------- -^^^ ^ REESE LIBRARY OF THK UNIVERSITY OF CALIFORNIA. -; Glass APPLIED MECHANICS. BY GAETANO LANZA, S.B., C. &M.E., ii PROFESSOR OF THEORETICAL AND APPLIED MECHANICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY. NINTH EDITION, REVISED. FIRST THOUSAND. UNIVERSITY OF NEW YORK: JOHN WILEY & SONS. LONDON : CHAPMAN & HALL, LIMITED. 1905. REESE c COPYRIGHT, 1885, 1900, 1905, BY GAETANO LANZA. RObBKT DRUMMOND, 1RIN1BK, NBW YORK. PREFACE. THIS book is the result of the experience of the writer in teaching the subject of Applied Mechanics for the last twelve years at the Massachusetts Institute of Technology. The immediate object of publishing it is, to enable him to dispense with giving to the students a large amount of notes. As, however, it is believed that it may be found useful by others, the following remarks in regard to its general plan are submitted. The work is essentially a treatise on strength and stabil- ity ; but, inasmuch as it contains some other matter, it was thought best to call it " Applied Mechanics," notwithstanding the fact that a number of subjects usually included in trea- tises on applied mechanics are omitted. It is primarily a text-book ; and hence the writer has endeav- ored to present the different subjects in such a way as seemed to him best for the progress of the class, even though it be at some sacrifice of a logical order of topics. While no attempt has been made at originality, it is believed that some features of the work are quite different from all pre- 147GG3 iv PREFACE. vious efforts ; and a few of these cases will be referred to, with the reasons for so treating them. In the discussion upon the definition of "force," the object is, to make plain to the student the modern objections to the usual ways of treating the subject, so that he may have a clear conception of the modern aspect of the question, rather than to support the author's definition, as he is fully aware that this, as well as all others that have been given, is open to objection. In connection with the treatment of statical couples, it was thought best to present to the student the actual effect of the action of forces on a rigid body, and not to delay this subject until dynamics of rigid bodies is treated, as is usually done. In the common theory of beams, the author has tried to make plain the assumptions on which it is based. A little more prominence than usual has also been given to the longi- tudinal shearing of beams. In that part of the book that relates to the experimental results on strength and elasticity, the writer has endeavored to give the most reliable results, and to emphasize the fact, that, to obtain constants suitable for use in practice, we must deduce them from tests on full-size pieces. This prin- ciple of being careful not to apply experimental results to cases very different from those experimented upon, has long been recognized in physics, and therefore needs no justifica- tion. The government reports of tests made at the Watertown Arsenal have been extensively quoted from, as it is believed PREFACE. that 'they furnish some of our most reliable information on these subjects. The treatment of the strength of timber will be found to be quite different from what is usually given ; but it speaks for itself, and will not be commented upon here. In the chapter on the " Theory of Elasticity," a combina- tion is made of the methods of Rankine and of Grashof. In preparing the work, the author has naturally consulted the greater part of the usual literature on these subjects ; and, whenever he has drawn from other books, he has endeavored to acknowledge it. He wishes here to acknowledge the assist- ance furnished him by Professor C. H. Peabody of the Massa- chusetts Institute of Technology, who has read all the proofs, and has aided him materially in other ways in getting out the work. GAETANO LANZA. MASSACHUSETTS INSTITUTE OF TECHNOLOGY, April, 1885. PREFACE TO THE FOURTH EDITION. THE principal differences between this and the earlier editions consist in the introduction of the results of a large amount of the experimental work that has been done during the last five years upon the strength of materials. The other changes that have been made in the book are not a great many, and have been suggested as desirable by the author's experience in teaching. September, 1890. PREFACE TO THE SEVENTH EDITION. THE principal improvements in this edition consist in the introduction, in Chapter VII, of the results of a considerable amount of the experimental work on the strength of materials that has been done during the last six years. A few changes have also been made in other parts of the book. October, 1896. PREFACE 'TO THE EIGHTH EDITION. IN this edition a considerable number of additional results of recent tests, especially upon full-size pieces, have been introduced, some of the older ones having been omitted to make room for them. September, 1900. PREFACE TO THE NINTH EDITION. THE principal improvements in the Ninth Edition consist in very extensive changes in Chapter VII, in order to bring the account of the experimental work that has been performed in various places up to date. Some changes have also been made in the mathematical portion of the book, especially in the Theory of Columns. TABLE OF CONTENTS. CHAPTER I. COMPOSITION AND RESOLUTION OF FORCES . , CHAPTER II. DYNAMICS . . 75 CHAPTER III. ROOF-TRUSSES **,* 138 CHAPTER IV. BBIDGE-TRUSSES 184 CHAPTER V. CENTRE OF GRAVITY 221 CHAPTER VI. STRENGTH OF MATERIALS 240 CHAPTER VII. STRENGTH OF MATERIALS AS DETERMINED BY EXPERIMENT ..... 350 viii TABLE OF CONTENTS. CHAPTER VIII. CONTINUOUS GIRDERS 743 r CHAPTER IX. EQUILIBRIUM CURVES. ARCHES AND DOMES . . 779 CHAPTER X. THEORY OF ELASTICITY, AND APPLICATIONS .... < 852 APPLIED MECHANICS, CHAPTER I. COMPOSITION AND RESOLUTION OF FORCES. i. Fundamental Conceptions. The fundamental con- ceptions of Mechanics are Force, Matter, Space, Time, and Motion. 2. Relativity of Motion. The limitations of our natures are such that all our quantitative conceptions are relative. The truth of this statement may be illustrated, in the case of motion, by the fact, that, if we assume the shore as fixed in position, a ship sailing on the ocean is in motion, and a ship moored in the dock is at rest ; whereas, if we assume the sun as our fixed point, both ships are really in motion, as both par- take of the motion of the earth. We have, moreover, no means of determining whether any given point is absolutely fixed in position, nor whether any given direction is an absolutely fixed direction. Our only way of determining direction is by means of two points assumed as fixed ; and the straight line joining them, we are accustomed to assume as fixed in direction. Thus, it is very customary to assume the straight line joining the sun with any fixed star as a line fixed in direction ; but if the whole visible universe were in motion, so as to change the absolute direction of this line, we should have no means of recognizing it. APPLIED MECHANICS. 3. Rest and Motion. - In order to define rest and motion, we have the following ; viz., When a single point is spoken of as having motion or rest, some other point is always expressed or understood, which is for the time being considered as a fixed point, and some direc- tion is assumed as a fixed direction : and we then say that the first-named point is at rest relatively to the fixed point, when the straight line joining it with the fixed point changes neither in length, nor in direction; whereas it is said to be in motion relatively to the fixed point, when this straight line changes in length, in direction, or in both. If, on the other hand, we had considered the first-named point as our fixed point, the same conditions would determine whether the second was at rest, or in motion, relatively to the first. A body is said to be at rest relatively to a given point and to a given direction, when all its points are at rest relatively to this point and this direction. 4. Velocity and Acceleration. When the motion of one point relatively to another, or of one body relatively to another, is such that it describes equal distances in equal times, however small be the parts into which the time is divided, the motion is said to be uniform and the velocity constant. The velocity, in this case, is the space passed over in a unit of time, and is to be found by dividing the space passed over in any given time by the time ; thus, if s represent the space passed over in time /, and v represent the velocity, we shall have When the motion is not uniform, if we divide the time into small parts, and then divide the space passed over in one of these intervals by the time, and then pass to the limit as these intervals of time become shorter, we shall obtain the velocity FORCE. Thus, if A.y represent the space passed over in the interval of time A^, then we shall have v = limit of as A/ diminishes, A/ or ds In this case the rate of change of velocity per unit of time is called the Acceleration, and if we denote it by/, we have 5. Force. We shall next attempt to obtain a correct defi- nition of force, or at least of what is called force in mechanics. It may seem strange that it should be necessary to do this ; as it would appear that clear and correct definitions must have been necessary in order to make correct deductions, and there- fore that there ought to be no dispute whatever over the mean- ing of the word force. Nevertheless, it is a fact in mechanics, as well as in all those sciences which attempt to deal with the facts and laws of nature, that correct definitions are only gradu- ally developed, and that, starting with very imperfect and often erroneous views of natural laws and phenomena, it is only after these errors have been ascertained and corrected by a long range of observation and experiment, and an increased range of knowledge has been acquired, that exactness and perspicuity can be obtained in the definitions. Now, this is precisely what has happened in the case of force. In ancient times rest was supposed to be the natural state of bodies ; and it was assumed that, in order to make them move, force was necessary, and that even after they had been set in motion their own innate inertia or sluggishness would cause them to come to rest unless they were constantly urgea APPLIED MECHANICS. on by the application of some force, the bodies coming to rest whenever the force ceased acting. It was under the influence of these vague notions that such terms arose as Force of Inertia, Moment of Inertia, Vis Viva or Living Force, etc. A number of these terms are still used in mechanics; but in all such cases they have been re -defined, such new mean- ings, having been attached to them as will bring them into accord with the more advanced ideas of the present time. Such definitions will be given in the course of this work, as the necessity may arise for the use of the terms. NEWTON'S FIRST LAW OF MOTION. Ideas becoming more precise, in course of time there was framed Newton's first law of motion ; and this law is as fol- lows : A body at rest will remain at rest, and a body in motion will continue to move uniformly and in a straight line, unless and until some external force acts upon it. The assumed truth of this law was based upon the observed facts of nature ; viz., When bodies were seen to be at rest, and from rest passed into a state of motion, it was always possible to assign some cause ; i.e., they had been brought into some new relationship, either with the earth, or with some other body: and to this cause could be assigned the change of state from rest to motion. On the other hand, in the case of bodies in motion, it wa<> seen, that, if a body altered its motion from a uniform rectilinear motion, there was always some such cause that could be .assigned. Thus, in the case of a ball thrown from the hand, the attraction of the earth and the resistance of the ait soon caused it to come to rest. In the case of a ball rolled along the ground, friction (i.e., the continual contact and collision with the ground) gradually destroyed its motion, and brought it to FORCE. rest ; whereas, when such resistances were diminished by rolling it on glass or on the ice, the motion always continued longer : hence it was inferred, that, were these resistances entirely removed, the motion would continue forever. In accordance with these views, the definition of force usually given was substantially as follows : Force is that which causes, or tends to cause, a body to change its state from rest to motion, from motion to rest, or to change its motion as to direction or speed. Under these views, uniform rectilinear motion was recog- nized as being just as much a condition of equilibrium, or of the action of no force or of balanced forces, as rest ; and the recognition of this one fact upset many false notions, destroyed many incorrect conclusions, and first rendered possible a science of mechanics. Along with the above-stated definition of force is ordinarily given the following proposition ; viz., Forces are proportional to the velocities that they impart, in a unit of time (i.e. to the accelerations that they impart), to the same body. The reasoning given is as follows : Suppose a body to be moving uniformly and in a straight line, and suppose a force to act upon it for a certain length of time t in the direction of the body's motion : the effect of the force is to alter the velocity of the body ; and it is only by this alteration of velocity that we recognize the action of the force. Hence, as long as the alteration continues at the same rate, we recognize the same force as acting. If, therefore,/ represent the amount of velocity which the force would impart in one unit of time, the total increase in the velocity of the body will be //; and, if the force now stop acting, the body will again move uniformly and in the same direction, but with a velocity greater by//. Hence, if we are to measure forces by their effects, it will follow that The velocity which a force will impart to a given (or standard) APPLIED MECHANICS. body in a unit of time is a proper measure of the force. And we shall have, that two forces, each of which will impart the same velocity to the same body in a unit of time, are equal to each other ; and a force which will impart to a given body twice the velocity per unit of time that another force will impart to the same body, is itself twice as great, or, in other words, Forces are proportional to the velocities that they impart, in a unit of time (i.e. to the accelerations that they impart), to the same body. MODERN CRITICISM OF THE ABOVE. The scientists and the metaphysicians of the present time are recognizing two other facts not hitherto recognized, and the result is a criticism adverse to the above-stated definition of force. Other definitions have, in consequence, been proposed ; but none are free from objection on logical grounds, and at the same time capable of use in mechanics in a quantitative way. The two facts referred to are the following ; viz., i. That all our ideas of space, time, rest, motion, and even of direction, are relative. 2. That, because two effects are identical, it does not follow that the causes producing those effects are identical. Hence, in the light of these two facts, it is plain, that, inas- much as we can only recognize motion as relative, we can only recognize force as acting when at least two bodies are con- cerned in the transaction ; and also that if the forces are simply the causes of the motion in the ordinary popular sense of the word cause, we cannot assume, that, when the effects are equal, the causes are in every way identical, although we have, of course, a perfect right to say that they are identical so far as the production of motion is concerned. I shall now proceed, in the light of the above, to deduce a definition of force, which, although not free from objection, seems as free as any that has been framed. It is one of the facts of nature, that, when bodies are by any FORCE. means brought under certain relations to each other, certain tendencies are developed, which, if not interfered with, will exhibit themselves in the occurrence of certain definite phe- nomena. What these phenomena are, depends upon the nature of the bodies concerned, and on the relationships into which they are brought. As an illustration, we know that if an apple is placed at a certain height above the surface of the earth, there is developed between the two bodies a tendency to approach each other ; and if there is no interference with this tendency, it exhibits itself in the fall of the apple. If, on the other hand, the apple were hung on the hook of a spring balance in the same posi- tion as before, the spring would stretch, and there would be developed a tendency of the spring to make the apple move upwards. This tendency to make the apple move upwards would be just equal to the tendency of the earth and apple to approach each other. This would be expressed by saying that the pull of the spring is just equal and opposite to the weight of the apple. As other illustrations of these tendencies developed in bodies when placed in certain relations to each other, we have the following cases : (a} When two bodies collide. (b} When two substances, coming together, form a chemical union, as sodium and water. (c) When the chemical union is entered into only by raising the temperature to some special point. Any of these tendencies that are developed by bringing about any of these special relationships between bodies might properly be called a force ; and the term might properly be, and is, used in the same sense in the mental and moral world, as well as in the physical. In mechanics, however, we have to deal only with the relative motion of bodies ; and hence we give the name force only to tendencies to change the relative 8 APPLIED MECHANICS. motion of the bodies concerned ; and this, whether these ten- dencies are unresisted, and exhibit themselves in the actual occurrence of a change of motion, or whether they are resisted by equal and opposite tendencies, and exhibit themselves in the production of a tensile, compressive, or other stress in the bodies concerned, instead of motion. DEFINITION OF FORCE. Hence our definition of force, as far as mechanics has to deal with it or is capable of dealing with it, is as follows; viz., - Force is a tendency to change the relative motion of the two bodies between which that tendency exists. Indeed, when, as in the illustration given a short time ago, the apple is hung on the hook of a spring balance, there still exists a tendency of the apple and the earth to approach each other ; i.e., they are in the act of trying to approach each other ; and it is this tendency, or act of trying, that we call the force of gravitation. In the case cited, this tendency is balanced by an opposite tendency on the part of the spring ; but, were the spring not there, the force of gravitation would cause the apple to fall. Professor Rarikine calls force "an action between two bodies, either causing or tending to cause change in their relative rest or motion ;" and if the act of trying can be called an action, my definition is equivalent to his. For the benefit of any one who wishes to follow out the discussions that have lately taken place, I will enumerate the following articles that have been written on the subject : (a) " Recent Advances in Physical Science," by P. G. Tait, Lecture XIV. (b) Herbert Spencer, "First Principles of Philosophy* (certain portions of the book). MEASURE OF FORCE. () Discussion by Messrs. Spencer and Tait, " Nature," Jan. 2, 9, 1 6, 1879. (d] Force and Energy, "Nature," Nov. 25, Dec. 2, 9, 16, 1880. 6. External Force. We thus see, that, in order that a force may be developed, there must be two bodies concerned in the transaction ; and we should speak of the force as that developed or existing between the two bodies. But we may confine our attention wholly to the motion or condition of one of these two bodies ; and we may refer its motion either to the other body as a fixed point, or to some body different from either; and then,' in speaking of the force, we should speak of it as the force acting on the body under consideration, and call it an external force. It is the tendency of the other body to change the motion of the body under con- sideration relatively to the point considered as fixed. 7. Relativity of Force. In adopting the above-stated definition of force, we acknowledge our incapacity to deal with it as an absolute quantity ; for we have defined it as a tendency to change the relative motion of a pair of bodies. Hence it is only through relative motion that we recognize force; and hence force is relative, as well as motion. 8. Newton's First Law of Motion. In the light of the above discussion, we might express Newton's first law of motion as follows : A body at rest, or in uniform rectilinear motion relatively to a given point assumed as fixed, will continue at rest, or in uni- form motion in the same direction, unless and until some external force acts either on the body in question, or on the fixed point, or on the body which furnishes us our fixed direction. This law is really superfluous, as it has all been embodied in the defini- tion. 9. Measure of Force. We next need some means of comparing forces with each other in magnitude ; and, subse- 10 APPLIED MECHANICS. quently, we need to select one force as our unit force, by means of which to estimate the magnitude of other forces. Let us suppose a body moving uniformly and in a straight line, relatively to some fixed point ; as long as this motion continues, we recognize no unbalanced force acting on it ; but, if the motion changes, there must be a tendency to change that motion, or, in other words, an unbalanced force is acting on the body from the instant when it begins to change its motion. Suppose a body to be moving uniformly, and a force to be applied to it, and to act for a length of time /, and to be so applied as not to change the direction of motion of the body, but to increase its velocity; the result will be, that the velocity will be increased by equal amounts in equal times, and if f represent the amount of velocity the force would impart in one unit of time, the total increase in velocity will 'be//. This results merely from the definition of a force ; for if the velocity pro- duced in one (a standard) body by a given force is twice as great as that produced by another given force, then is the ten- dency to produce velocity twice as great in the first case as in the second, or, in other words, the first force is twice as great as the second. Hence Forces are proportional to the velocities which they will impart to a given (or standard} body in a unit of time. We may thus, by using one standard body, determine a set of equal forces, and also the proportion between different forces. 10. Measure of Mass. After having determined, as shown, a set of equal (unit) forces, if we apply two of them to different bodies, and let them act for the same length of time on each, and find that the resulting velocities are unequal, these bodies are said to have unequal masses ; whereas, if the result- ing velocities are equal, they are said to have equal masses. Hence we have the following definitions : RELATION BETWEEN FORCE AND MOMENTUM. II I . Equal forces are those which, by acting 1 for equal times on tJie same or standard body, impart to it equal velocities. 2. Equal masses are those masses to which equal forces will impart equal velocities in equal times. 11. Suppose two bodies of equal mass moving side by side with the same velocity, and uniformly, let us apply to one of them a force F in the direction of the body's motion : the effect of this force is to increase the velocity with which the body moves ; and if we wish, at the same time, to increase the velocity of the other, so that they will continue to move side by side, it will be necessary to apply an equal force to that also. We are thus employing a force 2F to impart to the two bodies the required increment of velocity. If we unite them into one, it still requires a force 2F to impart to the one body resulting from their union the re- quired increment of velocity : hence, if we double the mass to which we wish to impart a certain velocity, we must double the force, or, in other words, employ a force which would impart to the first mass alone a velocity double that required. Hence Forces are proportional to the masses to which they will impart the same velocity in the same time. 12. Momentum. The product obtained by multiplying the number of units of mass in a body by its velocity is called the momentum of the body. 13. Relation between Force and Momentum. The number of units of momentum imparted to a body in a unit of time by a given force, is evidently identical with the number of units of velocity that would be imparted by the same force, in the same time, to a unit mass. Hence Forces are proportional to the momenta (or velocities per unit of mass) which they will generate in a unit of time. 12 APPLIED MECHANICS. Hence, if F represent a force which generates, in a unit of time, a velocity/" in a body whose mass is m, we shall have and, inasmuch as the choice of our units is still under our con- trol, we so choose them that F = mf; i.e., the force F contains as many units of force as mf contains units of momentum ; in other words, The momentum generated in a body in a unit of time by a. force acting in the direction of the body's motion, is taken as a measure of the force. 14. Statical Measure of Force. When the forces are prevented from producing motion by being resisted by equal and opposite fofces, as is the case in that part of mechanics known as Statics, they must be measured by a direct comparison with other forces. An illustration of this has already been given in the case of an apple hung on the hook of a spring balance. In that case the pull of the spring is equal in magni- tude to the weight of the apple : indeed, it is very customary to adopt for forces what is known as the gravity measure, in which case we take as our unit the gravitation, or tendemy to fall, of a given piece of metal, at a given place on the surface of the earth ; in other, words, its weight at a given place. The gravity unit may thus be the kilogram, the pound, or the ounce, etc. It is evident, moreover, from our definition of force, and the subsequent discussion, that whatever we take as our unit of mass, the statical measure of a force is proportional to its dynamical measure ; i.e., the numbers representing the magni- tudes of any two forces, in pounds, are proportional to the momenta they will impart to any body in a unit of time. 15. Gravity Measure of Mass. If we assume one pound as our unit of force, one foot as our unit of length, and NEWTON'S SECOND LAW OF MOTION 13 one second as our unit of time, the ratio between the number of pounds in any given force and the momentum it will impart to a body on which it acts unresisted for a unit of time, will depend on our unit of mass ; and, as we are still at liberty to fix this as we please, it will be most convenient so to choose it that the above-stated ratio shall be unity, so that there shall be no difference in the measure of a force, whether it is measured statically or dynamically. Now, it is known that a body falling freely under the action of its own weight acquires, every second, a velocity of about thirty-two feet per second : this number is denoted by g t and varies for different distances from the centre of the earth, as does also the weight of the body. Now, if W represent the weight of the body in pounds, and m the number of units of mass in its mass, we must have, in order that the statical and dynamical measures may be equal, W = mg. Hence m.y, g i.e., the number of units of mass in a body is obtained by divid- ing the weight in pounds, by the value of g at the place where the weight is determined. The values of W and of g vary for different positions, but the value of m remains always the same for the same body. UNIT OF MASS. If m = I, then W g; or, in words, The weight in pounds of the unit of mass (when the gravity measure is used} is equal to the value of g in feet per second for the same place. 16. Newton's Second Law of Motion. Newton's second law of motion is as follows : 14 APPLIED MECHANICS. " Change of momentum is proportional to the impressed mov- ing f rce > an d occurs along the straight line in which the force is impressed" Newton states further in his " Principia :" " If any force generate any momentum, a double force will generate a double, a triple force will generate a triple, momentum, whether simultaneously and suddenly, or gradually and successively impressed. And if the body was moving before, this momentum, if in the same direction as the motion, is added; if opposite, is subtracted; or if in an oblique direc- tion, is annexed obliquely, and compounded with it, according to the direction and magnitude of the two." Part of this law has reference to the proportionality between the force and the momentum imparted to the body ; and this has been already embodied in our definition of force, and illus- trated in the discussion on the measure of forces. The other part is properly a law of motion, and may be expressed as follows : If a body have two or more velocities imparted to it simulta- neously, it will move so as to preserve them all. The proof of this law depends merely upon a proper con- ception of motion. To illustrate this law when two velocities are imparted simultaneously to a body, let us suppose a man walking on the deck of a moving ship : he then has two motions in relation to the shore, his own and that of the ship. Suppose him to walk in the direction of motion of the ship at the rate of 10 feet per second, while the ship moves at 25 feet per second relatively to the shore : then his motion in relation to the shore will be 25 -|- 10 = 35 feet per second. If, on the other hand, he is walking in the opposite direction at the same rate, his motion relatively to the shore will be 25 10 15 feet per second. Suppose a body situated at A (Fig. i) to have two motions imparted to it simultaneously, one of which would carry it to B POLYGON OF MOTIONS 15 in one second, and the other to C in one second ; and that it is required to find where it will be at the end of one second, and what path it will have pursued. c Imagine the body to move in obedience to the first alone, during one second : it would thus arrive at B ; then suppose the second motion to be imparted to the body, instead of the first, it will arrive at the end of the next sec- ond at D, where BD is equal and parallel to AC. When the two motions are imparted simultaneously, instead of suc- cessively, the same point D will be reached in one second, instead of two; and by dividing AB and AC into the same (any) number of equal parts, we can prove that the body will always be situated at some point of the diagonal AD of the parallelogram, hence that it moves along AD. Hence follows the proposition known as the parallelogram of motions. PARALLELOGRAM OF MOTIONS. If there be simultaneously impressed on a body two velocities, which would separately be represented by the lines AB and AC, the actual velocity will be represented by the line AD. which is the diagonal of the parallelogram of which AB and AC are the adjacent sides. 17. Polygon of Motions. In all the above cases, the point reached by the body at the end of a second when the two motions take place simultaneously is the same as that which would be reached at the end of two seconds if the motions took place successively ; and the path described is the straight line joining the initial position of the body, with its position at the end of one second when the motions are simultaneous. The same principle applies whatever be the number of velocities that may be imparted to a body simultaneously. Thus, if we suppose the several velocities imparted to be (Fig. 2) AB, AC, AD, AE, and AF, and it be required to 1 6 APPLIED MECHANICS. determine the resultant velocity, we first let the body move with the velocity AB for one second ; at the end of that second it is found at B ; then let it move with the velocity AC only, and "at the end of another second it will be found at c ; then with AD only, and at the end of the third second it will be found at d; at the end of the fourth at e; at the end of the fifth at /. Hence the resultant velocity, when all are imparted simultaneously, is Af, or "the closing side of the polygon. This proposition is known as the polygon of motions. POLYGON OF MOTIONS. If there be simultaneously impressed on a body any number of velocities, the resulting velocity will be represented by the closing side of a polygon of which the lines representing tJie separate velocities form the other sides. 1 8. Characteristics of a Force A force has three characteristics, which, when known, determine it ; viz., Point of Application, Direction, and Magnitude. These can be repre- sented by a straight line, whose length is made proportional to the magnitude of the force, whose direction is that of the motion which the force imparts, or tends to impart, and one end of which is the point of application of the force ; an arrow-head being usually employed to indicate the direction in which the force acts. 19. Parallelogram of Forces. PROPOSITION. If two forces acting simultaneously at the same point be represented, in point of application, direction, and magnitude, by two adjacent sides of a parallelogram, their resultant will be represented by the diagonal of the parallelo- gram, drawn from the point of application of the two forces. PROOF. In the last part of 16 was proved the propo* PARALLELOGRAM OF FORCES. I/ sition known as the Parallelogram of Motions, for the state- ment of which the reader is referred to the close of that section. We have also seen that forces are proportional to the velo- cities which they impart, or tend to impart, in a unit of time, to the same body. Hence the lines representing the two impressed forces are coincident in direction with, and proportional to, the lines repre- senting the velocities they would impart in a unit of time to the same body ; and moreover, since the resultant velocity is represented by the diagonal of the parallelogram drawn with the component velocities as sides, the resultant force must coin- cide in direction with the resultant velocity, and the length of the line representing the resultant force will bear to the result- ant velocity the same ratio that one of the component forces bears to the corresponding velocity. Hence it follows, that the resultant force will be represented by the diagonal of the paral lelogram having for sides the two component forces. 20. Parallelogram of Forces : Algebraic Solution. PROBLEM. Given two forces F and F, acting at the same point A (Fig. 3), and inclined to each other at an angle ; required the magnitude and direction of the resultant force. Let AC represent F, AB represent F t , and let angle BAG ; then will R = AD A represent in magnitude and direction the resultant force. Also let angle DAC a; then from the tri~ angle DAC we have AD 2 = AC 2 + CD 2 - 2AC. CDcosACD. But ACD = 180 - .'. cosACD = -cos* .'. R 2 = F 2 + F 2 + 2FF, cos (9 + F 2 -f 2FF, cos (9. i8 APPLIED MECHANICS. This determines the magnitude of R. To determine its direc- tion, let angle CAD a. .'. angle BAD = a, and we shall have from the triangle DAC or and similarly CD : AD = sin CAD : smACD, F t : R = sin a : sin T? .*. sin a = -sin0, R sin(0-a) = sin0. R EXAMPLES. . Given F = 47-34, 75.46, = 73 14' 21"; find R and a. 2. Given ^ = 5.36, F l = 4.27, = 32 10' ; find R and a. 3. Given F = 42.00, F t = 31.00, = 150 ; find R and a. 4. Given F = 47.00, F t 75.00, 6 = 253 ; find R and a. 21. Parallelogram of Forces when 6 = 90. When the two given forces are at right angles to each other, the for- mulae become very much simplified, since the parallelogram becomes a rectangle. From Fig. 4 we at once deduce R = V^F* + ^;, sin a = ^, R COS a = . 1. Given ^ = 2. Given ^ = 3. Given ^ = 4. Given /? = 3.0, ^ = 3.0, F t = 5.0, F l = 23.2, F t = 5.0 ; find ^ and a. 5.0 ; find i? and a. 12.0 ; find ^ and a. 21.3 ; find R and a. DECOMPOSITION OF FORCES IN ONE PLANE. 19 22. Triangle of Forces. If three forces be represented* in magnitude and direction, by the three sides of a triangle taken in order, then, if these forces be simultaneously applied at one point, they will balance each other. Conversely, three forces which, when simultaneously applied at one point, balance each other, can be correctly represented in magnitude and direction by the three sides of a triangle taken in order. These propositions, which find a very extensive application, especially in the determination of the stresses in roof and bridge trusses, are proved as follows : If we have two forces, AC and AB (see Fig. 3), acting at the point A, their resultant is, as we have already seen, AD ; and hence a force equal in magnitude and opposite in direction to AD will balance the two forces AC and AB. Now, the sides of the triangle AC DA, if taken in order, represent in magnitude and direction the force AC, the force CD or AB, and a force equal and opposite to AD ; and these three forces, if applied at the same point, would balance each other. Hence follows the proposition. Moreover, we have AC : CD \ DA = sinAUC : sin CAD : smACZ>, or F-.F, \R = sin(0 -a) : sin a : sintf; or each force is, in this case, proportional to the sine of the angle between the other two. 23. Decomposition of Forces in one Plane. It is often convenient to resolve a force into two components, in two given directions in a plane containing the force. Thus, suppose we have the force R = AD (Fig. 3), and we wish to resolve it into two components acting respectively in the directions AC and AB ; i.e., we wish to find two forces acting respectively in these directions, of which AD shall be the resultant : we 20 APPLIED MECHANICS. determine these components graphically by drawing a parallelo- gram, of which AD shall be the diagonal, and whose sides shall have the directions AC and AB respectively. The algebraic values of the magnitudes of the compo- nents can be determined by solving the triangle ADC. In the case when the directions of the components are at right angles to each other, let the force R (Fig. 5), applied at O, make an angle a with OX. We may, by drawing the rect- angle shown in the figure, decompose R into two components, F and F u along OX and O Y respectively ; and we shall readily obtain from the figure, F = R cos a, Fi = R sin a. FIG. 6. EXAMPLES. i. The force exerted by the steam upon the piston of a steam-engine at the moment when it is in the position shown in the figure is AB = 1000 Ibs. The resistance of the guides upon the cross-head DE is vertical. Determine the force acting along the connecting-rod AC and the pressure on the guides ; also resolve the force acting along the connecting-rod into two components, one along, and the other at right angles to, the crank OC. 2. A load of 500 Ibs. is placed at the apex C of the frame ACB ': find the stresses in AC and CB respectively. 3. A load of 4000 Ibs. is hung at C, on the crane ABC: find the pressure in the boom BC, and the pull on the tie AC, where BC makes an angle of 60 with the horizontal, and AC an angle of 15. COMPOSITION OF FORCES IN ONE PLANE. 21 4. A force whose magnitude is 7 is resolved into two forces whose magnitudes are 5 and 3 : find the angles they make with the given; force. 24. Composition of any Number of Forces in One Plane, all applied at the Same Point. (a) GRAPHICAL SOLUTION. Let the forces be represented (Fig. 2) by AB, AC, AD, AE, and AF respectively. Draw Be || and = AC, cd || and = AD, de || and = AE, and ef j| and = AF; then will Af represent the resultant of the five forces. This solution is to be deduced from 17 in the same way as 19 is deduced from 1 6. c, (b) ALGEBRAIC SOLUTION. Let the given forces (Fig. 9), of which B, three are represented in the figure, be F, F t) F 2 , Fy F 4 , etc. ; and let the angles l made by these forces with the axis OX o 1 ^ < jj be a, a,, 02, a 3 , a 4 , etc., respectively. FlG -9- Resolve each of these forces into two components, in the directions OX and OY respectively. We shall obtain for the components along OX OA = Fcosa, OB = ^cosa,, OC F 2 cosa 2 , etc.; and for those along OY OA, = Fs'ma, OB, == ^sino,, OC, = J? 2 sma 2 , etc. These forces are equivalent to the following two ; viz., a force Fcos a -f F, cos a, + F 2 cos a 2 -f- F 3 cos a 3 + etc. along OX, and a force .Fsin a + F\ $ m a i + F* siri 2 + F z sin a 3 -f- etc. along OY. The first may be represented by ^Fcosa, and the second by ^Fsina, where 2, stands for algebraic sum. There remains only to find the resultant of these two, the magnitude of which is given by the equation R = V(2^cosa)2 -j- 22 APPLIED MECHANICS. and, if we denote by a^ the angle made by the resultant with OX, we shall have COS a r = sin OT R EXAMPLES. a 3 = 112 Find the result- ant force and its direction. Solution. F. a. COS a. sin a. F COS a. F sin a. 47 21 0-93358 o.35 8 37 43.87826 16.84339 73 4 8 0.66913 o-743i5 48.84649 54-24995 43 82 O.I39I7 0.99027 5-9843 1 42.58161 23 112 -0.37461 0.92718 -8.6l603 21.32414 90.09303 134.99909 *. 2^ cos a = 90.09303, -. R a = 134.99909, log ^F COS a = 1.954691 = 2.210331 + (2F sin a) 2 = 162.2976. log COS Or = 9.744360 Or = 56 I/- OBSERVATION. It would be perfectly correct to use the minus sign in extracting the square root, or to call R = 162.2976 ; but then we should have 050,= 90.09303 or 162.2976 i8o -f 56 - 134.99909 ? 162.2976 = 23 6 - 7'; COMPOSITION OF FORCES APPLIED AT SAME POINT. 2$ a result which, if plotted, would give the same force as when we call R = 162.2976 and a* == 56 if. Hence, since it is immaterial whether we use the plus or the minus sign in extracting the square root provided the rest of the computation be consistent with it, we shall, for convenience, use always plus. * = 77> 3. a,= 82, a 2 = 163, S= 275- a, = o, 2 = 90. 25. Polygon of Forces. If any number of forces be represented in magnitude and direction by the sides of a polygon taken in order, then, if these forces be simultaneously applied at one point, they will balance each other. Conversely, any number of forces which, when simultaneously applied at one point, balance each other, can be correctly repre- sented in magnitude and direction by the sides of a polygon taken in order. These propositions are to be deduced from 24 (a) in the same way as the triangle of forces is deduced from the parallelo- gram of forces. 26. Composition of Forces all applied at the Same Point, and not confined to One Plane. This problem can be solved by the polygon of forces, since there is nothing in the demonstration of that proposition that limits us to a plane rather than to a gauche polygon. The following method, however, enables us to determine algebraic values for the magnitude of the resultant and for its direction. 2 4 APPLIED MECHANICS. FIG. 10. We first assume a system of three rectangular axes, OX, OY, and OZ (Fig. 10), whose origin is at the common point of the given forces. Now, let OE = F be one of the given forces. First resolve it into two forces, OC and OD, the first of which lies in the z axis, and the second perpendicular to OZ, x or, as it is usually called, in the z plane ; the plane perpendicular to OX being the x plane, and that perpendicular to OY the y plane. Then resolve OD into two com- ponents, OA along OX, and OB along OY. We thus obtain three forces, OA, OB, and OC respectively, which are equivalent to the single force OE. These three components are the edges of a rectangular parallelepiped, of which OE = Fis the diagonal. Let, now, angle EOX = a, EOY = (3, and EOZ = y ; and we have, from the right-angled triangles EOA, EOB, and EOC respectively, OA = Fcosa, OB = Fcosp, OC = Fcosy. Moreover, OA 2 + OB 2 = OD 2 and OD 2 + OC 2 = OE 2 .'. OA 2 + OB 2 + OC 2 = OE 2 , and by substituting the values of OA, OB, and OC, given above, we obtain COS 2 a -j- COS 2 (3 + COS 2 y = I ; a purely geometrical relation existing between the three angles that any line makes with three rectangular co-ordinate axes. When two of the angles a, /3, and y are given, the third can be determined from the above equation. COMPOSITIOA r OF- FORCES APPLIED AT SAME POINT. 2$ Resolve, in the same way, each of the given forces into three components, along OX, OY, and OZ respectively, and we shall thus reduce our entire system of forces to the following three forces : i. A single force 2/7 cos a along OX. 2. A single force 2/7 cos ft along OY. 3. A single force 2/7 cosy along OZ. We next proceed to find a sin- gle resultant for these three forces. Let (Fig. ii) OA = 2/7 cos a OB = 2/7 cos ft OC = FIG. xx. Compounding OA and OB, we find OD to be their resultant ; and this, compounded with OC, gives OE as the resultant of the entire system. Moreover, OE 2 = OD 2 -4- OC 2 = OA 2 + OB 2 + OC 2 , or fc = (2/7 cos a)* 4- (S^cos^) 2 -h (2/? cosy)* ( 2.F cos 0) and if we let BOX a r , EOY = ft, and have (2/7 cosy)*; = y r , we shall cos a " = OA OE R 2/7 cos 8 2/7 co r = Y~^, and cosy r = ^ - This gives us the magnitude and direction of the resultant. The same observation applies to the sign of the radical for R as in the case of forces confined to one plane. 26 APPLIED MECHANICS. DETERMINATION OF THE THIRD ANGLE FOR ANY ONE FORCE. When two of the angles a, /3, and y are given, the cosine of the third may be determined from the equation, cos 2 a + cos 2 /? + cos 2 y = i ; but, as we may use either the plus or the minus sign in extract* ing the square root, we have no means of knowing which of the two supplementary angles whose cosine has been deduced is to be used. Thus, suppose a = 45, (3 = 60, then cosy = i -- i - J = f /. y = 60, or 1 20 ; but which of the two to use we have no means of deciding. This indetermination will be more clearly seen from the fol- lowing geometrical considerations : The angle a (Fig. 12), being given as 45, locates the line representing the force on a right circular cone, whose axis is OX, and whose semi-vertical angle is AOX-BOX = 4$. On the other hand, the statement that (3 = 60 locates the force on another right circular cone, having O Y for axis, and a semi-vertical angle of 60; both cones, of course, having their vertices at O. Hence, when a and (3 are given, we know that the line representing the force is an element of both cones ; and this is all that is given. (a) Now, if the sum of the two given angles is less than 90, the cones will not intersect, and the data are consequently inconsistent. DETERMINATION OF THE THIRD ANGLE. 2/ (b) If, on the other hand, one of the given angles being greater than 90, their difference is greater than 90, the cones will not intersect, and the data are again inconsistent. (c) If a + /? = 90, the cones are tangent to each other, and 7 = 90. (d) If a -f- J3 > 90, and a /? or /3 < 90, the cones intersect, and have two elements in common ; and we have no means of determining, without more data, which intersection is intended, this being the indetermination that arises in the algebraic solution. I. Given F = EXAMPLES. 63 a = 53 49 a = 8 7 2 = 70 ft = 42' 7 = 72' 7 = 45' Find the magnitude and direction of the resultant. Solution. p a. p. Y- COS a. cosp. COS Y . F COS a. /^COS/3. F COS y. 63 49 2 53 87 42 700 7 2 45 0.60182 0.05234 0.6l888 0.74314 0.94961 0.34202 0.29250 0.30902 0.70711 37.91466 2.56466 1.23776 46.81782 46.53089 0.68404 18.42750 15.14198 I.4I422 41.71708 2/^cos a 94-03275 2/^cos 3 34-98370 2.F cos y R = V(S^cosa) 2 -j- (XF cos/3) 2 + (S/? cosy) 2 = 108.6569. log 2/^cosa = 1.620314 log S^cos/^ = 1.973279 log 2/^cosy = 1.543866 log j? = 2.036057 log R = 2.036057 log R = 2.036057 log cos a r =9-584257 Iogcosj8 r =9.937222 log cos y r =9.507809 a r = 67 25' 20 X/ (3 r = 30 4' i4 /x =71 13' 5" 28 APPLIED MECHANICS. F. a. 0- F. a. V- 2. 4-3 47 2' 65 7' 3- 5 9 90 37.5 88 3' 10 5 ' 7 6.4 68 4' 8 3 2' 4 75 73 45 27. Conditions of Equilibrium for Forces applied at a Single Point. i. When the forces are not confined to one plane, we have already found, for the square of the resultant, But this expression can reduce to zero only when we have a = o, S/^cos (3 = o, and 2/^cos y = o ; for the three terms, being squares, are all positive quantities, and hence their sum can reduce to zero only when they are separately equal to zero. Hence : If a set of balanced forces applied at a single point be resolved into components along three directions at right angles to each other, the algebraic sum of the components of the forces along each of the three directions must be equal to zero, and con- versely. 2. When the forces are all confined to one plane, let that plane be the z plane ; then y = 90 in each case, and /. (3 = 90 - a /. cos (3 = sin a /. fc = (^F cos a) 2 4- Hence, for equilibrium we must have cos a) 2 4- CSJ? sin a) 2 = o; STATICS OF RIGID BODIES. 29 and, since this is the sum of two squares, o, and S/^sina = o. Hence : If a set of balanced forces, all situated in one plane \ and acting at one point, be resolved into components along two directions at right angles to each other, and in their own plane, the algebraic sum of the components along each of the tzvo given directions must be equal to zero respectively; and conversely. 28. Statics of Rigid Bodies. A rigid body is one that does not undergo any alteration of shape when subjected to the action of external forces. Strictly speaking, no body is absolutely rigid ; but different bodies possess a greater or less degree of rigidity according to the material of which they are composed, and to other circumstances. When a force is ap- plied to a rigid body, we may have as the result, not merely a rectilinear motion in the direction' of the force, but, as will be shown later, this may be combined with a rotary motion ; in short, the criterion by which we determine the ensuing motion is, that the effect of the force will distribute itself through the body in such a way as not to interfere with its rigidity. What this mode of distribution is, we shall discuss here- after ; but we shall first proceed to some propositions which can be proved independently of this consideration. 29. Principle of Rectilinear Transferrence of Force in Rigid Bodies. If a force be applied to a rigid body at the point A (Fig. 13) in the direction AB, whatever be the motion that this force would produce, it will be prevented from taking place if an equal and opposite force be applied at A, B, C, or D, or at FlG - I3 - any point along the line of action of the force : hence we have the principle that The point of application of a force acting on a rigid body, may be transferred to any other point which lies in the line of APPLIED MECHANICS. action of the force, and also in the body, without altering the resulting motion of the body, although it does alter its state of stress. 30. Composition of two Forces in a Plane acting at Different Points of a Rigid Body, and not Parallel to Each Other. Suppose the force F (Fig. 14) to be applied at A, and F, at B t both in the plane of the paper, and acting on the rigid body abcdef. Produce the lines of direction of the forces till they meet at <9, and suppose both F and F, to act at O. Con- struct the parallelogram ODHE, where OD = F and OE = F t ; then will OH R rep^ resent the resultant force in magnitude and in direction. Its point of application may be conceived at any point along the line OH, as at C, or any other point ; and a force equal and opposite to OH, applied at any point of the line OH, will balance F at A, and F, at B. The above reasoning has assumed the points A, B, C and O, all within the body : but, since we have shown, that when this is the case, a force equal and opposite to R at C will bal- ance Fat A, and F t at B, it follows, that were these three forces applied, equilibrium would still subsist if we were to remove the part bafeghc of the rigid body ; or, in other words, The same construction holds even when the point O falls out- side the rigid body. 31. Moment of a Force with Respect to an Axis Per- pendicular to the Force. DEFINITION. The moment of a force with respect to an axis perpendicular to the force, and not intersecting it, is the FIG. 14. EQUILIBRIUM OF THREE PARALLEL FORCES. FIG. 15. product 'of the force by the common perpendicular to (shortest distance between) the force and the axis. Thus, in Fig. 15 the moment of F about an axis through O and perpendicular to the plane of the paper is F(OA). The sign of the moment will depend on the sign attached to the force and that attached to the perpen- dicular. These will be assumed in this book in such a manner as to render the following true ; viz., The moment of a force with respect to an axis is called posi- tive when, if the axis were supposed fixed, the force would cause the body on which it acts to rotate around the axis in the direc- tion of the hands of a watch as seen by the observer looking at the face. It will be called nega- tive when the rotation would take place in the opposite direction. 32. Equilibrium of Three Parallel Forces applied at Different Points of a Rigid Body. Let it be required to find a force (Fig. 16) that will balance the two forces F at A, and -F t at B. Apply at A and B respectively, and in the line AB, the equal and opposite forces Aa and Bb. Their introduction will produce no alteration in the body's motion. The resultant of F and Aa is Af, that of F, and Bb is Bg. Compound these by the method of 30, and we obtain as result- ant ce. A force equal in magnitude and opposite in direction FIG. 16. 32 APPLIED MECHANICS. to cej applied at any point of the line cC, will be the force required to balance Fat A and F, at B ; and, as is evident from the construction, this line is in the plane of the two forces. Moreover, by drawing triangle fKl equal to Bbg, we can readily prove that triangles oce and Afl are equal : hence the angle oce equals the angle fAl, and R is parallel to /^and F t . Also R = ce = ch + he = ,4 AT + A7 = F + ^ __ _ AC fK Ad and CL~. M. =- -5.- BC~ Bb~ Bb' y .'. since ^4# = ^ BC F " BC AC AB where qr is any line passing through C. Hence we have the following propositions ; viz., If three parallel forces balance each other, 1. They must lie in one plane. 2. The middle one must be equal in magnitude and opposite in direction to the sum of the other two. 3. Each force is proportional to the fcj IB o distance between the lines of direction of the other two as measured on any line intersecting all of them. The third of the above-stated con- ditions may be otherwise expressed, thus : FIG. 17. The algebraic sum of the moments of the three forces about any axis perpendicular to the forces must be zero. RESULTANT OF A PAIR OF PARALLEL FORCES. 33 PROOF. Let F, F a and R (Fig. 17) be the forces ; and let the axis referred to pass through O. Draw OA perpendicular to the forces. Then we have F(OA) + Ft (OB) = F(OC + CA) + F t (OC - BC) = (F+F l )OC + F(AC) - But, from what we have already seen, F + F, = -R and JL^JH BC AC .-. F(AC) = ^(^C) .-. F(OA) + Ft(OB) = -R(OC) -f o F,(OB) + tf(0C) = o, or the algebraic sum of the moments of t\\Q forces about the axis through O is equal to zero. 33. Resultant of a Pair of Parallel Forces. In the preceding case, the resultant of any two of the three forces F y F iy and R, in Fig. 16 or Fig. 17, is equal and opposite to the third force. Hence follow the two propositions : I. If two parallel forces act in the same direction, their resultant lies in the plane of the forces, is equal to their sum, acts in the same direction, and cuts the line joining their points of application, or any common perpendicular to the two forces, at a point which divides it internally into two segments in- versely as the forces. II. If two unequal parallel forces act in opposite directions, their resultant lies in the plane of the forces, is equal to their difference, acts in the direction of the larger force, and cuts the line joining their points of application, or any common perpen- dicular to them, at a point which (lying nearer the larger force) 34 APPLIED MECHANICS. divides it externally into two segments which are inversely as the forces. Another mode of stating the above is as follows : i. The resultant of a pair of parallel forces lies in the plane of the forces. 2. It is equal in magnitude to their algebraic sum, and coin- cides in direction with the larger force. 3. The moment of the resultant about an axis perpendicu- lar to the plane of the forces is equal to the algebraic sum of the moments about the same axis. EXAMPLES. 1. Find the length of each arm of a balance such that i ounce at the end of the long arm shall balance i pound at the end of the short arm, the length of beam being 2 feet, and the balance being so propor- tioned as to hang horizontally when unloaded. 2. Given beam =28 inches, 3 ounces to balance 15. 3. Given beam = 36 inches, 5 ounces to balance 25 ounces. MODE OF DETERMINING THE RESULTANT OF A PAIR OF PARALLEL FORCES REFERRED TO A SYSTEM OF THREE RECTANGULAR AXES. Let both forces (Fig. 18) be parallel to OZ ' ; then we have, from what has preceded, F = = F_F> = be ab ac a But from the figure or .'. Fx 2 Fxt = FjX F^ 2 RESULTANT OF NUMBER OF PARALLEL FORCES. 35 and similarly we may prove that or i. The resultant of two parallel forces is parallel to the forces and equal to their algebraic sum. R=F+F, FIG. 18. 2. The moment of the resultant with respect to OX is equal to the algebraic sum of their moments with respect to OX ; and likewise when the moments are taken with respect to OY. 34. Resultant of any Number of Parallel Forces. Let it be required to find the resultant of any number of paral- lel forces. In any such case, we might begin by compounding two of them, and then compounding the resultant of these two with a third, this new resultant with a fourth, and so on. Hence, for the magnitude of any one of these resultants, we simply add to the preceding resultant another one of the forces ; and for the moment about any axis perpendicular to the forces, we add APPLIED MECHANICS. to the moment of the preceding resultant the moment of the new force. Hence we have the following facts in regard to the resultant of the entire system : I . The resultant will be parallel to the forces and equal to their algebraic sum. 2. The moment of the resultant about any axis perpendicular to the forces will be equal to the algebraic sum of the moments of the forces about the same axis. The above principles enable us to determine the resultant in all cases, except when the algebraic sum of the forces is equal to zero. This case will be considered later. 35. Composition of any System of Parallel Forces Y when all are in One Plane. Refer the forces to a pair of rect- angular axes, OX, OY (Fig. 19), and assume OY parallel to the forces. The forces and the co-ordinates of their lines of direction being as indicated in the figure, if we denote by R the resultant, and by X Q the co-ordinate of its line of direction, we shall have, from the preceding, R = ^F; ( i ) and if moments be taken about an axis through O, and perpendicular F, F, FIG. 19. to the plane of the forces, we shall also have Rx = -S.FX. Hence R = ^F and x (t = (2) determine the resultant in magnitude and in line of action, .except when %F = o, which case will be considered later. EQUILIBRIUM OF ANY SET OF PARALLEL FORCES. $? 36, Composition of any System of Parallel Forces not confined to One Plane. Refer the forces to a set of rect- angular axes so chosen that OZ is parallel to their direction. If we denote the forces by F iy F 2 , F y F 4 , etc., and the co-ordinates of their lines of direction by (* 7,), (x 2J jj> 2 ), etc., and if we denote their resultant by R, and the co-ordinates of its line of direction by (x m j^ ), we shall have, in accordance with what has been proved in 34, 1. The magnitude of the resultant is equal to the algebraic mm of the forces , or R = 2F. 2. The moment of the resultant about OY is equal to the mm of the moments of the forces about OY, or 3. The moment of the resultant about OX is equal to the of the moments about OX, or Hence determine the resultant in all cases, except when 2<F = o. 37. Conditions of Equilibrium of any Set of Parallel Ferces. If the axes be assumed as before, so that OZ is parallel to the forces, we must have ^F = o, ^Fx = o, and ^Fy o. To prove this, compound all but one of the forces. Then equilib- rium will subsist only when the resultant thus obtained is equal and directly opposed to the remaining force ; i.e, it must be equal, and act along the same line and in the opposite direction. Hence, calling R a the resultant above referred to, and (x a , y a ) the co-ordinates of its line of direction, and calling F H the 38 APPLIED MECHANICS. remaining force, and (x w y^ the co-ordinates of its line of direc- tion, we must have Ra = ~Fn, *a = *n, J^ = JK, . ' . R a + F n = O, R a X a + F n X n O, ^JFa + F n y n = O, .-. 2F = o, *ZFx =o, ^Fy = o. When the forces are all in one plane, the conditions become 2F = o, ^Fx = o. 38. Centre of a System of Parallel Forces. The resultant of the two parallel forces F and F t (Fig. 20), ap- plied at A and B respectively, is a force R = F -\- F lt whose line of action cuts the line AB at a point C, which divides it into two segments inversely as the forces. If the forces F and F, are turned through the same angle, and assume the posi- tions AO and BO l respectively, the line of action of the resultant will still pass through C, which is called the centre of the two parallel forces F and /v Inasmuch as a similar reasoning will apply in the case of any number of parallel forces, we may give the following definition : The centre of a system of parallel forces is the point through which the line of action of the resultant always passes, no matter how the forces are turned, provided only i. Their points of application remain the same. 2. Their relative magnitudes are unchanged. 3. They remain parallel to each other. Hence, in finding the centre of a set of parallel forces, we may suppose the forces turned through any angle whatever, and the centre of the set is the point through which the line of action of the resultant always passes. DISTRIBUTED FORCES. 39 39- Co-ordinates of the Centre of a Set of Parallel Forces. Let F l (Fig. 21) be one of the forces, and (x lt y u zj the co-ordinates of its point of application. Let F 2 be another, and (x 2t y 2t z 2 ) co- ordinates of its point of application. Turn all the forces around till they are parallel to OZ, and find the line of direction of the re- sultant force when they are in this position. The co- ordinates of this line are FIG. 21. and, since the centre of the system is a point on this line, the above are two of the co-ordinates of the centre. Then turn the forces parallel to OX, and determine the line of action of the resultant. We shall have for its co-ordinates y* = Hence, for the co-ordinates of the centre of the system, we have y = When 2F = o the co-ordinates would be oo, therefore such a system has no centre. 40. Distributed Forces While we have thus far as- sumed our forces as acting at single points, no force really acts at a single point, but all are distributed over a certain surface 40 APPLIED MECHANICS. or through a certain volume ; nevertheless, the propositions already proved are all applicable to the resultants of these distributed forces. We shall proceed to ' discuss distributed forces only when all the elements of the distributed force are parallel to each other. As a very important example of such a distributed force, we may mention the force of gravity which is distributed through the mass of the body on which it acts. Thus, the weight of a body is the resultant of the weights of the separate parts or particles of which it is composed. As another example we have the following : if a straight rod be subjected to a direct pull in the direction of its length, and if it be conceived to be divided into two parts by a plane cross- section, the stress acting at this section is distributed over the surface of the section. 41. Intensity of a Distributed Force. Whenever we have a force uniformly distributed over a certain area, we obtain its intensity by dividing its total amount by the area over which it acts, thus obtaining the amount per unit of area. If the force be not uniformly distributed, or if the intensity vary at different points, we must adopt the following means for rinding its intensity. Assume a small area containing the point under consideration, and divide the total amount of force that acts on this small area by the area, thus obtaining the mean intensity over this small area : this will be an approxima- tion to the intensity at the given point ; and the intensity is the limit of the ratio obtained by making the division, as the area used becomes smaller and smaller. Thus, also, the intensity, at a given point, of a force which is distributed through a certain volume, is the limit of the ratio of the force acting on a small volume containing the given point, to the volume, as the latter becomes smaller and smaller. 42. Resultant of a Distributed Force. i. Let the force be distributed over the straight line AB (Fig. 22), and RESULTANT OF A DISTRIBUTED FORCE. let its intensity at the point E where AE = x, be represented by EF ' = p <(*), a function of x ; then will the force acting on the por- tion Ee = A^r of the line be/A^r: and if we denote by R the magnitude of the resultant of the force acting on the entire line AB, and by x the distance of its point of application from A, we shall have R = 3/A.x approximately, or R = fpdx exactly ; and, by taking moments about an axis through A perpendicular to the plane of the force, we shall have XoR = ^x(pkx) approximately, or x R = fpxdx exactly ; whence we have the equations R = fpdx, Spdx Let the force be distributed over a plane area EFGff (Fig. 23), let this area be re- ferred to a pair of rectangular axes OX and OV, in its own plane, and let the intensity of the force per unit of area at the point P, whose co- B ordinates are x and y, be p = $(x, y) ; then will p&x&y be approximately the force act- ing on the small rectangular area A^rAj/. Then, if we rep- resent by R the magnitude of the resultant of the distributed force, and by x m y m the co-ordi- DC FIG. 23. 42 APPLIED MECHANICS. nates of the point at which the line of action of the resultant cuts the plane of EFGH, we shall have R 2/A^cAy approximately, x R = or, as exact equations, we shall have R = fSpdxdy, ffpxdxdy = ffpydxdy ~ ' ~ 3. Let the force be distributed through a volume, let this volume be referred to a system of rectangular axes, OX, O Y, and OZ, let A V represent the elementary volume, whose co- ordinates are x, y, z t and let p = <j>(x, y y z] be the intensity of the force per unit of volume at the point (x, y, z) ; then, if we represent by R the magnitude of the resultant, and by x , y , z m the co-ordinates of the centre of the distributed force, we shall have, from the principles explained in 38 and 39, the approx- imate equations R = and these give, on passing to the limit, the exact equations R - MV - - SpydV - ~ Jpd x ~' y ~' ~ 43. Centre of' Gravity. The weight of a body, or system of bodies, is the resultant of the weight of the separate parts or particles into which it may be conceived to be divided ; and the centre of gravity of the body, or system of bodies, is the centre of the above-stated system of parallel forces, i.e., the point through which the resultant always passes, no matter how the forces are turned. The weight of any one particle is the force which gravity exerts on that particle : hence, if we repre- FORCE APPLIED TO CENTRE OF STRAIGHT ROD. 43 sent the weight per unit of volume of a body, whether it be the same for all parts or not, by w, we shall have, as an approximation, and as exact equations, fwxdV fwydV fwzdV > (0 where W denotes the entire weight of the body, and x ot y m z , the co-ordinates of its centre of gravity. If, on the other hand, we let M = entire mass of the body, dM mass of volume dV t and m = mass of unit of volume, we shall have W = Mg, w = mg, wdV mgdV = gdM. Hence the above equations reduce to fxdM fydM fzdM Equations (i) and (2) are both suitable for determining the centre of gravity; one of the sets being sometimes most con- venient, and sometimes the other. 44. Centre of Gravity of Homogeneous Bodies __ If the body whose centre of gravity we are seeking is homogeneous, or of the same weight per unit of volume throughout, we shall have, that w ==. a constant in equations (i) ; and hence these reduce to 45. Effect of a Single Force applied at the Centre of a Straight Rod of Uniform Section and Material. If a straight rod of uniform section and material have imparted to it 44 APPLIED MECHANICS. a motion, such that the velocity imparted ima unit of time to each particle of the rod is the same, and if we represent this velocity by/, then if at each point of the rod, we lay off a line xy (Fig. 24) in the direction of the motion, and representing the velocity imparted to that point, the line bounding the other ends of the lines xy will be straight, and parallel to the rod. If we conceive the rod to be divided into any number of small equal parts, and denote the mass of one of these parts by <\M, then will contain as many units of momentum as there are units of force in the force required to impart to this particle the velocity f in a unit of time ; and hence f&M is the measure of this force. Hence the resultant of the forces which impart the velocity f to every particle of the rod will have for its measure fM, where M is the entire mass of the rod ; and its point of applica- tion will evidently be at the middle of the rod. It therefore follows that The effect of a single force applied at the middle of a straight rod of uniform section and material is to impart to the rod a motion of translation in the direction of the force, all points of , the rod acquiring equal velocities in equal times. 46. Translation and Rotation combined. Suppose that we.have a straight rod AB (Fig. 25), and suppose that such a force or such forces are applied to it as will impart to the point A in a unit of time the velocity Aa, and to the point B the (different) velocity Bb in a unit of time, both being perpendicu- lar to the length of the rod. It is required to determine the motion of any other point of the rod and that of the entire rod. TRANSLATION AND ROTATION COMBINED. 45 FIG. 25. Lay off Aa and Bb (Fig. 25), and draw the line ab t and pro- duce it till it meets AB produced in O : then, when these velocities Aa and Bb are imparted to the points A and B, the rod is in the act of rotating around an axis through O perpendicular to the plane of the paper ; for when a body is rotating around an axis, the linear velocity of any point of the body is perpendicular to the line joining the point in question with the axis (i.e., the perpendicular dropped from the point in question upon the axis), and proportional to the dis- tance of the point from the axis. Hence : If the velocities of two of the points in the rod are given, and if these are perpendicular to the rod, the motion of the rod is fixed, and consists of a rotation about some axis at right angles to the rod. Another way of considering this motion is as follows : Sup- pose, as before, the velocities of the points A and B to be represented by Aa and Bb respec- tively, and hence the velocity of any other point, as x (Fig. 26), to be represented by xy, or the length of the line drawn perpendicular to FIG. 26. AB, and limited by AB and ab. Then, if we lay off Aa, Bb, = \(Aa + Bb) = Cc, and draw a,b,, and if we also lay off Aa 2 a,a, and Bb 2 = bjb, we shall have the following relations ; viz., Aa = Aa, Aa 2 , Bb = Bb, + Bb 2 , xy = X y, xy 2 , etc., or we may say that the actual motion imparted to the rod in a unit of time may be considered to consist of the following two parts : 46 APPLIED MECHANICS. i. A velocity of translation represented by Aa J} the mean velocity of the rod ; all points moving with this velocity. 2. A varying velocity, different for every different point, and such that its amount is proportional to its distance from Cy the centre of the rod, as graphically shown in the triangles Aa 2 CBb 2 . In other words, the rod has imparted to it two motions : i. A translation with the mean velocity of the rod. 2. A rotation of the rod about its centre. 47. Effect of a Force applied to a Straight Rod of Uniform Section and Material, not at its Centre. If the force be not at right angles to the rod, resolve it into two com- ponents, one acting along the rod, and the other at right angles to it. The first component evidently produces merely a trans- lation of the rod in the direction of its length : hence the second component is the only one whose effect we need to study. To do this we shall proceed to show, that, when such a rod has imparted to it the motion described in 46, the single re- A cd B sultant force which is required to impart this motion in a unit of time is a force acting at right angles to the rod, at a point different from its centre ; and we shall de- (/ FIG. 27. termine the relation between the force and the motion imparted, so that one may be deduced from the other. Let A be the origin (Fig. 27), and let Ac = x, cd = dx. AB I = length of the rod. ce =f= velocity imparted per unit of time at distance x from A. Aa = / Bb = f 2 . w weight per unit of length. m mass per unit of length = ^!. g EFFECT OF FORCE APPLIED TO A STRAIGHT ROD. 47 W = entire weight of rod. M = entire mass of rod . g R = single resultant force acting for a unit of time to produce the motion. x distance from A to point of application of R. Then we shall have, Hence, from 42, AabB) = ^(/ +/,)/ = ^(/ + / 2 ). (i) 2 2 (2) . I /. + '/. , (-) " 3 /, +/, We thus have a force R, perpendicular to AB, whose mag- nitude is given by equation (i), and whose point of application is given by equation (3) ; the respective velocities imparted by the force being shown graphically in Fig. 27. EXAMPLES. i. Given Weight of rod = W = 100 Ibs., Length of rod = 3 feet, Assume g = 32 feet per second, Force applied = R = 5 Ibs., Point of application to be 2.5 feet from one end; determine the motion imparted to the rod by the action of the force for one second. 48 APPLIED MECHANICS. Solution. Equation (i) gives us, 5 = ( Equation (2) gives, < 2 '5)(5) - (^p) (3)C/ + /), or/ + 2/ 2 = 8 .-. / 2 = 4.8, / = -1.6. Hence the rod at the end nearest the force acquires a velocity of 4.8 feet per second, and at the other end a velocity of 1.6 feet per second. The mean velocity is, therefore, 1.6 feet per second; and we may consider the rod as having a motion of translation in the direc- tion of the force with a velocity of 1.6 feet per second, and a rotation about its centre with such a speed that the extreme end (i.e., a point | feet from the centre) moves at a velocity 4.8 1.6 = 3.2 feet per second. Hence angular velocity = ^ = 2.14 per second = 122. 6 per second. , 2. Given JF== 50 Ibs., /= 5 feet. It is desired to impart to it, in one second, a velocity of translation at right angles to its length, of 5 feet per second, together with a rotation of 4 turns per second : find the force required, and its point of application. 3. Assume in example 2 that the velocity of translation is in a direction inclined 45 to the length of the rod, instead of 90. Solve the problem. 4. Given a force of 3 Ibs. acting for one-half a second at a distance of 4 feet from one end of the rod, and inclined at 30 to the rod : determine its motion. 5. Given the same conditions as in example 4, and also a force of 4 Ibs., parallel and opposite in direction to the 3-lb. force, and acting also for one-half a second, and applied at 3 feet from the other end : determine the resulting motion. MOMENT OF THE FORCES CAUSING ROTATION. 49 6. Given two equal and opposite parallel forces, each acting at right angles to the length of the rod, and each equal to 4 Ibs., one being applied at i foot from one end, and the other at the middle of the rod ; find the motion imparted to the rod through the joint action of these forces for one-third of a second. 48. Moment of the Forces causing Rotation. Re- ferring again to Fig. 26, and considering the motion of the rod as a combination of translation and rotation, if we take moments about the centre C, and compare the total moment of the forces causing the rotation alone, whose accelerations are represented by the triangles aajbj), with the total moment of the actual forces acting, whose accelerations are represented by the trapezoid AabB, we shall find these moments equal to each other ; for, as far as the forces represented by the rectangle are concerned, every elementary force nt(xy^dx on one side of the centre C has its moment (Cx)\m(xy^dx\ equal and opposite to that of the elementary force at the same distance on the other side of C : hence the total moment of the forces represented graphically by the rectangle AaJb^B is zero, and hence The moment about C of those represented by the trapezoid equals the moment of those represented by the triangles. Hence, from the preceding, and from what has been pre- viously proved, we may draw the following conclusions : i. If a force be applied at the centre of the rod, it will impart the same velocity to each particle. 2. If a force be applied at a point different from the centre, and act at right angles to its length, it will cause a translation of the rod, together with a rotation about the centre of the rod. 3. In this latter case, the moment of the forces imparting the rotation alone is equal to the moment of the single resultant force about the centre of the rod, and the velocity of translation imparted in a unit of time is equal to the number of units of force in the force, divided by the entire mass of the rod. APPLIED MECHANICS. 49. Effect of a Pair of Equal and Opposite Parallel Forces applied to a Straight Rod of Uniform Section and Material. Suppose the rod to be AB (Fig. 28), and let the two equal and opposite parallel forces be Dd and Ee, each equal to F, applied at D and E respectively. The mean velocity imparted in a unit F of time by either force will be ; and, from what we have already seen, the trap- ezoid AabB will furnish us the means of representing the actual velocity imparted to any point of the rod by the force Dd. The relative magnitudes of Aa and Bb, the accelerations at the ends, will depend, of course, on the position of D ; but we shall 77 always have Cc l(Aa -f- Bb) = , a M quantity depending only on the magnitude of the force. So, likewise, the trapezoid AaJb^B will represent the velocities imparted by the force Ee ; and while the relative magnitude of Aa l and Bb l will depend upon the position of E, we shall always have Cc l = \(Aa l + Bb,) = . Hence, since Cc = Cc,, the centre C of the rod has no motion imparted to it by the given pair of forces, hence the motion of the rod is one of rotation about its centre C. The resulting velocity of any point of the rod will be the difference between the velocities imparted by the two forces ; and if these be laid off to scale, we shall have the second figure. Hence A pair of equal and opposite parallel forces, applied to a straight rod of uniform section and material, produce a rota- tion of the rod about its centre. Also, Such a rotation about the centre of the rod cannot be pro- FIG. 28. EFFECT OF STATICAL COUPLE ON STRAIGHT ROD. 51 duced by a single force, but requires a pair of equal and op- posite parallel forces. 50. Statical Couple. A pair of equal and opposite parallel forces is called a statical couple. 51. Effect of a Single Force applied at the Centre of Gravity of a Straight Rod of Non-Uniform Section and Material. In the case of a straight rod of non-uniform sec- tion and material, we may consider the rod as composed of a set of particles of unequal mass : and if we imagine each par- ticle to have imparted to it the same velocity in a unit of time,, then, using the same method of graphical representation as before (Fig. 24), the line ab, bounding the other ends of the lines representing velocities, will be parallel to AB ; but if we were to represent by the lines xy, not the velocities imparted, but the forces per unit of length, the line bounding the other ends of these forces would not, in this case, be parallel to AB. Moreover, since these forces are proportional to the masses, and hence to the weights of the several particles, their resultant would act at the centre of gravity of the rod. Hence A force applied at the centre of gravity of a straight rod will impart the same velocity to each point of the rod ; i.e., will im- part to it a motion of translation only. 52. Effect of a Statical Couple on a Straight Rod of Non-Uniform Section and Material. Let such a rod have imparted to it only a motion of rotation about its centre of gravity, and let us adopt the same modes of graphical repre- sentation as before. Let the origin be taken at O (Fig. 29), the centre of gravity of the rod. Let Aa = /, = velocity imparted to A. Bb = / 2 = velocity imparted to B. OA = a, OB = b, OC = x. CD = f = velocity imparted to C. dM = elementary mass at C. 52 APPLIED MECHANICS, Then, from similar triangles, we have /_4*-4r, a b and hence for the force acting on dM we have dF=(CE)dx = ^xdM. Hence the whole force acting on AO, and represented graph- ically by Aa^Oy is f (*x---a J - \ xdM, aj x = o and that acting on OB, and represented by B0b t , is / f*x = f (*x = o -V 2 / xdM = J - I xdM. bjx = -b ajx = -b Hence for the resultant, or the algebraic sum, of the two, we have But from 43 we have for the co-ordinate x of the centre of gravity of the rod f, x = a xdM M but, since the origin is at the centre of gravity, we have X = O, and hence \xdM = o .-. R = o. Jx=-6 Hence the two forces represented by Aa,O and Bb,O are equal in magnitude and opposite in direction : hence the rotation about the centre of gravity is produced by a Statical Couple. MEASURE OF THE ROTATORY EFFECT. 53 Now, a train of reasoning similar to that adopted in the case of a rod of uniform section and material will show that a single force applied at some point which is not the centre of gravity of the rod will produce a motion which consists of two parts ;. viz., a motion of translation, where all points of the rod have equal velocities, and a motion of rotation around the centre of gravity of the rod. 53. Moment of a Couple. The moment of a statical couple is the product of either force by the perpendicular dis- tance between the two forces, this perpendicular distance being called the arm of the couple. 54. Measure of the Rotatory Effect. Before proceed- ing to examine the effect of a statical couple upon any rigid body whatever, we will seek a means of measuring its effect in the cases already considered. The measure adopted is the moment of the couple ; and, in order to show that it is proper to adopt this measure, it will be necessary to show That the moment of the couple is proportional to the angu- lar velocity imparted to the same rod in a unit of time ; and from this it will follow That two couples in the same plane with equal moments will balance each other if one is right-handed and the other left-handed If we assume the origin of co-ordinates at C (Fig. 30), the centre of gravity of the rod, and if we denote by a the angular velocity imparted in a unit of time by the forces F and F, and let CD * CE = X M then we have for the linear velocity of a particle situated at a distance x from C the value CLT. FIG. 30. The force which will impart this velocity in a unit of time to the mass dM is axdM. 54 APPLIED MECHANICS. The total resultant force is afxtfM, which, as we have seen, is equal to zero. The moment of the elementary force about C is and the sum of the moments for the whole rod is and this, as is evident if we take moments about C, is equal t* Fxi - Fx 2 = F(x, - x,) = F(DE). Now, fx z dM is a constant for the same rod : hence any quan- tity proportional to F(DE) is also proportional to a. The above proves the proposition. Moreover, we have F(DE) = a whence it follows, that when the moment of the couple is given, and also the rod, we can find the angular velocity imparted in a unit of time by dividing the former by fx z dM. 55. Effect of a Couple on a Straight Rod when the Forces are inclined to the Rod. We shall next show that the effect of such a couple is the same as that of a couple of equal moment whose forces are perpen- r% =^^.^ dicular to the rod. /} ^"-^^^ In this case let AD and BC be the forces (Fig. 31). The moment of this couple is the product of AD by the per- c * pendicular distance between AD and BC, the graphical representation of this being the area of the parallelogram ADBC. EFFECT OF A STATICAL COUPLE ON A RIGID BODY. 55 Resolve the two forces into components along and at right angles to the rod. The former have no effect upon the motion of the rod : the latter are the only ones that have any effect upon its motion. The moment of the couple which they form is the product of Ad by AB, graphically represented by paral- lelogram AdBb ; and we can readily show that ADBC = AdBb. Hence follows the proposition. 56. Effect of a Statical Couple on any Rigid Body. Refer the body (Fig. 32) to three rectan- gular axes, OX, OY, and OZ, assuming the origin at the centre of gravity of the body, and OZ as the axis about which the body is rotating. Let the mass of the particle P be AJ/, and its co-ordinates be Then will the force that would impart FIG. 32. to the mass AJf the angular velocity a in a unit of time be where r =. perpendicular from P on OZ, or r ^x 2 + y 2 . This force may be resolved into two, one parallel to O Y an<f the other to OX; the first component being ax&M, and the second a/y&M. Proceeding in the same way with each particle, and finding the resultant of each of these two sets of parallel forces, we shall obtain, finally, a single force parallel to OY and equal to and another parallel to OX, equal to 56 APPLIED MECHANICS. But, since OZ passes through the centre of gravity of the body, we shall have = o and 2A M = o. Hence the resultant is in each case, not a single force, but a statical couple. Hence, to impart to a body a rotation about an axis passing through its centre of gravity requires the action of a statical couple ; and conversely, a statical couple so applied will cause such a rotation as that described. Further discussion of the motion of rigid bodies resulting from the action of statical couples is unnecessary for our pres- ent purpose, hence we shall pass to the deduction of the fol- lowing propositions, viz.: PROP. I. Two statical couples in the same plane balance each other when they have equal moments, and tend to pro- duce rotation in opposite directions. Let F 1 at a and F l at b represent one couple (left-handed in the figure), and let F t at d and F^ at e represent the other (right-handed in the figure), and let F\ab] = FJ&) ; then will these two couples balance each other. Proof. The resultant of F l at a and F^ FlG> 33> at d will be equal in amount, and directly opposed to the resultant of F l at b and F t at e and both will act along the diagonal fh of the parallelogram fchg. For we have (fg)(ab) = (fc)(de) 9 each being equal to the area of the parallelogram. . . -.&. fg " f< ^ ~f* ' hence follows the proposition. Hence follows that for a couple we may substitute another in the same plane, having the same moment, and tending to rotate the body in the same direction. COUPLES IN THE SAME OR PARALLEL PLANES. 57 FIG. 33 (a). PROP. II. Two couples in parallel planes balance each other when their moments are equal, and the directions in which they tend to rotate the body are opposite. Let (Fig. 33 (a)) the planes of both couples, be perpendicular to OZ. Reduce them both so as to have their arms equal and transfer them, each in its own plane, till their arms are in the X plane. Let ab be the arm of one couple, and dc that of the other. Then will the two couples form an equilibrate system. For the resultant of the force at a and that at c acts at e, and is twice either one of its com- ponents, and hence is equal and di- rectly opposed to the resultant of the force at b and that at d. Hence we may generalize all our propositions in regard to the effect of statical couples and we may conclude that In order that two couples may have the same effect, it is necessary i. That they be in the same or parallel planes. 2. That they have the same moment. 3. That they tend to cause rotation in the same direction (i.e., both right-handed or both left-handed when looked at from the same side}. It also follows, that, for a given statical couple, we may sub- stitute another having the magnitudes of its forces different, provided only the moment of the couple remains the same. 57. Composition of Couples in the Same or Parallel Planes. If the forces of the couples are not the same, reduce them to equivalent couples having the same force, transfer them to the same plane, and turn them so that their arms shall lie in the same straight line, as in Fig. 34; the first couple consisting of the force F at A and F at B, and the second of F at B and F at C. L t FIG. APPLIED MECHANICS. The two equal and opposite forces counterbalance each other, and we have left a couple with force F and arm AC = AB + PC .*. Resultant moment = F. AC = F(AB) + F(BC). Hence : The moment of the couple which is the resultant of two or more couples in the same or parallel planes is equal t~ the algebraic sum of the moments of the component couples. EXAMPLES. i. Convert a couple whose force is 5 and arm 6 to an equivalent couple whose arm is 3. Find the resultant of this and another coupk in the same plane and sense whose force is 7 and arm 8 ; also find the force of the resultant couple when the arm is taken as 5. Solution. Moment of first couple = 5 x 6 = 30 When arm is 3, force = ^- = 10 Moment of second couple = 7 x 8 = 56 Moment of resultant couple = 30 -f 56 = 86 When arm is 5, force = - 8 ^ = 17 1. Given the following couples in one plane : Force. Arm. 12 17 1 3 8 5 7 6 9 12 12 10 9 14 6 J Force. 5 Arm. Convert to equivalent couples having the < following : 8 20 The first and the last three are right-handed ; the second, third, and fourth are left-handed. Find the moment of the resultant couple, and also its force when it has an arm n. COUPLES IN PLANES INCLINED TO EACH OTHER. 59 58. Representation of a Couple by a Line. From the preceding we see that the effect of a couple remains the same as long as i . Its moment does not change. 2. The direction of its axis (i.e., of the line drawn perpen- dicular to tJie plane of the couple} does not change. 3. The direction in which it tends to make the body turn (right-handed or left-handed) remains the same. Hence a couple may be represented by drawing a line in the direction of its axis (perpendicular to its plane), and laying off on this line a distance containing as many units of length as there are units of moment in the couple, and indicating by a dot, an arrow-head, or some other means, in what direction one must look along the line in order that the rotation may appear right-handed. This line is called the Moment Axis of the couple. 59. Composition of Couples situated in Planes inclined to Each Other. Suppose we have two couples situated neither in the same plane nor in parallel planes, and that we wish to find their resultant couple. We may proceed as fol- lows : Substitute for them equivalent couples with equal arms, then transfer them in their own plane respectively to such posi- tions that their arms shall coincide, and lie in the line of intersection of the two planes. This having been done, let OO, (Fig. 35) be the common arm, F and F the forces of one couple, F l and F t those of the other. The forces F and F, have for their resultant R, and F and F, have R,. Moreover, we may readily show that R and R, are equal and 60 APPLIED MECHANICS. parallel, both being perpendicular to <9<9 2 . The resultant of the two couples is, therefore, a couple whose arm is OO^ and force R, the diagonal of the parallelogram on F and F lt so that R = \F 2 -h F* + 2FF l cos 0, where is the angle between the planes of the couples. Now, if we draw from O the line Oa perpendicular to OO I and to F, and hence perpendicular to the plane of the first couple, and if we draw in the same manner Ob perpendicular to the plane of the second couple, so that there shall be in Oa as many units of length as there are units of moment in the first couple, and in Ob as many units of length as there are units of moment in the second couple, we shall have i. The lines Oa and Ob are the moment axes of the two given couples respectively. 2. The lines Oa and Ob lie in the same plane with F and F T , this plane being perpendicular to OO lt 3. We have the proportion Oa-.0b = F. 00, -.F^OO^F: F,. 4. If on Oa and Ob as sides we construct a parallelogram, it will be similar to the parallelogram on F and /v We shall have the proportion Oc : R = Oa : F = Ob\F*\ and since the sides of the two parallelograms are respectively perpendicular to each other, the diagonals are perpendicular to each other ; and since we have also Oc = R ' Oa and Oa = F. OO, .'. Oc = R . OO t , F it follows that Oc is perpendicular to the plane of the resultant couple, and contains as many units of length as there are units of moment in the moment of the resultant couple; in other COUPLE AND SINGLE FORCE IN THE SAME PLANE. 6l words, Oc will represent the moment axis of the resultant couple, and we shall have Oc = \Oa* -f Ob* + 2Oa . or, if we let Oa = Z, Ob = M, Oc = G, aOb = 0, G = VZ 2 + J/ 2 + 2ZJ/cos6>. This determines the moment of the resultant couple ; and, for the direction of its moment axis, we have and sin a Oc = sin 6 G sin0. Hence we can compound and resolve couples just as we do forces, provided we represent the couples by their moment axes EXAMPLES. 1. Given L = 43, M 15, 6 = 65; find resultant couple. 2. Given Z = 40, M = 30, # = 30 ; find resultant couple. 3. Given L i, M ' = 5, # = 45; find resultant couple. 60. Resultant of a Couple and a Single Force in the Same Plane. Let M (Fig. 36 or 37) be the moment of the wF FIG. 36. FIG. 37. given couple, and let OF = F be the single force. For the given couple substitute an equivalent couple, one of whose forces is F at O, equal and directly opposed to the single 62 APPLIED MECHANICS, force F, these two counterbalancing each other, and leaving only the other force of the couple, which is equal and parallel to the original single force F, and acts along a line whose M distance from O is OA = . F Hence The resultant of a single force and a couple in the same plane is a force equal and parallel to the original force, having its line of direction at a perpendicular distance from the original force equal to the moment of the couple divided by the force. Fig. 36 shows the case when the couple is right-handed, and Fig. 37 when it is left-handed. 61. Composition of Parallel Forces in General. In each case of composition of parallel forces ( 34, 35, and 36) it was stated that the method pursued was applicable to all cases except those where ^F= o. We were obliged, at that time, to reserve this case, because we had not studied the action of a statical couple ; but now we will ad pt a method for the composition of parallel forces which will apply in all cases. (a) When all the forces are in one plane. Assume, as we did in 35, the axis OY to be parallel to the forces ; assume the forces and the co-ordinates of their lines of direction, as shown in the figure (Fig. 38). Now place at the origin O, along OY, two equal and opposite forces, each equal to x F, ; then these three forces, viz., F, at D, OA, and OB, produce the same effect as F, at D alone ; but F, at D and OR form a couple (left-handed in the figure) whose moment is F,*,. Hence the force Ft is equivalent to COMPOSITION OF PARALLEL FORCES. i. An equal and parallel force at the origin, and 2. A statical couple whose moment is P\x^. Likewise the force F 2 is equivalent to (i) an equal and par- allel force at the origin, and (2) a couple whose moment is -F 2 x 2 , etc. Hence we shall have, if we proceed in the same way with all the forces, for resultant of the entire system a single force R = ^F along OY, and a single resultant couple (Observe that downward forces and left-handed couples are to be accounted negative.) Now, there may arise two cases. i. When ^F o, and 2. When 2F><0. CASE I. When ^F = o, the resultant force along Y van- ishes, and the resultant of the entire system is a statical couple whose moment is CASE II. When %F > < o, we can reduce the resultant to a single force. Let (Fig. 39) OB represent the resultant force along OY, R = %F. With this, compound the couple whose moment is M 2<Fx, and we obtain as resultant ( 60) a single force FIG. 39. whose line of action is at a perpendicular distance from OY equal to AO = X r = 6 4 APPLIED MECHANICS. (b) When the forces are not confined to one plane. Assume, as before (Fig. 40), OZ parallel to the forces, and let F acting through A be one of the given forces, the co-ordinates of A be- ing x and y. Place at O two equal and opposite forces, each equal to F, and also at B two equal and opposite forces, each equal to F. These five forces produce the same effect as F alone at A, and they may be considered to con- sist of i. A single force F at the origin. 2. A couple whose 'forces are F at B and F at O, and whose moment is Fx acting in the y plane. 3. A couple whose forces are F at A and F at B, and whose moment is Fy acting in the x plane. Treating each of the forces in the same way, we shall have, in place of the entire system of parallel forces, the following forces and couples : FIG. 40. i. A single force R 2. A couple My = 3. A couple M x = Now, there may be two cases : i. When 2F >< O. 2. When $F = o. along OZ. in the y plane. in the x plane. CASE I. When < o, we can reduce to a single resultant force having a fixed line of direction. Lay off (Fig. 4 1 ) along OZ, OH $F. FIG. 4X . Combining this with the first of the above-stated couples, we R--SF COMPOSITION OF PARALLEL FOKCE.1. obtain a force R = 2<F at A, where OA = -' x r . Then 2F combine with this resultant force R 2F at A, the second couple, and we shall have as single resultant of the entire system a single force R = 2F acting through B, where Hence the resultant is a force whose magnitude is R 2/7, the co-ordinates of its line of direction being CASE II. When S/ 7 o, there is no single resultant force ; but the system reduces to two couples, one in the x plane and one in the y plane, and these two can be reduced to one single resultant couple. (Observe that couples are to be accounted positive when, on being looked at by the observer from the posi- tive part of the axis towards the origin, they are t right- handed ; otherwise they are negative.) The moment axis of the couple in the x plane will be laid off on the axis OX from the origin towards the positive side if the moment is positive, or towards the negative side if it is nega- FIG. 42. tive, and likewise for the couple in the y plane. -~ x 66 APPLIED MECHANICS. Hence lay off (Fig. 42) OB M x , OA = M y) and by completing the rectangle we shall have OD as the moment axis of the resultant couple ; hence the resultant couple lies in a plane perpendicular to OD, and its moment bears to OD the same ratio as M x bears to OB. Hence we may write OD = M r = cosDOX = = cos0. M r If My had been negative, we should have OR as the moment axis of the resultant couple. EXAMPLES. p. X. y- P. X. >' I. 5 4 3 2. 5 -4 3 3 2 i 2 2 i i 3 5 -3 3 5 Find the resultant in each example. 62. Resultant of any System of Forces acting at Dif- ferent Points of a Rigid Body, all situated in One Plane. Let CF = F (Fig. 43) be one of the given forces. Let all the forces be referred to a system of rectangular axes, as in the figure, and let a = angle made by F with FIG. 43. OX, etc. Let the co-ordi- M A nates of the point of application of F be AO = x, BO = SYSTEM OF FORCES ACTING ON RIGID BODY. 6? We first decompose CF = F into two components, parallel respectively to OX and O Y. These components are CD = Fcosa, CE = Fsina. Apply at O in the line O Y two equal and opposite forces, each equal to Fsin a, and at O in the line OX two equal and opposite forces, each equal to Fcosa. Since these four are mutually balanced, they do not alter the effect of the single force ; and hence we have, in place of Fat C, the six forces CD, OM, OK, CE, ON, OG. Of these six, CE and OG form a couple whose moment is (Fsma)x = Fxsina, CD and OK form a couple whose moment is (Fcosa)y = Fycosa. These two couples, being in the same plane, give as result- ant moment their algebraic sum, or F(y cos a x sin a) . We have, therefore, instead of the single force at C, the follow- ing: i. OM Fcos a along OX. 2. ON = Fsin a along O Y. 3. The couple M F(y cos a x sin a) in the given plane. Decompose in the same way each of the given forces ; and we have, on uniting the components along OX, those along OY, and the statical couples respectively, the following: i. A resultant force along OX, R x ^Fcos a. 2. A resultant force along OY, R y ^Fsm a . 3. A resultant couple in the plane, whose moment is M = %FLi> cos a x sin a}. 68 APPLIED MECHANICS. This entire system, on compounding the two forces at O t reduces to making with OX an angle a r , where ^F cos a cos OT = R 2. A resultant couple in the same plane, whose moment is M = *%F(y cos a x sin a) . Now compound this resultant force and couple, and we have, Y for final resultant, a single force equal and parallel to B E R, and acting along a line whose perpendicular dis- tance from O is equal to M R' G -\ ,-- A* R C I ^^ L E H FIG. 44 . Suppose (Fig. 44) the force OB = ^F cos a, 614 = OR = The equation of this line may be found as follows : + (S^ 1 sin a) 2 ; and let us suppose the resultant couple to be right-handed, and let then will the line ME parallel to OR be the line of direction of the single resultant force. CONDITIONS OF EQUILIBRIUM. 69 Assuming the force R to act at any point C (x r , y r ) of this line, if we decompose it in the same way as we did the single forces previously, we obtain i. The force R cos a r = 2^ cos a along OX. 2. The force R sina r = XFsina along OY. 3. The couple R(y r cos a r x r sin a^). Hence we must have R(y r cos a r x r sin a r ) = ^F(y cos a x sin a) = J/~. Hence for the equation of the line of direction we have M y r cos a r x r sin a r = . ( i j R Another form for the same equation is cos a) _ Xr (2J?sma) = M. (2) 63. Conditions of Equilibrium. If such a set of forces be in equilibrium, there must evidently be no tendency to h-an<r lation and none to rotation. Hence we must have R = o and M = o. Hence the conditions of equilibrium for any system of force? in a plane are three ; viz., 2/7 cos a = o, 2/7 sin a = o, 2/7(j>cosa ^sina) = o. Another and a very convenient way to state the conditions of equilibrium for this case is as follows : If the forces be resolved into components along two direction? at right angles to each other, then the algebraic sum of the com- ponents along each of these directions must be zero, and th* algebraic sum of the moments of the forces about any axis pendicular to the plane of the forces must equal zero. APPLIED MECHANICS. EXAMPLES. i. Given 2. Given p. X. y> 5 3 2 10 i 3 -7 4 2 P. X. * 12 27 3 -5 - 54 30 45 Find the resultant, and the equation of its line of direction. Find the resultant, and the equation of its line of direction. 64. Resultant of any System of Forces not confined to One Plane Suppose we have a number of forces applied at different points of a rigid body, and acting in different directions, of which we wish to find the resultant. Refer them all to a system of three rect- x angular axes, OX, OY, OZ (Fig. 45). Let PR = F be one of the given forces. Re- solve it into three components, PK, PH, and PG, parallel Let FIG. 45- respectively to the three axes. RPK = a, RPH = RPG Let OA x, OB = y, OC z, be the co-ordinates of the point of application of the force F. Now introduce at B and also at O two forces, opposite in direction, and each equal to PK. We now have, instead of the force PK, the five forces PK, BM, BN, OS, and OT. The two forces PK and BN form a couple in the y plane, whose axis is a line parallel to the axis OY, and whose moment is (PK)(EB) (Fcos a )z = Fzcosa. The FORCES NOT CONFINED TO ONE PLANE. fl forces Mand OT form a couple in the z plane, whose moment is (BM)(OB} = -Jycosa. Now do the same for the other forces PH and PG, and we shall finally have, instead of the force PR, three forces, F cos a, F cos ft F cos y, acting at O in the directions OX, O V, and OZ respectively, together with six couples, two of which are in the x plane, two in the y plane, and two in the z plane. They thus form three couples, whose moments are as fol- lows : Around OX, F(y cos y z cos /?) ; Around OY, F(zcosa #cosy); Around OZ, F(x cos fty cos a) . Treat each of the given forces in the same way, and we shall have, in place of all the forces of the system, three forces, ^F cos a along OX, ^F cos J3 along OY, along OZ; and three couples, whose moments are as follows : Around OX, M x ^F(y cos y z cos ft) ; Around O Y, M y = ^F(z cos a x cos y) ; Around OZ, M z 2F(xcos(3 jycosa). The three forces give a resultant at O equal to R = V(cosa) 2 -f (XFcos/3) 2 4- -&F cosy) 2 , (i) a ( . cosa r = - - , cos ft- = - ~ S cosy r = - - *-. ( 2 ) . K APPLIED MECHANICS. For the three couples we have as resultant --* COS /a = M' COS v = M z (3) (4) A, p, and v being the angles made by the moment axis of the resultant couple with OX, O Y, and OZ respectively. Thus far we have reduced the whole system to a single result- ant force at the origin, and a couple. Sometimes we can reduce the system still farther, and sometimes not. The following investigation will show when we can do so. Let (Fig. 46) OP R be the resultant force, and OC =M the moment axis of the resultant couple. Denote the angle between them by 6 (a quantity thus far undetermined). Pro- ject OP = R on OC. Its projection will be OD = RcosO; then project, in its stead, the broken line OABP on OC. By the principles of projections, the projection of this broken line will equal OD. Now OA, AB, and BP are the co-ordinates of P, and make with OC the same angle as the axes OX, OY, and OZ ; i.e., A., //,, and v respectively : hence the length of the projection is FIG. 46. But Hence OA = OAcosX + AB = R COS = R COS Or COS A. COS0 = COS Or COS A. + COS fi r COS fJL BP = cosy r . R cos p r cos p, -f ^cosy r cosv + cos y r cosv. (5) CONDITIONS OF EQUILIBRIUM. 73 This enables us to find the angle between the resultant force and the moment axis of the resultant couple. The following cases may arise : i. When cos o, or 6 90, the force lies in the plane of the couple, and we can reduce to a single force acting at a distance from O equal to , and parallel to R at O. R 2. When cos = I, or o, the moment axis of the couple coincides in direction with the force : hence the plane of the couple is perpendicular to the force, and no farther reduction is possible. 3. When is neither o nor 90, we can resolve the couple M into two component couples, one of which, McosO, acts in a plane perpendicular to the direction of R, and the other, J/sin 0, acts in a plane containing R. The latter, on being combined with the force R at the origin, gives an equal and parallel force whose line of action is at a distance from that of R at O, equal to MsmO R 4. When M = o, the resultant is a single force at O. 5. When R o, the resultant is a couple. 65. Conditions of Equilibrium. To produce equilibrium, we must have no tendency to translation and none to rotation. Hence we must have R = o and M = o. Hence we have, in general, six conditions of equilibrium ; viz., a = o, 2,J?cos/3 == o, ^F cos 7 == o. = o, My = o, M z = a 74 APPLIED MECHANICS. EXAMPLES. 1. Prove that, whenever three forces balance each other, they must lie in one plane. 2. Show how to resolve a given force into two whose sum is given, the direction of one being also given. 3. A straight rod of uniform section and material is suspended by two strings attached to its ends, the strings being of given length, and attached to the same fixed point : find the position of equilibrium of the rod. 4. Two spheres are supported by strings attached to a given point, and rest against each other : find the tensions of the strings. 5 . A straight rod of uniform section and material has its ends resting against two inclined planes at right angles to each other, the vertical plane which passes through the rod being at right angles to the line of intersection of the two planes : find the position of equilibrium of the rod, and the pressure on each plane, disregarding friction. 6. A certain body weighs 8 Ibs. when placed in one pan of a false balance of equal arms, and 10 Ibs. in the other : find the true weight of the body. 7. The points of attachment of the three legs of a three-legged table are the vertices of an isosceles right-angled triangle ; a weight of 100 Ibs. is supported at the middle of a line joining the vertex of one of the acute angles with the middle of the opposite side : find the pressure upon each leg. 8. A heavy body rests upon an inclined plane without friction : find the horizontal force necessary to apply, to prevent it from falling. 9. A rectangular picture is supported by a string passing over a smooth peg, the string being attached in the usual way at the sides, but one-fourth the distance from the top : find how many and what are the positions of equilibrium, assuming the absence of friction. 16. Two equal and weightless rods are jointed together, and form a right angle ; they move freely about their common point : find the ratio of the weights that must be suspended from their extremities, that one of them may be inclined to the horizon at sixty degrees. ii. A weight of 100 Ibs. is suspended by two flexible strings, one of which is horizontal, and the other is inclined at an angle of thirty degrees to the vertical : find the tension in each string. D YNAMICS. DEFINITIONS. ?$ CHAPTER II. DYNAMICS. 66. Definitions -- Dynamics is that part of mechanics which discusses the forces acting, when motion is the result. Velocity, in the case of uniform motion, is the space passed over by the moving body in a unit of time ; so that, if s repre- sent the space passed over in time t t and v represent the velocity, then Velocity, in variable motion, is the limit of the ratio of the space (AJ-) passed over in a short time (A/), to the time, as the latter approaches zero : hence r-* dt Acceleration is the limit of the ratio of the velocity ^A ; Im- parted to the moving body in a short time (A/), to the time, as the time approaches zero. Hence, if a represent the accelera- tion, * 76 APPLIED MECHANICS. 67. Uniform Motion In this case the acceleration is zero, and the velocity is constant ; and we have the equation s = vt. 68. Uniformly Varying Motion. In this case the ac- celeration is constant : hence a is a constant in the equation and we obtain by one integration ds v = - = +,,. where c is an arbitrary constant : to determine it we observe, that, if v represent the value of v when / = o, we shall have v = o -f c .'. c = v and by another integration s = -.vJiera s is the space passed over in time // the arbitrary con- s f ant vanishing, because, when / = o, s is also zero. 69. Measure of Force. It has already been seen, that, when a body is either at rest or moving uniformly in a straight line, there are either no forces acting upon it, or else the forces actr > upon it are balanced. If, on the other hand, the motion of -<>e body is rectilinear, but not uniform, the only unbalanced force acting is in the direction of the motion, and equal in mag- nitude to the momentum imparted in a unit of time in the direc- tion of the motion, or, in other words, to the limit of the ratio of the momentum imparted in a short time (A*), to the time, as the latter approaches zero. MECHANICAL WORK. UNIT OF WORK. // Thus, if F denote the force acting in the direction of the motion, m the mass, and a the acceleration, we shall have ,., dv d 2 s (i) F = ma = m = m . v ' dt dt 2 From (i) we derive mdv = Fdt; (2) and, if V Q be the velocity of the moving body at the time when / = 4, and 2/ x its velocity when / = t lt we shall have Xvi r> mdv = I Jto Fdt JV Q J t Q or m(v, - v ) = J *Fdt; (3) or, in words, the momentum imparted to the body during the time / = (/, / ) by the force F, will be found by integrating the quantity Fdt between the limits / x and t Q . 