVETED LAP-JOINTS. 

Repeating section containing two rivets may fail by 

nd 2 
i, shearing two rivet sections. Resistance = J 8 . 

2, tearing plate straight across. Resistance = }t(pd)t. 
3, compression on two rivets. Resistance = j c (2td). 

Equating i and 3 gives d= /. (7) 

7T / 



EXAMPLE OF A SPECIAL JOINT. 553 

A larger rivet will crush, a smaller one will shear. 

The diameter given by (7) would usually be found too large. 

Equating 2 and 3 gives p=d(i+-j?J. (8) 

Equating i and 2 gives p=d \i H -- ^/ . (9) 

The pitch should be determined by (8) for a larger and by (9) 
for a smaller rivet than that given by (7). 

For p d we should have, as in the last case, according to 
Kennedy, p d 



EXAMPLE OF A SPECIAL JOINT. 

The joint shown in the cut is one where a part of the rivets 
are in single and a part in double shear. 

Repeating section containing five rivet 
sections may fail by 

i, tearing on ab. 

Resistance = ft(p d)t. 

2, shearing five rivet sections. 

, 
Resistance = / a 




4 
3, tearing on ce, and shearing one rivet on ab. 

nd 2 
Resistance = j t (p zd)t + f 8 . 

4 
4, tearing on ce, and crushing one rivet. 

Resistance = j t (p- 2d) + fad. 
5, crushing two rivets and shearing one. 

Resistance =f c (2td) + /, . 
4 

6, crushing on three rivets. Resistance = f c (2td+tid). 
7, crushing three rivets, where /i ^ /. 

Resistance = 



554 APPLIED MECHANICS. 

In this case, we should so proportion the joint that its effi- 
ciency may be determined from its resistance to tearing along ab. 
Hence all its other resistances should be equal to or greater than 
this. 

Hence equate i and 3, and calculate the resulting diameter 
of rivet, which will generally be too small, and hence we select 
a larger rivet, so that 3 may be greater than i. 

Having fixed the diameter of rivet, determine the pitch in 
each of three ways, viz., by equating i and 2, by equating i 
and 6, and by equating i and 5, and adopt the least value of p. 

In this joint as used fji d > f 8 , and hence 6 is greater than 



LAP, 

To compute the lap, the following method is a good one. 
Consider the plate in front of the rivet as a rectangular beam 
fixed at the ends and loaded at the middle, whose span=d, 
breadth =/ (for cover-plate t]), depih = h=ld/2. Assume for 
modulus of rupture j t and for center load W, where 

i. When rivet fails by single shear W=j s . 

4 

2. When rivet fails by double shear W=f 8 . 

3. When rivet fails by crushing and lap in plate is sought 



4. When rivet fails by crushing and lap in cover-plate is 
sought W = j c tid. 



JOINTS IN THE WEB OF A PLA TE GIRDER. 



555 



JOINTS IN THE WEB OF A PLATE GIRDER. 

While no experiments on the strength of such joints have 
been published, the constants necessary for use in the ordinary 
method of calculating them are : i, the allowable outside 
fibre-stress ; 2, the allowable shearing-stress on the outer 
rivet ; and, 3, the allowable compression on the bearing- 
surface. 

As an example of the usual method of calculation of such 
a joint, let us consider a chain-riveted butt-joint with two 
covering strips (as shown in the cut) as being a joint in the 
web of a plate girder which has equal 
flanges, and let us determine the allow- 
able amount of bending-moment which 
the web alone (without the flanges) can 
resist. The modifications necessary 
when the flanges are unequal, and 
hence when the neutral axis is not at 
the middle of the depth, will readily 
suggest themselves. 

The stress on any one rivet is pro- 
portional to its distance from the 

neutral axis of the girder, and hence, in this case, from the 
middle of the depth. 

Use the following letters, viz.: 

f t allowable stress per sq. in. at outer edge of web-plate ; 
f t = allowable shearing-stress per sq. in. on outer rivet ; f c = 
allowable bearing-pressure per sq. in. on outer rivet ; / = thick- 
ness of plate ; h total depth of web-plate ; /^ = total depth 

of girder ; d= diameter of driven rivet ; a = area of 

4 

driven rivet section ; r = number of vertical rows on each side ; 
2n = number of rivets in each vertical row ; y l = distance from 





o o 









O 


O 
















o o 






















o o 






o o 


















556 APPLIED MECHANICS. 

neutral axis to centre of nearest rivet ; y t = distance from 
neutral axis to centre of second rivet, etc., etc.;j M = distance 
from neutral axis to centre of outer rivet. 

Then, for allowable bending-moment, we must take the 
least of the three following, viz : 

i, that determined from the shearing f s ; 

2, that determined from the compression f c \ 

3, that determined from maximum fibre-stress f t , observ- 
ing that if f = greatest allowable fibre-stress in girder, then 



To determine these proceed as follows : 



hence allowable stress on rivet at distance y m from neutral axis 



i/ 



and the moment of this stress is 



Hence greatest allowable moment on joint for shearing is 

(,) 



JOINTS IN THE WEB OF A PLATE GIRDER. 557 



2. Greatest allowable compression on outer rivet \sf c td\ 
hence allowable stress on rivet at distance y m from neutral axis is 



. 

y* ' 



and the moment of this stress is 



Hence greatest allowable moment on joint for compression is 

-. './i *r a i r * i i *r ' t \ / 

Jn 

3. The section of the plate is a rectangle, width / and 
height //, with the spaces where the rivet-holes are cut left out. 
It will be near enough to take for the stress to be deducted on 
account of the rivet-hole at distance y m from neutral axis 




and for its moment 



Hence greatest allowable moment on joint for tearing is 



558 APPLIED MECHANICS. 



This mode of calculation for (3) would seem to be war- 
ranted from the fact that the rivets do not fill the holes, 
although many deduct only the effect of the holes on the ten- 
sion side, and consider that those on the compression side do not 
weaken the metal. The greatest allowable bending-moment on 
the joint is the smallest of (i), (2), and (3), and it is plain that, in 
order to make the calculation, we need to know what to use 

o f s , and/ c , or, since f t =f-j-, what to use for /, f s , and 



f e \ and while /"should be determined from the tests on the 
transverse strength of the metal, whether wrought-iron or steel, 
the best evidence we have as to the proper values of f s and f e 
is furnished by the tests on tension-joints, which have already 
been discussed. 

Moreover, we might determine the diameter of rivet by 
equating (i) and (2), but we should generally find it desirable 
to use a larger rivet, and then we should determine the pitch 
by equating (2) and (3) if a larger, or (i) and (2) if a smaller, 
rivet is used. 

Moreover, the rivets in common use in such cases are either 
f" or I" in diameter. 



TESTS OF THE COMMITTEE OF THE BRITISH INSTITUTION OF 
MECHANICAL ENGINEERS. 

The Committee on Riveted Joints of the British Institu- 
tion of Mechanical Engineers consisted of Messrs. W. Boyd, 
W. O. Hall, A. B. W. Kennedy, R. N. J. Knight, W. Parker, 
R. H. Twedell, and W. C. Unwin. 



RI VE TED JOIN TS. 559 



Before beginning operations Prof. Unwin was asked to 
prepare a preliminary report, giving a summary of what had 
already been done by way of experiment, and also to make 
recommendations as to the course to be pursued in the tests. 

This preliminary report is contained in vol. xxix. of Engineer- 
ing, on the pages already cited. In regard to its recommenda- 
tions it is unnecessary to speak here, as the records of the tests 
show what was done ; but in regard to the summary of what 
had been done, it may be well to say that he gives a list of 
forty references to tests that had been made before 1880, be- 
ginning with those of Fairbairn in 1850, and ending with some 
made by Greig and Eyth in 1879, together with a brief account 
of a number of them. 

Almost all of this work was done, however, with small strips 
with but few rivets, and will not be mentioned here. Inas- 
much, however, as Fairbairn's proportional numbers have been 
very extensively published, and are constantly referred to by 
the books and by engineers, it may be well to quote a portion 
of what Unwin says in that regard, as follows : 

" The earliest published experiments on riveted joints, and 
probably the first experiments on the strength of riveting ever 
made, are contained in the memoir by Sir Wm. Fairbairn in the 
Transactions of the Royal Society. 

" The author first determined the tenacity of the iron, and 
found, for the kinds of iron experimented upon, a mean tenacity 
of 22.5 tons per square inch with the stress applied in the 
direction of the fibre, and 23 with the stress across it. That 
the plates were found stronger in a direction at right angles to 
that in which they were rolled is probably due to some error 
in marking the plates. 

" Making certain empirical allowances, Sir Wm. Fairbairn 
adopted the following ratios as expressing the relative strength 
of riveted joints : 



560 APPLIED MECHANICS. 

Solid plate 100 

Double-riveted joint 70 

Single-riveted joint 50 

These well-known ratios are quoted in most treatises on rivet- 
ing, and are still sometimes referred .to as having a considerable 
authority. 

" It is singular, however, that Sir Wm. Fairbairn does not 
appear to have been aware that the proportion of metal 
punched out in the line of fracture ought to be different in 
properly designed double and single riveted joints. These 
celebrated ratios would therefore appear to rest on a very 
unsatisfactory analysis of the experiments on which they are 
based. Sir Wm. Fairbairn also gives a well-known table of 
standard dimensions for riveted joints. It is not very clear 
how this table has been computed, and it gives proportions 
which make the ratio of tearing to shearing area different for 
different thicknesses of plate. There is no good reason for 
this." 

As to the tests which constitute the experimental work of 
the committee, these were made by or under the direction of 
Pi*of. A. B. W. Kennedy, of London. Steel plates and steel 
rivets were used throughout, the steel containing about 0.18 per 
cent of carbon, and having a tensile strength varying from 
about 62000 to about 70000 pounds per square inch, and hence 
being a little harder than would correspond to our American 
ideas of what is suitable for use in steam-boilers. The greater 
portion of the work was performed by the use of a testing- 
machine of looooo pounds capacity, and hence one which did 
not admit of testing wide strips with a sufficient number of 
rivets to correspond to the cases which occur in practice; 
indeed, only eighteen of the tests were made on such strips. 
Nevertheless, a brief summary of what was done will be given 
here, though some of the conclusions which he drew are aL 



RIVETED JOINTS. 561 



ready, and others are liable to be, proved untrue by tests of 
wide strips. The tests made by Prof. Kennedy up to 1885 
consisted of fourteen series numbered I to V, VA and VI to 
XIII, and covering 290 experiments, 64 on punched or drilled 
plates, 97 on joints, 44 on the tenacity of the plates used in 
the joints, 33 on the tenacity and shearing-resistance of the 
rivet-steel used in the joints, and the remaining 52 on various 
other matters. 

The first three series were upon the tenacity of the steel 
used, and showed it to be, as stated, from 62000 to 70000 pounds 
per square inch, with an ultimate elongation of 23 to 25 per 
cent in a gauged length of ten inches ; the tenacity of the 
rivet-steel being practically the same as that of the plates. 
The fourth series showed the shearing-strength of the rivet- 
steel to be about 55000 pounds per square inch when tested in 
one way, and 59000 pounds per square inch when tested in 
another way which corresponded, as Kennedy claims, better 
to the conditions of a rivet, though neither was by using a 
riveted joint. 

The tests of series V and VA were made upon pieces of 
plate which had been punched or drilled, in other words, on 
grooved specimens ; and, as might be expected, these specimens 
showed invariably an increase in tensile strength over the 
straight specimens. In the J" and fV' plates drilled with holes 
i inch in diameter and 2 inches pitch, the net metal between 
the holes had a tenacity 11 to 12 per cent greater than that of 
the untouched plate. Even with punched holes the metal had 
a similar excess of tenacity of over 6 per cent. The remaining 
eight series, VI to XIII inclusive, were made on riveted joints, 
the first five on single-riveted lap-joints, and the last three, 
or XI, XII, and XIII, on double-riveted lap and butt joints. 

Series VI was made on twelve joints in f-inch plates which 
contained only two rivets each, the proportions not being in- 
tended to be those of practice, but such as should give, to 



562 APPLIED MECHANICS. 

some extent, limiting values for the resistances of the plate to 
tearing, and of the rivets to shearing and pressure. The results 
were rather irregular; and the main conclusion which he drew, 
was, that if the joint is not to break by shearing, the ratio of 
the tearing to the shearing area must be computed on a much 
lower value of shearing-strength per square inch than the ex- 
periments of series IV had shown ; indeed, some of the joints 
of series VI gave way by shearing the rivets at loads no greater 
than 36000 pounds per square inch of shearing-area. 

Series VII was made upon six (single-riveted lap) joints in 
f-inch plate, with only three f-inch rivets in each joint, and 
with varying pitch and lap ; all these joints breaking by shear- 
ing the rivets. His conclusion from these tests was, that the 
lap need not be more than 1.5 times the diameter of the rivet. 

Series VIII was made on eighteen (single-riveted lap) joints 
in six sets of three each, and these are the only single-riveted 
lap-joints which he tested, having as many as seven rivets each. 
The results are given in the accompanying table. 

Before giving the table, it may be said that No. 652 was in- 
tended to have such proportions as to be equally likely to give 
way by tearing or by shearing, the intensity of the shearing- 
strength being assumed as two-thirds that of the tensile 
strength of the steel, while the bearing-pressure per square 
inch was intended to be about 7.5 per cent greater than the 
tension. No. 653 was proportioned with excess of shearing or 
rivet-area, No. 654 with defect of shearing-area, No. 655 with 
excess of tearing or plate area, No. 656 with defect of tearing- 
area, and No. 657 with excess of bearing-pressure, the different 
proportions being arrived at by varying the pitch and diameter 
of the rivets, and, in the case of 657, the thickness of the plate 
also. The margin (or lap minus radius of rivet) was f inch in 
each case. The following table will show how far these inten- 
tions were realized, and further comments will be deferred till 
later. 



RIVETED JOINTS. 



563 





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Shearing. 




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564 APPLIED MECHANICS. 

Series IX was made on twenty-one joints in f-inch plate 
(each containing only two rivets) designed in a manner similar 
to series VIII, while three were afterwards made from some 
of the broken plates, with as heavy rivets as it was deemed 
possible to make tight. 

From these tests Kennedy thinks it fair to conclude 

I . That the efficiency of a single-riveted lap-joint in a |-inch 
plate cannot be greater than 50 per cent, unless rivets larger 
than i.i inch are used ; and he also calls attention to the fact 
that, as he claims, strength is gained by putting more metal in 
the heads and ends of the rivets, claiming that it will make 
also a tighter joint for boiler-work. 

Series X was made on eight single-riveted lap-joints in 
J-inch and f-inch plate, made from the broken specimens of 
series V and VA ; they also had only two rivets each. These 
joints were made with a view of investigating the effect of 
more or less bearing-pressure. He claims that high bearing- 
pressure induces a low shearing-strength in the rivets, and that 
the bearing-pressure should not exceed about 96000 pounds per 
square inch ; also, that when a large bearing-pressure is used, 
the " margin " should be extra large to prevent distortion, and 
consequent local inequalities of stress ; also, that smaller bearing- 
pressures do not much affect the strength of the joint one way 
or the other. 

Series XI was made upon twelve specimens of double-riveted 
joints ; three being lap-joints in f-inch plate, three lap-joints in 
f-inch plate, three butt-joints with two equal covers in f-inch 
plate, and three butt-joints with two equal covers in f-inch 
plate. Kennedy designed these joints with a view to their 
being equally likely to fail by tearing or by shearing. His as- 
sumptions and the results of the tests are all given in the fol- 
lowing table : 



RIVETED JOINTS. 



565 



SERIES XI. DOUBLE-RIVETED LAP AND BUTT JOINTS AVERAGES. 







!| 


5 -Q 

II 






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Lbs. 


Lbs. 


Lbs. 


Lbs. 


% 


LAP-JOINTS. 


f 


0.8 


70560 


51970 


2.9 


2.15 


29 


89609 


75150 


53920 


91530 


80.8 


f 


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70560 


51970 


3-1 


2-45 


35 


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69910 


49710 


58910 


70.8 


BUTT-JOINTS WITH TWO COVERS. 


1 


0.7 


70560 


34720 


2-75 


2.00 


27 




68000 


33780 


94710 


80.2 


1 


i.i 


67200 


42560 


4.4 


3-18 


26 


100800 59290 


37650 


88460 


71-3 



Series XII contains the same joints as series XI, the strained 
ends having been cut off, and the rest redrilled and riveted by 
means of Mr. Twedell's hydraulic riveter; and series XIII con- 
tained the same joints treated a second time in the same way. 
These experiments, so far as they went, showed no gain in 
ultimate strength to result from hydraulic as compared with 
hand-riveting ; but it was found that, through a misunderstand- 
ing, they had been riveted up at a pressure much lower than 
that intended by Mr. Twedell. 

On the other hand, the load at which visible slips occurred 
was about twice as much greater with hydraulic as with hand 
riveting. 



$66 APPLIED MECHANICS. 



KENNEDY S CONCLUSIONS. 

The following are a portion of what he gives as his con- 
elusions : 

i. The metal between the rivet-holes had a considerably 
greater tensile resistance per square inch than the unperfo- 
rated metal. 

2. In single-riveted joints, with the metal that he used, he 
assumed about 22 tons (49280 Ibs.) per square inch as the shear- 
ing-strength of the rivet-steel when the bearing-pressure is 
below 40 tons (89600 Ibs.) per square inch. In double-riveted 
joints with rivets of about f-inch diameter we can generally 
assume 24 tons (53760 Ibs.) per square inch, though some fell 
to 22 tons (49280 Ibs.). 

3. He advises large rivet heads and ends. 

4. For ordinary joints the bearing-pressure should not ex- 
ceed 42 or 43 tons (94000 or 96000 Ibs.) per square inch. For 
double-riveted butt-joints a higher bearing-pressure may be 
allowed ; the effect of a high bearing-pressure is to lower the 
shearing-strength of the steel rivets. 

5. He advises for margin the diameter of the hole, except 
in double-riveted butt-joints, where it should be somewhat 
larger. 

6. In a double-riveted butt-joint the net metal, measured 
zigzag, should be from 30 to 35 per cent greater than that meas- 
ured straight across, i.e., the diagonal pitch should be p -\ > 

o j 
where/ transverse pitch and d-=- diameter of rivet-hole. 

7. Visible slip occurs at a point far below the breaking- 
load, and in no way proportional to that load. 

Kennedy thinks that these tests enable him to deduce rules 
for proportioning riveted joints, and the following are his rules, 
viz. : 



RIVETED JOINTS. 567 



(a) For single-riveted lap-joints the diameter of the hole 
should be 2\ times the thickness of the plate, and the pitch of 
the rivets 2f times the diameter of the hole, the plate-area being 
thus 71 per cent of the rivet-area. If smaller rivets are used, 
as is generally the case, he recommends the use of the follow- 
ing formula : 



where / = thickness of plate, d diameter of rivet, and/ = 

pitch. 

For 30-ton (67200 Ibs.) plate, and 22-ton (49280 Ibs.) rivets, a = 0.524 
For 28-ton (62720 Ibs.) plate, and 22-ton (49280 Ibs.) rivets, a = 0.558 
For 30-ton (67200 Ibs.) plate, and 24-ton (53760 Ibs.) rivets, a = 0.570 
For 28-ton (62720 Ibs.) plate, and 24 ton (53760 Ibs.) rivets, a = 0.606 

Or, as a mean, a = 0.56. 

(&) For double-riveted lap-joints he claims that it would be 
desirable to have the diameter of the rivet 2\ times the thick- 
ness of the plate, and that the ratio of pitch to diameter of 
hole should be 3.64 for 3<D-ton (67200 Ibs.) plate and 22-ton 
(49280 Ibs.) or 24-ton (53760 Ibs.) rivets, and 3.82 for 28-ton 
(62720 Ibs.) plate. 

Here, however, it is specially likely that this size of rivet 
may be inconveniently large, and then he says they should be 
made as large as possible, and the pitch should be determined 
from the formula to 

/ = "y + 4 
where, 

For 30-ton (67200 Ibs.) plate, and 24-ton (53760 Ibs.) rivets, = x.if 
For 28-ton (62720 Ibs.) plate, and 22-ton (49280 Ibs.) rivets, a = 1.16 
For 3o-ton (67200 Ibs.) plate, and 22-ton (49280 Ibs.) rivets, a = 1.06 
For 28-ton (62720 Ibs.) plate, and 24-ton (53760 Ibs.) rivets, a = 1.24 



568 APPLIED MECHANICS. 

(c) For double-riveted butt-joints he recommends that the 
diameter of the hole should be about 1.8 times the thickness 
of the plate, and the pitch 4.1 times the diameter of the hole, 
and that this latter ratio be maintained even when the former 
cannot be. 

Two of the principal participants in the discussion of the 
report were Mr. R. Charles Longridge and Prof. W. C. Unwin. 

Mr. Longridge was of the opinion that wider strips with 
more rivets should have been used ; that holding the specimens 
in the machine by means of a central pin at each end was not 
the best method ; that the results obtained from specimens 
which had been made from the remnants of other fractured 
specimens were at least questionable, for, even if the plate had 
not been injured, the ratio of the length to the width of the 
narrowest part was different after the strained ends were cut 
off from what it was before ; that machine-riveting should have 
been adopted throughout instead of hand-riveting, as it is not 
possible to secure uniformity with the latter even were it all 
done by the same man, as he would be more tired at one time 
than at another ; that experiments should be made to determine 
the effect of different sizes and different shapes of heads, as 
well as of different pressures upon the load causing visible slip , 
and that experiments should be made upon chain-riveting, as 
he thought the chain-riveted joint would show a greater effi- 
ciency than the staggered. 

Professor Unwin said : 

i. In examining the results to ascertain how far a variation 
from the best proportions was likely to affect the strength of 
the joint, he found that while the ratio of rivet diameter to 
thickness of plate varied 21 per cent, the ratio of shearing to 
tearing area 30 per cent, and the ratio of crushing to tearing 
area 34 per cent, the efficiency of the weakest joint was only 
six per cent less than that of the strongest, or, in other words, 



RIVETED JOINTS. 569 



the whole variation of strength was only 1 1 per cent of the 
strength of the weakest joint. 

2. With reference to the effect which the crushing-pres- 
sure on the rivet produced upon the strength of the joint, 
there were some old experiments, which showed that', when 
the bearing-pressure on the rivet became very large there was 
a great diminution in the apparent tenacity of the plate in 
the case of riveted joints in iron. Why should the crushing- 
pressure affect either the tenacity of the plate or the shearing 
resistance of the rivet? He believed that it did not really 
affect either. What happened was that, if the crushing-pres- 
sure exceeded a certain limit, there was a flow of the metal, 
and the section which was resisting the load was diminished. 
Either the section of the plate in front of the rivet, if the plate 
was soft, or the section of the rivet itself, if the rivet was soft, 
became reduced. 

3. He thought that the point at which visible slip began 
was the initial point at which the friction of the plates was 
overcome, and of course was greater the greater the grip 
upon the plates, and hence greater in machine than in hand 
riveting. In some cases with hydraulic riveting loads were got 
as high as 10 tons (22400 Ibs.) per square inch of rivet section 
before slipping began. 

4. In regard to the rules for proportioning riveted joints, 
he preferred to distinguish the joints as single-shear and double- 
shear joints, and then we have the following three equations : 
one by equating the load to the tearing-resistance of the plates, 
a second by equating it to the shearing-resistance of the rivets, 
and a third by equating it to the crushing resistance ; these 
three determining the thickness of the] plate, the diameter of 
the rivet, and the pitch. 

By taking the crushing as double the tenacity, we should 
obtain for single shear d = 2.57*, and for double-shear, d = 



57 APPLIED MECHANICS. 

In a single-shear joint the rivet cannot generally be made 
so big, and in the double-shear it could not always be made so 
small, hence the rivet diameter is chosen arbitrarily, and then 
the single-shear joint is proportioned by the equations for shear- 
ing and tearing, no attention being paid to the crushing, while 
the double-shear joint is proportioned by the equations for 
crushing and tearing, no attention being paid to the shearing. 

5. The general drift of the report was to advocate the use 
of larger rivets. Whether this could be done or not, he could 
not say. For lap-joints it would increase the strength, whereas 
for double-shear joints he was not sure that it would not be 
better to diminish the size of the rivet, and hence the crushing, 
pressure. 

This report has been given so fully because it emanates 
from a committee of the British Institution of Mechanical 
Engineers; but inasmuch as series VIII is the only one where 
wide strips were used, it seems to the writer that any conclu- 
sions which may be drawn from any of the other tests given 
in the report require confirmation by tests on wide strips with 
more rivets, before being accepted as true. 

Government Experiments. The references to these experi- 
ments have been mentioned on page ooo. 

Those included in the first five of the volumes mentioned 
may be divided into three parts: 

i. Those contained in the first two Executive Documents 
mentioned above. 

2. Those contained in the third and fourth. 



RIVETED JOINTS. 



3. Those contained in the fifth. 

Summaries of these sets of tests will be given here in their 
order, as each set was made with certain special objects in 
view, and, if not all, at any rate the i and 2, form, as has been al- 
ready stated, the first portion of a systematic series ; and it seems 
to the author that, although the series are not yet completed, 
yet these tests themselves furnish more reliable information in 
regard to the behavior and the strength of joints than any other 
experiments that have been made, and that the figures them- 
selves furnish the engineer with the means of using his judg- 
ment in many cases where he had no reliable data before. 

A perusal of the tables will give a good idea of the shear- 
ing-strength per square inch of the rivet iron, which is seen to 
be less than the tensile strength of the solid plate ; also the 
effect on strength of the plates due to the entire process of 
riveting, punching, drilling, and driving the rivets ; also the 
efficiencies of the joints tested. 

One of the strongest single-riveted joints tested was a single- 
riveted lap-joint with a single covering-strip. 

The apparent anomaly of the punched plates in a few cases, 
showing a greater strength than the drilled plates, is explained 
by Mr. Howard to be due to the strengthening effect of cold- 
punching combined with smallness of pitch, inasmuch as then 
the masses of hardened metal on the two sides re-enforce each 
other. 

Further than this, the student is left to study the figures 
themselves as to the effect of different proportions, etc. 

In regard to the first series, i.e., those contained in the first 
two Executive Documents mentioned, it is stated in the report 
that 

i. " The wrought-iron. plate was furnished by one maker 
out of one quality of stock." 

2 " The steel plates were supplied from one heat, cast in 
ingots of the same size; the thin plates differing from the 



5/2 APPLIED MECHANICS. 

thicker plates only in the amount of reduction given by the 
rolls." 

The modulus of elasticity of the metal was, iron plate^ 
31970000 Ibs. ; steel plate, 28570000 Ibs. 

In the tabulated results, the manner of fracture is shown 
by sketches of the joints, and is further indicated by heavy 
figures in columns headed " Maximum Strains on Joints, in Jbs., 
per Square Inch." 



RIVETED JOINTS. 



573 



juiof jo Xouaptjjg 



Q 

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vr> 



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574 



APPLIED MECHANICS. 



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ris 



t^ 

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II 

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2 

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s| 

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to a, 







RIVETED JOINTS. 



575 



juiof jo Aouaiotyg 





I 
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1-6 



C3 



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t* 



8 



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l 



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57 6 



APPLIED MECHANICS. 



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c d c c 






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RIVETED JOINTS. 



577 



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ro 

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% 

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in 



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578 



APPLIED MECHANICS. 



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10 vo 



VO vo 

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vo 

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S "2 g 



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sx jo -OK 



RIVETED JOINTS. 



579 



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g 

I ! 

c/5 



jj 

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> 



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= 8 
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to co 



2. 
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580 



APPLIED MECHANICS. 



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to m 
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1-1 00 

Tf ro 

CO CO 



o 3 o | HI! 

cccc cede 



vO 

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O 
10 
VO 

co 
vo 



.? -2 




RIVETED JOINTS. 



juiof jo Xouapiyg 


N ON 

t^. O 
vO l^. 


