Investigation of Locally Loaded Multilayer Shells by a Mixed Finite-Element Method. 2. Geometrically Nonlinear Statement

Based on mixed finite-element approximations, a numerical algorithm is developed for solving linear static problems of prestressed multilayer composite shells subjected to large displacements and arbitrarily large rotations. As the sought-for functions, six displacements and eleven strains of the shell faces are chosen, which allows us to use nonlinear deformation relationships exactly representing arbitrarily large displacements of the shell as a rigid body. The stiffness matrix of a shell element has a proper rank and is calculated based on exact analytical integration. The bilinear element developed does not allow false rigid displacements and is not subjected to the membrane, shear, or Poisson locking phenomenon. The results of solving the well-known test problem on a nonsymmetrically fixed circular arch subjected to a concentrated load and the problem on a locally loaded toroidal multilayer rubber-cord shell are presented.     

Stress-strain state in anisotropic multilayer shells with zones of nonideal contact between layers

Conclusions The theorem formulated here corresponds to the most general variational principle in the theory of elasticity. The equations and conditions derived from it constitute a complete system of relations necessary for defining and solving the problems which involve determining the stress-strain state in anisotropic multilayer shell structures. Assuming that some of the relations (2.2)–(2.9) are satisfied a priori, one can formulate other partial variational principles (Lagrange's, Reissner's, et al.).The result obtained here can be utilized for a correct derivation of two-dimensional equations for anisotropic multilayer shells of discrete structure, also as the starting point for devising approximate methods of solution of problems which involve determining the state of stress and strain in anisotropic multilayer shells.Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 832–836, September–October, 1981.     

Comparative analysis of two algorithms for numerical solution of nonlinearstatic problems for multilayered anisotropic shells of revolution 2. Account of transverse compression

Two algorithms for numerical solution of static problems for multilayer anisotropic shells of revolution are discussed. The first algorithm is based on a differential approach using the method of discrete orthogonalization, and the second one—on the finite element method with linear local approximation in the meridional direction. It is assumed that the layers of the shell are made of linearly elastic, anisotropic materials. As the unknown functions, six displacements of the shell are chosen, which often simplifies the definition of static problems for multilayer shells. The calculation of a cross-ply cylindrical shell stretched in the axial direction is considered. It is shown that taking account of the transverse compression, anisotropy, and geometrical nonlinearity is important for the given class of problems.Tambov State Technical University, Tambov, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 435–446, May–June, 1999.     

Comparative analysis of two algorithms for numerical solution of nonlinear static problems for multilayer anisotropic shells of revolution. 1. Account of transverse shear

The discussion focuses on two numerical algorithms for solving the nonlinear static problems of multilayer composite shells of revolution, namely the algorithm based on the discrete orthogonalization method and the algorithm based on the finite element method with a local linear approximation in the meridian direction. The material of each layer of the shell is assumed to be linearly elastic and anisotropic (nonorthotropic). A feature of this approach is that the displacements of the face surfaces of the shell are chosen as unknown functions, i.e., the functions which allows us to formulate the kinematic boundary conditions on these surfaces. As an example, a cross-ply cylindrical shell subjected to uniform axisymmetric tension is considered. It is shown that the algorithms elaborated correctly describe the local distribution of the stress tensor over the shell thickness without an expensive software based on the 3D anisotropic theory of elasticity.Tambov State Technical University, Tambov, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 3, pp. 347–358, May–June, 1999.     

Contact Problem for a Pneumatic Tire Interacting with a Rigid Foundation

Based on mixed finite-element approximations, a numerical algorithm is developed for solving a geometrically nonlinear contact problem for a prestressed multilayered Timoshenko-type shell undergoing arbitrarily large displacements and rotations. As unknowns, six displacements of faces of the shell are taken, which allows one to use principally new relationships for components of the Green–Lagrange strain tensor in curvilinear orthogonal coordinates, exactly representing arbitrarily large displacements of the shell as a rigid body. As an example, a tire interacting with a rigid foundation is considered.     

Investigation of the Stress State of Pressure Vessels of Inhomogeneous Structure Made of Composite Materials

The problem on the stress-strain state of layered cylindrical shells with bottoms of intricate shape under the action of internal pressure is considered. The elastic system examined is formed by spiral-circular winding. Two variants of the shell bottom structure are investigated. In the first variant, one spiral layer is installed, which leads to great variations in the bottom thickness along the meridian. In the second one, the bottoms are formed according to the zone-winding scheme. The stress state of the shell constructions of the classes considered is determined by solving boundary-value problems for systems of ordinary differential equations. The solution results for cylindrical shells with elliptic bottoms for the two types of winding are given. It is shown that the zone winding leads to smaller deflections and stresses than the conventional ways of reinforcing shell bottoms.     

