 ^^^
^
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 textbook ; 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 fullsize 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 fullsize 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.
ROOFTRUSSES **,* 138
CHAPTER IV.
BBIDGETRUSSES 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
firstnamed 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 firstnamed
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 abovestated 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 abovestated 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 abovestated
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 abovestated 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 thirtytwo 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 arrowhead
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(0a) = sin0.
R
EXAMPLES.
. Given F = 4734,
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 steamengine
at the moment when it is in the position shown in the figure is AB =
1000 Ibs. The resistance of the
guides upon the crosshead DE is
vertical. Determine the force acting
along the connectingrod AC and
the pressure on the guides ; also
resolve the force acting along the connectingrod 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
093358
o.35 8 37
43.87826
16.84339
73
4 8
0.66913
o743i5
48.84649
5424995
43
82
O.I39I7
0.99027
59843 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 rightangled 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 coordinate 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 semivertical angle is
AOXBOX = 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 semivertical 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
9403275
2/^cos 3
3498370
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 =9584257 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.
43
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 abovestated 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 coordinates
of their lines of direction being as
indicated in the figure, if we denote
by R the resultant, and by X Q the
coordinate 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 coordinates
of their lines of direction by (* 7,), (x 2J jj> 2 ), etc., and if we
denote their resultant by R, and the coordinates 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 coordinates of its line of direction, and calling F H the
38 APPLIED MECHANICS.
remaining force, and (x w y^ the coordinates 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 Coordinates 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 coordinates 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 coordinates of the centre. Then turn
the forces parallel to OX, and determine the line of action of
the resultant. We shall have for its coordinates
y* =
Hence, for the coordinates of the centre of the system, we
have
y =
When 2F = o the coordinates 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 coordi
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 coordinates 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 abovestated 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 coordinates 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 onehalf 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 3lb. force, and acting
also for onehalf 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 onethird 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 NonUniform Section and
Material. In the case of a straight rod of nonuniform 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
NonUniform 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 (*xa
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 coordinate 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 righthanded and the other lefthanded
If we assume the origin of coordinates 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 coordinates 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 (lefthanded in the figure), and
let F t at d and F^ at e represent the
other (righthanded 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 righthanded or both lefthanded 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 righthanded ; the second, third, and
fourth are lefthanded. 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
(righthanded or lefthanded) 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 arrowhead, or some other means, in what direction one
must look along the line in order that the rotation may appear
righthanded.
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 righthanded, and
Fig. 37 when it is lefthanded.
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
coordinates 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 (lefthanded 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 lefthanded 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 coordinates 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 abovestated couples, we
RSF
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 coordinates 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 coordi
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 righthanded, 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 han<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 coordinates 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 coordinates 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 threelegged table
are the vertices of an isosceles rightangled 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
onefourth 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 footpound.
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 lopound 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 footpounds, and it would be necessary
to do upon it 50 footpounds 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 horsepower,
which in English units is equal to 33000 footpounds per
minute, or 550 footpounds 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 lopound weight on one side and a 5pound
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 coefficients, 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 coordinates 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 lopound 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 drivingwheels 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 sealevel 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 (coordinates 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 coordinates 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 startingpoint, 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 lopound 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 lopound 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 rightangled 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 loopound 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 = Jf / 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 coordinates 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 footlbs.
(l>) Up grade, 2719200 + 6ooo ^ x 5 = 5219200 footlbs.
(c) Down grade, 2719200  6ooo X J x 5 = 219200 footlbs.
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 crosshead
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 connectingrod connecting
the drivewheel shaft of a stationary engine with the pistonrod,
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 lefthanded 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 midstroke) ; and its value is then
v = ar,
this being also the velocity of the crankpin at midstroke.
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 crosshead 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 footlbs.
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
ar.
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 coordi
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
(semiaxes 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 coordi
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 coefficient
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 coordinates 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 coordinates 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 (semiaxes 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
semiaxes 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 semiaxes 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 Reaction. One funda
mental principle that holds in all cases of central impact is the
equality of action and reaction ; 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,
COEFFICIEA 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. Coefficient 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 coefficient of restitution, then we shall so define it that
v v l
c* v v c 2
or, in words, the coefficient 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\vL + ^2^2 == m^ci 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 coefficient 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.
ROOFTRUSSES.
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 bendingaction,
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 reactions
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 loadline 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 rooftrusses, 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 righthanded or always in lefthanded rotation, we shall be
led to the simplest diagrams. Hereafter this notation will be
used exclusively in determining the stresses in rooftrusses.
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 lefthand support, and
proceed again in righthanded
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 lefthand 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, onehalf is
FIG. 77.
POLYGONAL FRAME.
the component at the support, and onehalf 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 lefthand 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 righthand 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 lefthand 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
righthand 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 lefthand 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 lefthand support and at the apex.
FIG. 81.
TRIANGULAR TRUSS: WIND PRESSURE.
149
Now proceed to the lefthand 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
righthand 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 rooftruss, 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, jackrafters, 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 ROOFTRUSSES.
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 . . .
\ " " " iinch boards .
475
6.75
Hemlock
2 C
JL u jl
730
Maple
^j
41
^ " " " laths . . .
700
Oak, live
CQ
& " " " iinch boards,
9OO
Oak white
4Q
& " " " 'i " "
955
Pine, white **
'Vy
2 C to "3O
i " " laths . . .
9 2 5
Pine, yellow, Southern . .
Spruce ... . . . .
45
2 C to 7O
i " " " iinch boards,
JL << TJL
11.25
1 1. 80
IRON.
With slatingfelt 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 pressureplates, 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
510
50
38.64
38.10
10
1359
9.60
55
3921
3940
15
19.32
14.20
60
3974
40.00
20
24.24
18.40
65
3982
40.00
25
28.77
22.6O
70
3991
40.00
30
32.00
26.50
75
39 9 6
40.00
35
3452
30.10
80
40.00
40.00
40
36.40
3330
85
40.00
40.00
45
3773
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 crosscurrents 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 pressureplate,
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 pressureplates 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 deadweight 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 indicatorneedle, 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 pressuregauges, 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 righthand 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 pressureboards 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 fastmoving 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 crosssection 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 roofweight 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 tiebeam . . . 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 bendingstress 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 crosssection in the figure. On these
purlins are supported the jackrafters parallel to the rafters, and
at sufficiently frequent intervals to support suitably the plank
and superincumbent roofingmaterials.
By this means we secure that the entire bendingstress comes
upon the jackrafters 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 bendingstress 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
ingstress 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 rooftrusses the third method.
In the case of bridgetrusses, on the other hand, I believe
the graphical not to be as convenient as a purely analytic
method.
135. RoofTrusses. 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.
ROOFTRUSSES.
I6 7
Now proceed to either support, say, the lefthand 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
ROOFTRUSSES.
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 lefthand end of the truss,
and none under the righthand end ; and we will proceed to
determine the stresses due to wind pressure.
First, suppose the wind to blow from the lefthand 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 lefthand rafter at its middle point, as shown
in Fig. 85.
The lefthand supporting force will be vertical : hence, pro
ducing the abovedescribed dotted line, and a vertical through
the roller to their intersection, and joining this point with the
righthand 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 righthand supporting
force coincides in direction with the righthand 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 lefthand 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,
ROOFTRUSSES 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. RoofTruss 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 righthanded 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 rooftrusses, 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
ROOFTRUSSES 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 jackrafters, 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 tierod 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.
HAMMERBEAM TRUSS. 177
When such a thrust is furnished (or were there a tierod),
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 bendingstress 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 tierod, 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.
SCISSORBEAM TRUSS.
139. HammerBeam Truss: Wind Pressure. Fig. 95
shows the stress diagram of the hammerbeam 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. ScissorBeam Truss. We have already discussed
two forms of scissorbeam truss
in Figs. 90 and 91. These
trusses having the right number
of parts, their diagrams present
no difficulty. Another form of A
the scissorbeam 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 lefthand support, and obtain
mabkgm ;
and so continue.
141. ScissorBeam Truss without Horizontal Tie.
Very often the scissorbeam 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
SCISSORBEAM TRUSS WITHOUT HORIZONTAL TIE. l8l
made in Fig. 97^, where abcdefga is the polygon of external
forces, gabkg the polygon of stresses for the lefthand 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 lefthand 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 lefthand 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)
SCISSORBEAM 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.
BRIDGETRUSSES.
143. Method of Sections. It is perfectly possible to
determine the stresses in the members of a bridgetruss
graphically, or by any methods that are used for rooftrusses.
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 bridgetruss (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 68 and 7~9
and the diagonal 78, 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., 
SHEARINGFORCE AND BENDINGMOMENT. .185
i. The algebraic sum of the vertical components of the
abovementioned 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. ShearingForce and BendingMoment. Assum
ing all the loads and supporting forces to be vertical, we shall
have the following as definitions.
The ShearingForce 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 BendingMoment 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 ShearingForce and BendingMoment.
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
shearingforce.
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 bendingmoment 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 BridgeTrusses. Figs. 1091 12 rep
resent two common kinds of bridgetrusses : 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 halflattice 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 BridgeTruss 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 BRIDGETRUSS. 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 shearingforce at the
section.
4. Find the bendingmoment 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 shearingforce.
(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 bendingmoment at the section. Hence the stress
in the horizontal will be found by dividing the bendingmoment
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 semigirder 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 810 and 910.
Assume a vertical section very near the joint 9, but to the
right of it, so that it shall cut both 810 and 910.
If, now, the truss were actually separated into two parts at
this section, the righthand 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 911 on the
part x\\ of the same bar.
2. The thrust exerted by the part 82 of the bar 810 on
the part ^10 of the same bar.
3. The pull exerted by the part 97 of the bar 910 on the
part yio of the same bar.
The shearingforce 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 910 = = 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 97 of the diagonal 910 must pull the part yio of
the same diagonal.
Next take moments about 9 : and, since the moment of the
stresses in 91 1 and 910 about 9 is zero, we must have that the
moment of the stress in 810; i.e., the product of this stress
by the height of the girder, must equal the bendingmoment.
DETERMINING THE STRESSES 2N A BRIDGETRUSS. 189
The bendingmoment about 9 is
8000 x 5 4 4000 x 10 = 80000 footlbs.
80000
Hence
Stress in 810
433
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 shearingforces by multiplying
by 1.1547, and the chord stresses from the bendingmoments
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.
footlbs.
* 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 BRIDGETRUSS,
Numbers of Diagonals.
Stresses
in Diagonals, in Ibs.
I 2
2829
70000 X
II547 =
80829
2 3
2728
+ 60000 X
I.T547 =
+ 69282
3 4
2627
60000 X
I.I547 =
69282
4 5
2526
+ 50000 X
I.I547 =
+ 57735
56
2425
50000 X
LI547 =
57735
6 7
2324
+40000 X
I.I547 =
+46188
7 8
2223
40000 X
LI547 =
46188
8 9
2122
+ 30000 X
LI547 =
+ 34641
910
2O2I
30000 X
LI547 =
34641
IOII
I92O
+ 20000 X
LI547 =
+ 23094
1112
1819
20000 X
I.I547 =
23094
1213
I7I8
+ IOOOO X
LI547 =
+ II547
I3H
1617
i oooo x
I.I547 =
H547
1415
I5I6
+
LOWER CHORDS.
IM umbers of Chords.
Stresses in Chords, in Ibs.
2 4
2628
65OOOO
X 0.11547 =
 755 6
4 6
2426
I2OOOOO
X 0.11547 =
138564
6 8
2224
I65OOOO
X 0.11547 =
190526
810
2022
2OOOOOO
X 0.11547 =
230940
1012
I 820
225OOOO
X 0.11547 =
259808
1214
1618
245OOOO
X 0.11547 =
277128
I4l6
245OOOO
X 0.11547 =
282902
1 9 2
APPLIED MECHANICS.
UPPER CHORDS.
Numbers of Chords.
Stresses in Chords, in Ibs.
' 3
2729
350000
X 0.11547 =
+ 40415
3 5
2527
950000
x 0.11547 =
+ 109697
5 7
2325
I45OOOO
X 0.11547 =
+ 167432
7 9
2123
1850000
X 0.11547 =
+ 213620
91 1
1921
2I5OOOO
X 0.11547 =
+248261
ii 13
1719
2350000
X 0.11547 =
+ 267355
'j '5
i5i7
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 semigirder, 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.
TRAVELLINGLOAD.
148. HalfLattice Girder: TravellingLoad. 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 travellingload, 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 SemiGirder. Wherever the
section be assumed in a semigirder, it is evident that any load
placed on the truss at any point between the section and the
free end increases both the shearingforce 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 shearingforce or the bendingmoment at that section.
Hence every member of a semigirder 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
bendingmoment 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 78 (Fig. 109), through which we take
our section ab. Now it is evident that any load placed on the
truss between ab and the lefthand (nearer) support will cause a
shearingforce 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 78 ; while any load between the section and the right
194 APPLIED MECHANICS.
hand (farther) support will cause a shearingforce of the oppo
site kind, and hence a tension in the bar 78.
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 78 is in tension.
Hence, any load placed upon the truss between the section
and the farther support tends to increase the shearingforce 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
ingforce at the section due to the dead load, or to produce a
shearingforce 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 BRIDGETRUSS,
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 = sSooofJ 133000.
Section at
BendingMoment, in footlbs.
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.
i3
I5I7
665000 X 0.11547
= + 76788
35
I3I5
1805000 X 0.11547
= +208423
57
1113
2565000 X 0.11547
= +296181
79
91 1
2945000 X 0.11547
= +340059
APPLIED MECHANICS.
Numbers of Chords.
Stresses
in Lower Chords.
2 4
1416
1330000 X
O.II547 =
153575
4 6
1214
2280000 X
O.II547 =
263272
6 8
IOI2
2850000 X
O.II547 =
329090
810
3040000 X
O.II547 =
351029
Next, as to the diagonals, take, for instance, the diagonal
78. When the dead load alone is on the bridge, the diagonal
78 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
shearingforce of (38000) = 9500 Ibs. at any section to the
10
left of 13; and this shearingforce 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 shearingforce of (38000) = 28500 Ibs.
16
at any section to the right of 13 ; and this shearingforce 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 78 causing tension in this bar ; the magnitude
of this shearingforce 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 ShearingForces producing Stresses of Same Kind as
Diagonals of
Diagonals.
Dead Load.
as those due
to Dead
Load.
12
1716
3^ (2 + 4+6+8+IO+I2+I4) = 133000
'53575
23
1615
^(2+4+6+8+10+12+14) = I33 ooo
+ '53575
34
'5H
3 ^(2+ 4 +6+8+io+ I2 )~(2) = 98750
114027
45
1413
^(2+4+6+8+10+12) ^(2) = 98750
+ i 14027
56
1312
^(2+4+6+8+10) ^(2+4) = 68250
 78808
67
I 21 1
^(2+4+6+8 + 10) ^(2+4)  68250
+ 78808
78
IIIO
^(2+4+6+8) ^(2+4+6) = 41500
 47920
89
io 9
^(2+4+6+8)  ^(2+4+6)  41500
+ 479 20
Greatest
Stresses in
Numbers of
Greatest ShearingForces producing Stresses of Kind Opposite
Diagonals ^pf
Diagonals.
from Dead Load.
Kind Oppo
site from
Dead Load.
89
io 9
^(2+ 4 +6)  ^(2+4+6+8)  18500
21362
78
IIIO
^(2+4+6)  ?^( 2 + 4 +6+8)  18500
+ 21362
The diagonals 78, 89, 910, and 1011 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
bridgetruss 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. BridgeTrusses 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 32 must be equal and opposite to
the vertical component of the stress in diagonal 12, as these
stresses are the only vertical forces acting at joint 2.
Vertical 54 has for its stress the vertical component of the
stress in 34, etc. Thus
Stress in 32 = shearingforce in panel 13,
Stress in 54 = shearingforce in panel 35, etc.
TRUSSES WITH VERTICAL AND DIAGONAL BRACING. 199
On the other hand, if the loads be applied at the lower
joints, then
Stress in 32 = shearingforce in panel 35,
Stress in 54 = shearingforce in panel 57, 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.
BendingMoment 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 bendingmoment 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
2728
+ 227500
2 4
2426
227500
3 5
2527
4420000
4 6
2224
420000
5 7
2 3 2 5
+ 5775
6 8
2022
5775
7 9
2123
+ 700000
810
1820
700000
91 1
1921
+ 787500
IOI2
1 61 8
787500
1113
1719
+ 840000
1214
1416
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 shearingforces 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 1 112,
1314, 1417, and 1619 would sometimes be called upon to
bear a thrust, and the verticals 1213 and 1716 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 1013, 1215, I 5~ I 6,
and 1718 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
2826
^ I+2 + 3 + ... +I3 )
= 227500
3 4
2724
3J^ (l + 2 + 3+ ... + I2) _I5^( l)
= I94H3
56
2522
3 ^P(l + 2 + 3+ ... + II) '^(1 + 2)
= 162429
78
2320
^(i + 2+3+ . . . + 10)  x ^?(i + 2+3)
J4 J4
= 132357
910
2II8
^(1 + 2+3+ ...+ 9)^p(i + 2+..
+ 4) = 103929
1112
1916
33222(1 + 2+3+...+ 8)^(i + 2+..
+ 5)= 77H3
I3T4
1714
3^ (l + 2 + 3 +...+ 7) _^ I + 2+ ..
+ 6)= 52000
Diagonals.
Greatest ShearingForces of the Opposite Kind to those produced by
Dead Load.
314
1714
^ (I+2+3+ ... + 6) _^ I+2+ ... +7)
= 28500
1112
1916
22f(i + 2+ ... + 5 )_H5p ( r + 2 + ... + 8)
= 6643
910
2II8
^(, + 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 shearingforces 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
2826
321685
1512
1516
40299
3 4
2724
274518
I3IO
I7I8
 9393
56
2522
229675
7 8
2320
187153
910
2118
146956
1112
1916
109080
i3~ I 4
1714
 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 shearingforce 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
2726
+ 2275OO
5 4
2524
+ I94M3
7 6
2322
+ 162429
9 8
2I2O
+ 132357
IIIO
I9I8
+ 103929
1312
I7l6
+ 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 rooftrusses.
This method is shown in the figure (Fig.
1 1 6); floorbeams 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 shearingforce, 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 35 of the last example.
In determining its greatest shearingforce, we considered a load
of 35000 Ibs. per panel point to rest on all the joints from the
righthand 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 shearingforce, 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 35 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 shearingforce due to the moving load on
panel 35 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 coefficient
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 shearingforce in panel 35, and hence the
stress in diagonal 34, is a maximum.
Panels.
Portion of ShearingForce
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
2728
iv ( I 3 *A
IO.OO
11500
3
11500
i4\ J 20 y
3 5
5 7
2527
2325
I4\ 20 /
923
8.46
3
5
9797
8230
5
7
11432
11227
I4\ 20 /
7 9
2123
I4\ 20 /
7.69
7
6801
9
10886
91 1
1921
H( 9X ~ ^f?)
6.92
9
5507
n
10409
1113
1315
1719
1517
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 ShearingForce of Same Kind as that due to Dead Load.
3 5
5 7
7 9
1113
2728
2527
2325
2I2 3
1921
1719
1517
3500Q,
14 l
227500
= 193385
= 161038
101654
(i+...+S)= 49345
Hence, for stresses in main braces, we have
Diagonals.
Stresses.
I 2
2826
321685
3 4
2724
273446
5 6
2522
227708
7 8
2320
184472
910
2118
143739
1112
1916
105507
i3 J 4
1714
69774
Moreover, for the shearingforces 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.
1715
/6*  ^}
4.62
15
2455
13
8171
I4\ 20 /
1113
1917
E( SX  l^f]
3.84
13
1695
II
7137
I4\ 20 /
Panels.
Greatest ShearingForces of Opposite Kind from those due to Dead
Load.
135
1715
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 :
CounterBraces.
Stresses.
1512
1310
1516
I7l8
 36546
5820
And, for the verticals, we have the new, instead of the old,
shearingforces.
208
APPLIED MECHANICS.
The following table compares the results :
Diagonals.
Stress, Ordinary
Method.
Stress, New Method.
Difference.
I 2
2826
321685
321685
3 4
2724
274518
273446
1072
5 6
2522
229675
227708
1967
7 8
2320
187153
184472
268l
910
2118
146956
 J 43739
3217
1112
1916
109080
105507
3573
i3 J 4
1714
 73528
 69774
3754
1512
1516
 40299
36546
3753
1310
1718
 9393
5820
3573
Verticals.
Stress, Ordinary
Method.
Stress, New Method.
Difference.
3 2
2726
f227500
+ 227500
O
5 4
2524
+ I94H3
+ 193385
758
7 6
2322
4162429
+ 161038
I39 1
9 8
2120
+ T 3 2 357
+ 130461
1896
IIIO
I9I8
4103929
+ 101654
2275
1312
I7l6
+ 77H3
+ 74616
2527
i5 J 4
4' 28500
+ 49345
2655
156. Compound BridgeTrusses 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 bridgetrusses 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 latticegirder 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 13 (Fig. 117) is the
same as the stress in 15 (Fig. I ija).
The stress in 35 = stress in 15 (Fig.
1170) + stress in 37 (Fig. 117^).
The stress in 57 = stress in 59 (Fig.
117*7) f stress in 37 (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 latticegirder
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
travellingload 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 63 and 32, and with the sup
port at 12 by means of the diagonals 67, 710, lon, and the
vertical 1112, 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 travellingload is on the bridge,
it is perfectly possible to divide it into the two trusses shown
in Figs. II&T and n&/, the diagonals 45, 710, 67, and 58
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 ShearingForce
of One Kind.
Greatest ShearingForce
of Opposite Kind.
Corresponding
Stresses.
Fig.
n8a.
Fig.
118*.
2 3
129
~~~(3 + J ) = 20160
O
+23279
36
93
~(3+.i) = 20160
23279
+ o
6 7
85
25200 7200,
z (2) = 2160
25200., . 7200 .....
(2) = 0040
+ 2494
9976
710
1011
54
41
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
BendingMoment.
