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