History of Strength of Materials

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HISTORY OF STRENGTH OF MATEHIALS

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Preface

JIISTORY OF STRENGTH OF MATERIALS Copyright, 1953, by the McGraw-Hill Book Con~~a~y, Inc. l'rinted in the United States of America. All rights reserved. 1 !us book, or parts thereof, may oot be reproduced in any form without permission of the publishers.

uibrar11 of Co11greaa Catd i11 ),t'ihnitz's p11blirnt ion "Acln .Eruditorum Lipsiee," 1694, he put his fin nl ,·e1-sio11 of the problem in the " llisloire de l'Acad6mie des Sciences de Paris," 1705. See also "Collected \forks of J, Bcrno11lli," vol. 2, p. !176, Gration of bars and a demonstration of the fact that large amplitudes

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for the maximum deflection and observes t.hat the second term in the bracket, representing the effect of shearing force, is of a practical importance only in comparatively short beams. In debating the question of selecting safe stresses, Poncelet favors the maximum strain theory and asserts that failure occurs when the maximum strain reaches a definite limit.. Thus the condition of rupture in the case of compression of such brittle materials a-S stone and cast iron rests upon the lateral expansion. The maximlLm strain theory was consistently used later by Saint-Venant and found extensive use on the Continent, whereas English writers continued to base their design calculations on that of maximum stress. Poncelet's rcsea.rches also embraced the theory of structures. In discussing the stability of retaining walls, he offers a graphic method of finding the maxi.mum pressure on the wall. 2 In dealing with the stresses in arches, he was the first to point out t hat a rational stress a nalysis can be 1 See p. 317 of the third edition of "Introduction a. la Mecaniquc Industrielle," Paris, 1870. • See his "Memoire s111· la stnhilit6 des revetements et de lours fondations" in ilfem. o.f/icier genie, 13, J s,10.

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I-Iislory of Slrenglh of Nlaleriab;

Slrenglh of Malerials between 1800 and 1833

developed only by considering an arch as au elastic cw·ved bar (see page 323) .

suppose t haLhe possessed it; but spoke as if he took it for granted thaL we all understood t he matter as well as he did. . . . His language was col'rect his utterance rapid, and his senten ces, t,hough without any aliectation'. never left unfinished. But his words were not Lhose in fami]jar use, and t,he arraugemcnt of his ideas seldom the same as those he conversed with. He was, t h erefol'e, worse calculated than a ny man I ever knew for the communication of knowledge. . . . It was difficult to say how he employed himself; he read liUle, and though he had access to the College an d University libraries, he was seldom seen in t hem. There were no books piled on his floor, no pape1·s scattered on his table, a nd his room had all the appearance of belonging to an idle man . . . . Re seldom gave an opinion, and never volun teered one. A philosophical fact, a difficult calculation, an ingenious instrument, or a new invent.ion, would engage his atten tion; but he never spoke oI morals, of metaphysics, or of religion." Very early Young started his original scientific wot-k. As early as 1793 a paper of his on t he theory of sight was presented to the Royal Soci~ty. "While at Cambridge (in 1798) Young became interesLed i.J.1 sound. Regarding this work he writes: " I have been studying, not the theory of the winds, but of tbe ail-, and I have made observations on harmonics which I believe arc new. Several circumstances unknown to the English mathematicians which I thought I had fast discovered, I since find Lo have been discovered and demonstrated by Lho foreign mathematicians; in fact., Britain is very much behind its neighbours in many branches of the mathematics: were I Lo apply deeply to them, I would become a disciple of the French and German school; but the field is too wide and too barren for me." The paper "Outlines and Experiments Respecting Sound and Light" was prepared at, Cambridge in the summer of 1799 and was read to the Royal Society in January of the following year. In 1801, Young made his famous discovery of the interference of light. His great knowledge of the physical sciences was recognized when, in L802, he was elected a member of the Roya l Society. The same year he was installed as a professor of 11atural philosophy by the Royal Institution. This institution was founded in 1799 "for diffusing tho knowledge and facilitating the general and speedy introduction of new and useful mechanical inventions and improvements and also for teaching, by regular courses of philosophical lectures and experiments, the applicatio ns of tho new discoveries in science to the improvements of arts and manufactures and in facilitating th e means of procuring the comforts and conveniences of life.'' As a lecturer at the Royal Institution, Young was a failure; fol· his presentation was usually too terse and he seemed unable to sense and dwell upon those parts of his subject t hat were t ho most cal-