"d I 

r^. t^ 


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8 S 
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5 82 



APPLIED MECHANICS. 



vo 

Co 



& 



c > 
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8, 2 

10 o 






J' ! 



^f t- N 

^8- ft S> 



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1 



&, .s &, .a 
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3}vj(f uo^j 



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S S 

42-0 2 p 

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d d d d 





9X JO 



RIVETED JOINTS. 



583 



?uiof jo XonaiojBg 



a s, 

t> NO 



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fO ro 



f 



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tx 00 



a 



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ft 



CC C G 



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3}VJ<f py}$ 




APPLIED MECHAA'ICS. 



JUIOf JO ADU3IOIJJ3 



9 

* 



1^ ON 



x; ^ 

IJT 

w 






a 



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\ 

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K1YETED JO J NTS. 



585 



juiof jo Xouapiya 

s, 

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ss 



s Si 



SS 



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vO vO 



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'C 'C 

en T3 en 'O 

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v;z%/ 7^5- 



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ill 



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586 



APPLIED MECHANICS. 



8, 
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2 3 



8 






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vr> 

CSO^ 



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5 ff J? ft 

CO CO CO CO 



vS 



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RIVETED JOINTS. 



587 



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4 

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1S9.I JO '0 N 






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588 



APPLIED MECHANICS. 



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RIVETED JOINTS. 



589 



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590 



APPLIED MECHANICS. 



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O 

\o 



3- 

10 



O O 

N fx 



O O 

N *r\ 

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of Rivets 
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'9ivtf 199}$ 




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RIVETED JOINTS. 



591 



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592 



APPLIED MECHANICS. 



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Ills 

MS-S! 



tr> M 

ON C\ 
10 vo 



o o o o o o o 

O O\ Oi r^ "^r c^ r** 

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RIVETED JOINTS. 



593 



GOVERNMENT TESTS OF GROOVED SPECIMENS. 



Tensile Tests of .J-in. 
Grooved Specimens 
Wroughl-Iron 


Punched. 


|jj 


"o 


P 


|2 


1| 


I Is 





H 


3 


Inch. 


Inch. 




0.48 


0.240 


48090 


0.46 


0.235 


46940 


0.46 


0.241 


49280 


0.49 


0.240 


55340 


0.44 


0.239 


51520 


0.47 


0.241 


49910 


0.97 


0.247 


49540 


0.98 


0.247 


49960 


0.94 


0.249 


50128 


0.96 


0.248 


46900 


0.98 


0.250 


46980 


0.96 


0.251 


46350 


i-47 


0.250 


37636 


1.50 


0.252 


37326 


1.48 


0.249 


41030 


1.48 


0.247 


39480 


i-47 


0.250 


37446 


i-45 


0.251 


39533 


1.96 


0.281 


43194 


i-95 


0.274 
0.282 


47499 
41360 


1.92 


0.279 


43080 


2.03 


0.250 


41140 


1.99 


0.248 


39575 


2.42 


0.280 


'36210 


2.40 


0.245 


42245 


2.47 


0.243 


42233 


2.46 


0.285 


42712 


2.48 


0.245 


38125 


2-44 


0.248 


41620 


2.97 


0.247 


38964 


2.98 


0.241 


41540 


2.96 


0.241 


39972 


2.92 


0.240 


41712 


2.98 


0.250 


40430 


3.95 


0.247 


40850 



Tensile Tests of $-in. 
Grooved Specimens 
Wrought-Iron 
Drilled. 


S 




^JZ 


o 




"bJD ^ 






c c 


M u 


*0 


g 


3 


3 


'wof j 


|0 


Jl 


i v~~ 





H 


i5 


Inch. 


Inch. 




0.51 


0.249 


55787 


0.52 


0.245 


55905 


0.52 


0.275 


57480 


0.52 


0.276 


56000 


0.49 


0.248 


49600 


0.50 


0.248 


56700 


0.47 


0.275 


54880 


0.51 


0.276 


57800 


I.OO 


0.276 


54300 


1.02 


0.273 


57700 


I.OO 


0.276 


53800 


I.OO 


0.280 


52430 


I.OO 


0.252 


49400 


1.02 


0.275 


54060 


I.OI 


0.247 


52770 


I.OO 


0.278 


54600 


1.50 


0.276 


49 J 3 


1-52 


0.273 


51300 


1.48 


0.251 


47220 


1.51 


0.273 


53400 


1-52 


0.275 


54l8o 


1.50 


o 276 


54600 


1.48 


0.274 


56250 


1.50 


0.249 


46260 


2.OI 


0.275 


459oo 


2.05 


0.279 


46820 


2.OO 


0.275 


47950 


2.00 


0.278 


49640 


2.00 


0.286 


44650 


2.00 


0.275 


50780 


2.02 


0.279 


48850 


2.OO 


0.277 


49840 


2-51 


0.244 


44980 


2.52 


0.280 


40150 


2.51 


0.282 


43150 


2.50 


0.244 


455oo 


2.51 


0.285 


46500 


2.49 


0.242 


49520 


2.49 


0.242 





2.50 


0.280 


44780 


3-02 


0.250 


45700 


3.02 


0.249 


44870 


3.00 


0.240 


46760 


3.00 


0.250 


45700 


2.93 


0.242 


47950 




0.250 


48740 


2.98 


0.279 


459 


3.01 


0.281 


44410 



Tensile Tests of -in. 
Grooved Specimens 


Steel Plate 


Punched. 


S 




.c > 


o 




s! 


-1 


1 


& cr 


5^ 


H 


P 


Inch. 
0.49 


Inch. 
0.250 


65120 


0.47 


0.249 


67010 


0.48 


0.249 


63420 


0.48 


0.248 


66550 


0.48 


0.247 


67060 


0.47 


o 248 


65300 


o-99 


0.249 


59840 


I.OO 


0.250 


62160 


I.OI 


0.249 


68246 


0.96 
0.96 


0.250 
0.248 


67330 
65966 


0.95 


0.245 


62700 


i-45 


0.248 


64080 


1-45 


0.252 


64000 


i-45 


0.249 


61025 


i5* 


o 251 


59420 


1.96 

i-93 


0.250 
0.252 


599oo 
63500 


1.98 
1.96 


0.250 
0.251 


59350 
59060 


2-49 


0.249 


58100 


2.47 


0.249 


63900 


2-43 


0.250 


61640 


2-95 


0.251 


56530 


3-oi 


0.249 


58780 


3-04 


0-253 


555oo 


2.97 


0.252 


60060 


2.98 


0.251 


54050 


2.97 


0.249 


56040 



Tensile Tests of J-in. 
Grooved Specimens 
Steel Plate 


Drilled. 


| 




si o 






C C 


W ** 


^o 


*" 




$ 


C/5 ,y 




v 


^ f- t/J 


4=0 


II 


E "- "~ 


I* 8 




-Mia 


* 


P 


u> 


Inch. 


Inch. 




0.52 

o-54 


0.246 
0.248 


67890 

67160 


0-53 


0.247 


66870 


0.50 


0.247 


65610 


0.51 


0.249 


66370 


0.51 


0.250 


67420 


0.52 


0.248 


67750 


0.52 


0.252 


61910 


1.03 


0.247 


57090 


1.02 


0.250 


66390 


1.02 


0.246 


66770 


1.02* 


0.250 


67730 


I.OI 


0.247 


66020 


I.OO 


0.251 


67010 


I.OO 


0.247 


64450 


I.OI 


0.250 


66090 


i-54 


0.250 


64390 


1.52 0.251 


63350 


1.5 


0.253 


64370 


i-54 


0.248 


64895 


2.O2 


0.252 


64320 


2.00. 


0.251 


62970 


2.00 


0.251 


60910 


2.50 


0.248 


59260 


2.50 


0.252 


63250 


2-53 


0.248 


59390 


3-03 


0.251 


61577 


3.00 


0.249 


59080 


3-02 


0.251 


59550 


3-02 


0.250 


59700 


3.00 


o 250 


63370 


3.00 


0.251 


58630 


3-03 


0.252 


63940 



594 



APPLIED MECHANICS. 



IRON POUCHED. 



IRON DRILLED. 



STEEL PUNCHED. 



STEEL DRILLED. 



Tensile Tests of 
Grooved Wrought- 
Iron Plates. 




Tensile Tests of 
Grooved Wrought- 
Iron Plates. 




Tensile Tests 
of 
Grooved Steel Plates. 




Tensile Tests 
of 
Grooved Steel Plates. 




% 


BM 

c'c 

~ 

^ 6- 

JJC/JJ 




1 


a* 

bJ3 o 
|J 

w sr 

WO! ' 




i 


If 

Jjl-H 

^ cr 

WC/2 ui 




, 


SJ 

c"e 
U 1 " 1 

OT rJ, 
JjdTjj 


*2 

? 


J 


ls.s 

p 







.a 

B 


IS.fi 




1 





lS.a 

t> 




1 


g 


ib 

5 


Inch. 


Inch. 






Inch. 


Inch. 






Inch. 


Inch. 






Inch. 


Inch. 




I.OI 


0-373 


47000 




0.98 


0.376 


50870 




1.99 


0-365 


61890 




1.97 


0.369 


63620 


0.98 


0.370 


47520 




0.98 


0-377 


52660 




0.99 


0-494 


70080 




I.OO 


0.498 


66220 


2.OO 


0.382 


39760 




1.98 


0-379 


49710 




I.OO 


0.492 


68130 




o-99 


0-495 


66800 


2. 02 


0.383 


36630 




2.OO 


0.380 


49830 




1.50 


0.497 


66340 




I.OO 


0.500 


67000 


2-39 


0.390 


37600 




2.50 


0.390 


50250 




i-5i 


0-494 


63810 




i-53 


0-497 


65930 


2. 9 8 


o-395 


36340 




3-00 


0.392 


45 I 5o 




1.99 


0.499 


55930 




1.50 


0.498 


66270 


2. 9 8 


0.392 


39210 




3-00 


0-393 


47540 




1.97 


0.500 


64260 




1.98 


0.504 


67510 


3-47 


0.390 


37680 




3-50 


0.392 


43940 




2-43 


0.502 


52050 




2.03 


0.502 


66730 


3-47 


0.389 


38340 




3-49 


0.390 


46490 




2-51 


0.504 


64360 




2.50 


0.497 


67950 


0.97 


0.467 


- 50820 




0.99 


0.477 


47140 




3-00 


0.503 


60320 




2.52 


0.501 


67440 


1.48 


0.506 


45090 




1. 00 


0.479 


48370 




2-99 


0.503 


62430 




3-oi 


0.502 


66310 


1.49 


0.506 


45050 




1.49 


0.510 


51240 




3-50 


0.503 


49430 




3-oi 


0.503 


66190 


1.91 


0.513 


42500 




1.49 


0.512 


51510 




3-50 


0.505 


48270 




3-49 


0.504 


64920 


1.97 


0.512 


43430 




1.98 


0.514 


50050 




4.00 


0.497 


48010 




3-50 


0.502 


65210 


2.47 


0.516 


39410 




1.98 


0.516 


47790 




4.00 


0.499 


55190 




3-99 


0-499 


64470 


2.41 


0.513 


39720 




2.51 


0.520 


4558o 




3-99 


0.501 


55780 




4.00 


0.498 


64810 


3.00 


0-515 


38950 




2.52 


0.516 


44960 




3-99 


0.498 


46250 




4.00 


0.503 


64690 


2.90 


0.517 


37290 




3-oo 


0-5*5 


44980 




I.OI 


0.613 


66720 




4.00 


0.498 


64140 


3-5 


0.520 


37800 




3.01 


0.519 


4703 




1-52 


0.612 


64800 




o-99 


0.619 


60290 


3-49 


0-513 


37770 




3-51 


o-5 I 3 


46170 




i-5 


0.615 


64400 




1.49 


0.614 


63610 


4.00 


0.515 


35730 




3-49 


0.514 


44760 




2.50 


0.618 


58060 




1.49 


0.616 


63450 


4-03 


0.516 


36690 




3-99 


0.510 


4533 




2.52 


0.619 


58780 




2-49 


0.620 


59170 


3-99 


0.511 


37000 




3.98 


-5 I 3 


45000 




2.99 ' 0.617 


57180 




2.50 


0.619 


59600 


4.03 


0.508 


37420 




4.00 


0.506 


46100 




3-46 


0.615 


58410 




3-oi 


0.617 


59270 


0.97 


0.614 


49770 




o-97 


0.628 


47220 




3-51 


0.615 


57190 




3-50 


0.614 


61610 


I.OI 


0.619 


52960 




1. 00 


0.626 


4835 




4.04 


0.612 


54450 




3-49 


0.617 


62060 


1.48 


0.618 


46320 




1.52 


0.625 


47170 




4-3 


0.614 


57380 




4.00 


0.615 


60330 


i.S 2 . 


0.620 


46750 




1.49 


0.629 


4653 




I.OI 


0.721 


67930 




4.01 


0.617 


61120 


2.99 


0.614 


40140 




2.98 


0.613 


48220 




I.OO 


0.718 


67620 




0.96 0.726 


58480 


3-5 


0.615 


37480 




3.46 


0.616 


4777 




1.50 


0.719 


62890 




I.OI 


0.727 


58790 


3-5 


0.616 


36940 




3-47 


0.617 


449 




3-50 


0-735 


56730 




i-5* 


0.726 


59290 


4.04 


0.619 


373 10 




3-9 1 


0.625 


44840 




3-51 


0-733 


54220 




3-5o 


0.736 


58700 


; 0.98 


0.678 


50840 




3.96 


0.626 


45ioo 












3-49 


0.729 


59 x 8o 


] 

I.OI 


0.682 


46590 




0-99 


0.695 


47500 


















i-49 


0.688 


4597 




0.99 


0.691 


52780 


















3.48 


0.691 


4035 




I-5I 


0.692 


48470 


















( 3-53 


0.692 


39380 




3-44 


0.700 


47750 


























3-49 


0.692 


46350 


















L 































TENSILE TESTS OF RIVETED JOINTS. 595 



Next will be given the two series of tests already referred, 
to, with Mr. Howard's analysis of them. 

TENSILE TESTS OF RIVETED JOINTS. 

" Earlier experiments on this subject made with single and 
double riveted lap and butt joints in different thicknesses of 
iron and steel plate, together with the tests of specimens pre- 
pared to illustrate the strength of constituent parts of joints, 
are recorded in the report of tests for 1882 and 1883. 

From the results thus obtained it appeared desirable to 
institute a synthetical series of tests, beginning with the most 
elementary forms of joints in which the stresses are found in 
their least complicated state. To meet these conditions, a 
series of joints have been prepared which may be designated as 
single-riveted butt-joints, in which the covers are extended so 
as to be grasped in the testing-machine ; thereby enabling one 
plate of the joint to be dispensed with, and securing the test of 
one line of riveting. 

Such a joint, made with carefully annealed mild steel plate 
of superior quality, with drilled holes, seems well adapted to 
demonstrate the influence on the tensile strength of the metal 
taken across the line of riveting, of variations in the width of 
the net section between rivets, and variations in the compres- 
sion stress on the bearing-surface of the rivets ; elements which 
are believed to be fundamental in all riveted construction. 

This series comprises 2 16 specimen joints, the thickness of the 
plate ranging from J" to | 7/ , advancing by eighths. The covers 
are from y 3 ^-" to yV'- The rivets are wrought-iron, and range from 
-3-$" to lyV diameter; they are machine-driven in drilled holes 
iV' l ai "g er in diameter than the nominal size of the rivets. Ten- 
sile tests of the material accompany the tests of the joints. 

From each sheet of steel two test-strips were sheared, one 
lengthwise and one crosswise. The strips were 2\" wide and 24" 
to 36" long; they were annealed with the specimen plates, and had 
their edges planed, reducing their widths to i|" before testing. 



596 APPLIED MECHANICS. 

Micrometer readings were taken in 10" along the middle of 
the length of each. 

The strength and ductility appear to be substantially the 
same in each direction. But the practice of the rolling-mill 
where these sheets were rolled is such that nearly the same 
amount of work may have been given the steel in each direc- 
tion ; that is, lengthwise and crosswise the finished sheet. 

The ingots of open-hearth metal are first rolled down to 
slabs about 6" thick, then reheated and rolled either length- 
wise or crosswise their former direction, as best suits the re- 
quired finished dimensions. 

The tensile tests show among the thinner plates a relatively 
high elastic limit as compared with the tensile strength ; in the 
f$' f plate the percentage is 72.2, while with the f" plate the 
percentage is found to be 53.3. 

It is noticeable that the thinner plates particularly exhibit 
a large stretch immediately following the elastic limit, and the 
stretching is continued at times under a load lower than that 
which has been previously sustained. It is characteristic of all 
the thicknesses that a considerable stretch takes place under 
loads approaching the tensile strength in some cases the 
stretch increases 5 to 6 per cent, while the stress advances 1000 
pounds per square inch or less. Herein is found a valuable 
property of this metal as a material for riveted construction. 
The stress from the bearing-surface of the rivets is distributed 
over the net section of plate between the rivets, due to the large 
stretch of the metal, with little elevation of the stress, and a 
nearer approximation of uniform stress in this section attained 
than is found in a brittle or less ductile metal. The joints were 
held for testing in the hydraulic jaws of the testing-machine, 
having 24'' exposure between them. A loose piece of steel 
the same thickness as the plate was placed between the covers 
to receive the grip of the jaws, and avoid bending the covers. 

Elongations were measured in a gauged length of 5", the 
micrometer covering the joint at the middle of its width. Loads 



TENSILE TESTS OF RIVETED JOINTS. 597 

were applied in increments of 1000 pounds per square inch of the 
gross section of the plate, the effect of each increment determined 
by the micrometer, and permanent sets observed at intervals. 

The progress of the test of a joint is generally marked 
by three well-defined periods. In the first period greatest 
rigidity is found, and it is thought that the joint is now held 
entirely by the friction of the rivet-heads, and the movement of 
the joint is principally that due to the elasticity of the metal. 

The second period is distinguished by a rapid increase in 
the stretch of the joint ; attributed to the overcoming of the 
friction under the rivet-heads and closing up any clearance 
about the rivets, bringing them into bearing condition against 
the fronts of the rivet-holes. Rivets which are said to fill the 
holes can hardly do so completely, on account of the contrac- 
tion of the metal of the rivet from a higher temperature than 
that of the plate, after the rivet is driven. 

After a brief interval the movement of the joint is retarded, 
and the third period is reached. The stretch of the joint is 
now believed to be due to the distortion of the rivet-holes and 
the rivets themselves. The movement begins slowly, and so 
continues till the elastic limit of the metal about the rivet-holes 
is passed, and general flow takes place over the entire cross-sec- 
tion, and rupture is reached. These stages in the test of a joint 
are well defined, except when the plates are in a warped condi- 
tion initially, when abnormal micrometer readings are observed. 

The difference in behavior of a joint and the solid metal 
suggests the propriety of arranging tension joints in boiler con- 
struction and elsewhere as nearly in line as practicable. 

The efficiencies of the joints are computed on the basis of 
the tensile strength of the lengthwise strips, this being the 
direction in which the metal of the joints is strained. The 
efficiencies here found are undoubtedly lowered somewhat by 
the contraction in width of the specimens, causing in most cases 
fractures to begin at the edges and extend towards the middle 
of the joint. Of the entire series, 88 joints have been tested ; 
tfr* 2", ", and f " plates yet remain." 



APPLIED MECHANICS. 



w o 



w 



o 
3 
u 

U 

a 

rt 

1 

.a 


| 

rt 
C 

1 
fl .1 

, w . e <" 

>i ^ 

;3 ri .:S . . S rt . . 
'55 ~~ o 'ir M ~ 
"xQ.yQQ >,>QQ>, 

gjd g ^^ JSd 


Do. 

ne silky. 
Iky, lamellar. , 
ne silky, 
ne silky, slight lamination. 
Iky. lamellar. 
Do. 
me silky. 
Do. 
Do. 
Do. 


Iky, slight lamination, 
ne silky, surface blister. 
Iky, slight lamination. 
Iky, stratified. 
Iky. 
ne sil'ky. 
Ikv, lamellar. 
Iky. slight lamination. 
Iky, laminated, stratified. 
Iky, laminated. 




fe j/5 fc c/5 j/5 c/5 


t, c/) fc fe i/) tt. 


(73ttic75c7}c7:)fc<i/3c/3c/;i75 


I ( c 

C'Ss., jj 

is rt 


pE?R i^li 


-*oo ooo 10 10 rooo O o 


2^^U o,^ 


Ssc 2| 


S^r.o-00. -^- 






S'^-S 


d 






^6 J 
S Sc 


5 


8 s 

1 


1 


W " u 


l v l|p;l > &^H 


OOOOQOOOOQO 
^NO OQNO o lOr^hx^-O f^ 




C 

rt 
u 


^ 

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00 

10 

o 


1 


*~ && 


OOOOOQ COOOO 
oor^Q^ONO Not^io-^--^- 


ooooooooooo 

i/> 10NO NONOO"M^-iO->J- 


t^-OO iO C O u-) t^. C>OO 


^^' S 








ilj 


^ t^ t-^oo t^ t^ 10 r^co ONOO NO 

CNlMNtNIPllN NCMININCNl 


l&SSs-H^&SS 


54feg6g4SS 


1/3 -^ rt 


w 






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00 OO ONOO 00 tv 00 OO ONOO OO 


NO w m M t- CNI O * * O 00 


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<| -s 


8^^858'S ^S^S; 


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c 
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S d d d d d ^ d d d d 
bf'o T3-a*UT3 tfl-a-o-a-a 

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^ d d d d d d d d 6 

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M <N| r Tf IONO t^OO ON O H 


W ro ^t- iovo txoo ON O w 



TABULATION OF 0. H. STEEL STRIPS. 



599 



c d 
.2 .2 

rt cS 

C C 

a .s 

rt 53 .2 

^Ij^ 
tag bfi 

3 .2 So" 



c 
g .2 






m 

r. 



edi 
t l 
lla 

la 
lla 
t l 



la 
fied 







i ^ 

cflc/ir^wr^xx^ 3 



fc te c/j (73 c/5 c/5 



C/)COCO C/3CCW3 C/5 



w H OOp ro M c> TJ-VO O 



o in ^- r**vo N N rf H ^t- io\o t-^ao t^ t^ moo -^-vo tnvo *OH\OVO\OIOOI S S ^cM-^-oor^oi 

rOMCSMM CSNMNIN NNOISNNNMNN NMMNWNNNMtM NNNNNMrOl 






ooo ooooooooo 



OOOOOOOO OOOOOOOOOO QOOOO 
t^ro*^-o>^t*t^rocn t^inoofoi^NOco^ ooin-^-M 



- ON t^ rOOO - 
mininm 



IN H 00 O 00 

r~> in f-oo in 



m\o M c> i 

>& %$M 



OOOO OOOOO OO 1 
^oo wvo o* in O M *o cow 1 

tx ro 0) vO VO <* -^-vo 10 fO TJ- 



O OOOOOOOOOO OOOOOOOO 

, IT) H M lOOO VO C* O <N rO tx O U~)VO LOCO N CO IO 



t^ i-^vo TJ- 10 u*> LO loco vo vo tx i-- -^f- 10 LOVO miomM w ^J-CON 



oooo -Sooooooooo ? ooooooooo 0000000 

T3 T3 T3 *o tiro 'O'O'a'a'a'O'O'O {g'O-o'O'aTJ'a'a'D'o bo^a -o *c o -a -o -o 

c o * c 

3 3 



invO <* N ro 
" {7 f? fT N 



600 



APPLIED MECHANICS. 



z 

Q 
W 

O 


pearance of fracture. 


c 
o 

w 
c c c c v 

1 -I 'I -s.- -s -i 

1 1 - II 1 8 

S -3 ~ 8 "*> S 

-: ll i- 1 II 1 ii| 

1 11 Jrlilf Is 1 III 

o O *QMCO n^^r^:^ <> en tnO 


lamellar, 
slight lamination, 
o. 

o. 
stratified, 
slight lamination. 


0, 


< 


>i >>Q Q>i>.> 4)X W( y >.>>>, t^.^^ 
^Jsj J^^J^ 'C-^CC * -^ -^ -^ -^ -^ >- 


>,>M Q >,>, >;>, 

Ji^ J!J<1 ^!^! 


s 




C/3t/3 C7)C/2C/3 bnc/jfefe C/2C/3C/2C/2 C/2t//t> 


C/2C/) t/2{/5 C/3C/3 


W 


i 


J 




J 




Vj r^ r^oo' vd 4- c^oo* r^vo" od M 't^ M >o o u^oo' 


M o <* M iovo * 




g rt 


a 




W 




^ 




o 


o-2 c'S 


ScTScTS^c? S^cTcT cT^^JT Rff'g 


00 m * ao t^ * Tt-t^ 


ll 


' .s ^ 

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ON *5> ^ f^ 


t^ m ui 


c/) 


1 53 S u 

H8 *B 


^^ in \T) too vo in 10 in mvo in om^nm ininin 


III MI !f 


SH 


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1 ^ s 1 


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1 1* H 


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1111 ill II! 


!!! HI If 


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^"jr as^? ^5- 




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^'ct^SS^!^^ i^Rcgo? cg^cSocT 82^ 


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JllillSS IH? HH S?? 


1H ?a 11 


fan 





|H W ,HHMH HMHM M M H MMM 






C 
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8 




p 


l* 


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^ d d 6 6 d 

en T3 -O &QT3 T3 Jg -Q 

s g s 

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>s* > N i > N 




O MM tn 

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O M CN fO -^- IOVO M CM CO M* t^OO O\ O IO\O t^ 


ifi if? &i 



RIVET METAL FOR RIVETED JOINTS. 



601 





cT o" ^ oo* o* w* "* oT ON -^ 

(NW Nt-*'-WC<WWW 


e? J? 


c 

.2 


""d^H *^' ? 8 - ! ! t *?^ 


*'i' * 




y-y ^? 5 " ? " li ? R 


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a 
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tx^ ' ocf^ * W^^O^O\ M^ C^ O * fO d 


^ vo" 


s 


^.- ^- ^.- 2 ' ct S S? J? ' -V S 


-?" 


gjl 

"0 


y-*M Tj-^O-^-OOOOOOv 


* M 


8 

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H'M N o mrom-frioin 


s 5 


23 * 


s* 




< r 5 S u 


-C ^- -^- iO u"> VO ^ VO t^. OO OO 


s- s 


Q V 






e 

is 


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UMW 4-MOiooovdr-.oo' 


* 00 


11 


a 




11 


g .ff ff ^ff-yasjj'g.fi 


S 12 








^ be </) ^ 


i tis^&^SS^ 

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VO CO 


1 s 

H *-* "rt 
o 
H 


ill | HI!! fl 



vg | 
10 10 


<y . 

Q A 3 fl 


^oo S- o? m2^5-Rv2 ^;J 


00 O 


J CA M 






rt 5 
S H 


oo oooooooo 
10^- 5^0-^^SooxoO 

2?C>ON f>.^J-CI N rOONVO tx 

^oooo 2"^ c?c?&o 


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ill 


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10 10 

cT o 


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i; | 



6O2 



APPLIED MECHANICS. 



TABULATION OF SINGLE 

i" STEEL 



No. 01 
Test. 


Sheet Letters. 


Pitch. 


No. of 
Rivets. 


Width 
of 
Joint. 


Nominal 
Thickness. 


Size of 
Rivets 
and 
Holes. 


Actual 
Thick- 
ness of 

Plate. 


Lap. 


Plate. 


Covers. 


Plate. 


Covers 










in. 




in. 


in. 


in. 


in. 


in. 


in. 


