Did S. P. Timoshenko and P. Ehrenfest Overestimate the Importance of the Fourth-Order Time Derivative in Their Beam Theory?

Abstract

Abstract In this study, we investigate the importance of the fourth-order time derivative that appears in the equations derived by Jacques Antoine Charles Bresse in 1859, as well as in equations that were derived by Stephen Prokofievich Timoshenko and Paul Ehrenfest during years 1912 and 1913 and reported by Timoshenko in the 1916 book on the theory of elasticity in the Russian language and then in two papers dated 1920 and 1921, in English. The first part of the study demonstrates that Timoshenko and Ehrenfest did not overestimate the importance of the fourth-order derivative term in their equations. The second part deals with the debate on the so-called second spectrum attendant in the original set of equations. It is shown that in the truncated Timoshenko—Ehrenfest equations—which is asymptotically consistent with elasticity theory—“the second spectrum” issue does not arise. Thus, the two parts of this study are intricately interrelated with each other.