Due to economic efficiency, members with variable cross-sections are widely used in steel structures, especially industrial sheds. But with the increase in temperature caused by thermal changes, the strength and hardness characteristics of industrial frame members decrease rapidly. It is necessary to check stability (thermal buckling) in industrial frames for a safe and complete design. In this article, thermal buckling and vibrations caused by thermal changes of non-radiative elastic columns are investigated by the Rayleigh-Ritz method. In the next step, the weak form of the differential equation is calculated and the Chebyshev series is used as the transverse displacement function and the weight function. In the last step, after extracting material stiffness matrices, geometric stiffness, and mass matrix, the eigenvalues of the equation are checked. The results show that the simultaneous increase of the slope of the section and the coefficient of thermal changes significantly affect the effective length coefficient and reduce the buckling load capacity in all the boundary conditions of different supports. Also, the simultaneous increase of the slope of the cross-section and the coefficient of thermal variation, depending on the type of boundary conditions of the support, cause an increase or decrease in the dimensionless natural frequency. Aligned curves are used to present results and display graphs to apply results in engineering calculations. The results of previous research are used for validation. There is an acceptable agreement between the present results and previous studies.