On the stability of elastic structures subjected to follower forces

Within the framework of a second-order theory, this study revisits classical stability problems characterized by critical loads associated with dynamic instability, previously explored by Feriani and Carini (in: Zingoni (ed) Insights and innovations in structural engineering, mechanics and computation, 2016) . The primary focus remains on systems exclusively comprising a single lumped mass. This simplification, together with the assumption of a negligible axial strain and the adoption of a second-order theory, reduces the analysed systems to problems featuring a single Lagrangian coordinate. Consequently, static methods become applicable, facilitating the derivation of analytical expressions for stiffness coefficients and enabling an investigation into dynamic stability. The study begins by examining a well-established case: a cantilever beam with a lumped mass positioned at its free end, subjected to a follower load, as presented in Panovko and Gubanova (Stability and oscillations of elastic systems: models, paradoxes and errors, Nunka Press, Moscow, 1967). In this current work, a novel lumped mass system is explored. The system consists of a straight-axis beam characterized by a constant cross-section area and stiffness. The distributed mass of the beam is neglected and the beam is hinged at one end, simply supported at an intermediate point, and left free at the other end, where a lumped mass is introduced. Various loading scenarios are scrutinized, including: (a) The application of a follower force to the free end; (b) The imposition of two forces-one conservative and one follower-both applied to the free end; and (c) The application of a uniformly distributed follower force along the length of the beam. As seen in the examples introduced in Feriani and Carini (2016), the new examples considered in the present paper reveal that the first asymptote of the stiffness coefficient corresponds to the critical load. This critical load corresponds to the phenomenon known as divergence at infinity, as described by Felippa (Nonlinear finite element methods, 2014). It is also confirmed that, in all cases, the first dynamical critical load equals the minimum value between the static buckling load of the original structure and the static buckling load of an auxiliary structure. This last differs from the original one because the concentrated mass is replaced by a constraint fixing the corresponding Lagrangian coordinate.