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STEPHEN P. TIMOSHENKO December 23' 1878-May 29' 1972 BY C. RICHARD SODERBERG THE MAJOR FACTS of the life of Stephen P. Timoshenko are by now well known. He was born as Stepen Prokof- yevich Timoshenko* in the village of Shpotovka in the Ukraine on December 23, IS78. Stephen's father, born a serf, had been brought up in the home of a landowner, who later married Stephen's aunt. His father subsequently received an education as a land surveyor and practiced this profession until he himself became a lanclowner of some means. Timoshenko's early life seems to have been a happy one, in pleasant rural surroundings. The conclucling decades of the nineteenth century were a periocI of relative tranquility in Russia, and the educational ideals of the micldie class were not much different from, and certainly not inferior to, those of their counterparts in Western Europe. He concluded his secondary education with a goicl mecial at the technical realschule~f in Romny, near Kiev. His father hacl rented an NOTE: The Academy would like to express its gratitude to Dr. I. P. Den Hartog for his help in the preparation of this memoir after the death of C. Richard Soderberg in 1979. *The spelling of Russian names and terms follows that of E. H. Mansfield and D. H. Young, "Stephen P. Timoshenko," in: Biographical Memoirs of Fellows of the Royal Society, vol. 19 (London: The Royal Society, 1973), pp. 679-94. t Professor Tichvinsky informs me that this designation was already used in Russia to indicate preparation for entry into technical universities. In most of Western 323

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324 BIOGRAPHICAL MEMOIRS apartment there in which young Stephen and a friend] lived, together with Stephen's grandmother. His outstanding sub- ject appears to have been mathematics. His enjoyment in helping classmates with their studies anticipated his subse- quent desire to become a teacher. Timoshenko had already cieveloped an intuition for goocT teaching, but his early ambi- tion was to become a railway engineer. His language studies were less successful, anct his Russian had a strong Ukrainian accent, but he appears to have been well read in the Russian classics. One of the principal objectives of the technical realschule was to prepare for the entrance examinations to institutions of higher learning. In IS96 Timoshenko took the exami- nation to enter the Institute of Engineers of Ways of Com- munication at St. Petersburg, which he seems to have passed with honor.* The trip to St. Petersburg, on which he was accompanied by his mother, opener] his eyes to the outside worIcl and began his indefatigable habit of traveling and visit- ing cathedrals, harbors, bridges, and the like, which was to continue to the enct of his life. His five years (1896-1901) at the Institute of Ways of Communication were spent in intensive and single-minded studies in the sequence of subjects in engineering. But this period also marked the beginning of the enc! of the years of political tranquility in Russia, anct soon there were many inci- dents of student unrest. Timoshenko always took the liberal view, but one gets the impression that he also regarded these incidents as obstacles to his own professional clevelopment. Europe distinction was made between the science-oriented realschule and the humanities-oriented latinschule. The technical realschule, which came later, was a "normal" school where science received greater emphasis than in a gymnasium. * To play safe he also applied to the Institute of Civil Engineers, which gave easier examinations. The admission rate was twenty to thirty students from among one thousand who had taken entrance examinations, and only good students dared to take these exams.

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STEPHEN P. TIMOSHENKO 325 He graduated in 1901; before this he had maple two trips to Western Europe (luring vacation periods. These trips stimu- latecI him greatly; they were the beginning of close as- sociations with outstanding professionals, particularly in Germany. His military service, which started in September 1901, after his second trip abroad, seems to have been a much more constructive experience than that in corresponding systems in most European nations at the time. He did not have to drop his professional contacts completely, and he actually started his career as a teacher at the Institute during this period. Following his military service, in 1902, he married ATex- andra Archangelskaya, a student in medical school and an acquaintance from his student days. At that time, he was serving in the Mechanics Laboratory of the Ways of Commu- nication Institute, where, in addition to his duties in testing of materials, he also participated in supplementary lectures in mathematics given by Professors Stanevich and Bobylev, among others. This noncompulsory program of studies seems to have been important: it brought him into contact with several young physicists, and he also began to attend the sessions of the Physical Society. It made him aware of an issue in engi- neering education that has remained important ever since. The engineers needed a much more mature background in science, particularly in mathematics, but the professional mathematicians of the period pursued very abstract lines of thought that often failed to attract the interest of the engi- neering students. The physicist Aleksey N. Krylov appears to have been one of the first to clarify this situation for Timoshenko. Later in his life, and at a more advanced stage, Felix Klein inspires] him the same way. Timoshenko, mean- while, developed his own utilitarian attitude toward] mathe- matics.

