Differential equations for physically orthotropic and isotropic cylindrical shells under action of longitudinal

Аuthors

Nerubailo B. V.^*, Vu X. D.^**

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

*e-mail: borisn@km.ru
**e-mail: ducmoscow@gmail.com

Abstract

The article is devoted to an actual problem from the field of strength of aviation and rocket-cosmic thin- walled constructions such as solution to the boundary problems for definition of their stress-deformed state on the basis of high definition of differential equations in the partial derivatives. An object for investigations is the circle cylindrical shells made from a material which has different elastic properties in the three mutually perpendicular directions, videlicet the physically orthotropic cylindrical shells.
Under action of arbitrarily changing on shell surface longitudinal load the high defining differential equations in the partial derivatives of general theory for the physically orthotropic shells which are most often meeting among anisotropic thin-walled constructions were obtained for the first time in this article. From sufficiently great varieties of initial equalities of shells theory the Vlassovs version was sel ected by the authors as the most consistent fr om the viewpoint of energy- static principles observance. In particular, the Bettis equalities for reciprocity of work are exactly executed and it causes the symmetry of three equations system in displacements for the theory of shells. Thus, in the article the Vlassovs general theory of isotropic cylindrical shells is firstly generalized for the case of the physically orthotropic shells and the new differential equations are obtained.
Since the integration of the eight defining differential equations in the partial derivatives of the general theory for the physically orthotropic shells is a great problem the simplification was made in the article on the basis of variability confrontation principle of stress-deformed states in axis and circle directions (Novogilovs criterion). As a result, there were obtained the new differential equations in the partial derivatives such as «gently sloping shells type» of equations passing into the Vlassov-Donnells equations in the case of isotropic shells, «semimomentless theory type» of equations passing into the widely well-known Vlassovs equations in the case of isotropic shells, as well as the equations for the tangential stress-deformed state with very high variability — plane problem analogue of elasticity theory.
On the basis of asymptotic errors for the mentioned here approximate equations three methods of asymptotic synthesis of the shells stress state were formulated for the concrete problem considering in the article and two of their give possibility to make the asymptotically exact solutions. On the basis of the obtained differential equations the boundary problems for the infinite long closed cylindrical shells under action of axial local loads are considering. The differential equations of the general theory and approximate theories of shells are applying for solution to these problems by Fouriers integral method. In the equations case of the semimomentless theory and tangential stress state the analytical solutions and simple formulae for important unknown quantities were obtained as results of improper integrals taking.
Some results of calculating represented in the graphic form showing the influence of loading kind and orthotropic parameter on the stress-deformed state of shells.

Keywords:

orthotropic shell, differential equation, asymptotic synthesis, local load, Fouriers integral

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