Pipes conveying fluid play an important role in the oil & gas industry. It is critical to determine the critical velocity    for structural stability design and safety evaluation of these pipes. Pipes conveying fluid under thermal load are also often used in    practical engineering fields, such as crude oil pipeline heating transportation and heated pipelines. Compared with the basic pipes    conveying fluid, the natural vibration frequency and critical velocity of a pipeline under thermal load are different from those    of the basic pipeline. Based on Hamilton’s principle, the partial differential equation of vibration of supported pipes conveying    fluid under thermal loads is derived and the equation is reduced to a univariate fourth-order homogeneous ordinary differential    equation by separating variables. A general solution is obtained according to the critical flow velocity conditions of supported    pipes conveying fluid. Furthermore, the analytical solutions of the critical velocity are obtained considering various boundary    conditions. Finally, numerical examples are presented for analyzing the influence of linear thermal stress and nonlinear thermal    stress on the critical velocity under various boundary conditions. The predictions using the proposed analytical solution are    compared with results using the differential quadrature method available in the literature. It is demonstrated that the proposed    analytical solution can give an accurate solution efficiently, which can be used in engineering practice. The critical flow rate of     the pipes conveying a fluid system under linear thermal stress and non-linear thermal stress decreases with an increase of thermal    load, and the decrease becomes larger and larger. In the same case, the critical velocity under nonlinear thermal stress is greater    than that under linear thermal stress, and the gap between them increases with an increase of thermal load. Comparing the bound   ary conditions, it is found that fixed boundary conditions can bear the largest thermal load. Therefore, applying fixed boundary    conditions to the pipes conveying fluid system under the thermal load is beneficial to improve the stability of the system. In this    paper, the analytical method for critical velocity of pipes conveying fluid under thermal load can be obtained conveniently and    quickly at the engineering site, which provides a reference for the design and safety evaluation of pipes conveying fluids under    thermal load.