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326 BIOGRAPHICAL MEMOIRS In 1903 Timoshenko was made an instructor in the newly organized St. Petersburg Polytechnic Institute; the following years at this Institute mark the beginning of his creative scientific work. He spent the summers in Europe, mostly in Germany, where he receives] important inspiration from August Fopp} in Munich, Ludwig Prandt! and Felix Klein at Gottingen, ant! others. In his autobiography,* Timoshenko occasionally criticizes] some of the lecture courses at the In- stitute in St. Petersburg, but he mentions several of the outstanding teachers, such as Prince Gagarin, the Director of the Institute, who gave English lessons using results in Love's Theory of Elasticity as exercises, and Viktor Kirpichev, who induced him to read G. Lame, Bernhard Riemann, F. Gras- hof, an(l, perhaps most important, Lord Rayleigh's "Theory of-Sound." Kirpichev's influence on Timoshenko was very important; through him he was introcluce(l to the Castigliano theorem and the Rayleigh-Ritz method. These influences finally induced Timoshenko to become a teacher rather than . . . a practicing engineer. The school year 190~1905 was much influenced by po- litical turmoil in Russia- the disastrous Japanese War in- creased student demonstrations and general unrest. The Polytechnic Institute in St. Petersburg was closed, and Timoshenko decided to use the time studying under Prandt} at Gottingen. PrandtI, already a professor at twenty-nine, had contributed to Timoshenko's main subject of interest at the *Stephen P. Timoshenko, As I Remember (New York: Van Nostrand, 1968). The original of this autobiography was written in Russian after Timoshenko's-trip to Russia in 1958. It was published in Paris in 1963. The foreword to the Russian edition was written by Eugene A. Vetchorine, chairman of the Association of Graduates of St. Petersburg Polytechnic Institute. The translation into English was by Robert Addis under the guidance of Professors`~. M. Gere and D. H. Young of Stanford University. The volume contains a complete listing of Timoshenko's publi- cations as of March 1967. It also contains a list of I~imoshenko's doctoral students in the United States.

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STEPHEN P. TIMOSHENKO 327 time: the buckling of beams. Here Timoshenko made his first creative discovery in connection with the buckling of I-beams, where the torsion of rectangular elements of the section had to be taken into account. But by this time, Prandt! had left the fielcI to concentrate on his epoch-making work in connection with boundary layers in fluicI flow. It is remarkable, and to some clegree characteristic of Timoshenko's single-minded devotion to his own studies, that he makes only a passing reference to this important event. The stay in Gottingen was important in many other re- spects. Felix Klein had succeeded in expressing his conviction of the necessity for strong links between abstract and appliecI sciences. The School of Philosophy at Gottingen had already establishect an Institute of AppliecI Mathematics (Car} Runge), Applier! Mechanics (PrandtI), and Electrical Engi- neering (Simon). The impressions from these developments patterned his attitude toward education in technology and contributed much to his future development as a teacher. The situation in Russia continued to be characterized by much political unrest ant] in the summer of 1906 Timo- shenko resumes! his studies at Gottingen, extending them to potential theory, thermodynamics, anti other areas, while continuing his work on elastic stability and buckling. In the fall of 1906 he was appointed to the Chair of Strength of Materials at the Polytechnic Institute in Kiev. The return to his native Ukraine turned out to be an important part of his career anti also influenced his future personal life. He was elected clean of the Division of Structural Engineering in 1909; he never ceased to regret the inevitable interference . , , . , , , . . . . ~ . - . - . with his own work that this position brought. Political unrest again began to be felt, however, and his icieas of academic freedom now came uncler scrutiny. In 1911 this conflict led to his dismissal from the school, together with two other professors. Ten professors then resigned their positions in