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
812
11639
13
91 1 i of 15
+ 17459
6
8
20160X10=201600! 3 7
5 9+23279
35
7 9> 7+15
+40738 :
7
5
20160X15 25200
X5 = i?6400
610
4 8 20369
57
3 7+59
+46558
10
4
30240X 5 = 151200
71 1
I 5 +17459 2 4
1012 2 6+24
11639
1012
2 4
O
46
8102 6+48
32008 !
68
610+48
40738 !
COMPOUND BRIDGETRUSSES.
2I 3
SECOND METHOD OF DIVISION (FIGS. nSc AND
Diagonals (Fig. n8<r).
Diagonals.
Maximum
Shear.
Corresponding
Stresses.
14
1011
252OO
29098
45
710
O
Fig. u&/.
Diagonals.
Maximum
Shear.
Corresponding
Stresses.
23
912
25200
+ 29098
36
89
25200
29098
67
58
O
O
Chords.
Each supporting force in either figure = 25200.
Fig. n8c.
Bendingmoment anywhere between 4 and 10 = (25200) (5) = 126000;
/. Stress in in = +14549,
.*. Stress in 410 = 14549.
Fig. n8d.
Bendingmoment at 3 or 9 = 126000,
Bendingmoment anywhere between 6 and 8 = 252000;
/. Stress in 39 = 429098,
Stress in 26 or 812 = 14549,
Stress in 68 = 29098.
214
APPLIED MECHANICS.
Hence we have for chord stresses, with this second divis
ion,
Chords.
Stresses.
i3
91 1
III 
+ 14549
35
7 9
in + 39
+ 43 6 47
57
 .
in + 39
+ 43647
24
IOI2
f 26
14549
46
810
410 4 26
29098
68
410 + 68
43 6 47
Hence, selecting for each bar the greatest, we shall have, as
the stresses which the truss must be able to resist,
14
IOII
+ o
34918
i3
91 1
+ 17459
2 3
129
+ 29098
35
7 9
+43647
36
98
+ o
29098
57

+46558
45
10 7
+ 9976
2494
24
IOI2
14549
58
7 6
+ 2494
 9976
46
810
32008
68
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 doublepanel 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
1816; then on each side
of 98 (the middle post of
this truss) is a secondary
truss (149 on tne left,
and 91216 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 travellingload 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 819 *'>
Let angle 415 = i n
Let angle 213 ** \
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
12
25
56
69
14
49
18
W + Wi
O
o
o
w + w l
O
W + Wi
o
o
o
w + Wi
o
o
o
3 wfw/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 25,
34, 47, 56, 78, 69,
81 1, and 910 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'
ingforce 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 coordinates 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 coordinates 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 coordinates 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 coordinates 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 coordinates 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 coordinate 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 coordinates 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 coordinate axes
in its own plane,
Sfxdxdy ffydxdy
(d) For a slender rod of uniform sectional area, a, if x, y, z,
be the coordinates 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 coordinate 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 wroughtiron beams of various sections, on
account of the thinness of the iron, a sufficiently close approxi
mation is often obtained by considering the crosssection 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.
() AngleIron 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) TIron (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) ChannelIron (Fig. 134). Area of flanges = A, area of web
= B, depth of flanges + J thickness of
Ah
FIG. 134.
(d} HBeam (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 coordinates, 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) QuarterArc of Circle AB, Radius r (Fig. 139).
r 2 I 2 cos OdB
Jo
2r
Semicircumference 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) TSection (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(Bt)\
$
 k(S 
whence we can readily derive x t .
(b) ISection (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 nonparallel 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 HalfSegment OAB (Fig. 145). Let OA ~ x,,
AB = jh ; let x ,y , be the coordinates 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 coordi
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 coordinates of the centre of gravity :
. . ^o^O,
Xo =
/r r*V r"* x* /rcos0j /*jr tan 0,
/ *<**#+ / / JCflkfljK
.ooggyV^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 HalfSegment 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(2rrr} (?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 semiaxes 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 (^45) < ^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 crosssection : then will the in
tensity of the stress on the crosssection 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 wroughtiron 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 castiron
under the same load, assuming the modulus of elasticity of castiron 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 workingstrength of the material ; this workingstrength
being a certain fraction of the breakingstrength 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 workingstrain ; 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 wellfitting joints, the first
case, or that of a uniformly distributed stress, can be practi
cally realized ; but with illfitting 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 bendingaction 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 crosssection A, we shall have, for the
intensity of the pull,
P
On the other hand, if the workingstrength of the material
per unit of area be /, we shall have, for the greatest admissible
load to be applied,
P = fA.
If / be the workingstrength 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 wroughtiron, and 28000000 Ibs. per square
inch as its modulus of elasticity, its workingstrain 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 abovementioned
account.
(a) CastIron.
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) WroughtIron.
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 inchlbs.
Hence, if we are to perform upon the rod 308.4 inchlbs. 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
00514"
= 6000 Ibs.
RESILIENCE OF A TENSIONBAR. 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 TensionBar. The resilience of a
tensionrod 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 wroughtiron rod is 1 2 feet long and i inch in diameter, and
is loaded in the direction of its length; the workingstrength of the
iron being 12000 Ibs. per square inch, and the modulus of elasticity
28000000 Ibs. per square inch.
Find the workingstrain.
Find the workingload.
Find the workingelongation.
Find the workingresilience.
From what height can a 5opound weight be dropped so as to produce
tension, without stretching it more than the working elongation?
2. Do the same fora castiron rod, where the workingstrength 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 breakingstrength 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 carryingstrength 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 = carryingstrength.
Hence, instead of using, as breakingstrength 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 workingstrength, Weyrauch divides this by 3 : thus
obtaining, for workingstrength per square inch,
( i least stress )
b 10000 < i H [ Ibs. per sq. in. ;
( 2 greatest stress )
lor Krupp's caststeel, 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 breakingstrength per square inch for a wroughtiron 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. SuspensionRod of Uniform Strength. In the
case of a long suspensionrod, 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 = workingstrength 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 wroughtiron tensionrod 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 crosssection, 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
halfring CED. PIG. 153.
The total upward force acting on this halfring, 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 crosssection
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
workingstrength of the material for tension ; hence, putting
fpr
/ / ,
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 steamboiler ; for in
that case the steampressure 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 steampressure. On the other hand, in the case of an
ordinary waterpipe, 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
Sheetiron ......... /* = 0.00086 pd f 0.12
Castiron ......... 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 crosssection 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 workingstrength 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 workingstrength 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 caststeel,
/ = 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 breakingstrength 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 givingway 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 singleriveted lapjoint is one
with a single row of rivets, as
shown in Fig. 154.
A singleriveted buttjoint with
covering plate is shown in
T C C
one
Fig. 155
A singleriveted buttjoint with
two covering, plates is shown in
Fig. 156.
FIG. 154.
FIG. 155.
FIG. 156.
25 8
APPLIED MECHANICS.
FIG. 157.
FIG. 158.
A doubleriveted lapjoint with
the rivets staggered is shown in
Fig. 157; one with chain riveting,
in Fig. 158.
Taking the case of the singleriveted lapjoint 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 rivetholes (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 rivethole to the outer edge of the plate.
f t the ultimate tensile strength of the iron.
/, the ultimate shearingstrength of the rivetiron.
f s > the ultimate shearingstrength of the plate.
f c the ultimate crushingstrength 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 rivetholes 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 doubleriveted
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 wedgeshaped 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 abovestated 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 coordinate 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 = $(x, y) be
264 APPLIED MECHANICS.
the intensity of the stress at the point (x, y), we shall have,
for the coordinates of the centre of stress,
ffxpdxdy J'Sypdxdy
:
(see 42), where the denominator, or ffpdxdy, represents the
total amount of the stress.
When the stress is positive and negative at different parts
of the surface, as in Fig. 164, the case may arise when the posi
tive and negative parts balance each other, and hence the
stress on the surface constitutes a statical couple. In that case
Sfpdxdy = o.
181. Uniform Stress. In the case of uniform stress, we
have
i. The intensity of the stress is constant, or / = a con
stant.
2. The volume which represents it graphically becomes a
cylinder with parallel and equal bases, as in Fig. 165.
3. The centre of stress is at the centre of gravity of the
surface of action ; for the formulae become, when / is constant,
_ pffxdxdy _ ffxdxdy _
Xl ~~~ pffdxdy ~~~' ~~
pffydxdy
=
pffdxdy ~' ffdxdy '
"where x , y , are the coordinates of the centre of gravity of the
surface.
Examples of uniform stress have already been given in the
cases of direct tension, direct compression, and, in the case of
riveted joints, for the shearingforce on the rivet.
UNIFORMLY VARYING STRESS. 26$
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 coordinates 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 coordinates 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 crosssection 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 crosssection,
is called the Neutral Axis of the crosssection. 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 crosssection be made, this crosssection 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
crosssection be assumed, the elongation or shortening per unit
of length of any fibre at the point where it cuts this crosssec
tion, is proportional to the distance of the fibre from the neutral
axis of the crosssection.
Proof. Imagine two originally parallel crosssections 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 crosssections 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 crosssection,
while r remains the same for the same section, it follows, that,
if a certain crosssection be assumed, the strain of any fibre at
the point where it cuts this crosssection 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
crosssection 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 crosssec
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 crosssection.
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 castiron 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 fibrestresses and of the de
flections are practically in agreement, while, on the other hand,
the intensities of the shearingforces 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. ShearingForce and BendingMoment. 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 crosssection, we
have imposed the second of the three lastmentioned 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 shearingforce 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.
SHEARINGFORCE AND BEND INGMOMENT. 2? 3
The bendingmoment 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 shearingforce at the section must be balanced
by the resistance opposed by the beam to shearing at the
section.
2. The bendingmoment 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 shearingforce and bending
moment at any point, and also the greatest shearingforce and
the greatest bendingmoment, 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 shearingforce, and also the greatest bendingmoment, 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
shearingforce is at either support, the greatest bendingmoment
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 bendingmoment is at the load ; the greatest shear
ingforce 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*
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^ "5 <u
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G "Q
rt K
~^
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oj^i 4)^i<u t>rt ^t> ^3'^g
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4>
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X rt<U ^rtS ^"3^ c3 2i "P
to <u w . Mo'' tt '3.2'i'aow
"Sec 'Sex 'J3Qo o'xuc^w
jjrtO _j,j rt * j.rt*. ^rtortofi
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>> G
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bJO C JU
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s  r >.
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.X3 J3 .t3
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^ "5 ^o
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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
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.22 
MOMENTS OF INERTIA OF SECTIONS.
28 3
2 a
aj **i w
O V
kM
CT 1
Cfl M
O
CJ
Sh
II
fe
!
ii
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st:
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284
APPLIED MECHANICS.
o o D
u H* a
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a is
rt x
S5 W
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+
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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. CrossSections 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
2nrt +
20.
Eight flanges
2irrt + 8a
(0
Square, four flanges,
r radius of cir
cumscribed circle
Six flanges
6a
REPRESENTATION OF B ENDINGMOMENTS.
287
189. Graphical Representation of BendingMoments.
The bendingmoment 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 bendingmoment 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 bendingmoment 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 bendingmoment, 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
bendingmoment, we obtain the curve CA, any ordinate of
which will represent the bendingmoment at the corresponding
point of the beam.
When we have more than one load on a beam, we must draw
the curve of bendingmoments for each load separately, and
then find the actual bendingmoment 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 bendingmoments 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 bendingmoments
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 bendingmoments 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 =
bendingmoment 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 bendingmo
ments for each individual load being
respectively AFE, AGE, and AHE,
FIG. 219. and the actual line of bendingmo
ments being AKLME.
REPRESENTATION OF BEND INGMOMENTS.
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
bendingmoments 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 bendingmoment
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 bendingmoment 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 bendingmoment 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 bendingmoment
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 bendingmoment ; while
on the left of A and on the right
The line of bending
FlG  225 '
of B we have a varying bendingmoment.
moments is, in this case, CabD.
We may, in a similar way, derive curves of bendingmoment
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 crosssection, we must pro
ceed as follows :
i. Find the actual bendingmoment (M) at that crosssec
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 bendingmoment 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.
bendingmoment 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 Ibeam 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 bendingmoment at a section one foot from the end ; (<) the
greatest bendingmoment ; (<r) the greatest intensity of the stress at
each of the above crosssections.
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 workingstrength 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 workingstrength of the material (or compression.
Thus, we must in this case first determine where is the
section of greatest bendingmoment (this determination some
limes involves the use of the Differential Calculus).
Next we must determine the magnitude of the greatest
bendingmoment, 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 bendingmoment, when the loads
are such that no fibre is strained beyond the elastic limit, 7 the
WORKINGSTRENGTH. . 2$$
moment of inertia of that section where this greatest bendingmoment
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 WORKINGLOAD.
If // = safe workingstrength per square inch for tension,
/ c '=safe workingstrength per square inch for compression, and
M Q = greatest safe working bendingmoment, 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
BREAKINGLOAD AND MODULUS OF RUPTURE.
If M is the greatest bendingmoment when the beam is
subjected to its breakingload, 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 crosssection.
Nevertheless, it is customary to compute the
breakingstrength 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.
WORKINGSTRENGTH.
The workingstrength 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.
WORKINGSTRENGTH. 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 lefthand support: find the position
of the section of greatest bendingmoment, and the value of the greatest
bendingmoment.
o A a Solution.
1
(i) Lefthand supporti ng force =  \ .
Righthand supportingforce = _ _j_ t
(2) Assume a section at a distance x from the lefthand support
(this support being the origin), and the bendingmoment 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 bendingmoment,
differentiate each, and put the first differential coefficient = 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
bendingmoment. But if the first is > a, and the second < 0, then the
greatest bendingmoment is at the concentrated load.
These conclusions will be evident on drawing a diagram representing
the bendingmoments graphically, as in Figs. 223 and 224; and the
greatest bendingmoment may then be found by substituting, in the cor
responding expression for the bendingmoment, the deduced value of x.
296 APPLIED MECHANICS.
2. Given an Ibeam, 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 bendingmoment at the middle, the greatest bendingmoment,
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 crosssection are varied in such a manner, that, at each
crosssection, 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 crosssection
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 crosssection, where M = bendingmoment
''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.
CROSSSECTION OF EQUAL STRENGTH.
A crosssection 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 breakingstrength per square inch for compression,
f t = breakingstrength 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 crosssection
of equal strength, we must have, according to the definition,
<=; but we have = = intensity of stress at a unit's
pt ft yc yt
CROSSSECTION 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 Tsection 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 crosssection 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 = bendingmoment
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 lefthand 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 lefthand 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 CrossSection.
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
0098 EI
0.010 EJ
306 APPLIED MECHANICS.
197. Deflection with Uniform BendingMoment If
the bendingmoment 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 bendingmoment 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 coefficient 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 breakingload, 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 breakingload.
3io
APPLIED MECHANICS.
2Oi. Deflection and Slope under WorkingLoad. 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 workingload, 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.
WorkingLoad.
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 workingloads, 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 bendingmoment due to this load would be of the same
amount at all points between the supports ; i.e., a uniform
bendingmoment. 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 bendingmoment 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 bendingmoment 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 bendingmoment at either fixed end ; and the
bendingmoment at any special section at a distance x from
the origin will be
where M is the bendingmoment 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 lefthand
support,
W W,
M=ocl. (5)
When x = , we obtain, as bendingmoment 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 bendingmoment 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 halfway 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 WORKINGLOAD.
If f represent the workingstrength of the material per
square inch, and if W represent the centre workingload, 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 bendingmoment M t is
()
Hence, since their sum equals zero,
Wl
12 '
which is the bendingmoment over either support.
The bendingmoment 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 bendingmoment 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 WORKINGLOAD.
For workingload we have
Wl fl
77 = 7
B ENDINGMOMENT AND SHEARINGFORCE.
3'7
EXAMPLES.
1. Given a 4inch by 12 inch yellowpine 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 4inch wide yellowpine beam, 20
feet span, fixed at the ends, may not deflect more than one fourhun
dredth of the span under a load of 5000 Ibs. centre load.
204. Variation of BendingMoment with Shearing
Force. If, in any loaded beam whatever, M represent the
b endingmoment, and F the shearingforce 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 bendingmoment 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 coordinate 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 lefthand
/^ I ^j,j 7\ end, and let the lefthand support
ingforce 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 bendingmoment at
that section be M. Let the section BD be
at a distance x + &>* from the origin, and
let the bendingmoment 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 shearingforce 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 shearingforce 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
shearingforce is desired, and multiply the quotient by the statical
moment of the portion of the crosssection 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 shearingforce 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 4inch by 1 2inch 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 9inch by 14inch 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 bendingmoment 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 bendingmoment, 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 breakingstrength of columns subjected to a load
whose resultant acts along the axis have been, until recently,
the Gordon formulae with Rankine's modifications, the socalled
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 fullsize 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
ingload.
Hence, all formulae for the breakingload are, of necessity
3 2 4 APPLIED MECHANICS.
empirical, and depend for their accuracy upon their agreement
with the results of experiments upon the breakingstrength of
such fullsize columns as are used in practice.
Nevertheless, the ordinary socalled deductions of the Gor
don, and of the socalled 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 breakingload, 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 breakingload, 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 halfway 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.
Wroughtiron
36000
36000
Castiron
80000
Dry timber .
72OO
3OOO
326
APPLIED MECHANICS.
2o8a. Socalled 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 breakingload on the assumption that the column
will give way by direct compression. This will be
PI=/A, CO
where /"= crushingstrength 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 crosssection, and /= length of column.
Then will the actual breakingstrength, 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/=bendingmoment 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 . Jdx 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 crushingload,
P~/A.
2. For the breakingload 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 crushingload,
2. For the breakingload by bending, put //3 for / in (6) ; hence
((/) Column fixed in Direction at Both Ends (Fig. 234).
i. For the crushingload,
P./4.
2. For the breakingload 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 castiron, 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 castiron, 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 breakingweight, the following formula:
where W= breaking weight that would be derived from the preceding
formula, W'= actual breaking weight,
BREAKINGLOAD OF FULLSIZE COLUMNS 329
For hollow castiron 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 wroughtiron pillars, Hodgkinson found the strength
to be 1.745 times that of a castiron pillar of the same dimensions; but,
for very short pillars, he found the strength of the wroughtiron pillar
very much less than that of the castiron one of the same dimensions.
With a length of 30 diameters and flat ends, the wroughtir on exceeded
the castiron by about 10 per cent.
210. Breakingload of Fullsize Columns. The tests
made upon fullsize 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 = breakingload, 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 breakingload may be computed by means 01
the formula P = f c A .
For greater values of l/p, the breakingload 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 fullsize columns.
(a) In the case of castiron columns no tests have been made
of fullsize 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 wroughtiron 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 wroughtiron 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 *
= 46000125.
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 pinends 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 = ~(//fl8o)2'
10000
($) For solid web, square, or box columns with pinends, and
l/P>6o,
P _ 31000
A (l/p 60)*
6000
The number of tests that have been made upon fullsize 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 fullsize 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 workingstrength.
(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 bendingmoment 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 crosssection 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 halfway 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 pinended 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 pinended 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 pinended 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 crosssections be taken
between these two pulleys, the por
tion of the shaft between these two
crosssections 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 crosssection, 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 crosssection.
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 crosssection 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 twistingmoment 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 shearingstrength 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 twistingmoment.
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 twistingmoment 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 horsepower transmitted and
the speed, rather than the twistingmoment.
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 horsespower 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 workingstrength for shearing of wroughtiron as 10000
Ibs. per square inch ; find proper diameter of shaft to transmit
2ohorse 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
twistingmoment 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 horsespower,
with a speed of 300 revolutions per minute, and factor of safety 6, break
ing shearingstrength 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 beltpulls,
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 bendingmoment,
M 2 = greatest twistingmoment,
r = external radius of shaft,
/ = moment of inertia of section about a diameter,
TTf 4
for a solid shaft / = ,
4
f = workingstrength 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 steamen
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 bendingmoment under load P ;
M= maximum bendingmoment 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 bendingmoment acts, the depth being
measured parallel to the load.
Then if / denote the moment of inertia of the section of
greatest bendingmoment 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 bendingmoment 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 bendingmoment throughout their length,
this bendingmoment 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 fullsize pieces,
and of introducing into the experiments the conditions of
practice, is pretty generally recognized today, nevertheless
there are some who have not yet learned to recognize the fact
that attempts to infer the behavior of fullsize 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 yieldpoint,
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 testingmachine are attached.
2. In the case of a round specimen of that kind there may
be a screwthread on the shoulders as shown in Fig. b.
In the case of a brittle material, as
1^ JJJII castiron or hard steel, it is desirable to
use a holder with a balljoint, 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
wroughtiron, of the ist or the
FlG  ' 5th shape, subjected to stress
in the testingmachine, 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 stressstrain 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 " yieldpoint " or "stretchlimit," 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
measuringinstruments at our command.
I After the elastic limit and the yieldpoint 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 breakingstress, the behavior is apparently some
what different when the piece is subjected to dead weight from
what it is when in a testingmachine. 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 testingmachine, 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 " breakingload " 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 breakingstrength and ductility.
CASTIRON. 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 breakingstrength per square inch of the material;
2. The limit of elasticity of the material ;
3. The yieldpoint or stretchlimit 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 allimportant matter, and a suit
able breakingstrength per square inch, both a lower and an
upper limit being generally prescribed for this last.
On the other hand, although castiron and hard steel are
brittle metals when compared with wroughtiron 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. Castiron 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, castiron 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.
PigIron is the result of the first smelting, being obtained
directly from the blastfurnace. 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 pigiron.
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 castiron.
Pigiron 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 castiron has been, and is sometimes classified in various
CASTIRON. 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 allimport
ant considerations, those grades are required which are neither
extremely soft nor extremely hard.
As to the adaptability of castiron 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 wroughtiron 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 wroughtiron, 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
wroughtiron 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 castiron should be of a bluishgray
color and closegrained texture.
