Multimode Interaction in Axially Stiffened Cylindrical Shells

ABSTRACT

A number of studies show that nonlinear interaction between a shortwave local panel buckling mode and a longwave overall shell buckling mode usually causes strong imperfection sensitivity in stiffened shells. Interaction is particularly strong for simultaneous or nearly simultaneous modes. Mode interaction between two overall buckling modes, as well as interaction between two overall and one local mode, has received much less attention. The present study indicates that for certain imperfection combinations, such mode interactions are important. On the other hand, in most cases, interaction between one overall mode and one local mode governs.     

Koiter circles in the buckling of axially compressed conical shells

As is well known, the elastic stability of shell structures under certain loading conditions is characterised by a severely unstable postbuckling behaviour. The presence of simultaneous buckling modes (‘competing’ modes corresponding to the same critical buckling load) is deemed to be largely responsible for such a behaviour. In the present paper, within the framework of the so-called classical theory (linear bifurcation eigenvalue analysis), the buckling behaviour of axially compressed cylindrical shells is firstly reviewed. Accordingly, doubly periodic eigenvectors (buckling modes) corresponding to the same eigenvalue (critical buckling load) can be determined, and their locus in a dimensionless meridional and circumferential buckling wavenumber space is described by a circle (known as the Koiter circle). In the case of axially compressed conical shells, no clear evidence of the existence of simultaneous buckling modes can be found in the literature. Then, such a problem is studied here via linear eigenvalue finite element analyses, showing that simultaneous doubly periodic modes do also occur for cones, and that their locus in a specifically defined dimensionless wavenumber space can be described by an ellipse (hereafter termed as the Koiter ellipse) whose aspect ratio is dependent on the tapering angle of the cone.     

Interactive buckling in long thin-walled rectangular hollow section struts

An analytical model describing the nonlinear interaction between global and local buckling modes in long thin-walled rectangular hollow section struts under pure compression founded on variational principles is presented. A system of nonlinear differential and integral equations subject to boundary conditions is formulated and solved using numerical continuation techniques. For the first time, the equilibrium behaviour of such struts with different cross-section joint rigidities is highlighted with characteristically unstable interactive buckling paths and a progressive change in the local buckling wavelength. With increasing joint rigidity within the cross-section, the severity of the unstable post-buckling behaviour is shown to be mollified. The results from the analytical model are validated using a nonlinear finite element model developed within the commercial package Abaqus and show excellent comparisons. A simplified method to calculate the local buckling load of the more compressed web undergoing global buckling and the corresponding global mode amplitude at the secondary bifurcation is also developed. Parametric studies on the effect of varying the length and cross-section aspect ratio are also presented that demonstrate the effectiveness of the currently developed models.     

Buckling of a beam subjected to electromagnetic force and its stabilization by controlling the perturbation of the bifurcation

For a beam subjected to electromagnetic force, magnetoelastic buckling due to the increase of such force is theoretically investigated by taking account of the nonlinearity of the electromagnetic force and the elastic force of the beam. Using Liapunov-Schmidt method and center manifold theory, the equilibrium space, the bifurcation set and the bifurcation diagram are theoretically derived. Also, the effect of the higher modes other than the buckling mode on the mode shape of the postbuckling state is discussed. Furthermore, a control method to stabilize the magnetoelastic buckling is proposed, and the unstable equilibrium state of the beam in the postbuckling state, i.e., the straight position of the beam, is stabilized by controlling the perturbation of the bifurcation.     

On the sensitivity of non-generic bifurcation of non-linear normal modes

It is shown that a non-generic bifurcation of non-linear normal modes may occur if the ratio of linear natural frequencies is near r-to-one, r=1,3,5,… . Non-generic bifurcations are explicitly obtained in the systems having certain symmetry, as observed frequently in literatures. It is found that there are two kinds of non-generic bifurcations, super-critical and sub-critical. The normal mode generated by the former kind is extended to large amplitude, but that by the latter kind is limited to small amplitude which depends on the difference between two linear natural frequencies and disappears when two frequencies are equal. Since a non-generic bifurcation is not generic, it is expected generically that if a system having a non-generic bifurcation is perturbed then the non-generic bifurcation disappears, and generic bifurcation appears in the perturbed system. Examples are given to verify the change in bifurcations and to obtain the stability behavior of normal modes. It is found that if a system having a super-critical non-generic bifurcation is perturbed, then two new normal modes are generated, one is stable, but the other unstable, implying a saddle-node bifurcation. If the system having a sub-critical non-generic bifurcation is perturbed, then no new normal mode is generated, but there is an interval of instability on a normal mode, implying two saddle-node bifurcations on the mode. Application of this study is discussed.     

