An improved numerical method evaluating exact static element stiffness matrices of thin-walled beam-columns on elastic foundations

An improved numerical method to obtain the exact static element stiffness matrices is newly proposed to perform the spatially coupled elastic and stability analyses of non-symmetric and open/closed thin-walled beam-columns with and without two types of elastic foundation. Particularly this method overcomes drawbacks of the previous method to evaluate the exact static stiffness matrix for the spatially coupled stability analysis of thin-walled beam-column. This numerical technique is firstly accomplished via a generalized eigenproblem associated with 14 displacement parameters by transforming equilibrium equations to a set of first order simultaneous ordinary differential equations. Next polynomial expressions as trial solutions are assumed for displacement parameters corresponding to zero eigenvalues and the eigenmodes containing undetermined parameters equal to the number of zero eigenvalues are determined by invoking the identity condition. And then the exact displacement functions are constructed by combining eigensolutions and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently an exact static stiffness matrix is evaluated by applying the member force–deformation relationships to these displacement functions. In order to illustrate the accuracy and the practical usefulness of this study, the numerical solutions are compared with results obtained from the thin-walled beam and shell elements.