70. Mechanical Work. Whenever a force is applied to a moving body, the force is either used in overcoming resist- ances (i.e., opposing forces, such as gravity or friction), and leaving the body free to continue its original motion undis- turbed, or else it has its effect in altering the velocity of the body. In either case, the work done by the force is the prod- uct of the force, by the space passed through by the body *n the direction of the force. Unit qf Work. The unit of work is that work which is done when a unit of force acts through a unit of distance in the same direction as the force ; thus, if one pound and one foot are our units of force and length respectively, the unit of work will be one foot-pound. If a constant force act upon a moving body in the direction of its motion while the body moves through the space s, the work done by the force is Fs; APPLIED MECHANICS. and this, if the force is unresisted, is the energy, or capacity for performing work, which is imparted to the body upon which the force acts while it moves through the space s. Thus, if a lo-pound weight fall freely through a height of 5 feet, the energy imparted to it by the force of gravity during this fall is 10 X 5 = 50 foot-pounds, and it would be necessary to do upon it 50 foot-pounds of work in order to destroy the velocity acquired by it during its fall. If, on the other hand, the force is a variable, the amount of work done in passing over any finite space in its own direction will be found by in- tegrating, between the proper limits, the expression The power which a machine exerts is the work which it performs in a unit of time. The unit of power commonly employed is the horse-power, which in English units is equal to 33000 foot-pounds per minute, or 550 foot-pounds per second. 71. Energy. The energy of a body is its capacity for performing work. Kinetic or Actual Energy is the energy which a body pos- sesses in virtue of its velocity ; in other words, it is the work necessary to be done upon the body in order to destroy its velocity. This is equal to the work which would have to be done to bring the body from a state of rest to the velocity with which it is moving. Assume a body whose mass is m, and sup- pose that its velocity has been changed from V Q to v v Then if F be the force acting in the direction of the motion, we shall have, from equation (2), 69, that Fvdt = mvdv; (i) but vdt = ds /. Fds = mvdv. (2) ATWOOD'S MACHINE. 79 Hence, by integration, I mvdv = / Fds *Jvo *J /. \m(v* - V 2 ) = fFds; (3) but fFds is the work that has been done on the body by the force, and the result of doing this work has been to increase its velocity from v to v t . It follows, that, in order to change the velocity from v to v u the amount of work necessary to per- form upon the body is *(*,* - *>o 2 ) = i (z> x * - *>o 2 ). (4) 6 If v = o, this expression becomes \mv*, or ^ (5) 2g which is the expression for the kinetic energy of a body of mass m moving with a velocity v t . 72. Atwood's Machine. A particular case of uniformly accelerated motion is to be found in Atwood's machine, in which a cord is passed over a pulley, and is loaded with unequal weights on the two sides. Were the weights equal, there would be no unbalanced force acting, and no motion would ensue ; but when they are unequal, we obtain as a result a uniformly accelerated motion (if we disregard the action of the pulley), because we have a constant force equal to the difference of the two weights acting on a mass whose weight is the sum of the two weights. Thus, if we have a lo-pound weight on one side and a 5-pound weight on the other, the unbalanced force acting is F = io- 5 = 5 Ibs. SO APPLIED MECHANICS. T O " i_ f The mass moved is M == - 3UL : hence the resulting ac- celeration is 73. Normal and Tangential Components of the Forces acting on a Heavy Particle. If a body be in motion, either in a straight or in a curved line, and if at a certain instant all forces cease acting on it, the body will continue to move at a uniform rate in a straight line tangent to its path at that point where the body was situated when the forces ceased acting. If an unresisted force be applied in the direction of the body's motion, the motion will still take place in the same straight line; but the velocity will vary as long as the force acts, and, from what we have seen, the equation F=m* (i) dt 2 will hold. If an unresisted force act in a direction inclined to the body's motion, it will cause the body to change its speed, and also its course, and hence to move in a curved line. Indeed, if a force acting on a body which is in motion be resolved into two components, one of which is tangent to its path and the other normal, the tangential component will cause the body to change its speed, and the normal component will cause it to change the direction of its motion. The measure of the tangential component is, as we have seen, and we will proceed to find an expression for the normal com- ponent otherwise known as the Deviating Force. For this CENTRIFUGAL FORCE. 8 1 purpose we may substitute, for a small portion of the curve, a portion of the circle of curvature ; hence we will proceed to find an expression for the centrifugal force of a body which moves uniformly with a velocity v in a circle whose radius is r. CENTRIFUGAL FORCE. Let AC (Fig. 47) be the space described in the time A/. Then we have A B AC = The motion AC may be approximately consid- ered as the result of a uniform motion AB = z/A/ nearly, .and a uniformly accelerated motion PIG. 47 . BC = itf(A/) 2 = s, where a = acceleration due to centrifugal force. But (AB) 2 = BC . BD, or (vkty = %a(MY(2r + s) t where AO = OC = r /. v 2 = %a(2r + s) approximately 2V 2 .*. a = -- approximately. 2r + s For its true value, pass to the limit where s = o. Hence we have, for the acceleration due to the centrifugal force, the expression r' Hence the centrifugal force is -equal to gr 82 APPLIED MECHANICS. DEVIATING FORCE. If a body is moving in a curved path, whether circular or not, and the unbalanced force acting on it be resolved into tan- gential and normal components, the tangential component will be, as has already been seen, and the normal component will be mv 2 _ m/dsV r '- \dt)' where r is the radius of curvature of the path at the point in question. RESULTANT FORCE. Hence it follows that the entire unbalanced force acting on the body will be or F = m 74. Components along Three Rectangular Axes of the Velocities of, and of the Forces acting on, a Moving Rociy. If we resolve the velocity into three components along OX, OY, and OZ, we shall have, for these components respectively, dx dy , dz - aDd ' this being evident from the fact that dx, dy, and dz are respec- COMPONENTS OF VELOCITIES AND FORCES. 83 tively the projections of ds on the axes OX, OY, and OZ ; and, from the differential calculus, we have ds_ dt On the other hand, dx dy A dz *' it' and 7/ are not only the components of the velocity in the directions OX, OY, and OZ, but they are also the velocities of the body in these directions respectively. Now, the case of the accelerations is different ; for, while d 2 x d 2 y , d 2 z - are the accelerations in the directions OX, OY, and OZ respec- tively, they are not the components of the acceleration dt 2 along the three axes. That they are the former is evident from the fact that , dt -f-, and are the velocities in the directions of the axes, and at at d 2 x d 2 v d 2 z , ~~, are their differential co-efficients, and hence repre- sent the accelerations along the three axes. But if we consider the components of the force acting on the body, we shall have 84 APPLIED MECHANICS. for its components along OX t OY, and OZ, if a, ft, and y are the angles made by F with the axes respectively, Fcosa = m, F cos ft = m ^ Fcosy = m -. dt 2 dt 2 ^ 2 .-. F and we found ( 73) for F, the value Hence, equating these values of F, and simplifying, we shall have the equation Hence it is plain that - , ^-, and - can only be the com- dt 2 df dt 2 ponents of the actual acceleration when the last term f J vanishes, or when r = oo , i.e., when the motion is rectilinear. Moreover, we have the two expressions (i) and (2) for the force acting upon a moving body. The truth of the proposition just proved may also be seen from the following considerations : If a parallelopiped be constructed with the edges dx dy dz CENTRIFUGAL FORCE OF A SOLID BODY. 85 the diagonal will be the actual velocity ds df and will, of course, coincide in direction with its path. On the other hand, if a parallelepiped be constructed with the edges d 2 x d 2 y d 2 z dt 2 ' dt 2 ' dt 2 ' its diagonal must coincide in direction with the force and can coincide in direction with the path, and hence with the actual acceleration d 2 s dt 2 ' only when the force is tangential to the path, and hence when the motion is rectilinear. 75. Centrifugal Force of a Solid Body. When a solid body revolves in a circle, the resultant centrifugal force of the entire body acts in the direction of the perpendicular let fell from the centre of gravity of the body on the axis of rotation, and its magnitude is the same as if its entire weight were con- centrated at its centre of gravity. PROOF. Let (Fig, 48) the angular velocity = a, and the *eta' weight = W. Assume the axis of rotation perpendicular t the plane of the paper and passing through O ; assume, as axis of ;r, the perpendicular dropped from the centre of gravity upon the axis of rotation. The co-ordinates of the centre of gravity will then be (r , _^ ), and y will be equal to zero. FIG 8 If, now, P be any particle of weight w, where r = perpendicular distance from P on axis of rotatsoo, 86 APPLIED MECHANICS. and x OA, y = AP, we shall have for the centrifugal force of the particle at P w , -a. 2 r; g but if we resolve this into two components, parallel respectively to OX and OY, we shall have for these components and o.', = -wy, g sr g \g /r g ' and, for the resultant for the entire body we shall have, parallel to OX, (i) g and F y 2wy = Wy Q = o. (2) g g Hence the centrifugal force of the entire body is F-*-W*.; (3) ani if we let v = o,x = linear velocity of the centre of gravity, we have F- Wv * * ~~~ ) wnuh 13 the same as though the entire weight of the body ;cic concentrated at its centre of gravity. EXAMPLES. H. A lo-pound weight is fastened by a rope 5 feet long to the centre, aroun 1 which it revolves at the rate of 200 turns per minute ; hrd the pull on the cord. 2. A locomotive weighing 50000 Ibs., whose driving-wheels weigh toe Ibs., is running at 60 miles per hour, the diameter of the drivers UNIFORMLY VARYING RECTILINEAR MOTION. 8/ being 6 feet, and the distance from the centre of the wheel to the centre of gravity of the same being 2 inches (the drivers not being properly balanced) \ find the pressure of the locomotive on the track (a) when the centre of gravity is directly below the centre of the wheel, and (b) when it is directly above. 3. Assume the same conditions, except- that the distance between centre of the wheel and its centre of gravity is 5 inches instead of 2. 76. Uniformly Varying Rectilinear Motion. We have already found for this case ( 68) the equations - = a = a constant. (it* and we may write for the force acting, which is, of course, coin- cident in direction with the motion, F = m = ma = a constant. dr 77. Motion of a Body acted on by the Force of Gravity only. A useful special case of uniformly varying motion is that of a body moving under the action of gravity only. The downward acceleration due to gravity is represented by g feet per second, the value of g varying at different points on the surface of the earth according to the following law : g = gi(i 0.00284 cos 2X)(i ^ feet per second, where g, = 32.1695 feet, A = latitude of the place, h = its elevation above mean sea-level in feet, R 20900000 feet. 88 APPLIED MECHANICS. If, now, we represent by h the height fallen through by a descending body in time /, we shall have the equations, v. v + gt, h = v Q t + \gt*, where v is the initial downward velocity. If, on the other hand, we represent by v the initial upward velocity, and by h the height to which the body will rise in time / under the action of gravity only, we must write the equa- tions When v = o, the first set of equations gives v = gt y h = &/, which express the law of motion of a body starting from rest and subject to the action of gravity only. Eliminate / between these equations, and we shall have or h is called the height due to the velocity v, and represents the height through which a falling body must drop to acquire the velocity v ; and v = \2gh UNRESISTED PROJECTILE. 89 is the velocity which a falling body will acquire in falling through the height h. Thus, if a body fall through a height of 50 feet, it will, by that fall, acquire a velocity of about V 2 (3 2 i) (5) = V32i6.66 = 56.7 feet per second. Again : if a body has a velocity of 40 feet per second, we shall have v 2 1600 r , h = - - = 24.8 feet ; *g 64.3 and we say that the body has a velocity due to the height 24.8 feet, i. e., a velocity which it would acquire by falling through a height of 24.8 feet. EXAMPLES. 1. A stone is dropped down a precipice, and is heard to strike the bottom in 4 seconds after it started : how high is the precipice ? 2. How long will a stone, dropped down a precipice 500 feet high, take to reach the bottom ? 3. What will be its velocity just before striking the ground? 4. A body is thrown vertically upwards with a velocity of 100 feet per second ; to what height will it rise ? 5. A body is thrown vertically upwards, and rises to a height of 50 feet. With what velocity was it thrown, and how long was it in its ascent ? 6. What will be its velocity in its ascent at a point 15 feet above the point from which it started, and what at the same point in its descent ? 78. Unresisted Projectile. In the case of an unresisted projectile, we have a body on which is impressed a uniform APPLIED MECHANICS. motion in a certain direction (the direction of its initial motion), and which is acted on by the force of gravity only. Let OPC be the path (Fig. 49), OA the initial di- rection, and v the initial velocity, and the angle -4 CUT = K e. Then we shall FlG 49 have, for the hori- zontal and vertical components of the unbalanced force acting, when the projectile is at P (co-ordinates x and j), m = o along OX, and m = mg W along O Y. dP dt* Hence ^ = ' ^ Tip = ~ g ' ^ Integrating, and observing, that, when t o, the horizontal and the vertical velocities were respectively z; cos and z> sin 0, we have dx n , , = V Q cos 0, (3) ^ n t \ i- ***'-* W These equations could be derived directly by observing that the horizontal component of the initial velocity is V Q cos 0, and that this remains constant, as there is no unbalanced force act- ing in this direction, also that v sin 9 is the initial vertical velocity ; and, since the body is acted on by gravity only, this velocity will in time / be decreased by gt. UNRESISTED PROJECTILE. 91 Integrating equations (3) and (4), and observing that for / o, x and y are both zero, we obtain X = V Q COS O.t, (5) y = V Q sin O.t - \gt\ (6) Eliminate /, and we have ^ = *tan0 -- $* - (7) 2V * COS 2 as the equation of the path, which is consequently a parabola. Equations (i), (2), (3), (4), (5), (6), and (7) enable us to solve any problem with reference to an unresisted projectile. Equation (7) may be written / v 2 sin 2 0\ g / V ~~' ~~ ~ 2 Vo *ca*0 \ P sin0cos0 which gives for the co-ordinates of the vertex _ v 2 sin 2 _ z/o 2 sin cos y\ ~~~ ) x\ - - 2g g EXAMPLES. i. An unresisted projectile starts with a velocity of 100 feet pei* second at an upward angle of 30 to the horizon ; what will be its velocity when it has reached a point situated at a horizontal distance of icou teet from its starting-point, and how long will be required for it to that point? Solution. v = 100, = 30, v cos = 86.6, v sis as 50, g = 3 2 -i6. Equation (5) gives us 1000 = 86.6 / .'. / = = 11.55 seconds. 86,6 9 2 APPLIED MECHANICS. e> sin<9 - gt = 50 - 371.5 = -3 2I -5> v = V^(86.6) 2 + (3 2I -5) 2 = V75 + 103362 = 333. Hence the point in question will be reached in nj seconds after start- ing, and the velocity will then be 333 feet per second. 2. An unresisted projectile is thrown upwards from the surface of the earth at angle of 39 to the horizontal : find the time when it will reach the earth, and the velocity it will have acquired when it reaches the earth, the velocity of throwing being 30 feet per second. 3. A lo-pound weight is dropped from the window of a car when travelling over a bridge at a speed of 25 miles an hour. How long will it take to reach the ground 100 feet below the window, and what will be the kinetic energy when it reaches the ground ? 4. With what horizontal velocity, and in what direction, must it be thrown, in order that it may strike the ground 50 feet forward of the point of starting? 5. Suppose the same lo-pound weight to be thrown vertically up- wards from the car window with a velocity of 100 feet a minute, how long will it take to reach the ground, and at what point will it strike the ground ? 79. Motion of a Body on an Inclined Plane without Friction. If a body move on an inclined plane along the line of steepest descent, subject to the action of gravity only, and if we resolve the force acting on it (i.e., its weight) into two components, along and perpen- dicular to the plane respec- tively, the latter component will be entirely balanced by the resistance of the plane, and the former will be the only unbalanced force acting on the body. MOTION OF A BODY ON AN INCLINED PLANE. 93 Suppose a body whose weight is represented (Fig. 50) by HF = W to move along the inclined path AB under the action of gravity only. Let 9 be the inclination of AB to the horizon. Resolve W into two components, and HE = ^cos 9, respectively parallel and perpendicular to the plane. The former is the only unbalanced force acting on the body, and will cause it to move down the plane with a uniformly accel- erated motion ; the acceleration being (i) If the body is either at rest or moving downwards at the beginning, it will move downwards ; whereas, if it is first mov- ing upwards, it will gradually lose velocity, and move upwards more slowly, until ultimately its upward velocity will be de- stroyed, and it will begin moving downwards. The equations for uniformly varying motion are entirely applicable to these cases. Thus, suppose that the body has an initial downward velocity v ot this velocity will, at the end of the time /, become z> = ^ = z> + Crsintf)/ (2) at .-. s = v t -f k sin B . t*, (3) and, for the unbalanced force acting, we have F=ml = !(gsmO) = WsmO. (4) at 2 g 94 APPLIED MECHANICS. If, on the other hand, the body's initial velocity is upward, and we denote this upward velocity by v of we shall have the equations v =|= v - (g*m$)t (5) s = vt - fesinfl ./ 2 (6) F= -WsinO. (7) Again, if the initial velocity is zero, equations (2) and (3) become (8) From these we obtain, for this case, 2S do) and, substituting this value of / in (8), we have v = \2g(s sin 6), (n) or, if we let s sin 6 = h the vertical distance through which the body has fallen, we have v 2gh. (12) Hence, When a body, starting from rest, falls, under the action of gravity only, through a height h, the velocity acquired is \/2gh, whether the path be vertical or inclined. EXAMPLES. i. A body moves from the top to the bottom of a plane inclined to the horizon at 30, under the action of gravity only : find the time required for the descent, and the velocity at the foot of the plane. MOTION ALONG A CURVED LINE. 95 FIG. 51. 2. In the right-angled triangle shown in the figure (Fig. 51), given AB = 10 feet, angle BAC = 30: find the time a A body would require, if acted on by gravity only, to fall from rest through each of the sides respectively, AB being vertical. 3. Given inclination of plane to the horizon = 0, length of plane = /. compare the time of falling down the plane with the time of falling down the vertical. 4. A loo-pound weight rests, without friction, on the plane of example 3. What horizontal force is required to keep it from sliding down the plane. 5. Suppose 5 pounds horizontal force to be applied (a) so as to oppose the descent, () so as to aid the descent : find in each case how long it will take the weight to descend from the top to the bottom plane. 80. Motion along a Curved Line under the Action of Gravity only. We shall consider two questions in this regard : (a) the velocity at any point of the curve (b) the time of descent through any part of the curve. (a) Velocity at any point. Let us suppose the body to have started from rest at A, and to have reached the point P in time /, where AB = x (Fig. 52). Then, since the curved line AP may be considered as the limit of a broken line running from A to P, and as it has already been seen that the velocity acquired by falling through c a certain height depends only upon the height, and not upon the incli- nation of the path, we shall have for a curved line also FIG. 52. where v is the velocity at P. APPLIED MECHANICS. (b) Time down a curve. Referring to the same figure, let / denote the time required to go from A to P, and &t the time to go from P to f, where PP' = AJ, and BB ! =. kx ; then, as we have seen that the velocity at P is \2gx, we shall have approx- imately for the space passed over in time A/, the equation or, passing to the limit, This equation gives tis ds or / = c = r J^2gX J (2) v/here, of course, the proper limits of integration must be used. If / denote the time from A to P, we have = (""-*= J * ,,VV FK, 53 EXAMPLE. A body acted on by gravity only is constrained to move in the arc of a circle from A to C (Fig. 53), radius 10 feet. Find the time of describing the arc (quadrant) and the velocity acquired by the body when it reaches SIMPLE CIRCULAR PENDULUM. 97 8i. Simple Circular Pendulum. To find the time occu- pied in a vibration of a simple circu- ^c lar pendulum, we take D (Fig. 54) as origin, and DC as axis of x, and the axis of jj/at right angles to DC. Let AC /and BD = //, we shall have for the time of a single oscillation trom A to E /-, f J * = Now, from the equation of the circle AFDE, y 2 = 2/X X 2 , we have dy_ = I - x dx y ds I y s/2 ix - & Idx - x 2 )\_2g{h - *)] dx - x 2 V2/- or This can only be integrated approximately. Expanding f i J we obtain (-3T- 7 + ~Ta 4/ 32 / 2 98 APPLIED MECHANICS. The greatest value of x is //; and if h is so small that we may omit , we shall have as our approximate result t = J-f / dx = \m vQ ~*^T\ k = nA o> * g J yhx x 2 V g( h ) o V ^ o If, however, the value of h as compared with / is too large to render it sufficiently accurate to omit , but so small that 4/ we can safely omit the higher powers of ^, we shall have xdx h ' 4 / t h 4/[_2 or ' = V^ 1 + ^ (2) a nearer approximatioa The formula is the most used, and is more nearly correct, the smaller the value of h. EXAMPLES. i . Find the length of the simple circular pendulum which is to beat seconds at a place where g = 32^. Solution. SIMPLE CYCLOIDAL PENDULUM. 99 2. What is the time of vibration of a simple circular pendulum 5 feet long? 82. Simple Cycloidal Pendulum. The equation of the cycloid is x x y = # versin \- (2ax Jf 3 )^ a . dy_ \/ 2a ~ x dx V x ds_ = /2^\5 dx \ x I Hence we shall have, for the time of a single oscillation, dx or This expression is independent of //, so that the time of vibra- tion is the same whether the arc be large or small. A body can be made to vibrate in a cycloidal arc by suspend- ing it by a flexible string between two cycloidal cheeks. This is shown from the fact that the evolute of the cycloid is another cycloid (Fig. 55). To prove this, we have, from the equation of the cycloid, y = a versin - -j- (2ax dy _ t / dx ~ V 2a x ds a &.^_ <& *^ia - x I00 APPLIED MECHANICS. Hence the radius of curvature is and since we have for the evolute the relation ds' = dp, where ds f is the elementary arc of the evolute, f*x = za .-. /= I *; i/.*r^* and, observing that when x 2a p = o, we have If x l is the abscissa of the point of the evolute, - - x + d y = a - x ds and, transforming co-ordinates to B by putting x 2 . + 2a for we obtain which is the equation of another cycloid just like the first. The motion along a vertical cycloid may also be obtained by letting a body move along a groove in the form of a cycloid acted on by gravity alone ; and in this case the time of descent of the body to the lowest point is precisely the same at what- ever point of the curve the body is placed. 83. Effect of Grade on the Tractive Force of a Rail- way Train. -Asa useful particular case of motion on an inclined plane, we have the case of a railroad train moving up or down a grade. It is necessary that a certain tractive force EFFECT OF GRADE ON TRACTIVE FORCE. IOI be exerted in order to overcome the resistances, and keep the train moving at a uniform rate along a level track. If, on the other hand, the track is not on a level, and if we resolve the weight of the train into components at right angles to and along the plane of the track, we shall have in the latter component a force which must be added to the tractive force above referred to when we wish to know the tractive force re- quired to carry it up grade, and must be subtracted when we wish to know the tractive force required to carry it down grade. The result of this subtraction may give, if the grade is suffi- ciently steep and the speed sufficiently slow, a negative quan- tity ; and in that case we must apply the brakes, instead of using steam, unless we wish the speed of the train to increase. EXAMPLES. i. A railroad train weighing 60000 Ibs., and running at 50 miles per hour, requires a tractive force of 618 Ibs. on a level ; what is the tractive force necessary when it is to ascend a grade of 50 feet per mile? What when it is to descend? Also what is the amount of work per minute in each case ? Solution. The resolution of the weight will give (Fig. 50, 7?^ tor the com- ponent along the plane, (60000)^ = 568.2 nearly. Hence Tractive force for a level = 618.0, Tractive force for ascent = 1186.2, Tractive force for descent 49.8. To ascertain the work done per minute in each case, we have (a) For a level track, 6l8 x 5 6 o x 528 = 2719200 foot-lbs. (l>) Up grade, 2719200 + 6ooo ^ x 5 = 5219200 foot-lbs. (c) Down grade, 2719200 - 6ooo X J x 5 = 219200 foot-lbs. 102 APPLIED MECHANICS, 2. Suppose the tractive force required for each 2000 Ibs. of weight of train to be, on a level track, for velocities of 5.0 miles per hour, 10.0 20.0 30.0 40.0 50.0 60 6.1 Ibs., 6.6 8.3 ii. 2 15.3 20.6 27; find the tractive force required to carry the train of example i (a) Up an incline of 50 feet per mile at 30 miles per hour. (^) Down an incline of 50 feet per mile at 30 miles per hour. (<:) Down an incline of 10 feet per mile at 20 miles per hour. (//) What must be the incline down which the train must run to require no tractive force at 40 miles per hour? 3. If in the first example the tractive force remains 618 Ibs. while the train is going down grade, what will be its velocity at the end of one minute, the grade being 10 feet per mile? 84. Harmonic Motion If we imagine a body to be moving in a circle at a uniform rate (Fig. 56), and a second body to oscillate back and forth in the diameter AB, both starting from B, and if when the first body is * at C the other is directly un- der it at G, etc., then is the second body said to FIG. 56. move in harmonic motion. A practical case of this kind of mo- tion is the motion of a slotted cross-head of an engine, as shown in the figure ig- 57) i the crank moving at a form rate. In the case of the ordinary crank, and connecting-rod connecting the drive-wheel shaft of a stationary engine with the piston-rod, FIG. 57. HARMONIC MOTION. 1 03 we have in the motion of the piston only an approximation to harmonic motion. We will proceed to determine the law of the force acting upon, and the velocity of, a body which is con- strained to move in harmonic motion. Let the body itself and the corresponding revolving body be supposed to start from B (Fig. 56), the latter revolving in left-handed rotation with an angular velocity a, and let the time taken by the former in reaching G be t: then will the angle BOC at; and we shall have, if s denote the space passed over by the body that moves with harmonic motion, s = BG OB - OCcosat, or, if r=O= OC t s = r rcosa/, (l) the velocity at the end of the time t will be V = = arsina/, (2) and the acceleration at the end of time / will be (3) Hence the force acting upon the body at that instant, in the direction of its motion, is F = m = ma 2 r cos at = ma 2 (OG). (4) dt* The force, therefore, varies directly as the distance of the body from the centre of its path. It is zero when the body is at the IO4 APPLIED MECHANICS. centre of its path, and greatest when it is at the ends of its travel, as its value is then W ma 2 r = o?r; S this being the same in amount as the centrifugal force of the revolving body, provided this latter have the same weight as the oscillating body. On the other hand, the velocity is greatest when at = - (i.e., at mid-stroke) ; and its value is then v = ar, this being also the velocity of the crank-pin at mid-stroke. EXAMPLE. Given that the reciprocating parts of an engine weigh 10000 Ibs., the length of crank being i foot, the crank making 60 revolutions per minute ; find the force required to make the cross-head follow the crank, (i) when the crank stands at 30 to the line of dead points, (2) when at 60, (3) when at the dead point. 85. Work under Oblique Force. If the force act in any other direction than that of the motion, we must resolve it into two components, the component in the direction of the motion being the only one that does work. Thus if the force F is variable, and 6 equals the angle it makes with the direction of the motion, we shall have as our expression for the work done fFcosOds. Thus if a constant force of 100 Ibs. act upon a body in a direc- tion making an angle of 30 with the line of motion, then wil! the work done by the force during the time in which it moves through a distance of 10 feet be (100) (0.86603) (10) = 866 foot-lbs. ROTATION OF RIGID BODIES. 1 05 86. Rotation of Rigid Bodies -- Suppose a rigid body (Fig. 58) to revolve about an axis perpendicular to the plane of the paper, and passing through O ; imagine a particle whose weight is w to be situated at a perpendicular distance OA = r from the axis of rotation, and let the angular accel- eration be a : let it now be required to find the moment of the force or forces required to impart this ac- celeration ; for we know that, if the axis of rotation pass through the centre of gravity of the body, the motion can be imparted only by a statical couple ; whereas if it do not pass through the centre of gravity, the motion can be imparted by a single force. We shall have, for the particle situated at A, Weight = w. Angular acceleration = a. Linear acceleration = o.r. Force required to impart this acceleration to this particle w a-r. g 7ff Moment of this force about the axis = ar 2 . g Hence the moment of the force or forces required to impart to the entire body in a unit of time a rotation about the axis through O, with an angular velocity a, is 8 8 S where / is used as a symbol to denote the limit of ^wr 2 , and is called the Moment of Inertia of the body about the axis through O. 106 APPLIED MECHANICS. 87. Angular Momentum. This quantity,, which ex- g presses the moment of the force or forces required to impart to the body the angular acceleration a about the axis in question is also called the Angular Momentum of the body when rotat- ing with the angular velocity about the given axis. 88. Actual Energy of a Rotating Body. If it be re- quired to find the actual energy of the body when rotating with the angular velocity w, we have, for the actual energy of the particle at A, g 2 2g and for that of the entire body <u* w 2 / Iwr 2 = - . *g zg This is the amount of mechanical work which would have to be done to bring the body from a state of rest to the velocity w, or the total amount of work which the body could do in virtue of its velocity against any resistance tending to stop its rotation. 89. Moment of Inertia. The term "moment of inertia" originated in a wrong conception of the properties of matter. The term has, however, been retained as a very convenient one, although the conceptions under which it originated have long ago vanished. The meaning of the term as at present used, in relation to a solid body, is as follows : The moment of inertia of a body about a given axis is the limit of the sum of the products of the weight of each of the ele- mentary particles that make tip the body, by the squares of their distances from the given axis. Thus, if w lt w 2 , w y etc., are the weights of the particles which are situated at distances r lf r r y etc., respectively from MOMENT OF INERTIA OF A PLANE SURFACE. IO/ the axis, the moment of inertia of the body about the given axis is / = limit of 90. Radius of Gyration. The radius of gyration of a body with respect to an axis is the perpendicular distance from the axis to that point at which, if the whole mass of the body were concentrated, the angular momentum, and hence the mo- ment of inertia, of the body, would remain the same as they are in the body itself. If p is the radius of gyration, the moment of inertia would be, when the mass is concentrated, hence we must have whence where W = entire weight of the body. 91. Moment of Inertia of a Plane Surface The term , "moment of inertia," when applied to a plane figure, must, of course, be defined a little differently, as a plane surface has no weight ; but, inasmuch as the quantity to which that name is given is necessary for the solution of a great many questions. The moment of inertia of a plane surface about an axis, either in or not in the plane, is the limit of the sum of the products of the elementary areas into which the surface may be conceived to be divided, by the squares of their distances from the axis in question. In a similar way, for the radius of gyration p of a plane figure whose area is A, we have 108 APPLIED MECHANICS. From this definition it will be evident, that, if the surface be referred to a pair of axes in its own plane, the moment of iner- tia of the surface about O Y will be (i) and the moment of inertia of the surface about OX will be J^fffdxdy. (2) The moment of inertia of the surface about an axis passing through the origin, and perpendicular to the plane XO Y, will be SS**dx*y, (3) where r=. distance from O to the point (x,y) ; hence r 2 =. x 2 -f y 2 , and the moment of inertia becomes ff(x 2 4- y 2 )dxdy = ffx 2 dxdy + ffy 2 dxdy = / + /. (4) This is called the "polar moment of inertia." If polar co-ordi- nates be used, this last becomes ffp 2 ( P dpdB) = ffptdpdO. (5) All these quantities are quantities that will arise in the discus- sion of stresses, and the letters /and./ are very commonly used to denote respectively ffx z dxdy and ffy*dxdy. Another quantity that occurs also, and which will be repre- sented by K, is ffxydxdy; (6) and this is called the moment of deviation. EXAMPLES OF MOMENTS OF INERl'IA. 109 EXAMPLES. The following examples will illustrate the mode of finding the moment of inertia : .x i. Find the moment of inertia of the rectangle ABCD about OY (Fig. 59). Solution. h FIG. 59. 2. Find the moment of inertia of the entire circle (radius r) about the diameter OY (Fig. 60). FIG. 60. Solution. _ 4 " " 64 =2 -xtf+ r * f Vr / 4V 3- Find the moment of inertia of the circular ring (outside radius r, inside radius r^ about OY (Fig. 61). Solution. "44 64 4. Find the moment of inertia of an ellipse (semi-axes a and b) about the minor axis OY. FIG. 61. no APPLIED MECHANICS. Solution. Equation of ellipse is -f- ^ 7ta*b On the other hand, I x 4 92. Moments of Inertia of Plane Figures about Parallel Axes. PROPOSITION. The moment of inertia of a plane figure about an axis not passing through its centre of gravity is equal to its moment of inertia about a parallel axis passing through its centre of gravity increased by the product obtained by multiply- ing the area by the square of the distance between the two axes. PROOF. Let A B CD (Fig. 62) be the surface ; let 0Fbe the axis not passing through the centre of grav- ity ; let P be an elementary area A^rAr, whose co-ordi- nates are OR x and RP y ; and let OO T a = a constant = distance be- tween the axes. Let O,R x, abscissa of P with reference to the axis passing through the centre of gravity, x = a -f- x 2 = x, 2 2ax t Ay POLAR MOMENT OF INERTIA OF PLANE FIGURES. Ill Hence, summing, and passing to the limit, we have fftfdxdy = fjxfdxdy + zaffxjxdy + a*ffdxdy ; ( i ) but if we were seeking the abscissa of the centre of gravity when the surface is referred to Y^OY lt and if this abscissa be denoted by x m we should have _ = ffdxdy ' and, since X Q = o, /. ffx^dxdy = o ; hence, substituting this value in (l), we obtain ffx 2 dxdy = ffxfdxdy -f- a 2 ffdxdy. . (2) If, now, we call the moment of inertia about O Y, 7, that about O, Y lt / and let the area = A = ffdxdy, we shall have 7=7, + a- A. (3) Q. E. D. 93. Polar Moment of Inertia of Plane Figures. The moment of inertia of a plane figure about an axis perpen- dicular to the plane is equal to the sum of its moments of inertia about any pair of rectangular axes in its plane passing through the foot of the perpendicular. PROOF. Let BCD (Fig. 63) be the surface, and P an ^ Y elementary area, and let OA x, AP = y, OP r; then the moment of inertia of the surface about OZ will be f ffdxdy = ff(x 2 +y*}dxdy = ffx*dxdy + f ffdxdy = / -f /. Q. E. D. FIG. 63. 112 APPLIED MECHANICS. Hence follows, also, that the sum of the moments of inertia of a plane surface relatively .to a pair of rectangular axes in its own plane is isotropic ; i.e., the same as for any other pair of rectangular axes meeting at the same point, and lying in its plane. EXAMPLES. i. To find the moment of inertia of the rectangle (Fig. 59) about an axis through its centre perpendicular to the plane of the rectangle. Solution. Moment of inertia about YY , 12 Moment of inertia about an axis through its hence centre and perpendicular to YY = 12 Polar moment of inertia = 1 = (h 2 + 12 12 12 2. To find the moment of inertia of a circle about an axis through its centre and perpendicular to its pla'ne (Fig. 60). Solution. Moment of inertia about OY = , 4 hence Moment of inertia about OX = 4 -r, , , . Trr 4 TIT 4 TTf Polar moment of inertia = - f- = 442 3. To find the moment of inertia of an ellipse about an axis passing through its centre and perpendicular to its plane. MOMENTS OF INERTIA ABO^JT DIFFERENT AXES. 113 Solution* From example 4, 91, we have / -- 7ra ^ 3 4 .: Polar moment of inertia = (a 2 4 94. Moments of Inertia of Plane Figures about Different Axes compared. Given the surface KLM (Fig. 64), suppose we have already determined the quantities / = ffx 2 dxdy, / = fffdxdy, K = ffxydxdy, it is required to determine, in terms of them, the quantities A the angles JTOFand X,OY, being both right angles, and YO Y, = a. We shall have, from the ordinary equations for the transformation of co- ordinates, to be found in any analytic geometry, the equations x t = x cos a -f- y sin a, y, = ycosa - *sina, FIG ^ x? = x 2 cos 2 a -f- y 2 sin 2 a -f- 2xy cos a sin a, jj 2 = ^ 2 sin 2 a -f- jy 2 cos 2 a 2^' cos a sin a, ^jjj = ^y(cos 2 a sin 2 a) (x 2 y 2 ) cos a since. 1 14 APPTIED MECHANICS. Hence = ffxfdxdy* = limit of Sxf&A = cos 2 a limit of 2x 2 &A + sin 2 a limit of 2 cos a sin a limit 2 (cos a sin a) ffxydxdy. J s = ffy l 2 dx l dy l = limit of lyf&A = (sin 2 a) limit of 2^A^ + (cos 2 a) limit of 2 (cos a sin a) limit of 2 (cos a sin a)ffxydxdy. K t = ffx l y 1 ^x j ^y l = limit of S^ij^iA^ = (cos 2 a sin 2 a) limit of 2<xy&A (cos a sin a) {limit of 2x 2 &A - limit of ^y 2 ^A\ = (cos 2 a sin 2 a)ffxydxdy (cos a sin a) \ffx 2 dxdy fffdxdy}. Or, introducing the letters /, J t and TsT, we have 7, = /cos 2 a + / sin 2 a -f 2^ cos a sin a, (i) y r = /sin 2 a + y COS 2 a 2 A" COS a sin a, (2) ^ = (J /) cos a sin a + ^(cos 2 a sin 2 a). (3) The equations (i), (2), and (3) furnish the solution of the problem. 95. Principal Moments of Inertia in a Plane. In every plane figure, a given point being assumed as origin, there is at least one pair of rectangular axes, about one of which the moment of inertia is a maximum, and a minimum about the other ; these moments of inertia being called principal moments of inertia, and the axes about which they are taken being called principal axes of inertia AXES OF SYMMETRY OF PLANE FIGURES. I 15 PROOF. In order that / equation (i), 94, may be a maxi- mum or a minimum, we must have, as will be seen by differen- tiating its value, and putting the first differential co-efficient equal to zero, 2/cos a sin a 4- 2/cos a sin a 4- 2^(cos 2 a sin 2 a) = o /. ^(cos 2 a sin 2 a) (/ /) cos a sin a = o (i) cos a sin a K iK , \ /. - = - .*. tan 2 a = - - -. (2) cos 2 a sin 2 a / J I J Hence, for the value of a given by (2), we have 7, a maximum or a minimum ; and as there are two values of 2a corresponding to the same value of tan 2a, and as these two values differ by 1 80, the values of a will differ by 90, one corresponding to a maximum and the other to a minimum. Moreover, when the value of a is so chosen, we have as is proved by equation (i). Indeed, we might say that the condition for determining the principal axes of inertia is K, = o. 96. Axes of Symmetry of Plane Figures. An axis which divides the figure symmetrically is always a principal axis. PROOF. Let us assume that the y axis divides the surface symmetrically ; then we shall have, with reference to this axis, K = And, since K is zero, the axis of y is one principal axis, and of course the axis of x is the other. The same method of reason- ing would show K = o if the x axis were the axis of symmetry. II 6 APPLIED MECHANICS. Hence, whenever a plane figure has an axis of symmetry, this axis is one of the principal axes, and the other is at right angles to it. Thus, for a rectangle, when the axis is to pass through its centre of gravity, the principal axes are par- allel to the sides respectively, the moment of inertia being- greatest about the shortest axis, and least about the longest. Thus in an ellipse the minor axis is the axis of maximum, and the major that of minimum, moment of inertia, etc. On the other hand, in a circle, or in a square, since the maximum and minimum are equal, it follows that the moments of inertia about all axes passing through the centre are the same. 97. Conditions for Equal Values of Moment of In- ertia. When the moments of inertia of a plane figure about three different axes passing through the same point are the same, the moments of inertia about all axes passing through this point are the same. PROOF. Let / be the moment of inertia about O Y, 7 l about OY lt I 2 about OY 2 , and let YOY, = a, YOY 2 = ft, and let /, = /* = /. Then, from equation (i), 94, we have 1=1 cos 2 a 4- J sin 2 a -f 2 K cos a sin a, /= /cos 2 /? +/sin 2 /3 + 2 A" cos/? sin/3. Hence (/ y)sin 2 a = 2 A' cos a sin a, (i) (7-/)sin 2 /3 == 2 K cos ft sin ft. (2) Hence (7-/)tana = 2 AT, (3) (7-/)tan0 = 2 AT. (4) And, since tan a is not equal to tan ft, we must have / J o and K = o. Hence, since K o and / = J t we shall have, from eqja- MOMENTS OF INERTIA ABOUT PARALLEL AXES. I I/ tion (i), 94, for the moment of inertia /' about an axis, making any angle with O Y, I' = /cos 2 + /sin 2 + o = /. (5) Hence all the moments of inertia are equal. 98. Components of Moments of Inertia of Solid Bodies. Refer the body to three rectangular axes, OX, OY, and OZ ; and let I x , I y , and I z represent its moment of inertia about each axis respectively. Then, if r denote the distance of any particle from OZ> we shall have I z = limit of ^wr 2 \ but r* = x 2 + y 2 .'. I z = limit of So/C* 2 + y 2 } = limit of Saw 2 + limit of So/? 2 , (i) In the same way we have 7* = limit of ^wy 2 + limit of ^wz 2 , (2) I y = limit of Sow 2 + limit of ^wz 2 . (3) 99. Moments of Inertia of Solids around Parallel Axes. The moment of inertia of a solid body about an axis not passing through its centre of gravity is equal to its moment of inertia about a parallel axis passing through the centre of gravity, increased by the product of the entire weight of the body by the square of the distance between the two axes. PROOF. Refer the body to a system of three rectangular axes, OX, OY, and OZ, of which OZ is the one about which the moment of inertia is taken. Let the co-ordinates of the centre of gravity of the body with reference to these axes be (^oi Jo, #<>) Through the centre of gravity of the body draw a system of rectangular axes, parallel respectively to OX, OY, and OZ. Then we shall have for the co-ordinates of any point X = X o -\~ Xi, y = y* +y Z = Z + *,. APPLIED MECHANICS. Hence 7 2 = limit of 2w(x 2 + y 2 ) = limit of *Zwx 2 4 limit of = limit of *2w(x 4- x,) 2 4- limit of %w(y + y t ) 2 = x 2 limit of 2o> 4- 2 limit of 2w 4- 2x limit of 4- limit of 2 limit of 2o> 4- y 2 limit of 2w -f 2y limit of Soy^ -f limit of -f limit of = (* 2 -f 7o 2 ) -f limit of = r 2 W 4- // 4 2^ limit of But, since 6^ x is the centre of gravity, /. ^wx I = o and Hence 4 limit of limit of o. which proves the proposition. 100. Examples of Moments of Inertia. i . To find the moment of inertia of a sphere whose radius is r and weight per unit of volume w, about the axis OZ drawn through its centre. Solution. Divide the sphere into thin slices (Fig. 65) by planes drawn perpen- dicular to OZ. Let the distance of the slice shown in the figure, above O be z, and its thickness dz : then will its radius be Vr 2 z 2 ; and we can readily see, from ex- ample 2, 93, that its moment of inertia about OZ will be dz. FIG. 65. Hence the moment of inertia of the entire sphere about OZ will be w - f V - 2 J_ r V EXAMPLES OF MOMENTS OF INERTIA. which easily reduces to I z = 15 2. To find the moment of inertia of an ellipsoid (semi-axes a, b, c) about OZ (Fig. 66). SOLUTION. The equa- tion of the ellipsoid is Divide it into thin slices perpendicular to OZ, and let the slice shown in the figure be at a distance z from O. Then will this slice be elliptical, and its semi-axes will be FIG. 66. - V<T 2 - Z 2 and C* and from example 3, 93, we readily obtain, for its moment of inertia about OZ, = Hence, for the moment of inertia of the ellipsoid about OZ, we have I U IS 3. Find the moment of inertia of a right circular cylinder, length a, radius r, about its axis. Ans. 120 APPLIED MECHANICS. 4. Find the moment of inertia of the same about an axis perpen- dicular to, and bisecting its axis. Ans wwar* I a*\ 4 V " 3/ 5. Find the moment of inertia of an elliptic right cylinder, length 2<r, transverse semi-axes a and b, about its longitudinal axis. Ans. ~(a 2 + b 2 }. 6. Find the moment of inertia of the same about its transverse axis 2b. Ans. + -) 3/ 7. Find the moment of inertia of a rectangular prism, sides 2a, 2b, 2c, about central axis 2C. Ans. %wabc(a 2 -f b 2 }. 101. Centre of Percussion. Suppose we have a body revolving, with an angular velocity a, about an axis perpendicu- lar to the plane of the paper, and passing through O. Join O with the centre of gravity, G, and take OG as axis of x ; the axis of y passing through (9, and lying in the plane of the paper. If, with a radius OA ;-, we describe an arc CA (Fig. 67), all particles situated in this arc have a linear velocity o.r. The force which would impart this velocity to any one of them, as that at A, in a unit of time, is g and this may be resolved into two, w , w ax and ay, S g respectively perpendicular and parallel to OG. The moment of this force about the axis is g hence the total moment of the forces which would impart to CENTRE OF PERCUSSION. \2l the body in a unit of time the angular velocity a, is, as has been shown already, g g The resultant of the forces acting on the body is g since, the centre of gravity being on OB, it follows that ^wy = o ; and hence -*2wy o. g Hence the perpendicular distance from O to the line of direc- tion of the resultant force is measured along OG, and is g 7 f \ g and the point of application of the resultant force may be con- ceived to be at a point on OG at a distance / from O ; and this point of application of the resultant of the forces which pro- duce the rotation is called the Centre of Percussion. If p = radius of gyration about the axis through (9, and if x := distance from (9 to the centre of gravity, we have XoSw Hence or, in words, The radius of gyration is a mean proportional between the distance 1, and the distance x , betiveen the axis of oscillation and the centre of gravity. The centre of percussion with respect to a given axis of oscillation O has been defined as the point of application of the 122 APPLIED MECHANICS. resultant of the forces which cause the body to rotate around the. point O. Another definition often given is, that it is the point at which) if a force be applied, there will be no shock on the axis of oscilla- tion ; and these two definitions are equivalent to each other. Let the particles of the body under consideration be con- ceived, for the sake of simplicity, to be distributed along a single line AB, and suppose a force F applied at D (Fig. 68). Conceive two equal and opposite forces, each equal to F, applied at C, the cen- tre of gravity of the body. \ Then these three forces are equivalent to a single force ^ applied at the centre of grav- ity C, which produces translation of the whole body ; and, secondly, a couple whose moment is F(CD), whose effect is to produce rotation FIG. 68. around an axis passing through the centre of gravity C. Under this condition of things it is evident that the centre of gravity C will have imparted to it in a unit of time a forward velocity equal to -, where M is the entire mass of the body ; the point D will have imparted to it a greater forward velocity ; while those points on the upper side of C will have imparted to them a less and less velocity as they recede from C, until, if the rod is sufficiently long, the particle at A will acquire a backward velocity. Hence there must be some point which for the instant in question is at rest; i.e., where the velocity due to rotation is just equal and opposite to that due to the translation, or about which, for the instant, the body is rotating : and if this point were fixed by a pivot, there would be no stress on the pivot caused by the force applied at D. An axis through this point is called the Instantaneous Axis. IMPACT OR COLLISION. 12$ 102. Interchangeability of the Centre of Percussion and Axis of Oscillation. If we take, as axis of oscillation, a line perpendicular to the plane of the paper, and passing through D, then will O be the new centre of percussion. PROOF. We have seen ( 101) that where / = OD, X Q = OC, and p = radius of gyration about an axis through O perpendicular to the plane of the paper. Moreover, if /o represent the radius of gyration about an axis through C perpendicular to the plane of the paper, we shall have P* = p 2 + x<* XQ Now if D is taken as axis of oscillation, we shall have for the distance l t to the corresponding centre of percussion, CD I- Xo ' where p I = radius of gyration about the axis of oscillation through D. / . Pi 2 po 2 + CD 2 p 2 r (j ~\ j I '-CD = -CD~ "D+ CJ >-**+ -*)-* Hence the new centre of percussion is at <9. Q. E. D. 103. Impact or Collision. Impact or collision is a pressure of inappreciably short duration between two bodies. The direction of the force of impact is along the straight line drawn normal to the surfaces of the colliding bodies at their point of contact, and we may call this line the line of impact. 124 APPLIED MECHANICS. The action that occurs in the case of collision may be de- scribed as follows : at first the bodies undergo compression ; the mutual pressure between them constantly increasing, until, when it has reached its maximum, the elasticity of the mate- rials begins to overpower the compressive force, and restore the bodies wholly or partially to their original shape and dimen- sions. Central impact occurs when the line joining the centres of gravity of the bodies coincides with the line of impact. Eccentric impact occurs when these lines do not coincide. Direct impact occurs when the line along which the relative motion of the bodies takes place, coincides with the line of impact. Oblique impact occurs when these lines do not coincide. CENTRAL IMPACT. 104. Equality of Action and Re-action. One funda- mental principle that holds in all cases of central impact is the equality of action and re-action ; in other words, we must have, that, at every instant of the time during which the impact is taking place, the pressure that one body exerts upon the other is equal and opposite to that exerted by the second upon the first. The direct consequence of this principle is, that the algebraic sum of the momenta of the two bodies before impact remains unaltered by the impact, and hence that this sum is just the same at every instant of, and after, the impact. If we let m lt m 2 , be the respective masses, c lt c 2 , their respective velocities before impact, v u v 2 , their respective velocities after impact, i/, v" , their respective velocities at any given instant during the time while impact is taking place, CO-EFFICIEA r T OF RESTITUTION. 125 then we must have the following two equations true ; viz., m 1 v l + m 2 v 2 = m l c l + m 2 c 2 , (i) mjf 4- m 2 v" = m l c l 4- m 2 c 2 . (2) 105. Velocity at Time of Greatest Compression. At the instant when the compression is greatest i.e., at the instant when the elasticity of the bodies begins to overcome the deformation due to the impact, and to tend to restore them to their original forms the values of v' and v" must be equal to each other; in other words, the colliding bodies must be moving with a common velocity v = v' = v". (i) To determine this velocity, we have, from equation (2), 104, combined with (i), v = m ^ + m * c \ (2) m l 4- m 2 106. Co-efficient of Restitution. In order to determine the values v lt v 2 , of the velocities after impact, we need two equations, and hence two conditions. One of them is fur- nished by equation (i), 104. The second depends upon the nature of the material of the colliding bodies, and we may dis- tinguish three cases : i. Inelastic Impact. In this case the velocity lost up to the time of greatest compression is not regained at all, and the velocity after impact is the common velocity ^ at the instant of greatest compression. In this case the whole of the work used up in compressing the bodies is lost, as none of it is restored by the elasticity of the material. 2. Elastic Impact. In this case the velocity regained after the greatest compression, is equal and opposite to that lost up to the time of greatest compression ; therefore v z/j = c v v. (i) v 2 v v c 2 . (2) 126 APPLIED MECHANICS. We may also define this case as that in which the work lost in compressing the bodies is entirely restored by the elasticity of the material, so that j . z2 , -- -- -- r 2222 Either condition will lead to the same result. 3. Imperfectly Elastic Impact. In this case a part only of the velocity lost up to the time of greatest compression is regained after that time. If, when the two bodies are of the same material, we call e the co-efficient of restitution, then we shall so define it that v v l c* v v c 2 or, in words, the co-efficient of restitution is the ratio of the velocity regained after compression to that lost previous to that time. In this case only a part of the work done in producing the compression is regained, hence there is loss of energy. Its amount will be determined later. Strictly speaking, all bodies belong to the third class ; the value of e being always a proper fraction, and never reaching unity, the value corresponding to perfect elasticity ; nor zero, the value corresponding to entire lack of elasticity. 107. Inelastic Impact. In this case the velocity after impact is the common velocity at the time of greatest com- pression ; hence v = v, = v 2 (i) (2) And for the loss of energy due to impact we have m 2 c 2 , ^v* 1 --- (M! -f- m 2 ) , 2 2 ELASTIC IMPACT. 12 J which, on substituting the value of v, reduces to ,,"'?) (' - <> (3) 2(m 1 +- m 2 ) 1 08. Elastic Impact. In this case we have, of course, the condition, equation (i), 104, m\v-L + ^2^2 == m^c-i -f* and, for second equation, we may use equation (3), 106 ; viz., w^! 2 , m 2 v 2 2 _ mj? m^c^ 2222 Combining these two equations, we shall obtain m l We can obtain the same result without having to solve an equation of the second degree, by using instead the equations (i) and (2) of 106, together with (i) of 104; i.e., m l v I -f- m z v z = m 1 f l + m^\ or and ( 105) l ~\~ As the result of combining these equations, and eliminating v, we should obtain equations (i) and (2), as above, for the values of z\ and v 2 . In this case the energy lost by the collision is zero. 128 APPLIED MECHANICS. 109. Special Cases of Inelastic Impact. (a) Let the mass m 2 be at rest. Then c 2 o, v = m ' c ' < + * .' . Loss of energy = m * m * . ( 2 ) () Let w 2 be at rest, and let m 2 = oo ; i.e., let the mass ;;/ r strike against another which is at rest, and whose mass is in- finite. We have m 2 = oo , c 2 = o, - = o, (3) m W. ^ r Wi^ r Loss of energy = - - -- - = -L, (4) or the moving body is reduced to rest by the collision, and all its energy is expended in compression. (c) Let m l c l = m 2 c 2 ; i.e., let the two bodies move towards each other with equal momenta : o, (5) and the loss of energy = ^^ -f ^2!, (6) 2 2 the entire energy being lost. 110. Special Cases of Elastic Impact. (a) Let the mass m 2 be at rest. Then c 2 =. o, EXAMPLES OF ELASTIC AND INELASTIC IMPACT. I2g (b) Let m 2 be at rest, and let m 2 oo . Then we have ' 2 =0, ^+ I m 2 V 2 = o. (4) Hence the moving body retraces its path in the opposite direc- tion with the same velocity. (c) Let m^, = m 2 c 2 . Then our equations of condition become WiVi + m 2 v 2 = o, 2222 and from these we readily obtain i.e., both bodies return on their path with the same velocity with which they approached each other. in. Examples of Elastic and of Inelastic Impact. 1. With what velocity must a body weighing 8 pounds strike one weighing 25 pounds in order to communicate to it a velocity of 2 feet per second, (a) when the bodies are perfectly elastic, (b) when wholly inelastic. 2. Suppose sixteen impacts per minute take place between two bodies whose weights are respectively 1000 and 1200 pounds, their initial velo- cities being 5 and 2 feet per second respectively : find the loss of energy, the bodies being inelastic. 112. Imperfect Elasticity. In this case we have the relations (see 106) V Vi _ v - c 2 130 APPLIED MECHANICS. where v = m + * f * ; and we have also m^, -h m 2 v 2 = m^ -f m 2 c 2 . Determining from them the values of z/ x and v z , we obtain Vs = (i + e) - ect, (i) v 2 = (i + - ec v (2) or, by substituting for v its value, These may otherwise be put in the form -- -'-(+) - O, (5) Moreover, we have for the loss of energy due to impact E = (^ 2 - ^ 2 ) + ^(r 2 2 - v 2 2 2 or but, from (5) and (6) respectively, f s - Vl = ( T + <?) ^2(^*1 Wj -f w 2 ^ - , a = _ (i + g),( W, -f- IMPERFECT ELASTICITY. 131 2(m l /. E = . m But, from (i) and (2), or When ^ = i, or the elasticity is perfect, this loss of energy becomes zero. When e = o, or the bodies are totally inelastic, then the loss of energy becomes / L ** \ v m f * V / *\rr*l ~T~ "*2/ as has been already shown in 107. An interesting fact in this connection is, that since (8) is the work expended in producing compression, and (7) is the work lost in all, therefore the work restored by the elasticity of the body is so that e 2 , or the square of the co-efficient of restitution, is the ratio of the work restored by the elasticity of the bodies, to the work expended in compressing the bodies up to the time of greatest compression. 132 APPLIED MECHANICS. 113. Special Cases. (a) Let m 2 be at rest, therefore 2 = o. Then we shall have ( _^(i + *) 1 = , ,*.-**. (I) ( #*i + * 2 ) m t + m 2 and for loss of energy When #z 2 = oo , and 2 = o, we have PI = ^i, (4) ^ 2 = o, ^ = (I _ , 2) ^!. (5) When m l c l = m 2 c 2 , then m,) ^ 1 14. Values of e as Determined by Experiment. Since we have _ & IMPERFECT ELASTICITY. 133 we shall have, when m 2 = oo and c 2 = o, m x 4- Hence Now, if we let a round ball fall vertically upon a horizontal slab from the height H y we shall have for the velocity of ap- proach and if we measure the height h to which it rises on its rebound, we shall have Hence In this way the value of e can be determined experimentally for different substances. Newton found for values of e: for glass, ||; for steel, |~ and Coriolis gives for ivory from 0.5 to 0.6. On the other hand, if we desired to adopt as our constant the ratio of the work restored, to the work spent in compres- sion, we should have for our constant ^ 2 , and hence the squares of the preceding numbers. EXAMPLES. i. If two trains of cars, weighing 120000 and 160000 Ibs., come into collision when they are moving in opposite directions with veloci- ties 20 and 15 feet per second respectively, what is the loss of mechan- ical effect expended in destroying the locomotives and cars ? 134 APPLIED MECHANICS. 2. Two perfectly inelastic balls approach each other with equal velocities, and are reduced to rest by the collision ; what must be the ratio of their weights ? 3. Two steel balls, weighing 10 Ibs. each, are moving with velocities 5 and 10 feet per second respectively, and in the same direction : find their velocities after impact, the fastest ball being in the rear, and over- taking the other ; also the loss of mechanical effect due to the impact, assuming e = 0.55. 115. Oblique Impact. Let m lt m 2 , be the masses of the colliding bodies ; c lt c 2 , their respective velocities before impact ; a n a 2 , the angles made by c lt c 2 , with the line of centres ; v u v 2y the components of the velocities after impact ; , cos a,, c 2 cos a 2 , the components of c lt c 2 , along the line of centres ; c l sin a,, c 2 sin a 2 , the components of c u c 2t at right angles to the line of centres ; v the common component of the velocity at the instant of greatest compression along line of centres ; i/ t v", actual velocities after impact ; a', a", angles they make with line of centres ; vj, v", actual velocities when compression is greatest ; a/, a/', angles they make with line of centres. Then we shall have, by proceeding in the same way as was done m 112, V, = ^COSa, (l + e) ~ (VjCOSa, ^ 2 COSa 2 ), (l) #*i H- ^2 V 2 = C 2 COS a 2 + (l OBLIQUE IMPACT. 135 (3) (4) (5) (6) (7) (8) (9) (10) tf = VW + ^sin'a,, if' = y^ 2 2 + ^2 2 sin 2 a,, COS a' = ^, COS a" = J&, V Vc = Vz' 2 + <Tj 2 sin 2 a x , v/' = ^v 2 H~ ^sin 2 ^, 00.*' -J, COS a/' = -^ /?* And for the energy lost in impact, we have , _ . 2(m l -f- w 2 ) When the bodies are perfectly elastic, g = i, and equations (i), (2), and (12) become respectively Vi = ^ COS a, -- 2 - ( fi cos af _ ^ cos ^ C 2 COS a 2 H -- ?^ (f s COS a, ^ 2 COSa,), m l + m 2 The rest remain the same in form. When the bodies are totally inelastic, * SB O, 136 APPLIED MECHANICS. and equations (i), (2), and (12) become respectively #, = fj COS a, - - - (^TjCOStt! ^ COS 03), m^ + m z V 2 = c 2 cos a 2 H -- - - (Cj. cos a x <r 2 cos 03), /, + w 2 , - r 2 cosa 2 ) 2 . 2(m l 4- m 2 ) The rest remain the same in form. 116. Impact of Revolving Bodies. Let the bodies A and B revolve about parallel axes, and impinge upon each other. Draw a common normal at the point of contact. This common normal will be the line of impact. Let c, angular velocity of A before impact, c 2 = angular velocity of B before impact, o> x = angular velocity of A after impact, <o 2 = angular velocity of B after impact, ^ perpendicular from axis of A on line of impact, a z = perpendicular from axis of B on line of impact, 7 X = moment of inertia of A about its axis, 7 2 = moment of inertia of B about its axis. Then we shall have a l e l = Ci = linear velocity of A at point of contact before impact ; a 2 2 = c 2 = linear velocity of B at point of contact before impact ; a l (j} l = Vj. = linear velocity of A at point of contact after impact ; a 2 <a 2 v 2 = linear velocity of^ at point of contact after impact; 2g // V 2 _ / _!_ \JL = actual energy of A before impact : \*i 2 hg / / \C 2 = ( ^ ]- L - actual energy of B before impact ; \a 2 *)2& -^- = ( - ) = actual energy of A after impact : 2g W/2- ^- = ( - j- 2 - = actual energy of B after impact ; 2 S \ a 2 2 / 2g IMPACT OF REVOLVING BODIES. 137 Hence it follows that we have the case explained in 112 for imperfectly elastic impact, provided only we write - instead of m^g and instead of m 2 g. a, 2 a 2 2 - instead a, 2 Hence we shall have o>, = e, - *,(,, - <* 2 e 2 ) 8 f 2 r ,(l + ') (0 /,#, -f/x W 2 = C 2 4- tfaOi*! ^2^2)7 - ' - (l + <?), (2) !&* + /I^2 2 The case of perfect elasticity is obtained by making e = i. The case of total lack of elasticity is obtained by making f O. In the latter case the loss of energy is 7 r / 2 x v as can be seen by substituting the proper values in equation (8), 112. 138 APPLIED MECHANICS. CHAPTER III. ROOF-TRUSSES. 117. Definitions and Remarks. The term "truss" may be applied to any framed structure intended to support a load. In the case of any truss, the external loads may be applied only at the joints, or some of the truss members may support loads at points other than the joints. In the latter case those members are subjected, not merely to direct tension or compression, but also to a bending-action, the determination of which we shall defer until we have studied the mode of ascertaining the stresses in a loaded beam ; and we shall at present confine ourselves to the consideration of the direct stresses of tension and compression. For this purpose any loads applied between two adjacent joints must be resolved into two parallel components acting at those joints, and the truss is then to be considered as loaded at the joints. By this means we shall obtain the entire stresses in the members whenever the loads are concentrated at the joints; and, when certain members are loaded at other points, our re- sults will be the direct tensions and compressions of these mem- bers, leaving the stresses due to bending yet to be determined. A tie is a member suited to bear only tension. A strut is a member suited to bear compression. 1 1 8. Frames of Two Bars. Frames of two bars may consist, (i) of two ties (Fig. 69), (2) of two struts (Fig. 70), (3) of a strut and a tie (Fig. 71). FRAMES OF TWO BARS. 139 CASE I. Two Ties (Fig. 69). Let the load be repre- sented graphically by CF = W. a Then if we resolve it into two components, CD and CE, acting along CB and CA re- spectively, CD will represent graphically the pull or tension in the tie CB, and CE that in the tie CA. The force acting on CB at B is equal and opposite to FIG. 69. CD, while that acting on CA at A is equal and opposite to CE. To compute these stresses analytically, we have CE = CF sin CFE = W sin 2 sin CEF sin(Y + /,)' CD = CF sin CFD sin CDF = W sin/, sin(Y + /,) CASE II. Two Struts (Fig. 70). Let the load be repre- sented graphically by CF= W. Then will the components CD and CE represent the thrusts in the struts CB and CA re- spectively, and the re-actions of the supports at B and A will be equal and opposite to them. For analytical solution, we derive from the figure FIG. 70. CE = W smi. sin(* -f- CD = W sin* sin(i -f *,) CASE III. A Strut and a Tie (Fig. 71). Let the load be represented graphically by CF = W. Resolve it, as before, into components along the members of the truss. Then will 140 APPLIED MECHANICS. CE represent the tension in the tie AC, and CD will represent the thrust in the strut BC ; and we may deduce the analytical formulae as before. 1 19. Stability for Lateral Deviations. -In Case I, if the joint C be moved a little out of the plane of the paper, the load at C has such a direction that it will cause the truss to rotate around AB so as to return to its former position ; hence such a frame is stable as regards lateral deviations. In Case II the effect of the load, if C were moved a little out of the plane of the paper, would be to cause rotation in such a way as to overturn the truss ; hence such a frame is unstable as regards lateral deviations. In Case III the stability for lateral deviations will depend upon whether the load CF = W is parallel to AB, is directed away from it or towards it. If the first is the case (i.e., if A is the point of suspension of the tie), the frame is neutral, as the load has no effect, either to restore the truss to its former posi- tion, or to overturn it ; if the second is the case (i.e., if A t is the point of suspension of the tie), the truss is stable ; and, if the third is the case (i.e., if A is the point of suspension of the tie), it is unstable as regards lateral deviations. 1 20. General Methods for Determining the Stresses in Trusses. In the determination of the stresses as above, it would have been sufficient to construct only the triangle CFD by laying off CF= W to scale, and then drawing CD parallel to CB, and FD parallel to CA, and the triangle CFD would have given us the complete solution of the problem. Moreover, the determination of the supporting forces of any truss, and of the stresses in the several members, is a question of equilibrium. Adopting the following as definitions, viz., External forces are the loads and supporting forces, TRIANGULAR FRAME. 14! Internal forces are the stresses in the members : we must have i. The external forces must form a balanced system; i.e., the supporting forces must balance the loads. 2. The forces (external and internal) acting at each joint of the truss must form a balanced system ; i.e., the external forces (if any) at the joint must be balanced by the stresses in the members which meet at that joint. 3. If any section be made, dividing the truss into two parts, the external forces which act upon that part which lies on one side of the section, must be balanced by the forces (internal) exerted by that part of the truss which lies on the other side of the section, upon the first part. The above three principles, the triangle, and polygon of forces, and the conditions of equilibrium for forces in a plane, enable us to determine the stresses in the different members of roof and bridge trusses. 121. Triangular Frame. Given the triangular frame ABC (Fig. 72), and given the load W at C in magnitude and direction, given also the N direction of the support- ing force at B, to find the magnitude of this support- ing force, the magnitude and direction of the other supporting force, and the stresses in the members. SOLUTION. Join A \ b with D, the point of inter- FIG. 72. section of the line of direction of the load and the line BE. Then will DA be the direction of the other supporting force ; for the three external forces, in order to form a balanced sys- tem, must meet in a point, except when they are parallel. Then draw ab to scale, parallel to CD and equal to W. From 142 APPLIED MECHANICS. a draw ac parallel to BD, and from b draw be parallel to AD ; then will the triangle abca be the triangle of external forces, the sides ab, be, and ca, taken in order, representing respectively the load W, the supporting force at A, and the supporting force at A Then from a draw ad parallel to BC, and from c draw cd parallel to AB ; then will the triangle acd be the triangle of forces for the joint B, and the sides ca, ad, and dc, taken in order, will represent respectively the supporting force at B, the force exerted by the bar BC at the point B, and the force exerted by the bar AB at the point B. Since, therefore, the force ad exerted by the bar CB at B is directed away from the bar, it follows that CB is in compres- sion ; and, since the force dc exerted by the bar AB at B is directed towards the bar, it follows that AB is in tension. In the same way bdc is the triangle of forces for the point A ; the sides be, cd, and db representing respectively the sup- porting force at A, the force exerted by the bar AB at A, and the force exerted by the bar AC at A. The bar AB is again seen to be in tension, as the force cd exerted by the bar AB at A is directed towards the bar. So likewise the triangle abd is the triangle of forces for the point C. Fig. 73 shows the case when the supporting forces meet the load-line above, instead of below, the truss. 122. Triangular Frame with Load and Supporting Forces Vertical. Fig. 74 shows the construction when the load and also the supporting forces are vertical. In this case \ FIG. 73. BOW'S NOTATION. 143 FIG. 74. the diagram becomes very much simplified, the triangle of external forces abd becom- ing a straight line. The diagram is otherwise con- structed just like the last one. 123. Bow's Notation. The notation devised by Robert H. Bow very much simplifies the construction of the stress diagrams of roof- trusses. This notation is as follows : Let the radiating lines (Fig. 75) represent the lines of action of a system of forces in equilib- rium, and let the polygon abcdefa be the polygon representing these forces in magnitude and direction ; then denote the sides of the polygon in the ordinary way, by placing small letters at the vertices, but denote the radiating lines by capital letters placed in the angles. Thus the line AB is the line of direction of the force ab y etc. In applying the notation to roof-trusses, we letter the truss with capital letters in the spaces, and the stress dia- gram with small letters at the vertices. If, then, in drawing the polygon of equilibrium for any one joint of the truss, we take the forces always in the same order, proceeding always in right-handed or always in left-handed rotation, we shall be led to the simplest diagrams. Hereafter this notation will be used exclusively in determining the stresses in roof-trusses. 124. Isosceles Triangular Frame: Concentrated Load (Fig. 76.) Let the load W act at the apex, the supporting FIG. 75. 144 APPLIED MECHANICS. FIG. 76. forces being vertical ; each will be equal to \ W : hence the polygon of external forces will be the triangle abc, the sides of which, ab, be, and ca, all lie in one straight line. Then begin at the left-hand support, and proceed again in right-handed rotation, and we have as the tri- angle of forces at this joint cad, the forces ca, ad, and dc, these being respectively the support- ing force, the stress in AD, and that in DC ; the directions of these forces being indicated by the order in which the letters follow each other : thus, ca is an upward force, ad is a downward force ; and, this being the force exerted by the bar AD at the left-hand support, we con- clude that the bar AD is in compression. Again : dc is directed towards the right, or towards the bar itself, and hence the bar DC is in tension. The triangle of forces for the other support is bed, and that for the apex abd. 125. Isosceles Triangular Frame: Distributed Load. Let the load W be uniformly distributed over the two rafters AF and FB (Fig. 77) ; then will these two rafters be subjected to a direct stress, and also to a bend- ing action : and if we resolve the load on each rafter into two com- ponents at the ends of the rafter, then, considering these components as the loads at the joints, we shall determine correctly by our diagram the direct stresses in all the bars of the truss. The load distributed over AF is ; and of this, one-half is FIG. 77. POLYGONAL FRAME. the component at the support, and one-half at the apex, and similarly for the other rafter. This gives as our loads, at 4 each support, and-- at the apex. The polygon of external forces is eabcde, where the sides are as follows : W W , W , W , W ea = , ab , be , cd = , de = . 42422 Then, beginning at the left-hand support, we shall have for the polygon of forces the quadrilateral deafd, where de = = sup- porting force, ea = -- = downward load at support, af 4 stress in AF (compression), fd stress in FD (tension). The polygon for the apex is abf, and that for the right-hand support cdfbc. 1 26. Polygonal Frame Given a polygonal frame (Fig. 78) formed of bars jointed together at the vertices of the angles, and free to turn on these joints, it is evident, that, in order that the frame may retain its form, it is necessary that the direc- tions of, and the proportions between, the loads at the dif- ferent joints, should be speci- ally adapted to the given form : otherwise the frame will change its form. We will proceed to solve the following problem : Given the form of the frame, the magnitude of one load as AB, and the direction of all the external forces (loads and supporting forces) except one, we shall have" sufficient data to determine the magnitudes of all, \ \ \ \ F FIG. 78. 146 APPLIED MECHANICS. and the direction of the remaining external forces, and also the stresses in the bars Let the direction of all the loads be given, and also that of the supporting force EF, that of the supporting force AF being thus far unknown ; and let the magnitude of AB be given. Then, beginning at the joint ABG, we have for triangle of forces abg formed by drawing ab || and = AB, then drawing ga || AG, and bg || BG ; ga and bg both being thrusts. Then, passing to the joint BCG, we have the thrust in BG already determined, and it will in this case be represented by gb. If, now, we draw be || BC, and gc || GC, we shall have determined the load BC as be, and we shall have eg and gb as the thrusts in CG and GB respectively. Continuing in the same way, we obtain the triangles gcd, gde, and gfe, thus determining the magnitudes of the loads cd, de, and of the supporting force ef; and then the triangle gaf, formed by joining a and/, gives us af for the magnitude and direction of the left-hand support. The polygon abcdefa of external forces is called the Force Polygon, while the frame itself is called the Equilibrium Polygon. 127. Polygonal Frame with Loads and Supporting Forces Vertical In this case (Fig. 79) we may give the form of the frame and the mag- nitude of one of the loads, to determine the other loads and the supporting forces, and also the stresses in the bars ; or we may give the form of the frame and the magnitude of the re- sultant of the loads, to find the loads and supporting forces. In the former case let the load AB be given. Then, proceeding in the same way as before, we find the diagram of Fig. 79 ; the polygon of external forces abcdefa falling all in one straight line. FIG. 79. FUNICULAR POLYGON. TRIANGULAR TRUSS. 147 If, on the other hand, the whole load ae be given, we observe that this is borne by the stresses in the extreme bars AG and GE ; hence, drawing ag || AG, and eg || EG, we find eg and ga as the stresses in EG and GA respectively. Then, proceeding to the joint ABG, we find, since ' ga is the force exerted by GA at this point, that, drawing gb || GB, we shall have ab as the part of the load acting at the joint ABG, etc. 128. Funicular Polygon. If the frame of Fig. 79 be inverted, we shall have the case of Fig. 80, where all the bars, except FG, are sub- jected to tension; FG itself being subjected to compres- sion. The construction of the diagram of stresses being en- \ tirely similar to that already explained for Fig. 79, the ex- planation will not be repeated here. If the compression piece be omitted, the case becomes that of a chain hung at the upper joints (the supporting forces then becoming iden- tical with the tensions in the two extreme bars), the line gf would then be omitted from the diagram, and the polygon of external forces would become abcdega. 129. Triangular Truss : Wind Pressure. Inasmuch as the pressure of the wind on a roof has been shown by experi- ment to be normal to the roof on the side from which it blows, we will next consider the case of a triangular truss with the load distributed over one rafter only, and normal to the rafter. There may be three cases : i. When there is a roller under one end, and the wind blows from the other side. FIG. 80. 148 APPLIED MECHANICS. 2. When there is a roller under one end, and the wind tlows from the side of the roller. 3. When there is no roller under either end. The last arrangement should always be avoided except in small and unimportant constructions ; for the presence of a roller under one end is necessary to allow the truss to change its length with the changes of temperature, and to prevent the stresses that would occur if it were confined. CASE I. Using Bow's notation, we have (Fig. 81) the whole load represented in the diagram by db. Its resultant acts at the middle of the rafter AE, whereas the sup- porting force at the right-hand end is (in consequence of the pres- ence of the roller) verti- cal. Hence, to find the line of action of the other supporting force, pro- duce the line of action of the load till it meets a vertical line drawn through the roller, and join their point of intersection with the support where there is no roller. We thus obtain CD as the line of action of the left-hand support. We can now determine the magnitude of the supporting forces be and cd by constructing the triangle bcdb of external forces. Now resolve the normal distributed force db into two single forces (equal to each other in this case), da and ab respectively, acting at the left-hand support and at the apex. FIG. 81. TRIANGULAR TRUSS: WIND PRESSURE. 149 Now proceed to the left-hand support. We find four forces in equilibrium, of which two are entirely known ; viz., cd and da: hence, constructing the quadrilateral cdaec, we have ae as the thrust in AE, and ec as the tension in EC. Next proceed to the apex, and construct the triangle of equilibrium abea, and we obtain be as the thrust in BE. The triangle bceb is then the triangle of equilibrium for the right-hand sup- port. CASE II. - In this case (Fig. 82) we fol- low the same method of pro- cedure, only the point of inter- section of the load and sup- porting forces is above, instead of below, the truss. The figure explains itself so fully that it is unnecessary to explain it here. CASE III. In this case the supports are capable of exerting resistance in any direction what- ever ; so that, if any circumstance should determine the direction of one of them, that of the other would be determined also. When there is no such circum- stance, it is customary to assume them parallel to the load (Fig. 83). Making this assumption, we begin by dividing the line db, which represents the load, into two parts, inversely FIG. 8 FIG. 83. 150 APPLIED MECHANICS. proportional to the two segments into which the line of action of the resultant of the load (the dotted line in the figure) divides the line EC. We thus obtain the supporting forces be and cd, and bcdb is the triangle of external forces. We then follow the same method as in the preceding cases. 130. General Determination of the Stresses in Roof- Trusses. In order to compute the stresses in the different members of a roof-truss, it is necessary first to know the amount and distribution of the load. This consists generally of i. The weight of the truss itself. 2. The weight of the purlins, jack-rafters, and superin- cumbent roofing, as the planks, slate, shingles, felt, etc. 3. The weight of the snow. 4. The weight of the ceiling of the room immediately below if this is hung from the truss, or the weight of the floor of the loft, and its load, if it be used as a room. 5. The pressure of the wind ; and this may blow from either side. 6. Any accidental load depending on the purposes for which the building is used. As an instance, we might have the case where a system of pulleys, by means of which heavy weights are lifted, is attached to the roof. In regard to the first two items, and the fourth, whenever the construction is of importance, the actual weights should be determined and used. In so doing, we can first make an approximate computation of the weight of the truss, and use it in the computation of the stresses ; the weights of the ceiling or of the floor below being accurately determined. After the stresses in the different members have been ascertained by the use of these loads, and the necessary dimensions of the mem- bers determined, we should compute the actual weight of the truss ; and if our approximate value is sufficiently different from the true value to warrant it, we should compute again STA'SSS IN ROOF-TRUSSES. the stresses. This second computation will, however, seldom be necessary. In making these computations, the weights of a cubic foot of the materials used will be needed ; and average values are given in the following table with sufficient accuracy for the purpose. WEIGHT OF SOME BUILDING MA- TERIALS PER CUBIC FOOT. Pounds. WEIGHT OF SLATING PER SQUARE FOOT. According to Trautwine. Pounds. TIMBER. 41 \ inch thick on laths . . . \ " " " i-inch boards . 4-75 6.75 Hemlock 2 C JL u jl 7-30 Maple ^j 41 ^ " " " laths . . . 7-00 Oak, live CQ & " " " i-inch boards, 9-OO Oak white 4Q & " " " 'i " " 9-55 Pine, white ** 'Vy 2 C to "3O i " " laths . . . 9- 2 5 Pine, yellow, Southern . . Spruce ... . . . . 45 2 C to 7O i " " " i-inch boards, JL << TJL 11.25 1 1. 80 IRON. With slating-felt add . . . With |-mch mortar add . . ilb. 3lbs. 4 Co Iron, wrought ...... 480 NUMBER OF NAILS IN ONE POUND No Steel 490 3 -penny . 4 CO OTHER SUBSTANCES. 80 to 90 4 " 6 " 340 I CO Mortar, hardened .... IOT 8 " IOO Snow, freshly fallen 5 to 12 10 " .... 60 Snow, compacted by rain . I C to CO 12 " 40 Slate 140 to 180 20 " . 2 c As to the weight of the snow upon the roof, Stoney recom- mends the use of 20 pounds per square foot in moderate climates ; and this would seem to the writer to be borne out by the experiments of Trautwine as recorded in his handbook, I5 2 APPLIED MECHANICS. although Trautwine himself considers 12 pounds per square foot as sufficient. 131. Wind Pressure. While a great deal of work has been done to ascertain the direction and the greatest intensity of the pressure of the wind upon exposed surfaces, as those of roofs and bridges, nevertheless the amount of information on the subject is very small, inasmuch as but few experiments have been under the conditions of practice. Before giving a summary of what has been done the following statements will be made : i. The pressure of the wind upon a roof, or other surface, is assumed to be normal tc the surface upon which it blows ; and what little experimenting has been done upon the subject tends to confirm this view. 2. Inasmuch as more attempts have been made to deter- mine experimentally the velocity of the wind than its pressure, hence there have been a good many experiments to determine the relation between the velocity and the pressure upon a sur- face to which the direction of the wind is normal. 3. A few experimenters have tried to determine the rela- tion between the intensity of the pressure on a surface normal to the direction of the wind and one inclined to its direction. 4. While the above have been the investigations most com- monly pursued, other subjects of experiment have been (a) The variation of pressure with density ; (b) with tem- perature ; (c) with humidity; (d) with the size of surface pressed upon ; (e) with the shape of surface pressed upon ; (/) whether the pressure corresponding to a certain velocity is the same whether the air moves against a body at rest, or whether the body moves in quiet air. By way of references to the literature of the subject may be given the following, as most of the work that has been done is included in them or in other references which they contain : WIND PRESSURE. 153 i. Proceedings of the British Institution of Civil Engineers, vol. Ixix., year 1882, pages 80 to 218 inclusive. 2. A. R. Wolff : Treatise on Windmills. 3. C. Shaler Smith : Proceedings American Society of Civil En- gineers, vol. x., page 139. 4. A. L. Rotch : Report of Work of the Blue Hill Meteorological Observatory, 1887. 5. Engineering, Feb. 28th, 1890 : Experiments of Baker. 6. Engineering, May 30, June 6, June 13, 1890: Experiments of O. T. Crosby. The first gives an account of a very full discussion of the subject, by a large number of Engineers. The second con- tains a recommendation that the temperature of the air be con- sidered in estimating the pressure. The fifth gives an account of Baker's experiments on wind pressure in connection with the building of the Forth Bridge. Before an account is given of the experimental work that has been done, the following statements will be made of what are some of the methods in most common use : 1. A great many engineers very commonly call from 40 to 55 pounds per square foot the maximum pressure on a vertical surface at right angles to the direction of the wind. One rather common practice, in the case of bridges, is to estimate 30 pounds per square foot on the loaded, or 50 pounds per square foot on the unloaded structure. Nevertheless pressures of 80 and 90 pounds per square foot have been registered and re- corded by the use of small pressure-plates, and by computation from anemometer records. 2. By way of determining the intensity of the pressure on an inclined surface in terms of that on a surface normal to the direction of the wind, four methods more or less used will be enumerated here : (a) Duchemin's formula, which Professor W. C. Unwin recommends, is as follows, viz. : 154 APPLIED MECHANICS. sin 6 -*\ + sin'0' where / = intensity of normal pressure on roof, /, = intensity of piessure on a plane normal to the direction of the wind. (b) Hutton's formula, p=p l (sin 0)*.*4 &-*. Unwin claims that this and Duchemin's formula give nearly the same results for all angles of inclination greater than 15. The following table gives the results obtained by the use of each, on the assumption that p l = 40 : e Duchemin. Hutton. 9 Duchemin. Hutton. 5 6.89 5-10 50 38.64 38.10 10 13-59 9.60 55 39-21 39-40 15 19.32 14.20 60 39-74 40.00 20 24.24 18.40 65 39-82 40.00 25 28.77 22.6O 70 39-91 40.00 30 32.00 26.50 75 39 -9 6 40.00 35 34-52 30.10 80 40.00 40.00 40 36.40 33-30 85 40.00 40.00 45 37-73 36.00 90 40.00 40.00 (c) A formula very commonly favored, but which does not agree with any experiments that have been made, is sn 6. It gives much lower results, as a rule, than either of the others, but it is favored by many because, if we assume the wind to blow in parallel lines till it strikes the surface, and then to get suddenly out of the way, forming no eddies on the back side of the surface and meeting no lateral resistance on the front WIND PRESSURE. 155 side, all of which are conditions that do not exist, we could then deduce it as follows: Assume a unit surface making an angle 6 with the direction of the wind, the total pressure on this surface in the direction of the wind would be^j sin #; and by resolving this into nor- mal and tangential components we should have, for the former, (d] Another rule which is sometimes used, but which has nothing to recommend it, is to consider the normal intensity of the wind pressure per square foot of roof surface as equal to the number of degrees of inclination of the roof to the hori- zontal. The wind pressure allowed for by this rule is very excessive, as it would be 90 pounds per square foot for a ver- tical surface. Taking up, now, the experimental work that has been done, we will begin with the attempts to determine velocities and pressures, and the relation between them. i. In regard to velocities, these are determined by using some kind of an anemometer, and in all these cases there are several difficulties and sources of error, as follows : (a) In many cases the anemometers have not even been graduated experimentally, but it has been assumed outright that the velocity of the air is just three times the linear velocity of the cups of a cup anemometer. (b) When they have been graduated, it has generally been done by attaching them to the end of the arm of a whirling machine, which, when the arm is long, and the velocity moder- ate, will do very well, but is the more inaccurate the shorter the arm and the higher the velocity of motion. (c) The wind always comes in gusts, and hence the ane- mometer does not register the average velocity of the wind at any one moment, but that of the particular portion that comes 156 APPLIED MECHANICS. in contact with it, and this is always a small portion, on ac- count of the small size of the anemometer. (d) In order to get an indication which is not affected by the cross-currents reflected from the surrounding buildings and chimneys, it is necessary to put the anemometer very high up, and then, of course, we obtain the indications corresponding to that height, which is greater than that of the buildings, and it is well known that the velocity of the wind increases very considerably with the height. Next, as to the direct determination of pressure, this has usually been done by means of some kind of pressure-plate, either round or square, but of small size, thus allowing the eddies formed on the back side of the plate to have a con- siderable effect. The results obtained by the use of different sizes and different shapes of .plates have therefore differed very considerably ; and while some have claimed that the pressure per square foot increases with the size of the surface pressed upon, it has been very thoroughly proved by the more modern investigations that the opposite is true, and that the pressure decreases with the size. While the records from small pressure-plates have fre- quently shown very high pressures per square foot, as 80, 90, or even over 100 pounds per square foot, it has become very generally recognized by engineers that by far the greater part of existing buildings and bridges would be overturned by winds of such force, or anywhere near such force, and it has not been customary among them to make use of such high figures for wind pressure on bridges and roofs in computing the stability of structures. While some of the figures in general use have already been given, nevertheless the tendency of modern inves- tigation seems to be to obtain rather lower figures. In this con- nection it is well to refer to the work done by Baker in connec- tion with the construction of the Forth Bridge. The following description is taken from " Engineering" of Feb. 28th, 1890: WIND PRESSURE. 157 " The wind pressure to be provided for in the calcu- lations for bridges in exposed positions is 56 Ibs. per square foot, according to the Board of Trade regulations, and this twice over the whole area of the girder surface exposed, the resistance to such pressure to be by dead-weight in the struc- ture alone. 44 The most violent gales which have occurred during the construction of the Forth Bridge are given, with the pressures recorded on the wind gauges, in the annexed table : Year. Month and Day. Pressure in pounds per square foot. Direction of Wind. Revolving Gauge. Small fixed Gauge. Large fixed Gauge. In centre of large Gauge. Right- hand top of large Gauge. 1883 Dec. ii, 33 39 22 s. w.* 1884 Jan. 26, 65 4i 35 s. w.* 1884 Oct. 27, 29 23 18 s. w. 1884 Oct. 28, 26 29 !9 s. w. 1885 Mar. 20, 30 25 17 w. 1885 Dec. 4, 25 27 19 w. 1886 Mar. 31, 26 3i J 9 s. w. 1887 Feb. 4, 26 4i 15 s. w. 1888 Jan. 5, 27 16 7 S. E. [888 Nov. 17, 35 41 27 w. 1889 " 2, 27 34 12 s. w. 890 Jan. 19, 27 28 16 s. w. 1890 " 21, 26 38 15 w. 1890 " 22, 27 24 18 231 22 S. W. by W. * These data are unreliable, owing to faulty registration by the indicator-needle, as will b' presently explained. They were altered after this date. The barometer fell to 27.5 inches ot, <hat occasion, over three quarters of an inch within an hour. 158 APPLIED MECHANICS. "The pressure-gauges, which were put up in the summer of 1882 on the top of the old castle of Inchgarvie, and from which daily records have been taken throughout, were of very simple construction. The maximum pressures only were taken. The most unfavorable direction from which the wind pressure can strike the bridge is nearly due east and west, and two out of the three gauges were fixed to face these directions, while a third was so arranged as to register for any direction of wind. "The principal gauge is a large board 20 feet long by 15 feet high, or 300 square feet area, set vertically with its faces east and west. The weight of this board is carried by two rods sus- pended from a framework surrounding the board, and so ar- ranged as to offer as little resistance as possible to the passage of the wind, in order not to create eddies near the edge of the board. In the horizontal central axis of the board there are fixed two pins which fit into the lower eyes of the suspension- rods, the object being to balance the board as nearly as pos- sible. Each of the four corners of the board is held between two spiral springs, all carefully and easily adjusted so that any pressure exerted on either face will push it evenly in the op- posite direction, but upon such pressure being removed the compressed springs will force the board back to its normal position. To the four corners four irons are attached, uniting in a pyramidal formation in one point, whence a single wire passes over a pulley to the registering apparatus below. In order to ascertain to some extent how far great gusts of wind are quite local in their action, and exert great pressure only upon a very limited area, two circular spaces, one in the exact centre and one in the right-hand top corner, about 18 inches in diameter, were cut out of the board and circular plates in- serted, which could register independently the force of the wind upon them. " By the side of the large square board, at a distance of WIND PRESSURE. 159 about 8 feet, another gauge, a circular plate of 1.5 square feet area, facing east and west, was fixed up with separate regis- tration. This was intended as a check upon the indications given by the large board. "Another gauge of the same dimensions as the last, but with the disc attached to the short arm of a double vane, so that it would face the wind from whatever direction it might come, was set up. " On one occasion the small fixed board appeared to regis- ter 65 pounds to the square foot a registration which caused no little alarm and anxiety. Mr. Baker found, upon inves- tigation, that the registering apparatus was not in good order, and after adjusting it the highest pressure recorded was 41 pounds. " In order to determine the effect of the wind upon surfaces like that of the exposed surface of the bridge, he devised an apparatus which consisted of a light wooden rod suspended in the middle, so as to balance correctly, by a string from the ceiling. At one end was attached a cardboard model of the surface, the resistance of which was to be tested, and at the opposite end was placed a sheet of cardboard facing the same way as the model, so arranged that by means of another and adjustable sheet, which would slide in and out of the first, the surface at that end could be increased or decreased at the will of the operator. The mode of working is for a person to pull it from its perpendicular position towards himself, and then gently release it, being careful to allow both ends to go together. If this is properly done, it is evi- dent that the rod will in swinging retain a position parallel to its original position, supposing that the model at one end and the cardboard frame at the other are balanced as to weight, and that the two surfaces exposed to the air pressure coming against it in swinging are exactly alike. Should one area be greater than the other, the model or card- 160 APPLIED MECHANICS. board sheet, whichever it may be, will b~ lagging behind, and twist t^e string." The experiments carried on in various ways by different people and at different times are generally in agreement with each other and with the results of more elaborate processes. The information specially desired was in regard to the wind pressure upon surfaces more or less sheltered by those imme- diately in front of them. In this regard Mr. Baker satisfied himself that, while the results differed very considerably ac- cording to the distance apart of the surfaces, in no case was the area affected by the wind, in any girder which had two or more surfaces exposed, more than 1.8 times the area of the surface directly fronting the wind, and, as the calculations had been made for twice this surface, the stresses which the struc- ture will receive from this cause will be less than those pro- vided for. Next, as to the relation between velocity and pressure, a great many formulae have been devised, to agree with the results of different experimenters. Most all of them make the pressure proportional to the square of the velocity; while some add a term proportional to the velocity itself, and when higher velocities are reached, as those usual in gunnery, terms have been introduced with powers of the velocity higher than the second. It is hardly worth while to consider these dif- ferent formulae, as it is rather the pressure than the velocity that the engineer is interested in, and correct information in this regard is to be obtained rather from pressure-boards than from anemometers. Nevertheless, it may be stated that one of the most usual formulae is that of Smeaton, and is 200 where P= pressure in pounds per square foot, and V '= velocity WIND PRESSURE. l6l in miles per hour. This formula agrees very well with a num- ber of experiments that have been made where anemometers have been used to determine the velocity, and small pressure- plates (say one square foot) to determine the pressure ; thus this formula satisfies very well the experiments made at the Blue Hill Meteorological Observatory, near Boston, Mass., U. S. A. It was originally deduced from some very old experiments of Rouse ; and it agrees with a good many, but disagrees with other experiments. It is probably the formula that has been more quoted than any other. A little ought also to be said in regard to the pressure of the wind on very high structures, as on the piers of high via- ducts and on tall chimneys. In this regard it is to be ob- served : i. The pressure, as well as the velocity of the wind, be- comes greater the higher up from the ground the surface ex- posed is situated. 2. From calculations on chimneys that have stood for a long time, Rankine deduced, as the greatest average wind pressure that could be realized in the case of tall chimneys, 55 pounds per square foot. 3. In making the piers of high viaducts, it would seem desirable not to make them solid, but to use only four up- rights at the corners connected by lattice work, in order to- expose a smaller surface to the wind. Nevertheless, as was ex- plained, it will not do to separate the structure into its com- ponent parts, and to estimate the pressure on each part separately and then add the results together to get the total effect ; but we really need some such experiments as those of Baker. 4. Some old experiments of Borda bear out the common practice of assuming the wind pressure on the surface of a cir- 1 62 APPLIED MECHANICS cular cylinder one half that which would exist on its projection on a plane normal to the direction of the wind, There remains now only to refer to a serial article by O. T. Crosby, in " Engineering" of May 30, June 6th, and June I3th, 1890, containining some experiments made by him on wind pressure near Baltimore, Md. The first two numbers contain rather a summary of what has been done by others, and it is in the copy of June I3th that is to be found the account of his own work, which was done in order to determine the resistances of the air to fast-moving trains. He used a whirling arrangement turning about a vertical axis, to the end of which was attached a car, the circumference through which the car moved being 36 feet. In order to determine whether the circular motion produced any disturbing effect, he ran a car having a cross-section of 5.1 square feet on a circular track about two miles in circumference, the speed of the car being about 50 miles per hour, and the results obtained in this way agreed very nearly with those ob- tained from his whirling table. The special peculiarity of his results is that he obtained, by plotting them, the law that the pressure varies directly as the first power of the velocity, and not as the square or some higher power; also, his pressures, after the velocity had passed 25 or 30 miles per hour, are much lower than those given by Smeaton and others, the pres- sure on a normal plane surface moving at 115 miles per hour being about 27 pounds per square foot. The cars used were generally about 3 feet long without the front. The fronts attached were: i. Normal plane surface; 2. Wedge, base i, height I ; 3. Pyramid, base I, height 2; 4. Wedge and cyma, base I, height 2; 5. Parabolic wedge, base i, height 2. His experiments covered a range of velocities from 30 to 130 miles per hour. DISTRIBUTION OF THE LOADS. 163 The law of the first powers of the velocities seems peculiar, and certainly ought not to be accepted without further cor- roborative evidence ; but the low values of the pressures agree with Baker's results and with the tendency of the more modern investigations. 132. Approximate Estimation of the Load. In all important constructions, the estimates of the loads should be made as above described. For smaller constructions, and for the purposes of a preliminary computation in all cases, we only estimate the roof-weight roughly ; and some authors even as- sume the wind pressure as a vertical force. Trautwine recommends the use of the following figures for the total load per square foot, including wind and snow, when the span is, 75 feet or less : Roof covered with corrugated iron, unbearded ... 28 Ibs. Roof plastered below the rafters 38 " Roof, corrugated iron on boards 31 " Roof plastered below the rafters 41 " Roof, slate, unboarded or on laths 33 " Roof, slate, on boards ij inches thick 35 " Roof, slate, if plastered below the rafters 46 " Roof, shingles on laths 30 " Roof plastered below rafters or below tie-beam . . . 40 " From 75 to 100 feet, add 4 Ibs. to each. 133. Distribution of the Loads. The methods for de- termining the stresses, which will be used here, give correct results only when the loads are concentrated at joints, and are not distributed over any members of the truss. In. constructions of importance, this concentration of the loads at the joints should always be effected if possible ; because, when this is the case, the stresses in the members of the truss can be, if properly fitted, obliged to resist only stresses of direct tension, or of direct compression ; but, when there is a load distributed over any member of the truss, that member, in addition to the direct compression or direct tension, is subjected to a bending-stress- The effect of this bending 164 APPLIED MECHANICS. cannot be discussed until we have studied the theory of beams. Nevertheless, it is a fact that we have no experimental knowl- edge of the behavior of a piece under combined compression and bending ; and we know that when certain pieces are to resist bending, in addition to tension, they must be made much larger in proportion than is necessary when they bear tension only. FIG. 84. The manner in which this concentration of the loads is effected, is shown in Fig. 84, which is intended to represent one of a series of trusses that supports a roof, the rafters being the two lower ones in the figure. Resting on two consecutive trusses, and extending from one to the other, are beams called purlins, which should be placed only above the joints of the truss, and which are shown in cross-section in the figure. On these purlins are supported the jack-rafters parallel to the rafters, and at sufficiently frequent intervals to support suitably the plank and superincumbent roofing-materials. By this means we secure that the entire bending-stress comes upon the jack-rafters and purlins, and that the rafters proper are subjected only to a direct compression. When, however, the load is distributed, i.e., when the roofing rests directly on the rafters, or when the purlins are placed at points other than the joints, the bending-stress should be taken into account; and while the methods to be developed here will give the stresses DIRECT DETERMINATION OF THE STRESSES. 165 in all the members that are not subjected to bending, the bend- ing-stress may be largely in excess of the direct stress in those pieces that are subjected to bending. How to take this into account will be explained later. Another important item to consider is, that, in constructions of importance, a roller should be placed under one end of the truss to allow it to move with the change of temperature ; as otherwise some of the members will be either bent, or at least subjected to initial stresses. The presence of a roller obliges the supporting force at that point to be vertical, whether the load be vertical or inclined. It is customary, and does not entail any appreciable error, to consider the weight of the truss itself, as well as that of the superincumbent load, as concentrated at the upper joints ; i.e., those on the rafters. In the case of a ceiling on the room below, or of a loft whose floor rests on the lower joints, we must, of course, ac- count the proper load as resting on the lower joints. 134. Direct Determination of the Stresses. This, as we have seen, is merely a question of equilibrium of forces in a plane, where certain forces acting are known, and others are to be determined. As to the methods of solution, we might adopt i. A graphical solution, laying off the loads to scale, and constructing the diagram by the use of the propositions of the polygon, and the triangle of forces, and then determining the results by measuring the lines representing the stresses to the same scale. 2. An analytical solution, imposing the analytical conditions of equilibrium, as given under the " Composition of Forces," between the known and unknown forces. 3. A third method is to construct the diagram as in the graphical solution, but then, instead of determining the results by measuring the resulting lines to scale, to compute the un- 1 66 APPLIED MECHANICS. known from the known lines of the diagram by the ordinary methods of trigonometry. .The first, or purely graphical, method, is one which has received a very large amount of attention of late years, and in which a great deal of progress has been made. It is, doubt- less, very convenient for a skilled draughtsman, and especially convenient for one who, though skilled in draughting, is not free with trigonometric work ; but it seems to me, that, when the results are determined by computation from the diagram, there is less chance of a slight error in some unfavorable tri- angle vitiating all the results. I am therefore disposed to recommend for roof-trusses the third method. In the case of bridge-trusses, on the other hand, I believe the graphical not to be as convenient as a purely analytic method. 135. Roof-Trusses. In what follows, the graphical solu- tions will be explained with very little reference to the trigono- metric work, as that varies in each special case, and to one who has a reasonable familiarity with the solution of plane triangles, it will present no difficulty ; whereas to deduce the formulae for each case would complicate matters very much. Proceed- ing to special examples, let us take, first, the truss shown in Fig. 85, and let the vertical load upon it be W uniformly dis- tributed over the top of the roof, the purlins being at the joints on the rafters. The loads at the several joints will then be as follows, viz. (Fig. 85*), ab = kl = ~, be = cd = de = ef = fg = gh = hk = ~. 16 8 Then the supporting forces will be lm = ma = . 2 We thus have, as polygon of external forces, abcdefghklma. ROOF-TRUSSES. I6 7 Now proceed to either support, say, the left-hand one ; and there we have the two forces ab and ma known, while by and ym are unknown. We then construct the quadrilateral maby in the figure, and thus determine by and ym. As to whether FIG. 850. FIG. 8s<5. dbcde FIG. 85. these represent thrust or tension, we need only remember that they are the forces exerted by the re- spective bars at the joints : and, since by is directed away from the bar BY, this bar is in compression; whereas, ym being directed towards the bar YM, that bar is in tension. l68 APPLIED MECHANICS. Having determined these two stresses, we next proceed to another joint, where we have only two unknown forces. Take the joint at which the load be acts, and we have as known quantities the load be, and also the force exerted by the bar YB, which is in compression. This force is now directed away from the bar, and hence is represented by yb. The unknown forces are the stresses in CX and XY. Hence we construct the quadrilateral cxybc ; and we thus determine the stresses in CX and XY as ex and xy t both being thrusts. Next proceed to the joint YXW, and construct the quadri- lateral myxwm, and thus determine the tension xw and the tension wm. Next proceed to the joint where cd acts, and so on. We thus obtain the diagram (Fig. 85*2) giving all the stresses. The truss in the figure was constructed with an angle of 30* at the base, and hence gives special values in accordance with that angle. In order to show how we may compute the stresses from the diagram, the computation will be given. From triangle bmy, we have bm = - W 10 ym = -Wcot 30 = 16 16 by = --^cosec 30 = w = ky. 1 6 8 From the triangle umc, we have cm W, 16 um = w 16 ROOF-TRUSSES. 