<![CDATA[The objective of this paper is to provide a novel method for developing the shape functions of a beam-column element with semi-rigid connection ends, thereby establishing a static analysis method for semi-rigid steel frames. This method takes into account the influence of the P-Delta effect, according to the finite element method based on displacement (FEM). The shape function is established directly from a third-order Hermitian displacement function polynomial combined with the bending element deflection differential equation. The linear elastic stiffness matrix, the geometric stiffness matrix of a semi-rigid connection beam-column, and the equilibrium equation of the element in a local coordinate system are simultaneously obtained by applying Castigliano’s theorem (Part 1) for elastic deformation potential energy expression. The computational program was developed using Matlab software, and the calculation results are verified against published research results, showing that the derived shape functions and the steel frame analysis method are reliable and trustworthy. In addition, this article also derives stiffness matrices and an equivalent nodal load vector for specific cases where the semi-rigid connection is fully rigid (FR) or a pin connection. The derived shape functions are polynomial expressions with coefficients that are simply calculated from the connection stiffness and the geometric and material characteristics of the element, making them highly convenient to use. Doi: 10.28991/CEJ-2025-011-01-021 Full Text: PDF]]>
