thermal buckling of functionally graded beams

in this article, thermal stability of beams made of functionally graded material (fgm) is considered. the derivations of equations are based on the one-dimensional theory of elasticity. the material properties vary continuously through the thickness direction. tanigawa's model for the variation of poisson's ratio, the modulus of shear stress, and the coefcient of thermal expansion is considered. the equilibrium and stability equations for the functionally graded beam under thermal loading are derived using the variational and force summation methods. a beam containing six different types of boundary conditions is considered and closed form solutions for the critical normalized thermal buckling loads related to the uniform temperature rise and axial temperature difference are obtained. the results are reduced to the buckling formula of beams made of pure isotropic materials.