Post-buckling behavior of a point-supported cylindrical liner shell of concrete pressure vessels

Using W. T. Koiter's initial post-buckling theory, this paper deals with the critical load and post-buckling behavior of a point-supported cylindrical liner shell encased in a concrete pressure vessel while the liner shell is subjected to axial and lateral compressions. The reasonable spacing of anchors is given. The results show that the point-supported cylindrical liner shell has a relatively complicated post-buckling behavior. The behavior is subject to the change of anchor spacing in both axial and circumferential directions. When the ratio of the anchor spacing in the two directions satisfies a certain condition, the liner shell will carry the maximum load.     

Buckling of elastic-plastic shells of revolution with discrete elastic-plastic ring stiffeners

The theory is summarized for axisymmetric prebuckling and nonsymmetric bifurcation buckling of ring-stiffened shells of revolution. The analysis is based on finite difference energy minimization in which moderately large meridional rotations, elastic-plastic effects, and primary or secondary creep are included. This theory is implemented in a computer program called BOSOR5, for the analysis of segmented and branched ring-stiffened shells of revolution of multi-material construction.Comparisons between test and theory are given for axisymmetric collapse and nonsymmetric bifurcation buckling of 69 machined ring-stiffened aluminum cylinders submitted to external hydrostatic pressure. Because most of the cylinders fail at an average stress which corresponds to the knee of the stress-strain curve, the analytical predictions are not very sensitive to modeling particulars such as nodal point density or boundary conditions. Agreement between test and theory is improved if the analytical model reflects the fact that the shell and rings intersect over finite axial lenths.     

AN EXPERIMENTAL STUDY ON POST-BUCKLING BEHAVIOR OF SLENDER COLUMN WITH FIBER REINFORCED COMPOSITE MATERIAL

Equilibrium paths of post-buckling are measured for large slenderness column specimensmade of the fiber reinforced composite material. The influence of the initial curvature is investigatedexperimentally and compared with the result of the initial post-buckling theory. Both the theoreticaland experimental results reveal that the column with the initial curvature has stable post-buckling be-haviors and is not sensitive to the imperfection in the form of initial curvature. The experimental re-sults show that when the lateral buckling displacement is less than 20 percent of the column length, theexperimental results agree with the results from the theory of initial post-buckling quite well, whilethey agree with the results from the large deflection theory in a quite large range.     

Inelastic behavior of thick-walled cylinders subjected to nonproportionate loading

Inelastic behavior of thick-walled cylinders subjected to nonproportionate loading was studied by the testing of specimens made of C1045 steel and of annealed copper. Several theories were reviewed. A closed-form solution proposed by Mendelson¹² was used to predict external strains for open-end and closed-end thick-walled cylinders. An incremental theory proposed by Chu¹³ was used to provide incremental solutions for open-end thick-walled cylinders, and for cylinders subjected to nonproportionate loading. Test data for open-end and closed-end thick-walled cylinders made of C1045 steel and of annealed copper were in excellent agreement with the incremental theory. Larger values were predicted by use of the closed-form solution for circumferential strains than actual test data for open-end thick-walled cylinders at large depth of yielding. For cylinders subjected to nonproportionate loading, excellent agreement was indicated between the incremental theory and the experiments for the plot of axial load vs. circumferential strain for specimens made of both metals. Agreement between the incremental theory prediction of axial strains for the specimens made of annealed copper and test data is quite satisfactory. Larger values were predicted by the incremental theory for axial strain than experimental data for specimens made of C1045 steel. The error was conservative.     

Thermomechanical Postbuckling of Imperfect Moderately Thick Plates on Nonlinear Elastic Foundations*

ABSTRACT

A postbuckling analysis is presented for a moderately thick rectangular plate subjected to combined axial compression and uniform temperature loading, and resting on a softening nonlinear elastic foundation. The cases of (1) thermal postbuckling of initially compressed plates and (2) compressive post-buckling of initially heated plates are considered. The initial geometrical imperfections of the plates are taken into account. Formulations are based on Reissner-Mindlin plate theory, considering first-order shear deformation effects, and including plate-foundation interaction and thermal effects. The analysis uses a deflection-type perturbation technique to determine buckling loads and postbuckling equilibrium paths. Numerical examples include the performance of perfect and imperfect, moderately thick plates resting on softening nonlinear elastic foundations. Typical results are presented in dimensionless graphical form.     

