Based on Kirchhoff’s assumption of straight normal line of beams and considering the effects of the axial elongation, the initial curvature and the stretching-bending coupling on the arch deformation, geometrically nonlinear governing equations of functionally graded arch subjected to mechanical and thermal loads are derived. In the analysis, it is assumed that the material properties of the arch vary through the thickness as a power function. As a numerical example, the critical buckling load and the corresponding mode shapes of a semicircle arch, with both of the ends fixed, subjected to normally uniform distributed follower force is obtained by the shooting method. The effects of the parameters of material gradient on the critical loads and the deformation of the structure are examined in detail. Equilibrium configurations for different values of the load or temperature rise are plotted. Analysis and numerical results show that the behavior of buckling of the arch is of bifurcation and the buckling modes corresponding to minimum buckling load is asymmetric. In other words, bifurcation buckling occurs prior to the snap-through buckling.