Analysis of dynamic stability of beam structures
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O R I G I NA L PA P E R
Hrvoje Smoljanovi´c · Ivan Bali´c Vlaho Akmadži´c · Boris Trogrli´c
· Ante Munjiza ·
Analysis of dynamic stability of beam structures
Received: 19 December 2019 / Revised: 30 June 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract This paper presents two numerical models (Model L and Model N) and its application in the analysis of dynamic stability of beam-type structures. Both numerical models use two-noded rotation-free finite elements and take into account the exact formulation for finite displacement, finite rotations, and finite strains. Model L was previously developed and is intended for linear elastic material behavior, whereas Model N is newly developed, considers laminar cross sections, and takes into account the nonlinear material behavior. Both models have been implemented into the open-source finite discrete element package Y-FDEM. Performance and conditions under which both numerical models can be used for the analysis of dynamic stability are presented by numerical examples which show good agreement in comparison with the analytical solutions.
1 Introduction The stability of beam-type structures under dynamic loading such as seismic loading, wind loading, and impact loading presents a substantially important issue in structural engineering. In 1774, Leonard Euler presented the relation for the critical buckling force of a compressed beam under static loading conditions. However, the behavior of the beam under dynamic loading is a more complicated task which significantly depends on the loading condition. Due to this reason, the determination of the dynamic buckling force has been the topic of interest of numerous studies for the past more than half a century. One of the earliest studies on this subject was performed by Hoff [1]. He analyzed the response of a simply supported, perfectly elastic beam with small initially deformed geometry under constant compression loading rates. He demonstrated that, even at very low loading rates compared to the speed of sound, the dynamic buckling force can exceed the Euler buckling force as much as by a factor of 100. These differences are caused by the lateral inertial forces of the beam. By assuming the first mode buckling and by neglecting the longitudinal vibration of the beam, Hoff showed that the dynamic buckling force depends on the amplitude of the beam’s imperfection and the similarity number depending on the length/thickness ratio of the beam and the compression rate. Hoff’s theoretical results were also supported by experimental results [2,3]. Based on Hoff’s study, many researchers analyzed the behavior of the compressed beam under constant compression rate [4–11] taking into account various boundary conditions [4], influence of axial inertial forces [5], random imperfections [7], and material nonlinearity [10]. Furthermore, many authors also investigated the various types of loading histories such as the triangular loading history [12] and the impact load [13]. H. Smoljanovi´c · I. Bali´c (B) · A. M
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