The frequency and buckling characteristics of functional gradient (FG) beams with asymmetric material distribution in the temperature field are analyzed in this paper. Generally, the asymmetrical material distribution of FG beams results in a non-zero neutral axis and non-zero thermal moment. However, some previous studies adopted the treatment of homogeneous beams in which the neutral axis and thermal moment were set as zero. To this end, a comprehensive FG beam model with thermal effect is developed based on the absolute nodal coordinate formulation, in which Euler–Bernoulli beam theory, Lagrangian strain, exact curvature, thermally induced strain, and neutral axis position are considered. For the convenience of comparisons, the presented model can be simplified into three models which do not consider the neutral axis or thermal moment. The numerical results indicate that the influence of the neutral axis on the thermal axial force is minimal while that on the thermal moment is significant. In the case of the high temperature difference, frequency, critical temperature difference, unstable state, and the buckling type of the FG beams are misjudged when the neutral axis or thermal moment is ignored.

This paper develops a geometrically nonlinear coupled finite element model for high-speed rotating AFG beams using the complete Green–Lagrange strain tensor. The model extends the complete Green–Lagrange formulation, previously used for homogeneous rotating beams, to AFG beams and retains all higher-order axial nonlinear terms. Instead of directly adding a density-dependent centrifugal stiffening term, it solves the steady-state axial deformation from dynamic equilibrium and uses it as an intermediate variable carrying the joint distribution of $E\left(x\right)$ and $\rho \left(x\right)$, so that both properties enter the geometric stiffening terms. The gyroscopic coupling matrix $\mathbf{G}$ is fully retained. Numerical results show that (i) under the present Green–Lagrange formulation, the model predicts a monotonic stiffening trend consistent with complete stiffening in the limiting case at high rotational speeds, whereas the conventional centrifugal-force model can exhibit nonphysical softening under the fully coupled formulation; (ii) the coupled model captures frequency veering and mode exchange; and (iii) the steady-state tip deformation and first bending frequency show opposite trends with k. These results provide a reference for gradient design of rotating AFG beams within the considered parameter range.