Finite Element Investigation of the Free Vibration Behavior of Functionally Graded Porous Timoshenko Beams on a Two-parameter Elastic Foundation with Shear Correction Effects

Abstract

This study investigates the free vibration behavior of porous functionally graded beams using Timoshenko theory on a Winkler–Pasternak foundation. Such beams are commonly used in practical engineering applications, including aerospace structures and civil engineering components, where vibration control and weight efficiency are critical. The main advantage of the proposed method lies in its ability to capture the coupled effects of porosity and material gradation on transverse shear behavior, which are neglected in classical models using constant shear factors. The effects of key parameters are analyzed, including the material gradation index material, the foundation coefficients, porosity distributions, and transverse shear deformation through an adjusted correction factor. The governing equations of motion are derived using Hamilton's principle. The formulation is generally applicable to different boundary conditions, material configurations, and elastic foundation parameters. The proposed model is validated through comparisons with results from the literature, demonstrating agreement and confirming its reliability. The numerical results indicate that increasing the gradation index materials leads to a reduction in natural frequencies due to decreased material stiffness, an effect that becomes more pronounced for higher vibration modes. Conversely, the combined influence of the foundation parameters results in an increase in natural frequencies, highlighting the stiffening effect of the foundation. The shear correction factor decreases with increasing index material. Finally, comparing the natural frequencies obtained using a shear correction factor for isotropic materials with those of corrected porous FGM materials highlights the need to include coupled porosity–gradation effects in the dynamic analysis of FGM beams.