169 yx yw sec 30 - 16 = ( ^ \i6 = = xv = vt, i6 256 256 , 8 ex = wm sec 256 256 vd = urn sec 30 = W\ -4 \ 16 / y/ 8 4 ' Hence we shall have for the stresses, RAFTERS (compression) . VERTICALS (tension). by = kn ex = ho = \W. xw = op W 16* dv = gq ct =fs = \Z. vu = qr : T' HORIZONTAL TIES (tension). _4/7 ts = 1 07 8 DIAGONAL BRACES (compression). my = mn 16 xy = - if 8 ' mw = mp = g .- o/z; = qp ~"i6 mu = mr 16 fu = sr ** 8 1/0 APPLIED MECHANICS. Next, as to the stresses due to wind pressure, we will sup- pose that there is a roller under the left-hand end of the truss, and none under the right-hand end ; and we will proceed to determine the stresses due to wind pressure. First, suppose the wind to blow from the left-hand side of rhe truss, and let the total wind pressure be (Fig. 8$b) af= W^. The resultant, of course, acts along the dotted line drawn per- pendicular to the left-hand rafter at its middle point, as shown in Fig. 85. The left-hand supporting force will be vertical : hence, pro- ducing the above-described dotted line, and a vertical through the roller to their intersection, and joining this point with the right-hand end of the truss, we have the direction of the right- hand supporting force. In this case, since the angle of the truss is 30, the line of action of the right-hand supporting force coincides in direction with the right-hand rafter. We now construct the triangle of external forces afm y and we obtain the supporting forces fm and ma. We then have, as the loads at the joints,' ab - = ef, be = - = cd = de. 4 Then proceed as before to the left-hand joint ; and we find that two of the four forces acting there are known, viz., ma and ab, and two are unknown, viz., the stresses in .Z? Fand YM. Then construct the quadrilateral mabym, and we have the stresses by and ym ; the first being compression and the second tension, as shown by reasoning similar to that previously adopted. Then pass to the next joint on the rafter, and construct the quadrilateral ybcxy, where yb and be are already known, and we obtain ex and xy ; and so proceed as before from joint to joint, ROOF-TRUSSES WITH LOADS AT LOWER JOINTS. I/T remembering, that, in order to be able to construct the polygon of forces in each case, it is necessary that only two of the forces acting should be unknown. When the wind blows from the other side, we shall obtain the diagram shown in Fig. 85^. After having determined the stresses from the vertical load diagram and those from the two wind diagrams, we should, in order to obtain the greatest stress that can come on any one member of the truss, add to the stress due to the vertical load the greater of the stresses due to the wind pressure. 136. Roof-Truss with Loads at Lower Joints. In Fig. 86 is drawn a stress diagram for the truss shown in Fig. 84 on the supposition that there is also X. a load on the lower joints. In this case let W be the whole load of the truss, except the ceiling, ^ the weight of the ceiling and below ; the latter being supported a,t the lower joints and on the two extreme vertical suspension FlG - 86 - rods. Then will the loads at the joints be as follows; viz., ab = be = \( cd = \W mn = rq = kl, = gh = de = jfe = on = qp = op. Observe that there is no joint at the lower end of either of the end suspension rods, but that whatever load is supported by these is hung directly from the upper joints, where be and hk act We have also for each of the supporting forces Im and ra 1/2 APPLIED MECHANICS. Hence we have, for the polygon of external forces, abcdefghklm nopqra, which is all in one straight line, and which laps over on itself. In constructing the diagram, we then proceed in the same way as heretofore. 137. General Remarks. As to the course to be pursued in general, we may lay down the following directions : I . Determine all the external forces ; in other words, the loads being known, determine the supporting forces. 2. Construct the polygon of forces for each joint of the truss, beginning at some joint where only two of the forces acting at that joint are unknown. This is usually the case at the support. Then proceed from joint to joint, bearing in mind that we can only determine the polygon of forces when the magnitudes of all but two sides are known. 3. Adopt a certain direction of rotation, and adhere to it throughout; i.e., if we proceed in right-handed rotation at one joint, we must do the same at all, and we shall thus obtain neat and convenient figures. 4. Observe that the stresses obtained are the forces exerted by the bars under consideration, and that these are thrusts when they act away from the bars, and tensions when they are directed towards the bars. We will next take some examples of roof-trusses, and con- struct the diagrams of some of them, calling attention only to special peculiarities in those cases where they exist. It will be assumed that the student can make the trigono- metric computations from the diagram. The scale of load and wind diagram will not always be the same ; and the stress diagrams will in general be smaller than is advisable in using them, and very much too small if the ROOF-TRUSSES WITH LOADS AT LOWER JOINTS. 173 results were to be obtained by a purely graphical process with- out any computation. The loads will in all cases be assumed to be distributed uniformly over the jack-rafters, or, in other words, concen- trated at the joints. Those cases where no stress diagram is drawn may be con- sidered as problems to be solved. FIG. 87. FIG. 870. FIG. 87*. 174 APPLIED MECHANICS. PIG. 88. FIG. 883. abcdl FIG. R O OF- TR USSES WITH LOADS A % L O WER JOINTS. I 7 5 FIG. 89. FIG. 8ga. FIG. 90. FIG. a b c A X FIG. 92. FIG. 92*. FIG. 93. FIG. 93 a. APPLIED MECHANICS. 138. Hammer- Beam Truss (Fig. 94). This form of truss is frequently used in constructions where architectural effect is the principal consideration rather than strength. It is not an advantageous form from the point of view of strength, FIG. FIG. 94. FIG. 944. FIG. 94,:. for the absence of a tie-rod joining the two lower joints causes a tendency to spread out at the base, which tendency is usually counteracted by *the horizontal thrust furnished by the but tresses against which it is supported. HAMMER-BEAM TRUSS. 177 When such a thrust is furnished (or were there a tie-rod), and the load is symmetrical and vertical, the bars are not all needed, and some of them are left without any stress. In the case in hand, it will be found that UV, VM, MQ, and QR are not needed. We must also observe that the effect of the curved members MY, MV, MQ, and MAT on the other parts of the truss is just the same as though they were straight, as shown in the dotted lines. The curved piece, of course, has to be subjected to a bending-stress in order to resist the stress acting upon it. If, as is generally the case, the abutments are capable of furnishing all the horizontal thrust needed, it will first be necessary to ascertain how much they will be called upon to furnish. To do this, observe that we have really a truss similar to that shown in Fig. 92, supported on two inclined framed struts, of which the lines of resistance are the dotted lines (Fig. 94) I 4 and 7 8, and that, under a symmetrical load, this polygonal frame will be in equilibrium, and, moreover, the curved pieces MV and MQ will be without stress, these only- being of use to resist unsymmetrical loads, as the snow or wind. Let the whole load, concentrated by means of the purlins at the joints of the rafters, be W. Then will the truss 467 have W to bear \ W, and this will give to be supported at each of 4 the points 4 and 7. Moreover, on the space 2 4 is distributed , which has, as far as overturning the strut is concerned, the 4 W W same effect as at 2, and at 4. Hence the load to be sup- 8 8 ported at 4 by the inclined strut is a vertical load equal to (i + J) w 1 w - We ma y then find the force that must be furnished by the abutment, or by the tie-rod, in either of the two following ways : 178 APPLIED MECHANICS i. By constructing the triangle ySe (Fig. 94*2), with 78 = | W, ye || 14, and eS parallel to the horizontal thrust of the abut- ment ; then will y& be the triangle of forces at I, and eS will be the thrust at i. 2. Multiply f W by the perpendicular distance from 4 to i 2, and divide by the height of 4 above I 8 for the thrust of the abutment ; in other words, take moments about the point i. Now, to construct the diagram of stresses, let, in Fig. 94^, the loads be ab, be, cd, </<?, ef,fg, gh, hk, and kl t and let lz = za = \W be the vertical component of the supporting force ; let zm be the thrust of the abutment : then will Im and ma be the real supporting forces ; and we shall have, for polygon of external forces, abcdefghklma. Then, proceeding to the joint i, we obtain, for polygon of forces, maym ; and, proceeding from joint to joint, we obtain the stresses in all the members of the truss, as shown in Fig. 94^. It will be noticed that UV and RQ are also free from stress. If we had no horizontal thrust from the abutment, and the supporting forces were vertical, the members MV and MQ would be called into action, and J/Fand MN would be inactive. To exhibit this case, I have drawn diagram 94/7, which shows the stresses that would then be developed. A Fand NL would become merely part of the supports. In this latter case the stresses are generally much greater than in the former, and a stress is developed in UV. SCISSOR-BEAM TRUSS. 139. Hammer-Beam Truss: Wind Pressure. Fig. 95 shows the stress diagram of the hammer-beam truss for wind pressure when there is no roller under either end, and when the wind blows from the left. A similar diagram would give the stresses when it blows from the right. FIG. 95. FIG. The cases when there is a roller are not drawn : the student may construct them for himself. 140. Scissor-Beam Truss. We have already discussed two forms of scissor-beam truss in Figs. 90 and 91. These trusses having the right number of parts, their diagrams present no difficulty. Another form of A the scissor-beam truss is shown in Fig. 96, and its diagram pre- sents no difficulty. The only peculiarity to be noticed is, that, after having coa structed the polygon of external forces, abcdefma, we cannot proceed to construct the polygon of equilibrium for one of the supports, because there are three unknown forces FlG - FlG - ^ 1 8o APPLIED MECHANICS. there. We therefore begin at the apex CD, and construct the triangle of forces cdl for this point ; then proceed to joint CB, and construct the quadrilateral bclkb; then proceed to the left-hand support, and obtain mabkgm ; and so continue. 141. Scissor-Beam Truss -without Horizontal Tie. Very often the scissor-beam truss is constructed without any horizontal tie, in which case it has the appearance of Fig. 97, where there is sometimes a pin at GKLH and sometimes not. FIG. gja. FIG. 97. FIG. 97<5. FIG. 97 c. In this case, if the abutments are capable of furnishing hori- zontal thrust to take the place of the horizontal tie of Fig. 96, we are reduced back to that case. If the abutments are not capable of furnishing horizontal thrust, we are then relying on the stiffness of the rafters to prevent the deformation of the truss ; for, were the points BC and DE really joints, with pins, the deformation would take place, as shown in Fig. 97^ or Fig. 97^, according as the two inclined ties were each made in one piece or in two (i.e., according as they are not pinned together at KH, or as they are pinned). This necessity of depending on the stiffness of the rafters, and the liability to deformation if they had joints at their middle points, become apparent as soon as we attempt to draw the diagram. Such an attempt is SCISSOR-BEAM TRUSS WITHOUT HORIZONTAL TIE. l8l made in Fig. 97^, where abcdefga is the polygon of external forces, gabkg the polygon of stresses for the left-hand support, kbclk that for joint BC. Then, on proceeding to draw the tri- angle of stresses for the vertex, we find that the line joining d and / is not parallel to DL, and hence that the truss is not stable. We ought, however, in this latter case, when the sup- porting forces are vertical, and when we rely upon the stiffness of the rafters to prevent deformation, to be able to determine the direct stresses in the bars ; and for this we will employ an analytical instead of a graphical method, as being the most con- venient in this case. Let us assume that there is no pin at the intersection of the two ties, and that the two rafters are inclined at an angle of 45 to the horizon. We then have, if W = the entire load, and a = angle between BK and KG, w w ab = cf = , be ^ cd = de = , 8 4 T 2 tana = 4, sin a = , cos a = , Vs ^s Let x be the stress in each tie, and let y = cl dl = thrust in each upper half of the rafters. Then we must observe that the rafter has, in addition to its direct stresses, a tendency to bend, due to a normal load at the middle, this normal load being equal to the sum of the normal components of be and of x> when these are resolved along and normal to the rafter. Hence normal load = x cos a -\ sin 45 4 This, resolved into components acting at each end of the rafter r gives a normal downward force at each end equal to -f- I 82 APPLIED MECHANICS. Hence, resolving all the forces acting at the left-hand support into components along and at right angles to the rafter, and imposing the condition of equilibrium that the algebraic sum of their normal components shall equal zero, we have, if we call upward forces positive, -f JFsin45 (%xcosa + ^-fFsin45) #sina = o; (i) but, since we have from (i) W 2#sina = sin 45 4 W o /. jfsma = sin 45 8 ( Then, proceeding to the apex of the roof, we have that the load , W cd = 4 gives, when resolved along the two rafters, a stress in each equal to 4 Hence the load to be supported in a direction normal to the rafter at the apex is sin 45 -f- (^ cos a -\ -- sin 45). 4 8 Hence, substituting for x its value, we have y = cl=dl= 5Tsin 4 5. (3) Then, proceeding to the left-hand support, and equating to zero the algebraic sum of the components along the rafter, we have bk = (ga 0)cos45 ~f~ -^cosa -f JWsii^ = f ^5^45. (4) SCISSOR-BEAM TRUSS WITHOUT HORIZONTAL TIE. 183 We have thus determined in (2), (3), and (4) the values of x y y, and bk eh. By way of verification, proceed to the middle of the left- hand rafter, and we find the algebraic sum of the components of be and x along the rafter to be and this is the difference between bk and cl, as it should be. We have thus obtained the direct stresses ; and we have, in addition, that the rafter itself is also subjected to a bending- moment from a normal load at the centre, this load being equal to xcosa H -- sin 45 = sin 45. 4 2 How to take this into account will be explained under the " Theory of Beams." 142. Examples. The following figures of roof -trusses may be considered as a set of examples, for which the stress diagrams are to be worked out. Observe, that, wherever there is a joint, the truss is to be supposed perfectly flexible, i.e., free to turn around a pin. FIG. 98. FIG. 99- FIG. too. FIG. 101. FIG. 102. FIG. 103. FIG. 104. FIG. FIG. 106. FIG. 107. FIG. 108 1 84 APPLIED MECHANICS. CHAPTER IV. BRIDGE-TRUSSES. 143. Method of Sections. It is perfectly possible to determine the stresses in the members of a bridge-truss graphically, or by any methods that are used for roof-trusses. In this work an analytical method will be used ; i.e., a method of sections. This method involves the use of the analytical con- ditions of equilibrium for forces in a plane explained in 63. These are as follows ; viz., If a set of forces in a plane, which are in equilibrium, be resolved into components in two directions at right angles to each other, then i. The algebraic sum of the components in one of these directions must be zero. 2. The algebraic sum of the components in the other of these directions must be zero. 3. The algebraic sum of the moments of the forces about any axis perpendicular to the plane of the forces must be zero. Assume, now, a bridge-truss (Figs. 109, no, in, 112, pages 186 and 187) loaded at a part or all of the joints. Conceive a vertical section ab cutting the horizontal members 6-8 and 7~9 and the diagonal 7-8, and dividing the truss into two parts. Then the forces acting on either part must be in equilibrium, in other words, the external forces, loads, and supporting forces, acting on one part, must be balanced by the stresses in the members cut by the section ; i.e., by the forces exerted by the other part of the truss on the part under consideration. Hence we must have the three following conditions ; viz., - SHEARING-FORCE AND BENDING-MOMENT. .185 i. The algebraic sum of the vertical components of the above-mentioned forces must be zero, 2. The algebraic sum of the horizontal components of these forces must be zero. 3. The algebraic sum of the moments of these forces about any axis perpendicular to the plane of the truss must be zero. 144. Shearing-Force and Bending-Moment. Assum- ing all the loads and supporting forces to be vertical, we shall have the following as definitions. The Shearing-Force at any section is the force with which the part of the girder on one side of the section tends to slide by the part on the other side. In a girder free at one end, it is equal to the sum of the loads between the section and the free end. In a girder supported at both ends, it is equal in magnitude to the difference between the supporting force at either end, and the sum of the loads between the section and that support- ing force. The Bending-Moment at any section is the resultant moment of the external forces acting on the part of the girder to one side of the section, tending to rotate that part of the girder around a horizontal axis lying in the plane of the section. In a girder free at one end, it is equal to the sum of the moments of the loads between the section and the free end, about a horizontal axis in the section. In a girder supported at both ends, it is the difference be- tween the moment of either supporting force, and the sum of the moments of the loads between the section and that sup- port ; all the moments being taken about a horizontal axis in the section. 145. Use of Shearing-Force and Bending-Moment. The three conditions stated in 143 may be expressed as fol- lows : i. The algebraic sum of the horizontal components of the stresses in the members cut by the section must be zero. 1 86 APPLIED MECHANICS. 2. The algebraic sum of the vertical components of the stresses in the members cut by the section must balance the shearing-force. 3. The algebraic sum of the moments of the stresses in the members cut by the section, about any axis perpendicular to the plane of the truss, and lying in the plane of the section, must balance the bending-moment at the section. As the conditions of equilibrium are three in number, they will enable us to determine the stresses in the members, pro- vided the section does not cut more than three ; and this determination will require the solution of three simultaneous equations of the first degree with three unknown quantities (the stresses in the three members). By a little care, however, in choosing the section, we can very much simplify the operations, and reduce our work to the solution of one equation with only one unknown quantity ; the proper choice of the section taking the place of the elimination. 146. Examples of Bridge-Trusses. Figs. 109-1 12 rep- resent two common kinds of bridge-trusses : in the first two the braces are all i 3 5 _7]ajk.n..i3 45 17 19 21 23 25 27 29 diagonal, in the last two they are partly vertical and partly diagonal. The first two are called Warren girders, or half-lattice girders ; since there is only one system of bracing, as in the figures. When, on the other hand, there are more than one system, so that the diagonals cross each other, they are called lattice girders. 147. General Outline of the Steps to be taken in determining the Stresses in a Bridge-Truss under a Fixed Load. i. If the truss is supported at both ends, find the sup- porting forces. VVV\/K/V\/\/\A/\/\/\A/ 2 4 6 b\ 8 *| 10 12 14 16 18 20 22 24 2628 FIG. 109. 1357 a 91 a- 11 13 vw 2157 2466 8 j 10 13 FIG. no. DETERMINING THE STRESSES IN A BRIDGE-TRUSS. 1 87 2. Assume, in all cases, a section, in such a manner as not to cut more than three members if possible, or, rather, three of those that 1 13 15 17 19 21 23 25 27 28 XlXIXbd/l/l/1// brought 7 a \ 10 12 14 16 18 20 22 24 26 FlG - 2 4 6 R 10 12 14 /\/]/\, / /MX 1357 9 11 13 FIG. are into action by the loads on the truss ; and it will save labor if we assume the section so as to cut two of the three very near their point of inter- section. 3. Find the shearing-force at the section. 4. Find the bending-moment at the section. 5. Impose the analytical conditions of equilibrium on all the forces acting on the part of the girder to one side of the section, the part between the section and the free end when the girder is free at one end, or either part when it is supported at both ends. In the cases shown in Figs. 109 and no, we may describe the process as follows ; viz., (a) Find the stress in the diagonal from the fact, that (since the stress in the diagonal is the only one that has a vertical component at the section) the vertical component of the stress in the diagonal must balance the shearing-force. (b) Take moments about the point of intersection of the diagonal and horizontal chord near which the section is taken ; then the stresses in those members will have no moment, so that the moment of the stress in the other horizontal must balance the bending-moment at the section. Hence the stress in the horizontal will be found by dividing the bending-moment at the section by the height of the girder. The above will be best illustrated by some examples. I 88 APPLIED MECHANICS. EXAMPLE I. Given the semi-girder shown in Fig. no, loaded at joint 13 with 4000 pounds, and at each of the joints l > 3> 5> 7> 9 an d ii with 8000 pounds. Suppose the length of each chord and each diagonal to be 5 feet. Required the stress in each member. Solution. For the purpose of explaining the method of procedure, we will suppose that we desire to find first the stresses in 8-10 and 9-10. Assume a vertical section very near the joint 9, but to the right of it, so that it shall cut both 8-10 and 9-10. If, now, the truss were actually separated into two parts at this section, the right-hand part would, in consequence of the loads acting on it, separate from the other part. This tendency to separate is counteracted by the following three forces : i. The pull exerted by the part <$-x of the bar 9-11 on the part x-\\ of the same bar. 2. The thrust exerted by the part 8-2 of the bar 8-10 on the part ^-10 of the same bar. 3. The pull exerted by the part 9-7 of the bar 9-10 on the part y-io of the same bar. The shearing-force at this section is 8000 -f- 4000 = 12000 Ibs., and this is equal to the vertical component of the stress in the diagonal. Hence T 2OOO Stress in 9-10 = = 12000(1.1547) = 13856 Ibs. This stress is a pull, as may be seen from the fact, that, in order to prevent the part of the girder to the right of the section from sliding downwards under the action of the load, the part 9-7 of the diagonal 9-10 must pull the part y-io of the same diagonal. Next take moments about 9 : and, since the moment of the stresses in 9-1 1 and 9-10 about 9 is zero, we must have that the moment of the stress in 8-10; i.e., the product of this stress by the height of the girder, must equal the bending-moment. DETERMINING THE STRESSES 2N A BRIDGE-TRUSS. 189 The bending-moment about 9 is 8000 x 5 4- 4000 x 10 = 80000 foot-lbs. 80000 Hence Stress in 8-10 4-33 80000(0.23094) = 18475 Proceed in a similar way for all the other members. The work may be arranged as in the following table ; the diagonal stresses being deduced from the shearing-forces by multiplying by 1.1547, and the chord stresses from the bending-moments by multiplying by 0.23094. 2_ Stresses in Diagonals cut Stresses in Chords opposite the JJ Shearing- by Section, in Ibs. Bending- respective Joints. c .S> Force Moment, in O * in Ibs. foot-lbs. * J Tension. Compression. Tension. Compression. I 44OOO 50806 72OOOO 166277 2 44OOO 50806 6lOOOO 140873 3 36000 4^69 500000 11547 4 36OOO 41569 4IOOOO 94685 5 28000 32331 \ 320000 73901 6 28OOO 32331 \ 250000 57735 7 2OOOO 23094 ' ISOOOO 41569 8 20000 23094 I3OOOO 30022 9 i 2000 13856 80000 18475 10 I2OOO 13856 5OOOO ,"547 II 4OOO 4619 2OOOO 4618 12 4OOO 4619 IOOOO 2309 EXAMPLE II. Given the truss (Fig. 109) loaded at each oi the lower joints with 10000 Ibs. : find the stresses in the members. The length of chord is equal to the length of diagonal = 10 ft. Throughout this chapter, tensions will be written with the minus, and compressions with the plus sign. Solution. Total load = 14(10000) = 140000 Ibs. Each supporting force = 70000 " The entire work is shown in the following tables: i go APPLIED MECHANICS. CO ^O O\ II II o o o to >-o to *t VO 00 CO * CO rt CO ^ CO x x to o ^o O to O H M ** M <-> o 4- + + + + + to o to o to O I I I I I I I I I I I I I O to o to O X X X X X X X 10 O to >-o O to O N co co Tj- X X X X X X 888 N CO CO 1 1 1 1 1 CO. CO N M >-> *- II II II II II II II II I I I I I I I I tt N ONOO t>sVO torfcoN "-I O ONOO txO MNNNNMNNMNH-,H4 i- M CO 00 ON O NH N CO DETERMINING THE STRESSES IN A BRIDGE-TRUSS, Numbers of Diagonals. Stresses in Diagonals, in Ibs. I- 2 28-29 70000 X I-I547 = 80829 2- 3 27-28 + 60000 X I.T547 = + 69282 3- 4 26-27 60000 X I.I547 = 69282 4- 5 25-26 + 50000 X I.I547 = + 57735 5-6 24-25 50000 X LI547 = -57735 6- 7 23-24 +40000 X I.I547 = +46188 7- 8 22-23 40000 X LI547 = -46188 8- 9 21-22 + 30000 X LI547 = + 34641 9-10 2O-2I 30000 X LI547 = 34641 IO-II I9-2O + 20000 X LI547 = + 23094 11-12 18-19 20000 X I.I547 = -23094 12-13 I7-I8 + IOOOO X LI547 = + II547 I3-H 16-17 i oooo x I.I547 = -H547 14-15 I5-I6 + LOWER CHORDS. IM umbers of Chords. Stresses in Chords, in Ibs. 2- 4 26-28 65OOOO X 0.11547 = - 755 6 4- 6 24-26 I2OOOOO X 0.11547 = 138564 6- 8 22-24 I65OOOO X 0.11547 = 190526 8-10 20-22 2OOOOOO X 0.11547 = -230940 10-12 I 8-20 225OOOO X 0.11547 = 259808 12-14 16-18 245OOOO X 0.11547 = -277128 I4-l6 245OOOO X 0.11547 = 282902 1 9 2 APPLIED MECHANICS. UPPER CHORDS. Numbers of Chords. Stresses in Chords, in Ibs. '- 3 27-29 350000 X 0.11547 = + 40415 3- 5 25-27 950000 x 0.11547 = + 109697 5- 7 23-25 I45OOOO X 0.11547 = + 167432 7- 9 21-23 1850000 X 0.11547 = + 213620 9-1 1 19-21 2I5OOOO X 0.11547 = +248261 ii 13 17-19 2350000 X 0.11547 = + 267355 'j -'5 i5-i7 245OOOO X 0.11547 = + 282902 EXAMPLE III. Given the same truss as in Example II., loaded at 2, 4, 6, 8, 10, and 12 with 10000 Ibs. at each point, the remaining lower joints being loaded with 50000 Ibs. at each joint : find the stresses in the members. EXAMPLE IV. Given a semi-girder, free at one end (Fig. 112), loaded at 2, 4, and 6 with 10000 Ibs., and at 8, 10, and 12 with 5000 Ibs. : find the stresses in the members. TRAVELLING-LOAD. 148. Half-Lattice Girder: Travelling-Load. When a girder is used for a bridge, it is not subjected all the time to the same set of loads. The load in this case consists of two parts, one, the dead load, including the bridge weight, together with any permanent load that may rest upon the bridge ; and the other, the moving or variable load, also called the travelling-load, such as the weight of the whole or part of a railroad train if it is a railroad bridge, or the weight of the passing teams, etc., if it is a common- road bridge. Hence it is necessary that we should be able to determine the amount and distribution of the loads upon the bridge which will produce the greatest tension or the greatest GREATEST DIAGONAL STRESSES IN GIRDER. 193 compression in every member, and the consequent stress pro- duced. 149. Greatest Stresses in Semi-Girder. Wherever the section be assumed in a semi-girder, it is evident that any- load placed on the truss at any point between the section and the free end increases both the shearing-force and the bending- momerit at that section, and that any load placed between the section and the fixed end has no effect whatever on either the shearing-force or the bending-moment at that section. Hence every member of a semi-girder will have a greater stress upon it when the entire load is on, than with any partial load. 150. Greatest Chord Stresses in Girder supported at Both Ends. Every load which is placed upon the truss, no matter where it is placed, will produce at any section whatever a bending-moment tending to turn the two parts of the truss on the two sides of the section upwards from the supports ; i.e., so as to render the truss concave upwards. Hence every load that is placed upon the truss causes com- pression in every horizontal upper chord, and tension in every horizontal lower chord. Hence, in order to obtain the greatest chord stresses, we assume the whole of the moving load to be upon the bridge. 151. Greatest Diagonal Stresses in Girder supported at Both Ends. To determine the distribution of the load that will produce the greatest stress of a certain kind (tension or compression) in any given diagonal, let us suppose the diag- onal in question to be 7-8 (Fig. 109), through which we take our section ab. Now it is evident that any load placed on the truss between ab and the left-hand (nearer) support will cause a shearing-force at that section which will tend to slide the part of the girder to the left of the section downwards with refer- ence to -the other part, and hence will cause a compressive stress in 7-8 ; while any load between the section and the right- 194 APPLIED MECHANICS. hand (farther) support will cause a shearing-force of the oppo- site kind, and hence a tension in the bar 7-8. Now, the bridge weight itself brings an equal load upon each joint ; hence, when the bridge weight is the only load upon the truss, the bar 7-8 is in tension. Hence, any load placed upon the truss between the section and the farther support tends to increase the shearing-force at that section due to the dead load (provided this is equally dis- tributed among the joints) ; whereas any load placed between the section and the nearer support tends to decrease the shear- ing-force at the section due to the dead load, or to produce a shearing-force of the opposite kind to that produced by the dead load at that section. Hence, if we assume the dead load to be equally distributed among the joints, we shall have the two following propositions true : (a) In order to determine the greatest stress in any diagonal which is of the same kind as that produced by the dead load, we must assume the moving load to cover all the panel points between the section and the farther abutment, and no other panel points. (b) In order to determine the greatest stress in any diagonal of the opposite kind to that produced by the dead load, we must assume the moving load to cover all the panel points between the section and the nearer abutment, and no others. This will be made clear by an example. EXAMPLE I. Given the truss shown in Fig. 113. Length of chord = length of diagonal = A A A g !Lu 10 feet. Dead load = 8000 Ibs. Y 4 Y Y Y Y Y Ypl applied at each upper panel point. FIG Moving load = 30000 Ibs. applied at each upper panel point. Find the greatest stresses in the members. EXAMPLE OF BRIDGE-TRUSS, 195 Solution, (a) Chord Stresses. Assume the whole load to be upon the bridge : this will give 38000 each (1) + 76788 (3) 4-. 20R423 (5) -f- 296181 (7; + 340059 (9) (2) -153575 (4) -263272(6) -329090 (8)- 3510251 Ibs. at eacn upper panel point ; i.e., omit- ting I and 17, where the load acts directly on the support, and not on the truss. FlG ' II4- Hence, considering the bridge so loaded, we shall have the fol- lowing results for the chord stresses : Each supporting force = sSooof-J 133000. Section at Bending-Moment, in foot-lbs. 2 16 133000 x 5 = 665000 3 15 133000 X IO = 1330000 4 14 133000 X 15 38000 x 5 = 1805000 5 13 133000 X 2O 38000 X 10 = 2280000 6 12 133000 X 25 - 3 8ooo( 5 -f- 15) = 2565000 7 ii 133000 X 30 - 38000(10 + 20) = 2850000 8 10 133000 x 35 38ooo( 5 + 15 -{- 25) = 2945000 9 133000 X 40 38000(10 4- 20 -j- 30) = 3040000 Numbers of Chords. Stresses in Upper Chords. i-3 I5-I7 665000 X 0.11547 = + 76788 3-5 I3-I5 1805000 X 0.11547 = +208423 5-7 11-13 2565000 X 0.11547 = +296181 7-9 9-1 1 2945000 X 0.11547 = +340059 APPLIED MECHANICS. Numbers of Chords. Stresses in Lower Chords. 2- 4 14-16 1330000 X O.II547 = -153575 4- 6 12-14 2280000 X O.II547 = -263272 6- 8 IO-I2 2850000 X O.II547 = 329090 8-10 3040000 X O.II547 = -351029 Next, as to the diagonals, take, for instance, the diagonal 7-8. When the dead load alone is on the bridge, the diagonal 7-8 is in tension. From the preceding, we see that the greatest tension is produced in this bar when the moving load is on the points 9, n, 13, and 15, and the dead load only on the points 3, 5, 7. Now, a load of 38000 Ibs. at 13, for instance, causes a shearing-force of (38000) = 9500 Ibs. at any section to the 10 left of 13; and this shearing-force tends to cause the part to the left of the section to slide upwards, and that to the right downwards. On the other hand, with the same load at the same place, there is produced a shearing-force of (38000) = 28500 Ibs. 16 at any section to the right of 13 ; and this shearing-force tends to cause the part to the left to slide downwards, and that to the right upwards. Paying attention to this fact, we shall have, when the loads are distributed as above described, a shearing- force at the bar 7-8 causing tension in this bar ; the magnitude of this shearing-force being 6 + 8) _ i6 16 Hence, we may arrange the work as follows : 6 ) = 41500. GREATEST DIAGONAL STRESSES IN GIRDER. 197 Greatest Stresses in Numbers of Greatest Shearing-Forces producing Stresses of Same Kind as Diagonals of Diagonals. Dead Load. as those due to Dead Load. 1-2 17-16 3^ (2 + 4+6+8+IO+I2+I4) = 133000 -'53575 2-3 16-15 ^(2+4+6+8+10+12+14) = I33 ooo + '53575 3-4 '5-H 3 ^(2+ 4 +6+8+io+ I2 )-~(2) = 98750 114027 4-5 14-13 ^(2+4+6+8+10+12) -^(2) = 98750 + i 14027 5-6 13-12 ^(2+4+6+8+10) -^(2+4) = 68250 - 78808 6-7 I 2-1 1 ^(2+4+6+8 + 10) -^(2+4) - 68250 + 78808 7-8 II-IO ^(2+4+6+8) -^(2+4+6) = 41500 - 47920 8-9 io- 9 ^(2+4+6+8) - ^(2+4+6) - 41500 + 479 20 Greatest Stresses in Numbers of Greatest Shearing-Forces producing Stresses of Kind Opposite Diagonals ^pf Diagonals. from Dead Load. Kind Oppo- site from Dead Load. 8-9 io- 9 ^(2+ 4 +6) - ^(2+4+6+8) - 18500 -21362 7-8 II-IO ^(2+4+6) - ?^( 2 + 4 +6+8) - 18500 + 21362 The diagonals 7-8, 8-9, 9-10, and 10-11 are the only ones that, under any circumstances, can have a stress of the kind opposite to that to which they are subjected under the dead load alone. I9 8 APPLIED MECHANICS. Fig. 114 exhibits the manner of writing the stresses on the diagram. 152. General Application of this Method. It is plain that the method used above will apply to any single system of bridge-truss with horizontal chords and diagonal bracing, what- ever be the inclination of the braces. When seeking the stress in a diagonal, the section must be so taken as to cut that diagonal ; and, as far as this stress alone is concerned, it may be equally well taken at any point, as well as near a joint, provided only it cuts that diagonal which is in action under the load that produces the greatest stress in this one, and no other. On the other hand, when we seek the stress in a horizontal chord, the section might very properly be taken through the joint opposite that chord. Taking it very near the joint, only serves to make one sec- tion answer both purposes simultaneously. 153. Bridge-Trusses with Vertical and Diagonal Bra- cing. When, as in Figs, in and 112, there are both vertical and diagonal braces, and also horizontal chords, we may deter- mine the stresses in the diagonals and in the chords just as before ; only we must take the section just to one side of a joint, and never through the joint. As to the verticals, in order to determine the stress in any vertical, we must impose the conditions of equilibrium between the vertical components of the forces acting at one end of that vertical: thus, if the loads are at the upper joints in Fig. in, then the stress in vertical 3-2 must be equal and opposite to the vertical component of the stress in diagonal 1-2, as these stresses are the only vertical forces acting at joint 2. Vertical 5-4 has for its stress the vertical component of the stress in 3-4, etc. Thus Stress in 3-2 = shearing-force in panel 1-3, Stress in 5-4 = shearing-force in panel 3-5, etc.- TRUSSES WITH VERTICAL AND DIAGONAL BRACING. 199 On the other hand, if the loads be applied at the lower joints, then Stress in 3-2 = shearing-force in panel 3-5, Stress in 5-4 = shearing-force in panel 5-7, etc. EXAMPLE. Given the truss shown in Fig. in. Given panel length = height of truss 10 feet, dead load per panel point = 12000 Ibs., moving load per panel point = 23000 Ibs. ; load applied at upper joints. Solution, (a) Chord Stresses. Assume the entire load on the bridge, i.e., 35000 Ibs. per panel point. Hence Total load on truss =13 (35000) = 455000 Ibs., Each supporting force = 227500 Ibs. Joint near which Section is taken. Bending-Moment at the Section very near the Joint, on Either Side of the Joint. I 28 3 27 227500 x 10 = 2275000 5 25 227500 X 20 35000 X 10 = 4200000 7 23 227500 X 30 35000(10 + 20) = 5775000 9 21 227500 X 40 35000 ( 10 + 20 + 30) 7000000 ir 19 227500 X 50 35000 (10 + 20 + 30 + 40) = 7875000 13 17 227500 X 60 35000(10 + 20 -h 30 + 40 + 50) = 8400000 IS 227500 X 70 35000 (10 + 20 + 30 + 40 + 50 -f- 60) = 8575000 To find any chord stress, divide the bending-moment at a section cutting the chord, and passing close to the opposite joint, by the height of the girder, which in this case is 10. Hence we have for the chord stresses (denoting, as before, com- pression by +, and tension by ) : 2OO APPLIED MECHANICS. Stresses in Upper Chords. Stresses in Lower Chords. i- 3 27-28 + 227500 2- 4 24-26 227500 3- 5 25-27 4-420000 4- 6 22-24 420000 5- 7 2 3- 2 5 + 5775 6- 8 20-22 -5775 7- 9 21-23 + 700000 8-10 18-20 700000 91 1 19-21 + 787500 IO-I2 1 6-1 8 -787500 11-13 17-19 + 840000 12-14 14-16 840000 i3-!5 iS- 1 ? + 8575 Diagonals. It is evident, that, for the diagonals, the same rule holds as in the case of the Warren girder : i.e., the greatest stress of the same kind as that produced by the dead load occurs when the moving load is on all the joints between the diagonal in question and the farther abutment ; whereas the greatest stress of the opposite kind occurs when the moving load covers all the joints between the diagonal in question and the nearer abutment. The work of determining the greatest shearing-forces may be arranged as in tables on p. 191. Counterbraces. If the truss were constructed with those diagonals only that slope downwards towards the centre, and which may be called the main braces, the diagonals 1 1-12, 13-14, 14-17, and 16-19 would sometimes be called upon to bear a thrust, and the verticals 12-13 and 17-16 a pull : this would necessitate making these diagonals sufficiently strong to resist the greatest thrust to which they are liable, and fixing the verticals in such a way as to enable them to bear a pull. In order to avoid this, the diagonals 10-13, 12-15, I 5~ I 6, and 17-18 are inserted, which are called counterbraces, and which come into action only when the corresponding main TRUSSES WITH VERTICAL AND DIAGONAL BRACING. 2OI braces would otherwise be subjected to thrust. They also prevent any tension in the verticals. Diagonals. Greatest Shearing- Forces of the Same Kind as those produced by Dead Load. I- 2 28-26 ^ I+2 + 3 + ... +I3 ) = 227500 3- 4 27-24 3J^ (l + 2 + 3+ ... + I2) _I5^( l) = I94H3 5-6 25-22 3 -^P(l + 2 + 3+ ... + II)- '-^(1 + 2) = 162429 7-8 23-20 ^(i + 2+3+ . . . + 10) - x -^?(i + 2+3) J-4 J-4 = 132357 9-10 2I-I8 ^(1 + 2+3+ ...+ 9)-^p(i + 2+.. + 4) = 103929 11-12 19-16 33222(1 + 2+3+...+ 8)-^(i + 2+.. + 5)= 77H3 I3T4 17-14 3^ (l + 2 + 3 +...+ 7) _^ I + 2+ .. + 6)= 52000 Diagonals. Greatest Shearing-Forces of the Opposite Kind to those produced by Dead Load. 3-14 17-14 ^ (I+2+3+ ... + 6) _^ I+2+ ... +7) = 28500 11-12 19-16 22f(i + 2+ ... + 5 )_H5p ( r + 2 + ... + 8) = 6643 9-10 2I-I8 ^(, + 2+ ... + 4) -^ (l + 2 + ... + 9) = ~i357i The main braces and counterbraces of a panel are never in action simultaneously. Hence we have, for the greatest stresses in the diagonals, the following results, obtained by multiplying the corresponding shearing-forces by - 1.414. cos 45 2O2 APPLIED MECHANICS. In the following I have used this number to three decimal places, as being sufficiently accurate for practical purposes. Stresses in Main Braces. Stresses in Counterbraces. I- 2 28-26 -321685 15-12 15-16 40299 3- 4 27-24 -274518 I3-IO I7-I8 - 9393 5-6 25-22 -229675 7- 8 23-20 -187153 9-10 21-18 146956 11-12 19-16 109080 i3~ I 4 17-14 - 735 28 Vertical Posts. Since the loads are applied at the upper joints, the conditions of equilibrium at the lower joints require that the thrust in any vertical post shall be equal to the vertical component of the tension in that diagonal which, being in action at the time, meets it at its lower end. Hence it is equal to the shearing-force in that panel where the acting diagonal meets it at its lower end. We therefore have, for the posts, the following as the greatest thrusts : STRESSES IN VERTICALS. 3- 2 27-26 + 2275OO 5- 4 25-24 + I94M3 7- 6 23-22 + 162429 9- 8 2I-2O + 132357 II-IO I9-I8 + 103929 13-12 I7-l6 + 77143 i5- J 4 + 52000 CONCENTRATING THE LOAD AT THE JOINTS. 203 X X X FIG. Fig. 115 shows the stresses marked on the diagram. 154. Manner of Concentrating the Load at the Joints. In using the methods given above, we are assuming that all the loads are concentrated at the joints, and that none are distributed over any of the pieces. As far as the mov- ing load is concerned, and also all of the dead load except the weight of the truss itself, this always is, or ought to be, effected ; and it is accomplished in a manner similar to that adopted in the case of roof-trusses. This method is shown in the figure (Fig. 1 1 6); floor-beams being laid across from girder to girder at the joints, on top of which are laid longi- tudinal beams, and on these the sleepers if it is a railroad bridge, or the floor if it is a road bridge. The weight of the truss itself is so small a part of what the bridge is called upon to bear, that it can, without appreciable error, be considered as concentrated at the joints either of the up- per chord, of the lower chord, or of both, according to the manner in which the rest of the load is distributed. 155. Closer Approxima- tion to Actual Shearing- Force. In our computations of greatest shearing-force, we FIG. 115. make an approximation which is generally considered to be APPLIED MECHANICS. sufficiently close, and which is always on the safe side. To illustrate it, take the case of panel 3-5 of the last example. In determining its greatest shearing-force, we considered a load of 35000 Ibs. per panel point to rest on all the joints from the right-hand support to joint 5, inclusive, and the dead load to rest on all the other joints of the~truss. Now, it is impossible, if the load is distributed uniformly on the floor of the bridge, to have a load of 35000 Ibs. on 5 and 12000 on 3 simultaneously ; for, if the moving load extended on the bridge floor only up to 5, the load on 5 would be only 12000 + ^-(23000) = 23500 Ibs., and that on 3 would then be 12000 Ibs. If, on the other hand, the moving load extends beyond 5 at all, as it must if the load on 5 is to be greater than 23500 Ibs., then part of it will rest on 3, and the load on 3 will then be greater than 12000 Ibs. ; for whatever load there is between 3 and 5 is supported at 3 and 5. Moreover, we know that the effect of increasing the load on 5 is to increase the shearing-force, provided we do not at the same time increase that on 3 so much as to destroy the effect of increasing that on 5. Hence, there must be some point between 3 and 5 to which the moving load must extend in order to render the shearing- force in panel 3-5 a maximum. Let the distance of this point from 5 be^r; then, if we let w = moving load per foot of length, Moving load on panel = wx, Part supported at 3 = -- , 20 Part supported at 5 = wx -. 20 Hence, portion of shearing-force due to the moving load on panel 3-5 equals CONCENTRATING THE LOAD AT THE JOINTS. I2/ WX 2 \ I WX 2 W I I*X 2 \ ( WX -- ) --- = ( I2X -- - ). i4\ 20 / 14 20 i4\ 20 / This becomes a maximum when its first differential co-efficient becomes zero, i.e., when therefore 12 - x = o X = 9.2 3 . Hence, when the moving load extends to a distance of 9.23 feet from 5, then the shearing-force in panel 3-5, and hence the stress in diagonal 3-4, is a maximum. Panels. Portion of Shearing-Force due to Moving Load on Panel. Value of x t in feet. Portion of Load at Joints named below. Portion of Load at Joints named below. i- 3 27-28 iv ( I 3 *A IO.OO 11500 3 11500 i4\ J 20 y 3- 5 5- 7 25-27 23-25 I4\ 20 / 9-23 8.46 3 5 9797 8230 5 7 11432 11227 I4\ 20 / 7- 9 21-23 I4\ 20 / 7.69 7 6801 9 10886 9-1 1 19-21 H( 9X ~ ^f?) 6.92 9 5507 n 10409 11-13 13-15 17-19 15-17 14 \ 20 / u<l ^ I3^" 2 \ 6.15 5.38 ii 13 4350 3329 13 15 9795 9045 1 I4\ '' 20 / To show how the adoption of this method would affect the resulting stresses in the diagonals and verticals, I have given the work above, and shown the difference between these and 206 APPLIED MECHANICS. the former results. In this table x = distance covered by load from end of panel nearest the centre. Panels. Greatest Shearing-Force of Same Kind as that due to Dead Load. 3- 5 5- 7 7- 9 11-13 27-28 25-27 23-25 2I-2 3 19-21 17-19 15-17 3500Q, 14 l 227500 = 193385 = 161038 -101654 (i+...+S)= 49345 Hence, for stresses in main braces, we have Diagonals. Stresses. I- 2 28-26 -321685 3- 4 27-24 -273446 5- 6 25-22 227708 7- 8 23-20 -184472 9-10 21-18 143739 11-12 19-16 105507 i3- J 4 17-14 69774 Moreover, for the shearing-forces of opposite kind from CONCENTRATING THE LOAD AT THE JOINTS. 2O/ those due to dead load, we have, if x = distance from end of panel nearest support which is covered by moving load, Panels. Portion of Shear due to Moving Load on Panel. Value otx. Portion of Load at Joints named below. Portion of Load at Joints named below. 17-15 /6* - ^} 4.62 15 2455 13 8171 I4\ 20 / 11-13 19-17 E( SX - l^f] 3.84 13 1695 II 7137 I4\ 20 / Panels. Greatest Shearing-Forces of Opposite Kind from those due to Dead Load. 13-5 17-15 S22(i+...+ 5) + l(3:6 7I )-f 4 , I4455 ,-" i 7 (l+ ... + 6) = 25846 M-.3 1^-17 35000 <i -j- + 4)+ l (3o637) _ f4(l3695) _^ (l+ ... +7) = 4116 Hence we have the following as the stresses in the counter- braces : Counter-Braces. Stresses. 15-12 13-10 15-16 I7-l8 - 36546 5820 And, for the verticals, we have the new, instead of the old, shearing-forces. 208 APPLIED MECHANICS. The following table compares the results : Diagonals. Stress, Ordinary Method. Stress, New Method. Difference. I- 2 28-26 -321685 -321685 3- 4 27-24 -274518 -273446 1072 5- 6 25-22 -229675 -227708 1967 7- 8 23-20 -187153 -184472 268l 9-10 21-18 -146956 - J 43739 3217 11-12 19-16 109080 105507 3573 i3- J 4 17-14 - 73528 - 69774 3754 1512 15-16 - 40299 36546 3753 13-10 17-18 - 9393 5820 3573 Verticals. Stress, Ordinary Method. Stress, New Method. Difference. 3- 2 27-26 -f227500 + 227500 O 5- 4 25-24 + I94H3 + 193385 758 7- 6 23-22 4-162429 + 161038 I39 1 9- 8 21-20 + T 3 2 357 + 130461 1896 II-IO I9-I8 4-103929 + 101654 2275 13-12 I7-l6 + 77H3 + 74616 2527 i5- J 4 4-' 28500 + 49345 2655 156. Compound Bridge-Trusses The trusses already discussed have contained but a single system of latticing, or COMPO UND BRID GE- TR USSES. 209 at least only one system that comes in play at one time ; so that a vertical section never cuts more than three bars that are in action simultaneously, the main brace having no stress upon it when the counterbrace is in action, and vice versa. We may, however, have bridge-trusses with more than- one system of lattices ; and, in determining the stresses in their members, we must resolve them into their component systems, and determine the greatest stress in each system separately, and then, for bars which are common to the two systems, add together the stresses brought about by each. In some cases, the design is such that it is possible to resolve the truss into systems in more than one way, and then there arises an uncertainty as to which course the stresses will actually pursue. In such cases, the only safe way is to determine the greatest stress in each piece with every possible mode of resolution of the systems, and then to design each piece in such a way as to be able to resist that stress. Generally, however, such ambiguity is an indication of a waste of material ; as it is most economical to put in the bridge only those pieces that are absolutely necessary to bear the stresses, as other pieces only add so much weight to the struc- ture, and are useless to bear the load. The mode of proceeding can be best explained by some examples. EXAMPLE I. Given the lattice-girder shown in Fig. 117, loaded at the lower panel points 1 TA i i i 11 1 3 5 7 9 11 13 13 17 19 21 23 only. Dead load = 7200 Ibs. per panel point, moving load 18000 Ibs per panel point; let the entire length of bridge FlG ' " 7 * be 60 feet ; let the angle made by braces with horizontal = 60. 210 APPLIED MECHANICS. +75600 FIG. ujc. 19 23 10 14 18 22 This truss evidently consists of the two single trusses shown in Figs. \\ja njb; and njb ; and we can compute the greatest stress of each kind in each member of these trusses, and thus obtain at once all the diag- onal stresses, and then, by E3 Ec4 .... FIG. 117*. addition, the greatest chord stresses. Thus the stress in 1-3 (Fig. 117) is the same as the stress in 1-5 (Fig. I ija). The stress in 3-5 = stress in 1-5 (Fig. 1170) + stress in 3-7 (Fig. 117^). The stress in 5-7 = stress in 5-9 (Fig. 117*7) -f stress in 3-7 (Fig. 117^). The results are given on the diagram (Fig. 117^); the work being left for the student, as it is similar to that done heretofore. EXAMPLE II. Given the lattice-girder shown in Fig. 1 1 8. Given, as before, Dead load = 7200 Ibs. per panel point, moving load = 18000 Ibs. per panel point, entire length of bridge = 25 feet ; load applied at lower panel points. Solution. In this case, there are two possible modes of resolving it into systems. The first is shown in Figs. uSa and n%b : and this is necessarily the mode of division that must hold whenever the load is unevenly distributed, or when the COMPO UND BRID GE- TR USSES. 211 travelling-load covers only a part of the bridge ; for a single load at 6 is necessarily put in communication with the support at 2 by means of the diagonals 6-3 and 3-2, and with the sup- port at 12 by means of the diagonals 6-7, 7-10, lo-n, and the vertical 11-12, and can cause no stress in the other diagonals 1 3 5 7 9 11 7 11 6 10 12 FIG. ii&r. 24 8 12 FIG. i i 83. 5 7 11 Z37I 10 12 FIG. II&T. 5 7 When, however, the whole travelling-load is on the bridge, it is perfectly possible to divide it into the two trusses shown in Figs. II&T and n&/, the diagonals 4-5, 7-10, 6-7, and 5-8 having no stress upon them. When the load is unevenly distributed, we have certainly the first method of division ; and when evenly, we are not sure which will hold. Hence we must compute the greatest stresses with each mode of division, and use for each member the greatest ; for thus only shall we be sure that the truss is made strong enough. We shall thus have the following results : 212 APPLIED MECHANICS. FIRST MODE OF DIVISION (FIGS. n8 AND Diagonals. Greatest Shearing-Force of One Kind. Greatest Shearing-Force of Opposite Kind. Corresponding Stresses. Fig. n8a. Fig. 118*. 2- 3 12-9 ~~~(3 + J ) = 20160 O +23279 3-6 9-3 ~(3+.i) = 20160 -23279 + o 6- 7 8-5 25200 7200, z (2) = 2160 25200., . 7200 ..... (2) = 0040 + 2494 9976 7-10 10-11 5-4 4-1 25200 7200, . -- - (2) = 2160 = 25200, , nroin - 2494 + 9976 34918 . ( 2 + 4; 3024 Chords. Supporting force at 2 (Fig. uSa) or 12 (Fig. = '-*? (3 + Supporting force at 12 (Fig. u8a) or 2 (Fig. = 20160, Section. Chords. Maxi- i mum Com- Bending-Moment. i Stresses in Chords. ponents of Greatest Resultant s S si i "2 00 H ! Separate Stresses. Stresses. M bib bi) bi Trusses. j i 1 i 3 9 20i6oX 5 = 100800 2- 6 8-12 11639 1-3 9-1 1 i o-f- 1-5 + 17459 6 8 20160X10=201600! 3- 7 5- 9+23279 3-5 7- 9> 7+1-5 +40738 : 7 5 20160X15 25200 X5 = i?6400 6-10 4- 8 20369 5-7 3- 7+5-9 +46558 10 4 30240X 5 = 151200 7-1 1 I- 5 +17459 2 -4 10-12 2- 6+2-4 -11639 10-12 2- 4 O 4-6 8-102- 6+4-8 32008 ! 6-8 6-10+4-8 -40738 ! COMPOUND BRIDGE-TRUSSES. 2I 3 SECOND METHOD OF DIVISION (FIGS. nSc AND Diagonals (Fig. n8<r). Diagonals. Maximum Shear. Corresponding Stresses. 1-4 10-11 252OO 29098 4-5 7-10 O Fig. u&/. Diagonals. Maximum Shear. Corresponding Stresses. 2-3 9-12 25200 + 29098 3-6 8-9 25200 29098 6-7 5-8 O O Chords. Each supporting force in either figure = 25200. Fig. n8c. Bending-moment anywhere between 4 and 10 = (25200) (5) = 126000; /. Stress in i-n = +14549, .*. Stress in 4-10 = 14549. Fig. n8d. Bending-moment at 3 or 9 = 126000, Bending-moment anywhere between 6 and 8 = 252000; /. Stress in 3-9 = 4-29098, Stress in 2-6 or 8-12 = 14549, Stress in 6-8 = 29098. 214 APPLIED MECHANICS. Hence we have for chord stresses, with this second divis- ion, Chords. Stresses. i-3 9-1 1 I-II -|- + 14549 3-5 7- 9 i-n + 3-9 + 43 6 47 5-7 - . i-n + 3-9 + 43647 2-4 IO-I2 -f- 2-6 -14549 4-6 8-10 410 4- 2-6 29098 6-8 4-10 + 6-8 43 6 47 Hence, selecting for each bar the greatest, we shall have, as the stresses which the truss must be able to resist, 1-4 IO-II + o -34918 i-3 9-1 1 + 17459 2 -3 12-9 + 29098 3-5 7- 9 +43647 3-6 9-8 + o 29098 5-7 - +46558 4-5 10- 7 + 9976 2494 2-4 IO-I2 -14549 5-8 7- 6 + 2494 - 9976 4-6 8-10 32008 6-8 43647 These results are recorded in Fiff. uSe. (1)+17459 (3)+ 43647(5)+ 46558(7)+ 43647(9^17459(11) )- 32008 (6) - 43647 (8)-32008(10)-14549(12) FIG. ii&?. 157. Other Trusses. In Figs. 119, 120, and 121, we have examples of the double-panel system with the load placed OTHER TRUSSES. 21 5 at the lower panel points only. When, as in 119 and 120, the number of panels is odd, the same ambiguity arises as took place in Fig. 118. When, on the other hand, the number of panels is even, as shown in Fig. 121, there is only one mode of division into systems possible. The diagrams speak for themselves, and need no explanation. 24 6 8 10 12 14 16 18 20 22 24 26 28 30 1 8 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 34 24 8 12 16 20 24 13 17 FIG. 25 29 33 34 22 26 30 32 7 11 15 19 23 27 31 38 34 FlG. no*. 2 4 22 26 30 32 5 9 13 17 19 23 27 31 33 31 FIG. ngc. 2 6 10 24 28 32 1 11 15 21 26 2 3S 14 Fro. norf. 216 APPLIED MECHANICS. 246 8 10 12 14 16 18 20 22 24 26 2 4 12 16 24 28 32 11 15 19 23 27 FIG. izoa. 35 6 10 14 18 22 26 30 S4 36 159 13 17 21 25 FIG. i2o. 33 35 6 10 14 24 28 32 36 2 4 26 30 34 36 13 7 11 " 15 21 25 29 33 35 FIG. izod. 2 4 6 8 10 12 14 16 18 1 3 5 7 9 11 13 15 17 19 20 FINK'S TRUSS. 217 2 6 10 14 18 11 15 FIG. i2i. 12 16 18 V / \ 5 9 13 17 FIG. i2i3. The trusses given above may be considered as examples, to be solved by the student by assuming the dead and the moving load per panel point respectively. 158. Fink's Truss. The description of this truss will be evident from the figure. There is, first, the primary truss 1-8-16; then on each side of 9-8 (the middle post of this truss) is a secondary truss (1-4-9 on tne left, and 9-12-16 on the right). Each of these secondary trusses contains a pair of smaller secondary trusses, and the division might be continued if the segments into which the upper chord is thus divided were too long. Of the inclined ties, there is none in which any load tends to produce compression ; in other words, every load either in- creases the tension in the tie, or else does not affect it. Hence 218 APPLIED MECHANICS. the greatest stresses in all the members will be attained when the entire travelling-load is on the truss, and we need only con- sider that case. The determination of the stress in any one member can readily be obtained by determining, by means of the triangle of forces, the stress in that member due to the presence of the total load per panel point, at each point, and then adding the results. This will be illustrated by a few diagonals. Let angle 8-1-9 *'> Let angle 4-1-5 = i n Let angle 2-1-3 ** \ we shall have, if w -{- w^ entire load per panel point, Designation of Ties. EFFECT OF LOADS AT Resultant Tensions. 3 5 7 9 11 13 15 1-2 2-5 5-6 6-9 1-4 4-9 1-8 W + Wi O o o w + w l O W + Wi o o o w + Wi o o o 3 w-fw/t o o o W + Wi O o W + Wi 8 sin i W + Wi 2 sin i 2 w + w l 2 sin z' 2 W + Wj 2 sin z' 2 o w -f -Wi 2 sin z' 2 w + w t 2 sin z' 2 W + Wi 2 sin /' 2 w + w x 2 sin / 2 W + Wi 2 sin /' 2 W + Wi 4 sin * x W -\~ Wi 2 sin *! 7f -}- W x 4 sin z'i W + Wi sin /! ze/ + Wi 4 sin /i W + Wi 2 sin z'i W + Wi 4 sin /'x SW+Wi sin /j , 2(/+W,) 8 sin * 4 sin / 8 sin z 2 sin 2 8 sin* 4 sin / sin 2 i i The stresses in all the other members may be found in a similar manner. GENERAL REMARKS. 2I 9 159. Bollman's Truss. The description of this truss is made sufficiently clear by the figure. The upper chord is made in separate pieces ; and 1 3 5 7 9 11 12 the short diagonals 2-5, 3-4, 4-7, 5-6, 7-8, 6-9, 8-1 1, and 9-10 are only needed to prevent a bending of the upper chord at the joints. FlG - I24> When this is their only object, the stress upon them cannot be calculated : indeed, it is zero until bending takes place ; and then it is the less, the less the bending. Hence, in this case, the stress is wholly taken up by the principal ties ; and these have their greatest stress when the whole load is on the bridge. The computation of the stresses is made in a similar man- ner to that used in the Fink. 1 60. General Remarks. The methods already explained are intended to enable the student to solve any case of a bridge- truss where there is no ambiguity as to the course pursued by the stresses. In cases where a large number of trusses of one given type are to be computed, it would, as a rule, be a saving of labor to determine formulae for the stresses in the members, and then substitute in these formulae. Such formulae may be deduced by using letters to denote the load and dimensions, instead of inserting directly their numerical values ; and then, having deduced the formulae for the type of truss, we can apply it to any case by merely sub- stituting for the letters their numerical values corresponding to that case. Such sets of formulae would apply merely to specific styles of trusses, and any variation in these styles would require the formulae to be changed. 220 APPLIED MECHANICS. In order to show how such formulas are deduced, a few will be deduced for such a bridge as is shown in Fig. 1 1 1. Let the load be applied at the upper panel points only ; let dead load per panel point = w, moving load per panel point = w,. Let the whole number of panels be N, N being an even number. Let the length of one panel = height of truss = /. Then length of entire span = Nl. Consider the (n + i) th panel from the middle. The stress in the main tie is greatest when the moving load is on all the panel points from the farther abutment up to the panel in question, (n + th - Hence, for the n tb panel from the middle, the greatest shear' ing-force that causes tension in the main tie is equal to w-\-w 1 Hence stress in main tie N For the counterbrace, we should obtain, in a similar way, the formula _N H T \ ^~ 1 I \ fl I 2N LA 2 / 2 ] - wN(n + ,) } , which represents tension when it is positive. Proceed in a similar way for the other members. When there is more than one system, we must divide the truss into its component systems; and when there is ambiguity, we must use, in determining the dimensions of each member, the greatest stress that can possibly come upon it. CENTRE OF GRAVITY. 221 CHAPTER V. CENTRE OF GRAVITY. 161. The centre of gravity of a body or system of bodies, is that point through which the resultant of the system of parallel forces that constitutes the weight of the body or system of bodies always passes, whatever be the position in which the body is placed with reference to the direction of the forces. 162. Centre of Gravity of a System of Bodies. If we have a system of bodies whose weights are W iy W 2y W y etc., the co-ordinates of their individual centres of gravity being fo, y *i), (* y **}> (* y Iv * 3 ) etc., respectively, and if we denote by x m y , z , the co-ordinates of the centre of gravity of the system, we should obtain, just as in the determination of the centre of any system of parallel forces, i. By turning all the forces parallel to OZ> and taking moments about OY, (W, + W 2 + W 3 + etc.)* = W,x, + W 2 x 2 + W t x 3 + etc., or and, taking moments about OX, etc., or 222 APPLIED MECHANICS. 2. By turning all the forces parallel to OX, and taking moments about OY, (W* + W, + W z + etc.K = W& + W 2 z 2 + W z z z + etc., or Hence we have, for the co-ordinates of the centre of gravity of the system, EXAMPLES. I. Suppose a rectangular, homogeneous plate of brass (Fig. 125), where AD = 1 2 inches, AB = 5 inches, and whose weight is 2 Ibs., to have weights attached at the points A, B, C, and D respectively, equal to 8, 6, 5, and x $ Ibs. ; find the centre of gravity of the system. 4- Solution. Assume the origin of co-ordinates at the centre of the rectangle, and we have W, = 2, W 2 = 8, W, = 6, W 4 = 5, W s = 3, *, =o, x 2 = 6, x z = 6, * 4 = -6, ^ s = -6, Ji =o, ^ 2 = f, J 3 = -f, y 4 = -|, j s = f ; = o -f 48 -f 36 30.0 18.0 = 36, = o -f- 20 15 12.5 4- 7.5 = o, = 2 -f- 8 -f 6 + 5.0 4- 3.0 = 24; _ 3^> _ ^ = 2 4 = = 2 4 = Hence the centre of gravity is situated at a point E on the line OX, where OE = 1.5. CENTRE OF GRAVITY OF . HOMOGENEOUS BODIES. 22$ 2. Given a uniform circular plate of radius 8, and weight 3 Ibs. (Fig. 126). At the points A, B, C, and D, weights are attached equal to 10, 15, 25, and 23 Ibs. respectively, also given AB = 45, BC = 105'', CD = 120 ; find the centre of gravity of the system. 163. Centre of Gravity of Homogeneous Bodies. For the case of a single homogeneous body, the formulae have been already deduced in 44. They are fxdV ~ JdV and for the weight of the body, W = wfdV, where x& y , z m are the co-ordinates of the centre of gravity of the body, W its weight, and w its weight per unit of volume. From these formulae we can readily deduce those for any special cases ; thus, (a) For a volume referred to rectangular co-ordinate axes, d V dxdydz. x _ = fffxdxdydz _ = fffydxdydz = fffzdxdydz SSfdxdydz' y " Sffdxdydz* * == fffdxdydz (b) For a flat plate of uniform thickness, t, the centre of grav- ity is in the middle layer; hence only two co-ordinates are required to determine it. If it be referred to a system of rect- angular axes in the middle plane, dV '= tdxdy, _ ffxdxdy _ ffydxdy 224 APPLIED MECHANICS. The centre of gravity of such a thin plate is also called the centre of gravity of the plane area that constitutes the middle plane section ; hence (c) For a plane area referred to rectangular co-ordinate axes in its own plane, Sfxdxdy ffydxdy (d) For a slender rod of uniform sectional area, a, if x, y, z, be the co-ordinates of points on the axis (straight or curved) of the rod, we shall have dV ads a^(dx) 2 + (dy)* + (dzf> (I)* fyds = l- = fds A* (e) For a slender rod whose axis lies wholly in one plane, the centre of gravity lies, of course, in the same plane ; and if our co-ordinate axes be taken in this plane, we shall have z = c -=- =r o, and also Z Q = o. Hence we need only two co ax CENTRE OF GRAVITY OF HOMOGENEOUS BODIES. 22$ ordinates to determine the centre of gravity, hence dV =. ads m fxds _ J AMI) + }<tx * 4 \dx (/) For a line, straight or curved, which lies entirely in one plane, we shall have, again, + Sds fds ' ^ } doc Whenever the body of which we wish to determine the centre of gravity is made up of simple figures, of which we already know the positions of the centres of gravity, the method explained in 162 should be used, and not the formulae that involve integration ; i.e., taking moments about any given line will give us the perpendicular distance of the centre of gravity from that line. In the case of the determination of the strength and stiff- ness of beams, it is necessary to know the distance of a hori- zontal line passing through the centre of gravity of the section, 226 APPLIED MECHANICS. from the top or the bottom of the section ; but it is of no prac- tical importance to know the position of the centre of gravity on this line. In most of the examples that follow, therefore, the results given are these distances. These examples should be worked out by the student. In the case of wrought-iron beams of various sections, on account of the thinness of the iron, a sufficiently close approxi- mation is often obtained by considering the cross-section as composed of its central lines ; the area of any given portion being found by multiplying the thickness of the iron by the corresponding length of line, the several areas being assumed to be concentrated in single lines. EXAMPLES. i. Straight Line AB (Fig* 127). The centre of gravity is evidently at the middle of the line, as this is a point of FIG. 127. - B symmetry. 2. Combination of Two Straight Lines. The centre of gravity in each case lies on the line OO I} Figs. 128, 129, 130, and 131. (a) Angle- Iron of Unequal Arms (Fig. 128). Length AB = b, length BC = h, area AB = A, area BC = B; q _^^ E;\ p, /. BE = DE = % . A D"C" \Jb 2 -f- h 2 FIG. 128. () Angle-Iron of Equal Arms (Fig. 129). Length AB = BC B = b; \^ b b FIG. 129. CENTRE OF GRAVITY OF HOMOGENEOUS BODIES. 22 / (c) Cross of Equal Arms (Fig. 130). AB = OO t = h; :. AC = BC c p, B FIG 130. (d) T-Iron (Fig. 131). Area AB = A, area CE = B, length A E B CE = h; Bh Fi?. 131. 2 ^ -}- &) 3. Combination of Three Lines. OO l = line passing through the centre of gravity. (a) Thin Isosceles Triangular Cell (Fig. 132). Length BC = a, length AC = b, area AB = BC A D c = ^ area ^4C = B; o XZ "7 /. DB B FIG. 132. -(- B BE - Same in Different Position (Fig. 133). BD = DC = - FIG. i J3 . 228 APPLIED MECHANICS. (c) Channel-Iron (Fig. 134). Area of flanges = A, area of web = B, depth of flanges + J thickness of Ah FIG. 134. (d} H-Beam (Fig. 135). Area of upper flange = A I9 area of lower flange = A 2 , area of web = B, height = h. 4 2 + B _ = k C B EOF FIG. 135. 2 A s + A 2 + B 4. Combination of Four Lines. OO^ = line passing through the centre of gravity. (a) Thin Rectangular Cell (Fig. 136). Length AB = h; /. AE = BE = - 2 FIG. 136. () Thin Square Cell (Fig. 137). Z? 77 /~* Z7 yj y^ ^^ OA> * 2 = BC = -O, FIG. 137. 5. Circular Arcs. (a) Circular Arc AB (Fig. 138). Angle A OB = 0,, radius = r, Use formula fyds FIG. i 3 i. " fds ' - fds ' but use polar co-ordinates, where ds = rdQ, sf = rcosO, y = rsinO, CENTRE OF GRAVITY OF HOMOGENEOUS BODIES. 229 r 2 cos OdO 2 f i Oi si sin OdB (i cos0,) Circular Arc AC (same figure). r sin #! , y<> = o. (c) Quarter-Arc of Circle AB, Radius r (Fig. 139). r 2 I 2 cos OdB Jo 2r Semi-circumference ABC (same figure). FIG- 13* 6. Combination of Circles and Straight Lines. Barlow Rail (Fig. 140). Two quadrants, radius r, and web, c , whose area = ^- the united area of the quadrants. Let united area of quadrants = A, area of web ; let AI IB FIG. 140. 230 APPLIED MECHANICS. 7. Areas. (a) T-Section (Fig. 141). Let length AB = B, EF = b, entire B height = H, GE = h. Let distance of centre x.*mmmmmm^i o f gravity below AB = x t ; therefore, taking moments about AB as an axis, -h(B-t)\ -$ - k(S - whence we can readily derive x t . (b) I-Section (Fig. 142). Let AB = B, GH = b, MN = b n entire height = H, BC = H h, EH = h t ; and let x t = distance of centre of gravity below AB. A Hence, taking moments about AB, we have Xl \B(H - h) B - h,) whence we can deduce x t (c} Triangle (Fig. 143). If we consider the triangle OBC as composed of an indefinite number of narrow strips parallel to the side CB, of which FLHK is one, the centre of gravity of each one of these strips will be on the line OD drawn from O to the middle point of the side CB ; hence the centre ^f gravity of the entire tri- c angle must be on the line OD. For a similar rea- son, it must be on the median line CE ; hence the centre of gravity mhst be at the intersection of the median lines, and hence BC . ODsv^ODC o FIG. 143. X Q = = ^OD. Moreover, area = CENTRE OF GRAVITY OF HOMOGENEOUS BODIES. (d) Trapezoid (Fig. 144). First Solution. Bisect AB in 6>, and CE in >; let g^ be the centre of gravity of CEJB, and g t that of ABC. Then will 6 1 , the centre of gravity of the trape- zoid, be on the line g^^ and Gg, -. Gg, CEB* But it must be on the line OD; hence it is at their intersectioa From the similarity of GG l g l GG^g^ we have GG, GG ~ ABC BEC ~ CE B m b ; GG, andsince OD Second Solution. Fig. 144 (a). Let O be the point of intersection of the non-parallel sides AC and BE. Let OF = x lt OD * OG = x n . Take moments about an axis through O, and perpen- dicular to OF) and we readily obtain Fie. 232 APPLIED MECHANICS. (e) Parabolic Half-Segment OAB (Fig. 145). Let OA ~ x,, AB = jh ; let x ,y , be the co-ordinates of the centre of gravity ; let the equation of the parabola be y 2 = Y '.r, /-' 2 *M f Xl 3 Jo xdocd y 2a J ** d X = /*' rya t/o t/o ' r<^ /.r, ^ 3 ^ t/o y ~~ s i ~ 3v i MI Area (/) Parabolic Spandril OBC (Fig. 145). Let x ,y , be co-ordi- nates of centre of gravity of the spandril. xdxdy x^ (*y\ I _ l_?y /*Jr, /*y t / / L 1/0 ^ _ > /*, /^ / / 1/0 ^ 2 * Area = ^j, - CENTRE OF GRA VI TY OF HOMOGENEOUS BODIES. 233 (g) Circular Sector OAC (Fig. 146). Let OA = r, AOX = 0,, be the co-ordinates of the centre of gravity : . . ^o^O, Xo = /r r*V r"* x* /rcos0j /*jr tan 0, / *<**#+ / / JCflkfljK .ooggy-V^rra y ^-rtanfl, ^ Area c FIG. 146. Second Solution. Consider the sector to be made up of an indefinite number of narrow rings ; let p be the variable radius, and dp the thickness : Elementary area = 2pft l dp, and centre of gravity of this elementary area is on OX, at a distance from O equal to p ^ 1 [see Example 5 ()] ; X = (ti) Circular Half-Segment ABX (Fig. 146), f" f Q " xdxdy f" xVr* - x*dx Sector minus triangle %r*S l |r 2 sin 0, cos 0, , sn <, cos r r^ ^rcosgyo '"""' = . 4sin B ig 1 -sin a 0.cosg 1 , sin 0, cos 0,) ~~ 0, sin 0, cos 0, 234 APPLIED MECHANICS. 164. Pappus's Theorems. The following two theorems serve often to simplify the determination of the centres of gravity of lines and areas. They are as follows : THEOREM I. If a plane curve lies wholly on one side of a straight line in its own plane, and, revolving about that line, generates thereby a surface of revolution, the area of the sur- face is equal to the product of the length of the revolving line, and of the path described by its centre of gravity. Proof. Let the curve lie in the xy plane, and let the axis of y be the line about which it revolves. We have, from what fxds precedes, 163 (e\ X Q =- -' .*. x fds = fxds, where x equals the perpendicular distance of the centre of gravity of the curve from O Y, ds = elementary arc, 2irx fds = f(2irx)ds; or, reversing the equation, f(2irx)ds = But f(2irx)ds = surface described in one revolution, while s =. length of arc, and 2irx Q = path described by the centre of gravity in one revolution. Hence follows the proposition. THEOREM II. If a plane area lying wholly on the same side of a straight line in its own plane revolves about that line, and thereby generates a solid of revolution, the volume of the solid thus generated is equal to the product of the revolving area, and of the path described by the centre of gravity of the plane area during the revolution, PAPPUS'S THEOREMS. 235 Proof. Let the area lie in the xy plane, and let the axis OY be the axis of revolution. We then have, from what has preceded, if x = perpendicular distance of the centre of gravity of the plane area from OY t the equation, 163 (b), Sfxdxdy *- ffdxdy' Hence Xo ffdxdy = ffxdxdy; /. (2irx ) ffdxdy or ff(2trx)dxdy But ff(2irx)dxdy = volume described in one revolution, and 2iex Q = path described by the centre of gravity in one revolu- tion. Hence follows the proposition. The same propositions hold true for any part of a revolution, as well as for an entire revolution, since we might have multi- plied through by the circular measure 6, instead of by 2ir. It is evident that the first of these two theorems may be used to determine the centre of gravity of a line, when the length of the line, and the surface described by revolving it about the axis, are known ; and so also that the second theorem may be used to determine the centre of gravity of a plane area whenever the area is known, and also the volume described by revolving it around the axis. EXAMPLES. i. Circular Arc AC (Fig. 138). Length of arc = s = 2rO, sur- face of zone described by revolving it about O Y = circumference of a great circle multiplied by the altitude = (ztrr) (2rsmO l ); x l = rsinfl, sm0 z r- 236 APPLIED MECHANICS. 2. Semicircular Arc (Fig. 139). Length of arc = nr, surface of sphere described = 4?rr 2 ; 2r .'. 2Trx (Trr) = 4?rr 2 .*. x = 7T 3. Trapezoid (Fig. 147). Let AD = b, BC b ; let it revolve around AD : it generates two cones and a cylinder. AD + BC Y Area of trapezoid = - BG, B Volume = ~ -(AG + HD) + 7r(G) 2 . BC \-HD+ 3 BC) FlG - J 47. = ^ ^(^Z) + BC + ^C) GBI BC \ GBI = KL 4. Circular Sector AGO (Fig. 146). Area of sector = r*0 lt volume described = -Jr (surface of zone) = \r(2-rrr} (?r sin 0,) = sin 0! 165. Centre of Gravity of Solid Bodies. The general formulae furnish, in most cases, a very complicated solution, and hence we generally have recourse to some simpler method. A few examples will be given in this and the next section. CENTRE OF GRAVITY OF SYMMETRICAL BODIES. 237 Tetrahedron ABCD (Fig. 148). The plane ABE, containing the edge AB and the middle point E of the edge CD, bisects all lines drawn parallel to CD, and terminating in the faces A ABD and ABC : hence a similar reasoning to that used in the case of the triangle will show that the cen- tre of gravity of the pyramid must be in the plane ABE ; in the same way it may be shown that it must lie in the plane ACF. Hence it must lie in their intersection, or in the line AG joining the vertex A with the centre of gravity (intersection of the medians) of the opposite face. FIG. 148. In the same way it can be shown that the centre of gravity of the triangular pyramid must lie in the line drawn from the vertex B to the centre of gravity of the face A CD. Hence the centre, of gravity of the tetrahedron will be found on the line AG at a distance from G equal to \A G. 1 66. Centre of Gravity of Bodies which are Symmet- rical with Respect to an Axis. Such solids may be gener- ated by the motion of a plane figure, as ABCD (Fig. 149), of variable dimensions, and of any form whose centre G remains upon the axis OX ; its plane being always perpendicular to OX, and its variable area X being a function of x, its distance from the origin. Here the centre of gravity will evidently FIG. i 49 . jj e on tne ax j s QX^ an d the elementary vol- ume will be the volume of a thin plate whose area is X and thickness A;r ; hence the elementary volume will be Take moments about OY, and we shall have or x fXdx = fXxdx and Volume = fXdx, fXxdx ** = 7xJ*" F =/^ APPLIED MECHANICS. EXAMPLES. x 2 y 2 z 2 I. Ellipsoid -f ^- + = i (Fig. 150). Find centre of gravity a D c of the half to the right of the x plane. Let OK = x. Now if, in the equation of the ellipsoid, v X 2 Z 2 we make y = o, we have H = I ; where z = Make z = o in the equation of the ellipsoid, and + ij = I > where ^ = .-. EK are the semi-axes of the variable ellipse EGFH, which, by moving along OX, generates the ellipsoid. Hence hence irbc Area EGFH = Tr(EK . GK) = (a* - x 2 ) = X; Elementary volume = (a 2 irbc (* a . ( a 2 x 2 x 4 ) I (a 2 x x*)dx < > a 2 Jo _ ( 2 4 ). trbc ^ J ( x*} a a 2 -x*)dx \a 2 x--\ 3)0 V = - - I a (a 2 - x 2 )dx = \irabc. d 2 t/0 a. Hemisphere. Make a = b = c, and x = f a, V CENTRE OF GRAVITY OF SYMMETRICAL BODIES. 239 If the section X were oblique to OX t making an angle 0. with it, the elementary volume would not be Xdx, but Xdx sin 0, and we should have 3. Oblique Cone (Fig. 151). Let OA = h; let area of base be and let the angle made by OX with the base be 6; X x> A FIG. 151. r h sin0 / ^^ ** o 4. Truncated Cone (Fig. 151). Let height of entire cone be h = OA ; let height of portion cut off be h l ; AT* h*- h* I x*dx 4 ,/^-A 4 ^TTSi 240 APPLIED MECHANICS. CHAPTER VI. STRENGTH OF MATERIALS. 167. Stress, Strain, and Modulus of Elasticity When a body is subjected to the action of external forces, if we imagine a plane section dividing the body into two parts, the force with which one part of the body acts upon the other at this plane is called the stress on the plane ; it may be a tensile, a compressive, or a shearing stress, or it may be a com- bination of either of the two first with the last. In order to know the stress completely, we must know its distribution and its direction at each point of the plane. If we consider a small area lying in this plane, including the point O, and represent the stress on this area by /, whereas the area itself is repre- sented by a, then will the limit of <- as a approaches zero be the a intensity of the stress on the plane under consideration at the point 0. When a body is subjected to the action of external forces, and, in consequence of this, undergoes a change of form, it will be found that lines drawn within the body are changed, by the action of these external forces, in length, in direction, or in both ; and the entire change of form of the body may be correctly described by describing a sufficient number of these changes. If we join two points, A and B, of a body before the external forces are applied, and find, that, after the application of the external forces, the line joining the same two points of the body has undergone a change of length &(AB), then is the STRESS, STRAIN, AND MODULUS OF ELASTICITY. 241 limit of the ratio ' as AB approaches zero called the strain of the body at the point A in the direction AB. If AB 4- &(AB) > AB, the strain is one of tension. If AB + A (^4-5) < ^4-#, the strain is one of compression. Suppose a straight rod of uniform section A to be subjected to a pull P in the direction of its length, and that this pull is uniformly distributed over the cross-section : then will the in- tensity of the stress on the cross-section be If P be measured in pounds, and A in square inches, then will / be measured in pounds per square inch. If the length of the rod before the load is applied be /, and its length after the load is applied be I ~\- e, then is e the elongation of the rod ; and if this elongation is uniform through- x> out the length of the rod, then is - the elongation of the rod per unit of length, or the strain. Hence, if a represent the strain due to the stress / per unit of area, we shall have The Modulus of Elasticity is commonly defined as the ratio of the stress per unit of area to the strain, or *-*-; a and this is expressed in units of weight per unit of area, as in pounds per square inch. This definition is true, however, only for stresses for which Hooke's law " The stress is proportional to the strain " holds. 242 APPLIED MECHANICS. For greater stresses the permanent set must first be deducted from the strain, and the remainder be used as divisor. The limit of elasticity of any material is the stress above which the stresses are no longer proportional to the strains. The modulus of elasticity was formerly defined as the weight that would stretch a rod one square inch in section to double its length, if Hooke's law held up to that point, and the rod did not break. EXAMPLES. 1. A wrought-iron rod 10 feet long and i inch in diameter is loaded in the direction of its length with 8000 Ibs. ; find (i) the intensity of the stress, (2) the elongation of the rod ; assuming the modulus of the iron to be 28000000 Ibs. per square inch. 2. What would be the elongation of a similar rod of cast-iron under the same load, assuming the modulus of elasticity of cast-iron to be 1 7000000 Ibs. per square inch ? 3. Given a steel bar, area of section being 4 square inches, the length of a certain portion under a load of 25000 Ibs. being 10 feet, and its length under a load of 100000 Ibs. being 10' o".o75 ; find the modulus of elasticity of the material. 4. What load will be required to stretch the rod in the first example Y 1 ^ inch ? 1 68. Resistance to Stretching and Tearing. The most- used criterion of safety against injury for a loaded piece is, that the greatest intensity of the stress to which any part of it is subjected shall nowhere exceed a certain fixed amount, called the working-strength of the material ; this working-strength being a certain fraction of the breaking-strength determined by practical considerations. The more correct but less used criterion is, that the great- est strain in any part of the structure shall nowhere exceed the working-strain ; the greatest allowable amount of strain being a fixed quantity determined by practical considerations. RESISTANCE TO STRETCHING AND TEARING. 243 This is equivalent to limiting the allowable elongation or compression to a certain fraction of its length, or the deflection of a beam to a certain fraction of the span. If the stress on a plane surface be uniformly distributed, its resultant will evidently act at the centre of gravity of the surface, as has been already shown in 42 to be the case with any uniformly distributed force. If a straight rod of uniform section and material be sub- jected to a pull in the direction of its length, and if the result- ant of the pull acts along a line passing through the centres of gravity of the sections of the rod, it is assumed in practice that the stress is uniformly distributed throughout the rod, and hence that for any section we shall obtain the stress per square inch by dividing the total pull by the number of square inches in the section. If, on the other hand, the resultant of the pull does not act through the centres of gravity of the sections, the pull is not uniformly distributed ; and while will express the mean stress per square inch, the actual inten- sity of the stress will vary at different points of the section, p being greater than at some points and less at others. How A to determine its greatest intensity in such cases will be shown later. With good workmanship and well-fitting joints, the first case, or that of a uniformly distributed stress, can be practi- cally realized ; but with ill-fitting joints or poor workmanship, or with a material that is not homogeneous, the resultant of the pull is liable to be thrown to one side of the line passing through the centres of gravity of the sections, and thus there 244 APPLIED MECHANICS. is set up a bending-action in addition to the direct tension, and therefore an unevenly distributed stress. It is of the greatest importance in practice to take cogni- zance of any such irregularities, and determine the greatest intensity of the stress to which the piece is subjected : though it is too often taken account of merely by means of a factor of safety ; in other words, by guess. Leaving, then, this latter case until we have studied the stresses due to bending, we will confine ourselves to the case of the uniformly distributed stress. If the total pull on the rod in the direction of its length be P, and the area of its cross-section A, we shall have, for the intensity of the pull, P On the other hand, if the working-strength of the material per unit of area be /, we shall have, for the greatest admissible load to be applied, P = fA. If / be the working-strength of the material per square inch, and E the modulus of elasticity, then is the greatest admissible strain equal to Thus, assuming 12000 Ibs. per square inch as the working tensile strength of wrought-iron, and 28000000 Ibs. per square inch as its modulus of elasticity, its working-strain would be 1 2000 28000000 7000 Hence the greatest safe elongation of the bar would be of its length. Hence a rod 10 feet long could safely be stretched ^ of a foot = 0.05 14". VALUES OF BREAKING AND WORKING STRENGTH. 245 169. Approximate Values of Breaking Strength, and of Modulus of Elasticity. In a later part of this book the attempt will be made to give an account of the experiments that have been made to determine the strength and elas- ticity of the materials ordinarily used in construction, in such a way as to enable the student to decide for himself, in any- special case, upon the proper values of the constants that he ought to use. For the present, however, the following will be given as a rough approximation to some of these quantities, which we may make use of in our work until we reach the above-mentioned account. (a) Cast-Iron. Breaking tensile strength per square inch, of common quali- ties, 14000 to 20000 Ibs. ; of gun iron, 30000 to 33000 Ibs. Modulus of elasticity for tension and for compression, about 17000000 Ibs. per square inch. (b) Wrought-Iron. Breaking tensile strength per square inch, from 40000 to* 60000 Ibs. Modulus of elasticity for tension and for compression, about 28000000. (c) Mild Steel. Breaking tensile strength per square inch, 55000 to 70000 Ibs. Modulus of elasticity for tension and for compression, from 28000000 to 30000000 Ibs. per square inch. (d) Wood. Breaking compressive strength per square inch : Oak, green 3000 Ibs. Oak, dry 3000 to 6000 Ibs. Yellow pine, green 3000 to 4000 Ibs. Yellow pine, dry 4000 to 7000 Ibs. 246 APPLIED MECHANICS. Modulus of elasticity for compression (average values) : Oak 1300000 Ibs. per square inch. Yellow pine 1600000 Ibs. per square inch. 170. Sudden Application of the Load. If a wrought- iron rod 10 feet long and I square inch in section be loaded with 12000 pounds in the direction of its length, and if the modulus of elasticity of the iron be 28000000, it will stretch 0.05 14" provided the load be gradually applied : thus, the rod begins to stretch as soon as a small load is applied ; and, as the load gradually increases, the stretch increases, until it reaches 0.05 14". If, on the other hand, the load of 12000 Ibs. be suddenly applied (i.e., put on all at once) without being allowed to fall through any height beforehand, it would cause a greater stretch at first, the rod undergoing a series of oscillations, finally settling down to an elongation of 0.05 14". To ascertain what suddenly applied load will produce at most the elongation 0.05 14", observe, that, in the case of the gradually applied load, we have a load gradually increasing from o to 12000 Ibs. Its mean value is, therefore, ^(12000) = 6000 Ibs. ; and this force descends through a distance of 0.05 14". Hence the amount of mechanical work done on the rod by the gradually applied load in producing this elongation is (6000) (0.0514) = 308.4 inch-lbs. Hence, if we are to perform upon the rod 308.4 inch-lbs. of work with a constant force, and if the stretch is to be 0.05 14", the magnitude of the force must be 308.4 0-0514" = 6000 Ibs. RESILIENCE OF A TENSION-BAR. 247 Hence a suddenly applied load will produce double the strain that would be produced by the same load gradually applied ; and, moreover, a suddenly applied load should be only half as great as one gradually applied if it is to produce the same strain. 171. Resilience of a Tension-Bar. The resilience of a tension-rod is the mechanical work done in stretching it to the same amount that it would stretch under the greatest allowable gradually applied load, and is found by multiplying the greatest allowable load by half the corresponding elongation. Thus, suppose a load of 100 Ibs. to be dropped upon the rod described above in such a way as to cause an elongation not greater than 0.05 14", it would be necessary to drop it from a height not greater than 3.08". EXAMPLES. 1 . A wrought-iron rod is 1 2 feet long and i inch in diameter, and is loaded in the direction of its length; the working-strength of the iron being 12000 Ibs. per square inch, and the modulus of elasticity 28000000 Ibs. per square inch. Find the working-strain. Find the working-load. Find the working-elongation. Find the working-resilience. From what height can a 5o-pound weight be dropped so as to produce tension, without stretching it more than the working- elongation? 2. Do the same fora cast-iron rod, where the working-strength is 5000 pounds per square inch, and the modulus of elasticity 17000000; the dimensions of the rod being the same. 172. Results of Wohler's Experiments on Tensile Strength. According to the experiments of Wohler, of which an account will be given later, the breaking-strength of a piece 248 APPLIED MECHANICS. depends, not only on whether the load is gradually or suddenly applied, but also on the extreme variations of load that the piece is called upon to undergo, and the number of changes to which it is to be submitted during its life. For a piece which is always in tension, he determines the following two constants ; viz., /, the carrying-strength per square inch, or the greatest quiescent stress . that the piece will bear, and u, the primitive safe strength, or the greatest stress per square inch of which the piece will bear an indefinite number of repetitions, the stress being entirely removed in the inter- vals. This primitive safe strength, u y is used as the breaking- strength when the stress varies from o to u every time. Then, by means of Launhardt's formula, we are able to determine the ultimate strength per square inch for any different limits of stress, as for a piece that is to be alternately subjected to 80000 and 6000 pounds. Thus, for Phoenix Company's axle iron, Wohler finds / = 3290 kil. per sq. cent. = 46800 Ibs. per sq. in., u 2100 kil. per sq. cent. = 30000 Ibs. per sq. in. Launhardt's formula for the ultimate strength per unit of area is t u least stress ) a = u{\ + u greatest stress)' Hence, with these values of t and u y we should have, for the ultimate strength per square inch, ( i least stress ) a 2100% i -h - ->kil. per sq. cent., ( 2 greatest stress) or ii least stress ) i -f- - -\ Ibs. per sq. in. 2 greatest stress) WOH LEWS EXPERIMENTS ON TENSILE STRENGTH. 249 Thus, if least stress = 6000, and greatest = 80000, we should have a = 30000$ i -f . -fo\ = 30000^1 + isul = 3 II2 5> if least stress = 60000, and greatest = 80000, a = 30000 | i -f i . f I = 30000^1 + fj = 41250; if least stress = greatest stress = 80000, a 30000^1 -j- \\ = 45000 = carrying-strength. Hence, instead of using, as breaking-strength per square inch in all cases, 45000, we should use a set of values varying from 45000 down to 30000, according to the variation of stress which the piece is to undergo. For working-strength, Weyrauch divides this by 3 : thus obtaining, for working-strength per square inch, ( i least stress ) b 10000 < i H [ Ibs. per sq. in. ; ( 2 greatest stress ) lor Krupp's cast-steel, notwithstanding the fact that Wohler finds / = 7340 kil. per sq. cent. = 104400 Ibs. per sq. in., u = 33 kil. per sq. cent. = 46900 Ibs. per sq. in., Weyrauch recommends ( q least stress ) a 3300 \ i + - -Vkil. per sq. cent., ( ii greatest stress) ( o least stress ) a = 46000 < i 4- >lbs. per sq. in., ( ii greatest stress) = I5633J o least stress ) i 4- - ; - - > Ibs. per sq. in. 1 1 greatest stress j EXAMPLES. Find the breaking-strength per square inch for a wrought-iron tension rod. 1. Extreme loads are 75000 and 6000 Ibs. 2. Extreme loads are 120000 and 100000 Ibs. 3. Extreme loads are 300000 and 10000 Ibs. Find the safe section for the rod in each case. 250 APPLIED MECHANICS. 173. Suspension-Rod of Uniform Strength. In the case of a long suspension-rod, the weight of the rod itself some- times becomes an important item. The upper section must, of course, be large enough to bear the weight that is hung from the rod plus the weight of the rod itself ; but it is sometimes desirable to diminish the sections as they descend. This is often accomplished in mines by making the rod in sections, each section being calculated to bear the weight below it plus its own weight. Were the sections gradually diminished, so that each section would be just large enough to support the weight below it, we should, of course, have a curvilinear form ; and the equation of this curve could be found as follows, or, rather, the area of any section at a distance from the bottom of the rod. Let W = weight hung at O (Fig. 152), Let w = weight per unit of volume of the rod, Let x = distance AO, Let 5 = area of section A, Let x + dx = distance BO, Let 5 + dS = area of section at B t Let f = working-strength of the mate- rial per square inch. i. The section at O must be just large enough to sustain the load W; *ff ' W FIG. 152. * S Q = f- 2. The area in dS must be just enough to sustain the weight of the portion of the rod between A and B. The weight of this portion is wSdx ; _ wSdx .'. <>= dS w iv .*. -= = -fdx /. log, S = -7X H- a constant > / / CYLINDERS SUBJECTED TO INTERNAL PRESSURE. .2$ I W When x = o, 5 = -^ ; W IW\ w .-. log* Y = the constant .% log,.S log,( -y- ) = -^ This gives us the means of determining the area at any dis- tance x from O. EXAMPLES. 1. A wrought-iron tension-rod 200 feet long is to sustain a load of 2000 Ibs. with a factor of safety of 4, and is to be made in 4 sections, each 50 feet long; find the diameter of each section, the weight of the wrought iron being 480 Ibs. per cubic foot. 2. Find the diameter needed if the rod were made of uniform section, also the weight of the extra iron necessary to use in this case. 3. Find the equation of the longitudinal section of the rod, assum- ing a square cross-section, if it were one of uniform strength, instead of being made in 4 sections. 174. Thin Hollow Cylinders subjected to an Internal Normal Pressure. Let/ denote the uniform intensity of the pressure exerted by a fluid which is confined within a hollow cylinder of radius r and of thickness / (Fig, 153), the thickness being small compared with the radius. Let us consider a unit of length of the cylinder, and c ( let us also consider the forces acting on the upper half-ring CED. PIG. 153. The total upward force acting on this half-ring, in conse- quence of the internal normal pressure, will be the same as that acting on a section of the cylinder made by a plane pass- ing through its axis, and the diameter CD. The area of this 252 APPLIED MECHANICS. section will be 2r X I = 2r : hence the total upward force will be 2r X / = 2pr; and the tendency of this upward force is to cause the cylinder to give way at A and B, the upper part separating from the lower. This tendency is resisted by the tension in the metal at the sections AC and BD ; hence at each of these sections, there has to be resisted a tensile stress equal to \(2pr) pr. This stress is really not distributed uniformly throughout the cross-section of the metal ; but, inasmuch as the metal is thin, no serious error will be made if it be accounted as distributed uniformly. The area of each section, however, is t X i = / / therefore, if T denote the intensity of the tension in the metal in a tangential direction (i.e., the intensity of the hoop tension), we shall have Hence, to insure safety, T must not be greater than f, the working-strength of the material for tension ; hence, putting f-pr /- / , we shall have / = 7 as the proper thickness, when/ = normal pressure per square inch, and radius = r. The above are the formulas in common use for the deter- mination of the thickness of the shell of a steam-boiler ; for in that case the steam-pressure is so great that the tension induced by any shocks that are likely to occur, or by the weight of the boiler, is very small in comparison with that induced by the steam-pressure. On the other hand, in the case of an ordinary water-pipe, the reverse is the case. RESISTANCE TO DIRECT COMPRESSION. To provide for this case, Weisbach directs us to add to the thickness we should obtain by the above formulae, a constant minimum thickness. The following -are his formulae, d being the diameter in inches, / the internal normal pressure in atmospheres, and / the thickness in inches. For tubes made of Sheet-iron ......... /* = 0.00086 pd -f 0.12 Cast-iron ......... t = 0.00238^ -f- 0.34 Copper ....... . . . / = o.ooi48/^/ + 0.16 Lead .......... /= 0.00507^+ 0.21 Zinc ........... / = 0.00242 pd -+- 0.16 Wood .......... / = 0.03230^ -f- 1.07 Natural stone ........ /= 0.03690/^4- 1.18 Artificial stone ....... t = 0.05380/^4- 1.58 175. Resistance to Direct Compression. When a piece is subjected to compression, the distribution of the compressive stress on any cross-section depends, first, upon whether the resultant of the pressure acts along the line containing the cen- tres of gravity of the sections, and, secondly, upon the dimen- sions of the piece ; thus determining whether it will bend or not. In the case of an eccentric load, or of a piece of such length that it yields by bending, the stress is not uniformly distributed ; and, in order to proportion the piece, we must determine the greatest intensity of the stress upon it, and so proportion it that this shall be kept within the working-strength of the ma- terial for compression. Either of these cases is not a case of direct compres- sion. In the case of direct compression (i.e., where the stress over each section is uniformly distributed), the intensity of the stress is found by dividing the total compression by the area of the 254 APPLIED MECHANICS. section ; so that, if P be the total compression, and A the area of the section, and / the intensity of the compressive stress, On the other hand, if f is the compressive working-strength oi the material per square inch, and A the area of the section in square inches, then the greatest allowable load on the piece subjected to compression is The same remarks as were made in regard to a suddenly applied load and resilience, in the case of direct tension, apply in the case of direct compression. 176. Results of Wohler's Experiments on Compressive Strength. Wohler also made experiments in regard to pieces subjected to alternate tension and compression, taking, in the experiments themselves, the case where the metal is subjected to alternate tensions and compressions of equal amount. The greatest stress of which the piece would bear an indefi- nite number of changes under these conditions, is called the vibration safe strength, and is denoted by s. Weyrauch deduces a formula similar to that of Launhardt for the greatest allowable stress per unit of area on the piece when it is subjected to alternate tensions and compressions of different amounts. Thus, for Phoenix Company's axle iron, Wohler deduces / = 3290 kil. per sq. cent. = 46800 Ibs. per sq. in., u = 2100 kil. per sq. cent. = 30000 Ibs. per sq. in., s = 1 1 70 kil. per sq. cent. = 1 6600 Ibs. per sq. in. EXPERIMENTS ON COMPRESSIVE STRENGTH. 2$$ Weyrauch's formula for the ultimate strength per unit of area is {u s least maximum stress | u greatest maximum stress \ ' and, with these values of u and s, it gives least maximum stress a = 2100 !i least maximum stress | 1 ~ 2 greatest maximum stress j ' per S( ** cent * 11 least maximum stress ) i - = - / Ibs. per sq. in. 2 greatest maximum stress ) With a factor of safety of 3, we should have, for the greatest admissible stress per square inch, ( i least maximum stress ) b = i oooo < i : Jibs. ( 2 greatest maximum stress) For Krupp's cast-steel, / = 7340 kil. per sq. cent. = 104400 Ibs. per sq. in., u = 3300 kil. per sq. cent. = 46900 Ibs. per sq. in. approximately, s = 2050 kil. per sq. cent. = 29150 Ibs. per sq. in. approximately. We have, therefore, for the breaking-strength per unit of area, according to Weyrauch's formula, least maximum stress a or a !c least maximum stress ) - il greatest maximum stress } kiL per Sq ' Cent ' ( c least maximum stress ) - 4 6 9 oo| , - fi greatestmaximumstress [lbs. per sq. .; 256 APPLIED MECHANICS. and, using a factor of safety of 3, we have, for the greatest admis- sible stress per square inch, !r least maximum stress ) , i f: - : - > Ibs. per sq. in. 1 1 greatest maximum stress j b 15630 The principles respecting an eccentric compressive load, and those respecting the giving-way of long columns so far as they are known, can only be treated after we have studied the resist- ance of beams to bending; hence this subject will be deferred until that time. EXAMPLES. Find the proper working and breaking strength per square inch to be used for a wrought- iron rod, the extreme stresses being 1. 80000 Ibs. tension and 6000 Ibs. compression. 2. i ooooo Ibs. tension and 100000 Ibs. compression. 3. 70000 Ibs. tension and 60000 Ibs. compression. Do the same for a steel rod. 177. Resistance to Shearing. One of the principal cases where the resistance to shearing comes into practical use is that where the members of a structure, which are themselves subjected to direct tension or compression or bending, are united by such pieces as bolts, rivets, pins, or keys, which are sub- jected to shearing. Sometimes the shearing is combined with tension or with bending ; and whenever this is the case, it is necessary to take account of this fact in designing the pieces. It is important that the pins, keys, etc., should be equally strong with the pieces they connect. Probably one of the most important modes of connection is by means of rivets. In order that there may be only a shearing action, with but little bending of the rivets, the latter must fit very tightly. The manner in which the riveting is done will necessarily affect very essentially the strength of the joints; RESISTANCE TO SHEARING. 257 hence the only way to discuss fully the strength of riveted joints is to take into account the manner of effecting the rivet- ing, and hence the results of experiments. These will be spoken of later ; but the ordinary theories by which the strength and proportions of .some of the simplest forms of riveted joints are determined will be given, which theories are necessary also in discussing the results of experiments thereon. The principle on which the theory is based, in these simple cases, is that of making the resistance of the joint to yielding equal in the first three, and also in either the fourth or the fifth of the ways in which it is possible for it to yield, as enumerated on pages 258 and 259. A single-riveted lap-joint is one with a single row of rivets, as shown in Fig. 154. A single-riveted butt-joint with covering plate is shown in T C C one Fig. 155 A single-riveted butt-joint with two covering, plates is shown in Fig. 156. FIG. 154. FIG. 155. FIG. 156. 25 8 APPLIED MECHANICS. FIG. 157. FIG. 158. A double-riveted lap-joint with the rivets staggered is shown in Fig. 157; one with chain riveting, in Fig. 158. Taking the case of the single-riveted lap-joint shown in Fig. 1 54, it may yield in one of five ways : i. By the crushing of the plate in front of the rivet (Fig. 159). FIG. 159. FIG. 160. 2. By the shearing of tne nvet (Fig. 160). RESISTANCE TO SHEARING. 259 3. By the tearing of the plate between the rivet-holes (Fig. 161). 1 FIG. 161. 4. By the rivet breaking through the plate (Fig. 162). 5. By the rivet shearing out the plate in front of it. Let us call d the diameter of a rivet. / the pitch of the rivets ; i.e., FlG - l62 - their distance apart from centre to centre. t the thickness of the plate. / the lap of the plate ; i e., the distance from the centre of a rivet-hole to the outer edge of the plate. f t the ultimate tensile strength of the iron. /, the ultimate shearing-strength of the rivet-iron. f s > the ultimate shearing-strength of the plate. f c the ultimate crushing-strength of the iron. We shall then have i. Resistance of plate in front of rivet to crushing =r f c td. 2. Resistance of one rivet to shearing = //- Y 3. Resistance of plate between two rivet-holes to tearing = /'(/ - d). 4. Resistance of plate to being broken through = a~ , d where a is a constant depending on the material, taken as empirical for the present. A reasonable value of this constant is /". This may be 260< APPLIED MECHANICS. 5. Resistance of plate in front of the rivet to shearing = 2/,7/. Assuming that we know the thickness of the plate to start with, we obtain, by equating the first two resistances, which determines the diameter of the rivet. Equating 3 and 2, we obtain which gives the pitch of the rivets in terms of the diameter of the rivet, and the thickness of the plate. Equating, next, 4 and i, we have which gives the lap of the plate needed in order that it may not break through. By equating 5 and i, we find the lap needed that it may not shear out in front of the rivet. A similar method of reasoning would enable us to determine the corresponding quantities in the cases of double-riveted joints, etc. There are a number of practical considerations which modify more or less the results of such calculations, and which can only be determined experimentally. A fuller account of this subject from an experimental point of view will be given later. 178. Intensity of Stress. Whenever the stress over a plane area is uniformly distributed, we obtain its intensity at each point by dividing the total stress by the area over which it acts, thus obtaining the amount per unit of area. When, how- ever, the stress is not uniformly distributed, or when its inten- INTENSITY OF STRESS. 26l sity varies at different points, we must adopt a somewhat differ- ent definition of its intensity at a given point. In that case, if we assume a small area containing that point, and divide the stress which acts on that area by the area, we shall have, in the quotient, an approximation to the intensity required, which will approach nearer and nearer to the true value of the intensity at that point, the smaller the area is taken. Hence the intensity of a variable stress at a given point is, -- The limit of the ratio of the stress acting on a small area containing that point, to the area, as the latter grows smaller and smaller. By dividing the total stress acting on a certain area by the entire area, we obtain the mean intensity of the stress for the entire area. 179. Graphical Representation of Stress A conven- ient mode of representing stress graphically is the following: Let AB (Fig. 163) be the plane surface upon which the stress acts ; let the axes OX and OY be taken in this plane, the axis OZ being at right angles to the plane. Conceive a portion of a cylinder whose elements are all parallel to OZ, bounded at one end by the given plane surface, and at the other by a surface whose ordinate many units of length as there are units of force in the intensity of the stress at that point of the given plane surface where the ordinate cuts it. The volume of such a figure will evidently be V = ffzdxdy = ffpdxdy, where z / = intensity of the stress at the given point. FIG. 163. at any point contains as 262 APPLIED MECHANICS. Hence the volume of the cylindrical figure will contain as many units of volume as the total stress contains units of force ; or, in other words, the total stress will be correctly repre- sented by the volume of the body. If the stress on the plane figure is partly tension and partly compression, the sur- face whose ordinates repre- sent the intensity of the stress will lie partly on one side of the given plane sur- face and partly on the other ; this surface and the plane surface on which the stress acts, cutting each other in some line, straight or curved, as shown in Fig. 164. In that FlG - case, the magnitude of the resultant stress P V will be equal to the difference of the wedge-shaped volumes shown in the figure. It will be observed that the above method of representing stress graphically represents, i, the intensity at each point of the surface to which it is applied ; and, 2, the total amount of the stress on the surface. It does not, however, represent its direction, except in the case when the stress is normal to the surface on which it acts. In this latter case, however, this is a complete representa- tion of the stress. The two most common cases of stress are, i, uniform stress, and, 2, uniformly varying stress. These two cases are repre- sented respectively in Figs. 165 and 166; the direction also being correctly represented when, as is most frequently the case, the stress is normal to the surface of action. In Fig. 165, AB is supposed to be the surface on which the stress GRAPHICAL REPRESENTATION OF STRESS. 263 acts ; the stress is supposed to be uniform, and normal to the surface on which it acts ; the bound- ing surface in this case becomes a plane parallel to AB ; the intensity of the stress at any point, as P, will be represented by PQ; while the whole cylinder will contain as many units of volume as there are units of force in the whole stress. Fig. i(56 represents a uniformly varying stress. Here, again, AB is the surface of action, and the stress is supposed to vary at a uniform rate FlG> l65 ' from the axis O Y. The upper bounding surface of the cylin- drical figure which represents the stress becomes a plane inclined to the XOY plane, and containing the axis O Y. In this case, if a represent the in- tensity of the stress at a unit's distance from O Y, the stress at a distance x from OY will be/ = ax, and the total amount of the stress will be FIG. 166. P = ffpdxdy = affxdxdy. When a stress is oblique to the surface of action, it may be represented correctly in all particulars, except in direction, in the above-stated way. 1 80. Centre of Stress. The centre of stress, or the point of the surface at which the resultant of the stress acts, often becomes a matter of practical importance. If, for con- venience, we employ a system of rectangular co-ordinate axes, of which the axes OX and OY are taken in the plane of the surface on which the stress acts, and if we let p = $(x, y) be 264 APPLIED MECHANICS. the intensity of the stress at the point (x, y), we shall have, for the co-ordinates of the centre of stress, ffxpdxdy J'Sypdxdy : (see 42), where the denominator, or ffpdxdy, represents the total amount of the stress. When the stress is positive and negative at different parts of the surface, as in Fig. 164, the case may arise when the posi- tive and negative parts balance each other, and hence the stress on the surface constitutes a statical couple. In that case Sfpdxdy = o. 181. Uniform Stress. In the case of uniform stress, we have i. The intensity of the stress is constant, or / = a con- stant. 2. The volume which represents it graphically becomes a cylinder with parallel and equal bases, as in Fig. 165. 3. The centre of stress is at the centre of gravity of the surface of action ; for the formulae become, when / is constant, _ pffxdxdy _ ffxdxdy _ Xl ~~~ pffdxdy ~~~' ~~ pffydxdy = pffdxdy ~' ffdxdy ' "where x , y , are the co-ordinates of the centre of gravity of the surface. Examples of uniform stress have already been given in the cases of direct tension, direct compression, and, in the case of riveted joints, for the shearing-force on the rivet. UNIFORMLY VARYING STRESS. 26$ 182. Uniformly Varying Stress. Uniformly varying stress has already been denned as a stress whose intensity varies uniformly from a given line in its own plane ; and this line will be called the Neutral Axis. Thus, if the plane be taken as the XOY plane (Fig. 166), and the given line be taken as OY, we shall have, if a denotes the intensity of the stress at a unit's distance from OY, and x the distance of any special point from O Y, that the intensity of the stress at the point will be p = ax. The total amount of the stress will be P= affxdxdy. The total moment of the stress about O Y will be found by multiplying each elementary stress by its leverage. This lever- age is, in the case of normal stress, x ; hence in that case the moment of any single elementary force will be and the total moment of the stress will be M - affx^dxdy = al. In the case of oblique stres x s, this result has to be modified, as the leverage is no longer x. Confining ourselves to stress normal to the plane of action, we have, for the co-ordinates of the centre of stress, _ ffpxdxdy _ affx*dxdy _ ffx^dxdy _ ffx*dxdy_ I ffpdxdy ~~ P = ffxdxdy '' x Q A ~ x A _ ffpydxdy _ affxydxdy _ ffxydxdy _ ffxydxdy ~~ ffpdxdy ~~ P ~~ ffxdxdy = x*A since P = affxdxdy = aXoA, where x m y m are the co-ordinates of the centre of gravity, and A is the area of the surface of action. 