Study of nonlinear deformation and stability of reinforced shells under combined loading by a bending moment and a transverse boundary force

We present a finite-element statement for the solution of stability problems for reinforced elliptic cylindrical shells with moment properties and nonlinearity in their precritical stressstrain state taken into account. Integrating the equations obtained by equating the linear strain components with zero, we find explicit expressions for the displacements of elements of noncircular cylindrical shells as rigid bodies. Using these expressions, we construct the shape functions of a fourangle finite element of natural curvature and develop an effective algorithm for studying nonlinear deformation and stability of shells. We study the stability of reinforced elliptic cylindrical shells under combined loading by a transverse boundary force and a bending moment and investigate how the ellipticity of the shells and the nonlinearity of deformation at the precritical stage affect the shell stability.     

On stability of orthotropic cylindrical shells

In contrast to [1–3], the present paper obtains a system of stability equations and the corresponding resolving equation for orthotropic cylindrical shells of any but very short length in the case where the precritical stress state cannot be treated as the zero-moment state. These equations are a generalization of the results obtained in [4]. On the basis of these equations, one can obtain both the well-known formulas [1–3] and, for medium-length shells, some new expressions of the critical load in longitudinal compression and that under the joint action of torsionalmoments, normal pressure, and longitudinal compression. Some estimates are performed and the determination of the domain of application of some formulas given in [2] and in the present paper is attempted. For an orthotropic shell, a relationship between the elastic parameters and the shear modulus is established for axisymmetric and nonaxisymmetric buckling mode shapes in longitudinal compression.     

Studies of nonlinear deformation and stability of oval and elliptic cylindrical shells under axial compression

The stability of noncircular shells, in contrast to that of circular ones, has not been studied sufficiently well yet. The publications about circular shells are counted by thousands, but there are only several dozens of papers dealing with noncircular shells. This can be explained on the one hand by the fact that such shells are less used in practice and on the other hand by the difficulties encountered when solving problems involving a nonconstant curvature radius, which results in the appearance of variable coefficients in the stability equations. The well-known solutions of stability problems were obtained by analytic methods and, as a rule, in the linear approximation without taking into account the moments and nonlinearity of the shell precritical state, i.e., in the classical approximation. Here we use the finite element method in displacements to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with noncircular contour of the transverse cross-section. We use quadrilateral finite elements of shells of natural curvature. In the approximations to the element displacements, we explicitly distinguish the displacements of elements as rigid bodies. We use the Lagrange variational principle to obtain a nonlinear system of algebraic equations for determining the unknown nodal finite elements. We solve the system by a step method with respect to the load using the Newton-Kantorovich linearization at each step. The linear systems are solved by the Kraut method. The critical loads are determined with the use of the Silvester stability criterion when solving the nonlinear problem. We develop an algorithm for solving the problem numerically on personal computers. We also study the nonlinear deformation and stability of shells with oval and elliptic transverse cross-section in a wide range of variations in the ovalization and ellipticity parameters. We find the critical loads and the shell buckling modes. We also examine how the critical loads are affected by the strain nonlinearity and the ovalization and ellipticity of shells.     

Stress-strain state of a parabolic shell of revolution subjected to external magnetic pressure

The paper investigates the stress-strain state of an axisymmetrically loaded shell which arises when a strong electric current flows in it. The shell with current is an element of a system intended for focusing π and K mesons in neutron experiments. The problem is solved by numerical integration on a computer of equations of the theory of shells by the two-sided matrix run-through method, and also by an approximate analytical solution. The algorithm being applied can be used to calculate an arbitrary shell of revolution of variable thickness. The results thus obtained are discussed.     

Weak formulation of mixed state equation and boundary value problem of laminated cylindrical shell

IntroductionBasedonthree-dimensionalelasticitytheory,exactsolutionofhomogeneousisotropic,orthotropic,andlaminatedplatesandshellshasbeenstUdiedll]-[61,respeChvely.Butallofabovepapersadoptedrigorousequilibriumandboundarycondihons,andtheirsolutioncanbereliedonlyonspecialtechnique.Thusthosemethodswouldbedifficulttobepopularized.Ref.[7]clarifiedtheimportanceofdriedstateequationofelasticity,andfirStgaveHndltoncanonicalequationbymodifyingHellinger-Reissuervariationalprinciple.AtthesametimeTangL'I…     

Determining the Axisymmetric Geometrically Nonlinear Thermoviscoelastoplastic State of Laminated Shells by the Theory of Deformation Along Paths of Small Curvature

