The buckling analysis of Euler beam made up offunctionally graded material with linear and quadraticvariations of elasticity modulus in axial direction is studied. Thesimply supported beam is rested on a rotational spring hinge atright end. The beam is ideal without initial curvature and it issubjected to axial compression without eccentricity. The linearelasticmaterial behavior and engineering axial strain areconsidered. The analytical formulas of linearized bucklingcharacteristic equations are calculated for linear and quadraticmaterial gradation in the axial direction of the beam. Thebuckling behaviors of these kinds of elements is a basic theoryfor more complex structural analyses. By approachingrotational stiffness of the rotational spring hinge to zero orinfinity, the critical load of simply supported or pinned-clampedaxially graded beam is calculated, respectively. The numericalexamples are conducted and critical load diagrams are depicted.