22. Thomas Yoimg (1773-1829) T h omas Young1 was born in a Quaker family at Milverton, Somerset. As a child he showed a remarkable capacity for learning, especially in mastering languages and mathematics. Before he was fomteen years old, he had a knowledge not only of modern languages buL also of Latin, Greek, Arabic, Persian, and Hebrew. \ From 1787 to 1792 he earned his living working for a rich family as a tutor. This position left him sufficient free time to continue his studies, and he worked hard in the fields of philosophy and mathematics. In 1792, Young started to study medicine, first in London and Edinburgh and lalser at Gottingen University, where he Fm. G3. 'I'homus Young. received his doctor's degree in 1796. Jn. J797, after his return to England, he was admitted as a Fellow Commoner of Emmanuel College, Cambridge, and spent some time studying there. A man who knew Young well at that Lime describes him in the following words: 2 "When the Master introduced You_ng to his tutors, ho jocularly said: 'I have brought you a pupil qualified to read l ectures to _ bis tutors.' This, however, he did not attempt, and the forbearance was mutual; be was never required to attend the common duties of the college. . . . The views, objects, character, and acquirements of our mathemat icians were very different then to what they are now, and Young, •. who was certainly beforehand with the world, perceived their defects. Certain it is, that be looked down upou t he science and would not cultivate the acquaintance of any of ow· philosophers. . . . Ile never obtruded his various learning in conversation; b ut if appealed to on tho most difficult subject, he answered in a quick, flippant, decisive way, as if he was speaking of the most easy; and in this mode of talking he differed from all the clever men that I ever saw. His reply never seemed to cost him an effort, and he did not appear to think there was any credit in being able to make it. He did not assert any superiority, or seem to 1 A very interesting biogi:aphy of Thomas Young was written by G. Peacock, London, 1855. 'Sec Lhc book of G. Peacock.

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Ilislory of Strength of Materials

culated to cause difficulty. He resigned his professorship in 1803, but continued to be interested in natural philosophy and prepared his course of lectures for publication. Young's principal contributions to mechanics of materials are to be found in this coursc. 1 In the chapter on passive strength and friction (vol. 1, p. 136), the principal types of deformation of prismatical bars arc discussed. In considering tension and compressiou, the notion of modulus of elasticity is introduced for the first time. The definition of this quantity differs from that which we now use to specify Young's modulus. It states: "·The modulus of the elasticity of any substance is a column of the same substance, capable of producing a pressure on its base which is to the weight causing a certain degree of compression as the length of the substance is \ to the climinuLion of its length." Young also speaks of the "weight of the modulus" and the "height of the modulus" and notes that the height of the modulus for a given material is independent of the crosssectional area. The weight of the modulus amounts to the product of the quantity wh ich we now call Young's modulus and the cross-sectional area of the bar. 2 Describing experiment-a on tension and compression of b1us, Young draws the attention of his readers to the fact that longitudinal deformations are always accompanied by some change in the lateral dimensions. Introducing Hooke's Jaw, he observes that it holds only up to a certain limit beyond which a part of the deformation is inelastic and constitutes permanent set. Regarding shear forces, Young remarks that no direct tests have been made to establish the relation between shearing forces and the deformations that they produce, and states: "It may be inferred, however, from the properties of twisted substances, that the force varies in the simple ratio of the distance of the par ticles from their natural position, and it must also be simply proportional to the magnitude of the sul'face to which it is applied." When circular shafts are twisted, Young points out that the applied torque is mainly balanced by shearing stresses which act in the cross-sectional planes and are proportiona l to distance from the a:-.is of the shaft and to the angle of twist. He also notes that an additional resistance to torque, proportional to the cube of the angle of twist, will be furnished by the longitudinal stresses in the fibers which will be bent 1

Thomas Young," A Cowsc of Lectures on Natural Philosophy and the Mechanical Arts," 2 vols., London, 1807. The most interesting part of the material dealing with the deform ation of beams was not included in the new edition which was edited by Kclland. t Young had determined the weight of the modulus of steel from the frequency of vibn1tiou of a tuning fork and found it equal to 29 X 10' lb per in.t See "Natural Philosophy," vol. 2, p. 86.

