Axisymmetric deformation of the reinforced by threads toroidal membrane

Abstract

Nonlinear axisymmetric deformation of a toroidal shell under action of internal pressure is considered. The shell is reinforced by two system of threads located on parallels and meridians. It is supposed, that threads are disposed frequently enough therefore after averaging we get the anisotropic elastic membrane. The basic feature of a membrane is that it can not hold the compression stresses. In state of equilibrium all meridians of the shell are stretched, but the part of parallels can be compressed. That leads to for mation of folds. For search of deformations and displacements of a shell the system of the ordinary differential equations of the fourth order is received. Geometrical and physical nonlinearity is taken into account. The method of numerical solution and also the method of asymptotical integration in the case when the meridian radius is much smaller than the parallel one are elaborated. Comparison of asymptotic and numerical results is performed. It is shown, that there is an ultimate pressure at which excess states of equilibrium are absent. If pressure is less that the ultimate one there are two states of equilibrium- subcritical and supercritical. Asymptotic and numerical solutions for the minimal value of pressure at which folds on the shell are not formed are found. Refs 3. Figs 2. Tables 3.

Keywords:

anisotropic membrane, geometrical and physical nonlinearity, internal pressure, asymptotic and numerical solutions