As an alternative method, the element stiffness matrix is modeling using stability parameters. The transverse deflections along the beam axis, including the axial- flexural effects in the beam-column element, are not adequately described by polynomials. In the static analysis of beam-column systems using matrix methods, polynomials are using as the shape functions. Two illustrative examples are presented to demonstrate the simplicity of the developed second-order MSA approach for a systematic analysis of imperfect frames. The established formulations and the intuitive plots are readily coded in an easy-to-use computer program given in the Appendix for analyzing the exact second-order effect of general frames with initial imperfections and intermediate loads. Some intuitive plots of the fixed-end force vector are presented for convenient applications to typical engineering situations. 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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