Flexural–Torsional Buckling Behavior of FGP Curved Beams Based on a Modified Variational Method

This study presents a modified variational modeling method for the flexural–torsional buckling of functionally graded porous (FGP) curved beams. Unlike conventional methods that rely on strictly admissible functions, the proposed framework accommodates arbitrary orthogonal polynomial basis functions and segment-wise discretization. Lagrange multipliers and least-squares residuals are introduced to rigorously enforce interfacial and boundary constraints, which enables the proposed method to adapt to arbitrary boundary conditions. The energy functional accounts for shear deformation and material gradient effects, with the critical buckling loads determined numerically through a discretized matrix formulation. Due to the high orders of admissible functions employed, the accuracy of the responses in stress fields is enhanced and, in turn, the solution accuracy of the buckling properties is also increased. The accuracy and computational efficiency of the method are validated against existing results in the literature. A comprehensive parametric investigation is conducted to examine the influences of the power-law exponent, porosity distribution, slenderness ratio, boundary conditions, and loading configurations on buckling behaviors of the curved beams. This research provides a unified and high-fidelity framework for the stability design of advanced FGP structures in civil, mechanical, and aerospace engineering applications.