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References andHistorical Notes

1-1 Timoshenko, S. P., History of Strength of Materials,Dover Publications, Inc., New York, 1983 (originally pub-lished by McGraw-Hill Book Co., Inc., New York, 1953).

Note: Stephen P. Timoshenko (1878–1972) was afamous scientist, engineer, and teacher. Born in Russia, hecame to the United States in 1922. He was a researcher withthe Westinghouse Research Laboratory, a professor at theUniversity of Michigan, and later a professor at StanfordUniversity, where he retired in 1944.

Timoshenko made many original contributions, boththeoretical and experimental, to the field of applied mechanics,and he wrote twelve pioneering textbooks that revolutionizedthe teaching of mechanics in the United States. These books,which were published in as many as five editions andtranslated into as many as 35 languages, covered the subjectsof statics, dynamics, mechanics of materials, vibrations,structural theory, stability, elasticity, plates, and shells.

1-2 Todhunter, I., and Pearson, K., A History of the Theoryof Elasticity and of the Strength of Materials, Vols. I and II,Dover Publications, Inc., New York, 1960 (originallypublished by the Cambridge University Press in 1886 and1893). Note: Isaac Todhunter (1820–1884) and Karl Pear-son (1857–1936) were English mathematicians and

educators. Pearson was especially noteworthy for his origi-nal contributions to statistics.

1-3 Love, A. E. H., A Treatise on the Mathematical Theoryof Elasticity, 4th Ed., Dover Publications, Inc., New York,1944 (originally published by the Cambridge UniversityPress in 1927); see “Historical Introduction,” pp. 1–31.

Note: Augustus Edward Hough Love (1863–1940)was an outstanding English elastician who taught at OxfordUniversity. His many important investigations included theanalysis of seismic surface waves, now called Love wavesby geophysicists.

1-4 Jacob Bernoulli (1654–1705), also known by thenames James, Jacques, and Jakob, was a member of thefamous family of mathematicians and scientists ofBasel, Switzerland (see Ref. 9-1). He did important workin connection with elastic curves of beams. Bernoulli alsodeveloped polar coordinates and became famous forhis work in theory of probability, analytic geometry, andother fields.

Jean Victor Poncelet (1788–1867) was a Frenchmanwho fought in Napoleon’s campaign against Russia andwas given up for dead on the battlefield. He survived, wastaken prisoner, and later returned to France to continue hiswork in mathematics. His major contributions to mathemat-ics are in geometry; in mechanics he is best known for hiswork on properties of materials and dynamics. (For thework of Bernoulli and Poncelet in connection with stress-strain diagrams, see Ref. 1-1, p. 88, and Ref. 1-2, Vol. I,pp. 10, 533, and 873.)

1-5 James and James, Mathematics Dictionary, Van Nos-trand Reinhold, New York (latest edition).

1-6 Robert Hooke (1635–1703) was an English scientistwho performed experiments with elastic bodies and devel-oped improvements in timepieces. He also formulated thelaws of gravitation independently of Newton, of whom hewas a contemporary. Upon the founding of the RoyalSociety of London in 1662, Hooke was appointed its firstcurator. (For the origins of Hooke’s law, see Ref. 1-1,pp. 17–20, and Ref. 1-2, Vol. I, p. 5.)

S. P. Timoshenko(1878–1972)

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R2 References and Historical Notes

1-7 Thomas Young (1773–1829) was an outstandingEnglish scientist who did pioneering work in optics, sound,impact, and other subjects. (For information about his workwith materials, see Ref. 1-1, pp. 90–98, and Ref. 1-2,Vol. I, pp. 80–86.)

1-8 Siméon Denis Poisson (1781–1840) was a greatFrench mathematician. He made many contributions toboth mathematics and mechanics, and his name is pre-served in numerous ways besides Poisson’s ratio. Forinstance, we have Poisson’s equation in partial differentialequations and the Poisson distribution in theory of proba-bility. (For information about Poisson’s theories of materialbehavior, see Ref. 1-1, pp. 111–114; Ref. 1-2, Vol. I,pp. 208–318; and Ref. 1-3, p. 13.)

