# Stability of cylindrical anisotropic composite shells under torsion in a three-dimensional formulation

## Authors

• Volodymyr Trach National University of Water and Environmental Engineering, Ukraine
• Andrii Podvornyi National University of Water and Environmental Engineering, Ukraine

## Keywords:

anisotropic cylindrical shell, three-dimensional setting, torsional stability

## Abstract

The article presents a calculation of the stability of non-thin cylindrical anisotropic layered shells under the action of end torsional moments in a spatial formulation. The anisotropy of the used material is characterized by one plane of elastic symmetry of characteristics. This is caused by the mismatch between the main elastic directions of the composite fibrous orthotropic material and the axes of the curvilinear cylindrical coordinate system.

A three-dimensional inhomogeneous system of partial differential equations describing the subcritical stress-strain state within the linear theory of elasticity is derived using the Hu-Washizu variational principle. Reducing the dimension of the problem under consideration from three-dimensional to one-dimensional is carried out by taking into account the axial symmetry of the deformation of the cylindrical shell and using the method of straight lines along the generatrix.

Based on the modified Hu-Washizu variational principle, a three-dimensional system of homogeneous partial differential stability equations is derived within the framework of the spatial theory of elasticity. The reduction of a three-dimensional system to a one-dimensional one is carried out along the generatrix and in the circular direction - by expanding the components of stresses and displacements into double trigonometric series when applying the procedure of the Bubnov-Galorkin method, as well as taking into account the periodicity of the resolving functions.

An algorithm has been developed, implemented in the form of application software packages for PCs. In it, in a single computational process using the numerical method of discrete orthogonalization in the direction normal to the middle surface of the shell, the establishment of the parameters of the subcritical stress-strain state and the solution on this basis of stability problems for non-thin anisotropic cylindrical shells under the influence of torsion are combined.

The problem of the influence on the stability of an anisotropic cylindrical non-thin shell of an increase in the number of cross-reinforced layers depending on the angle of rotation of the main directions of elasticity of the material and the direction of application of torque is considered. The obtained results of stability calculations according to the proposed approach were compared with critical torsion loads calculated using an orthotropic model for calculating anisotropic shells. It is shown that for single-layer cylindrical shells the difference between the compared results reaches 69%. An increase in the number of cross-reinforced layers leads to a decrease in this discrepancy, and with seven to eight layers, the difference between the critical loads obtained using the described approach and the orthotropic model is within 5%. This result is consistent with those obtained using classical or refined theories of calculations of both thin and non-thin anisotropic cylindrical shells.

## Author Biographies

### Volodymyr Trach, National University of Water and Environmental Engineering

doctor of technical sciences, professor, head of the department of bridges and tunnels, support of materials and construction mechanics

### Andrii Podvornyi, National University of Water and Environmental Engineering

candidate of technical sciences, associate professor of the department of bridges and tunnels, support of materials and construction mechanics

## References

Bazhenov V.A., Semenyuk M.P., Trach V.M. Nelinijne deformuvannia, stijkist' i zakrytychna povedinka anizotropnykh obolonok [Nonlinear deformation, stability and critical behavior of anisotropic shells]: Monograph. - K.: Karavela, 2010. - 352 s.(ukr).

Grigorenko Ya.M., Kryukov N.N. Chislennyie resheniya zadach statiki gibkih sloistyih obolochek s peremennyimi parametrami [Numerical solutions of problems of statics of flexible layered shells with variable parameters]. - K.: Naukova dumka, 1988, - 264 s.(rus).

Grigorenko Ya.M., Vasilenko A.T., Pankratova N.D. Zadachi teorii uprugosti neodnorodnyih tel [Problems of the theory of elasticity of inhomogeneous bodies]. – Kyiv, 1991. - 216 s.(rus).

