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.     

Nonlinear deformation and stability of oval cylindrical shells under combined loading

A study is made of the stability of cylindrical shells of oval cross section loaded by a shear force combined with torsional and bending moments. The variational method of finite elements in displacements is used. The subcritical stress-strain state of the shells is considered momental and nonlinear. The effects of the nonlinearity of shell deformation and shell ovalization on the critical load and buckling mode are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 134–138, January–February, 2008.     

Studies of nonlinear deformation and stability of noncircular cylindrical shells in transverse bending

The variational finite element method in displacements is used to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with a noncircular contour of the cross-section. Quadrangle finite elements of shells of natural curvature are used. In the approximations of element displacements, the displacements of elements as solids are explicitly separated. The variational Lagrange principle is used to obtain a nonlinear system of algebraic equations for the unknown nodal finite elements. The system is solved by the method of successive loadings and by the Newton-Kantorovich linearization method. The linear system is solved by the Crout method. The critical loads are determined in the process of solving the nonlinear problem by using the Sylvester stability criterion. An algorithm and a computer program are developed to study the problem numerically. The nonlinear deformation and stability of shells with oval and elliptic cross-sections are investigated in a broad range of variation of the elongation and ellipticity parameters. The shell critical loads and buckling modes are determined. The influence of the deformation nonlinearity, elongation, and ellipticity of the shell on the critical loads is examined.     

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.     

Nonlinear deformation and stability of oval cylindrical shells under pure bending and internal pressure

The stability problem of a cylindrical shell of oval cross section loaded by a bending moment and internal pressure is studied. The variational displacement finite-element method is used. For the prebuckling stress-strain state, the bending and nonlinearity are taken into account. The effects of the nonlinear nature of the deformation and the cross-sectional ovality of the shells on the critical loads and buckling modes are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 119–125, May–June, 2006.     

Study of nonlinear deformation and stability of discretely reinforced elliptic cylindrical shells in transverse bending

A finite-element method for solving problems of nonlinear deformation and stability of nonuniformly discretely reinforced noncircular cylindrical shells is considered. An effective computer algorithm for the study of shells is developed. Stability of stringer cylindrical shells with an elliptical cross section in transverse bending is examined. The effect of ellipticity, nonlinearity of shell deformation at the subcritical stage, reinforcement discreteness, and heterogeneity on shell stability is determined.     

Nonlinear deformation and stability of reinforced elliptical cylindrical shells loaded by internal pressure under twisting and bending

The problem of stability of cylindrical shells with an elliptical cross-sectional contour reinforced by a set of stringers under combined loading by bending and twisting moments, transverse force, and internal pressure is studied with the use of the variational method of finite elements in displacements. The subcritical stress-strain state of the shells is assumed to be moment and nonlinear. The effect of nonlinearity of deformation of the shells and their ellipticity on the critical loads and buckling type is determined.     

Repeated buckling and its influence on the geometrical imperfections of stiffened cylindrical shells under combined loading

The present experimental study aims at providing better inputs for improvement of the buckling load predictions of stiffened cylindrical shells subjected to combined loading. The work focuses on two main factors which considerably affect the combined buckling load of stiffened shells, namely geometric imperfections and boundary conditions. Six shells with nominal simple supports were tested under various combinations of axial compression and external pressure. The vibration correlation technique is employed to define the real boundary conditions. The geometric imperfections of the integrally stiffened shells are measured in the present experiments in situ and are used as inputs to a multimode analysis which yields the corresponding “knockdown” factor for various combinations of loading. Thus, when employing the repeated buckling procedure for obtaining interaction curves, each point on the curve is adjusted (using the multimode analysis) for the measured “new” surface of the shell and this results in more realistic interaction curves. The geometrical imperfections of the preloaded shells can also serve as an input to the International Imperfection Data Bank for future studies on the correlation between the manufacturing method of the shell and their geometric imperfections.     

Inelastic deformation of thin cylindrical shells under axisymmetric loading

Summary The response of a thin cylindrical shell of elastic-viscoplastic material to internal pressure is considered. Small displacements and the validity of the kinematical Love-Kirchhoff-hypothesis are presupposed. Further it is assumed that the inelastic behavior of the shell material is governed by a unified constitutive model with internal state variables, where the total strain tensor can be decomposed additively into an elastic and an inelastic part. Under these assumptions the governing differential equation for the radial displacement is derived. A general solution is obtained by the method of variation of parameters and adjusted to different boundary conditions.Solution of inelastic problems requires tracing of the entire loading path which leads to an initial value problem. This initial value problem is formulated for Hart's constitutive model and solved numerically by an implicit time integration procedure. Finally numerical results are presented.

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Inelastische Deformation dünner Kreiszylinderschalen bei axialsymmetrischer Belastung

Übersicht Es wird die Reaktion einer dünnen Kreiszylinderschale aus elastisch-viskoplastischem Werkstoff auf eine Belastung durch Innendruck untersucht. Es werden kleine Verschiebungen und die Gültigkeit der kinematischen Hypothese von Love-Kirchhoff vorausgesetzt. Ferner wird angenommen, daß das inelastische Verhalten des Werkstoffs durch ein einheitliches konstitutives Gesetz mit inneren Zustandsvariablen beschrieben wird, wobei der Tensor der Gesamtverzerrungen additiv in einen elastischen und einen inelastischen Anteil aufgespalten werden kann. Unter diesen Voraussetzungen wird die Differentialgleichung für die Radialverschiebung abgeleitet. Für sie wird eine allgemeine Lösung mittels Variation der Konstanten ermittelt und an verschiedene Randbedingungen angepaßt.Bei der Lösung inelastischer Probleme muß der gesamte Belastungspfad verfolgt werden. Dies führt auf ein Anfangswertproblem. Dieses Anfangswertproblem wird für das Werkstoffgesetz von Hart formuliert und numerisch mittels eines impliziten Zeitintegrationsverfahrens gelöst. Abschließend werden numerische Ergebnisse vorgestellt.

     

Stability of reinforced cylindrical shells under combined loading

A new approach to solving buckling problems for ribbed cylindrical shells subject to longitudinal force and pressure is proposed. Results for shells reinforced with stringers and rings are presented     

Postbuckling of imperfect stiffened cylindrical shells under combined external pressure and heating

1.IntroductionStiffenedcylindricalshellsarewidelyusedinmanytypesofstructures.Inpracticetheyoftensubjecttovarioustypesofcombinedthermalandmechanicalloadingandmayhavesignificantandunavoidableinitialgeometricalimperfections.Therefore,thepostbucklingbehaviorofimperfectstiITenedcylindricalshellsundercombinedexternalpress.ureandthermalloadingmustbewellunderstood.Manypostbucklingstudieshavebeenmadetbrstiffenedcylindricalshellsunderpureaxialcompression,uniformexternalpressureortheircombinations,where…     

Stability of reinforced cylindrical shells in bending by a moment with torsion

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 158–162, January–February, 1989.     

Buckling and postbuckling of stiffened cylindrical shells under axial compression

Buckling and postbuckling behaviors of perfect and imperfect,stringer andorthotropically stiffened cylindrical shells have been studied under axial compression.Based on the boundary layer theory for the buckling of thin elastic shells suggested in ref.[1],a theoretical analysis is presented.The effects of material properties of stiffeners andskin,which are made of different materials,on the buckling load and postbuckling behaviorof stiffened cylindrical shells have also been discussed.