Thermal deformation of a long elastic strip

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.