Abstract

 
 
 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.
 
 
Highlights
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
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 stress-strain state (SSS) and boundary conditions are functions of three coordinates
The mathematical theory (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
Summary
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. 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. 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.
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