Deformation of flexible anisotropic elastic systems of shells of revolution on a nonlinear foundation

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UDC 539.3 The problem of the interaction of the elastic elements of structures and their foundation is of central importance in the design of machines, space facilities, and engineering structures. A large number of studies have analyzed the work of structures without allowance for the elastic foundation. There have also been numerous investigations that have made allowance for same. However, most of them have used linear models both for the shells (the structural elements) and the foundation. For the latter, the model has usually been the classical Winkler model. The Winkler model presumes that the intensity of the reaction of the foundation is proportional to the deformation in the structure. The foundation is represented as a set of linear springs without mass that are deformed only in the normal direction [1, 2]. Also, researchers generally examine isotropic or orthotropic plates and cylindrical or spherical shells. At the same time, experience shows that in the case of large elastic displacements, the interaction of the structure with the foundation is described by a nonlinear function. This article analyzes the interaction of anisotropic shell systems with a nonlinear elastic foundation on the basis of the geometrically nonlinear theory of thin shells. We will examine the stress-strain state of elastic systems composed of a series of coaxial closed mnltilayered shells of revolution. The shells themselves are made up of an arbitrary number of anisotropic layers having a thickness that varies along the meridian. There is one plane of elastic symmetry at each point of the shell, this plane being parallel to its coordinate surface. The elastic characteristics do not change along the parallel. The elastic system is subjected to surface and contour loads such that it deforms while retaining its axial symmetry. We also assume that the system interacts with the elastic foundation without slip or separation and that the intensity of the reaction of the foundation is determined by the formula