Nonlinear vibrations of slightly curved Timoshenko beams with nonlinear boundary conditions

Official URL: https://dx.doi.org/10.1080/15376494.2026.2661862

In this study, nonlinear vibrations of a slightly curved Timoshenko beam with nonlinear boundary conditions are investigated. The formulation is developed based on Timoshenko beam theory, incorporating shear deformation and rotary inertia effects. Geometric nonlinearity is introduced through the von Karman strain–displacement relations, and the governing equations of motion are derived using Hamilton’s principle. The resulting partial differential equations are reduced to a set of ordinary differential equations via the Galerkin method, where the mode shapes of a linear straight Timoshenko beam with linear spring boundary conditions are employed as admissible functions. The temporal response of the system is obtained using the method of multiple scales. Validation of the proposed formulation is carried out by examining the limiting case of vanishing curvature and by comparison with the nonlinear response of straight Timoshenko beams and thick shallow arch models. The results reveal that both hardening and softening behaviors may occur due to the presence of second- and third-order nonlinear terms. The proposed semi-analytical framework provides an efficient tool for analyzing nonlinear vibrations of slightly curved Timoshenko beams, capturing the combined effects of geometric nonlinearity, shear deformation, rotary inertia, and boundary flexibility.