Postbuckling and Imperfection Sensitivity Analysis of Axially Stiffened Cylindrical Shells with Mode Interaction

Abstract

Interaction of nearly simultaneous buckling modes in the presence of imperfections is studied. The investigation is concerned with axially stiffened cylindrical shells under axial compression. In these structures, two modes are of particular interest, namely an overall long-wave and a local shortwave buckling mode. Numerical results show that in some cases bending of the stringers in the local mode postbuckling solution plays an important role. Exclusion of this effect, as was done in a previous study by Byskov and Hutchinson, may lead to an overestimation of the carrying capacity of the shell. Furthermore, it is found that apparently reasonable approximations to the postbuckling fields associated with both the local and the overall mode, as well as with the overall mode alone, may lead to inexact values of the buckling load.     

One and two-parameter bifurcations to divergence and flutter in the three-dimensional motions of a fluid conveying viscoelastic tube with D4-symmetry

The loss of stability of the trivial downhanging equilibrium position of a slender circular tube conveying incompressible fluid flow is studied. The tube is clamped at its upper end and free at its lower end. Inbetween the three-dimensional transversal motion is constrained by an elastic support which is considered to beD ₄-symmetric, that is, has the symmetry of the square (Figure 1). Kirchhoff's rod theory and the Kelvin-Voigt viscoelastic law are used to derive the tube equations under the assumption of large displacement but small strain.The stability analysis is performed making use of the methods of equivariant bifurcation theory, that is, making use of the symmetry properties of the original system in deriving the amplitude equations of the critical modes. All cases of loss of stability which are possible for generic one-parameter bifurcations and the coincident case of a zero root and a purely imaginary pair of roots are investigated.Dedicated to Professor P. R. Sethna on the Occasion of His 70th Birthday     

Mode Finding in Nonlinear Structural Eigenvalue Calculations

Abstract

The eigenvalue problems resulting from stiffness matrix formulations of structural vibration and buckling problems are nonlinear if substructures are analyzed exactly, or if classical frequency (vibration problems) or load factor (buckling problems) dependent member equations are used. This makes rapid calculation of accurate free vibration or buckling modes difficult. This paper presents several techniques which might overcome this difficulty, examines them theoretically and experimentally, and gives some of the ways in which the more successful techniques can be incorporated in mode finding methods. Coincident eigenvalues (i.e., natural frequencies or critical load factors) are included.     

Symmetry breaking in an initially curved pre-stressed micro beam loaded by a distributed electrostatic force

The symmetric and asymmetric buckling of an initially curved micro beam subjected to an axial pre-stressing load and transversal distributed electrostatic force is studied. The analysis is based on a reduced order (RO) model resulting from the Galerkin decomposition with buckling modes of a straight beam used as the base functions. The criteria of symmetric limit point buckling and of non-symmetric bifurcation are derived in terms of the geometric parameters of the beam and the axial load. Two symmetry breaking conditions, defining the relations between the axial load and the geometric parameters of beams for which an asymmetric response bifurcates from the symmetric one, are obtained. The necessary criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric response curve; the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the symmetric response curve. A comparison between the RO model results and those obtained by direct numerical analysis shows good agreement between the two and indicates that the obtained criteria can be used to predict symmetric and non-symmetric buckling in electrostatically actuated curved pre-stressed micro beams. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through criterion. The nature of the latter and the relations between it and the symmetry breaking criteria are found to go through a prominent qualitative change as the initial distance between the beam and the electrode, characterizing the electrostatic force, changes.     

