THE METHOD OF SUCCESSIVE APPROXIMATIONS IN THE MATHEMATICAL THE-ORY OF SHALLOW SHELLS OF ARBITRARY THICKNESS

The method of sequential approximations (MSA) in mathematical theory (MT) of transversal-isotropic shallow shells of arbitrary thickness is developed. MT takes into account all components of stress-strain state (SSS). SSS and boundary conditions are considered to be functions of three varia-bles. Three-dimensional problems are reduced to two-dimensional decompositions of all the compo-nents of the SSS into series in the transverse coordinate using Legendre polynomials and using the Reisner variational principle. The boundary conditions for stresses on the front surfaces of the shell are fulfilled precisely. Previous studies have shown the high efficiency of this MT. The boundary-value problem for a shallow shell is reduced to sequences of two boundary-value problems for the respective plates. One sequence describes symmetric deformation relative to the median plane, and the other sequence is skew symmetric. MSA makes it easier to find a common solution of differential equations (DE) for shallow shells. Highly accurate results for SSS are already in the first approxi-mation. MSA can be used when solving problems for shallow shells by other theories.


Introduction.
Problem solving for shells and plates is performed on the basis of classical and refining theories, using equations of three-dimensional elastic theory and on the basis of variants of mathematical theory.Classical and clarifying theories are based on various physico-geometric assumptions [1-3, 8, 12, 15, 17, 20, 22, 23].The limits of using these theories for different classes of problems require further research.The most common in practical use are the theories of the Tymoshenko-Reisner type [20,22,23] and their various modifications [3,8,12,17].Clarifying theories include theories based on specific deformation models [11].The main drawback of all the clarifying theories is the inability to increase the accuracy of the solution of the problems within these theories.
The use of three-dimensional elasticity theory in the analytical solution of boundary value problems for plates and shells [6,14] is too much of a problem for mathematical physics, since all the components of the SSS and boundary conditions are functions of three coordinates.At the same time, three-dimensional SSS occurs in thick plates and shells, in the field of local, discontinuous and nonsmooth loads, under the action of other SSS concentrators.And so there is an urgent need to develop

PHYSICS AND MATHEMATICS
and construct theories that take into account all the components of the SSS and boundary effects as a functions of the three variables.And so that these theories can be used to analytically solve boundary value problems.with the required accuracy.These qualities are satisfied by the variants of MT, which are based on a mathematical approach in the image of the components of the SSS with infinite rows in transverse coordinates.These theories are devoid of physico-geometric assumptions.Different mathematical series are used: tensor [9], power [13], using the Lezhran-dra polynomials [4,5,7,10,16,18,19,24].Three-dimensional problems are reduced to two-dimensional by different methods: operating [4,5,24], variational [7,10,16,18,19], others [15].The MT variants have different accuracy depending on the approach of reducing three-dimensional problems to two-dimensional ones and the method of representing the SSS in the form of mathematical series.
In this article, MSA is developed in solving boundary value problems for transversely isotropic shallow shells of arbitrary thickness based on the MT variant [25][26][27][28].Shells can be subjected to arbitrary transverse loads.All SSS components that are functions of three variables are taken into account.The MT is based on the representation of the SSS components in the form of infinite rows with a transverse coordinate using Legendre polynomials.The transverse normal and tangent stresses are approximated by taking into account the three-dimensional DE equilibrium theory of elasticity such that the boundary conditions in the stresses on the face surfaces are satisfied exactly.Three-dimensional problems for shells are reduced to two-dimensional problems based on the Reisner variational principle [21].This method of constructing the MT variant showed efficiency and high accuracy [25,26].As the number of additives in the mathematical series increases, the order of the systems of equations and the complexity of solving them increases, but the accuracy of the solution increases.The MSA makes it possible to reduce the complex boundary-value problem for the shell to simpler boundary-value problems for the corresponding plates with symmetric and oblique deformation relative to the median plane.

