Stress-strain state of the spherical shell based on the refined theory

Dynamics, strength of machines, instruments and equipment


Firsanov V. V.*, Pham V. T.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia



The classical theory does not produce a satisfactory compliance with practice while calculating the stress-strain state of structural elements in the areas of fixture and action of local and rapidly changing loads, as well as structural elements made of heterogeneous materials. To describe the volumetric stress-strain state, it is necessary to develop a refined, with respect to the classical theory of the Kirchhoff-Love type, theory based on the three-dimensional equations of the elasticity theory.

The presented work proposes an option of the refined theory of the stress-strain state calculation of isotropic shells. The mathematical model is built based on the 3D equations of the elasticity theory. The problem of 3D equations reduction to the 2D equations is being solved by representing the desired displacements with polynomials over the normal coordinate of two orders higher relative to the classical theory.

The system of differential equations of equilibrium in displacements with variable coefficients was obtained using the Lagrange variational principle. The aforesaid system of equations solution was being performed by methods of finite differences and matrix run. As a result, the displacements in the grid nodes were obtained, and splines were used for their approximation. The shell deformations are being found by dint of geometry relationships, and tangential stresses are determined from the Hook’s law relationships. Lateral stresses are obtained by direct integrating of the equilibrium equations of the 3D elasticity theory.

A hemispherical shell rigidly clamped along the lower contour of the base is considered as an example. Comparing the obtained results with the data of the classical theory allowed establishing that the refined theory should be used while studying the stress-strain state of the spherical shell near the zones of the stress state distortion. For example, it should be applied in the vicinity of a rigidly clamped edge, since the maximum stresses in this zone are being refined substantially.

Neglected in the classical theory, the transverse normal and tangential stresses in the border zone are of the same order as the maximum stress values corresponding to the classical theory. Such high levels of additional stresses should be accounted for while assessing the shell structures strength and durability.

With distancing from the edge the stresses, obtained by the refined and classical theories are practically concurring, which confirms the fidelity of the obtained results.


herical shell, refined theory, Lagrange variational principle, stress-strain state of the “boundary layer”, finite difference method, matrix run method, transverse normal stresses


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