Comparative vibration analysis of shear-deformable beams on a Winkler foundation using transfer matrix methods

Abstract Based on the unified shear deformation theory, this paper presents the free vibration analysis of linear, isotropic, homogeneous beams resting on a Winkler elastic foundation. Unlike existing transfer matrix formulations that operate on four-variable state-space systems, the present work establishes a six-variable state-space framework in which an arbitrary shear strain function is incorporated into the formulation solely through integral coefficients, enabling the unified treatment of seven distinct shear deformation models without theory-specific rederivation. The governing equations and boundary conditions are derived using energy principles. Natural frequencies are determined using the initial value method via both exact transfer matrix (ETM) and approximate transfer matrix (ATM). Since the analytical derivation of the ETM involves a sixth-order characteristic polynomial with prohibitively complex expressions, an ATM is constructed using a three-term truncated matricant series expansion, thereby eliminating the need for eigenvalue extraction. A correction methodology based on the chain property is proposed to ensure convergence to the exact solution. The formulation reduces the frequency determinant from a $$6 \times 6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>6</mml:mn> <mml:mo>×</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> system to a $$3 \times 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>×</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> system for each boundary condition. Systematic comparisons across different shear deformation theories, boundary conditions, slenderness ratios, and foundation stiffness values confirm the accuracy and reliability of the proposed approach, with ATM results converging to the ETM reference within less than 0.25% relative error for all examined configurations upon moderate interval refinement. These findings establish the proposed formulation as a unified, highly accurate, and versatile framework that enables consistent vibration analysis of shear-deformable beams without requiring theory-specific rederivation.