RUDN Journal of Engineering ResearchRUDN Journal of Engineering Research2312-81432312-8151Peoples’ Friendship University of Russia3029610.22363/2312-8143-2021-22-3-293-304Research ArticleThermal deformation of a long elastic stripZveryaevEvgeny M.<p>Doctor of Technical Sciences, Professor, leading researcher at the Keldysh Institute of Applied Mathematics; Professor of the Department of Construction, Academy of Engineering, Peoples’ Friendship University of Russia (RUDN University); Professor of the Department of Design of Complex Mechanical Systems, Moscow Aviation Institute</p>zveriaev@mail.ruhttps://orcid.org/0000-0001-8097-6684Keldysh Institute of Applied Mathematics, Russian Academy of SciencesPeoples’ Friendship University of Russia (RUDN University)Moscow Aviation Institute (National Research University)3012202122329330424022022Copyright © 2021, Zveryaev E.M.2021<p style="text-align: justify;">A general method is proposed for the formulation and solution of temperature problems of the theory of elasticity for thin-walled bodies for a given temperature distribution with the preservation of the order of differential equations and the fulfilment of all boundary conditions. The elasticity relations, taking into account temperature deformations, are transformed to a form that allows, in accordance with the Saint-Venant-Picard-Banach method, to perform iterative calculation of all the looking for unknowns of the problem. The procedure for constructing a solution is reduced to replacing four differential equations of the first order of the original system of elasticity theory with four corresponding integral Picard equations with a small factor of relative thinness. Seven unknowns of the original problem calculated by direct integration are expressed in terms of four basic unknowns. The fulfilment of the boundary conditions on the long sides of the strip leads to the solution of four ordinary differential equations for slowly varying and rapidly changing components of the main unknowns. Slowly changing components describe the classical stress-strain state. The rapidly changing ones determine the edge effects at the points of discontinuity of the slowly changing classical solution and the fulfilment of the unsatisfied boundary conditions due to the lowering of the order of the differential equations based on the Kirchhoff hypothesis. In the general case, the solution is represented in the form of asymptotic series in the small parameter of thinness with coefficients in the form of power series in the transverse coordinate. The presentation is illustrated by examples of warping of a free strip and of the occurrence of stresses and displacements of only the edge effect in a strip rigidly clamped at the ends with a linear temperature distribution along the height.</p>elasticitystripcomplete solutionSaint-Venant - Picard - Banach methodboundary conditionsboundary effectпринцип сжатых отображенийтеорема о неподвижной точкеупругостьполосаполное решениеметод Сен-Венана - Пикара - Банахаграничные условиякраевой эффектсontraction mapping principle[Love AEH. A treatise on the mathematical theory of elasticity. Cambridge: Cambridge University Press; 1927.][Friedrichs KO. Asymptotic phenomena in mathematical physics. Bull. Amer. Math. Soc. 1955;61(6):485–504.][Vasiliev VV. On the theory of thin plates. Izvestiya RAN. Mekhanika Tverdogo Tela. 1992;(3):26–47. (In Russ.)][Vasilev VV. Kirchhoff and Thomson – Tait transformations in the classical theory of plates. Mechanics of Solids. 2012;47:571–579. https://doi.org/10.3103/S0025654412050111][Vasilev VV. Torsion of a square isotropic plate by forces applied at the corners and by distributed torques. Mechanics of Solids. 2017;52:134–143. https://doi.org/10.3103/S0025654417020030][Grigolyuk EI, Selezov IT. Non-classical theory of oscillations of rods, plates and shells. Results of Science and Technology. Mechanics of Solid Deformable Bodies (vol. 5). Moscow: VINITI Publ.; 1973. (In Russ.)][Zveryaev EM. Saint-Venant – Picard – Banach method of integration of equations of the theory of elasticity of thin-walled systems. Prikladnaya Matematika i Mekhanika. 2019;83(5–6):823–833. (In Russ.)][Zveryaev EM. Interpretation of semi-invers Saint-Venant method as iteration asymptotic method. In: Pietraszkiewicz W, Szymczak C. (eds.) Shell Structures: Theory and Application. London: Taylor & Francis Group; 2006. p. 191–198.][Zveryayev EM. A consistent theory of thin elastic shells. Journal of Applied Mathematics and Mechanics. 2016;80(5):409–420. https://doi.org/10.1016/j.jappmathmech.2017.02.008][Zveryayev EM, Makarov GI. A general method for constructing Timoshenko-type theories. Journal of Applied Mathematics and Mechanics. 2008;72(2):197–207. https://doi.org/10.1016/j.jappmathmech.2008.04.004][Zveryaev EM, Olekhova LV. Reduction 3D equations of composite plate to 2D equations on base of mapping contraction principle. Keldysh Institute Preprints (issue 95). Moscow; 2014. (In Russ.) Available from: http://keldysh.ru/papers/2014/prep2014_95.pdf (accessed: 02.14.2021).][Zveryayev EM. Analysis of the hypotheses used when constructing the theory of beam and plates. Journal of Applied Mathematics and Mechanics. 2003;67(3):425–434.][Lebedev NN. Temperature stresses in the theory of elasticity. Moscow, Leningrad: ONTI. Glavnaya redaktsiya tekhniko-teoreticheskoi literatury Publ.; 1937. (In Russ.)][Zveryaev EM, Olekhova LV. Iterative interpretation of the semi-inverse Saint-Venant method when constructing equations for thin-walled structural elements made of composite material. Trudy MAI. 2015;(79):1‒27. (In Russ.)]