Strength of Materials between 1800 and 1833

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to helices. Because of this, the outer fibers will be in tension and the inner fibers in compression. Further, the shaft will be shortened during torsion "one fom·th as much as the external fibers would be extended if the lengt h remained undiminished." 1 In discussing bending of cantilevers and beams supported at their two ends, Young gives Lhe principal results regarding deflections and strength, without derivation. llis treatment of the lateral buckling of compressed columns includes the following interesting remark: "Oonsidern.ble irregularities may be observed in all the experiments which have been made on the flexure of columns and rafters exposed to longitudinal forces; and there is no doubt but that some of them were occasioned by the difficulty of applying the force precisely at the extremities of the a>..is, and others by the accidental inequalities of the substances, of which the fibres must often have been in such directions as to constitute originally rather bent than straight columns." When considering inelastic deformations, Young makes an important statement: "A permanent alteration of form .. . limits the strength of materials with regard to practical purposes, almost as much as fracture, since in general the force which is capable of producing this effect is sufficient, with a small addition, to increase it till fracture takes place." As we have seen, Navier came to the same conclusion and proposed that working stresses should be kept. much lower than the elastic limit of the material. In conclusion, Yow1g gives an interesting discussion of the fracture of elastic bodies produced by impact. In this case, not the weight of the striking body but the a mount of its kinetic energy must be considered. "Supposing the dfrection of the stroke to be horizontal, so that its effect may not be increased by t.he force of.gravity," he concludes that "i£ the pressure of a weight of 100 lb (applied statically) broke a given subs tance, after extending it through the space of an inch, the same weight would break it by striking it with the velocity that would be acquired by the fall of a heavy body from the height of half an inch, and a weight of one pound would break it by falling from a height of 50 in." Young states that, when a prismatical bar is subjected to longitudinal impact, the l'esilience of t.he bar "is pl'Oportional to its length, since a similar extension of a longer fiber produces a greater elongation." Further on, he finds that "there is, however, a limit beyond which tbe velocity of a body stl'iking another· cannot be increased without overcoming its resilience and breaking it, however small the bulk of the first body may be, and this limit depends on the inertia of the parts of the second body, which must not be 1 The correct value of the shortening is twice as large as that given by Young. "Strength of Materials," vol. 2, p. 300.)

(See

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Strength of Materials between 1800 and 1833

disregarded when they are impelled with a considerable velocity." Denoting t.hevelociiywit.h which a compr ession wave travels along a bar by l' and the velocity of the striking body by v, he concludes that the unit compression prndueed at the end of the bar at the instant of impact is equal to v/ V, and that the limiting value for the velocity vis obtained by equating the ratio v/ V to the unit compression at which fracture of tho material of the struck bar occurs i.11 static tests. Considering the effects of impact upon a rectangular beam, Young decides that, fot· a given maximum bending stress generated by the blow, the quantity of energy accumulated in the beam is proportional to its volume. This is because the maximum force P, produced on the beam by I.he striking body, and the deflection oat the point of impact are given by the formulas p = k bhz

tai.ns col'l'ect solutions of several important problems, which were new in Young's time. For example, we sec the solution oft.he problem of eccentric tension or compression of rectangular bars for the first time. Assuming that the stress distribution in this case is represented by two triangles, as shown in Fig. 64c, Young determines the position of the neutral axis from the condition that the resultant of these st.resses must pass through the point O of application of I.he external force (Fig. 64b). This gives

l

where b is the widt.h, h the depth, and l the length of the beam; k and k 1 are constants depending on the modulus of the material and ou the assu med magnitude of the maximum stress. Substituting into the expression for the strain energy U, we obtain Pli

