Approximate estimation of stresses and displacements of a thin-walled square cross-section beam after torsional buckling

It is important to estimate the strength of the vehicle structure in the initial design stage of the vehicle development. The buckling of the plate is known to occurs in thin-walled structures before yielding, however, the structures can withstand loads even after buckling. The load acting on the thin-walled beam constituting the vehicle frameworks may be not only a compressive force and a bending moment but also a torsional torque. The purpose of this paper is to derive the equation for the torsional post-buckling behavior of a thin-walled square cross-section beam. When a torsional torque acts on a square cross-section beam, in-plane shear stress acts on the four plates constituting the cross section, which may cause shear buckling. Assuming that the same buckling deformation occurs on the four plates, Timoshenko's approximate expression can be applied as an out-of-plane displacement after shear buckling. The post-buckling stress distributions are calculated as a function of the out-of-plane displacement amplitude. Also, the post-buckling torsional constant and relationships between the torsional torque and the stresses are derived on the basis of Karman's effective width theory. The validity of derived expressions is examined by comparing with the computation results by the finite element method (FEM). As a result, in the target dimensional range of the plate (width-to-length ratio 1/3 or less), the absolute difference values of the derived relationship between the torsional torque and the maximum Mises stress are within 10% for FEM results. However, it is found that instable phenomena such as secondary buckling may occur as the width-to-length ratio of the plate becomes smaller. From the above results, it is found that the relational expression derived in this paper is sufficiently useful for calculating the torsional strength of a thin-walled square cross-section beam.