Postbuckling Analysis of Nonconservative Elastic Systems

Abstract

Previous work on the postbuckling and imperfection-sensitivity of elastic structures has concentrated on conservative systems. The results of Koiterand others have led to a general theory of nonlinear stability behavior for these systems. The theory must be modified when nonconservative forces are present, and this is the aim of the present paper.

Discrete, nonconservative, elastic systems which exhibit static (divergence) instability are considered. The nonlinear behavior in the neighborhood of a critical point is analyzed by means of a perturbation procedure. When the critical point is simple, the results are similar to those for conservative systems. When a coincident critical point exists, however, different types of behavior occur. In many cases there is no bifurcation at all, with only the fundamental (trivial) equilibrium path passing through the critical point. Imperfection-sensitivity is more severe than for the typical bifurcation points and can even occur when the perfect system has no bifurcation. The results are illustrated with the use of a nonlinear double pendulum model subjected to a partial follower load.     

On axisymmetric/diamond-like mode transitions in axially compressed core–shell cylinders

Recent interests in curvature- and stress-induced pattern formation and pattern selection motivate the present study. Surface morphological wrinkling of a cylindrical shell supported by a soft core subjected to axial compression is investigated based on a nonlinear 3D finite element model. The post-buckling behavior of core–shell cylinders beyond the first bifurcation often leads to complicated responses with surface mode transitions. The proposed finite element framework allows predicting and tracing these bifurcation portraits from a quantitative standpoint. The occurrence and evolution of 3D instability modes including sinusoidally deformed axisymmetric patterns and non-axisymmetric diamond-like modes will be highlighted according to critical dimensionless parameters. Besides, the phase diagram obtained from dimensional analyses and numerical results could be used to guide the design of core–shell cylindrical systems to achieve the desired instability patterns.     

Analysis of cylindrical shells of nonuniform thickness using two-dimensional photoelastic models

An experimental method is described whereby symmetrically loaded cylinders of nonuniform thickness are analyzed using two-dimensional photoelastic models mounted on elastic foundations. The technique is most conveniently applied to ring-stiffened or notched cylinders. The particular model studied simulated a notched cylindrical pressure vessel which had been previously studied with three-dimensional photoelasticity. The stress-concentration factors at the base of the notch, found using both methods, showed excellent agreement. An analysis was also performed which allows estimation of the error involved when a beam-on-elastic-foundation model does not rigorously simulate a cylinder.     

Elastic instability of an orthotropic elliptic plate

On the basis of von Kármán equations and using the general bifurcation theory,theelastic instability of an orthotropic elliptic plate whose edge is subjected to a uniform planecompression is discussed.Following the well-known Liapunov-Schmidt process theexistance of bifurcation solution at a simple eigenvalue is shown and the asymptoticexpression is obtained by means of the perturbation expansion with a small parameter.Finally,by using the finite element method,the critical loads of the plate are computed andthe post-buckling behavior is analysed.And also the effect of material and geometric parameters on the stability is studied.     

Imperfection sensitivity of the post-buckling behavior of higher-order shear deformable functionally graded plates

This paper investigates the sensitivity of the post-buckling behavior of shear deformable functionally graded plates to initial geometrical imperfections in general modes. A generic imperfection function that takes the form of the product of trigonometric and hyperbolic functions is used to model various possible initial geometrical imperfections such as sine type, local type, and global type imperfections. The formulations are based on Reddy’s higher-order shear deformation plate theory and von Karman-type geometric nonlinearity. A semi-analytical method that makes use of the one-dimensional differential quadrature method, the Galerkin technique, and an iteration process is used to obtain the post-buckling equilibrium paths of plates with various boundary conditions that are subjected to edge compressive loading together with a uniform temperature change. Special attention is given to the effects of imperfection parameters, which include half-wave number, amplitude, and location, on the post-buckling response of plates. Numerical results presented in graphical form for zirconia/aluminum (ZrO₂/Al) graded plates reveal that the post-buckling behavior is very sensitive to the L2-mode local type imperfection. The influences of the volume fraction index, edge compression, temperature change, boundary condition, side-to-thickness ratio and plate aspect ratio are also discussed.     