At one time castiron was extensively used for all sorts
of structural work, but it was soon superseded by wroughtiron,
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 nonconducting 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, gearwheels, and various other parts of machinery of a
similar character are usually made of castiron, 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 bedplates 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 castiron, 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 CastIron. 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 CASTIRON. 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 Castiron. Trans. Am,
Soc. Mechl. Engrs., 1893.
(b) Relative Tests of Castiron. Trans. Am. Soc.
Mechl. Engrs., 1895.
(c) Transverse Strength of Castiron. Trans. Am.
Soc. Mechl. Engrs., 1895.
(d) Keep's Cooling Curves. Trans. Am. Soc.
Mechl. Engrs., 1895.
(e) Strength of Castiron. 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 Castiron. As the use of
castiron to resist tension has been almost entirely superseded by
that of wroughtiron and steel, results of tests of fullsize pieces
of castiron in tension are not available. Tensile tests, however,
have been extensively employed to determine the quality ; especially
so when castiron 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
castiron.
About 1840 Eaton Hodgkinson made a few experiments to
determine the laws of extension of castiron, 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
castiron 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
I05377I
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 castiron 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 wroughtiron 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
castiron. Hence, strictly speaking, castiron 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 castiron.
In making specifications intended to secure a good quality
of 'castiron 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) Testpieces 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 testpieces 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 castiron, 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 castiron 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 castiron 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 guniron, this having been a
requirement of the United States Government, in the days of
castiron cannon, for all castiron that was to be used in their
manufacture.
The following table is taken from a paper on the Strength
of CastIron, 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 castiron.
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
castiron from another source will also be given for the same
purpose as Mr. Keep's results :
CASTIRON.
363
CASTIRON TENSION.
a
.2
.S
c
.2
O.S
u
13 ""
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 castiron 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 highgrade 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 castiron.
219. CastIron Columns. In consequence of the high
compressive strength shown by castiron 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; castiron having been
displaced first by wroughtiron 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 castiron, that it
cannot be riveted, and that if it breaks it cannot be eas'ly
repaired. Castiron is, however, used to a very considerable
extent for the columns of buildings.
The Gordon, the socalled Euler, and the Hodgkinson
formulae for the breakingstrength of castiron 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 fullsize
columns as are used in practice. We will next consider,
therefore, the tests that have been made upon fullsize castiron
columns, and the conclusions that are warranted in the light of
these tests.
Two sets of tests of castiron mill columns have been made
on the Government testingmachine at Watertown Arsenal; an
account of these sets of tests is published in their reports of
1887 and of 1888.
CASTIRON COLUMNS. 365
The first lot consisted of eleven old castiron columns, which
had been removed from the Pacific Mills at Lawrence, Mass.,
during repairs and alterations.
The second lot consisted of five new castiron 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 testingmachine, 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 castiron, 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 castiron 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 castiron 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 fullsize castiron 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
CASTIKON COLUMNS.
367
Remarks.
rt T3 rt rt ^aJ
w nj w T3 W
iple flexure. Post con
t middle before testing,
iple flexure.
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3 68
APPLIED MECHANICS.
i
*
513
Mb!
tMAJsJi.
S'OI
Sg'6OI
CASTIRON 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
4903
4.92
38.67
27126
B*
190.25
15
\
1198000
4903
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
1737
1.8o
66.67
27236
CASTIRON 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
CASTIRON COLUMNS.
371
The cut on page 370 shows a graphical representation of
the preceding tests of fullsize castiron columns.
In Heft VIII (1896) of the Mitt. d. Materialpriifungsanstalt
in Zurich is an account of 296 castiron 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
castiron 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
29862
210
n. 4 8
59
91.9
0577
69
70
7*
9.84
9.84
8.20
563
5.6z
5.65
64.4
64.4
533
24174
32564
36546
211
212
213
9.84
9.84
8.20
56
.60
.56
78.9
777
654
194.12
19482
3*843
72
8.20
5.63
534
47353
3I 4
8.20
4.61
64.7
33 '33
86
9.84
6.69
532
32564
225
13.12
541
87.8
15216
87
9.84
6.67
533
34270
226
1312
543
87.5
17623
88
8.20
673
439
44224
227
11.48
541
76.7
22326
89
8.20
6.69
44.1
46642
228
229
11.48
9.84
539
539
773
66.4
2I3I
23748
230
9.84
541
66.1
23463
23 1
8.20
541
54 8
38110
2 3 2
8.20
541
545
36688
243
13.12
6.56
71.8
22041
244
13.12
654
71.9
24885
245
11.48
6.56
62.6
27729
2 4 6
11.48
6.56
62.5
28156
247
9.84
653
S39
355 20
2 4 8
9.84
654
539
31853
249
8.20
6.56
448
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 knifeedges.
Prof. Bauschinger of Munich made two series of tests of
fullsize cast and of wroughtiron columns to determine the
effect of heating them redhot and sprinkling them with water
while under load. They were loaded in his testingmachine
with their estimated safe load as calculated from the formulas.
For castiron,
19912^4
./"
For wroughtiron,
i  0.00065
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 Ushaped 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 meltingpoints ; 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 castiron posts of various
styles, and three wroughtiron posts, one of them being made
of channelirons and plates put together with screwbolts, one
of I irons and plates also put together with screwbolts, and
one hollow circular.
The details of the tests will not be given here, but only
Bauschinger's conclusions. He said :
That wroughtiron 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 castiron 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 castiron
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 wroughtiron 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 wroughtiron 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
wroughtiron posts were not properly made.
Bauscliinger therefore concluded to make a new set of tests,
and for this purpose he had made two castiron and five
wroughtiron columns the former being carefully cast, but on
the side, while the wroughtiron 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 wroughtiron posts are as well constructed as
the two referred to, they resist fire and sprinkling tolerably
well, though not as well as castiron ; 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 wroughtiron 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 wroughtiron in fires, considering
especially the burning of a large warehouse in Berlin, and
advocating the protection of ironwork 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 castiron 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 castiron 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 CASTIRON. 375
4. He claims that in either cast or wroughtiron 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 onesided heating and cooling.
6. Castiron is more likely to have at any one section a
collection of hidden flaws than wroughtiron.
220. Transverse Strength of Castiron. At one time
castiron 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 wroughtiron and steel.
A great many experiments have been made on the trans
verse strength of castiron ; 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.
Castiron 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 breakingstrength diminished in proportion to the amount
of these strains.
In the case of castiron 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 castiron beam be broken transversely, and
the modulus of rupture be computed by using the ordinary
formula,
fMy
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 castiron 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 castiron.
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 CASTIRON. 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 crosssection 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 castiron beams, we cannot use
one fixed value of f for all beams made of one given quality
of castiron, 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 castiron, 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
177
Planed
C. Bach.
Mean.
1.74
19470
34000
175
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
145
Planed
C. Bach.
16070
225OO
1.40
Planed
Considere.
31860
36640
II5
Planed
Considere.
25040
3I3IO
1.25
Rough
Robinson and Segundo,
Mean.
I3I
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 CASTIRON.
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 castiron 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 castiron, 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 breakingweights 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 breakingweights 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 CASTIRON.
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.
BreakingLoad,
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
castiron pulleys ; and also, incidentally, to determine the hold
ing power of keys and setscrews.
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 ijJ inches diameter, and keyseated 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 wroughtiron I
beams, resting upon a pair of jackscrews, 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 breakingload 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 bendingmoment 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 = breakingload determined by experiment.
TRANSVERSE STRENGTH OF CASTIRON.
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
<
tpj tbj
Mps otjeo
X X
X
p
*
X
Nm
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
83
iX
with
39939
84
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 CASTIRON. 385
STANDARD SPECIFICATIONS FOR CASTIRON, OF THE AMERICAN
SOCIETY FOR TESTING MATERIALS.
The standard specifications for castiron, of the American
Society for Testing Materials, contain specifications for i
Foundry Pigiron, 2 Gray Iron Castings, 3 Malleable Iron
Castings, 4 Locomotive Cylinders, 5 Castiron Pipe and Special
Castings, 6 Castiron Carwheels. 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 PIGIRON.
ANALYSIS.
It is recommended that all purchases be made by analysis.
SAMPLING.
In all contracts where pigiron 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 fracturesurface
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 pigirons, 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 . . .
275
0035
0.045
N >. 2 . . .
2.25
0.045
0055
No 3 ...
i75
055
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 breakingstrength 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 CASTIRON.
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 moldingsand, 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 testbar 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 CASTIRON. 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 testpieces 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 TestBar," 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.
TESTPIECES 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 testpiece, 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 testpiece for each cylinder shall be required.
CHARACTER OF CASTINGS.
Castings shall be smooth, well cleaned, free from blowholes, shrink
age cracks, or other defects, and must finish to blueprint 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.
CASTIRON 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 cinderiron 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.
WROUGHTIRON, 39!
221. Wrou giltIron. Wrought iron is obtained by melt
ing pigiron 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 castiron borings.
The process is commonly carried on in a puddling furnace,
where an oxidizing flame is passed over the melted pigiron.
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 muckbar, 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 muckbar is sometimes used
exclusively, a considerable part, and often the greater part, is
made of scrap.
Wroughtiron 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 castiron
from which it is made. Thus, the presence of sulphur makes
it redshort, or brittle when hot; and the presence of phosphorus
makes it coldshort, or brittle when cold.
It cannot, like castiron, be melted and run into moulds;
but it can be easily welded by the ordinary methods
Wroughtiron is much more capable of bearing a tensile or
transverse stress than castiron: it is tougher, it stretches more,
and gives more warning before fracture. At one time castiron
was the principal structural material, but it was soon displaced
by wroughtiron, which became the principal metal used in
construction, but now, since the modern methods of steelmaking
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.
Wroughtiron is also expected to withstand a great many
trials that would seriously injure castiron: thus, two pieces
of wroughtiron 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. Wroughtiron 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 Wroughtiron 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 Wroughtiron 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 wroughtiron and a great many of
them contain tests of fullsize pieces of wroughtiron.
9. A. Wohler: (a) Die Festigkeits versuche mit Eisen und Stahl.
(b) Strength and Determination of the Dimensions of Structures
TENSILE STRENGTH OF WROUGHTIRON. 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 Wroughtiron. 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 castiron 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 wroughtiron, 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 wroughtiron began to increase
at a rapid rate, the necessary appliances were introduced to roll
it into I beams, channelirons, angleirons, and other shapes,
and it began to displace castiron 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 allimportant 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 breakingstrengths and their ductility.
In the light of the results obtained by him, he proceeded
to draw up his famous sixtysix conclusions.
These sixtysix 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 wroughtiron 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. Breakingstrength
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. Breakingweight 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 socalled cold
crystallization, which will be mentioned later.
SPECIFICATIONS FOR WROUGHTIRON.
395
Tests of the tensile strength of wroughtiron 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
fullsize 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 wroughtiron of the American Society for
Testing Materials will be given first, as they refer to the kind
of wroughtiron that is in most common use, and then some
other tensile tests of various kinds of wroughtiron in small pieces
will be given. Subsequently tests of wroughtiron eyebars will
be quoted.
AMERICAN SOCIETY FOR TESTING MATERIALS.
SPECIFICATIONS FOR WROUGHTIRON.
PROCESS OF MANUFACTURE.
1. Wroughtiron shall be made by the puddling process or rolled
from fagots or piles made from wroughtiron scrap, alone or with
muckbar added.
PHYSICAL PROPERTIES.
2. The minimum physical qualities required in the four classes of
wroughtiron shall be as follows :
Staybolt
Iron.
Merchant
Iron.
Grade "A."
Merchant
Iron.
Grade "B."
Merchant
Iron,
Grade "C."
Tensile strength, pounds
per square inch .
46000
50000
48000
48000
Yieldpoint, 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) Staybolt 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) Staybolt iron, a piece of staybolt 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) Stayolt iron, shall bend through 180 flat on itself, without
SPECIFICATIONS FOR WROUGHTIRON. 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 ninetenths 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. Staybolt 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 onehalf 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 18about \
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 testingmachine.
TESTS OF COMMANDER BEARDSLEE. 399
FINISH.
13. All wroughtiron 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 wroughtiron 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 chaincable, and the chaincable 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 bariron 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 wroughtiron, and will be given here before giving a table of
the results of the tests.
i. Kirkaldy considers the breakingstrength 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 testingmachine.
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 crosssection is
practically zero (as is the case when a groove is cut around
the specimen), the breakingstrength 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
crosssection area is smallest, in order that the tensile strength
may not be greater than with a long specimen. The conclusion
reached was, that no testpiece should be less than onehalf 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 :
ijin. 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 "
II63%
(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 breakingstrength per square inch.
4. He gives evidence to show, that if a bar is underheated,
it will have an unduly high tenacity and elastic limit ; and that
if it is overheated, the reverse will be the case.
402
APPLIED MECHANICS
5. The discovery was made independently by Commander
Beardslee and Professor Thurston, that wroughtiron, after
having been subjected to its ultimate tensile strength without
breaking it, would, if relieved of its load and allowed to rest,
have its breakingstrength 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 94%
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 UnitedStates 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
castiron 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 chainwelder.
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 bariron 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 provingstrengths, as given
in the following table ; the Admiralty provingstrengths 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 provingstress 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
ProvingStrains.
Admiralty
ProvingStrains.
Diameter of
Iron,
in inches.
Recommended
ProvingStrains.
Admiralty
ProvingStrains.
2
121737
161280
I*
66138
83317
lit
114806
I5I357
If
60920
76230
I 8
108058
I4I75
iiV
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 wroughtiron for boilerplate,
and while steel I beams, channelbars, angleirons, and other
shapes, as well as eyebars, have, of late years, displaced
wroughtiron to a very great extent, nevertheless wroughtiron
is still very extensively used, and for a great variety of struc
tural purposes.
For wroughtiron 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
wroughtiron 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""
4r"!
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
576
29175000
75
46340
21160
62.7
30780000
75
50023
26030
498
30643000
75
48280
28030
62.6
29020000
76
47724
25350
476
30310000
77
45160
20400
68.8
27388000
77
46772
24700
452
28347000
75
46063
19240
68.6
27666000
77
46600
22550
46.2
29528000
77
44490
20510
675
28452000
77
47395
22550
46.2
28347000
74
43233
22079
70.5
29026000
77
47963
22695
48.6
29475000
75
43470
19400
755
26700000
77
47860
26948
46.4
26948000
73
38950
22030
72.3
30140000
77
475CQ
26927
423
28435000
74
43240
21970
752
27726000
76
47610
23036
53 l
29551000
74
44564
21970
73.8
28663000
77
49238
22725
492
27470000
74
43860
19658
75o
18000000
76
50037
27700
536
29251000
i .00
41620
15560
73
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
54
30438000
.76
4'574
14328
695
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
534
54i
29969000
33657000
75
41875
16978
70.1
30487000
.76
49690
27480
533
29614000
75
43396
19112
593
28000000
.76
47430
22950
518
29443000
74
39210
15216
73' 2
30294000
76
4795
23000
57o
29504000
74
12603
70.5
28810000
.76
22892
458
28779000
74
39896
15187
69.7
31153000
77
49411
18420
46.6
30112000
74
39156
16123
76.4
29807000
.76
49660
23186
513
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
512
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
743
31124000
74
39698
17638
705
30023000
49350
21503
66.5
31793000
73
43187
17846
68.6
28553000
.76
50083
20940
29097000
73
40669
17820
738
30159000
76
47019
23140
5 1 4
29978000
73
39348
16593
693
29518000
76
47504
20942
532
28527000
73
39671
12987
78.1
28861000
76
47747
20942
495
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
537
29293000
73
44470
16844
734
31114000
76
50083
23146
45 
29097000
74
41940
17523
78.5
29373000
75
48168
23767
556
29879000
74
42531
16449
7.S
30410000
76
49500
26500
55o
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
545
28200000
406
APPLIED MECHANICS.
Refined Iron.
Wroughtiron 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
335
27711000
Annealed wire
135
61100
39800
494
23000000
77
53850
29370
333
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
335
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
254
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
533
26600000
.76
47I5I
25982
754
21628000
Common wire
079
120000
73600
28.1
26100000
77
5035 1
35720
253
27477000
Common wire
.079
IO9OOO
54300
40.4
26400000
75
48202
28521
14.7
27888000
Common wire
.080
98300
438
27100000
75
50703
30558
13.0
23713000
Annealed wire
.081
99600
61 .9
26600000
75
49223
30517
J52
27126000
Annealed wire
.082
93500
50400
643
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
550
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
569
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
4ooX
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
239
26500000
.OI
50200
30000
325
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
325
27700000
.OI
50600
30000
275
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
334
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 wroughtiron
and mildsteel angles, tees, and channels. The following is a
summary of his results for wroughtiron shapes :
ANGLEIRONS.
a'
rt
g
u
J w"o
 v"5
*j' u> S
SU
Modulus of
5
Dimensions,
Inches.
V
a
Is
X g 3
*ls
(2S 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
95
28824000
4
2.76 X 2.76 X 0.51
27.62
49060
20190
32560
II. 7
28070000
6
354 X 354 X 0.35
26.21
50620
253 10
15.8
28269000
8
3 l 5 X 354 X 0.55
3851
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 X413 Xo.6 7
5524
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 512 X 0.87
102.61
48490
31140
12.3
TEEIRONS.
3
3". 60 X 3" 35
22.88
5 2 470
25880
3768o
14.2
27615000
4
"
49200
23610
34700
155
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
145
27402000
28255000
CHANNELIRONS.
I
4.13 X 2.56
28.43
50630
23329
35120
159
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
175
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 335
85.68
52050
22330
3456o
20.9
27701000
10
4k 4*T
'*
5347
24170
36260
17.0
28710000
la
8.46 X 350
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
RollingMill 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 rf5
a a
w
Ultimate Strength,
in Ibs., per
Square Inch.
$>
c
'*?
S5
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
333
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
H5
27.6
95
5
27826036
L 6
6.40
27500
50530
173
22.3
70
30
27118644
L 7
639
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
'33
243
100
Slightly
27586206
S 12
3.08
28000
50390
16.9
35i
100
o
27586206
s 13
305
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
157
177
85
!5
26490066
S 17
6.41
24500
48740
143
17.3
80
20
28119507
S 18
3i7
24600
49680
195
32.0
IOO
Slightly
27972027
Round.
S 19
317
25870
49338
18.3
38.0
IOO
29357798
"
Cinder
S 20
3i7
24920
48864
18.4
37o
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 153
379
100
o
2 7633 8 5 l
L 2O2
303
30000
52650 16.2
20.6
85
15
34042553
L 203
3.06
32500
53500 1 16.5
275
TOO
28169014
L 204
3.06
32500
54480 15.4
24.8
TOO
o
29090909
L 205
633
27000
51230 17.8
24.2
80
20
28119507
L 206
634
27500
50500 17.6
21. 1
100
Slightly
29629629
L 207
634
27000
51030 ; 21.4
319
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
45o
TOO
"
28622540
"
L 210
3.20

50188 19.9
43o
IOO
"
28985507
"
S 211
305
29500
5II50 J22.0
3i5
IOO
o
32989690
S 212
35
28500
5IIIO 22.0
36.1
IOO
o
25559105
S 213
3"
29500
51860 225
392
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
633
27000
51340
193
352
IOO
o
28070175
S 218
3i7
24610
50631
20.4
41.0
IOO
o
28622540
Round.
Cup
S 219
3i7
50915
255
44.0
IOO
shaped
28268551
"
S 220
3^7
50205
237
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 :
oo
5
8 .
a
v c2
&
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' ;
H.S * i.s *
ga . . fc so . .
si . _ r3_
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 eyebars 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
139
518
96.05
75
5103
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.
WROUGHTIRON PLATE.
The following table contains some tests of wroughtiron
plate and bars made on the Government testingmachine at
Watertown in 1883 and 1884 for the Supervising Architect at
Washington, D.C.
412
APPLIED MECHANICS.
.g '3
I" Mi
fii e . i3 _e fl ^ **
rf 11181 8
sssg.sg
**
.0   w
8s r
"1 81
to J3 (fl
EE tt,
HiHit
a; ".n ^ w C r* ** T3 *
'
a
</,
6*OOc*)NOOfi
fO NNNflO
gation
acture.
E
t
M O txO O rorotxd " ON M
O O vO O * iO*O O 00 <* ir> t^ t^\O OOOOO
satpui jqSp uj
saqoui
'
Is
Ultimat
Strengt
OO
ir)M
ro CO M \o ^ fO
00 00 <0 00 M 000 ON OOO r^ ON O
,s,s,
N? O O O O
o &.o <? 5oo N ?"
cKoo o o t* *o in co ti m u"> ^
O O
SS
VJNO ^"NO NO
CNlNNNCMNNNN
o oo oo 10 m 1000 10 M t~ o 8 oo m ON ON * N o5 8 <iv
ONON^'^'i' ONOOfO^ ON ONOO IO O IO ^ OO NO VO 1^*00 1
p
8OOOQOOOOOOQOOOOOOOOQOOOQO OOOOQQQS OOOOOO
^5^0ooOO^om5>oOooNoi5oooOOOvooOooooto %<% 00,,. JoOOOtOM
WNO HI ^ONfOtNNO IOOO OO Ht ONrOCNl CNN M M CN1 lot 1 t^fO O^W CN1 txONONfONU ^"MOO ^ONH
t^ rxNO NO 10 <J ^NO NO JO t>. fxNO *CNlC^m^*i(l<NOiOrOlO MMWIO 10NO NO T3 00 NO t^OO NO NO
B3JV
c<> r. ooo oo ^^ONIOQ O rot^o O NOOOO o ^*oo ^ o ooo ^J O ^ *^~ ^oo en t^ o t^ tx ro TO
C O l t^MONlOlOON ON NO N H ON ONNO IO ON ON ON ONOO OOMQOH ONONOOOHrOTj* J OONOOt < O
 ioO'^' l o^'^"'* p ^'^' l ^' l o i o^'^'f r ifOf r Nro^^*fOfOioiooio rororoioioiotONVS lo^fiovoio^o
oo'do'dddddddddddddddddoddddd ddo'ddddfidddddd
ddQdddo'ddQodoooooooo oooooo
O H O M M ONOO 00
o d d d odd
. ^"2 "fL n"* ^^ m l^vo 1 8 (? ON ^00*
COO^OONvNNromrsOOONO>ON
^SEgisI!? SEtifSS
' . 5(5 M . HM d d d d ^ H
uauipadc;
a a a s 8 9 s
NO t>.OO ONOMCS
^ tovo NO tx rwo NO _ N m ^ ^^ M ro * 10 IONO <O
OU_ )lJl j 1 _3 ( j ( j
voo^tx
ro (r> (rN^'^it'^^^4>^^ioioioio IONO NONONONO Fi^K 1010 io ir
NONONONONONONONONONONONONONONOVONONONONONONONONOVONO NONONONO
WROUGHTIRON AND STEEL EYEBARS.
