*Practical importance.* The analytical and numerical results obtained can be used in the study of the stress state and, respectively, to evaluate the strength and stiffness of curvilinear tubular structural elements, in particular, the pipeline bends and pipes of economizers.

*The purpose.* Investigation of the temperature field and the thermoelastic state of the curvilinear rod with a tubular cross-section caused by it; analysis of the influence of geometrical and thermophysical parameters on the characteristics of its temperature field, as well as on the forces and moments in it.

*The research method*. The object of research is the curvilinear rod with a tubular cross-section, which is in the conditions of convective heat exchange with the external and internal environments. The thermal conductivity equation for a large curvature rod was used to describe its temperature field. For determining the characteristics of the stress-strain state due to the described temperature field, the equations of thermal elasticity of curvilinear rods in displacements were used; the case was analyzed when radial and tangential displacements are absent in the extreme cross-sections of the curvilinear part of the rod, and there are no turns of these cross-sections.

*Results.* The studies have shown that the specific coefficients of heat transfer of the outer and inner surfaces of the rod affect its stress state in a different way; as the heat transfer coefficient from the outer surface increases, the absolute values of the longitudinal and transverse forces, as well as of the bending moment, decrease, and with the increase of the heat transfer coefficient from the inner surface, they increase. In contrast to the stress problem, in the case of heating, the value of the bending moment in a certain cross-section of a curvilinear rod depends on the geometrical parameters of the cross-section.

*The scientific novelty.* Analytical-numerical method shows for the first time that there is such a cross-section in which the heat-induced bending moment changes its sign; the position of this cross-section depends on the curvature of the axis, the thickness of the wall, the bend angle of the curvilinear part of the rod and practically does not depend on the specific coefficients of heat transfer from its inner and outer surfaces. There are values of the bend angle at which the bending moment and transverse force in the curvilinear part of the rod reach their maximums.

[1] B. F. Boley, “On thermal stresses end deflections on thin rings”, *Inter. J. Mech. Sci.*, vol. 11 (9),

pp. 781–789, 1969. https://doi.org/10.1016/0020-7403(69)90025-3

[2] H. Lok, H. D. Conway, “Thermal stresses in multi-layered curved bars”, *Fibre Sci. and Technol.*, vol. 9 (2), pp. 135–151, 1976. https://doi.org/10.1016/0015-0568(76)90016-6

[3] I. A. Birger, B. F. Shorr,* Termoprochnost’ detaley mashin** [**Thermal strength of machine parts**]*. Moscow, Russia: Mashinostroenie Publ., 1975. [in Russian].

[4] V. A. Knyazeva, “Raschet sostavnykh osesimmetrichnykh kol’tsevykh konstruktsiy” [“Calculation of composite axisymmetric ring structures”], *Izvestiya vuzov. Mashinostroyeniye [**News of Universities. Mechanical Engineering]*, vol. 4, pp. 10–15, 1979. [in Russian].

[5] G. N. Feodosiev, “O zadache termouprugosti dlya mnogosloynykh kolets” [“On the problem of thermoelasticity for multilayer rings”], *Izvestiya vuzov. Mashinostroyeniye [**News of Universities. Mechanical Engineering]*, vol. 8, pp. 13–17, 1970. [in Russian].

[6] D. L. Kostovetskiy, *Prochnost’ truboprovodnykh sistem energeticheskikh ustanovok** [**Durability of pipeline systems of power plants**]*. Leningrad, Russia: Energiya Publ., 1973. [in Russian].

[7] A. G. Kamerstein, V. V. Rozhdestvenskiy, M. N. Ruchimskiy, *Raschet truboprovodov na prochnost’. Spravochnaya kniga** [**Strength calculation of pipelines. Reference book**]*. Moscow, Russia: Nedra Publ., 1969. [in Russian].

[8] G. Elter, “Zur Bewertung von Spannungen in Rohrleitungen infolge Wärmetauschen des Systems” [“On the evaluation of stresses in pipelines caused by heat exchange of the system”], *T.V.*, vol. 15 (1), pp. 25–28, 1974. [in German].

[9] M. A. Axelrod, P. V. Chukanov, N. D. Mishura, “Korozionno-ustalostnoye razrusheniye gibov trub vodyanogo ekonomayzera” [“Corrosion-fatigue failure of water economizer pipe bends”], *Teploenergetika [**Heat Power Engineering]*, vol. 1, pp. 45–47, 1973. [in Russian].

[10] Ya. S. Podstrigach, Yu. A. Chernukha, N. I. Voitovich, “K opredeleniyu temperaturnykh poley i napryazheniy v obolochkakh, sopryazhennykh cherez sterzhen” [“On the determination of temperature fields and stresses in shells conjugated through a rod”], in *Matematicheskiye metody v termomekhanike** [**Mathematical methods in thermomechanics]*. Kiev, Ukraine: Naukova Dumka Publ., 1978, pp. 3–11. [in Russian].

[11] S. P. Timoshenko, *Soprotivleniye materialov** [**Strength of materials**]*. Kyiv, Ukraine: Vyshcha Shkola Publ., 1973. [in Russian].

[12] V. A. Svetlitskiy, *Mekhanika sterzhney. Chast 1 **[**Mechanics of rods. Part 1**]*. Moscow, Russia: Vysshaya Shkola Publ., 1987. [in Russian].