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328 BIOGRAPHICAL MEMOIRS the spirit of solidarity, so strong were their beliefs in the right of the cause. These difficulties notwithstancling, his scientific work matured, particularly the ideas of strain energy and the Rayleigh-Ritz method in their application to buckling. In his own lectures, begun in 1907, he had gradually developed the technique of beginning a course with the simplest and most elementary concepts, gradually leading to more complicated and sophisticated methods of analysis. His first textbook ap- pearecl in 191 I, a year that marked the beginning of a period of more than ten years of uncertainty, anxiety, and hardship. One bright event was his winning the ~ourowski Mecial in 191 l, along with 2,500 gold rubles.* In the fall of 1911, he went to St. Petersburg and suc- ceeded in getting part-time teaching work, meanwhile con- .. . . . . . ~` . ~ , ~ truing nits writing. During the summer of 1912, he and his wife decided to use the gold rubles from the Jourowski Prize for a trip abroad. This journey was extended to England, where Timoshenko attended a mathematical congress in Cambriclge. He met, for the first time, Lord Rayleigh, A. E. H. Love, Horace Lamb, and Levi Civita, among others. One of the lectures at the congress was given by the young representative from Gottingen, Theoclor von Karman. Timoshenko found himself hampered by lack of fluency in English, a lack he determined to remedy as soon as possible. He did not have a scientific discussion with Lorc! Rayleigh, chancellor of the University, except as part of the crowd at a reception in the University museum. Returning to Russia in the fall of 1912, he resumed part- time teaching in several schools in St. Petersburg. He was also invited to teach at the Ways of Communication Institute, * This illustrates again the courage and solidarity of the Russian men of science, who awarded their top prize to a man dismissed from his job.

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STEPHEN P. TI MOSHEN KO 329 where he took over some of the work in theoretical mechanics from Professor Krylov, who tract retired. But events overtook him before he really got started on this program; he clic] not resume this work until he had emigrated to the United States. During this period he also worked on applications of elastic stability to ship bulkheacls as a consultant to the Naval Minis- try in St. Petersburg. These activities improved his financial position, but the living conditions in St. Petersburg were ctif- f~cult, and his health was impaired. During this period he happened to meet Paul Ehrenfest, with whom he formed a lasting friendship. Ehrenfest hacI come from Germany to Russia in the hope that academic freedom there wouIc! offer him better opportunities. He was not successful in establish- ing himself in Russia, but the two met frequently; Ehrenfest would then enlarge upon the current icleas abroad in physics- relativity and quantum physics new to Timo- shenko at the time. Timoshenko's period of disgrace with the authorities ended in 1913, when he was confirmed in the position of professor at the Ways of Communication Institute, an ap- pointment later extenclec] to inclucle teaching at the Electrical Engineering Institute of the Polytechnic Institute. In the autumn of 1913 he was asker to reorganize the teaching of strength of materials, succeeding Professor Mitinsky. By the summer of 1914, he considered! this task completecl and went off with his family on a well-earned vacation at Khapsalw on the Baltic. There he completed the proofreading of Theory of Elasticity anti also continued his work on elastic stability relatecI to ship structures. This work constitutes the fore- runner of his books Theory of Elasticity, Elastic Stability, and Plates and Shells, published many years later in English. It was during this short vacation that WorIc;l War I began, and with it the beginning of the end of the oIct worIc! order. During the early war years, Timoshenko continued and

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330 BIOGRAPHICAL MEMOIRS intensifier] his consulting work, now clirectec! toward aspects of the war effort shipbuilcling problems for the navy ant] railroad] transportation- along with service to many military committees on technical matters. With the gradual clisin- tegration of the Russian social structure, living conditions worsened. In the spring of the year of the Revolution ~ 19 17), he sent his family to the Crimea, later joining them. But even there the events of the Revolution reached them. Subse- quently, he left his three children with his family in Kiev while his wife accompanied him to St. Petersburg. But she soon returned to Kiev to rejoin her family. During the Christ- mas holidays of ~ 9 ~ 7, he made a trip to Kiev to visit them; this turned out to be his final departure from St. Petersburg. Kiev was held by the Communists for a short while, but in March 1918 the German army took possession. Things im- provec! under German discipline, and Timoshenko was asked to resume his professorship at the Kiev Polytechnic Institute and to participate in the organization of the Ukrainian Acaclemy of Sciences. Before long, German society itself began to disintegrate, anal in 1919 there were rumors of the White Army advancing from the south. It arrived in Kiev at the ens! of August, and Timoshenko visited Rostov to negotiate further on matters of professional educa- tion with the leadership of the White Army. But it soon became apparent that this government lacked the necessary strength, and there was a great deal of mutual suspicion among Timoshenko's colleagues about the oIcI issue of a separate Ukrainian state. During this period Timoshenko was in frequent contact with many of his former students and colleagues, all despairing about the possibility of a return to order in Russia. This lect to his decision to leave Russia for good. Through his contacts in the Ukraine he was encouraged to flee to Yugoslavia, where there seemec} to be possibilities for a position. After many adventures he found himself pro-

