Finite-Element Formulation for the Linear Steady-State Response of Asymmetric Thin-Walled Members under Harmonic Forces

Abstract

A closed-form solution and finite-element formulation are developed for the dynamic analysis of thin-walled members with asymmetric open sections subjected to harmonic forces. The dynamic equations of motion and associated boundary conditions are derived from Hamilton’s principle. The formulation is based on a generalized Vlasov-Timoshenko beam theory and accounts for the effects of shear deformation caused by bending and warping and translational and rotary inertia effects. It also captures the effects of flexural-torsional coupling caused by cross-sectional asymmetry. From this a general closed-form solution is obtained. A family of shape functions is then developed based on the exact solution of the coupled field equations and is used to formulate a beam finite element. The new element has two nodes with six degrees of freedom per node and successfully captures the coupled bending-torsional static and steady-state responses of asymmetric thin-walled members under harmonic forces. Results based on the closed-form solution and finite-element formulation are assessed and validated against other well-established finite-element solutions.