The 100th Anniversary of the Timoshenko–Ehrenfest Beam Model

Abstract

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 Austrianborn 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 crosssection.” 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].