Beams are ubiquitous elements in numerous fields of engineering. “A beam is defined as a structure having one of its dimensions much larger than the other two. The axis of the beam is defined along that longer dimension and a cross section normal to this axis is assumed to smoothly vary along the span or length of the beam” [1]. It is understandable then that the theory of beams was the first one to be developed, with theories of plates, and shells following it much later. “Prevailing consensus is that Galileo Galilei (1564–1642) made the first attempts at developing a theory of beams, but recent studies argue that Leonardo da Vinci (1452–1519) was the first to make the crucial observations. Da Vinci lacked Hooke’s law and calculus to complete the theory, whereas Galileo was held back by an incorrect assumption he made” [2]. The first viable theory was suggested by Daniel Bernoulli (1700–1882) and Leonhard Euler (1707–1783), famous Swiss mathematicians. This is the simplest theory using an assumption that the normal to the cross section prior to the deformation remains normal to it also after the deformation takes place. It may appear paradoxical that mathematicians, rather than engineers, proposed the simplest theory. Further refinements, associated with complications of the classical beam theory, were furnished by engineers. Jacques Antoine Charles Bresse (1822–1883) [3], and John William Strutt (Lord Rayleigh) (1842–1919) [4] apparently independently took into account the effect of rotary inertia. Jean Victor Poncelet (1788–1867) in France, William John Macquorn Rankine (1820–1872) in England [5], Franz Grashof (1826–1893), and August Föppl (1854–1924) in Germany dealt with incorporation of shear deformation in beams in static settings. It turned out recently, that Bresse (1859) [3] incorporated both rotary inertia and shear deformation in his classic book in 1859, where he dealt with curved beams. Unfortunately, this book was never translated into English (see details in the paper by Challemel and Elishakoff [6]). Stephen Prokofievich Timoshenko (1978–1972) either did not know of this fact or somehow, he overlooked and never reported it. Be it as it may, during the years 1911–1912 he worked with Austrian-born Dutch physicist Paul Ehrenfest (1880–1933) who at a time temporarily lived in St. Petersburg, Russian Empire. For unknown reasons, they did not publish the work on the incorporation of rotary inertia and shear deformation in straight beams in a journal. Rather, Timoshenko included it in his book on theory of elasticity [7]. There, he noted as a footnote number 2 on page 206: “By us, jointly with Prof. Ehrenfest, also an exact solution was also obtained for the beam with rectangular cross-section.” The word “also” indicates that, Timoshenko and Ehrenfest developed what later became known as the Timoshenko beam theory, jointly. It appears that this theory ought to be more justifiably called Timoshenko–Ehrenfest beam theory. It should be emphasized that the Timoshenko–Ehrenfest version includes the shear correction coefficient whereas the Bresse one lacks it. Thus, Timoshenko–Ehrenfest beam theory reduces to that developed by Bresse if one puts shear correction factor as unity. The full story of how the name of Ehrenfest was not included in the name of the theory is fascinating and not yet fully uncovered. For the details, the interested reader can consult with Refs. [8,9].
In 1920, Timoshenko left Ukraine—the country in turmoil then—and arrived in the city of Zagreb in the Kingdom of Serbs, Croats, and Slovenes (now Croatia). There Timoshenko published the joint development of beam theory with Ehrenfest, exactly as it appeared in his book [7], as a paper [8]. He could not be accused harshly in the second, repeated publication of the results he and Ehrenfest obtained, since he was seeking international dissemination of the beam theory. At that time, he had an opportunity to include the name of Ehrenfest as that of a co-author, since he was publishing a paper rather than a monograph, as it was in 1916. But he chose not to, for unknown reasons.
Did Timoshenko deliberately fail to include the name of Ehrenfest, the latter living now in Leiden, the Netherlands, faraway from Zagreb where the paper was published? Was it given as sort of a parting “gift” to Timoshenko by Ehrenfest when the latter has left Petersburg in 1912, assuming his chair position in Leiden? Or was Ehrenfest no more interested in classical mechanics being himself consumed with dramatic developments of the 20th-century physics? It seems that we will never know the answers to these questions unless some new evidence surfaces.
Timoshenko apparently was not fully satisfied with the publication of his paper [8] in the English language. The journal was an obscure one, and on top of this. So, he published his and Ehrenfest’s equations yet again, in 1921, in the prestigious journal Philosophical Magazine [9]. However, in both of his papers the footnote that appeared in his 1916 book, mysteriously disappears. Details of this affair are to be found in the Refs. [10,11]. Additionally, Timoshenko [12] does not list his 1920 paper in his autobiography. Hence, it is rarely cited by researchers. Likewise, although Timoshenko [12] writes somewhat in detail about Ehrenfest, he remains silent about the collaboration they had.
Naturally, in the past 100 or so years numerous works employing this theory have been published in the literature. Due to lack of space, we indicate just some here, though many more would be worth mentioning. Dolph [13], Downs [14], and O’Reilly and Turcotte [15] deal with a subtle case of exact solution for vibration of a shear-deformable beam without transverse vibration. Huang [16] consider the effect of boundary conditions. Wang et al. [17] establish the intriguing connection between classical and refined beam theories. In 1985, Elishakoff and Lubliner [18] delete altogether the fourth-order time derivative based on observations made in works by Timoshenko in 1916 [7] and 1920 and 1921 [8,9] that it had a small effect on frequency determination. This simplified the theory tremendously, leading to closed-form expressions for random vibration response (Ref. [19]). Later on, Goldenveiser et al. [20], Kaplunov et al. [21], and Elishakoff et al. [22] establish that the fourth-order time derivative does not enter in the first-order asymptotic theory, in the first place. Definitive reviews were written by Laura et al. [23], Han et al. [24], and Rossikhin and Shitikova [25]. Most importantly, the Timoshenko–Ehrenfest beam theory was extended to plates by Uflyand [26] and Mindlin [27,28]. Then blissful generalizations to shells and laminated structures mushroomed. We are sure that the Timoshenko beam theory is not exhausted, and many papers will appear in the future if the previous publication rate can serve as a guide.
We are celebrating the Timoshenko–Ehrenfest beam’s theory’s 100th anniversary. Google cites the concept “Timoshenko beam” a whopping 970,000 times. This is much less than numbers associated, say, with “Kalman filter” (about 12 million) or “J integral” (466 million), but still extremely impressive. Both Timoshenko and Ehrenfest would have been glad to know that their collaboration led to such a spectacular result. In fact, Timoshenko [12, p. 119], in 1968, reminisces: “In the winter of 1911-12, I …investigated the influence of shear stresses on the transverse vibrations of beams…[it] served a point of departure for the research of many authors.” On the next page (Timoshenko [12, p. 120]), he admits: “My own mental and physical state during this transitional year of 1911–1912 was not too good.” This might partially explain why he omits the name of his collaborator Ehrenfest as a co-author. We are glad to be able to combine these names again, for the sake of truth and historic justice.
Timoshenko died in 1972, 60 years after the conception of this theory, making this year, 2022, a half-century anniversary of his passing. Despite being an extremely prolific book writer, he did not think it worthwhile to compose a book on this very theory that appears to be the most central contribution to his legacy. This was remedied by monographs of Wang et al. [17] and Elishakoff [11], in years 2000, and 2020, respectively. The Russian account of the refined theories of beams, plates, and shells was given by Grigolyuk and Selezov [29], their comprehensive book still awaiting the English translation.
Long live Timoshenko–Ehrenfest beam theory, and the better ones to come!