U = -

2

= -k 2k1- bhl 2

which proves the above statement. We see t hat Young conti-ibuted much to strength of. materials by int.roducing the notion of a modulus in tension and compression. He was also the pioneer in analyzing stresses bt;ought about by impact and gave a method of calculating them for perfectly elastic mate1fals which follow Hooke's law up to fracture. Some more complicated problems concerning the bending of bars are discussed in the chapter "Of the Equilibrium and Strength of Elastic Substances" in volume II of "Natural Philosophy." The following remarks regarding this chapter are to be found in the "History of the Theory of Elasticity" by I. Todhunter and K. Pearson: "This is a series of theorems which in some cases suffer under the old mistake as to the position of the neutral surface. . . . The whole section seems to 'm e very obscurn like most of the writings of its distinguished a uthor·; among his vast attainments in sciences and languages that of expressing himself clearly in the ordi nary dialect of mathematicians was unfortunately not included. The formulae of the section ,vere probably mainly new at tho time of their appearance, but they were little likely to gain attention in consequence of the unattractive form in which they were presented." I t is agreed that this chapte1· of Young's book is difficult to read; but it con-

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Ji2 a= 12e

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where e is the eccentricity of the force. When e = h/ 6, a = h/ 2. The stress distribution is represented by a triangle, and the maximum stress becomes twice as great as that caused by applying the load centrally. The correct value of the radius of the circle to which the axis of I.he bar in Fig. 64a will be bent, in the case of very small deflectious, is also derived in Young's analysis. As his next problem Young takes up the bending of a compressed prismatical column which is initially slightly FIG. 64. curved. Assuming that the initial curvatme may be represented by one-half wave of the sine curve, 50 sin (1rx/ l ), be finds that the deflection at the middle after application of the compressiveforceP will be

oo

0= 1 - (Pl2/Efr2)

From this result he concludes that if P = E!?r 2/ l 2, "the deflection becomes infinite, whatever may be the magnitude of oo , X and the force will overpower the beam, or will at least cause it to bend so much as to derange the operation of the forces concerned." Ilere we have the first. P derivation of the column equation allowing for lack of initial straightness. In using Euler's formula for determining the cross-sectional dimensions of a column, Young points out that its application is limited to 7hl77-i7J'?/;---Y slender columns and gives certain limiting values of the Fro. 66. ratio of the length of the column to its depth. For the smaller values of this ratio, failure of the column is brought about by crushing of the material rather than by buckling. If a slender column is built-in at one end and free at the other, Thomas Young shows that, when an axial force P acts eccentrically (as in Fig. 65), the deflection y is given by

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History of Strength of Materials

Strength of Materials between 1800 and 1833

_ e(l - cos p x) y cos pl

is governed by the magnitude of t he cross-sectional moment of inertia, the strength by the section modulus and the resilience by the cross-sectional area. Analogously he solves the problem for thin circular tubes. Young states: "Supposing a tube of evanescent thickness to be expanded into a similar tube of greater diameter, but of equal length, the quantity of matter remaining the same, the strength will be increased in the ratio of the diameter, and the stiffness in the ratio of the square of the diameter, but the resilience will remain unaltered." From this discussion we see t hat the chapter on mechanics of materials in the second volume of "~atural Philosophy" contains valid solutions of several important problems of strength of materials, which were completely new in Young's Lime. This work did not gain much atten tion from engineers because the a uthor's presentation was always brief and seldom clear. Before leaving the subject of this book, some remarks regarding Young's "Natmal Philosophy" will be q uoted. They come from the pen of Lord Rayleigh's biographcr. 1 In 1892, Lord Rayleigh was appointed as professor aL the Royal Institution and his lectures "followed somewhat closely the lines of those which had been given by Thomas Young in the same place nearly a century before, at the birth of the Royal Institution. These were fully written out in Young's Natural Philosophy, published in 1807, and many of the identical pieces of demonstration appa ratus figured in that work, which bad been preserved in t he museum of the Institution, were brought out and used. . . . Rayleigh had studied Young's Lectures and had found them a mine of interesting matter. The pencil marks in his copy show how closely he had gone over the book, and in the lecture he brought to notice some of the good but forgotten work which be had found there and also in other writings of the same author. One of the most striking points was Young's estimate of the size of molecules; (he gave) the molecular diameter as between t he two-thousand and the ten-thousand millionth of an inch. This is a wonderful anticipation of modern knowbdge, and antedates by more than fifty years the similar estimates of molecular dimensions ma de by Kelvin. Until Rayleigh drew attention to it, it had fallen completely out of notice, even if we suppose that it had ever attracted notice, and of this there is no evidence." Young showed his unusual ability not only in solving purely scientific problems but also in at tacking practical engineering difficulties. For instance, he presented a report to the Board of Admiralty dealing with the use of oblique 1ide1•5, and concerning other alterations in the construction of sh ips. 2 He t reats the hull of a ship as a beam, assumes a. definite