Experimental investigation of the buckling instability of monocoque shells

Buckling of a series of thin-metal, shallow spherical shells under a uniform hydrostatic pressure has been investigated. Stress and deformation histories, as well as the critical buckling pressure and the post-buckling behavior, have been determined. Comparisons with theoretical analyses for buckling of spherical caps are given. Results are presented for an initial phase of a stability study of truncated conical sections which have been subjected to combinations of axial load and lateral pressure. A series of roll-formed and butt-welded, truncated aluminum conical shells with a 75-deg base angle have been tested. Buckle modes for axial-load condition alone, laterial-pressure load alone and combinations of these loading conditions are described. Interaction curves for the conditions investigated are given.     

Large-deflection and post-buckling behavior of slender beam-columns with non-linear end-restraints

The large-deflection analysis and post-buckling behavior of laterally braced or unbraced slender beam-columns of symmetrical cross section subjected to end loads (forces and moments) with both ends partially restrained against rotation, including the effects of out-of-plumbness, are developed in a classical manner. The classical theory of the “Elastica” and the corresponding elliptical functions utilized herein are those presented previously by Aristizabal-Ochoa [1]. The proposed method can be used in the large-deflection analysis and post-buckling behavior of elastic slender beam-columns with rigid, semi-rigid, and simple flexural connections at both ends including linear and non-linear inelastic connections like those that suffer from flexural degradation (such as flexural cracking and elasto-plastic connections) or flexural stiffening. Only bending strains are considered in the proposed analysis. Results from the proposed method are theoretically exact from small to very large curvatures and transverse and longitudinal displacements for laterally braced or unbraced slender beam-columns under bending caused by end loads. The large-deflection analysis and post-buckling behavior of slender beam-columns with both supports partially restrained against rotation and with sway inhibited or uninhibited are complex problems requiring the simultaneous solution of two coupled non-linear equations with elliptical integrals whose unknowns are the limits of the integrals. The validity of the proposed method and equations are verified against solutions available in the technical literature. Three comprehensive examples are included that show the effects of linear and non-linear connections at both ends on the large-deflection analysis and post-buckling behavior of slender beam-columns.     

Influence of concentrated lateral loads on the elastic stability of cylinders in bending

The character of the instability and the degradation of the moment-carrying capacity are found by Mylar model experiments for cylinders in bending when subjected to concentrated lateral loads. Lateral loads can seriously degrade the moment capability of cylinders. Critical combinations of moment and lateral load cause two distinct modes of failure—collapse and snapping. Collapse modes exhibit buckles which cover the compression half of the cylinder and are critical for large values of moment and small values of lateral load. Snapping modes of failure involve a single dimple and exist for smaller values of moment and larger values of lateral load.     

Non-linear response of curved sandwich panels – extended high-order approach

The geometrical non-linear behavior a curved sandwich panel with a stiff or compliant core when subjected to a pressure load using the Extended High-Order Sandwich Panel theory (EHSAPT), is presented. The formulation follows the EHSAPT procedure where the in-plane. i.e circumferential rigidity of the core is considered and the distribution of the displacements through the depth of the core are presumed. These displacement distributions are the closed-form solutions of the 2D governing equations of the curved core without circumferential rigidity that appear in the HSAPT curved sandwich panel model. The mathematical formulation includes the field equations along with the appropriate boundary and continuity conditions that take into account the high-order stress resultants in the core due to the presumed distributions. Finally a numerical study is conducted for a panel loaded by a distributed pressure at the upper face sheet. It reveals that the post-buckling response of a curved sandwich panels is associated with shallow to deep wrinkling deformations of the upper face sheet in the case of a simply-supported panel or a general non-linear pattern without wrinkles in the case of pinned supports with a short span. In both cases a stable post-buckling response is observed similar to that of a plate one.     

Creep of thick-walled cylinders subjected to internal pressure and axial load

A creep theory is presented to predict deformations at any specified time for a thick-walled cylinder subjected to internal pressure and axial load. The theory is based on the usual assumptions that the deformations are infinitestimal, that the material is incompressible and that the total strain theory is valid. The stress-strain-time relation for the material is assumed to be represented by an isochronous stress-strain diagram which is approximated by an arc hyperbolic sine function. The experimental part of the investigation included tests of thick-walled cylinders made of high-density poly-ethylene whose ratio of outside to inside radii were either 1.5 or 2.0. The test cylinders were either tested as closed-ented cylinders with internal pressure or subjected to a combination of internal pressure and axial load. Also, the application of the theory for varying load conditions was studied. Good agreement was found between theory and experiment.     

