This study aims to obtain Bree's diagram for a Functionally Graded (FG) beam under cyclic thermal and constant mechanical loadings. For this purpose, the mapping algorithm is employed to specify the borders of ratcheting, shakedown, and elastic. Also, two parameters are considered to obtain Bree's diagram, and the iteration method is utilized to predict the beam's elastoplastic behavior uniaxially. Since the FG beam comprises two types of metal, namely 90% aluminum alloy (EN A W 6061 T4) and 10% carbon (S275), the procedure to develop Bree's diagram for FG structure is unique, and it can be implemented for other combinations. Bilinear isotropic hardening model and the FG beam with fxed-free boundary conditions are considered. The FG beam is constrained along the vertical direction to avoid any defection in the horizontal directions. The analysis conducted in this study is geared toward the Finite Element Method (FEM). Ansys Parametric Design Language (APDL) in Ansys 16 is employed. A novel method based on the FEM is proposed, which is on par with the other state-ofthe-art methods in this regard and is even able to outperform them in terms of accuracy and authenticity. Interestingly Bree's diagram present for the metal–metal FG beam can be generalized to other FG beams like ceramic–ceramic and metal–ceramic FG beams and so on, considering the analysis of this study. Notably, due to the available results in the literature, the Bree diagram obtained by the proposed numerical method is reliable.