In this formula, e denotes the eccentricity, p = vP/ El, and X is measured from the built-in end. fo this case then, Young is ahead of Navier.1 In his treatment of la teml buckling of columns having variable cross sections, Young shows that a column of constant depth \Vill bend to a circular curve (Fig. 66a) if its width (Fig. 66b) varies along the length as the ordinates y of a circula r arch. _Consid~l'ing the buckling of a column consisting of two triangular pnsms (Fig. 67) and assuming that the deflection is such that the radius p

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p (b)

(a)

Fro. 66.

p F ro. 67.

of cw-vature at the ~ddle is equal to half the over-all length, Young states that the deflection cmve is a cycloid. This follows from the equation

(a) The moment of inertia I of a ny cross sections is proportional to 8 a and the ordinates y of the cycloid arc proportional to 5 21 where s is defined as shown in Fig. 67. Now the left-hand side of Eq. (a) has the dimensions of a number divided bys. Since p = 8 for a cycloid, we can always select such a value for P that Eq. (a) will be satisfied. Wh,~n a rectan~ular beam~ cut out of a given circular cylinder, Young finds thaL the stiffest beam 18 that of which the depth is to the breadth as the square root of 3 to 1, and the strongest as t he square root of 2 to l · bu~ the most resilien t will be that which has its depth and breadth equal.': This follows from the fact that for a given length the stiffness of the beam 1

The solution was giYcn by Navier in a memoir presented to the Academy in 1819. See Samt-Venan t's " l fo1tory," p. 118.

1

"Life of Lord Rayleigh" by his son , foUJ·th Daron Rayleigh, p. 234, 1924. 7'ra11s., 1814, p. 303.

2 Pln'l.

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Slrenglh of Materials be/ween 1800 and 1833

distribution of weight and a definite shape of waves, and calculates the magnitudes of the shearing forces and bending moments at several cross sections. He also shows how the deflection of a ship can be calculated. He explains that the flexural rigidity of a ship depends, to a great extent, upon the connections between the parts and remarks: "A coach spring, consisting of ten equal plates, would be rendered ten times as strong, if it were united into one mass, and at the same time a hundred times as stiff, bending only one hundredth of an incb with the same weight that would bend it a wl10le inch in its usual state, although nothing would be gained by the union with respect to the power of resisting a ve1y rapid motion, which I have, on another occasion, ventured to call resilience. Now it appears to be extremely difficult to unite a number of parallel planks so firmly together, by pieces crossing them at right angles, as completely to prevent their sliding in any degree over each other: and a diagonal brace of sufficient strength, even if it did not enable the planks to bear a greater strain without giving way, might still be of advantage, in many cases, by diminishing the degree in which the whole structure would bend before it broke." Basing his arguments upon such considerations, Young favors the adoption of diagonal bracing in the constmction of wooden ships. It seems that this report constitutes the fil'st attempt at applying theoretical analysis to the structural design of ships. After this outline of Young's contributions to the theory of strength of materials, we can perhaps appreciate Lord Rayleigh's 1 remark: "Young . . . from various causes did oot succeed in gaining due attention from his contemporaries. Positions which be had already occupied were in more than one instance reconquered by his successors at a great expense of intellectual energy."

