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

1
Department of Civil Engineering, Tsinghua University, Beijing 100084, China
2
College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China
*
Author to whom correspondence should be addressed.
The co-first author due to his equal contribution with the first author.
Appl. Sci. 2019, 9(18), 3814; https://doi.org/10.3390/app9183814
Received: 13 August 2019 / Revised: 6 September 2019 / Accepted: 6 September 2019 / Published: 11 September 2019
(This article belongs to the Section Civil Engineering)
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.
Keywords: shear deformable beam–columns; matrix structural analysis; second-order analysis; element stability stiffness matrix; Timoshenko beam theory (TBT) shear deformable beam–columns; matrix structural analysis; second-order analysis; element stability stiffness matrix; Timoshenko beam theory (TBT)
MDPI and ACS Style

Hu, Z.-P.; Pan, W.-H.; Tong, J.-Z. Exact Solutions for Buckling and Second-Order Effect of Shear Deformable Timoshenko Beam–Columns Based on Matrix Structural Analysis. Appl. Sci. 2019, 9, 3814.

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