1308 


F 


A 


A 


if 


6 


9-75 


i/4 






.242 


2 


1309 


F 


A 


A 


44 


6 


44 


44 




44 


.242 


2 


1310 


F 


A 


A 


it 


6 


10.50 


44 




44 


.242 


2 


13" F 


A 


A 


11 


6 


4 ' 


44 




44 


.249 


2 


1312 F 


A 


A 


if 


6 


11.25 


44 




44 


.244 


2 


1313 


F 


A 


A 


" 


6 


" 


44 


" 


44 


-243 


2 


i3H 


F 


A 


A 


it 


6 


10.49 


44 




tt** 


.248 


2 


1315 F 


A 


A 


" 


6 


" 


44 




44 


.242 


2 


1316 F 


A A 


i| 


6 


11.27 


tt 


" 


44 


.244 


2 


1317 | F 


A A 


44 


6 


44 


44 


44 


44 


.246 


2 


1318 F 


B 1 B 


2 


6 


12.01 


44 


it 


44 


245 


2 


1319 G 


B B 


" 


6 


" 


44 


" 


44 


.240 


2 


1320 G 


B 


B 


2* 


6 


12.76 


44 


" 


' 4 


243 


2 


1321 G 


B 


B 


14 


6 


44 


44 


" 


44 


243 


2 


1322 H 


D 


D 


2} 


6 


I3-5I 


44 


44 


tt 


245 


2 


1323 : H 


D 


C 


" 


6 


" 


44 


" 


tt 


247 


2 


1324 F 


A 


A 


1* 


6 


11.26 


44 


" 


T* J 


.248 


2 


1325 F 


A 


A 


44 


6 


" 


44 


44 


" 


245 


2 


1326 G 


B 


B 


2 


6 


12. OO 


44 


" 


4 ' 


.241 


2 


1327 G 


B 


B 


44 


6 


44 


44 


44 





.242 


2 


1328 1 G 


B 


C 


2* 


6 


12.76 


" 


it 


tt 


.241 


2 


1329 ! G 


C 


C 


14 


6 


" 


44 


44 


44 


.242 


2 


'33 


L 


C 


D 


2* 


6 


33-50 


44 


44 


44 


.248 


2 


I33 t 


H 


C 


D 


44 


6 


44 


it 


" 


it 


.246 


2 


1332 


H 


D 


E 


2f 


6 


14.25 




u 


44 


.248 


2 


1333 


H 


D 


E 


" 


6 


44 


44 


44 


44 


.248 


2 


1334 


M 


E 


E 


a* 


6 


15.00 


ti 


ti 


44 


243 


2 


1335 








.... 


" 


6 


" 


it 


44 


44 


245 


2 


1336 


H 


C 


C 


2f 


5 


I3-I3 


<t 


it 


44 


238 


2 


1337 


L 


C 


C 


" 


5 


44 


44 


44 


it 


.252 


2 


1338 


G 


B 


B 


2 


6 


12.00 


44 


44 


tf *i 


.238 


2 


J 339 


F 


B 


B 


11 


6 


11 


44 


44 


44 


.248 


2 


1340 


G 


B 


C 


2* 


. 6 


12-75 


44 


ii 


" 


.240 


2 


i34i 


G 


C 


C 


11 


6 


44 


it 


it 


M 


.242 


2 


1342 


L 


C 


D 


2* 


6 


I3-5I 


44 


4i 


it 


.250 


2 


1343 


L 


D 


D 


" 


6 


" 


" 


it 


" 


.250 


2 



TABULATION OF RIVETED JOINTS. 



603 



RIVETED BUTT-JOINTS. 
PLATE. 



Sectional Area 
of Plate. 


Bearing 
Surface 
of 
Rivets. 


Shear- 
ing 
Area of 
Rivets. 


Tensile 
Strength 
of Plate 
per 
Sq. In. 


Maximum Stress on Joint per Sq. In. 


Effi- 
ciency 
of 
Joint. 


Tension 
on Gross 
Section of 
Plate. 


Tension 
on Net. 
Section of 
Plate. 


Comp. on 
Bearing 
Surface 
of Rivets. 


Shear- 
ing of 
Rivets. 


Gross. 


Net. 


sq. in. 

2.360 


sq. in. 

1-452 


sq in. sq. in. 
.908 3.682 


Ibs. 
61740 


Ibs. 
41690 


Ibs. 
67770 


Ibs. 
108370 


Ibs. 
26720 


67-5 


2.360 


1.452 


.908 ! 3.682 


61740 


42180 


68560 


109640 


27040 


68.3 


2 541 


1.634 


.907 


3.682 


61740 


42540 


66160 


119180 v 


29360 


68.9 


2.6l5 


1.681 


934 


3.682 


61740 


43 1 7 


67160 


119810 


30660 


69.9 


2-745 


1.830 


9*5 


3-682 


61740 


44920 


67380 


134750 


33490 


72.8 


2 -739 


1.827 


.912 


3.682 


61740 


44520 


66750 


133720 


33120 


72.1 


2.602 


1.486 


i . 116 


5.300 


61740 


40700 


71270 


94890 


19980 


65.9 


2.541 


1-452 


1.089 


5-300 


61740 


40000 


70000 


93330 


19180 


64.8 


2.750 


1-652 


1.098 


5.300 


61740 


40980 


68210 


102630 


21260 


66.4 


2.772 


1-665 


i . 107 


5.300 


61740 


4 I 43 


68980 


103750 


21670 


67.1 


2 942 


1.840 


I. 102 


5-300 


61740 


42180 


67450 


112610 


23420 


68.3 


2.882 


1.802 


I.oSo 


5.300 


62660 


43000 


68770 


"475 


23380 


68.6 


3.100 


2.007 


1.093 


5.300 


62660 


43060 


66520 


122140 


25190 


68.7 


3.100 


2.007 


1.093 


5.300 


62660 


44030 


68000 


124880 


2575 


70.3 


3-3 10 


2.207 


I.I03 


5.300 


59180 


43040 


64540 


129150 


26880 


72.7 


3-337 


2.225 


I . 112 


5-3 


59180 


43810 


65700 


131460 


27580 


74-o 


2.792 


1.490 


I .302 


7.216 


61740 


38650 


72480 


82870 


14950 


62.0 


2.756 


1.470 


1.286 


7.216 


61740 


39430 


73930 


84510 


15010 


63-8 


2.892 


1.627 


1.26 5 


7.216 


62660 


40340 


71700 


92210 


16170 


64-4 


2.909 


1-638 


I.27I 


7.216 


62660 


41280 


73310 


94480 


16640 


65-9 


3-075 


1.810 


1.265 


7.216 


62660 


42290 


71850 


102810 


18020 


67-5 


3.088 


1.817 


I.27I 


7.216 


62660 


4275 


72650 


103860 


18290 


68.2 


3.343 


2.046 


1.302 


7.216 


61470 


43100 


70530 


110830 


20000 


70.1 


3-321 


2.029 


I.2QI 


7.216 


59180 


4*45 


67840 


106620 


19080 


70.0 


3-534 


2.232 


1.302 


7.216 


59180 


41820 


66210 


113500 


20480 


70-7 


3-534 


2.232 


1.302 


7.216 


59 l8 o 


42760 


67710 


116070 


20940 


72-3 


3- 6 45 


2.369 


1.276 


7 216 


58170 


44650 


68700 


127550 


22550 


76.8 


3- 6 75 . 
5.125 


2.389 
2.084 


1.286 
I.O4T 


7.216 
6.013 


64170 
59*80 


43 5o 
41310 


66230 
61960 


123030 
124020 


21930 
21470 


67.1 
69.8 


3 39 


2.206 


I.I03 


6.013 


61470 


42000 


62990 


125990 


23IIO 


68.3 


2 .8 b 6 


1.428 


1.428 


9-425 


62660 


40620 


81230 


81230 


I23IO 


64.8 


2.976 


1.488 


1.488 


9-425 


61740 


36290 


72580 


72580 


11460 


58.8 


3.060 


i .620 


1.440 


9-425 


62660 


38660 


73020 


82150 


"550 


61.7 


3.088 


1.636 


1.452 


9-425 


62660 


38000 


71730 


80820 


12450 


60.6 


3.378 


1.878 


1.500 


9-425 


61470 


37800 


68000 


85130 


13550 


61.5 


3-375 


1-875 


1.500 


9-425 


61470 


39000 


70200 


87750 


13970 


63-4 



604 



APPLIED MECHANICS. 



TABULATION OF SINGLE- 
i" STEEL 



No. 
of 
Test. 


Sheet Letters. 


Pitch. 


No. 
of 
Rivets. 


Width 
of 
Joint. 


Nominal Thick- 
ness. 


Size of 
Rivets 
and 
Holes. 


Actual 
Thick- 
ness of 
Plate. 


Lap. 


Plate. 


Covers. 


Plate. 


Covers. 










in. 




in. 


in. 


in. 


in. 


in. 


in. 


1344 


L 


D 


E 


*t 


6 


14.24 


1/4 


3/i 6 


lf*i 


.250 


2 


J 345 


H 


D 


E 


14 


6 


" 


" 


44 


44 


.247 


2 


1346 


I 


E 


E 


2* 


6 


15.00 


" 


44 


44 


.251 


2 


1347 


I 


E 


E 


" 


6 


" 


" 


" 


* 


.252 


2 


1348 


H 


c 


C 


2f 


5 


13-13 


" 


it 


it 


.244 


2 


J 349 


G 


C 


C 


44 


5 


" 


" 


44 


* 


239 


2 


I35<> 


N 


D 


D 


a* 


5 


13-77 


" 


it 


M 


.250 


2 


I35i 


N 


D 


D 


44 


5 


" 


" 


44 


44 


.249 


2 


1352 


H 


D 


D 


2* 


5 


14-39 


" 


44 


II 


247 


2 


1353 


H 


E 


E 


ii 


5 


(i 


ii 


ii 


|| 


.248 


2 


T 354 


I 


E 


E 


3 


5 . 


15.00 


(i 


14 


II 


.252 


2 


J 355* 


I 


E 


E 


" 


5 


" 


44 


44 


(I 


25 1 


2 



STEEL 



I35 6 


A 


I 


I 


if 


6 


9-75 


3/8 


i/4 


A*i 


-365 


2 


1357 


A 


I 


I 


44 


6 


44 


44 


44 


44 


.364 


2 


1358 


A 


I 


J 


i* 


6 


10.49 


K 


44 


44 


365 


2 


1359 


A 


J 


J 


44 


6 


" 


44 


44 


44 


.366 


2 


1360 


A 


I 


J 


44 


6 


10.50 


44 


11 


tt*l 


.366 


2 


1361 


A 


I 


I 


44 


6 


" 


41 


11 


44 


367 


2 


1362 


A 


J 


J 


i* 


6 


".25 


44 


44 


44 


.366 


2 


*3>3 


A 


J 


J 


44 


6 


44 


(4 


41 


44 


365 


2 


1364 


B 


K 


K 


2 


6 


12.00 


II 


K 


44 


.388 


2 


1^65 


B 


K 


K 


II 


6 


44 


44 


44 


44 


39 


2 


1366 


C 


K 


K 


2* 


6 


12.76 


44 


44 


41 


.367 


2 


1367 


B 


K 


L 


44 


6 


44 


44 


11 


44 


387 


2 


1368 


A 


J 


J 


it 


6 


11.25 


44 


44 


13*1 


-369 


2 


1369 


A 


J 


J 


44 


6 


14 


11 


It 


44 


.366 


2 


1370 


B 


K 


K 


2 


6 


12. OO 


44 


It 


44 


-389 


2 


i37i 


B 


J 


J 


44 


6 


" 


44 


II 


41 


388 


2 


i37 2 


B 


L 


L 


at 


6 


"-77 


44 


" 


44 


-385 


2 


1373 


C 


K 


L 


44 


6 


41 


It 


(I 


ii 


-367 


2 


1374 


D 


H 


N 


2i 


6 


13.5 


44 


11 


44 


.376 


2 


1375 


E 


N 


N 


" 


6 


11 


K 


14 


44 


.380 


2 


1376 


D 


L 


M 


2| 


6 


14.23 


u 


11 


ii 


383 


2 


1377 


H 


L 


M 


44 


6 


" 


" 


" 


it 


37i 


2 



* Fractured two outside sections of plate at each edge along line 



TABULATION OF RIVETED JOINTS 



605 



RIVETED BUTT-JOINTS Continued. 



PLATE Continued. 



Sectional Area 
of Plate. 


Bearing 
Surface 
of 
Rivets. 


Shear- 
ing 
Area of 
Rivets. 


Tensile 
Strength 
of Plate 
per 
Sq. In. 


Maximum Stress on Joint per Sq. In. 


Effi- 
ciency 
of 
Joint. 


. Tension 
on Gross 
Section of 
Plate. 


Tension 
on Net 
Section of 
Plate. 


Comp. on 
Bearing 
Surface 
of Rivets. 


Shear- 
ing of 
Rivets. 


Gross. 


Net. 


sq. in. 


sq. in. 


sq. in. 


sq. in. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 




3-56o 


2.060 


1.500 


9-425 


61470 


3913 


67640 


92870 


14780 


63.6 


3-5 20 


2.038 


1.482 


9-425 


59180 


- 40450 


69860 


96070 


15110 


68.3 


3-765 


2.259 


1.506 


9-425 


60480 


4359 


72640 


108960 


17410 


72.1 


3.780 


2.268 


1.512 


9-425 


60480 


41420 


69030 


103540 


16610 


68.5 


3.204 


1.984 


i .220 


7-854 


59180 


38700 - 


62490 


101620 


!5790 


65.4 


3-i3i 


1.936 


1-195 


7-854 


62660 


42890 


69370 


112380 


17100 


68. 4 


3-442 


2. 192 


1.250 


7-854 


5374 


42960 


67460 


118300 


18830 


7 j.i 


3.426 


2.181 


1-245 


7-854 


55740 


41780 


65640 


114980 


18230 


74-9 


3-554 


2 -3!9 


1-235 


7-854 


59i8o 


435 10 


66690 


125220 


19690 


73-5 


3-569 


2.329 


1.240 


7-854 


59i8o 


4383 


67170 


126160 


19920 


74-1 


3.780 


2.520 


1.260 


7-854 

7 Rcj 


60480 
60480 


44580 


66870 

66610 


133730 


2I 450 j 73.7 


3' 7^5 






7 54 




444^0 




133230 


21290 73-4 



PLATE. 



3-559 


2.190 


1.369 


3.682 


54260 


40460 


65740 


105170 


39100 


74 .6 


3-549 


2.184 


1-365 


3.682 


54260 


39420 


64060 


102490 


38000 


72.6 


3-829 


2.460 


1.369 


3.682 


54260 


39780 


61910 


111250 


41360 


73-3 


3-843 


2.471 


1.372 


3.682 


54260 


39060 


60740 


109400 


40770 


72.0 


3-843 


2.196 


1.647 


5-300 


54260 


37000 


64750 


86330 


26830 


68.2 


3-854 


2. 2O2 


1.652 


5-300 


54260 


37050 


64840 


86430 


26940 


68.3 


4.118 


2.471 


1.647 


5.30 


54260 


37450 


62400 


93620 


29090 


69.0 


4.106 


2.464 


1.642 


5-300 


54260 


38040 


63390 


95130 


.29470 


70.1 


4-656 


2.910 


1.746 


5-3oo 


59730 


41820 


66910 


111510 


3673 


70.0 


4.680 


2.925 


1-755 


5-3oo 


59730 


42000 


67200 


II2OOO 


37090 


70-3 


4.683 


3-031 


1.652 


5.300 


57870 


41040 


63410 


116340 


36260 


70.9 


4-938 


3-197 


1.741 


5-3 


59730 


40910 


63180 


116030 


38110 


68.5 


4-i5i 


2.214 


J-937 


7.216 


54260 


35000 


65620 


75000 


20130 


64-5 


4.114 


2.192 


1.922 


7.2,6 


54260 


34180 


64140 


73150 


19480 


63.0 


4 .668 


2.6 2 6 


2.042 


7 216 


59730 


36870 


65540 


84280 


23850 


61.7 


4.656 


2 619 


2.037 


7.216 


59730 


38940 


69220 


SgOOO 


25160 


65-2 


4.916 


2.895 


2.021 


7.216 


59730 


38730 


65770 


94210 


26390 


64.8 


4.672 


2-745 


1.927 


7.216 


57870 


39010 


65660 


94580 


25260 


67.4 


5.076 


3.102 


1-974 


7.216 


53730 


37960 


62120 


97620 


26700 


70.6 


5-130 


3-^35 


1-995 


7.216 


5834 


39810 


65140 


102360 


28300 


68.2 


5-450 


3-439 


2. Oil 


7.216 


53730 


38920 


61670 


105470 


29390 


72-4 


5.290 


3-343 


1.948 


7.216 


56670 


39870 


6311O 


108260 


29230 


70.4 



of riveting ; the two middle sections sheared in front ^f rivets. 



6o6 



APPLIED MECHANICS. 



TABULATION OF SINGLE- 
S'' STEEL 





Sheet Letters. 








Nominal 


















Thickness. 








No. of 
Test. 




Pitch. 


No. of 
Riv- 
ets. 


Width 
of 
Joint. 




Size of 
Rivets 
and 
Holes. 


Actual 
Thick- 
ness of 
Plate. 


Lap. 










Plate. 


Covers. 








Plate. 


Covers. 








1378 


D 


M 


M 


in. 

2* 


.6 


in. 
15.00 


in. 

3/8 


in. 
i/4 


in. 

li** 


in. 

383 


in. 

2 


1379* 1 D 


M 


M 


" 


6 


" 


it 


" 


" 


.385 


2 


1380 


B 


J 


K 


2 


6 


12.00 


" 


" 


if* ' 


388 


2 


1381 


E 


K 


K 


" 


6 


" 


" 


" 


" 


.381 


2 


1382 


B 


K 


K 


2* 


6 


12.75 


" 


" 


" 


.388 


2 


i 3 8 3 t 


E 


G 


K 


" 


6 


" 


11 


11 


" 


383 


2 


1384 


F 


H 


H 


2 


6 


13-49 


M 


it 


" 


.381 


2 


1385 


E 


N 


N 


" 


6 


" 


" 


(i 


* 


.380 


2 


1386* 


H 


L 


L 


2f 


6 


14.25 


" 


it 


" 


.368 


2 


1387 


G 


L 


M 


" 


6 


tt 


" 


" 


" 


365 


2 


1388 


D 


M 


M 


** 


6 


15.00 


11 


" 


" 


385 


2 


1389 


D 


M 


M 


" 


6 


" 


" 


" 


" 


.386 


2 


1390 


C 


G 


G 


2| 


5 


13.12 


" 


" 


" 


372 


2 


MQI 


C 


H 


L 


" 


5 


" 


" 


" 


M 


369 


2 


1302 


C 


L 


L 


2j 


5 


13-75 


" 





" 


374 


2 


1393 


C 


L 


N 


" 


5 


" 


K 


" 


" 


372 


2 


J 394 


D 


I 


M 


2| 


5 


M-39 


tl 


it 


" 


.386 


2 


1395 


D 


M 


M 


" 


5 


" i " 


1C 


" 


383 


2 



* Test discontinued soon after passing maximum load. 
+ Test discontinued at maximum load. 
$ Test discontinued after passing maximum load. 
Test discontinued before fracture was complete. 



TABULATION OF RIVETED JOINTS. 



607 



RIVETED BUTT-JOINTS Continued. 



PLATE Continued. 



Sectional Area 








Maximum Stress on Joint per 




of Plate. 








Square Inch. 










Tensile 










Bearing 


Shear- 


Strength 








Effi- 






Surface 
of 


ing 
Area of 


of Plate 
per 


Tension 


Compres- 
Tension sion on 




ciency 
of 


Plate. 


Covers. 


Rivets. 


Rivets. 


Square 
Inch. 


on Gross 
Section 


on Net 
Section 


Bearing 
Surface 


Shear- 
ing of 


Joint. 












oi Plate. 


of Plate. 


of 


Rivets. 


















Rivets. 






sq. in. 


sq. in. 


sq. in. 


sq. in. 


Ibs. 


Ibs. 


Ibs. 


tos. 


Ibs. 




5 745 


3-734 


2. Oil 


7.216 


53730 


40560 


62400 


115860 


32290 


75-5 


5 775 


3-754 


2.021 


7.216 


5373 


40700 


62620 


116280 


3 2 57o 


75-7 


4656 


2.328 


2.328 


9-425 


5973 


345 


69010 


69010 


17050 


57-8 


4-57 2 


2.286 


2.286 


9.425 


58340 


33440 


66880 


66880 


16220 


57-3 


4-947 


2.619 


2.328 


9-425 


59730 


35590 


67230 


75640 


18680 


59-6 


4.883 


2-585 


2.298 


9-425 


58340 


34730 


65610 


73800 


17990 


59-5 


5.140 


2.854 


2.286 


9-425 


54290 


3549 


63930 


79810 


19360 


65.4 


5126 


2.846 


2.280 


9-425 


58340 


35840 


64550 i 80570 


19490 


61.4 


5-244 


3.036 


2.208 


9-425 


56670 


37010 


63930 


87910 


20590 


65-3 


5-205 


3-oi5 


2.190 


9.4 2 5 


53840 


36750 


63450 


87350 


20300 


68.2 


5-775 


3-465 


2.310 


9-425 


53730 


3749 


62480 


937io 


22970 


69.8 


5-79 


3-474 


2.3l6 


9-425 


53730 


3736o 


62260 


93390 


21890 


69.5 


4.881 


3-021 


1. 860 


7.854 


57870 


39000 


63010 


102340 


24240 


67-4 


4.841 


2.996 


1.845 


7-854 


57870 


39520 


63850 


103690 


24360 


68.3 


5-143 


3-273 


I.SjO 


7-854 


57870 


39840 


62590 


109540 


26080 


68.9 


5.111 


3251 


1.860 


7-854 


57870 


40420 


63550 


111070 


26310 


69.8 


5o55 


3-625 


1.930 


7-854 


53730 


3934 


60290 


113240 


37830 


73-2 


5-496 


3.58i 


I -9 I S 


7-854 


5373 


40300 


eiseo 


115660 


28200 


75-o 



6O8 APPLIED MECHANICS. 



SINGLE-RIVETED BUTT-JOINTS, STEEL PLATE. 
DESCRIPTION OF TESTS AND DISCUSSION OF RESULTS. 

" The following tests complete a series of two hundred and 
sixteen single-riveted butt-joints in steel plates, in which the 
thickness of the plates ranged from J" to f " , and the size of 
the rivets from -fa" to ly 3 ^" diameter. 

The plates were annealed after shearing to size, the edges 
opposite the joint milled to the finished width ; the holes were 
drilled and rivets machine-driven. Iron rivets were used 
throughout, except in some of the f x/ joints. 

Tensile tests of the plates and rivet-metal, together with 
the tests of the joints in " and f " plate, are contained in the 
Report of Tests of 1885, Senate Document No. 36, Forty-ninth 
Congress, first session. 

The tests herewith presented comprise the details and tab- 
ulation of joints in ", f", and f-" thickness of plate, a portion 
of which were tested hot. 

The gauged length in which elongations and sets were 
measured was 5"; 2\" each side of the centre line of the joint. 

During the progress of testing the same characteristics were 
displayed which were referred to in the previous report. The 
joints were very rigid under the early loads. This rigidity is 
overcome by loads which exceed the friction between the plate 
and covers, after which the stretching proceeded slowly with 
some fluctuations till elongation of the metal of the net section 
became general ; the metal under compression in front of the 
rivets yielding, also the rivets themselves. 

The behavior of joints in different thicknesses of plate is 
substantially the same, and an examination of the results shows 
that when exposed to similar conditions the strength per unit 



SIXGLE-RIVETED BUTT-JOINTS, STEEL PLATE. '609 

of fractured metal is nearly the same, whether J" or J" plate is 
used. 

It will not be understood from this, however, that as a con- 
sequence the same efficiency may be obtained in different 
thicknesses of plate for single-riveted work, because it will be 
seen that certain essential conditions change as we approach 
the stronger joints in different thicknesses of plate. 

A riveted joint of the maximum efficiency should fracture 
the plate along the line of riveting, for it is clear that if failure 
occurs in any other manner, as by shearing the rivets or tear- 
ing out the plate in front of the rivet-holes, there remains an 
excess of strength along the line of riveting, or in other words 
along the net section of metal if in a single-riveted joint 
which has not been made use of ; but when fracture occurs, 
along the net section an excess of strength in other directions 
is immaterial. 

If the strength per unit of metal of the net section was con. 
stant, it would be a very simple matter to compute the effi- 
ciency of any joint, as it would merely be the ratio of the net 
to the gross areas of the plates. 

The tenacity of the net section, however, varies, and this 
variation extends over wide limits. 

In the present series there is an excess in strength of the 
net section over the strength of the tensile test-pieces in all 
joints. 

Special tables have been prepared showing this behavior. 

The efficiencies shown in Table No. I are obtained by divid- 
ing the tensile stress on the gross area of plate by the tensile 
strength of the plate as represented by the strength of the ten- 
sile test-strip, stating the values in per cent; of the latter. 

Table No. 2 exhibits the differences between the efficien- 
cies of the joints and the ratios of net to gross areas of plate. 
If the tenacity of net section remained constant per unit of 



APPLIED MECHANICS. 



area, the efficiencies in Table No. I would, as above explained, 
be identical with the ratios of net to gross areas of plate, and 
the values in this table reduced to zero. 

Table No. 3 shows the excess in strength of the net section 
of the joint over the strength of the tensile test-strip in per 
cent of the latter. 

Table No. 4 exhibits the compression on the bearing-surface 
of the rivets in connection with the excess in tensile strength 
of the net section of plate. 

Table No. I is valuable in showing at once the value of 
different joints wherein the pitch of the rivets and their diame- 
ters vary. 

It is seen there is considerable latitude allowed in the choice 
of rivets and pitch without materially changing the efficiency 
of the joint ; thus in J" plate, 

f" rivets (driven), if" pitch, 72.4 per cent efficiency, 

f" rivets (driven), i\" pitch, 73.3 per cent efficiency, 

f-" rivets (driven), af" pitch, 71.5 per cent efficiency, 

i" rivets (driven), 2-3-" pitch, 70.3 per cent efficiency, 

i" rivets (driven), 2^" pitch, 73.8 per cent efficiency, 

give nearly the same results. 

In these examples the ratios of net to gross areas of plate 
range from 60 to 67 per cent, while the rivet-areas range from 
.3067 square inch to .7854 square inch. The actual areas of 
net sections of plate and rivets are as follows : 





|" rivets. 


\" rivets. 


\" rivets. 


i" rivet*. 


Rivets. 


sq. in. 
. ^067 


sq. in. 
.4418 


sq. in. 
.6013 


sq. in. 

.7854 


Plate . . 


1.486 


2. 2O7 


2.2^2 


j 2.259 










( 2.319 



SINGLE-RIVETED BUTT-JOINTS, STEEL PLATE. 6l I 



The areas of the rivets stand to each other as the following 
numbers : 



100 



144 



.96 



and the net areas of the plate to each other as 
100 149 150 



256 



( 152 
1 1 5 6 



From these illustrations it appears that to attain the same 
degree of efficiency in this quality of metal, although that 
efficiency is probably not the highest attainable, a fixed ratio 
between rivet metal and net section of plate is not essential. 

In \" plate with \" rivets the efficiencies of the joints tested 
cold are nearly constant over the range of pitches tested. 

The efficiencies and the ratio of net to gross areas of plate 
are as follows : 





Pitch. 


4" 


2" 


-4" 


2i" 




per cent. 


per cent. 


per cent. 


per cent. 


Efficiency. . 
Ratio of areas . . 


64.5 
53-4 


66.3 
56.3 


66.3 
58.9 


66.4 
6l.I 



In this we have illustrated a case which, in passing from 
the widest pitch, having 61.1 per cent of the solid plate left, to 
the narrowest pitch, which had 53.4 per cent of the solid plate, 
the gain or excess in strength in the net section almost exactly 
compensated for the loss of metal. 

In Table No. 3 the average of all the joints shows the high- 
est per cent of excess of strength in the narrowest pitch, and a 
tendency to lose this excess as the pitch increases. 

Tests of detached grooved specimens show the same kind 



612 



APPLIED MECHANICS. 



of behavior, but as they are not subject to all the conditions 
found in a joint, the analogy does not extend very far. 

