Damped free vibration analysis of classical and shear deformable beams resting on elastic foundations: Damping effects and design implications

Abstract This paper proposes a comprehensive analysis of the free vibration of SB, RB, and EBB with damping effects, resting over VHF, VFF, and VWF. Both numerical and analytical techniques are used to establish and solve the governing fourth‐order differential equations. The study provides details on the formulation of the governing equations via shear deformation theory, embedding damping, and considering the unique stiffness characteristics of each foundation type. The impact of foundation parameters, comprising spring constants and damping ratios, on the vibrational characteristics, such as natural frequencies and mode shapes, is thoroughly investigated. Numerical methods are conducted using FEM to validate analytical solutions, ensuring accuracy, and computational efficiency. An analogous analysis highlights the strengths and limitations of both methods, offering a comprehensive insight into beam dynamics. This research enhances the understanding of dynamic behavior in beam structures and gives a framework for future design optimizations, parameter sensitivity analysis, and real‐world engineering applications involving shear beams with complex support conditions.