Numerical solution of the problem of the stress-strain state in hollow cylinders using spline approximations

Abstract

The three-dimensional theory of elasticity is used for a study of the stress-strain state in a hollow cylinder with varying stiffness. The corresponding problem is solved by a method that is partly analytical and partly numerical in nature: Spline approximations and collocation are used to reduce the partial differential equations of elasticity to a boundary-value problem for a system of ordinary differential equations of higher order for the radial coordinate, which is then solved using the method of stable discrete orthogonalization. Results for an inhomogeneous cylinder for various types of stiffness are presented.The increasingly stringent requirements for the estimation of strength characteristics, the tendency toward a detailed consideration of real properties of structural materials, and the discovery and study of three-dimensional effects occurring in thick-walled elements require the treatment of hollow cylindrical structures in terms of a three-dimensional model. Finding a solution for the stress-strain state in thick-walled structures within the framework of spatial linear elasticity theory goes hand-in-hand with significant difficulties related to the complexity of initial systems and partial differential equations, as well as the necessity of satisfying the boundary conditions prescribed on the surfaces of the elastic body. These difficulties arise substantially during the calculation of structural elements such as cylinders made of anisotropic and inhomogeneous materials. The facts mentioned above are consistent with the relative sparseness of the number of publications addressing such questions (Kollar, Patterson, and Springer [11], Banerjee and Henry [3], Kollar [10], Shi, Zhang, and Xiang [12], Gal and Dvorkin [5], Collin, Caillerie, and Chamlon [4], and Tsurkov and Drach [13]).Along with universal approaches used for solving boundary-value problems in mechanics and mathematical physics, such as the finite-difference technique, finite-element method, and other discrete methods, the new technique now finds wide application for this particular class of problems [2,3,14]. It allows reducing the initial problem to a system of ordinary differential equations, based on an approximation of the solution with respect to other variables by analytical methods. The exact reduction of multidimensional problems to onedimensional ones and the solution of the latter by the stable numerical method of discrete orthogonalization gives reasons to believe that the obtained results are highly accurate. Due to cylindrical geometry, the method of finite elements, if used for calculating the mechanical behavior, is time-consuming and inefficient and requires large memory and processing speed of the computer.Recently, an approach based on spline approximations has been developed in several articles [6][7][8][9] in order to study the mechanical behavior of plate and shells. Its main advantages are the following [1]:• stability against local perturbations, i.e., the local behavior of splines in the neighborhood of a point d...