Stress-strain state of shallow shells of arbitrary thickness according to mathematical theory

Abstract

In the research the stress-strain state of shallow shells of arbitrary constant thickness is investigated from the point of the three-dimensional theory of elasticity. The investigation is conducted on the basis of the constructed variant of the mathematical theory of transversely isotropic shallow shells of arbitrary constant thickness. A variant of the theory is based on the image of the components of the stress-strain state and boundary conditions on the side surface in the form of mathematical series using Legendre polynomials on the transverse coordinate. The constructed variant of the theory describes the stress-strain state with high accuracy. The three-dimensional problem of the theory of elasticity is reduced to a two-dimensional one using the Reisner variational principle. The boundary conditions on the front surfaces of the shell are met precisely. This increases the effectiveness of the theory. The basic dependences and equations are obtained. Systems of differential equilibrium equations with partial derivatives have a high order. To solve them, the operator method, the method of single and double trigonometric series are generalized. The influence of the shell thickness, variability of the transverse load in the region and the values of mechanical and geometric parameters on the stress-strain state of the shell is investigated. Important theoretical and applied conclusions are made, which are essential in the calculations of structural elements for strength and rigidity in construction and various fields of technology.