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STEPHEN P. TIMOSHENKO 331 fessor at the newly establisher] School of Engineering in Zagreb; eventually he also succeeded in bringing his family there from Kiev. The years in Zagreb he regarded as pleasant in many ways, even though there were perpetual clifficulties clue to material shortages of all kinds. The stay in Zagreb from 1920 to 1922 also included visits to Western Europe and England, where he renewed his acquaintances with Love, R. V. Southwell, G. I. Taylor, and others. He also became acquainted with Piotr Kapitza cluring one of his trips to En- gland. On his return to Western Europe he visited Weimar to see his friend Ehrenfest, who now held the Chair of Physics at Leyden. Timoshenko's work in Zagreb required that he lecture in Croatian. His assistants translated his Russian lectures into Croatian, and he started the series by reacting them in Cro- atian. In this process the Timoshenko touch was lost. In the end he clecidect to deliver his lectures in Russian, using as many Croatian words as possible. Eventually his students could follow him without difficulty. During this period he also found it desirable to study more English. He and his English teacher started the task of translating some of his papers into English anc! sending them to Processor Love, who hac] them published in England. Through this process the name of Timoshenko began to be known to workers in ap- plied mechanics. To those of us who hearc! him lecture in English soon after his arrival in the United States, it became apparent that the Russian-Croatian combination was merely one example of his utilitarian approach to language. His career in Zagreb came to a sudden end in 1922, when he received "a letter from America from a pupil of mine at the Petersburg Polytechnic, one Zelov," ~ who was then work- *Viktor Zelov, whose original Russian name was Tselovalnikov, subsequently became a well-known industrialist in the United States and was founder and presi- dent of the Viz Manufacturing Company in Germantown, Pennsylvania.

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332 BIOGRAPHICAL MEMOIRS ing with the Vibration Specialty Company, whose president, Akimoff, was familiar with Timoshenko's work. Timoshenko was offered a position with this company and arrived in Philadelphia alone in June 1922. America depressed and frightened! him. His work at the Vibration Specialty Com- pany, although well paid, lacked focus, and his future in the new world clip not appear bright. After considerable hesita- tion he cleciclect to stay, however, and in the fall of 1922 he sent for his wife and youngest childleaving the other two children in Germany. He wanted them to get a good educa- tion, and by this time he knew "that there were no good engineering schools in America." From the vantage point in Philadelohiaq Timoshenko ~ ~ ~ , began looking for other jobs, naturally beginning with the well-known engineering schools. He records that he received no reply from any one of these. Eventually, one of his letters reached the engineering group at the Westinghouse Electric Corporation in East Pittsburgh. He was identified by L. S. Jacobsen as the author of many interesting papers in applier! mechanics, and it became this writer's pleasure to make the first contact with him. Soon after, Timoshenko received a formal offer to join the Westinghouse Research Laboratory. The details and background of this introduction of Timoshenko into American professional life will be clealt with in the following section. By 1927 Timoshenko had become well known in the Unites! States; in that year he joined the faculty at the Uni- versity of Michigan at Ann Arbor. His position was professor of graduate mechanics, and he soon had a large following. The years of his widening influence in applied mechanics hac! begun. He soon had as many doctoral students as he couIc! handle. At Michigan, he also had his first opportunity to realize his dream of joining applied and abstract sciences. One of his

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STEPHEN P. TIMOSHENKO 339 cireams and with the successors to the old institutions. It is significant, as observed earlier, that his accounts of his life following this trip were written in Russian this from the author of dozens of successful textbooks in English. It is also worth observing that circumstances prevented him from fully sharing in the wave of scientific revival that was part of the World War II scene. The experience of the first epoch of American superiority in scientific and technological develop- ments was somehow denied him. But whatever there was of bitterness was encased within his innermost being. The principal recollection on the part of those of us who were privileged to know him during those first years on the American scene is that of a man of great wisdom and a keen sense of humor, enhanced by his special version of the English language. This was usually clevoid of articles and retained the syntax of Russian, mixed with that of other languages. When confronted with examples of American colloquialisms, his favorite expression was a quiz- zical "What means this?" Some of us who tract the opportunity to accompany him to international conferences the Con- gress of Applied Mechanics in Zurich in 1926 is an exam- ple also were privileged to sense the enormous range of his acquaintances in the scientific world, acquired cluring his years of travel. The initial impression was of a remarkable linguistic versatility on his part, but this impression was tem- pered when it was discovered that he spoke only one lan- guage, modif~e<1 in the international circles with French or German phrases, depending on the makeup of his audience. His Russian was flawless but always retained the strong Ukrainian accent. While Timoshenko was well known in professional circles all over the world, the number of people admitted to his innermost sphere of affection was not large. His former students had a special position; so clic! a small group of his

