A unified framework is established for the buckling analysis of axially functionally graded (AFG) beams, integrating Euler–Bernoulli and Timoshenko kinematic formulations. For the Euler–Bernoulli limit, exact closed-form solutions for arbitrary power-law gradations are derived using Bessel functions, providing rigorous analytical benchmarks. A high-order Generalized Differential Quadrature Method (GDQM) is developed for the variable-coefficient Timoshenko eigenvalue problem, achieving relative errors below 3% against 3D finite element simulations. Results elucidate a boundary-dependent shear sensitivity hierarchy of C–C > S–S > C–F, with shearinduced load reductions reaching 24% in deep beams (L/H = 5). Under iso-stiffness constraints, axial gradation is found to degrade the stability of doubly-constrained beams by promoting mode migration toward compliant regions. Conversely, proximal stiffening in cantilevered configurations significantly enhances the critical buckling load by approximately 20% through optimized stiffness redistribution. Furthermore, applicability phase diagrams are constructed using a 5% error threshold to delineate regimes where shear-deformable theories are mandatory to preclude non-conservative predictions. This combination of benchmarks, numerical procedures, and design maps provides a robust toolkit for the optimized design of AFG thin-walled members.
Yu et al. (Fri,) studied this question.