A technique is developed for determining the thermoviscoelastoplastic geometrically nonlinear axisymmetric stress–strain state of laminar shells of revolution under loads that induce meridional stress and torsion. The technique is based on the hypotheses of rectilinear element for the whole stack of layers. The relations of the theory of deformations along paths of small curvature are used as equations of state. The solution is reduced to the numerical integration of a system of ordinary differential equations. The technique is tried out by a test example and illustrated by determining the geometrically nonlinear thermoviscoelastoplastic state of a corrugated shell     

Stability and efficient design of cylindrical shells of metal composites subject to combination loading

A study is made of the stability of boron-aluminum shells under a combination of axial compression and uniform external pressure. An approximate theoretical model is constructed to describe the deformation of a layer of a fiber composite consisting of elastoplastic components. The model is used to derive the equations of state of multilayered shells reinforced by different schemes. The nonlinear equation describing the subcritical state is solved by the method of discrete orthogonalization with the use of stepped loading. The homogeneous problem is also solved by discrete orthogonalization. It is shown that shells can be efficiently designed for combination loading by plotting the envelope of the boundary curves for specific reinforcement schemes. The envelope is convex for elastic shells and is of variable curvature for elastoplastic shells. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 6, pp. 67–73, June, 1999.     

Stability of Cylindrical Shells with Microdamages

Problems on bifurcational stability of cylindrical shells are formulated and solved within the framework of the Kirchhoff–Love hypotheses with regard for damageability in the precritical stress state. The damageability of the material is due to the inhomogeneity of its microstrength and is modeled by empty quasispherical pores whose distribution over the shell volume is statistically homogeneous and isotropic. The problems are solved for shells under axial and radial compression.     

Effect of localized imperfections on the critical loads of ribbed shells

A numerical approach to the assessment of the critical stresses for imperfect ribbed shells is developed. Initial deflections occupying a part of the shell surface (they are circumferentially bounded) are considered. The couple stresses and nonlinearity of the precritical state are taken into account. Numerical examples are given     

General solution of the equilibrium equations for open circular cylindrical shells reinforced with discrete longitudinal ribs

The exact solution for the inhomogeneous system of equilibrium equations for open circular cylindrical shells reinforced with a quasi-regular system of discrete ribs is obtained. The dependence of stress–strain state of semi-infinite shells on the distance from the loaded edge is analyzed.     

Strength calculations and optimal weight design of multilayer shell-shaped composite products under a set of loads

The stress-strain state of axisymmetric multilayer shells is analyzed using kinematic and static hypotheses that allow for the transverse shear stresses satisfying the necessary equations of state, continuity conditions at the boundaries between the layers and given boundary conditions. A numerical solution of the problem of the stress-strain state for a multilayer bar is compared with the Lekhnitskii solution (for a cantilever beam loaded by a concentrated force and moment) to asses the applicability of the employed bending equations of multilayer shells. It is shown that these solutions are in good agreement. The problem of the initial fracture of the shells considered is formulated using phenomenological strength criteria for each layer. A coordinate-wise descent method in the unit interval is proposed to solve weight optimization problems for multilayer shells of composite materials under combined loading. Regions of safe operating loads and the optimal weight distribution of layer thicknesses are determined for a multilayer bar acted upon by a uniformly distributed load and concentrated force.     

Approximate Methods for the Linear and Nonlinear Analysis of Plates and Shallow Shells

Abstract

Berger's equations for the large amplitude deformation of membranes are used to produce a simple approximate expression for the large amplitude deflection of plates. The deformation of shallow shells is also considered and two approximate methods are outlined. Several important problems are discussed, the obtained solution being in good agreement with both experimental data and other approximate results. The main advantage of this technique is its ease of application, as it requires comparatively little computational work. A simple approximate formula for computing the fundamental frequency of a vibrating shallow shell is also presented and is shown to yield very accurate values in the case of a shallow dome and a rectangular panel.     

Nonlinear dynamic responses of sandwich functionally graded porous cylindrical shells embedded in elastic media under 1:1 internal resonance

In this article, the nonlinear dynamic responses of sandwich functionally graded(FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the outer and inner skins are made of solid metal, while the core is FG porous metal foam. Partial differential equations are derived by utilizing the improved Donnell's nonlinear shell theory and Hamilton's principle. Afterwards, the Galerkin method is used to transform the governing equations into nonlinear ordinary differential equations, and an approximate analytical solution is obtained by using the multiple scales method. The effects of various system parameters,specifically, the radial load, core thickness, foam type, foam coefficient, structure damping,and Winkler-Pasternak foundation parameters on nonlinear internal resonance of the sandwich FG porous thin shells are evaluated.     