Thermoelasticity Problem for Orthotropic Spherical Shells Heated by Concentrated Heat Sources

A method is proposed for solving a thermoelasticity problem using a two-dimensional Fourier integral transform for thin, gently sloping orthotropic spherical shells heated by concentrated heat sources. A linear temperature distribution over the shell thickness and newtonian convective heat exchange with the surroundings are assumed. The effects of the curvature and orthotropic properties of the shall material on the internal force factors are estimated.     

Dynamic stability of a porous cylindrical shell

This paper is devoted to a closed cylindrical shell made of a porous-cellular material. The mechanical properties vary continuously on the thickness of a shell. The mechanical model of porosity is as described as presented by Magnucki, Stasiewicz. A shell is simply supported on edges. On the ground of assumed displacement functions the deformation of shell is defined. The displacement field of any cross section and linear geometrical and physical relationships are assumed in cylindrical coordinate system. The components of deformation and stress state were found. Using the Hamilton's principle the system of differential equations of dynamic stability is obtained. The forms of unknown functions are assumed and the system of a differential equations is reduced to a simple ordinary equation of dynamic stability of shell (Mathieu's equation). The derived equation are used for solving a problem of dynamic stability of porous-cellular shell with intensity of load directed in generators of shell. The critical loads are derived for a family of porous shells. The unstable space of family porous shells is found. The influence a coefficient of porosity on the stability regions in Figures is presented. The results obtained for porous shell are compared to a homogeneous isotropic cylindrical shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of a coupled problem of thermopiezoelectricity based on a geometrically exact shell element

Based on a 7-parameter shell model, a numerical algorithm has been developed for solving a coupled problem of thermoelectroelasticity for a laminated piezoelectric shell subjected to a thermoelectromechanical loading. As unknowns, six tangential and transverse displacements of outer surfaces and the transverse displacement of shell midsurface are chosen. This choice provides a possibility of utilizing the complete 3D constitutive equations of thermopiezoelectricity. A geometrically exact 3D hybrid piezoelectric shell element is formulated by using nonconventional analytical integration. With the help of this finite element, solutions of coupled problems of thermoelectroelasticity for laminated plates and shells with segmented and distributed piezoelectric sensors and actuators are obtained.     

Problems in dynamics of viscoelastic multilayer anisotropic shells and plates. Part 3: Optimization of vibration immunity characteristics of composite cylindrical shell

Conclusions Equations have been proposed which describe steady-state vibrations of multilayer shells made of composite materials. On the basis of these equations some simplest modes of deformation have been analyzed. The problem of optimizing the vibration immunity has been solved for a cylindrical shell serving as a component of a simple mechanical system.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 258–262, March–April, 1982.     

Calculation of composite structures subjected to follower loads by using a geometrically exact shell element

Based on a 7-parameter shell model, a numerical algorithm has been developed for solving the geometrically nonlinear problem of a multilayer composite shell subjected to a follower pressure and undergoing large displacements and rotations. As unknowns, six displacements of the outer surfaces and addition ally the transverse displacement of midsurface of the shell are chosen. This allows one to use the Green–Lagrange strain tensor, introduced earlier by the authors, which exactly represents arbitrarily large rigid-body displacements of the shell in curvilinear coordinates of a reference surface. A geometrically exact solid shell element is formulated, which permits one to solve the nonlinear deformation problem for thin-walled composite structures subjected to a follower pressure by using a very small number of load steps.     

Solving some problems in the theory of thin shells of revolution with complex boundary conditions

An approach is proposed to solving linear boundary-value problems for shells of revolution that are closed in the circumferential direction, with complex boundary conditions in which the coefficients of the solving functions depend on the circumferential coordinate. The approach relies on reduction of the boundary-value problem to a number of boundary-value problems for systems of ordinary differential equations and systems of algebraic equations. We solve a specific problem for the stressed state of a conical shell with one of its ends supported by an elastic foundation with a variable modulus.Institute of Mechanics, Ukrainian Academy of Sciences. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 85–93, 1989.     

Experimental-theoretical studies of the stability of orthotropic shells with a filler in axial compression

Conclusions 1. An analysis has been made of the solution to the problem of the stability of multilayer cylindrical shells having a filler and simple calculation formulas have been obtained for determining the critical forces.2. The stability of fiberglass-plastic shells with rubber-like fillers has been studied experimentally.3. Comparative experimental-theoretical studies of critical forces have been made, and the stability coefficients have been ascertained for the shell class under consideration.Translated from Mekhanika Polimerov, No. 3, pp. 485–489, May–June, 1978.     

