<p>Mechanical analyses of functionally graded materials (FGMs) are essential for accurately predicting structural performance and ensuring the reliability of FGM-based components. In this study, a unified form of the fundamental equations governing the behavior of FGM hollow cylinder and spherical shell is derived. A more comprehensive gradient model based on the generalized power-exponential function is developed to describe the variation of Young's modulus and magnetic permeability along the material's thickness. Moreover, through the selection of suitable parameters, this model can be reduced to the classical exponential and power-law gradient models. By solving the hypergeometric ordinary differential equations, general solutions for displacement and stress are obtained. Considering the six different combinations of displacement-stress boundary conditions, the analytical solutions for the mechanical response of FGM structures are derived under the combined influences of the magnetic field, external pressure, and a Winkler elastic foundation. The correctness of the proposed solution is validated by comparing it with existing analytical solutions for classical exponential and power-law FGM structures, which are special cases of the present model. Through detailed case studies, the research investigates the effects of various graded parameters, elastic foundation stiffness, and magnetic field strength on the displacements and stresses of the hollow cylinder and the spherical shell. The innovation of this study lies in proposing a more general gradient material model that can accurately describe the non-uniform variation of material properties. The findings provide valuable insights that can guide the optimal design of FGM structures using the proposed comprehensive gradient model.</p>