Iterative methods for constructing an equations of non-closed shells solution

Abstract

The elasticity relations are transformed to a form that allows, in accordance with the previously proposed Saint-Venant - Picard - Banach method, to iteratively calculate all the required unknowns of the problem. The procedure for constructing a solution is reduced to replacing eight first-order differential equations of the original system of shell theory with eight corresponding integral equations with a small parameter that has the meaning of the ratio of the shell width to its length or the variability of the stress-strain state in the transverse direction. The fifteen unknowns of the original problem calculated by direct integration are expressed in terms of five main unknowns. The fulfillment of the boundary conditions on the long sides of the strip leads to the solution of eight ordinary differential equations for slowly varying and rapidly varying components of the main unknowns. Slowly varying 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 fulfillment of the boundary conditions unsatisfied by them due to the lowering of the order of the differential equations of the classical theory based on the Kirchhoff hypothesis. In the general case, the solution is represented as asymptotic series in a small variability parameter with coefficients in the form of power series in the transverse coordinate. The presentation is illustrated by an example of constructing an iterative process for a long circular cylindrical panel. By virtue of the fixed-point theorem, the iterative process is convergent.