Bending–Extension Coupling at Restrained Ends and the Nonlinear Transient Response of Porous FGM Sandwich Beams in Thermal Environments

Porous functionally graded material (FGM) sandwich beams combine ceramic heat resistance with metal toughness, but their through-thickness grading and porosity make their response in thermal environments sensitive to modelling assumptions. This paper examines the nonlinear transient response of porous FGM sandwich beams with elastically restrained ends under suddenly applied loads in thermal environments. A refined higher-order shear deformation theory (HSDT) is used with temperature-dependent properties, even and uneven porosity distributions, and symmetric and asymmetric material layouts. In asymmetric layouts, bending–extension coupling and the thermal moment produce end moments that conventional one-term strong-form Galerkin approximations omit. A multi-term weak-form Galerkin approach is developed that retains these moments for arbitrary support locations, and it is validated against exact linear solutions and an independent nonlinear finite element model. The results show that the material layout through the thickness, rather than porosity or gradation alone, determines whether strong-form approximations are adequate. For asymmetric beams, neglecting thermal bending overestimates the peak deflection at elevated temperatures, and the deviation grows when the supports act at the neutral surface. Parametric studies clarify the effects of porosity, volume fraction index, face-sheet thickness, temperature, end restraint, foundation stiffness and initial imperfection.