in which the bending is taken into account. Mr. Fuss has treated the same subject with relation to carpent1y in a subsequent volume. But there is little in these papers besides a dry mathematical disquisition, proceeding on assumptions which (to speak favomably) are extr~mel_Y gratuitous. The most important consequence of lhe compression 1s wholly overlooked, as we shaU presently see. Our knowledge of the mechanism of cohesion is as yet far too imperfect to entitle us to a confident applicaLion . . . . " and further, "'-Ne are thus severe in our observations because his theory of the strength of columns is one of the strong~t evidences of this wanton kind of proceeding and because his followers in the Academy of St. Petersburgh, such as Mr. Fuss, Lexell and others, adopt hls conclusions, and merely echo his words." Such a criticism, as Todhunter remarks, is very idle in a writer, who on the very 'next page places the neutral line on the concave surface of a beam subjected to flexure. . . . It was Robison who perpetuated (in the English textbooks) this cnor that had already been corrected by Cottlomb. However, although English theoretical work on strength of materials was of such poor quality, British engineers had to solve many important engineering problems. The country was ahead of others in industrial development and the introduction of such materials as cast iron and wrought iron into structural and mechanical engineering presented many new problems and called for investigations of the mechanical properties of these new materials. A considerable amount of experimental work was completed and the results that were compiled found use not only in England but also in France. A book entitled" An Essay on the Strength and Stress of Timber" was written by Peter Barlow (1776 1862), the fast edition of which appeared in 1817. The book was very popular in England and was reedited many times. 1 At the beginning of the book, Barlow gives a history of strength of materials using material from Girard's book (see page 7); but while Girard devotes much space in his book lo Euler's study of elastic cmves, Barlow is of the opinion that Euler's "instruments of analysis" are "too delicate to operate successfully upon the materials to which Lhey have been applied; so that while they exhibit under the strongest point of view, the immense resources of analysis, and the transcendent talents of their author, they unfortunately furnish but little, very little, useful iuformation." In discussing the theory of bending, Barlow makes the same error as Duleau did (see page 81) and assumes that the neutral line divides the cross section of the beam in such a way that Lhe moment aboul this lino

23. Strength of Male1'ials in Enyland between L800 and 1833

During the period 1800-1833 England had no schools equivalent to the ll:colc Polytechnique and the ~colc des Ponts ct Chaussees in France. The country's engineers did not receive a broad scientific education and the level of its textbooks Oil strength of materials was much lower than that in France. In describing those textbooks, the English historian of theory of elasticity, Todhunter, states: "Nothing shews more clearly the depth to which English mechanical knowledge had sunk at the commencement of this century. " 2 To illustrate the standard of English authors of this time, T odhunter quotes the following criticism of Euler's work from Robison's book "A System of Mechanical Philosophy": "In the old Memofrs of the Academy of Petersburgh for 1778 there is a dissertation by Euler on the subject, but particularly limited to the strain on columnsI l

"Collected Papers," vol. l, p. 190. "History . . . , " vol. I, p. 88.

2 See

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1 The sixth edition of tbe book, prepared by Peter Barlow's two sons, was published in 1867. This edition is of considerable interest in so far as the history of otu· subject is concerned, for it contains a biography of Peter Barlow and, in an appendix, the work of WilUs entitled "Essay on the Effects Produced by Causing Weights to Travel Over Elastic Bars."

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of all tensile stresses is equal lo the moment of compressive stresses. This en-or is corrected in the 1837 edition of Lhe book. Barlow attribu tes the conect determination of the neul.ral line to Hodgkinson. Ile seems still to have been ignorant of the fact Lhat Coulomb had made this correction fifty years before. Of Barlow, Todhunter says: "As a theorist he is another striking example of that want of clear thinking, of scicul.ilic accuracy, and of knowledge of the work accomplished abt·oad, which rnnders the perusal of the English text-books on practical mechanics published in the first half of this century, such a dispiriting, if not hopeless, task to the historian of theory." It seems that Todhunter is too se\rerc in I.bis criticism, especially in view of the fact that, al that time, engineers of such caliber as Duleau and Navier were making similar nristakes in the determination of the neutral a.xis. , The book does not add anything to the theory of strength of materials, but it contains descriptions of many experiments I.hat were made in the first half of the nineteenth century and which are of historical interest. Thus we find a reporl. on the tests made on iron wires by Badow for Telford who was then planuing the construction of the Runcorn suspension bridge. The author's experiments with iron rails of different shapes are also very interesting. In them the deflections of rails under rapidly moving loads were measmed for the first l.ime and compared with statical results. In the last edition of Barlow's book (1867) the important fatigue tests on plate girders, carried out by Fairbairn, are mentioned. R. Willis's report on the dynamic action of a moving load on a beam is also included. These investigations will be discussed later. Another English engiuccr whose books were very popular in the first half of the nineteenth ceutmy was Thomas Tredgold (1788- 1829). His work "The Elementary Principles or"Carpent.ry" (1820) contains a chapter dealing with strength of materials. There is not. much new work to be found in it and all the theoretical results can be seen in Thomas Young's "Lectures" (see page 92). Another book by Tredgold "A Practical Essay on the Strength of Cast Iron" (1822) contains I.he 1·esults of the author's experiments and gives many practical rules for designers of cast-iron structures. It seems that Tredgold was the first to introduce a formula for calculating safe stresses for colu1ID1s (see page 209). Ju the period from 1800 to 1833 the use of cast iron in structures rapidly increased and Tredgold's book was widely used by practical engineers. It was translated into French, Italian, and German. 24. Other N otalJle Ew·opean Contributions to Strength of .l'vlaterials

