Generalized strain-based finite element for non-linear stability analysis of beams with thin-walled open cross-section

Abstract

Based on the generalized strain theory a shear-deformable finite element is developed for nonlinear stability analysis of thin-walled open-section beams in multibody systems. In this formulation a finite number of deformations, characterized as generalized strains, is defined which are related to dual stress resultants in a co-rotational frame. The stiffness formulation is based on a second-order approximation of the local elastic displacement field. Timoshenko’s beam theory and Vlasov’s modified thin-walled beam theory are used to include the shear strain effects due to non-uniform bending and restrained warping torsion and their mutual coupling effects. Axial shortening associated with the Wagner Hypothesis is taken into account such that the nonlinear behaviour of the beam is predicted accurately, especially under large torsion. Coupling between bending and torsional deformation due to non-coincident centroid and shear centre is modelled using a second-order cross-section transformation matrix. Cubic Hermitian polynomials are used as shape functions for the lateral displacements and twist rotation to derive the elastic and geometric stiffness matrices. Geometric nonlinearities associated with axial elongation and bending curvatures are described by additional torsion, bending and warping-related quadratic deformation terms yielding a set of modified deformations. The inertia properties of the beam are described using both consistent and lumped mass formulations. The latter is used to model rotary and warping inertias of the beam cross-section. The accuracy and the computational efficiency of the new beam element is demonstrated in several static and dynamic examples.