In the report of the Government tests for 1886 is given the
following table of tensile tests of wroughtiron eyebars. The
wroughtiron 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 WROUGHTIRON. 413
WROUGHT IRON EYEBARS.
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
'x2
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
499
1.14
21790
43320
78
3.0
26.4
27950000
48492
1 '
Fibrous, 70%
Granular, 30%
238.64
5.00
1. 16
22410
39550
5 i
48
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 WROUGHTIRON.
In regard to the compressive strength of wroughtiron, we
may wish to study it with reference to
1. The strength of wroughtiron columns;
2. The strength of wroughtiron 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 wroughtiron bridge columns, the compressive strength
per square inch of wroughtiron 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 fullsize 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 wroughtiron is tested with
out lateral . support, and with flat ends, the friction of the ends
against the platforms of the testingmachine 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 wroughtiron
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. Wroughtiron 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 fullsize wroughtiron 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 UGHTIRON 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 fullsize Phoenix columns, made
for them at the Watertown Arsenal, together with a comparison
of the actual breakingweights with those which would have
been obtained by using the common form of Gordon's formula
for wroughtiron. 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
ingloads, 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 fullsize 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 squareended columns . . . j =  n  ry
' + *>)
18000
P f
For phiended columns . . . . j =
7.8000
416
APPLIED MECHANICS.
frlil

vO ONfOONi^toior^> O vO VO r^ M TJ N t
Tjvo. rorfLOO\O O^OvOOO ^M N ro
MdOfCOOOMits t^O OO OO O O ro
1 1 1 1
tf r^ r
C\ O\<> O w N
^
w 11 M O ""If
J~> rj >o io LO Tj
O Q O O
?&S2
t~^ r^.vo n
lO to to ^j
oo ^o
vo vO
CJ "^ N O O O ^"
^ O ^ CO ^OSO 10
ro ro to ro to to to
10 II
fOVO OO fO N O ON
^" fO O\ M i i ^ i M O i i
N N S M 1 1 ii 1 w M 1
N fOM oo r>
" l\iJ TfJUNMll^lMtJi HNM t^VO
l NMMM>iM' l i l iitc l 'O O O 11 p
dddddd d do d dddd
ONOO VOVO>OfON'iOSON 1 to I fO N OO'i'^f
HtM'MMM'MMMlOO' ' O ' 'O O Opllp
do odd ddddd d do dddd
ioo too r^*" MOO Q Ooo
^NeiNcivi'fiM'cIci MM oo 06
_ M O O ON I I^O VO
^TjtOfOfOtOt)
to O
O "">
UOOOO
N ON ONVO vO totoo O r^
r^oo ON o M N to rf mvo t^oo
WROUGHTIRON 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 channelbars 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 fullsize wroughtiron columns, viz. :
i. The series made at Watertown Arsenal, this being the
most complete set of tests of fullsize wroughtiron columns in
existence.
2. A series of tests of Zbar 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
WROUGHTIRON 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 flatended, and 60 for pinended columns;
p
the value of (i.e., the breakingload 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 flatended Phcenix columns he recommends Cooper's
formula.
For lattice columns with pinends, 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 flatends r= 
A.
/ \ 2 '
80
p /
10000
p 31000
For pinends .......... ;=
6000
4 2 APPLIED MECHANICS.
In these formulae P = breakingload 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
flatended, and less than 60 in pinended 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 =46oooi2 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.
WROUGHTIRON COLUMNS SUBJECTED TO ECCENTRIC LOAD.
All the formulae given thus far for the breaking or for the
safe load on wroughtiron 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 breakingstrength per square
inch, i.e., the number of pounds per square inch which, multiplied
WROUGHTIRON COLUMNS. 421
by the area in square inches, gives the breakingload of the column;
the safe load per square inch being obtained by dividing the
breakingload 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
fibrestress due to the direct load, and to this add the greatest
fibre stress due to the bendingmoment, 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 FULLSIZE WROUGHTIRON 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.
jj
1 1 I
6
=
^0
1
rQ
it , ..,..,;,, ,"
1 1 1
s 4 s
"4} 3 .3 3 C
o
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b
rt sj
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C 3 u)OOOQOOOO''VO Lr > 1  r
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rOMrOMMiiNNNNNiHMMM
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WROUGHTIRON COLUMNS.
423
I 1
11*9
I.
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I...........I I". s i
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H OO OO O O
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424
APPLIED MECHANICS.
a
1 s
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fe Q
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pi
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T3 "3 "3
72 rt^ rt^ So 
ggsgs S
IsJsJ * H
g 1,, i
3: = 3 S 3 * IS
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WROUGHTIRON COLUMNS.
42 s
CX O &, O
SJ.C 3 .3
O vo *"O o co O f^
52 O ^ r^ O ON O ON
^ 10 10 i^\o ON N oo
(4 M ft ti N <*)(*
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426
APPLIED MECHANICS.
'rf
IS
rS
. C
*O O
QJ N
3'S
.s'S'S
6 o
g
t
3
a
u
.9j
Manner of '
t Deflected ho
( and upwar
1
o
je

t/3 c
~ S
Ultimate
1
3
4
Ij
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it
WR UGHTIR ON COL UMNS.
427
428
APPLIED MECHANICS.
WROUGHTIRON COLUMNS.
429
The next table taken from the volume for 1882 men
tioned above contains the results of some compressive tests
of wroughtiron Ibeams placed in the machine with the ends
vertical and tested with flatends ; also of some tensile speci
mens cut off from two of them.
TESTS OF IBEAMS 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
545
0.64
9.00
228
14.40
545 TOO
37854
2
15545
4.40
0.40
10.52
443
IO.26
207000
20170
3
191.90
356
0.40
9.08
365
6.85
85380
12460
4
191.90
359
043
9.09
38i
715
85200
11916
5
11985
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
045
794
355
6.65
83400
12540
8
192.90
3.60
0.44
7.98
353
659
92300
I40IO
9
21588
4.28
0.40
10.52
561
930
I49OOO
I6O2O
10
264.08
449
0.48
1053
747
10.19
II3IOO
IIIOO
ii
264.08
443
0.50
10.51
767
10.46
107800
10306
12
264.00
4.90
0.53
I5I5
1085
14.80
184700
I24OO
13
263.95
4.84
053
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.
300
0.65
195
103300
52970
IO
Web.
300
0.50
I5I
65400
43340
39
1
Flange.
4.00
0.75
301
146400
48640
19.6
1
Flange.
4.00
0.76
302
I47IOO
48640
159
f
Flange.
3.00
0.51
153
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 WROUGHTIRON COLUMNS, LATTICED, BOX,
AND SOLID WEB.
" This series of tests comprises seventyfour 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
" Channelbars with solid webs ;
" Channelbars and plates ;
" Plates and angles ;
" Channelbars latticed, with straight and swelled sides ;
" Channelbars, 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 crosssection ; 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 testingmachine the posts occupied a horizontal
position.
" They were counterweighted at the middle.
" Castiron bolsters for pinseats were used between the ends
WROUGHTIRON COLUMNS. 43!
of the columns and the flat compression platforms of the test
ingmachine.
" The sectional areas were obtained from the weights of the
channelbars, angles, and plates, which were weighed before
any holes were punched, calling the sectional area, in square
inches, onetenth 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
nelbar or plate, always using a length shorter than the dis
tance between the eyeplates, 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 eyeplates along the continuous plates
and channelbars 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 eyeplate, and one end to the
channelbar. 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 pinholes for
placing the pins either parallel or perpendicular to the webs of
the channelbars.
" 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 crosssection, 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 UGHT1RON COL UMNS.
433
TABULATION OF EXPERIMENTS ON WROUGHTIRON COLUMNS
WITH 3JINCH PINENDS.
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
9831
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
9977
251000
25160
Deflected perpendicular
to axis of pins.
756
r
~~n~\
180.00
9977
210000
21050
Do. do.
753
*~~z
6 r>
240.00
9732
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 OUGHTIRON COLUMNS
WITH 3^INCH PINENDS.
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.
15920
7.628
258700
339X0
Deflected perpendicular
to axis of pins.
747
?^i
f Tt
15927
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
77
243000
of pins.
1649
3^985
7 '673
229200
29870
Deflected in diagonal
direction.
740
I
i599o
7645
262500
34340
Deflected perpendicular
Set I.
to axis of pins.
74 l
(swelled.)
i599o
7.624
255650
33530
Do. do.
739
~~
i ~~t
239.70
75 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
JK
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
WROUGHT1RON COLUMNS.
435
TABULATION OF EXPERIMENTS ON WROUGHTIRON COLUMNS
WITH 3JINCH PINENDS.
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
19925
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
p10
I
.2
4
300.00
1757
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 seventyfour
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 pinends.
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 channelbars, giving an eccentric bear
ing to these columns, on account of the continuous plate on
one side of the channelbars."
TABULATION OF EXPERIMENTS ON WROUGI ITIRON 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
Bucklingplate D be
tween the riveting.
Bucklingplates.
"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
Bucklingplates.
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
1556
536900
3455
Bucklingplates.
348
T 3 "75
21 .02
708000
33682
Bucklingplates.
ll ^ ^
1
349
; i" ? 
1
X 3 "75
21 .46
709500
33061
Triple flexure.
343
SetG. jfcKa"
_> 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 WROUGHTIRON 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
20 794
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!
25 775
25 788
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 WROUGHTIRON COLUMNS
WITH 3^INCH PINENDS.
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
944
290000
30720
ling between rivets.
Do. do.
354
'*
20 o.o
9.24
267500
28950
Triple flexure.
365
20 o.o
936
279700
29879
Hor. deflection.
438
APPLIED MECHANICS.
TABULATION OF EXPERIMENTS ON WROUGHTIRON COLUMNS
WITH 3^INCH PINENDS.
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
1534
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
channelbars and latticing, these results of tests on fullsize
columns furnish us the best data upon which to base our con
clusions.
WROUGHTIRON COLUMNS.
439
In the Trans. Am. Soc. C. E. for April, 1888, Mr. C. L. Stro
bel gives an account of his tests on wroughtiron Zbar columns,
from which the following is condensed, viz.: The Zirons 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
9435
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
945 6
28000
I 4 6
27750
300
9.516
28400
I 4 6
27750
336
9375
27700
164
25500
336
9643
28000
l6 4
25500
336
9375
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 wroughtiron pipe columns.
They were tested with the ordinary castiron flangecoupling
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 fullsize wroughtiron 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 wroughtiron and steel columns of the following
forms, viz.: i. Angleirons; 2. Tee iron; 3. Channel bars ;
4*. Two angleirons riveted together ; 5. Four angleirons
riveted together; 6. Two channelbars 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 abovedescribed
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
237
7
I
51
30000
i. 08
27800
24300000
88.8
2
2.04
239
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
344
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
239
23000
25200000
78.2
32
359
399
9i
1:051
100.5
65000
239
27200
246OOOOO
78.5
4
4.07
4'53
9fr
100.5
80000
3"
25700
258OOOOO
771
4
4.09
450
"7ft
100.5
69000
2.76
25000
24900000
773
224. Transverse Strength of Wroughtiron.
Wroughtiron 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 WROUGHTIRON. 441
the means to Eaton Hodgkinson to make extensive experiments
on castiron 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 firstnamed treatise he urges very strongly
the use of wroughtiron, instead of castiron, to bear a trans
verse load.
Fairbairn tested a number of wroughtiron builtup beams,
but they were of small dimensions and are hardly comparable
with those used in practice.
In the light of the tests made upon wroughtiron columns,
it is evident that the compressive strength of wroughtiron 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
crosssection, 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 breakingpoint.
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 wroughtiron 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 breakingloads 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
handbooks.
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 deckbeams, 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 WROUGHTIRON. 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
393
59i
7.87
945
n.8i
52.04
52.04
4.13
1785
5 T 95
103. ..4
62.96
62.96
3i44
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 deckbeams 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.)
493
19.88
70.86
56170
25112500
4.26
938
59.06
48920
25823500
352
533
47.24
553 20
25596000
3.48
4.71
3937
54180
26804700
2.36
1.30
31.50
52760
24202400
'93
0.60
23.62
58160
Tensile tests of specimens cut from these deckbeams
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.)
iS75
iS75
19.69
19.69
51480
53180
SH? 6
52610
26449200
25539100
24813900
25605500
23.62
23.62
27.56
2756
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 flangeplates was 51050 pounds per square inch, with a
contraction of area of 17 percent. The tensile strength of the
angleirons 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 wroughtiron 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 WROUGHTIRON 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
435
14 o
14100
48082
28457000
< a
126
5
12.47
12
6450
46624
29549000
< t t 1 !
209
7
4493
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
4305
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
4596
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
7I25
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
Englishspeaking 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 "wroughtiron," shall be called weldiron
(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 weldsteel (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 ingotiron (Flusseisen).
4. That all such compounds, when they will, from any cause,
so harden, shall be called ingotsteel (Flussstahl).
On the other hand, in Englishspeaking countries, those
compounds which have been aggregated from a pasty mass,
usually in the puddling furnace, and which contain slag, are
generally called wroughtiron, 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 ingotiron 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 openhearth 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 remelting
blistersteel in crucibles; the blistersteel being made by the
cementation process, in which bars of very pure wroughtiron,
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 highcarbon 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 castiron, 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 pigiron
that is low in phosphorus.
Openhearth Steel. In the openhearth process, a charge of
pigiron 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 openhearth 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 openhearth 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 wroughtiron, is fusible; unlike castiron, 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 openhearth 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,
448 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 steamboilers, steel boilerplate displacing almost entirely
wroughtiron boilerplate. 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 wroughtiron was formerly employed. Thus the eyebars
and the struts of bridges are almost exclusively made of steel,
also such shapes as angleirons, channelbars, 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 crankpins, 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, highcarbon 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 armorplates, 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 bicyclespokes, for shafts for ocean steam
ships, for pistonrods, 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 openhearth, 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
Yieldpoint, 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 onehalf 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 onehalf 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 sinkheads (in case heads of sufficient
size are used). The coupon or sinkhead must receive the same treat
ment as the casting or castings before the specimen is cut out, and
before the coupon or sinkhead is removed from the casting.
10. One specimen for bending test one inch by onehalf inch (i"X i")
shall be cut cold from the coupon or sinkhead 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 yieldpoint specified in paragraph No. 4 shall be deter
mined by the careful observation of the drop of the beam
Yieldpoint. . J .
or halt in the gauge of the testingmachine.
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. Bearingsurfaces 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 openhearth, 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
Lowcar
bon Steel.
Forgings
of Carbon
Steel not
Annealed.
Forgings of
Carbon Steel
Oiltempered
or Annealed.
Loco
motive
Forg
ings.
Forgings of
Nickel Steel,
Oiltempered
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
differentsized 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 LOWCARBON 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 OILTEMPERED.
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, OILTEMPERED.
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 onehalf 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, onehalf 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 fullsized prolongation of same parallel
to the axis of the forging and halfway 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, halfway between the inner and outer
surface of the wall of the forging.
7. The specimen for bending test one inch by onehalf 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.
OPENHEARTH BOILER PLATE AND RIVET STEEL. 455
8. The yieldpoint 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. Yieldpoint
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.
OPENHEARTH BOILER PLATE AND RIVET STEEL.
Adopted 1901.
PROCESS OF MANUFACTURE.
1. Steel shall be made by the openhearth process.
CHEMICAL PROPERTIES.
2. There shall be three classes of openhearth boiler plate and
rivet steel; namely, flange, or boiler steel, firebox steel, and extra
soft steel, which shall conform to the following limits in chemical
composition:
APPLIED MECHANICS.
Flange or
Boiler Steel.
Per Cent.
Firebox
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
Boilerrivet
Steel.
3. Steel for boiler rivets shall be of the extrasoft
class as specified in paragraphs Nos. 2 and 4.
PHYSICAL PROPERTIES.
4. The three classes of openhearth boiler plate and rivet steel
Tensile
Tests.
ities :
namely, flange or boiler steel, firebox steel, and extra
soft steel shall conform to the following physical qual
Flange or
Boiler Steel.
Firebox
Steel.
Extra Soft
Steel.
Tensile strength, pounds
per square inch .
Yieldpoint, 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 fivesixteenths inch (&") and more than
threefourths 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 oneeighth inch (") in thick
ness above threefourths inch (f") a deduction of one
per cent (i%) shall be made from the specified elongation.
(b). For each decrease of onesixteenth inch (&") in thickness
below fivesixteenths inch (&") a deduction of two and onehalf per
cent (2^%) shall be made from the specified elongation.
6. The three classes of openhearth 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 onehalf inches
(ii") wide, if possible, and for all material threefourths inch (}") or
less in thickness the test specimen shall be of the same thickness as that
OPENHEARTH BOILER PLATE AND RIVET STEEL.
of the finished material from which it is cut, but for material more than
threefourths inch }") thick the bendingtest 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 cherryred, as seen in the dark,
and quenched in water the temperature of which is between 80 and
90 Fahrenheit.
(d). Flange or boiler steel, firebox 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 firebox steel a sample taken from a broken tensiletest
specimen shall not show any single seam or cavity more Homogeneity
than onefourth 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 tensiletest specimen will be furnished from each plate as
it is rolled, and two tensiletest 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 threefourths inch (") or less in thickness the
bendingtest specimen shall have the natural rolled surface
mens S ?or CI ~ on two opposite sides. The bendingtest specimens cut
from plates shall be one and onehalf inches (ij") wide,
and for material more than threefourths inch (") thick the bending
test specimens may be onehalf inch (J") thick. The sheared edges
of bendingtest 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 coldbending specimen and one quenchedbending specimen
will be furnished from each plate as it is rolled. Two
Bending coldbending specimens and two quenchedbending speci
mens will be furnished from each melt of rivet rounds.
The homogeneity test for firebox steel shall be made on one of the
broken tensiletest specimens.
12. The homogeneity test for firebox steel is made as follows: A
portion of the broken tensiletest specimen is either nicked
Homogeneity with a chisel or grooved on a machine, transversely about
Phibox 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
Yieldpoint. ^ eam or na j t m tne g au g e of the testing machine.
OPENHEARTH 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
firebox steel. In the case of locomotive firebox 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 openhearth 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 .
Yieldpoint, 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 fivesixteenths inch (h") and more than
threefourths inch (}") in thickness the following modifica
tions shall be made in the requirements for elongation: Modifications
(a) For each increase of oneeighth inch (") in thick f^* and"
ness above threefourths inch (f") a deduction of one Thick Material 
per cent (i%) shall be made from the specified elongation.
(b) For each decrease of onesixteenth inch (T&") in thickness
below fivesixteenths inch (&") a deduction of two and onehalf 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 onehalf inches (ij") wide, if Tests 
possible, and for all material threefourths ( 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 threefourths inch (f ")
thick the bendingtest specimen may be onehalf 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 eightinch (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 tensiletest 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 threefourths inch (f ") and less in thickness
this specimen shall have the natural rolled surface on two
opposite sides. The bendingtest specimen shall be one and one
half inches (ij") wide, if possible; and for material more than three
fourths inch (I") thick the bendingtest specimen may be onehalf
inch (I") thick. The sheared edges of bendingtest 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 fullsized 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 yieldpoint shall be
determined by the careful observaton of the drop of the Yieldpoint.
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. Seventyfive 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 openhearth 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 yieldpoint, 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 sinkheads, if the heads are of
sufficient size. The coupon or sinkhead, 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 516 inch and more than f inch in
thickness the following modifications will be allowed in the
Elongation. . r , .
requirements for elongation:
(a) For each 116 inch in thickness below 516 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. Fullsized material for eyebars 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 crosssection 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) Seventyfive 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
516
38
716
12
9~l6
58
Over 58 '
ch.
10.20 It
12.75
1530
1785
20.40
22.95
2550
)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",
18" up to 532"
532" " 3i6"
316'" " 14"
5.10 to 6.37
637 " 765
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 openhearth 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 fivesixteenths inch (&") and more than
threefourths inch (}") in thickness the following modifi
Modifications cations shall be made in the requirements for elongation :
for Thin and (#) For each increase of oneeighth inch ( J") in thickness
'above threefourths inch (}") a deduction of one per cent
(i%) shall be made from the specified elongation.
(b) For each decrease of onesixteenth inch (TS") in thickness
below fivesixteenths inch (&") a deduction of two and onehalf 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. Eyebars shall be of medium steel. Fullsized tests shall show
Tensile Tests I2 i P er cent elongation in fifteen feet of the body of the
of Eyebars, eyebar, and the tensile strength shall not be less than
55,000 pounds per square inch. Eyebars shall be required to break
in the body; but, should an eyebar break in the head, and show twelve
and onehalf per cent (12^%) elongation in fifteen feet and the tensile
strength specified, it shall not be cause for rejection, provided that not
more than onethird (J) of the total number of eyebars 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 onehalf inches wide,
if possible, and for all material threefourths 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
threefourths inch (f") thick the bendingtest 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 tensiletest 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 threefourths inch (f ") and less in thickness nSns for CI "
this specimen shall have the natural rolled surface on Bendmg 
two opposite sides. The bendingtest specimen shall be one and one
half inches (i J") wide, if possible, and for material more than three
fourths inch (I") thick the bendingtest specimen may be onehalf
inch (J") thick. The sheared edges of bendingtest 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 fullsized section of tensile
test, specimen length, shall be similarly treated before cutting the
tensiletest specimen therefrom.