The maximum gain in strength on the net section, not for 
the time being regarding the hot joints, and disregarding the 
exceptionally high value of joint No. 1339, J-" plate, was 21.2 
per cent, the minimum value 2.5 per cent of the tensile test- 
:strip. In other forms of joints, and with punched holes in both 
iron and steel plate, illustrations are numerous in which there 
have been large deficiencies, the metal of the net section fall- 
ing far below the strength of the plate. 

It is believed to have been amply shown that increasing 
the net width diminishes the apparent tenacity of the plate, 
although other influences may tend to counteract this tendency 
in some joints. 

In order to compare the excess in strength of one thickness 
of plate with another having the same net widths, we have the 
following table, rejecting those joints that failed otherwise 
than along the line of riveting in making these averages: 



Thickness of Plate. 


Width of plate between rivet-holes. 


i" 


Ii" 


i}" 


if" 


I*" 


If" 


If" 


i*" 


* 2" 


y 

i-' 


P. ct. 

16.7 
18.4 
16.7 
17.7 
11.4 


P. ct. 

12.6 

13-7 
14-3 

16.3 

JS-i 


P. ct. 

11.4 
12.7 

9-3 
14.2 
13.8 


P. ct. 

12.0 
13-5 
10-7 
14-5 
I4.I 


P. Ct. 

13-4 
I 4 .6 
Q.I 
I 4 .6 
7.6 


P. Ct. 

8.9 

12.9 

8.8 
12.7 

IT. 8 


P. ct. 
"5 

9.0 

8.2 

9-9 

IO.O 


P. Ct. 

13-1 

13.6 

12.2 

0.8 
10. i 

ii. 8 


P. Ct. 

10.6 

3-5 


' 


i' 


!/ 


Average of all thick- 
nesses 


16.2 


14.4 


12.3 


12. Q 


II. 9 


II. 


9-7 


7.0 



The excess in strength is generally well maintained in each 
of the several thicknesses, and were it possible to retain the 
same ratio of net to gross areas of plate, and at the same time 



SINGLE-RIVETED BUTT-JOINTS, STEEL PLATE. 613 



equal net widths between rivets, it would seem from this point 
of view feasible to obtain the same degree of efficiency in thick 
as in thin plates. 

The following causes, however, tend to prevent such a con 
summation. 

For equal net widths thick plates require larger rivets to 
avoid shearing than thin ones, the diameters of the rivets being 
somewhat increased for this cause, and again because it has 
become necessary to increase the metal of the net section in 
order to retain a suitable ratio of net to gross areas of plate. 

There results from these considerations such an increase in 
net width of plate that the excess in strength displayed by 
narrower sections is lost, and consequently the result is a joint 
of lower efficiency. 

The data relating to the influence of compression on the 
bearing-surface of the rivets, on the tensile strength of the 
plate, as shown by Table No. 4 are more or less conflicting. 
However, in the J" plate, in which the most intense pressures 
are found, there is seen a pronounced increase in tensile strength 
as the pressures diminish in intensity. 

It is probable that the effects of intense compression would 
be more conspicuous in a less ductile metal, or one in which 
the ductility had been impaired by punched holes or otherwise. 

A number of joints were tested at temperatures ranging 
between 200 and 703 Fahr. 

The heating was done after the joints were in position for 
testing, by means of Bunsen burners, arranged in a row par- 
allel to and under the line of riveting. 

The temperature was determined with a mercurial ther- 
mometer, the bulb of which was immersed in a bath of oil, 
contained in a pocket drilled in the middle rivet of the joint. 

When at the required temperature the thermometer was 
removed from the joint, a dowel was driven into the pocket to 




6 14 APPLIED MECHANICS. 

compensate for the metal of the rivet which had been removed 
by the drill, and then loads applied and gradually increased up 
to the time of rupture. 

Three joints, Nos. 1423, 1426, and 1430, were tested with- 
out dowels in the oil-pockets. 

The method of heating was to raise the temperature of the 
joint, as shown by the thermometer, a few degrees above the 
temperature at which the test was made, shut off the gas- 
burners, and allow the temperature to fall to the required limit. 
The temperature fell slowly, draughts of cold air being excluded 
from the under side of the joint by the hood which covered 
the gas-burners ; the upper side and edges of the joint were 
covered with fine dry coal-ashes. 

The results show an increase in tensile strength when heated 
over the duplicate cold joints at each temperature except 200 
Fahr. 

From 200 there was a gain in strength up to 300, when 
the resistance fell off some at 350, increased again at 400, and 
reached the maximum effect observed at 500 Fahr. ; from this 
point the strength fell rapidly at 600 and 700. 

In per cent of the cold joint there was a loss at 200 of 3.2 
per cent, the average of three joints ; at 500 the gain was 22.6 
per cent, the average of four joints. The maximum and mini- 
mum joints at this temperature showed gains of 27.6 per cent, 
and 18.3 per cent, respectively. 

The highest tensile strength on the net section of plate was 
found in joint No. 1433, tested at 500 Fahr., where 81050 
pounds per square inch was reached against a strength of 58000 
pounds per square inch in the cold tensile test-strip. 

The hot joints showed less ductility than the cold ones, 
those tested at 200 Fahr. not being exempt from this behav- 
ior, although there was no near approach to brittleness in any. 

Three joints, Nos. 1418, 1420, and 1424, were heated; 



SINGLE-RIVETED BUTT-JOINTS, STEEL PLATE. 615 

strained when hot with loads exceeding the ultimate strength 
of their duplicate cold joints; the loads were released, and 
after having cooled to the temperature of the testing-room 
(No. 1424 cooled to 150 Fahr.) were tested to rupture, and 
were found to have retained substantially the strength due 
their temperature when hot. 

In order to ascertain that the time intervening between hot 
straining and final rupture did not contribute towards the ele- 
vation in strength, joint No. 1434 was strained in a similar 
manner with a load approaching rupture, after which a period 
of rest was allowed and then ruptured without material gain 
in strength. 

A peculiarity of the joints fractured at 400 and higher tem- 
peratures was the comparatively smooth surface of the frac- 
tured sections, and which took place in planes making angles 
of about 50 with the rolled surface of the plate. 

The shearing-strength of the iron rivets was also increased 
by an elevation of temperature. 

The rivets in joint No. 1410 at the temperature of 350 
sheared at 43060 pounds per square inch, while in the dupli- 
cate cold joint No. 1411 they sheared at 38530 pounds per 
square inch, and the rivets in pint No. 1398 at 300 Fahr. 
were loaded with 46820 pounds per square inch and did not 
shear. 

Other examples, where some of the rivets sheared and the 
plate fractured in part, showed corresponding gains in shearing- 
strength. 

The almost entire absence of granular fractures in these 
tests is a feature too important to pass by without special men 
tion." 



6i6 



APPLIED MECHANICS. 



TABULATION OF SINGLE- 
STEEL PLATE. 



No. 
of 
Test. 


Sheet Letters. 


Pitch. 


No. 
of 
Rivets. 


Width 
of 
Joint. 


Nominal Thick- 
ness. 


Size of 
Rivets 
and 
Holes. 


Actual 
Thick- 
ness of 
Plate. 


Lap. 


Plate. 


Covers. 


Plate. 


Covers. 










in. 




in. 


in. 


in. 


in. 


in. 


in. 


1396 


R 


R 


R 


'* 


6 


10.50 


1/2 


5/i6 


tt** 


.481 


2 


J397 


R 


R 


R 


" 


6 


" 


" 


" 


" 


.484 


2 


1398 


R 


R 


R 


t 


6 


11.25 


" 


" 


" 


484 


2 


1399 


R 


R 


R 


" 


6 


" 


11 


it 


11 


-483 


2 


1400 


R 


R 


R 


2 


6 


13.00 





" 


ti 


.486 


2 


1401 


R 


S 


T 


" 


6 


" 


" 


" 


44 


-483 


2 


1402 


R 


R 


R 


4 


6 


11.25 


" 


" 


ii*l 


.481 


2 


1403 


R 


R 


R 


" 


6 


" 


" 


11 


" 


.486 


2 


1404 


R 


R 


R 


2 


6 


12. OO 


' 


" 


" 


.486 


2 


1405 


R 


S 


S 


" 


6 


" 


" 


44 


" 


487 


2 


1406 


S 


S 


S 


a* 


6 


12.75 


" 


M 


11 


.470 


2 


1407 


S 


S 


S 


" 


6 


" 


n 


" 


ti 


.471 


2 


1408 


T 


Q 


Q 


at 


6 


I3-SO 


" 


it 


it 


.486 


2 


1409 


T 


Q 


Q 


' 


6 


" 


11 





" 


.482 


2 


1410 


T 








af 


6 


14.25 


44 


it 


" 


.481 


2 


1411 


T 





o 


" 


6 


It 


it 


tt 


" 


.485 


2 


1412 


R 


R 


R 


2 


6 


12.00 


" 


" 


IS* i 


.484 


2 


1413 


R 


R 


R 


" 


6 


" 


it 


" 


11 


.481 


2 


14.4 


S 


S 


S 


at 


6 


12-75 


44 


it 


11 


472 


2 


4'5 


S 


S 


S 


" 


6 


11 


*' 


11 


" 


.468 


2 


,416 


S 


Q 


Q 


at 


6 


JS-SO 


" 


" 


" 


.468 


2 


1417 


T 


Q 


Q 


" 


6 


u 


44 


it 


it 


.482 


2 


1418 


T 








a| 


6 


14-25 


" 


" 


it 


.481 


2 


14-9 


T 


o 





" 


6 


" 


11 


ti 


11 


.482 


2 


1420 


U 


p 


p 


2* 


6 


15.00 


11 


" 


it 


479 


2 


1421 


u 


p 


p 


" 


6 


" 


" 


M 


" 


.483 


2 


1422 


S 


S 


S 


a* 


5 


I3-I3 


" 


it 


11 


.469 


2 


1423 


S 


S 


S 


" 


5 


" 


l< 


" 


it 


473 


2 


1424 


T 








a* 


5 


13-75 


" 


44 


' 


.484 


2 


1425 


T 


o 


o 


" 


5 


" 


" 


" 


" 


-483 


2 


1426 


S 


S 


S 


at 


6 


12.75 


11 


" 


xA*ii 


474 


2 


1427 


R 


S 


S 


** 


6 


" 


" 


ct 


44 


475 


2 


1428 


T 


Q 





at 


6 


I3-50 


" 


" 


11 


479 


2 


1429 


S 





Q 


" 


6 


" 


" 


14 


" 


465 


2 


'43 


U 


o 


o 


f 


6 


14-25 


" 


44 


44 


-484 


2 


M3i 


U 





o 


" 


6 


" 


11 


" 


" 


483 


2 

I 



TABULATION OF RIVETED JOINTS. 



6l 7 



RIVETED BUTT-JOINTS. 



STEEL PLATE. 



Sectional Area 
of Plate. 


Bearing 

Surface 
n f 


Shear- 
ing 


Tensile 
Strength 
of Plate 


Max. Stress on Joint per Sq. In. 


J| 


|.E| 


c ^ ^o 


!- s !s 


^ 


I'ga 


Gross. 


Net. 


OI 

Rivets. 


Area of 
Rivets. 


per 
Sq. In. 


|c|l 


^ c o 5"! 

C-N O Q> 


all 5 


J3 


1? 


T*'c 












c/) 


C/J 


O "Q 






11 


sq in. 


sq. in. 


sq. in. 


sq. in. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 






5-051 


2.886 


2.165 


5-3 01 


57180 


37750 


66070 


88080 


35970 


66.1 


2OO 


5.082 


2.904 


2.178 


5-3* 


57180 


38980 


68220 


90960 


37370 


68.1 




5-445 


3.267 


2.178 


5-301 


57180 


45560. 


75960 


113960 


46820 


79-6 


3 00 


5-439 


3.266 


2-173 


5-3o: 


57180 


39260 


65400 


98290 


40290 


68.6 




5.842 


3-655 


2.187 


5-301 


57180 


37000 


59*40 


98830 


40770 


64-7 




S-79 6 


3.622 


2.174 


5-3 01 


57180 


39420 


63080 


105100 


43100 


68 9 


35 


5-416 


2.891 


2-525 


7.216 


57180 


36890 


69110 


79*3 


27690 


64-5 




5-477 


2.926 


2-551 


7.216 


57180 


38250 


71600 


81730 


29030 


66.9 


250 


5-832 


3.281 


2-55* 


7.216 


57180 


379*0 


67380 


86670 


30640 


66.3 




5-854 


3-297 


2-557 


7.216 


57180 


43730 


77650 


IOOI2O 


3548o 


76-4 


300 


5-997 


3-529 


2.468 


7.216 


59050 


44790 


76110 


108830 


37220 


76.0 


400 


6.010 


3-537 


2-473 


7.216 


59050 


39210 


66630 


953*0 


32660 


66.3 




6.561 


4.010 


2-55 1 


7.216 


60000 


39850 


65210 


102500 


36230 


66.4 




6.512 


3.982 


2-530 


7.216 


60000 


46610 


76220 


i 19980 


42060 


77 .6 


500 


6.859 


4 334 


2-525 


7.216 


60000 


45300 


71690 


123050 


43060 


75-5 


350 


6.916 


4-37 


2.546 


7.216 


60000 


40050 


63390 


108800 


38530 


66.7 




5-813 


2.909 


2.904 


9-425 


57*80 


35920 


71770 


71900 


22150 


62.8 


250 


5-772 


2.886 


2.886 


9-425 


57*8o 


34390 


68780 


68780 


21060 


60. i 




6.023 


3-i9i 


2.832 


9-425 


59050 


35000 


66020 


7443 


22360 


59-2 




5-967 


3-159 


2.808 


9-425 


59050 


40250 


76030 


85540 


25480 


68.1 


300 


6.327 


3-5 T 9 


2.808 


9-425 


59050 


34660 


62320 


78090 


23160 


60.3 


200 


6.512 


3 . 620 


2.892 


9-425 


60000 


36950 


66480 


83220 


25530 


61.5 




6.859 


3-973 


2.886 


9-425 


60000 


437*o 


75460 


103880 


31810 


72.8 


(*) 


6.883 


3-99 1 


2.892 


9-425 


60000 


38720 


66770 


92*50 


28270 


64-5 




7-194 


4-320 


2-874 


9-425 


58000 


44840 


74670 


112250 


34230 


77-3 


(t) 


7- 2 45 


4-347 


2.898 


9-425 


58000 


38740 


64570 


96850 


2 97 Po 


66.9 




6. .67 


3 822 


2-345 


7-854 


59050 


39730 


64110 


104490 


3T200 


67.. 




6 220 


3-855 


2-365 


7-854 


59050 


45420 


73250 


119450 


35840 


76.9 


400 


6.660 


4.240 


2.420 


7-854 


60000 


48950 


76890 


137110 


41510 


81.5 


(*) 


6 632 


4.217 


2-4*5 


7-854 


60000 


40600 


63860 


111490 


34280 


67.6 




6-053 


2-853 


3.200 


11.928 


59050 


35070 


74410 


66340 


18630 


59-3 


3 00 


6.042 


2.836 


3.206 


11.928 


59050 


30420 


64810 


5733 


15410 


51.5 




6.471 


3-238 


3-233 


11.928 


60000 


40330 


80620 


80730 


21880 


67-2 


350 


6.278 


3-*39 


3-139 


11.928 


595 


33420 


66840 


66840 


*759 


56-5 




6.897 


3.620 


3-277 


11.928 


58000 


36390 


65150 


76590 


21040 


62.7 


700 


6.852 


3.632 


3.260 


11.928 


58000 


3366o 


63870 


71160 


1945 


58.0 





* Strained while at temperature of 400 Fahr 
i Strained while at temperature of 500 Fahr. 

* Strained while at temperature of 500 Fahr 



,, and allowed to cool before rupture. 
, and allowed to cool before rupture. 
,, then cooled to 150 Fahr., and ruptured. 



6i8 



APPLIED MECHANICS. 



TABULATION OF SINGLE- 
STEEL PLATE- Continued. 





Sheet 








Nominal 










Letters. 








Thickness. 








No. 
of 
Test. 




5 itch. 


No. 
of 
Riv- 
ets. 


Width 
of 
Joint. 




Size of 
Rivets 
and 
Holes. 


Actual 
Thick- 
ness of 
Plate. 


Lap. 












Plate. 


Covers. 








Plate. 


Covers. 








1432 


u 


P 


P 


in. 

ai 


6 


in. 
15.00 


in. 

1/2 


in. 

5/i6 


in. 
iA* il 


in. 

.484 


in. 

2 


M33 


U 


P 


P 


41 


6 


41 


44 


14 


44 


.481 


2 


M34 


R 


S 


Q 


2g 


5 


13-13 


44 


41 


44 


.472 


2 


*435 


S 


S 


S 


" 


5 


" 


44 


44 


44 


475 


2 


1436 


T 





o 


*! 


5 


13-75 


44 


44 


44 


.482 


2 


1437 


T 








11 


S 


" 


44 


" 


11 


.482 


2 


1438 


U 


P 


P 


aj 


5 


14-38 


it 


44 


f 


.484 


2 


1439 


U 


P 


P 


it 


5 


it 


44 


<t 


44 


.485 


2 


1440 


U 


P 


P 


3 


5 


15.00 


II 


44 


44 


.482 


2 


1441 


U 


P 


P 


it 


5 


44 


44 


" 


44 


483 


2 


1442 


u 


P 


P 


34 


5 


15.68 


11 


41 


41 


483 


2 


M43 


u 


P 


P 


" 


5 


" 


41 


" 


41 


.484 


2 


1444 


V 


E 


E 


I* 


6 


11.25 


5/8 


3/8 


11*5 


.621 


2 


1445 


V 


B 


E 


" 


6 


44 


it 


44 


41 


.624 


2 


1446 


V 


E 


E 


2 


6 


12. OO 


44 


44 


44 


.616 


2 


M47 


V 


E 


E 


11 


6 


44 


44 


44 


** 


.624 


2 


1448 


V 


E 


E 


4 


6 


12 -75 


" 


44 


*' 


.621 


2 


M49 


V 


E 


E 


11 


6 


" 


44 


44 


44 


.624 


2 


145 


w 


F 


G 


si 


6 


13-50 


44 


4 


41 


.610 


2 


MS 1 


w 


G 


G 


" 


6 


44 


44 


44 


44 


.611 


2 


1452 


V 


E 


E 


2 


6 


12.00 


44 


44 


if* i 


.624 


2 


I4S3 


V 


E 


E 


'* 


6 


41 


" 


" 


** 


.620 


2 


M54 


V 


E 


E 


ai 


6 


"75 


44 


44 


4< 


.622 


1 


1455 


V 


E 


E 


" 


6 


" 


tt 


44 


41 


.618 


2 


I45 6 


w 


G 


F 


*i 


6 


I3.50 


44 


' 


tt 


.612 


2 


1457 


w 


G 


G 


" 


6 


" 


4< 


44 


44 


.611 


2 


1458 


w 


I 


I 


a| 


6 


M.25 


.1 


r tt 


4 


.610 


2 


1459 


w 


H 


H 


** 


6 


44 


11 


44 


44 


.608 


2 


1460 


X 


I 


I 


ai 


6 


15.00 


it 


44 


44 


.617 


2 


1461 


X 


J 


I 


" 


6 


" 


ti 


44 


44 


.618 


2 


1462 


V 


F 


F 


i 


5 


I3.I3 


it 


44 


it - 


.630 


2 


I4 6 3 


w 


F 


F 


lk 


5 


14 


44 


44 


* 


.608 


2 


1464 


V 


E 


E 


ai 


6 


12-75 


41 


tt 


iA* i* 


.624 


2 


14^5 


V 


D 


E 


" 


6 


44 


" 


44 


44 


.623 


2 


1466 


w 


G 


G 


aj 


6 


I3-50 


it 


44 


44 


.613 


2 


1467 


w 


N,V 


G 


" 


6 


" 


" 


" 


" 


.606 


2 



TABULATION OF RIVETED JOINTS. 



619 



RIVETED BUTT-JOINTS Continued. 

STEEL PLATE Continutd. 









* 


Maximum Stress on Joint per 




c c 
._ 4, 


Sectional Area 






3 


Square Inch. 


^ 


O w 


of Plate. 






h*/i O* 




"o 


>~"c3 




Bearing 


Shear- 


Sf*> 










> 


Ofc, 

at ,_ 




Surface 
of 


ing 
Area of 


4) u 


t/3 < O 


&g 


! 





"o 
> 


>- 
22 










Rivets. 


Rivets. 


-Sj3 


i rt j 


* 


lra-3* 


c 5 


c 

AJ 


w w . 


Gross. 


Net. 






||c 


8 

gO-sS 


' 5 


P.** i- > 


js 


i 


I. si 










h 




H 


u 


C/5 


W 


H 


sq. in 


sq. in. 


sq.in. 


sq. in. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 






7.270 


4-43 


3-227 


11.928 


58000 


36140 


65000 


81430 


22030 


62.3 




7-215 


3.968 


3-247 


11.928 


58000 


44490 


81050 


99040 


26960 


76.7 


500 


6.193 


3.538 


2-655 


9.940 


59050 


36670 


64190 


85540 


22850 


62 .0 





6.232 


3.560 


2.672 


9.940 


59050 


36200 


63370 


84420 


22690 


61.2 




6.632 


3.921 


2.711 


9.940 


60000 


4243 


71610 


103580 


28250 


70.5 


600 


6.632 


3-921 


2.711 


9-940 


60000 


38720 


65440 


94730 


25840 


64.5 




6.965 


4-243 


2.722 


9.940 


58000 


46630 


76550 


119320 


32680 


80.3 


500 


6.974 


4.246 


2.728 


9.940 


58000 


38900 


63890 


99400 


27290 


67.0 




7.230 


4-519 


2.711 


9.940 


58000 


39180 


62690 


104500 


28500 


67-5 


200 


7.250 


4.528 


2.722 


9.940 


58000 


40360 


65060 


108220 


29630 


70.0 




7-564 


4-837 


2.717 


9-940 


57410 


38570 


60450 


107610 


29410 


67.2 




7-565 


4-843 


2.722 


9.940 


57410 


3^160 


61170 


108830 


29800 


68.2 




6.986 


3.726 


3.260 


7.216 


55000 


3375 


63280 


72330 


32670 


60. i 




7.020 


3-744 


3.276 


7.216 


55000 


3453 


64740 


74000 


3359 


62.7 




7-393 


4.158 


3-234 


7.216 


55000 


36760 


65340 


84010 


37650 


66.0 




7.488 


4.212 


3.276 


7.216 


55000 


35120 


62440 


80280 


36440 


63-8 




7.918 


4-658 


3.260 


7.216 


55000 


41930 


71270 


101840 


46010 


76.2 


300 


7-956 


4.680 


3.276 


7.216 


55000 


36800 


62560 


89370 


40570 


66.9 




8.241 


5-039 


3.202 


7.216 


57290 


39320 


64290 


101 180 


44900 


68.6 


400 


8.249 


5.042 


3-207 


7.216 


57290 


36850 


60290 


94790 


42130 


64-3 


600 


7.488 


3-744 


3-744 


9-425 


55000 


32080 


64150 


64150 


25480 


58-3 




7.440 


3.720 


3.720 


9-425 


55000 


32060 


64110 


64110 


25300 


58.3 




7-93 1 


4.199 


.3-732 


9-425 


55000 


34120 


64440 


72510 


28710 


60.0 




7.880 


4.172 


3.708 


9-425 


55000 


34000 


64220 


72250 


28420 


61.8 




8.262 


4-59 


3.672 


9-425 


57290 


36490 


65680 


82110 


32000 


63.6 




8.249 


4-S83 


3.666 


9-425 


57290 


36020 


64830 


81040 


31520 


62.8 




8 662 


5.002 


3.660 


9-425 


57290 


37720 


65310 


89260 


38490 


65.8 




8.664 


5.016 


3-648 


9-425 


57290 


37540 


64850 


89170 


345'o 


65-5 




9-255 


5-553 


S-? 02 


9-425 


55940 


37300 


62160 


9325 


36630 


66.6 




9.282 


5-574 


3.708 


9-425 


55940 


37000 


61610 


92620 


36440 


66.1 




8.259 


5.109 


3-I50 


7-854 


55000 


3578o 


57840 


93810 


37620 


65.0 




7-965 


4-925 


3 040 


7-854 


57290 


36960 


59770 96840 


37500 


64-5 




7-950 


3.738 


4.212 


11.928 


55000 


31000 


.66130 58690 


20720 


56-5 





7-949 


3-744 


4-205 


ii .928 


55000 


31090 


66020 58780 


20720 


56-5 





8.269 


4- I 3 I 


4-138 


i i . 928 


57290 


33*5 


66350 66240 


22980 


57-8 




8.181 


4.090 


4.091 


11.928 


55940 


33240 


66250 


66240 


22720 


58.0 





620 



APPLIED MECHANICS. 



TABULATION OF SINGLED 
STEEL PLATE Continued. 



No. 

of 
Test. 


Sheet Letters. 


Pitch. 


No. 
of 
Rivets. 


Width 
of 
Joint. 


Nominal 
Thickness. 


Size of 
Rivets 
and 
Holes. 


Actual 
Thick- 
ness of 
Plate. 


Lap. 


Plate 


Covers. 


Plate. 


Covers. 






in. 




in. 


in. 


in. 


in. 


in. 


in. 


1468 W 


H 


I 


at 


6 


M-25 


5/8 


3/8 


iA * X B 


-613 


2 


1469 W 


I 


N&" 


" 


6 


" 


44 


44 


* 


.609 


2 


1470 X 


J 


I 


2i 


6 


15.00 


44 


44 


44 


.619 


2 


1471 X 


J 


I 


" 


6 


44 


44 


44 


ii 


.616 


2 


1472 V 


K 




at 


5 


13-13 


44 


44 


it 


.628 


2 


1473 : W 


F 


F 


44 


5 


" 


ii 


44 


it 


.609 


2 


i 474 i W 


H 


D 


at 


5 


13-75 


44 


41 


it 


.609 


2 


1475 ; w 


G 







5 


1 








.610 


2 


1476 ; W 


I 


I 


2* 


5 


14.38 


it 


44 


44 


.610 


2 


1477 W 


I 


I 


41 


5 


44 


ii 


44 


44 


.6oy 


2 


1478 X 


I 


J 


3 


5 


15.00 


44 


44 


44 


.616 


2 


1479 X 


I 


J 


14 


5 


44 


44 


44 


44 


.623 


2 


1480 G 


E 


E 


3* 


4 


12.50 


44 


44 


44 


.625 


2 


1481 H 


E 


E 





4 


" 


44 


44 


44 


.621 


2 


1482 Z 


K 


K 


2 


6 


12.00 


3/4 


7/16 


tt* 


736 


2 


1483 


Z 


K 


K 


" 


6 


44 


44 


44 


41 


757 


2 


! 1484 Z 


K 


K 


a* 


6 


12.75 


44 


44 


ii 


.742 


2 


1485 


Z 


P 


P 


K 


6 


44 


44 


44 


" 


.762 


2 


1486 


Z 


L 


M 


2* 


6 


I3-50 


44 


11 


ii 


749 


2 


1487 


z 


L 


M 


44 


6 


44 


11 


* 4 


it 


.764 


2 


14 8 


z 


N 


N 


2f 


6 


I4-25 


44 


ii 


ii 


745 


2 


1489 


z 


'o 


N 


" 


6 


" 


44 


44 


it 


735 


2 


1490 


z 


K 




2i 


6 


12.75 


44 


44 


A * i\ 


723 


2 


1491 


z 


P 


P 


" 


6 


" 


44 


ti 


44 


752 


2 


1492 


z 


M 


L 


ai 


6 


I3-50 


44 


44 


44 


736 


2 


1493 


z 


M 


M 


44 


6 


44 


44 


44 


it 


754 


2 


1494 j Z 


N 





at 


6 


14.25 


44 


44 


44 


.760 


2 


1 ~ 
H95 z 








44 


6 


44 


44 


11 


44 


.760 


2 i 


1496 


Y 


Q 


P 


a* 


6 


15.00 


44 


44 


it 


745 


2 


M97 


Y 


P 


P 


44 


6 


44 


44 


44 


44 


725 


2 


1498 


Z 


L 


L 


at 


5 


I3-I3 


II 


ii 


44 


733 


2 


1499 


z 


L 


L 


44 


5 


44 


44 


44 


44 


744 


2 


1500 


z 


N 


M 


2* 


5 


13-75 


44 


44 


it 


.762 


2 


1501 


z 


N 


N 


44 


5 


44 


44 


it 


44 


727 


2 


1502 


z 


O 


P 


at 


5 


14.38 


II 


4< 


44 


.722 


2 


1503 


z 


O 


O 


" 


5 


II 








.741 


2 



TABULATION OF RIVETED JOINTS. 



