Estimation of stresses and displacements of a rectangular plate after shear buckling using approximate expression by Timoshenko

In the initial design stage of vehicle development, it is important to estimate the strength of the structure against various loads. It is known that buckling occurs in thin-shell structures before yielding as the load increases, and that the structures can withstand loads even after buckling. The load acting on the beam constituting the vehicle structures may be not only a compressive force and a bending moment but also a torsional torque. Due to this torsion torque, a shear stress acts on the thin shell constituting the beam. In this paper, after shear buckling of a rectangular plate, the approximate expression of buckling deformation presented by Timoshenko as an out-of-plane displacement is used, and the relationship between the out-of-plane displacement amplitude (maximum out-of-plane displacement) and the shear force is determined based on Karman's effective width theory. Using the obtained out-of-plane displacement amplitude, three stress distributions and the maximum value of Mises equivalent stress are derived as a function of shear force. Then, the derived expressions are compared with the computation results of the finite element method (FEM) using shell elements under the boundary condition that does not constrain the in-plane displacement of both sides in the longitudinal direction, and the applicability is examined. As a result, it turned out that the derived relations between the out-of-plane displacement amplitude, the maximum value of Mises stress, and the shear load are less than the absolute value error within 10% in the discussed load range, compared to the FEM computation results. Therefore, the derived expressions are sufficiently effective in determining the yield strength based on the Mises stress.