2-1 Timoshenko, S. P., and Goodier, J. N., Theory of Elas-ticity, 3rd Ed., McGraw-Hill Book Co., Inc., New York,1970 (see p. 110). Note: James Norman Goodier(1905–1969) was well known for his research contributions

to theory of elasticity, stability, wave propagation insolids, and other branches of applied mechanics. Born inEngland, he studied at Cambridge University and later atthe University of Michigan. He was a professor at CornellUniversity and subsequently at Stanford University, wherehe headed the program in applied mechanics.

2-2 Leonhard Euler (1707–1783) was a famous Swissmathematician, perhaps the greatest mathematician of alltime. Ref. 11-1 gives information about his life and works.(For his work on statically indeterminate structures, seeRef. 1-1, p. 36, and Ref. 2-3, p. 650.)

2-3 Oravas, G. A., and McLean, L., “Historical develop-ment of energetical principles in elastomechanics,” AppliedMechanics Reviews, Part I, Vol. 19, No. 8, August 1966,pp. 647–658, and Part II, Vol. 19, No. 11, November 1966,pp. 919–933.

2-4 Louis Marie Henri Navier (1785–1836), a famousFrench mathematician and engineer, was one of thefounders of the mathematical theory of elasticity. He con-tributed to beam, plate, and shell theory, to theory ofvibrations, and to the theory of viscous fluids. (See Ref. 1-1,p. 75; Ref. 1-2, Vol. I, p. 146; and Ref. 2-3, p. 652, for hisanalysis of statically indeterminate structures.)

2-5 Piobert, G., Morin, A.-J., and Didion, I., “Commissiondes Principes du Tir,” Mémorial de l’Artillerie, Vol. 5,1842, pp. 501–552.

Note: This paper describes experiments made by firingartillery projectiles against iron plating. On page 505appears the description of the markings that are the slipbands. The description is quite brief, and there is no indica-tion that the authors attributed the markings to inherentmaterial characteristics. Guillaume Piobert (1793– 1871)was a French general and mathematician who made manystudies of ballistics; when this paper was written, he was acaptain in the artillery.

2-6 Lüders, W., “Ueber die Äusserung der elasticität anstahlartigen Eisenstäben und Stahlstäben, und über einebeim Biegen solcher Stäbe beobachtete Molecularbewe-gung,” Dingler’s Polytechnisches Journal, Vol. 155, 1860,pp. 18–22.

Note: This paper clearly describes and illustrates thebands that appear on the surface of a polished steelspecimen during yielding. Of course, these bands are onlythe surface manifestation of three-dimensional zones ofdeformation; hence, the zones should probably be charac-terized as “wedges” rather than bands.

3-1 The relationship between torque and angle of twistin a circular bar was correctly established in 1784 byCharles Augustin de Coulomb (1736–1806), a famous

Thomas Young(1773–1829)

S. D. Poisson(1781–1840)



French scientist (see Ref. 1-1, pp. 51–53, 82, and 92, andRef. 1-2, Vol. I, p. 69). Coulomb made contributions inelectricity and magnetism, viscosity of fluids, friction,beam bending, retaining walls and arches, torsion andtorsional vibrations, and other subjects (see Ref. 1-1,pp. 47–54).

Thomas Young (Ref. 1-7) observed that the appliedtorque is balanced by the shear stresses on the cross sec-tion and that the shear stresses are proportional to thedistance from the axis. The French engineer AlphonseJ. C. B. Duleau (1789–1832) performed tests on bars intorsion and also developed a theory for circular bars (seeRef. 1-1, p. 82).

5-1 A proof of the theorem that cross sections of a beam inpure bending remain plane can be found in the paper byFazekas, G. A., “A note on the bending of Euler beams,”Journal of Engineering Education, Vol. 57, No. 5, January1967. The validity of the theorem has long been recognized,and it was used by early investigators such as Jacob Bernoulli(Ref. 1-4) and L. M. H. Navier (Ref. 2-4). For a discussion ofthe work done by Bernoulli and Navier in connection withbending of beams, see Ref. 1-1, pp. 25–27 and 70–75.

5-2 Galilei, Galileo, Dialogues Concerning Two NewSciences, translated from the Italian and Latin into Englishby Henry Crew and Alfonso De Salvio, The MacmillanCompany, New York, 1933 (translation first published in1914.)