Optimization of reinforced cylindrical shells with nonuniform thickness

An optimum multilayer shell is designed whose stack of elementary layers has a nonuniform thickness. This optimization problem is solved numerically for the special cases of three-layer cylindrical shells with dynamic and static stability. The optimum variants of layer distribution in this model are compared with the optimum solutions in [1].Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 298–303, March–April, 1976.     

Application of nonclassical models of shell theory to study mechanical parameters of multilayer nanotubes

Stress-strain state of multilayer anisotropic cylindrical shells under a local pressure is studied. Such a problem may model the bending of an asbestos nanotube under the action of a research probe. In earlier works, these authors showed that the application of classical shell theories yields results far from experimental data. More accurate results are obtained by taking into account additional factors, such as the change of the transverse displacement magnitude (according to the Timoshenko-Reissner theory) or the layered structure of asbestos and cylindrical anisotropy (according to the Rodinova-Titaev-Chernykh theory). In the present paper, yet another shell theory, the Palii-Spiro theory, is applied to solve the problem; this theory was developed for shall of average thickness and is based on the following assumptions: (a) the rectilinear fibers of the shell perpendicular to its middle surface before deformation remain rectilinear after deformation; (b) the cosine of the angle between the shell of such fibers and the middle surface of the deformed shell equals the averaged angle of the transverse displacement. Deformation field are studied with the use of nonclassical (the Rodinova-Titaev-Chernykh and Palii-Spiro) shell theories; a comparison with results obtained for three-dimensional models with the use of the Ansys 11 package is performed.     

Axisymmetric contact of anisotropic shells of rotation with rigid and elastic foundations

A class of problems are investigated on determining the stressed-strained state of anisotropic shells of rotation that are in axisymmetric one-sided contact with rigid and elastic surfaces. The shells are under the action of surface and contour loads. For some combinations of these quantities the shell may break away from the surface. To determine the contact zone, the method of successive approximations is utilized. In contrast to most investigations in which the contact zone is first determined, the method proposed makes use of a special quantity characterizing the size of the contact zone. The load on contours is determined from the solution to the problem on the stressed state of the shell and the condition specified on the boundary of the contact zone. Some examples of solving concrete problems are given. Bibliography: 5 titles. Translated fromObchyslyuval’ na ta Prykladna Matematyka, No. 76, 1992, pp 70–74.     

Buckling Instability of Sectional Noncircular Cylindrical Composite Shells under Axial Compression

The problem of buckling instability of cylindrical shells under axial compression is considered. The shells consist of cylindrical sections of smaller radius. The geometrical parameters of the shells are approximated by Fourier series on a discrete point set. A Timoshenko-type shell theory is used. The solution is obtained in the form of trigonometric series. It is shown that shells consisting of cylindrical sections have considerable advantages over circular ones. At a constant shell weight, the choice of suitable parameters of shell sections leads to a significant increase in the critical load. The composite shells considered possess higher efficiency indices in comparison with isotropic ones.     

Analysis of the stress-strain state of a ring-reinforced cylindrical section of an underground pipe

A method is presented for solving the problem of determining the stress-strain state of closed circular cylindrical shells in an elastic medium. The problem relates to the design of underground pipelines. The work of cylindrical shells is examined from the viewpoint of the theory of thin-walled three-dimensional systems, with allowance being made for the unilateral character of the interaction with the elastic medium. The stress-strain state of a cylindrical section of an underground pipe reinforced in the middle by a ring is investigated. It is shown that different factors influence the stress-strain state of the shell of the pipe.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 66–72, 1987.     

Contact of shells with a sharp linear stamp of finite length

The action of a plane, absolutely rigid stamp on a transversely isotropic shell is investigated. The use of the equations of shells with finite shear stiffness enables the correct formulation of the problem of the action on a shell by a stamp of fixed length. The problem is reduced to an integral equation. Applying the Fourier transform, the kernel of the integral equation is represented in the form of an expansion with respect to Chebyshev polynomials. By the representation of the solution of the integral equation in the form of a product, of a series of Chebyshev polynomials and a function that takes into account the singularities of the solution at the boundary of the contact zone, the considered problem is reduced to the solving of an infinite system of linear algebraic equations, whose coefficients have been determined by the methods of numerical integration. As an example a problem for a cylindrical shell has been solved.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 59–63, 1989.