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.     

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.     

Investigation of Locally Loaded Multilayer Shells by a Mixed Finite-Element Method. 1. Geometrically Linear Statement

The Hu-Washizu functional is constructed for analyzing prestressed multilayer anisotropic Timoshenko-type shells. As unknown functions, six displacements and eleven strains of the faces of the shells are chosen. Based on mixed finite-element approximations, a numerical algorithm is developed for solving linear static problems of prestressed multilayer composite shells. The results of solving the well-known test problem on a cylindrical shell subjected to two opposite point forces and the problem on local loading of a toroidal multilayer rubber-cord shell are presented.     

Refined linear theory of elastic anisotropic multilayer shells. Part 2

On the basis of the refined linear theory of elastic anisotropic multilayer shells of arbitrary shape derived in [1] it is established that a number of theorems of the linear theory of elasticity have analogues in the theory of multilayer anisotropic shells.For Part 1 see [1].Institute of Fluid Mechanics, Bucharest. Translated from Mekhanika Polimerov, No. 1, pp. 100–109, January–February, 1976.     

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.     

Thermoelasticity of a regularly inhomogeneous curved layer with wavy surfaces

A curved inhomogeneous anisotropic layer of variable thickness is considered that has wavy surfaces. It is assumed that the elastic, thermo-physical characteristics of the layer material and the shape of its upper and lower surfaces are periodic in structure with a single periodicity cell (PC). The period of the structure is here comparable in magnitude with the layer thickness, which is assumed to be much less than the other linear dimensions of the body and the radius of curvature of its middle surface.On the basis of a general scheme for taking the average of processes in periodic media /1, 2/, a method is developed which enables a transition to be made from a spatial quasistatic thermoelasticity problem to a system of thermoelasticity equations for an average shell whose effective and thermophysical coefficients are determined from the solution of local problems in a PC. Results obtained for the static theory of elasticity in /3/ are used. The heat conduction problem is averaged to determine the temperature components occurring in the equation of motion.The model constructed enables thermoelastic strains, stresses and the temperature distribution to be obtained in shells and plates of composite or porous materials with a different kind of reinforcement of the periodic structure (waffle, ribbed, corrugated) in reinforced and grid-like shells and plates. In the limiting case of “smooth” surfaces and a homogeneous material, the thermoelasticity equations are obtained for shallow anisotropic shells and the heat conduction equations of anisotropic shells assuming a linear temperature distribution law over the thickness.     

Deformation of cylindrical composite shells under combined dynamic loading

Conclusions A procedure has been shown for calculating the stress-strain state of cylindrical multilayer shells made from composite materials under the combined action of dynamic axial compression and dynamic external pressure, as well as with different variants of combined loading with static and dynamic forces. An investigation has been made of the effect on the mode of the buckled shell surface of the ratio of the application rate of dynamic loads; ranges of loading rates have been established in which stresses predominate caused either by axial compression or external pressure. It has been shown that, as a result of preliminary static loading, a marked change occurs in the initial imperfections of the shell mode which affects subsequent dynamic buckling. To calculate the time when the first defect occurs and its location in the shell body, a procedure has been devised for layer-by-layer strength analysis employing a tensor-polynomial criterion. It was demonstrated that the level of preliminary static loading noticeably affects the time until the first failure of the layer, not only a reduction of this time being possible with an increase in the static loads, but also an increase in it.We should also point out the work in [10] where it is shown that it is possible to weaken the susceptibility of the shell to initial imperfections when internal pressure is applied.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 461–473, May–June, 1981.     

Solution of two-dimensional boundary-value problems of the theory of thin composite shells

Discrete analogues of the boundary-value problems of a two-dimensional refined theory of anisotropic shells taking into account the transverse shear deformation are presented. The systems of resolving equations in the general form are obtained for arbitrary nonshallow shells of variable curvature whose coordinate lines of the reduction surface may not coincide with the lines of principal curvatures. The algebraic problems of determining the stress-strain state in shells made of composite materials with stress concentrators under various kinds of loads are obtained as particular cases of the schemes presented. The results of calculating the stress concentration near a nonsmall circular hole in a transversely isotropic nonshallow spherical shell under internal pressure are presented. The dependences of stress concentration factors on the hole dimension and on a change in the shear stiffness of the shells are studied. A comparison between the calculation results obtained within the framework of the theories of shallow and nonshallow shells is given.Presented at the 11th International Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000).Timoshenko Institute of Mechanics, Ukranian National Academy of Sciences, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 465–472, July–August, 2000.     

Nonlinear equations of a timoshenko-type theory of laminated anisotropic shells

In recent years analysis of the stress—strain state of shell structures made out of composite materials has been based on refined shell theories which take into account strains in the direction normal to the reference surface. There are several approaches to the formulation of the refined theories. One can point to shell theories developed on the basis of variational principles (e.g., [1, 2]) as well as theories created with the help of iterational processes (e.g., [3–6]). A resolving system of nonlinear equations for laminated anisotropic shells has been derived in the proposed research based on the Reissner variational principle [7, 8]. A similar linear theory which takes into account the strain e₃₃ also has been developed in [1]. If the shear stiffnesses of the layers differ greatly from each other in the transverse direction, then one can treat the shell structure as a single-layer shell of nonuniform structure. In this case it is advisable to solve a problem of the type of a uniform shell with minimal stiffnesses.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 501–507, May–June, 1979.     

