Geometric nonlinear analysis of Timoshenko beams with variable cross-section using co-rotational formulation

<p>The geometrically nonlinear analysis of Timoshenko beams with variable cross-sections remains a challenging task in engineering practice, particularly for structures subjected to large deformations. While co-rotational (CR) formulations have been widely adopted for geometric nonlinear analysis, most existing CR-based beam models assume constant cross-sectional properties, limiting their applicability to beams with variable geometries. To overcome this limitation, this study introduces a novel co-rotational formulation specifically tailored for variable cross-section Timoshenko beams. The proposed approach integrates two key innovations: (1) the development of an improved spatial Timoshenko beam element employing analytical displacement shape functions to accurately capture bending deformation in variable cross-sections and (2) the introduction of an efficient Gaussian integration scheme for computing stiffness and mass matrices, eliminating the need for explicit moment-of-inertia evaluations at each cross-section. The tangent stiffness matrix is systematically derived within the co-rotational framework. The method is validated through five benchmark examples, including comparisons with experimental data and numerical results from the literature. Results demonstrate that the proposed model achieves superior computational accuracy and efficiency in handling large deformations, dynamic responses, and nonlinear behaviors of beams with irregular or proportionally graded cross-sections, offering a robust alternative to existing variable cross-section beam formulations.</p>