12. For the purpose of this specification the yieldpoint 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 openhearth 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
Tendertruck
Axles.
Per Cent.
Driving and
Engine truck
Axles.
(Carbon Steel.)
Per Cent.
Drivingwheel
Axles.
(Nickelsteel.)
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
drivingwheel axles shall be as follows :
Tensile Tests.
Driving and
Enginetruck
Axles.
(Carbon Steel.)
Driving and
Enginetruck
Axles.
(Nickel steel.)
Tensile strength pounds per square inch ....
80,000
40,000
20
25
80,000
50,000
25
45
Yieldpoint 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. Carbonsteel and nickelsteel drivingwheel axis shall not
subject to the above drop test.
be
STEEL AXLES. 475
TEST PIECES AND METHODS OF TESTING.
7. The standard test specimen onehalf 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 enginetruck axles one longitudinal test specimen
shall be cut from one axle of each melt. The center of Numberand
this test specimen shall be halfway 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 yieldpoint specified in paragraph No. 4 shall be determined
by the careful observation of the drop of the beam or halt
. J . Yieldpoint.
in the gauge of the testing machine.
4/6
APPLIED MECHANICS.
11. Turnings from the tensiletest specimen of driving and engine
truck axles, or drillings taken midway between the center
Sample for and outside of car, engine, and tendertruck 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 openhearth 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.
Freightengine
and
Carwheels.
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
Carwheels.
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, onehalf 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 openhearth 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
. 380 . 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 droptest 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 anvilblock 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 twentyfour 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 onesixtyfourth of an inch ( T V) less and onethirtysecond
of an inch (fa") greater than the specified height will be permitted. A
perfect fit of the splicebars, 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 onehalf 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 twentyfour feet
(24'). A variation of onefourth of an inch (J") in length from that
specified will be allowed.
9. Circular holes for splicebars 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 sawcutting 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 SPLICEBARS. 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 SPLICEBARS.
Adopted 1901.
PROCESS OF MANUFACTURE.
1. Steel for splicebars may be made by the Bessemer, or open
hearth process.
CHEMICAL PROPERTIES.
2. Steel for splicebars 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 . 300 . 60
PHYSICAL PROPERTIES.
*. Splicebar 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 splicebar 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
splicebar, 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 splicebar, shall be used to determine
the physical properties specified in paragraph No. 3.
6. One tensiletest 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 splicebar 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 splicebar
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 yieldpoint shall be de
termined by the careful observation of the drop of the beam
Yieldpoint. 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 splicebars 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 splicebar.
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 fullsize pieces, such as columns for bridges or
buildings, beams, large riveted joints, fullsize 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 wroughtiron, and the greater part of them contain
also experiments on steel.
References to such fullsize 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 openhearth 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 testingmachine 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 openhearth 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,
InchLbs.
Mechanical Work
Tensile Streng
in InchLbs.
Pounds per Sq. In.
Ruptured Sectio
753
0.09
O.II
1.009
0.80
21000
3
30000
52475
236
635
1585
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
435
3727
10660.77
126640
7560.37
0.70
0.798
0.50
25000
6
5OOOO
85160
175
453
42.50
1093548
134600

0.58
0.02
0.798
0.50
30000
6
58000
98760
14.9
41.6
58.00
11380.62
152380
758
057
o93
0.07
0.798
0.50
30000
6
55000
117440
10. I
14.0
5243
11169.34
134880
759
0.71
0.58
0.08
0757
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
5o
54
8435
7872.20
158140
761
0.89
o57
0.19
0757
045
45000
12
75000
141290
43
44
9500
6418.53 147860
762
o97
0.80
0.28
0757
045
50000
12
79000
'52550
43
58
108.62
755023 161910
486
APPLIED MECHANICS.
The following tables include sets of miscellaneous tests of
various kinds of steel.
Bessemer Steel.
Openhearth Steel.
Q u 
' W
 ^
i.
 ,1
O ^v
S!
5 .
c
S fc
3 Q.^,
t*
la?
ojj
w"G
If
1sjjS
'S' in C '
3 o
si
%
S6 ,.=
^ *^ C
Ps
2 '^
*j B
u s
!ssT
$3*
*t
o
W
!l
l^esr
J^sr
Is
.7426
70983
40397
547
29139000
.7600
64169
47395
56.7
29392600
.7481
57700
33000
535
28885000
.7500
63083
44137
64.0
30179000
7463
58408
28575
589
32799000
.7600
64477
47394
60. i
30780000
.7285
62761
34787
652
32135000
.7700
62449
46171
63.5
30481000
.7476
50505
19364
725
29479000
.7700
62556
46171
64.3
29073000
7442
51230
r 9550
697
30653000
.7700
62857
46171
595
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
579
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
45o
30058000
7550
65089
41320
61.3
291340:0
.7400
87356
49991
38.6
30090000
.87
43300
21900
756
28500000
75
.7420
86720
48665
49650
46.2
472
28868000
29887000
73
72
44900
46300
21500
22IOO
757
736
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
750
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
Machinesteel.
Boilerplate.
<u
ffjj
Si
W
i_
y *j . c
C !
_o rt  c
3 <3 u
"o'S*
II
Section.
. C
"2 >'"
0i .!U
!<
*o
1S
I 5
3dSf
Jsdsr
1
s
esr
;3d5f

.7608
91795
62693
533
29316000
379 i458
5945
31670
473
20459000
.7629
96256
66723
442
29391000
.384 148
58770
39590
56.3
30270000
7633
96767
65561
44.1
29586000
.365 1.65
32380
456
29305000
7520
92087
59665
404
30482000
.369 1.49
61657
37284
30135000
7593
92091
62940
30848000
.398 1.511
5437
30760
58.0
28608000
7598
92191
58445
504
28968000
.376 1.496
32889
571
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
516
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
513
28884000
.4258x1.0123
54984
67.7
7597
86678
58460
502
29867000
.4225 x 1.0025
60680
56.3
.7580
96119
54292
41.8
29818000
.4125 x 1.0025
61500
553
. 7600
86741
58415
457
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
456
26643000
.49 X 1.02
60024
29012
58.8
26677000
.7699
94513
5907 1
46.6
28148000
.50 X 1.02
59803
29012
505
30012000
.7622
9H35
64654
542
27164000
. 49 x i . 02
61024
29012
58.8
30012000
.7613
86775
60413
478
29291000
5GX 1.02
60393
29412
60.2
29412000
7567
9.6249
61150
452
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
598
26168000
7554
84990
45752
555
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
559
30400000
.1288
68300
1OIO
77500
61.4
30900000
.I2yr
67200
970
74100
635
30000000
.1283
66500
996
77000
571
28500000
.1283
69600
IO2I
79000
57  1
30000000
.1289
69700
I00 5
77000
60.5
3 i 200000
.1281
71400
I43
80900
639
29200000
.128,
65800
1004
77700
62.6
30700000
.1286
68500
IOOO
77000
63.2
31000000
BESSEMER SPRINGSTEEL WIRE.
.0911
76000
9^0
146000
346
24500000
.0910
69900
974
149000
51 .6
25900000
.0905
79600
969
150000
42.0
23000000
.0911
72900
95
146000
375
24200000
.0905
93i
143000
396
25400000
TESTS OF STEEL EYEBARS.
Tests of Steel Eyebars made on the Government Machine.
In the Tests of Metals at Watertown Arsenal for 1883 is
the record of the tests of six eyebars of steel, presented by
the president of the Keystone Bridge Company.
The following is an extract from the report in regard to
these eyebars :
" The eyebars were made of Pernot openhearth steel, fur
nished by the Cambria Iron Company of Johnstown, Penn.
"The furnace charges, about 15 tons each of castiron,
magnetic ore, spiegeleisen, and railends, preheated in an aux
iliary furnace, required six and onehalf hours for conversion.
" All these bars were rolled from the same ingot.
" Samples were tested at the steelworks 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
redhot, 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.
finch round rolled bar .
48040
73 I 5
%
457
28210000
%
0.27
finch round rolled and
annealed bar ....
422IO
69470
542
292IOOOO
0.27
" The billets measured 7 inches by 8 inches, and were
bloomed down from 14inch square ingot.
" They were rolled down to barsection in grooved rolls at
the Union Iron Mills, Pittsburgh.
" The reduction in the roughingrolls was from 7 inches by
8 inches to 6J inches by 4 inches ; and in the finishingrolls, to
6^ inches by I inch.
"The eyebar 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 hammerdie.
"The bars are next annealed, which is done in a gasfurnace
longer than the bars. They are placed on edge on a car in the
annealingfurnace, 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 pinholes 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 testingmachine, the stem from neck
to neck was laid off into loinch sections, to determine the
uniformity of the stretch after the bar had been fractured.
"A number of intermediate loinch 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 testingmachine, 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 PinHoles, 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 eyebars, 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 Pin1
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
74
73
31400000
74173
308.00
514
I .02
34730
69330
10.4
10.3
29279000
84093
308.00
515
I .02
37330
70286' 11.7
ii. 5
29017000
80043
308. 10
514
I .02
35000
70229
ii. 6
ii4
134
79826
Stem
Granular, radi
ating from a
button of
hard metal.
308.00
513
I .02
35950
71680
ii. 8
ii. 5
81323
307.95
5iS
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 fullsize stee eyebars 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 openhearth 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 fullsize steel eyebars, 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 fullsize 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.
FULLSIZE EYEBARS.
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
995
173
358.93
3768o
70160
9.98
i75
36123
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
994
099
287.88
3399
63220
10.05
2.20
222.88
2933
63100
10. 12
1.86
464.03
31970
53860
712
1.17
314.04
30270
51500
IO.07
2.20
338.73
28080
55160
10.03
I.8l
25L58
29670
62140
997
i37
250.28
32700
65400
7.02
385.73
28980
52010
7.01
i! 2 6
385.78
28410
54740
999
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
183
221 .98
32290
62270
10.04
o99
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
i5 c
I*
"+J
e~
P **
"3

Is
So*
If
fs
1
W
Js
w
1
O,
Sq. In.
Lbs.
Lbs.
9500
275
41580
73050
.027
.9918
24.4
42650
75620
015
.9520
28.8
40280
70280
.062
.9500
275
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
313
34190
58460
039
.0170
28.1
41400
67840
.OIO
9338
25.0
40910
70360
.014
.9700
255
40410
69900
.063
954
27.0
40400
70490
.023
5557
295
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 fullsize eyebars
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 fullsize Bessemersteel 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
Pinholes.
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'
63075
20.86
40070
28398000
7
9
1323
25'7T
5 s 795
2734
35570
26557000
8
9
1323
257i"
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
webplate.
2. Failed by bending upwards at rivet in latticing at center,
buckling flange angles and webplate. 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 webplate.
Failed by bending upwards between latticing, 4 inches from.
center, buckling flange angles and webplate.
Failed by bending upwards between latticing, 9^ inches
from center, buckling flange angles and webplate.
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 webplate.
No. 8. Failed by bending upwards at rivet in latticing, i foot
from center, buckling flange angles and webplate.
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 connectingrods 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
8938
100.5
7.19
7 .6o
57730
80280
258
30.9
28000000
38700
36700
B
9838
109.4
7.19
7.78
45 6 5o
78830
20.8
34.1
28300000
40600
37500
C
107.38
118.5
673
7.21
43900
77840
20.4
42.5
30000000
39300
36700
D
in75
125 .0
7.27
7.78
4.7560
79270
22.3
432
28500000
36100
33700
E
116.25
130.0
7.38
7.96
45820
79250
30500000
39300
36400
F
120.63
134.8
7.21
755
49440
81660
24.1
399
28800000
39300
37500
G
12513
1397
7.06
3959
79690
24.4
455
30300000
38000
35000
H
13413
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
5711
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
1575
I4L73
55174
26662500
1575
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 wroughtiron 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 wroughtiron 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 sixtysix 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 today, 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 today.
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 today 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 wellknown 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 testingmachine at Watertown Arsenal for 1885 is given
an account of a portion of a series of tests upon wroughtiron
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 doublerolled
muckbars, 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 tendertrucks of express loco
motives of the Boston and Albany Railroad. Mr. A. B. Under
hill, superintendent of motivepower, 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 wheelflanges 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
subnumber and letter to indicate from what part of the axle
each was taken.
" The tensile testpieces showed fibrous metal, and generally
free from granulation.
" The muckbar 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 muckbar 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 testingmachine 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 muckbar 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 siderods of a passenger locomotive which had been
in service about twelve years.
2. One siderod of a passenger engine which had been run
twentyeight years and eight months.
3. One mainrod which had been run thirtytwo 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 188283. 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 wellknown 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 beamstrap of the Kaaterskill, and of the
connectingrod of the chaincable testingmachine 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 connectingrod of the chaincable testingmachine
we find the crystalline appearance of the fracture less, if any
thing, than that of the beamstrap, 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
wellselected 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 ' coldcrystallization '
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
ferrocarbide, 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
3inch by 3inch square bar, onehalf 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 abovementioned 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
icewater, 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.
Wroughtiron.
.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 cherryred 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,
wroughtirons marked A and B, a muckbar railway axle, and
castiron specimens from a slab of guniron.
The specimens were o".798 diameter, and 5" length of stem,
having threaded ends \ fl '.25 diameter.
Wroughtiron 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
sheetiron muffle, through the ends of which passed auxiliary
bars screwed to the specimens, the auxiliary bars being secured
to the testingmachine.
EFFEC7" OF TEMPERATURE ON IRON AND STEEL. $11
The heating was done by means of gasburners arranged
below the specimen and within the muffle.
The temperature of the testbar 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 testingmachine, 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 testingroom 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.
Castiron 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 strawcolored : after 400, deep
straw ; from 500 to 600, purple, bronzecolored, 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 boilerplates 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 shearingstrength 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 wroughtiron 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 stretchlimit (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 wroughtiron 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 abovestated
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 wroughtiron, 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 ColdRolling 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 coldrolling or
colddrawing, 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 coldrolling:
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 coldrolling, 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 eyebars 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 eightfoot 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 wroughtiron 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
392
0.564
104160
60000
24.6
0.564
93160
47000
392
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
394
0.798
IOI6OO
53000
394
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 breakingstrength 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 onefourth) 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 onefourth 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
ingweight 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 smoothrunning 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 oftrepeated stresses, and of changes of
stress, upon wroughtiron 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 tensionrod
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 breakinglimit. The differences of
the stresses which limit the variations of stress determine the
breakingstrength. 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 breakingstrength 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 carryingstrength (Tragfestigkeit) of the material
per unit of area, which is the same as the tensile strength as
determined by the ordinary tensile testingmachine.
2. u, the primitive breakingstrength (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 breakingstrength 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 caststeel, have already been given in 172.
WEYRAUCH'S FORMULA. $2$
In the same paragraph are given the corresponding values of
by the safe workingstrength, 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 breakingstrength, which has been already
defined.
2. s, the vibration breakingstrength (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(us), (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
axleiron and for Krupp caststeel, are given in 176.
GENERAL REMARKS.
In each case the value of a given by the formula (3) or (6)
is the breakingstrength per unit of area.
If either of these values of a be divided by 3, we have, accord
ing to Weyrauch, the safe workingstrength.
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 axlesteel,
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 axlesteel,
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 breakingload, 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.
SHEARINGSTRENGTH 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
breakingstrengths, 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, testpieces 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 testpieces, 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 breakingstrength.
2. With oftrepeated 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 TENSIONMACHINE. 533
4. The tensile strength is not diminished with a million
repetitions, but rather increased, when the testpiece 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 wroughtiron plates :
/ = 49500 Ibs. per sq. in.
u 28450 " " " "
2. For mildsteel plates (Bessemer) :
t = 62010 Ibs. per sq. in.
= 34140 " " " "
> i
3. For bar wroughtiron, 80 mm. by 10 mm. :
t = 57600 Ibs. per sq. in.
u= 31290 " " " "
4. For bar wroughtiron, 40 mm. by 10 mm. :
/ == 57180 Ibs. per sq. in.
u = 34140 " " " "
5. For Thomassteel axle :
t = 87050 Ibs. per sq. in.
u = 42670 " " " "
6. For Thomassteel rails :
/ = 84490 Ibs. per sq. in.
u = 39820 " " " "
534 APPLIED MECHANICS.
7. For Thomassteel boilerplate :
=57600 Ibs. per sq. in.
u = 34140 " " " "
For Thomassteel axle, and Thomassteel rails, Bauschinger's
obtained for the vibration breakingstrength the same values as
those for primitive breakingstrength. 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 wroughtiron, 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
castiron, wroughtiron, hot and cold rolled metal, and steels
ranging in carbon from o.i per cent to i.i per cent, also milled
steels. The fibrestresses have ranged from 10000 pounds
per square inch on the castiron bars up to 60000 pounds pel
square inch on the higher tensilestrength 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
335
519
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, propellershafts, crankshafts, 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. Shearingstrength 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 bridgepins, crankpins, etc.
3. In the case of riveted joints.
The socalled apparent outside fibrestress 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 shearingstrength, for the same reasons as
render the modulus of rupture greater than the actual outside
fibrestress at fracture in transverse tests.
Moreover, the shearing strength of wroughtiron rivets is
shown by experiment to be about f the tensile strength of the
rivet metal.
In regard to castiron, Bindon Stoney found the shearing and
tensile strength about equal.
The cases where shearing comes in play in wroughtiron and
steel will therefore be treated separately.
230. Torsional Strength of Wroughtiron 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 twistingmoment 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 breakingstrength of
wroughtiron and steel the socalled apparent outside fibrestress
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 beltpulls 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^gg
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 socalled
"apparent outside fibrestress 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 threeeighths to two
fifths of the tensile modulus of elasticity.
TORSIONAL STRENGTH OF WROUGHTIRON 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 bendingmoment ;
M% = greatest twistingmoment ;
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 beltpull sustained by a
shaft in actual use.
4th. Tests are made not only of breakingstrength, 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
375
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
4331
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
85925
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
37815
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 WROUGHTIRON, 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 :
Breakingstrength, 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 bendingmoments actually used were
generally less than such as might easily be realized in practice
with the twistingmoments used.
4th. It seems fair to conclude that, in the greater part of
cases where shafting is used to transmit power, as in lineshaft
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 WROUGHTIRON.
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
950
533oo
11300000
.02
72.00
74970
16.00
46600
13215000
.01
590
86400
8.63
54200
11500000
3
7 1 3
72000
14.00
43757
12902000
.01
53o
84510
6.87
53000
11200000
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8.01
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71.80
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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 UGHTIRON AND S TEEL. 545
ogy upon the torsional strength of various kinds of wroughtiron.
The figures in the column headed " Apparent outside fibrestress "
Mr
are obtained from the formula / = j 9 where M = maximum
twistingmoment, 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
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11.40
546 APPLIED MECHANICS,
232. Riveted Joints. The most common way of uniting
plates of wroughtiron 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 testingmachine 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 testingmachine 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 testingmachine 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 steamboiler, from the case when it is
in a girder or other part of a structure.
In the case of boilerwork, the joint must be steamtight, and
hence the pitch of the rivets must be small enough to render
it so : whereas in girderwork 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 steamtight joint with considerably greater pitches
than those commonly used in boilerwork ; and now and then
some boilermaker 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 boilerwork is now drilled, and drilling is
also used to a very considerable extent in girderwork. 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 rivetholes, 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 softsteel plates
RIVETED JOINTS. 549
than to wroughtiron 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 testingmachine
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 shearingstrength 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 shearingstrength 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 coverplate in inches ;
f s = shearingstrength of rivet per square inch ;
f t tearingstrength of plate per square inch ;
f c = crushingstrength 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 tensionjoint 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.
SINGLERIVETED LAPJOINT. 55 1
Singleriveted Lapjoint. 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.
SINGLERIVETED DOUBLESHEAR BUTTJOINT.
The combined thickness of the two coverplates 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.
DOUBLERIVETED LAPJOINTS.
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 coverplate 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 coverplate 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
fibrestress ; 2, the allowable shearingstress 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 chainriveted buttjoint 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 bendingmoment 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 webplate ;
f t = allowable shearingstress per sq. in. on outer rivet ; f c =
allowable bearingpressure per sq. in. on outer rivet ; / = thick
ness of plate ; h total depth of webplate ; /^ = 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 bendingmoment, 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 fibrestress f t , observ
ing that if f = greatest allowable fibrestress 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 rivetholes are cut left out.
It will be near enough to take for the stress to be deducted on
account of the rivethole 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 bendingmoment 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 =fj, 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 wroughtiron or steel,
the best evidence we have as to the proper values of f s and f e
is furnished by the tests on tensionjoints, 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
Doubleriveted joint 70
Singleriveted joint 50
These wellknown 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 wellknown 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 steamboilers. 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 shearingresistance of the
rivetsteel 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
rivetsteel being practically the same as that of the plates.
The fourth series showed the shearingstrength 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 singleriveted lapjoints, and the last three,
or XI, XII, and XIII, on doubleriveted lap and butt joints.
Series VI was made on twelve joints in finch 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 shearingstrength 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 shearingarea.
Series VII was made upon six (singleriveted lap) joints in
finch plate, with only three finch 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 (singleriveted lap) joints
in six sets of three each, and these are the only singleriveted
lapjoints 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 twothirds that of the tensile
strength of the steel, while the bearingpressure 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
rivetarea, No. 654 with defect of shearingarea, No. 655 with
excess of tearing or plate area, No. 656 with defect of tearing
area, and No. 657 with excess of bearingpressure, 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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564 APPLIED MECHANICS.
Series IX was made on twentyone joints in finch 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 singleriveted lapjoint 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 boilerwork.
Series X was made on eight singleriveted lapjoints in
Jinch and finch 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 bearingpressure. He claims that high bearing
pressure induces a low shearingstrength in the rivets, and that
the bearingpressure should not exceed about 96000 pounds per
square inch ; also, that when a large bearingpressure 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 doubleriveted
joints ; three being lapjoints in finch plate, three lapjoints in
finch plate, three buttjoints with two equal covers in finch
plate, and three buttjoints with two equal covers in finch
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. DOUBLERIVETED LAP AND BUTT JOINTS AVERAGES.