Problem statement.
We study the transversal isotropic shallow shell of constant arbitrary thickness h h in a rectangular coordinate system z y x , , .The surface of the isotropy coincides with the median surface.The axes x, y belong to the plan of the shell, and the axis z is perpendicular to the plane of the plan of the shell and is directed in the direction of the convexity (up ).On the upper and lower surfaces of the shell there is a static transverse load ) , ( 1 y x q and ) , ( 2 y x q directed downwards.All SSS components are functions of three coordinates.Boundary conditions on the front surfaces: The transverse loads on the upper and lower surfaces are depicted as the sum of two additions: oblique symmetric 2 / q and symmetric 2 / p loads relative to the middle surface: . The boundary conditions on the side surface may be different.The displacement components are represented by the Fourier-Legendre series in the coordinate z : -sought components in displacements.If in (2)  Since the shell is of arbitrary thickness, tangential displacements are taken into account in the shear deformations of yz xz   , [1] (in the theory of thin shells they are neglected): is the principal radii of curvature of the middle surface of the shell.Clarifying additives in the expressions for the transverse angular deformations contain 2 1 , k k   .Here are general structural formulas for stress components [25], which derive from the DE system of the spatial theory of elasticity and the Reisner variational equation: The stress components according to (3): The transverse normal and tangent stresses satisfy exactly the conditions (1).For the approximations K0-3 and K0-5, the functions are given in [26].3.2.Boundary conditions.The boundary conditions are obtained from the Reisner variation equation: where n Z must balance the transverse load on the upper and lower surfaces of the shell.Equations ( 6) and ( 7) yield different boundary conditions.Here are some of them.

1) Boundary conditions in displacements. Only the displacement components
are known on the side surface Г of the shell.Boundary conditions: , ( 2) Boundary conditions in stresses.Only the external load is specified on the side surface.Then we have the following boundary conditions: 3) The boundary conditions for the freely fixed at the edges of the shells: ). ,...
In the approximations K01, K0- is the mechanical parameters of the transversely isotropic material.The DE system (12) is not divided into two systems that describe independently symmetric and oblique deformation.This indicates the interdependence of symmetric and oblique deformation of the shells.For plates, DE systems are separated.

Approximation K013.
In approximation K013, the DE (16th order) system consists of the first-fourth, seventh-ninth, and eleventh equations (12).In addition, you need to put 0 ) , ( = y x p and consider only the functions of ), where 0 Equations 1, 2 of systems ( 16) and ( 17) describe the symmetric deformation of the plates, and 3, 4, 7-9, 11skew symmetry.Similarly, DE systems are obtained for other approximations.
The systems DE ( 15), ( 17) coincide for the corresponding plates.,And ( 14) and ( 16) are structurally different only in the right parts.Therefore, each of the systems ( 14)-( 17) can be divided into systems that separately describe the vortex boundary effect, the internal SSS, and the potential boundary effect.Methods of transformation and decoupling of such systems are given in [26].
In MSA, at each approximation, the general solutions must satisfy the same set boundary conditions.In the method of perturbations of geometric parameters [27] in a null approximation by a small parameter, the general solutions must satisfy the given boundary conditions, and in subsequent approximations the corresponding homogeneous boundary conditions.4.3.Numerical results.The effectiveness of MSA was investigated in a boundary value problem for transversally isotropic shallow shells, freely fixed on the lateral surface (10) . Numerical results show that in the zero approximation of the difference between the SSS components and the results obtained by the direct solution of the DE equilibrium system, for ) , ( / ) , , ( y x q z y x x  is less than 3.9%, for ) ) , ( /( ) , , ( h y x q E z y x W -less than 1.1%.In the first approximation for the difference does not exceed 1%.This indicates a high convergence of MSA.

Conclusions.
1) The method of sequential approximations in MT of transversely isotropic shallow shells of arbitrary thickness is developed.In the zero approximation of MSA, the systems of equations for shells coincide with the equations for the corresponding plates.In the following approximations, the left parts are the same and coincide with the equations for the plates, and the right parts of the equations depend on the curvatures and components in the displacement components of the previous approximations.
2) By this method, the boundary value problem for the shell is reduced to a sequence of boundary value problems for the corresponding plates with symmetric and oblique deformation.Then inhomogeneous high-order DE can be reduced to low-order equations.MSA makes it easier to find a common solution for shallow shells.
3) Numerous studies have shown a high convergence of results.4) MSA can be used to solve problems for shallow shells based on other theories.

4 )
Boundary conditions for rigidly secured shells: that the differential matrix of the DE (12) system is symmetric ( in tangential displacements we take into account terms with indices