This paper is inspired by two articles on the title subject, namely by Koiter and by Sugiyama, Langthjem and Ryu. The former warned the engineering community to beware of the above forces. The latter maintained that these forces are realistic. It is hoped that this review sheds some additional light on an issue that seems to perplex many students of dynamic stability. The paper does not contain any new information unknown to researchers; it represents a critical review of pertinent papers dedicated to the topic of dynamic stability of structures under so called “follower” forces. The attempt here is to present an account of the literature in a manner that is both objective and humble. This paper reviews both the theoretical and experimental contributions to the theory of nonconservative problems with a single objective in mind, to attempt to answer a nagging question “Is the model of the statically applied follower forces useful?,” that arose due to the papers by Koiter and Sugiyama, Langthjem and Ryu. This article explores the static and dynamic stability criteria as pertaining to the nonconservative problems; the experimental side of the problem; the results pertaining to Beck’s column placed on homogenous or inhomogeneous elastic foundations; and criticisms expressed in the literature about the result on the immunity of the instability load to the Winkler foundation modulus: The paper then discusses Koiter’s ideas on nonconservative instability problems, and attempts to provide insights on the abovementioned question. Special emphasis is placed on pipes conveying fluid with or without an elastic foundation. It then summarizes the literature on the “follower forces.” Such a summary is inevitably incomplete because of the huge literature accumulated so far on the nonconservative problems in the theory of elastic stability. There are 202 references cited in this review article, and a supplementary bibliography is provided.

The use of the Google Scholar produces about 78,000 hits on the term “Timoshenko beam.” The question of priority is of great importance for this celebrated theory. For the first time in the world literature, this study is devoted to the question of priority. It is that Stephen Prokofievich Timoshenko had a co-author, Paul Ehrenfest. It so happened that the scientific work of Timoshenko dealing with the effect of rotary inertia and shear deformation does not carry the name of Ehrenfest as the co-author. In his 2002 book, Grigolyuk concluded that the theory belonged to both Timoshenko and Ehrenfest. This work confirms Grigolyuk’s discovery, in his little known biographic book about Timoshenko, and provides details, including the newly discovered letter of Timoshenko to Ehrenfest, which is published here for the first time over a century after it was sent. This paper establishes that the beam theory that incorporates both the rotary inertia and shear deformation as is known presently, with shear correction factor included, should be referred to as the Timoshenko-Ehrenfest beam theory.

This study revisits Timoshenko beam theory (TBT). It discusses at depth a more consistent and simpler governing differential equation. The so-called second spectrum is also addressed. Then, we provide the asymptotic justification of the aforementioned differential equation along with detailed discussion of the boundary and initial conditions. The paper also presents remarks of historical character, in the context of other pertinent studies.