Experimental techniques employed for photoelastic analysis of cellular shells

The search for a shell construction superior to the usual ring-stiffened shells in strength and stability under external pressure with minimum weight has led to consideration of several other shell wall constructions.¹ The cellular-shell structure is one of the most promising designs of shells because of its ability to withstand high-pressure loading while maintaining a high degree of material efficiency. The analytical treatment of cellular shells has been undertaken only recently² and limited experimental study of these shells has been conducted. Thus, for obtaining reliable design formulas for the cellular-shell construction, these studies were undertaken. The cellular-shell construction may be visualized as two concentric thin cylinders spaced radially by a series of thin rings along their common longitudinal axis. The optimum wall thickness, rib thickness and rib spacing for a cellular shell of a given diameter and material which will result in the most efficient utilization of the material when the shell is placed under external pressure is the information required for shell design. The experimental techniques described in this paper have been employed to assist in the determination of the necessary design parameters.     

Thermo-mechanical post-buckling of FGM cylindrical panels with temperature-dependent properties

This paper presents thermo-mechanical post-buckling analysis of cylindrical panels that are made of functionally graded materials (FGMs) with temperature-dependent thermo-elastic properties that are graded in the direction of thickness according to a simple power law distribution in terms of the volume fractions of the constituents. The panel is initially stressed by an axial load, and is then subjected to a uniform temperature change. The theoretical formulations are based on the classical shell theory with von-Karman–Donnell-type nonlinearity. The effect of initial geometric imperfection is also included. A differential quadrature (DQ) based semi-analytical method combined with an iteration process is employed to predict the critical buckling load (where it is applicable) and to trace the post-buckling equilibrium path of FGM cylindrical panels under thermo-mechanical loading. Numerical results are presented for panels with silicon nitride and nickel as the ceramic and metal constituents. The effects of temperature-dependent properties, volume fraction index, axial load, initial imperfection, panel geometry and boundary conditions on the thermo-mechanical post-buckling behavior are evaluated in detail through parametric studies.     

Post-buckling analysis of FGM annular sector plates based on three dimensional elasticity graded finite elements

In this article, post-buckling and non-linear bending analysis of functionally graded annular sector plates based on three dimensional theory of elasticity in conjunction with non-linear Green strain tensor is considered. In-plane normal compressive loads have been applied to either radial, circumferential, or all edges of annular sector plates. Material properties are graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents while Poisson׳s ratio is assumed to be constant. The governing equations are developed based on the principle of minimum total potential energy and solved based on graded finite element method. Non-linear equilibrium equations are solved based on iterative Newton–Raphson method. The effects of material gradient exponent, different sector angles, thickness ratio, loading condition and two different boundary conditions on the post-buckling behavior of FGM annular sector plates have been investigated. Results denote that due to the stretching–bending coupling effects of the FGMs, the post-buckling behavior of movable simply supported FGM plates is not of the bifurcation-type buckling. Moreover, FGM annular sector plates subjected to uniaxial compression at radial edges show a non-linear bending behavior with unique and stable equilibrium paths following a flattening feature.     

Geometric Nonlinearity

The nonlinear behavior or structures has attracted considerable attention in the last decade. This paper is specifically concerned with three-dimensional skeletal structures wherein nonlinearities are attributable to change of geometry. The frameworks that are considered possess particular relevance to roofing systems.

The outline theoretical analysis incorporates: (a) effects of finite joint deformation; (b) change of member lateral stiffness; (c) change of member length due to bowing. Cable systems are discussed simply as a particular case of the more general solution.

Theoretical and experimental results are compared for a relatively complex rigid jointed structure.

An example of a shallow dome is presented in which the salient design considerations that must be made for this form of structure are clearly demonstrated.     

Catastrophe fracture of thin-wall pressure tubes

Catastrophe theory was used to investigate the fracture behavior of thin-wall cylindrical tubes subjected to internal explosive pressure. Based on the energy theory and catastrophe theory, a cusp catastrophe model for the fracture was established, and a critical condition associated with the model is given. Contributed by YANG Gui-tong Biographies: WEI De-min (1955-); YANG Gui-tong (1931-)