In this doctoral dissertation, a numerical approach for solving the problem of nonlinear stability of beam structures made of FG (functionally graded) materials under conditions of variable temperature is developed. The geometric nonlinear algorithm was developed using a 1D numerical model based on the application of a spatial beam finite element. The effects of large spatial displacements and spatial rotations are included. Deformations are considered small and the Green-Lagrange deformation tensor is used to define them. The finite element model is based on the Euler-Bernoulli beam theory for bending and the Vlasov theory for torsion. In the nonlinear analysis, the UL (updated Lagrangian) incremental formulation was implemented using the principle of virtual works. The proposed beam model enables the implementation of buckling analysis for cases of uniform, linear and nonlinear temperature distribution through the thickness of the crosssectional walls as well as linear temperature distribution along the beam length will be conducted. The analysis also includes the temperature dependence of the mechanical properties of the material. The computer program THINWALL FG was created on the basis of the above. The problem was solved in two stages. In the first phase, the response of the structure to a given mechanical load is determined. In the second phase, while the mechanical load is kept constant, a temperature load is applied and the temperature at which thermal buckling occurs is determined. The accuracy and reliability of the numerical algorithm was tested by comparison with the results available from the literature and those obtained using 2D and/or 3D finite element models from established software packages.