Exact dynamic and static element stiffness matrices of nonsymmetric thin-walled beam-columns

An improved numerical method to exactly evaluate 14 × 14 dynamic and static element stiffness matrices is proposed for the spatial free vibration and stability analysis of nonsymmetric thin-walled straight beams subjected to eccentrically axial loads. Firstly equations of motion and force-deformation relations are rigorously derived from the total potential energy for a uniform beam element with nonsymmetric thin-walled cross-section. Next a system of linear algebraic equations with nonsymmetric matrices is constructed by introducing 14 displacement parameters and transforming the higher order simultaneous differential equation into the first order simultaneous equation. And then explicit expressions for displacement parameters are exactly evaluated by solving a generalized eigenproblem with complex eigenvalues. Finally exact element stiffness matrices are determined using force-deformation relations. Particularly straightforward application of the present method may not give the exact static stiffness because of existence of multiple zero eigenvalues in case of static buckling problems. Accordingly, a modified numerical method to resolve this difficulty is developed for two cases depending on the initial state of stress resultants. In order to demonstrate the validity and the accuracy of this method, the natural frequencies and buckling loads of nonsymmetric thin-walled beam-columns having bending-torsional deformation modes are evaluated and compared with analytical and F.E. solutions or results analyzed by ABAQUS’s shell element.