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340 BIOGRAPHICAL MEMOIRS early acquaintances in the United States. One has the impres- sion, however, that real intimacy was reserved for his own family* and his Russian-speaking friends. Among those who might be mentioned are G. B. Karelitz, who passed away in 1940, and VIaclimir Zworykin. Professor Leonict M. Tichvin- sky, in a personal communication, observed that "Timo- shenko was the last person who knew my parents; he was my best man when ~ married my first wife, coming from Ann Arbor to Pittsburgh for this occasion in 1935." In a later communication Professor Tichvinsky observed that Timo- shenko, while leading a comfortable life in the United States, dicl not accumulate any substantial wealth. He left a modest legacy to be cliviclecl among his chilciren. The royalties from all his books were assigned to Stanford University. In the perspective of more than a half century, Timoshenko's great influence upon applied science and tech- nology in America resulted less from his original, creative discoveries than from his ideals of engineering education, his superb skill as a teacher, and his highly cleveloped pragmatic skill in using fragments of exact solutions for a variety of approximate solutions to difficult problems in applied mechanics. Examples of this are his skillful use of the solu- tions for beams on elastic foundations to problems such as railroad rails anti to details of machinery such as highly stressed dovetail joints. Another example is his frequent use of the Boussinesq solutions to a variety of intractable prob- lems in machine structures. He also cIarif~ecl the premises of the Rayleigh-Ritz iteration method, extending it to a variety of problems in elasticity and dynamics. Throughout all his * He was a member of a remarkable family, which contributed much to his world- wide views and connections. Of his two younger brothers, Serhij (an architect) was Secretary of Transportation in the Ukrainian government in the early 1920's, while Vladimir (an economist) was Secretary of Commerce of the Ukraine and later Chief of Statistics of the AAA under President Franklin D. Roosevelt in Washington and at Stanford University.

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STEPHEN P. TIMOSHENKO 341 work there is a pragmatic attitude toward mathematics, and even to theoretical mechanics, which was regarded as one of his great sources of strength. This charming pragmatism was new to most of us and seemed of immense value to the practicing engineers. It is well to remember that even elementary computer aids, which we now take for granted, were then many years distant in the future. Iteration procedures had to be worked out with brute strength anti untoIcl man-hours. With the development of modern computers and new methods such as the finite- element method, many problems once beyonc! our reach have now become routine. The pragmatic approach may now seem less essential, but I believe this is only a temporary phase. In any case, the effectiveness and charm of Timoshenko's teaching will always be a treasured memory to his students. A A FACTUAL ACCOUNT of the career of Stephen P. Timoshenko has been condensed from his book As I Remember, aided by tributes and memoirs by former students and associates, particularly those of Professor D. H. Young of Stanford University (Donovan H. Young, "Stephen P. Timoshenko 1878-1972," Applied Mechanics Review, July 1972. 5 pp.) and Professor Chia-Shun Yih of the University of Michigan (Chia-Shun Yih, "Stephen P. Timoshenko: A Portrait in Miniature." A note to the faculty of the College of Engineering, October 30, 1972.~. These tributes emphasize his years in American . . . un~vers~hes. Since this writer and a few remaining colleagues were part of the group that first became associated with Timoshenko on his arrival, in 1923, at the Westinghouse Electric Corporation in East Pittsburgh, Pennsylvania, we wished to record some of the back- ground and impressions from his early years there. In the prepara- tion of this memoir I have been greatly assisted by the members of the "International Hiking Club" in Wilkinsburg- Professors I. P. Den Hartog, J. Ormondroyd, and L. S. Jacobsen as well as by other members of the group, specifically R. E. Peterson, M. Stone, and Leonid M. Tichvinsky, whose contributions and criticisms I gratefully acknowledge.

