A full modal decomposition of thin-walled, single-branched open cross-section members via the constrained finite strip method

Abstract

This paper derives a new method for fully decomposing the elastic stability solution, of a thin-walled single-branched open cross-section member, into mechanically consistent buckling classes associated with global, local, distortional, and shear and transverse extension buckling modes. The method requires a set of formal mechanical definitions for each of the buckling classes. For global and distortional buckling the definitions employed successfully by generalized beam theory are utilized herein, while for local and other (shear and transverse extension) buckling, new definitions are provided. The mechanical definitions for a given buckling class represent a series of constraint conditions on the general deformations that the thin-walled cross-section may undergo. These constraint conditions are derived as explicit constraint matrices within the context of the finite strip method, and provide the desired decomposition of the buckling deformations of the member. The decomposition is full in the sense that the union of the deformation spaces of the decomposed buckling classes is the same as the general deformation space in the original finite strip method. The resulting method is termed the constrained finite strip method (cFSM). The two primary applications for cFSM are modal decomposition and modal identification. Modal decomposition reduces the general finite strip solution to a desired set of buckling classes and performs a useful model reduction that allows the results to focus on a particular buckling class, e.g., distortional buckling. Modal identification provides a means to quantify the extent to which a given buckling class is contributing to a general buckling deformation. Application of cFSM, including graphical representation of the buckling classes, and the advantages of modal decomposition and modal identification, are provided in a series of numerical examples.