Nonlinear finite element analysis of elastic systems under nonconservative loading—natural formulation part II. Dynamic problems

Abstract The paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces. Attention is focused on the stability behaviour of such systems. This leads mathematically to non-self-adjoint boundary-value problems which are of great theoretical and practical interest, in particular in connection with alternative modes of instability like divergence of flutter. Only quasistatic effects are however considered in the present part. The methodology of our theory is general, but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement-dependent nonconservative (follower) forces. In a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives, in general, rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix. As a result, the total tangent stiffness matrix becomes unsymmetric - an indication of the non-self-adjoint character of the problem. Our theory is based on the natural mode technique [1, 2, 3]and permits i.a. a simple but elegant derivation of the load correction matrix. The application of the general theory as evolved in this paper is demonstrated on the beam element in space. A number of numerical examples are considered including divergence and flutter types of instability, for which exact analytic solutions are known. The problems demonstrate the efficiency of the present finite element formulation. The paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in [2, 3].

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