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342 BIOGRAPHICAL MEMOIRS BIBLIOGRAPHY 1905 Torsional vibrations of shafts. Proc. St. Petersburg Polytech. Inst., 3:55-106. Krylov's method for integration of ordinary differential equations. Proc. St. Petersburg Polytech. Inst., 3:397~06. Formulas for combined stresses from various strength theories. Proc. St. Petersburg Polytech. Inst., 3:415-55. 1906 Lateral buckling of I-beams under the influence of forces acting in the plane of largest rigidity. Proc. St. Petersburg Polytech. Inst., 4:151-219; 5~1-2):3-34, 262-92. 1907 Stresses in a plate with a circular hole. Proc. Kiev Polytech. Inst., 9:95-113. Secondary stresses caused by rigidity of joints. Proc. St. Petersburg Polytech. Inst., 7: 135~4. Buckling of a bar on an elastic foundation. Proc. St. Petersburg Polytech. Inst., 7: 145-57. Forced vibration of prismatic bars. Proc. Kiev Polytech. Inst. 9: 201-52. Stability of Plates under Compression. Kiev: Kulzhenko Typography. 60 pp. 1908 On the subject of buckling. Proc. Kiev Polytech. Inst., 8:181-212. Strength of Materials. Part 1. Kiev: Kiev Polytechnic Institute. 373 PP Collection of Problems in Strength of Materials. Kiev: Kiev Polytechnic Institute. 1909 A Course in Elasticity Theory. Part 1. Kiev: Kiev Polytechnic Institute. Stresses in a circular ring compressed by two opposing forces. Proc. Kiev Polytech. Inst., 9:21-37. 1910 Application of normal coordinates in analyzing the bending of bars and plates. Proc. Kiev Polytech. Inst., 10:1-19.

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STEPHEN P. TIMOSHENKO 343 Stability of elastic systems. Proc. Kiev Polytech. Inst., 10:147-67. Stability of elastic systems. Proc. Kiev Polytech. Inst., 10:375-560. Einige Stabilitatsprobleme der Elastizitatstheorie. Z. Math. Phys., 58:337-85. 19 An Approximate Methodfor Investigating the Stability of Elastic Systems. Kiev: St. Vladimir University. 25 pp. A Course in Strength of Materials. Kiev: Kiev Polytechnic Institute. 517 pp. Erzwungene Schwingungen prismatischer Stabe. Z. Math. Phys. 59: 163-203. 1912 Problems of strength of steam turbines. I. Soc. Technol. St. Peters- burg, 7: 266-79. Effect of impact on a beam. Proc. St. Petersburg Polytech. Inst., 17:407-25. 1913 Bending of bars having a slight curvature. I. Soc. Technol. St. Petersburg, 13:411-14. Bending of spherical shells. l. Soc. Technol. St. Petersburg, 17: 549-57. Plotting the deflection curve. I. Soc. Technol. St. Petersburg, 21: 241-42. Use of stress functions to study flexure and torsion of prismatic bars. St. Petersburg. 21 pp. (Reprinted in: Mem. Inst. Ways Commun., 82:1-21.) Sur la stabilite des systemes elastiques. Ann. Ponts Chaussees, 1st part, Memoires et Documents, 9th Ser., 13:496-566; 16:72-132, 373-412. Zur Frage nach der Wirkung eines Stosses auf einen Balken. Z. Math. Phys., 62: 198-209. 1914 Stability of a cylindrical shell. J. Soc. Technol. St. Petersburg, 21:785-92. Effect of initial curvature on the bending of a rectangular plate. Petrograd.4 pp. (Reprinted in: Mem. Inst. Ways Commun., 89.)

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344 BIOGRAPHICAL MEMOIRS Etude de la flexion des barres au moyen d'une methode approxi- mative. Ann. Trav. Publics. Belg., 71 Ser., 219:263-96. 1915 Strength of rails. Proc. Inst. Ways Commun. 42 pp. Large deflections of circular plates. Petrograd. 10 pp. (Reprinted in: Mem. Inst. Ways Commun., 89.) Stability of plates with stiffening ribs. Petrograd. (Reprinted in: Mem. Inst. Ways Commun., 89.) Vibration of rails. Proc. Inst. Electr. Eng., Petrograd, 13. 1916 The effect of clearances between the rail and the tie on bending of rails. Mem. Inst. Ways Commun. 14 pp. A Course in Elasticity Theory. Part 2. St. Petersburg. 1917 Allowable stresses in railway bridges. Proc. Inst. Ways Commun.,7. 1919 Use of trigonometric series for calculation of suspension bridges. Eng. J., 8:239-43. 1921 .. Uber die Stabilitat versteifter Platten. Eisenbau, 12:147-63. Etude de ['action des charges roulantes sur les rails. Genie Civ., 79:555-56. On the correction for shear of the differential equation for trans- verse vibrations of prismatic bars. Philos. Mag., 41: 74~46. 1922 A Course of Statics of Structures. Parts 1 and 2. Petrograd. 344 pp. Uber dies Biegung der allseitig unterstutzten rechteckigen Platte unter Wirkung einer Einzellast. Bauingenieur, 3:51-54. Beams with loads irregularly distributed. Engineering, 113: 196-97. Calcul des Arcs Elastiques, ed. Ch. Beranger. Paris: Librairie Poly- technique. 77 pp. On the transverse vibrations of bars of uniform cross-sections. Philos. Mag., 43: 125-31.

