Mathematical Theory of Transversally Isotropic Shells of Arbitrary Thickness at Static Load

摘要

The construction variant of the mathematical theory of elastic shells of small curvature of arbitrary constant thickness under static loading is considered. The theory variant is based on the use of three-dimensional differential equations of equilibrium and taking into account all components of the stress-strain state (SSS) as a function of three coordinates. Transverse stresses are determined from three-dimensional equations so that the boundary conditions on the lower and upper surfaces are precisely satisfied. The three-dimensional problem is reduced to two-dimensional using the method of decomposition of displacement components in the Fourier-Legendre series on transverse coordinates. The Reissner variational principle is applied. In this case, other components of the SSS are also depicted in the form of series using Legendre polynomials. The boundary value problem is formulated in terms of the three-dimensional theory of elasticity. For different approximations, the main dependences, differential equations of equilibrium, boundary conditions on the lateral surface are deduced. By means of the operator method, solvable equations and forms of general solutions are obtained. The analysis of the obtained theoretical results and the formulated conclusions are carried out.