A method is proposed to solve the contact problem for laminated anisotropic shells of revolution. The method is based on a two-dimensional model that accounts for transverse shears and reduction. Also the method is based on the method of successive approximations, the generalized pseudo-force method, and a numerical-analytical method of solving boundary-value problems. The results obtained for a cylindrical shell of complex thickness structure are compared with those obtained in three-dimensional formulation Keywords: contact problem, anisotropic shell, laminated structure Introduction. In analyzing the stress-strain state of thin-walled structures on rigid or elastic foundations, it is necessary to describe the deformation of a shell and the mechanism of its interaction with the foundation [1-3, 6-10, 13].The model chosen to describe the deformation of a laminated shell should strike a reasonable balance between accuracy and implementability. The spatial theory of elasticity can adequately describe processes of interest, but may involve certain computational difficulties because of the high dimension of the corresponding problems. Approximate two-dimensional models of shells materially simplify the way the final result is obtained, but describe the contact interaction of bodies with different degrees of adequacy. For example, the classical theory of shells requires introducing concentrated forces at the contact boundary, which distort the contact pressure distribution. Allowing for transverse-shear strains eliminates the discontinuity of the shearing forces, but does not make the normal reactions at the boundary vanish. Additional tricks, such as an elastic layer between contacting surfaces, allow us to describe contact interaction more accurately, though describing the properties of such a layer involves difficulties [6,7]. A natural way to solve the contact problem for thin-walled shells is to allow for all kinds of transverse strains.In describing the interaction of contacting elements, the contact conditions are formulated as inequalities reflecting the nonnegativity of the constraint reactions in the contact region. In the absence of tangential forces, the geometrical contact condition is usually equality of displacements (strains) or curvatures of these elements [8].Expanding upon the studies [1, 2, 13], we propose here a method to solve static problems for laminated anisotropic shells of revolution contacting with rigid or elastic planes. The method is based on the nonclassical theory of shells and accounts for transverse-shear strains and reduction. To test the method, we will compare the results it produces with the exact solutions of some problems.
The stationary bending of a two-dimensional elastic system of joined rectangular plates with different mechanical properties and boundary conditions is studied. A technique for solving the corresponding problem is proposed. It is based on approximate approaches in combination with a generalization of the force method. It is established that the static strain state of the system is essentially dependent on the difference between the elastic moduli of the constituent plates. It is also shown that with certain boundary conditions, the dependence of the first resonant frequency on the relative position of plates along the contact line is nonmonotonic Keywords: system of rectangular plates, static strain state, arbitrary boundary conditions, generalization of the force methodThe stress-strain state of various structural elements can be determined within the framework of the general bending problem for elastic systems of joined rectangular plates. Such elements are, for example, walls and floors with openings, corner balconies, and other building structures.Different ways of combining rectangular plates into a single system necessitates formulating conditions for their combined action on the contact line. What these conditions are depends on the functionality of a specific system. In particular, they may be perfect or slipping contact of plates or local reinforcing elements such as beams or point supports [2,3]. Designing such a system is a challenge because of the necessity of joining solutions in subdomains of the system and the complex configuration of its outer boundary, which, generally, is a nonconvex polygon. The design approach is mainly based on discrete methods, which describe well the class of objects under consideration (see, e.g., [1,6]).Continuum or discrete-continuum methods in combination with a generalization of the force method may appear to perform well in solving bending problems for specific plate-like systems. This is especially true of hinged systems [3,[7][8][9].The present paper sets forth a technique for designing two-dimensional systems of joined inhomogeneous rectangular plates. The technique is based on the generalized Kantorovich-Vlasov method (method of complete systems) and the force method well known in structural mechanics. We will use the technique to analyze the influence of the mechanical characteristics of materials and the boundary conditions on the stationary strain state of the system. Problem Formulation and Solution Method. Consider an elementary system consisting of two joined (over a plane)
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