Зеленський А. A variant of the mathematical theory of non-thin elastic plates and shallow shells under static loading

The actual scientific problem of mechanics of deformed solids is solved in the work: construction of new effective variants of mathematical theory (MT) of physically linear and nonlinear according to Kauderer homogeneous and layered plates and shallow shells at arbitrary static transverse loading on the basis of three-dimensional theory of elasticity; development of approximate methods for solving systems of high-order differential equations (DE); obtaining partial and general solutions of boundary value problems of MT variants of the specified elements and numerical dependences of the stress-strain state (SSS) on mechanical-geometrical characteristics, type of loading, boundary conditions and approximations in the constructed variants. Important in solving the scientific problem is that the constructed versions of MT provide a real opportunity to analytically solve the boundary value problems for these elements and determine the SSS with high accuracy. The three-dimensional problem of theory of elasticity for plates and shells is re-duced to the two-dimensional one using Reissner’s variational principle, method of series expansion of SSS components on lateral coordinate with the use of Legendre-Polynom for homogeneous plates and shells or com-bination of both. The three-dimensional boundary value problem for these elements is re-duced using perturbation methods and successive approximations to an infinite recurrent sequence of two-dimensional linear boundary value problems. A new method for integrating inhomogeneous systems DE of high-order equilibrium equations of the MT of non-thin plates has been developed, based on mathematical transformations of equations and their reduction to homogeneous and inhomogeneous DE of the 2nd order; developed new approximate methods for solving the obtained systems of equations for plates and shallow shells. General solutions are obtained using the method of operators. Boundary value problems for determination of SSS of nonfine homogeneous physically linear and nonlinear plates, shallow shells and two-layer and three-layer trans-verse-isotropic plates are solved. New conclusions on the influence of mechanical and geometric parameters, types of load and boundary conditions on the SSS are obtained. Keywords: new variants of mathematical theory, nonfine linearly and nonlinearly elastic homogeneous and layered plates and shallow shells, Legendre polynomials, Reisner variational principle, high-order differential equations, methods, general solutions, stress-strain state, boundary effects.