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.     

Nonlinear contact problems for thin-walled elements of constructions

Some one-dimensional contact problems for plates and shells are considered for one-side contact with a rigid base. Contrary to analogous papers about the zone of contact, we use applied theories of contraction of Winkler type, which are obtained from equations of elasticity theory by asymptotic methods together with bending equations of thin-walled elements. The possibility of deviation of shells needs a definition of a contact zone in the process of solution of the problem from the condition of continuity of bending and its derivatives up to the second order inclusive. Some conclusions are made with respect to the optimal projects of reinforcement of shells taking into account their deviation.Translated from Dinamicheskie Sistemy, No. 8, pp. 40–45, 1989.     

Nonlinear nonaxisymmetric deformation of composite shells of revolution

We discuss the results of the determination of the stress and displacement fields in nonaxisymmetrically loaded nonlinear-elastic shells of revolution. The original nonlinear system of equations is linearized in accordance with the method of variation of elastic parameters. The two-dimensional linear boundary-value problem is reduced to a sequence of one-dimensional problems, which are solved using a numerical method. We carry out an analysis of the stress-strain state of a conical shell made of a composite material of granular structure. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 80–83.     

A refined problem on the strain of orthotropic noncircular cylindrical shells

Based on the refined Timoshenko theory, a semi-analytic method for solving problems of statics of orthotropic noncircular cylindrical shells is developed. The essence of this method consists in the spline-approximation of a solution in one coordinate direction and utilization of the collocation method and numerical solution of a high-order one-dimensional boundary-value problem by the discrete orthogonalization method in the second direction. The state of stress and strain of an open elliptic cylindrical shell under external load is investigated in the case where three contours rest upon supports and the fourth contour is rigidly fixed. Bibliography: 4 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 67–70.     

Sound insulation of composite cylindrical shells

We study problems involving the acoustic insulation of cylindrical shells of finite length made of a composite material. The motion of the medium (a gas) is described by the usual wave equation of acoustic approximation, and the equations of the applied theory of composite shells are used to describe the vibrations of the shell. To determine the levels of sound suppression the finite-element method is applied for both the medium and the shell. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 47–50.     

Application of the principle of possible displacements to problems of the dynamics of a “shell-fluid” system

We state in general form the principle of possible displacements for a “shell-fluid” mechanical system, on the basis of which it is possible to solve dynamic problems taking account of a geometrically nonlinear process of deformation of the shell and nonpotential motions of a viscous fluid. It is shown that this principle yields the equations of motion of the shell and fluid as components of this system, confirming the reliability of the principle. The conditions of force contact are taken into account as a load term in the equations of motion of the shell. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 117–123.     

The influence of the stiffness of reinforcing elements on the deformed state of shells with a hole

We propose a method of experimental estimation of the influence of the stiffness of elements that reinforce the contour of a hole on the deformed state of shells under surface pressure. We obtain the experimental data for the deflections on the contour of a lateral hole in a cylindrical shell as a function of the stiffness characteristics and the caps. We compare the results of experiments with numerical computations and give an analysis of the theoretical and experimental data obtained. Three figures. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 67–71, 1991.     

Influence of geometric parameters on the stress concentration in an elastoplastic cylindrical shell

The elastoplastic stress-strain state of cylindrical shells weakened by a curvilinear (circular) hole is investigated. Numerical results are given for shells with a reinforced hole under the action of an internal pressure of specified intensity. The influence of the geometric parameters of the shell on the stress distribution in the concentration zone is investigated in the elastic and inelastic stages of deformation. The variation in flexure along the hole contour is given for linear and nonlinear problems, and the distribution of the maximum stress-concentration coefficients is also given for various geometric parameters.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 100–103, 1988.     

On a method of studying the stress state of orthotropic shells and plates weakened by a hole with cracks extending to its edge

On the basis of the Timoshenko kinematic hypothesis for shells we give a formulation of the problem of studying the stress-strain state of orthotropic plates and shells weakened by a combined stress concentrator (a hole with two symmetric cracks extending to its edge). We propose a method of solving such problems on the basis of the finite-element method. To simulate the singularity of the stresses and displacements in a neighborhood of the tip of a crack we apply special finite elements with degenerate faces and nodes displaced by 1/4 the length of an edge. The stress intensity factors are found in terms of the displacements of the nodes of such elements. We give the results of computation of the concentration coefficients and the stress intensity factors for spherical and cylindrical shells loaded by internal pressure and for a cylindrical shell and a plate under the action of a distending load with various concentrators: a circular hole, an isolated crack, and a combined concentrator. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 48–54.     

The method of decomposition over velocities in the dynamics of extended systems

We propose a new method of solving multimodal problems of the dynamics of nonlinear extended shells, plates, and bars in a field of distributed mass and surface forces. We study the mode of motion of a towed system with the body in the steady state and under acceleration. We propose a new version of the method of decomposition based on identifying the characteristic times of propagation of various modes in the system, which can be used with success in the early stage of design of shell and bar structures, and also in problems of control of towed systems. Translated fromDinamicheskie Sistemy, No. 13, 1994. pp. 73-2-80.     