Pseudo-nonlinear dynamic analysis of buckled pipes

In this study, the post-divergence behavior of fluid-conveying pipes supported at both ends is investigated using the nonlinear equations of motion. The governing equation exhibits a cubic nonlinearity arising from mid-plane stretching. Exact solutions for post-buckling configurations of pipes with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions are investigated. The pipe is stable at its original static equilibrium position until the flow velocity becomes high enough to cause a supercritical pitchfork bifurcation, and the pipe loses stability by static divergence. In the supercritical fluid velocity regime, the equilibrium configuration becomes unstable and bifurcates into multiple equilibrium positions. To investigate the vibrations that occur in the vicinity of a buckled equilibrium position, the pseudo-nonlinear vibration problem around the first buckled configuration is solved precisely using a new solution procedure. By solving the resulting eigenvalue problem, the natural frequencies and the associated mode shapes of the pipe are calculated. The dynamic stability of the post-buckling configurations obtained in this manner is investigated. The first buckled shape is a stable equilibrium position for all boundary conditions. The buckled configurations beyond the first buckling mode are unstable equilibrium positions. The natural frequencies of the lowest vibration modes around each of the first two buckled configurations are presented. Effects of the system parameters on pipe behavior as well as the possibility of a subcritical pitchfork bifurcation are also investigated. The results show that many internal resonances might be activated among the vibration modes around the same or different buckled configurations.     

Secondary buckling of an elastic column with spring-supports at clamped ends

Summary The postbuckling behavior of an elastic column with spring supports of equal stiffness of extensional type at both clamped ends is studied. Attention is focused on those of spring stiffnesses near the critical value at which, under axial load, the column becomes critical with respect to two buckling modes simultaneously. By using the Liapunov-Schmidt-Koiter approach, we show that there are precisely two secondary bifurcation points on each primary postbuckling state for the spring stiffness greater than the critical value. The bifurcation takes place at one of the two least buckling loads. The corresponding secondary postbuckling states connect all the secondary bifurcation points in a loop. For the spring stiffness less than the critical value, no secondary bifurcation occurs. Asymptotic expansions of the primary and secondary postbuckling states are constructed. The stability analysis indicates that the primary postbuckling state for the spring stiffness greater than the critical value is bifurcating from the first buckling load and becomes unstable from a stable state via the secondary bifurcation, i.e., secondary buckling occurs. Received 22 April 1997; accepted for publication 22 December 1997     

Influence of shear deformation on the buckling of elastically supported beams

Summary The influence of shear deformation on the buckling behavior of a beam supported laterally by a Winkler elastic foundation is studied. A full investigation of the bifurcation points at which, under axial load, the beam becomes critical with respect to one or two simultaneous buckling modes is made. The configurations and stabilities of the equilibrium paths that bifurcate from the critical points are derived. From the results of theoretical analysis, it becomes evident that shear deformation has a considerable effect upon the equilibriums and stabilities of the post-buckling of the beam. The results for the Bernoulli-Euler beam can be obtained as a limiting case for those of the present beam by letting the shear stiffness tend to infinity.Supported by the National Natural Science Foundation of China     

The viscous curtain: General formulation and finite-element solution for the stability of flowing viscous sheets

The stability of thin viscous sheets has been studied so far in the special case where the base flow possesses a direction of invariance: the linear stability is then governed by an ordinary differential equation. We propose a mathematical formulation and a numerical method of solution that are applicable to the linear stability analysis of viscous sheets possessing no particular symmetry. The linear stability problem is formulated as a non-Hermitian eigenvalue problem in a 2D domain and is solved numerically using the finite-element method. Specifically, we consider the case of a viscous sheet in an open flow, which falls in a bath of fluid; the sheet is mildly stretched by gravity and the flow can become unstable by ‘curtain’ modes. The growth rates of these modes are calculated as a function of the fluid parameters and of the geometry, and a phase diagram is obtained. A transition is reported between a buckling mode (static bifurcation) and an oscillatory mode (Hopf bifurcation). The effect of surface tension is discussed.     