621 



RIVETED BUTT-JOINTS Continued. 
STEEL PLATE Continued. 



Sectional Area 
of Plate. 


Bearing 
Surface 
of 
Rivets. 


Shear- 
ing 
Area of 
Rivets. 


Tensile 
Str'gth 
of Plate 
per 
Sq. In. 


Max. Stress on Joint per Sq. In. 


Efficiency of 
Joint. 


Temperature 
of Joint in 
Deg. Fahr^ 


gill 

l15 

g</>0 


c- cS 

o w o j 

afc-gcL 



bC U 

. c o <u 

cx'C<2 > 

sg^S 

Sfflu-g 


be 

C w 

'- oj 

$*0> 

% * 


Gross. 


Net. 


sq. in. 


sq. in. 


sq. in. 


sq. in. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 






3-735 


4-597 


4.138 


11.928 


572>o 


34260 


65110 


72330 


25090 


59-8 




8.690 


4-579 


4. in 


11.928 


" 


34790 


66030 


73540 


25350 


60.7 




9.285 


5-i7 


4.178 


ii .928 


55940 


34980 


63600 


77740 


27230 


62.5 




9.240 


5.082 


4.158 


11.928 


" 


34770 


63220 


77270 


26930 


62.1 




8.239 


4.706 


3-533 


9.940 


55oo 


36350 


63640 


84770 


30130 


66.1 




7.978 


4-552 


3.426 


9.940 


57290 


37100 


65020 


86400 


29780 


64-7 




8.362 


4-936 


3.426 


9-940 


" 


38150 


64630 


93110 


32090 


66.5 




8 381 


4-950 


3-43i 


9-94 


" 


38120 


64540 


93120 


32140 


66.5 




8.833 


5.402 


3-431 


9.940 


" 


38620 


63140 


99410 


343io 


67.4 




8-739 


5-3'3 


3.426 


9.940 


" 


38180 


62830 


97430 


3358o 


66.6 




9.240 


5-775 


3-465 


9.940 


55940 


38480 


61570 


102630 


35770 


68.7 




9-345 


5.841 


3-504 


9.940 


" 


38410 


61430 


102440 


36110 


68.6 




7-8i3 


5.000 


2.813 


7-952 


55000 


37340 


58360 


103730 


36690 


67.9 




7-763 


4.968 


2-795 


7-952 


" 


38440 


60060 


106760 


37520 


69-9 




8.847 


4-431 


4.416 


9-425 


59000 


31990 


63870 


64090 


30030 


54-2 




9.099 


4-557 


4-542 


9.425 


" 


31980 63860 


64070 


30870 


54-2 




9-475 


5-023 


4-452 


9.425 


" 


34440 64960 


73920 


34620 


58.3 




9-723 


5-*5i 


4.572 


9 425 


41 


34700 67340 


73790 


35800 


58.8 




IO.II2 


5.618 


4-494 


9-42S 


14 


35000 63000 


78750 


37550 


59-3 




10.329 


5-745 


4-584 


9-425 


" 


36780 I 66130 


82870 


40310* 


62.3 




10.624 


6-154 


4.470 


9-425 


" 


38120 65810 


90600 


42970* 


64.6 




10.488 


6.078 


4.410 


9-425 


11 


34000 58670 


80860 


37830 


57-6 




9-233 


4-353 


4.880 


11.928 


" 


31050 65860 


58750 


24030 


52.6 




9.596 


4.520 


5.076 


11.928 


4 


32000 \ 67940 


65340 


25740 


54-2 




9 .951 


4-983 


4.968 


ii .928 


" 


34270 68430 


68640 


28590 


58-0 




| 10.179 


5 082 


5-090 


11.928 


" 


33770 | 67540 


67520 


28810 


57-2 




10 845 


5-7*5 


5-130 


11.928 





34900 66230 


73780 


31730 


59-1 




10.838 


5.708 


5-!30 


ii .928 


" 


35810 67990 


75650 


3254 


60.6 




i-*75 


6.146 


5-029 


11.928 


60420 


38470 


69940 


85480 


36040 


63.6 




10. 890 


5-996 


4.894 


11.928 


" 


37740 68650 


83980 


34460 


62.4 




9.624 


5-501 


4- I2 3 


9.940 


59000 


35000 61230 


81700 


33890 


57- 




9 776 


5-572 


4.204 


10.030 


" 


36470 63990 


84810 


35550 


61.7 




10.478 


6. 192 


4.286 


9.940 


11 


38760 65590 


94750 


40850* 


65-7 




10.004 


5-9!5 


4.089 


9.940 


It 


36740 62130 


89880 


36970 


62.2 




10.390 


6 329 


4.061 


9-940 


" 


3793 


62270 


97050 


39650* 


64-3 




10.663 


6-495 


4.168 | 9.940 


II 


40630 


65810 


90600 


42970* 


68.8 





* Steel rivets. 



622 



APPLIED MECHANICS. 



TABULATION OF SINGLE- 
STEEL PLATE Continued. 



No. 

of 
Test. 


Sheet Letters. 


Pitch. 


No. 
of 
Rivets. 


Width 
of 
Joint. 


Nominal 
Thickness. 


Size of 
Rivets 
and 
Holes. 


Actual 
Thick- 
ness of 
Plate. 


Lap. 


Plate. 


Covers. 


Plate. 


Covers. 










in. 




in. 


in. 


in. 


in. 


in. 


in. 


J b4 


Z 


M 


L 


a* 


6 


13-50 


3/4 


7/16 


iA*il 


.722 


2 


*505 


Z 


M 


M 


" 


6 


11 





" 


44 


.762 


2 


1506 


Y 


N 





at 


6 


14-25 


11 


" 


44 


.727 


2 


1 57 


Y 


N 





" 


6 


" 


11 


it 


44 


735 


2 


1508 


Y 


Q 


Q 


a* 


6 


15.00 


" 


" 


44 


737 


2 


T 59 


Y 


Q 


Q 


" 


6 


" 


" 


" 


K 


753 


2 


1510 


Y 


L 


L 


8| 


5 


13-13 


" 


" 


44 


.748 


2 


'5" 


Y 


L 


L 


" 


5 


" 


" 


" 


(1 


755 


2 


1512 


Z 




.... 


at 


5 


13-75 


" 


" 


44 


.750 


2 


1513 


Z 


N 


N 


" 


5 


" 


44 


" 


41 


.764 


2 


JSH 


Y 


o 





3| 


5 


M-iS 


" 


" 


" 


.760 


2 


1515 


Y 








K 


5 


" 


" 


" 


44 


.746 


2 


*5'6 





PS 


P 


3 


5 


15.00 


" 


11 


44 


749 


2 


1517 


Y 


P; 


Q 


" 


S 


" 


" 


" 


" 


.741 


2 


1518 


Z 


K 


K 


3t 


4 


12.50 


14 


" 


44 


756 


2 


1519 


Z 


K 


K 


" 


4 


41 


11 


it 


44 


741 


2 


1520 


Z 


K 


K 


3i 


4 


13.00 


it 


41 


ii 


-763 


2 


1521 


Z 


K 


K 


" 


4 


11 


" 


41 


41 


.718 


2 


1522 


Z 


M 


M 


3t 


4 


13-50 


11 


44 


(I 


.742 


2 


1523 


Z 


M 


M 


" 


4 


" 


" 


M 




754 


2 



TABULATION OF RIVETED JOINTS 



623 



RIVETED BUTT-JOINTS Continued. 

STEEL PLATE -Continued. 



Sectional Area 
of Plate. 


Bearing 
Surface 
of 
Rivets. 


Shear- 
ing 
Area of 
Rivets. 


Tensile 
Str'gth 
of Plate 
per 
Sq. In. 


Max. Stress on Joint per Sq. In. 


Efficiency of 
Joint. 


Ill 
pi 


% u 



|o|s 

cc^ 


OC/2 > 


o b 2 

. C U 4J 

Ills 

c3 ffi "^ 


U> 
C en 

|o| 


Gross. 


Net. 


sq. in. 


sq. in. ' 


sq. in. 


sq. in. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


\ 




0.761 


4-346 - 


5-415 


14.726 


59000 


29090 


65350 


52460 


19280 


49-3 




10.287 


4 572 


5-7I5 


14.726 


59000 


30010 


67520 


54010 


20960 


50.8 




10.367 


4.914 


5-453 


14-726 


60420 


33610 


70900 


f389o 


23660 


55-6 




10.474 


4.961 


5-513 


14.726 


60420 


33660 


71070 


63960 


23940 


55-7 




i i . 070 


5-542 


5-528 


14 726 


60420 


3478o 


69470 


69650 


26140 


57-5 




11.310 


5-662 


5-648 


14.726 


60420 


34670 


69250 


69420 


26620 


57-4 




9 918 


5-243 


5-675 


12.272 


60420 


36120 


68380 


76680 


29210 


59-7 




9.928 


5.209 


4.719 


12.272 


60420 


36940 


70400 


77710 


29880 


61.1 




10.328 


5 6 4 


4.688 


12.272 


59000 


33730 


61770 


74320 


28390 


57.0 




10 505 


5-73 


4-775 


12.272 


59coo 


35260 


64640 


77570 


30100 


59-7 




10.929 


6.179 


4-75 


12.272 


60420 


3793 


67080 


87260 


36220 


62.7 




io-735 


6.072 


4.663 


12.272 


60420 


38720 


68460 


89150 


33870 


64.0 




T i . 205 


6.524 


4.681 


12.272 


55520 


36530 


62740 


87440 


36610 


65.8 




11.108 


6.477 


4-631 


12.272 


60430 


38740 


66440 


92920 


35060 


64.1 




9-465 


5.685 


3.780 


9.818 


59000 


3756o 


62360 


9378o 


36110 


63 6 




9- 2 /7 
9-934 


5-572 
6. 119 


3-705 


9.818 
9.818 


59000 
59000 


39000 
37600 


64930 
61040 


97650 
97900 


36850* 
38040* 


66.1 
63-7 




9 348 


5.753 


3 590 


9.818 


59000 


36000 


58440 


9374 


34280 


61 .0 




10.032 


6.322 


3.710 


9.818 


59000 


40040 


63540 


108270 


40^10* 


67-7 




10 . 187 


6.417 


3.770 


9.818 


59000 


39720 


63050 


107320 


41210* 


67-3 





* Steel rivets. 



624 



APPLIED MECHANICS. 



TABLE 

TABLE OF EFFICIENCIES OP 



STEEL PLATE. 



Plate. 


No. of 
Test. 


Pitch of Rivets. 


if" 


if" 


it" 


2" 


*" 






per cent. 


per cent. 


per cent. 


per cent. 


per cent. 




1308 


67-5 


68.9 


72.8 









1313 


68.3 


69.9 


72.1 







t" . . . .- 


1314 
1323 
1324 




65-9 
64.8 


66.4 
67.1 
62.6 


68.3 
68.6 
64.4 


I' 3 
67-5 




1337 






63-9 


65-9 


68.2 




1338 


.... 






64.8 


61.7 




I35S 








58.7 


60.6 




1356 


74-6 


73.3 










1359 


72.7 


72.0 




.... 






1360 




68.2 


69.0 


70.0 


70.9 


I" . . . . - 


^67 
1368 




68.3 


70.1 
64-5 


70.3 
61.7 


68.5 

64.8 




1379 






63.0 


65.2 


67.4 




1380 










58.7 


59-6 




1395 


.... 






57-3 


59-5 








300 


300 








1396 


.... 


66.1 


79.6 


64.7 


.... 




1401 




68.1 


68.6 


350 
68.9 




- 


1402 




.... 


64-5 


66. 3 


400 

76.0 


w 


1411 




.... 


250 
66.9 


300 
76.4 


66.3 




1412 


.... 


.... 


.... 


260 
62.8 


59-2 




1425 


... 


.... 


.... 


60. 1 


68 i 














300 




1426 


.... 


.... 


.... 


.... 


59-3 


, 


1443 


.... 




.... 


.... 


51-5 




1444 


.... 


.... 


60. 1 


66.0 


300 

7 6.2 


// 


M5i 






62. 7 


63.8 


66.9 


. . . . 


1452 




. . . 




58.3 


60.0 




1463 








58-3 


61.8 




1464 


... 


.. 


. 




56.5 




1481 










56-5 




1482 




. . . 


, 


54-2 


58.3 




1489 






. . 


54-2 


58.8 




1490 
1503 




. . . 




.... 


52.6 
54-2 




1523 






' ' 




.... 



NOTES. Figures in heavy- face type denote that 
Super numbers state the temperature of 



TABULATION OF RIVETED JOINTS. 



62 5 



IsO. i. 

SINGLE-RIVETED BUTT-JOINTS. 



STEEL PLATE. 



Pitch of Rivets. 


Diam- 
eter of 
Rivet- 
holes. 


*" 


a|" 


2*" 


af" 


2 J// 


at" 


3" 


3*" 


3*" 


3*" 


per ct. 


per ct. 


per ct. 


per ct. 


per ct. 


per ct. 


per ct. 


per ct. 


pr ct. 


per ct. 


in. 

i 

I 

i 
i 

f 
* 

J 
i 

f i 
* 
i 
i 
it 
* 

1 
i 

t 

i 

Ii 










72.7 
74.0 
70.1 
70.0 
61.5 
63-4 






































70.7 

72.3 
63-7 
68.4 


76.8 
67.1 
72.1 
68.5 


69.8 
68.3 
6s-4 
68.4 


























77.1 
75-o 


73-5 
74- * 


m 












































































70.7 
68.2 
65.4 
64.1 


72.4 
70.4 
65-3 
68.3 


75-5 
75 6 
69.8 
69.5 
















6 7 :; 
68. 3 


6S!8 
69.8 


73-2 
75- 




















































66.4 
too 

77.6 

a oo 
60.3 

61.5 

360 

6 7 .2 

56.5 

6 4 8 6 

64 3 
63.6 
62.8 
57-8 
58.0 

59.3 

62.3 
58.0 
57 2 
49-3 
50.8 


75 6 .5 
66.7 

400 

72.8 

64-5 
700 

62.7 

58.0 


















600 

77-3 
66.9 
62.3 

500 

76.7 
















6 7 .2 

400 
76.9 

62.O 
6l.2 


600 

81.5 

67.6 

800 
70-5 

64.5 






















600 

80.3 
67.0 


200 

67-5 

70.0 


67.2 
68.2 


.... 


.... 


65*8 
65-5 
59-8 
60.7 

64.6 
57.6 

59-1 
60.6 
55 6 
55 7 


66.6 
66.1 

62.5 
62.1 

6 3 "6 
62.4 

57-5 
' 57-4 


65.0 
64.5 

66.1 
64.7 

67.0 

61.7 

59-7 
61.1 






















6^5 
66. 5 


67 .'i 

66.6 


68.7 
68.6 


67^9 
69.9 


" 


!;;; 


65~7 

62.2 

57.0 
59 7 


64' 3 

68.8 
62.7 
64.0 










65-8 
64.1 


63.6 

66.1 


ai 


67.7 
67.3 



joint did not fracture along- line ol riveting, 
'oints tested at temperatures above atmospheric. 



626 



APPLIED MECHANICS. 



TABLE NO. 2. 

TABLE OF DIFFERENCES BETWEEN THE EFFICIENCIES AND RATIOS OF NET TO 
GROSS AREAS. SINGLE-RIVETED BUTT-JOINTS, STEEL PLATE. 



Plate. 


No. of 
Test. 


Width of Plate between Rivet Holes. 


Diameter 
of Rivet 
Holes. 


i" 


ii" 


ii" 


if" 


*T 


if" 


if" 


i*" 


2" 


in. 
i - 

r 
i 

r 

* - 

i - 
r 
* 


in. 

1308 
1313 
i3H 
1323 
i3 2 4 
1337 
1338 
1355 

1356 
1359 
1360 
1367 
1368 
1379 
1380 
1395 

1396 
1401 
1402 
1411 
1412 
H25 
7426 
1443 

1444 

i45i 
1452 
1463 
1464 
1481 

1482 
1489 
1490 
I5Q3 
1504 
1521 


perct. 

6.0 
6.8 
8.8 

K 

,fl 

18.8 

13- 
11. 
ii. 
ii. 
ii. 
9-7 
6.8 
7-3 

200 
9-0 

II. I 

II . I 
250 
13-5 
850 
12.8 

10. I 

300 
12.2 

4.6 

6.4 

9-4 
8-3 
8-3 
9-5 
9-4 

4- 1 
4- 1 

5-5 
7- 1 
4.8 
6.4 


perct. 

4-6 
5-6 
6-3 
7.0 
8.1 
9.6 
8.8 
7.6 

9.1 
7.7 

9.0 

IO. I 

5-4 

I' 9 
6.7 

6.6 

300 
19.6 

8.6 

IO.O 

300 

2O. I 

6.2 
300 
15-2 
360 
I 7 .2 

6-5 

9.7 

7-5 
7- 1 
8.9 
7-8 
8.0 

\\ 

7-9 
7.2 

8.2 

8-3 


per ct. 

6.1 

5-4 
7.2 
6.1 
8.6 
9-4 
5-9 
7-8 


per ct. 


per ct. 


perct. 


per ct. 


perct. 


per ct. 


in. 

i 

i 

i 
i 

i 

* 

i 
i 

i 

* 
i 
I 
i 
i 
ii 
ii 

1 

i 

i 

3 

i 
i 

i! 














4.0 
5-6 
xi. 7 

8.9 
5-8 
10.5 


8 7 :i 

7.6 

9-2 
12. I 

8-5 


















ii. 8 

2.1 

3-5 
6.6 


i:i 

13-4 

"3 






8. '2 
8.8 


'i : ? 














7-5 

7.8 

11 

9-9 
5-9 

2. I 

350 
6.4 

400 
17.2 

7-4 

200 

4-7 
5-9 

700 
10.2 

5-3 

300 

17 4 

8.1 
8.0 
7-2 
7.2 
8.0 

3.7 

6.6 
6.4 

7-9 

8 


6.2 

3.8 

9-6 
7-i 
7-4 
10.4 




















9.4 

7-2 

9.8 
9-5 


10.5 
10.6 

i:? 








5.2 

6.2 


7-9 
9.8 


.... 


5-3 

500 
16.4 

400 
14-9 

6-5 
6-9 

600 
21.7 

400 

7-5 

600 
3-2 

8.1 
7-6 
7-5 
7- 1 

67 
-0.4 

8.6 

1:1 

8.6 


ll! 3 
3.5 

500 
I 7 .2 

6.9 
4-9 
4-i 


5-2 

400 
14.9 

600 
II.4 

5-4 


500 

17.8 

4.0 

500 

19.4 

6.1 


.... 


200 

5- 
7-5 


3.2 
4.2 













6.6 
6.0 

9.0 

7-6 


3.1 
2.7' 

7-5 
7-4 






6.2 

5.8 


6.2 

6.1 


3.9 
5.9 


-.02 

4-7 
2.4 

5-i 


"6.6 
3.1 

6.2 

7-4 


3-4 
7-9 

7.6 

5.8 


3.5 

6.0 


2.1 

-0.6 


2i" 


perct. 
4.7 
4.3 





































NOTES. Figures in heavy-faced type denote that joint did not fracture along line of riveting 
Super numbers state the temperature of joints tested at temperatures above 
atmospheric. 



TABULATION OF RIVETED JOINTS. 



62 7 



TABLE NO. 3. 

EXCESS IN STRENGTH OF NET SECTION IN JOINT OVER STRENGTH OF TENSILE 
TEST-STRIP. SINGLE-RIVETED BUTT-JOINTS, STEEL PLATE. 



Plate. 

in. 
i 

t 
* 

t 

i 

A 


No. of 
t Test. 


Width of Plate between Rivet Holes. 


Diameter 
of Rivet 
Holes. 


x" 


ii" 








if 


ij" 


if" 


2" 








1308 
1313 

y*3*4 

1323 
1324 
1337 

1338 

1355 
1356 

1359 
1360 

1367 
1368 
1379 
1380 
1395 

1396 

1401 
1402 
1411 
1412 
1425 

1426 

1443 
1444 
1451 

1452 

M 6 3 
1464 
1481 

1482 
1489 
1490 
1503 
1504 
1521 


3er ct. 

9-8 
n. i 
!5-4 
13-4 
17.4 
19.7 
29.6 
17.6 

21.2 

18.1 

19-3 
19-5 
20.9 
18.2 
15-5 
14.6 

200 

15.6 

T 5 .8 

20.9 
260 
25-2 
350 
25.5 

20.3 
300 
2O. O 

9 .8 

IS-I 
17.7 

16.6 
16.6 

20.2 
20.0 

8-3 

8.2 

ii. 6 
15.2 
10.8 
14.4 


per ct. 

7-2 

8.8 
10.5 
11.7 
14.4 
17.0 
16.5 
14-5 
14.1 
11.9 
15.0 
16.8 
9-7 
15-9 

12.6 

12.5 

300 

32-8 

14.4 

17.8 

300 

35-8 
ii. 8 
2 8 8 8 

360 

34-4 
13.2 

18.8 

i3-5 
17.2 
16.8 
15.8 
18.4 

10. I 

14.1 

16.0 
14-5 
17-4 
17.6 


perct. 

9.1 
8.1 

9-2 

9-7 
M.7 
15-9 
10.6 
14.2 

12.0 
12-5 
10. 1 

13-5 
17.8 
10.6 

3-4 

360 
10.3 

400 

28.9 

12.8 
200 

5-5 
10.8 

700 
I2. 3 

10. 1 

300 

20. 6 

13.7 

14-6 
13.2 
13-6 
15-3 
6.8 

12. 1 
I2. 3 
15-2 
I 5 .0 

I 4 .6 


perct. 


perct. 


perct. 


perct. 


per ct. 


perct. 

10.6 
10.1 


in. 
f 

* 

! 

i 
i 

i 

t 

* 

X 
X 

i 
1 

* 
i 

X 

Ii 

It 

* 
i 

t 

i 
i 

Ii 

:i 


6.2 

8-5 
14.7 
14.6 

10. 

18.0 


9-i 
ii .0 
11.9 
14.4 

20.1 
I4.I 


ie.'i 

1:1 

10.7 


4-7 
2-5 

21.0 

17.7 


12.7 
13-5 










9.6 
5.8 
15-6 
11.7 

12.8 

17.9 






















14.9 
II.4 
I6. 3 
15-9 


16.1 
16.5 
8.9 
10.3 








8.2 

9-8 


12.2 
IS-* 


'.'..'. 














8.7 

600 
27.0 

400 

2 5 .8 

"3 

12. I 
600 

39-7 

400 
12.2 

600 
5-2 
14.0 
I 3 .2 

!3-7 
13.0 

11.5 
-0.6 

15.8 

13-6 
13.2 
16.5 


19 6 .5 
5.6 

600 
28.7 

"3 

8.7 

7-3 


















8.6 

400 

24.0 

600 
19.4 

9-i 


500 
28.1 

6.4 
600 
32.0 

10. 1 






a 8i 

12.2 


5.3 
6.5 












11.1 
10.1 

15-7 
13-5 


5.2 
4.3 

12.8 

12.7 


10.2 

9-7 


lo.'i 

9 .8 


e.'i 

9.2 












3.8 

8.5 
4-7 
9.6 


II. 2 

5.3 

II. 

13-3 


5-5 
"5 
13.0 
10.0 










5.7 

10. 1 


4i 


2i" 


per ct. 
7.7 
6.9 


















verage of 
all joints. 


















16.2 


14.4 


12.3 


12.9 


11.9 


II. 


9-7 


ii. 8 


7.0 



Norms. Figures in heavy-faced type denote that joint did not fracture along line of riveting. 
Super numbers state the temperature of joints tested at temperatures abcrc 
atmospheric. 



628 



APPLIED MECHANICS. 



In the Report of Tests made at Watertown Arsenal during the fiscal year 
ended June 30, 1891, is the following account of another series of tests on riveted 
joints: 

44 Comprised in the present report are 113 tests made with steel plates of 
1/4", 5/16", 3/8", and 7/16" thickness with iron rivets machine driven in drilled 
or punched holes. 

44 The plates used were from material used in earlier tests, the results of 
which have been published in previous reports. 

41 In the use of metal once before tested, such plates were selected as had 
not been overstrained previously, or those in which the elastic limit had been 
but very slightly exceeded. 



SINGLE-RIVETED 



STEEL PLATE. 





Sheet Letters. 








Nominal 
Thickness. 




















Size 


Actual 




No. of 
Test. 






Pitch. 


No. of 
Rivets. 


Width 
of 
Joint. 






and 
Kind 
of 


Thick- 
ness 
of 


Lap. 




Plate. 


Covers. 








Plate. 


Covers. 


Holes. 


Plate. 




49i3 


H 


C 


D 


in. 

2* 


5 


in. 
i3-7 2 


in. 

i/4 


in. 
3/i6 


in. 

7/8 d 


in. 
247 


in. 

2 


49M 


L 


D 


D 


" 


5 


13.69 


" 


" 


" " 


.248 


" 


49'5 


M 


E 


E 


8* 


5 


14.32 


i 


it 


.4 tt 


.247 


44 


4916 


M 


D 


D 


it 


5 


14-33 


11 


" 


tt tl 


.247 


44 


49i7 


M 


E 


E 


3 


5 


15.00 


11 


tt 


tt tt 


.246 


14 


4918 


M 


E 


E 


" 


5 


14.98 


it 


it 


tt It 


.247 


11 


4985 


Q 


D 


D 


3* 


4 


14.00 


A 


" 


I " 


39 


I 


4987 


s 


C 




" 


4 


14.01 


" 


14 


" " 


.310 


li 


499* 


Q 


1 




" 


4 


14.05 





" 


it U 


-308 


If 


55 


R 


A 


A 


I 


10 


IO.02 


5/'6 


11 


1/2 " 


.306 


I* 


5"6 


R 


A 


.... 


I* 


8 


10.02 


" 


tt 


tt It 


-34 


" 


5127 


R 


B 




I* 


7 


10.51 


" 


" 


tt tt 


.310 


" 


5143 


L 


P 




2 


7 


14.03 


7/16 


5/i6 


7/8 P- 


.440 


I* 


5M4 


L 


O 





" 


7 


14.01 


it 


11 


" d. 


.440 


" 


SMS 








Q 


2* 


6 


I3-50 


44 


44 


it tt 


-434 


" 


5H6 


M 


O 





" 


6 


I3-5I 


u 


it 


" P- 


.421 


14 


5M7 


O 


P 


P 


2* 


6 


15-02 


" 


44 


41 d. 


413 


44 


5148 


N 


P 


.... 


14 


6 


15.02 


" 


H 


" P- 


.411 


" 


SiSS 


K 


S 


.... 


2* 


5 


13.75 


41 


It 


M d. 


425 





TABULATION OF RIVETED JOINTS. 



629 



"The present tests are supplementary to those of earlier reports, and occupy 
a place intermediate between the elementary forms of joints and the more elab- 
orate types of joints which have been investigated. 

" Wide variation has been given the pitches, and rivets of extreme diameters 
have been used for the purpose of including joints in which these features have 
been carried to their extreme limits. 

" The efficiencies of the joints are stated in per cent of strength of the solid 
plate." 



BUTT-JOINTS. 



STEEL PLATE. 



Sectional Area 
of Plate. 