266 APPLIED MECHANICS. 183. Case of a Uniformly Varying Stress which amounts to a Statical Couple. Whenever P = o, we have affxdxdy = o /. ffxdxdy = o .'. x^A = o .*. x = o. In this case, therefore, we have i. There is no resultant stress, and hence the whole stress amounts to a statical couple. 2. Since X Q = o, the centre of gravity of the surface of action is on the axis OY, which is the neutral axis. Hence follows the proposition : When a uniformly varying stress amounts to a statical couple, the neutral axis contains (passes through) the centre of gravity of the surface of action. In this case there is no .single resultant of the stress ; but the moment of the couple will be, as has been already shown, M = affx 2 dxdy. 184. Example of Uniformly Varying Stress. One of the most common examples of uniformly varying stress is that of the pressure of water upon the sides of the vessel contain- ing it. Thus, let Fig. 167 represent the vertical cross-section of a reservoir wall, the water pressing against the vertical face AB. It is a fact established by experiment, that the intensity of the pressure of any body of water at any point is propor- tional to the depth of the point below the free upper level of the water, and normal to the surface pressed upon. Hence, if we sup- pose the free upper level of the water to be even with the top of the wall, the intensity of the pressure there will be zero ; and if we represent by CB the intensity of the pressure at the bottom, then, joining^ and STRESSES IN BEAMS UNDER TRANSVERSE LOAD. 267 C we shall have the intensity of the pressure at any point, as D t represented by ED, where ED : CB = AD : AB. Here, then, we have a case of uniformly varying stress nor- mal to the surface on which it acts. 185. Fundamental Principles of the Common Theory of the Stresses in Beams under a Transverse Load. Fig. 168 shows a beam fixed at one end and loaded at the other, while Fig. 169 shows a beam supported at the ends and loaded at the middle. Let, in each case, the plane of the paper contain a vertical longi- tudinal section of the beam. In Fig. 1 68, it is evi- dent that the upper fibres are lengthened, while the lower ones are shortened, and vice versa in Fig. 169. In either case, there is, somewhere between the upper and lower fibres, a fibre which is neither elongated nor com- pressed. Let CN repre- sent that fibre, Fig. .168, and CP, Fig. 169. This line may be called the neutral FIG. i6g. line of the longitu- dinal section ; and, if a section be made at any point at right 268 APPLIED MECHANICS. angles to this line, the horizontal line which lies in the cross- section, and cuts the neutral lines of all the longitudinal sec- tions, or, in other words, the locus of the points where the neutral lines of the longitudinal sections cut the cross-section, is called the Neutral Axis of the cross-section. In the ordinary theory of the stresses in beams, a number of assumptions are made, which will now be enumerated. ASSUMPTIONS MADE IN THE COMMON THEORY OF BEAMS. ASSUMPTION No. I. If, when a beam is not loaded, a plane cross-section be made, this cross-section will still be a plane after the load is put on, and bending takes place. From this assumption, we deduce, as a consequence, that, if a certain cross-section be assumed, the elongation or shortening per unit of length of any fibre at the point where it cuts this cross-sec- tion, is proportional to the distance of the fibre from the neutral axis of the cross-section. Proof. Imagine two originally parallel cross-sections so near to each other that the curve in which that part of the neutral line between them bends may, without appreciable error, be accounted circular. Let ED and GH (Fig. 168 or Fig. 169) be the lines in which these cross-sections cut the plane of the paper, and let O be the point of intersection of the lines ED and GH. Let OF = r, FL = y, FK = /, LM = / + a/, in which a is the strain or elongation per unit of length of a fibre at a distance y from the neutral line, y being a variable ; then, because FK and LM are concentric arcs subtending the same angle at the centre, we shall have the proportion r + y I -\- ol y ^ = -y- or i + a = i + y .'.a = - Or a = r ASSUMPTIONS IN THE COMMON THEORY OF BEAMS. 269 but as y varies for different points in any given cross-section, while r remains the same for the same section, it follows, that, if a certain cross-section be assumed, the strain of any fibre at the point where it cuts this cross-section is proportional directly to the distance of this fibre from the neutral axis of the cross- section. ASSUMPTION No. 2. This assumption is that commonly known as Hooke s Law. It is as follows : " Ut tensio sic vis ; " i.e., The stress is proportional to the strain, or to the elonga- tion or compression per unit of length. As to the evidence in favor of this law, experiment shows, that, as long as the mate- rial is not strained beyond safe limits, this law holds. Hence, making these two assumptions, we shall have : At a given cross-section of a loaded beam, the direct stress on any fibre varies directly as the distance of the fibre from the neutral axis. Hence it is a uniformly varying stress, and we may repre- sent it graphically as follows : Let ABCD, Fig. 170, be the cross-sec- tion of a beam, and KL the neutral axis. Assume this for axis OY, and draw the other two axes, as in the figure. If, now, EA be drawn to represent the intensity of the direct (normal) stress at A, then will the pair of wedges AEFBKL and DCHGKL represent the stress graphically, since it is uni- formly varying. POSITION OF NEUTRAL AXIS. ASSUMPTION No. 3. It will next be shown that, on the two assumptions made above, and from the further assumption that the deformation of each fibre of the beam parallel to its* longitudinal axis is due to the forces acting on its ends FIG. 170. 2~0 APPLIED MECHANICS. and that it suffers no traction from neighboring fibres, it fol- lows that the neutral axis must pass through the centre of gravity of the cross-section. D N 1 1 |I C E 1 A A iff * ""vi B ' FIG. 171. FIG. 172. Since the curvatures in Figs. 168 and 169 are exaggerated in order to render them visible, Figs. 171 and 172 have been drawn. If, now, we assume a section DE, such that AD = x (Fig. 171) and NE = x (Fig. 172), and consider all the forces acting on that part of the beam which lies to the right of DE (i.e., both the external forces and the stresses which the other parts of the beam exert on this part), we must find them in equilibrium. The external forces are, in Fig. 172, i. The loads acting between B and E ; in this case there are none. 2. The supporting force at B ; in this case it is equal to W , and acts vertically upwards. In Fig. 171 they are, The loads between D and N ' ; in this case there is only the one, W at N. The internal forces are merely the stresses exerted by the other parts of the beam on this part : they are, i. The resistance to shearing at the section, which is a vertical stress. 2. The direct stresses, which are horizontal. Now, since the part of the beam to the right of DE is at rest, the forces acting on it must be in equilibrium ; and, since POSITION OF NEUTRAL AXIS. 2 7 l they are all parallel to the plane of the paper, we must have the three following conditions ; viz., i. The algebraic sum of the vertical forces must be zero. 2. The algebraic sum of the horizontal forces must be zero. 3. The algebraic sum of the moments of the forces about any axis perpendicular to the plane of the paper must be zero. But, on the above assumptions, the only horizontal forces are the direct stresses : hence the algebraic sum of these direct stresses must be zero ; or, in other words, the direct stresses must be equivalent to a statical couple. Now, it has already been shown, that, whenever a uniformly varying stress amounts to a statical couple, the neutral axis must pass through the centre of gravity of the surface acted upon. Hence in a loaded beam, if the three preceding assump- tions be made, it follows that the neutral axis of any cross- section must contain the centre of gravity of that section. By way of experimental proof of this conclusion, Barlow has shown by experiment, that, in a cast-iron beam of rectangu- lar section, the neutral axis does pass through the centre of gravity of the section. RESUME. The conclusions arrived at from the foregoing are as fol- lows : i. That at any section of a loaded beam, if a horizontal line be drawn through the centre of gravity of the section, then the fibres lying along this line will be subjected neither to tension nor to compression ; in other words, this line will be the neutral axis of the section. 2. The fibres on one side of this line will be subjected to tension, those on the other side being subjected to compres- sion ; the tension or compression of any one fibre being proper tionai to its distance from the neutral axis. 2/2 APPLIED MECHANICS. The first of the three assumptions of the common theory was not accepted by St. Venant, who developed by means of the methods of the Theory of Elasticity a theory of beams based upon the second and third assumptions only. A study of. St. Venant's theory involves, however, far more complica- tion, and requires a good previous knowledge of the Theory of Elasticity. Moreover the results of the two theories as far as the determination of the outside fibre-stresses and of the de- flections are practically in agreement, while, on the other hand, the intensities of the shearing-forces as computed by the two theories are not in agreement. The St. Venant theory may be found in several treatises upon the Theory of Elasticity. 1 86. Shearing-Force and Bending-Moment. In deter- mining the strength of a beam, or the proper dimensions of a beam to bear a certain load, when we assume the neutral axis to pass through the centre of gravity of the cross-section, we have imposed the second of the three last-mentioned conditions of equilibrium. The remaining two conditions may otherwise be stated as follows : i. The total force tending to cause that part of the beam that lies to one side of the section to slide by the other part, must be balanced by the resistance of the beam to shearing at the section. 2. The resultant moment of the external forces acting on that part of the beam that lies to one side of the section, about a horizontal axis in the plane of the section, must be balanced by the moment of the couple formed by the resisting stresses. The shearing-force at any section is the force with which the part of the beam on one side of the section tends to slide by the part on the other side. In a beam free at one end, it is equal to the sum of the loads between the section and the free end. In a beam supported at both ends, it is equal in magnitude to the difference between the supporting force at either end, 'and the sum of the loads between the section and that support. SHEARING-FORCE AND BEND ING-MOMENT. 2? 3 The bending-moment at any section is the resultant moment of the external forces acting on the part of the beam to one side of the section, these moments being taken about a hori- zontal axis in the section. In a beam free at one end, it is equal to the sum of the moments of the loads between the section and the free end, about a horizontal axis in the section. In a beam supported at both ends, it is the difference be- tween the moment of either supporting force, and the sum of the moments of the loads between the section and that sup- port ; all the moments being taken about a horizontal axis in the section. Hence the two conditions of equilibrium may be more briefly stated as follows : i. The shearing-force at the section must be balanced by the resistance opposed by the beam to shearing at the section. 2. The bending-moment at the section must be balanced by the moment of the couple formed by the resisting stresses. It is necessary, therefore, in determining the strength of a beam, to be able to determine the shearing-force and bending- moment at any point, and also the greatest shearing-force and the greatest bending-moment, whatever be the loads. A table of these values for a number of ordinary cases will now be given ; but I should recommend that the table be merely considered as a set of examples, and that the rules already given for finding them be followed in each individual case. Let, in each case, the length of the beam be /, and the total load W. When the beam is fixed at one end and free at the other, let the origin be taken at the fixed end ; when it is supported at both ends, let it be taken directly over one support. Let x be the distance of any section from the origin. Then we shall have the results given in the following table : 274 APPLIED MECHANICS. At Dista from O ifeh. ft Distance m Origin. ' " pq Si I T? & g '-3 >, IS 11 4'g ! It W iJ .g r- 4> u PQ MOMENTS OF INERTIA OF SECTIONS. 2/5 In a beam fixed at one end and free at the other, the great- est shearing-force, and also the greatest bending-moment, are at the fixed end. In a beam supported at both ends, and loaded at the middle, or with a uniformly distributed load, the greatest shearing-force is at either support, the greatest bending-moment being at the middle. In the last case (i.e., that of a beam sup- ported at the ends,- and having a single load not at the middle),. the greatest bending-moment is at the load ; the greatest shear- ing-force being at that support where the supporting force is greatest. 187. Moments of Inertia of Sections. In the usual methods of determining the strength of a beam or column, it is necessary to know, i, the distance from the neutral axis of the section to the most strained fibres ; 2, the moment of in- ertia of the section about the neutral axis. The manner of finding the moments of inertia has been explained in Chap. II. In the following table are given the areas of a large number of sections, and also their moments of inertia about the neutral axis, which is the axis YY in each case. These results should be deduced by the student. 276 APPLIED MECHANICS. Distance of YY from Extreme Fibres. S. HP | 1! II MOMENTS OF INERTIA OF SECTIONS. 277 fa ^r S- SS 5ia 5lx 13 w rt -o CQ 278 APPLIED MECHANICS. o a 1 rt X w W Q vO f I I -S -8 S 3 ^ ^H ^ > MH 55 "5 g -s ^ bfl tfl ^ bo !* MOMENTS OF INERTIA OF SECTIONS. '279 + + 4- ^ IN *?! II- ^ I H ^ c/T > | -S'S 4 s,-^ 's ^ s N 2 " s * * K if fi ii + S o < H o '^3 rt oS rt Vj (U (L) o i- >_ i- JA, ^ U 280 APPLIED MECHANICS. o g 4> c SJIN u II ** rj- fp 1 8" V V. II II i2 x a J5 131 ' 11 I '~ rt i MOMENTS OF INERTIA OF SECTIONS. 28l 1 -^1 ^ 8 8 u 3 1 V* - 1 . - 1* * ^ "5 <u i; -G "-Q rt K ~^ s w f 1 T5 yj <U G nj w ^J o o ^ III C s !*l rt bJO c/3 -' I 1 ^ fi c ^ '1 ^ j^ o + I' (J^ ^.| ^J 1|l 8 S -g 1 " <U i^ O N sli ^ ^ ^ ^ I, O ^^"^ CT3 ||) g|) | | ||||^ c/3 '53 ^ ^J ^J ^^ J *5"^ s ^ n G 3 g ^v ^^ ^ w to 2 ^ "^ -2 ""* 'ft oj^i 4)^i<u t>rt ^-t-> ^3'^g <U O G -G o,^ 4-> 1 ~ .2 " '?:! '-= ^ ^ | ^ f X rt<U ^rtS ^"3^ c3 2i "P to <u w . Mo'' tt '3.2'i'aow "Sec 'Sex 'J3-Qo o'xuc^w j-jrtO _j,j rt *- j-.rt*. ^rtortofi "rt *hO ^ C/2 C/2 C/2 U >> G O "-> N ro bJO C JU 2" 2" 2 s 2" ^ O C4 jS>- < crj *"O o>- \ ^ 'o ^ c w <---i \ m ? ^' s - r >. <U O 00 ^ 1 V i x fe % x .X3 J3 .t3 ;t-|| J- L-N "FT ^ "5 ^o >fo 282 APPLIED MECHANICS. .5 + C <L> T3 rt X! fi it!! ^ P S <u . M_ S M U) rt e 3 c a S .g rt T3 ^1 ^ oq .5P - *- u ,e <u < bfl H O 13 C 1-. O nj . .22 - MOMENTS OF INERTIA OF SECTIONS. 28 3 2 a aj **-i w O V k|M CT 1 Cfl M O CJ Sh II fe ! ii I 5 st: VM " O fi si S 3 ^ 284 APPLIED MECHANICS. o o D u H* a u <u a is rt x S5 W Q + ^ I . ^ k w S "i '-3 1 < 1 3 C 42 - MOMENTS OF INERTIA OF SECTIONS. 285 .< a K JO a rl IN s | N s I N <i I ( II II II H ripti -1 1 -1 z I Egg < U J < U in Oq ?< Q CQ > < : i 286 APPLIED MECHANICS. 188. Cross-Sections of Phoenix Columns considered as made of Lines. It is to be observed that the moments of inertia are the same for all axes passing through the centre. Thickness /, radius of round ones = r, area of each flange = a, length of each flange = /. Figure. Description. A. Y 2 II Y2I2 Four flanges 2-nrt + 20. Eight flanges 2irrt + 8a (-0 Square, four flanges, r radius of cir- cumscribed circle Six flanges 6a REPRESENTATION OF B ENDING-MOMENTS. 287 189. Graphical Representation of Bending-Moments. The bending-moment at each point of a loaded beam may be represented graphically by lines laid off to scale, as will be shown by examples. I. Suppose we have the cantilever shown in Fig. 215, loaded at D with a load W ': then will the bending-moment at any section, as at F t be obtained by multiplying W by FD ; that at AC being W X (AB). If, now, we lay off CE to scale to represent this, i.e., having as many units of length FIG. 215. as there are units of moment in the product W X (AB), and join E with D, then will the ordinate FG of any point, as G, represent (to the same scale) the bending-moment at a section through F. II. If we have a uniformly distributed load, we should have, for the line corresponding to CE in Fig. 215, a curve. This is shown in Fig. 216, where we have the uniformly distributed load EIGF. If we take the origin at D, as before, we have, for the bending-moment, at a distance ^ from the origin, as has been W shown, -(/ xY ; and by giving x dif- ferent values, and laying off the corresponding value of the bending-moment, we obtain the curve CA, any ordinate of which will represent the bending-moment at the corresponding point of the beam. When we have more than one load on a beam, we must draw the curve of bending-moments for each load separately, and then find the actual bending-moment at any point of the beam FIG. 216. 283 APPLIED MECHANICS. by taking the sum of the ordinates (drawn from that point) of each of these separate curves or straight lines. If we then draw a new curve, whose ordinates are these sums, we shall have the actual curve of bending-moments for the beam as loaded. Some examples will now be given, which will explain them- selves. III. Fig. 217 shows a cantilever with three concentrated loads. The line of bending-moments for the load at C is CE, that for the load at O is OF, and for the load at P is PG. They are combined above the beam by laying off AH DE, HK = DF, and KL = DG, and thus obtaining the broken line LMNB, which is the line of bending-moments of the beam loaded with all three loads. FIG. 217. IV. Fig. 218 shows the case of a beam supported at both ends, and loaded at a single . point D; ALB is the line of bending- moments when the weight of the beam is disregarded, so that xy = bending-moment at x. FIG. 218 V. Fig. 219 shows the case of a beam supported at the ends, and loaded with three concentrated loads at the points B, C, and D re- spectively; the lines of bending-mo- ments for each individual load being respectively AFE, AGE, and AHE, FIG. 219. and the actual line of bending-mo- ments being AKLME. REPRESENTATION OF BEND ING-MOMENTS. 289 VI. Fig. 220 shows the case of a beam supported at the ends, and loaded with a uniformly dis- A E F B tributed load ; the line of bending- moments being a curve, ACDB, as shown in the figure. FlG . 220> VII. In Fig. 221 we have the case of a beam, over a part of which, viz., EF, there is a distributed load ; the rest of the beam being unloaded. The line of bending-moments is curvilinear be- tween E and F, and straight outside Xjs^/tt y^ of these limits. It isAGSHB; and, when the curve is plotted, we can N/ find the greatest bending-moment graphically by finding its greatest ordinate. We can also determine it analytically by first determining the bending- moment at a distance x from the origin, and on the side towards the resultant of the load, and then differentiating. This process is shown in the following: Let A (Fig. 222) be the point where the resultant of the load acts, and O the middle of the beam, and let w be the c OA B load per unit of length ; let OA = a, AB = AC = b, and ED = 2^, so that the whole load = 2wb : there- a A- c wb(a 4- c] fore supporting force at D = 2ivb = - -. If we take a section at a distance x from O to the right, we shall have, for the bending-moment at that section, wb(a -+ c) w (c x) (a -f b x) 2 = a maximum. Differentiate, and we have wb(a -\- c\ a(c &) I tioi ( /T j h */\ /~\ * c T V -r ; ... - c 290 APPLIED MECHANICS. hence the greatest bending-moment will be ^ \ C ) 2 \ C ) 7 ( a ac VIII. In Figs. 223 and 224 we have the case of a beam supported at the ends, and 1 J J <.!_ loaded with a uniformly dis- tributed load, and also with a c o n c e n- trated load. In the first Fi G .2 24 . FlG - 22 3- figure, the greatest bending-moment is at/?, and in the second at C. IX. In Fig. 225 we have a beam supported at A and B, and loaded at C and D with equal weights; the lengths of AC and BD being equal. We have, con- sequently, between A and , a uniform bending-moment ; while on the left of A and on the right The line of bending- FlG - 225 ' of B we have a varying bending-moment. moments is, in this case, CabD. We may, in a similar way, derive curves of bending-moment for all cases of loading and supporting beams. AT DIFFERENT PARTS OF A BEAM. 2QT 190. Mode of Procedure for Ascertaining the Stresses^ at Different Parts of a Beam when the Loads and the Di- mensions are given, and when no Fibre at the Cross- section under Consideration is Strained beyond the Elastic Limit. When the dimensions of a beam, the load and its distribution, and the manner of supporting are given, and it is desired to find the actual intensity of the stress on any particular fibre at any given cross-section, we must pro- ceed as follows :-- i. Find the actual bending-moment (M) at that cross-sec- tion. 2. Find the moment of inertia (/) of the section about its neutral axis. 3. Observe, that, from what has already been shown, the moment of the couple formed by the tensions and compressions is al, where a = intensity of stress of a fibre whose distance from the neutral axis is unity, and that this moment must equal the bending-moment at the section in order to secure equilib- rium. Hence we must have Moreover, if / denote the (unknown) intensity of the stress of the fibre where the* stress is desired, and if y denote the distance of this fibre from the neutral axis, we shall have from which equation we can determine/. EXAMPLES. i. Given a beam 18 feet span, supported at both ends, and loaded uniformly (its own weight included) with 1000 Ibs. per foot of length. The cross- section is a T, where area of flange = 3 square inches, area of web = 4 square inches, height = 10 inches. Find (a) the 292 APPLIED MECHANICS. bending-moment at 3 feet from one end ; (b) the greatest bending- moment; (c) the greatest intensity of the tension at each of the above sections ; (d) the greatest intensity of the compression at each >of these sections. 2. Given an I-beam with equal flanges, area of each flange = 3 square inches, area of web = 3 square inches, height = 10 inches; the beam is 1 2 feet long, supported at the ends, and loaded uniformly (its own weight included) with a load of 2000 Ibs. per foot of length. Find J(a) the bending-moment at a section one foot from the end ; (<) the greatest bending-moment ; (<r) the greatest intensity of the stress at each of the above cross-sections. 191. Mode of Procedure for Ascertaining the Dimen- sions of a Beam to bear a Certain Load, or the Load that a Beam of Given Dimensions and Material is Capable of Bearing. If we wish to determine the* proper dimensions of the beam when the load and its distribution, as well as the manner of supporting, are given, so that it shall nowhere be strained beyond safe limits, or if we wish to determine the greatest load consistent with safety when the other quantities are given, we must impose the condition that the greatest intensity of the tension to which any fibre is subjected shall not exceed the safe working-strength for tension of the mate- rial of which the beam is made, and the greatest intensity of the compression to which any fibre is subjected shall not exceed the safe working-strength of the material (or compression. Thus, we must in this case first determine where is the section of greatest bending-moment (this determination some- limes involves the use of the Differential Calculus). Next we must determine the magnitude of the greatest bending-moment, absolutely if the load and length of the beam are given (if not, in terms of these quantities), and then equate this to the moment of the resisting couple. Thus, if MQ is the greatest bending-moment, when the loads are such that no fibre is strained beyond the elastic limit, 7 the WORKING-STRENGTH. . 2$$ moment of inertia of that section where this greatest bending-moment acts, and if } t = greatest tensile fibre stress per square inch, f c = greatest compressive fibre strength per square inch, y t = distance of most stretched fibre from the neutral axis, and y c = distance of most compressed fibre from the neutral axis, then will be yt the greatest tension per square inch, at a unit's distance from the neutral axis, and the greatest compression per square inch, at a unit's distance from the neutral axis. Moreover, in this case, these two ratios are equal, and hence l/f f* T ? c T MQ = I = I. yt y c SAFE OR WORKING-LOAD. If // = safe working-strength per square inch for tension, / c '=safe working-strength per square inch for compression, and M Q = greatest safe working bending-moment, then the ratios, and , are not equal. yt yc f r f ' f ' Hence, when is less than we have M o' = /, and when. yt y c yt It' ]c fc is greater than we have MQ =I. yt y c y c BREAKING-LOAD AND MODULUS OF RUPTURE. If M is the greatest bending-moment when the beam is subjected to its breaking-load, the formulae given above do not apply, inasmuch as a portion of the fibres are strained beyond the elastic limit, and Hooke's law no longer holds, since, after the elastic limit is passed, the ratio of stress to strain decreases when the stress increases. Indeed, the stresses in the different fibres are no longer pro- 294 APPLIED MECHANICS. portional to the distances of those fibres from the neutral axis. A graphical representation of the stress at different points of any given section AB would be of the character shown in the figure, o , the form of the curve CDE varying with the shape of the cross-section. Nevertheless, it is customary to compute the breaking-strength of a beam by means of the My formula /= = , where y is taken as the distance from the neutral axis to that outer fibre which gives way first, i.e., to the most stretched fibre if the beam breaks by tension, or to the most com- pressed fibre, if it breaks by compression. The quantity /, which may thus be computed from the formula My is defined as the Modulus oj Rupture. Inasmuch as this formula would give the outside fibre stress, if the stress were uniformly varying, it follows that, in the case of materials for which the tensile is less than the compressive strength, the modulus of rupture is greater than the tensile strength, while in that of materials for which the compressive is less than the tensile strength the modulus of rupture is greater than the compressive strength. For experimental work bearing upon this matter, see an article by Prof. J. Sondericker, in the Technology Quarterly for October, 1888. WORKING-STRENGTH. The working-strength per square inch of a material for trans- verse strength is the greatest stress per square inch to which it is safe to subject the most strained fibre of the beam. It is usually obtained by dividing the modulus of rupture by some factor of safety, as 3 or 4. WORKING-STRENGTH. 2$$ 192. EXAMPLES. i. Given a beam (Fig. 226) supported at both ends, and loaded, i, with w pounds per unit of length uniformly, and 2, with a single load Wat a. distance a from the left-hand support: find the position of the section of greatest bending-moment, and the value of the greatest bending-moment. o A a Solution. 1 (i) Left-hand supporti ng- force = - -\ . Right-hand supporting-force = _ _j_ t (2) Assume a section at a distance x from the left-hand support (this support being the origin), and the bending-moment at that sec- tion is, (wl W(l - a)\ wx* when x < a, -\ 2 ) 2 and when x > a, wx* To find the value of x for the section of greatest bending-moment, differentiate each, and put the first differential co-efficient = zero. We shall thus have, in the first case, W l W(l -a) I W(l - a) -- 1 -- ~ - - w# = o, or x = - H -- - -. - : 2 / 2 wl and in the second case, wl W(l -a) I W(l -a) W - + , - - -wx- W** o, or x = - + v . - -- . 2 / 2 Wl W Now, whenever the first is < #, or the second is > a, we shall have in that one the value of x corresponding to the section of greatest bending-moment. But if the first is > a, and the second < 0, then the greatest bending-moment is at the concentrated load. These conclusions will be evident on drawing a diagram representing the bending-moments graphically, as in Figs. 223 and 224; and the greatest bending-moment may then be found by substituting, in the cor- responding expression for the bending-moment, the deduced value of x. 296 APPLIED MECHANICS. 2. Given an I-beam, 10 feet long, supported at both ends, and loaded, at a distance 2 feet to the left of the middle, with 20000 pounds. Find the bending-moment at the middle, the greatest bending-moment, also the greatest intensity of the tension, and that of the compression at each of these sections. Given Area of upper flange = 8 sq. in. Area of lower flange = 5 sq. in. Area of web = 7 sq. in. Total depth = 14 in. 193. Beams of Uniform Strength. Abeam of uniform strength (technically so called) is one in which the dimensions of the cross-section are varied in such a manner, that, at each cross-section, the greatest intensity of the tension shall be the same, and so also the greatest intensity of the com- pression. -Such beams are very rarely used ; and, as the cross-section varies at different points, it would be decidedly bad engineering to make them of wood, for it would be necessary to cut the wood across the grain, and this would develop a tendency to split. In making them of iron, also, the saving of iron would gen- erally be more than offset by the extra cost of rolling such a beam. Nevertheless, we will discuss the form of such beams in the case wlien the section is rectangular. In all cases we have the general equation y applying at each cross-section, where M = bending-moment ''section at distance x from origin), / = moment of inertia of same section, ' y = distance from neutral axis to most strained fibre, and p intensity of stress on most strained fibre ; the condition for this case being that / is a constant for all values of x (i.e., for all positions of the section), while M, /, and y are functions of x. BEAMS OF UNIFORM STRENGTH. 297 As we are limiting ourselves to rectangular sections, if we let b = breadth and h = depth of rectangle (one or both vary- ing with x\ we shall have as the condition for such a beam, with/ a constant for all values of x, when the same load remains on the beam. We must, therefore, have bk 2 proportional to M. Hence, assuming the origin as before, i. Fixed at one end, load at the other, bh* =(-) W(l *). 2. Fixed at one end, uniformly loaded, bh 2 = ( - ) (/ x) 2 . \ 21' , 3. Supported at ends, loaded at I 2 \/ 2 middle - | 2 p 2 4. Supported at ends, uniformly loaded, bh 2 = ( --- }(lx x 2 ). \j> 2l ' Now, this variation of section may be accomplished in one of two ways: ist, by making h constant, and letting b vary; and 2d, by making b constant, and letting h vary. Thus, in the first case above mentioned, if h is constant, we have, for the plan of the beam, and if one side be taken parallel to the axis of the beam, this will be the equation of the other side ; and, as this is the equa- tion of a straight line, the plan will be a triangle. APPLIED MECHANICS. If, on the other hand, b be constant, and h vary, we shall nave, for the vertical longitudinal section of the beam, and, if one side be taken as a straight line in the direction of the axis, the other will be a parabola. A similar reasoning will give the plan or elevation respect- ively in each case ; and these can be readily plotted from their equations. CROSS-SECTION OF EQUAL STRENGTH. A cross-section of equal strength (technically so called) is one so proportioned that the greatest intensity of the tension shall bear the same ratio to the breaking tensile strength of the material as the greatest intensity of the compression bears to the breaking compressive strength of the material. This is accomplished, as will be shown directly, by so arranging the form and dimensions of the section that the distance of the neutral axis from the most stretched fibre shall bear to its distance from the most compressed fibre the same ratio that the tensile bears to the compressive strength of the material. Let f c breaking-strength per square inch for compression, f t = breaking-strength per square inch for tension, y c = distance of neutral axis from most compressed fibre, y t = distance of neutral axis from most stretched fibre. If p c = actual greatest intensity of compression, and p t = actual greatest intensity of tension, then, for a cross-section of equal strength, we must have, according to the definition, <=; but we have = = intensity of stress at a unit's pt ft yc yt CROSS-SECTION OF EQUAL STRENGTH. 2Q9 distance from the neutral axis. Hence, combining these two, we obtain y - = 7 y t ft EXAMPLE. Suppose we have^ = 80000 Ibs. per square inch, and/ = 20000 Ibs. per square inch. : find the proper proportion between the flange A t and the web A 2 of a T-section whose depth is h. 194. Deflection of Beams. We have already seen ( 185), that, in the case of a beam which is bent by a transverse load, we have -'oo, a r where (having assumed a certain cross-section whose distance from the origin is x) a = the strain of a fibre whose distance from the neutral axis is y, and r = radius of curvature of the neutral lamina at the section in question. Hence follows the equation but from the definition of E t the modulus of elasticity, we shall have V* where / = intensity of the stress, at a distance y from the neutral axis. Hence it follows, assuming Hooke's law, that r Ey E y We have already seen, that, disregarding signs, M = - / 3^0 APPLIED MECHANICS. (making, of course, the two assumptions already spoken of when this formula was deduced), where M = bending-moment at, and / = moment of inertia of, the section in question ; i.e., of that section whose distance from the origin is x. This gives - = , if, denoting tension by the + sig n > and taking y positive upwards, we call M positive when it tends to cause tension on the lower, and compression on the upper, side; these being the conventions in regard to signs which we shall adopt in future. Hence, by substitution, we have 1 - p - M (i\ ~r~Ey- El Now, if we assume the axis of x coincident with the neutral line of the central longitudinal section of the beam, and the axis of v at right angles to this, and v positive upwards, no matter where the origin is taken, we shall always have, as is shown in the Differential Calculus, ' (+(1)7 Hence equation (i) becomes (*) M and / being functions of x : and, when we can integrate this equation, we can obtain v in terms of x, thus having the equation of the elastic curve of the neutral line ; and, by com- puting the value of v corresponding to any assumed value of x, we can obtain the deflection at that point of the beam. FORMULA FOR SLOPE AND DEFLECTION. 3OI The above equation (2) is, as a rule, too complicated to be integrated, except by approximation ; and the approximation usually made is the following : Since in a beam not too heavily loaded, the slope, and con- sequently the tangent of the slope (or angle the neutral line makes with the horizontal at any point), is necessarily small, it follows that is very small, and hence (--) is also very small, dx \dxl and i + ( ) is nearly equal to unity. Making this substitu- tion, we obtain, in place of equation (2), -. d* ~ EI* and this is the equation with which we always start in com- puting the slope and deflection of a loaded beam, or in finding the equation of the elastic line. By one integration (suitably determining the arbitrary con- stant) we obtain the slope whose tangent is , and by a second dx integration we obtain the deflection v at a distance x from the origin ; and thus, by substituting any desired value for x, we can obtain the deflection at any point. 195. Ordinary Formulae for Slope and Deflection. We may therefore write, if i is the circular measure of the slope at a distance x from the origin, since i = tan i = -j- dx nearly, <v = M_ dx 2 ~~~ Ef f M 3O2 APPLIED MECHANICS. In these equations, of course, E is taken as a constant, M must ALWAYS be expressed in terms of x, and so also must / whenever the section varies at different points. When, how- ever, the section is uniform, / is constant, and the formulae reduce to "Jiff- 196. Special Cases -- i. Let us take a cantilever loaded with a single load at the free end. Assume the origin, as before, at the fixed end, and let the beam be one of uniform section. We then have M = W(l x\ *'~<> -*< ?- To determine c, observe that when x = o, i = o ; c = o is the slope at a distance x from the origin. The deflection at the same point will be , . f,** = - y& = -^- 2 - ^w J EIJ \ 2 ) EI\ 2 6 ) but when x = o, v = o .*. ^ = o /. the deflection at a distance x from the origin will be The equations (i) and (2) give us the means of finding the slope and deflection at any point of the beam. To find the greatest slope and deflection, we have that both expressions are greatest when x = I. Hence, if * and V Q rep- resent the greatest slope and deflection respectively, Wl 2 SPECIAL CASES. 33 2. Next take the case of a beam supported at both ends and loaded uniformly, the load per unit of length being w. Assume the origin at the left-hand end ; then wl wx 2 w M x --- = Ux x 2 ) and W= wl 2 2 2 V w /w /lx 2 x*\ (lx - x ^ x = _(___) + ,. 2EI ^ To determine c, we have that when x = -, then i = o; W // 3 / 3 \ 2// 3 V 8 " V + - } - W ^ W (6/x 2 - /3^ f ff '/ ' V -5 "?/ r> A 7? T <y A Ji ' 7 ^ ^ ' V / /w /* to = ^7j (6 ^ " ^ ~ /3) ^ 24^7 But when x = o, v = o ; /. r = o For the greatest slope, we have ;tr = o, or x = I; 24^*7 24^7 For the greatest deflection, x = - ; w 5/ 4 5 a// 4 384^7 304 APPLIED MECHANICS. 3. Take the case of a beam supported at both ends, and loaded at the middle with a load W. Assume, as before, the origin at the left-hand support. Then we shall have W I W I M = x, x < -, and M (/ x} when x > Therefore, for the slope up to the middle, we have w r w x 2 i = ^r, I xdx = =- T h c. 2EIJ 2EI 2 When x ~ , then i = o ; wr W . and w /v , r\. w (x> rx\ v - I \x lax = I AJcl'J \ 4/ 4.fi/\3 4/ But when x o, v = o ; c = o. i 4 The slope is greatest when x o ; .'. z' = The deflection is greatest when x = -; 4. In the following table 7 denotes the moment of inertia of the largest section : SPECIAL CASES. 305 Uniform Cross-Section. Greatest Slope. Greatest Deflection. Fixed at one end, loaded at the other Fixed at one end, loaded uniformly . . Supported at ends, load at middle . . . Supported at ends, uniformly loaded . . i Wl 2 2 7?7 i Wl 2 i Wl* ZEI i Wl* *>EJ i Wl 2 *EI i Wl* '6 El i Wl* ^EI 48^7 5 #73 384^/ Uniform Strength and Uniform Depth, Rectangular Section. Fixed at one end, load at the other . . Fixed at one end, uniformly loaded . . Supported at both ends, load at middle . Supported at both ends, uniformly loaded, Wl 2 ~EI i Wl 2 2 El i Wl* *~EI i Wl 2 i Wl* ~*~EI iWZ* 4^7 i Wl* V~EI i m* & EI 64^7 Uniform Strength and Uniform Breadth, Rectangular Section. Fixed at one end, loaded at the other, Supported at both ends, load at middle, Supported at both ends, uniformly loaded wr 2 ~EI i wr 2 wr 3 EI i wr 4 El o wr 24 EI 0018 Wr 0-098 EI 0.010 EJ 306 APPLIED MECHANICS. 197. Deflection with Uniform Bending-Moment If the bending-moment is uniform, then M is constant ; and, if / is also constant, we have _ ^L f - Mx but when x = -, then i = o; Ml 2EI Ml l\ dv " t = T^rl x -- ) = -r EI\ 2] dx lx the constant disappearing because v = o when x = o. Hence, for a beam where the bending-moment is uniform, we have _ J ^\ M Y* 2 ^ and for greatest slope and deflection, we have -Ml Mil* / 2 \ i Ml 2 1 .71 ^O * ~~ r\ T * V<~\ -r- 198. Resilience of a Beam. 7^ resilience of a beam is the mechanical work performed in deflecting it to the amount it would deflect under its greatest allowable gradually applied load. In the case of a concentrated load, if W is the greatest allowable gradually applied load, and v l the corresponding deflection at the point of application of the load, then will the W mean value of the load that produces this deflection be W and the resilience of the beam will be z/,. 2 SLOPE AND DEFLECTION OF A BEAM. 3O/ i99- Slope and Deflection of a Beam with a Con- centrated Load not at the Middle. Take, as the next A a case, a beam (Fig. 228). Let < a the load at A be W, and dis- tance OA = a y and let a > -. 2 FIG. 228. W(l - a) x < a M = ^ '- x, ^f - a) - x<a ' = When x = o, * = 4 = undetermined slope at = . = W(l - a and When x =. o, v = o ; To determine r, observe that when ^r ^, this value of i and that deduced from (i) must be identical. Waf _^!\, W(l - a)a* . Wa* . TEI\ a 2 ) H 2/^7 2^7 308 APPLIED MECHANICS. Wat , x*\ Wa* , . , 2 / 2EI or and v = Jlx -**- ld)dx To determine c, observe that when x = a, this value of v and (2) must be identical ; Wa f 3 \ . W(l a) / a ( \ i 4 4- tf 4 ) = 61EI 6EI + ^ 2 ) H- ^- (4) To determine 4> we have that when x /, v = o ; Substituting this value of i in the equations (i), (2), (3), and (4), we obtain for SLOPE AND DEFLECTION OF A BEAM. 309 Wa , r Wa (4) v = To find the greatest deflection, differentiate (2), and place the first differential co-efficient equal to zero : or, which is the same thing, place i = o in (i), and find the value of x ; then substitute this value in (2), and we shall have the greatest deflection. We thus obtain (/-*)*> = ^( 3 / -*/-*>) .-. = "-(* ~ & + a -\ 3 3\ ' ' / or ' *=' ' a ~'> and the greatest deflection becomes Wa(l - a)(2l - a) _ 2OO. EXAMPLES. 1. In example i, p. 294, find the greatest deflection of the beam when it is loaded with \ of its breaking-load, assuming E = 1200000. 2. In the same case, find what load will cause it to deflect ^J^ of its span. 3. What will be the stress at the most strained fibre when this occurs- 4. In example 3, p. 294, find the load the beam will bear without deflecting more than ^J^ of its span, assuming E = 24000000. 5. Find the stress at the most strained fibre when this occurs. 6. In example 6, p. 295, find the greatest deflection under a load J the breaking-load. 3io APPLIED MECHANICS. 2Oi. Deflection and Slope under Working-Load. If we take the four cases of deflection given in the first part of the table on p. 305, and calling/ the working strength of the material, and y the distance of the most strained fibre from the neutral axis, and if we make the applied load the working- load, we shall have respectively ! m =*L W=^ y ty Wl // 2/7 2 . = /. W 2 y ly m = fj 4 '" y wi fl 4- T -T *y : W= ^ And the values of slope and deflection will become respectively, Slope. Deflection. Slope. Deflection. / 2. From these values, and those given on p. 305, we derive the following two propositions : i. If we have a series of beams differing only in length; and we apply the same load in the same manner to each, their greatest slopes will vary as the squares of their lengths, and their greatest deflections as the cubes of their lengths. SLOPE AND DEFLECTION OF RECTANGULAR BEAMS. 31! 2. If, however, we load the same beams, not with the same load, but each one with its working-load, as determined by allowing a given greatest fibre stress, then will their greatest slopes vary as the lengths, and their greatest deflections as the squares of their lengths. 202. Slope and Deflection of Rectangular Beams bfc h If the beams are rectangular, so that / = and y = -, the values of slope and deflection above referred to become further simplified, and we have the following tables : Given Load W. Working-Load. Greatest Fibre Stress =/. * Slope. Deflection. Slope. Deflection. 1. 2. 3. 4- 6W1 2 4 #73 I 2 / 2 Ebfc 2W* Ebte 3 Wl* Eh 2_fl 3 Eh i Ebte 3 Wl* 46fc i Wl* *Ebh* i Wl* 3 Eh i fl 2 Eh i/' lEbfc 5 wr* *Eh 2ft 1 Eh (>Eh sfi* ^Eh * Ebfc 32 Ebh* So that, in the case of rectangular beams similarly loaded and supported, we may say that Under a given load W, the slopes vary as the squares of the lengths, and inversely as the breadths and the cubes of the depths ; while the deflections vary as the cubes of the lengths, and inversely as the breadths and the cubes of the depths. 312 APPLIED MECHANICS. On the other hand, under their working-loads, the slopes vary directly as the lengths, and inversely as the depths ; while the deflections vary as the squares of the lengths, and inversely as the depths. 203. Beams Fixed at the Ends. The only cases which we shall discuss here are the two following ; viz., i. Uniform section loaded at the middle. 2. Uniform section, load uniformly distributed. CASE I. Uniform Section loaded at the Middle. The fixing at the ends may be effected by building the beam for some distance into the wall, as shown in Fig. 229. The same result, as far as the effect on |w the beam is concerned, might be effected as follows : Hav- ing merely supported it, and placed upon it the loads it has to bear, load the ends outside of the supports just enough to make the tangents at the sup- ports horizontal. These loads on the ends would, if the other load was re- moved, cause the beam to be convex upwards : and, moreover, the bending-moment due to this load would be of the same amount at all points between the supports ; i.e., a uniform bending-moment. Moreover, since the effect of the central load and the loads on the ends is to make the tangents over the supports horizontal, it follows that the upward slope at the support due to the uniform bending-moment above described must be just equal in amount to the downward slope due to the load at the middle, which occurs when the beam is only sup- ported. Hence the proper method of proceeding is as follows : i. Calculate the slope at the support as though the beam were supported, and not fixed, at the ends ; and we shall if we represent this slope by i u the equation BEAMS FIXED AT THE ENDS. 313 w* 2. Determine the uniform bending-moment which would produce this slope. To do this, we have, if we represent this uniform bending- moment by M lf that the slope which it would produce would be and, since this is equal to * we shall have the equation _^/_ ;w_. M TEI -*"- V3) .: Jf, = ~ (4) This is the actual bending-moment at either fixed end ; and the bending-moment at any special section at a distance x from the origin will be where M is the bending-moment we should have at that sec- tion if the beam were merely supported, and not fixed. Hence, when it is fixed at the ends, we shall have, for the bending- moment at a distance x from O, where O is at the left-hand support, W W, M=oc-l. (5) When x = -, we obtain, as bending-moment at the middle, *-?; (6) o and, since M l = M , it follows that the greatest bending- moment is W 8 ; 3H APPLIED MECHANICS. this being the magnitude of the bending-moment at the middle and also at the support. POINTS OF INFLECTION. The value of M becomes zero when x = - and when x = ; 4 4' hence it follows that at these points the beam is not bent, and that we thus have two points of inflection half-way between the middle and the supports. SLOPE AND DEFLECTION UNDER A GIVEN LOAD. We shall have, as before, W& Wlx . /M , EI dX = and since, when x = o, i = o, .. c ' o """ * = ~dx = W I 2X l IX 2 v I 3 2 the constant vanishing because v = o when x = o. The slope becomes greatest when x = -, and the deflection when x = -. 4 Hence for greatest slope and deflection, we have Wl 2 f . 64^7' BEAMS FIXED AT THE ENDS. 315 SLOPE AND DEFLECTION UNDER THE WORKING-LOAD. If f represent the working-strength of the material per square inch, and if W represent the centre working-load, we shall have fiP7 = /7 8 " y CASE II. Uniform Section, Load uniformly Distributed. Pursuing a method entirely similar to that adopted in the former case, we have i. Slope at end, on the supposition of supported ends, is Wl* 2 4 ^/ 2. Slope at end under uniform bending-moment M t is () Hence, since their sum equals zero, Wl 12 ' which is the bending-moment over either support. The bending-moment at distance x from one end is W Wl M - \lx x 2 ) . ^4) 2/ 12 Wl This is greatest when x = o, and is then . Hence great- 1 2 est bending-moment is, in magnitude, (5) 12 316 APPLIED MECHANICS. POINTS OF INFLECTION. M becomes zero when x = - =. (6) 2 2 ^3 Hence the two points of inflection are situated at a distance / on either side of the middle. SLOPE AND DEFLECTION. , M 7 t s= I -~dx = the constant vanishing because z = o when x = Oi W v = the constant vanishing because ^ = o when .r = o. Hence for greatest slope and deflection we have, t is greatest when x = -f i zt -y=\ and z; is greatest when * = - ; SLOPE AND DEFLECTION UNDER WORKING-LOAD. For working-load we have Wl fl 77 = 7 B ENDING-MOMENT AND SHEARING-FORCE. 3'7 EXAMPLES. 1. Given a 4-inch by 12 -inch yellow-pine beam, span 20 feet, fixed at the ends ; find its safe centre load, its safe uniformly distributed load, and its deflection under each load. Assume a modulus of rupture 5000 Ibs. per square inch, and factor of safety 4. Modulus of elasticity, 1200000. 2. Find the depth necessary that a 4-inch wide yellow-pine beam, 20 feet span, fixed at the ends, may not deflect more than one four-hun- dredth of the span under a load of 5000 Ibs. centre load. 204. Variation of Bending-Moment with Shearing- Force. If, in any loaded beam whatever, M represent the b ending-moment, and F the shearing-force at a distance x from the origin, then will *-<* > Proof (a). In the case of a cantilever (Fig. 230), assume the origin at the fixed end ; then, if M represent the bending-moment at a distance x from the origin, and M '+ ^M that at a distance x + tx from the origin, we shall have the following equations : x = l M= S W(a- x), X = X x = l M + AJ/ = -2 W(a- x A#) nearly. X = X a being the co-ordinate of the point of application of W, x = l AJ/= AjeS W nearly ====- = 2 W: 3 I 8 APPLIED MECHANICS. and, if we pass to the limit, and observe that we shall obtain (b) In the case of a beam supported at the ends (Fig. 231), , A.,*. assume the origin at the left-hand /|^ I ^j,j 7\ end, and let the left-hand support- ing-force be S ; then, if a represent FIG. 231. the distance from the origin to the point of application of W, we shall have the equations M = Sx - 2 W(x - a), M 4- bM = S(* + A#) - S ^(tf d5 4- A*) nearly. Hence, by subtraction, X- X = S . tec 2 WA* nearly JT = o = o 2 ^nearly; mt-' x = o and, if we pass to the limit, and observe that p ^ g 2 iff Jf = O we shall obtain as before. LONGITUDINAL SHEARING OF BEAMS. 319 205. Longitudinal Shearing of Beams. The resistance of a beam to longitudinal shearing sometimes becomes a mat- ter of importance, especially in timber, where the resistance to shearing along the grain is very small. We will therefore pro- ceed to ascertain how to compute the intensity of the longi- tudinal shear at any point of the beam, under any given load ; as this should not be allowed to exceed a certain safe limit, to be determined experimentally. Assume a A section AC (Fig. 232) at a distance x from V the origin, and let the bending-moment at that section be M. Let the section BD be at a distance x + &>* from the origin, and let the bending-moment at that section be FIG. 232 . M + kM. Let y be the distance of the outside fibre from the neutral axis ; and let ca y^ be the distance of a, the point at which the shearing-force is required, from the neutral axis. Consider the forces acting on the portion ABba, and we shall have i. Intensity of direct stress at A = -j^. 2. Intensity of direct stress at a unit's distance from neu- M tral axis = -j. My 3. Intensity of direct stress at ^, where ce = y, is =-. (M + So, likewise, intensity of direct stress at / is Therefore, if z represent the width of the beam at the point ?, we shall have M (*y Total stress on face Aa = -j- I yzdy, 1 Jyi M + ^M r?o Total stress on face Bb = - j - I yzdy ; l J yi 320 APPLIED MECHANICS. ^.rr bM C y ,*. Difference I yzdy : / J yi and this is the total horizontal force tending to slide the piece AabB'on the face ab. Area of face ab, if #, is its width, is therefore intensity of shear at a is approximately &M C? -r I yzJy or exactly (by passing to the limit) /dM\ dM And, observing that F = -T-, this intensity reduces to (0 We may reduce this expression to another form by observ- ing, that, if y z represent the distance from c to the centre of gravity of area Aa, and A represent its area, we have /Vo J yzdy=y 2 A; therefore intensity of shear (at distance j/j from neutral axis) at point a = ( -M } - (a) This may be expressed as follows : LONGITUDINAL SHEARING OF BEAMS. 321 Divide the shearing-force at the section of the beam under consideration, by the product of the moment of inertia of the section and its width at the point where the intensity of the shearing-force is desired, and multiply the quotient by the statical moment of the portion of the cross-section between the point in question and the outer fibre ; this moment being taken about the neutral axis. The result is the required intensity of shear. The last factor is evidently greatest at the neutral axis ; hence the intensity of the shearing-force is greatest at the neutral axis. LONGITUDINAL SHEARING OF RECTANGULAR BEAMS. For rectangular beams, we have th* /=-, *, = *. Hence formula (2) becomes ^)- (3) For the intensity at the neutral axis, we shall have, therefore, I2F /h bh\ 3 F b*h?> \4 2 / 2 bh? since for the neutral axis we have h bh v a = - and A = . 4 2 EXAMPLES. i . What is the intensity of the tendency to shear at the neutral axis of a rectangular 4-inch by 1 2-inch beam, of 14 feet span, loaded at the middle with 5000 (bs. 3 22 APPLIED MECHANICS. 2. What is that of the same beam at the neutral axis of the cross- section at the support, when the beam has a uniformly distributed load of T 2000 Ibs. 3. What is that of a 9-inch by 14-inch beam, 20 feet span, loaded with 15000 Ibs. at the middle. 206. Strength of Hooks. The following is the method to be pursued in determining the stresses in a hook due to a given load ; or, vice versa, the proper dimensions to use for a given load. Suppose (Fig. 233) a load hung at E; the load being P, and the distances AB n\ OF=y y O being the centre of gravity of this section, conceive two equal and opposite forces, each equal and parallel to P, acting at O. Let A = area of section, and let 7 = its moment of inertia about CD (BCDF represents the section revolved into the plane of the paper) ; then i . The downward force at O causes a uniformly distributed stress over the section, whose intensity is 2. The downward force at E and the upward force at O constitute a couple, whose moment is and this is resisted, just as the bending-moment in a beam, by a uniformly varying stress, producing tension on the left, and compression on the right, of CD. COLUMNS. 323 If we call p^ the greatest intensity of the tension due to this bending-moment, viz., that at B, we have and if / 3 denote the greatest intensity of the compression due to the bending moment, viz., that at F, we have therefore the actual greatest intensity of the tension is and this must be kept within the working strength if the load is to be a safe one ; and so also the actual greatest intensity of the compression, viz., that at F, is, when/, >/,, ,-, _, -A*+*)y, ^ A -A A- 7 ^, which must be kept within the working strength for compression. 207. Strength of Columns. The formulae most commonly employed for the breaking-strength of columns subjected to a load whose resultant acts along the axis have been, until recently, the Gordon formulae with Rankine's modifications, the so-called Euler formulae, and the avowedly empirical formulae of Hodg- kinson. These formulae do not give results which agree with those obtained from tests made upon such full-size columns as are used in practice. The deductions of the first two are not logical, certain assump- tions being made which are not borne out by the facts. When a column is subjected to a load which strains any fibre beyond the elastic limit, the stresses are not proportional to the strains, and hence there can be no rational formula for the break- ing-load. Hence, all formulae for the breaking-load are, of necessity 3 2 4 APPLIED MECHANICS. empirical, and depend for their accuracy upon their agreement with the results of experiments upon the breaking-strength of such full-size columns as are used in practice. Nevertheless, the ordinary so-called deductions of the Gor- don, and of the so-called Euler formula? will be given first. 208. Gordon's Formulae for Columns. (a) Column fixed in fc Direction at Both Ends. Let CAD be the central axis of the column, P the breaking-load, and v the greatest deflection, AB. Conceive at A two equal and opposite forces, each equal to P; then i. The downward force at A causes a uniformly distributed stress over the section, of intensity, ^P_ [ D 2. The downward force at C and the upward force at A Fio.234. constitute a bending couple whose moment is M=Pv. If p 2 = the greatest intensity of the compression due to this bending, where 7= distance from the neutral axis to the most strained fibre of the section at A. Then will the greatest intensity of stress at A be and, since P is the breaking-load, p must be equal to the breaking- strength for compression per square inch=/'. (i) where p= smallest radius of gyration of section at A. Thus far the reasoning appears sound; but in the next step it is assumed that GORDON'S FORMULA FOR COLUMNS. 325 where c is a constant to be determined by experiment. Hence, sub- stituting this, and solving for P, P = fA , 9 , (2) which is the formula for a column fixed in direction at both ends. (b) Column hinged at the Ends. It is assumed that the points of inflection are half-way between the middle and the ends, and -jr~ hence that, by taking the middle half, we have the case of bending of a column hinged at the ends (Fig. 235). Hence, to obtain the formula suitable for this case, substitute, in (2), 2/ for /, and we obtain * FIG. 235. " M <3) (c) Column fixed at One End and hinged at the Other (Fig. 236). ~~r In this case we should, in accordance with these assumptions, take J of the column fixed in direction at both ends; hence, to obtain the formula for this case, substitute, in (2), J/ for /, and we thus obtain j- 1 (4) i6/ 2 ' gcp 2 FIG. 236. >. 236. Rankine gives, for values of / and c, the following, based upon Hodgkinson's experiments: f (Ibs. per sq. in.). c. Wrought-iron 36000 36000 Cast-iron 80000 Dry timber . 72OO 3OOO 326 APPLIED MECHANICS. 2o8a. So-called Euler Formulae for the Strength of Columns. (a) Column fixed in Direction at One End only, which bends, as shown in the Figure, i. Calculate the breaking-load on the assumption that the column will give way by direct compression. This will be PI=/A, CO where /"= crushing-strength per square inch, and A = area of cross- section in square inches. 2. Calculate the load that would break the column if it were to give way by bending, by means of the following formula : f, = where E= modulus of elasticity of the material, 7= smallest moment of inertia of the cross-section, and /= length of column. Then will the actual breaking-strength, according to Euler, be the smaller of these two results. To deduce the latter formula, assume the origin at the upper end, and take x vertical and y horizontal. Let p= radius of curvature at point (x, y), and let 3/=bending-moment at the same point. Then we have, with compression plus and tension minus, PIG. as?. M_ El Py El' (3) But d*y P "dx* = "El*' dy d*y , P f dy, JL . -J-dx I y-~dx dx dx* EIJ J dx dx ^\ = --^y 2 4- c; and, since for y Er ~dx EULER FORMULA FOR STRENGTH OF COLUMNS. 327 dy J y ' Sm - a = And since, when x=o, y=> " c=o, we have When y=a, x=l\ hence, substituting in (5), and solving for P, >-(=) (b) Column hinged at Both Ends (Fig. 235). i. For the crushing-load, P~/A. 2. For the breaking-load by bending, put 1/2 for I in (6) ; hence (7) (c) Column fixed in Direction at One End, and hinged at the other <Fig. 236). i. For the crushing-load, 2. For the breaking-load by bending, put //3 for / in (6) ; hence ((/) Column fixed in Direction at Both Ends (Fig. 234). i. For the crushing-load, P.-/4. 2. For the breaking-load by bending, :his being obtained from (2) or (6) by substituting 1/4 for /. 328 APPLIED MECHANICS. (e) In order to ascertain the length wnere incipient flexure occurs, according to this theory we should place the two results equal to each other, and from the resulting equation determine /. We should thus obtain, for the three cases respectively, <) /- 4/, (10) (r) l= Hence all columns whose length is less than that given in these formulae will, according to Euler, give way by direct crushing; and those of greater length, by bending only. 209. Hodgkinson's Rules for the Strength of Columns. Eaton Hodgkinson made a large number of tests of small columns, especially of cast-iron, and deduced from these tests certain empirical formulas. The strength of pillars of the ordinary sizes used in practice has been computed by means of Hodgkinson's formulae, and tabulated by Mr. James B. Francis; and we find in his book the following rules for the strength of solid cylindrical pillars of cast-iron, with the ends flat, i.e., "finished in planes perpendicular to the axis, the weight being uniformly distributed on these planes": For pillars whose length exceeds thirty times their diameter, ^=99318^, (.) where D= diameter in inches, /= length in feet, W= breaking- weight in Ibs. If, on the other hand, the length does not exceed thirty times the diameter, he gives, for the breaking-weight, the following formula: where W= breaking- weight that would be derived from the preceding formula, W'= actual breaking- weight, BREAKING-LOAD OF FULL-SIZE COLUMNS 329 For hollow cast-iron pillars, if D= external diameter in inches, d= internal diameter in inches, we should have, in place of (i), 7*. 5S _^ ( ^ /'? and in place of (3), c = i 0080 i 4 For very long wrought-iron pillars, Hodgkinson found the strength to be 1.745 times that of a cast-iron pillar of the same dimensions; but, for very short pillars, he found the strength of the wrought-iron pillar very much less than that of the cast-iron one of the same dimensions. With a length of 30 diameters and flat ends, the wrought-ir on exceeded the cast-iron by about 10 per cent. 210. Breaking-load of Full-size Columns. The tests made upon full-size columns are not as many as would be desir- able. The details will be given in Chapter VII, but a few of the empirical formulae which represent their results will be given here. If P = breaking-load, A= area of smallest section, / = length of column, p = least radius of gyration of section, and } e = crushing- strength of the material per unit of area, it will be found that for values of I/ p less than a certain amount, the column remains straight, and the breaking-load may be computed by means 01 the formula P = f c A . For greater values of l/p, the breaking-load is smaller than that given by this formula, and may be computed by mean;.; of the formula P = fA, by using for / a value smaller than f c , this value varying with the value of l/p t and being determined empirically from the results of tests of full-size columns. (a) In the case of cast-iron columns no tests have been made of full-size columns of the second class, while those made upon the first class indicate that the value of } c suitable for use in practice is from 25,000 to 30,000 Ibs. per square inch. (b) In the case of wrought-iron columns, the tests of the first class indicate that the value of } c suitable for use in practice is from 30,000 to 35,000 Ibs. per square inch. 330 APPLIED MECHANICS (c) In the case of wrought-iron columns of the second class, the formula of Mr. C. L. Strobel for bridge columns with either flat or pin ends, when l/p> 90, is p A * = 46000-125-. A p On the other hand, those recommended by Prof. J. Sonde- ricker, of which the first was devised by Mr. Theodore Cooper, are as follows: (a) For Phoenix columns with flat ends l/p > So, P_ 36000 A 18000 For lattice columns with pin-ends and l/p>6o, P_ = 340QO A 12000 (7-) For solid web, square, or box columns with flat ends, and l/P>8o, P 33000 Z = ~(//fl-8o)2' 10000 ($) For solid web, square, or box columns with pin-ends, and l/P>6o, P _ 31000 A (l/p -60)* 6000 The number of tests that have been made upon full-size steel columns is very small, hence no formulae will be given here, but the subject will be discussed in Chapter VII. The number of tests that have been made upon full-size timber columns is con- siderable, but this subject will also be discussed in Chapter VII. 211. Columns subjected to Loads which do not Strain any Fibre beyond the Elastic Limit. Under this head will be discussed, first, the mode of determining the greatest fibre THEORY OF COLUMNS. 331 stress in a straight column subjected to an eccentric load, and, secondly, the general theory of columns. (a) Straight column, under eccentric load. Let O' be the centre of gravity of the lower section, and let A'O' = x , where A' is the point of application of the resultant of the eccentric load. Conceive two equal and opposite forces at O', each equal and parallel to P. Then we have: i. Downward force along OO r causes uniform P stress of intensity p\ = -r . 2. The other two form a couple whose moment is Px , and the greatest intensity of the stress due to this couple is p2= - r > where a = O'B'. Hence, FIG. 238. 1 the greatest intensity of the stress is P Px a and this should be kept within the limits of the working-strength. (b) Theory of columns. The theory of columns is that of the Inflectional Elastica, and is explained in several treatises, among which is that of A. E. H. Love on the Theory of Elasticity. It is as follows: Let the curve OP be an elastic line, on which O is a point of inflection. It follows that there is no bending-moment at this point, and hence we may assume that at O a single force R acts. Take Y| the origin at O, and axis of X along the line of action of the force R. Let E\ = modulus of elasticity of the material, 7 = moment of inertia of section about an axis through its centre of gravity, and perpen- 33 2 APPLIED MECHANICS. dicular to the plane of the curve, (> = angle between OX and the tangent at any point P whose coordinates are x and y, a = value of <j) at point O, r= radius of curvature of the curve at P, s length of arc OP, / = length of one bay, i.e., measured from O to the next point of inflection, 0=T> -4= area of section, p = -*rr> R = ' Then we have for any such elastic line, when compressions are plus and tensions minus, i M_ p ET Moreover, since = 3 and M = Ry, we have, for a column p ds d6 R of the same cross-section throughout its length, ~y~ = ~~7>'> ID where the quantity j^j is a constant. By differentiation we obtain R dy R Integrating, and observing that at O, ~^J =O an ^ <i> =a > obtain The integration of this equation requires the use of elliptic integrals, hence only the results will be given here. THEORY OF COLUMNS. 332* They are : (2) and (3) (4) where E denotes the elliptic integral of the second kind, and K the complete elliptic integral of the first kind. Moreover, for the determination of the load R, we obtain from equation (4) K=- and hence 4K2 (6) From these equations, we can, by using a table of elliptic functions, deduce the following results for the coordinates of points on the inflectional elastica, for various values of a : a 5 T X / y I 10 o.oo o . oooo . OOOO 0.25 0.50 0.2476 o . 4962 o .0392 -554 20 o.oo o . oooo . OOOO 0.25 o. 50 0.2376 o . 4849 0.0773 o. 1079 30 o.oo . OOOO . OOOO 0.25 0.50 0.2224 o . 4662 0.1135 o. 1620 Moreover, these results agree with those which we obtain by APPLIED MECHANICS. experiment, and thus we can, by making use of our calculations, compute the load required to produce a given elastica, determined by the slope at the points of inflection, which, in the case of pin- ended columns, are at the ends, and, in the case of columns fixed in direction at the ends, are half-way between -the middle and the ends. All this can be done, and can be verified by experiment, provided that the load is not so great that any fibre is strained beyond the elastic limit of the material, and provided the value of l/p is not so small that the curvilinear form is unstable. For smaller values of l/p the only stable form is a straight line, and the column does not bend. To ascertain the least value of l/p for which a curved form is stable, observe that K cannot be less than 71/2, and since this cor- responds to one bay, and hence to the case of a pin-ended column, we have in that case, by substituting n/2 for K in equation (6) , 7T 2 n and, since I=AfP and -7=<7, we have for the line of demarcation between the straight and curved form in a pin-ended column -; (7) and for that in the case of a column fixed in direction at the ends As an example, if a= 10,000 and 1=30,000,000 we should find that a pin-ended column would not bend unless l/p were greater than 172, and that a column fixed in direction at the ends STRENGTH OF SHAFTING. 333 would not bend unless l/p were greater than 344. Columns with smaller values of l/p would remain straight when the resultant of the load acts along the axis, and no fibre is strained beyond the elastic limit. 212. Strength of Shafting. The usual criterion for the strength of shafting is, that it shall be sufficiently strong to resist the twisting to which it is exposed in the transmission of power. Proceeding- in this way, let EF (Fig. 239) be a shaft, AB the driving, and CD the following, pulley. Then, if two cross-sections be taken between these two pulleys, the por- tion of the shaft between these two cross-sections will, during the trans- mission of power, be in a twisted con- F riG. 239. dition ; and if, when the shaft is at rest, a pair of vertical parallel diameters be drawn in these sec- tions, they will, after it is set in motion, no longer be parallel, but will be inclined to each other at an angle depending upon the power applied. Let GH be a section at a distance x from O, and let KI be another section at a distance x -f- dx from O. Then, if di represent the angle at which the originally parallel diameters of these sections diverge from each other, and if r = the radius of the shaft, we shall have, for the length of an arc passed over by a point on the outside, rdi; and for the length of an arc that would be passed over if the sections were a unit's distance apart, instead of dx apart, rdi _ di dx dx This is called the strain of the outer fibres of the shaft, as it is the distortion per unit of length of the shaft. 334 APPLIED MECHANICS. In all cases where the shaft is homogeneous and symmet- rical, if i is the angle of divergence of two originally parallel diameters whose distance apart is x, we shall have the strain, di i v = r = r-. dx x This also is the tangent of the angle of the helix. A fibre whose distance from the axis of the shaft is unity, will have, for its strain, dt_ = / dx x A fibre whose distance from the axis of the shaft is p, will have, for its strain, di i v = p - = p-. dx x Fixing, now, our attention upon one cross-section, GH, we have that the strain of a fibre at a distance p from the axis (p varying, and being the radius of any point whatever) is where - is a constant for all points of this cross-section. X Hence, assuming Hooke's law, " Ut tensio sic vis" we shall have, if C represent the shearing modulus of elasticity, that the stress of a fibre whose distance from the axis is p, is which quantity is proportional to p, or varies uniformly from the centre of the shaft. The intensity at a unit's distance from the axis is 0- STRENGTH OF SHAFTING. 335 and if we represent this by a, we shall have for that at a dis- tance p from the axis, Hence we shall have (Fig, 240), that, on a small area, V J dA = dp( P dB) _ pdpdO, ^^ the stress will be pdA = apdA = ap 2 dpd9. The moment of this stress about the axis of the shaft is ppdA = ap z dA = ap^dpdO, and the entire moment of the stress at a cross-section is afp*dA = affpidpdO = al, where / = fp 2 dA is the moment of inertia of the section about the axis of the shaft. This moment of the stress is evidently caused by, and hence must be balanced by, the twisting-moment due to the pull of the belt. Hence, if M represent the greatest allowable twisting- moment, and a the greatest allowable intensity of the stress at a unit's distance from the axis, we shall have M = al = - /. P If / is the safe working shearing-strength of the material per square inch, we shall have / as the greatest safe stress per square inch at the outside fibre, and hence M=- I r will be the greatest allowable twisting-moment. 33^ APPLIED MECHANICS. For a circle, radius r t 2 " ~ * ~~2~ ~ J ~i6~* For a hollow circle, outside radius r v inside radius r M Moreover, if the dimensions of a shaft are given, and the actual twisting-moment to which it is subjected, the stress at a fibre at a distance p from the axis will be found by means of the formula The more usual data are the horse-power transmitted and the speed, rather than the twisting-moment. If we let P = force applied in pounds and R = its leverage in inches, as, for instance, when P = difference of tensions of belt, and R = radius of pulley, we have and if HP number of horses-power transmitted, and N = number of turns per minute, then TT T) _ \ *' *_ /_ . 12 X 33000 ' 12 X -l^OOoIfP ^r^r. ^ Jyi 271 N EXAMPLE. Given working-strength for shearing of wrought-iron as 10000 Ibs. per square inch ; find proper diameter of shaft to transmit 2o-horse power, making 100 turns per minute. TRANSVERSE DEFLECTION OF SHAFT. 337 Mp Angle of Torsion. From the formula, page 336, p~ =- % combined with we have = ap = Cp-, oc . _ MX " ~' which gives the circular measure of the angle of divergence of two originally parallel diameters whose distance apart is x ; the twisting-moment being M, and the modulus of shearing elas- ticity of the material, C. EXAMPLES. 1. Find the angle of twist of the shaft given in example i, 212, when the length is 10 feet, and C = 8500000. 2. What must be the diameter of a shaft to carry 80 horses-power, with a speed of 300 revolutions per minute, and factor of safety 6, break- ing shearing-strength of the iron per square inch being 50000 Ibs. 213. Transverse Deflection of Shafts. In determining the proper diameter of shaft to be used in any given case, we ought not merely to consider the -resistance to twisting, but also the deflection under the transverse load of the belt-pulls, weights of pulleys, etc. This deflection should not be allowed to exceed y^- of an inch per foot of length. Hence the de- flection should be determined in each case. The formulae for computing this deflection will not be given here, as the methods to be pursued are just the same as in the case of a beam, and can be obtained from the discussions on that subject. APPLIED MECHANICS. 214. Combined Twisting and Bending. The most com- mon case of a shaft is for it to be subjected to combined twisting and bending. The discussion of this case involves the theory of elasticity, and will not be treated here ; but the formulae com- monly given will be stated, without attempt to prove them until a later period. These formulae are as follows : Let M l = greatest bending-moment, M 2 = greatest twisting-moment, r = external radius of shaft, / = moment of inertia of section about a diameter, TTf 4 for a solid shaft / = , 4 f = working-strength of the material = greatest al- lowable stress at outside fibre ; then i. According to Grashof, /= LjfJ/i + fVJ/x 2 + M*\. (i) 2. According to Rankine, / = j M, + \!M* + M; j . (2) 215. Springs. The object of this discussion is to enable us to answer the following three questions : (a) Given a spring, to determine the load that.it can bear without producing in the metal a maximum fibre stress greater than a given amount. (&) Given a spring, to determine its displacement (elongation, compression, or deflection) under any given load, (c) Given a load P and a displacement & t ; a spring is to be made of a given material such that the load P shall produce the displace- ment 6 I , and that the metal shall not, in that case, be subjected to more than a given maximum fibre stress. Determine the proper dimensions of the spring. SPRINGS. 339 There are practically only two cases to be considered as far as the manner of resisting the load is concerned. In the first, the spring is subjected to transverse stress, and is to be calcu- lated by the ordinary rules for beams. In the second, the spring is subjected to torsion, and the ordinary rules for re- sistance to torsion apply. It is true that in most cases where the spring is subjected to torsion there is also a small amount of transverse stress in addition to the torsion ; but in a well- made spring this transverse stress is of very small amount, and we may neglect it without much error. We will begin with those cases where the spring is subjected to torsion, and for all cases we shall adopt the following nota tion : P = load on spring producing maximum fibre stress/; f = greatest allowable maximum fibre stress for shearing ; C = shearing modulus of elasticity ; x = length of wire forming the spring ; M l = greatest twisting moment under load P\ L = any load less than the limit of elasticity ; M = twisting moment under this load ; p = maximum fibre stress under load L ; p distance from axis of wire to most strained fibre ; / = moment of inertia of section about axis of wire ; z'j = angle of twist of wire under load P- i = angle of twist of wire under load L ; V = volume of spring ; #j = displacement of point where load is applied when load isP; d displacement of point where load is applied when load isZ. Then from pages 335 and 337 we obtain the following four formulae : *=/, (i) 34O APPLIED MECHANICS. MX '=C7' (3) These four formulae will enable us to solve all the cases of springs subjected to torsion only. Moreover, in the cases which we shall discuss under this head, the wire will have either a circular or a rectangular section : in the former case we will denote its diameter by d, and we shall then have net* d /= -- and p = ; 32 2 while in the latter case we will denote the two dimensions of the rectangle by b and h, respectively, and we shall then have We will now proceed to determine the values of P, #, S l , and V in each of the following four cases, all of which are cases of torsion : CASE i. Simple round torsion wire. Let AB, the leverage of the load about the axis, be R ; then we shall have M = LR, M, = PR ; and we readily obtain from the formulae (i), (2), (3), and (4) ^\ ,f C- <" (7) SPRINGS. 341 and from these we readily obtain (8) CASE 2.' Simple rectangular torsion wire. In this case we readily obtain (9) , D . , . = =ri ' (IC >=*'> = "' CASES 3 and 4. Helical springs made of round and of rec- tangular wire respectively. A helical spring may be used either in tension or in compression. In either case it is important that the ends should be so guided that the pair of equal and opposite forces acting at the ends of the spring should act ex- actly along the axis of the spring. This is of especial importance when the spring is used for making accurate measurements of forces, as in the steam-en- gine indicator, in spring balances, etc. Moreover, it is generally safer, as far as accuracy is con- cerned, to use a helical spring in tension rather than in com- pression, as it is easier to make sure that the forces act along 34 2 APPLIED MECHANICS. the axis in the case of tension than in the case of compres- sion. Whichever way the spring is used, however, provided only the two opposing forces act along the axis of the spring, the resistance to which the spring is subjected is mainly torsion, inasmuch as the amount of bending is very slight. This bending, however, we will neglect, and will compute the spring as a case of pure torsion, the same notation being used as before, except that we will now denote by R the radius of the spring, and we shall have M = LR, M, = and now formulae (5), (6), (7), and (8) become applicable to a spring made of round wire, and formulae (9) and (10), (n) and (12), to one made of rectangular wire. We must bear in mind, however, that x denotes the length of the wire composing the spring, and not the length of the spring, d and d l now denote the elongations or compressions of the spring. GENERAL REMARKS. By comparing equations (8) and (12), it will be seen that if a spring is required for a given service, its volume and hence its weight must be 50 per cent greater if made of rectangular than if made of round wire. Again, it is evident that when the kind of spring required is given. SPRINGS. 343 and the values of C and f for the material of which it is to be made are known, the volume and hence the weight of the spring depends only on the product Pd lt and that as soon as P and d\ are given, the weight of the spring is fixed inde- pendently of its special dimensions. If, however, we fix any one dimension arbitrarily, the others must be so fixed as to satisfy the equations already given. Next, as to the values to be used for /and C, these will depend upon the nature of the special material of which the spring is made, and these can only be determined by experiment. Confining ourselves now to the case of steel springs, it is plain that /and 7 should be values corresponding to tempered steel. As an example, suppose we require the weight of a helical spring, which is to bear a safe load of 10000 Ibs. with a deflec- tion of one inch, assuming C= 12600000 and/= 80000 Ibs. per sq. in., and as the weight of the steel 0.28 Ib. per cubic inch. From formula (8) we obtain _, 2 X 12600000 X 10000 X i =39.4cu.m. Hence the weight of the spring must be (39.4) (0.28) = II Ibs. We may use either a single spring weighing 1 1 Ibs., or else two or more springs either side by side or in a nest, whose com- bined weight is 1 1 Ibs. Of course in the latter case they must all deflect the same amount under the portion of the load which each one is expected to bear, and this fact must be taken into account in proportioning the separate springs that compose the nest. FLAT SPRINGS. Let P, L, V, d, and d^ have the same meanings as before, and let 344 APPLIED MECHANICS, f= greatest allowable fibre stress for tension or compres- sion : R = modulus of elasticity for tension or compression ; /= length of spring; M l = maximum bending-moment under load P ; M= maximum bending-moment under load L. Moreover, the sections to be considered are all rectangular, and we will let b = breadth and h = depth at the section where the greatest bending-moment acts, the depth being measured parallel to the load. Then if / denote the moment of inertia of the section of greatest bending-moment about its neutral axis, we shall have f= M 12 We will now consider six cases of flat springs, and will de- termine P, tf, tf z , and V for each case, and for this purpose we only need to apply the ordinary rules for the strength and de- flection of beams. CASE i. Simple rectangular spring, fixed at one end and loaded at the other. I 3 L i*> I* f (24) E E (26) SPRINGS. 345 CASE 2. Spring- of uniform depth and uniform strength, tri- in plan , fixed at one end and loaded at the other. (27) (28) (29) (30) CASE 3. Spring of uniform breadth and uniform strength, parabolic in elevation, fixed at one end and loaded at the other. (31) (33) (34) CASE 4. Compound wagon spring, made up of n simple rec- tangular springs laid one above the other, fixed at one end and loaded at the other. 346 APPLIED MECHANICS. Let the breadth be b, and the depth of each separate layer be h. Then '- n bh* 6 / /' (35) i N^ 6 = 4 /* L nbh* E* (30 i =3=^ /'/ (37) i \ i y^ ^fi 1 * (38) CASE 5. Compound spring composed of n triangular springs laid one above the other, fixed at one end and loaded at the other. *-\r=r ^ 6 = -nW L E> < 4 > *=ji ; (4I) CASE 6. This case differs from the last in that in order to economize material we superpose springs of different lengths, SPRINGS. 347 and make them of such a shape that by the action of a single force at the free end they are bent in arcs of circles of nearly or exactly the same radius. The force P bends the lowest triangular piece AA in the arc of a circle. length of this piece is -. In order that the re- maining parallelopipedical portion may bend into an arc of the same circle it is necessary that it should have acting on it a uniform bending-moment throughout, and this is attained if it exerts a pressure at A l upon the succeeding spring equal to the force P, and following this out we should find that the entire spring would bend in an arc of a circle. The values of P, 6, d z , and Fare correctly expressed for this case by (39), (40), (41), and (42). For any flat springs which are supported at the ends and loaded at the middle, or where two springs are fastened to- gether, it is easy to compute, by means of the formulae already developed, by making the necessary alterations, the quantities P. 3 d z , and V, and this will be left to the student. COILED SPRINGS SUBJECTED TO TRANSVERSE STRESS. Three cases of coiled springs will now be given as shown in the figures, and the values of P, 3, d lt and Fwill be deter- mined for each. In each of these cases let R be the leverage, of the load, and let GO = angle turned through under the load. Then we may observe that all the three cases are cases of beams sub- jected to a uniform bending-moment throughout their length, this bending-moment being LR for load L and PR for load P. 348 APPLIED MECHANICS. CASES I and 2. Coiled spring, rectangular in section. f b>i i \ ^ = i/-^> (43) UP L , , (44) (45) f Rl CASE 3. Coiled spring, cir- cular in section. f = -^f^> (47) 64 l^_L . . ~* j* z^> \4w (49) E TIME OF OSCILLATION OF A SPRING. (46) (So) Since in any spring the load producing any displacement is proportional to the displacement, it follows that when a spring oscillates, its motion is harmonious. SPRINGS. 349 Suppose the load on the spring to be P t and hence its nor- mal displacement to be <S\. Now let the extreme displacements on the two sides of #, be # , and the force producing it />, so that the actual displacement varies from # x -f- tf to <$, <? , and the force acting varies from P -\- p to P p. Now. from the properties of the spring we must have =; /.*. = *.. (so Moreover, in the case of harmonic motion the maximum value of the force acting is - (see p. 104). But the load o oscillating is P instead of W, and the extreme displacement is 6 e instead of r. Hence we have (52) S d (S3) Hence the time of a double oscillation (54) g 35 APPLIED MECHANICS. CHAPTER VII. STRENGTH OF MATERIALS AS DETERMINED BY EXPERIMENT. 216. Whatever computations are made to determine the form and dimensions of pieces that are to resist stress and strain should be based upon experiments made upon the mate- rials themselves. The most valuable experiments are those made upon pieces of the same quality, size, and form as those to which the results are to be applied, and under conditions entirely similar to those to which the pieces are subjected in actual practice. From such experiments the engineer can learn upon what he can rely in designing any structure or machine, and this class of tests must be the final arbiter in deciding upon the quality of material best suited for a given service. An attempt will be made in this chapter to give an account of the most important results of experiments on the strength of materials, and to explain the modes of using the results. While the importance of making tests upon full-size pieces, and of introducing into the experiments the conditions of practice, is pretty generally recognized to-day, nevertheless there are some who have not yet learned to recognize the fact that attempts to infer the behavior of full-size pieces under practical conditions from the results of tests on small models, made under conditions which are, as a rule, necessarily, quite different from those of practice, are very liable to lead to con- clusions that are entirely erroneous. GENERAL REMARKS. 351 Such a proceeding is in direct violation of a principle that the physicist is careful to observe throughout his work, viz.: not to apply the results to cases where the conditions are essentially different from those of the experiments. When the quality of the material suited for a given service is known, tests of the material furnished must be made to determine its quality. Such tests, made upon small samples, should be of such a kind that there may be a clear understanding, as to the quality desired, between the maker of the specifications and the producer. Whenever possible, standard forms of specimens and standard methods of tests should be used. The determination of standards is occupying the at- tention of the Int. Assoc. for Testing Materials, the British Standards Committee, the Am. Soc. for Testing Materials, and others. To ascertain the quality of the material tensile tests are most frequently employed, their objects being to determine the tensile strength per square inch, the limit of elasticity, the yield-point, the ultimate contraction of area per cent, the ultimate elongation per cent in a certain gauged length, and sometimes the modulus of elasticity. While the standard forms and dimensions will be given later, the following general classification of the forms in use will be given here, viz. i. The specimen may be provided with a shoulder at each end, having a larger sectional area than the main body of the specimen, the section of this being uniform throughout as shown in Fig. a, the latter being of so great a length in proportion to the diameter that the stretch of i i i i i i i i i i i the specimen is not essentially I different from what it would FIG. a. be if the section were uniform throughout. The shoulders are, 35 2 APPLIED MECHANICS. of course, the portions of the specimen where the holders (or clamps) of the testing-machine are attached. 2. In the case of a round specimen of that kind there may be a screw-thread on the shoulders as shown in Fig. b. In the case of a brittle material, as 1^ JJJII cast-iron or hard steel, it is desirable to use a holder with a ball-joint, and to screw the specimen into the holder. 3. The specimen may be provided with a shoulder at each end, the main body of the specimen being, however, so short in proportion to the diameter that the stretch is essentially modified. Such a form is shown in Fig. c. FIG. c. FIG. d. 4. The specimen may be a grooved specimen as shown in Fig. d, where the length of the smallest section is zero. 5. The section of the specimen may be uniform through- out, the length between the holders being so great in propor- tion to the diameter that the stretching of the fibres is not interfered with. This form of specimen is shown in Fig. e. Assume a specimen of duc- I I 'tile material, as mild steel or wrought-iron, of the ist or the FlG - ' 5th shape, subjected to stress in the testing-machine, or else by direct weight, and suppose that we mark off upon the main body, i.e., the parallel section of the specimen, a gauged length of 8 or 10 inches (preferably 8 inches), and measure, by means of some form of extensom- eter, the elongations in the gauged length, corresponding to the stresses applied ; then plot a stress-strain diagram as shown in Fig. /, having stresses per square inch for abscissae, and the corresponding strains for ordinates. GENERAL REMARKS. 353 CO 111 5- 2 0" </ f ^-- 'B "eo -too Z .m I O .002 h u o -- **-*" .--* ^ -""" ^~~- ^-- ~^~ ^ - -- r^" -"" OC V 2000 6000 KM CO XX) 14( XX) 18 LOAD XX) 22 PER SQ. K)0 20000 30< IN, XX) 31000 3800C FIG. /. We shall find that the strains begin by being proportional to the stresses, but when a certain stress is reached, called the " limit of elasticity " or " elastic limit," shown at A, the strains increase more rapidly than the stresses, but the rate of increase in the ratio of the strain to the stress is not large until a stress is reached called the " yield-point " or "stretch-limit," shown at B, which is usually a little larger than the elastic limit ; and then the rate of increase of the ratio of strain to stress becomes much larger. Observe, also, that if a small load be applied to the piece under test, and then removed, the deformation or distortion caused by the application of the load apparently vanishes, and the piece resumes its original form and dimensions on the removal of the load ; in other words, no permanent set takes place. When the load, however, is increased beyond a certain point, the piece under test does not return entirely to its original dimensions on the removal of the load, but retains a certain permanent set. While permanent set that is easily determined begins at or near the elastic limit, and while the permanent sets corresponding to stresses greater than the elastic limit are much greater than the corresponding recoils, and hence form the greater part of the strains corresponding to such stresses, nevertheless experiments show that even a very small load will often produce a permanent set, and that the apparent return of the piece to its original dimensions is, 354 APPLIED MECHANICS. in a number of cases, only due to the want of delicacy in the measuring-instruments at our command. I- After the elastic limit and the yield-point have been passed, the ratio of the strain to the stress is much greater than before, the stretch becomes local, with a local contraction of area, this being due to the plasticity of the metal. Finally, when the maximum stress is applied, or, in other words, the breaking-stress, the behavior is apparently some- what different when the piece is subjected to dead weight from what it is when in a testing-machine. In the former case, when the maximum load is reached, the specimen continues to stretch rapidly, without increase in the load, until the specimen breaks. In the case of the testing-machine, however, the application of the maximum load causes, of course, the specimen to stretch, but this stretch naturally reduces the load applied, and the actual load under which the specimen separates into two parts is less, and often very considerably less, than the maxi- mum or breaking stress. Observe that the terms " breaking-load " and " breaking- stress " are always used to mean the " maximum load " and "maximum stress " respectively, and are never used to denote the load or the stress under which the specimen separates into two parts when the latter differs from the former. If the stretch of the specimen, as described above, is in any way interfered with, the behavior of the specimen will not be a proper criterion of the properties of the material ; the per- centage contraction of area at fracture will vary with the amount of interference with the stretch, and hence with the proportions of the specimen ; and the maximum or breaking strength will be greater than the real maximum or breaking strength per square inch of the material. Hence it follows that the 3d and 4th forms of specimen do not indicate cor- rectly the quality of the material, furnishing, as they do, erroneous values for both breaking-strength and ductility. CAST-IRON. 355 The quantities sought in such tests as those described above (with specimens of the 1st, 2d or 5th forms) are, as already stated : i. The breaking-strength per square inch of the material; 2. The limit of elasticity of the material ; 3. The yield-point or stretch-limit of the material; 4. The ultimate contraction of area per cent : 5. ^he ultimate elongation per cent in a given gauged length ; 6. The modulus of elasticity. The first gives, of course, the tensile str :h of the ma- terial ; the second and third ought both to be determined, but many content themselves with the third alone, since it is much easier to obtain. While they are commonly not far apart, it is a fact that certain kinds of stress to which the piece may be subjected may cause them to become very different from each other. The fourth and fifth are the usual ways of measuring the ductility of the metal ; and while the fourth is the most definite, the fifth is very much employed, and finds favor with most iron and steel manufacturers. The sixth is not often determined for commercial work, but it is one of the important properties of the metal. Of these six properties the two most universally insisted upon in specifications for material to be used in the construc- tion of structures or of machines are ductility, which is universally recognized as an all-important matter, and a suit- able breaking-strength per square inch, both a lower and an upper limit being generally prescribed for this last. On the other hand, although cast-iron and hard steel are brittle metals when compared with wrought-iron and mild steel, nevertheless it is true that the third and fourth forms of specimen will show too high results for tensile strength even in these materials on account of the interference with the stretch of the metal. APPLIED MECHANICS. 217. Cast- Iron. Cast-iron is a combination of iron with carbon, the most usual quantity being from 3 to 4 per cent. The large amount of carbon which it contains is its distinguishing feature, and determines its behavior in most respects. Besides carbon, cast-iron contains such substances as silicon, phosphorus, sulphur, manganese, and others. A considerable amount (more than 1.37 per cent as stated by Prof. Howe) of silicon forces carbon out of v combination and into the graphitic form, thus lowering the strength. Pig-Iron is the result of the first smelting, being obtained directly from the blast-furnace. The ore and fuel (usually coke, though anthracite coal is used to some extent, and some- times charcoal) are put into the furnace, together with a flux, which is usually limestone, in suitable proportions. The mass is brought to a high heat, a strong blast of heated air being intro- duced. The mass is thus melted, the fluid metal settling to the bottom, while slag, which is the result of the combination of the flux with impurities of the ore and fuel, rises to the top. The iron is drawn off in the liquid state and run into moulds, the result being pig-iron. The metal usually contains from 3 to 4 per cent of carbon, a part being chemically combined with the iron, and a part in the form of graphite. The larger the proportion of combined carbon, the whiter the fracture, and the harder and more brittle the product, while the larger the proportion of graphite, the darker the fracture, and the softer and less brittle the product. That which has most of its carbon in combination is called white iron, while that which contains a large proportion of graphite is called gray cast-iron. Pig-iron also contains silicon, sulphur, phosphorus, etc. The quantity of the first two can, to a certain extent, be controlled in the furnace, but not that of the last, so that if low phosphorus is desired, the ore and the fuel used must both be low in phosphorus. Gray cast-iron has been, and is sometimes classified in various CAST-IRON. 357 ways, according to the proportions of the combined carbon, and of the graphite, but the most modern practice is to sell, buy, and specify the iron by means of its chemical composition, and not by brands. That which contains the largest amount of carbon in mechan- ical mixture is, as a rule, soft and fusible, and hence suitable for making castings where precision of form is the chief desidera- tum, as its fusibility causes it to fill the mould well. For general use in construction, where strength and toughness are all-import- ant considerations, those grades are required which are neither extremely soft nor extremely hard. As to the adaptability of cast-iron to construction, it presents certain advantages and certain disadvantages. It is the cheapest form of iron. It is easy to give it any desired form. It resists oxidation better than either wrought-iron or steel. Its com- pressive strength is comparatively high when the castings are small and perfect. On the other hand, its tensile strength is much less than that of wrought-iron, or that of steel, averaging in common varieties from 16000 or 17000 to about 26000 pounds per square inch. It cannot be riveted or welded. It is a brittle and not a ductile material, it does not give much warning before fracture, and, while the stretch under any given load per square inch is decidedly larger than that of wrought-iron or steel, its total stretch before fracture is small when compared with wrought iron and steel. One of the dif- ficulties in the use of cast -iron in construction is its liability to initial strains from inequality in cooling. Thus if one part of the casting is very thin and another very thick, the thin part cools first, and the other parts, in cooling afterwards, cause stresses in the thin part. The fracture of good cast-iron should be of a bluish-gray color and close-grained texture. At one time cast-iron was extensively used for all sorts of structural work, but it was soon superseded by wrought-iron, and later by steel. Thus it is no longer used in bridgework, nor for floor- APPLIED MECHANICS. beams of a building, though it is still used to a considerable extent for the columns of buildings; and for this purpose it has in its favor the fact that it resists the action of a fire better than wrought iron or steel. Thus, in the present day, when the steel skeleton construction of buildings is so extensively employed, it is very necaesary to protect the steel beams and columns by covering them with some non-conducting material, as, otherwise, they would be liable to collapse in case of fire. It is used in cases where the form of the piece is of more importance than strength, and also where, on account of its form, it would be difficult or expensive to forge ; thus hangers, pulleys, gear-wheels, and various other parts of machinery of a similar character are usually made of cast-iron, as well as a great many other pieces used in construction. It is also used where mass and hence weight is an important consideration, as in the bed-plates and the frames of machines, etc. Malleable Iron. When a casting, in which toughness is required is to be made of a rather intricate form, it is frequently the .custom to malleableize the cast-iron, i.e., to remove a part of its carbon, and the result is provided the casting is small a product that can be hammered into any desired shape wher* c old, but is brittle when hot. A list of references to some of the principal experimental works on the strength and elasticity of cast iron will be given. i. Eaton Hodgkinson : (a) Report of the Commissioners on the Application of Iron to Railway Structures. (b) London Philosophical Transactions. 1840. (c) Experimental Researches on the Strength and other Prop- erties of Cast-Iron. 1846. 2. W. H. Barlow : Barlow's Strength of Materials. 3. Sir William Fairbairn : On the Application of Cast and Wrought Iron to Building Purposes. 4. Major Wade (U.S.A.) : Report of the Ordnance Department on the Experiments on Metals for Cannon. 1856. 5. Capt. T. J. Rodman : Experiments on Metals for Cannon. 6. Col. Rosset: Resistenzadei Principal! Metalli daBocchidi Fuoco. TENSILE STRENGTH OF CAST-IRON. 359 7. Tests of Metals made on the Government Testing Machine at Watertown Arsenal, 1887, 1888, 1889, 1890, 1891, 1892, 1893, l8 94 1896, 1897, 1898. 8. Transactions Am. Soc. Mechl. Engrs. for 1889, p. 187 et seq. 9. W. J. Keep: (a) Transverse Strength of Cast-iron. Trans. Am, Soc. Mechl. Engrs., 1893. (b) Relative Tests of Cast-iron. Trans. Am. Soc. Mechl. Engrs., 1895. (c) Transverse Strength of Cast-iron. Trans. Am. Soc. Mechl. Engrs., 1895. (d) Keep's Cooling Curves. Trans. Am. Soc. Mechl. Engrs., 1895. (e) Strength of Cast-iron. Trans. Am. Soc. Mechl. Engrs., 1896, 10. Bauschinger: Mittheilungenausdem Mech. Tech. Lab. Miinchen. Heft 12, 1885; Heft 15, 1887; Heft 27, 1902; Heft 28. 1902. ii. Tetmajer; Mittheilungen der Materialpriifungsanstalt Zurich. Heft 3, 1886; Heft 4, 1890; Hefte 5 and 9, 1896. 12. Technology Quarterly. October 1888, page 12 et seq. 13. Technology Quarterly. Vol. 7, No. 2; VoL 10. No. 3. 14. Transactions of the American Foundrymen's Association. 15. Transactions of the American Society for Testing Materials. 218. Tensile Strength of Cast-iron. As the use of cast-iron to resist tension has been almost entirely superseded by that of wrought-iron and steel, results of tests of full-size pieces of cast-iron in tension are not available. Tensile tests, however, have been extensively employed to determine the quality ; especially so when cast-iron cannon were in use; and tensile tests of cast- iron are still made, to a certain extent, for the determination of quality. For such tests standard specimens should be used, and attempts are being made to reduce their number. As the strength that should be attained in such specimens will become evident from the Standard Specifications of the Am. Soc. for Testing Materials, on page 385 et seq., only a few tensile tests will be quoted here, and those, for the purpose of 360 APPLIED MECHANICS. acquainting the reader with the results of some tensile tests of cast-iron. About 1840 Eaton Hodgkinson made a few experiments to determine the laws of extension of cast-iron, and for this purpose used rods lofeet long and i square inch in section. The tables of average results are given below. These tables show that the ratio of the stress to the strain of cast-iron varies with the load, growing gradually smaller as the load increases, that with moderate loads the ratio of stress to RESULTS OF NINE TENSILE TESTS. RESULTS OF EIGHT COMPRESSIVE TESTS. Weights Strains in Ratio of Weights Strains in Ratio of Laid on Fractions of Stress to Laid on Fractions of Stress to in Pounds. the Length. Total Strain. in Pounds. the Length. Total Strain. 1053.77 O.OOCO7 14050320 2064.75 ocoo. i 6 13214400 1580.65 o . ooo 1 1 13815720 4129.49 0.00032 12778200 2107.54 o . ooo i 6 13597080 6194.24 0.00050 12434040 3161.31 0.00024 13218000 8258.98 0.00066 12578760 4215.08 0.00033 12936360 10323.73 o . 00083 12458280 5268.85 o . 00042 12645240 12388.48 O.OOIOO 12357600 6322.62 0.00051 12377040 14453.22 0.00188 12245880 7376.39 0.00061 12059520 16517.97 0.00136 12132240 8430.16 0.00072 11776680 18582.71 0.00154 12050400 9483 . 94 0.00083 11437920 20647.46 0.00172 12013680 I0537-7I 0.00095 11314440 24776.95 0.00208 11911560 11591.48 0.00107 10841640 28906.45 0.00247 11679720 12645.25 O.OOI2I 10479480 33030 . 80 0.00295 11215560 13699.83 0.00139 9855960 14793.10 O.OOI55 9549120 strain for tension of cast-iron does not differ materially from that for compression, and that the difference increases as the load becomes greater. The agreement is even closer in the case of wrought-iron and steel. The gradual decrease of the ratio of stress to strain with the increase of load shows that Hooke's law, " Ut tensio sic vis" (the stress is proportional to the strain), does not hold true in RESULTS OF TESTS. 361 cast-iron. Hence, strictly speaking, cast-iron has no elastic limit and no modulus of elasticity, nevertheless we are accustomed to call the ratio of the stress to the strain under moderate loads the modulus of elasticity of the cast-iron. In making specifications intended to secure a good quality of 'cast-iron it is very common to call for a transverse test. Indeed the resolutions of the international conferences relative to uniform methods of testing recommend, in the case of cast- iron: (a) Test-pieces to be of the shape of prismatic bars no cm. standard length (43") and to have a section of 3 cm. square (i".i8), one having an addition on one end, from which cubes can be cut for compression tests. (b) Three such specimens to be tested for transverse strength. (c) The tensile strength to be determined from turned test- pieces 20 mm. (o".785) diameter and 200 mm. (7". 85) long, cut from the two ends of the test-pieces broken by flexure. (d) The compressive strength to be determined from cubes 3 cm. (i".i8) on a side cut from the first specimens, pressure to be applied in the direction of the axis of the original bar. These requirements, while calling for transverse tests, call also for tensile and compressive tests. T .* Siandard Specifications of the Am. Soc. for Testing Mater.;. Js w.ll be found on page 385 et seq. Inasmuch as the tensile strength has been, and is also made the basis of specifications for cast-iron, it is important to con- sider what should be attained in this regard. For this purpose a few tables of comparatively modern tests will be given here, and it will be seen that in the ordinary varieties of cast-iron it is easy to secure tensile strengths from 16,000 to 25,000 pounds per square inch, and that more can be secured by taking proper precautions in the manufacture. Indeed cast-iron which, when tested in the form of a grooved specimen, shows a tensile strength of at least 30,000 5 62 APPLIED MECHANICS. pounds per square inch is called gun-iron, this having been a requirement of the United States Government, in the days of cast-iron cannon, for all cast-iron that was to be used in their manufacture. The following table is taken from a paper on the Strength of Cast-Iron, by Mr. W. J. Keep, published in the Transactions of the American Society of Mechanical Engineers for 1896, and it gives the averages of the tensile strengths of the fifteen different series of tests recorded in the paper. This table is given here merely as an example of the results that can be obtained by tension tests upon usual varieties of cast-iron. The table is as follows : AVERAGES OF TENSION TESTS OF ROUND BARS. Area of Section, Area of Section Area of Section Area of Section, 0.375 Sq. In. 1.12 Sq In. o 375 Sq. In. 1. 12 Sq. In. . No. of Breaking Load per Sq. Inch. Breaking Load per Sq. Inch Series. Breaking Load per Sq Inch. Breaking Load per Sq. Inch. J 2OOOO 1 57OO 14800 2 20580 22500 IO oeo^O 2O4CO 1 1 I7OOO 4 21850 19350 12 17700 17500 5 22425 19750 13 I4OOO 2I3OO 6 25550 17200 H 24400 2O3OO 7 18950 17700 15 23525 20500 8 17700 15350 The following table of results of tension tests of ordinary cast-iron from another source will also be given for the same purpose as Mr. Keep's results : CAST-IRON. 363 CAST-IRON TENSION. a .2 |.S c .2 O.S u 1-3 "" u J <" Dimensions. *3 ll Modulus of Elasticity. Dimensions. c S v 3 a 8 en Modulus of Elasticity. .2 .Q f" Ir 3d .03 X .04 .06 19340 14857000 .00 X .00 .00 17100 13333000 .03 X .02 .05 23910 15481000 .00 X .02 .02 19068 13680000 .00 X 98 .98 21180 15238000 .00 X .00 .00 1 8000 13333000 .00 X 97 97 23227 15881000 .00 X .02 .02 19299 12057000 .ot X .06 .08 19830 14539000 .06 X .98 o? 17488 13249000 . X .03 03 20413 17632000 .00 X .98 .98 19500 13250000 .93 X .00 93 16774 14337000 .02 X .02 03 20747 14543000 .00 X -oo .00 18600 15383000 .03 X .03 .06 18620 13434000 .00 X .00 .00 18000 16666000 .00 X .00 .00 18910 13043000 .00 X .00 .00 19400 17911000 .00 X .00 .00 20950 15789000 .00 X .00 .00 19900 15000000 .00 X .00 .00 22900 15000000 .00 X .02 .02 19594 13373000 .00 X .00 .00 22400 15564000 .01 x .03 .04 16341 13108000 .00 X .00 .00 21300 15384000 .01 X .03 .04 '3844 13640000 .00 X .02 .02 19692 i 5966000 .02 X .08 .01 13798 11840000 .ot X .03 .05 21005 15075000 .00 X .C2 .02 17647 12787000 08 < 2; 33 20600 11900000 .03 X .03 .06 14025 12^68000 .05 X .03 03 17067 12676000 .04 X .02 .06 15083 13466000 00 X .03 .03 19900 12929000 .02 X .04 06 16874 9751900 .08 X .03 .02 16404 12577000 .00 X .00 .00 aoooo 13043000 i .co X .02 .02 16450 12570000 Colonel Rosset, of the Arsenal at Turin, made a series of experiments upon the influence of the shape of the specimen upon the tensile strength. For this purpose he used specimens with shoulders ; and, among other tests, he compared the strength of the same iron by using specimens the lengths of whose smallest parts were respectively i metre, 30 millimetres, and o millimetres, with the following results : Length of Specimen. Tensile Strength, in Ibs., per Square Inch. ist Cannon. 2d Cannon. 3d Cannon. i metre . . 30 millimetres . o millimetres . 