Application of discrete Fourier series to the stress analysis of shell structures

Abstract

A method for the solution of problems associated with determination of the stress state of shells of revolution using a discrete Fourier-series method is proposed. In the present work the classical theory based on Kirchhoff-Lova hypotheses is used to describe the stress state of shells. The problem of determination of the stress state of a shell with variable parameters along a generatrix is reduced to the solution of the system of partial differential equations. Discrete Fourier series are the set of functions defined on a discrete set of points. This method allows one to reduce the dimension of the problem and to solve it by the numerical Godunov’s method of discrete orthogonalization. As the approach based on application of a discrete Fourier-series for the functions set on a discrete ensemble of points is used, the curvilinear grid with an equal step on a meridian is plotted on the shell surface. The circles obtained are broken into equal amounts. This yields a set of virtual shell elements. It is recognized that the value of distributed loads on this set of virtual elements is known. We suggest to approximate loads using a discrete Fourier series expansion in cosines and sines, which makes it possible to describe any asymmetric loading. Examples of application of analytical and discrete Fourier series to calculation of isotropic and the orthotrophic of shells are given. The errors in approximating the given functions are defined by discrete series. For all the problems the necessary number of harmonics is determined. On necessary retention of the quantity of summable harmonics, the approximating function describes rather precisely various external superficial loading modes. The approximation error obtained by the Fourier series expansion method proposed here is insignificant, and this provides a description of any asymmetric loading. Application of discrete Fourier series makes it possible to reduce the dimension of the problem and to determine the stress state of shells of revolution.