Comment on wes-2025-180

Abstract. The geometrically nonlinear analysis of Timoshenko beam structures with variable cross-sections is a common challenge in engineering practice. However, traditional nonlinear analysis methods for such structures often suffer from limited accuracy and computational inefficiency. To address these challenges, this study proposes an efficient geometrically nonlinear analysis framework for variable cross-section Timoshenko beams based on the co-rotational formulation. First, the novel Timoshenko beam element with a variable cross-section, based on analytical displacement shape functions, is developed to enhance the computational accuracy of the co-rotational formulation. The Gaussian integration method is employed to compute the stiffness and mass matrices of variable cross-section elements, thereby improving computational efficiency. Then, the tangent stiffness matrix of the variable cross-section beam element is derived based on co-rotational formulation and the proposed variable cross-section beam element. Finally, the dedicated finite element program is developed and validated through four benchmark examples and comparisons with experimental data from the literature. The results demonstrate that the proposed method achieves both high computational efficiency and accuracy in handling large deformations and nonlinear behavior. The proposed method is particularly suitable for analyzing structures with irregular or proportionally graded cross-sections and demonstrates advantages over existing co-rotational approaches.