Note: This book was published in 1638 by LouisElzevir in Leida, now Leiden, Netherlands. Two NewSciences represents the culmination of Galileo’s work ondynamics and mechanics of materials. It can truly be saidthat these two subjects, as we know them today, began withGalileo and the publication of this famous book.

Galileo Galilei was born in Pisa in 1564. He mademany famous experiments and discoveries, including those

References and Historical Notes R3

on falling bodies and pendulums that initiated the scienceof dynamics. Galileo was an eloquent lecturer and attractedstudents from many countries. He pioneered in astronomyand developed a telescope with which he made many astro-nomical discoveries, including the mountainous characterof the moon, Jupiter’s satellites, the phases of Venus, andsunspots. Because his scientific views of the solar systemwere contrary to theology, he was condemned by thechurch in Rome and spent the last years of his life in seclu-sion in Florence; during this period he wrote Two NewSciences. Galileo died in 1642 and was buried in Florence.

5-3 The history of beam theory is described in Ref. 1-1,pp. 11–47 and 135–141, and in Ref. 1-2. Edme Mariotte(1620–1684) was a French physicist who made develop-ments in dynamics, hydrostatics, optics, and mechanics. Hemade tests on beams and developed a theory for calculatingload-carrying capacity; his theory was an improvement onGalileo’s work, but still not correct. Jacob Bernoulli(1654–1705), who is described in Ref. 1-4, first determinedthat the curvature is proportional to the bending moment.However, his constant of proportionality was incorrect.

Leonhard Euler (1707–1783) obtained the differentialequation of the deflection curve of a beam and used it tosolve many problems of both large and small deflections(Euler’s life and work are described in Ref. 11-1). The firstperson to obtain the distribution of stresses in a beam andcorrectly relate the stresses to the bending moment proba-bly was Antoine Parent (1666–1716), a French physicistand mathematician. Later, a rigorous investigation ofstrains and stresses in beams was made by Saint-Venant(1797–1886); see Ref. 2-10. Important contributions werealso made by Coulomb (Ref. 3-1) and Navier (Ref. 2-4).

5-4 Manual of Steel Construction (ASD/LRFD), publishedby the American Institute of Steel Construction, Inc., OneEast Wacker Drive, (Suite 3100), Chicago, Illinois 60601.

C. A. de Coulomb(1736–1806)

Galileo Galilei(1564–1642)




(For other publications and additional information, go totheir website: www.aisc.org.)

5-5 Aluminum Design Manual, published by the AluminumAssociation, Inc., 900 19th Street NW, Washington, D.C.20006. (For other publications and additional information,go to their website: www.aluminum.org.)

5-6 National Design Specification for Wood Construction(ASD/LRFD), published by the American Wood Council, adivision of the American Forest and Paper Association,1111 19th Street NW, (Suite 800), Washington, D.C. 20036.(For other publications and additional information, go totheir websites: www.awc.org and www.afandpa.org.)

5-7 D. J. Jourawski (1821–1891) was a Russian bridge andrailway engineer who developed the now widely usedapproximate theory for shear stresses in beams (seeRef. 1-1, pp. 141–144, and Ref. 1-2, Vol. II, Part I, pp. 641–642). In 1844, only two years after graduating from theInstitute of Engineers of Ways of Communication inSt. Petersburg, he was assigned the task of designing andconstructing a major bridge on the first railway line fromMoscow to St. Petersburg. He noticed that some of thelarge timber beams split longitudinally in the centers of thecross sections, where he knew the bending stresses werezero. Jourawski drew free-body diagrams and quickly dis-covered the existence of horizontal shear stresses in thebeams. He derived the shear formula and applied his theoryto various shapes of beams. Jourawski’s paper on shear inbeams is cited in Ref. 5-8. His name is sometimes translit-erated as Dimitrii Ivanovich Zhuravskii.

5-8 Jourawski, D. J., “Sur la résistance d’un corps prisma-tique . . . ,” Annales des Ponts et Chaussés, Mémoires etDocuments, 3rd Series, Vol. 12, Part 2, 1856, pp. 328–351.

5-9 Zaslavsky, A., “On the limitations of the shearingstress formula,” International Journal of Mechanical Engi-neering Education, Vol. 8, No. 1, 1980, pp. 13–19. (Seealso Ref. 2-1, pp. 358–359.)