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STEPHEN P. TIMOSHENKO 345 On the forced vibration of bridges. Philos. Mag., 43: 1018-19. On the distribution of stresses in a circular ring compressed by two forces acting along a diameter. Philos. Mag., 44: 1014-19. On the torsion of a prism, one of the cross-sections of which re- mains plane. Proc. Lond. Math. Soc., Ser. 2, 20:389-97. A membrane analogy to flexure. Proc. Lond. Math. Soc., Ser. 2, 20:398-407. Elasticity of pipe bends. Trans. Am. Soc. Mech. Eng., 44:585-93. Torsion of crankshafts. Trans. Am. Soc. Mech. Eng., 44:653-67. Berechnung der Schubspannungen im gebogenen Balken. Z. Angew. Math. Mech. 2:160-61. 1923 Determination of the modulus of elasticity. Mech. Eng., 45:259-60. The pendulum hardness tester. Engineer, 136:21. Bending stresses in curved tubes of rectangular cross-section. Trans. Am. Soc. Mech. Eng., 45:135-40. The bending and torsion of multi-throw crankshafts on many sup- ports. Trans. Am. Soc. Mech. Eng., 45:449-70. Kippsicherheit des gekrummten Stabes mit kreisRormiger Mittel- linie. Z. Angew. Math. Mech., 3:358-62. 1924 .. Uber die Biegung van Staben, die eine kleine anfangliche Krum- mung haben. In: Festschrift zum siebzigsten Geburtstage August Foppls, pp. 7~81. Berlin: Springer-Verlag. On stresses in a plate with a circular hole. }. Franklin Inst., 197: 505-16. An approximate method of solution of two dimensional problems in elasticity. Philos. Mag., 47: 1095- 104. Beams without lateral support. Trans. Am. Soc. Civ. Eng., 87: 1247-62. Deflections of a uniformly loaded circular plate with clamped edges. Sci. Pap. No. 162, Westinghouse Electric and Manu- facturing Co. 1925 Analysis of bi-metal thermostats. }. Opt. Soc. Am., 11 :233-55. Properties of matter under high pressure. Mech. Eng., 47:513- 15. With W. Dietz. Stress concentration produced by holes and fillets.

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346 BIOGRAPHICAL MEMOIRS Trans. Am. Soc. Mech. Eng., 47:199-220. With }. M. Lessells. Applied Elasticity. New York: D. Van Nostrand Co. 1926 With R. V. Baud. The strength of gear teeth. Mech. Eng., 48: 1105-9. Method of analysis of statical and dynamical stresses in rail. In: Proceedings of the Second International Congress for Applied Me- chanics, pp. 407-18. Zurich and Leipzig: Orell Fussli Verlag. Stress concentration produced by fillets and holes. In: Proceedings of the Second Congressfor Applied Mechanics, pp. 419-26. Zurich and Leipzig: Orell Fussli Verlag. 1927 Vibration of bridges. Trans. Am. Soc. Mech. Eng., 49-50(part 2~: 53-61. 1928 Vibration problems in engineering. N.Y.: D. Van Nostrand Co. The stiffness of suspension bridges. Atti Congr. Int. Mat. (Proc. Int. Congr. Math.), Bologna, 6:305-6. Steifigkeit von Hangebrucken. Z. Angew. Math. Mech., 8:1-10. 1929 Teaching of advanced mechanics in engineering schools. Mech. Eng., 51 :609-10. 1930 Vibration problems. In: Marks' Mechanical Engineering Handbook, pp. 489-502. New York: McGraw-Hill. The theory of elasticity. Mech. Eng., 52:49~96. Recent developments in the application of mechanics to machine design. Mech. Eng., 52:607-10. Strength of Materials. 2 vols. Princeton, N.T.: D. Van Nostrand Co. Problems concerning elastic stability in structures. Trans. Am. Soc. Civ. Eng., 94: 1000- 1020. 1931 Stability and strength of thin-walled constructions. In: Proceedings of the Third International Congress for Applied Mechanics, vol. 3, pp.