Bearing 


Shear- 
ing 


Tensile 
Str'gth 


Maximum Stress on Joint per Sq. In. 


Effi- 












Surface 
of 


Area 
of 


OI 

Plate 


Tension 
on 


Tension 

on Net 


Compres- 
sion on 


Shearing 


ciency 
of 






Gross. 


Net. 


Rivets. 


Rivets. 


per 
Sq. In. 


Gross 
Section 


Section 
of 


Bearing 
Surface 


of 
Rivets. 


Joint. 












of Plate. 


Plate. 


of Rivets. 






sq. in. 


sq. in. 


sq. in. 


sq. in. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 




3-39 


2.31 


i. 08 


6.01 


59180 


44180 


65760 


140650 


25270 


75-7 


3-40 


2.31 


1.09 


6.01 


61470 


435oo 


64030 


135690 


24610 


70.8 


3-54 


2.46 


i. 08 


6.01 


58170 


46300 


66630 


151780 


27270 


79-6 


3-54 


2.46 


i. 08 


6.01 


58170 


435oo 


62590 


142570 


25620 


74.8 


3-69 


2.61 


i. 08 


6.01 


58170 


46290 


65440 


158150 


28420 


79.6 


3-70 


2.62 


i. 08 


6.01 


58170 


44400 


62700 


152110 


27330 


76.3 


4-33 


3-9 


1.24 


6.28 


56760 


24040 


33690 


83950 


16580 


42-3 


4-34 


3.10 


1.23 


6.28 


57000 


26770 


3748o 


93710 


18500 


46.9 


4-33 


3.10 


1.23 


6.28 


56760 


33940 


47410 


119500 


23400 


59-8 


3. 7 


i. 54 


i-53 


3-92 


61130 


35930 


71620 


72090 


28140 


58.8 


3-5 


1.83 


1.22 


3-i4 


61130 


41280 


68800 


103200 


40100 


67-5 


3.26 


2.17 


I.OQ 


2.74 


61130 


39250 


58960 


117380 


46690 


64.2 


6.17 


3.38 


2.79 


8.41 


59390 


32540 


59410 


71970 


23880 


54-8 


6.16 


3.48 


2.69 


8.41 


59390 


23360 


41350 


53490 


17110 


30. -\ 


5-86 


3-58 


2.28 


7.21 


52910 


42250 


60160 


108600 


34340 


79.8 


5-69 


3-4 


2.29 


7.21 


61650 


39740 


66500 


98730 


31360 


64.5 


6.20 


4-03 


2.1 7 


7.21 


52910 


44150 


67920 


126130 


37960 


83-4 


6.17 


3-94 


2.23 


7.21 


61650 


36660 


57410 


101430 


3*370 


60.0 


5-84 


3.98 


1.86 


6.01 


59000 


40270 


59100 126450 


39130 


68.3 



630 



APPLIED MECHANICS. 



TABULATION OF SINGLE- 
STEEL PLATE. 



No. of 
Test. 


Sheet Letters. 


Pitch. 


No. of 
Rivets. 


Width 
of 
Joint. 


Nominal 
Thickness. 


Size and 
Kind 
of 
Holes. 


Actual 
Thick- 
ness o: 
Plate. 


Lap. 


Plate. 


Plate. 


Plate. 


Plate. 








in. 




in. 


in. 


in. 


in. 


in. 


in. 


4933 


I 


J 


H 


5 


10.62 


i/4 


i/4 


id. 


.252 


2 


4934 


J 


J 


" 


5 


10.65 


" 


" 


41 


253 


2 


4939 


L 


K 


4 


4 


11.50 


" 


" 


i d. 


.250 


2 


494* 


J 


J 


it 


4 


11.50 





ti 


44 


.256 


2 


494 1 


K 


J 


44 


4 


ii Si 


(i 


44 


iid. 


.252 


2 


4942 


K 


J 


" 


4 


11.52 


" 


ti 


41 


.250 


2 


4943 


K 


K 


(l 


4 


11.50 


K 


44 


iid. 


.252 


2 


4944 


E 


J 


K 


4 


ii .50 


" 


" 


44 


253 


2 


4945 


K 


K 


3i 


4 


12.52 


" 


" 


it 


.248 


2 


4946 


L 


G 


44 


4 


12.55 


t> 


44 


ii 


253 


2 


4947* 


N 


H 


44 


4 


13-52 


(i 


" 


44 


.247 


2 


4948* 


N 


H 


" 


4 


13-52 


" 


it 


ii 


247 


2 


4949 


M 


L 


3* 


4 


14-51 


44 


ii 


ii 


.248 


2 


495* 


M 


L 


44 


4 


14-51 


" 


it 


44 


.247 


2 


4961 


B 


E 


i* 


6 


10.52 


3/8 


3/8 


id. 


.388 


2 


4979 


E 


E 


2f 


5 


11.84 


44 


44 


i d. 


.384 


2 


5131 


K 


K 


I 


8 


12. OO 


7/16 


7/16 


Id. 


427 


i-75 


5132 


N 


O 


44 


8 


12.00 


ii 


ii 


IP. 


415 


i-7S 


5133* 


K 


K 


If 


8 


13.00 


< 


ii 


|d. 


.427 


i-75 


5134 


N 


N 


" 


8 


13.00 


" 


it 


IP- 


4*3 


i-75 


5135 


M 


M 


I* 


8 


M-03 


" 


ii 


Id. 


.422 


i-75 


5 T 36 






" 


8 


13-99 


" 


ii 


IP- 


.420 


i-75 


5137 


L 


M 


3 


7 


14.02 


44 


" 


Id. 


44 


1-75 


5138 


P 


M 


" 


7 


14.05 


" 


ii 


IP- 


.420 


i-75 


5139 


O 


K 


II 


6 


12.06 


< 


ii 


iid. 


.428 


2 


5140 


M 


M 


2t 


6 


14.28 


" 


4 * 


it 


.421 


2 


5Mi* 


L 


L 


2} 


5 


13-73 


" 


4 * 


M 


.438 


2 


5142* 


Q 





3* 


5 


15-67 


44 


it 


M 


.422 


2 



* Pulled off rivet-heads. 
t Pulled off 3 rivet-heads. 
$ Pulled off 2 rivet-heads. 



TABULATION OF RIVETED JOINTS. 



631 



RIVETED LAP-JOINTS. 



STEEL PLATE. 



Sectional Area 
of Plate. 


Bear- 
ing 
Surface 
of 
Rivets. 


Shear- 
ing 
Area 
of 
Rivets. 


Tensile 
Strength 
of Plate 
per 
Sq. In. 


Maximum Stress on Joint per Sq. In. 


Effi- 
ciency 
of 
Joint. 


Tension 
on Gross 
Section 
of Plate. 


Tension 
on Net 
Section 
of Plate. 


Comp. on 
Bearing 
Surface 
of Rivets. 


Shear- 
ing of 
Rivets. 


Gross. 


Net. 


sq. in. 


sq. in. 


sq. in. 


sq. in. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 




2.68 


i-57 


I.IO 


3-00 


61000 


39750 


67850 


96840 


35510 


65.1 


2.70 


i-59 


i. ii 


3-00 


61000 


39660 


67360 


96490 


35700 


65.0 


2.87 


1.87 


1. 00 


3-M 


58150 


40560 


62250 


116400 


37070 


69.7 


2.94 


1.92 


1.02 


3-i4 


61000 


37010 


56670 


106670 


34650 


60.6 


2.90 


1.77 


1-13 


3-98 


61000 


43280 


70900 


111060 


31530 


70-9 


2.88 


i-75 


1.13 


3.98 


58150 


42770 


73880 


119010 


30950 


73-5 


2.90 


1.64 


1.26 


4.91 


58150 


41130 


72730 


94660 


24290 


70.7 


2.91 


1.64 


1.27 


4.91 


58150 


40200 


71330 


92110 


23820 


69.1 


3.10 


1.86 


1.24 


4.91 


58150 


40030 


66720 


100080 


25270 


69.1 


3-i7 


i. 91 


I 26 


4.91 


61470 


41770 


69320 


105080 


26970 


68.0 


3-34 


2.10 


1.24 


4.91 


55740 


42240 


67180 


112970 


28730 


75-7 


3-34 


2.10 


1.24 


4.91 


59180 


42600 


67760 


114760 


28980 


71.9 


3-6o 


2.36 


1.24 


4.91 


61470 


41390 


63140 


120180 


30350 


67.3 


3.58 


2-35 


1.23 


4.91 


58170 


42150 


64210 


122680 


30730 


72.4 


4.08 


2-33 


J -7S 


2.65 


5834 


25950 


45440 


60500 


39950 


44.4 


4-55 


2.6 3 


1.92 


3-93 


58340 


33050 


57 I 9 


78330 


38270 


56.6 


5-12 


2.13 


2-99 


4.81 


59000 


31740 


76290 


54350 


33780 


53-8 


4 99 


1.98 


3.01 


4.81 


52910 


27820 


70100 


46110 


28860 


52.6 


5-55 


2. 5 6 


2.99 


4.81 


59000 


31100 


67420 


57730 


35880 


52-7 


5-37 


2-37 


2-99 


4.81 


61140 


30370 


68820 


54550 


339*0 


49-7 


5-90 


2-95 


2.95 


4-81 


61650 


29240 


58490 


58490 


35870 


47-4 


5-89 


2-95 


2-94 


4.81 





31870 


63630 


63840 


39020 




5-94 


3-35 


2.60 


4.21 


5939 


27580 


48900 


63000 


38910 


46.4 


5-9 


3-24 


2.66 


4.21 


52910 


28530 


51940 


63270 


39980 


53-9 


5-16 


i-95 


3-21 


7.36 


58090 


28190 


74610 


45320 


19770 


48-5 


6.01 


2.85 


3.16 


7.36 


61650 


34850 


73490 


66280 


28460 


56.5 


6.01 


3.28 


2.74 


6.14 


59390 


3356o 


61490 


73610 


32850 


56.5 


6.61 


3-97 


2.64 


6.14 


56960 


30420 


50650 


76170 


32750 


53-4 



6 3 2. 



APPLIED MECHANICS. 



TABULATION OF DOUBLE- 
CHAIN-RIVETING-STEEL PLATE. 









o 














No. 
of 
Test 


Sheet Letters. 


Pitch. 


mce Apart o 
ws, Centre t 
ntre. 


Total 
Num- 
ber o 
Riv- 
ets. 


Width 
of 
Joint. 


Nominal 
Thickness. 


Size 
and 
Kind o 
Holes. 


Actual 
Thick- 
ness o 
Plate. 


Lap. 












Plate. 


Covers. 




* o v 






Plate 


Covers. 
















Q 
























in. 


in. 




in. 


in. 


in. 


in. 


in. 


in. 


4911 


K 


C 


C 


*f 


2i 


10 


13.10 


i/4 


3/i6 


5/8 d. 


253 


J T! 


4912 


L 


C 


C 


" 


" 


IO 


13.10 


44 


44 


44 


253 


14 


49i9 


L 


E 


D 


af 


2* 


10 


14-32 


44 


K 


7/8 d. 


.247 


i}| 


4920 


L 


E 


D 


" 


44 


10 


14-32 


44 


44 


" 


.249 


" 


4921 


K 


C 


B 


3* 


44 


8 


12.52 


44 


44 


44 


.252 





4922 


K 


C 


C 


44 


44 


8 


12.49 


it 


44 


" 


.252 


" 


49 2 3 


J 


A 


A 


3* 


44 


6 


II-S7 


44 


44 


44 


257 


44 


4924 


J 


A 




44 


44 


6 


"53 


44 


44 


44 


255 


11 


49 2 5 


L 


B 





4* 


44 


6 


13.09 


' 


41 


44 


.251 





4926 


K 


C 


B 


44 


41 


6 


13. 10 


*' 


44 


44 


.230 


" 


5128 


R 

/"v 


C 


C 


I* 


2 


M 


12.27 


5/i6 


3/16 


1/2 d. 


34 


if 


5 I2 9 
5130 


Q 
S 


D 




*i 





12 


I3-58 


. 


u 


.. 


305 
307 





4993 


Q 


E 





3* 


I* 


S 


14.05 


44 


44 


i d. 


309 


i$ 


4995 


Q 


E 


.... 


11 


I* 


8 


14.06 


44 


44 


44 


35 


i* 


4997 


Q 


E 





44 


2 


8 


14.08 


44 


K 


44 


.308 


t j 


495i 


B 


I 


I 


2* 


2* 


8 


8.52 


3/8 


1/4 


3/4 d. 


392 


i* 


4952 


E 


R 




44 


" 


8 


8.51 


41 


5/6 


44 


-383 


" 


4953 


E 


R 




af 


44 


8 


10.51 


44 





it 


.388 


" 


4954 


E 


N 


H 


fci 


44 


8 


10.03 


44 


i/4 


44 


384 


" 


4955 


E 


S 




3* 


44 


8 


12.50 


44 


5/16 





383 





4957 


H 





O 


3t 


44 


8 


I4-5I 


" 


44 


44 


-369 


" 


49.S8 


H 


M 


M 


4 ' 


44 


8 


I4-52 


44 


i/4 


44 


369 


" 


4959 


B 


S 


S 


4* 


44 


6 


12.42 


* 


5/i6 


44 


.388 


it 


4960 


E 


L 


N 


i4 


44 


6 


12.42 


44 


i/4 


44 


384 





4967 


C 


M 


M 


at 


2* 


10 


14-38 


44 


" 


i d. 


375 


2 


4969 


E 


M 




3l 


44 


8 


13-50 


" 


4 ' 


44 


.382 





4970 


F 







44 


44 


8 


13-58 


44 


5/i 6 


44 


.380 


" 


4971 


J 


P 


P 


3* 


K 


8 


15.46 


44 


44 


44 


379 


" 


4973 


E 


S 


S 


4t 


44 


6 


13-50 


44 


44 


44 


.385 





4975 


I 


O 


P 


4f 


44 


6 


14-65 


44 


44 





373 





4977 


J 


P 


P 


5f 


44 


6 


16.08 


" 


44 





379 


K 


4956 


K 


N 


N 


3* 


a* 


8 


12.48 


7/16 


1/4 


3/4 d. 


.427 


!J 


4968 


N 








** 


2* 


10 


14.41 




5/i 6 


id. 


.409 


2 



TABULATION OF RIVETED JOINTS. 



633 



RIVETED BUTT-JOINTS. 

CHAIN-RIVETINGSTEEL PLATE. 



Sectional Area 
of Plate. 


Bear- 
ing 
Surface 
of 
Rivets. 


Shear- 
ing 
Area 
of 
Rivets. 


Tensile 
Strength 
of 
Plate 
per 
Square 
Inch. 


Maximum Stress on Joint per Sq. In. 


Effi- 
ciency 
of 
Joint. 


Gross. 


Net. 


Tension 
on 
Gross 
Section 
of 
Plate. 


i Tension 
on 
Net 
Section 
of 
Plate. 


Compres 
sion on 
Bearing 
Surface 
of 
Rivets. 


Shearing 
on 
Rivets. 


sq. in. 


sq. in. 


sq. in. 


sq. in. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 




3-3i 


2.52 


1.58 


6.14 


58150 


49090 


64480 


102850 


26470 


84-4 


3-3i 


2.52 


1.58 


6.14 


61470 


51960 


68250 


108860 


28010 


84.5 


3*54 


2.46 


2.16 


12.02 


61470 


46810 


67370 


76720 


1379 


76.1 


3-57 


2.48 


2.18 


12.02 


61470 


457o 


65790 


74840 


13570 


74-3 


3-16 


2.27 


1.76 


9.62 


58150 


46330 


64490 


83180 


15220 


79.6 


3-'5 


2.27 


1-76 


9.62 


58150 


46730 


64850 


83640 


15300 


80.3 


2.97 


2.30 


r -35 


7 .2I 


61000 


49520 


63940 


108930 


20400 


81.2 


2.94 


2.27 


J -34 


7.21 


61000 


49460 


64050 


108510 


20170 


8i.x 


3-2Q 


2.63 


1.32 


7.21 


61470 


51440 


64350 


128210 


23470 


83-7 


3-oi 


2.41 


1. 21 


7 21 


58150 


55500 


69320 


138070 


23170 


95-4 


3-73 


2.66 


2.1 3 


5-49 


61130 


46690 


66650 


83240 


32300 


76.4 


4.27 


3-20 


2. 14 


5-49 


56760 


49040 


65430 


97850 


38140 


86. 4 


4-15 


3-23 


1.8 4 


4.70 


57000 


46480 


59720 


104840 


41040 


81.5 


4-34 


3-" 


2-47 


12.57 


56760 


44740 


62430 


78600 


iS45o 


78.8 


4.29 


3-7 


2.44 


12.57 


56760 


45490 


63570 


7998o 


15530 


80. i 


4-34 


3-io 


2.46 


12.57 


56760 


45530 


63740 


80330 


15720 


80.2 


3-34 


2.16 


2-35 


7.07 


5973 


43290 


66940 


61520 


20450 


72.4 


3-26 


2. II 


2.30 


7.07 


58340 


42380 


65470 


60070 


19540 


72.6 


4.08 


2. 9 t 


2 -33 


7.07 


58340 


46590 


65330 


81590 


26890 


79.8 


3-85 


2.70 


2-30 


7.07 


58340 


49130 


70060 


82240 


26750 


84.2 


4-79 


3-64 


2.30 


7.07 


58340 


48610 


63970 


101250 


32940 


83-3 


5-35 


4.25 


2.21 


7.07 


56670 


48500 


61060 


117420 


36700 


85.5 


5.36 


4-25 


2.21 


7.07 


56670 


47700 


60160 


115700 


36170 


84-1 


4.82 


3-95 


r -75 


5.30 


59730 


43070 


52560 


118630 


39170 


72.1 


4-77 


3-91 


i-73 


5-30 


58340 


42520 


51870 


117230 


38260 


72.8 


5-39 


3-52 


3-75 


I5-7I 


57870 


42890 


65680 


61650 


14720 


74- i 


5.16 


3-63 


3-o6 


12-57 


58340 


44263 


62920 


74640 


18170 


75-9 


5.16 


3-64 


3-4 


".57 


54290 


43240 


61290 


73390 


17750 


79-6 


5.86 


4-34 


3-03 


".57 


5713 


44910 


60650 


86860 


20940 


78.6 


5.20 


4.04 


2.31 


9.42 


58340 


45980 


59180 


103510 


25380 


78.8 


5-46 


4-35 


2.24 


9.42 


59030 


46720 


58640 


113880 


27080 


79.1 


6.09 


4.96 


2.27 


9.42 


5713 


44650 


54830 


119800 


28870 


78.1 


5-33 


4-05 


2.56 


7.07 


59000 


48120 


63300 


100190 


36280 


83-3 


5-89 


3.85 


4.09 


I5-7I 


61140 


433 


66340 


62440 


16260 


70.9 



634 



APPLIED MECHANICS. 



TABULATION OF RIVETED 
DOUBLE-RIVETED LAP-JOINTS. 





09 

c 




c 










Nominal 




8 




1 

Q 


Letters ol 
>and Cov 




part of Ro 
ang. to 1 
vets. 


Rivets in 
t Row. 


Rivets in 
id Row. 


Rivets in 
d Row. 


a 

'o 

o 


Thickness. 


"o 
o 

C 

"5 


i 








o 


oJ 


JS 


rt . tfS 


s-, C 


'o 8 


o;s 


-8 


u 


i2 

u 


1 


is 




1 


jE 


o 


.2 rt'o 


dfc 


o'c/ii 


ciH 


a 




> 

o 


SE 


t) O 


a 

rt 




in 


(X 










fc 


* 


OH 


U 


c/5 


^ 








in. 


in. 








in. 


in. 


in. 


in. 


in. 


in. 


4935 


J-J 


2 i 


2t 


5 


5 




10.68 


1/4 




7/8 d. 


257 


1 ii 


4936 


I-J 


4 * 


44 


5 


5 




10.53 


44 




44 44 


251 


k4 


4937 


L-M 


2 i 


** 


5 


5 




14.38 


" 




44 44 


.248 




4938 


I-L 


u 


** 


5 


5 




14.40 


44 




44 44 


.249 


44 


4999 
5000 

49 6 3 
4965 


ft? 

E-E 
E-E 


3* 

2 


2i 


4 

i 

6 


4 

4 
6 
6 




14.02 
14.00 
10.50 

12. OO 


5 /, ( 6 
3/8 




I " 
I P. 

3 / 4 <! 


305 
.306 
-387 
-384 


2 

4! 


498i 
4983 
5149 


E-D 
D-H 

M-L 


2t 
2* 
2 


ft 


5 
5 
7 


5 
5 
7 




11.83 
I4-3 6 
14.00 


7 /,6 




7/8 " 


385 
370 
-425 


2 


5*5 


M-M 


44 




7 


7 




I4.OO 






'* P- 


423 


44 




K-K 


2 1 


44 


6 


6 




JT . ^7 


k * 




" d. 


.428 


4 


5 I 52 


L-0 


4 - 


44 


6 


6 




*3-5 






" P. 


.440 


,| 


5153 


0-0 


2* 


" 


6 


6 




15.01 


44 




" d. 


.409 


Ij 7 B 


5154 


O-O 

K-K 


2t 


*' 


6 
5, 


6 
5 




15.02 


" 




" t 


.412 
.422 


44 



DOUBLE-RIVETED BUTT-JOINTS. 



4927 


MDE 


2* 


2t 


5 


4 




I4-36 


i/4 


3/16 


7/8 d. 


250 


'.H 


4928 


LDE 






5 


4 




I 4-35 








.247 




4929 


KBC 


H 


44 


4 


3 




12.51 


44 




4 


255 


" 


493 


KG 




" 


4 


3 




12.50 


44 




' 


.251 


44 


493i 
4932 


HDD 
HCC 


?,* 


" 


3 
3 


2 

2 




13-12 
13-12 


" 




4 


.248 
.246 


44 



DOUBLE-RIVETED LAP-JOINTS. 



5"9 
5120 


0-0 

p-p 


# 


t 


4 

4 


3 
3 




14.00 
14.03 


S/i6 




d. 
P- 


303 
305 


i* 


52I 


R-R 


" 


It 


4 


3 




14.03 


" 




d. 


.302 


i* 


5122 


O-O 


" 




4 


3 




14.03 


14 




P- 


.304 


" 


5123 


O-O 


44 


2 


4 


3 




14.02 


44 




d. 


.302 


it 


5124 


P-P 






4 


3 




14.02 






P- 


.307 


4 v 



TREBLE-RIVETED LAP-JOINTS 



5157 


KK 


2t 


a| 


5 


5 


5 


iS-M 


7/16 




7/8 d. 


'432 


i 


5158 


OP 


3 




5 


5 


5 


!5-05 


44 




" 4t 


.412 




5159 

5100 


PP 
LL 


3 


" 


4 
4 


4 

4 


4 
4 


12.78 
I3-50 


tt 




o 11 


'43 2 
.438 


ii 



TABULATION OF RIVETED JOINTS. 



635 



JOINTS. STEEL PLATE. 



CHAIN-RIVETING. 



Sectional Area 


0) 


<-. 


2 


Maximum Stress on Joint per Sq. In. 


c 


of Plate. 


J| 


1 


ft. 




"5 


















o 


been 




Tension 


Tension 


Compres- 




>> 


Gross. 


Net. 


b ^ 

.5 


II 


nj _rt G 
~ p l *"* 


on Gross 
Section ot 


on Net 
Section 


sion on 
Bearing 


Shearing 
on 


c 

V 






2~o 




"o c/} Plate 


of Plate . 


Surface 


Rivets. 


sg 






CQ 


C/3 


H 






of Rivets. 




H 


sq. in. 


sq. in. 


sq. in. 


sq. in. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 


Ibs. 




2-74 


1.62 


2.25 


6.01 


61000 


42770 


72350 


52090 


19500 


70.1 


2.64 


1.54 


2. 2O 


6.01 


62300 


42350 


72600 


58180 


18600 


i 67.9 


3-57 


2.49 


2.17 


6.01 


61470 ' 47870 


68630 


78760 


28440 


77.0 


3-58 
4.28 


2.49 
3-06 


2.l8 
2.44 


6.01 
6.28 


61470 48530 
56760 : 46070 


69780 
64440 


79700 
80820 


28910 
31400 


78.9 
80. i 


4.28 


3-3 


2-53 


6.28 


593o . 43900 


62100 


74370 


29960 


74.1 


4-06 


2.32 


3.48 


5-30 


58340 | 40570 


70900 


47330 




69.5 


4.61 


2.88 


3.46 


5-30 


58340 i 42150 


67470 


56160 


36660 


72.2 


4-55 


2.63 


3-85 


7-8 5 


53730 ! 38790 


67100 


45840 


22480 


72.2 


5-31 


3-46 


3-70 


7.85 56670 4 }c 5 o 


66070 


61780 


29120 


76.0 


5-95 


3-35 


5-21 


8.41 ; 61650 40620 


72150 


4 fi l-9o 


28740 


i 65.8 


5-93 


3-24 




8.41 61650 379'o 


69380 


41940 


26730 


: 61.4 


5-79 


3-54 


4.49 


7.21 


59000 43150 


70570 






73 - 1 


5-94 


3-55 


4.78 


7.21 59390 38870 


65040 


48310 


12O2O 


! 65.4 


6.14 


3-99 


4.29 


7.21 52910 43530 


6f QQO 


62310 


37070 


82.3 


6. 19 

5-81 


3-95 
3.96 


4.48 
3.69 


7-21 j 52910 40380 

6.01 59000 38850 


63290 

56990 


55800 

61170 


34670 
37550 


1 76.3 
; 65.8 



ZIGZAG-RIVETING. 



3 59 


So 


97 


10.82 


58170 


48150 


69140 


87740 


15980 


80.3 


3-54 


.46 


95 


10.82 


61470 


47420 


68240 


86090 


15520 


77-i 


3-!9 


3 


56 


8.41 


58150 


46610 


64650 


95320 


17680 


80.2 


3-'4 


.26 


.56 


8.41 


58150 


47520 


66020 


95640 


17740 


81.7 


3- 2 5 


.60 


05 


6.01 


59180 


47640 


59550 


M745 


25760 


80.5 


3- 2 3 


.58 


.08 


6.01 


59180 


46720 


5S490 


139720 


25110 


78.9 



ZIGZAG-RIVETING. 



4.24 

4.28 

4-23 

4.27 


3-03 
3.02 
3.02 
3 01 


. 12 
.20 
.IT 
19 


5-50 

5 50 
5 50 
5-50 


56760 
5930 
54350 
54350 


42750 
40630 
42990 
44^40 


59830 

57580 

60220 
6^000 


855^0 
79050 
86180 
86450 


32960 
31620 
33060 
34420 


75-3 
68. S 
79.1 
81 6 


4-23 


3.02 


. II 


5-5 


54350 


44870 


62850 


89050 


345 10 


82.5 


4-33 


3-03 


.22 


5-50 


593 


43490 


62150 


84820 


31400 


73-3 



CHAIN-RIVETING. 



5-93 

6.20 

5-52 
5-91 


4.04 
4.40 
4.01 
4.38 


5-66 
5-4 
4-54 
4.60 


9.02 
9.02 
7.21 
7.21 


59000 
52910 
58090 
59390 


4^720 
48710 
48040 
46430 


67100 
68630 
66130 

62650 


47900 
55820 
584- 
59650 


30060 
3348o 
36780 
38060 


77 5 
92.1 
82.7 
78.2 



636 APPLIED MECHANICS. 

In the design of a riveted tension-joint the problem usually 
presents itself in the following form : 

Given, in all particulars, the two plates to be united, to 
design the joint ; i.e., to determine, i, the diameter of rivet to 
be used ; 2, the spacing of the rivets, centre to centre ; and, 3, 
the lap. 

In regard to the determination of the lap, the common 
practice has been already explained and very little has been 
done experimentally. 