5-10 Maki, A. C., and Kuenzi, E. W., “Deflection andstresses of tapered wood beams,” Research Paper FPL 34,U. S. Forest Service, Forest Products Laboratory, Madison,Wisconsin, September 1965, 54 pages.

6-1 Augustin Louis Cauchy (1789–1857) was one of thegreatest mathematicians. Born in Paris, he entered theÉcole Polytechnique at the age of 16, where he studiedunder Lagrange, Laplace, Fourier, and Poisson. He wasquickly recognized for his mathematical prowess, and atage 27 he became a professor at the École and a member ofthe Academy of Sciences. His major works in pure mathe-matics were in group theory, number theory, series,integration, differential equations, and analytical functions.

In applied mathematics, Cauchy introduced the con-cept of stress as we know it today, developed the equationsof theory of elasticity, and introduced the notion ofprincipal stresses and principal strains (see Ref. 1-1,pp. 107–111). An entire chapter is devoted to his work ontheory of elasticity in Ref. 1-2 (see Vol. I, pp. 319–376).

6-2 See Ref. 1-1, pp. 229–242. Note: Saint-Venant was apioneer in many aspects of theory of elasticity, and Tod-hunter and Pearson dedicated their book, A History of theTheory of Elasticity (Ref. 1-2), to him. For further informa-tion about Saint-Venant, see Ref. 2-10.

6-3 William John Macquorn Rankine (1820–1872) was bornin Edinburgh, Scotland, and taught engineering at GlasgowUniversity. He derived the stress transformation equations in1852 and made many other contributions to theory of elastic-ity and applied mechanics (see Ref. 1-1, pp. 197–202, andRef. 1-2, Vol. II, Part I, pp. 86 and 287–322). His engineeringsubjects included arches, retaining walls, and structural theory.

Rankine also achieved scientific fame for his workwith fluids, light, sound, and behavior of crystals, and he isespecially well known for his contributions to molecularphysics and thermodynamics. His name is preserved by theRankine cycle in thermodynamics and the Rankine absolutetemperature scale.

6-4 The famous German civil engineer Otto ChristianMohr (1835–1918) was both a theoretician and a practicaldesigner. He was a professor at the Stuttgart Polytechnikumand later at the Dresden Polytechnikum. He developed thecircle of stress in 1882 (Ref. 7-5 and Ref. 1-1, pp. 283– 288).

Mohr made numerous contributions to the theory ofstructures, including the Williot-Mohr diagram for trussdisplacements, the moment-area method for beam deflec-tions, and the Maxwell-Mohr method for analyzingstatically indeterminate structures. (Note: Joseph VictorWilliot, 1843–1907, was a French engineer, and JamesClerk Maxwell, 1831–1879, was a famous British scientist.)

6-5 Mohr, O., “Über die Darstellung des Spannungs-zustandes und des Deformationszustandes eines Körperele-mentes,” Zivilingenieur, 1882, p. 113.

8-1 The work of Jacob Bernoulli, Euler, and many otherswith respect to elastic curves is described in Ref. 1-1,pp. 27 and 30–36, and Ref. 1-2. Another member of theBernoulli family, Daniel Bernoulli (1700–1782), proposedto Euler that he obtain the differential equation of thedeflection curve by minimizing the strain energy, whichEuler did. Daniel Bernoulli, a nephew of Jacob Bernoulli, isrenowned for his work in hydrodynamics, kinetic theory ofgases, beam vibrations, and other subjects. His father, JohnBernoulli (1667–1748), a younger brother of Jacob, was anequally famous mathematician and scientist who first



References and Historical Notes R5

formulated the principle of virtual displacements, andsolved the problem of the brachystochrone.

John Bernoulli established the rule for obtaining the lim-iting value of a fraction when both the numerator anddenominator tend to zero. He communicated this last rule toG. F. A. de l’Hôpital (1661–1704), a French nobleman whowrote the first book on calculus (1696) and included this the-orem, which consequently became known as L’Hôpital’s rule.

Daniel’s nephew, Jacob Bernoulli (1759–1789), alsoknown as James or Jacques, was a pioneer in the theory ofplate bending and plate vibrations.

Much interesting information about the many promi-nent members of the Bernoulli family, as well as otherpioneers in mechanics and mathematics, can be found inbooks on the history of mathematics.