31291 3 2 57I 33993 25601 345 62 36411 28019 30011 30011 APPLIED MECHANICS. It will thus be seen that, before we can decide upon the quality of cast-iron as affected by the tensile strength, it is necessary to know the length of that part of the specimen which has the smallest area. Colonel Rosset's tests of cast iron were almost entirely confined to high-grade irons, suitable to use in cannons. He deduced, for mean value of the modulus of elasticity of the specimens i metre in length, 20419658 Ibs. per square inch : this, of course, is a modulus only adapted to these high grades, and is not applicable to common cast-iron. 219. Cast-Iron Columns. In consequence of the high compressive strength shown by cast-iron when tested in small pieces, and in pieces free from imperfections, it was once considered a very suitable material for all kinds of columns. Nevertheless, its use for the compression members of bridge and roof trusses has been abandoned; cast-iron having been displaced first by wrought-iron and subsequently by steel, which is the substance now in use for these purposes. The principal reasons for the change are the lack of ductility, and the consequent brittleness of cast-iron, that it cannot be riveted, and that if it breaks it cannot be eas'ly repaired. Cast-iron is, however, used to a very considerable extent for the columns of buildings. The Gordon, the so-called Euler, and the Hodgkinson formulae for the breaking-strength of cast-iron columns, have all been given in paragraphs 208, 2080, and 209. They are, however, all based upon tests made upon very small columns, and do not give results agreeing with the tests of such full-size columns as are used in practice. We will next consider, therefore, the tests that have been made upon full-size cast-iron columns, and the conclusions that are warranted in the light of these tests. Two sets of tests of cast-iron mill columns have been made on the Government testing-machine at Watertown Arsenal; an account of these sets of tests is published in their reports of 1887 and of 1888. CAST-IRON COLUMNS. 365 The first lot consisted of eleven old cast-iron columns, which had been removed from the Pacific Mills at Lawrence, Mass., during repairs and alterations. The second lot consisted of five new cast-iron columns cast along with a lot that was to be used in a new mill. Of these five, the strength of two was greater than the capacity of the testing-machine, hence only three were broken ; while in the case of the other two the test was discontinued when a load of 800000 Ibs. was reached. All the columns con- tained a good deal of spongy metal, which of course rendered their strength less than it would otherwise have been ; never- theless, inasmuch as this is just what is met with in building, it is believed that these tests furnish reliable information as to what we should expect in practice, and that this information is much more reliable than any that can be derived from test- ing small columns. In all the tests the compressions were measured under a large number of loads less than the ultimate strength ; but in- asmuch as it is not possible, in the case of cast-iron, to fix any limits within which the stress is proportional to the strain, no attempt will here be made to compute the modulus of elas- ticity. Hence there will be given here a table showing the dimensions of the columns tested, their ultimate strengths, and, in those cases where they were measured, the horizontal and vertical components of their deflections, measured at the time when their ultimate strengths were reached, as the Govern- ment machine is a horizontal machine. A glance at the table will make it evident that we Cannot, in the case of such columns, rely upon a crushing strength any greater than 25000 or 30000 Ibs. per square inch of area of section. Hence it would seem to the writer that, in order to proportion a cast-iron column to bear a certain load in a building, we should determine the outside diameter in such a way as to avoid an excessive ratio of length to diameter ; if this ratio is not much in excess APPLIED MECHANICS. of twenty, the extra stress produced by any eccentricity of the load due to the deflection of the column will be very slight. At the same time see that the thickness of metal is sufficient to insure a good sound casting. Now, having figured the column in this way, compute the outside fibre stress (using the method of 207) that would occur with the loading of the floors assumed to be such as to give as great an eccentricity as it is possible to bring upon the column. If this distribution of the load is one that is likely to occur, then the maximum fibre stress in the column due to it ought not to be greatly in excess of 5000 Ibs. per square inch ; but if it is one which there is scarcely a chance of realiz- ing, then the maximum fibre stress under it might be allowed to reach 10000 Ibs. per square inch. If by adopting the di~ mensions already chosen these results can be obtained, we may adopt them ; but if it is necessary to increase the sectional area in order to accomplish them, we should increase it. Another matter that should be referred to here is the fact that a long cap on a column is more conducive to the produc- tion of an eccentric loading than a short one ; hence, that a long cap is a source of weakness in a column. Other sources of weakness in cast-iron columns are spongy places in the casting (which correspond in a certain way with knots in wood), and also an inequality in the thickness of the two sides of the column, the result of this being the same as that of eccentric loading; and it is especially liable to occur in consequence of the fact that it is the common practice to cast columns on their side, and not on end. The engineer should, however, inspect all columns to be used in a building, and reject any that have the thickness of the shell differing in different parts by more than a very small amount. A series of tests of full-size cast-iron columns was made by the Department of Buildings of New York City, under the direction of Mr. W. W. Ewing, in December, 1897, upon the CAST-IKON COLUMNS. 367 Remarks. rt T3 rt rt ^aJ w nj w T3 W iple flexure. Post con- t middle before testing, iple flexure. C U V en - ^ H 2 |i o ,_, ^ ** j ID U aanx; e flexure t 3 X U,^ *u* *>1'X X 3 3 X 3 X c g2 c 1 JfS S 1C oS^o-S "S-e-c cx CX CX ex a. a. .0^ -5 . w . _^tc ^ ^ " S "3 T3 - - 1 -o a 3 '3 j5' 3 'rt js '3 *o 2f 'O <u <u (3 (3 i 'rt rt rt rt rt rt 'rt "rt '1 lid 3 "5 -2 o 0^ CO rfr C^ HH M M en O t i a s > .2 c/5 -i- 8 in 6 m en 6 6 m oo O 'S a s Q 2 B D o 6 in N O co en oo M 6 CO en " -S r rt tuo a . 5 w ^ O O Q O O co m O "^ r^ oo en m co en co en en o O en ^ en w en C-O cr- C) co CO o in CO en Cl CO o en en R 2 m r^ 8 lip p w 1 1 1 1 N *, rf in w M en O M o in in ^- en en 352OOO in in 1 i CO 00 m en 8 O CM *! O in r^* m rt o CO en ,, vn co m en co H CO o 8 Cl 8 -f 4- -t r^ Tf to ^ t*** QO ! f|f s 5 tJ < cr i^ .be <i ' V in O rt rt rt *^ ^ eg . ST CN o m ft IO CO M en * 2 S " " " co n in O ^ co en r^ en r*^ t^ \o ^ ^ 5 in O -1- CM * M g 1 11 U O M N en *f in 8: r-. $ 1 1 1 sutunjoD M3JSI sutunjoo PIO 3 68 APPLIED MECHANICS. i * 513 Mb! --tMAJsJi. S-'OI- Sg-'6-OI CAST-IRON COLUMNS. 369 370 APPLIED MEGHAN rCS. hydraulic press of the Phoenix Bridge Works. This press weighs the load on the specimen plus the friction of the piston, the latter being, of course, a variable quantity. Nevertheless great pains were taken to determine this friction, and hence the results are doubtless substantially correct. The results are, it will be seen, similar to those obtained in the Watertown tests. The table of results is given below, and no farther comments are needed. Subsequently tests were made to determine the strength of the brackets. For this, however, the reader is referred to the Report itself, or to Engineering News of January 2O, 1898, and for further details of the tests of the columns, to the Report itself, or to Engi- neering News of January 13, 1898. Column Number. Length, Inches. Outside Diameter, Inches. Average Thick- ness, Inches. Breaking Load, Lbs. Average Area Sec- tion, Sq. In. Inches / P Break- ing Load per sq. in., IDS. I 190.25 15 1356000 43.98 4.96 .38.36 30832 II 190.25 15 ^ 1330000 49-03 4.92 38.67 27126 B* 190.25 15 \ 1198000 49-03 4.92 38.67 24434 B< 190.25 15* \ 1246000 49.48 4.98 38.20 25l8l 5 190.25 15 H 1632000 50.91 4.91 38.75 3 20 57 over over 6 I90.2 15 'A 2082000 51.52 4,90 38.73 ** ' +* 40411 XVI 1 60 81 to 7 f i 651000 21.99 2.50 64.00 29604 XVII 1 60 8 '& 645600 22.87 2.48 64.52 28229 7 t20 6 T V *& 455200 17.64 1.78 67.41 25805 8 1 2O 6ft i*V 474100 17-37 1.8o 66.67 27236 CAST-IRON COLUMNS. 50000 40000 30000 20000 70 80 90 100 110 120 Abscissge= length di- vided by radius of gyration of small- est section. Ordinates:= breaking strengths per square inch of smallest section. 130 no 150 CAST-IRON COLUMNS. 371 The cut on page 370 shows a graphical representation of the preceding tests of full-size cast-iron columns. In Heft VIII (1896) of the Mitt. d. Materialpriifungsanstalt in Zurich is an account of 296 cast-iron struts tested by Prof. Tetmajer; 46 being 3 cm. (i". 1 8) square will not be men- tioned farther. The other 250 were hollow circular, the inside diameters being 10 cm. (3". 94), 12 cm. (4". 72), or 15 cm. (5".9i); the thicknesses being i cm. (o".39) oro.8 cm. (0^.31). The lengths varied from 4 m. (13'. 12) to 20 cm. (?".g). They are not the most usual thicknesses of columns for buildings, though used to a considerable extent. They might be called cast-iron pipe columns. The following table contains all those 250 cm. (8'.2) long and over, and i cm. thick, and one set of those 0.8 cm. thick. This will exhibit the character of the results for such columns of usual lengths. In computing the actual Thickness o". 39. Thickness o".3i. Outside Ultimate Outside Ultimate No. of Test. Length, Feet. Diame- ter, I p Strength, Pounds No. of Test. Length, Feet. Diame- ter, / p Strength, Pounds Inches. per sq. in. Inches, per sq. in. 55 56 9.84 9.84 S3 7 7?: 7 18481 20761 207 208 13.12 13.12 4.62 .61 103.9 103.9 11518 11660 57 8.20 4.76 63.4 28156 209 n. 4 8 58 91.1 16922 58 8.20 4.78 639 2986-2 210 n. 4 8 59 91.9 -0577 69 70 7* 9.84 9.84 8.20 5-63 5.6z 5.65 64.4 64.4 53-3 24174 32564 36546 211 212 213 9.84 9.84 8.20 56 .60 .56 78.9 77-7 65-4 194.12 19482 3*843 72 8.20 5.63 53-4 47353 3I 4 8.20 4.61 64.7 33 '33 86 9.84 6.69 53-2 32564 225 13.12 5-41 87.8 15216 87 9.84 6.67 53-3 34270 226 13-12 5-43 87.5 17623 88 8.20 6-73 43-9 44224 227 11.48 5-41 76.7 22326 89 8.20 6.69 44.1 46642 228 229 11.48 9.84 5-39 5-39 77-3 66.4 2I3I 23748 230 9.84 5-41 66.1 23463 23 1 8.20 5-41 54- 8 38110 2 3 2 8.20 5-41 54-5 36688 243 13.12 6.56 71.8 22041 244 13.12 6-54 71.9 24885 245 11.48 6.56 62.6 27729 2 4 6 11.48 6.56 62.5 28156 247 9.84 6-53 S3-9 355 20 2 4 8 9.84 6-54 53-9 31853 249 8.20 6.56 44-8 4*949 2 5 8.20 6.56 44.8 453^2 372 APPLIED MECHANICS. length of the strut has been used, whereas Tetmajer adds to this 9 ".84, the thickness of the platforms of the machines, as they bore o;i knife-edges. Prof. Bauschinger of Munich made two series of tests of full-size cast- and of wrought-iron columns to determine the effect of heating them red-hot and sprinkling them with water while under load. They were loaded in his testing-machine with their estimated safe load as calculated from the formulas. For cast-iron, 19912^4 ./" For wrought-iron, i -|- 0.0006-5 11378^4 *' r , i -f- 0.00009 2 where P = safe load (factor of safety five), A = area of section, / = length, p least radius of gyration, pounds and inches being the units. A fire was made in a U-shaped receptacle under the post, so arranged that the flames enveloped the post. The tem- perature was determined from time to time by means of alloys of different melting-points ; and the horizontal and vertical components of the deflections were read off on a dial as indi- cated by a hand attached to the post by a long wire. The post was also examined for cracks or fractures. In the 1884 series he tested six cast-iron posts of various styles, and three wrought-iron posts, one of them being made of channel-irons and plates put together with screw-bolts, one of I irons and plates also put together with screw-bolts, and one hollow circular. The details of the tests will not be given here, but only Bauschinger's conclusions. He said : That wrought-iron columns, even under the most favorable CAST- 1 RON- COLUMNS. 373 adjustment of their ends and of the manner of loading, bend so much that they cannot hold their load, sometimes with a temperature less than 600 Centigrade, and always when they are at a red heat ; and this bending is accelerated by sprink- ling on the opposite side, even when only the ends of the post are sprinkled. . That under similar circumstances cast-iron posts bend, and this bending is increased by sprinkling ; but it does not exceed certain limits, even when the post is red for its entire length and the stream of water is directed against the middle, and the post does not cease to bear its load even when cracks are de- veloped by the sprinkling. Only when both ends of a cast-iron post are free to change their directions does sprinkling them at the middle of the opposite side when they are red make them break, but such an unfavorable case of fastening the ends hardly ever occurs in practice. That the cracks in the columns tested occurred in the smooth parts, and not at corners or projections. That the result of these tests warns us to be much more prudent in regard to the use of wrought-iron in building. If posts which are subjected to a longitudinal pressure bend so badly when subjected to heat on one side that they lose the power of bearing their load, how much more must this be the case with wrought-iron beams ; and he urges the importance of making more experiments. In Heft XV of the Mittheilungen he says that the results were criticised in two ways, viz. : Moller claiming that he should have used different constants, and Gerber that the wrought-iron posts were not properly made. Bauscliinger therefore concluded to make a new set of tests, and for this purpose he had made two cast-iron and five wrought-iron columns the former being carefully cast, but on the side, while the wrought-iron ones were made by a bridge company of very good reputation, and four of them were similar to those made at the time for a new warehouse in Hamburg. 374 APPLIED MECHANICS. The tests were made just as before, and the following are his conclusions : That when wrought-iron posts are as well constructed as the two referred to, they resist fire and sprinkling tolerably well, though not as well as cast-iron ; but that posts con- structed like the other three, even with the fire alone, and before the sprinkling begins, get so bent that they can no longer hold their load. Good construction requires that the rows of rivets shall extend through the entire length of the post, and the rivets should be quite near each other ; but the tests are not extensive enough to show what are the necessary requirements to make wrought-iron posts able to stand fire and sprinkling ; in order to know this more experiments are needed. In Dingler's Polytechnisches Journal for 1889, page 259 et seq., is an article by Professor A. Martens, of Berlin, uprn the behavior of cast- and wrought-iron in fires, considering especially the burning of a large warehouse in Berlin, and advocating the protection of iron-work by covering it with cement. He says that there are two series of tests upon this subject, one of which is the tests of Bauschinger already explained, and the other a set of tests made by Moller and Luhmann. No detailed account of these tests will be given here, but only Holler's conclusions, as stated by Prof. Martens, which are as follows: i. With ten cast-iron posts he could not get any cracks by sprinkling at a red heat; but it is to be noted that his were new posts, while those used in Bauschinger's first series were old ones, and that those in Bauschinger's second series, which were new and very carefully cast, did not show cracks either. 2. He claims that while the cracks would allow the post still to bear a centre load, it could not bear an eccentric load or a transverse load. 3. "He claims that the load on a cast-iron post should be limited to one which shall not produce sufficient bending to bring about a tensile stress anywhere when the post is bent by the heat and sprinkling. TRANSVERSE STRENGTH OF CAST-IRON. 375 4. He claims that in either cast- or wrought-iron posts, if the ends are not fixed, the ratio of length to diameter should not exceed I o, whereas if they are it should not exceed 17; also, that there is no such thing as absolute safety from fire with iron. 5. A covering of cement delays the action of the fire, and that therefore such a covering is a protection to the post against excessive one-sided heating and cooling. 6. Cast-iron is more likely to have at any one section a collection of hidden flaws than wrought-iron. 220. Transverse Strength of Cast-iron. At one time cast-iron was very largely used for beams and girders in build- ings to support a transverse load. Its use for this purpose has now been almost entirely abandoned, as it has been superseded by wrought-iron and steel. A great many experiments have been made on the trans- verse strength of cast-iron ; the specimens used in some cases being small, and in others large. The records of a great many experiments of this kind are to be found in the first four books of the list already enumerated in 217. The details of these tests will not be considered here, but an outline will be given of some of the main difficulties that arise in applying the results and in using the beams. Cast-iron is treacherous and liable to hidden flaws ; it is brittle. It is also a fact that in casting any piece where the thickness varies in different parts, the unequal cooling is liable to establish initial strains in the metal, and that therefore those parts where such strains have been established have their breaking-strength diminished in proportion to the amount of these strains. In the case of cast-iron also, the ratio of the stress to the strain is not constant, even with small loads, and is far from constant with larger loads ; also, inasmuch as the compressive strength is far greater than the tensile, it follows that, in a transversely loaded beam which is symmetrical above and be- low the middle, the fibres subjected to tension approach their 37 6 APPLIED MECHANICS. full tensile strength long before those subjected to compression are anywhere near their compressive strength. The result of all this is, that if a cast-iron beam be broken transversely, and the modulus of rupture be computed by using the ordinary formula, f-My 7 " I ' we shall find, as a rule, a very considerable disagreement be- tween the modulus of rupture so calculated and either the tensile or compressive strength of the same iron. Indeed, Rankine used to give, as the modulus of rupture for rectangu- lar cast-iron beams, 40000 Ibs. per square inch, and for open- work beams 17000 Ibs. per square inch, which latter is about the tensile strength of fairly good common cast-iron. A great deal has been said and written, and a good many experiments have been made, to explain this seeming disagree- ment between the modulus of rupture as thus computed, and the tensile strength of the iron. Barlow proposed a theory based upon the assumption of the existence of certain stresses in addition to those taken account of in the ordinary theory of beams, but his theory has no evidence in its favor. Rankine claimed that the fact that the outer skin is harder than the rest of the metal would serve to explain matters, but this would not explain the fact that the discrepancy exists in the case of planed specimens also. Neither Barlow nor Rankine seems to have attempted to find the explanation in the fact that the formula assumes the proportionality of the stress to the strain, and hence that is less and less applicable the greater the load, and hence the nearer the load is to the breaking load. An article by Mr. Sondericker in the Technology Quarterly of October, TRANSVERSE STRENGTH OF CAST-IRON. 377 1888, gives an account of some experiments made by him to test the theory that " the direct stress, tension, or compression, at any point of a given cross-section of a beam, is the same function of the accompanying strain, as in the case of the cor- responding stress when uniformly distributed," and the results bear out the theory very well ; hence it follows that, if we use the common theory of beams, determining the stresses as such multiples of the strains as they show themselves to be in direct tensile and compressive tests, the discrepancies largely vanish, and those that are left can probably be accounted for by initial stresses due to unequal rate of cooling, and by the skin, or by lack of homogeneity. In the same article he quotes the results of other tests bearing more or less on the matter, and there will be quoted here the table on page 378. If, therefore, we wish to make use of the formula y in calculating the strength of cast-iron beams, we cannot use one fixed value of f for all beams made of one given quality of cast-iron, but we shall have to use a very varying modulus of rupture, varying especially with the form, and also with the size of the beam under consideration. Now, in order to do this, and obtain reasonably correct results, we need, wherever possible, to use values of f that have been deduced from ex- periments upon pieces like those which we are to use in prac- tice, and under, as nearly as possible, like conditions. There are not very many records of such experiments avail- able, and, in cases where we cannot obtain them, it will prob- ably be best to use a value of f no greatei than the tensile strength for complicated forms, and forms having thin webs. For pieces of rectangular or circular section we might probably use, for good fair cast-iron, 25000 to 30000 Ibs. per square inch. A few tests of the character referred to have been made in the engineering laboratories of the Massachusetts Institute of 3 ;8 APPLIED MECHANICS. Modulus oil Form of Beam Sec- Tensile Strength, Rupture , My J ~7~< Ratio. Condition of Experimenter. tion, \bs. per Sq. In. / Ibs. perSq. Specimen. In. 19850 41320 2.08 Turned C. Bach.* x?H^, 16070 35500 2.21 Turned Considere.f 34420 63330 1.84 Turned Considere. \Hx 24770 54390 2.19 Turned Robinson and Segundo. J 25040 46280 1.8 5 Rough Robinson and Segundo. Mean. 2.03 16070 29250 1.82 Planed Considere. ijlflp 36270 58760 1.62 Planed Considfere. 19090 33740 1-77 Planed C. Bach. Mean. 1.74 19470 34000 1-75 Planed C. Bach. in 31430 49030 1.56 Planed Considere. Hi 19880 33860 1.70 Planed Sondericker. 24770 42340 1.71 Planed Robinson and Segundo. g^^ 25040 42IIO 1.68 Rough Robinson and Segundo. Mean. 1.68 19470 28150 1-45 Planed C. Bach. 16070 225OO 1.40 Planed Considere. 31860 36640 I-I5 Planed Considere. 25040 3I3IO 1.25 Rough Robinson and Segundo, Mean. I-3I 16070 23780 1.48 Planed Considere. n 31290 34730 I. II Planed Considere. n 18050 24550 1.36 Planed Sondericker. ^^ 22470 26150 1.16 Rough Burgess and Viel6. Mean. 1.28 See Zeitschrift des Vereines Deutscher Ingenieure, Mar. 3d and loth, 1888. t See Annales des Fonts et Chausse"es, 1885. t See Proceedings Institute of Civil Engineers, Vol 86. 5 Sec Proceedings Am. Soc. Mecbl. Engrs. 1889, pp. 187 et seq. TRANSVERSE STRENGTH OF CAST-IRON. 379 Technology, and a brief statement of" them will be given here. The first that will be referred to here is a series of experiments made by two students of the Institute, an account of which is given in the Proceedings of the American Society of Mechani- cal Engineers for 1889, pp. 187 et seq. The object of this investigation was to determine the trans- verse strength of cast-iron in the form of window lintels, and also the deflections under moderate loads, and from the latter to deduce the modulus of elasticity of the cast-iron, and to compare it with the modulus of elasticity of the same iron, as determined from tensile experiments ; also the tensile strength and limit of elasticity of specimens taken from different parts of che lintel were determined. The iron used was of two qualities, marked P and 5 respec- tively. The tensile specimens were cast at the same time, and from the same run as the lintels. Besides this, one of each kind of window lintels was cut up into tensile specimens, and the specimens were so marked as to show from what part of the lintel they were cut. The tables of tests will now be given, and the following ex- planation of the symbolism employed. P and S are used, as already stated, to denote the quality of the iron. A and B are used to denote, respectively, that the specimen was unplaned or planed. I, 2, 3, etc., denote the number of the test made on that particular kind and condition. 380 APPLIED MECHANICS. I., II., III., denote that the piece has been taken from a lintel, and also from what part, as will easily be seen by the sketch on page 379. Thus P. B. 3 would signify that the specimen was of quality P 9 had been planed, and was the third test of this class. On the other hand, P. B. 3 II., would signify in addition that it had been taken from a lintel, and was a piece of one of the strips marked II. in the sketch. The following is a summary of the breaking-weights per square inch of the specimens not cut from the lintels : P. A. i 23757 S. A. i 24204 P. A. 2 21423 S. A. 2 25258 P. A. 3 18938 S. A. 3 24706 P. A. 4...., 21409 3)74168 24723 21382 P. B. i 21756 P. B.3 25207 2)46963 S. B. i S. B, 2 29574 23201 2)52775 23482 26388 The following are the breaking-weights per square inch of the specimens cut from the iinteis : P B. ' 5 I 19651 6 I 20715 9 I 21076 10 I 21483 4 II 19016 7 II 19376 ii II 22146 12 II 20552 2 III 10594 (Broke at a flaw.) 13 III 16141 I 8 IV 10616 S. B. 6 I 29124 7 I 28372 8 I 25425 3 II 24704 4 II 29414 5 II 23610 9 III 27523 10 III 18301 4 IV 19616 TKAffSVERSE STRENGTH OF CAST-IRON. 381 All the window lintels tested were of the form shown in the figure, and all were supported at the ends and loaded at the middle, the span in every case being 52". From the cut it will be seen that the web varied in height, being 4 inches high above the flange in the centre, and decreasing to 2.5 inches at the ends over the supports. The following are the results of the separate tests, where tensile modulus of rupture means the outside fibre stress per square inch on the tension side, and compressive modulus of rupture that on the compression side, both being calculated from the actual breaking load by the formula f- J ~ ' ' Mark on Lintel. Breaking-Load, Lbs. Tensile Modulus of Rupture, Ibs. per Sq. In. Compressive Modulus of Rupture, Ibs. per Sq. In. P. I 27220 26648 81578 P. 2 30520 29879 91467 P. 3 27200 26659 81608 S. i 26750 26198 80164 S. 2 19850 19433 59490 S. 3 28670 28068 85924 S. 4 25120 24592 75285 The second series of experiments was made by two other students, and an account of the work is given in the same article as the former one. The object was to determine the constants suitable to use in the formulae for determining the strength of the arms of cast-iron pulleys ; and also, incidentally, to determine the hold- ing power of keys and set-screws. Some old pulleys with curved arms, which had been in use at the shops, were employed for these tests. They were all 382 APPLIED MECHANICS. about fifteen inches in diameter, and were bored for a shaft I T ^ inches in diameter. Inasmuch as this size of shaft would not bear the strain necessary to break the arms, the hubs were bored out to a diameter of i-j-J- inches diameter, and key-seated for a key one- half an inch square. In order to strengthen the hubs sufficiently, two wrought- iron rings were shrunk on them, so as to make it a test of the arms and not of the hub. The pulley under test is keyed to a shaft which, in its turn, is keyed to a pair of castings supported by two wrought-iron I- beams, resting upon a pair of jack-screws, by means of which the load is applied. A wire rope is wound around the rim of the pulley, and leaves it in a tangential direction vertically. This rope is connected with the weighing lever of the machine, and weighs the load applied. In a number of the experiments one arm gave way first, and then the unsupported part of the rim broke. The breaking-load of the separate pulleys was, of course, determined, and then it was sought to compute from this the value of f from the formula which is the one most commonly given for the strength of pulley arms, and which is based upon several erroneous assump- tions, one of which is that the bending-moment is equally divided among the several arms. In this formula /= moment of inertia of section, n = number of arms, y = half depth of each arm = distance from neutral axis to outside fibre, x = length of each arm in a radial direction, P = breaking-load determined by experiment. TRANSVERSE STRENGTH OF CAST-IRON. 383 The results are given in the following table, the units being inches and pounds : the the o o H rt cl II tly inc broke. H !i ii sad subsequently when the rim brok .sg ~| S "5 - II S*53 j- -S:r c 3 13 33 la 5*1 8JS u i !! iflSrf" 2 S^ 13 s2 is^-sl. O H C- J3- O H rt | 1 I =3 s- U^3 O I .2 "u Q S XXX P x L-^C X X < t-pj t-bj Mps otjeo X X X -p * X N|m X X stnay jo aaqtnnj^ | souy a " qn H co co co jo ssaujpiqi, -C -C -^ M H W H co co CO Xannj jo -oreia IT rt 2" "? iT ? APPLIED MECHANICS. In the cases of numbers 5, 7, 8, 9, and 10 some of the arms were not broken, the rims were now broken off, and the re- maining arms were tested separately, the pull being exerted by a yoke hung over the end of the arm, the lower end being at- tached to the link of the machine. The arms were always placed so that the direction of the pull was tangent to the curve of the rim at the end of the arm. The actual modulus of rupture was then determined by calcula- tion from the experimental results, and is recorded in the following table, the units being inches and pounds : Number of Arms. Dimensions of Sec- tion at Fracture : all elliptical. Bend of Arm with or against Load. Modulus of Rupture. AverageModulus of Rupture for each Pulley. 5 i i* X iJ against 4539 6 45396 7 i i* X* against 36802 7 2 'if Xf against 39537 7 3 itfxi with 46407 40915 8 i itt x H against 35503 8 2 iH x|| against 36091 8-3 iX with 39939 8-4 iHxtt with 42469 38500 9 1 i*Xf against 41899 9 2 '*XH against 44148 9 3 i*xf with 55442 47163 10 I if XH against 54743 10 2 IXH against 5943 io 3 ix against 38605 10 4 if x ft with 55 2 29 49880 -^^ STANDARD SPECIFICATIONS FOR CAST-IRON. 385 STANDARD SPECIFICATIONS FOR CAST-IRON, OF THE AMERICAN SOCIETY FOR TESTING MATERIALS. The standard specifications for cast-iron, of the American Society for Testing Materials, contain specifications for i Foundry Pig-iron, 2 Gray Iron Castings, 3 Malleable Iron Castings, 4 Locomotive Cylinders, 5 Cast-iron Pipe and Special Castings, 6 Cast-iron Car-wheels. Of these, i, 2, and 4 will be quoted in full, and extracts will be given from 5. For the remainder see the proceedings of the Society. AMERICAN SOCIETY FOR TESTING MATERIALS. SPECIFICATIONS FOR FOUNDRY PIG-IRON. ANALYSIS. It is recommended that all purchases be made by analysis. SAMPLING. In all contracts where pig-iron is sold by chemical analysis, each car load, or its equivalent, shall be considered as a unit. At least one pig shall be selected at random from each four tons of every car load, and so as to fairly represent it. Drillings shall be taken so as to fairly represent the fracture-surface of each pig, and the sample analysed shall consist of an equal quantity of drillings from each pig, well mixed .and ground before analysis. In case of disagreement between buyer and seller, an independent analyst, to be mutually agreed upon, shall be engaged to sample and analyze the iron. In this event one pig shall be taken to represent every two tons. The cost of this sampling and analysis shall be borne by the buyer if the shipment is proved up to specifications, and by the seller if other- wise. ALLOWANCES AND PENALTIES. In all contracts, in the absence of a definite understanding to the contrary, a variation of 10 per cent in silicon, either way, and of o.oi sulphur, above the standard, is allowed. A deficiency of over 10 per cent and up to 20 per cent, in the silicon, subjects the shipment to a penalty of 4 per cent of the contract price. 386 APPLIED MECHANICS. BASE ANALYSIS OF GRADES. In the absence of specifications, the following numbers, known to the trade, shall represent the appended analyses for standard grades of foundry pig-irons, irrespective of fracture, and subject to allowances and penalty as above: Grade. Per Cent Silicon. Per Cent Sulphur (Volumetric). Per Cent Sulphur (Gravimetric). No i . . . 2-75 0-035 0.045 N >. 2 . . . 2.25 0.045 0-055 No 3 ... i-75 0-55 0.065 No 4 ... 1.25 0.065 0.075 PROPOSED SPECIFICATIONS FOR GRAY IRON CASTINGS. PROCESS OF MANUFACTURE. Unless furnace iron is specified, all gray castings are understood to be made by the cupola process. CHEMICAL PROPERTIES. The sulphur contents to be as follows: Light castings not over o . 08 per cent. Medium castings . . . . " " o.io " " Heavy castings " " 0.12 " " DEFINITION. In dividing castings into light, medium, and heavy classes, the following standards have been adopted : Castings having any section less than \ of an inch thick shall be known as light castings. Castings in which no section is less than 2 ins. thick shall be known as heavy castings. Medium castings are those not included in the above definitions. PHYSICAL PROPERTIES. Transverse Test. The minimum breaking-strength of the "Arbi- tration Bar " under transverse load shall not be under: Light castings 2500 Ibs. Medium castings 2900 " Heavy castings 3300 " STANDARD SPECIFICATIONS t-'OR CAST-IRON. 387 In no case shall the deflection be under .10 of an inch. Tensile Test. Where specified, this shall not run less than: Light castings 18000 Ibs. per square inch. Medium castings .... 21000 " " " " Heavy castings 24000 " ' ' ' ' " THE " ARBITRATION BAR" AND METHODS OF TESTING. The quality of the iron going into castings under specification shall be determined by means of the " Arbitration Bar." This is a bar ij ins. in diameter and 15 ins. long. It shall be prepared as stated further on and tested transversely. The tensile test is not recommended, but in case it is called for, the bar as shown in Fig. i, and turned up from any of the broken pieces of the transverse test, shall be used. The expense of the tensile test shall fall on the purchaser. Two sets of two bars shall be cast from each heat, one set from the first and the other set from the last iron going into the castings. Where i :r the heat exceeds twenty tons, an additional set of two bars shall be cast for each twenty tons or fraction thereof above this amount. In case of a change of mixture during the heat, one set of two bars shall also be cast for every mixture other than the regular one. Each set of two bars is to go into a single mold. The bars shall not be rumbled or otherwise treated, being simply brushed off before testing. $88 APPLIED MECHANICS. The transverse test shall be made on all the bars cast, with supports 12 ins. apart, load applied at the middle, and the deflection at rupture noted. One bar of every two of each set made must fulfill the re- quirements to permit acceptance of the castings represented. The mold for the bars is shown in Fig. 2 (not shown here). The bottom of the bar is iV of an inch smaller in diameter than the top, to allow for draft and for the strain of pouring. The pattern shall not be rapped before withdrawing. The flask is to be rammed up with green molding-sand, a little damper than usual, well mixed and put through a No. 8 sieve, with a mixture of one to twelve bituminous facing. The mold shall be rammed evenly and fairly hard, thoroughly dried and not cast until it is cold. The test-bar shall not be removed from the mold until cold enough to be handled. SPEED OF TESTING. The rate of application of the load shall be thirty seconds for a deflection of .10 of an inch. * SAMPLES FOR CHEMICAL ANALYSIS. Borings from the broken pieces of the " Arbitration Bar " shall be used for the sulphur determinations. One determination for each mold made shall be required. In case of dispute, the standards of the American Foundrymen's Association shall be used for comparison. FINISH. Castings shall be true to pattern, free from cracks, flaws, and ex- cessive shrinkage. In other respects they shall conform to whatever points may be specially agreed upon. INSPECTION. The inspector shall have reasonable facilities afforded him by the manufacturer to satisfy him that the finished material is furnished in accordance with these specifications. All tests and inspections shall, as far as possible, be made at the place of manufacture prior to ship- ment. STANDARD SPECIFICATIONS FOR CAST-IRON. 389 SPECIFICATIONS FOR LOCOMOTIVE CYLINDERS. PROCESS OF MANUFACTURE. Locomotive cylinders shall be made from good quality of close- grained gray iron cast in a dry sand mold. CHEMICAL PROPERTIES. Drillings taken from test-pieces cast as hereafter mentioned shall conform to the following limits in chemical composition : Silicon from 1.25 to i . 75 per cent Phosphorus not over .9 ' ' " Sulphur " " .10 " " PHYSICAL PROPERTIES. The minimum physical qualities for cylinder iron shall be as follows : The ''Arbitration Test-Bar," ij ins. in diameter, with supports 12 ins. apart shall have a transverse strength not less than 30x50 Ibs., centrally appliedj and a deflection not less than o.io of an inch. TEST-PIECES AND METHOD OF TESTING. The standard test shall be ij ins. in diameter, about 14 ins. long, cast on end in dry sand. The drillings for analysis shall be taken from this test-piece, but in case of rejection of the manufacturer shall have option of analyzing drillings from the bore of the cylinder, upon which analysis the acceptance or rejection of the cylinder shall be based. One test-piece for each cylinder shall be required. CHARACTER OF CASTINGS. Castings shall be smooth, well cleaned, free from blow-holes, shrink- age cracks, or other defects, and must finish to blue-print size. Each cylinder shall have cast on each side of saddle manufacturer's mark, serial number, date made, and mark showing order number. INSPECTOR. The inspector representing the purchaser shall have all reasonable facilities afforded to him by the manufacturer to satisfy himself that the finished material is furnished in accordance with these specifications. All tests and inspections shall be made at the place of the manufacturer. 39 APPLIED MECHANICS. CAST-IRON PIPE AND SPECIAL CASTINGS. This specification is divided into the following sections, viz.: i Description of Pipes, 2 Allowable Variation in Diameter of Pipes and Sockets, 3' Allowable Variation in Thickness, 4 Defective Spigots may be Cut, 5 Special Castings, 6 Marking, 7 Allowable Percentage of Variation in Weight, 8 Quality of Iron, 9 Tests of Material, 10 Cast- ing of Pipes, 11 Quality of Castings, 12- Cleaning and Inspection, 13 Coating, 14 Hydrostatic Test, 15 Weighing, 16 Contractor to Furnish Men and Materials, 17 Power of Engineer to Inspect, 18 Inspector to Report, 19 Castings to be Delivered Sound and Perfect, 20 Defi- nition of the Word Engineer. Of these, only sections 8 and 9 will be quoted here, as follows: QUALITY OF IRON. SECTION 8. All pipes and special castings shall be made of cast- iron of good quality, and of such character as shall make the metal of the castings strong, tough, and of even grain, and soft enough to satisfactorily admit of drilling and cutting. The metal shall be made without any admixture of cinder-iron or other inferior metal, and shall be remelted in a cupola or air furnace. TESTS OF MATERIAL. SECTION 9. Specimen bars of the metal used, each being 26 inches long by 2 inches wide and i inch thick, shall be made without charge as often as the engineer may direct, and, in default of definite instruc- tions, the contractor shall make and test at least one bar from each heat or run of metal. The bars, when placed flatwise upon supports 24 inches apart and loaded in the centre, shall for pipes 12 inches or less in diameter support a load of 1900 pounds and show a deflection of not less than .30 of an inch before breaking, and for pipes of sizes larger than 12 inches shall support a load of 2000 pounds and show a deflection of not less than .32 of an inch. The contractor shall have the right to make and break three bars from each heat or run of metal, and the test shall be based upon the average results of the three bars. Should the dimensions of the bars differ from those above given, a proper allowance therefor shall be made in the results of the tests. WROUGHT-IRON, 39! 221. Wrou gilt-Iron. Wrought -iron is obtained by melt- ing pig-iron in contact with iron ore, oxidizing, and burning out, as far as may be, the carbon, the phosphorus, and the silicon. In many cases, however, the charge consists largely of wrought- iron or. steel scrap, and cast-iron borings. The process is commonly carried on in a puddling furnace, where an oxidizing flame is passed over the melted pig-iron. As the heat is not sufficiently intense to melt the wrought- iron produced, the metal is left in a plastic condition, full of bubbles and holes, which contain considerable slag. It is then squeezed, and rolled or hammered, to eliminate, as far as possible, the slag, and to weld the iron into a solid mass. The result of this first rolling is known as muck-bar, and must be "piled," heated, and rolled or hammered at least once more before it is suitable for use in construction. In making the piles, while muck-bar is sometimes used exclusively, a considerable part, and often the greater part, is made of scrap. Wrought-iron is thus, throughout its manufacture, a series of welds. Moreover, wherever slag is present, these welds cannot be perfect. It is also subject to the impurities of the cast-iron from which it is made. Thus, the presence of sulphur makes it red-short, or brittle when hot; and the presence of phosphorus makes it cold-short, or brittle when cold. It cannot, like cast-iron, be melted and run into moulds; but it can be easily welded by the ordinary methods Wrought-iron is much more capable of bearing a tensile or transverse stress than cast-iron: it is tougher, it stretches more, and gives more warning before fracture. At one time cast-iron was the principal structural material, but it was soon displaced by wrought-iron, which became the principal metal used in construction, but now, since the modern methods of steel-making supply a more homogeneous product at a cheaper price, wrought- 'iron has been superseded by mild steel in most pieces used in construction. 39 2 APPLIED MECHANICS. Wrought-iron is also expected to withstand a great many trials that would seriously injure cast-iron: thus, two pieces of wrought-iron are generally united together by riveting; the holes for the rivets have to be punched or drilled, and then the rivets have to be hammered; the entire process tending to injure the iron. Wrought-iron has to withstand flanging, and is liable to severe shocks when in use; as, for instance, those that occur from the changes of temperature in the different parts of a steam- boiler. The following references to a large number of tests of wrought- iron will be given : i. Eaton Hodgkinson: (a) Report of Commissioners on the Applica- tion of Iron to Railway Structures. (b) London Philosophical Transactions. 1840. 2. William H. Barlow: Barlow's Strength of Materials. 3. Sir William Fairbairn: On the Application of Cast and Wrought Iron to Building Purposes. 4. Franklin Institute Committee: Report of the Committee of the Franklin Institute. In the Franklin Institute Journal of 5. L. A. Beardslee, Commander U.S.N. : Experiments on the Strength of Wrought-iron and of Chain Cables. Revised and enlarged by William Kent, M.E., or Executive Document 98, 45th Congress, as stated below. 6. David Kirkaldy: Experiments on Wrought-iron and Steel. 7. G. Bouscaren: Report on the Progress of Work on the Cincinnati Southern Railway, by Thomas D. Lovett. Nov. i, 1875. 8. Tests of Metals made at Watertown Arsenal. Of these the first two volumes were published before 1881, and since that time one volume has been published every year. Nearly all of them contain tests of wrought-iron and a great many of them contain tests of full-size pieces of wrought-iron. 9. A. Wohler: (a) Die Festigkeits versuche mit Eisen und Stahl. (b) Strength and Determination of the Dimensions of Structures TENSILE STRENGTH OF WROUGHT-IRON. 393 of Iron and Steel, by Dr. Phil. Jacob J. Weyrauch. Translated by Professor Dubois. 10. Technology Quarterly, Vol. VII. No. 2, Vol. VIII. No. 3, Vol. IX. Nos. 2 and 3, and Vol. X. No. 4. 11. Mitt, der Materialpriifungsaustalt in Zurich. 12. Mitt, aus dem Mech. Tech. Lab. in Berlin. 13. Mitt, aus dem Mech. Tech. Lab. in Miinchen. 222. Tensile Strength of Wrought-iron. About the year 1840 was published the report of the Commission appointed by the British Government to investigate the application of iron to railway structures. While a number of tests of iron had been previously made, this work may properly be regarded as having been the first investigation of the kind that was at all thorough. At that time cast-iron was the metal most used in construction, and hence the greater part of the work of the Commission was devoted to a study of that metal. They made, however, a number of tests of wrought-iron, which, though they were of the greatest value at the time, and still have some value, will not be quoted here. At about that time the use of wrought-iron began to increase at a rapid rate, the necessary appliances were introduced to roll it into I beams, channel-irons, angle-irons, and other shapes, and it began to displace cast-iron for one after another purpose until it came to be the metal most extensively used in construction, both in the case of structures and machines. At first the chief desideratum was assumed to be that it should have a high tensile strength, and scarcely any attention was paid to its ductility. About 1865, however, engineers began to realize that duc- tility is an all-important property of a metal to be used in construction, and that this is not necessarily and not generally obtainable with a very high tensile strength. The most 394 APPLIED MECHANICS. prominent advocate, at that time, of the importance of duc- tility was David Kirkaldy, who published a book, entitled " Experiments on Wrought Iron and Steel," containing the results of his tests down to 1866. In the early part of his book will be found a summary of what had been done by earlier experimenters in this line. Kirkaldy tested a large number of English irons, determin- ing both their breaking-strengths and their ductility. In the light of the results obtained by him, he proceeded to draw up his famous sixty-six conclusions. These sixty-six conclusions will not be quoted here, but the following statement will be made regarding the main results of his work : i. He proved that the results obtained by testing grooved specimens (or specimens of such form as to interfere with the flow of the metal while under test) did not indicate correctly the quality of the metal, but that such specimens should be used as did not interfere with the flow of the metal.. 2. He advocated, with all the earnestness of which he was capable, the conclusion that it was of the greatest importance that all wrought-iron used in construction should have a good ductility, and, in his tests, he adopted five different methods of measuring ductility. These methods are : i. Contraction of area at fracture per cent ; 2. Ultimate elongation per cent ; 3. Breaking-strength per square inch of fractured area ; 4. Contraction of stretched area per cent, i.e., the contraction of area attained when the maximum load is first reached; 5. Breaking-weight per square inch of stretched area. Of these only two are used at the present time, the first and second, and they serve as measures of ductility. These two are the principal conclusions from Kir- kaldy's tests, though he cites a great many more, one of the principal of them being his conclusion regarding so-called cold crystallization, which will be mentioned later. SPECIFICATIONS FOR WROUGHT-IRON. 395 Tests of the tensile strength of wrought-iron may be divided into two classes: i those made mainly for the purpose of deter- mining the quality of the material, and 2 those made upon such full-size pieces as are used in practice to resist tension. The tests of the first class are made upon small specimens, and, in order that the results may be comparable, the use of standard forms and dimensions is, generally, a desideratum. The specifications for wrought-iron of the American Society for Testing Materials will be given first, as they refer to the kind of wrought-iron that is in most common use, and then some other tensile tests of various kinds of wrought-iron in small pieces will be given. Subsequently tests of wrought-iron eye-bars will be quoted. AMERICAN SOCIETY FOR TESTING MATERIALS. SPECIFICATIONS FOR WROUGHT-IRON. PROCESS OF MANUFACTURE. 1. Wrought-iron shall be made by the puddling process or rolled from fagots or piles made from wrought-iron scrap, alone or with muck-bar added. PHYSICAL PROPERTIES. 2. The minimum physical qualities required in the four classes of wrought-iron shall be as follows : Stay-bolt Iron. Merchant Iron. Grade "A." Merchant Iron. Grade "B." Merchant Iron, Grade "C." Tensile strength, pounds per square inch . 46000 50000 48000 48000 Yield-point, pounds per square inch 25000 25000 25000 25000 Elongation, per cent in 8 inches 28 25 20 2O 3. In sections weighing less than 0.654 pound per lineal foot, the percentage of elongation required in the four classes specified in para- 39^ APPLIED MECHANICS. graph No. 2 shall be 12 per cent., 15 per cent., 18 per cent., and 21 per cent., respectively. 4. The four classes of iron when nicked and tested as described in paragraph No. 9 shall show the following fracture : (a) Stay-bolt iron, a long, clean, silky fibre, free from slag or dirt and wholly fibrous, being practically free from crystalline spots. (b) Merchant iron, Grade "A," a long, clean, silky fibre, free from slag or dirt or any course crystalline spots. A few fine crystalline spots may be tolerated, provided they do not in the aggregate exceed 10 per cent of the sectional area of the bar. (c) Merchant iron, Grade "B," a generally fibrous fracture, free from coarse crystalline spots. Not over 10 per cent of the fractured surface shall be granular. (d) Merchant iron, Grade "C," a generally fibrous fracture, free from coarse crystalline spots. Not over 15 per cent of the fractured surface shall be granular. 5. The four classes of iron, when tested as described in paragraph No. 10, shall conform to the following bending tests: (e) Stay-bolt iron, a piece of stay-bolt iron about 24 inches long, shall bend in the middle through 180 flat on itself, and then bend in the middle through 180 flat on itself in a plane at a right angle to the former direction without a fracture on outside of the bent portions. Another specimen with a thread cut over the entire length shall stand this double bending without showing deep cracks in the threads. (/) Merchant iron, Grade "A," shall bend cold 180 flat on itself, without fracture on outside of the bent portion. (g) Merchant iron, Grade "B," shall bend cold 180 around a diameter equal to the thickness of the tested specimen, without fracture on outside of bent portion. (h) Merchant iron, Grade "C," shall bend cold 180 around a diameter equal to twice the thickness of the specimen tested, without fracture on outside of the bent portion. 6. The four classes of iron when tested as described in paragraph No. n, shall conform to the following hot bending tests: (i) Stay-olt iron, shall bend through 180 flat on itself, without SPECIFICATIONS FOR WROUGHT-IRON. 397 showing cracks or flaws. A similar specimen heated to a yellow heat and suddenly quenched in water between 80 and 90 F. shall bend, without hammering on the bend, 180 flat on itself, without showing cracks or flaws. (/) Merchant iron, Grade "A," shall bend through 180 flat on itself, without showing cracks or flaws. A similar specimen heated to a yellow heat and suddenly quenched in water between 80 and 90 F. shall bend, without hammering on the bend, 180 flat on itself, without showing cracks or flaws. A similar specimen heated to a bright- red heat shall be split at the end and each part bent back through an angle of 180. It will also be punched and expanded by drifts until a round hole is formed whose diameter is not less than nine-tenths of the diameter of the rod or width of the bar. Any extension of the original split or indications of fracture, cracks, or flaws developed by the above tests will be sufficient cause for the rejection of the lot rep- resented by that rod or bar. (k) Merchant iron, Grade "B," shall bend through 180 flat on itself, without showing cracks or flaws. (/) Merchant iron, Grade "C," shall bend sharply to a right angle, without showing cracks or flaws. 7. Stay-bolt iron shall permit of the cutting of a clean sharp thread and be rolled true to gauges desired, so as not to jam in the threading dies. TEST PIECES AND METHODS OF TESTING. 8. Whenever possible, iron shall be tested in full size as rolled, to determine the physical qualities specified in paragraphs Nos. 2 and 3, the elongation being measured on an eight inch (8") gauged length. In flats and shapes too large to test as rolled, the standard test specimen shall be one and one-half inches (ii") wide and eight inches (8") gauged length. In large rounds, the standard test specimen of two inches (2") gauged length shall be used; the center of this specimen shall be half- way between the center and outside of the round. Sketches of these two standard test specimens are as follows: 39* APPLIED MECHANICS. . 4*: --- \ I " jt -18-about \ PIECE TO BE OF SAME THICKNESS AS T&E PLATE. 9. Nicking tests shall be made on specimens cut from the iron as rolled. The specimen shall be slightly and evenly nicked on one side and bent back at this point through an angle of 180 by a succession of light blows. 10. Cold bending tests shall be .made on specimens cut from the bar as rolled. The specimen shall be bent through an angle of 180 by pressure or by a succession of light blows. 11. Hot bending tests shall be made on specimens cut from the bar as rolled. The specimens, heated to a bright red heat, shall be bent through an angle of 180 by pressure or by a succession of light blows and without hammering directly on the bend. If desired, a similar bar of any of the four classes of iron shall be worked and welded in the ordinary manner without showing signs of red shortness. 12. The yield -point specified in paragraph No. 2 shall be deter- mined by the careful observation of the drop of the beam or halt in the gauge of the testing-machine. TESTS OF COMMANDER BEARDSLEE. 399 FINISH. 13. All wrought-iron must be practically straight, smooth, free from cinder spots or injurious flaws, buckles, blisters or cracks. In round iron, sizes must conform to the Standard Limit gauge as adopted by the Master Car Builders' Association in November, 1883. INSPECTION. 14. Inspectors representing the purchasers shall have all reason- able facilities afforded them by the manufacturer to satisfy them that the finished material is furnished in accordance with these specifications. All tests and inspections shall be made at the place of manufacture prior to shipment. TESTS OF COMMANDER BEARDSLEE. One of the most valuable sets of tests of wrought-iron is that obtained by committees D, H, and M of the Board appointed by the United States Government to test iron and steel; the special duties of these committees being to test such iron as would be used in chain-cable, and the chain-cable itself. The chairman of these three committees, which were consolidated into one, was Commander L .A. Beardslee of the United States Navy. The full account of the tests is to be found in Executive Document 98, 45th Congress, second session; and an abridged account of them was published by William Kent, as has been already mentioned. The samples of bar-iron tested were round, and varied from one inch to four inches in diameter. AOO APPLIED MECHANICS. Certain conclusions which they reached refer to all kinds- of wrought-iron, and will be given here before giving a table of the results of the tests. i. Kirkaldy considers the breaking-strength per square inch of fractured area as the main criterion by which to deter- mine the merits of a piece of iron or steel. Commander Beardslee, on the other hand, thinks that a better criterion is what he calls the "tensile limit;" i.e., the maximum load the piece sustains divided by the area of the smallest section when that load is on, i.e., just before the load ceases to increase in the testing-machine. 2. Kirkaldy had already called attention to the fact that the tensile strength of a specimen is very much affected by its shape, and that, in a specimen where the shape is such that the length of that part which has the smallest cross-section is practically zero (as is the case when a groove is cut around the specimen), the breaking-strength is greater than it is when this portion is long ; the excess being in some cases as much as 33 per cent. Commander Beardslee undertook, by actually testing speci- mens whose smallest areas varied in length, to determine what must be the least length of that part of the specimen whose cross-section area is smallest, in order that the tensile strength may not be greater than with a long specimen. The conclusion reached was, that no test-piece should be less than one-half inch in diameter, and that the length should never be less than four diameters ; while a length of five or six diameters is necessary with soft and ductile metal in order to insure correct results. The following results of testing steel are given in Mr. Kent's book, as confirming the same rule in the case of steel. The tests were made upon Bessemer steel by Col. Wilmot at the Woolwich arsenal. TESTS OF COMMANDER BEARDSLEE. 4OI Tensile Strength. Pounds per Square Inch. Highest Lowest 162974 Average . . . . . Highest T owpst 153677 123165 Average 10 3 2 55 114460 3. Commander Beardslee also noticed that rods of certain diameters of the same kind of iron bore less in proportion than rods of other diameters ; and, after searching carefully for the reason, he found it to lie in the proportion between the diam- eter of the rod and the size of the pile from which it is rolled. The following examples are given : ij-in. diameter, 6.62% of pile, 56543 Ibs. per sq. in. tensile strength. If I* If 'I 8.i8% tt 56478 ' 9.90% " 54277 " 11.78% tt 5355 " 7.68% " 56344 " 8.90% tt 55018 " 10.22% (i 54034 " II-63% (i 51848 " He therefore claims, that, in any set of tests of round iron, it is necessary to give the diameter of the rod tested, and not merely the breaking-strength per square inch. 4. He gives evidence to show, that if a bar is under-heated, it will have an unduly high tenacity and elastic limit ; and that if it is over-heated, the reverse will be the case. 402 APPLIED MECHANICS 5. The discovery was made independently by Commander Beardslee and Professor Thurston, that wrought-iron, after having been subjected to its ultimate tensile strength without breaking it, would, if relieved of its load and allowed to rest, have its breaking-strength and its limit of elasticity increased. His experiments show that the increase is in irons of a fibrous and ductile nature, rather than in brittle and steely ones ; hence the latter class would be but little benefited by the action of this law. The most characteristic table regarding this matter is the following : EFFECT OF EIGHTEEN HOURS' REST ON IRONS OF WIDELY DIFFER ENT CHARACTERS. I Ultimate Strength i i per Square Inch. i "O 1 First Second JxGXEl&rJCS* Strain. Strain. Boiler iron . . . 48600 56500 Not broken. tt tt 49800 57000 Broken \ tt (t 49800 58000 Broken 1 Average gain, H ft 48100 54400 Broken f 15.8%. (I 11 48150 5555 Broken J Contract chain iron, 50200 54000 Broken *j (t ii 50250 53200 Not broken 1 Average d ti tt 50700 553oo Not broken j* gain, n t( ft 49600 52900 Not broken j 6.4%. ft tt a 51200 52800 Not broken ) Iron K . . . . ft ft 58800 64500 59000 ! 65800 Broken ^ Broken I Avera g e ^ ft tt 56400 j 60600 Broken J 9-4%- j CHAIN CABLE. 403 233. Chain Cable. The most thorough set of tests of the strength of chain cable is that made by Commander Beardslee for the United-States government, an account of which may be found either in the report already referred to, or in the abridg- ment by William Kent. In this report are to be found a number of conclusions, some of which are as follows : i. That cables made of studded links (i.e., links with a cast-iron stud, to keep the sides apart) are weaker than open- link cables. 2. That the welding of the links is a source of weakness ; the amount of loss of strength from this cause being a very uncertain quantity, depending partly on the suitability of the iron for welding, and partly on the skill of the chain-welder. 3. That an iron which has a high tensile strength does not necessarily make a good iron for cables. Of the irons tested, those that made the strongest cables were irons with about 51000 Ibs. tensile strength. 4. The greatest strength possible to realize in a cable per square inch of the bar from which it is made being 200 per cent of that of the bar-iron from which it was made, the cables tested varied from 155 to 185 per cent of that of the bar- iron. 5. The Admiralty rule for proving chain cables, by which they are subjected to a load in excess of their elastic limit, is objected to, as liable to injure the cable : and the report suggests, in its place, a lower set of proving-strengths, as given in the following table ; the Admiralty proving-strengths being ilso given in the table. In these recommendations, account is taken of the different proportion of strength of different size bars as they come from th: rolls, also no proving-stress is recommended greater than 50 per cent of the strength of the weakest link, and 45.5 per cent rf the strongest ; v/hereas in the Admiralty tests, 66.2 404 APPLIED MECHANICS. per cent of the strength of the weakest, and 60.3 per cent of the strongest, is sometimes used. For the details of this investigation, see the report, Execu- tive Document No. 98, 45th Congress, second session, or the abridgment already referred to. Diameter of Iron, in inches. Recommended Proving-Strains. Admiralty Proving-Strains. Diameter of Iron, in inches. Recommended Proving-Strains. Admiralty Proving-Strains. 2 121737 161280 I* 66138 83317 lit 114806 I5I357 If 60920 76230 I 8 108058 I4I75 i-iV 55903 69457 lit 101499 I3 2 457 li 5I08 4 63000 If 95128 123480 IA 46468 56857 Itt 88947 114817 It 42053 5 I0 3 If 82956 106470 nV 37820 455*7 I* 77159 98437 i 33 8 40 40320 it 7I55 90720 While steel long ago displaced wrought-iron for boiler-plate, and while steel I beams, channel-bars, angle-irons, and other shapes, as well as eye-bars, have, of late years, displaced wrought-iron to a very great extent, nevertheless wrought-iron is still very extensively used, and for a great variety of struc- tural purposes. For wrought-iron to be used in construction, ductility, homogeneity, and often weldability are the great desiderata, together with as large a tensile strength as is consistent with these. As to the requirements made by different engineers for wrought-iron for structural purposes, the minimum tensile strength called for varies from about 46000 to about 50000 pounds per square inch, with ultimate elongations varying from 15$ to 30$ in 8 inches, according to the purpose for which it is wanted. It is also very common, when good iron is wanted, CHAIN CABLE. 405 to insist that it shall not be made of scrap. The following tables of tensile tests of wrought iron of various kinds will show what results can be obtained. Norway Iron. Burden's Best. od c c e"a d a o"". -.'". 4> o"" 4-r"! -8 u . 1& o a 1 5 If it i 1? o a o u c w .2 5 jf " >> & G 11 Is' u || "5 ! i! u 11 Is 5 s s 5 s 5 OS s 75 48390 23620 62.6 30090000 .76 53566 27554 57-6 29175000 75 46340 21160 62.7 30780000 75 50023 26030 49-8 30643000 75 48280 28030 62.6 29020000 76 47724 25350 47-6 30310000 77 45160 20400 68.8 27388000 77 46772 24700 45-2 28347000 75 46063 19240 68.6 27666000 77 46600 22550 46.2 29528000 77 44490 20510 67-5 28452000 77 47395 22550 46.2 28347000 74 43233 22079 70.5 29026000 77 47963 22695 48.6 29475000 75 43470 19400 75-5 26700000 77 47860 26948 46.4 26948000 73 38950 22030 72.3 30140000 77 475CQ 26927 42-3 28435000 74 43240 21970 75-2 27726000 76 47610 23036 53- l 29551000 74 44564 21970 73.8 28663000 77 49238 22725 49-2 27470000 74 43860 19658 75-o 18000000 76 50037 27700 53-6 29251000 i .00 41620 15560 7-3 27295000 76 48538 27224 48.8 29355000 75 42215 68.6 29292000 76 50060 23201 S3- 31028000 75 42033 19239 62.4 29729000 .76 49143 23240 5-4 30438000 .76 4'574 14328 69-5 27450000 .76 48655 23414 49.6 30062000 .76 75 41574 426^6 16531 19240 68.7 59- 29098000 31785000 76 76 47220 47090 22880 23020 53-4 54-i 29969000 33657000 75 41875 16978 70.1 30487000 .76 49690 27480 53-3 29614000 75 43396 19112 59-3 28000000 .76 47430 22950 51-8 29443000 74 39210 15216 73' 2 30294000 76 4795 23000 57-o 29504000 74 12603 70.5 28810000 .76 22892 45-8 28779000 74 39896 15187 69.7 31153000 77 49411 18420 46.6 30112000 74 39156 16123 76.4 29807000 .76 49660 23186 51-3 30160000 74 41030 17490 69.8 29310000 76 48055 20940 56.7 28809000 75 41180 18000 72.5 31073000 77 49026 22578 40.5 27292000 74 42320 19660 68.0 30834000 76 47220 23060 51-2 33710000 74 43913 198:53 69.8 26970000 .76 5 OI 49 20940 41.5 27450000 74 42102 191.81 78.3 29127000 75 48553 23767 74-3 31124000 74 39698 17638 70-5 30023000 49350 21503 66.5 31793000 73 43187 17846 68.6 28553000 .76 50083 20940 29097000 73 40669 17820 73-8 30159000 76 47019 23140 5 1 -4 29978000 73 39348 16593 69-3 29518000 76 47504 20942 53-2 28527000 73 39671 12987 78.1 28861000 76 47747 20942 49-5 29874000 75 39951 16886 77.2 30020000 .76 50927 23453 46.9 28350000 74 41093 16400 74- 1 28634000 75 51269 21182 51 -9 32551000 74 40192 14053 76.3 28627000 75 50930 23770 53-7 29293000 73 44470 16844 73-4 31114000 76 50083 23146 45 - 29097000 74 41940 17523 78.5 29373000 75 48168 23767 55-6 29879000 74 42531 16449 7.S 30410000 76 49500 26500 55-o 3 i 600000 75 48400 27200 46.2 28700000 75 47600 27200 50.1 29300000 77 47200 23600 56.1 27800000 .76 46700 24200 41.8 29700000 77 . 45600 23600 54-5 28200000 406 APPLIED MECHANICS. Refined Iron. Wrought-iron Wire. |a a c u "rt'.S ,H C V % J a- f sr o "- 1 x Jj cr 'iif c <> "" u S <n fc !* 8. o ^ 01 JO Kind of Wire. feg i .2 s - ^~S fjj S ^ o a, 2 f i ' i> By 8 ^ .H ^~ is "5.ti 3 a 5 || "S-< "83 E9 g.S 1 j "^ <5 5 5"" 5 & 5"* w~~ (S s* 1 56270 28293 33 *9 28618000 Annealed wire ^3800 g, j 77 53450 28990 22.0 26997000 Annealed wire ii3 61500 4jJCH_XJ 03. i 75- .76 55880 29758 33-5 27711000 Annealed wire 135 61100 39800 49-4 23000000 77 53850 29370 33-3 28718000 Annealed wire .136 59500 39200 71.2 25500000 77 52770 33722 14.8 27355000 Annealed wire 45100 35800 76.8 23500000 74 52770 28829 33-5 29273000 Annealed wire 136 59800 34000 72.7 77 51320 29294 22 6 28082000 Annealed wire J 35 62400 77 74 53778 , 4888* 27138 28822 25-4 13 8 28659000 28137000 Common wire Common wire . no .109 90900 103000 64000 .1 30300000 27500000 75 49240 28190 14.0 27520000 Common wire . no 104000 60000 51.0 22900000 75 50190 30590 17.8 26237000 Common wire "3 93700 68200 60.5 25200000 77 75 51460 47495 29256 30387 22.4 12.2 25680000 27613000 Common wire Common wire .080 .080 113000 113000 45800 56700 41.9 51.0 27000000 26500000 48352 30574 17.3 27177000 Common wire .079 IT2OOO 54300 53-3 26600000 .76 47I5I 25982 75-4 21628000 Common wire 079 120000 73600 28.1 26100000 77 5035 1 35720 25-3 27477000 Common wire .079 IO9OOO 54300 40.4 26400000 75 48202 28521 14.7 27888000 Common wire .080 98300 43-8 27100000 75 50703 30558 13.0 23713000 Annealed wire .081 99600 61 .9 26600000 75 49223 30517 J5-2 27126000 Annealed wire .082 93500 50400 64-3 75 49120 29000 17.8 28290000 Annealed wire .082 86300 50400 68.5 27100000 75 47060 31700 j c 4 Annealed wire .082 89900 54OOO ej 7 24900000 :3 47830 51300 29400 26000 17.8 29.1 29290000 30100000 Annealed wire Annealed wire .082 .082 97100 93500 57600 39600 55-0 26100000 76 52400 35000 29.1 25400000 Annealed wire .082 71000 50400 67 .'i 27000000 :8 53400 52IOO 29000 26000 24.9 29. i 28200000 Common wire Annealed wire .167 .081 57200 45100 65.6 60.4 CJTOO 29000 24.9 Annealed wire .082 959 553 76 04- l *" K -' 51500 26500 24.6 33100000 Common wire .163 935OO 67400 40100 56-9 76 52500 242OO 22 3 Common wire 6l5OO .___- 52.8 77 77300 34400 26.5 26800000 Piano wire, 3 75 75 53100 52900 31700 24.9 27.2 27200000 26100000 No. 13 Piano wire, 031 345000 29500000 76 51600 28700 22 . 3 26000000 No, 23 .048 ofi^enn ononoooo 4-ooX j " 1. 01 40700 14.4 76 53100 28700 24.6 .76 52200 33100 26.9 31000000 75 50100 31700 22.6 28700000 76 49400 26500 26.9 .02 50300 31800 16.9 27200000 .01 47000 32500 20.6 27700000 .01 50400 30000 25.8 28300000 .02 49600 31800 23-9 26500000 .OI 50200 30000 32-5 26800000 .02 50500 29400 30.6 26200000 .01 51400 30000 29.2 28300000 .02 50400 20.4 28200000 .02 50200 31800 I 5- 1 27200000 .01 48100 30000 32-5 27700000 .OI 50600 30000 27-5 25800000 77 48600 25800 52.6 28000000 74 53900 27900 38.6 29700000 74 54000 25600 18.0 29700000 76 AI 8 .70 .76 53500 30900 41.0 33-4 27600000 TENSILE TESTS OF WROUGHT IRON. 407 In Heft IV (1890) of the Mitt. d. Materialpriifungsanstalt in Zurich is an account of a set of tensile tests of wrought-iron and mild-steel angles, tees, and channels. The following is a summary of his results for wrought-iron shapes : ANGLE-IRONS. a' rt g u J w"o - v-"5 *j' u> S SU Modulus of 5 Dimensions, Inches. V a I|s X g 3 *ls (2-S jj *O 3 2 Ifc 1* Elasticity, Pounds per Square Inch. 3 & J o a" 13 o o* j^CXyj T3 u Lbs. 2 2.76 X 2.76 X 0.31 l6 -53 49910 25020 37680 9-5 28824000 4 2.76 X 2.76 X 0.51 27.62 49060 20190 32560 II. 7 28070000 6 3-54 X 3-54 X 0.35 26.21 50620 253 10 15.8 28269000 8 3- l 5 X 3-54 X 0.55 38-51 5II90 25310 32000 l6. 4 27786000 1O 4.13 X 4'*3 X 0.47 35.48 49200 28010 33*30 12. O 28537000 12 4.13 X4-13 Xo.6 7 55-24 46070 22750 32280 10.2 28554000 M 5.12 X 5.12 X 0.67 62.09 47780 22610 3043 12. 27985000 16 5.12 X 5-12 X 0.87 102.61 48490 31140 12.3 TEE-IRONS. 3 3". 60 X 3" -35 22.88 5 2 470 25880 3768o 14.2 27615000 4 " 49200 23610 34700 15-5 27672000 7 3". 94 X 3"-94 31.75 51760 21610 38820 11.7 27857000 8 54040 18630 354*0 21.3 27743000 9 10 5". 90 X S'^94 46.^91 53 6l o 52900 23600 22890 36970 3598o 19.0 14-5 27402000 28255000 CHANNEL-IRONS. I 4.13 X 2.56 28.43 50630 23329 35120 15-9 27544000 a 44 49200 24170 33700 12.7 27885000 4 4.13 X 2.64 31.65 543 2 o 23040 36690 20.6 27772000 5 6.93 X 2.83 48.89 51760 24460 35550 19.9 27658000 6 " " " 54610 23320 34270 17-5 27999000 7 8 6.93 X 2.^91 S4 ti 43 51900 52900 19620 24320 35690 30860 20.4 14.5 27487000 29663000 9 8.46 X 3-35 85.68 52050 22330 3456o 20.9 27701000 10 4k 4*T '* 5347 24170 36260 17.0 28710000 la 8.46 X 3-50 92.33 52760 32040 34840 ix. 9 28568000 408 APPLIED MECHANICS. TENSILE TESTS MADE SUBSEQUENTLY AT THE WATERTOWN ARSENAL. Here will next be given, in tabulated form, the results of a number of tensile tests made on the government machine at the Watertown Arsenal. The following tables of results on rolled bars, from the Elmira Rolling-Mill Company (mark L) and from the Passaic Rolling- Mills (mark S), are given in Executive Document 12, ^.Jth Con- gress, 1st session, and in Executive Document /, <ffth Congress, 2d session. SINGLE REFINED BARS. 1 c ^ rt % Sectional Area, in square inches. . U id J 8, . 2 rf-5 a a w Ultimate Strength, in Ibs., per Square Inch. $> c '*? S-5 I' S Contraction of Area, %. Appearance of Fracture. Modulus of Elas- ticity at Load of 20000 Lbs. per Square Inch. 1^ ta t* II L i 3.06 28500 52710 18.4 33-3 95 5 26981450 L 2 3 06 29500 53630 16.4 36.0 92 8 27826036 L 3 3.06 29000 52090 21.4 34.6 95 5 28419182 L 4 3.06 29000 5 '440 15.0 20.3 90 IO 30888030 L 5 6.46 27500 505 00 H-5 27.6 95 5 27826036 L 6 6.40 27500 50530 17-3 22.3 70 30 27118644 L 7 6-39 27000 50200 18.0 22.5 95 5 27444253 L 8 3- 2 4 - 51667 22.0 36.0 70 3 28318584 Round. L 9 3.20 - 50844 I6. 3 22.0 '5 85 27972027 u L 10 3.20 - 53062 2I.O 4O.O 95 5 28119507 S ii 3.08 28500 48640 '3-3 24-3 100 Slightly 27586206 S 12 3.08 28000 50390 16.9 35-i 100 o 27586206 s 13 3-05 28500 47050 9.0 22.0 IOO o 27874564 S i S 6.40 26000 49700 17.1 19.2 85 15 29906542 S 16 6.40 24000 49280 15-7 177 85 !5 26490066 S 17 6.41 24500 48740 14-3 17.3 80 20 28119507 S 18 3-i7 24600 49680 19-5 32.0 IOO Slightly 27972027 Round. S 19 3-17 25870 49338 18.3 38.0 IOO 29357798 " Cinder S 20 3-i7 24920 48864 18.4 37-o IOO at centre 27729636 DOUBLE REFINED BARS. 409 DOUBLE REFINED BARS. Mark on Bar. Sectional Area, in square inches. S ~- H, ! j? 3 I . u - -c llJ H Ultimate Strength, in Ibs., per Square Inch. S g g^ 11 I -s w Contraction of Area, %. Appearance of Fracture. Modulus of Elas- ticity at Load of 20000 Ibs. per Square Inch. 1 ^ fa si ^ \ i u - L 20 1 3.06 29000 5356o 15-3 37-9 100 o 2 7633 8 5 l L 2O2 3-03 30000 52650 16.2 20.6 85 15 34042553 L 203 3.06 32500 53500 1 16.5 27-5 TOO 28169014 L 204 3.06 32500 54480 15.4 24.8 TOO o 29090909 L 205 6-33 27000 51230 17.8 24.2 80 20 28119507 L 206 6-34 27500 50500 17.6 21. 1 100 Slightly 29629629 L 207 6-34 27000 51030 ; 21.4 31-9 100 o 27826086 C^ up- L 208 i 3.20 50156 '22.7 43 IOO shaped 28021015 Round. L 209 3.20 - 49937 22.6 45-o TOO " 28622540 " L 210 3.20 - 50188 19.9 43-o IOO " 28985507 " S 211 3-05 29500 5II50 J22.0 3i-5 IOO o 32989690 S 212 3-5 28500 5IIIO 22.0 36.1 IOO o 25559105 S 213 3-" 29500 51860 22-5 39-2 IOO o 26446280 S 215 6.31 27500 50980 19.1 23.6 95 5 29357798 S 216 6.38 27OOO 50770 20.7 29.6 IOO 28268551 S 217 6-33 27000 51340 19-3 35-2 IOO o 28070175 S 218 3-i7 24610 50631 20.4 41.0 IOO o 28622540 Round. Cup- S 219 3-i7 50915 25-5 44.0 IOO shaped 28268551 " S 220 3-^7 50205 23-7 44.0 IOO 28070175 The moduli of elasticity had not been computed in the report, but have been computed in these tables from the elon- gations under a load of 20000 Ibs. per square inch in each case, as recorded in the details of the tests. In these reports are also to be found tensile tests of iron from other companies, as the Detroit Bridge Company, the Phoenix Company, the Pencoyd Company, etc. Some of these 4io APPLIED MECHANICS. tests were made to determine the effect of rest upon the bar after it had been strained to its ultimate strength, also to determine the strength after annealing. The following table shows these latter results : o-o 5 8| . a v c-2 & li! c a c c rt rt rt rt 1 S S o far hfl ** M M hi) & fci hli u 5^ . (c" g.^ vC i5C * w iJC "*C *C <3 fits 3 R>* 3 8. J 2 ^ v &*!,: 8* vg^ &* g p en w ^ ^S CB' "> g" gjftif E&8 2'2o 0.0*5 0^2*3 iJ 2^ ^ 3J3 1 ; - 10 wwrt i !i III il & ,?3. . & g 2 2 S" 1 I S" ? ja"3 ja g ^ .0 3 a o fc Q ___ E E ' o -d 'uopoas . , . B9JV JO M 1 VO CO M * tx 1 ^ S S co * * " " 3 'd 'qiSuaq \vu\BuQ cuoaj uoiiBxfuo^g M-^-Min^-O coco MM MM HM H M ro vo co N sn C\ O^ VO H M M H M C1 ti d 5 ti S> 8 2 vg 8>S\ 8 (8 MO HONCOCO OOv 8 5- ft & 5- vS WVO VO H IN 10 R. 8 8 v8 S O S> ij- in co o O (J "" S 5 ri ** & n vo m vo invo in n co * in vo * in vo in vo n vo vo a .H -- v c {/) tC w Q P * H ~ I S O Q O Q o o ^ o o o o I >n * I m m J. 8 8 | O Q O S o o 0> Jo f in oo | o s^lcfl I? ^ S5 -S ST ^ in N \ft in co in jlta S 1 . ff i 'Si ^ i ^0 vg ^ =0^ CO vo tv rx i I o vo vo w? - "^cJf to ro ro vo in co ei N co M ro N *| ^g d 181*81 ET S S 8.8 jn w M ^t* w u c c"" 1 H j I" 1 M M M " K * 5. T3 C S , o" . S 1 ! 5* R ? <? <2 ? <^ ff , S 1 * o? m ^_ fr> _^ ^ * ro M ro M M ro d M Condition of Bar when Tested. :p 11 :, : i 11 - a l -I 2 || 'l| "S ' llgl^ll? 1 s Hitiln * s O&< OPS Ooi O OS og ' ' -a" S ' ' & ; IJ' ; H.S * i-.s * g-a . . fc s-o . . si . _ r-3-_ il 4 nl 4 o.h .SP w c.h .5P s rtrt o E rtrt o / 1 !!!i i i = !ls s s s 2~s - s -5 73 *?, T) 1- c -0 -0 8 818558 S 4) OJCJDrt'CvD OJ pi D: as o oi rt i g; i I s R ** R oo oo' P. oo oo" co co s co co DOUBLE REFINED BARS. 411 Some tests were made to determine the values of the modulus of elasticity of the same iron for tension and for compression ; and these were found experimentally to be almost identical, as was to be expected. For these tests the student is referred to the reports themselves ; and only cer- tain tests on eye-bars of the Phcenix Company will be appended here. Arsenal Number. Outside Length, Inches. Gauged Length, Inches. Sectional Area, Sq. In. Ultimate Strength, Pounds per sq. in. Contraction of Area at Fracture, per cent. 511 67.75 50 1.478 40600 16.8 513 67.80 50 1.940 39480 13-9 518 96.05 75 5-103 46720 8.1 Quite a number of tests of the iron of different American companies are to be found in the "Report on the Progress of Work on the Cincinnati Southern Railway," by Thomas D. Lovett, Nov. i, 1875. For these the student is referred to the report named. WROUGHT-IRON PLATE. The following table contains some tests of wrought-iron plate and bars made on the Government testing-machine at Watertown in 1883 and 1884 for the Supervising Architect at Washington, D.C. 412 APPLIED MECHANICS. .g '3 I" Mi fi-i e . i3 _e fl ^ ** rf 11181 8 s-s-sg.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 -<i-v ONON^'-^'i-' ONOOfO-^ ON ONOO IO O IO ^ OO NO VO 1^*00 1 p 8OOOQOOOOOOQOOOOOOOOQOOOQO OOOOQQQ-S 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 o-o'do'dddddddddddddddddoddddd ddo'ddddf-idddddd 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!? S-EtifSS -' . 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 F-i^K 1010 io ir NONONONONONONONONONONONONONONOVONONONONONONONONOVONO NONONONO WROUGHT-IRON AND STEEL EYE-BARS. In the report of the Government tests for 1886 is given the following table of tensile tests of wrought-iron eye-bars. The wrought-iron ones were furnished by the General Manager of the Boston and Maine Railroad, and the steel ones by the Chief Engineer for the American Committee of the Statue of Liberty. CO M PRESS IV E STRENGTH OF WROUGHT-IRON. 413 WROUGHT- IRON EYE-BARS. Dimensions. K Elongation. 1 O "*"* 0) 01 in . Fracture. |-s , *| bcx i ( *o J3S l O C oj $ V 0> . HH SHH T) u.S o ^co ^.^ fill 1 "!! W g r=3 if t^s "o 1 W3 4; ia? G _0 Appearance. c2 8 H He/2 cw Ojj Isl -d^^c 'x-2 1 J 5 H 2 H c J S S 3 Ins. Ins. Ins. Lbs. Lbs. % % % Lbs. Lbs. 238.55l5.oo 1.14 22456 45105 II .7 ii .6 31 .2 28037000 52763 Stem Fibrous, traces of granulation. 238.60 5.00 1.15 22610 44540 9.4 9.4 29.6 28125000 50588 " Fibrous, 70% Granular, 30% 238.57 4-99 1.14 21790 43320 7-8 3.0 26.4 27950000 48492 1 ' Fibrous, 70% Granular, 30% 238.64 5.00 1. 16 22410 39550 5 i 4-8 9.8 27355000 54013 1 ' Fibrous, 70% Granular, 30% 238.62 6.05 i .44 19750 43260 12.05 I 2 .06 24.8 28800000 43166 1 ' Granular, 80% Fibrous, 20% 238.62 6.05 i .44 22730 42020 6.5 6.6 19. 2 28301000 41929 " Granular 5% at one edge, fi- brous for bal- ance of fracture. The gauged length of the bars was 180 inches. The moduli of elasticity computed between 5000 and 10,000 pounds per square inch. COMPRESSIVE STRENGTH OF WROUGHT-IRON. In regard to the compressive strength of wrought-iron, we may wish to study it with reference to 1. The strength of wrought-iron columns; 2. The strength of wrought-iron beams ; 3. The effects of a crushing force upon small pieces not laterally supported ; 4. The effects of a crushing force upon small pieces laterally supported. i. In this case it may be said that, by reference to the tests of wrought-iron bridge columns, the compressive strength per square inch of wrought-iron in masses of such sizes is given by the tests of the shorter lengths of such columns, i.e., by those that are short enough not to acquire, when the maximum load is just reached, a deflection sufficient to throw any appreci- ably greater stress per square inch on any part of the column in consequence of the eccentricity of the load due to the deflec- 4H APPLIED MECHANICS. tion. The results thus obtained are naturally lower than we should expect to obtain in smaller masses. 2. In this case the evidence that there is goes to show that the compressive strength is the same as in the case of i, and hence that it is less than the tensile strength. Indeed, the results of tests of full-size beams show a modulus of rupture greater than the compressive strength, less than the tensile strength in I sections, and greater in circular sections; all this being what would naturally be expected. 3. If a small cylinder of ductile wrought-iron is tested with- out lateral . support, and with flat ends, the friction of the ends against the platforms of the testing-machine comes in to interfere with the flow of the metal; and if, besides this, the ratio of length to diameter is so small as to prevent buckling, then the specimen will gradually flatten out, and it becomes impossible to find any maximum load, because the area of the central part is constantly increasing. 4. In this case the crushing strength per square inch that causes continuous flow, and also the maximum strength per square inch, is greater than that where the specimen has no lateral support. Hence follows, that in the case of wrought-iron rivets it is .entirely safe to allow a bearing pressure in the neighbor- hood of 90,000 or 100,000 pounds per square inch, according to circumstances. 223. Wrought-iron Columns. Until after about 1880 there was but little experimental knowledge on this subject beyond the experiments of Hodgkinson, which have furnished the con- stants for Hodgkinson's, and also for Gordon's formula, as already given in 208 and 209. These formula have been in very general use, and it is only of late years that we have been able to test their accuracy by tests on full-size wrought-iron columns. The disagreement of the formulae already referred to, with the results of the tests, has led to the proposal of a large number of similar formulae, WRO UGHT-IRON COL UMNS. 4 1 5 each having its constants determined to suit a certain definite set of tests, and hence all these formulae thus proposed, which are, of course, empirical, and can only be applied with safety within the range of the cases experimented upon. A few of these will now be enumerated; and then will follow tables of the actual tests, which furnish the best means of deter- mining the strength of these columns ; and it would appear that it is these tables themselves which the engineer would wish to use in designing any structure. On the 1 5th of June, 1881, Mr. Clark, of the firm of Clark, Reeves & Co., presented to the American Society of Civil Engineers a report of a number of tests on full-size Phoenix columns, made for them at the Watertown Arsenal, together with a comparison of the actual breaking-weights with those which would have been obtained by using the common form of Gordon's formula for wrought-iron. The table is shown on page 416. The very considerable disagreement between the breaking- loads as calculated by Gordon's formula, and the actual break- ing-loads, led a number of people to propose empirical formulae of one form or another which should represent this set of tests, and also others which should represent some other tests on full-size bridge columns, which had been previously made in other places. Of these I shall only give those proposed by Mr. Theodore Cooper, which are as follows : p f For square-ended columns . . . -j = - -n - ry ' + -*>) 18000 P f For phi-ended columns . . . . -j = 7.8000 416 APPLIED MECHANICS. frlil - vO ONfOONi^toior^>- O vO VO r^ M -TJ- N t Tj-vo-. rorfLOO\O O^OvOOO -^-M N ro Md-OfCOOOM-its t^O OO OO O O ro 1 1 1 1 tf r^ r C\ O\<-> O w N ^ w 1-1 M O ""If J~> rj- >-o i-o 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 i-i 1 w M 1 N fOM oo r> " l\iJ TfJUNMll^lMtJi HNM t^VO l NMMM>-iM' l -i l i-it-c l 'O O O 1-1 p dddddd d do d dddd ONOO VOVO>-O|fON'-iOSON| 1 to I fO N OO'-i'^f HtM'MMM'MMMlOO' ' O ' 'O O Opl-lp 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 ^Tj-tOfOfOtOt) to O O ""> UOOOO N ON ONVO vO totoo O r^ r^oo ON o M N to rf mvo t^oo WROUGHT-IRON COLUMNS. 417 418 APPLLED MECHANICS. And he gives, for the values off, For Phoenix columns /*= 36000; " American Company's columns . . . ./"= 30000; " box and open columns /= 31000. He deduces these values of f from some tests made in 1875 by Mr. Bouscaren, combined with those, already referred to, made at the Watertown Arsenal. The box and open columns were made of channel-bars and latticing. The tables or dia- grams presented to justify the formulae proposed can be found in the "Transactions of the American Society of Civil Engineers " for 1882. Besides the above there will be given here tables of three sets of tests of full-size wrought-iron columns, viz. : i. The series made at Watertown Arsenal, this being the most complete set of tests of full-size wrought-iron columns in existence. 2. A series of tests of Z-bar columns made by Mr. C, L. Strobel. 3. A few tests made at the Mass. Institute of Technology. Reference will also be made to the tests of Mr. G. Bouscaren, and to those made by Prof. Tetmajer, at the Materialprufungs- anstalt in Zurich. Graphical rep esentations, however, will first be given of the 10000 results of those tested at Watertown Arsenal, with the correspond- ing curves, representing (a) the formulae of Prof. Sondericker (see WROUGHT-IRON COLUMNS. 419 page 417), and (b) that of Mr. Strobel (see page 418). These diagrams will be preceded by the corresponding formulae. A perusal of them will show that, for values of less than P a certain quantity, which Mr. Strobel assumes as 90, arid Prof. Sondericker as 80 for flat-ended, and 60 for pin-ended columns; p the value of (i.e., the breaking-load divided by the area) is A I P constant. For greater values of the value of decreases, and P A for this portion of the curve, Prof. Sondericker's formulae are as follows : For flat-ended Phcenix columns he recommends Cooper's formula. For lattice columns with pin-ends, reported in Exec. Doc. 12, 47th Congress, ist session, and Exec. Doc. 5, 48th Congress, ist session, he recommends the formula P_ = 34000 A = 60 12000 For the box and solid web columns reported in Exec. Doc. 5, 48th Congress, ist session, and Exec. Doc. 35, 49th Congress, ist session, taken together with Bouscaren's results on box and on American Bridge Company's columns, he recommends P 33000 For flat-ends -r= - A. / \ 2 ' -80 p / 10000 p 31000 For pin-ends .......... -;-= 6000 4 2 APPLIED MECHANICS. In these formulae P = breaking-load in pounds, A= sectional area in square inches, / = length in inches, and p = least radius of gyration of section in inches. Moreover, the numerator in each of these formulae is the *P / value of -j- corresponding to the case when is less than 80 in A p flat-ended, and less than 60 in pin-ended columns. Instead of the above Mr. Strobel recommends for value of P I -r when - is less than 90, 35000 pounds per square inch, and, A P for values of greater than 90, the formula r =46ooo-i2 5 -. Moreover, if P'=safe load, in pounds, he recommends / P' (a)For-<90, -^- = 8000; p A. I P' I (b) For > 90, r = 1 0600 30. p A p While Gordon's formula, or a modification of it, is still in use in many bridge specifications, quite a number of them have substituted the Strobel formula, or a modification of it. WROUGHT-IRON COLUMNS SUBJECTED TO ECCENTRIC LOAD. All the formulae given thus far for the breaking or for the safe load on wrought-iron columns are only applicable when the load is so applied that its resultant acts along the axis of the column, and either the diagrams on pages 417 and 418, or the corresponding formulae, give us the breaking-strength per square inch, i.e., the number of pounds per square inch which, multiplied WROUGHT-IRON COLUMNS. 421 by the area in square inches, gives the breaking-load of the column; the safe load per square inch being obtained by dividing the breaking-load per square inch by a suitable factor of safety. On the other hand, whenever the resultant of the load on the column does not act along the axis of the column, we must determine the fibre-stress due to the direct load, and to this add the greatest fibre stress due to the bending-moment, the sum of the two being the actual greatest fibre stress, and the column must be so pro- portioned that this greatest fibre stress shall not exceed the safe strength per square inch, as determined by dividing the breaking- strength per square inch by the proper factor of safety; and this proceeding should be followed whatever be the cause of the eccentric load whether it be due to the beams supported by the column on one side being more heavily loaded than those on the other, whether it be due to the load transmitted from the columns above being eccentric, whether it be due to the mode of connection of the column to the other parts of the structure, whether it be due to poor fitting, or to any other cause. TESTS OF FULL-SIZE WROUGHT-IRON COLUMNS. The tests made at the Watertown Arsenal will next be given, together with cuts showing the form of the columns ; these being taken from the Tests of Metals for 1881, 1882, 1883, 1884, and 1885. The following tables are taken from the volume for 1881.: 422 APPLIED MECHANICS. jj 1 1 I 6 |= ^0 1 rQ it , ..,..,;,, ,|" 1 1 1 s 4 s "4} 3 .3 3 C o 13 C b rt sj D^ O c 5 a i . 0> 5 3 JS N t^OO O Tj-\O OO OO M 1 ~ ^^gcgasN^^vSaa^ss ^ *-o r^* ^^ I-H Q\ to o fO toco oo r^ ONOO c^ 2 "' OQOOOOOOO C 3 u)OOOQOOOO'-'VO Lr > 1 - r '**? *o T^- i-o o ^- o\\O ^O ro ^^ OOOOQOOOOOOOOOO vn\O ^O O "XXD t^O ^OLON rONcO f"O u W) o C [-OOOOOOOOOOOOOOOOOO 000000000000000000000000000000 J.& "rt .OOOOOOOOO Is .s^ic^s^^ Iooooooooooooooo VO vO ^vO O t^'O "- 1 fOCTNt^O ro M O F -^ ^^Tj-Tj-^TrTj-TJ-^TtTf^Tt^r^ S - M M N C4 vs- OVOVOVOOVOVOOOVOVOOO^^- "w|3 g-vOvOvOvOvOVDvOvOvO VOVOVOVOVOVOVOVOVOVOVOVOVOOOOO "3 ' * <u . . fi - d I* 2 . ."o- Is - -S, S s g tS. g. c E O" S a *o *: ON Q ""^O t^oO M N 1-1 d 1^88^22 ff N ON O txOO ONO"WOCOON-'N rOMrOMMi-iNNNNNiHMMM J5 M M M M WROUGHT-IRON COLUMNS. 423 I 1 11*9 I. I u 3 I...........I I". s i 5 ii jg i O '1 c i <U3 S 3 3 A Q U ll fO ^O fO ro CO ^O 000000000000000000000000 ( ^ c^ ^~ ^ovo CTvoo ^ O *^ O ^ ^^ O ""< f^ ^O ^) ^ Q O O ^O ^-Ti- N M .ooooooooooogooovooooooooooo^ooo c oo oo ^o - o\ t^oo "^"-1 Ooo o\"">i-< ^o Tt-g o r^ mvo N i^oo -^f <* oo "" ^ -^ ^o 10 c\"O r^. r^oo oo vO V>VI\Q 1^.00 OQfO vr >>-iO'^ T *'OSfO'-iON^ 1 -^ H OO OO O O OOVOVOOOOOOOOOC^MOOOOOOOOOO eOOOOOOOOOOOOOOOOOOOOOOOOOOOOOONNNNNMNNMN 424 APPLIED MECHANICS. a 1 s .S fe Q ^. pi .S* T3 "3 "3 72 rt^ rt^ So- - ggsgs S IsJsJ * -H g 1,, i 3: = 3 S 3 * IS II = - i ^OOQQOOOOOOQOO O\ r^ O VO "O TJ-OO w00^*fiew OOOOOOOOOO fn ""KQ -^ TJ- ON vo rf vn -^f O O O N n u-> ro O Q C3 j < ^ooo O\ "> ^ -^- <*S r~vO ^O fO>-ivo M 'J'O'Ooo "^w r coro t^.oo S **> g ^ < 00001^1^ VO ^ >>>2 " " J^^'S'S W N N N N S N n C fO f"> t^ fO fO t-^00 to "-) ro fO ro ro fOOO OO VO * -fvo MD r>>i>.tON M\OON *** co ro to w N ^O fOoO OO t^ r^x t^ r^ i^ i^ i^* r^ O O t*x t^ ON ON ON O O ON ^ ^ 5 crNNNNWc5cirof'5fOfOfOfOfO'<i-'^-'<f l o i OTi-Ti-ioio iovd NO *ovd "^ < x' c - \O vO t^ t^ fO rOOO OO 00 f^ ^5 ^^ ONOO O O t^ ^O fO ONOO w N t*>. fO to ONOO g _ _ M N rj-^ MNNNT}-i-iH<>-iWNN'<$-i-ii-i l -iN N N-'*- < EG u *H .Sj o ^ c/, g w H - : a" ^ V. g . M *$ *r\t~*. t< t*VD *O'~ "~ - - O |S WROUGHT-IRON 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 <*)(* . y r^ i-< vo M o ON ON ""> >-o r^^. O^ON fC. ON O\ ^ N .BOOOOOOOO rfvO \OOOOO O O N N 5 5 S 5 5 S a | H J5 bo if 1 S | ^ S Jl I If 1 I 1 I 00*0 00 oo N " -oooooooooooooo S 3 3 S S S J3s 3 3 3 S 3 E 426 APPLIED MECHANICS. 'rf IS rS . C *O O QJ N 3'S .s'S'S 6 o g t 3 a u .9-j Manner of ' t Deflected ho ( and upwar 1 o |je | t/3 c ~ S Ultimate 1 3 4 Ij .s^ <t! c^^ " ~ -C .5 J N o g" -s c ID .s a o "rt V C ^^ & it- WR UGHT-IR ON COL UMNS. 427 428 APPLIED MECHANICS. WROUGHT-IRON COLUMNS. 429 The next table taken from the volume for 1882 men- tioned above contains the results of some compressive tests of wrought-iron I-beams placed in the machine with the ends vertical and tested with flat-ends ; also of some tensile speci- mens cut off from two of them. TESTS OF I-BEAMS BY COMPRESSION. Width of Thick- Total *J C tx Sectional. Ultimate Strength. Length. Flange. Web. Depth. (U Area. Actual. PerSq. In In. In. In. In. L s. Sq. In. Lbs. Lbs. I 57.06 5-45 0.64 9.00 228 14.40 545 TOO 37854 2 155-45 4.40 0.40 10.52 443 IO.26 207000 20170 3 191.90 3-56 0.40 9.08 365 6.85 85380 12460 4 191.90 3-59 0-43 9.09 38i 7-15 85200 11916 5 119-85 2.98 0.28 6. ii 139 4.18 IOI200 24210 6 i8o.33 3.60 0.42 6.96 303 6.05 84650 13990 7 192.04 3.58 0-45 7-94 355 6.65 83400 12540 8 192.90 3.60 0.44 7.98 353 6-59 92300 I40IO 9 215-88 4.28 0.40 10.52 561 9-30 I49OOO I6O2O 10 264.08 4-49 0.48 10-53 747 10.19 II3IOO IIIOO ii 264.08 4-43 0.50 10.51 767 10.46 107800 10306 12 264.00 4.90 0.53 I5-I5 1085 14.80 184700 I24OO 13 263.95 4.84 0-53 14.74 1081 14.74 I87OOO 12686 TESTS OF SPECIMENS FROM NOS. I AND 2 BY TENSION. Cut from Flange or Web. Width. In. Depth. In. Sectional Area. Sq. In. Ultimate Strength. Contrac- tion of Area. Per Cent. Actual. Lbs. Per Sq. In. Lbs. f Web. 3-00 0.65 1-95 103300 52970 IO Web. 3-00 0.50 I-5I 65400 43340 3-9 1 Flange. 4.00 0.75 3-01 146400 48640 19.6 1 Flange. 4.00 0.76 3-02 I47IOO 48640 15-9 f Flange. 3.00 0.51 1-53 55400 36210 n. I I Web. 3.00 0.40 I.I 9 52900 44640 16.5 43 APPLIED MECHANICS. Next will be given the set of tests which is reported in the volumes for 1883 and 1884. The following is quoted from the first of the two : " COMPRESSION TESTS OF WROUGHT-IRON COLUMNS, LATTICED, BOX, AND SOLID WEB. " This series of tests comprises seventy-four columns, forty of the number having been tested, the results of which are herewith presented. "The columns were made by the Detroit Bridge and Iron Company. " The styles of posts represented are those composed of " Channel-bars with solid webs ; " Channel-bars and plates ; " Plates and angles ; " Channel-bars latticed, with straight and swelled sides ; " Channel-bars, latticed on one side, and with continuous plate on one side. " All the posts were tested with 3^-inch pins placed in the centre of gravity of cross-section ; except two posts of set N y which had the pins in the centre of gravity of the channel- bars. " This gave an eccentric loading for these columns, on ac- count of the continuous plate on one side of the channel- bars. " The pins were used in a vertical position, unless other- wise stated in the details of the tests. " In the testing-machine the posts occupied a horizontal position. " They were counterweighted at the middle. " Cast-iron bolsters for pin-seats were used between the ends WROUGHT-IRON COLUMNS. 43! of the columns and the flat compression platforms of the test- ing-machine. " The sectional areas were obtained from the weights of the channel-bars, angles, and plates, which were weighed before any holes were punched, calling the sectional area, in square inches, one-tenth the weight in pounds per yard of the iron. " Compressions and sets were measured within the gauged length by a screw micrometer. " The gauged length covered the middle portion of the post, and was taken along the centre line of the upper chan- nel-bar or plate, always using a length shorter than the dis- tance between the eye-plates, to obtain gaugings unaffected by the concentration of the load at those points. " The deflections were measured at the middle of the post. The pointer, moving over the face of a dial, indicated the amount and direction of the deflection. " Loads were gradually applied, measuring the compressions and deflections after each increment ; returning at intervals to the initial load to determine the sets. " The maximum load the column was capable of sustaining was recorded as the ultimate strength, although, previous to reaching this load, considerable distortion may have been pro- duced. " Observations were made on the behavior of the posts after passing the maximum load, while the pressure was fall- ing, showing, in some cases, a tendency to deflect with a sudden spring, accompanied by serious loss of strength. " The slips of the eye-plates along the continuous plates and channel-bars during the tests were measured for certain posts in sets F, G, H, and /. The measurements of slip were taken in a length of 10 inches or 20 inches, one end of the micrometer being secured to the eye-plate, and one end to the channel-bar. The readings include both the compression movement of the material and the slip of the plates. 43 2 APPLIED MECHANICS. " Columns H, 7, Z, and J/ were provided with pin-holes for placing the pins either parallel or perpendicular to the webs of the channel-bars. " After the ultimate strength had been determined with the pins in their first position, a supplementary test was made, if the condition of the column justified it, with the pins at right angles to their former position ; thus changing the moment of inertia of the cross-section, taken about the pin as an axis. " The experiments with columns N show how much strength is saved by employing pins in the centre of gravity of the cross- section. Where such was not the case, the columns showed the effect of the eccentric loading by deflections perpendicular to the axis of the pins, from the initial loads, which resulted in their early failure." WKO UGHT-1RON COL UMNS. 433 TABULATION OF EXPERIMENTS ON WROUGHT-IRON COLUMNS WITH 3J-INCH PIN-ENDS. Ultimate Length, Strength. Centre Sec- No. of Test. Style of Column. to Centre tional Area. Total, Lbs. Manner of Failure. of Pins. Lbs. per In. Sq. In. Sq. In. Set! A. 752 u _J 126.20 9-831 297100 30220 Deflected perpendicular to axis of pins. 757 *- " 120.07 10.199 320000 31380 Sheared rivets in eye- if ' t; 1 plates. 755 X> 10 180.00 9-977 251000 25160 Deflected perpendicular to axis of pins. 756 r ~~n~\ 180.00 9-977 210000 21050 Do. do. 753 *~~z 6 r> 240.00 9-732 188600 19380 Do. do. 754 i 240.10 9.762 158300 16220 Do. do. i SetlD, 1642 240.00 240.00 16.077 16.281 425000 367000 26430 22540 Deflected perpendicular to axis of pins. Do. do. ~] r =! 1646 n? 320.00 16.179 3I8800 19700 Do. do. 1647 \ 320.10 16.141 283600 Do. do. Hj, ' 8 > . Set a ,653 */ 8 I 320.00 17.898 474000 26480 Deflected perpendicular to axis of pins. 1654 i 320.00 19.417 49IOOO 25290 Do. do. Le^ 1 1 Setjp. ,645 e s' T r 319.95 16.168 453000 28020 Deflected parallel to axis of pins. 1*50 "*'" IL 320.00 16.267 454000 27910 Deflected perpendicular to axis of pins. ? 434 A PPLIE /) ME CHA A ICS. TABULATION OF EXPERIMENTS ON WR OUGHT-IRON COLUMNS WITH 3^-INCH PIN-ENDS. Length, Ultimate Strength. Centre Sec- No. of Test. Style of Column. to Centre tional Area. T/-\t<i 1 Lbs. Manner of Failure. of Pins. i otai. Lbs. per In. Sq. In. Sq. In. Set- G. 1651 320.00 20.954 540000 25770 Deflected m diagonal direction. 1652 l^6 T 320.10 20.613 535000 25950 Sheared rivets in eye- ' 2* ' \ plates. 746 SetiH. 159-20 7.628 258700 339X0 Deflected perpendicular to axis of pins. 747 ?^i- f Tt 159-27 8.056 294700 36580 Do. do. 748 ^- 8 *44 239.60 7.621 260000 34120 | Do. do. 749 i 239.60 7.621 254600 j 33410 Deflected in diagonal direction. 1648 ffi / * x- 31610 1 Deflected narallel to ?xis 319.90 7-7 243000 of pins. 1649 3^9-85 7 '673 229200 29870 Deflected in diagonal direction. 740 I i59-9o 7-645 262500 34340 Deflected perpendicular Set I. to axis of pins. 74 l (swelled.) i59-9o 7.624 255650 33530 Do. do. 739 ~~ i |~~t 239.70 7-5 1 ? 251000 33390 Deflected parallel to axis 1 of pins. 75 ( | 3 '$ "U 239.70 7 -S3 1 259000 34390 Deflected perpendicular . to axis oi pins. 1643 _ 1 _l 319.80 7.691 237200 30840 Deflected parallel to axis rf rin<* 1644 1 319.92 7.702 237000 30770 Deflected in diagonal direction. < 1640 J-K 199.84 "944, 403000 33740 Deflected perpendicular ~i p* to axis of pins. 1641 1634 4- - 200.00 300.00 12.302 12.148 426500 408000 34670 33630 Deflected in diagonal direction. Deflected perpendicular 7 to axis of pins. 1635 _i 3OO.OO 12.175 395000 32440 Do. do. r WROUGHT-1RON COLUMNS. 435 TABULATION OF EXPERIMENTS ON WROUGHT-IRON COLUMNS WITH 3J-INCH PIN-ENDS. Length, Centre Sec- Ultimate Strength. - No. of Test. Style of Column. to Centre tional Area. Total, Lbs. Manner of Failure. of Pins. Lbs. per In. Sq. In. Sq. In. Sejt M. (swelled.) 1638 1 199-25 12.366 385000 330 Deflected perpendicular to axis of pins. 1639 n *"o 199.50 12.659 405000 31990 Do. do. 1636 I 300.20 11.920 391400 32830 Deflected in diagonal direction. l6 37 . J 300.15 11.932 390700 32740 Do. do. 1630 1 *rf ik s! s_l 300.00 17.622 461500 26190 Deflected perpendicular to axis of pins. itS 1631 1 300.00 17.231 485000 28150 Do. do. 1632 |p-10 I .2 4 300.00 17-57 306000 17420 Do. do. ^| 1 *^o 1633 jr^-f 300.00 17.721 307000 17270 Do. do. K I 1 " The remainder of the tests of this series of seventy-four columns is reported in the volume for 1884. The only portion of the description that it is worth while to quote is the following, as the tests were made in a similar way to what has been already described : " Sixteen posts were tested with flat ends ; eighteen were tested with 3^-inch pin-ends. 436 APPLIED MECHANICS. " The pins were placed in the centre of gravity of cross- section, except two posts of set K, which had the pins in the centre of gravity of the channel-bars, giving an eccentric bear- ing to these columns, on account of the continuous plate on one side of the channel-bars." TABULATION OF EXPERIMENTS ON WROUGI IT-IRON COLUMNS WITH FLAT ENDS. Ultimate Total Sec- Strength. No. of Test. Style of Column. Length. tional Area. Total, Per Number of Failure. Sq. In., Ft. In. Sq. In. Lbs. 377 378 SetB. 1 *" 10 7.90 10 7.90 12.08 ii. ii 383200 372900 31722 33564 Buckling-plate D be- tween the riveting. Buckling-plates. "4 I 379 SetE. - 8 4 ^ 13 i i. 80 13 11.80 17.01 17.80 633600 34950 35595 Buckling - plates be' tween the riveting. Triple flexure. 346 13 11.9 15.74 517000 32846 Buckling-plates. v^" "T" 347 1 , f 13 11.65 15.84 555200 35050 Do. do. Set F. .a//' ^J 7j6 342 < 7^ 20 7 . 63 15.68 517500 33003 Deflecting upward. 344 w 20 7 . 80 15-56 536900 3455 Buckling-plates. 348 T 3 "-75 21 .02 708000 33682 Buckling-plates. ll ^ ^ 1 349 ; i" ? | 1 X 3 "-75 21 .46 709500 33061 Triple flexure. 343 SetG. jfc-Ka" _>- 6.90% it> 20 7.60 20 7.63 21.20 21.49 700000 729450 330'9 33943 Deflecting upward. Deflecting downward. ^ 8 " WRO UC,H 7 '-IRON COL UMNS. 437 TABULATION OF EXPERIMENTS ON WROUGHT-IRON COLUMNS WITH FLAT ENDS. Ultimate Total Sec- Strength. No. of Test. Style of Column. Length. tional Area. Manner of Failure. Total, Per Sq. Ft. In. Sq. In. Lbs. In., Ibs. 339 20 7-94 12.64 412900 32666 Deflecting upward. " 7* 1 T *** H SetK. T 340 er= =LI 20 7.94 12.74 431400 33862 Do. do. Latticed %,'plate. 71 337 338 SetN. -"I! 25 7-75 25 7-88 16.99 17.40 582400 580000 34279 33333 Deflecting downward and sideways. Deflecting diagonally channel B and lattic- ing on the concave Latticed side. TABULATION OF EXPERIMENTS ON WROUGHT-IRON COLUMNS WITH 3^-INCH PIN-ENDS. Length, Ultimate Centre Sec- Strength. No. of Test. Style of Column. to i tional Centre Area, of Pins. Manner o( Failure. Total, Per Sq. Ft. In. Sq. In. Lbs. In., Ibs. l 368 1 15 o.i 11.42 379200 33205 Hor. deflection perpen- dic. to plane of pins. 356 Set B. jji if 20 0.0 11.42 ii .42 342000 29947 Do. do. 357 20 0.0 11.31 330100 29186 Do. do. % < 371 9 ir -9 9.14 286100 31302 Buckling - plates be- tween rivets. 372 J IO O.O 10.07 319200 31698 Do. do. 370 kf i 4 a 15 o.o 9.21 291500 31650 I Hor. deflec. and buck- 369 Set C. J (. o. 4 L_ 15 o.o 9-44 290000 30720 ling between rivets. Do. do. 354 '* 20 o.o 9.24 267500 28950 Triple flexure. 365 20 o.o 9-36 279700 29879 Hor. deflection. 438 APPLIED MECHANICS. TABULATION OF EXPERIMENTS ON WROUGHT-IRON COLUMNS WITH 3^-INCH PIN-ENDS. Style of Column. Ultimate Length, Centre Sec- Strength. No. of Test. to Centre tional Area. Total, Per Manner of Failure. of Pins. Sq. In., Ft. In. Sq. In. Lbs. I l _l 360 361 <-^t> SetD. l| <* *3 4- I 3 13 4.00 15-34 15.40 475000 485000 30965 3*494 Deflecting upward in plane of pins. Hor. deflection perpen- dicular to plane of pins. 358 Sot E. , h 20 o.o 17.77 570000 32077 Hor. deflection perpen- dicular to plane of ." "jj 1 l pins. 359 - s"\ 4* 20 o.o 17.22 5554oo 32253 Do. do. 350 ^ C 20 0.25 12.48 202700 16242 Hor. deflection, con- C0| o- cave on lattice side. 351 SetK. !l IJ 00 1 20 0.00 10.84 208200 10207 Do. do. 352 i "\ Lt 5 2O O.25 12.65 350000 27668 Do. do. 353 51 E 2O O.25 12.7 390400 30596 Hor. deflection perpen- . dicular to plane of -I- pins, convex on lat- tice side. Besides the above, there are four tests of lattice columns reported in Exec. Doc. 36, 49th Congress, 1st session, but as these columns were rather poorly constructed and form rather special cases they will not be quoted here. In determining the strength of a bridge column made of channel-bars and latticing, these results of tests on full-size columns furnish us the best data upon which to base our con- clusions. WROUGHT-IRON COLUMNS. 439 In the Trans. Am. Soc. C. E. for April, 1888, Mr. C. L. Stro- bel gives an account of his tests on wrought-iron Z-bar columns, from which the following is condensed, viz.: The Z-irons used in making the columns were 2^X3X2^ inches in size, and & inch thick. Two columns were about n ft. long, two 15 ft., two 19 ft., three 22 ft., three 25 ft., and three 28 ft., a total of fifteen columns. The table of results follows: Ultimate Ultimate Strength, Length, Sectional Area, Strength by Tests / Q Strobel's per Sq. In. Formula per Sq. In. Inches. Sq. Ins. Lbs. Lbs. 13'i 9-435 36800 64 131* 34600 64 180 9.480 34600 88 35000 180 9.280 36600 88 35000 228f 9.241 33800 112 32200 228f 10. 104 33700 112 32200 264 9.286 30700 129 29900 264' 9.286 29500 129 29900 264 9.286 30700 129 29900 300 9.156 28100 I 4 6 27750 3 00 9-45 6 28000 I 4 6 27750 300 9.516 28400 I 4 6 27750 336 9-375 27700 164 25500 336 9-643 28000 l6 4 25500 336 9-375 27600 164 25500 The following table shows the results of compression tests made in the engineering laboratories of the Massachusetts In- stitute of Technology upon some wrought-iron pipe columns. They were tested with the ordinary cast-iron flange-coupling screwed on to the ends, bearing against the platforms of the testing- machine, which were adjustable, inasmuch as they were provided with spherical joints. 440 APPLIED MECHANICS. The tests of full-size wrought-iron columns made by Mr. G. Bouscaren, are given in the Report of the Progress of Work on the Cincinnati Southern Railway, by Thos. D. Lovett, Nov. i, 1875- In Heft IV (1890) of the Mittheilungen der Materialprii- fungsanstalt in Zurich is given an account of a large number of tests of wrought-iron and steel columns of the following forms, viz.: i. Angle-irons; 2. Tee iron; 3. Channel- bars ; 4*. Two angle-irons riveted together ; 5. Four angle-irons riveted together; 6. Two channel-bars riveted together; 7. Two tee irons riveted together ; also quite a number of tests of columns of some of these forms subjected to eccentric loads, the eccentricity of the load being, in some cases, as much as 8 cm. (3". 15). The columns tested were of a variety of lengths, the longest ones being 560 cm. (18.37) ^ eet l n g- In Heft VIII (1896) of the same Mittheilungen is an ac- count of another set of tests of columns of the above-described forms. The results of these valuable tests will not be quoted here, but for them the reader is referred to the Mittheilungen themselves. "o u V V I _H 2 i a a 5 c ominal Size Pipe. iside Diame utside Diam eter. iameter of Flanges. li "0 auge Lengtl aximum Lo; rea of Cross section. 'aximum Lo per Sq. In. ompression, Modulus of Elasticity. o o 2 u c'o * M C O s < 2 U ^ TQ. In. In. In. In. In. In. Lbs. Sq. In. Lbs. 2 2.06 2-37 7 I 51 30000 i. 08 27800 24300000 88.8 2 2.04 2-39 7 69 1 51 29800 1 .22 24500 222OOOOO 89.1 2 5 2.50 2.89 8 93 86 34500 1.65 20900 25200000 98.1 2 i 2.48 2.88 8 93 : 86 37000 1.68 22000 259OOOOO 98.4 3 3.06 3-44 8f 93 | 86 45500 1.94 23500 27700000 81.4 3 3.48 8| 93' 86 51000 2.01 25300 25IOOOOO 80.5 3 '.60 4.00 105! 100.5 55000 2-39 23000 25200000 78.2 32 3-59 3-99 9i 1:051 100.5 65000 2-39 27200 246OOOOO 78.5 4 4.07 4'53 9fr 100.5 80000 3-" 25700 258OOOOO 77-1 4 4.09 4-50 "7ft 100.5 69000 2.76 25000 24900000 77-3 224. Transverse Strength of Wrought-iron. Wrought-iron owes its extensive introduction into con- struction as much or more to the efforts of Sir William Fairbairn than to anyone else; and while he was furnishing TRANSVERSE STRENGTH OF WROUGHT-IRON. 441 the means to Eaton Hodgkinson to make extensive experiments on cast-iron columns, and while he made experiments himself on cast iron beams, which were in use at that time, he also carried on a large number of tests on beams built of wrought- iron, more especially those of tubular form, and those having an I or a T section, and made of pieces riveted together. In his book on the " Application of Cast and Wrought Iron to Building Purposes " he gives an account of a large number of these experiments, including those made for the purpose of designing the Britannia and Conway tubular bridges, a fuller account of which will be found in his book entitled " An Ac- count of the Construction of the Britannia and Conway Tubular Bridges." In the first-named treatise he urges very strongly the use of wrought-iron, instead of cast-iron, to bear a trans- verse load. Fairbairn tested a number of wrought-iron built-up beams, but they were of small dimensions and are hardly comparable with those used in practice. In the light of the tests made upon wrought-iron columns, it is evident that the compressive strength of wrought-iron is less than the tensile strength. Hence we should naturally ex- pect that the modulus of rupture would be, in all cases, greater than the compressive strength, and that it might or might not be greater than the tensile strength of the iron. Of course the modulus of rupture varies very much with the shape of the cross-section, for the same reasons as were explained in the paragraph 191, i.e., that the formula M = f assumes Hooke's law, "the stress is proportional to the strain," to hold, and that this is not true near the breaking-point. The value of the modulus of rupture is also influenced by the reduction in the rolls, and hence somewhat by the size of the beam. Small round or rectangular bars tested for transverse strength show a modulus of rupture very much in excess of the compressive strength per square inch of the iron, and ex- ceeding very considerably even the tensile strength. While a great many tests of such specimens have been 44 2 APPLIED MECHANICS. made, none will be quoted here, but the last five tests of the table on page 542 show that for a wrought-iron having a ten- sile strength per square inch from 58700 to 60250 pounds, mo- duli of rupture were obtained from Soooo to 90000 pounds, as, the number of turns of these rotating shafts being com- paratively small, the breaking-loads were not far below the quiescent breaking loads. On the other hand the moduli of rupture of I beams and other shapes used in building have very much lower values, but for these, tests will be cited. As to experiments on large beams, we have : i. Some tests made by Mr. William Sooy Smith and jy Col. Laidley at the Watertovvn Arsenal. 2. Some tests made in Holland on iron and steel beams, an account of which is given in the Proceedings of the Brit- ish Institute of Civil Engineers for 1886, vol. Ixxxiv. p. 412 et seq. 3. Some tests made in the laboratory of Applied Mechan- ics of the Massachusetts Institute of Technology, on iron and steel I beams. 4. Tests made by the different iron companies upon beams of their own manufacture, and recorded in their respective hand-books. Mr. Smith's tests are recorded in Executive Document 23, 46th Congress, second session. 5. In Heft IV (1890) of the Mittheilungen der Material- priifungsanstalt in Zurich will be found accounts of tests made by Prof. Tetmajer upon the transverse strength of I beams, of deck-beams, and of plate girders. The results of these tests will be given in the table on top of page 443- Specimens cut from the flanges, and also from the webs of the last seven of these beams, were tested for tension. In the case of those cut from the flanges, the tensile strength varied TRANSVERSE STRENGTH OF WROUGHT-IRON. 443 Depth. (Inches.) Moment of Inertia. (Inches)*. Span. (Inches.) Modulus of Rupture. ^Lbs. per Sq. In.) Modulus of Elasticity. (Lbs. per Sq. In.) 7.87 7.87 3-93 5-9i 7.87 9-45 n.8i 52.04 52.04 4.13 17-85 5 T -95 103. .-.4 62.96 62.96 3i-44 47.28 62.96 75.60 94.48 51190 56453 62852 56453 53894 51619 CT t8^? 27501500 28937700 28767101 28212500 28226700 273735 o 4 34 from 50200 in the 1 5". 