Anisotropic Cylindrical Shells Research Articles

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

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.

Three-dimensional stability of non-thin anisotropic cylindrical shells under axial pressure

Based on the modified Hu-Washizu variational principle, a three-dimensional system of homogeneous linearized differential equations of stability of non-thin anisotropic cylindrical shells is obtained. The anisotropy type of shells is characterized by one plane of elastic symmetry. Reducing the system to a one-dimensional one is carried out using the Bubnov-Gal'orkin method, which approximates the functions of unknown systems of equations along the generatrix and takes into account their periodicity in the circumferential direction. The numerical solution of the obtained one-dimensional system of equations is carried out using the discrete orthogonalization method. 
 The problem of stability of a non-thin anisotropic cylindrical shell for a different number of cross-laid composite layers against the action of axial pressure is solved. The nature of the dependence of critical forces on the number of layers is analyzed.

Transient Deformation of an Anisotropic Cylindrical Shell with Structural Features

In this work, a new transient function of normal displacements is obtained in a thin elastic anisotropic cylindrical shell of constant thickness with structural features. Structural features form of two sequences in angular direction of weld spots or rivets. Structural features are mathematically described by point boundary conditions of rigid pinching or point boundary conditions of hinge support, accordingly. A load with time variable amplitude and impact boundaries influences on the outer surface of the shell along the normal. Such movement of the shell is considered in the cylindrical coordinate system as connected with the shell axis. Kirchhoff–Love hypothesis are confirmed as a shell model. The solution of this problem can be formed with the help of Green function method and lift compensation method. The required function of transient normal displacements is represented as the sum of integral operators of Green convolution for an infinite shell with functions of current transient load and compensating loads out of point boundary conditions. Compensating loads satisfying to boundary conditions can be found based on the solution of Volterra integral equations of the first kind with the difference kernel. Green function is such kernel. The system solution can be executed including preliminary discretization of compensating time loads. The convolution integrals can be taken numerically with the help of quadrature formula by the method of rectangles. The verification of the method is represented by comparing the solution of specific case with the known solution for a hinged isotropic cylindrical shell.

Investigation on Plastic Instability of a Thin-Walled Anisotropic Cylindrical Shell Subjected to Internal Impulsive Loading

Investigation on Plastic Instability of a Thin-Walled Anisotropic Cylindrical Shell Subjected to Internal Impulsive Loading

Стійкість нетонких анізотропних циліндричних оболонок в просторовій постановці під розподіленим боковим тиском

A three-dimensional approach to solving the problem of stability of non-thin cylindrical anisotropic layered shells under distributed lateral pressure is proposed. Based on the modified Hu-Washizu variational principle, a three-dimensional system of homogeneous differential stability equations is obtained for the calculation of shells, the anisotropy of which is characterized by a material with one plane of elastic symmetry. The solution of the three-dimensional system was carried out using the Bubnov-Galerkin methods and numerical discrete orthogonalization. The influence of an increase in the number of cross-laid layers of the same thickness on the stability of an anisotropic cylindrical shell is studied. The results of the solution are presented by graphs and their analysis is given.

Calculation of the stressed-strained state of rotating anisotropic cylindrical shells with a hole based on variational RVR-method

This paper proposes a new variational RVR-method for calculating the three-dimensional stressed-strained state of statically loaded shell elements of structures with holes of arbitrary shapes and sizes. The scientifically substantiated RVR method is based on the use of the Reissner variational principle, the Vekua method, the theory of R-functions by Rvachev, and the general equations of the spatial theory of elasticity. The use of mixed Reissner variational principle leads to an increase in the accuracy of solving boundary value problems due to the independent variation of the displacement vector and the stress tensor. The Vekua method makes it possible to replace the solution to a three-dimensional problem with a regular sequence of solutions to two-dimensional problems. The theory of R-functions at the analytical level takes into account the geometric information of boundary value problems, which is necessary for the construction of solution structures that accurately satisfy all boundary conditions. At the same time, the developed algorithm for bilateral integral accuracy assessment makes it possible to automate the search for such a number of approximations in which the process of convergence of solutions becomes stable. The possibilities of the RVR method are shown in numerous examples of solving boundary problems of calculating cylindrical shells with an elliptical hole when setting centrifugal loads according to a deformed scheme. Calculations according to the specified load scheme of the anisotropic cylinder lead (at certain values of the angular velocity of rotation) to a significant increase in stresses. Therefore, to obtain reliable results, it is necessary to set a centrifugal load that takes into account the change in the size of the body in the process of its deformation. The characteristic features of the proposed RVR-method, which can be used effectively in the manufacture of shell elements of structures in various branches of technology, are discussed.

