Orthotropic foundation effect on nonlinear bending, buckling, and post-buckling of porous orthotropic laminated beams under non-uniform axially and transversely varying loadings

This study develops a unified nonlinear framework to clarify how orthotropic Pasternak foundation effects interact with porosity patterns, lamination, and varying load distributions. A higher-order shear deformation theory (HSDT) is combined with von Kármán-type geometric nonlinearity to formulate the governing equations of porous orthotropic laminated beams resting on an orthotropic Pasternak foundation. Non-uniform axial compression patterns (A-DTG, A-DTR, A-UL, A-ITR, A-ITG, and point load) and linearly varying transverse loadings (T-ITG, T-ITR, and T-UL) are considered. The resulting nonlinear ODEs are solved using the Galerkin method to obtain equilibrium paths for nonlinear bending, critical buckling, and post-buckling responses. The results demonstrate that orthotropic foundation support is most influential (about 30–40%) near the onset of nonlinearity and instability but becomes secondary at large deflections where membrane stretching governs the response. Increasing the shear-layer stiffness Kx¯ elevates nonlinear bending and post-buckling equilibrium paths and increases the critical buckling load. The strong sensitivity to foundation orientation indicates that aligning the stiff foundation axis with the beam axis is essential for maximizing support efficiency, particularly for porous configurations. The combined influence of non-uniform axial/transverse load patterns, along with porosity distribution, provides actionable guidance for tailoring lightweight beam elements that operate in complex mechanical environments.