Elastic buckling analysis of 3-D trusses and frames with thin-walled members

Abstract

A finite element method is presented for the determination of the elastic buckling load of three-dimensional trusses and frames with rigid joints. The beam element stiffness matrix is constructed on the basis of the exact solution of the governing equations describing the coupled flexural-torsional buckling behaviour of a threedimensional beam with an open thin-walled section in the framework of a small deformation theory. Large deformation effects are taken into account approximately through consideration of P À D effects. The structural stiffness matrix is obtained by an appropriate superposition of the various element stiffness matrices. The axial force distribution in the members is obtained iteratively for every value of the externally applied loading and the vanishing of the determinant of the structural stiffness matrix is the criterion used to numerically determine the elastic buckling load of the structure. The effect of initial member imperfections is also included in the formulation. Comparisons of accuracy and efficiency of the present exact finite element method against the conventional approximate finite element method are made. Cases where the axial force distribution determination can be done without iterations are also identified. The effect of neglecting the warping stiffness of some mono-symmetric sections is also investigated. Numerical examples involving simple and complex three-dimensional trusses and frames are presented to illustrate the method and demonstrate its merits. IntroductionStability analysis of steel trusses and frames is a very important aspect of analysis and design of these structures. Today, all major steel design codes include the use of second-order elastic analysis through which a stability analysis can also be accomplished (Galambos 1998).Second-order analysis in the elastic region is usually done approximately by considering P À d effects at the element level and P À D effects at the structure level. Thus the effect of the axial compressive force on the member flexure and the effect of the vertical load on the laterally displaced structure, respectively, can be taken into account. An accurate second-order elastic analysis of steel trusses and frames including buckling load determination and taking into account flexural-axial coupling as well as that of large deflections can only be done by numerical methods such as the finite element method (FEM).The FEM has been successfully used for the elastic stability analysis of beam-columns and plane and space trusses and frames composed of members with doublysymmetric cross-section in the framework of either a linear analysis involving small deformations (where only P À d effects are taken into account) or a nonlinear analysis involving large deformations (where in addition P À D effects are taken into account). One can mention here, e.g., the works of Livesley (in connection with nonlinear stability analysis using an exact FEM based on displacement functions which are the exact solutions of the governing equations of member sta...