Coupled bidirectional bending and torsional vibrations of axially loaded non-symmetrical thin-walled Timoshenko–Ehrenfest beams

Abstract This paper introduces an analytical framework for examining the coupled bidirectional bending and torsional vibrations of non-symmetric, axially loaded thin-walled Timoshenko–Ehrenfest beams. By integrating axial loads, shear deformation, rotational inertia, and warping stiffness into the traditional Timoshenko–Ehrenfest beam theory, we enhance its ability to address complex bending-torsion interactions. Utilizing Hamilton’s principle, we derive five coupled differential equations and twelve boundary conditions to accurately describe the beam’s dynamic behavior. The normal mode method is used to derive closed-form expressions of frequency responses under arbitrary harmonic loads, and orthogonality conditions are established to obtain precise modal impulse and frequency response functions. Our framework provides accurate and computationally efficient solutions and examines the impact of axial loads on natural frequencies, offering practical guidance for engineering design. These findings contribute to the dynamic analysis of thin-walled Timoshenko–Ehrenfest beams, providing useful insights for engineers in designing and optimizing structures under complex loading conditions.