Stress-strain state of thick-walled anisotropic cylindrical shells under thermal power load, protected by the functionally graded material

In the article the stress-strain state of thick-walled structurally anisotropic composite cylindrical shells under thermal power load, which are protected by a functionally graded material, are analysed. Based on the interrelations of the spatial theory of elasticity, a system of inhomogeneous differential equations in three-dimensional formulation, which describes the stress-strain state of thick-walled anisotropic cylindrical shells, was obtained. To reduce the dimensionality of this system, the Bubnov-Galerkin analytical method was used. Thus, the obtained one-dimensional system of twelve equations of normal Cauchy form was implemented using the numerical method of discrete orthogonalization. To represent the possibilities of the proposed approach, there were used stress-strain states of two, four and five-layered anisotropic cylindrical shells of fibrous composites, protected from temperature by a layer of transversely isotropic functionally graded material.

The stability of anisotropic cylindrical shells under torsion in spatial position

The stability of anisotropic cylindrical shells under torsion in spatial position

Three-temperature nonlinear generalized magneto-thermoelastic stresses in anisotropic cylindrical shells

The main purpose of this paper is to investigate the transient three-temperature nonlinear generalized magneto-thermoelastic stresses in an anisotropic cylindrical shell. The system of governing equations is solved by means of a boundary element method (BEM) and numerical calculations are carried out for temperature and thermal stresses. The numerical results indicate that the effects of three-temperature on the thermal stresses are very pronounced, where we used two-dimensional three temperature nonlinear radiative heat conduction equations coupled with electron, ion and phonon temperatures. Also, these numerical results demonstrate the validity and accuracy of the present numerical method.

Асимптотическое решение одной смешанной краевой задачи анизотропной двухслойной цилиндрической оболочки

Асимптотическое решение одной смешанной краевой задачи анизотропной двухслойной цилиндрической оболочки

Toward Numerical Solution of the Problem on Stressed State of Thermoelastic Physically Orthotropic Cylindrical Shells

To investigate the stressed-strained state of thermoelastic anisotropic cylindrical shells the original system of nineteen differential equations of eighth order in partial derivatives of the general theory of shells is transformed into a system of eight differential equations of first order. It is suggested to carry out numerical realization by the method of orthogonal marching. Numerical results are presented for the case of temperature drop over the shell thickness.

Non-stationary influence function for an unbounded anisotropic Kirchhoff-Love shell

The purpose of this article is to investigate the process of the influence of a nonstationary load on an arbitrary region of an elastic anisotropic cylindrical shell. The approach to the study of the propagation of forced transient oscillations in the shell is based on the method of the influence function, which represents normal displacements in response to the action of a single load concentrated along the coordinates. For the mathematical description of the instantaneous concentrated load, the Dirac delta functions are used. To construct the influence function, expansions in exponential Fourier series and integral Laplace and Fourier transforms are applied to the original differential equations. The original integral Laplace transform is found analytically, and for the inverse integral Fourier transform, a numerical method for integrating rapidly oscillating functions is used. The convergence of the result in the Chebyshev norm is estimated. The practical significance of the work is that the obtained results can be used by scientists or students to solve new problems of dynamics of cylindrical shells on an elastic basis under pulse loads. The found non-stationary influence function opens up possibilities for studying the stress-strain state, solving nonstationary inverse and contact problems for anisotropic shells, studying nonstationary dynamics in the case of nonzero initial conditions, and also when constructing integral equations of the boundary element method.

Wave-Based Method for Free Vibration Analysis of Orthotropic Cylindrical Shells with Arbitrary Boundary Conditions

The wave-based method (WBM) is a feasible method which investigates the free vibration characteristics of orthotropic cylindrical shells under general boundary conditions. Based on Reissner–Naghid’s shell theory, the governing motion equation is established, and the displacement variables are transformed into wave functions formed to satisfy the governing equations. On the basis of the kinematic relationship between the force resultant and displacement vector, the overall matrix of the shell is established. Comparison studies of this paper with the solutions in the literatures were carried out to validate the accuracy of the present method. Furthermore, by analyzing some numerical examples, the free vibration characteristics of orthogonal anisotropic cylindrical shells under classical boundary conditions, elastic boundary conditions, and their combinations are studied. Also, the effects of the material parameter and geometric constant on the natural frequencies for the orthotropic circular cylindrical shell under general boundary conditions are discussed. The conclusions obtained can be used as data reference for future calculation methods.

Correction to: Stability and initial postbuckling behavior of anisotropic cylindrical shells under external pressure

The second author’s affiliation should read: National University of Water and Environmental Engineering, Rivne, Ukraine.

