Analytical Formulations for Coupled Axial-Bending-Shear-Torsion Vibrations of Asymmetric Laminated Timoshenko Beams

This paper presents the exact closed-form solutions for the vibration analysis of extension-bending-shear-torsion coupled responses in asymmetric laminated Timoshenko beams. The study derives four governing differential equations and corresponding boundary conditions using Hamilton’s variational principle. The formulation, based on first-order shear deformation theory (FSDT), accounts for rotary inertia, Poisson’s ratio, and extensional-flexural-torsional-shear coupling effects arising from material anisotropy. The exact solutions are obtained for various harmonic bending and twisting excitations under different boundary conditions, capturing the fully coupled axial, bending, and torsional responses. This approach offers a robust analytical method for understanding the complex dynamic behavior of laminated composite beams subjected to dynamic loads, with potential applications in aerospace, mechanical, and civil engineering structures, such as aircraft components, turbine blades, and bridges. Keywords: Closed-form solution; extension-bending-shear-torsion coupled response; harmonic excitations; asymmetric laminated beam.