A variational principle in a geometrically nonlinear theory of heterogeneous viscoelastic shells

The use of the hereditary theory for shells heterogeneous across their thickness is considered. A variational method is formulated for calculating thin anisotropic shells made of a material whose deformation behavior can be described by relations of the linear theory of viscoelasticity. In order to transform the corresponding functional into a form suitable for shells, some assumptions related to concepts of the theory of thin shells are introduced. In the capacity of Euler equations, physical relations, nonlinear equilibrium equations, and nonlinear boundary conditions are derived. The state equations are deduced for a multilayered shell. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 231–240, March–April, 2009.     

Determination of the boundary of the plastic zone in a mineral vein weakened by a three-dimensional hole

We propose a method of computing the boundary of the plastic zone formed in a neighborhood of the hole during the mining of a mineral. The problem is studied in a three-dimensional formulation. The boundary of the plastic zone is determined from the condition of continuity of the vertical normal stresses acting on the surface of contact of an elastic half-space and an elastoplastic layer. The computation is carried out for a hole having a parallelepipedal shape. One figure. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 6–10, 1991.     

Numerical solution of problems in the dynamics of ribbed shells of revolution

We consider dynamic processes in shells of revolution reinforced with discrete ribs. The stress—strain state of the shell is determined in the framework of the linear theory of Timoshenko elastic thin shells. The ribs are described using the theory of curvilinear rods. The system of differential equations is solved by applying the variational principle to nonstationary problems.S. I. Subbotin Institute of Geophysics of the Ukrainian Academy of Sciences. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 45–48, 1992;     

Application of the method of partitioning the state of stress to the analysis of thermoelastic shells : PMM vol. 40, n 6, 1976, pp. 1085–1092

By using an asymptotic approach [1], the method of partitioning the state of stress is extended to thermoelastic shells. It is examined in detail in [2] forun-heated shells subjected to the effect of external forces, and consists of representing the total state of stress of the shell as the sum of those simpler states of stress for each of which the simplest methods for their construction can be given.Partitioning of the state of stress was performed in [3] for shells with a constant temperature over the thickness. It was noted in [4] in an analysis of a circular cylindrical shell by bending theory that integrals extended over the whole middle surface, which describe the fundamental state of stress, and integrals which damp out with distance from the edges and represent edge effects are contained in the general solution. In a number of papers, [5] for example, partitioning is performed on the basis of graphic physical representations for simple examples of analyzing circular cylindrical shells.A general approach to the analysis of rigid thermoelastic shells by the partitioning method is described below.     

A study of the strength of an elastoplastic orthotropic shell of arbitrary curvature with a surface crack

Using an analog of the δ _c -model, we have obtained a solution of the problem of the stress-strain state of an elastoplastic orthotropic shell, having an arbitrary curvature, with a surface crack. Here, additional constraints on the elastic parameters of the material are not imposed. Furthermore, we have studied the dependence of the length of the plastic zone and surface-crack opening on the level of load, the shell and crack geometrical parameters, and the mechanical properties of the material.     

The linear theory of dislocations and disclinations in elastic shells

A stress state of a thin linearly elastic shell containing both isolated as well as continuously distributed dislocations and disclinations is considered using the classical Kirchhoff–Love model. A variational formulation of the problem of the equilibrium of both a multiply connected shell with Volterra dislocations as well as shells containing dislocations and disclinations distributed with a known density is given. The mathematical equivalence between the boundary-value problem of the stress state of a shell caused by distributed dislocations and disclinations and the boundary-value problem of the equilibrium of a shell under the action of specified distributed loads is established. A number of problems on dislocations and disclinations in a closed spherical shell is solved. The problem of infinitesimally deformations of a surface when there are distributed dislocations is formulated.     

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.     

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.     

Analysis of contact stresses in structural joints of composite shell with steel banding

Based on the generalized Timoshenko-type shell theory, a numerical-analytical procedure for determining contact stresses from the interaction between a cylindrical composite shell and rigid bandings is proposed. Specific cases of loading and contact interaction (ideal contact through an adhesive interlayer) are considered. The contact problems are reduced to the solution of a Fredholm integral equation of the second-kind. A calculation analysis is performed. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 1, pp. 109–120, January–February, 2000.     

Some general methods for constructing the theory of shells

The aim of the present short review is the exposition of the fundamental results obtained by Academician I. N. Vekua (1907–1977) in the theory of shells. The review deals with questions of constructing different versions of shell theory, questions of the infinitesimal bending of a surface of positive curvature and equilibrium membrane states of stress of a convex shell, and also the statically determinable problems and questions of existence of a neutral surface of the shell, i.e. the questions which Vekua investigated in different periods of his versatile scientific actively. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.