!
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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
handriveting ; 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 rivetholes had a considerably
greater tensile resistance per square inch than the unperfo
rated metal.
2. In singleriveted joints, with the metal that he used, he
assumed about 22 tons (49280 Ibs.) per square inch as the shear
ingstrength of the rivetsteel when the bearingpressure is
below 40 tons (89600 Ibs.) per square inch. In doubleriveted
joints with rivets of about finch 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 bearingpressure should not ex
ceed 42 or 43 tons (94000 or 96000 Ibs.) per square inch. For
doubleriveted buttjoints a higher bearingpressure may be
allowed ; the effect of a high bearingpressure is to lower the
shearingstrength of the steel rivets.
5. He advises for margin the diameter of the hole, except
in doubleriveted buttjoints, where it should be somewhat
larger.
6. In a doubleriveted buttjoint 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 rivethole.
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 singleriveted lapjoints 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 platearea being
thus 71 per cent of the rivetarea. 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 30ton (67200 Ibs.) plate, and 22ton (49280 Ibs.) rivets, a = 0.524
For 28ton (62720 Ibs.) plate, and 22ton (49280 Ibs.) rivets, a = 0.558
For 30ton (67200 Ibs.) plate, and 24ton (53760 Ibs.) rivets, a = 0.570
For 28ton (62720 Ibs.) plate, and 24 ton (53760 Ibs.) rivets, a = 0.606
Or, as a mean, a = 0.56.
(&) For doubleriveted lapjoints 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<Dton (67200 Ibs.) plate and 22ton
(49280 Ibs.) or 24ton (53760 Ibs.) rivets, and 3.82 for 28ton
(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 30ton (67200 Ibs.) plate, and 24ton (53760 Ibs.) rivets, = x.if
For 28ton (62720 Ibs.) plate, and 22ton (49280 Ibs.) rivets, a = 1.16
For 3oton (67200 Ibs.) plate, and 22ton (49280 Ibs.) rivets, a = 1.06
For 28ton (62720 Ibs.) plate, and 24ton (53760 Ibs.) rivets, a = 1.24
568 APPLIED MECHANICS.
(c) For doubleriveted buttjoints 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 machineriveting should have
been adopted throughout instead of handriveting, 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 chainriveting, as
he thought the chainriveted 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 crushingpres
sure on the rivet produced upon the strength of the joint,
there were some old experiments, which showed that', when
the bearingpressure 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 crushingpres
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 singleshear and double
shear joints, and then we have the following three equations :
one by equating the load to the tearingresistance of the plates,
a second by equating it to the shearingresistance 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 doubleshear, d =
57 APPLIED MECHANICS.
In a singleshear joint the rivet cannot generally be made
so big, and in the doubleshear it could not always be made so
small, hence the rivet diameter is chosen arbitrarily, and then
the singleshear joint is proportioned by the equations for shear
ing and tearing, no attention being paid to the crushing, while
the doubleshear 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 lapjoints it would increase the strength, whereas
for doubleshear 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
ingstrength 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 singleriveted joints tested was a single
riveted lapjoint with a single coveringstrip.
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 reenforce 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 wroughtiron. 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
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APPLIED MECHANICS.
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APPLIED MECHANICS.
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APPLIED MECHAA'ICS.
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APPLIED MECHANICS.
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APPLIED MECHANICS.
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APPLIED MECHANICS.
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APPLIED MECHANICS.
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RIVETED JOINTS.
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GOVERNMENT TESTS OF GROOVED SPECIMENS.
Tensile Tests of .Jin.
Grooved Specimens
WroughlIron
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
i47
0.250
37636
1.50
0.252
37326
1.48
0.249
41030
1.48
0.247
39480
i47
0.250
37446
i45
0.251
39533
1.96
0.281
43194
i95
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
244
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
WroughtIron
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
152
0.273
51300
1.48
0.251
47220
1.51
0.273
53400
152
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
251
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
302
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
o99
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
i45
0.248
64080
145
0.252
64000
i45
0.249
61025
i5*
o 251
59420
1.96
i93
0.250
0.252
599oo
63500
1.98
1.96
0.250
0.251
59350
59060
249
0.249
58100
2.47
0.249
63900
243
0.250
61640
295
0.251
56530
3oi
0.249
58780
304
0253
555oo
2.97
0.252
60060
2.98
0.251
54050
2.97
0.249
56040
Tensile Tests of Jin.
Grooved Specimens
Steel Plate
Drilled.

si o
C C
W **
^o
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$
C/5 ,y
v
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4=0
II
E " "~
I* 8
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Inch.
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0.52
o54
0.246
0.248
67890
67160
053
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
i54
0.250
64390
1.52 0.251
63350
1.5
0.253
64370
i54
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
253
0.248
59390
303
0.251
61577
3.00
0.249
59080
302
0.251
59550
302
0.250
59700
3.00
o 250
63370
3.00
0.251
58630
303
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
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5
Inch.
Inch.
Inch.
Inch.
Inch.
Inch.
Inch.
Inch.
I.OI
0373
47000
0.98
0.376
50870
1.99
0365
61890
1.97
0.369
63620
0.98
0.370
47520
0.98
0377
52660
0.99
0494
70080
I.OO
0.498
66220
2.OO
0.382
39760
1.98
0379
49710
I.OO
0.492
68130
o99
0495
66800
2. 02
0.383
36630
2.OO
0.380
49830
1.50
0.497
66340
I.OO
0.500
67000
239
0.390
37600
2.50
0.390
50250
i5i
0494
63810
i53
0497
65930
2. 9 8
o395
36340
300
0.392
45 I 5o
1.99
0.499
55930
1.50
0.498
66270
2. 9 8
0.392
39210
300
0393
47540
1.97
0.500
64260
1.98
0.504
67510
347
0.390
37680
350
0.392
43940
243
0.502
52050
2.03
0.502
66730
347
0.389
38340
349
0.390
46490
251
0.504
64360
2.50
0.497
67950
0.97
0.467
 50820
0.99
0.477
47140
300
0.503
60320
2.52
0.501
67440
1.48
0.506
45090
1. 00
0.479
48370
299
0.503
62430
3oi
0.502
66310
1.49
0.506
45050
1.49
0.510
51240
350
0.503
49430
3oi
0.503
66190
1.91
0.513
42500
1.49
0.512
51510
350
0.505
48270
349
0.504
64920
1.97
0.512
43430
1.98
0.514
50050
4.00
0.497
48010
350
0.502
65210
2.47
0.516
39410
1.98
0.516
47790
4.00
0.499
55190
399
0499
64470
2.41
0.513
39720
2.51
0.520
4558o
399
0.501
55780
4.00
0.498
64810
3.00
0515
38950
2.52
0.516
44960
399
0.498
46250
4.00
0.503
64690
2.90
0.517
37290
3oo
05*5
44980
I.OI
0.613
66720
4.00
0.498
64140
35
0.520
37800
3.01
0.519
4703
152
0.612
64800
o99
0.619
60290
349
0513
37770
351
o5 I 3
46170
i5
0.615
64400
1.49
0.614
63610
4.00
0.515
35730
349
0.514
44760
2.50
0.618
58060
1.49
0.616
63450
403
0.516
36690
399
0.510
4533
2.52
0.619
58780
249
0.620
59170
399
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
346
0.615
58410
3oi
0.617
59270
0.97
0.614
49770
o97
0.628
47220
351
0.615
57190
350
0.614
61610
I.OI
0.619
52960
1. 00
0.626
4835
4.04
0.612
54450
349
0.617
62060
1.48
0.618
46320
1.52
0.625
47170
43
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
35
0.615
37480
3.46
0.616
4777
1.50
0.719
62890
I.OI
0.727
58790
35
0.616
36940
347
0.617
449
350
0735
56730
i5*
0.726
59290
4.04
0.619
373 10
39 1
0.625
44840
351
0733
54220
35o
0.736
58700
; 0.98
0.678
50840
3.96
0.626
45ioo
349
0.729
59 x 8o
]
I.OI
0.682
46590
099
0.695
47500
i49
0.688
4597
0.99
0.691
52780
3.48
0.691
4035
I5I
0.692
48470
( 353
0.692
39380
344
0.700
47750
349
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
singleriveted buttjoints, in which the covers are extended so
as to be grasped in the testingmachine ; 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 bearingsurface 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 wroughtiron, and range from
3$" to lyV diameter; they are machinedriven in drilled holes
iV' l ai "g er in diameter than the nominal size of the rivets. Ten
sile tests of the material accompany the tests of the joints.
From each sheet of steel two teststrips were sheared, one
lengthwise and one crosswise. The strips were 2\" wide and 24"
to 36" long; they were annealed with the specimen plates, and had
their edges planed, reducing their widths to i" before testing.
596 APPLIED MECHANICS.
Micrometer readings were taken in 10" along the middle of
the length of each.
The strength and ductility appear to be substantially the
same in each direction. But the practice of the rollingmill
where these sheets were rolled is such that nearly the same
amount of work may have been given the steel in each direc
tion ; that is, lengthwise and crosswise the finished sheet.
The ingots of openhearth metal are first rolled down to
slabs about 6" thick, then reheated and rolled either length
wise or crosswise their former direction, as best suits the re
quired finished dimensions.
The tensile tests show among the thinner plates a relatively
high elastic limit as compared with the tensile strength ; in the
f$' 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 bearingsurface 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 testingmachine,
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 welldefined 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 rivetheads, 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 rivetheads and closing up any clearance
about the rivets, bringing them into bearing condition against
the fronts of the rivetholes. 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 rivetholes and
the rivets themselves. The movement begins slowly, and so
continues till the elastic limit of the metal about the rivetholes
is passed, and general flow takes place over the entire crosssec
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
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rt
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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
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is rt
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oor^Q^ONO Not^io^^
ooooooooooo
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^ t^ t^oo t^ t^ 10 r^co ONOO NO
CNlMNtNIPllN NCMININCNl
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6i
S d d d d d ^ d d d d
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5 6 d o 6 6 6 6 d o d
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y
s~^ '
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fc os~* JSZ ~
fc o S ^ JS2
l~2
iii? s??
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
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rt cS
C C
a .s
rt 53 .2
^Ij^
tag bfi
3 .2 So"
c
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m
r.
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t l
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la
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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> TJVO 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 foo 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
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t^ i^vo TJ 10 u*> LO loco vo vo tx i ^f 10 LOVO miomM w ^JCON
oooo Sooooooooo ? ooooooooo 0000000
T3 T3 T3 *o tiro 'O'O'a'a'a'O'O'O {g'Oo'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
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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
^
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o2 c'S
ScTScTS^c? S^cTcT cT^^JT Rff'g
00 m * ao t^ * Ttt^
ll
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t^ m ui
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^^ in \T) too vo in 10 in mvo in om^nm ininin
III MI !f
SH
n ^
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ro m
1 1* H
co W
H
1 4i
en
1111 ill II!
!!! HI If
J U
W >
ls
C iovo O * M M * Hroc>ro t)ooio i ^jvo
^"jr as^? ^5
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^ ^ ^ H [:
a
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^'ct^SS^!^^ i^Rcgo? cg^cSocT 82^
JT? ^S? ^?
T
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3 <fl ^
o
.l 1
JllillSS IH? HH S??
1H ?a 11
fan
H W ,HHMH HMHM M M H MMM
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O
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.2  .2 
8
p
l*
u J5 u ^
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en T3 O &QT3 T3 Jg Q
s g s
U J U
J
S1315 en'
^
03
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c
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li
^JS^OCUOJ f^tyiH^ pit/jH^ t>^^
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O MM tn
S5
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' *
yy ^? 5 " ? " li ? R
' ^ S^
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s
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y*M Tj^O^OOOOOOv
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8
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s 5
23 *
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< r 5 S u
C ^ ^ iO u"> VO ^ VO t^. OO OO
s s
Q V
e
is
tJfOlO OOrOlOMVONOOM
UMW 4MOiooovdr.oo'
* 00
11
a
11
g .ff ff ^ffyasjj'g.fi
S 12
^ be </) ^
i tis^&^SS^
'ITJIA mmiou^ioioioio
VO CO
1 s
H ** "rt
o
H
ill  HI!! fl
vg 
10 10
<y .
Q A 3 fl
^oo S o? m2^5Rv2 ^;J
00 O
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rt 5
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oo oooooooo
10^ 5^0^^SooxoO
2?C>ON f>.^JCI N rOONVO tx
^oooo 2"^ c?c?&o
II
ill
.sf.f s 1 1 Is f i 1
10 10
cT o
13 A
3ifc
33*
^^ \O*OON^N NVOVO
E ^
111 si
c w ^ * <* * *  ^g s
"R "R
M M
M
#3*
* ? 1 f & S, & S S
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
975
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
I35I
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
3350
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
I3I3
<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
1275
44
ii
"
.240
2
i34i
G
C
C
11
6
44
it
it
M
.242
2
1342
L
C
D
2*
6
I35I
44
4i
it
.250
2
1343
L
D
D
"
6
"
"
it
"
.250
2
TABULATION OF RIVETED JOINTS.
603
RIVETED BUTTJOINTS.
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.
1452
sq in. sq. in.
.908 3.682
Ibs.
61740
Ibs.
41690
Ibs.
67770
Ibs.
108370
Ibs.
26720
675
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
2745
1.830
9*5
3682
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
1452
1.089
5300
61740
40000
70000
93330
19180
64.8
2.750
1652
1.098
5.300
61740
40980
68210
102630
21260
66.4
2.772
1665
i . 107
5.300
61740
4 I 43
68980
103750
21670
67.1
2 942
1.840
I. 102
5300
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
33 10
2.207
I.I03
5.300
59180
43040
64540
129150
26880
72.7
3337
2.225
I . 112
53
59180
43810
65700
131460
27580
74o
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
638
2.892
1.627
1.26 5
7.216
62660
40340
71700
92210
16170
644
2.909
1638
I.27I
7.216
62660
41280
73310
94480
16640
659
3075
1.810
1.265
7.216
62660
42290
71850
102810
18020
675
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
3321
2.029
I.2QI
7.216
59180
4*45
67840
106620
19080
70.0
3534
2.232
1.302
7.216
59180
41820
66210
113500
20480
707
3534
2.232
1.302
7.216
59 l8 o
42760
67710
116070
20940
723
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
9425
62660
40620
81230
81230
I23IO
64.8
2.976
1.488
1.488
9425
61740
36290
72580
72580
11460
58.8
3.060
i .620
1.440
9425
62660
38660
73020
82150
"550
61.7
3.088
1.636
1.452
9425
62660
38000
71730
80820
12450
60.6
3.378
1.878
1.500
9425
61470
37800
68000
85130
13550
61.5
3375
1875
1.500
9425
61470
39000
70200
87750
13970
634
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
1313
"
it
it
.244
2
J 349
G
C
C
44
5
"
"
44
*
239
2
I35<>
N
D
D
a*
5
1377
"
it
M
.250
2
I35i
N
D
D
44
5
"
"
44
44
.249
2
1352
H
D
D
2*
5
1439
"
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
975
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 BUTTJOINTS 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.
356o
2.060
1.500
9425
61470
3913
67640
92870
14780
63.6
35 20
2.038
1.482
9425
59180
 40450
69860
96070
15110
68.3
3765
2.259
1.506
9425
60480
4359
72640
108960
17410
72.1
3.780
2.268
1.512
9425
60480
41420
69030
103540
16610
68.5
3.204
1.984
i .220
7854
59180
38700 
62490
101620
!5790
65.4
3i3i
1.936
1195
7854
62660
42890
69370
112380
17100
68. 4
3442
2. 192
1.250
7854
5374
42960
67460
118300
18830
7 j.i
3.426
2.181
1245
7854
55740
41780
65640
114980
18230
749
3554
2 3!9
1235
7854
59i8o
435 10
66690
125220
19690
735
3569
2.329
1.240
7854
59i8o
4383
67170
126160
19920
741
3.780
2.520
1.260
7854
7 Rcj
60480
60480
44580
66870
66610
133730
2I 450 j 73.7
3' 7^5
7 54
444^0
133230
21290 734
PLATE.
3559
2.190
1.369
3.682
54260
40460
65740
105170
39100
74 .6
3549
2.184
1365
3.682
54260
39420
64060
102490
38000
72.6
3829
2.460
1.369
3.682
54260
39780
61910
111250
41360
733
3843
2.471
1.372
3.682
54260
39060
60740
109400
40770
72.0
3843
2.196
1.647
5300
54260
37000
64750
86330
26830
68.2
3854
2. 2O2
1.652
5300
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
5300
54260
38040
63390
95130
.29470
70.1
4656
2.910
1.746
53oo
59730
41820
66910
111510
3673
70.0
4.680
2.925
1755
53oo
59730
42000
67200
II2OOO
37090
703
4.683
3031
1.652
5.300
57870
41040
63410
116340
36260
70.9
4938
3197
1.741
53
59730
40910
63180
116030
38110
68.5
4i5i
2.214
J937
7.216
54260
35000
65620
75000
20130
645
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
652
4.916
2.895
2.021
7.216
59730
38730
65770
94210
26390
64.8
4.672
2745
1.927
7.216
57870
39010
65660
94580
25260
67.4
5.076
3.102
1974
7.216
53730
37960
62120
97620
26700
70.6
5130
3^35
1995
7.216
5834
39810
65140
102360
28300
68.2
5450
3439
2. Oil
7.216
53730
38920
61670
105470
29390
724
5.290
3343
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
1349
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
1375
"
"
374
2
1393
C
L
N
"
5
"
K
"
"
372
2
J 394
D
I
M
2
5
M39
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 BUTTJOINTS 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
3734
2. Oil
7.216
53730
40560
62400
115860
32290
755
5 775
3754
2.021
7.216
5373
40700
62620
116280
3 2 57o
757
4656
2.328
2.328
9425
5973
345
69010
69010
17050
578
457 2
2.286
2.286
9.425
58340
33440
66880
66880
16220
573
4947
2.619
2.328
9425
59730
35590
67230
75640
18680
596
4.883
2585
2.298
9425
58340
34730
65610
73800
17990
595
5.140
2.854
2.286
9425
54290
3549
63930
79810
19360
65.4
5126
2.846
2.280
9425
58340
35840
64550 i 80570
19490
61.4
5244
3.036
2.208
9425
56670
37010
63930
87910
20590
653
5205
3oi5
2.190
9.4 2 5
53840
36750
63450
87350
20300
68.2
5775
3465
2.310
9425
53730
3749
62480
937io
22970
69.8
579
3474
2.3l6
9425
53730
3736o
62260
93390
21890
69.5
4.881
3021
1. 860
7.854
57870
39000
63010
102340
24240
674
4.841
2.996
1.845
7854
57870
39520
63850
103690
24360
68.3
5143
3273
I.SjO
7854
57870
39840
62590
109540
26080
68.9
5.111
3251
1.860
7854
57870
40420
63550
111070
26310
69.8
5o55
3625
1.930
7854
53730
3934
60290
113240
37830
732
5496
3.58i
I 9 I S
7854
5373
40300
eiseo
115660
28200
75o
6O8 APPLIED MECHANICS.
SINGLERIVETED BUTTJOINTS, STEEL PLATE.
DESCRIPTION OF TESTS AND DISCUSSION OF RESULTS.
" The following tests complete a series of two hundred and
sixteen singleriveted buttjoints 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 machinedriven. Iron rivets were used
throughout, except in some of the f x/ joints.
Tensile tests of the plates and rivetmetal, together with
the tests of the joints in " and f " plate, are contained in the
Report of Tests of 1885, Senate Document No. 36, Fortyninth
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
SIXGLERIVETED BUTTJOINTS, 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 singleriveted 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 rivetholes, there remains an
excess of strength along the line of riveting, or in other words
along the net section of metal if in a singleriveted 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 testpieces 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 teststrip, 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 teststrip in per
cent of the latter.
Table No. 4 exhibits the compression on the bearingsurface
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), 23" 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 rivetareas 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
SINGLERIVETED BUTTJOINTS, 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
534
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 rivetholes.
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
137
143
16.3
JSi
P. ct.
11.4
12.7
93
14.2
13.8
P. ct.
12.0
135
107
145
I4.I
P. Ct.
134
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
99
IO.O
P. Ct.
131
13.6
12.2
0.8
10. i
ii. 8
P. Ct.
10.6
35
'
i'
!/
Average of all thick
nesses
16.2
14.4
12.3
12. Q
II. 9
II.
97
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
SINGLERIVETED BUTTJOINTS, 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
bearingsurface 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 oilpockets.
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 gasburners ; the upper side and edges of the joint were
covered with fine dry coalashes.
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 teststrip.
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;
SINGLERIVETED BUTTJOINTS, 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 testingroom
(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 shearingstrength 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
I3SO
"
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
1275
44
it
11
472
2
4'5
S
S
S
"
6
11
*'
11
"
.468
2
,416
S
Q
Q
at
6
JSSO
"
"
"
.468
2
1417
T
Q
Q
"
6
u
44
it
it
.482
2
1418
T
a
6
1425
"
"
it
.481
2
149
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
I3I3
"
it
11
.469
2
1423
S
S
S
"
5
"
l<
"
it
473
2
1424
T
a*
5
1375
"
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
I350
"
"
11
479
2
1429
S
Q
"
6
"
"
14
"
465
2
'43
U
o
o
f
6
1425
"
44
44
484
2
M3i
U
o
"
6
"
11
"
"
483
2
I
TABULATION OF RIVETED JOINTS.
6l 7
RIVETED BUTTJOINTS.
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.
cl
^ c o 5"!
CN 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.
5051
2.886
2.165
53 01
57180
37750
66070
88080
35970
66.1
2OO
5.082
2.904
2.178
53*
57180
38980
68220
90960
37370
68.1
5445
3.267
2.178
5301
57180
45560.