DEFORMATION OF THE HELICAL TYPE WIRE STRUCTURES

Analysis of overhead power transmission lines (OPL) involves the simulation of statics and oscillation. Solving such problems strictly requires the proper accounting of the internal conductor structure, in particular for power safety and reliability systems of information-telecommunication supply of aerodromes and aircraft systems, as well as for overhead transmission lines subjected to intense wind, especially in icing conditions. Due to the complexity of wire structures, known issues arise in the estimates of their deformations, stiffness, bearing capacity, etc. For instance, the bending stiffness of the conductor can vary considerably. Consequently, the stiffness can vary along the conductor axis as well as in time. This paper proposes a new deformation model of wire structures similar to typical OPL conductors. Such structures include not only conductors and cables, but spiral clamps intended for tension, suspension, joining, protection, and repair of conductors. Based on energy averaging each wire layer is considered as an equivalent elastic anisotropic cylindrical shell. Therefore a conductor or a spiral clamp can be modeled as a system of shells nested into each other and interacting by means of pressure and friction forces. In the process of the study, calculations were made that made it possible to formulate equations for the matrices of flexibility and cruelty of spiral structures. The interaction problem for a tension clamp with an external wire layer of a conductor has been formulated and solved. Finally, the mechanism of the force transfer from the clamp on the conductor has been investigated.

Spatial Stability of Layered Anisotropic Cylindrical Shells Under Compressive Loads

An approach to solving spatial problems of the stress–strain state and stability of layered anisotropic cylindrical shells is developed. The anisotropy is characterized by one plane of elastic symmetry. The numerical implementation of the approach is based on the combination of the Bubnov–Galerkin, discrete Fourier transform, and discrete-orthgonalization methods. The results obtained are in good agreement with the solutions of simpler problems. A new stability problem for a sandwich shell is solved.

Stability of Thick-Walled Elastic Anisotropic 3-Dimensional Cylindrical Shells Under Axial Pressure Load

On the basis of the refined Timoshenko-Mindlin beam theory, an approach to calculate the stability of three-dimensional anisotropic cylindrical shells of geometrically nonlinear subcritical stress-strain state, in which the extreme layers are of higher rigidity than the average, is presented. The shell material has symmetry properties in a plane that could match the median surface. Numerical solutions are performed with the use of discrete orthogonalization methods. In the paper the plots illustrating the influence of stacking sequence and lay-up angle of layered fibrous composites on magnitude of critical values of axial compression are presented.

Analysis of deformability of composite laminated anisotropic cylindrical shells in the conceptual design of mechanical transducers and actuators

Analysis of axisymmetric deformability of mechanical transducers and actuators made in the form of composite anisotropic cylindrical shells subjected to axial tension or internal pressure is presented in the paper. The shells are modelled as cantilevers with one end fully clamped and another one free hanging with a rigid solid disk attached. Based on the analytical solution obtained and formulas derived, the conversion of the axisymmetric loading (axial load or internal pressure) to the shell torsion is analysed. Effects of material elastic characteristics, angle of the reinforcement orientation, and shell dimensions on the shell deformability and the value of the reactive shear force are illustrated using the analysis of anisotropic single-layered shells. It is shown that the angle of rotation and the reactive shear force have their maximum values at certain angles of the fibre orientation. Relationships between the shells’ lengths, thicknesses and the angles of the disk rotation or corresponding reactive shear forces are identified. It is shown, based on these equations, how the parameters of an anisotropic shell, delivering a required angle of the disk rotation, can be calculated.

Stability of Inhomogeneous Cylindrical Shells Under Distributed External Pressure in a Three-Dimensional Statement

The problem of the stability of anisotropic cylindrical shells is numerically solved in a three-dimensional statement. The shells are made of a material with one plane of elastic symmetry. By using the Bubnov–Galerkin method to approximate unknown functions with respect to the longitudinal coordinate by trigonometric series, the problem is reduced to a one-dimensional one that is solved by the discrete-orthogonalization method. The results are tested.

A Study of the Effect of Friction Between Delaminated Layers on the Stress-Strain State of Thick Laminated Anisotropic Cylindrical Shells by the Semianalytical Finite Element Method

An approach to studying the stress-strain state of axisymmetrical cylindrical shells with allowance for the delamination of layers by friction between them has been constructed within the framework of the three-dimensional theory of elasticity. The approach is based on the separation of shell across its thickness into a number of constituent shells with fixed curvature of each layer at the level of its median surface. To approximate the unknown functions in plan, linear polynomials are used, and their distribution over the thickness is found on the basis of the analytical solution of the corresponding system of differential equations. The proposed approach may be classified as a semianalytical finite element method. The effect of friction in the delamination zone on the stress-strain state of the shell has been analyzed.