![]() The equilibrium relationship between the element-end displacements and forces is developed in a matrix form, which includes a second-order element stiffness matrix and the fixed-end force vector. ![]() ![]() Based on the equilibrium analysis of an imperfect axial-loaded beam–column, the element-end displacements and forces are solved exactly from the governing differential equation. To develop the second-order MSA method, the fixed-end force vector is formulated in simplified forms to account for the effect of the member initial imperfections and the intermediate loads. Based on the matrix structural analysis (MSA), this paper presents a method for the exact second-order solutions of imperfect frame structures, which allows the use of one element per member for the exact solution. Initial member imperfections play important roles in the second-order effect of structures, and thus it is helpful to find ways for engineers to easily model the member imperfections in the analysis.
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