75 beam to 57300 pounds per square inch in the 3^.93 beam. On the other hand, in the case of the specimens cut from the web, the tensile strengths varied from 44900 in the I i^.Si beam to 54400 pounds per square inch in the 3"-93 beam, the contraction of area per cent varying from 23.6 to 32.1 per cent in the flanges, and from 12.5 to 15.9 per cent in the web. The results obtained with the deck-beams are as follows : Depth. (Inches.) Moment of Inertia. (Inches) 4 . Span. (Inches.) Modulus of Rupture. (Lbs. per Sq. In.) Modulus of Elasticity. (Lbs. per Sq. In.) 4-93 19.88 70.86 56170 25112500 4.26 9-38 59.06 48920 25823500 3-52 5-33 47.24 553 20 25596000 3.48 4.71 39-37 54180 26804700 2.36 1.30 31.50 52760 24202400 '93 0.60 23.62 58160 Tensile tests of specimens cut from these deck-beams showed tensile strengths of from 47540 in the i".93 beam to 54750 pounds per square inch in the 2". 36 beam, and contrac- tions of area of from 14.1 per cent to 18.4 per cent. The results obtained with the plate girders are as follows, viz. : Depth of Web. (Inches.) Modulus of Rupture. (Lbs. per Sq. In.) Modulus of Elasticity. (Lbs. per Sq. In.) Depth of Web. (Inches.) Modulus of Rupture. (Lbs. per Sq. In.) Modulus of Elasticity. (Lbs. per Sq. In.) iS-75 iS-75 19.69 19.69 51480 53180 SH? 6 52610 26449200 25539100 24813900 25605500 23.62 23.62 27.56 27-56 52760 48490 47780 46500 26321200 26548700 25667100 26776300 444 APPLIED MECHANICS. The tensile strength of the material of the webs varied from 29860 to 41240 pounds per square inch, while the contraction of area was only 0.4 per cent. The tensile strength of the material of the flange-plates was 51050 pounds per square inch, with a contraction of area of 17 percent. The tensile strength of the angle-irons was 46357 pounds per square inch, with a contraction of area of 14 per cent. The following table gives the results that have been obtained in the tests that have been made upon wrought-iron I beams in the laboratory of Applied Mechanics of the Massachusetts Insti- tute of Technology. This table will give a fair idea of the strength and elasticity of such beams. TESTS OP WROUGHT-IRON BEAMS MADE IN THE LABORATORY OF APPLIED MECHANICS OF THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY, ALL LOADED AT THE CENTER. No. of Test. Ins. Moment of Inertia. Span. Ft. Ins. Break- ing Load. Lbs. Moduli of Rupture. Lbs. per Sq. In. Moduli of Elasticity. Lbs. per Sq. In. Remarks. 121 6 24.41 12 9500 42386 26679000 From Phoenix Co. 124 7 43-5 14 o 14100 48082 28457000 < a 126 5 12.47 12 6450 46624 29549000 < t t 1 ! 209 7 44-93 13 8 1 2 2OO 39670 31057000 t 1 I 211 8 67.32 13 8 I7OOO 42000 28532000 < ( 4 215 9 no. 78 14 8 23000 41680 27165000 ' . 1 ' ' 227 8 61 .20 13 6 18300 49640 27397000 From Belgium. 230 9 86.41 13 8 21300 45850 27365000 < * < > 235 9 40.91 13 8 13800 49140 27923000 E < 253 7 43-05 14 8 II3I9 40660 28045000 From Phoenix Co. 256 8 66.56 14 8 14547 38460 28187000 i 263 9 108.67 14 8 19694 36160 27050000 291 7 45-96 14 6 IO7OO 36340 26790000 i < 292 8 66.39 14 6 14300 38200 27380000 294 9 92.89 14 6 I92OO 41470 27050000 i '- 338 , 6 25.92 i4 7 72OO 37800 27860000 N. T. Steel & Iron Co. 34i 7 46.73 12 II 13600 40600 27410000 I It H 345 8 7I-25 14 7 15400 38400 26940000 i ,: T, , t ,, 379 7 48.84 12 IO 15500 44300 26170000 STEEL. 445 225. Steel. While steel is a malleable compound of iron, with less than 2 per cent of carbon and with other substances, the definition recommended by an international committee of metallurgists in 1876, and used to some extent in German and Scandanavian countries, is different from that in general use in English-speaking countries, and in France. The definition recommended by the international committee may be found in the Trans. Am. Inst. Min. Engrs. for October, 1876, and is in the following: i. That all malleable compounds of iron with its ordinary ingredients, which are aggregated from pasty masses, or from piles, or from any form of iron not in a fluid state, and which will not sensibly harden and temper, and which generally resemble what is called "wrought-iron," shall be called weld-iron (Schweisseisen). 2. That such compounds, when they will, from any cause, harden and temper, and which resemble what is now called puddled steel, shall be called weld-steel (Schweissstahl). 3. That all compounds of iron, with its ordinary ingredients which have been cast from a fluid state into malleable masses, and which will not sensibly harden by being quenched in water while at a red heat, shall be called ingot-iron (Flusseisen). 4. That all such compounds, when they will, from any cause, so harden, shall be called ingot-steel (Flussstahl). On the other hand, in English-speaking countries, those compounds which have been aggregated from a pasty mass, usually in the puddling -furnace, and which contain slag, are generally called wrought-iron, while those which have been cast from a molten state into a malleable mass are generally called steel. While this classification is not perfect, it states the most common practice in a general way. Exceptions, two of which are that it does not include the cases of cementation steel and of puddled steel, will not be discussed here. In view of the above, it will be plain that what is commonly 44 APPLIED MECHANICS. called mild steel in America, would be called ingot-iron under the definition of the international committee. Steel is usually made by one of three processes, viz. : the crucible process, the Bessemer process, or the open-hearth process. While other processes, as the cementation process and others, are sometimes used, the three enumerated above are in most common use at the present time. Crucible Steel. This is very commonly made by re-melting blister-steel in crucibles; the blister-steel being made by the cementation process, in which bars of very pure wrought-iron, especially low in phosphorus, are heated in contact with charcoal until they have absorbed the necessary amount of qarbon. A cheaper process, and one much used at the present day, is to melt a mixture of charcoal and crude bar-;ron in a crucible. Crucible steel, which is always high-carbon steel, is used for the finest cutlery, tools, etc., and "wherever a very pure and homogeneous quality of steel is required. Bessemer Steel. In the Bessemer process, a blast of air is blown into melted cast-iron, removing the greater part of its carbon and burning out more or less of the other ingredients. The process is conducted in a converter, which is usually so arranged that, when the operation is complete, it can be rotated around a horizontal axis to such an extent that the tuyeres are above the surface of the molten steel, and the blast is shut off. In the acid Bessemer process, the lining of the converter is made of some silicious substance, the burning of silicon being relied upon to develop a sufficiently high temperature to keep the metal fluid. In the basic Bessemer process, the lining of the converter is of such a nature as to resist the action of basic slags. It is usually made of dolomite, or of some kind of limestone. Burned lime is added to the charge to seize the silicon and phosphorus, the latter serving to develop a sufficiently high temperature. STEEL. 447 In the latter part of the operation, the phosphorus is largely burned out, whereas in the acid process, in order to produce a steel that is low in phosphorus, it is necessary to use a pig-iron that is low in phosphorus. Open-hearth Steel. In the open-hearth process, a charge of pig-iron and scrap is placed on the bed of a regenerative furnace, and exposed to the action of the flame, and is thus converted into steel. In the acid open-hearth process, the lining of the furnace is of a silicious nature, and is covered with sand, while in the basic it is usually of dolomite, or of some kind of limestone. Bessemer and open-hearth steel contain more impurities than crucible steel, but they are very much cheaper, and are just as suitable for many purposes. It is only in consequence of their introduction that steel can be extensively used on the large scale, as crucible steel would be too expensive for many purposes. Steel, unlike wrought-iron, is fusible; unlike cast-iron, it can be forged; and, with the exception of the harder grades, it can be welded by heating and hammering, the welding of high -carbon steel in large masses being a very uncertain operation, though small masses can be welded by taking proper care. The special characteristic, however, is, that, with the exception of the milder grades, when raised to a red heat and suddenly cooled, it becomes hard and brittle, and that, by subsequent heating and cooling, the hardness may be reduced to any desired degree. The first process is called hardening and the second tempering. The principal element in the steels that are ordinarily used is carbon; nevertheless, both Bessemer and open-hearth steel con- tain also silicon, manganese, sulphur, phosphorus, etc., which have more or less effect upon the resisting properties of the metal. Sulphur, silicon, and phosphorus usually come from the ore, the fuel, and the flux, while manganese, which is added, operates, 44-8 APPLIED MECHANICS. among other things, to render the steel ductile while hot, and therefore workable, and to absorb oxygen from the melted mass. Sulphur is injurious by causing brittleness when hot, and phosphorus by causing brittleness when cold. Phosphorus is the most harmful ingredient in steel, so that when steel is to be used for structural purposes, it is important to have as little phosphorus as possible, and any excess of phosphorus is not to be tolerated. The injury done to steel plates by punching is greater than that done to iron plates: this injury can, however, be removed by annealing. Steel requires greater care in working it than iron, whether in punching, flanging, riveting, or other methods of working; otherwise it may, if overheated, burn, or receive other injury from careless workmanship. The chemical composition of steel is one important element in its resisting properties; but, on the other hand, the mode of working also has a great influence on the quality. The introduction of the Bessemer process was quickly fol- lowed by the general use of steel rails, and later, as this and the other processes for making steel for structural purposes have been developed, there has been a constant increase in the pur- poses for which steel has been used. One of the earlier applications was to the construction of steam-boilers, steel boiler-plate displacing almost entirely wrought-iron boiler-plate. Of late years the development of the steel manufacture has so perfected, and at the same time cheapened, structural steel that it is now used in most cases where wrought-iron was formerly employed. Thus the eye-bars and the struts of bridges are almost exclusively made of steel, also such shapes as angle-irons, channel-bars, Z bars, tee iron, I beams, etc., are almost exclusively made of steel, and while steel has long been used for many parts of machinery, never- theless it is now generally used in many cases where a con- siderable fear of it formerly existed, as in main rods, parallel ^ TEEL. 449 rods, and crank-pins, and in a large number of parts of machinery subjected to more or less vibration. On the other hand, the steel used for tools is, of course, high-carbon steel. Tools are almost always made of crucible steel, and they have of course a high percentage of carbon, a high tensile strength, and especially should they be capable of being well hardened and taking a good temper. The usual steel of commerce may be called carbon steel, because, although it always contains small percentages of other ingredients, nevertheless carbon is the ingredient that princi- pally determines its properties. When iron or steel is alloyed with large percentages of certain substances, the resulting al- loys enjoy certain special properties, and these alloys still bear the name of steel. Two of the most prominent of these are manganese steel and nickel steel. Regarding the first it may be said that although carbon steel becomes practically useless when the manganese reaches about ij per cent, nevertheless with manganese exceeding about 7 per cent we obtain manganese steel which is so hard that it is exceed- ingly difficult to machine it. The alloy that has come into most prominent notice recently is nickel steel, which consists most commonly of a carbon steel with from 0.2 to 0.4 per cent of carbon and with from 3 to 5 per cent of nickel. With this amount of nickel the tensile strength is very much increased, but more especially is the limit of elasticity increased by a very large amount; and while the contraction of area at fracture and the ultimate elongation per cent are a little less than that of carbon steel with the same percentage of carbon, they are not less than those of carbon steel of the same tensile strength. It is used for armor-plates, for which it is specially suitable on account of the fact that the nickel renders the steel more sensitive to hardening. It is finding, also, a great many other uses to which it is specially adapted by its peculiar properties. 450 APPLIED MECHANICS. It has been used for bicycle-spokes, for shafts for ocean steam- ships, for piston-rods, and for various other purposes. Among the many examples given by Mr. D. H. Browne in a paper before the American Institute of Mining Engineers is a case where the presence of 3.5 per cent nickel increased the ultimate strength of 0.2 per cent carbon steel from 55000 to 85000, and the elastic limit from 28000 to 48000 pounds per square inch, while the contraction of area at fracture was only decreased from 60 per cent to 55 per cent. The quality of steel to be used for different purposes differs, and while the specifications for any one purpose, made by different engineers, and by different engineering societies, often differ, the work of the American Society for Testing Materials is tending to harmonize them as far as possible. The result of their efforts is shown in the following set of specifications. AMERICAN SOCIETY FOR TESTING MATERIALS. SPECIFICATIONS FOR STEEL. STEEL CASTINGS. Adopted 1901. Modified 1905. PROCESS OF MANUFACTURE. 1. Steel for castings may be made by the open-hearth, crucible, or Bessemer process. Castings to be annealed unless otherwise specified. CHEMICAL PROPERTIES. 2. Ordinary castings, those in which no physical requirements are Ordinary specified, shall not contain over 0.40 per cent of carbon, Castings. nor ove] . Q Q g p er cent Q f ph os p norus> 3. Castings which are subjected to physical test shall not contain Tested oy er 0.05 per cent of phosphorus, nor over 0.05 per cent Castings. of sulphun PHYSICAL PROPERTIES. 4. Tested castings shall be of three classes: "hard," " medium," Tensile & n d "soft." The minimum physical qualities required in Tests * each class shall be as follows : STEEL CASTINGS. 451 Hard Medium Soft Castings. Castings. Castings. Tensile strength, pounds per square inch . 85000 70000 60000 Yield-point, pounds per square inch . 38250 3I5 00 27000 Elongation, per cent in 2 inches .... 15 18 22 Contraction of area, per cent 20 25 3 5. A test to destruction may be substituted for the tensile test in the case of small or unimportant castings by selecting three castings from a lot. This test shall show the material to be ductile and free from injurious defects and suitable for the pur- poses intended. A lot shall consist of all castings from the same melt or blow, annealed in the same furnace charge. 6. Large castings are to be suspended and hammered all over. No cracks,- flaws, defects, nor weakness shall appear after Percussive such treatment. Test - 7. A specimen one inch by one-half inch (i"Xi") shall bend cold around a diameter of one inch (i") without fracture on Bending outside of bent portion, through an angle of 120 for "soft" Test ' castings and of 90 for "medium " castings. TEST PIECES AND METHODS OF TESTING. 8. The standard turned test specimen one-half inch (J") diameter and two inch (2") gauged length shall be used to determine Test Speci- the physical properties specified in paragraph No. 4. It Tensiie"Test. is shown in Fig. i. (See page 398.) 9. The number of standard test specimens shall depend upon the character and importance of the castings. A test piece shall be cut cold from a coupon to be moulded and cast on some portion of one or more castings from each melt JSens? Spec " or blow or from the sink-heads (in case heads of sufficient size are used). The coupon or sink-head must receive the same treat- ment as the casting or castings before the specimen is cut out, and before the coupon or sink-head is removed from the casting. 10. One specimen for bending test one inch by one-half inch (i"X i") shall be cut cold from the coupon or sink-head of the cast- jest specimen ing or dastings as specified in paragraph No. 9. The for Bending - bending test may be made by pressure or by blows. 452 APPLIED MECHANICS. n. The yield-point specified in paragraph No. 4 shall be deter- mined by the careful observation of the drop of the beam Yield-point. . J . or halt in the gauge of the testing-machine. 12. Turnings from tensile specimen, drillings from the bending specimen, or drillings from the small test ingot, if preferred chemical by the inspector, shall be used to determine whether or not the steel is within the limits in phosphorus and sulphur specified in paragraphs Nos. 2 and 3. FINISH. 13. Castings shall be true to pattern, free from blemishes, flaws, or shrinkage cracks. Bearing-surfaces shall be solid, and no porosity shall be allowed in positions where the resistance and value of the casting for the purpose intended will be seriously affected thereby. INSPECTION. 14. The inspector, representing the purchaser, shall have all reasonable facilities afforded to him by the manufacturer to satisfy him that the finished material is furnished in accordance with these specifications. All tests and inspections shall be made at the place of manufacture, prior to shipment. STEEL FORCINGS. Adopted 1901. Modified 1905. PROCESS OF MANUFACTURE. 1. Steel for forgings may be made by the open-hearth, crucible, or Bessemer process. CHEMICAL PROPERTIES. 2. There shall be four classes of steel forgings which shall conform to the following limits in chemical composition: Forgings of Soft or Low-car- bon Steel. Forgings of Carbon Steel not Annealed. Forgings of Carbon Steel Oil-tempered or Annealed. Loco- motive Forg- ings. Forgings of Nickel Steel, Oil-tempered or Annealed. Phosphorus shall not exceed Sulphur Manganese " " " Nickel Per Cent. 0. 10 0. 10 Per Cent. 0.06 0.06 Per Cent. 0.04 0.04 Per Ct, 0.05 0.05 0.60 Per Cent. 0.04 0.04 3 . o to 4 . o Tensile Tests. PHYSICAL PROPERTIES. 3. The minimum physical qualities required of the different-sized forgings of each class shall be as follows: STEEL FORCINGS. 453 Tensile Strength. Lbs. per Sq. In. Yield- point. Lbs. per Sq. In. Elonga- tion in 2 Inches. Per Cent. Contrac- tion of Area. Per Cent. SOFT STEEL OR LOW-CARBON STEEL. 58000 29000 28 35 For solid or hollow f orgings, no diameter - or thickness of section to exceed 10". CARBON STEEL NOT ANNEALED. 75000 375 18 30 For solid or hollow forgings, no diameter or thickness of section to exceed 10". Elastic Limit. CARBON STEEL ANNEALED. 80000 40000 22 35 For solid or hollow forgings, no diameter or thickness of section to exceed 10". 75000 375 23 35 For solid forgings, no diameter to exceed 20" or thickness of section 15". 7OOOO 35000 24 3 For solid forgings, over 20" diameter. CARBON STEEL OIL-TEMPERED. 90000 55000 2O 45 For solid or hollow forgings, no diameter or thickness of section to exceed 3". 85000 50000 22 45 For solid forgings of rectangular sections not exceeding 6" in thickness or hol- low forgings, the walls of which do not exceed 6" in thickness. 80000 45000 23 40 For solid forgings of rectangular sections not exceeding 10" in thickness or hol- low forgings, the walls of which do not exceed 10" in thickness. 80000 40000 20 25 LOCOMOTIVE FORCINGS. NICKEL STEEL ANNEALED. 80000 50000 2 5 45 For solid or hollow forgings, no diameter or thickness of section to exceed 10". 8OOOO 45000 25 45 For solid forgings, no diameter to exceed 20" or thickness of section 15". 80000 45000 24 40 For solid forgings, over 20" diameter. NICKEL STEEL, OIL-TEMPERED. 95000 65000 21 50 For solid or hollow forgings, no diameter or thickness of section to exceed 3". 9OOOO 60000 22 50 For solid forgings of rectangular sections not exceeding 6" in thickness or hol- low forgings, the walls of which do not exceed 6" in thickness. 85000 55ooo 24 45 For solid forgings of rectangular sections not exceeding 10" in thickness or hol- low forgings, the walls of which do not exceed 10" in thickness. 454 APPLIED MECHANICS. 4. A specimen one inch by one-half inch (i"Xj") shall bend cold 180 without fracture on outside of the bent portion, Test. as follows: Around a diameter of J n ', for forgings of soft steel. Around a diameter of ij", for forgings of carbon steel not annealed. Around a diameter of ij", for forgings of carbon steel annealed, if 20" in diameter or over. Around a diameter of i", for forgings of carbon steel annealed, if under 20" diameter. Around a diameter of i", for forgings of carbon steel, oil tempered. Around a diameter of J", for forgings of nickel steel annealed. Around a diameter of i", for forgings of nickel steel, oil tempered, For locomotive forgjngs no bending tests will be required. TEST PIECES AND METHODS OF TESTING. 5. The standard turned test specimen, one-half inch (J") diameter Test sped- and two (2") gauged length, shall be used to determine site Test. the physical properties specified in paragraph No. 3. It is shown in Fig. i. (See page 398.) 6. The number and location of test specimens to be taken from a melt, blow, or a forging, shall depend upon its character LoSx>n a of d and importance, and must therefore be regulated by hSensf pec " individual cases. The test specimens shall be cut cold from the forging or full-sized prolongation of same parallel to the axis of the forging and half-way between the centre and outside, the specimens to be longitudinal; i.e., the length of the specimen to correspond with the direction in which the metal is most drawn out or worked. When forgings have large ends or collars, the test specimens shall be taken from a prolongation of the same diameter or section as that of the forging back of the large end or collar. In the case of hollow shafting, either forged or bored, the specimen shall be taken within the finished section prolonged, half-way between the inner and outer surface of the wall of the forging. 7. The specimen for bending test one inch by one-half inch. Test Specimen ( I// Xj // ) shall be cut as specified in paragraph No. 6. for Bending. The bending test may be made by pressure or by blows. OPEN-HEARTH BOILER PLATE AND RIVET STEEL. 455 8. The yield-point specified in paragraph No. 3 shall be determined by the careful observation of the drop of the beam, or halt in the gauge of the testing machine. Yield-point 9. The elastic limit specified in paragraph No. 3 shall be determined by means of an extensometer, which is to be attached to Elastic the test specimen in such manner as to show the change Limit. in rate of extension under uniform rate of loading, and will be taken at that point where the proportionality changes. 10. Turnings from the tensile specimen or drillings from the bend- ing specimen or drillings from the small test ingot, if pre- ferred by the inspector, shall be used to determine whether or not the steel is within the limits in chemical composition na ysis * specified in paragraph No. 2. FINISH. 11. Forgings shall be free from cracks, flaws, seams, or other injurious imperfections, and shall conform to the dimensions shown on drawings furnished by the purchaser, and be made and finished in a workmanlike manner. INSPECTION. 12. The inspector, representing the purchaser, sha.ll have all reason- able facilities afforded him by the manufacturer to satisfy him that the finished material is furnished in accordance with these specifications. All tests and inspections shall be made at the place of manufacture, prior to shipment. OPEN-HEARTH BOILER PLATE AND RIVET STEEL. Adopted 1901. PROCESS OF MANUFACTURE. 1. Steel shall be made by the open-hearth process. CHEMICAL PROPERTIES. 2. There shall be three classes of open-hearth boiler plate and rivet steel; namely, flange, or boiler steel, fire-box steel, and extra- soft steel, which shall conform to the following limits in chemical composition: APPLIED MECHANICS. Flange or Boiler Steel. Per Cent. Fire-box Steel. Per Cent. Extra Soft Steel. Per Cent. Phosphorus shall not exceed . Sulphur " " " . . Manganese ("Acid 0.06 \ Basic o . 04 0.05 o 30 to o 60 Acid o . 04 Basic 0.03 0.04 o 30 to o co Acid o . 04 Basic 0.04 0.04 Boiler-rivet Steel. 3. Steel for boiler rivets shall be of the extra-soft class as specified in paragraphs Nos. 2 and 4. PHYSICAL PROPERTIES. 4. The three classes of open-hearth boiler plate and rivet steel- Tensile Tests. ities : namely, flange or boiler steel, fire-box steel, and extra- soft steel shall conform to the following physical qual- Flange or Boiler Steel. Fire-box Steel. Extra Soft Steel. Tensile strength, pounds per square inch . Yield-point, in pounds per square inch, shall not be less than Elongation, per cent in 8 inches shall not be less than 55000 to 65000 JT.S. 2C 52000 to 62000 IT. s. 26 45000 to 55000 JT.S. 28 5. For material less than five-sixteenths inch (&") and more than three-fourths inch ( j") in thickness the following modifica- inEio'n*ation t * ons s ^ a ^ ^ e ma( ^ e m t ^ le requirements for elongation: for Thin and (#) For each increase of one-eighth inch (") in thick- ness above three-fourths inch (f") a deduction of one per cent (i%) shall be made from the specified elongation. (b). For each decrease of one-sixteenth inch (&") in thickness below five-sixteenths inch (&") a deduction of two and one-half per cent (2^%) shall be made from the specified elongation. 6. The three classes of open-hearth boiler plate and rivet steel B nd'n shall conform to the following bending tests; and for this Tests. purpose the test specimen shall be one and one-half inches (ii") wide, if possible, and for all material three-fourths inch (}") or less in thickness the test specimen shall be of the same thickness as that OPEN-HEARTH BOILER PLATE AND RIVET STEEL. of the finished material from which it is cut, but for material more than three-fourths inch }") thick the bending-test specimen may be one- half inch (J") thick: Rivet rounds shall be tested of full size as rolled. (c). Test specimens cut from the rolled material, as specified above, shall be subjected to a cold bending test, and also to a quenched bending test. The cold bending test shall be made on the material in the con- dition in which it is to be used, and prior to the quenched bending test the specimen shall be heated to a light cherry-red, as seen in the dark, and quenched in water the temperature of which is between 80 and 90 Fahrenheit. (d). Flange or boiler steel, fire-box steel, and rivet steel, both before and after quenching, shall bend cold one hundred and eighty degrees (180) flat on itself without fracture on the outside of the bent portion. 7. For fire-box steel a sample taken from a broken tensile-test specimen shall not show any single seam or cavity more Homogeneity than one-fourth inch (J") long in either of the three fractures Tests - obtained on the test for homogeneity as described below in paragraph 12. TEST PIECES AND METHODS OF TESTING. 8. The standard specimen of eight inch (8") gauged length shall be used to determine the physical properties specified in paragraphs Nos. 4 and <. The standard shape of the men for . , , ,, , , T^- Tensile Test. test specimen for sheared plates shall be as shown in rig. 2. (See page 398.) For other material the test specimen may be the same as for sheared plates, or it may be planed or turned parallel throughout its entire length ; and in all cases, where possible, two opposite sides of the test specimens shall be the rolled surfaces. Rivet rounds and small rolled bars shall be tested of full size as rolled. 9. One tensile-test specimen will be furnished from each plate as it is rolled, and two tensile-test specimens will be furnished from each melt of rivet rounds. In case any one of these develops flaws or breaks outside of the middle third of its gauged length, it may be discarded and another test specimen sub- stituted therefor. APPLIED MECHANICS. 10. For material three-fourths inch (-") or less in thickness the bending-test specimen shall have the natural rolled surface mens S ?or CI ~ on two opposite sides. The bending-test specimens cut from plates shall be one and one-half inches (ij") wide, and for material more than three-fourths inch (") thick the bending- test specimens may be one-half inch (J") thick. The sheared edges of bending-test specimens may be milled or planed. The bending- test specimens for rivet rounds shall be of full size as rolled. The bending test may be made by pressure or by blows. 11. One cold-bending specimen and one quenched-bending specimen will be furnished from each plate as it is rolled. Two Bending cold-bending specimens and two quenched-bending speci- mens will be furnished from each melt of rivet rounds. The homogeneity test for fire-box steel shall be made on one of the broken tensile-test specimens. 12. The homogeneity test for fire-box steel is made as follows: A portion of the broken tensile-test specimen is either nicked Homogeneity with a chisel or grooved on a machine, transversely about Phi-box a sixteenth of an inch (;&") deep, in three places about two inches (2") apart. The first groove should be made on one side, two inches (2") from the square end of the specimen; the second, two inches (2") from it on the opposite side; and the third, two inches (2") from the last, and on the opposite side from it. The test specimen is then put in a vise, with the first groove about a quarter of an inch (J") above the jaws, care being taken to hold it firmly. The projecting end of the test specimen is then broken off by means of a hammer, a number of light blows being used, and the bending being away from the groove. The specimen is broken at the other two grooves in the same way. The object of this treatment is to open and render visible to the eye any seams due to failure to weld up, or to foreign interposed matter, or cavities due to gas bubbles in the ingot. After rupture, one side of each fracture is examined, a pocket lens being used, if necessary, and the length of the seams and cavities is determined. 13. For the purposes of this specification the yield -point shall be determined by the careful observation of the 'drop of the Yield-point. ^ eam or na j t m tne g au g e of the testing machine. OPEN-HEARTH BOILER PLATE AND RIVET STEEL. 459 14. In order to determine if the material conforms to the chemical limitations prescribed in paragraph 2 herein, analysis shall be made of drillings taken from a small test ingot. aiemicai r An additional check analysis may be made from a tensile Anal * sis - specimen of each melt used on an order, other than in locomotive fire-box steel. In the case of locomotive fire-box steel a check analysis may be made from the tensile specimen from each plate as rolled. VARIATION IN WEIGHT. 15. The variation in cross section or weight of more than 2j per cent from that specified will be sufficient cause for rejection, except in the case of sheared plates, which will be covered by the following per- missible variations: (e) Plates 12 J pounds per square foot for heavier, up to 100 inches wide when ordered to weight, shall not average more than 2^ per cent variation above or 2\ per cent below the theoretical weight. When 100 inches wide and over, 5 per cent above or 5 per cent below the theoretical weight. (/) Plates under 12 J pounds per square foot, when ordered to weight, shall not average a greater variation than the following: Up to 75 inches wide, 2 J per cent above or 2 J per cent below the theo- retical weight. Seventy -five inches wide up to 100 inches wide, 5 per cent above or 3 per cent below the theoretical weight. When 100 inches wide and over, 10 per cent above or 3 per cent below the theoretical weight. (g) For all plates ordered to gauge there will be permitted an average excess of weight over that corresponding to the dimensions on the order equal. in amount to that specified in the following table: TABLE OF ALLOWANCES FOR OVERWEIGHT FOR RECTANGULAR PLATES WHEN ORDERED TO GAUGE. Plates will be considered up to gauge if measuring not over -^ inch less than the ordered gauge. The weight of one cubic inch of rolled steel is assumed to be 0.2833 pound. 460 APPLIED MECHANICS. PLATES \ INCH AND OVER IN THICKNESS. Width of Plate. Thickness of Plate. Inch. Up to 75 Inches. 75 to 100 Inches. Over 100 Inches. Per Cent. Per Cent. Per Cent. i IO 14 18 A 8 12 16 1 7 IO 13 tk 6 8 10 1 5 7 9 A 4* 6J 8i ^ 4 6 8 Overf 3* 5 6* PLATES UNDER INCH IN THICKNESS. Thickness of Plate. Inch. Width of Plate. Up to 50 Inches. Per Cent. 50 Inches and Above. Per Cent. up to ^j 5 ' i t t 3 32 16 3 < ( "1 16 1 10 7 Si IO FINISH. 1 6. All finished material shall be free from injurious surface defects and laminations, and must have a workmanlike finish. BRANDING. 17. Every finished piece of steel shall be stamped with the melt number, and each plate and the coupon or test specimen cut from it shall be stamped with a separate identifying mark or number. Rivet steel may be shipped in bundles securely wired together with the melt number on a metal tag attached. INSPECTION. 1 8. The inspector, representing the purchaser, shall have all reason- able facilities afforded to him by the manufacturer to satisfy him that the finished material is furnished in accordance with these specifica- tions. All tests and inspections shall be made at the place of manu- facture, prior to shipment. STRUCTURAL STEEL FOR BUILDINGS. 461 STRUCTURAL STEEL FOR BUILDINGS. Adopted 1901. PROCESS OF MANUFACTURE. 1. Steel may be made by either the open-hearth or Bessemer process. CHEMICAL PROPERTIES. 2. Each of the two classes of structural steel for buildings shall not contain more than o. 10 per cent of phosphorus. PHYSICAL PROPERTIES. 3. There shall be two classes of structural steel for buildings, namely, rivet steel and medium steel, which shall con- Classes, form to the following physical qualities: 4. Tensile Tests. Rivet Steel. Medium Steel. Tensile strength, pounds per square inch . Yield-point, in pounds per square inch, shall not be less than 50000 to 60000 *T. S. 60000 to 70000 *T. S. Elongation, per cent in 8 inches shall not be less than 26 22 5. For material less than five-sixteenths inch (h") and more than three-fourths inch (}") in thickness the following modifica- tions shall be made in the requirements for elongation: Modifications (a) For each increase of one-eighth inch (") in thick- f^* and" ness above three-fourths inch (f") a deduction of one Thick Material - per cent (i%) shall be made from the specified elongation. (b) For each decrease of one-sixteenth inch (T&") in thickness below five-sixteenths inch (&") a deduction of two and one-half per cent (2^%) shall be made from the specified elongation. (c) For pins the required elongation shall be five per cent (5%) less than that specified in paragraph No. 4, as determined on a test specimen the centre of which shall be one inch (i") from the surface. 6. The two classes of structural steel for buildings shall conform to the following bending tests ; and for this purpose the test Bending> specimen shall be one and one-half inches (ij") wide, if Tests - possible, and for all material three-fourths ( J") or less in thickness the test specimen shall be of the same thickness as that of the finished material 462 APPLIED MECHANICS. from which it is cut, but for material more than three-fourths inch (f ") thick the bending-test specimen may be one-half inch (J") thick. Rivet rounds shall be tested of full size as rolled. (d) Rivet steel shall bend cold 180 flat on itself without fracture on the outside of the bent portion. (e) Medium steel shall bend cold 180 around a diameter equal to the thickness of the specimen tested, without fracture on the outside of the bent portion. TEST PIECES AND METHODS OF TESTING. 7. The standard test specimen of eight-inch (8") gauged length shall be used to determine the physical properties specified men for Ten- in paragraphs Nos. 4 and 5. The standard shape of the test specimen for sheared plates shall be as shown by Fig. 2. (See page 398.) For other material the test specimen may be the same as for sheared plates or it may be planed or turned parallel throughout its entire length and, in all cases where possible, two oppo- site sides of the test specimen shall be the rolled surfaces. Rivet rounds and small rolled bars shall be tested of full size as rolled. 8. One tensile-test specimen shall be taken from the finished Number of material of each melt or blow ; but in case this develops Tensile Tests. fl aws> or breaks outside of the middle third of its gauged length, it may be discarded and another test specimen substituted therefor. 9. One test specimen for bending shall be taken from the finished material of each melt or blow as it comes from the rolls, and Test men for for material three-fourths inch (f ") and less in thickness this specimen shall have the natural rolled surface on two opposite sides. The bending-test specimen shall be one and one- half inches (ij") wide, if possible; and for material more than three- fourths inch (I") thick the bending-test specimen may be one-half inch (I") thick. The sheared edges of bending-test specimens may be milled or planed. Rivet rounds shall be tested of full size as rolled. (/) The bending test may be made by pressure or by blows. 10. Material which is to be used without annealing or further Annealed treatment shall be tested for tensile strength in the con- dition in which it comes from the rolls. Where it is STRUCTURAL STEEL FOR BUILDINGS. 463 impracticable to secure a test specimen from material which has been annealed or otherwise treated, a full-sized section of tensile- test specimen length shall be similarly treated before cutting the tensile- test specimen therefrom. 11. For the purposes of this specification the yield-point shall be determined by the careful observaton of the drop of the Yield-point. beam or halt in the gauge of the testing machine. 12. In order to determine if the material conforms (o the chemical limitations prescribed in paragraph No. 2 ch?micai r herein, analysis shall be made of drillings taken from a AnJ small test ingot. VARIATION IN WEIGHT. 13. The variation in cross section or weight of more than 2\ per cent from that specified will be sufficient cause for rejection, except in the case of sheared plates, which will be covered by the following per- missible variations : (g) Plates 12^ pounds per square foot or heavier, up to 100 inches wide, when ordered to weight, shall not average more than 2j per cent variation above or 2\ per cent below the theoretical weight. When 100 inches wide, and over 5 per cent above or 5 per cent below the theoretical weight. (h) Plates under 12 J pounds per square foot, when ordered to weight, shall not average a greater variation than the following : Up to 75 inches wide, 2^ per cent above or 2\ per cent below the theoretical weight. Seventy-five inches wide up to 100 inches wide, 5 per cent above or 3 per cent below the theoretical weight. When 100 inches wide and over, 10 per cent above or 3 per cent below the theoretical weight. , (i) For all plates ordered to gauge, there will be permitted an average excess of weight over that corresponding to the dimensions on the order equal in amount to that specified in the following table : TABLE OF ALLOWANCES FOR OVERWEIGHT FOR RECTANGULAR PLATES WHEN ORDERED TO GAUGE. Plates will be considered up to gauge if measuring not over T ^ inch less than the ordered gauge. The weight of i cubic inch of rolled steel is assumed to be 0.2833 pound. 464 APPLIED MECHANICS. PLATES J INCH AND OVER IN THICKNESS. Width of Plate. Plate. Inch. Up to 75 75 to 100 Over 100 Inches. Per Cent. Inches. Per Cent. Inches. Per Cent. t 10 8 14 12 18 16 I 7 10 *3 Tff 6 8 10 I 5 7 Q TS 4i 6| 8* 4 6 8 Over 3* 5 6* PLATES UNDER \ INCH IN THICKNESS. Width of Plate. Plate. Inch. Up to 50 Inches. 50 Inches and Above. Per Cent. Per Cent. \ up to & 10 15 & " " & 8J za| A " " i 7 10 FINISH. 14. Finished material must be free from injurious seams, flaws, or cracks, and have a workmanlike finish. BRANDING. 15. Every finished piece of steel shall be stamped with the melt or blow number, except that small pieces may be shipped in bundles securely wired together with the melt or blow number on a metal tag attached. INSPECTION. 1 6. The inspector, representing the purchaser, shall have all reason- able facilities accorded to him by the manufacturer to satisfy him that the finished material is furnished in accordance with these specifications. All tests and inspections shall be made at the place of manufacture, prior to shipment. SPECIFICATIONS FOR STEEL FOR BRIDGES. 465 STRUCTURAL STEEL FOR BRIDGES. Adopted 1905. 1. Steel shall be made by the open-hearth process. Manufacture. 2. The chemical and physical properties shall conform chemical and to the following limits: Physical Properties. Elements Considered. Structural Steel. Rivet Steel. Steel Castings. _ f Basic . o 04 per cent o 04 per cent Phosphorus Max. | Acid Sulphur Max 0.08 0.05 " 0.04 " 0.04 " 0.08 0.05 " Lit tensile strength Desired D esired Not less than Pounds per sq in 60,000 50,000 6c ooo Elong.: Min. per cent, in 8 in. (Fig. i) f i, =500,000* 1,^00,000 Uong.: Min. per cent, in 2 in. (Fig 2) . \ Ult. tens. str. 22 Ult. tens. str. 18 Character of fracture Silky Silky Cold bend without fracture 1 80 flat f 1 80 flat % granular. 00 d 3t Retests. * See par. n. \ See par. 12, 13 and 14. \ See par. 15. The yield-point, as indicated by the drop of beam, shall be recorded in the test reports. 3. If the ultimate strength varies more than 4,000 Ibs. from that desired, a retest may be made, at the discretion of the inspec- tor, on the same gauge, which, to be acceptable, shall be within 5,000 Ibs. of the desired ultimate. 4. Chemical determinations of the percentages of carbon, phos- phorus, sulphur, and manganese shall be made by the Chemica , r^. manufacturer from a test ingot taken at the time of the terminations. pouring of each melt of steel and a correct copy of such analysis shall be furnished to the engineer or his inspector. Check analyses shall be made from finished material, if called for by the purchaser, in which case an excess of 25 per cent above the required limits will be allowed. 5. Specimens for tensile and bending tests for plates, shapes, and bars shall be made by cutting coupons from the finished pjates shapes product, which shall have both faces rolled and both edges and Bar *- milled to the form shown by Fig. 2, page 398; or with both edges 466 APPLIED MECHANICS. parallel; or they may be turned to a diameter of f inch for a length of at least 9 inches, with enlarged ends. Rivets. 6. Rivet rods shall be tested as rolled. 7. Specimens shall be cut from the finished rolled or forged bar in Pins and such manner that the centre of the specimen shall be i Rollers. i ncn f rom the surface of the bar. The specimen for tensile test shall be turned to the form shown by Fig. i, page 398. The specimen for bending test shall be i inch by J inch in section. 8. The number of tests will depend on the character and import- steel Cast- ance of the castings. Specimens shall be cut cold from ings. coupons moulded and cast on some portion of one or more castings from each melt or from the sink-heads, if the heads are of sufficient size. The coupon or sink-head, so used, shall be annealed with the casting before it is cut off. Test specimens to be" of the form prescribed for pins and rollers. 9. Material which is to be used without annealing or further treat- Conditions m ent shall be tested in the condition in which it comes for Tests. from the rolls. When material is to be annealed or other- wise treated before use, the specimens for tensile tests, representing such material, shall be cut from properly annealed or similarly treated short lengths of the full section of the bar. 10. At least one tensile and one bending test shall be made from Number of each melt of steel as rolled. In case steel differing f inch Tests. an d more in thickness is rolled from one melt, a test shall be made from the thickest and thinest material rolled. 11. For material less than 5-16 inch and more than f inch in thickness the following modifications will be allowed in the Elongation. . r , . requirements for elongation: (a) For each 1-16 inch in thickness below 5-16 inch, a deduction of 2^ will be allowed from the specified percentage. (b) For each J inch in thickness above } inch, a deduction of i will be allowed from the specified percentage. 12. Bending tests may be made by pressure or' by blows. Plates, Bendin shapes, and bars less than i inch thick shall bend as called Tests. f or m paragraph 2. 13. Full-sized material for eye-bars and other steel i inch thick SPECIFICATIONS FOR STEEL FOR BRIDGES. 467 and over, tested as rolled, shall bend cold 180 around F d a pin the diameter of which is equal to twice the thickness Bends of the bar, without a fracture on the outside of bend. 14. Angles f inch and less in thickness shall open flat, and angles J inch and less in -thickness shall bend shut, cold, under Testson blows of a hammer, without sign of fracture. This test Angles, will be made only when required by the inspector. 15. Rivet steel, when nicked and bent around a bar of the same diameter as the rivet rod, shall give a gradual break and a Tests on fine, silky, uniform fracture. Rivet stee! - 1 6. Finished material shall be free from injurious seams, flaws, cracks, defective edges, or other defects, and have a smooth uniform, workmanlike finish. Plates 36 inches in width IS * and under shall have rolled edges. 17. Every finished piece of steel shall have the melt number and the name of the manufacturer stamped or rolled upon it. Markin Steel for pins and rollers shall be stamped on the end. Rivet and lattice steel and other small parts may be bundled with the above marks on an attached metal tag. 1 8. Material which, subsequent to the above tests at the mills and its acceptance there, develops weak spots, brittleness, cracks or other imperfections, or is found to have injurious ejecl defects, will be rejected at the shop and shall be replaced by the manu- facturer at his own cost. 19. A variation in cross-section or weight of each piece of steel of more than 2\ per cent from that specified will be suffi- p ei . miss j b | e cient cause for rejection, except in case of sheared plates, Variations, which will be covered by the following permissible variations, which are to apply to single plates. WHEN ORDERED TO WEIGHT. 20. Plates 12^ pounds per square foot or heavier: Variations? (a) Up to 100 inches wide, 2\ per cent above or below the pre- scribed weight. (b) One hundred inches wide and over, 5 per cent above or below. 21. Plates under i2\ pounds per square foot: (a) Up to 75 inches wide, 2j per cent above or below. 468 APPLIED MECHANICS. (b) Seventy-five inches and up to 100 inches wide, 5 per cent above or 3 per cent below. (c) One hundred inches wide and over, 10 per cent above or 3 per cent below. Permissible Variations. WHEN ORDERED TO GAUGE. 22. Plates will be accepted if they measure not more than o.oi inch below the ordered thickness. 23. An excess over the nominal weight corresponding to the dimen- sions on the order, will be allowed for each plate, if not more than that shown in the following tables, one cubic inch of rolled steel being assumed to weigh 0.2833 pound. 24. Plates i inch and over in thickness. Thickness Ordered. Nominal Weights. Width of Plate. Up to 75". 75" and up to too". ioo"and up to 115'. Over 1 1 5". 14 in 5-16 3-8 7-16 1-2 9~l6 5-8 Over 5-8 ' ch. 10.20 It 12.75 15-30 17-85 20.40 22.95 25-50 )S. 10 p 8 6 & 4 3* er ce nt. 14 p 12 IO 8 i j 5 er ce nt. 18 p i6 13 IO !> 6* er ce nt. 17 per cent. 13 ' 12 ' ii ' 10 ' 9 ' 25. Plates under J inch in thickness. Thickness Ordered. Nominal Weights. Pounds per Square Feet. Width of Plate. Up to 50". 50" and up to 70". Over 70", 1-8" up to 5-32" 5-32" " 3-i6" 3-16'" " 1-4" 5.10 to 6.37 6-37 " 7-65 7.65 " IO.2O 10 per cent. SJ " 7 " " 15 per cent. 12* " " 10 " " 20 per cent. 17 " " 15 " " 26. The purchaser shall be furnished complete copies of mill orders ins ction an d no material shall be rolled, nor work done, before the and Testing, purchaser has been notified where the orders have been placed, so that he may arrange for the inspection. STRUCTURAL STEEL FOR SHIPS. 469 27. The manufacturer shall furnish all facilities for inspecting and testing the weight and quality of all material at the mill where it is manufactured. He shall furnish a suitable testing machine for testing the specimens, as well as prepare the pieces for the machine, free of cost. 28. When an inspector is furnished by the purchaser to inspect material at the mills, he shall have full access, at all times, to all parts of mills where material to be inspected by him is being manufactured. STRUCTURAL STEEL FOR SHIPS. Adopted 1901 for bridges and ships. Restricted to ships, 1905. PROCESS OF MANUFACTURE. 1. Steel shall be made by the open-hearth process. CHEMICAL PROPERTIES. 2. Each of the three classes of structural steel for ships shall con- form to the following limits in chemical composition : Steel Made by the Acid Process. Per Cent. Steel Made by the Basic Process. Per Cent. Phosphorus shall not exceed . Sulphur O.o8 0.06 O.o6 0.06 PHYSICAL PROPERTIES. 3. There shall be three classes of structural steel for ships, namely, rivet steel, soft steel, and medium steel, which shall conform to the following physical qualities: 4. Tensile Tests. Classes. Rivet Steel. Soft Steel. Medium Steel. Tensile strength, pounds per square inch . 50000 to 60000 52000 to 62000 60000 to 70000 Yield -point, in pounds per square inch shall not be less than IT.S. JT. S. JT.S. Elongation, per cent in 8 inches shall not be less than 26 25 22 4/0 APPLIED MECHANICS. 5. For material less than five-sixteenths inch (&") and more than three-fourths inch (}") in thickness the following modifi- Modifications cations shall be made in the requirements for elongation : for Thin and (#) For each increase of one-eighth inch ( J") in thickness 'above three-fourths inch (}") a deduction of one per cent (i%) shall be made from the specified elongation. (b) For each decrease of one-sixteenth inch (TS") in -thickness below five-sixteenths inch (&") a deduction of two and one-half per cent (2 J%) shall be made from the specified elongation. (c) For pins made from any of the three classes of steel the required elongation shall be five per cent (5%) less than that specified in para- gaph No. 4, as determined on a test specimen, the center of which shall be one inch (i") from the surface. 6. Eye-bars shall be of medium steel. Full-sized tests shall show Tensile Tests I2 i P er cent elongation in fifteen feet of the body of the of Eye-bars, eye-bar, and the tensile strength shall not be less than 55,000 pounds per square inch. Eye-bars shall be required to break in the body; but, should an eye-bar break in the head, and show twelve and one-half per cent (12^%) elongation in fifteen feet and the tensile strength specified, it shall not be cause for rejection, provided that not more than one-third (J) of the total number of eye-bars tested break in the head. 7. The three classes of structural steel for ships shall conform Bendin * ^ e fM wm g bending tests; and for this purpose Tests. the test specimen shall be one and one-half inches wide, if possible, and for all material three-fourths inch (f ") or less in thick- ness the test specimen shall be of the same thickness as that of the finished material from which it is cut, but for material more than three-fourths inch (f") thick the bending-test specimen may be one- half inch (") thick. Rivet rounds shall be tested of full size as rolled. (d) Rivet steel shall bend cold 180 flat on itself without fracture on the outside of the bent portion. (e) Soft steel shall bend cold 180 flat on itself without fracture on the outside of the bent portion. (/) Medium steel shall bend cold 180 around a diameter equal to the thickness of the specimen tested, without fracture on the outside of the bent portion. STRUCTURAL STEEL FOR SHIPS. 47 1 TEST PIECES AND METHODS OF TESTING. 8. The standard test specimen of eight inch (8") gauged length shall be used to determine the physical properties specified in paragraphs Nos. 4 and 5. The standard shape of the test men specimen for sheared plates shall be as shown by Fig. 2, page 398. For other material the test specimen may be the same as for sheared plates, or it may be planed or turned parallel throughout its entire length; and, in all cases where possible, two opposite sides of the test specimens shall be the rolled surfaces. Rivet rounds and small rolled bars shall be tested of full size as rolled. 9. One tensile-test specimen shall be taken from the finished material of each melt; but in case this develops flaws, or breaks Numb ^ outside of the middle third of its gauged length, it may Tensile Tests. be discarded, and another test specimen substituted therefor. 10. One test specimen for bending shall be taken from the finished material of each melt as it comes from the rolls, and for material three-fourths inch (f ") and less in thickness nSns for CI " this specimen shall have the natural rolled surface on Bendmg - two opposite sides. The bending-test specimen shall be one and one half inches (i J") wide, if possible, and for material more than three- fourths inch (I") thick the bending-test specimen may be one-half inch (J") thick. The sheared edges of bending-test specimens may be milled or planed. (g) The bending test may be made by pressure or by blows. 11. Material which is to be used without annealing or further treatment shall be tested for tensile strength in the con- dition in which it comes from the rolls. Where it is imprac- Test Speci- ticable to secure a test specimen from material which has been annealed or otherwise treated, a full-sized section of tensile test, specimen length, shall be similarly treated before cutting the tensile-test specimen therefrom. 12. For the purpose of this specification the yield-point shall be determined by the careful observation of the drop of the beam or halt in the gauge of the testing machine. 13. In order to determine if the material conforms to the chemical limitations prescribed in paragraph No. 2 Ch?mica? r herein, analysis shall be made of drillings taken from a small test ingot. 472 APPLIED MECHANICS. VARIATION IN WEIGHT. 14. The variation in cross section or weight of more than 2j per cent from that specified will be sufficient cause for rejection, except in the case of sheared plates, which will be covered by the following per- missible variations: (h) Plates i2j pounds per square foot or heavier, up to 100 inches wide, when ordered to weight, shall not average more than 2 J per cent variation above or 2^ per cent below the theoretical weight. When 100 inches wide and over, 5 per cent above or 5 per cent below the theoretical weight. (i) Plates under 12^ pounds per square foot, when ordered to weight, shall not average a greater variation than the following: Up to 75 inches wide, 2j per cent above or 2\ per cent below the theoretical weight. 75 inches wide up to 100 inches wide, 5 per cent above or 3 per cent below the theoretical weight. When 100 inches wide and over, 10 per cent above or 3 per cent below the theoretical weight. (j) For all plates ordered to gauge there will be permitted an average excess of weight over that corresponding to the dimensions on the order equal in amount to that specified in the following table : TABLE OF ALLOWANCES FOR OVERWEIGHT FOR RECTANGULAR PLATES WHEN ORDERED TO GAUGE. Plates will be considered up to gauge if measuring not over T ^- inch less than the ordered gauge. The weight of i cubic inch of rolled steel is assumed to be 0.2833 pound. PLATE INCH AND OVER IN THICKNESS. Width of Plate. Plate. Inch. Up to 75 Inches. 75 to TOO Inches. Over 100 Inches. Per Cent. Per Cent. Per Cent. 10 14 18 & 8 12 16 | 7 10 13 A 6 8 IO i 5 7 Q & 4* 6* 8i f 4 6 8 Overf 3i 5 6* STEEL AXLES. 473 PLATES UNDER \ INCH IN THICKNESS. Thickness of Plate. Inch. Width of Plate. Up to 50 Inches. Per Cent. 50 Inches and Above. Per Cent. \ up to & A " "A & " "i* 10 8* 7 11, 10 FINISH. 15. Finished material must be free from injurious seams, flaws, or cracks, and have a workmanlike finish. BRANDING. 1 6. Every finished piece of steel shall be stamped with the melt number, and steel for pins shall have the melt number stamped on the ends. Rivets and lacing steel, and small pieces for pin plates and stiffeners, may be shipped in bundles, securely wired together, with the melt number on a metal tag attached. INSPECTION. 17. The inspector, representing the purchaser, shall have all reason- able facilities afforded to him by the manufacturer to satisfy him that the finished material is furnished in accordance with these specifica- tions. All tests and inspections shall be made at the place of manu- facture, prior to shipment. STEEL AXLES. Adopted 1901. Modified 1905. PROCESS OF MANUFACTURE. 1. Steel for axles shall be made by the open-hearth process. CHEMICAL PROPERTIES. 2. There shall be three classes of steel axles, which shall conform to the following limits in chemical composition: 474 APPLIED MECHANICS. Car and Tender-truck Axles. Per Cent. Driving and Engine -truck Axles. (Carbon Steel.) Per Cent. Driving-wheel Axles. (Nickel-steel.) Per Cent. Phosphorus shall not exceed 0.06 O.o6 O O4 Sulphur " " " o 06 c 06 o 04 Manganese " " " Nickel 0.60 3O to 4 O PHYSICAL PROPERTIES. 3. For car and tender- truck axles, no tensile test shall be required. 4. The minimum physical qualities required in the two classes of driving-wheel axles shall be as follows : Tensile Tests. Driving and Engine-truck Axles. (Carbon Steel.) Driving and Engine-truck Axles. (Nickel steel.) Tensile strength pounds per square inch .... 80,000 40,000 20 25 80,000 50,000 25 45 Yield-point pounds per square inch Elongation per cent in two inches Contraction of area per cent . 5. One axle selected from each melt, when tested by the drop test described in paragraph No. 9, shall stand the number of blows at the height specified in the following table without rupture and without exceeding, as the result of the first blow, the deflec- tion given. Any melt failing to meet these requirements will be rejected. Diameter of Axle at Center. Inches. Number of Blows. Height of Drop. Feet Deflection. Inches. 4i 5 24 81 4 5 26 8i 4^T5 5 28} 81 4i 5 31 , 8 4f 5 34 8 5l 5 43 7, 5i 7 43 Si . 6. Carbon-steel and nickel-steel driving-wheel axis shall not subject to the above drop test. be STEEL AXLES. 475 TEST PIECES AND METHODS OF TESTING. 7. The standard test specimen one-half inch (J") diameter and two inch (2") gauged length shall be used to determine the physical properties specified in paragraph No. 4. It men fo?Ten- , -,-,. /c , \ sile Tests. is shown in rig. i. (See p. 398.) 8. For driving and engine-truck axles one longitudinal test specimen shall be cut from one axle of each melt. The center of Numberand this test specimen shall be half-way between the center T^/e of ci _ and outside of the axle. mens - 9. The points of supports on which the axle rests during tests must be three feet apart from center to center; the tup must DropTest weigh 1,640 pounds; the anvil, which is supported on Described. springs, must weigh 17,500 pounds; it must be free to move in a ver- tical direction; the springs upon which it rests must be twelve in number, of the kind described on drawing; and the radius of supports and of the striking face on the tup in the direction of the axis of the axle must be five (5) inches. When an axle is tested, it must be so placed in the machine that the tup will strike it midway between the ends; and it must be turned over after the first and third blows, and, when required, after the fifth blow. To measure the deflection after the first blow, prepare a straight edge as long as the axle, by reinforcing it on one side, equally at each end, so that, when it is laid on the axle, the reinforced parts will rest on the collars or ends of the axle, and the balance of the straight edge not touch the axle at any place. Next place the axle in position for test, lay the straight edge on it, and measure the distance from the straight edge to the axle at the middle point of the latter. Then, after the first blow, place the straight edge on the now bent axle in the same manner as before, and measure the distance from it to that side of the axle next to the straight edge at the point farthest away from the latter. The difference between the two measurements is the de- flection. The report of the drop test shall state the atmospheric tem- perature at the time the tests were made. 10. The yield-point specified in paragraph No. 4 shall be determined by the careful observation of the drop of the beam or halt . J . Yield-point. in the gauge of the testing machine. 4/6 APPLIED MECHANICS. 11. Turnings from the tensile-test specimen of driving and engine- truck axles, or drillings taken midway between the center Sample for and outside of car, engine, and tender-truck axles, or Analysis! drillings from the small test ingot, if preferred by the inspector, shall be used to determine whether the melt is within the limits of chemical composition specified in paragraph No. 2. FINISH. 12. Axles shall conform in sizes, shapes, and limiting weights to the requirements given on the order or print sent with it. They shall be made and finished in a workmanlike manner, and shall be free from all injurious cracks, seams, or flaws. In centering, sixty- (60) degree centers must be used, with clearance given at the point to avoid dulling the shop lathe centers. BRANDING. 13. Each axle shall be legibly stamped with the melt number and initials of the maker at the places marked on the print or indicated by the inspector. INSPECTION. 14. The inspector, representing the purchaser, shall have all reason- able facilities afforded to him by the manufacturer to satisfy him that the finished material is furnished in accordance with these specifications. All tests and inspections shall be made at the place of manufacture, prior to shipment. STEEL TIRES. Adopted 1901. PROCESS OF MANUFACTURE. 1. Steel for tires may be made by either the open-hearth or crucible process. CHEMICAL PROPERTIES. 2. There will be three classes of steel tires which shall conform to the following limits in chemical composition: Passenger Engines. Per Cent. Freight-engine and Car-wheels. Per Cent. Switching- engines. Per Cent. Manganese shall not exceed Silicon shall not be less than 0.80 o. 20 0.8o O.2O 0.8o O.2O Phosphorus shall not exceed 0.05 O Os O.O5 O O? O.O5 O O ^ STEEL TIRES. 477 PHYSICAL PROPERTIES. 3. The minimum physical qualities required in each of the three classes of steel tires shall be as follows : Tensile Tests. Passenger- engines. Freight- engine and Car-wheels. Switching- engines. Tensile strength, pounds per square inch. Elongation, per cent in two inches .... 100,000 12 110,000 IO 120,000 g Drop Test. 4. In the event of the contract calling for a drop test, a test tire from each melt will be furnished at the purchaser's expense, provided it meets the requirements. This test tire shall stand the drop test described in paragraph No. 7, without breaking or cracking, and shall show a minimum deflection equal to D 2 -r- (4oT 2 +2D), the letter "D" being internal diameter and the letter "T " thickness of tire at center of tread. TEST PIECES AND METHODS OF TESTING. 5. The standard turned test specimen, one-half inch (J") diameter and two inch (2") gauged length, shall be used to determine Test Speci- the physical properties specified in paragraph No. 3. It {ensile Tests. is shown in Fig. i. (See p. 398.) 6. When the drop test is specified, this test specimen shall be cut cold from the tested tire at the point least affected by the drop Location of test. If the diameter of the tire is such that 'the whole "SSSf Speci " circumference of the tire is seriously affected by the drop test, or if no drop test is required, the test specimen shall be forged from a test ingot cast when pouring the melt, the test ingot receiving, as nearly as pos- sible, the same proportion of reduction as the ingots from which the tires are made. 7. The test tire shall be placed vertically under the drop in a run- ning position on solid foundation of at least ten tons in Dro Test weight and subjected to successive blows from a tup weigh- Described, ing 2,240 pounds, falling from increasing heights until the required deflection is obtained. 8. Turnings from the tensile specimen, or drillings from the small test ingot, or turnings from the tire, if preferred by the inspector, shall be used to determine whether the melt is chemical* within the limits of chemical composition specified in ' paragraph No. 2. 478 APPLIED MECHANICS. FINISH. 9. All tires shall be free from cracks, flaws, or other injurious im- perfections, and shall conform to dimensions shown on drawings fur- nished t>y the purchaser. BRANDING. 10. Tires shall be stamped with the maker's brand and number in such a manner that each individual tire may be identified. INSPECTION. 11. The inspector representing the purchaser shall have all reason- able facilities afforded to him by the manufacturer to satisfy him that the finished material is furnished in accordance with these specifications. All tests and inspections shall be made at the place of manufacture, prior to shipment. STEEL RAILS. Adopted 1901. PROCESS OF MANUFACTURE. 1. (a) Steel may be made by the Bessemer or open-hearth process. (b) The entire process of manufacture and testing shall be in accord- ance with the best standard current practice, and special care shall be taken to conform to the following instructions : (c) Ingots shall be kept in a vertical position in pit heating furnaces. (d) No bled ingots shall be used. (e) Sufficient material shall be discarded from the top of the ingots to insure sound rails. CHEMICAL PROPERTIES. 2. Rails of the various weights per yard specified below shall con- form to the following limits in chemical composition: 50 to 59 + Pounds. Per Cent. 60 to 69 + Pounds. Per Cent. 70 to 79 + Pounds. Per Cent. 80 to 89 + Pounds. Per Cent. 90 to 100 Pounds. Per Cent. Carbon .... o. 3S 0.45 . 38-0 . 48 o . 400 . so 0.430 S3 O 4S O SS Phosphorus shall not exceed O. IO O. IO O. IO O IO 'IO u Oo O IO Silicon shall not ex- ceed o 20 o 20 o 20 o 20 M^anganese o 701 oo o . 701 . oo O 7S I OS o 80 i 10 o 80 i 10 STEEL RAILS. 479 PHYSICAL PROPERTIES. 3. One drop test shall be made on a piece of rail not more than six feet long, selected from every fifth blow of steel. The rail shall be placed head upwards on the supports, and the various sections shall be subjected to the following impact tests: Drop Test. Weight of Rail. Pounds per Yard. Height of Drop. Feet. 45 to and including 55"" 15 More than 55 " 65.... 16 < < < < 55 75- - 17 < 75 " 85.... 18 85 " 100. . . . J 9 If any rail break when subjected to the drop test, two additional tests will be made of other rails from the same blow of steel, and, if either of these latter tests fail, all the rails of the blow which they represent will be rejected ; but, if both of these additional test pieces meet the require- ments, all the rails of the blow which they represent will be accepted. If the rails from the tested blow shall be rejected for failure to meet the requirements of the drop test, as above specified, two other rails will be subjected to the same tests, one from the blow next preceding, and one from the blow next succeeding the rejected blow. In case the first test taken from the preceding or succeeding blow shall fail, two additional tests shall be taken from the same blow of steel, the accept- ance or rejection of which shall also be determined as specified above; and, if the rails of the preceding or succeeding blow shall be rejected, similar tests may be taken from the previous or following blows, as the case may be, until the entire group of five blows is tested, if necessary. The acceptance or rejection of all the rails from any blow will depend upon the result of the tests thereof. TEST PIECES AND METHODS OF TESTING. 4. The drop-test machine shall have a tup of two thousand (2,000) pounds weight, the striking face of which shall have a Drop _ testing radius of not more than five inches (5"), and the test rail Machine. shall be placed head upwards on solid supports three test (3') apart. The anvil-block shall weigh at least twenty thousand (20,000) pounds, and the supports shall be a part of, or firmly secured to, the anvil. 480 APPLIED MECHANICS. The report of the drop test shall state the atmospheric temperature at the time the tests were made. 5. The manufacturer shall furnish the inspector daily with carbon determinations of each blow, and a complete chemical ChTm!ca? r ^analysis every twenty-four hours, representing the average Analysis. of the Qther elements conta ined in the steel. These analy- ses shall be made on drillings taken from a small test ingot. FINISH. 6. Unless otherwise specified, the section of rail shall be the Amer- ican Standard, recommended by the American Society of Civil Engineers, and shall conform, as accurately as possible, to the templet furnished by the railroad company, consistent with paragraph No. 7, relative to specified weight. A variation in height of one-sixty-fourth of an inch ( T V) less and one-thirty-second of an inch (fa") greater than the specified height will be permitted. A perfect fit of the splice-bars, however, shall be maintained at all times. 7. The weight of the rails shall be maintained as nearly as possible, after complying with paragraph No. 6, to that specified in contract. A variation of one-half of one per cent (4%) for an entire order will be allowed. Rails shall be accepted and paid for according to actual weights. 8. The standard length of rails shall be thirty feet (30'). Ten pe cent (10%) of the entire order will be accepted in shorter lengths, varying by even feet down to twenty-four feet (24'). A variation of one-fourth of an inch (J") in length from that specified will be allowed. 9. Circular holes for splice-bars shall be drilled in accordance with the specifications of the purchaser. The holes shall ac- curately conform to the drawing and dimensions furnished in every respect, and must be free from burrs. 10. Rails shall be straightened while cold, smooth on head, sawed square at ends, and prior to shipment shall have the burr occasioned by the saw-cutting removed, and the ends made clean. No. i rails shall be free from injurious defects and flaws of all kinds. BRANDING. 11. The name of the maker, the month and year of manufacture, STEEL SPLICE-BARS. 481 shall be rolled in raised letters on the side of the web, and the number of the blow shall be stamped on each rail. INSPECTION. 12. The inspector, representing the purchaser, shall have all reason- able facilities afforded to him by the manufacturer to satisfy him that the finished material is furnished in accordance with these specifications. All tests and inspections shall be made at the place of manufacture, prior to shipment. No. 2 RAILS. 13. Rails that possess any injurious physical defects, or which for any other cause are not suitable for first quality, or No. i rails, shall be considered as No. 2 rails, provided, however, that rails which contain any physical defects which seriously impair their strength shall be rejected. The ends of all No. 2 rails shall be painted in order to distinguish them. STEEL SPLICE-BARS. Adopted 1901. PROCESS OF MANUFACTURE. 1. Steel for splice-bars may be made by the Bessemer, or open- hearth process. CHEMICAL PROPERTIES. 2. Steel for splice-bars shall conform to the following limits in chemical composition: Per Cent. Carbon shall not exceed - I 5 Phosphorus shall not exceed o . 10 Manganese o . 30-0 . 60 PHYSICAL PROPERTIES. *. Splice-bar steel shall conform to the following physi- . ... Tensile Tests. cal qualities: Tensile strength, pounds per square inch 54,ooo to 64,000 Yield -point, pounds per square inch ._.... 32,000 Elongation, per cent in eight inches shall not be less than. 25 482 APPLIED MECHANICS. 4. (a) A test specimen cut from the head of the splice-bar shall bend 180 flat on itself without fracture on the outside of Bending Tests. the bent portion. (b) If preferred, the bending tests may be made on an unpunched splice-bar, which, if necessary, shall be first flattened, and shall then be bent 1 80 flat on itself without fracture on the outside of the bent por- tion. TEST PIECES AND METHODS OF TESTING. Test Sped- 5- A test specimen of eight inch (8") gauged length, cut Te e nsi f ie r Tests. fr m the head of the splice-bar, shall be used to determine the physical properties specified in paragraph No. 3. 6. One tensile-test specimen shall be taken from the rolled splice- bars of each blow or melt ; but in case this develops flaws, Number of or breaks outside of the middle third of its gauged length, Tensile ests. ^ mav ^ discarded, and another test specimen substituted therefor. 7. One test specimen cut from the head of the splice-bar shall be taken from a rolled bar of each blow or melt, or, if preferred, nfen foi^ 1 " the bending test may be made on an unpunched splice-bar Bending. which, if necessary, shall be flattened before testing. The bending test may be made by pressure or by blows. 8. For the purposes of this specification the yield-point shall be de- termined by the careful observation of the drop of the beam Yield-point. or nalt j n the g auge o f tne testing machine. 9. In order to determine if the material conforms to the Sample for chemical limitations prescribed in paragraph No. 2 herein, Analysis! analysis shall be made of drillings taken from a small test ingot. FINISH. 10. All splice-bars shall be smoothly rolled and true to templet. The bars shall be sheared accurately to length and free from fins and cracks, and shall perfectly fit the rails for which they are intended. The punching and notching shall accurately conform in every respect to the drawing and dimensions furnished. A variation in weight of more than 2 J per cent from that specified will be sufficient cause for rejection. STRENGTH OF STEEL. 483 BRANDING. 11. The name of the maker and the year of manufacture shall be rolled in raised letters on the side of the splice-bar. INSPECTION. 12. The inspector, representing the purchaser, shall have all reason- able facilities afforded to him by the manufacturer, to satisfy him that the finished material is furnished in accordance with these specifications. All tests and inspections shall be made at the place of manufacture, prior to shipment. 226. Strength of Steel. The literature upon steel is exceedingly voluminous, and many books and articles written upon the metallurgy of steel, such as "Metallurgy of Steel," by Henry M. Howe, and "The Manufacture and Properties of Iron and Steel," by H. H. Campbell, contain a great many tests, which have, as a rule, to do with its properties and the effects of different compositions and treatments. They do not often contain, how- ever, tests upon full-size pieces, such as columns for bridges or buildings, beams, large riveted joints, full-size parts of machinery, etc. The greater part of this latter class of tests are to be found in the reports of the various testing laboratories, such as those "of the laboratories at Munich, at Berlin, and at Zurich in Europe, and the Watertown Arsenal reports and the Technology Quarterly in America; and also in various articles in the Proceedings of the various Engineering Societies in Europe and America. A number of these have already been mentioned among the refer- ences to tests of wrought-iron, and the greater part of them contain also experiments on steel. References to such full-size tests of steel as are quoted here will be given in connection with the tests themselves. A detailed study of the effect of the different ingredients and combinations of ingredients, upon the strength, elasticity, and ductility of steel, is a very complicated matter; it belongs to the study of Metallurgy and is beyond the scope of this work. Nevertheless, the engineer needs, of course, some general 484 APPLIED MECHANICS. knowledge of these matters, and especially of the effect, within certain limits, of different percentages of carbon. This subject has been dealt with by Mr. Wm. R. Webster in the Trans. Am. Inst. Mining Engineers, of October, 1892, August, 1893, and October, 1898, and in the Journal of the Iron and Steel Institute, No. i, 1894; also by Mr. A. C. Cunningham in the Trans. Am. Soc. Civil Engineers of December, 1897; and by Mr. H. H. Campbell, in his book, " Metallurgy of Iron and Steel." Of course none of them claims anything more than approximation for their various rules and formulae, and then only in the case of what they call normal steel, i.e., such steel as is most frequently manufactured by the mills. Mr. Webster made an investigation of the effects of carbon, phosphorus, manganese, and sulp'hur upon the tensile strength of the steel. He gives a set of tables from which to determine, approximately, the tensile strength of normal steel, of a given chemical composition. His investigations were principally made upon basic Bessemer, and basic open-hearth steel. Mr. Campbell gives a formula for the tensile strength of acid, and another for the tensile strength of basic steel, and states that they represent the facts with a good degree of accuracy. His formulae are as follows : For acid steel, 38600 + 1 2iC + 89? + R = ultimate strength; For basic steel, = ultimate strength; where C indicates carbon, P phosphorus, and Mn manganese, in units of o.ooi per cent, and R depends upon the finishing temperature, and may be plus or minus. Mr Cunningham gives the following rule: To find the approx- imate tensile strength of structural steel; to a base of 40000 add 1000 pounds for every o.oi per cent of carbon, and 1000 pounds for STRENGTH OF STEEL. 485 every o.oi per cent of phosphorus, neglecting all other elements in normal steel. In this connection a set of tests will be quoted which were made on the government testing-machine at Watertown Arsenal, upon specimens of steel containing different percentages of carbon, the tests themselves forming a portion of a series denominated in the government report as the "Temperature Series." The account of the tests to be quoted is to be found in their report for 1887. Ten grades of open-hearth steel are here represented, in which the carbon ranges from 0.09 to 0.97 per cent, varying by tenths of a per cent as nearly as was practicable to obtain the steel. The other elements do not follow any regular succession. TENSILE TESTS OF STEEL BARS TEMPERATURE SERIES. Tests at Atmospheric Temperature. c i) a I $ Is a ss 3.S M n 1 1 d Carbon, Per Cent Manganese, Per C Silicon, Per Cent. Diameter, Inches. Sectional Area, Sq i. Length of Rest, Months. |a ojf li fe C/3 . &~4 | W Elongation in 30 Per Cent. Contraction of A at Fracture, Cent. Mechanical Work Elastic Limit, Inch-Lbs. Mechanical Work Tensile Streng in Inch-Lbs. Pounds per Sq. In. Ruptured Sectio 753 0.09 O.II 1.009 0.80 21000 3 30000 52475 23-6 63-5 15-85 9808.36 106434 754 0.20 0.45 1.009 0.80 25000 3 395oo 68375 21.2 49.1 26.40 10651.90 "3704 755 0.31 0.57 0.798 0.50 25000 6 46500 80600 18.0 43-5 37-27 10660.77 126640 7560.37 0.70 0.798 0.50 25000 6 5OOOO 85160 17-5 45-3 42.50 10935-48 134600 | 0.58 0.02 0.798 0.50 30000 6 58000 98760 14.9 41.6 58.00 11380.62 152380 758 0-57 o-93 0.07 0.798 0.50 30000 6 55000 117440 10. I 14.0 52-43 11169.34 134880 759 0.71 0.58 0.08 0-757 0.45 35000 12 57000 116000 8.8 26.2 56.53 9231.21 151510 760 0.81 0.56 0.17 0.798 0.50 40000 12 70000 149600 5-o 5-4 84-35 7872.20 158140 761 0.89 o-57 0.19 0-757 0-45 45000 12 75000 141290 4-3 4-4 95-00 6418.53 147860 762 o-97 0.80 0.28 0-757 0-45 50000 12 79000 '52550 4-3 5-8 108.62 7550-23 161910 486 APPLIED MECHANICS. The following tables include sets of miscellaneous tests of various kinds of steel. Bessemer Steel. Open-hearth Steel. Q u - ' W | ^ i. | ,1 O ^v S! 5 . c S fc 3 Q.^, t* la? ojj w"G If 1-sjjS 'S'- in C ' 3 o si % S-6 ,.= ^ *^ C Ps 2 '^ *j B u s !-ssT $3* |*t o W !l l^esr J^sr Is .7426 70983 40397 54-7 29139000 .7600 64169 47395 56.7 29392600 .7481 57700 33000 53-5 28885000 .7500 63083 44137 64.0 30179000 7463 58408 28575 58-9 32799000 .7600 64477 47394 60. i 30780000 .7285 62761 34787 65-2 32135000 .7700 62449 46171 63.5 30481000 .7476 50505 19364 72-5 29479000 .7700 62556 46171 64.3 29073000 7442 51230 r 9550 69-7 30653000 .7700 62857 46171 59-5 29073000 75 51110 21503 7 1 '5 28457000 .7600 643 '5 45189 64.1 29843000 75 51518 21503 50.1 27665000 .7650 63527 44600 64.6 29008000 .7300 73865 46584 56.8 29600000 .7600 64830 42984 61.8 28527000 .7500 .7400 .7400 50294 97655 87086 26029 54641 47666 27.0 44.8 46.8 18055000 30539000 30090000 755 .7600 7575 65020 65140 65240 45790 45*90 43270 57-9 64.9 62.3 29338000 31288000 30040000 .7600 87508 65235 50673 49598 48.0 62.5 30057000 30310000 7575 .7600 65125 64500 45487 40780 59 9 61.9 29912000 30060000 7350 87014 50673 45-o 30058000 7550 65089 41320 61.3 291340:0 .7400 87356 49991 38.6 30090000 .87 43300 21900 75-6 28500000 75 .7420 86720 48665 49650 46.2 47-2 28868000 29887000 73 72 44900 46300 21500 22IOO 75-7 73-6 29900000 29900000 .7691 60465 35526 61.5 29149000 .60 46800 24800 73 '3 30800000 7730 66077 35 J 59 61.5 30244000 .60 46700 24800 75-0 30000000 .7690 66745 35526 62.9 30075000 .7690 66445 39832 62.5 30560000 .7690 66142 35526 60.7 27864000 .7680 66530 356i8 60.9 29225000 .7690 67068 61.5 30075000 Machine-steel. Boiler-plate. <u ffjj Si W i_ y *j . c C -! _o rt - c 3 <3 u "o'S* II Section. . C "2 >'" 0i .!U !< *o 1-S I 5 |3dSf Jsdsr 1 |s esr |;3d5f | .7608 91795 62693 53-3 29316000 379 i-458 5945 31670 47-3 20459000 .7629 96256 66723 44-2 29391000 .384 1-48 58770 39590 56.3 30270000 7633 96767 65561 44.1 29586000 .365 1.65 32380 45-6 29305000 7520 92087 59665 40-4 30482000 .369 1.49 61657 37284 30135000 7593 92091 62940 30848000 .398 1.511 5437 30760 58.0 28608000 7598 92191 58445 50-4 28968000 .376 1.496 32889 57-1 28511000 7625 86941 62413 48.7 28340000 .4095x1.3647 5493 33" 29826000 .7623 9575 62445 46.2 30604000 375 Xi. 494 55173 29451 58.5 28849000 .7620 .7620 96045 91220 62495 62495 44.6 51-6 28802000 32706000 .3737x1.4974 .475 x 1.0295 54220 47954 31270 g; 27490000 .7560 96684 63490 42.8 29400000 4702X 1.0064 51035 67.5 .7634 96567 66638 43 -3 30518000 .4292x1.0235 5556o 64.9 .7609 92804 60755 51-3 28884000 .4258x1.0123 54984 67.7 7597 86678 58460 50-2 29867000 .4225 x 1.0025 60680 56.3 .7580 96119 54292 41.8 29818000 .4125 x 1.0025 61500 55-3 . 7600 86741 58415 45-7 28738000 40X 1.02 5719 26552 58.5 25800000 75 T 3 99635 62032 27.4 21813000 . 50 x i .02 60352 28431 58.8 36199000 7613 106980 60413 48.4 29291000 .49 X 1.02 63825 31010 52.0 25273000 759 96142 54149 45-6 26643000 .49 X 1.02 60024 29012 58.8 26677000 .7699 94513 5907 1 46.6 28148000 .50 X 1.02 59803 29012 50-5 30012000 .7622 9H35 64654 54-2 27164000 . 49 x i . 02 61024 29012 58.8 30012000 .7613 86775 60413 47-8 29291000 5GX 1.02 60393 29412 60.2 29412000 7567 9.6249 61150 45-2 28776000 49X 1.02 63625 30012 46.1 31866000 7579 95303 67606 44.9 30005000 39X 1.28 50480 26041 62.5 32051000 754 54870 40.8 29416000 . 3 8x 1.27 53543 29009 59-8 26168000 7554 84990 45752 55-5 29751000 .41x1.27 58144 27846 50.1 35455000 TENSILE STRENGTH OF STEEL. 487 BESSEMER STEEL WIRE. Diameter of Cross- section. (Inches.) Elastic Limit. (Lbs. per sq. in.) Maximum Load. (Pounds.) Maximum Load. (Lbs. per sq. in.) Reduction of Area. (Per cent.) Modulus of Elasticity. (Lbs, per sq. in. . 1290 1013 77500 64.4 30OCOOOO .1280 66100 1021 79400 55-9 30400000 .1288 68300 1OIO 77500 61.4 30900000 .I2yr 67200 970 74100 63-5 30000000 .1283 66500 996 77000 57-1 28500000 .1283 69600 IO2I 79000 57 - 1 30000000 .1289 69700 I00 5 77000 60.5 3 i 200000 .1281 71400 I43 80900 63-9 29200000 .128, 65800 1004 77700 62.6 30700000 .1286 68500 IOOO 77000 63.2 31000000 BESSEMER SPRING-STEEL WIRE. .0911 76000 9^0 146000 34-6 24500000 .0910 69900 974 149000 51 .6 25900000 .0905 79600 969 150000 42.0 23000000 .0911 72900 95 146000 37-5 24200000 .0905 93i 143000 39-6 25400000 TESTS OF STEEL EYE-BARS. Tests of Steel Eye-bars made on the Government Machine. In the Tests of Metals at Watertown Arsenal for 1883 is the record of the tests of six eye-bars of steel, presented by the president of the Keystone Bridge Company. The following is an extract from the report in regard to these eye-bars : " The eye-bars were made of Pernot open-hearth steel, fur- nished by the Cambria Iron Company of Johnstown, Penn. "The furnace charges, about 15 tons each of cast-iron, magnetic ore, spiegeleisen, and rail-ends, preheated in an aux- iliary furnace, required six and one-half hours for conversion. " All these bars were rolled from the same ingot. " Samples were tested at the steel-works taken from a test ingot about one inch square, from which were rolled |-inch round specimens. 488 APPLIED MECHANICS. "The annealed specimen was buried in hot ashes while still red-hot, and allowed to cool with them. " The following results were obtained by tensile tests : Elastic Limit, in Ibs., per Sq. In. Ultimate Strength, in Ibs., per Sq. In. Contrac- tion of Area. Modulus of Elasticity. Carbon. f-inch round rolled bar . 48040 73 I 5 % 45-7 28210000 %- 0.27 f-inch round rolled and annealed bar .... 422IO 69470 54-2 292IOOOO 0.27 " The billets measured 7 inches by 8 inches, and were bloomed down from 14-inch square ingot. " They were rolled down to bar-section in grooved rolls at the Union Iron Mills, Pittsburgh. " The reduction in the roughing-rolls was from 7 inches by 8 inches to 6J inches by 4 inches ; and in the finishing-rolls, to 6^ inches by I inch. "The eye-bar heads were made by the Keystone Bridge Company, Pittsburgh, by upsetting and hammering, proceeding as follows : "The bar is heated bright red for a length of (approxi- mately) 27 inches, and upset in a hydraulic machine ; after which the bar is reheated, and drawn down to the required thickness, and given its proper form in a hammer-die. "The bars are next annealed, which is done in a gas-furnace longer than the bars. They are placed on edge on a car in the annealing-furnace, separated one from another to allow free circulation of the heated gases. They are heated to a red heat, when the fires are drawn, and the furnace allowed to cool. Three or four days, according to conditions, are required before the bars are withdrawn. TENSILE STRENGTH OF STEEL. 489 " The pin-holes are then bored. "The analyses of the heads before annealing were: " Carbon, by color 0.270 per cent Silicon 0.036 " Sulphur -75 " Phosphorus 0.090 " Manganese 0.380 " Copper Trace. " The bars were tested in a horizontal position, secured at the ends, which were vertical. " To prevent sagging of the stem, a counterweight was used at the middle of the bar. " Before placing in the testing-machine, the stem from neck to neck was laid off into lo-inch sections, to determine the uniformity of the stretch after the bar had been fractured. "A number of intermediate lo-inch sections were used as the gauged length, obtaining micrometer measurements of elongation, and the elastic limit for that part of the stem -which was not acted upon during the formation of the heads. Elon- gations were also measured from centre to centre of pins, taken with an ordinary graduated steel scale. " The moduli of elasticity were computed from elongations taken between loads of 10000 and 30000 Ibs. per square inch, deducting the permanent sets. "The behavior of bars Nos. 4582 and 4583, after having been strained beyond the elastic limit, is shown by elongations of the gauged length measured after loads of 40000 and 50000 Ibs. per square inch had been applied ; and with bar No. 4583, after its first fracture under 64000 Ibs. per square inch, a rest of five days intervening between the time of fracture and the time of measuring the elongations. "Considering the behavior between loads of roooo and 30000 Ibs. per square inch, we observe the elongations for the 49 APPLIED MECHANICS. primitive readings are nearly in exact proportion to the incre- ments of load. " Loads were increased to 40000 Ibs. per square inch, passing the elastic limit at about 37000 Ibs. per square inch ; the respec- tive permanent stretch of the bars being 1.31 and 1.26 per cent. " Elongations were then immediately redetermined, which show a reduction in the modulus of elasticity, as we advanced with each increment, of 5000 Ibs. per square inch. " Corresponding measurements after the bars had been loaded with 50000 Ibs. per square inch reach the same kind of results. "The first fracture of bar No. 4583, under 64000 Ibs. per square inch, occurred at the neck, leaving sufficient length to grasp in the hydraulic jaws of the testing-machine, and con- tinue observations on the original gauged length. This was done after the fractured bar had rested five days. " The elongations now show the modulus of elasticity con- stant or nearly so, the only difference in measurements being in the last figures, up to 50000 Ibs. The readings were then immediately repeated, and the same uniformity of elongations obtained. " An illustration of the serious influence of defective metal in the heads is found in the first fracture of bar No. 4583. " There was about 27 per cent excess of metal along the line of fracture over the section of the stem." TENSILE STRENGTH OF STEEL. 49 I z s * 2 W* W^ ^* ft S !?*(. 8. 8. w g S 3- fr * f* r* J* 1 1 r* ^ =r 5 ,S W N <s? 3 f S 8 S, 8 8 8 8- - r Gauged Length, in inches. C 1" i C 1 1 ON 00 Width, in inches. p p vO <O V| V| p p 1 1 P P N! j< . w PT W W OJ (A) w w ^w 5V_w O> M CO 1 1 oJ ^ o C "ft -; ^ 8 s j" -" s 1 If i 1 "3 ^ <8 o* f ! [fvril OJ Ov M M 8 M 10 ^ OJ 1 1 In Gauged W Length. J <2 < M M 1 ^* Centre to Cen- S 5- in M w ' vb tre of Pins. ' ? * -f" % S 00 ON W Contraction of Area, per * *" w M *. <J\ ^ cent. NO ON f S 1 I it W ff 3 f s 1 1 2 Maximum Compression on Pin-Holes, in Ibs., 8 s 8 s 6 o o 8 per Square Inch. W ed w s s 8 5 * 1 n 1 1 *J* s s S S R 5' S 5' 3 I- 5 5 a 8 & 8 s H S- 3 ?. fi ^^ ^ O Jrt ^ 8 Z p JC < 1 8 i I s 2 II 5r vj 1 w ff. < P I f c rj ? g. i I ? t> i 1 S s o- c 3, i ^ $ 2, ? 55 3 1 o 3 W B O ? ? P p 492 APPLIED MECHANICS. ELONGATIONS OF No. 4582 FOR EACH INCREMENT OF 5000 LBS. PER SQUARE INCH. Loads, in Ibs., per Square Inch. Elongations. Primitive Load- ing. After Load of 40000 Ibs. per Square Inch. After Load of 50000 Ibs. per Square Inch. IOOOO _ . _ 15000 20000 25000 0.0274 0.0269 0.0269 0.0300 0.0305 0.0320 0.03II 0.0322 0.0337 30000 0.0269 0.0330 0.0341 ELONGATIONS OF No. 4583 FOR EACH INCREMENT OF 5000 LBS. PER SQUARE INCH. Loads, in Elongations. Elongations after 64000 Ibs. per Square Inch. Ibs., per Square Inch. Primitive Load- After 40000 Ibs. After 50000 Ibs. First Second ing. per Square Inch. per Square Inch. Reading. Reading. IOOOO _ . . . . 15000 0.0272 0.0291 0.0302 0.0311 0.0310 2OOOO 0.0272 0.0305 0.0315 0.0308 0.0310 25OOO 0.0268 0.0314 0.0325 0.0311 0.0310 3OOOO 0.0267 0.0326 0.0340 0.0312 0.0310 35000 - - - 0.0311 - 40000 - - - 0.0312 - 45000 - - - 0.0310 - 50000 " ~ " 0.0315 In the Tests of Metals for 1886 is given the following table of tensile tests of steel eye-bars, furnished by the Chief Engineer of the Statue of Liberty. TENSILE STRENGTH OF STEEL. 493 Dimensions. ft ft Elongation. i . *l- Fractxire. o , f< . +3 o Q, rj O || V M *o c J~ cS(^ -S bo "0 F Width. Thickness Elastic Square Tensile S1 Square it fj Center to of Pin-1 Contracti Modulus c per Sqx Maximun sion on per Sqx Location. Appearance. Ins. Ins. Ins. Lbs. Lbs. % % % Lbs. Lbs. 308.00 5- 16 I .02 34610 64870 7-4 7-3 31400000 74173 308.00 5-14 I .02 34730 69330 10.4 10.3 29279000 84093 308.00 5-15 I .02 37330 70286' 11.7 ii. 5 29017000 80043 308. 10 5-14 I .02 35000 70229 ii. 6 ii-4 13-4 79826 Stem Granular, radi- ating from a button of hard metal. 308.00 5-13 I .02 35950 71680 ii. 8 ii. 5 81323 307.95 5-iS I .02 35000 70895 12. I ii. 8 30162000 80737 The gauged length of the bars was 260 inches. The moduli of elasticity computed between 25000 and 30000 pounds per square inch. In connexion with the work upon the bridge over the Missis- sippi at Memphis, Mr. Geo. S. Morison, the Chief Engineer, had 56 full-size stee eye-bars tested. The results are given in his Report, dated March, 1894, and furnish valuable information regarding the behavior of the steel, and the design, and con- struction of the bars. Only the following table (see page 494) will be given here, containing a portion of the results of the tests upon 31 of the bars, all made of basic open-hearth steel, and all of which broke in the body. This table will aid the reader in comparing the tensile strength and the limit of elasticity of full-size steel eye-bars, with those obtained from the tests of small samples of the steel. In Engineering News of Feb. 2, 1905, 'is an article containing a comparison of full-size and specimen tests of eleven steel eye- bars, made at the Phoenix Iron Co. Each of these bars was 15 inches wide; two of them were ij inches thick; one was i& inches thick, six were 2 inches thick, and two were 2& inches thick. The specimen tests gave tensile strengths varying from 60310 to 67000 pounds per square inch, and limits of elasticity varying from 31550 to 41760 pounds per square inch. 494 APPLIED MECHANICS. FULL-SIZE EYE-BARS. SAMPLE BARS FROM SAME MELT. u u J. i Jl Ii . jd a 44 o Js^ ff c O W & *& P '1 JT 3 Ins. Ins. Ins. Lbs. Lbs. 10.07 1.50 160.63 35 ICO 67490 9-95 1-73 358.93 3768o 70160 9.98 i-75 361-23 39700 65500 10.05 1.50 162.38 33140 65060 6.08 291.26 29690 56700 10.07 iley 287.37 32860 65600 9.92 284.28 31110 61060 9-94 0-99 287.88 3399 63220 10.05 2.20 222.88 2933 63100 10. 12 1.86 464.03 31970 53860 7-12 1.17 314.04 30270 51500 IO.07 2.20 338.73 28080 55160 10.03 I.8l 25L58 29670 62140 9-97 i-37 250.28 32700 65400 7.02 385.73 28980 52010 7.01 i! 2 6 385.78 28410 54740 9-99 1.62 249.98 30500 58870 9.96 2.05 341.28 3336o 7355 10.13 1.30 249.48 32520 60710 9.98 1.81 284.82 28000 58720 10. 15 1-83 221 .98 32290 62270 10.04 o-99 361.68 29970 58680 7.01 1.27 258.68 28640 56830 7.98 i .20 254.63 31930 63870 8.03 2.32 338.58 32840 62400 7.00 1.18 258.68 27870 53520 9.09 1.25 206.58 3259 574io 8. ii 1.79 279.98 28940 58010 7.00 i .00 289.23 31380 59850 t 3d i-5 c I* -"+J e~ P *-* "3 | Is So* If fs 1 W Js w 1 O, Sq. In. Lbs. Lbs. 9500 27-5 41580 73050 .027 .9918 24.4 42650 75620 015 .9520 28.8 40280 70280 .062 .9500 27-5 41580 73050 .027 9756 28.1 40490 69700 .026 1.1590 20. o 43750 75000 .021 1.0140 28.8 42210 69730 .046 .9868 28.1 40230 69720 .025 9635 28.8 38090 71300 .017 I .O2OI 27.0 40200 71860 .017 I .0180 28.8 33400 57170 .014 . I22O 24.2 38320 70220 .023 .O20O 26.3 40200 71080 .028 .0670 25.0 3936o 69360 .041 . I70O 31-3 34190 58460 039 .0170 28.1 41400 67840 .OIO 9338 25.0 40910 70360 .014 .9700 25-5 40410 69900 .063 954 27.0 40400 70490 .023 5557 29-5 40000 66800 .008 .9746 21.3 40530 72240 .056 .1720 27.0 40610 70480 .O6O .0200 28.! 4079 68730 .030 .0100 21.9 40900 69800 .024 .0620 23.1 41710 71000 .066 .0560 32480 58050 .027 9734 28^7 38110 60920 .014 .114 23.0 40480 66880 .030 .020 28.1 40790 68730 .030 The decrease of tensile strength in the full-size eye-bars varied from 6.3 per cent to 11.9 per cent, while the decrease in elastic limit varied from 8.3 per cent to 17 per cent. STEEL COLUMNS. In the Trans. Am. Soc. C. E., of June, 1889, will be found a paper by Mr. J. G. Dagron, giving an account of a set of tests of eight full-size Bessemer-steel bridge columns, made for the Sus- STEEL COLUMNS. 495 quehanna River Bridge of the Baltimore and Ohio R.R. The steel varied in tensile strength from 83680 to 84440 pounds per square inch, in elastic limit from 51190 to 53890 pounds per square inch, in elongation in 8 inches from 18.75 P er cent to 2O -75 P er cent, and in contraction of area from 30.55 per cent to 39.7 per cent. The columns were made by the Keystone Bridge Company and tested in their hydraulic press, with the columns in a horizontal position, and with the pins horizontal. The results obtained are given by the accompanying table : No. of Column. Depth. Inches. Sectional Area. Sq. Ins. Length Center to Center Pin-holes. Ratio of Length to Radius of Gyration. Square of Radius of Gyration. Ultimate Strength, in Lbs. per Sq. In. Modulus of Elasticity. Lbs. per Sq. In. I 8 8.24 i6'o' 42.05 20.86 41020 27705000 2 8 8.24 i6'o' 42.05 20.86 41650 27705000 3 8 8.24 20' 0' 5 2 -5 6 4 20.86 39440 26113000 4 8 8.24 20' 52.5 6 4 20.86 41050 25816000 5 8 8.24 2 4 '0' 63.075 20.86 40230 29504000 6 8 8.24 24'0' 63-075 20.86 40070 28398000 7 9 13-23 25'7-T 5 s -795 27-34 35570 26557000 8 9 13-23 25-7-i" 58.795 27.34 38810 29478000 The columns failed as follows : i. No. No. No. 3. No. 4 No. 5 Failed by bending downwards at rivet in latticing, i foot loj inches from the center, buckling flange angles and web-plate. 2. Failed by bending upwards at rivet in latticing at center, buckling flange angles and web-plate. One angle was fractured at point of buckling, and also at the two adjacent rivets in latticing Failed by bending upwards between latticing, 3 feet from center, buckling flange angles and web-plate. Failed by bending upwards between latticing, 4 inches from. center, buckling flange angles and web-plate. Failed by bending upwards between latticing, 9^ inches from center, buckling flange angles and web-plate. 49<5 APPLIED MECHANICS. No. 6. Failed by bending upwards between latticing, i foot 5! inches from center, buckling flange angles and web plate. No. 7. Failed by bending upwards at rivet in latticing, 3 inches from center, buckling flange angles and web-plate. No. 8. Failed by bending upwards at rivet in latticing, i foot from center, buckling flange angles and web-plate. In every case, after test, the rivets of each column were found by hammer test to be perfectly right. The following table gives the results of a set of tests by direct compression, of eight connecting-rods specially made for these tests, by the Baldwin Locomotive Works, and tested in the Labora- tory of Applied Mechanics of the Mass. Institute of Technology. Breaking- Area. Tensile Properties of the Steel. strength per Sq. o *. In. of the Rod. ,3 oo !3 J_ g g '(3 bo g Modulus d o c -i< P "o "o Id- *w ll SO" J 'li of Elasticity per 1 o 1 f<3 1 re J"^ % ft |& C Sq. In. 1 1 M M H w O M Ins. Sq.In. Sq.In. Lbs. Lbs. Pr.Ct. Pr. Ct. Lbs. .Lbs. Lbs. A 89-38 100.5 7.19 7 .6o 57730 80280 25-8 30.9 28000000 38700 36700 B 98-38 109.4 7.19 7.78 45 6 5o 78830 20.8 34.1 28300000 40600 37500 C 107.38 118.5 6-73 7.21 43900 77840 20.4 42.5 30000000 39300 36700 D in-75 125 .0 7.27 7.78 4.7560 79270 22.3 43-2 28500000 36100 33700 E 116.25 130.0 7.38 7.96 45820 79250 30500000 39300 36400 F 120.63 134.8 7.21 7-55 49440 81660 24.1 39-9 28800000 39300 37500 G 125-13 139-7 7.06 3959 79690 24.4 45-5 30300000 38000 35000 H 134-13 149.4 7.28 7*78 39470 78650 21.0 28.3 30800000 37400 35oo TRANSVERSE STRENGTH OF STEEL. The following table gives the results of tests of a number of steel I beams, made in the Laboratory of Applied Mechanics of the Mass. Institute of Technology. ^S o E V B E ^ UNIVERSITY OF TRANSVERSE S: T GTH OF STEEL. 497 Mo- Modulus Mo- Break- dulus of of No. of Test. Depth. Inches. ment of Inertia. Span. Feet and ing Load. Rup- ture Elasticity per Remarks. Ins. Inches. Lbs. per Sq. In. Sq. In. Lbs. Lbs. 290 7 38.00 14' 6" 10500 42874 29030000 From Phoenix Co. 2 93 8 57-11 14' 6" 14200 44270 29410000 t t 1 1 it 2 95 9 81.34 14' 6" 16700 40200 29890000 n 1 1 it 337 6 24.86 14' 7" 8200 44900 28170000 N. J. Iron & Steel Co. 340 7 39.63 12' II" I2OOO 42100 27480000 < < if < < it 343 8 5 J -67 14' 7" 14900 46400 29040000 11 I ( < C (I 63ia 10 i34.oo 14' o" 24200 3 8 500 28400000 Carnegie Steel Co. 638 10 134.00 14' o" 25100 395 29300000 ( ( i i t ( 674 IO 129.00 14' o" 249OO 41300 27450000 1C C C t I 675 IO 131 .20 14' o" 25600 41700 27850000 C ( t ( t I In Heft IV of the Mitth. der Materialprtifungsanstalt in Zurich are given the following results of tests of the transverse strength of ten steel plate girders : Depth of Web. (Inches.) Span. (Inches.) Modulus of Rupture. (Lbs. per sq. in.) Modulus of Elasticity. (Lbs. persq.in.) 19.76 177.17 53325 29193660 19.76 I77.I7 55316 27430380 15-75 I4L73 55174 26662500 15-75 I4L73 55316 28738620 19.69 177.17 53325 29193560 19.69 177.17 55316 27430380 23.62 2I2.6O 57591 28795500 23.62 2I2.6O 52472 28155600 27.56 248 . 03 54320 27529920 27.56 248.03 53041 28752840 498 APPLIED MECHANICS. COLD CRYSTALLIZATION OF IRON AND STEEL. The question of cold crystallization of wrought-iron and steel is one that has been agitated from the earliest times, and, although Kirkaldy tried to dispose of it finally by offering evi- dence showing that it does not exist, nevertheless we find the same old question cropping out every little while, and although the bulk of the evidence is admitted to be against it, and, as it seems to the writer, there is no evidence in its favor, we find every now and then some one who thinks that certain observed phenomena can be explained in no other way. The most usual phenomenon which cold crystallization is called upon to explain is the crystalline appearance of the fracture of some piece of wrought-iron or steel that has been in service for a long time, and which has, as a rule, been sub- jected to more or less jars or shocks. The cases most fre- quently cited are those of axles of some sort which have been broken, and, in the case of which, the fracture has had a crys- talline appearance, and where samples cut from the other parts of the axle and tested have shown a fibrous fracture. The assumption has therefore been made that the iron was origi- nally fibrous, and that crystallization has been caused by the shocks or the jarring to which it has been subjected in the natural service for which it was intended. Kirkaldy showed (see his sixty-six conclusions) that when fibrous iron was broken suddenly, or when the form of the piece was such as not to offer any opportunity for the fibres to stretch, the fibres always broke off shorthand the fracture was at right angles to their length, and hence followed the crystal- line appearance ; whereas if the breaking was gradual, and the fibres had a chance to stretch, they produced a fibrous appear- ance : in short, he claimed that the difference between the crys- talline or the fibrous appearance of the fracture was only a COLD CRYSTALLIZATION OF IRON AND STEEL. 499 difference of appearance, and not a change of internal structure from fibrous to crystalline. The facts that Kirkaldy showed in this regard are generally acknowledged to-day, and doubtless answer by far the greater part of those who claimed cold crystallization at the time that he wrote, and also a great many of those who claim its exist- ence to-day. But it is easy, if suitable means be taken, to distinguish cases of crystalline appearance of fracture from cases where there are actual crystals in the piece ; and it is rather about those cases where the iron near the fracture actually contains distinct crystals that what discussion there is to-day that is worth considering takes place. The number of such cases is, of course, small, but every once in a while some one is cited, and the claim is put forward that the iron was originally fibrous, and that these crystals must therefore have been produced without heating the iron to a temperature where chemical change is known to occur. Inasmuch as the one who claims the existence of cold crys- tallization is announcing a theory which is manifestly opposed to the well-known chemical law that crystallization requires freedom of molecular motion, and hence can only take place from solution, fusion, or sublimation, it follows that the burden of proof rests with him, and before he can substantiate his theory in any single case he must prove beyond the possibility of doubt, i, that the iron or steel was originally fibrous, i.e., not only that fibrous iron was used in manufacturing the pieces, but also that it had not been overheated during its manufac- ture, and, 2, that it has never been overheated during its period of service. It is because the writer is not aware of any case where these two circumstances have been proved to hold that he says that he knows of no evidence for cold crystallization. In this con- nection it is not worth while to quote very much of the exten- 500 APPLIED MECHANICS. sive literature on the subject ; hence only a little of the most modern evidence will be given here. On page 1007 et seq. of the report of tests on the govern- ment testing-machine at Watertown Arsenal for 1885 is given an account of a portion of a series of tests upon wrought-iron railway axles, and the following is quoted from that report : " This series of axle tests, begun September, 1883, is carried on for the purpose of determining whether a change in struc- ture takes place in a metal originally ductile and fibrous to a brittle, granular, or crystalline state, resulting from exposure to such conditions as are met with in the ordinary service of a railway axle. " Twelve axles were forged from one lot of double-rolled muck-bars, and in their manufacture were practically treated alike. Each axle was made from a pile composed of nine bars, each 6 in. wide, f in. thick, and 3 ft. 3 in. long, and was finished in four heats, two heats for each end. " The forging was done by the Boston Forge Company in their improved hammer dies, which finish the axle very nearly to its final dimensions. "Two axles were taken for immediate test, to show the -quality of the finished metal before it had performed any rail- way service, and serve as standards to compare with the remaining ten axles, to be tested after they had been in use. " The axles are in use in the tender-trucks of express loco- motives of the Boston and Albany Railroad. Mr. A. B. Under- hill, superintendent of motive-power, contributes the axles and furnishes the record of their mileage." The results of some measurements of deflection are given concerning one of the axles in tender 134, after it had run 95000 miles ; and then follows : " Regarding the axle for the time being as cylindrical, 3.96 COLD CRYSTALLIZATION OF IRON AND STEEL. 50 1 inches diameter, the modulus of elasticity by computation will be 28541000 pounds. " Applying this modulus to the deflections observed in rear axle of the rear trucks of tender No. 150, the maximum fibre strain is found to be 9935 pounds per square inch when the tender was partially loaded, and 14900 pounds per square inch when fully loaded. " Taken together, the tensile and compressive stresses, which are equal, amount to 19870 and 29800 pounds per square inch respectively, as the range of stresses over which the metal works. " This definition of the limits of stresses must be regarded as approximate. There are influences which tend to increase the maximum fibre strain, such as unevenness of the track, the side thrust of the wheel-flanges against the rails. On the other hand, the inertia of the axle, particularly under high rates of speed, would exert a restraining influence on the total deflec- tion. " Nine tensile specimens were taken from each axle ; three from each end, including the section of axle between the box and wheel bearings, and three from the middle of its length. They are marked M.B., with the number of the axle ; also a sub-number and letter to indicate from what part of the axle each was taken. " The tensile test-pieces showed fibrous metal, and generally free from granulation. " The muck-bar had a higher elastic limit and lower tensile strength, and less elongation than the axles. The moduli of elasticity of the two are almost identical. " Between loads of 15000 and 25000 pounds per square inch the muck-bar had a modulus of elasticity of 29400000 pounds, the axles (average of all specimens) between 5000 and 20000 pounds per square inch was 29367000 pounds. Individually 502 APPLIED MECH AXILS. the axles showed the modulus of elasticity to be substantially the same in each." Two specimens were subjected to their maximum load and removed from the testing-machine before breaking in order to see whether the straining followed by rest will cause any change. " It does not appear from these tests that 95000 miles run has produced any effect on the quality of the metal." On page 1619 et seq. of the Report for 1886 is given an account of the tests made on some more of these axles which had run 163138 miles, and the following is quoted from that account : " Specimens from muck-bar axle No. 4 after the axle had run 163138 miles. " Comparing these results with earlier tests of this series, the tensile strength of the metal in this axle is lower, and the modulus of elasticity less than shown by the preceding axles. " The variations in strength, elasticity, and ductility are no greater, however, than those met in different specimens of new iron of nominally the same grade, and while apparently there is a deterioration in quality, it needs confirmation of a more decisive nature from the remaining axles before attributing this result to the influence of the work done in service." Another set of tests made at Watertown Arsenal is to be found on page 1044 et seq. of the Report for 1885. There were tested - i. Two side-rods of a passenger locomotive which had been in service about twelve years. 2. One side-rod of a passenger engine which had been run twenty-eight years and eight months. 3. One main-rod which had been run thirty-two years and eight months in freight and five years in passenger service. In none of these tests were there any evidences of crystal- lization, as the metal was in all cases fibrous when fractured. COLD CRYSTALLIZATION OF IRON AND STEEL. 503 In the report is said : " There are no data at command telling what the original qualities of the metal of these bars were : it is sufficient, how- ever, to find toughness and a fibrous appearance in the iron to prove that brittleness or crystallization has not resulted from long exposure to the stresses and vibrations these bars have sustained." The only other evidence that will be referred to is the paper of Mr. A. F. Hill upon the " Crystallization of Iron and Steel," contained in the Proceedings of the Society of Arts of the Massachusetts Institute of Technology for 1882-83. In this article Mr. Hill covers the ground very fully, and distinctly asserts that " The fact is that there is at present not a single well- authenticated instance of iron or steel ever having become crystallized from use under temperatures below 900 F." He claims to have investigated a great many cases where cold crystallization has been claimed, and to have found, in every case where crystals existed, that at some period of its manufacture or working the metal was overheated. He says : " That the crystalline appearance of a fracture is not neces- sarily an indication of the presence of genuine crystals is proven by the well-known fact that a skilful blacksmith can fracture fine fibrous iron or steel in such a manner as to let it appear either fibrous and silky, or coarse and crystalline, according to his method of breaking the bar. On the other hand, where there is genuine crystallization, no skill of manipulation will avail to hide that fact in the fracture. The most striking illustrations of this that have come under my notice are the fractures of the beam-strap of the Kaaterskill, and of the connecting-rod of the chain-cable testing-machine at the Wash- ington Navy Yard. The photographs of both fractures are submitted to you, and the similarity of their appearance is 5O4 APPLIED MECHANICS. most singular. Yet what a difference in the development of the longitudinal sections by acid treatment, which are also presented to you. " In the Kaaterskill accident the fractures of both the upper and lower arms of the strap were found to be short and square. The appearance of the fractured faces showed no trace of fibre, and was altogether granular. Yet the longitudi- nal section, taken immediately through the break, and devel- oped by acid treatment, shows the presence of but few and small crystals, and the generally fibrous character of the iron used in the strap. " In the connecting-rod of the chain-cable testing-machine we find the crystalline appearance of the fracture less, if any- thing, than that of the beam-strap, while the development of the longitudinal section by acid treatment reveals most beauti- fully, in this case, the thoroughly crystalline character of the metal. As is well known, this rod, after many years of service, finally broke under a comparatively light strain, and having all along been supposed to have been carefully made, and from well-selected scrap, its intensely crystalline structure, as re- vealed by the fractures, has done service for quite a number of years as piece de resistance in all the ' cold-crystallization ' arguments which have been served up in that time." He then goes on to say that he cut the rod in a longitudinal direction, and treated the section with acid ; that some of the crystals shown are so large as to be discernible with the naked eye ; that the treated section furnished incontrovertible evi- dence that 'the rod, aside from the fact of being badly dimen- sioned anyhow, was made of poor material, badly heated, and msufrlciently hammered, all records, suppositions, ^and asser- tfons to the contrary notwithstanding ; that there are a large number of crystals composed of a substance, presumably a ferro-carbide, which is not soluble in nitric acid, and is found in steel only ; that the deduction from the large amount of this COLD CRYSTALLIZATION OF IRON AND STEEL. 