In order to determine the diameter and the spacing of the 
rivets by the usual methods of calculation, it becomes neces- 
sary to know the three following kinds of resistance of the 
metals, viz.: 

i. The tensile strength per square inch of the plate along 
the line or lines of rivet-holes ; 

2. The shearing-strength of the rivet metal ; 

3. The resistance to compression on the bearing-surface of 
either plate or rivet. 

Hence we need to ascertain what the tests cited show in 
regard to these three quantities. 

Tension. The tensile strength of the plate used should, 
of course, be determined by means of tests made on specimens 
cut from it. Further than this, questions arise as to the 
excess tenacity due to the grooved specimen form, and as to 
any injury due to punching when the holes are punched. 

The excess tenacity is, of course, greater with small than 
with large spaces between the rivet-holes ; hence, inasmuch as 
the tendency is toward the use of large rivets, and, conse- 
quently, large pitches, the excess tenacity applicable in practi- 
cal cases becomes small, and would be better disregarded in 
the design of most riveted joints. In cases where the holes 
are drilled, therefore, we should use for tensile strength per 
square inch of the plate along the line of rivet-holes, the tensile 
strength per square inch of the plate itself. 



COMPRESSION. 637 



The better and more ductile the plate the less is the 
injury done by punching; but, while more or less punching is 
done, the better class of work is drilled. A study of the results 
in the cases of punched plates will show approximately what 
allowance to make for the weakening due to punching different 
qualities of plate. 

Shearing". A study of the results of the government tests 
show that it is fair to assume the shearing-strength of the 
wrought-iron rivets used, to be about 38000 pounds per square 
inch, which is about two thirds of the tensile strength of the 
same rivet metal. 

For steel rivets, of the kinds now prescribed in most spec- 
ifications, the shearing-strength appears to be about 45000 
pounds per square inch. 

Compression. To determine what we should estimate as 
the ultimate compression on the bearing-surface is a more 
difficult problem ; for if a joint fails in consequence of too 
great compression on the bearing-surface the cause of the 
failure does not exhibit itself directly, but in some indirect 
manner probably by decreasing the resisting properties of 
either the plate or the rivets, and hence by causing either the 
joint to break by tearing the plate or by shearing either the 
rivets or the plate in front of the rivets, but at a lower load 
than that at which it would have broken had the compression 
not been excessive; and hence when such breakage occurs it is 
difficult to say whether it is due to excessive compression re- 
ducing the tensile or the shearing strength, or whether its full 
tensile or shearing strength was really reached. 

Observe, moreover, that in the tables of Government tests 
the heavy numbers in the column marked " Compression on 
the bearing-surface of the rivets " indicate that the plate broke 
out in front of the rivets, which might be due to excessive 
compression or to a deficiency of lap. 

While more experiments are needed, it would seem proba- 
ble that we might deduce some conclusions, at least, of a gen- 
eral nature, in regard to the ultimate compression by a study 



638 APPLIED MECHANICS. 

of the relations existing between the compression per square 
inch on the bearing-surface at fracture and the efficiency of 
the joint as shown by the Government tests. 

For this purpose the following diagrams (see pages 631 and 
632) have been plotted, with the efficiencies as abscissae and 
the compression per square inch on the bearing-surface at 
fracture as ordinates. If similar diagrams were plotted with 
the efficiencies as abscissae and the ratio of the compression per 
square inch on the bearing-surface at fracture to the tensile 
strength of the plate as ordinates the character of the diagrams 
would be substantially the same, as the plates used in the tests 
were all of mild steel of approximately the same quality, and 
hence the difference in tensile strength of different samples 
was not great. 

A study of these diagrams shows that in the case of the 
i-inch plates experiments were made with compressions up to 
about 158,000 pounds per square inch, but that the highest 
compression reached with any other thickness of plate was 
about 120,000 pounds per square inch. 

Inasmuch as Kennedy advises the use of 96,000 pounds per 
square inch, and as this is higher than the values that have 
been customarily advocated, it would hardly seem wise to 
adopt a much higher value unless the tests furnish us sufficient 
evidence for such a procedure. Considering the facts stated 
above, and also the fact that in the cases of the double-riveted 
joints some of the highest compressions were accompanied by 
a decrease in efficiency, it would seem best to limit our esti- 
mate of the ultimate compression on the bearing-surface to 
from 90,000 to 100,000 pounds per square inch until we have 
further light on the subject derived from experiment ; and it is 
not at all improbable that when we do obtain further light we 
may find ourselves warranted in using a somewhat higher 
value. 

The reasoning which leads to the above conclusion is, of 
course, based on evidence which is not conclusive, because of 
the lack of tests with higher compressions on the bearing sur- 



COMPRESSION. 



6380 




APPLIED MECHANICS. 



W/TH TWO 

STEEL PLJTE. $' 'STEEL PLJTE. 







ilnTOMmiilllfflt 








///> f\ s> si ::: 


;i;;;;J:i:;i;;:i;;!i;i;i;;N;; /soooo 


{ 










j |j J_[ |j_[_[ U'i 1 r H t ini* LL1J li 1 1 /?/?/?/?/) 
::;;::;:!;:::::;:::: 7W0 







70 



<ft7 



7<? W ^ /^ 

face, with plates thicker than one quarter of an inch. On the 
other hand, the quarter-inch plates show higher efficiencies 
with compressions above 100000 pounds than they do with 
compressions of 100000 pounds or less, and the author knows 
of tests upon riveted joints in T 7 -inch plates which tend to 
show that, with good wrought-iron rivets, it would be perfectly 
safe to use a considerably larger number for compression on 
the bearing-surface, in designing riveted joints at least 
Iioooo pounds per square inch, and probably more. 



COMPRESSION. 



It will be observed that no reference has been made to the 
friction, and it is safer to leave this out of account, as the tests 
show that slipping takes place at all loads, and as there. is no 
friction at the time of fracture. 

By far the greater part of the tests at Watertown Arsenal 
were made with wrought-iron rivets in mild-steel plates, this 
being, at the time, the most usual practice, although steel 
rivets were sometimes used. At the present time, notwith- 
standing the fact that steel long ago superseded wrought- 
iron for boiler-plate, and that it has, to-day, superseded 
wrought-iron for structural shapes, as I beams, channel-bars, 
angles, etc., and that the use of steel rivets has become very 
extensive, nevertheless a great many still adhere to the use 
of wrought-iron rivets, and feel more confidence in them than 
they do in steel rivets. Whereas the use of wrought-iron 
rivets had been practically universal, the qualifications for a 
good wrought-iron rivet metal became pretty well known, and 
while sometimes specifications were drawn up giving the 
requirements of the rivet metal for tensile strength, ductility, 
etc., which of course would vary more or less, nevertheless 
the variations would not be large. A study of the Watertown 
tests shows that the wrought-iron rivet metal used in those 
tests had a tensile strength of from about 52000 to about 
59000 pounds per square inch, with a percentage contraction 
of area at fracture of from about 30 to about 45. With this 
metal the shearing strength per square inch seems to be about 
f of the tensile strength per square inch. Of course other 
tests are necessary to show whether the metal can be properly 
worked, and whether it is red-short or not, such as that the 
metal should bend double, whether cold or hot, without cracks, 
and that cracks should not develop when the shank is ham- 
mered down, cold or hot, to a length considerably less than 
the diameter. 

When steel rivets were first used, the steel employed was 



638 d APPLIED MECHANICS. 

not an extremely soft steel, as shown by the few cases of steel 
rivets included in the Watertown Arsenal tests already quoted, 
where the shearing-strength per square inch varied from about 
50000 pounds per square inch up to as high a figure as 65000 
pounds per square inch; and by Kennedy's tests, where he ap- 
parently fixes on from about 49000 to about 54000 pounds per 
square inch as the shearing-strength of steel rivets. 

Now it would seem that metal with these shearing-strengths 
would have a tensile strength per square inch which would not 
warrant us in classifying it as very soft steel. 

On the other hand, it is evident that brittleness should not 
in any way be tolerated in rivet metal, and hence it would seem 
that at least soft steel should be used for rivets. 

The specifications proposed by the American Society for 
Testing Materials prescribe for tensile strength per square inch 
of steel for structural rivets from 50000 to 60000 pounds per 
square inch, and for boiler-rivets from 45000 to 55000 pounds. 

While the number of tests that have been made upon joints 
constructed with steel rivets is not large, the shearing-strength 
of such steel rivets as are in use to-day is not very far from 45000 
pounds per square inch, as a rule. 

The number of tests of joints constructed with steel rivets is 
not sufficiently large to warrant drawing from them definite 
conclusions regarding the ultimate compression on the bearing 
surface in such joints. Meanwhile, it would be advisable to use 
for it the same values as are suitable in the case of joints made 
with steel plates and wrought-iron rivets. 

The following table contains the joints tested at Watertown 
Arsenal, which were made with steel plate and wrought-iron 
rivets, and in which the plate broke out in front of the rivet. It 
is evident that only four of them, viz., 4915, 4916, 4917, 4918, 
failed in consequence of excessive compression on the bearing 
surface, and that the breaking out of the plate in the other cases 

was due to insufficiency of lap. The calculated -j was obtained 



WIRE AND WIRE ROPE. 



639 



by the method described on page 554, assuming ^ = 55000, and 
) 8 = 38000, and j c = 96000. 







"3 
- C 

V 


o 




II II 


'3 




s! 


u 


1 


d , A 

O i o 


t/j 

\i 




d 




ji* 


Kind of 

Joint. 


^ 


o 



jj 




o 
e 

^ 


a; 

"5 


M 3, 

01 1 

C P 


E 


S \ 


liil 


! 

S'S 

3 fe 


d ' 


1 


IH 




1 


P 
5: 




CQ 







^(3 


a 




5 ftffl J 


5*0 




6 






2 


a 




OH 







^ 


J 


J> 


^3 







113 








in 


5. 




II 


s. 


Ins. 


Ins. 


Ins. 


Lbs. 








7 i8 


Single lap 


Iron 


n 


l o 


P 


1 


f 




1.25 




795io 


Tore and 


1.18 


i.SS 


























sheared 






719 


' 






V 


P 








25 




80200 


Tore 


.18 


.55 


4947 


* 








D 








.00 




i i 2970 




.60 


. 75 


A 

4948 










D 








. oo 




114760 


Tore 


.60 


-75 


4949 


' 








D 








.00 




120180 




.60 


77 


767 
1442 


Single butt 




- 




P 






i 


25 

.00 


1.25 

2.00 


95210 
107610 


Tore 


.67 

.50 


.65 


1443 














. 


r 


5 


.00 


2 . OO 


108830 




5 


. 70 


49 1 5 






^ 


- 


D 






JL 


. oo 


2 . OO 


151780 




29 


93 


4916 






' 


- 


D 




: 


J- 


.00 


2.00 


142570 




2 9 


.89 


4917 
4918 






- 


: 


D 
D 




1 


ft 


. oo 

.00 


2 . OO 
2.00 


158150 
152110 




. 29 
29 


.96 
93 


4985 






r 




D 


1 


if 


A 


.00 


I . OO 


83950 




. oo 


57 


4987 






[ 




D 


1 




A 


.25 


1 25 


937io 




25 


.63 


4991 






I 




D 


J 


V 


3, 


75 


I ~ ~ 


i 19500 




75 


77 


298 


Reinforced 
lap 




\ 


i 


D 




i 


i 


j -15 

1 .00 


[l.!2 


67300 


Sheared 
rivets 


.19 


.66 


299 






\ 


* 


D 




1 


i 


J .10 
1 .12 


[l.I2 


68040 


,, 


.19 


.67 


5121 


Double lap 




I 




D 


1 


V 




50 




86180 




5 


.56 


5122 






1 




P 


1 


V 




50 




86450 






59 



234. Wire and Wire Rope. It is well known that the 
process of making wire by cold drawing greatly increases the 
strength of the metal. Annealing, on the other hand, decreases 
the strength, and increases the ductility. It is not the purpose 
of this article to discuss the various qualities of wire required and 
used for different purposes. Hence, inasmuch as results of tests 
of wrought-iron, and of steel wire, have already been given, there 
will be given here only a few tests of hard-drawn, of semi-hard- 
drawn, and of soft copper wire. 

Wire rope. Wire rope is used for a great many purpose:, as 
in suspension bridges, in hoisting, in haulage, in the transmission 
of power, etc. 

While flat wire rope is used for some purposes, and while 
wire rope made of parallel wires is used in large suspension 
bridges, the greater part is made by twisting a number of wire 



640 



APPLIED MECHANICS. 



HARD-DRAWN COPPER WIRE. 



SOFT COPPER WIRE. 



Diameter. 


Tensile 
Strength 


Elastic 
Limit per 


Contrac- 
tion of 




per 
Sq. In. 


Sq. In. 


Area. 


Inches. 


Lbs. 


Lbs. 


Per Cent. 


0.166 


53050 


37100 




0.138 


60350 


22800 




o-i35 


56300 


28150 




o-i34 


5 I0 5 


2,7140 




o. 105 


61800 


41000 


5 r 


o. 105 


57100 


35000 


49 


o. 105 


58900 


34000 


33 


o . 106 


60300 


36000 


42 


o. 106 


59500 


34000 


39 


0.086 


58170 


27870 




0.086 


58620 


29310 




o . 08 > 


6 1 5 10 


2 T 7OO 








* 1 oV w 




o 083 


66536 


206^0 




w w<kj O 

0.083 


65060 


*y w o 

37334 




0.083 


66536 


29630 














Diameter. 
Inches. 


Tensile 
Strength 
per 
Sq. In. 

Lbs. 


Elastic 
Limit per 
Sq. In. 

Lbs. 


Contrac- 
tion of 
Area. 

Per Cent. 


0.163 
o. 162 
o. 162 


35730 
35770 
36640 


13760 
I2QQO 




0.083 




IO5OO 




0.083 
0.081 

o .080 
o .080 


29500 
33200 
33100 


IJ2CO 


70 

45 



SEMI-HARD-DRAWN COPPER WIRE. 



o. 106 


44300 


30000 


60 


o. 106 


45100 


29000 


65 


0.106 


455oo 


29000 


55 


o. 106 


45100 


31000 


67 


o. 106 


44900 


30000 


64 



strands around a central core, which may be of tarred hemp, or 
which may be, itself, a wire strand, the wire strands being made 
of wires twisted together. 

In the case of a wire core, the strength of the rope is a little 
greater, but the resistance to wearing is less. J 

The most usual number of strands is six, each strand contain- 
ing seven, eighteen, or nineteen wires, though other numbers of 
wires are sometimes used. 

The strength that can be realized in practice is always less 
than the strength of the rope, and is determined by the method of 
holding the ends, as the junction point of the rope and the holder 
is the weakest point. 

The usual methods of holding the ends are as follows: splic- 
ing, as in the case of the transmission of power, passing the rope 
around a pulley, or around a thimble, fastening it in a socket, or 
in a clamp. 

The diameter of the drum or sheave around which a rope 



WIRE AND WIRE ROPE. 



641 



passes, should not be so small as to cause too much stress to be 
exerted upon some of the wires, in consequence of the bending- 
moment introduced by the curvature. 

Inasmuch as it may be a matter of convenience to have here 
some tables giving the strength of rope as claimed by some makers, 
there will follow here two tables of the strength of different sizes, 
as given by the Roebling Company for their rope. 

The following explanations are given by the Roebling Com- 
pany, about the quality of the metal used : 

Iron, open-hearth steel, crucible steel, and plough steel pos- 
sess qualities which cover almost every demand upon the material 
of a wire rope. Copper, bronze, etc., are, however, used for a 
few special purposes. 

The strength of iron wire ranges from 45000 to 100000 pounds 
per square inch; open-hearth steel, from 50000 to 130000 pounds 

SEVEN-WIRE ROPE. 
Composed of 6 Strands and a Hemp Center, 7 Wires to the Strand. 











Approximate Breaking-strain in Tons of 










2000 Lbs. 


Trade No. 


Diameter 
in Inches. 


Approxi- 
mate 
Circum- 
ference 


Weight 
per Foot in 
Pounds. 


Transmission or 
Haulage Rope. 


Extra 








in Inches. 




Swedish 


Cast 


Strong 
Cast Steel. 


Plough 
Steel. 










Iron. 


Steel. 






ii 


l| 


4f 


3-55 


34 


68 


79 


91 


12 


! 


4i 


3.00 


29 


58 


68 


78 


13 


J i 


4 


2.45 


24 


48 


56 


64 


14 


I| 


3* 


2 .00 


20 


40 


46 


53 


15 


I 


3 


1-58 


16 


3 2 


37 


42 


16 


1 


ai 


I . 20 


12 


24 


28 


3 2 


17 


f 


2i 


0.89 


9-3 


18.6 


21 


24 


18 


t* 


4 


0-75 


7-9 


15.8 


l8. 4 


21 


19 


f 


2 


0.62 


6.6 


13.2 


I5-I 


17 


20 


A 


if 


0.50 


5-3 


10.6 


12-3 


14 


21 


j 


l 


-39 


4.2 


8.4 


9.70 


II 


22 


iV 


l| 


0.30 


3-3 


6.6 


7-50 


8-55 


23 


1 


1* 


0.22 


2.4 


4-8 


5.58 


6-35 


24 


f 


I 


0-15 


i-7 


3-4 


3-88 


4-35 


25 


A 


4 


o. 125 


1.4 


2.8 


3.22 


3-65 



642 



APPLIED MECHANICS. 



NINETEEN-WIRE ROPE. 
Composed of 6 Strands and a Hemp Center, 19 Wires to a Strand. 











Approximate Breaking-strain in Tons of 










2000 Lbs. 






Approxi- 


Weight. 








Trade No. 


Diameter 
in Inches. 


mate 
Circum- 
ference 
in Inches. 


per Foot in 
Pounds. 


Standard Hoisting 
Rope. 


Extra 
Strong 
Cast Steel. 


Plough 
Steel. 


Swedish 


Cast 










Iron. 


Steel. 









2| 


8f 


n-95 











305 





2* 


7l 


9-85 











254 


i 




7i 


8.00 


78 


156 


182 


208 


2 


2 


^i 


6.30 


62 


124 


144 


165 


3 


if 


si 


4-85 


48 


96 


112 


128 


4 


If 


5 




42 


84 


97 


ill 


5 


I* 


4f 


3-55 


36 


72 


84 


96 


5* 


if 




3.00 


31 


62 


72 


82 


6 


I* 


4 


2.45 


25 


5 


58 


67 


7 




3* 


2 .00 


2 I 


42 


49 


56 


8 


I 


3 


I. 5 8 


17 


34 


39 


44 


9 


1 




I .20 


13 


26 


30 


34 


10 


f 


2 4 


0.89 


9-7 


19.4 


22 


25 


10* 


f 


2 


O.62 


6.8 


13-6 


15-8 


18 




A 


if 


0.50 


5-5 


II .0 


12-7 


14-5 


lof 






o. 39 


4-4 


8.8 


10. I 


11.4 


ioa 


A 


i* 


0.30 


3-4 


6.8 


7-8 


8.85 


lob 


| 


it 


0. 22 


2-5 


5- 


5-78 


6-55 


IOC 


T^> 


i 


0.15 


i . 7 


3-4 


4-05 


4-5 


lod 


i 


f 


0. 10 


I .2 


2 -4 


2.70 


3.00 



per square inch; crucible steel from 130000 to 190000 pounds 
per square inch; and plough steel from 190000 to 350000 pounds 
per square inch. Plough steel wire is made from a high grade of 
crucible cast-steel. 

235. Other Metals and Alloys. Copper is, next to iron 
and steel, the me'al most used in construction, sometimes in the 
pure state, especially in the form of sheets or wire, but 
more frequently alloyed with tin or zinc; those metals where 
the tin predominates over the zinc being called bronze, and 
those where zinc predominates over tin, brass. Copper in the 
pure state was used not long ago for the fire-box plates of loco- 



IRON AND STEEL WIRE. 643 

motive and other steam-boilers, "as it was believed to stand better 
the great strains due to the changes of temperature that come 
upon these plates, than iron or steel ; but now steel or iron has 
almost entirely superseded it for this purpose, except in some 
cases where the feed-water is very impure, and where the 
impurities are such as corrode iron. 

The alloys of copper, tin, and zinc which are used most 
where strength and toughness are needed, are those where the 
tin predominates over the zinc ; and the composition, mode of 
manufacture, and resisting properties of these metals, together 
with the effect of other ingredients, as phosphorus, have been 
very extensively investigated with reference to their use as a 
material for making guns, instead of cast-iron. 

Accounts of tests made on these alloys will be found as 
follows : 

Major Wade : Ordnance Report, 1856. 

T. J. Rodman : Experiments on Metals for Cannon. 

Executive Document No. 23, 46th Congress, 2d session. 

Materials of Engineering : Thurston. 

No attempt will be made to give a complete account of the 
results of these tests ; but a table will be given on page 639 for 
convenience of use, showing rough average values of the resist- 
ing powers of some metals and alloys other than iron. 

236. Timber. However extensively iron and steel may 
have superseded timber in construction, nevertheless, there are 
many cases in which iron is entirely unsuitable, and where 
timber is the only material that will answer the purpose ; and 
in many cases where either can be used, timber is much the 
cheaper. Hence it follows that the use of timber in construc- 
tion is even now, and as it seems always will be, a very impor- 
tant item. 

Another advantage possessed by timber is, that, on yielding, 
it gives more warning than iron, thus affording an opportunity 
to foresee and to prevent accident. 

If we make a section across any of the exogenous trees, as 



6 4 4 



APPLIED MECHANICS. 





Specific 
Gravity. 


Tensile 
Strength 
per Sq. In. 


Modulus 
of 
plasticity. 


Brass cast . . . .... 


8.7C-6 


18000 


9I7OOOO 






49OOO 


14230000 


Bronze unwrought : 
84.29 copper + 15.71 tin (gun metal) 
82.81 " + 17.19 " 
81.10 " + 18.90 " 
78.97 " +21.03 " (brasses). . . 
34.92 " + 65.08 " (small bells) . 
15.17 " +84.83 " (speculum metal) 
Tin . . 


8.561 
8.462 

8-459 
8.728 
8.056 
7447 

7 2QI 


36060 
34048 
39648 
30464 

3 T 3 6 
6944 
c6oo 




Zinc . . .... 


686l 


7SOO 




Copper cast 


8.712 


24138 






8.878 


77OOO 




Copper wire . 




60000 


I7OOOOOO 


Gold cast 


IQ 2C8 


2OOOO 






IO.476 


40000 






22.069 


56000 




Lead cast 


1 1 7 C2 


1800 













the oak, pine, etc., we shall find a series of concentric layers ; 
these layers being called annual rings, because one is generally 
deposited every year. 

Radiating from the heart outwards will be found a series of 
radial layers, these being known as the medullary rays. 

Of the annual rings, the outer ones are softer and lighter in 
color than the inner ones ; the former forming the sap-wood, and 
the latter the heart-wood. When the log dries, and thus tends 
to contract, it will be found that scarcely any contraction takes 
place in the medullary rays ; but it must take place along 
the line of least resistance, viz., along the annual rings, thus 
causing radiating cracks, and drawing the rays nearer together 
on the side away from the crack. This action is exhibited in 
Fig. 241, where a log is shown with two saw-cuts at right 
angles to each other ; when this log bf ^mes dry, the four 



STRENG7*H OF TIMBER. 



645 




FIG. 241. 



right angles all becoming acute 
through the shrinkage of the 
rings. 

If the log be cut into planks by 
parallel saw-cuts, the planks will, 
after drying, assume the forms 
shown in Fig. 242, as is pointed 
out in Anderson's " Strength of 
Materials," from which these two 
cuts are taken. 

This internal construction of a 
plank has an important influence 
upon the side which should be uppermost when it is used for 

flooring ; for, if the heart side is up- 
permost, there will be a liability to 
having layers peel off as the wood 
dries : indeed, boards for flooring 
should be so cut as to have the an- 
nual rings at right angles to the 
side of the plank. Before discuss- 
ing any other considerations which 
affect the adaptability of timber to 
use in construction, we will con- 
sider the question of its strength. 
237. Strength of Timber. In this regard we must 
observe, that, whereas the strength and elasticity and other 
properties of iron and steel vary greatly with its chemical com- 
position and the treatment it has received during its manufac- 
ture, the strength, etc., of timber is much more variable, being 
seriously affected by the soil, climate, and other accidents of its 
growth, its seasoning, and other circumstances ; and that over 
many of these things we have no control : hence we must not 
expect to find that all timber that goes by one name has the 
same strength, and we shall find a much greater variation and 




FIG. 242. 



646 APPLIED MECHANICS. 

irregularity in timber than in iron. The experiments that have 
been made on strength and elasticity of timber may be divided 
into the following classes : 

i. Those of the older experimenters, except those made 
on full-size columns by P. S. Girard, and published in 1798. 
A fair representation of the results obtained by them, all of 
which were deduced from experiments on small pieces, is to 
be found in the tables given in Professor Rankine's books, 
" Applied Mechanics," " Civil Engineering," and " Machinery 
and Millwork." 

2. Tests made by modern experimenters on small pieces. 
Such tests have been made by 

(a) Trautvvine : Engineers' Pocket-Book. 
(b} Hatfield : Transverse Strains. 
(c) Laslett : Timber and Timber Trees. 
(</) Thurston : Materials of Construction. 

(e) A series of tests on small samples of a great variety of American 
woods, made for the Census Department, and recorded in 
Executive Document No. 5, 48th Congress, ist session. 
Timber Physics, Division of Forestry, U. S. Department of Agri- 
culture. For a fairly complete bibliography of tests of tim- 
ber see a paper by G. Lanza, Trans. Am. Soc. C. E., 1905. 

3. Tests made by Capt. T. J. Rodman, U.S.A., the results 
of which are given in the " Ordnance Manual." 

4. All tests that have been made on full-size pieces. 

In regard to tests on small pieces, such as have commonly 
been used for testing, it is to be observed, that, while a great 
deal of interesting information may be derived from such tests 
as to some of the properties of the timber tested, nevertheless, 
such specimens do not furnish us with results which it is safe 
to use in practical cases where full-size pieces are used. Inas- 
much as these small pieces are necessarily much more perfect 
(otherwise they would not be considered fit for testing), having 
less defects, such as knots, shakes, etc., than the full-size pieces. 



Sl^RENGTff OF TIMBER. 



647 



they have also a far greater homogeneity. They also season 
much more quickly and uniformly than full-size pieces. In 
making this statement, I am only urging the importance of 
adopting in this experimental work the same principle that the 
physicist recognizes in all his work ; viz., that he must not 
apply the results to cases where the conditions are essentially 
different from those he has tested. 

Moreover, it will be seen in what follows, that, whenever 
full-size pieces have been tested, they have fallen far short of 
the strength that has been attributed to them when the basis 
in computing their strength has been tests on small pieces ; 
and, moreover, the irregularities do not bear the same propor- 
tion in all cases, but need to be taken account of. 

The results of the first class of experiments named in the 
following table are taken from Rankine's " Applied Mechanics ;" 
and, inasmuch as the table contains also the strengths of some 
other organic fibres, it will be inserted in full. The student 
may compare these constants with those that will be given 
later. 



Kind of Material. 


Tenacity 
or Resist- 
ance to 
Tearing. 


Modulus of 
Tensile 
Elasticity. 


Resist- 
ance to 
Crush- 
ing. 


Modulus 
of 
Rupture. 


Resist- 
ance to 
Shearing 
along 
Grain. 


Modulus 
oi 
Shearing 
Elasticity 
along the 
Grain. 


Ash 


I7OOO 


1600000 


QOOO 


( 12000 


I4OO 


76000 


Bamboo . . 








( I4OOO 






Beech 


II5OO 


I-KOOOO 


0160 


( 9000 


1 _ 




Birch . . 


1 5OOO 


1 64 sooo 


6400 


( I2OOO 
I I7OO 


' 




Blue cum 






8800 


( IbOOO 


1 _ 




Box 


2OOOO 




10300 


( 2OOOO 


f 




Bullet-tree .... 
Cedar of Lebanon . . 


II4OO 


486000 


14000 
5860 


(15900 
| 16000 
7400 


': 


- 



648 



APPLIED MECHANICS. 