Work on strength of materials was started in Germany as late as t he beginning of the nineteenth century, and the first important contribution

Strength of .Materials between 1800 and 1833

101

to this science was made by Franz Joseph Gerstner (1756- 1832) who graduated from Prague Universit.y in 1777. After several years of practical engineering work, Gerstner became an assistant at the astronomy observatory of his old u niversity. In 1789, ho became professor of mathematics. While in this position, he continued to be interested in practical applications of science in engineering and planned the organization of an engineering school in Pmgue. In 1806, "das Bomische technische Institut" was opened in t hat city and Gerstner was active up to 1832 as a professor of mechanics and as t he director. Gerstnor's main work in mechanics is incorporated in his three-volume book "Handbuch der Mechanik," Prague, 1831. The third chaptel" of t he fu·st volume of this work deali, with strength of materials. Discussing tension and compression, Gerstner gives the results of his experimental work on tho tension of pianoforte wires. He finds that. the 1·elation existing between the tensile force P and the elongation o is

P = ao - bo 2

(a)

where a and b are two constan t..s. For small elongatio11s, the second term on the right-hand side of Eq. (a) can be neglected and we obtain H ooke's law. Gerstner also studied the effect of permanent set on the property of tensile-test specimens. He showed Lhat, if a wire is stretched so that a certain permanent set is produced and then unloaded, it will follow Hooke's law up to the load which produced the initial permanent set during a second loading. Only for higher loads than that will deviation from the straight-line law occur. In view of this fact, Gerstner recommends that wires to be used in suspension bridges should be stretched to a certain Limit before use. Gerstner did important work in other branches of engineering mechanics. Of this the best known is his theory of trochoidal waves on water surfaces. 1 Another German engineer who contributed to our knowledge of engineering mechanics in the early days of the ni11eteenth centmy, was Johann Albert Eytelwein (1764-1848). As a boy of fifteen he became a bombardier in the artillery regiment in Berlin. In his spare time he studied mathematics and engineering science and was able, in 1790, to pass the examination for the engineer's degree (architect). Very quickly he gathered a reputation as an engineer with great. theoretical knowledge. In 1793 his book "Aufgaben grosstentheils a.us der angewandten Mathematik II was published; it contained discussions of many problems of practical importance in strnctural and mechanical engineer' See his "Theorie der Wellen," Prague, 1804.

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flislory of Strength of Materials

