Exact Solutions for Buckling and Second-Order Effect of Shear Deformable Timoshenko Beam–Columns Based on Matrix Structural Analysis

Shear deformable beams have been widely used in engineering applications. Based on the matrix structural analysis (MSA), this paper presents a method for the buckling and second-order solutions of shear deformable beams, which allows the use of one element per member for the exact solution. To develop the second-order MSA method, this paper develops the element stability stiffness matrix of axial-loaded Timoshenko beam–columns, which relates the element-end deformations (translation and rotation angle) and corresponding forces (shear force and bending moment). First, an equilibrium analysis of an axial-loaded Timoshenko beam–column is conducted, and the element flexural deformations and forces are solved exactly from the governing differential equation. The element stability stiffness matrix is derived with a focus on the element-end deformations and the corresponding forces. Then, a matrix structural analysis approach for the elastic buckling analysis of Timoshenko beam–columns is established and demonstrated using classical application examples. Discussions on the errors of a previous simplified expression of the stability stiffness matrix is presented by comparing with the derived exact expression. In addition, the asymptotic behavior of the stability stiffness matrix to the first-order stiffness matrix is noted.