Computational and Analytical Modeling of Damped Beams on Visco-Winkler and Visco-Pasternak Foundations Subject to Axial Forces

This investigation studies how compression and tensile forces (TFs) affect the vibration responses and damped frequencies of Euler–Bernoulli beam (EBB) and shear beam supported by two and three-parameters (visco–Winkler (VW) and visco–Pasternak (VP)) foundations. The study examines how different damping coefficients, as well as compression and TFs, affect these beams’ vibrational behavior and frequency response. The spatial matrices for displacements, slopes, and the corresponding derivatives are developed analytically by implementing an extended separation of variables technique with corresponding eigenvalues. An effective and straight forward solution is offered by the Galerkin finite element method (GFEM), demonstrating freeshear locking convergence to accurate solutions. The equations of motion for damped EB and SB are solved using the state-space formulation (SSF), Further, the damped system’s response is plotted using the Runge–Kutta methods (RK4) approach. These methods offer an opportunity to investigate more intricate beam dynamics within the framework of the study. The key findings of this study reveal the influence of axial forces on vibrational behavior. As tensile forces (TFs) increase from [Formula: see text] to [Formula: see text], the first four damped frequencies rise by 2%, 0.5%, 0.29%, and 0.207%, respectively. Conversely, increasing compressive forces from [Formula: see text] to [Formula: see text] reduces the frequencies by 2.05%, 0.5%, 0.292%, and 0.291%. These results are observed by letting Notably, axial forces affect fundamental modes more significantly than higher-order modes. Graphical representations are employed to demonstrate how the system behaves to various damping conditions, and the results are validated by comparing analytical techniques with finite element method (FEM). These results have significance for the design and optimisation of structures having viscoelastic supports in engineering applications.