Strength of Materials between 1800 and 1833

ing. In 1799, he and several other engineers organized the Bauakadcmie in Berlin and he became the director of that institute and its professor of engineering mechanics. Eytclwein's course " H andbuch der Mechimik fester K arper" was published in 1800, a nd the " Ifandbu ch der Statik fester Korper" was published in 1808. In the later h andbook, problems of strength of materials and t heory of structures are discussed. R egarding this strength of materials section, Todhunter remarks in his "History": "There is nothin,,. original in this section, but its author possesses t he advantage over B:nks 1 and Gregory• of being abreast of the mathematical knowledge of his day." In theory of s tructures, Eytelwei11 makes some additions to the theory of arches and to the theory of retaining walls. Ile also d evelops a useful method for determining the allowable load on a pile. After the · apoleonic wars an intensive movement developed for building up the various branches of industry in Germany. This required a considerable improvement in engineering education and, to this end, several schools were buill. 1n 1815, the P olytccbnical Institute in Vienna was opened under the directorship of J ohan J oseph P rechtl (1778- 1854), an d the Gewerheinstitut in Berlin was organized in 1821. These were followed by the founding of polytcchnical institutes in Karlsrnhe (1825), Munich (1827), Dresden (1828), H anover (1831), and Stuttgart (1840) . These new engineering schools greatly affected the development of German industry and indeed of engineering sciences in general. In no other country did there exist such close contact bet ween industry and engineering education as in Germany, and doubtless this cont11cL contributed much to the industrial progress of that country and to the establishment of its str ong position in engineering sciences at the end of the nineteenth century. British , German, and French progress in the field of strength of nmterials was followed by a similar developmen t in Sweden. The iron industry is of great. importauce in that coun try and the first experimental i1JVestigations there dealt with that metal. In the van of t his activity, we find the Swedish physicist. P. L agerhjelm. 2 Ile was acquainted wiLh t he theoretical work of Thomas Young, Duleau, and Eytclwcin and was guided in his investigations not only by practical but also by t heoretical considerations. The tensile-testing machine constructed by Broiling to his design proved very saLisfactory; so much so in fact t hat later a similar

machine was constructed for Lame's experiments in St. Petersburg (see page 83). Lagerhjelm's tests showed t hat t he modulus of elasticity in tension is about the same for all kinds of iron and is independent of such technological processes as rolling, hammering, and t he various kinds of heat-treatment but that at the same time the elastic limit and ultimate strength can be changed radically by these processes. lie found that t he ultimate strength of iron is usually propor tional to its elastic limit. He found that the density of it-on at t he point of rupture is somewh at, diminished. Lagerhjclm compared the modulus of elasticity obtained from static tests with that calculated from the observed frequency of lateral v ibrations of a bar and found good agreement. H e also demonstrated that if Lwo tuning forks give the same note, they continue to do so after one has been hardened.

L02

Contemporary English authors of books on engineering mechanics. Lagerhjclm's wo,-k was published in J em,- Kontorets Ann., 1826. A short account of his results was given in Poyg. Ann. Physik 11. Chem., Bd. 13, p. 4011, 1828. Later Swedish experimenters refer to Lagerhjcln1's work. For example, see IC Styffe, who gives a very complete description of Lagcrhjelm's machine for tensile test.~ in J em• Konlcrels Ann., 1866. 1

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105

The Beginning of lhe Malhemalical Theory of Elasticity

CJIAPTER V

The Beginning of t he Mathematical Theory of Elasticity i _

25. Equations of Equilibrium in the Theory of Elasticity

By the theory of strength of materials, the history of which we have so far traced, solutions which deal with prnblcms of t he deflection and stresses in beams were usually obtained on the assumption that cross sections of the beams remain plane during deformation and that the material follows Hooke's law. At the beginning of the nineteenth century, attempts were made to give the mechanics of elastic bodies a more fundamental basis. The idea bad existed 2 since Newton's time, that the elastic property of bodies can be explained in terms of some attractive and repu lsive forces bet.ween their ultimate pa1tides. This notion was expatiated by Boscovich 3 who assumed that between every t wo ultimate particles and along the line connecting them forces act which are attractive for some distances and repulsive for others. Fm-thermore, there are distances of equilibrium for which these forces vanish. Using this theory, with the added requirement that the molecular forces diminish rapidly with incrP,ase of the distances between the molecules, Laplace was able to develop hls theory of C!\pillarity. 4 The application of Boscovicb's theory in analyzing the deformations of elastic bodies was initiated by Poisson 6 in his investigation of bending of plates. He considers the plate as a system of particles distributed in the middle plano of the plate. Howevor, such a system will resist exten1 A more com pleto djscussion of the history of t he theory of elasticity can be foun.d in Saint-Venant's notes to t he third edition of Navier's book, " R~ um6 des Le90ns . . . ," 1864, in the notes to Saint.-Vennnt's translation of Cle bsch's '"fheorie de l'61astieit6 des corps solides," 1883, and in S1tint-Venant's two chap ters in Moigno's book, "Le