75960
113960
46820
796
3 00
5439
3.266
2173
53o:
57180
39260
65400
98290
40290
68.6
5.842
3655
2.187
5301
57180
37000
59*40
98830
40770
647
S79 6
3.622
2.174
53 01
57180
39420
63080
105100
43100
68 9
35
5416
2.891
2525
7.216
57180
36890
69110
79*3
27690
645
5477
2.926
2551
7.216
57180
38250
71600
81730
29030
66.9
250
5832
3.281
255*
7.216
57180
379*0
67380
86670
30640
66.3
5854
3297
2557
7.216
57180
43730
77650
IOOI2O
3548o
764
300
5997
3529
2.468
7.216
59050
44790
76110
108830
37220
76.0
400
6.010
3537
2473
7.216
59050
39210
66630
953*0
32660
66.3
6.561
4.010
255 1
7.216
60000
39850
65210
102500
36230
66.4
6.512
3.982
2530
7.216
60000
46610
76220
i 19980
42060
77 .6
500
6.859
4 334
2525
7.216
60000
45300
71690
123050
43060
755
350
6.916
437
2.546
7.216
60000
40050
63390
108800
38530
66.7
5813
2.909
2.904
9425
57*80
35920
71770
71900
22150
62.8
250
5772
2.886
2.886
9425
57*8o
34390
68780
68780
21060
60. i
6.023
3i9i
2.832
9425
59050
35000
66020
7443
22360
592
5967
3159
2.808
9425
59050
40250
76030
85540
25480
68.1
300
6.327
35 T 9
2.808
9425
59050
34660
62320
78090
23160
60.3
200
6.512
3 . 620
2.892
9425
60000
36950
66480
83220
25530
61.5
6.859
3973
2.886
9425
60000
437*o
75460
103880
31810
72.8
(*)
6.883
399 1
2.892
9425
60000
38720
66770
92*50
28270
645
7194
4320
2874
9425
58000
44840
74670
112250
34230
773
(t)
7 2 45
4347
2.898
9425
58000
38740
64570
96850
2 97 Po
66.9
6. .67
3 822
2345
7854
59050
39730
64110
104490
3T200
67..
6 220
3855
2365
7854
59050
45420
73250
119450
35840
76.9
400
6.660
4.240
2.420
7854
60000
48950
76890
137110
41510
81.5
(*)
6 632
4.217
24*5
7854
60000
40600
63860
111490
34280
67.6
6053
2853
3.200
11.928
59050
35070
74410
66340
18630
593
3 00
6.042
2.836
3.206
11.928
59050
30420
64810
5733
15410
51.5
6.471
3238
3233
11.928
60000
40330
80620
80730
21880
672
350
6.278
3*39
3139
11.928
595
33420
66840
66840
*759
565
6.897
3.620
3277
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
1313
44
41
44
.472
2
*435
S
S
S
"
5
"
44
44
44
475
2
1436
T
o
*!
5
1375
44
44
44
.482
2
1437
T
11
S
"
44
"
11
.482
2
1438
U
P
P
aj
5
1438
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
1350
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
1275
41
tt
iA* i*
.624
2
14^5
V
D
E
"
6
44
"
44
44
.623
2
1466
w
G
G
aj
6
I350
it
44
44
.613
2
1467
w
N,V
G
"
6
"
"
"
"
.606
2
TABULATION OF RIVETED JOINTS.
619
RIVETED BUTTJOINTS 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
*
lra3*
c 5
c
AJ
w w .
Gross.
Net.
c
8
gOsS
' 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
443
3227
11.928
58000
36140
65000
81430
22030
62.3
7215
3.968
3247
11.928
58000
44490
81050
99040
26960
76.7
500
6.193
3.538
2655
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
3921
2.711
9940
60000
38720
65440
94730
25840
64.5
6.965
4243
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
4519
2.711
9.940
58000
39180
62690
104500
28500
675
200
7.250
4.528
2.722
9.940
58000
40360
65060
108220
29630
70.0
7564
4837
2.717
9940
57410
38570
60450
107610
29410
67.2
7565
4843
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
3744
3.276
7.216
55000
3453
64740
74000
3359
62.7
7393
4.158
3234
7.216
55000
36760
65340
84010
37650
66.0
7.488
4.212
3.276
7.216
55000
35120
62440
80280
36440
638
7.918
4658
3.260
7.216
55000
41930
71270
101840
46010
76.2
300
7956
4.680
3.276
7.216
55000
36800
62560
89370
40570
66.9
8.241
5039
3.202
7.216
57290
39320
64290
101 180
44900
68.6
400
8.249
5.042
3207
7.216
57290
36850
60290
94790
42130
643
600
7.488
3744
3744
9425
55000
32080
64150
64150
25480
583
7.440
3.720
3.720
9425
55000
32060
64110
64110
25300
58.3
793 1
4.199
.3732
9425
55000
34120
64440
72510
28710
60.0
7.880
4.172
3.708
9425
55000
34000
64220
72250
28420
61.8
8.262
459
3.672
9425
57290
36490
65680
82110
32000
63.6
8.249
4S83
3.666
9425
57290
36020
64830
81040
31520
62.8
8 662
5.002
3.660
9425
57290
37720
65310
89260
38490
65.8
8.664
5.016
3648
9425
57290
37540
64850
89170
345'o
655
9255
5553
S? 02
9425
55940
37300
62160
9325
36630
66.6
9.282
5574
3.708
9425
55940
37000
61610
92620
36440
66.1
8.259
5.109
3I50
7854
55000
3578o
57840
93810
37620
65.0
7965
4925
3 040
7854
57290
36960
59770 96840
37500
645
7950
3.738
4.212
11.928
55000
31000
.66130 58690
20720
565
7949
3744
4205
ii .928
55000
31090
66020 58780
20720
565
8.269
4 I 3 I
4138
i i . 928
57290
33*5
66350 66240
22980
578
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
M25
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
1313
44
44
it
.628
2
1473 : W
F
F
44
5
"
ii
44
it
.609
2
i 474 i W
H
D
at
5
1375
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
I350
44
11
ii
749
2
1487
z
L
M
44
6
44
11
* 4
it
.764
2
14 8
z
N
N
2f
6
I425
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
I350
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
I3I3
II
ii
44
733
2
1499
z
L
L
44
5
44
44
44
44
744
2
1500
z
N
M
2*
5
1375
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 BUTTJOINTS 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
afcgcL
bC U
. c o <u
cx'C<2 >
sg^S
Sfflug
be
C w
' oj
$*0>
% *
Gross.
Net.
sq. in.
sq. in.
sq. in.
sq. in.
Ibs.
Ibs.
Ibs.
Ibs.
Ibs.
3735
4597
4.138
11.928
572>o
34260
65110
72330
25090
598
8.690
4579
4. in
11.928
"
34790
66030
73540
25350
60.7
9.285
5i7
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
3533
9.940
55oo
36350
63640
84770
30130
66.1
7.978
4552
3.426
9.940
57290
37100
65020
86400
29780
647
8.362
4936
3.426
9940
"
38150
64630
93110
32090
66.5
8 381
4950
343i
994
"
38120
64540
93120
32140
66.5
8.833
5.402
3431
9.940
"
38620
63140
99410
343io
67.4
8739
53'3
3.426
9.940
"
38180
62830
97430
3358o
66.6
9.240
5775
3465
9.940
55940
38480
61570
102630
35770
68.7
9345
5.841
3504
9.940
"
38410
61430
102440
36110
68.6
78i3
5.000
2.813
7952
55000
37340
58360
103730
36690
67.9
7763
4.968
2795
7952
"
38440
60060
106760
37520
699
8.847
4431
4.416
9425
59000
31990
63870
64090
30030
542
9.099
4557
4542
9.425
"
31980 63860
64070
30870
542
9475
5023
4452
9.425
"
34440 64960
73920
34620
58.3
9723
5*5i
4.572
9 425
41
34700 67340
73790
35800
58.8
IO.II2
5.618
4494
942S
14
35000 63000
78750
37550
593
10.329
5745
4584
9425
"
36780 I 66130
82870
40310*
62.3
10.624
6154
4.470
9425
"
38120 65810
90600
42970*
64.6
10.488
6.078
4.410
9425
11
34000 58670
80860
37830
576
9233
4353
4.880
11.928
"
31050 65860
58750
24030
52.6
9.596
4.520
5.076
11.928
4
32000 \ 67940
65340
25740
542
9 .951
4983
4.968
ii .928
"
34270 68430
68640
28590
580
 10.179
5 082
5090
11.928
"
33770  67540
67520
28810
572
10 845
57*5
5130
11.928
34900 66230
73780
31730
591
10.838
5.708
5!30
ii .928
"
35810 67990
75650
3254
60.6
i*75
6.146
5029
11.928
60420
38470
69940
85480
36040
63.6
10. 890
5996
4.894
11.928
"
37740 68650
83980
34460
62.4
9.624
5501
4 I2 3
9.940
59000
35000 61230
81700
33890
57
9 776
5572
4.204
10.030
"
36470 63990
84810
35550
61.7
10.478
6. 192
4.286
9.940
11
38760 65590
94750
40850*
657
10.004
59!5
4.089
9.940
It
36740 62130
89880
36970
62.2
10.390
6 329
4.061
9940
"
3793
62270
97050
39650*
643
10.663
6495
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
1350
3/4
7/16
iA*il
.722
2
*505
Z
M
M
"
6
11
"
44
.762
2
1506
Y
N
at
6
1425
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
1313
"
"
44
.748
2
'5"
Y
L
L
"
5
"
"
"
(1
755
2
1512
Z
....
at
5
1375
"
"
44
.750
2
1513
Z
N
N
"
5
"
44
"
41
.764
2
JSH
Y
o
3
5
MiS
"
"
"
.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
1350
11
44
(I
.742
2
1523
Z
M
M
"
4
"
"
M
754
2
TABULATION OF RIVETED JOINTS
623
RIVETED BUTTJOINTS 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
os
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
4346 
5415
14.726
59000
29090
65350
52460
19280
493
10.287
4 572
57I5
14.726
59000
30010
67520
54010
20960
50.8
10.367
4.914
5453
14726
60420
33610
70900
f389o
23660
556
10.474
4.961
5513
14.726
60420
33660
71070
63960
23940
557
i i . 070
5542
5528
14 726
60420
3478o
69470
69650
26140
575
11.310
5662
5648
14.726
60420
34670
69250
69420
26620
574
9 918
5243
5675
12.272
60420
36120
68380
76680
29210
597
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
573
4775
12.272
59coo
35260
64640
77570
30100
597
10.929
6.179
475
12.272
60420
3793
67080
87260
36220
62.7
io735
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
4631
12.272
60430
38740
66440
92920
35060
64.1
9465
5.685
3.780
9.818
59000
3756o
62360
9378o
36110
63 6
9 2 /7
9934
5572
6. 119
3705
9.818
9.818
59000
59000
39000
37600
64930
61040
97650
97900
36850*
38040*
66.1
637
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*
677
10 . 187
6.417
3.770
9.818
59000
39720
63050
107320
41210*
673
* 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
675
68.9
72.8
1313
68.3
69.9
72.1
t" . . . .
1314
1323
1324
659
64.8
66.4
67.1
62.6
68.3
68.6
64.4
I' 3
675
1337
639
659
68.2
1338
....
64.8
61.7
I35S
58.7
60.6
1356
746
73.3
1359
72.7
72.0
....
1360
68.2
69.0
70.0
70.9
I" . . . . 
^67
1368
68.3
70.1
645
70.3
61.7
68.5
64.8
1379
63.0
65.2
67.4
1380
58.7
596
1395
....
573
595
300
300
1396
....
66.1
79.6
64.7
....
1401
68.1
68.6
350
68.9

1402
....
645
66. 3
400
76.0
w
1411
....
250
66.9
300
76.4
66.3
1412
....
....
....
260
62.8
592
1425
...
....
....
60. 1
68 i
300
1426
....
....
....
....
593
,
1443
....
....
....
515
1444
....
....
60. 1
66.0
300
7 6.2
//
M5i
62. 7
63.8
66.9
. . . .
1452
. . .
58.3
60.0
1463
583
61.8
1464
...
..
.
56.5
1481
565
1482
. . .
,
542
58.3
1489
. .
542
58.8
1490
1503
. . .
....
52.6
542
1523
' '
....
NOTES. Figures in heavy face type denote that
Super numbers state the temperature of
TABULATION OF RIVETED JOINTS.
62 5
IsO. i.
SINGLERIVETED BUTTJOINTS.
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
634
70.7
72.3
637
68.4
76.8
67.1
72.1
68.5
69.8
68.3
6s4
68.4
77.1
75o
735
74 *
m
70.7
68.2
65.4
64.1
72.4
70.4
653
68.3
755
75 6
69.8
69.5
6 7 :;
68. 3
6S!8
69.8
732
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
578
58.0
59.3
62.3
58.0
57 2
493
50.8
75 6 .5
66.7
400
72.8
645
700
62.7
58.0
600
773
66.9
62.3
500
76.7
6 7 .2
400
76.9
62.O
6l.2
600
81.5
67.6
800
705
64.5
600
80.3
67.0
200
675
70.0
67.2
68.2
....
....
65*8
655
598
60.7
64.6
57.6
591
60.6
55 6
55 7
66.6
66.1
62.5
62.1
6 3 "6
62.4
575
' 574
65.0
64.5
66.1
64.7
67.0
61.7
597
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
658
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. SINGLERIVETED BUTTJOINTS, 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.
97
6.8
73
200
90
II. I
II . I
250
135
850
12.8
10. I
300
12.2
4.6
6.4
94
83
83
95
94
4 1
4 1
55
7 1
4.8
6.4
perct.
46
56
63
7.0
8.1
9.6
8.8
7.6
9.1
7.7
9.0
IO. I
54
I' 9
6.7
6.6
300
19.6
8.6
IO.O
300
2O. I
6.2
300
152
360
I 7 .2
65
9.7
75
7 1
8.9
78
8.0
\\
79
7.2
8.2
83
per ct.
6.1
54
7.2
6.1
8.6
94
59
78
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
56
xi. 7
8.9
58
10.5
8 7 :i
7.6
92
12. I
85
ii. 8
2.1
35
6.6
i:i
134
"3
8. '2
8.8
'i : ?
75
7.8
11
99
59
2. I
350
6.4
400
17.2
74
200
47
59
700
10.2
53
300
17 4
8.1
8.0
72
7.2
8.0
3.7
6.6
6.4
79
8
6.2
3.8
96
7i
74
10.4
9.4
72
9.8
95
10.5
10.6
i:?
5.2
6.2
79
9.8
....
53
500
16.4
400
149
65
69
600
21.7
400
75
600
32
8.1
76
75
7 1
67
0.4
8.6
1:1
8.6
ll! 3
3.5
500
I 7 .2
6.9
49
4i
52
400
14.9
600
II.4
54
500
17.8
4.0
500
19.4
6.1
....
200
5
75
3.2
4.2
6.6
6.0
9.0
76
3.1
2.7'
75
74
6.2
5.8
6.2
6.1
3.9
5.9
.02
47
2.4
5i
"6.6
3.1
6.2
74
34
79
7.6
5.8
3.5
6.0
2.1
0.6
2i"
perct.
4.7
4.3
NOTES. Figures in heavyfaced 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
TESTSTRIP. SINGLERIVETED BUTTJOINTS, 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.
98
n. i
!54
134
17.4
19.7
29.6
17.6
21.2
18.1
193
195
20.9
18.2
155
14.6
200
15.6
T 5 .8
20.9
260
252
350
25.5
20.3
300
2O. O
9 .8
ISI
17.7
16.6
16.6
20.2
20.0
83
8.2
ii. 6
15.2
10.8
14.4
per ct.
72
8.8
10.5
11.7
14.4
17.0
16.5
145
14.1
11.9
15.0
16.8
97
159
12.6
12.5
300
328
14.4
17.8
300
358
ii. 8
2 8 8 8
360
344
13.2
18.8
i35
17.2
16.8
15.8
18.4
10. I
14.1
16.0
145
174
17.6
perct.
9.1
8.1
92
97
M.7
159
10.6
14.2
12.0
125
10. 1
135
17.8
10.6
34
360
10.3
400
28.9
12.8
200
55
10.8
700
I2. 3
10. 1
300
20. 6
13.7
146
13.2
136
153
6.8
12. 1
I2. 3
152
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
85
14.7
14.6
10.
18.0
9i
ii .0
11.9
14.4
20.1
I4.I
ie.'i
1:1
10.7
47
25
21.0
17.7
12.7
135
9.6
5.8
156
11.7
12.8
17.9
14.9
II.4
I6. 3
159
16.1
16.5
8.9
10.3
8.2
98
12.2
IS*
'.'..'.
8.7
600
27.0
400
2 5 .8
"3
12. I
600
397
400
12.2
600
52
14.0
I 3 .2
!37
13.0
11.5
0.6
15.8
136
13.2
16.5
19 6 .5
5.6
600
28.7
"3
8.7
73
8.6
400
24.0
600
19.4
9i
500
28.1
6.4
600
32.0
10. 1
a 8i
12.2
5.3
6.5
11.1
10.1
157
135
5.2
4.3
12.8
12.7
10.2
97
lo.'i
9 .8
e.'i
9.2
3.8
8.5
47
9.6
II. 2
5.3
II.
133
55
"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.
97
ii. 8
7.0
Norms. Figures in heavyfaced 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.
SINGLERIVETED
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.
i37 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
1433
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
I350
44
44
it tt
434
"
5H6
M
O
"
6
I35I
u
it
" P
.421
14
5M7
O
P
P
2*
6
1502
"
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."
BUTTJOINTS.
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.
339
2.31
i. 08
6.01
59180
44180
65760
140650
25270
757
340
2.31
1.09
6.01
61470
435oo
64030
135690
24610
70.8
354
2.46
i. 08
6.01
58170
46300
66630
151780
27270
796
354
2.46
i. 08
6.01
58170
435oo
62590
142570
25620
74.8
369
2.61
i. 08
6.01
58170
46290
65440
158150
28420
79.6
370
2.62
i. 08
6.01
58170
44400
62700
152110
27330
76.3
433
39
1.24
6.28
56760
24040
33690
83950
16580
423
434
3.10
1.23
6.28
57000
26770
3748o
93710
18500
46.9
433
3.10
1.23
6.28
56760
33940
47410
119500
23400
598
3. 7
i. 54
i53
392
61130
35930
71620
72090
28140
58.8
35
1.83
1.22
3i4
61130
41280
68800
103200
40100
675
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
548
6.16
3.48
2.69
8.41
59390
23360
41350
53490
17110
30. \
586
358
2.28
7.21
52910
42250
60160
108600
34340
79.8
569
34
2.29
7.21
61650
39740
66500
98730
31360
64.5
6.20
403
2.1 7
7.21
52910
44150
67920
126130
37960
834
6.17
394
2.23
7.21
61650
36660
57410
101430
3*370
60.0
584
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
1352
(i
"
44
.247
2
4948*
N
H
"
4
1352
"
it
ii
247
2
4949
M
L
3*
4
1451
44
ii
ii
.248
2
495*
M
L
44
4
1451
"
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
i75
5132
N
O
44
8
12.00
ii
ii
IP.
415
i7S
5133*
K
K
If
8
13.00
<
ii
d.
.427
i75
5134
N
N
"
8
13.00
"
it
IP
4*3
i75
5135
M
M
I*
8
M03
"
ii
Id.
.422
i75
5 T 36
"
8
1399
"
ii
IP
.420
i75
5137
L
M
3
7
14.02
44
"
Id.
44
175
5138
P
M
"
7
14.05
"
ii
IP
.420
i75
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
1373
"
4 *
M
.438
2
5142*
Q
3*
5
1567
44
it
M
.422
2
* Pulled off rivetheads.
t Pulled off 3 rivetheads.
$ Pulled off 2 rivetheads.
TABULATION OF RIVETED JOINTS.
631
RIVETED LAPJOINTS.
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
i57
I.IO
300
61000
39750
67850
96840
35510
65.1
2.70
i59
i. ii
300
61000
39660
67360
96490
35700
65.0
2.87
1.87
1. 00
3M
58150
40560
62250
116400
37070
69.7
2.94
1.92
1.02
3i4
61000
37010
56670
106670
34650
60.6
2.90
1.77
113
398
61000
43280
70900
111060
31530
709
2.88
i75
1.13
3.98
58150
42770
73880
119010
30950
735
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
3i7
i. 91
I 26
4.91
61470
41770
69320
105080
26970
68.0
334
2.10
1.24
4.91
55740
42240
67180
112970
28730
757
334
2.10
1.24
4.91
59180
42600
67760
114760
28980
71.9
36o
2.36
1.24
4.91
61470
41390
63140
120180
30350
67.3
3.58
235
1.23
4.91
58170
42150
64210
122680
30730
72.4
4.08
233
J 7S
2.65
5834
25950
45440
60500
39950
44.4
455
2.6 3
1.92
393
58340
33050
57 I 9
78330
38270
56.6
512
2.13
299
4.81
59000
31740
76290
54350
33780
538
4 99
1.98
3.01
4.81
52910
27820
70100
46110
28860
52.6
555
2. 5 6
2.99
4.81
59000
31100
67420
57730
35880
527
537
237
299
4.81
61140
30370
68820
54550
339*0
497
590
295
2.95
481
61650
29240
58490
58490
35870
474
589
295
294
4.81
31870
63630
63840
39020
594
335
2.60
4.21
5939
27580
48900
63000
38910
46.4
59
324
2.66
4.21
52910
28530
51940
63270
39980
539
516
i95
321
7.36
58090
28190
74610
45320
19770
485
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
397
2.64
6.14
56960
30420
50650
76170
32750
534
6 3 2.
APPLIED MECHANICS.
TABULATION OF DOUBLE
CHAINRIVETINGSTEEL 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
1432
44
K
7/8 d.
.247
i}
4920
L
E
D
"
44
10
1432
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
IIS7
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
I358
.
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
I45I
"
44
44
369
"
49.S8
H
M
M
4 '
44
8
I452
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
1438
44
"
i d.
375
2
4969
E
M
3l
44
8
1350
"
4 '
44
.382
4970
F
44
44
8
1358
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
1350
44
44
44
.385
4975
I
O
P
4f
44
6
1465
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 BUTTJOINTS.
