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

Abstract

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.

Timoshenko Beam Theory Exact Solution For Bending, Second-Order Analysis, and Stability

“…The results are identical. In fact, both formulas are identical since following equivalences exist between the parameters considered by Hu et al [8] (, ), and those considered in the present study (k,, 1, 2 ):  = 1+ k,  = 1,  = 2, (/L)2 = -k/L2 The equations for the determination of the buckling loads (see Appendix D) are also identical to those of Hu et al [8] (Table 1 of [8]). However, the formula presented by Hu et al [8] only applies for compressive forces with k  -1.…”

“…Details of the results are presented in the supplementary material "Spreadsheet S9". The calculation of the element stiffness matrix of this Timoshenko beam yields the following:(78)Let us now calculate the element stiffness matrix of the beam with the formula presented by Hu et al[8]:(79)The aforementioned characteristics become P = 1.5  EI/L²,  = 1-P/(ksGA) = 1-1.5  0.05 = 0.925,…”

Timoshenko Beam Theory:First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

“…Hu et al [5] presented the following closed-form expression of the buckling load of a fixed−free beam:…”

“…Let us calculate the element stiffness matrix of a beam with the following characteristics: Let us now calculate the stiffness matrix of the beam with the following formula presented by Hu et al [5]:…”

“…Hu et al [5] used matrix structural analysis to derive a closed-form solution of the second-order element stiffness matrix; the buckling loads of single-span beams were also determined. Fogang [6] presented a material law describing the relationship between curvature, bending moment, and shear force; based on this material law, closed-form expressions of efforts and deformations are derived, as well as first-and second-order element stiffness matrices.…”