Kind of Material. 


Tenacity 
or Resist- 
ance to 
Tearing. 


Modulus of 
Tensile 
Elasticity. 


Resist- 
ance to 
Crush- 
ing. 


Modulus 
of 
Rupture. 


Resist- 
ance to 
Shearing 
along 
Grain. 


of 1 
Shearing I 
Elasticity 
along the 
Grain. 




( IOOOO 


) 










Chestnut 


to 


/ 1140000 


- 


10660 


- 


- 




( 13000 


) 










Cowrie 


- 


- 


- 


IIOOO 


- 


- 


Ebony .... 


_ 


_ 


I9OOO 


77000 










( 700000 


) 


( 6000 


) 




Elm . . 


14000 


5 to 




) to 

< IU 




76000 






\ LV -' 

( 1840000 


) 30C 


( 9700 


) M 




Fir, Red pine . . . 


( 12000 

to 
f 14000 


1460000 
to 
1900000 


5375 
to 
6200 


r 7100 
^ 9540 


500 
800 


62OOO 
II6OOO 


" Yellow pine (Am.) 


- 


- 


5400 


- 


- 


- 


" Spruce .... 


12400 


( 1400000 
to 
( 1800000 


| - 


j 9900 
) 12300 


| 600 


- 


" Larch .... 


( 9000 

to 

( IOOOO 


900000 
to 
1360000 


| 5570 


1 IOOOO 


970 
I7OO 


\ - 


Hoxen yarn .... 


25000 


- 


- 


- 


- 


- 


Hazel 


16000 














( I2OOO 


) 










Hempen rope . . . 


to 


( 


_ 


_ 


_ 


_ 




( 16000 


) 










Ox-hide, undressed 


6300 


_ 


- 


- 


- 


- 


Hornbeam .... 


20000 


_ 


- 


- 


- 


- 


Lancewood .... 


23400 


- 


- 


- 


- 


- 


Ox-leather .... 


42OO 


24300- 


- 


- 


- 


- 


Lignum-vitae .... 


IISOO 


- 


9900 


I2OOO 


- 


- 


Locust ... . 


l6oOO 




_ 








Mahogany 


( 8000 

< to 


\ i 2 e CQOO 


8200 


j 7600 








) " 














( 21800 


) 




-> 






Maple 


10600 












Oak, British .... 




- 


IOOOO 


( IOOOO 

] 13600 






" Dantzic . . . 


- 


- 


7700 


8700 






" European . . . 


( IOOOO 

to 

( 19800 


f 1200000 

C 1750000 


} - 


- 


2300 


82OOO 


" American red 


10250 


2I5OOOO 


6000 


10600 







STRENGTH OF TIMBER. 



649 













Resist- 


Modulus 


Kind of Material. 


Tenacity 
or Resist- 
ance to 
Tearing. 


Modulus of 
Tensile 
Elasticity. 


Resist- 
ance to 
Crush- 
ing. 


Modulus 
of 
Rupture. 


ance to 
Shearing 
along 
Grain. 


of 
Shearing 
Elasticity 
along the 














Grain. 


Silk fibre 


52OOO 


I3OOOOO 












I 7OOO 


I O4OOOO 




0600 




















Teak, Indian . . . 


I5OOO 


24OOOOO 


I2OOO 


( I2OOO 
\ IQOOO 


1- 


- 


African . . . 


2IOOO 


2300000 


- 


14980 


- 


- 


Whalebone .... 


7700 


- 


- 


- 


- 


- 


Willow .... 








6600 






Yew . ..... 


8000 



























in regard to the tests of the second class, a few comments 
are in order : 

i. These experiments, like those of the first class, were all 
made upon small pieces ; and the results are correspondingly 
high. 

The usual size of the specimens for crushing being one or 
two square inches in section, and of those for transverse 
strength being about two inches square in section and four or 
five feet span, those for tension had even a much smaller sec- 
tion than those for compression ; as it is necessary, in order to 
hold the wood in the machine, to give it very large shoulders. 

The only exception to this is the tests of Sir Thomas Las- 
lett, an account of which is given in his "Timber and Timber 
Trees," and also in D. K. Clark's " Rules and Tables." In these 
tests he gives very much lower tensile strengths than those 
given above ; and he states that his specimens were three inches 
square, but does not say how he managed to hold them in such 
a way as to subject them to a direct tensile stress. His results 
for crushing and transverse strength are about as great as 



650 APfLIED MECHANICS. 



those given in Rankine's tables, and as were obtained by the 
other experimenters on small pieces, as his specimens were of 
about the same dimensions as those used by the others. The 
figures obtained by these experimenters will only be given inci- 
dentally, as 

(a) They are very similar to those given in Rankine's table. 

(b) They are not suitable for practical use on the large 
scale. 

(c) While they have been used, it has only been done by 
employing a very large factor of safety for timber. 

The series of tests made for the Census Department, and 
recorded in Executive Document No. 5, 48th Congress, first 
session, form a very interesting series of experiments upon 
small specimens of an exceedingly large number of .America , 
woods. In order to have working figures, we should need to 
test large pieces of the same ; as the proportion between the 
strengths of the different kinds would be liable to be different 
in the latter case. 

The work done by the Division of Forestry of the U. S. 
Dept. of Agriculture before 1898 was mostly of this class, but 
little having been done with full-size pieces, and that with 
imperfect apparatus. 

The only record of Rodman's experiments available is a 
table of results in the " Ordnance Manual." These are lower, 
as a rule, than those obtained by the experimenters of the first 
or second class. This is to be accounted for by the fact that, 
while he did not experiment on full-size pieces, he used much 
larger pieces than those heretofore employed ; his specimens 
for transverse strength, many of which are still stored at the 
Watertown Arsenal, being 5f inches deep, 2$ inches thick, and 
5 feet span. 

The fourth class of tests are those which furnish reliable 
data for use in construction ; 'and we will proceed to a consid- 
eration of these, taking up (1) tension, (2) compression, (3) 
transverse strength, and (4) shearing along the grain. 



TENSION. 65 I 



TENSION. 

In all cases where the attempt has been made to experiment 
upon the tensile strength of timber, a great deal of difficulty 
has-been encountered in regard to the manner of holding the 
specimens. In all cases it has been found necessary to pro- 
vide them with shoulders, each shoulder being five or six times 
as long as the part of the specimen to be tested, and to bring 
upon these shoulders a powerful lateral pressure, to prevent 
the specimen from giving way by shearing along the grain, and 
pulling out from the shoulder, instead of tearing apart. 

The specimens tested have generally had a sectional area 
less than one square inch, and it seems almost impossible to 
provide the means of holding larger specimens. This being 
the case, it is plain, that, whenever timber is used as a tie-bar 
in construction (except in exceedingly rare and out-of-the- 
way cases), it will give way by some means other than direct 
tension ; i.e., either by the pulling-out of the bolts or fastenings, 
and the consequent shearing of the timber, or else by bending 
if there is a transverse stress upon the piece ; and, this being 
the case, these other resistances should be computed, instead 
of the direct tension. Hence, while the direct tensile strength 
of timber may be an interesting subject of experiment, it can 
serve hardly any purpose in construction ; and the conclusion 
follows, that the resistances of timber to breaking we may 
expect to meet in practice are its crushing, transverse, and 
shearing strength. Indeed, the use of timber for a tie-bar 
should be avoided whenever it is possible to do so ; and, when 
it is used, the calculations for its strength should be based 
upon the pulling-out of the fastenings, the shearing or splitting 
of the wood, etc., and not on the tensile resistance of the solid 
piece. 

Moreover, when a wooden tie-bar is subjected not merely 
to direct tension, but also to a bending-moment, whether the 
latter is caused by a transverse load, or by an eccentric pull, as 
it generally is in the case of timber joints, we must compute 



652 APPLIED MECHANICS. 

the greatest tension per square inch at the outside fibre due to 
the bending, and to that add the direct tension per square inch: 
and this sum must be less than the modulus of rupture if the 
piece is not to give way; i.e., the modulus of rupture and not 
the ultimate tensile strength per square inch must be our criterion 
of breaking in such a case, the working- strength per square inch 
being the modulus of rupture divided by a suitable factor of safety. 

COMPRESSIVE STRENGTH. 

Tests of the compressive strength of full-size wooden columns, 
with the exception of one set of tests, date from about 1880. 

TESTS OF FULL-SIZE COLUMNS. 

The following are references to tests of full-size timber columns: 

i. Trautwine, in his "Handbook," speaks of some tests of 
wooden pillars 20 feet long and 13 inches square, made by David 
Kirkaldy, which, as he says, gave results agreeing with Mr. C. 
Shaler Smith's rule. 

2. A series of tests made at the Watertown Arsenal for the 
Boston Manufacturers' Mutual Fire Insurance Company, under 
the direction of the author. 

3. Eleven tests of old spruce pillars made at the Watertown 
Arsenal, for the Jackson Company, under the direction of Mr. 
J. R. Freeman, and reported in the Journal of the Assoc. Eng. 
Societies for November, 1889. 

4. The tests that have been made at the Watertown Arsenal 
on the government testing-machine. 

5. Tests made in the Laboratory of Applied Mechanics of 
the Massachusetts Institute of Technology. 

6. A series of tests of full-size columns of oak and fir, made 
by P. S. Girard in 1798. 

In regard to the first, no details or results are given: hence 
nothing will be said about them. 

In regard to the second, a summary only will be presented 
here. 



TESTS OF YELLOW-PINE POSTS AND BLOCKS. 653 



cq S ^ 5 <n co eo 

" -4-> 'O "D t3 'u "o 

c,acc; i *'"ccGcccro.Crt4> c c c c 

*j^>**jtc*j^+j*j*j*JDx:73-w4)c: *j ** -M .w 

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PQ PQ P3 



;as ;si -jas pz -}3s pz 



654 



APPLIED MECHANICS. 



H 



S i 



t4 a 
< t 





,2 JS *4 ^ 

C G C C 


|| 

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*> c c c 




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vo oo vo oo 

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00 ON O 
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TJ- vO to 





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CO to CO 
N to ON 
00 N VO 


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c < "" - S 










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3*8.5 


vO vO vO VO 


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Q $:s 


d d o\ 06 


1 1 1 


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r^ 


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TESTS OF OLD AND SEASONED WHITE-OAK POSTS. 655 



8 

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rt r* n i-* rt *-" X ^C , i ~i <-* -4- 

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jad'-sqjui'Ajpp 
-sei3 jo snjnpop^ 



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rf ro >^> *o 

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1-1 N 1-1 N 



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^ 5. ^^^^ = 



saqoui arenbs 
ui ' 



asa 



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gS ii q rj-r^Ntv-HHvq o^ro LOCO q 

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N M w oooocooor^co VOVOVOVON 



ssqoui ui 

'3JCT) JO J3J3UIEIQ 



lOQO !O o N O OO^OOO O L oOOQn 

qvONq ON qv qv q q\q>ON q\cq r^. ON ON q\ ONVO 



ssqom 
ui 'pug; 3JBq 

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cooo r^t^oq f> ^. \ \ t \ \ i ill 

3 VO VO VO 



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. 



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ft CO 



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CT\ONON O ON ON ON < OG O Q UI U_ 



656 



APPLIED MECHANICS. 



In all the experiments enumerated in the tables given 
above, the columns gave way by direct crushing, and hence 
the strength of columns of these ratios of length to diameter 
can properly be found by multiplying the crushing-strength per 
square inch of the wood by the area of the section in square 
inches. 

This conclusion is deduced from the fact that the deflections 
were measured in every case, and found to be so small as not to 
exert any appreciable effect. 

In regard to other tests of this same set, there were eight 
tests made, in addition to those already enumerated ; and in 
five the loads were off centre. A summary of the results is 
appended, together with a comparison of their actual strength 
with that which would be computed on the basis of 4400 per 
square inch for yellow pine, and 3000 for oak. The first three 
tests were made on yellow-pine columns, and the last two on 
oak. 





Weight, 
in Ibs. 


Length, in 
feet and 
inches. 


Diameter 
of 
Column. 


Diam- 
eter of 
Core. 


Sectional 
Area, in 
square 
inches. 


Eccen- 
tricity, 
in 
inches. 


Ultimate 
Strength. 


Computed 
Ultimate 
Strength. 






ft. in. 














2, 2d series 


3 20 


II 11.27 


9.92 


i-53 


7545 


2-33 


265000 


331980 


5, 3d series 


298 


12 6.8 


( 8. 3 0) 

\ X \ 
I 7-60 ) 


- 


63-1 


2.07 


240000 


277640 


i, 3d series 


386 


12 9.3 


( 8.75) 

\ X ( 

( 8.92) 


- 


76.04 


2.25 


280000 


334576 


i, 2d series 


451 


II II.4 


10.95 


i. 80 


92.16 


2-75 


170000 


276480 


3, 2d series 


236 


II II. 2 


8.2 


i-55 


50.92 


1.91 


100000 


152760 



These results exhibit a great falling-off. of strength due to 
the eccentricity of the load ; and emphasizes the importance 
of taking into account eccentric loading in our calculations in 
a manner similar to that already mentioned on pages 370, 371, 
and 448. 



STRENGTH OF TIMBER. 6$/ 

The remaining experiments were : (i) Two tests of white- 
wood columns, average strength 3000 pounds per sq. in., and 
very brittle. (2) One yellow-pine square column (sectional area 
68.8 sq. in., length 12' 6".85) with one end resting against a 
thick yellow-pine bolster. 

The maximum load was 120,000 Ibs. = 1744 Ibs. per sq. 
in., the post beginning to split due to eccentricity of bearing 
caused by uneven yielding of the bolster. The bolster was 
then removed, the post cut off i^ in. at the end and tested 
without the bolster. Ultimate strength 375,000 Ibs. = 5451 
Ibs. per sq. in. 

The table of results of the tests on old and seasoned oak 
columns were made upon columns that had been in use for a 
number of years in different mills, from which they were re- 
moved, and replaced by new ones. Ten of them had been in 
use about twenty-five years, and the remainder for shorter 
periods. An inspection of this table will, I think, convince the 
reader that it would not be safe to calculate upon a higher 
breaking-strength per square inch in these than in the green 
ones. 

TESTS FOR THE JACKSON COMPANY. 

Eleven tests of old spruce pillars, which had been in use 
in a cotton-mill of the Jackson Company, were tested on the 
government machine at Watertown, under the direction of 
Mr. J. R. Freeman. The manner of making them was as fol- 
lows: 

In the first two the ends were brought to an even bear- 
ing. 

In the third the ends came to an even bearing under a load 
of 60000 pounds. 

In the fourth, fifth, ninth, tenth, and eleventh, the cap, 
and also the base-plate, were planed off on the back to a slope 
of i in 24, and placed with their inclinations opposite. 

In the eighth they had their inclinations the same way one 
as the other. 



658 



APPLIED MECHANICS. 



In the sixth and seventh the base-plate was not used, the 
larger end of the post having a full bearing on the platform of 
the machine. 

The results are given in the following table : 







Diameter 


Diameter 






Ultimate 




Length 
in Feet and 
Inches. 


at 
Small End, 
Inches. 


at 
Large End, 
Inches. 


Area at 
Small End, 
Sq. In. 


Ultimate 
Strength, 
Lbs. 


Strength per 
Sq. In., 
Lbs. 




ft. in. 












I 


10 4-75 


5.82 


7.78 


31.87 


142000 


4088 


2 


10 4. 


5-85 


7-49 


27.15 


192800 


6225 


3 


10 5-75 


5.85 


7-74 


32.17 


166100 


4900 


4 


10 5-5 


5-70 


7-77 


31.67 


108200 




5 


10 5.1 


5-70 


7.70 


30.78 


105000 




6 


TO 5.2 


5.80 


7.61 


30.39 


168000 




7 


9 7 


5-74 


7-81 


32.88 


194100 




8 


10 5.4 


5-85 


7.90 


34-21 


T55000 




9 


10 4.9 


5-82 


7-77 


40.72 


96100 




10 


10 5-13 


5-73 


7.78 




125000 




ii 10 4.38 


5-74 


7.81 




60000 





All but the first three of the tests were made with inclined 
bearings of one kind or another, hence the ultimate strength 
per square inch is only given here for the first three ; which, as 
Mr. Freeman says, were of " well-seasoned spruce, of excellent 
quality." Hence the average crushing-strength of spruce is 
doubtless considerably lower than the average of these three. 



TESTS MADE ON THE GOVERNMENT MACHINE. 

In Executive Document 12, 47th Congress, first session, 
will be found a series of tests of white and yellow pine posts 
made at the Watertown Arsenal ; and these tests probably fur- 
nish us the best information that we possess in regard to the 
strength of wooden columns. 

The summary of results is appended : 



COMPRESSION OF WHITE-PINE POSTS. 



659 



If 



1 i 

' 



S - 5 



Q 






^ 

'"". d c> c> 

crrorOfO 



OOOO U "> L '-> 

i-Hr^r^cxDMw 
d CNONcKd c5 

fONNNCOrO 



Ooooo M r^ to ON w to vo O N t^oo ro to oo to vo r~> 

v c*'^"'^" L 9'^" T ?" T ?"'^"'^" T ?' ij r' v c >T ?"'^" f r 5r P f / >r 7 > ^ i 



Ooooo O r^OOOOO tovo Ooo r--oo rotoo moo a 

^^^^iOTj-tO-^--^--^-Tj-to-^-Ti--^-fOfO^-fON Cl 



^r 



o o cJ 

O O O 



s- m^ 

odd 



d d 
8 



< O 



1 1 1 1 1 1 1 1 1 1 1 1 1 



O voo tovoo 



o to q q ? o q 



d<~ to 



S ox ax 



O\OvO r^.OO ro-^-toi-i N ro-^- 



660 



APPLIED MECHANICS. 






o 5 5 5 
- o 

2 5 " 

sr:: $^2 * 



s < 



'J5 . ** 

S O 'S O 3 3 - 

W) 5 hO 

'S "53 'S 

S S 3 S S S 3 



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s 



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ro i^* ^ ON 



M M M M M 



r^ N >S O H 
M M i- ro M 



vo 



ION N 



OO 
N 



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fO\O NOMD 



5 S 



il 

M M 



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vo 



OQQQOOOOOQQQO 

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M M M r^tv.r^t^r>.t^t^t^^O t^ 



M w N M M roforovo t^vOMD r^r^t^t-^M 



3, 



M M M rorotOt-^r^ 



dvdvdvd d d d 



oo O r^O f 

rO'^J - fOO | - | i-i p -i'-it-i 

vdvdvdododod d d d ^^ 



MMNMMNNNN 



O O O 



a, 

W 
, 



| 1 1 1 1 



qq 

mO 



qo 

rood 



qqoqooooiooooqooo 

r^tor^ONfOt^rJ-fodvdvO 

M N N\O r^'OONONWVOvOV 



COMPRESSION OF WHITE-PINE POSTS. 



66 1 



-d 
1 



. 



11 
fl 



I|I|.S 

-S .S g jg 



s 



rovOvO w r^N 






OOOOOQOOOOOOOQ 

^NiooqqqNWLOi-cwNqwvq 
r^t^r^^O r^.'Ooooooo O O O roro 



oo "nM 
. iot^vO 



w OMD -*r^oo O O -<trtQ 
^o lOfOroro^ovovOvOvOvO 



loinvnvdvdvdooodod 



1-1 



& w 4 

l$*i 



^qqqqqqqqqqqqqqqqqqq 
O ^foo ^O^ON i^Q r^r^^O t^t^.vO'-ovO r-^r^tx 
M r^r^oo O\"^"^^-N ONOO OvOvO o 
N N N N N rofOrO-^-fOrorON M fJ 



i^-OO N fOTj- 
rofOO O O 



O OO O 
tv.t^rxi>. 



662 



APPLIED MECHANICS. 



COMPRESSION OF WHITE PINE. SINGLE STICKS AND BUILT POSTS. 
In the multiple ones, dimensions of each stick are given. 



No. 
of 
Test. 


Weight. 


*5 

1 

Si* 

22& 
O.S 
<j 


Dimensions of Post. 


Sectional" 
Area. 


S |R 

.2 ^ 
1" 8- 

i*r* 

M II in 


Ultimate Strength. 


i 
j 


1 


s 

a 
Q 


a 

3 

S 

< 


^ 
fi 




Ibs. 




in. 


in. 


in. 


sq. in. 


in. 


Ibs. 




664 


i53 


ii 


i77'5o 


4.48 


11.65 


52.2 


0-0545 


IIOOOO 


2107 


665 


i43 


IO 


180.00 


4.48 


11.64 


52-1 


O.IOIO 


81500 


^64 


666 


163 


5 


179.97 


4-47 


11.63 


52.0 


0.0895 


70000 


1346 


667 


228 


13 


180.00 


5-40 


11.30 


61.0 


0.0505 


160000 


2623 


668 


i93 


5 


179-93 


5-6i 


n-73 


6 5 .8 


0.0622 


156300 


2375 


669 


253 


5 


180.00 


5-64 


11.76 


66. 3 


0.0608 


152300 


2297 


638 


ISI 


6 
8 


180.00 
180.00 


4-50 
4-5 


n. 60 
"59 


?::(- 


( 0.0670 1 
( 0.0750 i 


200000 


I9l6 


639 


IS}". 


7 
5 


180.00 
180.00 


4-52 
4.49 


11.66 
11.62 


?:;K 


i 'A 45 i 

/ 0.0670 ) 


212000 


2021 


640 


131- 


7 
5 


180.00 
180.00 


4-53 
4-52 


"59 
"59 


?!>- 


i 0.1060 | 
1 0.0955 } 


149000 


1419 


642 


ISSl* 


6 
7 


179.98 
179.98 


5-57 
5-58 


ii. 61 
n. 61 


&K 


( 0.0770) 
( 0.0390 j 


215000 


1661 


643 


IS!*- 


{'S 


179.92 
179.92 


5-65 
5-6i 


ii. 61 
11.62 


&SI-3- 


( 0.0440 | 
( 0.0600 ) 


261000 


J 995 


644 


i3!* 


i; 


179.96 
179.96 


5.60 
5-60 


11.63 
11.62 


&I-3-- 


| 0.0596 | 
I 0.0690 ) 


257800 


1980 


648 


l=Si* 


\- 


180.00 
180.00 


5.60 
5-6i 


11.72 
11.72 


SSI-*.. 


| 0.0590 | 
( 0.0700 J 


268000 


2042 


649 


l:Ui' 


9 
4 


180.00 
180.00 


5.60 
5-61 


11.71 
11.74 


:$!*> 


( 0.0600 | 
J 0.0705 \ 


277000 


2107 


650 


lisi* 


12 

9 


180.00 
180.00 


5-6i 
5-61 


"75 
11.71 


:?!"' 


\ 0-0530 | 
} 0.0885 i 


24OOOO 


1824 


645 


laf* 


5 
7 


179.97 
179.97 


5-59 
S-6o 


"59 
ii. 60 


:;!- 


| 0.0560 | 
/ 0.0540 i 


263200 


2028 


646 


!! 


6 

12 


180.00 
180.00 


5-59 
5.60 


IT. 61 

11.62 


&?i'3"> 


0.0493 | 
0.0620 5 


249000 


i9 J 5 


647 


151* 


" 

1*3 


180.00 
180.00 


5.62 
5-62 


11.62 
11.62 


:!}*" 


0.0630 

/ 0.0700 ) 


248000 


1899 


678 


IS!* 6 


16 
8 


179.94 
179.94 


5.58 
5-57 


11.47 
"45 


tll""* 


0.0529) 
0.0642 ) 


245500 


1921 


679 


I3I 


11 


180.00 
180.00 


5.62 
5.62 


11.76 
11.72 


fi.l\ 


0.0664 | 
0.0705 \ 


249000 


1886 


680 


IS!* 


ii 


180.00 
180.00 


5.60 
S-6i 


11.72 
"73 


g:S!-3- 


0.0650 1 

0.0495 $ 


278000 


2116 


663 


|3| 


it 


180.00 
180.00 


5.60 
S-63 


"75 
"75 


S:S!-3- 


0.0621 ) 
0.0657 i 


300000 


2273 


676 


Isil* 


( 12 
< ^ 


179.94 
179.94 


5.60 
5-6i 


11.71 
"73 


g:t|'3..4 


\ 0.0530 1 
1 0.0593 i 


274500 


2089 


677 


IS!** 


i ^ 

i 6 


180.00 
180.00 


5-6i 
5-68 


11.72 
11.72 


:Jl 


j 0.0551 1 
/ 0.0625 j 


255000 


<945 



COMPRESSION OF WHITE PINE. 



663 



COMPRESSION OF WHITE PINE.- Concluded. 
SINGLE STICKS AND BUILT POSTS. 



No. 
of 
Test. 


Weight. 


_c 

22 So 
w^ c 


Dimensions of Post. 


Sectional 
Area. 


"ft 

c3 


Ultimate Strength. 


M 


| 


Hi 

Q 


Actual. 







Ibs. 




in. 


in. 


in. 


sq. in. 


in. 


Ibs. 






(175 


(18 


180.00 


4-52 


11.62 


52-5) 


I 0.0460 ) 






690 


< 226 600 





180.00 


5-56 


ii .70 


65.0} 169.3 


I o.o58o[ 


310000 


1831 




' T 99 


(18 


180.00 


4.46 


11.62 


51-8) 


( 0.0480 ) 


% 






(164 


( 9 


179.98 


4-48 


ii .60 


52.0) 


( 0.0526 ) 






691 


< 197 520 


> I4 


179.98 


5-56 


1 1. 60 


64.5 J 168.2 


] 0.0430 } 


372500 


2215 




(i59 


( 12 


179.98 


4-45 


ii .61 


51-7) 


( 0.0390 ) 








(248 


(13 


i77- 2 5 


5-62 


1 1. 60 


65-2) 


( 0.0580 ) 






692 


{ 153 6l 4 


12 




4-50 


ii. 60 


52. 2 > l82.2 


I 0.0641 \ 


363000 


1992 




(213 


( 5 


177-25 


5-6o 


ii-57 


6 4 .8) 


( 0.0768 ) 








( I 5 I 


(ii 


180.00 


4-5 


ii. 60 


S 2.2) 


( 0.0460 ) 






687 


J2i8 536 


8 


180.00 


5-58 


ii .62 


6 4 .8 169.4 


o.o 5 8 7 [ 


325500 


1919 




( 167 


( 9 


180.00 


4-52 


11-59 


52.4) 


(0.0533) 








(176 


(10 


180.00 


4-5 


ii .60 


52-2) 


( 0.0565 ) 






688 


} 236 575 


10 


180.00 


4.62 


1 1. 60 


65.2 > 169.6 


j 0.0645 J 


306000 


1804 




(163) 


(ii 


180.00 


4-50 


1 1. 60 


52.2) 


( 0.0703 ) 








(188 


( 7 


180.05 


4-48 


n. 60 


52.0) 


{ 0.0510 ) 






689 


{ 203 550 


11 


180.05 


5-6o 


11.62 


65.1 > 168.7 


o.o66oj 


340000 


2015 




(i59 


I 9 


180.05 


4.46 


"57 


51-6) 


( 0.0789 ) 








fi93l 


r s 


179-95 


4-47 


ii. 60 


5I-91 


{0.0700"! 






681 


38 * 

U57J 


1 

I 9 


179-95 
179-95 
179-95 


4-52 
4.48 
4-51 


11.65 
11.65 
11.65 


52-sJ 


0.0714 1 
0-0531 [ 
0.0762 J 


362000 


1734 




fi6i-| 


f 7 


180.00 


4-5 


11.63 


52-31 


{0.0612") 






682 


!:r 


14 
\l 


180.00 
180.00 
180.00 


4-5 
4-49 
4.50 


ii .64 
ii. 60 
11.63 


S'sJ 


0.0546 1 
0.0542 f 
0.0916 J 


414000 


1980 




f 169") 


fio 


180.03 


4.46 


11.63 




{0.0570^ 






683 


\l\l\rt 


J" 

1 9 


18