505 substance is that the pile was formed of rather poorly selected scrap, with steel scrap mixed in ; that evidences of bad heating are abundant throughout ; and that the strongest evidence against the presumption that these crystals were formed during the service of the rod, or while the metal was cold, is found in the groupings of the crystals during their formation, as shown in the tracing developed by the acid ; that they are not of the same chemical composition, the lighter parts containing much more carbon than the darker ones ; it is therefore pretty evi- dent that with the grouping of the crystals a segregation of like chemical compounds took place, and this of course would have been impossible in the solid state. He then cites an experiment he made, in which he took a slab of best selected scrap weighing about 200 pounds and forged it down to a 3-inch by 3-inch square bar, one-half being properly forged, and the other half being exposed to a sharp flame bringing it quickly to a running heat, keeping it at this heat some time, and then hammering lightly and then treating it a second time in a similar manner ; the result being, that while no difference was discernible in the appearance of the two portions, when cut and treated with acid the portion that was properly made showed itself to be a fair representative of the best quality of iron, while in the other portion the crystallization was strongly marked, the majority of the crystals being large and well developed. He also says : " The fact is, all hammered iron or steel is more or less crystalline, the lesser or greater degree of crystallization de- pending altogether upon the greater or lesser skill employed in working the metal, and also largely upon the size of the forging. Crystallization tends to lower very sensibly the elastic limit of iron and steel, and therefore hastens the deterioration of the metal under strain. It is for this reason that large a:id heavy forgings ought to be, and measurably are, excluded as 5O6 APPLIED MECHANICS. much as possible from permanent structures. In machine con- struction we cannot do without them, and must therefore accept the necessity of replacing more or less frequently the parts doing the heaviest work." The evidence given above seems to the writer to be suffi- cient, and to warrant the conclusions stated on pages 475, 476. EFFECT OF TEMPERATURE UPON THE RESISTING PROPER- TIES OF IRON AND STEEL. Much the best and most systematic work upon this subject has been done at the Watertown Arsenal, and an account of it is to be found in " Notes on the Construction of Ordnance, No. 50," published by the Ordnance Department at Washing- ton, D. C, U.S.A. Other references are the following: Sir William Fairbairn: Useful Information for Engineers. Committee of Franklin Institute: Franklin Institute Journal. Knutt Styffe and Christer P. Sandberg: Iron and Steel. Kollman: Engineering, July 30, 1880. Massachusetts R. R. Commissioners' Report of 1874. Bauschinger: Mittheilungen, Heft 13, year 1886. A summary of the Watertown tests, largely quoted from the above-mentioned report, will be given here, and then a few remarks will suffice for the others. The subjects upon which experiments were made at Water town were the effect of temperatures upon i. The coefficient of expansion. 2. The modulus of elasticity. 3. The tensile strength. 4. The elastic limit. 5. The stress per square inch of ruptured section- 6. The percentage contraction of area. 7. The rate of flow under stress. 8. The specific gravity. EPFECT OF TEMPERATURE ON TRON AND STEEL. S7 9. The strength when strained hot and subsequently rup- tured cold. 10. The color after cooling. 11. Riveted joints. 1. THE COEFFICIENTS OF EXPANSION. These were determined from direct measurements upon the experimental bars, first measuring their lengths on sections 35 inches long, while the bars were immersed in a cold bath of ice-water, and again measuring the same sections after a period of immersion in a bath of hot oil. The range of temperature employed was about 210 degrees Fahr., as shown by mercurial thermometers. Observations were repeated, and again after the steel bars had been heated and quenched in water and in oil. The average values are exhibited in the following : TABLE I. First Series of Bars. Metal. Chemical Composition. Coefficients of Expansion per Degree Fahr., per Unit of Length. C. Mn. Si. Wrought-iron. .0000067302 Steel. .09 . II .0000067561 " .20 45 .0000066259 " 31 57 .0000065149 37 .70 .0000066597 4 ' .51 58 .02 .OOOOO662O2 tt 57 93 .07 .0000063891 " 71 58 .08 .0000064716 " .81 56 17 .0000062167 11 .89 57 .19 .0000062335 * * 97 .80 .28 .0000061700 Cast- (gun) iron. .0000059261 Drawn copper. 0000091286 APPLIED MECHANICS Subsequent determinations of the coefficient of expansion of a second series of steel bars gave TABLE II. Jnemical L omposition Coefficients of Expansion c. Mn. Si. S. P. Cu. per Degree Fahr., per Unit of Length. 17 I.I3 .023 .122 .079 .04 .0000067886 .20 .69 037 13 .078 .26 .0000068567 .21 .26 .08 .14 -.059 .00 .0000067623 .26 .07 .11 .096 .08 047 s .0000067476 .26 .26 .07 .112 .06 .038 .0000067102 .26 .28 .07 ."5 .062 035 .0000067175 .28 23 .09 .168 .09 .178 .0000067794 43 97 05 .08 .096 .024 .0000066124 43 i. 08 037 .08 .114 233 .0000066377 53 75 .10 .078 .087 .174 .0000064181 55 1.02 .05 .078 .12 15 .OOOOo66l22 .72 .70 .18 .07 13 23 .0000064330 .72 .76 .20 .056 .086 .186 . 0000063080 79 .86 .21 .084 093 .096 .0000063562 .07 .07 13 .01 .018 .006 .0000061528 .08 .12 .I 9 .Oil .02 trace .0000061702 .12 .10 .09 .013 .018 trace .0000060716 .14 .IO 15 trace .018 trace .0000062589 17 .10 .10 trace .018 o .0000061332 31 13 .19 .Oil .026 trace .0000061478 Ten bars of the first series were now heated a bright cherry- red and quenched in oil at 80 Fahr., the hot bars successively raising the temperature of the oil to about 240 Fahr., the bath being cooled between each immersion. The behavior of the bars under rising temperature, when examined for coefficients of expansion, seemed somewhat erratic, the highest temperature reached being 235 ; but this behavior was subsequently explained by the permanent changes in length found when the bars were returned to the cold bath. EFFECT OF TEMPERATURE ON IRON AND STEEL. $09 Generally the bars were found permanently shortened at the close of these observations. The bars were again heated bright cherry-red and quenched in water at 50 to 55 Fahr., the water being raised by the quenching to 1 10 to 125 Fahr. After resting 72 hours, measurements were taken in the cold bath, followed by a rest of 18 hours, when they were heated and measured in the hot bath, after which they were measured in the cold bath ; the maximum temperature reached with the hot bath being 233. 7 Fahr., erratic behavior occurring still. They were next heated in an oil bath at 300 Fahr., and kept at this temperature 6 hours, then cooled in the bath ; 15 hours later they were heated to 243 Fahr., and again measured hot, and then cold. These downward readings showed the quenched in water bars to have their coefficients elevated above the normal, as shown in the following table, these being the same steel bars as in Table I, and in the sarr.e order : TABLE III. Coefficients of Expansion Apparent Shortening of Bars Due to Six Hours at 300 per Degree per Unit of Length. Fahr., and the following Immersion in the Hot Bath. .0000067641 .0006 .0000066622 .0002 .0000066985 .OOl6 .0000067377 .OO23 .0000069776 .0004 .0000067041 .0082 .0000066939 .0064 .0000068790 .0054 .0000072906 .0055 .0000071578 .0048 510 APPLIED MECHANICS. Finally the bars were annealed by heating bright red and cooling in pine shavings, the effect of which was to approxi- mately restore the rate of expansion to the normal, as shown by Table I for these ranges of temperature. 2. MODULUS OF ELASTICITY. These were obtained with the first series of bars at atmos^ pheric temperatures, and at higher temperatures, up to 495 Fahr. There occurred invariably a decrease in the modulus of elasticity with an increase in temperature, and, in the case of the specimens tested, the low carbon steels showed a greater reduction in the modulus than the high carbon steels, the first specimen having a modulus of elasticity at the minimum temperature 30612000, and at the maximum 27419000, while the last specimen had at the minimum temperature 29126000, and at the maximum 27778000. 3. TENSILE STRENGTH. The tests were made upon the first series of steel bars, wrought-irons marked A and B, a muck-bar railway axle, and cast-iron specimens from a slab of gun-iron. The specimens were o".798 diameter, and 5" length of stem, having threaded ends \ fl '.25 diameter. Wrought-iron A was selected because it was found very hot short at a welding temperature. It had been strained with a tensile stress of 42320 pounds per square inch seven years previous to being cut up into specimens for the hot tests. The specimens while under test were confined within a sheet-iron muffle, through the ends of which passed auxiliary bars screwed to the specimens, the auxiliary bars being secured to the testing-machine. EFFEC7" OF TEMPERATURE ON IRON AND STEEL. $11 The heating was done by means of gas-burners arranged below the specimen and within the muffle. The temperature of the test-bar was estimated from the expansion of the metal, observed on a specimen length of six inches, using the coefficients which were determined at lower temperatures, as hereinbefore stated, assuming there was a uniform rate of expansion. Access to the specimen for the purpose of measuring the expansion was had through holes in the top of the muffle. The temperature was regulated by varying the number of gas- burners in use, the pressure of the gas, and also by means of diaphragms placed within the muffle for diffusing the heat. The approximate elongations under different stresses were determined during the continuance of a test from measurements made on the hydraulic holders of the testing-machine, at a convenient distance from the hot muffle, correcting these measurements from data obtained by simultaneous micrometer readings made on the specimen and the hydraulic holders at atmospheric temperatures. While it does not seem expedient in one series of tests to obtain complete results upon the tensile properties at high temperatures, yet, incidentally, much additional valuable infor- mation may be obtained while giving prominence to one or more features. From these elongations the elastic limits were established where the elongations increased rapidly under equal incre- ments of load. Proceeding with the test until the maximum stress was reached, recorded as the tensile strength, observing the elongation at the time, then, when practicable, noting the stress at the time of rupture." For the detailed tables of tests the student is referred to the " Notes on the Construction of Ordnance." The elastic limits and tensile strengths are computed in pounds per square inch, both on original sectional areas of the APPLIED MECHANICS. specimens and on the minimum or reduced sections, as meas- ured at the close of the hot tests. From the results it appears that the tensile strength of the steel bars diminishes as the temperature increases from zero Fahr., until a minimum is reached between 200 and 300 Fahr., the milder steels appearing to reach the place of minimum strength at lower temperatures than the higher carbon bars. From the temperature of this first minimum strength the bars display greater tenacity with increase of temperature, until the maximum is reached between the temperatures of about 400 to 650 Fahr. The higher carbon steels reach the temperature of maximum strength abruptly, and retain the highest strength over a lim- ited range of temperature. The mild steels retain the increased tenacity over a wider range of temperature. From the temperature of maximum strength the tenacity diminishes rapidly with the high carbon bars, somewhat less so with mild steels, until the highest temperatures are reached, covered by these experiments. The greatest loss observed in passing from 70 Fahr. to the temperature of first minimum strength was 6.5 per cent at 295 Fahr. The greatest gain over the strength of the metal at 70 was 25.8 per cent at 460 Fahr. The several grades of metal approached each other in tenacity as the higher temperatures were reached. Thus steels differing in tensile strength nearly 90000 pounds per square inch at 70, when heated to 1600 Fahr. appear to differ only about loooo pounds per square inch. The rate of speed of testing which may modify somewhat the results with ductile material at atmospheric temperatures has a very decided influence on the apparent tenacity at high temperatures. A grade of metal which, at low temperatures, had little EFFECT OF TEMPERATURE ON IRON AND STEEL. $13 ductility, displayed the same strength whether rapidly or slowly fractured from the temperature of the testing-room up to 600 Fahr. ; above this temperature the apparent strength of the rapidly fractured specimens largely exceeded the others. At 1410 Fahr. the slowly fractured bar showed 33240 pounds per square inch tensile strength, while a bar tested in two seconds showed 63000 pounds per square inch. Cast-iron appeared to maintain its strength with a tendency to increase until 900 Fahr. is reached, beyond which the strength diminishes. Under the higher temperatures it devel- oped numerous cracks on the surface of the specimens preced- ing complete rupture. 4. ELASTIC LIMIT. The report says of this that it appears to diminish with in- crease of temperature. Owing to a period of rapid yielding with- out increase of stress, or even under reduced stress, the elastic limit is well defined at moderate temperatures with most of the steels. Mild steel shows this yielding point up to the vicinity of 500; in hard steels, if present, it appears at lower temperatures. The gradual change in the rate of elongation at other times often leaves the definition of the elastic limit vague and doubt- ful, especially so at high temperatures. The exclusion of de- terminable sets would in most cases place the elastic limit below the values herein given. In approaching temperatures at which the tensile proper- ties are almost eliminated exact determinations are correspond- ingly difficult, the tendency being to appear to reach too high values. 5. STRESS ON THE RUPTURED SECTION. This, generally, follows with and resembles the curve of tensile strength. 514 APPLIED MECHANICS. Specimens of large contraction of area, tested at high temperature, have given evidence on the fractured ends of having separated at the centre of the bar before the outside metal parted. Elongation under Stress^ Although the metal is capable of being worked under the hammer at high temperatures, it does not then possess sufficient strength within itself to develop much elongation, general elongation being greatest at lower temperatures. Greater rigidity exists under certain stresses at intermedi- ate temperatures than at either higher or lower temperatures. Thus one of the specimens tested at 569 Fahr. showed less elongation under stresses above 50000 pounds per square inch than the bars strained at higher or lower temperatures. Two other specimens showed a similar behavior at 315 and 387 respectively, and likewise other specimens. In bars tested at about 200 to 400 Fahr. there are dis- played alternate periods of rigidity and relaxation under in- creasing stresses, resembling the yielding described as occur- ring with some bars immediately after passing the elastic limit. The repetition of these intervals of rigidity and relaxation is suggestive of some remarkable change taking place within the metal in this zone of temperature. 6. PERCENTAGE CONTRACTION OF AREA. This varies with the temperature of the bar ; it is somewhat less in mild and medium hard steels at 400 to 600 than at atmospheric temperatures. Above 500 or 600 the contraction increases with the temperature of the metal ; with three exceptions, which showed diminished contraction at 1100 Fahr., until at the highest temperatures some of them were drawn down almost to points. EFFECT OF TEMPERATURE ON IRON AND STEEL. 7. RATE OF FLOW UNDER STRESS. The full effect of a load superior to the elastic limit is not immediately felt in the elongation of a ductile metal, and the same is true at higher temperatures. The flow caused by a stress not largely in excess of the elastic limit has a retarding rate of speed, and eventually ceases altogether ; whereas under a high stress the rate of flow may accelerate, and end in rupture of the metal. Hence the apparent tensile strength maybe modified within limits by the time employed in producing fracture. 8. SPECIFIC GRAVITY. In general, the specific gravity is materially diminished in the vicinity of the fractured ends of tensile specimens, and this diminution takes place in the different grades of steel, in bars ruptured under different conditions of temperature, stress, and contraction of area. 9 BARS STRAINED HOT, AND SUBSEQUENTLY RUPTURED COLD. The effect of straining hot on the subsequent strength cold appears to depend upon the magnitude of the straining force and the temperature in the first instance. There is a zone of temperature in which the effect of hot straining elevates the elastic limit above the applied stress, and above the primitive value, and if the straining force approaches the tensile strength, there is also a material elevation of that value when ruptured cold. These effects have been observed within the limits of about 335 and 740 Fahr. After exposure to higher temperatures there occurs a gradual loss in both the elastic limit and tensile strength, and generally a noticeable increase in the contraction of area. 5l6 APPLIED MECHANICS. This was not sensibly changed by temperatures below 200. After 300 the metal was light straw-colored : after 400, deep straw ; from 500 to 600, purple, bronze-colored, and blue ; after 700, dark blue and blue black. After 800 some specimens still remained dark blue. After heating above about 800 the final color affords less satisfactory means of approximately judging of the temperature, the color remaining a blue black, and darker when a thick magnetic oxide is formed. At about 1100 the surface oxide reaches a tangible thick- ness, a heavy scale of o".ooi to o".oo2 thickness forming as higher temperatures are reached. The red oxide appears at about 1500. 11. IN THE TESTS OF RIVETED JOINTS of steel boiler-plates at temperatures ranging from 70 to about 700 Fahr. the indications of the tensile tests of plain bars were corroborated. Joints at 200 Fahr. showed less strength than when cold ; at 250 and higher temperatures the strength exceeded the cold joints ; and when overstrained at 400 and 500 there was found, upon completing the test cold, an increase in strength. Rivets which sheared cold at 40000 to 41000 pounds per square inch, at 300 Fahr. sheared at 46000 pounds per square inch ; and at 600 Fahr., the highest temperature at which the joints failed in this manner, the shearing-strength was 42130 pounds per square inch. In addition to the work at Watertown which has just been detailed two other matters will be referred to here. EFFECT OF TEMPERATURE ON IRON AND STEEL. $1? 1. It is well known that wrought-iron and steel are very brittle at a straw heat and a pale blue, as shown by the fact that when the attempt is made to bend a specimen at these tempera- tures it results in cracking it some time before a complete bend- ing can be effected, even in the case of metal which is so ductile that it can be bent double cold, red hot, or at a flanging heat, without showing any signs of cracking. 2. Bauschinger defines the elastic limit as the load at which the stress is no longer proportional to the strain ; whereas he calls stretch-limit (Streckgrenze) the load at which the strain diagram makes a sudden change in its direction ; i.e., where instead of showing a gradually increasing ratio of strain to stress it shows a sudden and rapid increase. From his experiments (see Heft 13 of the Mittheilungen, year 1886) he draws the following conclusions : (a) That the effect of heating and subsequent cooling in lowering both the elastic and the stretch limits in mild steel begins at about 660 Fahr. when the cooling is sudden, and at about 840 Fahr. when it is slow, and for wrought-iron at about 750 with either rapid or slow cooling. (b) That the operation of heating above those temperatures, and of subsequent slow or quick cooling, is that both the elastic and the stretch limit are lowered, and the more so the greater the heating ; also, that this effect is greater on the elastic than on the stretch limit. (c) Quick cooling after heating higher than the above-stated temperatures lowers the elastic and the stretch limit, especially the first, much more than slow cooling, dropping the elastic limit almost immediately at a heat of about 930 and certainly at a red heat to nothing or nearly nothing in wrought-iron, and in both mild and hard steel, while slow cooling cannot bring about such a great drop of the elastic limit, even from more than a red heat. APPLIED MECHANICS. Effect of Cold-Rolling on Iron and Steel. It has already been stated, p. 410, that it was discovered independently by Commander Beardslee and Professor Thurston, that if a load were gradually applied to a piece of iron or steel which exceeded its elastic limit, and the piece then allowed to rest, the elastic limit and the ultimate strength would thus be increased. This may be accomplished with soft iron and steel by cold-rolling or cold-drawing, but cannot be taken advantage of in hard iron or steel. Professor Thurston, who has investigated this matter at great length, and made a large number of tests on the subject, gives the following as the results of cold-rolling: Increase in Per Cent. Tenacity 2C to 4.0 Transverse stress .... CQ to 80 Elastic limit (tension, torsion, and transverse), 80 to 125 300 to 400 Elastic resilience (transverse) I CO to A.2 C He also says, in regard to the modulus of elasticity, " Collating the results of several hundred tests, the author [Professor Thurston] found that the modulus of elasticity rose, in cold-rolling, from about 25000000 Ibs. per square inch to 26000000, the tenacity .from 52000 Ibs. to nearly 70000, the elastic limit from 30000 Ibs. to nearly 60000 Ibs. ; and the ex- tension was reduced from 25 to ioj per cent. " Transverse loads gave a reduction of the modulus of elas- ticity to the extent of about 1000000 Ibs. per square inch, an increase in the modulus of rupture from 73600 to 133600, and reduction of deflection at maximum load of about 25 per cent. The resistance of the elastic limit was doubled, and occurred at a much greater deflection than with untreated iron." On the other hand, the two steel eye-bars referred to on FACTOR OF SAFETY. 519 p. 472 show a decrease of modulus of elasticity with increasing overstrain. Whitworth's Compressed Steel. Sir Joseph Whitworth pro- duces steel of great strength by applying to the molten metal, directly after it leaves the furnace, a pressure of about 14000 Ibs. per square inch; this being sufficient to reduce the length of an eight-foot column by one foot. He claims, according to D. K. Clark, to be able to obtain with certainty a strength of 40 English tons with 30 per cent ductility, and mild steel of a strength of 30 English tons with 33 or 34 per cent ductility. The following tests were made on the Watertown machine, upon some specimens of Whitworth steel taken from a section of a jacket which was shrunk upon a wrought-iron tube, and removed from shrinkage by the application of high furnace heat : TENSILE TESTS. Diameter, Inches. Tensile Strength, Ibs. per Sq. In. Elastic Limit, Ibs. per Sq. In. Contraction of Area, per cent, o 564 103960 55000 41.9 0.564 90040 48000 47.2 0.564 104200 57000 24.6 0.564 IOOI20 57000 44.6 0.564 93040 53000 39-2 0.564 104160 60000 24.6 0.564 93160 47000 39-2 COMPRESSIVE TESTS. Length, Inches. Diameter, Inches. Compressive Strength, Ibs. per Sq. In. Elastic Limit, Ibs. per Sq. In. 5 0.798 IO2IOO 61000 5 0.798 89000 57000 3-94 0.798 IOI6OO 53000 3-94 0.798 IOI6OO 54000 227. Factor of Safety In order to determine the proper dimensions of any loaded piece, it becomes necessary 52O APPLIED MECHANICS. to fix, in some way, upon the greatest allowable stress per square inch to which the piece shall be subjected. The most common practice has been to make this some fraction of the breaking-strength of the material per square inch. As to how great this factor should be, depends upon i. The use to which the piece is to be subjected ; 2. The liability to variation in the quality of the material ; 3. The question whether we are considering, as the load upon the piece, the average load, or the greatest load that can by any possibility come upon it ; 4. The question as to whether the structure is a temporary or a permanent one; 5. The amount of injury that would be done by breakage of the piece ; and other considerations. The factors most commonly recommended are, 3 for a dead or quiescent load, and 6 for a live or moving load. A common American and English practice for iron bridges is to use a factor of safety of 4 for both dead and moving load. In machinery a factor as large as 6 is desirable when there is no liability to shocks ; and when there is, a larger factor should be used. A method sometimes followed for tension and compression pieces is, to prescribe that the stretch under the given load should not exceed a certain fixed fraction of the length. This requires a knowledge of the modulus of elasticity of the mate- rial. In the case of a piece subjected to a transverse load, it is the most common custom to determine its dimensions in accord- ance with the principle of providing sufficient strength ; and for this purpose a certain fraction (as one-fourth) of the mod- ulus of rupture is prescribed as the greatest allowable safe stress per square inch at the outside fibre. Thus, for wrought- iron from 10000 to 12000 Ibs. per square inch is often adopted REPEATED STRESSES. $21 as the greatest allowable stress at the outside fibre, this being about one-fourth of the modulus of rupture. The other method for dimensioning a beam is, to prescribe its stiffness ; i.e., that it shall not deflect under its load more than a certain fraction of the span. This fraction is taken as rb- to 7TT<7- This latter method depends upon the modulus of elasticity of the beam ; and while it is the most advisable method to follow, and as a rule would be safer than the other method, nevertheless, in the case of very stiff and brittle material it might be dangerous ; hence we ought to know also the break- ing-weight and the limit of elasticity of the beam we are to use, and not allow it to approach either of these. This precaution will be especially important to observe in the case of steel beams, which are only now being introduced. On the other hand, in moving machinery a factor of safety of six is usually required when there is no unusual exposure to shocks, as in smooth-running shafting, etc. ; and when there are irregular shocks liable to come upon the piece, a greater factor is used. WOHLER'S RESULTS. 228. Repeated Stresses. The extensive experiments of Wohler for the Prussian government, which were subsequently carried on by his successor, Spangenberg, were made to deter- mine the effect of oft-repeated stresses, and of changes of stress, upon wrought-iron and steel. In the ordinary American and English practice, it is cus- tomary, in determining the dimensions of a piece, as of a bridge member, to ascertain the greatest load which the piece can ever be called upon to bear, and to fix the size of the piece in accordance with this greatest load. Wohler called attention to the fact that the load that would break a piece depends upon both the greatest and least load that it would ever be called upon to bear. Thus, a tension-rod 522 APPLIED MECHANICS. which is subjected to alternate changes of load extending from 20000 to 80000 Ibs. would require a greater area for safety than one which was subjected to loads varying only between the limits of 60000 and 80000 Ibs. ; and this would require more area than one which was subjected to a steady load of 80000 Ibs. Wohler expresses this law as follows, in his " Festigkeits versuche mit Eisen und Stahl." "The law discovered by me, whose universal application for iron and steel has been proved by these experiments, is as follows : The fracture of the material can be effected by variations of stress repeated a great number of times, of which none reaches the breaking-limit. The differences of the stresses which limit the variations of stress determine the breaking-strength. The absolute magnitude of the limiting stresses is only so far of influence as, with an increasing stress, the differences which bring about fracture grow less. " For cases where the fibre passes from tension to compres- sion and vice versa, we consider tensile strength as positive and compressive strength as negative ; so that in this case the difference of the extreme fibre stresses is equal to the greatest tension plus the greatest compression." Besides the ordinary tests of tensile, compressive, shearing, and torsional strength, he made his experiments mainly on the following two cases : i. Repeated tensile strength; the load being applied and wholly removed successively, and the number of repetitions required for fracture counted. 2. Alternate tension and compression of equal amounts successively applied, the number of repetitions required for fracture being counted. In making these two sets of tests, he made the first set in two ways : (a) By applying direct tension. LAUNHARDT'S FORMULA. 523 (b) By applying a transverse load, and determining the greatest fibre stress. The second set of tests was made by loading at one end a piece of shaft fixed in direction at the other, and then causing it to revolve rapidly, each fibre passing alternately from tension to an equal compression, and vice versa. He also tried a few experiments where the lower limit of stress was neither zero nor equal to the upper limit, with a minus sign, also some experiments on torsion, on shearing, and on repeated torsion. When Wohler had made his experiments, and published his results, there were a number of attempts made by different persons to deduce formulae which should depend upon these experiments for their constants, and which should serve to deter- mine the breaking-strength for any given variation of stresses. Only two of these formulae will be given here, viz. : i That of Launhardt for one kind of stress, 2 That of Weyrauch for alternate tension and compression. LAUNHARDT'S FORMULA. The constants used in this formula are : i. /, the carrying-strength (Tragfestigkeit) of the material per unit of area, which is the same as the tensile strength as determined by the ordinary tensile testing-machine. 2. u, the primitive breaking-strength (Ursprungsfestigkeit), i.e., the greatest stress per unit of area of which the piece can bear, without breaking, an unlimited number of repetitions, the load being entirely removed between times. These two quantities have been determined experimentally by Wohler; and it is the object of Launhardt's formula to deduce, in terms of /, u, and the ratio between the greatest and least loads to which the piece is ever subjected, the value a of the breaking-strength per unit of area when these loads are applied. 524 APPLIED MECHANICS. Let the greatest stress per unit area be a. the least stress per unit area be c. Plot the values of - as abscissae, and those of a as ordinates, making OA = u (since when - = o, a = u), OC=i, and CB = t (since when -=i, a = /). Then will any curve , E B which passes through the points A and B have \\^~\~\ ^ or * ts or( ^ mates values of a that will satisfy the conditions that when c = o, a u, and when c = /, a = t. By assuming for this curve, the straight line AB we obtain DE = AO + FE = AO + (BG)^ , and hence a=w + (/-w)~, (i) which is Launhardt's formula. Moreover, if we denote by max L the greatest load on the en- tire piece, and by min L the least, we shall have c_ min L a max L' Hence min L - r, max L (2) this being in such a form as can be used. Or we may write it thus: !/ u min L i+ j this being the more common form. The values of the constants as determined by Wohler's experi- ments, and the resulting form of the formula for Phcenix axle- iron and for Krupp cast-steel, have already been given in 172. WEYRAUCH'S FORMULA. $2$ In the same paragraph are given the corresponding values of by the safe working-strength, the factor of safety being three. WEYRAUCH'S FORMULA FOR ALTERNATE TENSION AND COMPRESSION. The constants used in this formula are : i. u, the primitive breaking-strength, which has been already defined. 2. s, the vibration breaking-strength (Schwingungsfestigkeit) i.e., the greatest stress per unit of area, of which the piece can bear, without breaking, an unlimited number of applications, when subjected alternately to a tensile, and to a compressive stress of the same magnitude. He lets a = greatest stress per unit of area, c= greatest stress of the opposite kind per unit of area. If a is tension, c is com- pression, and vice versa. Plot the values of - as abscissae, and those of a as ordinates, making OA=u (since when - = i, a=w), OC = i, and CB=s (since when = i , a =s) . Then will any curve which passes through the points A and B have for its ordinates values of a that will satisfy the conditions that when c=o, a=u, and when c=s, a=s. By assuming for this curve the straight line AB we obtain t and hence a=u-(u-s)-, (4) which is the Weyrauch formula. 526 APPLIED MECHANICS. Moreover, if we write c max U a max L ' where max L= greatest load on the piece, and max Z/= greatest load of opposite kind, so that, if L is tension, L' shall be com- pression, and vice versa, we shall have .max L' this being in a form suitable to use, the more common form being (u s max L' } i - --- r \ - (6) u max L J The values of the constants as determined from Wohler's experiments, and the resulting form of the formulae for Phoenix axle-iron and for Krupp cast-steel, are given in 176. GENERAL REMARKS. In each case the value of a given by the formula (3) or (6) is the breaking-strength per unit of area. If either of these values of a be divided by 3, we have, accord- ing to Weyrauch, the safe working-strength. WOHLER'S EXPERIMENTAL RESULTS. Wohler himself made his tests upon the extremes of fibre stresses of which a piece could bear, without breaking, an unlimited number of applications. He gives, as a summary of these results, the following: In iron, Between +16000 Ibs. per sq. in. and 16000 Ibs. per sq. in. + 30000 " " " o " " +44000 " " " +24000 " " In axle-steel, Between +28000 Ibs. per sq. in. and 28000 Ibs. per sq. in. " +48000 " " " o " " " +80000 " " " +35000 " " WOHLER'S EXPERIMENTAL RESULTS. 527 In untempered spring steel, Between +50000 Ibs. per sq. in. and o Ibs. per sq. in. -f 70000 " " " +25000 " " + 80000 " " " +40000 " " + 90000 " " " +60000 " " For shearing in axle-steel, Between +22000 Ibs. per sq. in. and 22000 Ibs. per sq. in. + 38000 " " o " This table would justify the use, in Launhardt's and Wey- rauch's formulae, of the following values of u and s ; viz., In iron, u = 30000 Ibs. per sq. in., s = 16000 Ibs. per sq. in. In axle steel, u = 48000 Ibs. per sq. in., s = 28000 Ibs. per sq. in. In untempered spring steel, u = 50000 Ibs. per sq. in. And it would require, that if, with these values of u, and the values of / given in 172 and 176, we put c 24000 in Launhardt's formula for iron, we ought to obtain approxi- mately a = 44000 ; and if we put c = 35000 in that for steel, we should obtain approximately a = 80000. 528 APPLIED MECHANICS. FACTOR OF SAFETY. We have seen that Weyrauch recommends, to use with Wohler's results, a factor of safety of three for ordinary bridge work and similar constructions. Wohler himself, however, in his " Festigkeits versuche mit Eisen und Stahl," says, i. That we must guard against any danger of putting on the piece a load greater than it is calculated to resist, by assum- ing as its greatest stress the actually greatest load that can ever come upon the piece ; and 2. This being done, that the only thing to be provided for is the lack of homogeneity in the material. 3. That any material which requires a factor of safety greater than two is unfit for use. This advice would hardly be accepted by engineers, however. He also claims that the reason why it is safe to load car- springs so much above their limit of elasticity, and so near their breaking-load, is, that the variation of stress to which they are subjected is very inconsiderable compared with the greatest stress to which they are subjected. GENERAL REMARKS. It is to be observed, i. 'The tests were all made on a good quality of iron and of steel, consequently on materials that have a good degree of homogeneity. 2. The specimens were all small, and the repetitions of load succeeded each other very rapidly, no time being given for the material to rest between them. 3. No observations were made on the behavior of the piece during the experiment before fracture. SHEARING-STRENGTH OF IRON AND STEEL. 529 4. As long as we are dealing only with tension, we can say without error that c_ _ min L t a max L ' but as soon as both stresses or either become compression, if the piece is long compared with its diameter, we cannot assert with accuracy the above relation, nor that c max U = a maxZ and hence results based on these assumptions must be to a certain extent erroneous. 5. When a piece is subjected to alternate tension and com- pression, it must be calculated so as to bear either : thus, if sufficient area is given it to enable it to bear the tension, it may not be able to bear the compression unless the metal is 'so dis- tributed as to enable it to withstand the bending that results from its action as a column. While Wohler's tests were mostly confined to ascertaining breaking-strengths, the later experimenters upon this subject, especially Prof. Bauschinger at Munich, Mr. Howard at the Watertown Arsenal, and Prof. Sondericker at the Mass. Institute of Technology, have all undertaken to study the elastic change.3 developed in the material by repeated stresses, and also, to some extent, the effect upon resistance to repeated stress, of flaws, of indentations, and of sudden changes of section, including sharp corners. They all agree in the conclusion that flaws and indentations (even though very slight) and sharp corners, including keyways, reduce the resistance to repeated stress very considerably. A brief account will be given of some of their principal con- clusions. 53 APPLIED MECHANICS. BAUSCHINGER'S TESTS ON REPEATED STRESSES. Bauschinger's tests upon repeated stress include work upon the properties of metals at or near the elastic limit. Of the properties which he enumerates, the following will be quoted here: (a) The sets within the elastic limit are very small, and in- crease proportionally to the load, while above that point they increase much more rapidly. (b) With repeated loading, inside of the elastic limit, dropping to zero between times, we find each time the same total elongations. (c) While within the elastic limit the elongations remain constant as long as the load is constant; with a load above the elastic limit the final elongations under that load are only reached after a considerable length of time. (d) If by subjecting a rod to changing stresses between an upper and lower limit, of which at least the upper is above the original elastic limit, the latter were either unchanged or lowered, or i f , in the case of its being raised, it were to remain below the upper limit, then the repetition of such stresses must finally end in rupture, for each new application of the stress increases the strain; but if both limits of the changing stress are and remain below the elastic limit, the repetition will not cause breakage. (e) Bauschinger says that by overstraining, the stretch limit is always raised up to the load with which the stretching was done; but in the time of rest following the unloading the stretch limit rises farther, so that it becomes greater than the max- imum load with which the piece was stretched, and this rising continues for days, months, and years; but, on the other hand, that the elastic limit is lowered by the overstraining, often to zero; and that a subsequent rest gradually raises it until it reaches, after several days, the load applied, and in time EXPERIMENTS WITH A REPEATED TENSION MACHINE. 531 rises above this ; that, as a rule, the modulus of elasticity is also lowered under the same circumstances, and is also restored by rest, and rises after several years above its original magnitude. (/) By a tensile load above the elastic limit the elastic limit for compression is lowered, and vice versa for a compres- sive load ; and a comparatively small excess over the elastic limit for one kind of load may lower that for the opposite kind down to zero at once. Moreover, an elastic limit which has been lowered in this way is not materially restored by a period of rest at any rate, of three or four days. (<") With gradually increasing stresses, changing from tension to compression, and vice versa, the first lowering of the elastic limit occurs when the stresses exceed the original elastic limit. (//) If the elastic limit for tension or compression has been lowered by an excessive load of the opposite kind, i.e., one ex- ceeding the original elastic limit, then, by gradually increasing stresses, changing between tension and compression, it can again be raised, but only up to a limit which lies considerably below the original elastic limit. EXPERIMENTS WITH A REPEATED TENSION MACHINE. Bauschinger states that in 1881 he acquired a machine similar to that used by Wohler for repeated application of a tensile stress. The plan of the experiments which he made with it, and which are detailed in the I3th Heft of the Mittheilungen, is as follows : From a large piece of the material there were cut at least four, and sometimes more, test-pieces for the Wohler machine. One of them was tested in the Werder machine to determine its limit of elasticity and its tensile strength ; the others were 532 APPLIED MECHANICS. tested in the Wohler machine, so arranged that the upper limit of the repeated stress should be, for the first specimen, near the elastic limit ; for the second, somewhat higher, etc., the lower limit being in all cases zero. From time to time the test-pieces, after they had been sub- jected to some hundred thousands, or some millions, of repeti- tions, were taken from the Wohler machine and had their limits of elasticity determined in the Werder machine. The tables of the tests are to be found in the Mittheilungen, and from them Bauschinger draws the following conclusions : i. With repeated tensile stresses, whose lower limit was zero, and whose upper limit was near the original elastic limit, breakage did not occur with from 5 to 16 millions of repeti- tions. Bauschinger says that in applying this law to practical cases we must bear in mind two things : (a) that it does not apply when there are flaws, as several specimens which contained flaws, many of them so small as to be hardly discoverable, broke with a much smaller number of repetitions ; (b) another caution is that we should make sure that we know what is really the origi- nal elastic limit, as this varies very much with the previous treatment of the piece, especially the treatment it received during its manufacture, and it may be very small, or it may be very near the breaking-strength. 2. With oft-repeated stresses, varying between zero and an upper stress, which is in the neighborhood of or above the original elastic limit, the latter is raised even above, often far above, the upper limit of stresses, and the higher the greater the number of repetitions, without, however, its being able to exceed a known limiting value. 3. Repeated stresses between zero and an upper limit, which is below the limiting value of stress which it is possible for the elastic limit to reach, do not cause rupture ; but if the upper limit lies above this limiting value, breakage must occur .after a limited number of repetitions. EXPERIMENTS WITH A REPEATED TENSION-MACHINE. 533 4. The tensile strength is not diminished with a million repetitions, but rather increased, when the test-piece after hav- ing been subjected to repeated stresses is broken with a steady load. 5. He discusses here the probability of the time of forma- tion of what he considers to be a change in the structure of the metal at the place of the fracture. Besides the above will be given the numerical values which Bauschinger obtained for carrying strength and for primitive safe strength as average values. i. For wrought-iron plates : / = 49500 Ibs. per sq. in. u 28450 " " " " 2. For mild-steel plates (Bessemer) : t = 62010 Ibs. per sq. in. = 34140 " " " " > i 3. For bar wrought-iron, 80 mm. by 10 mm. : t = 57600 Ibs. per sq. in. u= 31290 " " " " 4. For bar wrought-iron, 40 mm. by 10 mm. : / == 57180 Ibs. per sq. in. u = 34140 " " " " 5. For Thomas-steel axle : t = 87050 Ibs. per sq. in. u = 42670 " " " " 6. For Thomas-steel rails : / = 84490 Ibs. per sq. in. u = 39820 " " " " 534 APPLIED MECHANICS. 7. For Thomas-steel boiler-plate : =57600 Ibs. per sq. in. u = 34140 " " " " For Thomas-steel axle, and Thomas-steel rails, Bauschinger's obtained for the vibration breaking-strength the same values as those for primitive breaking-strength. His experiments on the other five niaterials, however, give lower values for 5 than for u. These values will not be quoted here, however, because they were obtained from experiments upon rotating bars of rectangular section transversely loaded. EXPERIMENTS UPON ROTATING SHAFTING SUBJECTED TO TRANS- VERSE LOADS, BY PROF. SONDERICKER. Accounts of these tests are to be found in the Technology Quarterly of April, 1892, and of March, 1899. In every case the (transverse) loads were so applied, that a certain portion, greater than ten inches in length, was subjected to a uniform bending- moment. At various times, the shaft was stopped, the load was removed, then replaced, and again removed, and measurements made of the strains and sets. The diameter of the shaft was, in every case, approximately one inch. Some extracts from the paper of March, 1899, will be given. The investigations were conducted along two lines. i. The determination of elastic changes, resulting from the repeated stresses, and the influence of such changes in pro- ducing fracture. 2. The influence of form, flaws, and local conditions generally in causing fracture. Accurate measurements of the elastic strains, and sets were made at intervals during each test. Characteristic curves of set indicate the general character of the changes which occurred in the set, the abscissa? being the number of revo- lutions, and the ordinates the amount of the set, a is the characteristic curve EXPERIMENTS WITH A REPEATED TENSION MACHINE. 535 for wrought-iron, and also occurred in one kind of soft steel. No change is produced until the elastic limit is reached, and then the change consists in a decrease of set. b is the characteristic curve for all the steels tested with the single exception mentioned. It is the reverse of the preceding, beginning commonly below the elastic limit, and consisting of an increase of set; rapid at first, but finally ceasing. Under heavy loads, the increase of set L very rapid, and ceases comparatively quickly. Accompanying the change of set there is a change in the elastic strain in the same direction but much smaller in amount. From the fact that these changes finally cease, we conclude that, if of sufficiently small magnitude, they do not necessarily result in fracture. The table on page 536 gives a number of his results. Regarding these results he says : i. In several cases, changes would have been detected under smaller stresses had observations been taken. 2. Changes of set may be expected to begin at stresses vary- ing from J to J of the tensile strength. 3. The set does not appear to have a notable influence in causing fracture until it reaches o".ooi or o".oo2 in a length of ten inches. 4. The effect of rest is to decrease the amount of set. In most cases, however, the set lost is soon regained, when the bar is again subjected to repeated stress, especially in the case of the harder steels. Prof. Sondericker also cites a few experiments to determine the loss of strength due to indentations, grooves, and key ways. In one case, the result of cutting a groove around the steel shaft about o".cx>3 deep was a loss of strength of about 40 per cent, while similar results were obtained with indentations, and with square shoulders. He also cites the case of two pieces of steel shafting united by a coupling, where the result of cutting the necessary keyways in the shafts caused, apparently, a loss of about 50 per cent. 536 APPLIED MECHANICS. Tensile Prop- i ! erties of the Revolu- Metal. tions at Maxi- X Material. Elastic Limit Tensile St'gth Stress per Sq. In. which Change was First Ob- Revolu- tions. mum Observed Sets. Remarks. rt B per Sq. In. Lbs. per Sq. In. Lbs. Lbs. served. Inches. D Wt.-Iron 15700 45080 30000 42300 86400 < .01200 Broke at one end at shoulder, and at other where arm was attached. 40 1 24000 50700 24000 1500000 1500000 26000 2427000 Broke. i ' 25900 51390 26000 2214000 2285000 .00026 Broke near center. 2 " 25900 5139 32000 486000 486000 .00136 Broke at mark burn- . ed by electric cur- rent. 3 1 ' 23400 50510 24000 6593000 6593000 .00037 24000 4059000 . OOOI I . 25000 8962000 .00016 26000 3932000 .00022 27000 8155000 .00037 28000 589000 .00038 Broke at shoulder. 4 33 Steel 23400 24800 50510 47400 28000 32000 2506000 85900 2506000 89750 .00042 .00771 Broke at center. Broke outside of arm near bearing; color blue black. 34 24800 47400 32000 103500 116600 .00832 Do. 21 ' ' 30400 62590 32000 4395000 4395000 . OOO29 34000 8339000 .00032 36000 4627000 .00041 36000 1428000 .OO008 After resting unload- ed 1 8 days. 38000 3769000 .00023 40000 4523000 .0005 4 42000 505000 .00072 Broke near shoulder. 54 42000 63130 45000 163000 163000 .00312 Broke at shoulder. 50 4 23200 73760 30000 339000 339000 .OOIOO 35000 16400 .00282 Not broken. 25 ' ' 38300 78010 40000 5031000 5031000 .00028 42000 2483000 .00046 Broke at shoulder. 26 " 38300 78010 40000 20838000 20838000 .00037 42000 3311000 .00044 Not broken. 18 " 50000 81010 36000 6463000 6982000 .00052 Broke where arm was attached. iQ 1 50000 81010 36000 7252000 7686000 .00069 Broke at shoulder. 20 ' 50000 81010 34000 2I22IOOO 21 22IOOO .00028 36000 13577000 .00067 38000 2263OOO .00113 38000 9237000 .00116 After resting 6 mos. unloaded. 40000 932000 .00177 Broke at shoulder. S3 58000 96580 50000 24000 146500 .00249 Broke at shoulder. 58 58000 96580 45000 50000 50100 50100 156900 .00020 .00289 Broke near middle. 29 " 54000 104480 40000 5257000 5257000 .00046 42000 7125000 .00067 44000 4626000 .00100 46000 6/60000 .00145 48000 4965000 .00196 50000 50000 I 7OOOO 1000 .00197 .00203 After 24 days rest un- loaded ; not broken. 55 " 50000 104830 35000 276900 276900 .00060 40000 237900 .00274 50000 22530 .00615 Broke near shoulder; color dark straw. 57 50000 104830 60000 14300 14900 .00768 Broke near shoulder; color dark blue. EXPERIMENTS WITH A REPEATED TENSION MACHINE. 537 TESTS OF ROTATING SHAFTING UNDER TRANSVERSE LOAD, BY MR. HOWARD AT THE WATERTOWN ARSENAL. A large number of tests of this character have been made at the Watertown Arsenal. A few extracts will be given from the remarks of Mr. Howard upon the subject, which may be found in the Technology Quarterly of March, 1899, as follows: "In the Watertown tests, two principal objects have been in view, namely, to ascertain the total number of repetitions of stresses necessary to cause rupture, and to observe through what phases the physical properties of the metal pass prior to the limit of ultimate endurance. The Watertown tests have included cast-iron, wrought-iron, hot and cold rolled metal, and steels ranging in carbon from o.i per cent to i.i per cent, also milled steels. The fibre-stresses have ranged from 10000 pounds per square inch on the cast-iron bars up to 60000 pounds pel- square inch on the higher tensile-strength steel bars. The speed of rotation was from 400 per minute up to 2200 per minute, in different experiments. Observations were made on the deflection of the shafts, and on the sets developed. It was early observed that intervals of rest were followed by tem- porary reduction in the magnitude of the sets. In the Report of Tests of Metals of 1888, he says the deflections tend to diminish under high speeds of rotation, when the loads exceed the elastic limit of the metal, and tend to cause permanent sets; but, on the other hand, when the elastic limit is not passed, the deflections are the same within the range of speeds yet experi- mented upon. Efforts were inaugurated at this time to ascertain the effect of repeated alternate stresses on the tensile properties of the metal, and it appeared that such treatment tended to raise the tensile strength of the metal before rupture ensued. Concerning the limit of indefinite endurance to repeated stress we know but very little. In most experiments rupture occurs after a few thousand repetitions, so high have been the 533 APPLIED MECHANICS. applied stresses. Examples are not uncommon in railway prac- tice of axles having made 200000000 rotations. In order to establish a practical limit of endurance, indefinite endurance, if we choose to call it so, our experimental stresses will need to be somewhat lowered, or new grades of metal found. The following table which accompanied the Watertown Arsenal Exhibit at the Louisiana Purchase Exposition gives a summary of some of the repeated stress tests upon three different grades of steel: STEEL BARS. Tensile Tests and Repeated Stress Tests on Different Carbon Steels. Tensile Tests. Repeated Stress Tests. Mechan- Maxi- Elastic Ten- Elon- Con- ical mum Mechan- Description. Limit per sile St'gth gation in 4 trac- tion of Work at Rupture Fiber Stress Number of Rota- ical Work at Rup- Sq. In. Lbs. per Ins. Per ct. Area Per ct. per Cu.In. per Sq. In. tions at Rupture. ture per Cu. In. Lbs. Ft.-lbs. Lbs. Ft.-lbs. (60000 6470 32835 50000 17790 62635 0.17 Carbon steel . 51000 68000 33-5 51-9 982 45000 j 40000 70400 293500 201960 665290 1 35000 5757920 9992390 30000 *236ooooo *295ooooo 60000 12490 63387 50000 93160 328000 0.5 5 Carbon steel. 57ooo 106100 16.2 18.7 1.047 45000 40000 166240 455350 476900 1032130 35000 9007 20 1563125 30000 *i 9870000 *24838ooo 60000 37250 189044 55000 93790 399780 0.82 Carbon steel. 63000 142250 8.5 6.5 888 50000 45000 213150 605460 750465 1736910 40000 *i756oooo *409730oo 35000 *I9220000 *33635ooo * Not ruptured. GENERAL REMARKS. That the amount of detailed information regarding repeated stresses is small compared with what is needed will be evident when we consider the number of cases in which metal is subjected to such stresses in practice, among which are shafting, connecting- rods, parallel rods, propeller-shafts, crank-shafts, railway axles, rails, riveted and other bridge members, etc. In the case of TORSIONAL STRENGTH OF WROUGHT IRON AND STE^L. 539 some of them, notably, railway axles, attempts have been made to base specifications for the material upon such tests as have become available upon repeated stresses. 229. Shearing-strength of Iron and Steel. Some of the most common cases where the shearing resistance of iron and steel is brought into play are : i. In the case of a torsional stress, as in shafting. 2. In the case of pins, as in bridge-pins, crank-pins, etc. 3. In the case of riveted joints. The so-called apparent outside fibre-stress at fracture, as determined from experiments on torsional strength, is found to be not far from the tensile strength of the metal, and is, of course, greater than the shearing-strength, for the same reasons as render the modulus of rupture greater than the actual outside fibre-stress at fracture in transverse tests. Moreover, the shearing strength of wrought-iron rivets is shown by experiment to be about f the tensile strength of the rivet metal. In regard to cast-iron, Bindon Stoney found the shearing and tensile strength about equal. The cases where shearing comes in play in wrought-iron and steel will therefore be treated separately. 230. Torsional Strength of Wrought-iron and Steel. The method formerly followed, and in use by some at the present day, was to compute the strength of a shaft from the twisting- moment only, neglecting the bending, but varying the working- strength per square inch to be used according to the character of the service. It is generally the fact, however, that when shafting is running the pulls of the belts create a bending back- wards and forwards, bringing the same fibre alternately into tension and compression; and this is combined with the shearing- stresses developed due to the twisting-moment alone. At the two extremes of these general cases are : i. The case when the portion of a shaft between two hangers 540 APPLIED MECHANICS. has no pulleys upon it, and when the pulls on the neighboring spans are not so great as to deflect this span appreciably. That is a case of pure torsion: and if the shaft is running smoothly, with no jars or shocks, and no liability to have a greater load thrown upon it temporarily, we may compute it by the usual torsion formula, given in 212; using for breaking-strength of wrought-iron and steel the so-called apparent outside fibre-stress at fracture as determined from torsional tests, and a factor of safety six, and such a proceeding will probably give us a reasonable degree of safety. 2. The case when, pulleys being placed otherwise than near the hangers, the belt-pulls are so great that the torsion becomes insignificant compared with the bending, and then it would be proper to compute our shaft so as not to deflect more than y^-g-g- of its span under the load, or better, not more than yeV o : f course we- should compute also the breaking transverse load, and see that we have a good margin of safety. In other cases, the methods pursued, the first two of which are incorrect, have been i. By using the ordinary torsion formula combined with a large factor of safety. 2. By computing the shaft also for deflection, and providing that its deflection shall not exceed rsV<r or Trinr f i ts span. This, however, neglects the torsion, and also the rapid change of stress upon each fibre from tension to compression. 3. By using the formula of Grashof or of Rankine for com : bined bending and twisting, with the constants that have been derived from experiments on simple tension or simple torsion. The results given on pages 544 and 545 are from pieces of shafting of considerable length. As has been stated, the so-called "apparent outside fibre-stress at fracture" appears to be not very far from the tensile strength of the material, and the torsional modulus of elasticity appears to be from three-eighths to two- fifths of the tensile modulus of elasticity. TORSIONAL STRENGTH OF WROUGHT-IRON AND STEEL. 541 Under certain circumstances the bending may have the greatest influence, while the twisting may be predominant in others, or their influence may be equally divided. Which of these is the case will depend upon the location of the hangers and of the pulleys, the width of the belts, etc., etc. As to the formulae which take into account both twisting and bending, there are two, both of which are based upon the theory of elasticity. The first, which is the most correct from a theo- retical point of view, is that given by Grashof and other writers on the theory of elasticity, and is where Mi='greatest bending-moment ; M% = greatest twisting-moment ; r = external radius of shaft; / = moment of inertia of section about a diameter; / = greatest allowable stress at outside fibre; w = a constant depending on the nature of the material. In the case of iron or steel the value of m is often taken as 4, though it is, in most cases, nearer 3. When m = 4 we have f __r '-/ The other formula, which is also based upon the theory of elasticity, but which is not as correct, is that given by Rankine, and is With a view to determine the behavior of shafting under a combination of twisting and bending, suitable machinery was erected in the engineering laboratories of the Mass. Institute of Technology, and a number of tests were made. 542 APPLIED MECHANICS. The principal points of the method of procedure are the following, viz.: i st. The shaft under test is in motion, and is actually driving an amount of power which is weighed on a Prony brake. 2d. A tr nsverse load is applied which may be varied at the option of the experimenter, and which is weighed on a platform scale. 3d. The proportion between the torsional and transverse loads may be adjusted to correspond with the proportion be- tween the power transmitted and the belt-pull sustained by a shaft in actual use. 4th. Tests are made not only of breaking-strength, but also angle of twist and deflection under moderate loads are measured. The following table will give the results oT the tests on iron shafts, and they will then be discussed : Time ji/i, *., A, /a, No. of Total H. P. max. max. max. max. j- of Test. run- ning, min- revolu- tions. trans- mitted. bending moment. twisting moment. bend, fibre twist, fibre Grashof. Ran- kine. Diam. ins. utes. In.-lbs. In.-lbs. stress. stress. 8 37-5 7040 11.717 11514.1 3926.4 60024 10234 62162 6i755 "25 9 200 38839 8.181 10507.8 2656.8 54777 6925 55876 55671 ".25 10 l62 31641 5.291 9891.0 1714.6 5*562 4469 52062 5*976 ".25 ii 553 108002 4-331 9241.7 1399.2 48179 3647 48539 48769 "25 12 408 80694 6.276 9241.7 2027.6 48179 5287. 48911 48769 ".25 13 98 19333 6.342 8917.1 2028.2 46485 5287 47245 47105 "25 14 423 82741 6.283 8917.1 2029.7 46485 5290 47246 47106 "25 15 565 108739 6.192 8592-5 2031.6 44793 5295 45582 45436 "25 10 443 88208 6.338 8267.8 2026.8 43100 5283 439H 437*3 ".25 17 95i 185233 6.283 3781-5 2029.7 38503 10333 41768 41117 " 14 874 8218 84185 0X028 68 It 2O *T"JT 7, 562 7976 2394 82112 I2l88 8 o i 21 9.972 /y/^ 8917 3232 90793 l6454 03716 22 T C T CQ 8017 2468l O86l2 *-> A jy 2 . 955 y L / 7652 945 77OI7 48ll 82 II //y 1 j T- OI> L TOR SIGNAL STRENGTH OF WROUGHT-IRON, ETC. 543 In 19 to 23 inclusive the number of revolutions was small and the outside fibre stress at fracture was correspondingly large. Two specimens of the \' '.25 shafting and two of the i" were tested for tension, the results being as follows : Breaking-strength, per sq. in. ,, ,. ( No. i . 46800 ,".35 diameter ] NQ 2 ...... Average ... Average .... 60250 As to conclusions : 1st. It is plain from these results that a shaft whose size is determined by means of the results of a quick test would be too weak, and that our constants should be obtained from tests which last for a considerable length of time. 2d. A perusal of the tables will show that the results ob- tained apply more to the bending than to the twisting of a shaft, as the transverse load used in these tests was so large compared with the twist as to exert the controlling influence. This will be plain by a comparison of the values of f lt f t , and/. 3d. Nevertheless, the bending-moments actually used were generally less than such as might easily be realized in practice with the twisting-moments used. 4th. It seems fair to conclude that, in the greater part of cases where shafting is used to transmit power, as in line-shaft- ing or in most cases of head shafting, the breaking is even 'more liable to occur from bending back and forth than from twist- ing, and hence 'that in no such case ought we to omit to make a computation for the bending of the shaft as well as the twist. 5th. As to the precise value of the greatest allowable out- side fibre stress to be used in the Grashof formula, it is plain 544 APPLIED MECHANICS. that it is not correct to use a value as great as the tensile strength of the iron, and while the tests show that this figure should not for common iron exceed 40000 Ibs. per square inch, it is probable that tests where a longer time is allowed for fracture will show a smaller result yet. TORSIONAL TESTS OF WROUGHT-IRON. Norway Iron. Burden's Best. x C " -*- ^ c (0 O J= <' --. be C A *-> M rt 01 '"' V OJ _C (fl C is, 3 C "3 .'". ir. o 8* ! g& ."S -E 2 -E Diameter. (In< Distance betw Grips. (Incl Maximum Twi Moment. (I Lbs.) Number of Tu between Gri Fracture. sarent Out ibre Stress, -bs. per sq. a >,cr o.ti tn Diameter of C section. (In< | u p MaximumTwi Moment. (] Lbs.) Number of Tu between Gri Fracture. Apparent Out Fibre Stress (Lbs. per sq. Shearing Modi of Elasticity (Lbs. per sq au^ j= ^ 1/3 .00 70.40 72360 16.50 46065 11406000 .01 63.8 85050 9-50 533oo 11300000 .02 72.00 74970 16.00 46600 13215000 .01 59-0 86400 8.63 54200 11500000 3 7 1 -3 72000 14.00 43757 12902000 .01 53-o 84510 6.87 53000 11200000 02 70.40 74520 17.00 46321 12247000 .00 58-8 87480 8. 4 o 557oo II600000 .02 69.80 72000 14-25 45837 12738000 .00 65-5 85410 8.52 544oo 11600000 03 70.30 74880 15-50 456Qa 11361000 .01 60.2 8559 8.82 537 1 1200000 03 74880 20.00 44658 11957000 .02 58.5 85140 8.05 52600 II3OOOOO 03 70.20 7956o 16.00 48437 11554000 .OO 57-0 82650 7-31 52600 II500000 03 84 74880 15-5 45590 11900000 .02 57-8 86580 8.54 535oo II2OOOOO 54 54 35100 12. 48950 9840000 .02 59-5 86040 8.61 53200 II200000 52 49 34200 11.25 49600 11410000 .02 60.0 87840 8-93 543 11300000 53 53 33840 8.50 48120 Il6oOOOO .OX 60.0 88200 8.48 544oo I 1200000 3-384.0 II . IO 48110 .OI 53- 3 87480 7.85 1 I 300000 53 49 6 JT- U 34920 14.56 49650 11840000 .OI 59-5 83970 8.01 52700 11400000 52 53 34200 8.98 49600 12480000 .01 59-5 84780 8.32 53200 II200000 25 70 111960 5-8o 50060 11830000 03 61.0 83520 8.98 50900 I I IOOOOO 27 75 106920 12-30 46600 10900000 .00 63.0 84050 9.24 535o 11700000) 25 70 108360 10.90 48500 IlSooooo .02 60.3 85950 7-94 53*00 1I2OOOOO 25 76 109800 9.90 49100 11700000 .OI 60.0 84600 8.62 53100 11400000 23 70 113670 11.00 52200 12000000 .02 61.0 83520 8.50 51600 IIOOOOOO .26 7 107640 IO.OO 47500 II400000 .OO 60.5 86040 8.80 54000 11600000! Refined Iron. .01 2.01 3o!8 85680 87480 siii 53700 54900 11300000 11500000 .03 72.10 84240 4.00 51285 I247IOOO 2.01 47-8 85860 7.24 539o 11300000 .03 71 .20 66960 40646 12576000 2.OI 59-4 85050 8.92 533 11500000 03 70.50 70200 3-50 41743 II372000 2.01 60.9 86400 8.75 54200 11300900 72.10 72000 2.30 43834 10960000 2.00 59-5 86400 9.08 55oo 11700000 03 71.80 61920 2.80 36820 11393000 2.01 58.5 85650 8.30 537 11500000 03 7!-3 68760 2.50 40887 H36OOOO 2.00 58.5 84870 7-87 54200 11800000 69.30 78120 2.80 46453 I287IOOO 2.OI 58.3 87300 9-45 54800 11500000 30 22320 14.60 52127 II436000 2.01 58-4 86490 8-73 54200 11900000 50 7I-25 36360 10.30 54867 II482000 2.OI 58.4 87120 8.09 54600 11600000 33* 71-75 45360 6.70 50852 12359000 2.00 59-8 84870 8.80 54000 11700000 So 79-5 32760 14.60 49435 IO7IOOO 2 7 62 23800 4.71 53920 2720000 .26 63 22950 5.60 54250 223OOOO .29 64 24600 5-24 52840 2510000 25 60 16640 3-70 52150 2190000 2 5 61 2159 4.40 5437 2510000 27 66 23750 5-oo 53890 2840000 77 7 1 38970 2. IO 35790 I2OOOOO 75 61.1 56150 7-1 534oo 1200000 75 64.0 55350 8.7 52600 12OOOOO 7 2 64-5 45090 5-4 44900 1800000 75 61.0 53360 6. i 50700 II500000 - /4 63.0 557* 1.49 52900 IT3OOOOO The above tables show the results of tests made in the engi- .leering laboratories of the Massachusetts Institute of Technol- TO RSI ON A L S TRENG TH OF IV RO UGHT-IRON AND S TEEL. 545 ogy upon the torsional strength of various kinds of wrought-iron. The figures in the column headed " Apparent outside fibre-stress " Mr are obtained from the formula / = -j- 9 where M = maximum twisting-moment, r= outside radius of shaft, and / = polar moment of inertia of section. Of course it is not the outside fibre -stress. TORSIONAL TESTS OF BESSEMER STEEL. 1 O ri J8 o c 1 +i (-1 S M |jti & *| 0) a 1 1 jj f +>" d IjJ iF! I 5 f: S 1 bo J S CJ ^1 23 rt * s 1u 11 rt o 'S .3 +3 <u- c 'O W5 Sf" **"* S g|S ^ 1 8 ri !> 3 J ^S ^ ^ G CX^^g SM ^ E & p S 3 S o 1 < w S I*" I 7C 60 .00 66960 63632 12418000 ii .88 - 1 /o en 7r 30600 "O w 3'* 7^031 I I243OOO I - "3Q oy i j fQ CQ oy 3762O /O^v)- 1 - 712 CO A i^ if ^^WV-* 12 CQ4OOO I C .OO 2 .OO oy o w 56 .00 o / 101520 / o w 64630 A^ ^yifwwvy 1 1 82 OOOO 7 ^5 2 .OO 55-Qo 30 100260 63830 10320000 7.87 2 O2 c6 . oo ?lS I I 1960 *"o^S?r> I IQQOOOO T &C A A Ort 2 4 81240 A - 2 -/-\ I O2 COOOO * R^7 jo OI 55 .00 36 112 COO 'roT) i n I34IOOOO <; 76 1 * ' w 2 O2 190 . oo o 144, oy I 34IOO 82860 I l83OOOO D * / 6 77 96.00 A if if 75 20160 30400 'O ' 56000 84500 12200000 u / / 16.30 5 94-00 75 19800 29900 5328o 80400 I220OOOO 1-570 52 93.00 75 21600 31300 53280 77300 IO70000O 16.10 53 94.00 75 21600 30800 52560 7470 I09OOOOO 15.80 49 60.00 40 43920 66280 IlSoOOOO 8.60 5 58.00 40 14400 21700 44640 67400 H7OOOOO 13.30 5 57 -5P 40 17000 25600 44820 67600 II900000 ii .50 5 57.60 40 17000 25600 45810 69100 II90OOOO 9.90 5 59-oo 40 16000 24100 4545 68600 Il6oOOOO 10. 80 5 58.80 40 18000 27200 44460 67100 II700000 10.50 5 59.10 40 l6j2OO 24400 45000 67900 II500000 13.20 5 58.20 40 18000 27200 ' 44920 67800 I I 7OOOOO 10.80 5 58.00 40 18000 27200 45540 68700 I I 7OOOOO 11.40 546 APPLIED MECHANICS, 232. Riveted Joints. The most common way of uniting plates of wrought-iron or steel is by means of rivets. It is, therefore, a matter of importance to know the strength of such joints, and also the proportions which will render their efficie,n- cies greatest; i.e., that will bring their strength as near as possible to the strength of the solid plate. In 177 was explained the mode of proportioning riveted joints usually taught, based upon the principle of making all the resistances to giving way equal, and assuming, as the modes of giving way, those there enumerated. This theory does not, however, represent the facts of the case, as i. The stresses which resist the giving- way are of a more complex nature than those there assumed, so that the efficiency of a joint constructed in the way described above may not be as great as that of one differently constructed ; 2. The effects of punching, drilling, and riveting, come in to modify further the action ; and 3. The purposes for which the joint is to be used, often fix some of the dimensions within narrow limits beforehand. In order to know, therefore, the efficiency of any one kind of joint, we must have recourse to experiment. And here again we must not expect to draw correct conclusions from experi- ments made upon narrow strips of plate riveted together with one or two rivets ; but we need experiments upon joints in wide plates containing a sufficiently long line of rivets to bring into play all the forces that we have in the actual joint. The greater part of the experiments thus far made have been made upon narrow strips, with but few rivets. The number of tests of the other class is not large, and of those that have been made, the greater part merely furnish us information as to the behavior of the particular form of joint tested, and do not teach us how to proportion the best or strongest joint in any given plates, as no complete and systematic series of tests has thus far been carried out, though such a series has been begun on the govern- ment testing-machine at the Watertown Arsenal. RIVETED JOINTS. The only tests to which it seems to the writer worth while to make reference here are : i. A portion of those made by a committee of the British Institution of Mechanical Engineers, inasmuch as, although a very large part were made upon narrow strips with but few rivets, nevertheless a portion were made upon wide strips. 2. The tests on riveted joints that have been made on the' government testing-machine at Watertown Arsenal. i. The account of this series is to be found at intervals from 1880 to 1885 inclusive, with one supplementary set in 1888, in the proceedings of the British Institution of Mechanical Engineers; but as all except the supplementary set has also been published in London Engineering, these latter references will be given here as follows : Engineering for 1880, vol. 29, pages no, 128, 148, 254, 300, 350. " 1881, vol. 31, " 427, 436, 458, 508, 588. " 1885, vol. 39, " 524. " 1885, vol. 40, " 19,43. Also, Proc. Brit. Inst. Mechl. Engrs., Oct. 1888. 2. The second series, referred to above, or those made on the government testing-machine at Watertown Arsenal, are to be found in their reports of the following years, viz., 1882, 1883, 1885, 1886, 1887, and 1895. 3. Report of tests of structural material made at the Watertown Arsenal, Mass., June, 1891. While it is from tests upon long joints that we can derive correct and reliable information to use in practice, and hence while the experiments already made give us a considerable amount of information, nevertheless as the tests have not yet been carried far enough to furnish all the information we need, and to settle cases that we are liable to be called upon to decide, therefore, before quoting the above experiments, a few of the rules and proportions more or less used at the present 54$ APPLIED MECHANICS. time, and the modes of determining them, will be first ex- plained. In this regard we must observe that practical considerations render it necessary to make the proportions different when the joint is in the shell of a steam-boiler, from the case when it is in a girder or other part of a structure. In the case of boiler-work, the joint must be steam-tight, and hence the pitch of the rivets must be small enough to render it so : whereas in girder-work this requirement does not exist ; and hence the pitch can, as far as this requirement goes, be made greater. It is probable, that, with good workmanship, we might be able to secure a steam-tight joint with considerably greater pitches than those commonly used in boiler-work ; and now and then some boiler-maker is bold enough to attempt it. Some years ago punching was the most common practice , but now drilling has displaced punching to such an extent that all the better class of boiler-work is now drilled, and drilling is also used to a very considerable extent in girder-work. When drilling is used, the plates, etc., to be united should be clamped together and the holes drilled through them all together. In this regard it should be said : i. When the holes are drilled, and hence no injury is done to the metal between the rivet-holes, this portion of the plate comes to have the properties of a grooved specimen, and hence has a greater tensile strength per square inch than a straight specimen of the same plate, as the metal around the holes has not a chance to stretch. This excess tenacity may amount to as much as 25 per cent in some cases, though it is usually nearer 10 or 12 per cent, depending not only on the nature of the material, but also on the proportions. 2. When the holes are punched, we have, again, a grooved specimen, but the punching injures the metal around the hole, and this injury is greater the less the ductility of the metal : thus, much less injury is done by the punch to soft-steel plates RIVETED JOINTS. 549 than to wrought-iron ones, and less to thin than to thick plates. This injury may reach as much as 35 per cent, or it may be very small. Besides this, in punching there is liability of crack- ing the plate, and of not having the holes in the two plates that are to be united come exactly opposite each other. A number of tests on the tenacity of punched and drilled plates of wrought- iron, and of mild steel, made on the government testing-machine at Watertown Arsenal, are given on page 564 tt seq. The hardening of the metal by punching also decreases the ductility of the piece. The injury done by punching may be almost entirely re- moved in either of the following ways : i. By annealing the plate. 2. By reaming out the injured portion of the metal around the hole ; i.e., by punching the hole a little smaller than is de- sired, and then reaming it out to the required size. There is a certain friction developed by the contraction of the rivets in cooling, tending to resist the giving way of the joint ; and some have advocated the determination of the safe load upon a riveted joint on the basis of the friction developed, instead of on the basis of strength notably M. Dupuy in the Annales des Fonts et Chausees for January, 1895 ; but this seems to the author an erroneous and unsafe method of pro- ceeding: i, because tests show that slipping occurs at all loads, beginning at loads much smaller than the safe loads on the joint ; 2, because all friction disappears before the break- ing load is reached. Hence it is safer to disregard friction in designing a tensile riveted joint. The shearing-strength of the rivets would appear to be about two thirds the tensile strength of the rivet metal. Before proceeding to give an account of Kennedy's tests, and of those made at the Watertown Arsenal, which form the principal basis for determining the constants, i.e., the tearing- strength of the plate, the shearing-strength of the rivet iron, 550 APPLIED MECHANICS. and the ultimate compression on the bearing surface, it will be best to outline the proper method of designing a riveted joint, and for this purpose a discussion of a few cases of tensile riveted joints, as given by Prof. Peter Schwamb, will be given by way of illustration. The letters used will be as follows, viz. : d = diameter of driven 'rivet in inches ; t = thickness of plate in inches ; /! thickness of one cover-plate in inches ; f s = shearing-strength of rivet per square inch ; f t tearing-strength of plate per square inch ; f c = crushing-strength of rivet or plate per square inch; / pitch of rivets in inches ; p d = diagonal pitch in inches ; / = lap in inches. In every case of a tension-joint we begin by selecting a repeating section and noting all the ways in which it may fail. It would seem natural, then, to determine the diameter of the rivet to be used by equating the resistance to shearing and the resistance to crushing, and in some cases it is desirable to adopt the resulting diameter of rivet ; but there are also many cases where there is good reason for adopting either a larger or a smaller rivet, and others where there is good reason for determining the trial diameter in some other way. Thus we may find that the rivet which presents equal re- sistance to shearing and crushing may be too large to be suc- cessfully worked, or it may require a pitch too large for the purposes for which the joint is to be used; or, on the other hand, it may be so small that it would lead to a pitch too small to be practicable ; or it might, in a complicated joint, where there are a good many ways of possible failing, lead to a low efficiency. In all cases, a commercial diameter must be selected. SINGLE-RIVETED LAP-JOINT. 55 1 Single-riveted Lap-joint. Repeating section containing one rivet may fail by 1, shearing one rivet. Resistance =^ t 4 2, tearing the plate. Resistance =f t (pd)t. 3, compression. Resistance f c td. Equating i and 3 gives d = y (i) 71 ft A larger rivet will crush, a smaller one will shear. The diameter given by (i) will frequently be found to be larger than can be successfully worked. Equating 2 and 3 gives p d(\ + y ). (2) Equating i and 2 gives p = d (i + ^ }. (3) 4* ft/ If the value of d given in (i) is used, then (2) and (3) give the sameVesult. If, however, a different value of d is used, then the pitch should be determined by (2) for a larger and by ($) for a smaller rivet. It may be well to note that whenever compression fixes the pitch, the computed efficiency P~d_ f< P -f t +f. is independent of the diameter of the rivet, and that this is the maximum efficiency obtainable with this style of joint. SINGLE-RIVETED DOUBLE-SHEAR BUTT-JOINT. The combined thickness of the two cover-plates should always be greater than /, and, this being the case, we proceed as follows : 55 2 APPLIED MECHANICS. Repeating section containing one rivet may fail by i , shearing one rivet in two places. Resistance = /, . 2, tearing the plate. Resistance = f t (p d)t. 3, compression Resistance =f e td. Equating i and 3 gives d= j (4) 71 ft A larger rivet will crush, a smaller one will shear. The diameter given by (4) is just one half that given by (i), f nd will frequently be found to lead to a pitch too small to use in practice. In such cases we should use a larger rivet. Equating 2 and 3 gives p=d[i + j). (5) v It' Equating i and 2 gives p=d(i+~j}. (6) If the value of d given by (4) be used, then (5) and (6) give the same result. If, however, a different value of d be used, then the pitch should be determined by (5) for a larger and by (6) for a smaller rivet. For the diagonal pitch, in the case of staggered riveting, we should have, at least, according to Kennedy's sixth conclusion (see page 566) 2(p d d)=^(pd) and hence p d = $p + $d. DOUBLE-RIVETED LAP-JOINTS. Repeating section containing two rivets may fail by nd 2 i, shearing two rivet sections. Resistance = J 8 . 2, tearing plate straight across. Resistance = }t(pd)t. 3, compression on two rivets. Resistance = j c (2td). Equating i and 3 gives d= /. (7) 7T / EXAMPLE OF A SPECIAL JOINT. 553 A larger rivet will crush, a smaller one will shear. The diameter given by (7) would usually be found too large. Equating 2 and 3 gives p=d(i+-j?J. (8) Equating i and 2 gives p=d \i H -- ^/ . (9) The pitch should be determined by (8) for a larger and by (9) for a smaller rivet than that given by (7). For p d we should have, as in the last case, according to Kennedy, p d EXAMPLE OF A SPECIAL JOINT. The joint shown in the cut is one where a part of the rivets are in single and a part in double shear. Repeating section containing five rivet sections may fail by i, tearing on ab. Resistance = ft(p d)t. 2, shearing five rivet sections. , Resistance = / a 4 3, tearing on ce, and shearing one rivet on ab. nd 2 Resistance = j t (p zd)t + f 8 . 4 4, tearing on ce, and crushing one rivet. Resistance = j t (p- 2d) + fad. 5, crushing two rivets and shearing one. Resistance =f c (2td) + /, . 4 6, crushing on three rivets. Resistance = f c (2td+tid). 7, crushing three rivets, where /i ^ /. Resistance = 554 APPLIED MECHANICS. In this case, we should so proportion the joint that its effi- ciency may be determined from its resistance to tearing along ab. Hence all its other resistances should be equal to or greater than this. Hence equate i and 3, and calculate the resulting diameter of rivet, which will generally be too small, and hence we select a larger rivet, so that 3 may be greater than i. Having fixed the diameter of rivet, determine the pitch in each of three ways, viz., by equating i and 2, by equating i and 6, and by equating i and 5, and adopt the least value of p. In this joint as used fji d > f 8 , and hence 6 is greater than LAP, To compute the lap, the following method is a good one. Consider the plate in front of the rivet as a rectangular beam fixed at the ends and loaded at the middle, whose span=d, breadth =/ (for cover-plate t]), depih = h=ld/2. Assume for modulus of rupture j t and for center load W, where i. When rivet fails by single shear W=j s . 4 2. When rivet fails by double shear W=f 8 . 3. When rivet fails by crushing and lap in plate is sought 4. When rivet fails by crushing and lap in cover-plate is sought W = j c tid. JOINTS IN THE WEB OF A PLA TE GIRDER. 555 JOINTS IN THE WEB OF A PLATE GIRDER. While no experiments on the strength of such joints have been published, the constants necessary for use in the ordinary method of calculating them are : i, the allowable outside fibre-stress ; 2, the allowable shearing-stress on the outer rivet ; and, 3, the allowable compression on the bearing- surface. As an example of the usual method of calculation of such a joint, let us consider a chain-riveted butt-joint with two covering strips (as shown in the cut) as being a joint in the web of a plate girder which has equal flanges, and let us determine the allow- able amount of bending-moment which the web alone (without the flanges) can resist. The modifications necessary when the flanges are unequal, and hence when the neutral axis is not at the middle of the depth, will readily suggest themselves. The stress on any one rivet is pro- portional to its distance from the neutral axis of the girder, and hence, in this case, from the middle of the depth. Use the following letters, viz.: f t allowable stress per sq. in. at outer edge of web-plate ; f t = allowable shearing-stress per sq. in. on outer rivet ; f c = allowable bearing-pressure per sq. in. on outer rivet ; / = thick- ness of plate ; h total depth of web-plate ; /^ = total depth of girder ; d= diameter of driven rivet ; a = area of 4 driven rivet section ; r = number of vertical rows on each side ; 2n = number of rivets in each vertical row ; y l = distance from o o O O o o o o o o 556 APPLIED MECHANICS. neutral axis to centre of nearest rivet ; y t = distance from neutral axis to centre of second rivet, etc., etc.;j M = distance from neutral axis to centre of outer rivet. Then, for allowable bending-moment, we must take the least of the three following, viz : i, that determined from the shearing f s ; 2, that determined from the compression f c \ 3, that determined from maximum fibre-stress f t , observ- ing that if f = greatest allowable fibre-stress in girder, then To determine these proceed as follows : hence allowable stress on rivet at distance y m from neutral axis i/ and the moment of this stress is Hence greatest allowable moment on joint for shearing is (,) JOINTS IN THE WEB OF A PLATE GIRDER. 557 2. Greatest allowable compression on outer rivet \sf c td\ hence allowable stress on rivet at distance y m from neutral axis is . y* ' and the moment of this stress is Hence greatest allowable moment on joint for compression is -. './i *r a i r * i i *r ' t \ / Jn 3. The section of the plate is a rectangle, width / and height //, with the spaces where the rivet-holes are cut left out. It will be near enough to take for the stress to be deducted on account of the rivet-hole at distance y m from neutral axis and for its moment Hence greatest allowable moment on joint for tearing is 558 APPLIED MECHANICS. This mode of calculation for (3) would seem to be war- ranted from the fact that the rivets do not fill the holes, although many deduct only the effect of the holes on the ten- sion side, and consider that those on the compression side do not weaken the metal. The greatest allowable bending-moment on the joint is the smallest of (i), (2), and (3), and it is plain that, in order to make the calculation, we need to know what to use o f s , and/ c , or, since f t =f-j-, what to use for /, f s , and f e \ and while /"should be determined from the tests on the transverse strength of the metal, whether wrought-iron or steel, the best evidence we have as to the proper values of f s and f e is furnished by the tests on tension-joints, which have already been discussed. Moreover, we might determine the diameter of rivet by equating (i) and (2), but we should generally find it desirable to use a larger rivet, and then we should determine the pitch by equating (2) and (3) if a larger, or (i) and (2) if a smaller, rivet is used. Moreover, the rivets in common use in such cases are either f" or I" in diameter. TESTS OF THE COMMITTEE OF THE BRITISH INSTITUTION OF MECHANICAL ENGINEERS. The Committee on Riveted Joints of the British Institu- tion of Mechanical Engineers consisted of Messrs. W. Boyd, W. O. Hall, A. B. W. Kennedy, R. N. J. Knight, W. Parker, R. H. Twedell, and W. C. Unwin. RI VE TED JOIN TS. 559 Before beginning operations Prof. Unwin was asked to prepare a preliminary report, giving a summary of what had already been done by way of experiment, and also to make recommendations as to the course to be pursued in the tests. This preliminary report is contained in vol. xxix. of Engineer- ing, on the pages already cited. In regard to its recommenda- tions it is unnecessary to speak here, as the records of the tests show what was done ; but in regard to the summary of what had been done, it may be well to say that he gives a list of forty references to tests that had been made before 1880, be- ginning with those of Fairbairn in 1850, and ending with some made by Greig and Eyth in 1879, together with a brief account of a number of them. Almost all of this work was done, however, with small strips with but few rivets, and will not be mentioned here. Inas- much, however, as Fairbairn's proportional numbers have been very extensively published, and are constantly referred to by the books and by engineers, it may be well to quote a portion of what Unwin says in that regard, as follows : " The earliest published experiments on riveted joints, and probably the first experiments on the strength of riveting ever made, are contained in the memoir by Sir Wm. Fairbairn in the Transactions of the Royal Society. " The author first determined the tenacity of the iron, and found, for the kinds of iron experimented upon, a mean tenacity of 22.5 tons per square inch with the stress applied in the direction of the fibre, and 23 with the stress across it. That the plates were found stronger in a direction at right angles to that in which they were rolled is probably due to some error in marking the plates. " Making certain empirical allowances, Sir Wm. Fairbairn adopted the following ratios as expressing the relative strength of riveted joints : 560 APPLIED MECHANICS. Solid plate 100 Double-riveted joint 70 Single-riveted joint 50 These well-known ratios are quoted in most treatises on rivet- ing, and are still sometimes referred .to as having a considerable authority. " It is singular, however, that Sir Wm. Fairbairn does not appear to have been aware that the proportion of metal punched out in the line of fracture ought to be different in properly designed double and single riveted joints. These celebrated ratios would therefore appear to rest on a very unsatisfactory analysis of the experiments on which they are based. Sir Wm. Fairbairn also gives a well-known table of standard dimensions for riveted joints. It is not very clear how this table has been computed, and it gives proportions which make the ratio of tearing to shearing area different for different thicknesses of plate. There is no good reason for this." As to the tests which constitute the experimental work of the committee, these were made by or under the direction of Pi*of. A. B. W. Kennedy, of London. Steel plates and steel rivets were used throughout, the steel containing about 0.18 per cent of carbon, and having a tensile strength varying from about 62000 to about 70000 pounds per square inch, and hence being a little harder than would correspond to our American ideas of what is suitable for use in steam-boilers. The greater portion of the work was performed by the use of a testing- machine of looooo pounds capacity, and hence one which did not admit of testing wide strips with a sufficient number of rivets to correspond to the cases which occur in practice; indeed, only eighteen of the tests were made on such strips. Nevertheless, a brief summary of what was done will be given here, though some of the conclusions which he drew are aL RIVETED JOINTS. 561 ready, and others are liable to be, proved untrue by tests of wide strips. The tests made by Prof. Kennedy up to 1885 consisted of fourteen series numbered I to V, VA and VI to XIII, and covering 290 experiments, 64 on punched or drilled plates, 97 on joints, 44 on the tenacity of the plates used in the joints, 33 on the tenacity and shearing-resistance of the rivet-steel used in the joints, and the remaining 52 on various other matters. The first three series were upon the tenacity of the steel used, and showed it to be, as stated, from 62000 to 70000 pounds per square inch, with an ultimate elongation of 23 to 25 per cent in a gauged length of ten inches ; the tenacity of the rivet-steel being practically the same as that of the plates. The fourth series showed the shearing-strength of the rivet- steel to be about 55000 pounds per square inch when tested in one way, and 59000 pounds per square inch when tested in another way which corresponded, as Kennedy claims, better to the conditions of a rivet, though neither was by using a riveted joint. The tests of series V and VA were made upon pieces of plate which had been punched or drilled, in other words, on grooved specimens ; and, as might be expected, these specimens showed invariably an increase in tensile strength over the straight specimens. In the J" and fV' plates drilled with holes i inch in diameter and 2 inches pitch, the net metal between the holes had a tenacity 11 to 12 per cent greater than that of the untouched plate. Even with punched holes the metal had a similar excess of tenacity of over 6 per cent. The remaining eight series, VI to XIII inclusive, were made on riveted joints, the first five on single-riveted lap-joints, and the last three, or XI, XII, and XIII, on double-riveted lap and butt joints. Series VI was made on twelve joints in f-inch plates which contained only two rivets each, the proportions not being in- tended to be those of practice, but such as should give, to 562 APPLIED MECHANICS. some extent, limiting values for the resistances of the plate to tearing, and of the rivets to shearing and pressure. The results were rather irregular; and the main conclusion which he drew, was, that if the joint is not to break by shearing, the ratio of the tearing to the shearing area must be computed on a much lower value of shearing-strength per square inch than the ex- periments of series IV had shown ; indeed, some of the joints of series VI gave way by shearing the rivets at loads no greater than 36000 pounds per square inch of shearing-area. Series VII was made upon six (single-riveted lap) joints in f-inch plate, with only three f-inch rivets in each joint, and with varying pitch and lap ; all these joints breaking by shear- ing the rivets. His conclusion from these tests was, that the lap need not be more than 1.5 times the diameter of the rivet. Series VIII was made on eighteen (single-riveted lap) joints in six sets of three each, and these are the only single-riveted lap-joints which he tested, having as many as seven rivets each. The results are given in the accompanying table. Before giving the table, it may be said that No. 652 was in- tended to have such proportions as to be equally likely to give way by tearing or by shearing, the intensity of the shearing- strength being assumed as two-thirds that of the tensile strength of the steel, while the bearing-pressure per square inch was intended to be about 7.5 per cent greater than the tension. No. 653 was proportioned with excess of shearing or rivet-area, No. 654 with defect of shearing-area, No. 655 with excess of tearing or plate area, No. 656 with defect of tearing- area, and No. 657 with excess of bearing-pressure, the different proportions being arrived at by varying the pitch and diameter of the rivets, and, in the case of 657, the thickness of the plate also. The margin (or lap minus radius of rivet) was f inch in each case. The following table will show how far these inten- tions were realized, and further comments will be deferred till later. RIVETED JOINTS. 563 P!lS J 5 U3 3 a3 d 5 u ! o f jo q^Susj^s iBuopaodoaj M in 00 in in oo" in CO CO ] Shearing. rt rt O be bi> ice H'-'~ Shearing. Shearing. L ^ c c ill Shearing. Shearing. pajjno pBoq-Sui3iB3jg jo uopaodoaj .j U CO o. ^ OJ in J I N q O *j r *' 3 N qoiq 3JBnbg -lad spunoj juiof uaqM. djnssdj j SuiJBsg 1 S o tn 1 o rl- eo o in 0? 00 ^ 8 g s 'B3JV SuiJBSg CT | N 2 CO M in I 1 ^ in U3 rt .j sjjojq luiof usqAv B3jy SuiJBsqs jo 'uj 'bg Jad ssaa^s if I o in T S, 1 5 X co tNERAl B3jy 3uiJB3qs .a ! O CO O CO M CO CO CO rt- CO 2 HH asjoaq ?u;of usqM. BSJV SUUBSXJO 'ui 'bs Jad ss3j;s 4 1 1 vO O 1 oo 1 01 O 01 > W -"*-> O *"** CO oi R in 01 M CO 04 H V} B3J V 3uuB3X 01 SujjBsg jo op-e^[ M CO o* 6 CO CO o O CO d 'B3jy S\Jl -JB3X Ol 3uiJB3qS JO OpB^ ^ tn vO CO ? rl I~N. - jl CO 0* 0* O 00 CO 1 d " S3 IH J ^ 3 5Jd S ' M- 00 M ^ rj- 5 ' S3 1H P 3 II! J Q J JajatQBiQ d vO CO in 6 d 00 d d .^ awoi c ? M g 8 oi i-i tn d oJ snarapads jo -o N CO CO co CO CO CO JS3X JO '0>I tn CO m tn tn in tn tn O 564 APPLIED MECHANICS. Series IX was made on twenty-one joints in f-inch plate (each containing only two rivets) designed in a manner similar to series VIII, while three were afterwards made from some of the broken plates, with as heavy rivets as it was deemed possible to make tight. From these tests Kennedy thinks it fair to conclude I . That the efficiency of a single-riveted lap-joint in a |-inch plate cannot be greater than 50 per cent, unless rivets larger than i.i inch are used ; and he also calls attention to the fact that, as he claims, strength is gained by putting more metal in the heads and ends of the rivets, claiming that it will make also a tighter joint for boiler-work. Series X was made on eight single-riveted lap-joints in J-inch and f-inch plate, made from the broken specimens of series V and VA ; they also had only two rivets each. These joints were made with a view of investigating the effect of more or less bearing-pressure. He claims that high bearing- pressure induces a low shearing-strength in the rivets, and that the bearing-pressure should not exceed about 96000 pounds per square inch ; also, that when a large bearing-pressure is used, the " margin " should be extra large to prevent distortion, and consequent local inequalities of stress ; also, that smaller bearing- pressures do not much affect the strength of the joint one way or the other. Series XI was made upon twelve specimens of double-riveted joints ; three being lap-joints in f-inch plate, three lap-joints in f-inch plate, three butt-joints with two equal covers in f-inch plate, and three butt-joints with two equal covers in f-inch plate. Kennedy designed these joints with a view to their being equally likely to fail by tearing or by shearing. His as- sumptions and the results of the tests are all given in the fol- lowing table : RIVETED JOINTS. 565 SERIES XI. DOUBLE-RIVETED LAP AND BUTT JOINTS AVERAGES. !| 5 -Q II | 1! Mi U *- J*J 8 |l & S & a 1 Ss Iv, CO V PH rt c W 4^ 3 be 3 O V C/) ^ _C % J2 > "^ C/T .S# O he 2 u 'O o J5 bo g U E "o "S h jU o Q > c II s -G v SJ3 W CL G o U *Q 53 cti Is o g -0 S T3 * S2 ^ H * C/2 u rt fr V c J4 fl' W > a 5 > g I s s-S S s ^ 2 3 a --; cfl rt rt 3 w "3 jj e u rt w O 2 P > X O S ^ 5 3 cr o o C/3 '3 H jr ^ o. 2 H Q < < H Q PQ <5 <; < H In. In. Lbs. Lbs. In. In. Lbs. Lbs. Lbs. Lbs. % LAP-JOINTS. f 0.8 70560 51970 2.9 2.15 29 89609 75150 53920 91530 80.8 f i.i 70560 51970 3-1 2-45 35 Low 69910 49710 58910 70.8 BUTT-JOINTS WITH TWO COVERS. 1 0.7 70560 34720 2-75 2.00 27 68000 33780 94710 80.2 1 i.i 67200 42560 4.4 3-18 26 100800 59290 37650 88460 71-3 Series XII contains the same joints as series XI, the strained ends having been cut off, and the rest redrilled and riveted by means of Mr. Twedell's hydraulic riveter; and series XIII con- tained the same joints treated a second time in the same way. These experiments, so far as they went, showed no gain in ultimate strength to result from hydraulic as compared with hand-riveting ; but it was found that, through a misunderstand- ing, they had been riveted up at a pressure much lower than that intended by Mr. Twedell. On the other hand, the load at which visible slips occurred was about twice as much greater with hydraulic as with hand riveting. $66 APPLIED MECHANICS. KENNEDY S CONCLUSIONS. The following are a portion of what he gives as his con- elusions : i. The metal between the rivet-holes had a considerably greater tensile resistance per square inch than the unperfo- rated metal. 2. In single-riveted joints, with the metal that he used, he assumed about 22 tons (49280 Ibs.) per square inch as the shear- ing-strength of the rivet-steel when the bearing-pressure is below 40 tons (89600 Ibs.) per square inch. In double-riveted joints with rivets of about f-inch diameter we can generally assume 24 tons (53760 Ibs.) per square inch, though some fell to 22 tons (49280 Ibs.). 3. He advises large rivet heads and ends. 4. For ordinary joints the bearing-pressure should not ex- ceed 42 or 43 tons (94000 or 96000 Ibs.) per square inch. For double-riveted butt-joints a higher bearing-pressure may be allowed ; the effect of a high bearing-pressure is to lower the shearing-strength of the steel rivets. 5. He advises for margin the diameter of the hole, except in double-riveted butt-joints, where it should be somewhat larger. 6. In a double-riveted butt-joint the net metal, measured zigzag, should be from 30 to 35 per cent greater than that meas- ured straight across, i.e., the diagonal pitch should be p -\ > o j where/ transverse pitch and d-=- diameter of rivet-hole. 7. Visible slip occurs at a point far below the breaking- load, and in no way proportional to that load. Kennedy thinks that these tests enable him to deduce rules for proportioning riveted joints, and the following are his rules, viz. : RIVETED JOINTS. 567 (a) For single-riveted lap-joints the diameter of the hole should be 2\ times the thickness of the plate, and the pitch of the rivets 2f times the diameter of the hole, the plate-area being thus 71 per cent of the rivet-area. If smaller rivets are used, as is generally the case, he recommends the use of the follow- ing formula : where / = thickness of plate, d diameter of rivet, and/ = pitch. For 30-ton (67200 Ibs.) plate, and 22-ton (49280 Ibs.) rivets, a = 0.524 For 28-ton (62720 Ibs.) plate, and 22-ton (49280 Ibs.) rivets, a = 0.558 For 30-ton (67200 Ibs.) plate, and 24-ton (53760 Ibs.) rivets, a = 0.570 For 28-ton (62720 Ibs.) plate, and 24 ton (53760 Ibs.) rivets, a = 0.606 Or, as a mean, a = 0.56. (&) For double-riveted lap-joints he claims that it would be desirable to have the diameter of the rivet 2\ times the thick- ness of the plate, and that the ratio of pitch to diameter of hole should be 3.64 for 3<D-ton (67200 Ibs.) plate and 22-ton (49280 Ibs.) or 24-ton (53760 Ibs.) rivets, and 3.82 for 28-ton (62720 Ibs.) plate. Here, however, it is specially likely that this size of rivet may be inconveniently large, and then he says they should be made as large as possible, and the pitch should be determined from the formula to / = "y + 4 where, For 30-ton (67200 Ibs.) plate, and 24-ton (53760 Ibs.) rivets, = x.if For 28-ton (62720 Ibs.) plate, and 22-ton (49280 Ibs.) rivets, a = 1.16 For 3o-ton (67200 Ibs.) plate, and 22-ton (49280 Ibs.) rivets, a = 1.06 For 28-ton (62720 Ibs.) plate, and 24-ton (53760 Ibs.) rivets, a = 1.24 568 APPLIED MECHANICS. (c) For double-riveted butt-joints he recommends that the diameter of the hole should be about 1.8 times the thickness of the plate, and the pitch 4.1 times the diameter of the hole, and that this latter ratio be maintained even when the former cannot be. Two of the principal participants in the discussion of the report were Mr. R. Charles Longridge and Prof. W. C. Unwin. Mr. Longridge was of the opinion that wider strips with more rivets should have been used ; that holding the specimens in the machine by means of a central pin at each end was not the best method ; that the results obtained from specimens which had been made from the remnants of other fractured specimens were at least questionable, for, even if the plate had not been injured, the ratio of the length to the width of the narrowest part was different after the strained ends were cut off from what it was before ; that machine-riveting should have been adopted throughout instead of hand-riveting, as it is not possible to secure uniformity with the latter even were it all done by the same man, as he would be more tired at one time than at another ; that experiments should be made to determine the effect of different sizes and different shapes of heads, as well as of different pressures upon the load causing visible slip , and that experiments should be made upon chain-riveting, as he thought the chain-riveted joint would show a greater effi- ciency than the staggered. Professor Unwin said : i. In examining the results to ascertain how far a variation from the best proportions was likely to affect the strength of the joint, he found that while the ratio of rivet diameter to thickness of plate varied 21 per cent, the ratio of shearing to tearing area 30 per cent, and the ratio of crushing to tearing area 34 per cent, the efficiency of the weakest joint was only six per cent less than that of the strongest, or, in other words, RIVETED JOINTS. 569 the whole variation of strength was only 1 1 per cent of the strength of the weakest joint. 2. With reference to the effect which the crushing-pres- sure on the rivet produced upon the strength of the joint, there were some old experiments, which showed that', when the bearing-pressure on the rivet became very large there was a great diminution in the apparent tenacity of the plate in the case of riveted joints in iron. Why should the crushing- pressure affect either the tenacity of the plate or the shearing resistance of the rivet? He believed that it did not really affect either. What happened was that, if the crushing-pres- sure exceeded a certain limit, there was a flow of the metal, and the section which was resisting the load was diminished. Either the section of the plate in front of the rivet, if the plate was soft, or the section of the rivet itself, if the rivet was soft, became reduced. 3. He thought that the point at which visible slip began was the initial point at which the friction of the plates was overcome, and of course was greater the greater the grip upon the plates, and hence greater in machine than in hand riveting. In some cases with hydraulic riveting loads were got as high as 10 tons (22400 Ibs.) per square inch of rivet section before slipping began. 4. In regard to the rules for proportioning riveted joints, he preferred to distinguish the joints as single-shear and double- shear joints, and then we have the following three equations : one by equating the load to the tearing-resistance of the plates, a second by equating it to the shearing-resistance of the rivets, and a third by equating it to the crushing resistance ; these three determining the thickness of the] plate, the diameter of the rivet, and the pitch. By taking the crushing as double the tenacity, we should obtain for single shear d = 2.57*, and for double-shear, d = 57 APPLIED MECHANICS. In a single-shear joint the rivet cannot generally be made so big, and in the double-shear it could not always be made so small, hence the rivet diameter is chosen arbitrarily, and then the single-shear joint is proportioned by the equations for shear- ing and tearing, no attention being paid to the crushing, while the double-shear joint is proportioned by the equations for crushing and tearing, no attention being paid to the shearing. 5. The general drift of the report was to advocate the use of larger rivets. Whether this could be done or not, he could not say. For lap-joints it would increase the strength, whereas for double-shear joints he was not sure that it would not be better to diminish the size of the rivet, and hence the crushing, pressure. This report has been given so fully because it emanates from a committee of the British Institution of Mechanical Engineers; but inasmuch as series VIII is the only one where wide strips were used, it seems to the writer that any conclu- sions which may be drawn from any of the other tests given in the report require confirmation by tests on wide strips with more rivets, before being accepted as true. Government Experiments. The references to these experi- ments have been mentioned on page ooo. Those included in the first five of the volumes mentioned may be divided into three parts: i. Those contained in the first two Executive Documents mentioned above. 2. Those contained in the third and fourth. RIVETED JOINTS. 3. Those contained in the fifth. Summaries of these sets of tests will be given here in their order, as each set was made with certain special objects in view, and, if not all, at any rate the i and 2, form, as has been al- ready stated, the first portion of a systematic series ; and it seems to the author that, although the series are not yet completed, yet these tests themselves furnish more reliable information in regard to the behavior and the strength of joints than any other experiments that have been made, and that the figures them- selves furnish the engineer with the means of using his judg- ment in many cases where he had no reliable data before. A perusal of the tables will give a good idea of the shear- ing-strength per square inch of the rivet iron, which is seen to be less than the tensile strength of the solid plate ; also the effect on strength of the plates due to the entire process of riveting, punching, drilling, and driving the rivets ; also the efficiencies of the joints tested. One of the strongest single-riveted joints tested was a single- riveted lap-joint with a single covering-strip. The apparent anomaly of the punched plates in a few cases, showing a greater strength than the drilled plates, is explained by Mr. Howard to be due to the strengthening effect of cold- punching combined with smallness of pitch, inasmuch as then the masses of hardened metal on the two sides re-enforce each other. Further than this, the student is left to study the figures themselves as to the effect of different proportions, etc. In regard to the first series, i.e., those contained in the first two Executive Documents mentioned, it is stated in the report that i. " The wrought-iron. plate was furnished by one maker out of one quality of stock." 2 " The steel plates were supplied from one heat, cast in ingots of the same size; the thin plates differing from the 5/2 APPLIED MECHANICS. thicker plates only in the amount of reduction given by the rolls." The modulus of elasticity of the metal was, iron plate^ 31970000 Ibs. ; steel plate, 28570000 Ibs. In the tabulated results, the manner of fracture is shown by sketches of the joints, and is further indicated by heavy figures in columns headed " Maximum Strains on Joints, in Jbs., per Square Inch." RIVETED JOINTS. 573 juiof jo Xouaptjjg Q ^O vr> 9 VO s I *8.1 eg i 1 8 j! R. 5 00 VO rt >-i ON w oo Q VO VO N ro > $ I > a" fi F. 5 S H " SN ON s J o .Is 3J, JO - ro to co ^* 574 APPLIED MECHANICS. juiof jo XDiwpyjg I*. J3 Q 8 II B e < JS I a -3 a ris t^ g II <u c 2 w A .S.S s| CA & .S.S to a, RIVETED JOINTS. 575 juiof jo Aouaiotyg I .s 1-6 C3 I II t* 8 iuv 1 l 1 x jo -ON & jS'o <u ^- E 4 N 00 f) CI O N VO O O O ID r^ vo oo f^ r^ # ^ vS 3" eo to vo vo eg S "T r~* el 00 00 O* " 1 I as R C? Jo ff * o 10 3; s VO vo vo VO VO VO vo t^ r^ 10 10 en en en -" -" -0 -a ^13 SJ > oj > j .C 4> .5 C^'C^ J^-g^- -" CU-i-i CU V) OH v; OH * ^O **^ ^O c/3 *X? c/} *^3 .S.S.2.S .S.2.2.S .2.2.2.2 .2.2.2.2 57 6 APPLIED MECHANICS. ? By is S S M 1 f In "U c "O c d c c CO TJ C C II d c ro ?O RIVETED JOINTS. 577 jinof jo Aouapiya ro .8 -3, % 'S-s I in Si 1 00 1 J a 2 e < ^ 8 O $ PH IPJ'1 13 .S d, c c .S d, C G JS3X JO I j, r^. vS 578 APPLIED MECHANICS. mof jo totwiDiya C/D 2 II' Il fO ro S3 Si i l-s 10 vo VO vo VO vo vo vo r>. vo vo - - S "2 g PS 2 3 . &, .fa eu .. -C-a ^ JB SL'T sx jo -OK RIVETED JOINTS. 579 jtnof jo Aairapiga g I ! c/5 jj a s I 1 JS9J. JO '0 N > oo = 8 II II .s.s .s.s II .s.s , ft I! to co 2. 2 s 3 v 580 APPLIED MECHANICS. juiof jo Xouapiyg; O ro Q* vo - ft o o r^ N ON N NO IT) LO a < .2 S 8. 