CHAINRIVETINGSTEEL 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.
33i
2.52
1.58
6.14
58150
49090
64480
102850
26470
844
33i
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
357
2.48
2.18
12.02
61470
457o
65790
74840
13570
743
316
2.27
1.76
9.62
58150
46330
64490
83180
15220
79.6
3'5
2.27
176
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
32Q
2.63
1.32
7.21
61470
51440
64350
128210
23470
837
3oi
2.41
1. 21
7 21
58150
55500
69320
138070
23170
954
373
2.66
2.1 3
549
61130
46690
66650
83240
32300
76.4
4.27
320
2. 14
549
56760
49040
65430
97850
38140
86. 4
415
323
1.8 4
4.70
57000
46480
59720
104840
41040
81.5
434
3"
247
12.57
56760
44740
62430
78600
iS45o
78.8
4.29
37
2.44
12.57
56760
45490
63570
7998o
15530
80. i
434
3io
2.46
12.57
56760
45530
63740
80330
15720
80.2
334
2.16
235
7.07
5973
43290
66940
61520
20450
72.4
326
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
385
2.70
230
7.07
58340
49130
70060
82240
26750
84.2
479
364
2.30
7.07
58340
48610
63970
101250
32940
833
535
4.25
2.21
7.07
56670
48500
61060
117420
36700
85.5
5.36
425
2.21
7.07
56670
47700
60160
115700
36170
841
4.82
395
r 75
5.30
59730
43070
52560
118630
39170
72.1
477
391
i73
530
58340
42520
51870
117230
38260
72.8
539
352
375
I57I
57870
42890
65680
61650
14720
74 i
5.16
363
3o6
1257
58340
44263
62920
74640
18170
759
5.16
364
34
".57
54290
43240
61290
73390
17750
796
5.86
434
303
".57
5713
44910
60650
86860
20940
78.6
5.20
4.04
2.31
9.42
58340
45980
59180
103510
25380
78.8
546
435
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
533
405
2.56
7.07
59000
48120
63300
100190
36280
833
589
3.85
4.09
I57I
61140
433
66340
62440
16260
70.9
634
APPLIED MECHANICS.
TABULATION OF RIVETED
DOUBLERIVETED LAPJOINTS.
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
JJ
2 i
2t
5
5
10.68
1/4
7/8 d.
257
1 ii
4936
IJ
4 *
44
5
5
10.53
44
44 44
251
k4
4937
LM
2 i
**
5
5
14.38
"
44 44
.248
4938
IL
u
**
5
5
14.40
44
44 44
.249
44
4999
5000
49 6 3
4965
ft?
EE
EE
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
ED
DH
ML
2t
2*
2
ft
5
5
7
5
5
7
11.83
I43 6
14.00
7 /,6
7/8 "
385
370
425
2
5*5
MM
44
7
7
I4.OO
'* P
423
44
KK
2 1
44
6
6
JT . ^7
k *
" d.
.428
4
5 I 52
L0
4 
44
6
6
*35
" P.
.440
,
5153
00
2*
"
6
6
15.01
44
" d.
.409
Ij 7 B
5154
OO
KK
2t
*'
6
5,
6
5
15.02
"
" t
.412
.422
44
DOUBLERIVETED BUTTJOINTS.
4927
MDE
2*
2t
5
4
I436
i/4
3/16
7/8 d.
250
'.H
4928
LDE
5
4
I 435
.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
1312
1312
"
4
.248
.246
44
DOUBLERIVETED LAPJOINTS.
5"9
5120
00
pp
#
t
4
4
3
3
14.00
14.03
S/i6
d.
P
303
305
i*
52I
RR
"
It
4
3
14.03
"
d.
.302
i*
5122
OO
"
4
3
14.03
14
P
.304
"
5123
OO
44
2
4
3
14.02
44
d.
.302
it
5124
PP
4
3
14.02
P
.307
4 v
TREBLERIVETED LAPJOINTS
5157
KK
2t
a
5
5
5
iSM
7/16
7/8 d.
'432
i
5158
OP
3
5
5
5
!505
44
" 4t
.412
5159
5100
PP
LL
3
"
4
4
4
4
4
4
12.78
I350
tt
o 11
'43 2
.438
ii
TABULATION OF RIVETED JOINTS.
635
JOINTS. STEEL PLATE.
CHAINRIVETING.
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.
274
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
357
2.49
2.17
6.01
61470 ' 47870
68630
78760
28440
77.0
358
4.28
2.49
306
2.l8
2.44
6.01
6.28
61470 48530
56760 : 46070
69780
64440
79700
80820
28910
31400
78.9
80. i
4.28
33
253
6.28
593o . 43900
62100
74370
29960
74.1
406
2.32
3.48
530
58340  40570
70900
47330
69.5
4.61
2.88
3.46
530
58340 i 42150
67470
56160
36660
72.2
455
2.63
385
78 5
53730 ! 38790
67100
45840
22480
72.2
531
346
370
7.85 56670 4 }c 5 o
66070
61780
29120
76.0
595
335
521
8.41 ; 61650 40620
72150
4 fi l9o
28740
i 65.8
593
324
8.41 61650 379'o
69380
41940
26730
: 61.4
579
354
4.49
7.21
59000 43150
70570
73  1
594
355
4.78
7.21 59390 38870
65040
48310
12O2O
! 65.4
6.14
399
4.29
7.21 52910 43530
6f QQO
62310
37070
82.3
6. 19
581
395
3.96
4.48
3.69
721 j 52910 40380
6.01 59000 38850
63290
56990
55800
61170
34670
37550
1 76.3
; 65.8
ZIGZAGRIVETING.
3 59
So
97
10.82
58170
48150
69140
87740
15980
80.3
354
.46
95
10.82
61470
47420
68240
86090
15520
77i
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
ZIGZAGRIVETING.
4.24
4.28
423
4.27
303
3.02
3.02
3 01
. 12
.20
.IT
19
550
5 50
5 50
550
56760
5930
54350
54350
42750
40630
42990
44^40
59830
57580
60220
6^000
855^0
79050
86180
86450
32960
31620
33060
34420
753
68. S
79.1
81 6
423
3.02
. II
55
54350
44870
62850
89050
345 10
82.5
433
303
.22
550
593
43490
62150
84820
31400
733
CHAINRIVETING.
593
6.20
552
591
4.04
4.40
4.01
4.38
566
54
454
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 tensionjoint 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 rivetholes ;
2. The shearingstrength of the rivet metal ;
3. The resistance to compression on the bearingsurface 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 rivetholes ; 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 rivetholes, 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 shearingstrength of the
wroughtiron 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 shearingstrength appears to be about 45000
pounds per square inch.
Compression. To determine what we should estimate as
the ultimate compression on the bearingsurface is a more
difficult problem ; for if a joint fails in consequence of too
great compression on the bearingsurface 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 bearingsurface 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 bearingsurface 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 bearingsurface 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 bearingsurface 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
iinch 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 doubleriveted
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 bearingsurface 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 quarterinch 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 wroughtiron rivets, it would be perfectly
safe to use a considerably larger number for compression on
the bearingsurface, 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 wroughtiron rivets in mildsteel 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 boilerplate, and that it has, today, superseded
wroughtiron for structural shapes, as I beams, channelbars,
angles, etc., and that the use of steel rivets has become very
extensive, nevertheless a great many still adhere to the use
of wroughtiron rivets, and feel more confidence in them than
they do in steel rivets. Whereas the use of wroughtiron
rivets had been practically universal, the qualifications for a
good wroughtiron 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 wroughtiron 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 redshort 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 shearingstrength 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 shearingstrength of steel rivets.
Now it would seem that metal with these shearingstrengths
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 boilerrivets from 45000 to 55000 pounds.
While the number of tests that have been made upon joints
constructed with steel rivets is not large, the shearingstrength
of such steel rivets as are in use today 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 wroughtiron rivets.
The following table contains the joints tested at Watertown
Arsenal, which were made with steel plate and wroughtiron
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 wroughtiron, and of steel wire, have already been given, there
will be given here only a few tests of harddrawn, of semihard
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.
HARDDRAWN 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
oi35
56300
28150
oi34
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
SEMIHARDDRAWN 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, openhearth 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; openhearth steel, from 50000 to 130000 pounds
SEVENWIRE ROPE.
Composed of 6 Strands and a Hemp Center, 7 Wires to the Strand.
Approximate Breakingstrain 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
355
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
158
16
3 2
37
42
16
1
ai
I . 20
12
24
28
3 2
17
f
2i
0.89
93
18.6
21
24
18
t*
4
075
79
15.8
l8. 4
21
19
f
2
0.62
6.6
13.2
I5I
17
20
A
if
0.50
53
10.6
123
14
21
j
l
39
4.2
8.4
9.70
II
22
iV
l
0.30
33
6.6
750
855
23
1
1*
0.22
2.4
48
5.58
635
24
f
I
015
i7
34
388
435
25
A
4
o. 125
1.4
2.8
3.22
365
642
APPLIED MECHANICS.
NINETEENWIRE ROPE.
Composed of 6 Strands and a Hemp Center, 19 Wires to a Strand.
Approximate Breakingstrain 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
n95
305
2*
7l
985
254
i
7i
8.00
78
156
182
208
2
2
^i
6.30
62
124
144
165
3
if
si
485
48
96
112
128
4
If
5
42
84
97
ill
5
I*
4f
355
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
97
19.4
22
25
10*
f
2
O.62
6.8
136
158
18
A
if
0.50
55
II .0
127
145
lof
o. 39
44
8.8
10. I
11.4
ioa
A
i*
0.30
34
6.8
78
8.85
lob

it
0. 22
25
5
578
655
IOC
T^>
i
0.15
i . 7
34
405
45
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 caststeel.
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 firebox plates of loco
IRON AND STEEL WIRE. 643
motive and other steamboilers, "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 feedwater 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 castiron.
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 . . . ....
8.7C6
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
8459
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 sapwood, and
the latter the heartwood. 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 sawcuts 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 sawcuts, 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 fullsize 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' PocketBook.
(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 fullsize 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 fullsize 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 fullsize pieces.
Sl^RENGTff OF TIMBER.
647
they have also a far greater homogeneity. They also season
much more quickly and uniformly than fullsize 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
fullsize 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
IKOOOO
0160
( 9000
1 _
Birch . .
1 5OOO
1 64 sooo
6400
( I2OOO
I I7OO
'
Blue cum
8800
( IbOOO
1 _
Box
2OOOO
10300
( 2OOOO
f
Bullettree ....
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
)
Oxhide, undressed
6300
_




Hornbeam ....
20000
_




Lancewood ....
23400





Oxleather ....
42OO
24300




Lignumvitae ....
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 fullsize 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 fullsize 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
hasbeen 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 tiebar
in construction (except in exceedingly rare and outofthe
way cases), it will give way by some means other than direct
tension ; i.e., either by the pullingout 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 tiebar
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 pullingout 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 tiebar is subjected not merely
to direct tension, but also to a bendingmoment, 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 fullsize wooden columns,
with the exception of one set of tests, date from about 1880.
TESTS OF FULLSIZE COLUMNS.
The following are references to tests of fullsize 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 testingmachine.
5. Tests made in the Laboratory of Applied Mechanics of
the Massachusetts Institute of Technology.
6. A series of tests of fullsize 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 YELLOWPINE POSTS AND BLOCKS. 653
cq S ^ 5 <n co eo
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654
APPLIED MECHANICS.
H
S i
t4 a
< t
,2 JS *4 ^
C G C C

^'f ^ s
*> c c c
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c c c
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en ! >>
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CO
ft
III!
O to vO
TJ vO to
1 " s  s s
3, ^ ^ 3
^ oo
*> Q *
VO to CO
CO to CO
N to ON
00 N VO
.
c < ""  S
ill
3*8.5
vO vO vO VO
? '
1 1 1
Q $:s
d d o\ 06
1 1 1
1 1 1
.
* 1 8
jb1
o\ 06 tx vo
O i^ ON
r^
oo oo co
ON 06 t^.
III
.l
. r^ O
e IH oj o M
"dodo
8 8 8
odd
vo O oo
O O OO
d d
N N *H
fl
to >O ON *O
ON M oo >>
n 8 .
ON w CO
V
bJO
A a 1
a 6
!*
M N f) Tf
C/5 C/3 C/5 C/5
O O O O
PH (X, Ps PH
*J 4J CJ
Oi fO Q
(S c2 5
CJ CO '^f
CJ O CJ
PQ P3 PQ
>
49SJSI
}9S pz
jas pj
TESTS OF OLD AND SEASONED WHITEOAK POSTS. 655
8
CCHC "
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11 N 11 N
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ro O l o O "^J" "^" r^4 oo *"* ro ^ vO t^.
N LovOoOOQQvO roco roco O r}
^ 5. ^^^^ =
saqoui arenbs
ui '
asa
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gS ii q rjr^NtvHHvq o^ro LOCO q
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ssqoui ui
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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
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cooo r^t^oq f> ^. \ \ t \ \ i ill
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ssqoui pus
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ro ro ro ro ro w
sqi ui
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SuiqsmSuiisiQ
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 crushingstrength 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 yellowpine 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
i53
7545
233
265000
331980
5, 3d series
298
12 6.8
( 8. 3 0)
\ X \
I 760 )

631
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
275
170000
276480
3, 2d series
236
II II. 2
8.2
i55
50.92
1.91
100000
152760
These results exhibit a great fallingoff. 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 yellowpine square column (sectional area
68.8 sq. in., length 12' 6".85) with one end resting against a
thick yellowpine 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 twentyfive 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
breakingstrength 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 cottonmill 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 baseplate, 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 baseplate 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
10 475
5.82
7.78
31.87
142000
4088
2
10 4.
585
749
27.15
192800
6225
3
10 575
5.85
774
32.17
166100
4900
4
10 55
570
777
31.67
108200
5
10 5.1
570
7.70
30.78
105000
6
TO 5.2
5.80
7.61
30.39
168000
7
9 7
574
781
32.88
194100
8
10 5.4
585
7.90
3421
T55000
9
10 4.9
582
777
40.72
96100
10
10 513
573
7.78
125000
ii 10 4.38
574
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 " wellseasoned spruce, of excellent
quality." Hence the average crushingstrength 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 WHITEPINE POSTS.
659
If
1 i
'
S  5
Q
^
'"". d c> c>
crrorOfO
OOOO U "> L '>
iHr^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 roo rotoo moo a
^^^^iOTjtO^^^Tjto^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^toii N ro^
660
APPLIED MECHANICS.
o 5 5 5
 o
2 5 "
sr:: $^2 *
s <
'J5 . **
S O 'S O 3 3 
W) 5 hO
'S "53 'S
S S 3 S S S 3
S55353523
s
CO f> v> LO
ro i^* ^ ON
M M M M M
r^ N >S O H
M M i ro M
vo
ION N
OO
N
v ooo r~^oo
fO\O NOMD
5 S
il
M M
M M
Q
vo
OQQQOOOOOQQQO
TJ\O^O ONt^^vo fofovqvo q co
M M M r^tv.r^t^r>.t^t^t^^O t^
M w N M M roforovo t^vOMD r^r^t^t^M
3,
M M M rorotOt^r^
dvdvdvd d d d
oo O r^O f
rO'^J  fOO    ii p i'iti
vdvdvdododod d d d ^^
MMNMMNNNN
O O O
a,
W
,
 1 1 1 1
qq
mO
qo
rood
qqoqooooiooooqooo
r^tor^ONfOt^rJfodvdvO
M N N\O r^'OONONWVOvOV
COMPRESSION OF WHITEPINE POSTS.
66 1
d
1
.
11
fl
II.S
S .S g jg
s
rovOvO w r^N
OOOOOQOOOOOOOQ
^NiooqqqNWLOicwNqwvq
r^t^r^^O r^.'Ooooooo O O O roro
oo "nM
. iot^vO
w OMD *r^oo O O <trtQ
^o lOfOroro^ovovOvOvOvO
loinvnvdvdvdooodod
11
& w 4
l$*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
00545
IIOOOO
2107
665
i43
IO
180.00
4.48
11.64
521
O.IOIO
81500
^64
666
163
5
179.97
447
11.63
52.0
0.0895
70000
1346
667
228
13
180.00
540
11.30
61.0
0.0505
160000
2623
668
i93
5
17993
56i
n73
6 5 .8
0.0622
156300
2375
669
253
5
180.00
564
11.76
66. 3
0.0608
152300
2297
638
ISI
6
8
180.00
180.00
450
45
n. 60
"59
?::(
( 0.0670 1
( 0.0750 i
200000
I9l6
639
IS}".
7
5
180.00
180.00
452
4.49
11.66
11.62
?:;K
i 'A 45 i
/ 0.0670 )
212000
2021
640
131
7
5
180.00
180.00
453
452
"59
"59
?!>
i 0.1060 
1 0.0955 }
149000
1419
642
ISSl*
6
7
179.98
179.98
557
558
ii. 61
n. 61
&K
( 0.0770)
( 0.0390 j
215000
1661
643
IS!*
{'S
179.92
179.92
565
56i
ii. 61
11.62
&SI3
( 0.0440 
( 0.0600 )
261000
J 995
644
i3!*
i;
179.96
179.96
5.60
560
11.63
11.62
&I3
 0.0596 
I 0.0690 )
257800
1980
648
l=Si*
\
180.00
180.00
5.60
56i
11.72
11.72
SSI*..
 0.0590 
( 0.0700 J
268000
2042
649
l:Ui'
9
4
180.00
180.00
5.60
561
11.71
11.74
:$!*>
( 0.0600 
J 0.0705 \
277000
2107
650
lisi*
12
9
180.00
180.00
56i
561
"75
11.71
:?!"'
\ 00530 
} 0.0885 i
24OOOO
1824
645
laf*
5
7
179.97
179.97
559
S6o
"59
ii. 60
:;!
 0.0560 
/ 0.0540 i
263200
2028
646
!!
6
12
180.00
180.00
559
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
562
11.62
11.62
:!}*"
0.0630
/ 0.0700 )
248000
1899
678
IS!* 6
16
8
179.94
179.94
5.58
557
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
S6i
11.72
"73
g:S!3
0.0650 1
0.0495 $
278000
2116
663
3
it
180.00
180.00
5.60
S63
"75
"75
S:S!3
0.0621 )
0.0657 i
300000
2273
676
Isil*
( 12
< ^
179.94
179.94
5.60
56i
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
56i
568
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
452
11.62
525)
I 0.0460 )
690
< 226 600
180.00
556
ii .70
65.0} 169.3
I o.o58o[
310000
1831
' T 99
(18
180.00
4.46
11.62
518)
( 0.0480 )
%
(164
( 9
179.98
448
ii .60
52.0)
( 0.0526 )
691
< 197 520
> I4
179.98
556
1 1. 60
64.5 J 168.2
] 0.0430 }
372500
2215
(i59
( 12
179.98
445
ii .61
517)
( 0.0390 )
(248
(13
i77 2 5
562
1 1. 60
652)
( 0.0580 )
692
{ 153 6l 4
12
450
ii. 60
52. 2 > l82.2
I 0.0641 \
363000
1992
(213
( 5
17725
56o
ii57
6 4 .8)
( 0.0768 )
( I 5 I
(ii
180.00
45
ii. 60
S 2.2)
( 0.0460 )
687
J2i8 536
8
180.00
558
ii .62
6 4 .8 169.4
o.o 5 8 7 [
325500
1919
( 167
( 9
180.00
452
1159
52.4)
(0.0533)
(176
(10
180.00
45
ii .60
522)
( 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
450
1 1. 60
52.2)
( 0.0703 )
(188
( 7
180.05
448
n. 60
52.0)
{ 0.0510 )
689
{ 203 550
11
180.05
56o
11.62
65.1 > 168.7
o.o66oj
340000
2015
(i59
I 9
180.05
4.46
"57
516)
( 0.0789 )
fi93l
r s
17995
447
ii. 60
5I91
{0.0700"!
681
38 *
U57J
1
I 9
17995
17995
17995
452
4.48
451
11.65
11.65
11.65
52sJ
0.0714 1
00531 [
0.0762 J
362000
1734
fi6i
f 7
180.00
45
11.63
5231
{0.0612")
682
!:r
14
\l
180.00
180.00
180.00
45
449
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
180.03
180.03
5.62
56o
11.69
ii .70
2349
0.0530 I
0.0494 f
501000
2133
Ii8ij
I 7
180.03
446
II. 01
o.osgoj
ri 4 6
fio
180.00
45
11.64
5241
f 0.0664!
684
180.00
180.00
564
56o
ii59
11.58
J 0.0610 !
I 0.0548 f
529000
2255
li66J
1 10
180.00
4.48
n. 61
52 oj
(,o.05i4j
f 145^
fto
180.00
45
1 1. 60
6 2 ' 2 ]
(0.0645!
6*5
ixo
1 12
180,00
180.00
5.63
56i
11.62
11.62
65.2 J 2 34'
0.0650 1
0.0506 f
430000
1831
Ii66j
I 9
180.00
4.48
11.62
52.0]
0.0546J
686
fi45l
II*
UsoJ
(S
I'l
180.00
180.00
180.00
180.00
4.48
450
45
452
ii. 61
11.56
11.36
ii. 61
52. oj
525J
To. 0500"!
J 0.0315 !
 0.0543 f
395000
1903
664
APPLIED MECHANICS.
u
S
T3
C 3 J 'O
.s 1 g g
iifilil
.^^^
Q fn Q
< "" Ost^oo O\T!VQ OVO rot^ooo
1OVO ON^Oii rOiiVO 'OO\fO L OO "^
vOiir^cxDvOr^w'iONNfONO^O
MOO O\ic >>
^
<&
a.
S
\OvOvO\OvovOvO
unio^^u^vo>ou^
COMPRESSION OF YELLOWPINE POSTS.
665
* .
13 >^
Mil
r fe M 'C
.s s  .s i .s S .s
oooo g t^">^5 o
T3
5 3 3 3 3
t^ro
ONf^
N N HH O
n vo N vo
^" ro fO CO
oooo
N fO
e  oq in o