Variant of the geometrically nonlinear theory of anisotropic shells with allowance for transverse shear

A very simple variant of the geometrically nonlinear theory of anisotropic shells with allowance for the high compliance of the material in transverse shear is proposed. From this theory there follow, as a special case, the equations for an isotropic shell; these differ from the relations of [2] with respect to terms of the order of the ratio of the thickness of the shell to the radii of curvature small as compared with unity. The equations obtained are used to solve the problem of the stability of orthotropic shells of revolution relative to the starting axisymmetric state of stress.Translated from Mekhanika Polimerov, No. 5, pp. 863–871, September–October, 1969.     

Effect of stabilizing additives on the strength characteristics of rigid polyvinyl chloride

The authors studied the effect of some stabilizing additives on the strength and durability (static fatigue) of stiff polyvinyl chloride (PVC) under uniaxial tension. They found these stabilizers to exert a significant effect on the durability of the compositions under study.Tambov Institute of Chemical Engineering. Scientific-Research Institute of Chemicals for Polymer Materials, Tambov. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 550–552, May–June, 1973.     

Construction of fundamental solutions of two-dimensional problems of the nonstationary theory of elasticity

We propose a method of constructing the images of the fundamental solutions in the space of the Laplace transform with respect to time, leading to simple formulas. The method is illustrated using three dynamical problems: planar deformation for an anisotropic body; flexural vibrations of an anisotropic plate; and vibrations of a shallow isotropic shell of arbitrary Gaussian curvature. Quadrature formulas are given for computing the values of the fundamental solutions. We give a new interpretation and a new method of computing the values of the special functions used in the construction of singular solutions in problems of the static theory of shells. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 86–92.     

Rationalization of the shape of wound membrane shells of revolution composed of high-modulus material

A method of designing composite membrane shells of revolution under axisymmetric loading is described. The properties of the shell material are analyzed. It is shown that for shells of high-modulus material in the presence of tensile membrane stresses the fibers fail in the matrix. A fiber arrangement and shell geometry ensuring isotensoid properties are proposed for this case. A technological and weight analysis is presented.S. Ordzhonikidze Moscow Aviation Institute. Translated from Mekhanika Polimerov, No. 5, pp. 822–828, September–October, 1975.     

Optimization of a cylindrical shell of reinforced plastic with a filler under dynamic loading

Conclusions 1. The numerical investigation of profiles of functions of the physical constraints carried out in this work allows us to assume that problems of optimal design of shells of reinforced plastics, strengthened by an elastic filler, for purposes of stability (static and dynamic) under axial compression, in the given formulation, are problems of convex programming. This guarantees uniqueness of their solution and allows us to use gradient methods for a numerical realization.2. Optimal shells of a composite material have a smaller mass than equivalent shells of high-strength metal alloys. The gain in the expenditure of the material is ensured not only as a result of higher specific characteristics of the composite, but basically as a result of optimizing the reinforcement structure of the pack of orthotropic layers of the shell.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 879–885, September–October, 1977.     

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.     

Shear buckling modes of cylindrical sandwich shells under external pressure, tension-compression of bearing layers with unequal forces, and inhomogeneous across-the-thickness temperature

Equations are set up for describing, in a correct statement and with an accuracy sufficient in actual practice, the shear buckling modes (BMs) of cylindrical sandwich shells with a transversely soft core of arbitrary thickness. Based on them, solutions are obtained to a number of problems on the buckling instability according to shear modes under some force and thermal loadings. It is found that the BMs occur in the shell along the circumferential and axial directions if, in the precritical state, a normal compressive stress arises in the transverse direction. It is shown that this condition is fulfilled in the following cases: in axial tension of the shell with unequal forces applied to the end faces of bearing layers (the parameter of critical load is maximum if the tensile forces are equal); under external (internal) pressure; on cooling the outer and heating the inner layers. The results obtained are presented in the form of simple analytical formulas for determining the corresponding critical parameters of the force and thermal actions.Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 1, pp. 37–48, January–February, 2005.     

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.     

An experimental study of the deformation of multilayer cylindrical shells

Experimental results are presented on the deformation of two-layer and four-layer reinforced shells subject to static loading. The strains on the outside and inside surfaces are unequal at points remote from the region of the edge effect when the shell is subject to an internal pressure and an axial force.P. I. Baranov Central Institute for Aviation Engine Design. Translated from Mekhanika Polimerov, No. 5, pp. 943–944, September–October, 1973.     

Boundary-Value Problems of the Theory of Thin and Nonthin Orthotropic Shells with Account of Nonlinearly Elastic Properties and Low Shear Rigidity of Composite Materials

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.     

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.