Determination of the stressed state of thin-construction structures using the methods of the theory of shells

Abstract

Introduction. With the development of numerical methods and computational complexes, it is quite easy to evaluate the stress state of thin-walled structures in the form of rotation bodies. However, when solving such problems by the finite element method, it is necessary to choose such finite element grid to "grasp" all possible singularities of the stressed state. To correctly take them into account, you must reduce the size of the finite elements. Reducing the size of the elements leads to an increase in the required computing power. Formulation of the problem. When solving applied problems, even with a sufficiently coarse grid, the number of elements can exceed hundreds of thousands. When solving problems for real constructions in a three-dimensional setting, the amount of computation can be quite large and not every supercomputer can even handle such a solution. Objective. The purpose of this paper is to use the well-known approach used in shell theory, which allows us to reduce the three-dimensional problem to the solution of a onedimensional problem, which substantially reduces the requirements for computing power. Method (methodology). The problem of determining the stress state of shell structures in the form of bodies of revolution is considered. The approach is based on the integration of the equations of the theory of shells and the expansion of functions into Fourier series for separation of variables. The expansion into a discrete Fourier series in cosines and sines is used in this paper, which describes arbitrary asymmetric mechanical loads. Results. A thin-walled cylindrical structure hinged at the ends is considered. The structure is loaded in three places by a distributed force acting normal to the surface of the shell. After integrating the system of equations for the shell, the found stress-strain state of the shell is determined by the stress components on the outer and inner surfaces of the shell and the displacement components. The paper compares the calculation results with the proposed methodology and the finite element method. The conclusion. It is shown that the use of methods of shell theory, and the proposed expansion of resolving functions and loads in a Fourier series, allows solving problems using small computing resources. At the same time, the necessary accuracy of calculation for all components of the stress-strain state of the structure is ensured.