Abstarct: In this thesis, the thermal buckling of the functional variable cylindrical shell with a variable wall is studied. The mechanical and thermal properties of the shell in the direction of the thickness and axis of the shell are continuously changed. Strain-displacement equations are baxsed on the shear deformation theory. Two kinds of temperature distribution consist of uniform and linear distribution is considered. The critical temperature of the buckling is calculated with the semi-analytical finite element method using the minimum potential energy principle. The boundary conditions are simply supported and fully clamped. The effect of different parameters including the distribution of properties of functional variable materials, changes in the thickness of the wall, different boundary conditions, length ratio to radius, radius ratio to thickness and changes in the exponential-law coefficient on the buckling of the variable cylindrical shell has been studied.