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.
A method of studying the natural vibrations of highly inhomogeneous shells of revolution is developed. The method is based on a nonclassical theory of shells that allows for transverse shear and reduction. By separating variables, the two-dimensional problem is reduced to a sequence of one-dimensional eigenvalue problems. The inverse iteration method is used to reduce these problems to a sequence of inhomogeneous boundary-value problems solved by the orthogonal sweep method. The capabilities of the method are illustrated by solving certain representative problems and comparing their solutions with those obtained using the three-dimensional theory of elasticity, the classical theory of shells, and the refined Timoshenko model Keywords: highly inhomogeneous shells of revolution, natural frequencies, transverse shear, reduction, solution technique, inverse iteration method, three-dimensional theoryIntroduction. General analysis of the frequency spectrum of elastic structures, as any linear system, has two interrelated subtasks: qualitative analysis of the behavior of this spectrum and quantitative dynamic analysis of a specific class of structures [1-3, 5, 7, 9, 10, 11, 16]. The present paper is concerned with the latter task and naturally continues the developments intended to determine the natural frequencies and modes of the elastic objects mentioned in [3]. Most studies in this area are based on two shell models: classical Kirchhoff-Love model and refined Timoshenko model. These models made it possible to accurately calculate the lowest frequencies for a wide class of thin and medium-thickness shells made of shearable materials. However, these models may fail to ensure required accuracy for shells highly inhomogeneous across the thickness, such as laminated shells with layers whose mechanical properties differ by more than an order of magnitude. In this connection, this paper proposes a method to calculate the dynamic characteristics of shells based on a nonclassical model that allows for all transverse strains. The capabilities of the method will be demonstrated by solving representative problems for a wide scope of assumptions on the inhomogeneity of shells across the thickness. The frequencies obtained by the three-dimensional theory of elasticity and by two-dimensional shell models of different degrees of accuracy will be compared. The nonclassical shell model that accounts for transverse shears and reduction was earlier tested by solving stress-strain problems for laminated shells under local loads and contact problems for shells of revolution on a rigid foundation [12,14,15]. Problem Formulation and SolutionTechnique. The subject of study is the class of inhomogeneous shells of revolution described, as three-dimensional bodies, in an orthogonal curvilinear coordinate system s, q, V (s is the meridional arc length, q is the cross-sectional central angle, and V is the thickness coordinate reckoned from some coordinate surface V = const). Shells of this class include an arbitrary number J of isotropic and ort...
The torsion problem for a rectangular prism with general anisotropy loaded on the lateral surface is solved using the advanced Kantorovich-Vlasov method, which reduces the original three-dimensional problem to three coupled one-dimensional problems, each for one of the variables of the domain. The warping of the cross-section and the deformation of the axis of the prism for different types of anisotropy are analyzed Introduction. The torsion problem for elastic prismatic bodies has been addressed by engineers and mathematicians for the last 100 years. The unremitting interest in this problem is due to practical needs (many structural elements are designed to resist torsion) and theoretical needs (the deformation of such elements should be studied to reveal the mechanism of their failure).Only two classical special cases of torsion of a rod acted upon by opposite torques at its ends are completely understood: pure torsion described by the rigorous theory developed by Saint Venant [5, 6] and generalized torsion theoretically studied by Voight [16] and Lekhnitskii [3]. In the former case, only two tangential stresses appear nonzero, the prism axis remains straight, and cross-sectional warping is constant throughout the height. Such a behavior is typical of isotropic, orthotropic, and anisotropic materials with the plane of elastic symmetry perpendicular to the prism axis. In the latter case, all the stresses are nonzero yet constant throughout the height, the prism axis bends, and the cross-sectional warping linearly varies throughout the height. This case of torsion is observed in materials with general anisotropy or anisotropic materials with one plane of elastic symmetry nonorthogonal to the prism axis.The classical torsion problem was generalized in different ways such as inclusion of large strains or stress concentration around cracks, analysis of stability and vibrations under various torsional loads, complication of the body shape, material properties (viscosity), and loading and boundary conditions [10][11][12][13][14][15]17].The present paper addresses the torsion problem for a rectangular prism made of anisotropic materials with low order of symmetry and the stress-strain state varying throughout the height. Use will be made of the general problem statement in linear elasticity. The problem-solving method to be used reduces the original three-dimensional problem to three coupled one-dimensional problems, each for one of the variables of the domain. The deformation of the cross-section and axis of a square-based prism will be studied for different types of anisotropy of the material.
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