Nonaxisymmetric Vibrations of Discretely Reinforced Inhomogeneous Multilayer Cylindrical Shells under Nonstationary Loads

The equations of nonaxisymmetric vibrations of discretely reinforced multilayer cylindrical shells are analyzed. A refined Timoshenko model of shells and beams is used to analyze elements of an elastic structure. The vibration equations for an inhomogeneous elastic system are derived using the Reissner variational principle. The numerical solver of the dynamic equations is based on the integro-interpolation method used to construct finite-difference schemes for equations with discontinuous coefficients. The dynamic behavior of a five-layer cylindrical shell under distributed nonstationary loading is analyzed     

Analyzing the stress–strain state of thick cylindrical shells

Two approaches to the analysis of the stress–strain state of thick cylindrical shells are elaborated. The shell is divided by concentric cross-sectional circles into several coaxial cylindrical shells. Each of these shells has its own curvature determined on its midline. The stress–strain state of the original shell is described by satisfying the interface conditions between the component shells. The distribution of unknown functions throughout the thickness is determined by finding the analytic solution of a system of differential equations in the first approach and is approximated by polynomial functions in the second approach. The stress–strain state of thick shells is analyzed. It is revealed that the effect of reduction becomes stronger with increasing curvature     

Determining the axisymmetric elastoplastic state of thin shells with allowance for the third invariant of the stress deviator

A technique to determine the axisymmetric elastoplastic state of thin shells with allowance for the third invariant of the stress deviator is developed. The technique is based on the theory of thin shells that takes into account transverse shear and torsional strains. Plastic equations that relate the components of the stress tensor in Eulerian coordinates with the linear components of the finite-strain tensor are used as constitutive equations. The nonlinear scalar functions in the constitutive equations are found from base tests on tubular specimens under proportional loading for different stress modes. The boundary-value problem is solved by numerically integrating a system of ordinary differential equations     

Stability of multilayered shells of revolution with allowance for geometric nonlinearity of the subcritical state

The finite-difference method and the Trefftz-Reissner variational principle are used to obtain a system of equations in mixed from to describe the stability and geometric nonlinearity of composite shells of revolution. Methods are developed and an algorithm is proposed to calculate the components of the geometrically nonlinear subcritical stress-strain state and to use those components to determine the “upper” critical values for shells with zero Gaussian curvature loaded by uniform external pressure, an axisymmetric load, or a combination of these loads. The stability of cylindrical, conical, and compound shells under uniform pressure is examined for different support conditions. Linear and nonlinear methods of determining the subcritical stress-strain state are compared and their effect on the critical loads is estimated. Ukrainian Transportation Institute and the Ukrainian Academy of Water Management, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 6, pp. 60–66, June, 1999.     

Allowance for geometrical nonlinearity in dynamic problems of the theory of discretely stiffened cylindrical shells under nonstationary loading

A variant of the two-dimensional equations of the motion of a discretely stiffened cylindrical shell is considered within the framework of the elastic nonlinear Timoshenko-type theory of shells and rods. The initial system of equations of motion is derived based on the Hamilton-Ostrogradskii variation principle. A numerical algorithm for solution of such problems with allowance for discrete nonuniformities is constructed. Some aspects of equation approximation are studied. The effect of geometrically nonlinear factors on the stress-strain state of a structure is analyzed. The scientific results of the present work were obtained during implementation of Project No. 182 of the Ukrainian Scientific and Technological Center. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 120–124, April, 2000.     

The iterative theory of deformation of laminated shells

The equations of the iteration theory of nonshallow transversally isotropic laminated shells, which account for all the components of the stress-strain state (SSS) and describe the inner SSS, potential, and vortex boundary effects, are obtained. The equations are based on the method of expansion of SSS into series in transverse coordinate and the method of variation with respect to the state being determined. The order of the equations does not depend on the number of layers and expansion terms that approximate the displacement and stress. The accuracy of the solution for the inner SSS and boundary effects is estimated. Pridneprovskaya State Academy of Building and Architecture, Dnepropetrovsk, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 11, pp. 40–45, November, 1999.     

Stress Analysis of Laminated Shells of Revolution with an Imperfect Interlayer Contact

An approach is proposed to solve problems on the contact interaction of layers in structurally inhomogeneous shells of revolution with allowance for both the normal and tangential stresses in the contact zones. A system of algebraic equations is constructed by using a method for solution of static problems for arbitrary shells of revolution. From this system, the contact zones and contact stresses are determined by the iteration method. The restricted behavior of the adhesion layer in cleavage and shear is taken into account. As an example, the stress state of a two-layer cylindrical shell under a concentrated load is determined