Porous Orthotropic Shallow Shells: Nonlinear Vibration and Post-Buckling Under Non-Uniform Edge Loads

Abstract

Porous orthotropic doubly-curved shallow shells (PODSSs) are widely utilized in aerospace and structural applications due to their high stiffness-to-weight ratios and adaptability to advanced material gradation. Capturing their nonlinear mechanical responses under realistic loading and geometric conditions is essential, particularly when higher-order effects, material porosity, and edge load non-uniformities are present. This study aims to develop an analytical framework to investigate the nonlinear free vibration and post-buckling responses of PODSSs subjected to non-uniformly distributed edge compressive loads. The governing equations are formulated using higher-order shear deformation theory (HSDT) integrated with von K & aacute;rm & aacute;n geometric nonlinearity. Porosity-dependent orthotropic material properties are graded through the shell thickness via cosine and sine functions. Galerkin's method is employed to derive frequency-amplitude and load-deflection relationships for different porosity distributions and geometrical configurations. The accuracy of the present theoretical formulation is verified by comparing it with available results in the literature. The influence of porosity coefficient, porosity distribution patterns, orthotropy, non-uniform edge loadings, and geometrical characteristics on nonlinear free vibration and post-buckling responses is discussed. Among load patterns, the triangular loading pattern (TGL) produces the highest post-buckling loads, while uniform loading (UL) results in the lowest, with deflection-dependent variations between spherical and hyperbolic geometries. NUDP3 yields the lowest nonlinear frequencies, while NUDP2 significantly affects both frequencies and post-buckling loads. These insights contribute to the optimal design and mechanical performance prediction of porous orthotropic shells under realistic service conditions.