Differential equations for the physically orthotropic cylindrical shells under action of circle loads

Аuthors

Nerubailo B. V.^1*, Kargaev M. V.^2**

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
2. National Helicopter Center Mil & Kamov, 26/1, Garshina str., Tomilino, Moscow region, 140070, Russia

*e-mail: borisn@km.ru
**e-mail: kargaev_mv@mail.ru

Abstract

The article is devoted to an actual problem from a field of strength of aviation and rocket-cosmic thin- walled constructions such as solution to boundary problems for a definition of their stress-deformed state on the basis of high defining differential equations in the partial derivatives. An object for investigations is the circle cylindrical shells made from material, which has different elastic properties in the three mutually perpendicular directions, i.e. the physically orthotropic cylindrical shells.
Under the action of arbitrarily changing on shell surface circle loads in the article for the first time there were obtained differential equations of the eighth order in the partial derivatives of general theory for the physically orthotropic shells most often meeting among anisotropic thin-walled constructions. From sufficiently great variety of initial equalities of the shells theory the authors sel ected the Vlassovs version as the most coherent fr om the point of view of the energy-static principles compliance. In particular, the Betti works return theorem is exactly executed, that causes the symmetry of the three equations system in displacements for the theory of shells. Thus, in the article the Vlassovs theory of isotropic cylindrical shells is generalized to the case of the physically orthotropic shells, i.e. the new differential equations are obtained.
Because the integration of the differential equations of eighth order in the partial derivatives of the general theory for the physically orthotropic shells is a great problem, a simplification is made in the article on the basis of confrontation principle of the stress-deformed states variability in axis and circle directions (Novogilovs criterion). As a result there were obtained the new differential equations in the partial derivatives of «gently sloping shells type», which for the isotropic shells transient to the Vlassov-Donnells equations; «semimomentless theory type » for the isotropic shells transient to the widely well-known equations; and the equations for the tangential stress-deformed state with very high variability, i.e. the analogue to the plane problem of the elasticity theory.
On the basis of obtained differential equations the boundary problems for the finite long closed cylindrical shell with hinge fixed edges under action of circle local loads are considered. The differential equations of the general theory are applied for solution to this problem by Fourier trigonometric series method.
Some results of calculation are represented in the graphic form showing the influence of loading type and orthotropic parameter on the stress-deformed state of shells.

Keywords:

orthotropic shell, differential equations, asymptotic synthesis, local load, Fourier series

References

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