Grigorenko Ya.M., Vlaikov H.G., Grigorenko A.Ya. Chislenno-analiticheskoye resheniye zadach mekhaniki obolochek na osnove razlichnykh modeley [Numerical-analytical solution of shell mechanics problems based on various models] : Monograph. - K.: Akademperiodika, 2006. - 472 s.(rus).

Guz A.N. Osnovy trekhmernoy teorii ustoychivosti deformiruyemykh tel [Basics of the three-dimensional theory of stability of deformable bodies]. - K.: Vyshcha Shk., 1986. – 511 s.(rus).

Guz A.N., Babych I.Yu. Prostranstvennyye zadachi teorii uprugosti i plastichnosti. T.4. Trekhmernaya teoriya ustoychivosti deformiruyemykh tel [Spatial problems of the theory of elasticity and plasticity. T.4. Three-dimensional theory of stability of deformable bodies]. - Kyiv: Nauk. dumka, 1985. - 280 s.(rus).

Kostromin V.P., Myachenkov V.I. Ustoychivost' mnogosloynykh obolochek s tsilindricheski-anizotropnymi neodnorodnymi sloyami [Stability of multilayer shells with cylindrically anisotropic inhomogeneous layers] // Soprotivleniye materialov i teoriya sooruzheniy, 1973. - Vyp. 21. – S. 11-16.(rus).

Lekhnytsky S.G. Teoriya uprugosti anizotropnogo tela [Theory of elasticity of an anisotropic body]. - 2nd ed., ed. and additional. - M.: Nauka, 1977. - 415 s.(rus).

Novozhilov V.V. Osnovy nelineynoy teorii uprugosti [Fundamentals of the nonlinear theory of elasticity]. - L.-M.: OGIZ, 1948. - 211 s.(rus).

Podvornyi A.V., Semenyuk N.P., Trach V.M. Stability of inhomogeneous cylindrical shells under distributed external pressure in a three-dimensional statement // Int. Appl. Mech., 2017, Volume 53, Issue 6. - P. 623-638.

Semenyuk N.P., Trach V.M., Podvornyi A.V. Spatial stability of layered anisotropic cylindrical shells under compressive loads // Int. Appl. Mech., 2019, Volume 55, Issue 2. - P. 211-221.

Semenyuk N.P., Trach V.M., Podvornyi A.V. Stability of cylindrical anisotropic shells under axial pressure in three-dimensional statement // Strength of Materials and Theory of Structures, issue 94, KNUBA, 2015. - P. 192-206.

Semenyuk M.P., Trach V.M., Podvornyi A.V. Stress–strain state of a thick-walled anisotropic cylindrical shell // Int. Appl. Mech., 2023, Volume 59, Issue 1. - P. 79-89.

Trach V.M., Podvornyi A.V. Prostorovi rivnyannya stiykosti anizotropnykh tovstykh tsylindrychnykh obolonok pid diyeyu osʹovoho tysku [Spatial stability equations of anisotropic thick cylindrical shells under the action of axial pressure] // Resursoekonomni materialy, konstruktsiyi, budivli ta sporudy: Zbirnyk naukovykh pratsʹ. – NUVHP. – Rivne, 2022. – Vyp. 41. – S. 197-212.(ukr).

Trach V.M., Podvornyi A.V., Khoruzhiy M.M. Deformuvannya ta stiykistʹ netonkykh anizotropnykh obolonok [Deformation and stability of thin anisotropic shells]: Monograph. - K.: Karavela, 2019. - 273 s.(ukr).

Vanin G.A., Semenyuk N.P. Ustoychivost' obolochek iz kompozitsionnykh materialov s nesovershenstvami [Stability of shells made of composite materials with imperfections]. - K.: Nauk. dumka, 1987. - 199 s.(rus).

Wasizu K. Variatsionnyye metody v teorii uprugosti i plastichnosti [Variational methods in the theory of elasticity and plasticity]. - M.: Mir, 1987. - 542 s.(rus).

2023-11-24

Статті