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STEPHEN P. TIMOSHENKO 347 3-15. Stockholm: Kungl. Boktryckeriet, P. A. Norstedt and S.. oner. Stabilitatsprobleme der Elastizitat. In: Handbuch der physikalischen and technischen Mechanik, ed. F. Auerbach and W. Hort, vol. 4 part 1, pp. 81-145. Leipzig: J. A. Barth. Festigkeitsprobleme im Maschinenbau. In: Handbuch der physika- lischen und technischen Mechanik, ed. F. Auerbach and W. Hort, vol. 4, part 1, pp. 14~98. Leipzig: J. A. Barth. Theory of elasticity. Mech. Eng., 53 :889-90. 1932 Stability of plate girders subjected to bending. Congr. Int. Assoc. Bridge Struct. Eng., Zurich, pp. 129-47. With B. F. Langer. Stresses in railroad track. Trans. Am. Soc. Mech. Eng., 54:277-302. 1933 Working stresses for columns and thin-walled structures. Trans. Am. Soc. Mech. Eng., 55:APM 173-83. Uber die Biegung van Tragerrosten. Z. Angew. Math. Mech., 13: 153-59. With S. Way. Suspension bridges with a continuous stiffening truss. Publ. Int. Assoc. Bridge Struct. Eng., Zurich, 2:452-66. 1934 . The stability of the webs of plate girders. Engineering, 138:207-9. I heory of Elasticity. New York: McGraw-Hill. 1935 Elements of Strength of Materials. Princeton, N.~.: D. Van Nostrand Co. Buckling of flat curved bars and slightly curved plates. J. Appl. Mech., 2:A 17-A20. 1936 Theory of Elastic Stability. N.Y - McGraw-Hill. 1937 With D. H. Young. Engineering Mechanics. New York: McGraw-Hill.

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348 BIOGRAPHICAL MEMOIRS 1938 Bending of rectangular plates with clamped edges. In: Proceedings of the Fifth International Congress of Applied Mechanics, pp. 40-43. New York: John Wiley & Sons. 1940 Theory of Plates and Shells. New York: McGraw-Hill. 1941 The forced vibrations of tie-rods. In: Theodore von Karman Anni- versary Volume, Contributions in Applied Mechanics, pp. 226-30. Friends of Theodore van Karman. 1942 Teaching dynamics. I. Eng. Educ., 32:463-66. 1943 Theory of suspension bridges. I. Franklin Inst., 235:213-38, 327- 49. 1945 Theory of bending torsion and buckling of thin-walled members of open cross-section. I. Franklin Inst., 239:201-19, 249-68, 343-61. With D. H. Young. Theory of Structures. New York: McGraw-Hill. 1947 Stress concentration and fatigue failures. Proc. Inst. Mech. Eng., London, 157: 163-69. 1948 With D. H. Young. Advanced Dynamics. New York: McGraw-Hill. 1950 D. J. Jourawski and his contribution to theory of structures. In: Federhofer-Girkmann-Festschrift, Beitrage our angewandten Mecha- nik, pp. 115-23. Vienna: Franz Deutickle Verlag.

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STEPHEN P. TIMOSHENKO 1953 349 History of the development of strength of materials in Russia. Academia Nazionale dei Lincei, Rome, no. 29. 8 pp. History of Strength of Materials. New York: McGraw-Hill. 1954 Stress concentration in the history of strength of materials. Proc. Soc. Exp. Stress Anal., 12: 1- 12. 1956 The development of engineering education in Russia. Russian Rev., 15:173-85. 1958 The background of engineering education in Russia. i. Eng. Educ., 49: 122-25. 1959 Engineering Education in Russia. New York: McGraw-Hill. 1968 As I Remember (Autobiography). New York: D. Van Nostrand Co. RELATED BOOKS Stephen Timoshenko 60th Anniversary Volume, Contributions to the Mechanics of Solids Dedicated to Stephen Timoshenko by His Friends. New York: The Macmillan Co., 1938. The Collected Papers of Stephen P. Timoshenko. New York: McGraw- Hill, 1953.