Euler’s differential equation, Euler’s equation of a varia-tional problem, Euler’s quadrature formula, the Eulersummation formula, Euler’s theorem on homogeneousfunctions, Euler’s integrals, and even Euler squares (squarearrays of numbers possessing special properties).

In applied mechanics, Euler was the first to derive theformula for the critical buckling load of an ideal, slendercolumn and the first to solve the problem of the elastica.This work was published in 1744, as cited previously. Hedealt with a column that is fixed at the base and free at thetop. Later, he extended his work on columns (Ref. 11-2).Euler’s numerous books include treatises on celestialmechanics, dynamics, and hydromechanics, and his papersinclude subjects such as vibrations of beams and plates andstatically indeterminate structures.

In the field of mathematics, Euler made outstandingcontributions to trigonometry, algebra, number theory,differential and integral calculus, infinite series, analyticgeometry, differential equations, calculus of variations, andmany other subjects. He was the first to conceive of trigono-metric values as the ratios of numbers and the first to presentthe famous equation eiu � cos u � i sin u. Within his bookson mathematics, all of which were classical references formany generations, we find the first development of thecalculus of variations as well as such intriguing items as theproof of Fermat’s “last theorem” for n � 3 and n � 4. Euleralso solved the famous problem of the seven bridges ofKönigsberg, a problem of topology, another field in whichhe pioneered.

Euler was born near Basel, Switzerland, and attendedthe University of Basel, where he studied under JohnBernoulli (1667–1748). From 1727 to 1741 he lived andworked in St. Petersburg, where he established a great

Jacob Bernoulli(1654–1705)

Leonhard Euler(1707–1783)

9-1 Euler, L., “Methodus inveniendi lineas curvas maximiminimive proprietate gaudentes . . . ,” Appendix I, “Decurvis elasticis,” Bousquet, Lausanne and Geneva, 1744.(English translation: Oldfather, W. A., Ellis, C. A., andBrown, D. M., Isis, Vol. 20, 1933, pp. 72–160. Also,republished in Leonhardi Euleri Opera Omnia, series 1,Vol. 24, 1952.)

Note: Leonhard Euler (1707–1783) made manyremarkable contributions to mathematics and mechanics,and he is considered by most mathematicians to be the mostproductive mathematician of all time. His name,pronounced “oiler,” appears repeatedly in present-day text-books; for instance, in mechanics we have Euler’s equationsof motion of a rigid body, Euler’s angles, Euler’s equationsof fluid flow, the Euler load in column buckling, and muchmore; and in mathematics we encounter the famous Eulerconstant, as well as Euler’s numbers, the Euler identity(eiu � cos u � i sin u), Euler’s formula (eip � 1 � 0),




reputation as a mathematician. In 1741 he moved to Berlinupon the invitation of Frederick the Great, King of Prussia.He continued his mathematical research in Berlin until theyear 1766, when he returned to St. Petersburg at the requestof Catherine II, Empress of Russia.

Euler continued to be prolific until his death in St.Petersburg at the age of 76; during this final period of hislife he wrote more than 400 papers. In his entire lifetime,the number of books and papers written by Euler totaled886; he left many manuscripts at his death and they contin-ued to be published by the Russian Academy of Sciences inSt. Petersburg for 47 years afterward. All this in spite of thefact that one of his eyes went blind in 1735 and the other in1766. The story of Euler’s life is told in Ref. 1-1,pp. 28–30, and some of his contributions to mechanics are

described in Ref. 1-1, pp. 30–36 (see also Refs. 1-2, 1-3, 2-2, and 5-3).

9-2 Euler, L., “Sur la force des colonnes,” Histoire de L’A-cadémie Royale des Sciences et Belles Lettres, 1757,published in Memoires of the Academie, Vol. 13, Berlin,1759, pp. 252–282. (See Ref. 11-3 for a translation anddiscussion of this paper.)

9-3 Van den Broek, J. A., “Euler’s classic paper ‘On thestrength of columns,’” American Journal of Physics,Vol. 15, No. 4, July–August 1947, pp. 309–318.

9-4 Keller, J. B., “The shape of the strongest column,”Archive for Rational Mechanics and Analysis, Vol. 5, No. 4,1960, pp. 275–285.

