In the buckling issue of pipelines,the structure is usually considered as a plane stress ring,but if there exists axial displacement constrain,the status may be closer to the status of a plane strain ring.In this paper,the elastic buckling problem of the plane-stress and plane-strain functionally graded material rings under hydrostatic pressure is theoretically investigated.It is assumed that the elastic material properties along the thickness direction continually changes according to the power law distribution.In this case,by combining the first shear deformation theory and the Kárman geometry relation and by employing the virtual displacement principle,a basic equilibrium equation of structures is deduced.Then,the buckling governing equation and the analytic solution to the structural buckling critical load are obtained by means of the eigenvalue method.Finally,the theoretical buckling perfor- mances of the structure in the plane-stress and plane-strain states are compared,and the influences of the structural radius-to-thickness ratio and the material constituent parameter on the performance are discussed.The results show that the buckling critical load of the plane-stain functionally graded material ring is slightly higher than that of the plane-stress case,and that as the inhomogeneous parameter of the material ranges from 0.1 to 10,the buckling criti- cal loads of the two kinds of rings decrease more obviously.