Nonlinear Dynamics of Flexible Partially Collapsed Structures

Abstract

The aim of this work is to present an implementation of some of the latest theories and developments in the finite element method involving the beam element and its potential for the analysis of structures undergoing sudden changes during its service, such as local structural collapses and/or instabilities or abrupt actions such as impacts. Geodesic domes, cable structures and cable supported bridges are some of the types of structures which may experiment such effects. The large deflection theory of linear elastic rods is used to model the structural elements, using a procedure that is geometrically exact and strain invariant, in which the Petrov-Galerkin method leads to non-symmetric stiffness matrices preserving the strain measures objectivity and path independence. The non-symmetry is in fact much more pronounced due to the gyroscopic (Coriolis and centrifugal) effects. The formulation uses Euler parameters matrix rotation which does not lead to singular transformation between inertial, material and space frames, undergoing arbitrary large rotations. The problems are solved using the finite element method in an implicit finite difference scheme in time that exactly preserves energy and momentum, using the Cayley transform despite of the Rodrigues formula in a mid-point rule integration. Several examples using the strain-invariant formulation for the static analysis are presented. The nonlinear dynamic using the energy momentum conserving algorithm, that doesn’t preserve the invariance of strains, is used to analyse simple examples as cantilever beams in free vibration, post collapsed cables and a beam-cable system subjected to a quasi-harmonic load. The cases analysed show the efficiency of the algorithmic procedure for both static and dynamics dealing with arbitrary large rotation. The Reissner-Simo finite-strain theory and the Simo-Tarnow-Doblare conserving time integration algorithm have shown to be effective in analyzing highly flexible structures. The robustness of the time stepping procedure and the effectiveness of the strain-invariant approach introduced by Jelenic and Crisfield in the Reissner-Simo rod element seem to allow their use in problems involving sudden changes in structures, such as local structural collapses and instabilities and extreme actions as impacts and shock phenomena. The accurateness of the algorithms in the presence of large displacements and arbitrary rotations as well as finite strains seems to be an advantage in face of other finite element procedures.