8 2 t^ S t^ if) MM M 00 ^- ^- 1O u-) j $ a, o o 00 O N ^o OO W jsaj, jo -o N ft c/> "o . 'o 0305 o 3 o 5 .^H Q-)*^4 P t . ^ P^.^H Q- .5.S.S.S .S.S.S.5 J:H-^S*** ' - ' -fc vo \O !- Tf to m m in 00 i- VO 00 if) LTI O Ti- N t^ vo Tf o Co Q O VO Tf 1-1 00 Tf ro CO CO o 3 o | HI! cccc cede vO -O O 10 VO co vo .? -2 RIVETED JOINTS. juiof jo Xouapiyg N ON t^. O vO l^. "d I r^. t^ rJ *? 8. | -o | 8 S $ % IO IT) O Tf CO VO m co a ' . , 68 ti jj 9< 6 5 - 1 e J ' 5 * jj 2 8 &' ^ s OO 00 o> >o Sot % 8 1 -i | 1^41 rt cr, S o S i: ^ w C. 1O Q V? ^f S _ rt ! i 1ll CO OO ^O O ("O __- Z "5 r fi S "~ t/3 c o o o o R ro 5 82 APPLIED MECHANICS. vo Co & c > o c 8, 2 10 o J' ! ^f t- N ^8- ft S> c S <8 o CO I IS?! 1 &, .s &, .a cede 3}vj(f uo^j -witf uojf S S 42-0 2 p II I? = | = .S 13 .ST3 d d d d 9X JO RIVETED JOINTS. 583 ?uiof jo XonaiojBg a s, t> NO S fO ro f > tx 00 a R ma lO ft CC C G '. I . I .S.S .S.S II II .S.S .S.S 3}VJ<f py}$ APPLIED MECHAA'ICS. JUIOf JO ADU3IOIJJ3 9 * 1^ ON x; ^ IJT w a ^ S Is \ if CO O O < oo. en 00 CO J8 "g ~ C/3 C C ^ HH 7 ^> icjx. O o a o 5 ->5 S C ^3 'I .g^ i a I ii 24 it! JS9 X JO 'ON K1YETED JO J NTS. 585 juiof jo Xouapiya s, .5 ss s Si SS O 0\ ro fO & I^> OO vO vO u-> oo I S oq q\ 8 to m 04 t^ 5 $ CO CO CO ro en en 'C 'C en T3 en 'O C d d d v;z%/ 7^5- 'V.'T jo ill Q 586 APPLIED MECHANICS. 8, I h s 2 3 8 O vr> CSO^ 8 S ^ eg 5 ff J? ft CO CO CO CO vS O to CO vo ON ON J* a o CO fill I O VO ^t 1 JjJ B | a | C| E|-S| ; 3 2S c c S 'C 43 C^3 8 w 00 ft RIVETED JOINTS. 587 jmof jo Aatrapgjg 4 i 1S9.I JO '0 N i in c/i g-g .b a .bo. O Q 00 \O M n C g S3 rj P s 2s .53 ex .be, c c d d V? V o fO vo 6 So R R t^ in c e c .3 O g .be* .b a .S.S .S.S ajfO M i^io ^ 'WWfucuj v;z?// ;^;5 588 APPLIED MECHANICS. juiof jo Xouiogj3 s, II I* C/D II II ^ vo fO o o I 1 in n CO CO 1 1 .5 2-S c c ' T a . c-^ SM .S.S -b i- V 18 ? $ jo -ON r?) RIVETED JOINTS. 589 juiof jo Xouapigg a" . i < PM a tf- 3 2"3 ,8 a ? J 91 in < >f . v ^ R 3- 3" 5? ro S J I m m os-g c 1 <o C C 00 Si .M Pi c c 590 APPLIED MECHANICS. juiof jo AouaiDtga O \o 3- 10 O O N fx O O N *r\ ^ ^. N N Q Q S rj- rf of Rivets es. .h tx .h a cd * 7 j en t cddc ' r > "'"t, it 6 w "i 8 M '9ivtf 199}$ " v v RIVETED JOINTS. 591 juiof O r^ ON J2 in w rt o * O HI r^ O oooooo rOi-i M >-< r^Tj- $ !> fO M HI Q t^ VO VO xO 000 \rt ^ CO O 5 M O Id M CO 00 M IO IT) ID VO ro^'-o \o<-oroOfO^ - "> 21 w 1-1 \O O ^ 12 3 oj5 S3 oooo oooooooo 03030303 0303030303030303 OOOOOOOO jsax jo -OM fOVO r^ 592 APPLIED MECHANICS. juiof l a Ills MS-S! tr> M ON C\ 10 vo o o o o o o o O O\ Oi r^ "^r c^ r** co O* co O N O ro o a\ a\ <-> _ M o ^- co o co co ro M VO W \O VO TJ- rj- o o o *tf- M M o^ N t^ IT) n a\ a\ \o OOOO ir> CO >-i ^> ^roivooo oooooo t^M c MOO o Q Q O O O 9, 8 < ct ^ t i ro ^* CO t^* -83J . 1P1 O O "~> ro ro M to ro w to ro r^ vo u-, 10 O o io>-n CO CO H-l >-< M >-i I-H o o o o (U O OJ <uo;<u<U <L)0)<U<L) .S .S . .S COJOO KpO r-pl | lit co*C ill o - 5 5 & RIVETED JOINTS. 593 GOVERNMENT TESTS OF GROOVED SPECIMENS. Tensile Tests of .J-in. Grooved Specimens Wroughl-Iron Punched. |jj "o P |2 1| I Is H 3 Inch. Inch. 0.48 0.240 48090 0.46 0.235 46940 0.46 0.241 49280 0.49 0.240 55340 0.44 0.239 51520 0.47 0.241 49910 0.97 0.247 49540 0.98 0.247 49960 0.94 0.249 50128 0.96 0.248 46900 0.98 0.250 46980 0.96 0.251 46350 i-47 0.250 37636 1.50 0.252 37326 1.48 0.249 41030 1.48 0.247 39480 i-47 0.250 37446 i-45 0.251 39533 1.96 0.281 43194 i-95 0.274 0.282 47499 41360 1.92 0.279 43080 2.03 0.250 41140 1.99 0.248 39575 2.42 0.280 '36210 2.40 0.245 42245 2.47 0.243 42233 2.46 0.285 42712 2.48 0.245 38125 2-44 0.248 41620 2.97 0.247 38964 2.98 0.241 41540 2.96 0.241 39972 2.92 0.240 41712 2.98 0.250 40430 3.95 0.247 40850 Tensile Tests of $-in. Grooved Specimens Wrought-Iron Drilled. S ^JZ o "bJD ^ c c M u *0 g 3 3 'wof j |0 Jl i v~~ H i5 Inch. Inch. 0.51 0.249 55787 0.52 0.245 55905 0.52 0.275 57480 0.52 0.276 56000 0.49 0.248 49600 0.50 0.248 56700 0.47 0.275 54880 0.51 0.276 57800 I.OO 0.276 54300 1.02 0.273 57700 I.OO 0.276 53800 I.OO 0.280 52430 I.OO 0.252 49400 1.02 0.275 54060 I.OI 0.247 52770 I.OO 0.278 54600 1.50 0.276 49 J 3 1-52 0.273 51300 1.48 0.251 47220 1.51 0.273 53400 1-52 0.275 54l8o 1.50 o 276 54600 1.48 0.274 56250 1.50 0.249 46260 2.OI 0.275 459oo 2.05 0.279 46820 2.OO 0.275 47950 2.00 0.278 49640 2.00 0.286 44650 2.00 0.275 50780 2.02 0.279 48850 2.OO 0.277 49840 2-51 0.244 44980 2.52 0.280 40150 2.51 0.282 43150 2.50 0.244 455oo 2.51 0.285 46500 2.49 0.242 49520 2.49 0.242 2.50 0.280 44780 3-02 0.250 45700 3.02 0.249 44870 3.00 0.240 46760 3.00 0.250 45700 2.93 0.242 47950 0.250 48740 2.98 0.279 459 3.01 0.281 44410 Tensile Tests of -in. Grooved Specimens Steel Plate Punched. S .c > o s! -1 1 & cr 5^ H P Inch. 0.49 Inch. 0.250 65120 0.47 0.249 67010 0.48 0.249 63420 0.48 0.248 66550 0.48 0.247 67060 0.47 o 248 65300 o-99 0.249 59840 I.OO 0.250 62160 I.OI 0.249 68246 0.96 0.96 0.250 0.248 67330 65966 0.95 0.245 62700 i-45 0.248 64080 1-45 0.252 64000 i-45 0.249 61025 i5* o 251 59420 1.96 i-93 0.250 0.252 599oo 63500 1.98 1.96 0.250 0.251 59350 59060 2-49 0.249 58100 2.47 0.249 63900 2-43 0.250 61640 2-95 0.251 56530 3-oi 0.249 58780 3-04 0-253 555oo 2.97 0.252 60060 2.98 0.251 54050 2.97 0.249 56040 Tensile Tests of J-in. Grooved Specimens Steel Plate Drilled. | si o C C W ** ^o *" $ C/5 ,y v ^ f- t/J 4=0 II E "- "~ I* 8 -Mia * P u> Inch. Inch. 0.52 o-54 0.246 0.248 67890 67160 0-53 0.247 66870 0.50 0.247 65610 0.51 0.249 66370 0.51 0.250 67420 0.52 0.248 67750 0.52 0.252 61910 1.03 0.247 57090 1.02 0.250 66390 1.02 0.246 66770 1.02* 0.250 67730 I.OI 0.247 66020 I.OO 0.251 67010 I.OO 0.247 64450 I.OI 0.250 66090 i-54 0.250 64390 1.52 0.251 63350 1.5 0.253 64370 i-54 0.248 64895 2.O2 0.252 64320 2.00. 0.251 62970 2.00 0.251 60910 2.50 0.248 59260 2.50 0.252 63250 2-53 0.248 59390 3-03 0.251 61577 3.00 0.249 59080 3-02 0.251 59550 3-02 0.250 59700 3.00 o 250 63370 3.00 0.251 58630 3-03 0.252 63940 594 APPLIED MECHANICS. IRON POUCHED. IRON DRILLED. STEEL PUNCHED. STEEL DRILLED. Tensile Tests of Grooved Wrought- Iron Plates. Tensile Tests of Grooved Wrought- Iron Plates. Tensile Tests of Grooved Steel Plates. Tensile Tests of Grooved Steel Plates. % BM c'c ~ ^ 6- JJC/JJ 1 a* bJ3 o |J w sr WO! ' i If Jjl-H ^ cr WC/2 ui , SJ c"e U 1 " 1 OT rJ, JjdTjj *2 ? J ls.s p .a B IS.fi 1 lS.a t> 1 g ib 5 Inch. Inch. Inch. Inch. Inch. Inch. Inch. Inch. I.OI 0-373 47000 0.98 0.376 50870 1.99 0-365 61890 1.97 0.369 63620 0.98 0.370 47520 0.98 0-377 52660 0.99 0-494 70080 I.OO 0.498 66220 2.OO 0.382 39760 1.98 0-379 49710 I.OO 0.492 68130 o-99 0-495 66800 2. 02 0.383 36630 2.OO 0.380 49830 1.50 0.497 66340 I.OO 0.500 67000 2-39 0.390 37600 2.50 0.390 50250 i-5i 0-494 63810 i-53 0-497 65930 2. 9 8 o-395 36340 3-00 0.392 45 I 5o 1.99 0.499 55930 1.50 0.498 66270 2. 9 8 0.392 39210 3-00 0-393 47540 1.97 0.500 64260 1.98 0.504 67510 3-47 0.390 37680 3-50 0.392 43940 2-43 0.502 52050 2.03 0.502 66730 3-47 0.389 38340 3-49 0.390 46490 2-51 0.504 64360 2.50 0.497 67950 0.97 0.467 - 50820 0.99 0.477 47140 3-00 0.503 60320 2.52 0.501 67440 1.48 0.506 45090 1. 00 0.479 48370 2-99 0.503 62430 3-oi 0.502 66310 1.49 0.506 45050 1.49 0.510 51240 3-50 0.503 49430 3-oi 0.503 66190 1.91 0.513 42500 1.49 0.512 51510 3-50 0.505 48270 3-49 0.504 64920 1.97 0.512 43430 1.98 0.514 50050 4.00 0.497 48010 3-50 0.502 65210 2.47 0.516 39410 1.98 0.516 47790 4.00 0.499 55190 3-99 0-499 64470 2.41 0.513 39720 2.51 0.520 4558o 3-99 0.501 55780 4.00 0.498 64810 3.00 0-515 38950 2.52 0.516 44960 3-99 0.498 46250 4.00 0.503 64690 2.90 0.517 37290 3-oo 0-5*5 44980 I.OI 0.613 66720 4.00 0.498 64140 3-5 0.520 37800 3.01 0.519 4703 1-52 0.612 64800 o-99 0.619 60290 3-49 0-513 37770 3-51 o-5 I 3 46170 i-5 0.615 64400 1.49 0.614 63610 4.00 0.515 35730 3-49 0.514 44760 2.50 0.618 58060 1.49 0.616 63450 4-03 0.516 36690 3-99 0.510 4533 2.52 0.619 58780 2-49 0.620 59170 3-99 0.511 37000 3.98 -5 I 3 45000 2.99 ' 0.617 57180 2.50 0.619 59600 4.03 0.508 37420 4.00 0.506 46100 3-46 0.615 58410 3-oi 0.617 59270 0.97 0.614 49770 o-97 0.628 47220 3-51 0.615 57190 3-50 0.614 61610 I.OI 0.619 52960 1. 00 0.626 4835 4.04 0.612 54450 3-49 0.617 62060 1.48 0.618 46320 1.52 0.625 47170 4-3 0.614 57380 4.00 0.615 60330 i.S 2 . 0.620 46750 1.49 0.629 4653 I.OI 0.721 67930 4.01 0.617 61120 2.99 0.614 40140 2.98 0.613 48220 I.OO 0.718 67620 0.96 0.726 58480 3-5 0.615 37480 3.46 0.616 4777 1.50 0.719 62890 I.OI 0.727 58790 3-5 0.616 36940 3-47 0.617 449 3-50 0-735 56730 i-5* 0.726 59290 4.04 0.619 373 10 3-9 1 0.625 44840 3-51 0-733 54220 3-5o 0.736 58700 ; 0.98 0.678 50840 3.96 0.626 45ioo 3-49 0.729 59 x 8o ] I.OI 0.682 46590 0-99 0.695 47500 i-49 0.688 4597 0.99 0.691 52780 3.48 0.691 4035 I-5I 0.692 48470 ( 3-53 0.692 39380 3-44 0.700 47750 3-49 0.692 46350 L TENSILE TESTS OF RIVETED JOINTS. 595 Next will be given the two series of tests already referred, to, with Mr. Howard's analysis of them. TENSILE TESTS OF RIVETED JOINTS. " Earlier experiments on this subject made with single and double riveted lap and butt joints in different thicknesses of iron and steel plate, together with the tests of specimens pre- pared to illustrate the strength of constituent parts of joints, are recorded in the report of tests for 1882 and 1883. From the results thus obtained it appeared desirable to institute a synthetical series of tests, beginning with the most elementary forms of joints in which the stresses are found in their least complicated state. To meet these conditions, a series of joints have been prepared which may be designated as single-riveted butt-joints, in which the covers are extended so as to be grasped in the testing-machine ; thereby enabling one plate of the joint to be dispensed with, and securing the test of one line of riveting. Such a joint, made with carefully annealed mild steel plate of superior quality, with drilled holes, seems well adapted to demonstrate the influence on the tensile strength of the metal taken across the line of riveting, of variations in the width of the net section between rivets, and variations in the compres- sion stress on the bearing-surface of the rivets ; elements which are believed to be fundamental in all riveted construction. This series comprises 2 16 specimen joints, the thickness of the plate ranging from J" to | 7/ , advancing by eighths. The covers are from y 3 ^-" to yV'- The rivets are wrought-iron, and range from -3-$" to lyV diameter; they are machine-driven in drilled holes iV' l ai "g er in diameter than the nominal size of the rivets. Ten- sile tests of the material accompany the tests of the joints. From each sheet of steel two test-strips were sheared, one lengthwise and one crosswise. The strips were 2\" wide and 24" to 36" long; they were annealed with the specimen plates, and had their edges planed, reducing their widths to i|" before testing. 596 APPLIED MECHANICS. Micrometer readings were taken in 10" along the middle of the length of each. The strength and ductility appear to be substantially the same in each direction. But the practice of the rolling-mill where these sheets were rolled is such that nearly the same amount of work may have been given the steel in each direc- tion ; that is, lengthwise and crosswise the finished sheet. The ingots of open-hearth metal are first rolled down to slabs about 6" thick, then reheated and rolled either length- wise or crosswise their former direction, as best suits the re- quired finished dimensions. The tensile tests show among the thinner plates a relatively high elastic limit as compared with the tensile strength ; in the f$' f plate the percentage is 72.2, while with the f" plate the percentage is found to be 53.3. It is noticeable that the thinner plates particularly exhibit a large stretch immediately following the elastic limit, and the stretching is continued at times under a load lower than that which has been previously sustained. It is characteristic of all the thicknesses that a considerable stretch takes place under loads approaching the tensile strength in some cases the stretch increases 5 to 6 per cent, while the stress advances 1000 pounds per square inch or less. Herein is found a valuable property of this metal as a material for riveted construction. The stress from the bearing-surface of the rivets is distributed over the net section of plate between the rivets, due to the large stretch of the metal, with little elevation of the stress, and a nearer approximation of uniform stress in this section attained than is found in a brittle or less ductile metal. The joints were held for testing in the hydraulic jaws of the testing-machine, having 24'' exposure between them. A loose piece of steel the same thickness as the plate was placed between the covers to receive the grip of the jaws, and avoid bending the covers. Elongations were measured in a gauged length of 5", the micrometer covering the joint at the middle of its width. Loads TENSILE TESTS OF RIVETED JOINTS. 597 were applied in increments of 1000 pounds per square inch of the gross section of the plate, the effect of each increment determined by the micrometer, and permanent sets observed at intervals. The progress of the test of a joint is generally marked by three well-defined periods. In the first period greatest rigidity is found, and it is thought that the joint is now held entirely by the friction of the rivet-heads, and the movement of the joint is principally that due to the elasticity of the metal. The second period is distinguished by a rapid increase in the stretch of the joint ; attributed to the overcoming of the friction under the rivet-heads and closing up any clearance about the rivets, bringing them into bearing condition against the fronts of the rivet-holes. Rivets which are said to fill the holes can hardly do so completely, on account of the contrac- tion of the metal of the rivet from a higher temperature than that of the plate, after the rivet is driven. After a brief interval the movement of the joint is retarded, and the third period is reached. The stretch of the joint is now believed to be due to the distortion of the rivet-holes and the rivets themselves. The movement begins slowly, and so continues till the elastic limit of the metal about the rivet-holes is passed, and general flow takes place over the entire cross-sec- tion, and rupture is reached. These stages in the test of a joint are well defined, except when the plates are in a warped condi- tion initially, when abnormal micrometer readings are observed. The difference in behavior of a joint and the solid metal suggests the propriety of arranging tension joints in boiler con- struction and elsewhere as nearly in line as practicable. The efficiencies of the joints are computed on the basis of the tensile strength of the lengthwise strips, this being the direction in which the metal of the joints is strained. The efficiencies here found are undoubtedly lowered somewhat by the contraction in width of the specimens, causing in most cases fractures to begin at the edges and extend towards the middle of the joint. Of the entire series, 88 joints have been tested ; tfr* 2", ", and f " plates yet remain." APPLIED MECHANICS. w o w o 3 u U a rt 1 .a | rt C 1 fl .1 , w . e <" >i ^ ;3 ri .:S . . S rt . . '55 ~~ o 'ir M ~ "xQ.yQQ >,>QQ>, gjd g ^^ JSd Do. ne silky. Iky, lamellar. , ne silky, ne silky, slight lamination. Iky. lamellar. Do. me silky. Do. Do. Do. Iky, slight lamination, ne silky, surface blister. Iky, slight lamination. Iky, stratified. Iky. ne sil'ky. Ikv, lamellar. Iky. slight lamination. Iky, laminated, stratified. Iky, laminated. fe j/5 fc c/5 j/5 c/5 t, c/) fc fe i/) tt. (73ttic75c7}c7:)fc<i/3c/3c/;i75 I ( c C'Ss., jj is rt pE?R i^li -*oo ooo 10 10 rooo O o 2^^U o,^ Ssc 2| S^r.o-00. -^- S'^-S d ^6 J S Sc 5 8 s 1 1 W " u l v l|p;l > &^H OOOOQOOOOQO ^NO OQNO o lOr^hx^-O f^ C rt u ^ Tf Tf 00 10 o 1 *~ && OOOOOQ COOOO oor^Q^ONO Not^io-^--^- ooooooooooo i/> 10NO NONOO"M^-iO->J- t^-OO iO C O u-) t^. C>OO ^^' S ilj ^ t^ t-^oo t^ t^ 10 r^co ONOO NO CNlMNtNIPllN NCMININCNl l&SSs-H^&SS 54feg6g4SS 1/3 -^ rt w s c )S <u 00 OO ONOO 00 tv 00 OO ONOO OO NO w m M t- CNI O * * O 00 H&5I??!J <| -s 8^^858'S ^S^S; O- ON ONOO ON ON ON O> ON ON O S;^o3 s; R SN^ ^ 5 ^ g T . . T "? pE c o "3 C I | "5 jj |6i S d d d d d ^ d d d d bf'o T3-a*UT3 tfl-a-o-a-a iJ U 5 6 d o 6 6 6 6 d o d |^]^ ^ d d d d d d d d 6 y s~-^ ' jl o c -n w |-BCH|5"R-B <SSS || OOQ 1 <u^ 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 .2 .2 rt cS C C a .s rt 53 .2 ^Ij^ tag bfi 3 .2 So" c g .2 m r. edi t l lla la lla t l la fied i ^ cflc/ir^wr^xx^ 3 fc te c/j (73 c/5 c/5 C/)COCO C/3CCW3 C/5 w H OOp ro M c> TJ-VO O o in ^- r**vo N N rf H ^t- io\o t-^ao t^ t^ moo -^-vo tnvo *OH\OVO\OIOOI S S ^cM-^-oor^oi rOMCSMM CSNMNIN NNOISNNNMNN NMMNWNNNMtM NNNNNMrOl ooo ooooooooo OOOOOOOO OOOOOOOOOO QOOOO t^ro*^-o>^t*t^rocn t^inoofoi^NOco^ ooin-^-M - ON t^ rOOO - mininm IN H 00 O 00 r~> in f-oo in m\o M c> i >& %$M OOOO OOOOO OO 1 ^oo wvo o* in O M *o cow 1 tx ro 0) vO VO <* -^-vo 10 fO TJ- O OOOOOOOOOO OOOOOOOO , IT) H M lOOO VO C* O <N rO tx O U~)VO LOCO N CO IO t^ i-^vo TJ- 10 u*> LO loco vo vo tx i-- -^f- 10 LOVO miomM w ^J-CON oooo -Sooooooooo ? ooooooooo 0000000 T3 T3 T3 *o tiro 'O'O'a'a'a'O'O'O {g'O-o'O'aTJ'a'a'D'o bo^a -o *c o -a -o -o c o * c 3 3 invO <* N ro " {7 f? fT N 600 APPLIED MECHANICS. z Q W O pearance of fracture. c o w c c c c v 1 -I 'I -s.- -s -i 1 1 - II 1 8 S -3 ~ 8 "*> S -: ll i- 1 II 1 ii| 1 11 Jrlilf Is 1 III o O *QMCO n^^r^:^ <> en tnO lamellar, slight lamination, o. o. stratified, slight lamination. 0, < >i >>Q Q>i>.> 4)X W( y >.>>>, t^.^^ ^Jsj J^^J^ 'C-^CC * -^ -^ -^ -^ -^ >- >,>M Q >,>, >;>, Ji^ J!J<1 ^!^! s C/3t/3 C7)C/2C/3 bnc/jfefe C/2C/3C/2C/2 C/2t//t> C/2C/) t/2{/5 C/3C/3 W i J J Vj r^ r^oo' vd 4- c^oo* r^vo" od M 't^ M >o o u^oo' M o <* M iovo * g rt a W ^ o o-2 c'S ScTScTS^c? S^cTcT cT^^JT Rff'g 00 m * ao t^ * Tt-t^ ll ' .s ^ i ^ M ON *5> ^ f^ t^ m ui c/) 1 53 S u H8 *B ^^ in \T) too vo in 10 in mvo in om^nm ininin III MI !f SH n ^ 1 ^ s 1 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- -^j-vo ^"jr as^? ^5- e/>.2 J5 1? ^ ^ ^ H [: a .. | ^'ct^SS^!^^ i^Rcgo? cg^cSocT 82^ JT? ^S? ^? T "3.1 EC 3 <fl ^ o .l 1 JllillSS IH? HH S?? 1H ?a 11 fan |H W ,HHMH HMHM M M H MMM C O J c .2 | .2 | 8 p l* u J5 u ^ ^ d d 6 6 d en T3 -O &QT3 T3 Jg -Q s g s U J U J S1315 en' ^ 03 |-S2 c c H li ^JS^OCUOJ f^tyiH^ pit/jH^ t>^^ >s* > N i > N 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' * y-y ^? 5 " ? " li ? R ' ^ S^ a o 1 ^V o ;\ 5^ a \ M ^^ tx^ ' ocf^ * W^^O^O\ M^ C^ O * fO d ^ vo" s ^.- ^- ^.- 2 ' ct S S? J? ' -V S -?" gjl "0 y-*M Tj-^O-^-OOOOOOv * M 8 ** V -i3 H'M N o mrom-frioin s 5 23 * s* < r 5 S u -C ^- -^- iO u"> VO ^ VO t^. OO OO s- s Q V e is tJfOlO OOrOlOMVONOOM UMW 4-MOiooovdr-.oo' * 00 11 a 11 g .ff ff ^ff-yasjj'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? m2^5-Rv2 ^;J 00 O J CA M rt 5 S H oo oooooooo 10^- 5^0-^^SooxoO 2?C>ON f>.^J-CI N rOONVO tx ^oooo 2"^ c?c?&o 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 9-75 i/4 .242 2 1309 F A A 44 6 44 44 44 .242 2 1310 F A A it 6 10.50 44 44 .242 2 13" F A A 11 6 4 ' 44 44 .249 2 1312 F A A if 6 11.25 44 44 .244 2 1313 F A A " 6 " 44 " 44 -243 2 i3H F A A it 6 10.49 44 tt** .248 2 1315 F A A " 6 " 44 44 .242 2 1316 F A A i| 6 11.27 tt " 44 .244 2 1317 | F A A 44 6 44 44 44 44 .246 2 1318 F B 1 B 2 6 12.01 44 it 44 245 2 1319 G B B " 6 " 44 " 44 .240 2 1320 G B B 2* 6 12.76 44 " ' 4 243 2 1321 G B B 14 6 44 44 " 44 243 2 1322 H D D 2} 6 I3-5I 44 44 tt 245 2 1323 : H D C " 6 " 44 " tt 247 2 1324 F A A 1* 6 11.26 44 " T* J .248 2 1325 F A A 44 6 " 44 44 " 245 2 1326 G B B 2 6 12. OO 44 " 4 ' .241 2 1327 G B B 44 6 44 44 44 .242 2 1328 1 G B C 2* 6 12.76 " it tt .241 2 1329 ! G C C 14 6 " 44 44 44 .242 2 '33 L C D 2* 6 33-50 44 44 44 .248 2 I33 t H C D 44 6 44 it " it .246 2 1332 H D E 2f 6 14.25 u 44 .248 2 1333 H D E " 6 44 44 44 44 .248 2 1334 M E E a* 6 15.00 ti ti 44 243 2 1335 .... " 6 " it 44 44 245 2 1336 H C C 2f 5 I3-I3 <t it 44 238 2 1337 L C C " 5 44 44 44 it .252 2 1338 G B B 2 6 12.00 44 44 tf *i .238 2 J 339 F B B 11 6 11 44 44 44 .248 2 1340 G B C 2* . 6 12-75 44 ii " .240 2 i34i G C C 11 6 44 it it M .242 2 1342 L C D 2* 6 I3-5I 44 4i it .250 2 1343 L D D " 6 " " it " .250 2 TABULATION OF RIVETED JOINTS. 603 RIVETED BUTT-JOINTS. PLATE. Sectional Area of Plate. Bearing Surface of Rivets. Shear- ing Area of Rivets. Tensile Strength of Plate per Sq. In. Maximum Stress on Joint per Sq. In. Effi- ciency of Joint. Tension on Gross Section of Plate. Tension on Net. Section of Plate. Comp. on Bearing Surface of Rivets. Shear- ing of Rivets. Gross. Net. sq. in. 2.360 sq. in. 1-452 sq in. sq. in. .908 3.682 Ibs. 61740 Ibs. 41690 Ibs. 67770 Ibs. 108370 Ibs. 26720 67-5 2.360 1.452 .908 ! 3.682 61740 42180 68560 109640 27040 68.3 2 541 1.634 .907 3.682 61740 42540 66160 119180 v 29360 68.9 2.6l5 1.681 934 3.682 61740 43 1 7 67160 119810 30660 69.9 2-745 1.830 9*5 3-682 61740 44920 67380 134750 33490 72.8 2 -739 1.827 .912 3.682 61740 44520 66750 133720 33120 72.1 2.602 1.486 i . 116 5.300 61740 40700 71270 94890 19980 65.9 2.541 1-452 1.089 5-300 61740 40000 70000 93330 19180 64.8 2.750 1-652 1.098 5.300 61740 40980 68210 102630 21260 66.4 2.772 1-665 i . 107 5.300 61740 4 I 43 68980 103750 21670 67.1 2 942 1.840 I. 102 5-300 61740 42180 67450 112610 23420 68.3 2.882 1.802 I.oSo 5.300 62660 43000 68770 "475 23380 68.6 3.100 2.007 1.093 5.300 62660 43060 66520 122140 25190 68.7 3.100 2.007 1.093 5.300 62660 44030 68000 124880 2575 70.3 3-3 10 2.207 I.I03 5.300 59180 43040 64540 129150 26880 72.7 3-337 2.225 I . 112 5-3 59180 43810 65700 131460 27580 74-o 2.792 1.490 I .302 7.216 61740 38650 72480 82870 14950 62.0 2.756 1.470 1.286 7.216 61740 39430 73930 84510 15010 63-8 2.892 1.627 1.26 5 7.216 62660 40340 71700 92210 16170 64-4 2.909 1-638 I.27I 7.216 62660 41280 73310 94480 16640 65-9 3-075 1.810 1.265 7.216 62660 42290 71850 102810 18020 67-5 3.088 1.817 I.27I 7.216 62660 4275 72650 103860 18290 68.2 3.343 2.046 1.302 7.216 61470 43100 70530 110830 20000 70.1 3-321 2.029 I.2QI 7.216 59180 4*45 67840 106620 19080 70.0 3-534 2.232 1.302 7.216 59180 41820 66210 113500 20480 70-7 3-534 2.232 1.302 7.216 59 l8 o 42760 67710 116070 20940 72-3 3- 6 45 2.369 1.276 7 216 58170 44650 68700 127550 22550 76.8 3- 6 75 . 5.125 2.389 2.084 1.286 I.O4T 7.216 6.013 64170 59*80 43 5o 41310 66230 61960 123030 124020 21930 21470 67.1 69.8 3 39 2.206 I.I03 6.013 61470 42000 62990 125990 23IIO 68.3 2 .8 b 6 1.428 1.428 9-425 62660 40620 81230 81230 I23IO 64.8 2.976 1.488 1.488 9-425 61740 36290 72580 72580 11460 58.8 3.060 i .620 1.440 9-425 62660 38660 73020 82150 "550 61.7 3.088 1.636 1.452 9-425 62660 38000 71730 80820 12450 60.6 3.378 1.878 1.500 9-425 61470 37800 68000 85130 13550 61.5 3-375 1-875 1.500 9-425 61470 39000 70200 87750 13970 63-4 604 APPLIED MECHANICS. TABULATION OF SINGLE- i" STEEL No. of Test. Sheet Letters. Pitch. No. of Rivets. Width of Joint. Nominal Thick- ness. Size of Rivets and Holes. Actual Thick- ness of Plate. Lap. Plate. Covers. Plate. Covers. in. in. in. in. in. in. in. 1344 L D E *t 6 14.24 1/4 3/i 6 lf*i .250 2 J 345 H D E 14 6 " " 44 44 .247 2 1346 I E E 2* 6 15.00 " 44 44 .251 2 1347 I E E " 6 " " " * .252 2 1348 H c C 2f 5 13-13 " it it .244 2 J 349 G C C 44 5 " " 44 * 239 2 I35<> N D D a* 5 13-77 " it M .250 2 I35i N D D 44 5 " " 44 44 .249 2 1352 H D D 2* 5 14-39 " 44 II 247 2 1353 H E E ii 5 (i ii ii || .248 2 T 354 I E E 3 5 . 15.00 (i 14 II .252 2 J 355* I E E " 5 " 44 44 (I 25 1 2 STEEL I35 6 A I I if 6 9-75 3/8 i/4 A*i -365 2 1357 A I I 44 6 44 44 44 44 .364 2 1358 A I J i* 6 10.49 K 44 44 365 2 1359 A J J 44 6 " 44 44 44 .366 2 1360 A I J 44 6 10.50 44 11 tt*l .366 2 1361 A I I 44 6 " 41 11 44 367 2 1362 A J J i* 6 ".25 44 44 44 .366 2 *3>3 A J J 44 6 44 (4 41 44 365 2 1364 B K K 2 6 12.00 II K 44 .388 2 1^65 B K K II 6 44 44 44 44 39 2 1366 C K K 2* 6 12.76 44 44 41 .367 2 1367 B K L 44 6 44 44 11 44 387 2 1368 A J J it 6 11.25 44 44 13*1 -369 2 1369 A J J 44 6 14 11 It 44 .366 2 1370 B K K 2 6 12. OO 44 It 44 -389 2 i37i B J J 44 6 " 44 II 41 388 2 i37 2 B L L at 6 "-77 44 " 44 -385 2 1373 C K L 44 6 41 It (I ii -367 2 1374 D H N 2i 6 13.5 44 11 44 .376 2 1375 E N N " 6 11 K 14 44 .380 2 1376 D L M 2| 6 14.23 u 11 ii 383 2 1377 H L M 44 6 " " " it 37i 2 * Fractured two outside sections of plate at each edge along line TABULATION OF RIVETED JOINTS 605 RIVETED BUTT-JOINTS Continued. PLATE Continued. Sectional Area of Plate. Bearing Surface of Rivets. Shear- ing Area of Rivets. Tensile Strength of Plate per Sq. In. Maximum Stress on Joint per Sq. In. Effi- ciency of Joint. . Tension on Gross Section of Plate. Tension on Net Section of Plate. Comp. on Bearing Surface of Rivets. Shear- ing of Rivets. Gross. Net. sq. in. sq. in. sq. in. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. 3-56o 2.060 1.500 9-425 61470 3913 67640 92870 14780 63.6 3-5 20 2.038 1.482 9-425 59180 - 40450 69860 96070 15110 68.3 3-765 2.259 1.506 9-425 60480 4359 72640 108960 17410 72.1 3.780 2.268 1.512 9-425 60480 41420 69030 103540 16610 68.5 3.204 1.984 i .220 7-854 59180 38700 - 62490 101620 !5790 65.4 3-i3i 1.936 1-195 7-854 62660 42890 69370 112380 17100 68. 4 3-442 2. 192 1.250 7-854 5374 42960 67460 118300 18830 7 j.i 3.426 2.181 1-245 7-854 55740 41780 65640 114980 18230 74-9 3-554 2 -3!9 1-235 7-854 59i8o 435 10 66690 125220 19690 73-5 3-569 2.329 1.240 7-854 59i8o 4383 67170 126160 19920 74-1 3.780 2.520 1.260 7-854 7 Rcj 60480 60480 44580 66870 66610 133730 2I 450 j 73.7 3' 7^5 7 54 444^0 133230 21290 73-4 PLATE. 3-559 2.190 1.369 3.682 54260 40460 65740 105170 39100 74 .6 3-549 2.184 1-365 3.682 54260 39420 64060 102490 38000 72.6 3-829 2.460 1.369 3.682 54260 39780 61910 111250 41360 73-3 3-843 2.471 1.372 3.682 54260 39060 60740 109400 40770 72.0 3-843 2.196 1.647 5-300 54260 37000 64750 86330 26830 68.2 3-854 2. 2O2 1.652 5-300 54260 37050 64840 86430 26940 68.3 4.118 2.471 1.647 5.30 54260 37450 62400 93620 29090 69.0 4.106 2.464 1.642 5-300 54260 38040 63390 95130 .29470 70.1 4-656 2.910 1.746 5-3oo 59730 41820 66910 111510 3673 70.0 4.680 2.925 1-755 5-3oo 59730 42000 67200 II2OOO 37090 70-3 4.683 3-031 1.652 5.300 57870 41040 63410 116340 36260 70.9 4-938 3-197 1.741 5-3 59730 40910 63180 116030 38110 68.5 4-i5i 2.214 J-937 7.216 54260 35000 65620 75000 20130 64-5 4.114 2.192 1.922 7.2,6 54260 34180 64140 73150 19480 63.0 4 .668 2.6 2 6 2.042 7 216 59730 36870 65540 84280 23850 61.7 4.656 2 619 2.037 7.216 59730 38940 69220 SgOOO 25160 65-2 4.916 2.895 2.021 7.216 59730 38730 65770 94210 26390 64.8 4.672 2-745 1.927 7.216 57870 39010 65660 94580 25260 67.4 5.076 3.102 1-974 7.216 53730 37960 62120 97620 26700 70.6 5-130 3-^35 1-995 7.216 5834 39810 65140 102360 28300 68.2 5-450 3-439 2. Oil 7.216 53730 38920 61670 105470 29390 72-4 5.290 3-343 1.948 7.216 56670 39870 6311O 108260 29230 70.4 of riveting ; the two middle sections sheared in front ^f rivets. 6o6 APPLIED MECHANICS. TABULATION OF SINGLE- S'' STEEL Sheet Letters. Nominal Thickness. No. of Test. Pitch. No. of Riv- ets. Width of Joint. Size of Rivets and Holes. Actual Thick- ness of Plate. Lap. Plate. Covers. Plate. Covers. 1378 D M M in. 2* .6 in. 15.00 in. 3/8 in. i/4 in. li** in. 383 in. 2 1379* 1 D M M " 6 " it " " .385 2 1380 B J K 2 6 12.00 " " if* ' 388 2 1381 E K K " 6 " " " " .381 2 1382 B K K 2* 6 12.75 " " " .388 2 i 3 8 3 t E G K " 6 " 11 11 " 383 2 1384 F H H 2 6 13-49 M it " .381 2 1385 E N N " 6 " " (i * .380 2 1386* H L L 2f 6 14.25 " it " .368 2 1387 G L M " 6 tt " " " 365 2 1388 D M M ** 6 15.00 11 " " 385 2 1389 D M M " 6 " " " " .386 2 1390 C G G 2| 5 13.12 " " " 372 2 MQI C H L " 5 " " " M 369 2 1302 C L L 2j 5 13-75 " " 374 2 1393 C L N " 5 " K " " 372 2 J 394 D I M 2| 5 M-39 tl it " .386 2 1395 D M M " 5 " i " 1C " 383 2 * Test discontinued soon after passing maximum load. + Test discontinued at maximum load. $ Test discontinued after passing maximum load. Test discontinued before fracture was complete. TABULATION OF RIVETED JOINTS. 607 RIVETED BUTT-JOINTS Continued. PLATE Continued. Sectional Area Maximum Stress on Joint per of Plate. Square Inch. Tensile Bearing Shear- Strength Effi- Surface of ing Area of of Plate per Tension Compres- Tension sion on ciency of Plate. Covers. Rivets. Rivets. Square Inch. on Gross Section on Net Section Bearing Surface Shear- ing of Joint. oi Plate. of Plate. of Rivets. Rivets. sq. in. sq. in. sq. in. sq. in. Ibs. Ibs. Ibs. tos. Ibs. 5 745 3-734 2. Oil 7.216 53730 40560 62400 115860 32290 75-5 5 775 3-754 2.021 7.216 5373 40700 62620 116280 3 2 57o 75-7 4656 2.328 2.328 9-425 5973 345 69010 69010 17050 57-8 4-57 2 2.286 2.286 9.425 58340 33440 66880 66880 16220 57-3 4-947 2.619 2.328 9-425 59730 35590 67230 75640 18680 59-6 4.883 2-585 2.298 9-425 58340 34730 65610 73800 17990 59-5 5.140 2.854 2.286 9-425 54290 3549 63930 79810 19360 65.4 5126 2.846 2.280 9-425 58340 35840 64550 i 80570 19490 61.4 5-244 3.036 2.208 9-425 56670 37010 63930 87910 20590 65-3 5-205 3-oi5 2.190 9.4 2 5 53840 36750 63450 87350 20300 68.2 5-775 3-465 2.310 9-425 53730 3749 62480 937io 22970 69.8 5-79 3-474 2.3l6 9-425 53730 3736o 62260 93390 21890 69.5 4.881 3-021 1. 860 7.854 57870 39000 63010 102340 24240 67-4 4.841 2.996 1.845 7-854 57870 39520 63850 103690 24360 68.3 5-143 3-273 I.SjO 7-854 57870 39840 62590 109540 26080 68.9 5.111 3251 1.860 7-854 57870 40420 63550 111070 26310 69.8 5o55 3-625 1.930 7-854 53730 3934 60290 113240 37830 73-2 5-496 3.58i I -9 I S 7-854 5373 40300 eiseo 115660 28200 75-o 6O8 APPLIED MECHANICS. SINGLE-RIVETED BUTT-JOINTS, STEEL PLATE. DESCRIPTION OF TESTS AND DISCUSSION OF RESULTS. " The following tests complete a series of two hundred and sixteen single-riveted butt-joints in steel plates, in which the thickness of the plates ranged from J" to f " , and the size of the rivets from -fa" to ly 3 ^" diameter. The plates were annealed after shearing to size, the edges opposite the joint milled to the finished width ; the holes were drilled and rivets machine-driven. Iron rivets were used throughout, except in some of the f x/ joints. Tensile tests of the plates and rivet-metal, together with the tests of the joints in " and f " plate, are contained in the Report of Tests of 1885, Senate Document No. 36, Forty-ninth Congress, first session. The tests herewith presented comprise the details and tab- ulation of joints in ", f", and f-" thickness of plate, a portion of which were tested hot. The gauged length in which elongations and sets were measured was 5"; 2\" each side of the centre line of the joint. During the progress of testing the same characteristics were displayed which were referred to in the previous report. The joints were very rigid under the early loads. This rigidity is overcome by loads which exceed the friction between the plate and covers, after which the stretching proceeded slowly with some fluctuations till elongation of the metal of the net section became general ; the metal under compression in front of the rivets yielding, also the rivets themselves. The behavior of joints in different thicknesses of plate is substantially the same, and an examination of the results shows that when exposed to similar conditions the strength per unit SIXGLE-RIVETED BUTT-JOINTS, STEEL PLATE. '609 of fractured metal is nearly the same, whether J" or J" plate is used. It will not be understood from this, however, that as a con- sequence the same efficiency may be obtained in different thicknesses of plate for single-riveted work, because it will be seen that certain essential conditions change as we approach the stronger joints in different thicknesses of plate. A riveted joint of the maximum efficiency should fracture the plate along the line of riveting, for it is clear that if failure occurs in any other manner, as by shearing the rivets or tear- ing out the plate in front of the rivet-holes, there remains an excess of strength along the line of riveting, or in other words along the net section of metal if in a single-riveted joint which has not been made use of ; but when fracture occurs, along the net section an excess of strength in other directions is immaterial. If the strength per unit of metal of the net section was con. stant, it would be a very simple matter to compute the effi- ciency of any joint, as it would merely be the ratio of the net to the gross areas of the plates. The tenacity of the net section, however, varies, and this variation extends over wide limits. In the present series there is an excess in strength of the net section over the strength of the tensile test-pieces in all joints. Special tables have been prepared showing this behavior. The efficiencies shown in Table No. I are obtained by divid- ing the tensile stress on the gross area of plate by the tensile strength of the plate as represented by the strength of the ten- sile test-strip, stating the values in per cent; of the latter. Table No. 2 exhibits the differences between the efficien- cies of the joints and the ratios of net to gross areas of plate. If the tenacity of net section remained constant per unit of APPLIED MECHANICS. area, the efficiencies in Table No. I would, as above explained, be identical with the ratios of net to gross areas of plate, and the values in this table reduced to zero. Table No. 3 shows the excess in strength of the net section of the joint over the strength of the tensile test-strip in per cent of the latter. Table No. 4 exhibits the compression on the bearing-surface of the rivets in connection with the excess in tensile strength of the net section of plate. Table No. I is valuable in showing at once the value of different joints wherein the pitch of the rivets and their diame- ters vary. It is seen there is considerable latitude allowed in the choice of rivets and pitch without materially changing the efficiency of the joint ; thus in J" plate, f" rivets (driven), if" pitch, 72.4 per cent efficiency, f" rivets (driven), i\" pitch, 73.3 per cent efficiency, f-" rivets (driven), af" pitch, 71.5 per cent efficiency, i" rivets (driven), 2-3-" pitch, 70.3 per cent efficiency, i" rivets (driven), 2^" pitch, 73.8 per cent efficiency, give nearly the same results. In these examples the ratios of net to gross areas of plate range from 60 to 67 per cent, while the rivet-areas range from .3067 square inch to .7854 square inch. The actual areas of net sections of plate and rivets are as follows : |" rivets. \" rivets. \" rivets. i" rivet*. Rivets. sq. in. . ^067 sq. in. .4418 sq. in. .6013 sq. in. .7854 Plate . . 1.486 2. 2O7 2.2^2 j 2.259 ( 2.319 SINGLE-RIVETED BUTT-JOINTS, STEEL PLATE. 6l I The areas of the rivets stand to each other as the following numbers : 100 144 .96 and the net areas of the plate to each other as 100 149 150 256 ( 152 1 1 5 6 From these illustrations it appears that to attain the same degree of efficiency in this quality of metal, although that efficiency is probably not the highest attainable, a fixed ratio between rivet metal and net section of plate is not essential. In \" plate with \" rivets the efficiencies of the joints tested cold are nearly constant over the range of pitches tested. The efficiencies and the ratio of net to gross areas of plate are as follows : Pitch. 4" 2" -4" 2i" per cent. per cent. per cent. per cent. Efficiency. . Ratio of areas . . 64.5 53-4 66.3 56.3 66.3 58.9 66.4 6l.I In this we have illustrated a case which, in passing from the widest pitch, having 61.1 per cent of the solid plate left, to the narrowest pitch, which had 53.4 per cent of the solid plate, the gain or excess in strength in the net section almost exactly compensated for the loss of metal. In Table No. 3 the average of all the joints shows the high- est per cent of excess of strength in the narrowest pitch, and a tendency to lose this excess as the pitch increases. Tests of detached grooved specimens show the same kind 612 APPLIED MECHANICS. of behavior, but as they are not subject to all the conditions found in a joint, the analogy does not extend very far. The maximum gain in strength on the net section, not for the time being regarding the hot joints, and disregarding the exceptionally high value of joint No. 1339, J-" plate, was 21.2 per cent, the minimum value 2.5 per cent of the tensile test- :strip. In other forms of joints, and with punched holes in both iron and steel plate, illustrations are numerous in which there have been large deficiencies, the metal of the net section fall- ing far below the strength of the plate. It is believed to have been amply shown that increasing the net width diminishes the apparent tenacity of the plate, although other influences may tend to counteract this tendency in some joints. In order to compare the excess in strength of one thickness of plate with another having the same net widths, we have the following table, rejecting those joints that failed otherwise than along the line of riveting in making these averages: Thickness of Plate. Width of plate between rivet-holes. i" Ii" i}" if" I*" If" If" i*" * 2" y i-' P. ct. 16.7 18.4 16.7 17.7 11.4 P. ct. 12.6 13-7 14-3 16.3 JS-i P. ct. 11.4 12.7 9-3 14.2 13.8 P. ct. 12.0 13-5 10-7 14-5 I4.I P. Ct. 13-4 I 4 .6 Q.I I 4 .6 7.6 P. Ct. 8.9 12.9 8.8 12.7 IT. 8 P. ct. "5 9.0 8.2 9-9 IO.O P. Ct. 13-1 13.6 12.2 0.8 10. i ii. 8 P. Ct. 10.6 3-5 ' i' !/ Average of all thick- nesses 16.2 14.4 12.3 12. Q II. 9 II. 9-7 7.0 The excess in strength is generally well maintained in each of the several thicknesses, and were it possible to retain the same ratio of net to gross areas of plate, and at the same time SINGLE-RIVETED BUTT-JOINTS, STEEL PLATE. 613 equal net widths between rivets, it would seem from this point of view feasible to obtain the same degree of efficiency in thick as in thin plates. The following causes, however, tend to prevent such a con summation. For equal net widths thick plates require larger rivets to avoid shearing than thin ones, the diameters of the rivets being somewhat increased for this cause, and again because it has become necessary to increase the metal of the net section in order to retain a suitable ratio of net to gross areas of plate. There results from these considerations such an increase in net width of plate that the excess in strength displayed by narrower sections is lost, and consequently the result is a joint of lower efficiency. The data relating to the influence of compression on the bearing-surface of the rivets, on the tensile strength of the plate, as shown by Table No. 4 are more or less conflicting. However, in the J" plate, in which the most intense pressures are found, there is seen a pronounced increase in tensile strength as the pressures diminish in intensity. It is probable that the effects of intense compression would be more conspicuous in a less ductile metal, or one in which the ductility had been impaired by punched holes or otherwise. A number of joints were tested at temperatures ranging between 200 and 703 Fahr. The heating was done after the joints were in position for testing, by means of Bunsen burners, arranged in a row par- allel to and under the line of riveting. The temperature was determined with a mercurial ther- mometer, the bulb of which was immersed in a bath of oil, contained in a pocket drilled in the middle rivet of the joint. When at the required temperature the thermometer was removed from the joint, a dowel was driven into the pocket to 6 14 APPLIED MECHANICS. compensate for the metal of the rivet which had been removed by the drill, and then loads applied and gradually increased up to the time of rupture. Three joints, Nos. 1423, 1426, and 1430, were tested with- out dowels in the oil-pockets. The method of heating was to raise the temperature of the joint, as shown by the thermometer, a few degrees above the temperature at which the test was made, shut off the gas- burners, and allow the temperature to fall to the required limit. The temperature fell slowly, draughts of cold air being excluded from the under side of the joint by the hood which covered the gas-burners ; the upper side and edges of the joint were covered with fine dry coal-ashes. The results show an increase in tensile strength when heated over the duplicate cold joints at each temperature except 200 Fahr. From 200 there was a gain in strength up to 300, when the resistance fell off some at 350, increased again at 400, and reached the maximum effect observed at 500 Fahr. ; from this point the strength fell rapidly at 600 and 700. In per cent of the cold joint there was a loss at 200 of 3.2 per cent, the average of three joints ; at 500 the gain was 22.6 per cent, the average of four joints. The maximum and mini- mum joints at this temperature showed gains of 27.6 per cent, and 18.3 per cent, respectively. The highest tensile strength on the net section of plate was found in joint No. 1433, tested at 500 Fahr., where 81050 pounds per square inch was reached against a strength of 58000 pounds per square inch in the cold tensile test-strip. The hot joints showed less ductility than the cold ones, those tested at 200 Fahr. not being exempt from this behav- ior, although there was no near approach to brittleness in any. Three joints, Nos. 1418, 1420, and 1424, were heated; SINGLE-RIVETED BUTT-JOINTS, STEEL PLATE. 615 strained when hot with loads exceeding the ultimate strength of their duplicate cold joints; the loads were released, and after having cooled to the temperature of the testing-room (No. 1424 cooled to 150 Fahr.) were tested to rupture, and were found to have retained substantially the strength due their temperature when hot. In order to ascertain that the time intervening between hot straining and final rupture did not contribute towards the ele- vation in strength, joint No. 1434 was strained in a similar manner with a load approaching rupture, after which a period of rest was allowed and then ruptured without material gain in strength. A peculiarity of the joints fractured at 400 and higher tem- peratures was the comparatively smooth surface of the frac- tured sections, and which took place in planes making angles of about 50 with the rolled surface of the plate. The shearing-strength of the iron rivets was also increased by an elevation of temperature. The rivets in joint No. 1410 at the temperature of 350 sheared at 43060 pounds per square inch, while in the dupli- cate cold joint No. 1411 they sheared at 38530 pounds per square inch, and the rivets in pint No. 1398 at 300 Fahr. were loaded with 46820 pounds per square inch and did not shear. Other examples, where some of the rivets sheared and the plate fractured in part, showed corresponding gains in shearing- strength. The almost entire absence of granular fractures in these tests is a feature too important to pass by without special men tion." 6i6 APPLIED MECHANICS. TABULATION OF SINGLE- STEEL PLATE. No. of Test. Sheet Letters. Pitch. No. of Rivets. Width of Joint. Nominal Thick- ness. Size of Rivets and Holes. Actual Thick- ness of Plate. Lap. Plate. Covers. Plate. Covers. in. in. in. in. in. in. in. 1396 R R R '* 6 10.50 1/2 5/i6 tt** .481 2 J397 R R R " 6 " " " " .484 2 1398 R R R t 6 11.25 " " " 484 2 1399 R R R " 6 " 11 it 11 -483 2 1400 R R R 2 6 13.00 " ti .486 2 1401 R S T " 6 " " " 44 -483 2 1402 R R R 4 6 11.25 " " ii*l .481 2 1403 R R R " 6 " " 11 " .486 2 1404 R R R 2 6 12. OO ' " " .486 2 1405 R S S " 6 " " 44 " 487 2 1406 S S S a* 6 12.75 " M 11 .470 2 1407 S S S " 6 " n " ti .471 2 1408 T Q Q at 6 I3-SO " it it .486 2 1409 T Q Q ' 6 " 11 " .482 2 1410 T af 6 14.25 44 it " .481 2 1411 T o " 6 It it tt " .485 2 1412 R R R 2 6 12.00 " " IS* i .484 2 1413 R R R " 6 " it " 11 .481 2 14.4 S S S at 6 12-75 44 it 11 472 2 4'5 S S S " 6 11 *' 11 " .468 2 ,416 S Q Q at 6 JS-SO " " " .468 2 1417 T Q Q " 6 u 44 it it .482 2 1418 T a| 6 14-25 " " it .481 2 14-9 T o " 6 " 11 ti 11 .482 2 1420 U p p 2* 6 15.00 11 " it 479 2 1421 u p p " 6 " " M " .483 2 1422 S S S a* 5 I3-I3 " it 11 .469 2 1423 S S S " 5 " l< " it 473 2 1424 T a* 5 13-75 " 44 ' .484 2 1425 T o o " 5 " " " " -483 2 1426 S S S at 6 12.75 11 " xA*ii 474 2 1427 R S S ** 6 " " ct 44 475 2 1428 T Q at 6 I3-50 " " 11 479 2 1429 S Q " 6 " " 14 " 465 2 '43 U o o f 6 14-25 " 44 44 -484 2 M3i U o " 6 " 11 " " 483 2 I TABULATION OF RIVETED JOINTS. 6l 7 RIVETED BUTT-JOINTS. STEEL PLATE. Sectional Area of Plate. Bearing Surface n f Shear- ing Tensile Strength of Plate Max. Stress on Joint per Sq. In. J| |.E| c ^ ^o !- s !s ^ I'ga Gross. Net. OI Rivets. Area of Rivets. per Sq. In. |c|l ^ c o 5"! C-N O Q> all 5 J3 1? T*'c c/) C/J O "Q 11 sq in. sq. in. sq. in. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. 5-051 2.886 2.165 5-3 01 57180 37750 66070 88080 35970 66.1 2OO 5.082 2.904 2.178 5-3* 57180 38980 68220 90960 37370 68.1 5-445 3.267 2.178 5-301 57180 45560. 75960 113960 46820 79-6 3 00 5-439 3.266 2-173 5-3o: 57180 39260 65400 98290 40290 68.6 5.842 3-655 2.187 5-301 57180 37000 59*40 98830 40770 64-7 S-79 6 3.622 2.174 5-3 01 57180 39420 63080 105100 43100 68 9 35 5-416 2.891 2-525 7.216 57180 36890 69110 79*3 27690 64-5 5-477 2.926 2-551 7.216 57180 38250 71600 81730 29030 66.9 250 5-832 3.281 2-55* 7.216 57180 379*0 67380 86670 30640 66.3 5-854 3-297 2-557 7.216 57180 43730 77650 IOOI2O 3548o 76-4 300 5-997 3-529 2.468 7.216 59050 44790 76110 108830 37220 76.0 400 6.010 3-537 2-473 7.216 59050 39210 66630 953*0 32660 66.3 6.561 4.010 2-55 1 7.216 60000 39850 65210 102500 36230 66.4 6.512 3.982 2-530 7.216 60000 46610 76220 i 19980 42060 77 .6 500 6.859 4 334 2-525 7.216 60000 45300 71690 123050 43060 75-5 350 6.916 4-37 2.546 7.216 60000 40050 63390 108800 38530 66.7 5-813 2.909 2.904 9-425 57*80 35920 71770 71900 22150 62.8 250 5-772 2.886 2.886 9-425 57*8o 34390 68780 68780 21060 60. i 6.023 3-i9i 2.832 9-425 59050 35000 66020 7443 22360 59-2 5-967 3-159 2.808 9-425 59050 40250 76030 85540 25480 68.1 300 6.327 3-5 T 9 2.808 9-425 59050 34660 62320 78090 23160 60.3 200 6.512 3 . 620 2.892 9-425 60000 36950 66480 83220 25530 61.5 6.859 3-973 2.886 9-425 60000 437*o 75460 103880 31810 72.8 (*) 6.883 3-99 1 2.892 9-425 60000 38720 66770 92*50 28270 64-5 7-194 4-320 2-874 9-425 58000 44840 74670 112250 34230 77-3 (t) 7- 2 45 4-347 2.898 9-425 58000 38740 64570 96850 2 97 Po 66.9 6. .67 3 822 2-345 7-854 59050 39730 64110 104490 3T200 67.. 6 220 3-855 2-365 7-854 59050 45420 73250 119450 35840 76.9 400 6.660 4.240 2.420 7-854 60000 48950 76890 137110 41510 81.5 (*) 6 632 4.217 2-4*5 7-854 60000 40600 63860 111490 34280 67.6 6-053 2-853 3.200 11.928 59050 35070 74410 66340 18630 59-3 3 00 6.042 2.836 3.206 11.928 59050 30420 64810 5733 15410 51.5 6.471 3-238 3-233 11.928 60000 40330 80620 80730 21880 67-2 350 6.278 3-*39 3-139 11.928 595 33420 66840 66840 *759 56-5 6.897 3.620 3-277 11.928 58000 36390 65150 76590 21040 62.7 700 6.852 3.632 3.260 11.928 58000 3366o 63870 71160 1945 58.0 * Strained while at temperature of 400 Fahr i Strained while at temperature of 500 Fahr. * Strained while at temperature of 500 Fahr ,, and allowed to cool before rupture. , and allowed to cool before rupture. ,, then cooled to 150 Fahr., and ruptured. 6i8 APPLIED MECHANICS. TABULATION OF SINGLE- STEEL PLATE- Continued. Sheet Nominal Letters. Thickness. No. of Test. 5 itch. No. of Riv- ets. Width of Joint. Size of Rivets and Holes. Actual Thick- ness of Plate. Lap. Plate. Covers. Plate. Covers. 1432 u P P in. ai 6 in. 15.00 in. 1/2 in. 5/i6 in. iA* il in. .484 in. 2 M33 U P P 41 6 41 44 14 44 .481 2 M34 R S Q 2g 5 13-13 44 41 44 .472 2 *435 S S S " 5 " 44 44 44 475 2 1436 T o *! 5 13-75 44 44 44 .482 2 1437 T 11 S " 44 " 11 .482 2 1438 U P P aj 5 14-38 it 44 f .484 2 1439 U P P it 5 it 44 <t 44 .485 2 1440 U P P 3 5 15.00 II 44 44 .482 2 1441 U P P it 5 44 44 " 44 483 2 1442 u P P 34 5 15.68 11 41 41 483 2 M43 u P P " 5 " 41 " 41 .484 2 1444 V E E I* 6 11.25 5/8 3/8 11*5 .621 2 1445 V B E " 6 44 it 44 41 .624 2 1446 V E E 2 6 12. OO 44 44 44 .616 2 M47 V E E 11 6 44 44 44 ** .624 2 1448 V E E 4 6 12 -75 " 44 *' .621 2 M49 V E E 11 6 " 44 44 44 .624 2 145 w F G si 6 13-50 44 4 41 .610 2 MS 1 w G G " 6 44 44 44 44 .611 2 1452 V E E 2 6 12.00 44 44 if* i .624 2 I4S3 V E E '* 6 41 " " ** .620 2 M54 V E E ai 6 "75 44 44 4< .622 1 1455 V E E " 6 " tt 44 41 .618 2 I45 6 w G F *i 6 I3.50 44 ' tt .612 2 1457 w G G " 6 " 4< 44 44 .611 2 1458 w I I a| 6 M.25 .1 r tt 4 .610 2 1459 w H H ** 6 44 11 44 44 .608 2 1460 X I I ai 6 15.00 it 44 44 .617 2 1461 X J I " 6 " ti 44 44 .618 2 1462 V F F i 5 I3.I3 it 44 it - .630 2 I4 6 3 w F F lk 5 14 44 44 * .608 2 1464 V E E ai 6 12-75 41 tt iA* i* .624 2 14^5 V D E " 6 44 " 44 44 .623 2 1466 w G G aj 6 I3-50 it 44 44 .613 2 1467 w N,V G " 6 " " " " .606 2 TABULATION OF RIVETED JOINTS. 619 RIVETED BUTT-JOINTS Continued. STEEL PLATE Continutd. * Maximum Stress on Joint per c c ._ 4, Sectional Area 3 Square Inch. ^ O w of Plate. h*/i O* "o >~"c3 Bearing Shear- Sf*> > Ofc, at ,_ Surface of ing Area of 4) u t/3 < O &g ! "o > >- 22 Rivets. Rivets. -Sj3 i rt j * lra-3* c 5 c AJ w w . Gross. Net. ||c 8 gO-sS ' 5 P.** i- > js i I. si h H u C/5 W H sq. in sq. in. sq.in. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. 7.270 4-43 3-227 11.928 58000 36140 65000 81430 22030 62.3 7-215 3.968 3-247 11.928 58000 44490 81050 99040 26960 76.7 500 6.193 3.538 2-655 9.940 59050 36670 64190 85540 22850 62 .0 6.232 3.560 2.672 9.940 59050 36200 63370 84420 22690 61.2 6.632 3.921 2.711 9.940 60000 4243 71610 103580 28250 70.5 600 6.632 3-921 2.711 9-940 60000 38720 65440 94730 25840 64.5 6.965 4-243 2.722 9.940 58000 46630 76550 119320 32680 80.3 500 6.974 4.246 2.728 9.940 58000 38900 63890 99400 27290 67.0 7.230 4-519 2.711 9.940 58000 39180 62690 104500 28500 67-5 200 7.250 4.528 2.722 9.940 58000 40360 65060 108220 29630 70.0 7-564 4-837 2.717 9-940 57410 38570 60450 107610 29410 67.2 7-565 4-843 2.722 9.940 57410 3^160 61170 108830 29800 68.2 6.986 3.726 3.260 7.216 55000 3375 63280 72330 32670 60. i 7.020 3-744 3.276 7.216 55000 3453 64740 74000 3359 62.7 7-393 4.158 3-234 7.216 55000 36760 65340 84010 37650 66.0 7.488 4.212 3.276 7.216 55000 35120 62440 80280 36440 63-8 7.918 4-658 3.260 7.216 55000 41930 71270 101840 46010 76.2 300 7-956 4.680 3.276 7.216 55000 36800 62560 89370 40570 66.9 8.241 5-039 3.202 7.216 57290 39320 64290 101 180 44900 68.6 400 8.249 5.042 3-207 7.216 57290 36850 60290 94790 42130 64-3 600 7.488 3-744 3-744 9-425 55000 32080 64150 64150 25480 58-3 7.440 3.720 3.720 9-425 55000 32060 64110 64110 25300 58.3 7-93 1 4.199 .3-732 9-425 55000 34120 64440 72510 28710 60.0 7.880 4.172 3.708 9-425 55000 34000 64220 72250 28420 61.8 8.262 4-59 3.672 9-425 57290 36490 65680 82110 32000 63.6 8.249 4-S83 3.666 9-425 57290 36020 64830 81040 31520 62.8 8 662 5.002 3.660 9-425 57290 37720 65310 89260 38490 65.8 8.664 5.016 3-648 9-425 57290 37540 64850 89170 345'o 65-5 9-255 5-553 S-? 02 9-425 55940 37300 62160 9325 36630 66.6 9.282 5-574 3.708 9-425 55940 37000 61610 92620 36440 66.1 8.259 5.109 3-I50 7-854 55000 3578o 57840 93810 37620 65.0 7-965 4-925 3 040 7-854 57290 36960 59770 96840 37500 64-5 7-950 3.738 4.212 11.928 55000 31000 .66130 58690 20720 56-5 7-949 3-744 4-205 ii .928 55000 31090 66020 58780 20720 56-5 8.269 4- I 3 I 4-138 i i . 928 57290 33*5 66350 66240 22980 57-8 8.181 4.090 4.091 11.928 55940 33240 66250 66240 22720 58.0 620 APPLIED MECHANICS. TABULATION OF SINGLED STEEL PLATE Continued. No. of Test. Sheet Letters. Pitch. No. of Rivets. Width of Joint. Nominal Thickness. Size of Rivets and Holes. Actual Thick- ness of Plate. Lap. Plate Covers. Plate. Covers. in. in. in. in. in. in. in. 1468 W H I at 6 M-25 5/8 3/8 iA * X B -613 2 1469 W I N&" " 6 " 44 44 * .609 2 1470 X J I 2i 6 15.00 44 44 44 .619 2 1471 X J I " 6 44 44 44 ii .616 2 1472 V K at 5 13-13 44 44 it .628 2 1473 : W F F 44 5 " ii 44 it .609 2 i 474 i W H D at 5 13-75 44 41 it .609 2 1475 ; w G 5 1 .610 2 1476 ; W I I 2* 5 14.38 it 44 44 .610 2 1477 W I I 41 5 44 ii 44 44 .6oy 2 1478 X I J 3 5 15.00 44 44 44 .616 2 1479 X I J 14 5 44 44 44 44 .623 2 1480 G E E 3* 4 12.50 44 44 44 .625 2 1481 H E E 4 " 44 44 44 .621 2 1482 Z K K 2 6 12.00 3/4 7/16 tt* 736 2 1483 Z K K " 6 44 44 44 41 757 2 ! 1484 Z K K a* 6 12.75 44 44 ii .742 2 1485 Z P P K 6 44 44 44 " .762 2 1486 Z L M 2* 6 I3-50 44 11 ii 749 2 1487 z L M 44 6 44 11 * 4 it .764 2 14 8 z N N 2f 6 I4-25 44 ii ii 745 2 1489 z 'o N " 6 " 44 44 it 735 2 1490 z K 2i 6 12.75 44 44 A * i\ 723 2 1491 z P P " 6 " 44 ti 44 752 2 1492 z M L ai 6 I3-50 44 44 44 736 2 1493 z M M 44 6 44 44 44 it 754 2 1494 j Z N at 6 14.25 44 44 44 .760 2 1 ~ H95 z 44 6 44 44 11 44 .760 2 i 1496 Y Q P a* 6 15.00 44 44 it 745 2 M97 Y P P 44 6 44 44 44 44 725 2 1498 Z L L at 5 I3-I3 II ii 44 733 2 1499 z L L 44 5 44 44 44 44 744 2 1500 z N M 2* 5 13-75 44 44 it .762 2 1501 z N N 44 5 44 44 it 44 727 2 1502 z O P at 5 14.38 II 4< 44 .722 2 1503 z O O " 5 II .741 2 TABULATION OF RIVETED JOINTS. 621 RIVETED BUTT-JOINTS Continued. STEEL PLATE Continued. Sectional Area of Plate. Bearing Surface of Rivets. Shear- ing Area of Rivets. Tensile Str'gth of Plate per Sq. In. Max. Stress on Joint per Sq. In. Efficiency of Joint. Temperature of Joint in Deg. Fahr^ gill l15 g</>0 c- cS o w o j afc-gcL bC U . c o <u cx'C<2 > sg^S Sfflu-g be C w '- oj $*0> % * Gross. Net. sq. in. sq. in. sq. in. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. 3-735 4-597 4.138 11.928 572>o 34260 65110 72330 25090 59-8 8.690 4-579 4. in 11.928 " 34790 66030 73540 25350 60.7 9.285 5-i7 4.178 ii .928 55940 34980 63600 77740 27230 62.5 9.240 5.082 4.158 11.928 " 34770 63220 77270 26930 62.1 8.239 4.706 3-533 9.940 55oo 36350 63640 84770 30130 66.1 7.978 4-552 3.426 9.940 57290 37100 65020 86400 29780 64-7 8.362 4-936 3.426 9-940 " 38150 64630 93110 32090 66.5 8 381 4-950 3-43i 9-94 " 38120 64540 93120 32140 66.5 8.833 5.402 3-431 9.940 " 38620 63140 99410 343io 67.4 8-739 5-3'3 3.426 9.940 " 38180 62830 97430 3358o 66.6 9.240 5-775 3-465 9.940 55940 38480 61570 102630 35770 68.7 9-345 5.841 3-504 9.940 " 38410 61430 102440 36110 68.6 7-8i3 5.000 2.813 7-952 55000 37340 58360 103730 36690 67.9 7-763 4.968 2-795 7-952 " 38440 60060 106760 37520 69-9 8.847 4-431 4.416 9-425 59000 31990 63870 64090 30030 54-2 9.099 4-557 4-542 9.425 " 31980 63860 64070 30870 54-2 9-475 5-023 4-452 9.425 " 34440 64960 73920 34620 58.3 9-723 5-*5i 4.572 9 425 41 34700 67340 73790 35800 58.8 IO.II2 5.618 4-494 9-42S 14 35000 63000 78750 37550 59-3 10.329 5-745 4-584 9-425 " 36780 I 66130 82870 40310* 62.3 10.624 6-154 4.470 9-425 " 38120 65810 90600 42970* 64.6 10.488 6.078 4.410 9-425 11 34000 58670 80860 37830 57-6 9-233 4-353 4.880 11.928 " 31050 65860 58750 24030 52.6 9.596 4.520 5.076 11.928 4 32000 \ 67940 65340 25740 54-2 9 .951 4-983 4.968 ii .928 " 34270 68430 68640 28590 58-0 | 10.179 5 082 5-090 11.928 " 33770 | 67540 67520 28810 57-2 10 845 5-7*5 5-130 11.928 34900 66230 73780 31730 59-1 10.838 5.708 5-!30 ii .928 " 35810 67990 75650 3254 60.6 i-*75 6.146 5-029 11.928 60420 38470 69940 85480 36040 63.6 10. 890 5-996 4.894 11.928 " 37740 68650 83980 34460 62.4 9.624 5-501 4- I2 3 9.940 59000 35000 61230 81700 33890 57- 9 776 5-572 4.204 10.030 " 36470 63990 84810 35550 61.7 10.478 6. 192 4.286 9.940 11 38760 65590 94750 40850* 65-7 10.004 5-9!5 4.089 9.940 It 36740 62130 89880 36970 62.2 10.390 6 329 4.061 9-940 " 3793 62270 97050 39650* 64-3 10.663 6-495 4.168 | 9.940 II 40630 65810 90600 42970* 68.8 * Steel rivets. 622 APPLIED MECHANICS. TABULATION OF SINGLE- STEEL PLATE Continued. No. of Test. Sheet Letters. Pitch. No. of Rivets. Width of Joint. Nominal Thickness. Size of Rivets and Holes. Actual Thick- ness of Plate. Lap. Plate. Covers. Plate. Covers. in. in. in. in. in. in. in. J b4 Z M L a* 6 13-50 3/4 7/16 iA*il .722 2 *505 Z M M " 6 11 " 44 .762 2 1506 Y N at 6 14-25 11 " 44 .727 2 1 57 Y N " 6 " 11 it 44 735 2 1508 Y Q Q a* 6 15.00 " " 44 737 2 T 59 Y Q Q " 6 " " " K 753 2 1510 Y L L 8| 5 13-13 " " 44 .748 2 '5" Y L L " 5 " " " (1 755 2 1512 Z .... at 5 13-75 " " 44 .750 2 1513 Z N N " 5 " 44 " 41 .764 2 JSH Y o 3| 5 M-iS " " " .760 2 1515 Y K 5 " " " 44 .746 2 *5'6 PS P 3 5 15.00 " 11 44 749 2 1517 Y P; Q " S " " " " .741 2 1518 Z K K 3t 4 12.50 14 " 44 756 2 1519 Z K K " 4 41 11 it 44 741 2 1520 Z K K 3i 4 13.00 it 41 ii -763 2 1521 Z K K " 4 11 " 41 41 .718 2 1522 Z M M 3t 4 13-50 11 44 (I .742 2 1523 Z M M " 4 " " M 754 2 TABULATION OF RIVETED JOINTS 623 RIVETED BUTT-JOINTS Continued. STEEL PLATE -Continued. Sectional Area of Plate. Bearing Surface of Rivets. Shear- ing Area of Rivets. Tensile Str'gth of Plate per Sq. In. Max. Stress on Joint per Sq. In. Efficiency of Joint. Ill pi % u |o|s cc^ OC/2 > o b 2 . C U 4J Ills c3 ffi "^ U> C en |o| Gross. Net. sq. in. sq. in. ' sq. in. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. \ 0.761 4-346 - 5-415 14.726 59000 29090 65350 52460 19280 49-3 10.287 4 572 5-7I5 14.726 59000 30010 67520 54010 20960 50.8 10.367 4.914 5-453 14-726 60420 33610 70900 f389o 23660 55-6 10.474 4.961 5-513 14.726 60420 33660 71070 63960 23940 55-7 i i . 070 5-542 5-528 14 726 60420 3478o 69470 69650 26140 57-5 11.310 5-662 5-648 14.726 60420 34670 69250 69420 26620 57-4 9 918 5-243 5-675 12.272 60420 36120 68380 76680 29210 59-7 9.928 5.209 4.719 12.272 60420 36940 70400 77710 29880 61.1 10.328 5 6 4 4.688 12.272 59000 33730 61770 74320 28390 57.0 10 505 5-73 4-775 12.272 59coo 35260 64640 77570 30100 59-7 10.929 6.179 4-75 12.272 60420 3793 67080 87260 36220 62.7 io-735 6.072 4.663 12.272 60420 38720 68460 89150 33870 64.0 T i . 205 6.524 4.681 12.272 55520 36530 62740 87440 36610 65.8 11.108 6.477 4-631 12.272 60430 38740 66440 92920 35060 64.1 9-465 5.685 3.780 9.818 59000 3756o 62360 9378o 36110 63 6 9- 2 /7 9-934 5-572 6. 119 3-705 9.818 9.818 59000 59000 39000 37600 64930 61040 97650 97900 36850* 38040* 66.1 63-7 9 348 5.753 3 590 9.818 59000 36000 58440 9374 34280 61 .0 10.032 6.322 3.710 9.818 59000 40040 63540 108270 40^10* 67-7 10 . 187 6.417 3.770 9.818 59000 39720 63050 107320 41210* 67-3 * Steel rivets. 624 APPLIED MECHANICS. TABLE TABLE OF EFFICIENCIES OP STEEL PLATE. Plate. No. of Test. Pitch of Rivets. if" if" it" 2" *" per cent. per cent. per cent. per cent. per cent. 1308 67-5 68.9 72.8 1313 68.3 69.9 72.1 t" . . . .- 1314 1323 1324 65-9 64.8 66.4 67.1 62.6 68.3 68.6 64.4 I' 3 67-5 1337 63-9 65-9 68.2 1338 .... 64.8 61.7 I35S 58.7 60.6 1356 74-6 73.3 1359 72.7 72.0 .... 1360 68.2 69.0 70.0 70.9 I" . . . . - ^67 1368 68.3 70.1 64-5 70.3 61.7 68.5 64.8 1379 63.0 65.2 67.4 1380 58.7 59-6 1395 .... 57-3 59-5 300 300 1396 .... 66.1 79.6 64.7 .... 1401 68.1 68.6 350 68.9 - 1402 .... 64-5 66. 3 400 76.0 w 1411 .... 250 66.9 300 76.4 66.3 1412 .... .... .... 260 62.8 59-2 1425 ... .... .... 60. 1 68 i 300 1426 .... .... .... .... 59-3 , 1443 .... .... .... 51-5 1444 .... .... 60. 1 66.0 300 7 6.2 // M5i 62. 7 63.8 66.9 . . . . 1452 . . . 58.3 60.0 1463 58-3 61.8 1464 ... .. . 56.5 1481 56-5 1482 . . . , 54-2 58.3 1489 . . 54-2 58.8 1490 1503 . . . .... 52.6 54-2 1523 ' ' .... NOTES. Figures in heavy- face type denote that Super numbers state the temperature of TABULATION OF RIVETED JOINTS. 62 5 IsO. i. SINGLE-RIVETED BUTT-JOINTS. STEEL PLATE. Pitch of Rivets. Diam- eter of Rivet- holes. *" a|" 2*" af" 2 J// at" 3" 3*" 3*" 3*" per ct. per ct. per ct. per ct. per ct. per ct. per ct. per ct. pr ct. per ct. in. i I i i f * J i f i * i i it * 1 i t i Ii 72.7 74.0 70.1 70.0 61.5 63-4 70.7 72.3 63-7 68.4 76.8 67.1 72.1 68.5 69.8 68.3 6s-4 68.4 77.1 75-o 73-5 74- * m 70.7 68.2 65.4 64.1 72.4 70.4 65-3 68.3 75-5 75 6 69.8 69.5 6 7 :; 68. 3 6S!8 69.8 73-2 75- 66.4 too 77.6 a oo 60.3 61.5 360 6 7 .2 56.5 6 4 8 6 64 3 63.6 62.8 57-8 58.0 59.3 62.3 58.0 57 2 49-3 50.8 75 6 .5 66.7 400 72.8 64-5 700 62.7 58.0 600 77-3 66.9 62.3 500 76.7 6 7 .2 400 76.9 62.O 6l.2 600 81.5 67.6 800 70-5 64.5 600 80.3 67.0 200 67-5 70.0 67.2 68.2 .... .... 65*8 65-5 59-8 60.7 64.6 57.6 59-1 60.6 55 6 55 7 66.6 66.1 62.5 62.1 6 3 "6 62.4 57-5 ' 57-4 65.0 64.5 66.1 64.7 67.0 61.7 59-7 61.1 6^5 66. 5 67 .'i 66.6 68.7 68.6 67^9 69.9 " !;;; 65~7 62.2 57.0 59 7 64' 3 68.8 62.7 64.0 65-8 64.1 63.6 66.1 ai 67.7 67.3 joint did not fracture along- line ol riveting, 'oints tested at temperatures above atmospheric. 626 APPLIED MECHANICS. TABLE NO. 2. TABLE OF DIFFERENCES BETWEEN THE EFFICIENCIES AND RATIOS OF NET TO GROSS AREAS. SINGLE-RIVETED BUTT-JOINTS, STEEL PLATE. Plate. No. of Test. Width of Plate between Rivet Holes. Diameter of Rivet Holes. i" ii" ii" if" *T if" if" i*" 2" in. i - r i r * - i - r * in. 1308 1313 i3H 1323 i3 2 4 1337 1338 1355 1356 1359 1360 1367 1368 1379 1380 1395 1396 1401 1402 1411 1412 H25 7426 1443 1444 i45i 1452 1463 1464 1481 1482 1489 1490 I5Q3 1504 1521 perct. 6.0 6.8 8.8 K ,fl 18.8 13- 11. ii. ii. ii. 9-7 6.8 7-3 200 9-0 II. I II . I 250 13-5 850 12.8 10. I 300 12.2 4.6 6.4 9-4 8-3 8-3 9-5 9-4 4- 1 4- 1 5-5 7- 1 4.8 6.4 perct. 4-6 5-6 6-3 7.0 8.1 9.6 8.8 7.6 9.1 7.7 9.0 IO. I 5-4 I' 9 6.7 6.6 300 19.6 8.6 IO.O 300 2O. I 6.2 300 15-2 360 I 7 .2 6-5 9.7 7-5 7- 1 8.9 7-8 8.0 \\ 7-9 7.2 8.2 8-3 per ct. 6.1 5-4 7.2 6.1 8.6 9-4 5-9 7-8 per ct. per ct. perct. per ct. perct. per ct. in. i i i i i * i i i * i I i i ii ii 1 i i 3 i i i! 4.0 5-6 xi. 7 8.9 5-8 10.5 8 7 :i 7.6 9-2 12. I 8-5 ii. 8 2.1 3-5 6.6 i:i 13-4 "3 8. '2 8.8 'i : ? 7-5 7.8 11 9-9 5-9 2. I 350 6.4 400 17.2 7-4 200 4-7 5-9 700 10.2 5-3 300 17 4 8.1 8.0 7-2 7.2 8.0 3.7 6.6 6.4 7-9 8 6.2 3.8 9-6 7-i 7-4 10.4 9.4 7-2 9.8 9-5 10.5 10.6 i:? 5.2 6.2 7-9 9.8 .... 5-3 500 16.4 400 14-9 6-5 6-9 600 21.7 400 7-5 600 3-2 8.1 7-6 7-5 7- 1 67 -0.4 8.6 1:1 8.6 ll! 3 3.5 500 I 7 .2 6.9 4-9 4-i 5-2 400 14.9 600 II.4 5-4 500 17.8 4.0 500 19.4 6.1 .... 200 5- 7-5 3.2 4.2 6.6 6.0 9.0 7-6 3.1 2.7' 7-5 7-4 6.2 5.8 6.2 6.1 3.9 5.9 -.02 4-7 2.4 5-i "6.6 3.1 6.2 7-4 3-4 7-9 7.6 5.8 3.5 6.0 2.1 -0.6 2i" perct. 4.7 4.3 NOTES. Figures in heavy-faced type denote that joint did not fracture along line of riveting Super numbers state the temperature of joints tested at temperatures above atmospheric. TABULATION OF RIVETED JOINTS. 62 7 TABLE NO. 3. EXCESS IN STRENGTH OF NET SECTION IN JOINT OVER STRENGTH OF TENSILE TEST-STRIP. SINGLE-RIVETED BUTT-JOINTS, STEEL PLATE. Plate. in. i t * t i A No. of t Test. Width of Plate between Rivet Holes. Diameter of Rivet Holes. x" ii" if ij" if" 2" 1308 1313 y*3*4 1323 1324 1337 1338 1355 1356 1359 1360 1367 1368 1379 1380 1395 1396 1401 1402 1411 1412 1425 1426 1443 1444 1451 1452 M 6 3 1464 1481 1482 1489 1490 1503 1504 1521 3er ct. 9-8 n. i !5-4 13-4 17.4 19.7 29.6 17.6 21.2 18.1 19-3 19-5 20.9 18.2 15-5 14.6 200 15.6 T 5 .8 20.9 260 25-2 350 25.5 20.3 300 2O. O 9 .8 IS-I 17.7 16.6 16.6 20.2 20.0 8-3 8.2 ii. 6 15.2 10.8 14.4 per ct. 7-2 8.8 10.5 11.7 14.4 17.0 16.5 14-5 14.1 11.9 15.0 16.8 9-7 15-9 12.6 12.5 300 32-8 14.4 17.8 300 35-8 ii. 8 2 8 8 8 360 34-4 13.2 18.8 i3-5 17.2 16.8 15.8 18.4 10. I 14.1 16.0 14-5 17-4 17.6 perct. 9.1 8.1 9-2 9-7 M.7 15-9 10.6 14.2 12.0 12-5 10. 1 13-5 17.8 10.6 3-4 360 10.3 400 28.9 12.8 200 5-5 10.8 700 I2. 3 10. 1 300 20. 6 13.7 14-6 13.2 13-6 15-3 6.8 12. 1 I2. 3 15-2 I 5 .0 I 4 .6 perct. perct. perct. perct. per ct. perct. 10.6 10.1 in. f * ! i i i t * X X i 1 * i X Ii It * i t i i Ii :i 6.2 8-5 14.7 14.6 10. 18.0 9-i ii .0 11.9 14.4 20.1 I4.I ie.'i 1:1 10.7 4-7 2-5 21.0 17.7 12.7 13-5 9.6 5.8 15-6 11.7 12.8 17.9 14.9 II.4 I6. 3 15-9 16.1 16.5 8.9 10.3 8.2 9-8 12.2 IS-* '.'..'. 8.7 600 27.0 400 2 5 .8 "3 12. I 600 39-7 400 12.2 600 5-2 14.0 I 3 .2 !3-7 13.0 11.5 -0.6 15.8 13-6 13.2 16.5 19 6 .5 5.6 600 28.7 "3 8.7 7-3 8.6 400 24.0 600 19.4 9-i 500 28.1 6.4 600 32.0 10. 1 a 8i 12.2 5.3 6.5 11.1 10.1 15-7 13-5 5.2 4.3 12.8 12.7 10.2 9-7 lo.'i 9 .8 e.'i 9.2 3.8 8.5 4-7 9.6 II. 2 5.3 II. 13-3 5-5 "5 13.0 10.0 5.7 10. 1 4i 2i" per ct. 7.7 6.9 verage of all joints. 16.2 14.4 12.3 12.9 11.9 II. 9-7 ii. 8 7.0 Norms. Figures in heavy-faced type denote that joint did not fracture along line of riveting. Super numbers state the temperature of joints tested at temperatures abcrc atmospheric. 628 APPLIED MECHANICS. In the Report of Tests made at Watertown Arsenal during the fiscal year ended June 30, 1891, is the following account of another series of tests on riveted joints: 44 Comprised in the present report are 113 tests made with steel plates of 1/4", 5/16", 3/8", and 7/16" thickness with iron rivets machine driven in drilled or punched holes. 44 The plates used were from material used in earlier tests, the results of which have been published in previous reports. 41 In the use of metal once before tested, such plates were selected as had not been overstrained previously, or those in which the elastic limit had been but very slightly exceeded. SINGLE-RIVETED STEEL PLATE. Sheet Letters. Nominal Thickness. Size Actual No. of Test. Pitch. No. of Rivets. Width of Joint. and Kind of Thick- ness of Lap. Plate. Covers. Plate. Covers. Holes. Plate. 49i3 H C D in. 2* 5 in. i3-7 2 in. i/4 in. 3/i6 in. 7/8 d in. 247 in. 2 49M L D D " 5 13.69 " " " " .248 " 49'5 M E E 8* 5 14.32 i it .4 tt .247 44 4916 M D D it 5 14-33 11 " tt tl .247 44 49i7 M E E 3 5 15.00 11 tt tt tt .246 14 4918 M E E " 5 14.98 it it tt It .247 11 4985 Q D D 3* 4 14.00 A " I " 39 I 4987 s C " 4 14.01 " 14 " " .310 li 499* Q 1 " 4 14.05 " it U -308 If 55 R A A I 10 IO.02 5/'6 11 1/2 " .306 I* 5"6 R A .... I* 8 10.02 " tt tt It -34 " 5127 R B I* 7 10.51 " " tt tt .310 " 5143 L P 2 7 14.03 7/16 5/i6 7/8 P- .440 I* 5M4 L O " 7 14.01 it 11 " d. .440 " SMS Q 2* 6 I3-50 44 44 it tt -434 " 5H6 M O " 6 I3-5I u it " P- .421 14 5M7 O P P 2* 6 15-02 " 44 41 d. 413 44 5148 N P .... 14 6 15.02 " H " P- .411 " SiSS K S .... 2* 5 13.75 41 It M d. 425 TABULATION OF RIVETED JOINTS. 629 "The present tests are supplementary to those of earlier reports, and occupy a place intermediate between the elementary forms of joints and the more elab- orate types of joints which have been investigated. " Wide variation has been given the pitches, and rivets of extreme diameters have been used for the purpose of including joints in which these features have been carried to their extreme limits. " The efficiencies of the joints are stated in per cent of strength of the solid plate." BUTT-JOINTS. STEEL PLATE. Sectional Area of Plate. Bearing Shear- ing Tensile Str'gth Maximum Stress on Joint per Sq. In. Effi- Surface of Area of OI Plate Tension on Tension on Net Compres- sion on Shearing ciency of Gross. Net. Rivets. Rivets. per Sq. In. Gross Section Section of Bearing Surface of Rivets. Joint. of Plate. Plate. of Rivets. sq. in. sq. in. sq. in. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. 3-39 2.31 i. 08 6.01 59180 44180 65760 140650 25270 75-7 3-40 2.31 1.09 6.01 61470 435oo 64030 135690 24610 70.8 3-54 2.46 i. 08 6.01 58170 46300 66630 151780 27270 79-6 3-54 2.46 i. 08 6.01 58170 435oo 62590 142570 25620 74.8 3-69 2.61 i. 08 6.01 58170 46290 65440 158150 28420 79.6 3-70 2.62 i. 08 6.01 58170 44400 62700 152110 27330 76.3 4-33 3-9 1.24 6.28 56760 24040 33690 83950 16580 42-3 4-34 3.10 1.23 6.28 57000 26770 3748o 93710 18500 46.9 4-33 3.10 1.23 6.28 56760 33940 47410 119500 23400 59-8 3. 7 i. 54 i-53 3-92 61130 35930 71620 72090 28140 58.8 3-5 1.83 1.22 3-i4 61130 41280 68800 103200 40100 67-5 3.26 2.17 I.OQ 2.74 61130 39250 58960 117380 46690 64.2 6.17 3.38 2.79 8.41 59390 32540 59410 71970 23880 54-8 6.16 3.48 2.69 8.41 59390 23360 41350 53490 17110 30. -\ 5-86 3-58 2.28 7.21 52910 42250 60160 108600 34340 79.8 5-69 3-4 2.29 7.21 61650 39740 66500 98730 31360 64.5 6.20 4-03 2.1 7 7.21 52910 44150 67920 126130 37960 83-4 6.17 3-94 2.23 7.21 61650 36660 57410 101430 3*370 60.0 5-84 3.98 1.86 6.01 59000 40270 59100 126450 39130 68.3 630 APPLIED MECHANICS. TABULATION OF SINGLE- STEEL PLATE. No. of Test. Sheet Letters. Pitch. No. of Rivets. Width of Joint. Nominal Thickness. Size and Kind of Holes. Actual Thick- ness o: Plate. Lap. Plate. Plate. Plate. Plate. in. in. in. in. in. in. in. 4933 I J H 5 10.62 i/4 i/4 id. .252 2 4934 J J " 5 10.65 " " 41 253 2 4939 L K 4 4 11.50 " " i d. .250 2 494* J J it 4 11.50 ti 44 .256 2 494 1 K J 44 4 ii Si (i 44 iid. .252 2 4942 K J " 4 11.52 " ti 41 .250 2 4943 K K (l 4 11.50 K 44 iid. .252 2 4944 E J K 4 ii .50 " " 44 253 2 4945 K K 3i 4 12.52 " " it .248 2 4946 L G 44 4 12.55 t> 44 ii 253 2 4947* N H 44 4 13-52 (i " 44 .247 2 4948* N H " 4 13-52 " it ii 247 2 4949 M L 3* 4 14-51 44 ii ii .248 2 495* M L 44 4 14-51 " it 44 .247 2 4961 B E i* 6 10.52 3/8 3/8 id. .388 2 4979 E E 2f 5 11.84 44 44 i d. .384 2 5131 K K I 8 12. OO 7/16 7/16 Id. 427 i-75 5132 N O 44 8 12.00 ii ii IP. 415 i-7S 5133* K K If 8 13.00 < ii |d. .427 i-75 5134 N N " 8 13.00 " it IP- 4*3 i-75 5135 M M I* 8 M-03 " ii Id. .422 i-75 5 T 36 " 8 13-99 " ii IP- .420 i-75 5137 L M 3 7 14.02 44 " Id. 44 1-75 5138 P M " 7 14.05 " ii IP- .420 i-75 5139 O K II 6 12.06 < ii iid. .428 2 5140 M M 2t 6 14.28 " 4 * it .421 2 5Mi* L L 2} 5 13-73 " 4 * M .438 2 5142* Q 3* 5 15-67 44 it M .422 2 * Pulled off rivet-heads. t Pulled off 3 rivet-heads. $ Pulled off 2 rivet-heads. TABULATION OF RIVETED JOINTS. 631 RIVETED LAP-JOINTS. STEEL PLATE. Sectional Area of Plate. Bear- ing Surface of Rivets. Shear- ing Area of Rivets. Tensile Strength of Plate per Sq. In. Maximum Stress on Joint per Sq. In. Effi- ciency of Joint. Tension on Gross Section of Plate. Tension on Net Section of Plate. Comp. on Bearing Surface of Rivets. Shear- ing of Rivets. Gross. Net. sq. in. sq. in. sq. in. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. 2.68 i-57 I.IO 3-00 61000 39750 67850 96840 35510 65.1 2.70 i-59 i. ii 3-00 61000 39660 67360 96490 35700 65.0 2.87 1.87 1. 00 3-M 58150 40560 62250 116400 37070 69.7 2.94 1.92 1.02 3-i4 61000 37010 56670 106670 34650 60.6 2.90 1.77 1-13 3-98 61000 43280 70900 111060 31530 70-9 2.88 i-75 1.13 3.98 58150 42770 73880 119010 30950 73-5 2.90 1.64 1.26 4.91 58150 41130 72730 94660 24290 70.7 2.91 1.64 1.27 4.91 58150 40200 71330 92110 23820 69.1 3.10 1.86 1.24 4.91 58150 40030 66720 100080 25270 69.1 3-i7 i. 91 I 26 4.91 61470 41770 69320 105080 26970 68.0 3-34 2.10 1.24 4.91 55740 42240 67180 112970 28730 75-7 3-34 2.10 1.24 4.91 59180 42600 67760 114760 28980 71.9 3-6o 2.36 1.24 4.91 61470 41390 63140 120180 30350 67.3 3.58 2-35 1.23 4.91 58170 42150 64210 122680 30730 72.4 4.08 2-33 J -7S 2.65 5834 25950 45440 60500 39950 44.4 4-55 2.6 3 1.92 3-93 58340 33050 57 I 9 78330 38270 56.6 5-12 2.13 2-99 4.81 59000 31740 76290 54350 33780 53-8 4 99 1.98 3.01 4.81 52910 27820 70100 46110 28860 52.6 5-55 2. 5 6 2.99 4.81 59000 31100 67420 57730 35880 52-7 5-37 2-37 2-99 4.81 61140 30370 68820 54550 339*0 49-7 5-90 2-95 2.95 4-81 61650 29240 58490 58490 35870 47-4 5-89 2-95 2-94 4.81 31870 63630 63840 39020 5-94 3-35 2.60 4.21 5939 27580 48900 63000 38910 46.4 5-9 3-24 2.66 4.21 52910 28530 51940 63270 39980 53-9 5-16 i-95 3-21 7.36 58090 28190 74610 45320 19770 48-5 6.01 2.85 3.16 7.36 61650 34850 73490 66280 28460 56.5 6.01 3.28 2.74 6.14 59390 3356o 61490 73610 32850 56.5 6.61 3-97 2.64 6.14 56960 30420 50650 76170 32750 53-4 6 3 2. APPLIED MECHANICS. TABULATION OF DOUBLE- CHAIN-RIVETING-STEEL PLATE. o No. of Test Sheet Letters. Pitch. mce Apart o ws, Centre t ntre. Total Num- ber o Riv- ets. Width of Joint. Nominal Thickness. Size and Kind o Holes. Actual Thick- ness o Plate. Lap. Plate. Covers. * o v Plate Covers. Q in. in. in. in. in. in. in. in. 4911 K C C *f 2i 10 13.10 i/4 3/i6 5/8 d. 253 J T! 4912 L C C " " IO 13.10 44 44 44 253 14 49i9 L E D af 2* 10 14-32 44 K 7/8 d. .247 i}| 4920 L E D " 44 10 14-32 44 44 " .249 " 4921 K C B 3* 44 8 12.52 44 44 44 .252 4922 K C C 44 44 8 12.49 it 44 " .252 " 49 2 3 J A A 3* 44 6 II-S7 44 44 44 257 44 4924 J A 44 44 6 "53 44 44 44 255 11 49 2 5 L B 4* 44 6 13.09 ' 41 44 .251 4926 K C B 44 41 6 13. 10 *' 44 44 .230 " 5128 R /"v C C I* 2 M 12.27 5/i6 3/16 1/2 d. 34 if 5 I2 9 5130 Q S D *i 12 I3-58 . u .. 305 307 4993 Q E 3* I* S 14.05 44 44 i d. 309 i$ 4995 Q E .... 11 I* 8 14.06 44 44 44 35 i* 4997 Q E 44 2 8 14.08 44 K 44 .308 t j 495i B I I 2* 2* 8 8.52 3/8 1/4 3/4 d. 392 i* 4952 E R 44 " 8 8.51 41 5/6 44 -383 " 4953 E R af 44 8 10.51 44 it .388 " 4954 E N H fci 44 8 10.03 44 i/4 44 384 " 4955 E S 3* 44 8 12.50 44 5/16 383 4957 H O 3t 44 8 I4-5I " 44 44 -369 " 49.S8 H M M 4 ' 44 8 I4-52 44 i/4 44 369 " 4959 B S S 4* 44 6 12.42 * 5/i6 44 .388 it 4960 E L N i4 44 6 12.42 44 i/4 44 384 4967 C M M at 2* 10 14-38 44 " i d. 375 2 4969 E M 3l 44 8 13-50 " 4 ' 44 .382 4970 F 44 44 8 13-58 44 5/i 6 44 .380 " 4971 J P P 3* K 8 15.46 44 44 44 379 " 4973 E S S 4t 44 6 13-50 44 44 44 .385 4975 I O P 4f 44 6 14-65 44 44 373 4977 J P P 5f 44 6 16.08 " 44 379 K 4956 K N N 3* a* 8 12.48 7/16 1/4 3/4 d. .427 !J 4968 N ** 2* 10 14.41 5/i 6 id. .409 2 TABULATION OF RIVETED JOINTS. 633 RIVETED BUTT-JOINTS. CHAIN-RIVETINGSTEEL PLATE. Sectional Area of Plate. Bear- ing Surface of Rivets. Shear- ing Area of Rivets. Tensile Strength of Plate per Square Inch. Maximum Stress on Joint per Sq. In. Effi- ciency of Joint. Gross. Net. Tension on Gross Section of Plate. i Tension on Net Section of Plate. Compres sion on Bearing Surface of Rivets. Shearing on Rivets. sq. in. sq. in. sq. in. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. 3-3i 2.52 1.58 6.14 58150 49090 64480 102850 26470 84-4 3-3i 2.52 1.58 6.14 61470 51960 68250 108860 28010 84.5 3*54 2.46 2.16 12.02 61470 46810 67370 76720 1379 76.1 3-57 2.48 2.18 12.02 61470 457o 65790 74840 13570 74-3 3-16 2.27 1.76 9.62 58150 46330 64490 83180 15220 79.6 3-'5 2.27 1-76 9.62 58150 46730 64850 83640 15300 80.3 2.97 2.30 r -35 7 .2I 61000 49520 63940 108930 20400 81.2 2.94 2.27 J -34 7.21 61000 49460 64050 108510 20170 8i.x 3-2Q 2.63 1.32 7.21 61470 51440 64350 128210 23470 83-7 3-oi 2.41 1. 21 7 21 58150 55500 69320 138070 23170 95-4 3-73 2.66 2.1 3 5-49 61130 46690 66650 83240 32300 76.4 4.27 3-20 2. 14 5-49 56760 49040 65430 97850 38140 86. 4 4-15 3-23 1.8 4 4.70 57000 46480 59720 104840 41040 81.5 4-34 3-" 2-47 12.57 56760 44740 62430 78600 iS45o 78.8 4.29 3-7 2.44 12.57 56760 45490 63570 7998o 15530 80. i 4-34 3-io 2.46 12.57 56760 45530 63740 80330 15720 80.2 3-34 2.16 2-35 7.07 5973 43290 66940 61520 20450 72.4 3-26 2. II 2.30 7.07 58340 42380 65470 60070 19540 72.6 4.08 2. 9 t 2 -33 7.07 58340 46590 65330 81590 26890 79.8 3-85 2.70 2-30 7.07 58340 49130 70060 82240 26750 84.2 4-79 3-64 2.30 7.07 58340 48610 63970 101250 32940 83-3 5-35 4.25 2.21 7.07 56670 48500 61060 117420 36700 85.5 5.36 4-25 2.21 7.07 56670 47700 60160 115700 36170 84-1 4.82 3-95 r -75 5.30 59730 43070 52560 118630 39170 72.1 4-77 3-91 i-73 5-30 58340 42520 51870 117230 38260 72.8 5-39 3-52 3-75 I5-7I 57870 42890 65680 61650 14720 74- i 5.16 3-63 3-o6 12-57 58340 44263 62920 74640 18170 75-9 5.16 3-64 3-4 ".57 54290 43240 61290 73390 17750 79-6 5.86 4-34 3-03 ".57 5713 44910 60650 86860 20940 78.6 5.20 4.04 2.31 9.42 58340 45980 59180 103510 25380 78.8 5-46 4-35 2.24 9.42 59030 46720 58640 113880 27080 79.1 6.09 4.96 2.27 9.42 5713 44650 54830 119800 28870 78.1 5-33 4-05 2.56 7.07 59000 48120 63300 100190 36280 83-3 5-89 3.85 4.09 I5-7I 61140 433 66340 62440 16260 70.9 634 APPLIED MECHANICS. TABULATION OF RIVETED DOUBLE-RIVETED LAP-JOINTS. 09 c c Nominal 8 1 Q Letters ol >and Cov part of Ro ang. to 1 vets. Rivets in t Row. Rivets in id Row. Rivets in d Row. a 'o o Thickness. "o o C "5 i o oJ JS rt . tfS s-, C 'o 8 o;s -8 u i2 u 1 is 1 jE o .2 rt'o dfc o'c/ii ciH a > o SE t) O a rt in (X fc * OH U c/5 ^ in. in. in. in. in. in. in. in. 4935 J-J 2 i 2t 5 5 10.68 1/4 7/8 d. 257 1 ii 4936 I-J 4 * 44 5 5 10.53 44 44 44 251 k4 4937 L-M 2 i ** 5 5 14.38 " 44 44 .248 4938 I-L u ** 5 5 14.40 44 44 44 .249 44 4999 5000 49 6 3 4965 ft? E-E E-E 3* 2 2i 4 i 6 4 4 6 6 14.02 14.00 10.50 12. OO 5 /, ( 6 3/8 I " I P. 3 / 4 <! 305 .306 -387 -384 2 4! 498i 4983 5149 E-D D-H M-L 2t 2* 2 ft 5 5 7 5 5 7 11.83 I4-3 6 14.00 7 /,6 7/8 " 385 370 -425 2 5*5 M-M 44 7 7 I4.OO '* P- 423 44 K-K 2 1 44 6 6 JT . ^7 k * " d. .428 4 5 I 52 L-0 4 - 44 6 6 *3-5 " P. .440 ,| 5153 0-0 2* " 6 6 15.01 44 " d. .409 Ij 7 B 5154 O-O K-K 2t *' 6 5, 6 5 15.02 " " t .412 .422 44 DOUBLE-RIVETED BUTT-JOINTS. 4927 MDE 2* 2t 5 4 I4-36 i/4 3/16 7/8 d. 250 '.H 4928 LDE 5 4 I 4-35 .247 4929 KBC H 44 4 3 12.51 44 4 255 " 493 KG " 4 3 12.50 44 ' .251 44 493i 4932 HDD HCC ?,* " 3 3 2 2 13-12 13-12 " 4 .248 .246 44 DOUBLE-RIVETED LAP-JOINTS. 5"9 5120 0-0 p-p # t 4 4 3 3 14.00 14.03 S/i6 d. P- 303 305 i* 52I R-R " It 4 3 14.03 " d. .302 i* 5122 O-O " 4 3 14.03 14 P- .304 " 5123 O-O 44 2 4 3 14.02 44 d. .302 it 5124 P-P 4 3 14.02 P- .307 4 v TREBLE-RIVETED LAP-JOINTS 5157 KK 2t a| 5 5 5 iS-M 7/16 7/8 d. '432 i 5158 OP 3 5 5 5 !5-05 44 " 4t .412 5159 5100 PP LL 3 " 4 4 4 4 4 4 12.78 I3-50 tt o 11 '43 2 .438 ii TABULATION OF RIVETED JOINTS. 635 JOINTS. STEEL PLATE. CHAIN-RIVETING. Sectional Area 0) <-. 2 Maximum Stress on Joint per Sq. In. c of Plate. J| 1 ft. "5 o been Tension Tension Compres- >> Gross. Net. b ^ .5 II nj _rt G ~ p l *"* on Gross Section ot on Net Section sion on Bearing Shearing on c V 2~o "o c/} Plate of Plate . Surface Rivets. sg CQ C/3 H of Rivets. H sq. in. sq. in. sq. in. sq. in. Ibs. Ibs. Ibs. Ibs. Ibs. 2-74 1.62 2.25 6.01 61000 42770 72350 52090 19500 70.1 2.64 1.54 2. 2O 6.01 62300 42350 72600 58180 18600 i 67.9 3-57 2.49 2.17 6.01 61470 ' 47870 68630 78760 28440 77.0 3-58 4.28 2.49 3-06 2.l8 2.44 6.01 6.28 61470 48530 56760 : 46070 69780 64440 79700 80820 28910 31400 78.9 80. i 4.28 3-3 2-53 6.28 593o . 43900 62100 74370 29960 74.1 4-06 2.32 3.48 5-30 58340 | 40570 70900 47330 69.5 4.61 2.88 3.46 5-30 58340 i 42150 67470 56160 36660 72.2 4-55 2.63 3-85 7-8 5 53730 ! 38790 67100 45840 22480 72.2 5-31 3-46 3-70 7.85 56670 4 }c 5 o 66070 61780 29120 76.0 5-95 3-35 5-21 8.41 ; 61650 40620 72150 4 fi l-9o 28740 i 65.8 5-93 3-24 8.41 61650 379'o 69380 41940 26730 : 61.4 5-79 3-54 4.49 7.21 59000 43150 70570 73 - 1 5-94 3-55 4.78 7.21 59390 38870 65040 48310 12O2O ! 65.4 6.14 3-99 4.29 7.21 52910 43530 6f QQO 62310 37070 82.3 6. 19 5-81 3-95 3.96 4.48 3.69 7-21 j 52910 40380 6.01 59000 38850 63290 56990 55800 61170 34670 37550 1 76.3 ; 65.8 ZIGZAG-RIVETING. 3 59 So 97 10.82 58170 48150 69140 87740 15980 80.3 3-54 .46 95 10.82 61470 47420 68240 86090 15520 77-i 3-!9 3 56 8.41 58150 46610 64650 95320 17680 80.2 3-'4 .26 .56 8.41 58150 47520 66020 95640 17740 81.7 3- 2 5 .60 05 6.01 59180 47640 59550 M745 25760 80.5 3- 2 3 .58 .08 6.01 59180 46720 5S490 139720 25110 78.9 ZIGZAG-RIVETING. 4.24 4.28 4-23 4.27 3-03 3.02 3.02 3 01 . 12 .20 .IT 19 5-50 5 50 5 50 5-50 56760 5930 54350 54350 42750 40630 42990 44^40 59830 57580 60220 6^000 855^0 79050 86180 86450 32960 31620 33060 34420 75-3 68. S 79.1 81 6 4-23 3.02 . II 5-5 54350 44870 62850 89050 345 10 82.5 4-33 3-03 .22 5-50 593 43490 62150 84820 31400 73-3 CHAIN-RIVETING. 5-93 6.20 5-52 5-91 4.04 4.40 4.01 4.38 5-66 5-4 4-54 4.60 9.02 9.02 7.21 7.21 59000 52910 58090 59390 4^720 48710 48040 46430 67100 68630 66130 62650 47900 55820 584- 59650 30060 3348o 36780 38060 77 5 92.1 82.7 78.2 636 APPLIED MECHANICS. In the design of a riveted tension-joint the problem usually presents itself in the following form : Given, in all particulars, the two plates to be united, to design the joint ; i.e., to determine, i, the diameter of rivet to be used ; 2, the spacing of the rivets, centre to centre ; and, 3, the lap. In regard to the determination of the lap, the common practice has been already explained and very little has been done experimentally. In order to determine the diameter and the spacing of the rivets by the usual methods of calculation, it becomes neces- sary to know the three following kinds of resistance of the metals, viz.: i. The tensile strength per square inch of the plate along the line or lines of rivet-holes ; 2. The shearing-strength of the rivet metal ; 3. The resistance to compression on the bearing-surface of either plate or rivet. Hence we need to ascertain what the tests cited show in regard to these three quantities. Tension. The tensile strength of the plate used should, of course, be determined by means of tests made on specimens cut from it. Further than this, questions arise as to the excess tenacity due to the grooved specimen form, and as to any injury due to punching when the holes are punched. The excess tenacity is, of course, greater with small than with large spaces between the rivet-holes ; hence, inasmuch as the tendency is toward the use of large rivets, and, conse- quently, large pitches, the excess tenacity applicable in practi- cal cases becomes small, and would be better disregarded in the design of most riveted joints. In cases where the holes are drilled, therefore, we should use for tensile strength per square inch of the plate along the line of rivet-holes, the tensile strength per square inch of the plate itself. COMPRESSION. 637 The better and more ductile the plate the less is the injury done by punching; but, while more or less punching is done, the better class of work is drilled. A study of the results in the cases of punched plates will show approximately what allowance to make for the weakening due to punching different qualities of plate. Shearing". A study of the results of the government tests show that it is fair to assume the shearing-strength of the wrought-iron rivets used, to be about 38000 pounds per square inch, which is about two thirds of the tensile strength of the same rivet metal. For steel rivets, of the kinds now prescribed in most spec- ifications, the shearing-strength appears to be about 45000 pounds per square inch. Compression. To determine what we should estimate as the ultimate compression on the bearing-surface is a more difficult problem ; for if a joint fails in consequence of too great compression on the bearing-surface the cause of the failure does not exhibit itself directly, but in some indirect manner probably by decreasing the resisting properties of either the plate or the rivets, and hence by causing either the joint to break by tearing the plate or by shearing either the rivets or the plate in front of the rivets, but at a lower load than that at which it would have broken had the compression not been excessive; and hence when such breakage occurs it is difficult to say whether it is due to excessive compression re- ducing the tensile or the shearing strength, or whether its full tensile or shearing strength was really reached. Observe, moreover, that in the tables of Government tests the heavy numbers in the column marked " Compression on the bearing-surface of the rivets " indicate that the plate broke out in front of the rivets, which might be due to excessive compression or to a deficiency of lap. While more experiments are needed, it would seem proba- ble that we might deduce some conclusions, at least, of a gen- eral nature, in regard to the ultimate compression by a study 638 APPLIED MECHANICS. of the relations existing between the compression per square inch on the bearing-surface at fracture and the efficiency of the joint as shown by the Government tests. For this purpose the following diagrams (see pages 631 and 632) have been plotted, with the efficiencies as abscissae and the compression per square inch on the bearing-surface at fracture as ordinates. If similar diagrams were plotted with the efficiencies as abscissae and the ratio of the compression per square inch on the bearing-surface at fracture to the tensile strength of the plate as ordinates the character of the diagrams would be substantially the same, as the plates used in the tests were all of mild steel of approximately the same quality, and hence the difference in tensile strength of different samples was not great. A study of these diagrams shows that in the case of the i-inch plates experiments were made with compressions up to about 158,000 pounds per square inch, but that the highest compression reached with any other thickness of plate was about 120,000 pounds per square inch. Inasmuch as Kennedy advises the use of 96,000 pounds per square inch, and as this is higher than the values that have been customarily advocated, it would hardly seem wise to adopt a much higher value unless the tests furnish us sufficient evidence for such a procedure. Considering the facts stated above, and also the fact that in the cases of the double-riveted joints some of the highest compressions were accompanied by a decrease in efficiency, it would seem best to limit our esti- mate of the ultimate compression on the bearing-surface to from 90,000 to 100,000 pounds per square inch until we have further light on the subject derived from experiment ; and it is not at all improbable that when we do obtain further light we may find ourselves warranted in using a somewhat higher value. The reasoning which leads to the above conclusion is, of course, based on evidence which is not conclusive, because of the lack of tests with higher compressions on the bearing sur- COMPRESSION. 6380 APPLIED MECHANICS. W/TH TWO STEEL PLJTE. $' 'STEEL PLJTE. ilnTOMmiilllfflt ///> f\ s> si ::: ;i;;;;J:i:;i;;:i;;!i;i;i;;N;; /soooo { j |j J_[ |j_[_[ U'i 1 r H t ini* LL1J li 1 1 /?/?/?/?/) ::;;::;:!;:::::;:::: 7W0 70 <ft7 7<? W ^ /^ face, with plates thicker than one quarter of an inch. On the other hand, the quarter-inch plates show higher efficiencies with compressions above 100000 pounds than they do with compressions of 100000 pounds or less, and the author knows of tests upon riveted joints in T 7 -inch plates which tend to show that, with good wrought-iron rivets, it would be perfectly safe to use a considerably larger number for compression on the bearing-surface, in designing riveted joints at least Iioooo pounds per square inch, and probably more. COMPRESSION. It will be observed that no reference has been made to the friction, and it is safer to leave this out of account, as the tests show that slipping takes place at all loads, and as there. is no friction at the time of fracture. By far the greater part of the tests at Watertown Arsenal were made with wrought-iron rivets in mild-steel plates, this being, at the time, the most usual practice, although steel rivets were sometimes used. At the present time, notwith- standing the fact that steel long ago superseded wrought- iron for boiler-plate, and that it has, to-day, superseded wrought-iron for structural shapes, as I beams, channel-bars, angles, etc., and that the use of steel rivets has become very extensive, nevertheless a great many still adhere to the use of wrought-iron rivets, and feel more confidence in them than they do in steel rivets. Whereas the use of wrought-iron rivets had been practically universal, the qualifications for a good wrought-iron rivet metal became pretty well known, and while sometimes specifications were drawn up giving the requirements of the rivet metal for tensile strength, ductility, etc., which of course would vary more or less, nevertheless the variations would not be large. A study of the Watertown tests shows that the wrought-iron rivet metal used in those tests had a tensile strength of from about 52000 to about 59000 pounds per square inch, with a percentage contraction of area at fracture of from about 30 to about 45. With this metal the shearing strength per square inch seems to be about f of the tensile strength per square inch. Of course other tests are necessary to show whether the metal can be properly worked, and whether it is red-short or not, such as that the metal should bend double, whether cold or hot, without cracks, and that cracks should not develop when the shank is ham- mered down, cold or hot, to a length considerably less than the diameter. When steel rivets were first used, the steel employed was 638 d APPLIED MECHANICS. not an extremely soft steel, as shown by the few cases of steel rivets included in the Watertown Arsenal tests already quoted, where the shearing-strength per square inch varied from about 50000 pounds per square inch up to as high a figure as 65000 pounds per square inch; and by Kennedy's tests, where he ap- parently fixes on from about 49000 to about 54000 pounds per square inch as the shearing-strength of steel rivets. Now it would seem that metal with these shearing-strengths would have a tensile strength per square inch which would not warrant us in classifying it as very soft steel. On the other hand, it is evident that brittleness should not in any way be tolerated in rivet metal, and hence it would seem that at least soft steel should be used for rivets. The specifications proposed by the American Society for Testing Materials prescribe for tensile strength per square inch of steel for structural rivets from 50000 to 60000 pounds per square inch, and for boiler-rivets from 45000 to 55000 pounds. While the number of tests that have been made upon joints constructed with steel rivets is not large, the shearing-strength of such steel rivets as are in use to-day is not very far from 45000 pounds per square inch, as a rule. The number of tests of joints constructed with steel rivets is not sufficiently large to warrant drawing from them definite conclusions regarding the ultimate compression on the bearing surface in such joints. Meanwhile, it would be advisable to use for it the same values as are suitable in the case of joints made with steel plates and wrought-iron rivets. The following table contains the joints tested at Watertown Arsenal, which were made with steel plate and wrought-iron rivets, and in which the plate broke out in front of the rivet. It is evident that only four of them, viz., 4915, 4916, 4917, 4918, failed in consequence of excessive compression on the bearing surface, and that the breaking out of the plate in the other cases was due to insufficiency of lap. The calculated -j was obtained WIRE AND WIRE ROPE. 639 by the method described on page 554, assuming ^ = 55000, and ) 8 = 38000, and j c = 96000. "3 - C V o II II '3 s! u 1 d , A O i o t/j \i d ji* Kind of Joint. ^ o jj o e ^ a; "5 M 3, 01 1 C P E S \ liil ! S'S 3 fe d ' 1 IH 1 P 5: CQ ^(3 a 5 ftffl J 5*0 6 2 a OH ^ J J> ^3 113 in 5. II s. Ins. Ins. Ins. Lbs. 7 i8 Single lap Iron n l o P 1 f 1.25 795io Tore and 1.18 i.SS sheared 719 ' V P 25 80200 Tore .18 .55 4947 * D .00 i i 2970 .60 . 75 A 4948 D . oo 114760 Tore .60 -75 4949 ' D .00 120180 .60 77 767 1442 Single butt - P i 25 .00 1.25 2.00 95210 107610 Tore .67 .50 .65 1443 . r 5 .00 2 . OO 108830 5 . 70 49 1 5 ^ - D JL . oo 2 . OO 151780 29 93 4916 ' - D : J- .00 2.00 142570 2 9 .89 4917 4918 - : D D 1 ft . oo .00 2 . OO 2.00 158150 152110 . 29 29 .96 93 4985 r D 1 if A .00 I . OO 83950 . oo 57 4987 [ D 1 A .25 1 25 937io 25 .63 4991 I D J V 3, 75 I ~ ~ i 19500 75 77 298 Reinforced lap \ i D i i j -15 1 .00 [l.!2 67300 Sheared rivets .19 .66 299 \ * D 1 i J .10 1 .12 [l.I2 68040 ,, .19 .67 5121 Double lap I D 1 V 50 86180 5 .56 5122 1 P 1 V 50 86450 59 234. Wire and Wire Rope. It is well known that the process of making wire by cold drawing greatly increases the strength of the metal. Annealing, on the other hand, decreases the strength, and increases the ductility. It is not the purpose of this article to discuss the various qualities of wire required and used for different purposes. Hence, inasmuch as results of tests of wrought-iron, and of steel wire, have already been given, there will be given here only a few tests of hard-drawn, of semi-hard- drawn, and of soft copper wire. Wire rope. Wire rope is used for a great many purpose:, as in suspension bridges, in hoisting, in haulage, in the transmission of power, etc. While flat wire rope is used for some purposes, and while wire rope made of parallel wires is used in large suspension bridges, the greater part is made by twisting a number of wire 640 APPLIED MECHANICS. HARD-DRAWN COPPER WIRE. SOFT COPPER WIRE. Diameter. Tensile Strength Elastic Limit per Contrac- tion of per Sq. In. Sq. In. Area. Inches. Lbs. Lbs. Per Cent. 0.166 53050 37100 0.138 60350 22800 o-i35 56300 28150 o-i34 5 I0 5 2,7140 o. 105 61800 41000 5 r o. 105 57100 35000 49 o. 105 58900 34000 33 o . 106 60300 36000 42 o. 106 59500 34000 39 0.086 58170 27870 0.086 58620 29310 o . 08 > 6 1 5 10 2 T 7OO * 1 oV w o 083 66536 206^0 w w<kj O 0.083 65060 *y w o 37334 0.083 66536 29630 Diameter. Inches. Tensile Strength per Sq. In. Lbs. Elastic Limit per Sq. In. Lbs. Contrac- tion of Area. Per Cent. 0.163 o. 162 o. 162 35730 35770 36640 13760 I2QQO 0.083 IO5OO 0.083 0.081 o .080 o .080 29500 33200 33100 IJ2CO 70 45 SEMI-HARD-DRAWN COPPER WIRE. o. 106 44300 30000 60 o. 106 45100 29000 65 0.106 455oo 29000 55 o. 106 45100 31000 67 o. 106 44900 30000 64 strands around a central core, which may be of tarred hemp, or which may be, itself, a wire strand, the wire strands being made of wires twisted together. In the case of a wire core, the strength of the rope is a little greater, but the resistance to wearing is less. J The most usual number of strands is six, each strand contain- ing seven, eighteen, or nineteen wires, though other numbers of wires are sometimes used. The strength that can be realized in practice is always less than the strength of the rope, and is determined by the method of holding the ends, as the junction point of the rope and the holder is the weakest point. The usual methods of holding the ends are as follows: splic- ing, as in the case of the transmission of power, passing the rope around a pulley, or around a thimble, fastening it in a socket, or in a clamp. The diameter of the drum or sheave around which a rope WIRE AND WIRE ROPE. 641 passes, should not be so small as to cause too much stress to be exerted upon some of the wires, in consequence of the bending- moment introduced by the curvature. Inasmuch as it may be a matter of convenience to have here some tables giving the strength of rope as claimed by some makers, there will follow here two tables of the strength of different sizes, as given by the Roebling Company for their rope. The following explanations are given by the Roebling Com- pany, about the quality of the metal used : Iron, open-hearth steel, crucible steel, and plough steel pos- sess qualities which cover almost every demand upon the material of a wire rope. Copper, bronze, etc., are, however, used for a few special purposes. The strength of iron wire ranges from 45000 to 100000 pounds per square inch; open-hearth steel, from 50000 to 130000 pounds SEVEN-WIRE ROPE. Composed of 6 Strands and a Hemp Center, 7 Wires to the Strand. Approximate Breaking-strain in Tons of 2000 Lbs. Trade No. Diameter in Inches. Approxi- mate Circum- ference Weight per Foot in Pounds. Transmission or Haulage Rope. Extra in Inches. Swedish Cast Strong Cast Steel. Plough Steel. Iron. Steel. ii l| 4f 3-55 34 68 79 91 12 ! 4i 3.00 29 58 68 78 13 J i 4 2.45 24 48 56 64 14 I| 3* 2 .00 20 40 46 53 15 I 3 1-58 16 3 2 37 42 16 1 ai I . 20 12 24 28 3 2 17 f 2i 0.89 9-3 18.6 21 24 18 t* 4 0-75 7-9 15.8 l8. 4 21 19 f 2 0.62 6.6 13.2 I5-I 17 20 A if 0.50 5-3 10.6 12-3 14 21 j l -39 4.2 8.4 9.70 II 22 iV l| 0.30 3-3 6.6 7-50 8-55 23 1 1* 0.22 2.4 4-8 5.58 6-35 24 f I 0-15 i-7 3-4 3-88 4-35 25 A 4 o. 125 1.4 2.8 3.22 3-65 642 APPLIED MECHANICS. NINETEEN-WIRE ROPE. Composed of 6 Strands and a Hemp Center, 19 Wires to a Strand. Approximate Breaking-strain in Tons of 2000 Lbs. Approxi- Weight. Trade No. Diameter in Inches. mate Circum- ference in Inches. per Foot in Pounds. Standard Hoisting Rope. Extra Strong Cast Steel. Plough Steel. Swedish Cast Iron. Steel. 2| 8f n-95 305 2* 7l 9-85 254 i 7i 8.00 78 156 182 208 2 2 ^i 6.30 62 124 144 165 3 if si 4-85 48 96 112 128 4 If 5 42 84 97 ill 5 I* 4f 3-55 36 72 84 96 5* if 3.00 31 62 72 82 6 I* 4 2.45 25 5 58 67 7 3* 2 .00 2 I 42 49 56 8 I 3 I. 5 8 17 34 39 44 9 1 I .20 13 26 30 34 10 f 2 4 0.89 9-7 19.4 22 25 10* f 2 O.62 6.8 13-6 15-8 18 A if 0.50 5-5 II .0 12-7 14-5 lof o. 39 4-4 8.8 10. I 11.4 ioa A i* 0.30 3-4 6.8 7-8 8.85 lob | it 0. 22 2-5 5- 5-78 6-55 IOC T^> i 0.15 i . 7 3-4 4-05 4-5 lod i f 0. 10 I .2 2 -4 2.70 3.00 per square inch; crucible steel from 130000 to 190000 pounds per square inch; and plough steel from 190000 to 350000 pounds per square inch. Plough steel wire is made from a high grade of crucible cast-steel. 235. Other Metals and Alloys. Copper is, next to iron and steel, the me'al most used in construction, sometimes in the pure state, especially in the form of sheets or wire, but more frequently alloyed with tin or zinc; those metals where the tin predominates over the zinc being called bronze, and those where zinc predominates over tin, brass. Copper in the pure state was used not long ago for the fire-box plates of loco- IRON AND STEEL WIRE. 643 motive and other steam-boilers, "as it was believed to stand better the great strains due to the changes of temperature that come upon these plates, than iron or steel ; but now steel or iron has almost entirely superseded it for this purpose, except in some cases where the feed-water is very impure, and where the impurities are such as corrode iron. The alloys of copper, tin, and zinc which are used most where strength and toughness are needed, are those where the tin predominates over the zinc ; and the composition, mode of manufacture, and resisting properties of these metals, together with the effect of other ingredients, as phosphorus, have been very extensively investigated with reference to their use as a material for making guns, instead of cast-iron. Accounts of tests made on these alloys will be found as follows : Major Wade : Ordnance Report, 1856. T. J. Rodman : Experiments on Metals for Cannon. Executive Document No. 23, 46th Congress, 2d session. Materials of Engineering : Thurston. No attempt will be made to give a complete account of the results of these tests ; but a table will be given on page 639 for convenience of use, showing rough average values of the resist- ing powers of some metals and alloys other than iron. 236. Timber. However extensively iron and steel may have superseded timber in construction, nevertheless, there are many cases in which iron is entirely unsuitable, and where timber is the only material that will answer the purpose ; and in many cases where either can be used, timber is much the cheaper. Hence it follows that the use of timber in construc- tion is even now, and as it seems always will be, a very impor- tant item. Another advantage possessed by timber is, that, on yielding, it gives more warning than iron, thus affording an opportunity to foresee and to prevent accident. If we make a section across any of the exogenous trees, as 6 4 4 APPLIED MECHANICS. Specific Gravity. Tensile Strength per Sq. In. Modulus of plasticity. Brass cast . . . .... 8.7C-6 18000 9I7OOOO 49OOO 14230000 Bronze unwrought : 84.29 copper + 15.71 tin (gun metal) 82.81 " + 17.19 " 81.10 " + 18.90 " 78.97 " +21.03 " (brasses). . . 34.92 " + 65.08 " (small bells) . 15.17 " +84.83 " (speculum metal) Tin . . 8.561 8.462 8-459 8.728 8.056 7447 7 2QI 36060 34048 39648 30464 3 T 3 6 6944 c6oo Zinc . . .... 686l 7SOO Copper cast 8.712 24138 8.878 77OOO Copper wire . 60000 I7OOOOOO Gold cast IQ 2C8 2OOOO IO.476 40000 22.069 56000 Lead cast 1 1 7 C2 1800 the oak, pine, etc., we shall find a series of concentric layers ; these layers being called annual rings, because one is generally deposited every year. Radiating from the heart outwards will be found a series of radial layers, these being known as the medullary rays. Of the annual rings, the outer ones are softer and lighter in color than the inner ones ; the former forming the sap-wood, and the latter the heart-wood. When the log dries, and thus tends to contract, it will be found that scarcely any contraction takes place in the medullary rays ; but it must take place along the line of least resistance, viz., along the annual rings, thus causing radiating cracks, and drawing the rays nearer together on the side away from the crack. This action is exhibited in Fig. 241, where a log is shown with two saw-cuts at right angles to each other ; when this log bf ^mes dry, the four STRENG7*H OF TIMBER. 645 FIG. 241. right angles all becoming acute through the shrinkage of the rings. If the log be cut into planks by parallel saw-cuts, the planks will, after drying, assume the forms shown in Fig. 242, as is pointed out in Anderson's " Strength of Materials," from which these two cuts are taken. This internal construction of a plank has an important influence upon the side which should be uppermost when it is used for flooring ; for, if the heart side is up- permost, there will be a liability to having layers peel off as the wood dries : indeed, boards for flooring should be so cut as to have the an- nual rings at right angles to the side of the plank. Before discuss- ing any other considerations which affect the adaptability of timber to use in construction, we will con- sider the question of its strength. 237. Strength of Timber. In this regard we must observe, that, whereas the strength and elasticity and other properties of iron and steel vary greatly with its chemical com- position and the treatment it has received during its manufac- ture, the strength, etc., of timber is much more variable, being seriously affected by the soil, climate, and other accidents of its growth, its seasoning, and other circumstances ; and that over many of these things we have no control : hence we must not expect to find that all timber that goes by one name has the same strength, and we shall find a much greater variation and FIG. 242. 646 APPLIED MECHANICS. irregularity in timber than in iron. The experiments that have been made on strength and elasticity of timber may be divided into the following classes : i. Those of the older experimenters, except those made on full-size columns by P. S. Girard, and published in 1798. A fair representation of the results obtained by them, all of which were deduced from experiments on small pieces, is to be found in the tables given in Professor Rankine's books, " Applied Mechanics," " Civil Engineering," and " Machinery and Millwork." 2. Tests made by modern experimenters on small pieces. Such tests have been made by (a) Trautvvine : Engineers' Pocket-Book. (b} Hatfield : Transverse Strains. (c) Laslett : Timber and Timber Trees. (</) Thurston : Materials of Construction. (e) A series of tests on small samples of a great variety of American woods, made for the Census Department, and recorded in Executive Document No. 5, 48th Congress, ist session. Timber Physics, Division of Forestry, U. S. Department of Agri- culture. For a fairly complete bibliography of tests of tim- ber see a paper by G. Lanza, Trans. Am. Soc. C. E., 1905. 3. Tests made by Capt. T. J. Rodman, U.S.A., the results of which are given in the " Ordnance Manual." 4. All tests that have been made on full-size pieces. In regard to tests on small pieces, such as have commonly been used for testing, it is to be observed, that, while a great deal of interesting information may be derived from such tests as to some of the properties of the timber tested, nevertheless, such specimens do not furnish us with results which it is safe to use in practical cases where full-size pieces are used. Inas- much as these small pieces are necessarily much more perfect (otherwise they would not be considered fit for testing), having less defects, such as knots, shakes, etc., than the full-size pieces. Sl^RENGTff OF TIMBER. 647 they have also a far greater homogeneity. They also season much more quickly and uniformly than full-size pieces. In making this statement, I am only urging the importance of adopting in this experimental work the same principle that the physicist recognizes in all his work ; viz., that he must not apply the results to cases where the conditions are essentially different from those he has tested. Moreover, it will be seen in what follows, that, whenever full-size pieces have been tested, they have fallen far short of the strength that has been attributed to them when the basis in computing their strength has been tests on small pieces ; and, moreover, the irregularities do not bear the same propor- tion in all cases, but need to be taken account of. The results of the first class of experiments named in the following table are taken from Rankine's " Applied Mechanics ;" and, inasmuch as the table contains also the strengths of some other organic fibres, it will be inserted in full. The student may compare these constants with those that will be given later. Kind of Material. Tenacity or Resist- ance to Tearing. Modulus of Tensile Elasticity. Resist- ance to Crush- ing. Modulus of Rupture. Resist- ance to Shearing along Grain. Modulus oi Shearing Elasticity along the Grain. Ash I7OOO 1600000 QOOO ( 12000 I4OO 76000 Bamboo . . ( I4OOO Beech II5OO I-KOOOO 0160 ( 9000 1 _ Birch . . 1 5OOO 1 64 sooo 6400 ( I2OOO I I7OO ' Blue cum 8800 ( IbOOO 1 _ Box 2OOOO 10300 ( 2OOOO f Bullet-tree .... Cedar of Lebanon . . II4OO 486000 14000 5860 (15900 | 16000 7400 ': - 648 APPLIED MECHANICS. Kind of Material. Tenacity or Resist- ance to Tearing. Modulus of Tensile Elasticity. Resist- ance to Crush- ing. Modulus of Rupture. Resist- ance to Shearing along Grain. of 1 Shearing I Elasticity along the Grain. ( IOOOO ) Chestnut to / 1140000 - 10660 - - ( 13000 ) Cowrie - - - IIOOO - - Ebony .... _ _ I9OOO 77000 ( 700000 ) ( 6000 ) Elm . . 14000 5 to ) to < IU 76000 \ LV -' ( 1840000 ) 30C ( 9700 ) M Fir, Red pine . . . ( 12000 to f 14000 1460000 to 1900000 5375 to 6200 r 7100 ^ 9540 500 800 62OOO II6OOO " Yellow pine (Am.) - - 5400 - - - " Spruce .... 12400 ( 1400000 to ( 1800000 | - j 9900 ) 12300 | 600 - " Larch .... ( 9000 to ( IOOOO 900000 to 1360000 | 5570 1 IOOOO 970 I7OO \ - Hoxen yarn .... 25000 - - - - - Hazel 16000 ( I2OOO ) Hempen rope . . . to ( _ _ _ _ ( 16000 ) Ox-hide, undressed 6300 _ - - - - Hornbeam .... 20000 _ - - - - Lancewood .... 23400 - - - - - Ox-leather .... 42OO 24300- - - - - Lignum-vitae .... IISOO - 9900 I2OOO - - Locust ... . l6oOO _ Mahogany ( 8000 < to \ i 2 e CQOO 8200 j 7600 ) " ( 21800 ) -> Maple 10600 Oak, British .... - IOOOO ( IOOOO ] 13600 " Dantzic . . . - - 7700 8700 " European . . . ( IOOOO to ( 19800 f 1200000 C 1750000 } - - 2300 82OOO " American red 10250 2I5OOOO 6000 10600 STRENGTH OF TIMBER. 649 Resist- Modulus Kind of Material. Tenacity or Resist- ance to Tearing. Modulus of Tensile Elasticity. Resist- ance to Crush- ing. Modulus of Rupture. ance to Shearing along Grain. of Shearing Elasticity along the Grain. Silk fibre 52OOO I3OOOOO I 7OOO I O4OOOO 0600 Teak, Indian . . . I5OOO 24OOOOO I2OOO ( I2OOO \ IQOOO 1- - African . . . 2IOOO 2300000 - 14980 - - Whalebone .... 7700 - - - - - Willow .... 6600 Yew . ..... 8000 in regard to the tests of the second class, a few comments are in order : i. These experiments, like those of the first class, were all made upon small pieces ; and the results are correspondingly high. The usual size of the specimens for crushing being one or two square inches in section, and of those for transverse strength being about two inches square in section and four or five feet span, those for tension had even a much smaller sec- tion than those for compression ; as it is necessary, in order to hold the wood in the machine, to give it very large shoulders. The only exception to this is the tests of Sir Thomas Las- lett, an account of which is given in his "Timber and Timber Trees," and also in D. K. Clark's " Rules and Tables." In these tests he gives very much lower tensile strengths than those given above ; and he states that his specimens were three inches square, but does not say how he managed to hold them in such a way as to subject them to a direct tensile stress. His results for crushing and transverse strength are about as great as 650 APfLIED MECHANICS. those given in Rankine's tables, and as were obtained by the other experimenters on small pieces, as his specimens were of about the same dimensions as those used by the others. The figures obtained by these experimenters will only be given inci- dentally, as (a) They are very similar to those given in Rankine's table. (b) They are not suitable for practical use on the large scale. (c) While they have been used, it has only been done by employing a very large factor of safety for timber. The series of tests made for the Census Department, and recorded in Executive Document No. 5, 48th Congress, first session, form a very interesting series of experiments upon small specimens of an exceedingly large number of .America , woods. In order to have working figures, we should need to test large pieces of the same ; as the proportion between the strengths of the different kinds would be liable to be different in the latter case. The work done by the Division of Forestry of the U. S. Dept. of Agriculture before 1898 was mostly of this class, but little having been done with full-size pieces, and that with imperfect apparatus. The only record of Rodman's experiments available is a table of results in the " Ordnance Manual." These are lower, as a rule, than those obtained by the experimenters of the first or second class. This is to be accounted for by the fact that, while he did not experiment on full-size pieces, he used much larger pieces than those heretofore employed ; his specimens for transverse strength, many of which are still stored at the Watertown Arsenal, being 5f inches deep, 2$ inches thick, and 5 feet span. The fourth class of tests are those which furnish reliable data for use in construction ; 'and we will proceed to a consid- eration of these, taking up (1) tension, (2) compression, (3) transverse strength, and (4) shearing along the grain. TENSION. 65 I TENSION. In all cases where the attempt has been made to experiment upon the tensile strength of timber, a great deal of difficulty has-been encountered in regard to the manner of holding the specimens. In all cases it has been found necessary to pro- vide them with shoulders, each shoulder being five or six times as long as the part of the specimen to be tested, and to bring upon these shoulders a powerful lateral pressure, to prevent the specimen from giving way by shearing along the grain, and pulling out from the shoulder, instead of tearing apart. The specimens tested have generally had a sectional area less than one square inch, and it seems almost impossible to provide the means of holding larger specimens. This being the case, it is plain, that, whenever timber is used as a tie-bar in construction (except in exceedingly rare and out-of-the- way cases), it will give way by some means other than direct tension ; i.e., either by the pulling-out of the bolts or fastenings, and the consequent shearing of the timber, or else by bending if there is a transverse stress upon the piece ; and, this being the case, these other resistances should be computed, instead of the direct tension. Hence, while the direct tensile strength of timber may be an interesting subject of experiment, it can serve hardly any purpose in construction ; and the conclusion follows, that the resistances of timber to breaking we may expect to meet in practice are its crushing, transverse, and shearing strength. Indeed, the use of timber for a tie-bar should be avoided whenever it is possible to do so ; and, when it is used, the calculations for its strength should be based upon the pulling-out of the fastenings, the shearing or splitting of the wood, etc., and not on the tensile resistance of the solid piece. Moreover, when a wooden tie-bar is subjected not merely to direct tension, but also to a bending-moment, whether the latter is caused by a transverse load, or by an eccentric pull, as it generally is in the case of timber joints, we must compute 652 APPLIED MECHANICS. the greatest tension per square inch at the outside fibre due to the bending, and to that add the direct tension per square inch: and this sum must be less than the modulus of rupture if the piece is not to give way; i.e., the modulus of rupture and not the ultimate tensile strength per square inch must be our criterion of breaking in such a case, the working- strength per square inch being the modulus of rupture divided by a suitable factor of safety. COMPRESSIVE STRENGTH. Tests of the compressive strength of full-size wooden columns, with the exception of one set of tests, date from about 1880. TESTS OF FULL-SIZE COLUMNS. The following are references to tests of full-size timber columns: i. Trautwine, in his "Handbook," speaks of some tests of wooden pillars 20 feet long and 13 inches square, made by David Kirkaldy, which, as he says, gave results agreeing with Mr. C. Shaler Smith's rule. 2. A series of tests made at the Watertown Arsenal for the Boston Manufacturers' Mutual Fire Insurance Company, under the direction of the author. 3. Eleven tests of old spruce pillars made at the Watertown Arsenal, for the Jackson Company, under the direction of Mr. J. R. Freeman, and reported in the Journal of the Assoc. Eng. Societies for November, 1889. 4. The tests that have been made at the Watertown Arsenal on the government testing-machine. 5. Tests made in the Laboratory of Applied Mechanics of the Massachusetts Institute of Technology. 6. A series of tests of full-size columns of oak and fir, made by P. S. Girard in 1798. In regard to the first, no details or results are given: hence nothing will be said about them. In regard to the second, a summary only will be presented here. TESTS OF YELL