The Effect of Multiple Buckling Modes on the Postbuckling Behavior of Plane Elastic Frames. Part I. Symmetric Frames

The postbuckling behavior of a one-bay, two-storey frame with built-in edges and symmetric with respect to midspan is analyzed. Columns are assumed to be inextensible and shear-undeformable, and beams are rigid. Then two buckling modes are possible, that is, sidesway of the lower floor with rigid horizontal displacement of the top floor and sidesway of the top floor with the lower floor undergoing no displacement. Obviously, the two buckling modes occur simultaneously if the ratios EI/h⁷ (EI being the bending stiffness of a column and h its length) are properly selected. Within the framework of a Koiter-type energy approach a suitable perturbation formulation is derived from a “hybrid” functional which is obtained by adding to the potential energy certain extra terms which account for the nonlinear energy associated with the internal forces applied to the beam at the joints. Results show that the postbuckling behavior of a single buckling mode can be stable or unstable according to the value of the ratio h/l, where l is the frame span. In the case of simultaneous buckling modes the structural behavior in the postbuckling range never improves, but no severe changes are noticed in comparison with the preceding case.     

Nonlinear three-dimensional oscillations of elastically constrained fluid conveying viscoelastic tubes with perfect and broken O(2)-symmetry

The loss of the stability of the trivial downhanging equilibrium position of a slender circular tube conveying incompressible fluid flow is studied. The tube is clamped at its upper end and is free at its lower end. Inbetween, the three-dimensional transversal motion is constrained by an elastic support considered to be rotationally symmetric. Tube equations valid for large displacement but small strain based on Kirchhoff's rod theory and the Kelvin-Voigt viscoelastic law are used.The stability analysis is performed by making use of the methods of the equivariant bifurcation theory; that is, but using the symmetry properties of the original system to drrive the amplitude equations of the critical modes. Two different types of results are given: First, for the perfect O(2)-symmetric system all three generic coincident eigenvalue cases of loss of stability in two-parameter families. Second, for the system with broken O(2)-symmetry due to imperfections, three special cases of loss of stability at simple eigenvalues.     

Resonance regimes in the neighborhood of bifurcation points of codimension 2 in the Couette-Taylor problem

The bifurcation in a dynamical system with cylindrical symmetry dependent on several parameters is studied with reference to the Couette-Taylor problem. Points at which two neutral curves intersect (bifurcation points of codimension 2) corresponding to several independent neutral modes are found. In the neighborhood of the bifurcation points of codimension 2 the interaction of these modes can be described by a system of amplitude equations on the central manifold. If the neutral modes are nonrotationally symmetrical, there exist seven different resonance states that influence the cubic terms of the amplitude system. For the resonances Res 0 and Res 3 the results of calculating the intersection points are presented and the conditions under which stationary regimes exist and are stable are analyzed.     

Non-linear in-plane buckling of rotationally restrained shallow arches under a central concentrated load

This paper investigates the non-linear in-plane buckling of pin-ended shallow circular arches with elastic end rotational restraints under a central concentrated load. A virtual work method is used to establish both the non-linear equilibrium equations and the buckling equilibrium equations. Analytical solutions for the non-linear in-plane symmetric snap-through and antisymmetric bifurcation buckling loads are obtained. It is found that the effects of the stiffness of the end rotational restraints on the buckling loads, and on the buckling and postbuckling behaviour of arches, are significant. The buckling loads increase with an increase of the stiffness of the rotational restraints. The values of the arch slenderness that delineate its snap-through and bifurcation buckling modes, and that define the conditions of buckling and of no buckling for the arch, increase with an increase of the stiffness of the rotational end restraints.     

The shape of the strongest column

The problem of determining that shape of column which has the largest critical buckling load is solved, assuming that the length and volume are given and that each cross section is convex. The strongest column has an equilateral triangle as cross section, and it is tapered along its length, being thickest in the middle and thinnest at its ends. Its buckling load is 61.2% larger than that of a circular cylinder. For columns all of whose cross sections are similar and of prescribed shape-not necessarily convex—the best tapering is found to increase the buckling load by one third over that of a uniform column. This result, which was independently obtained by H. F. Weinberger, is originally due to Clausen (1851). For a uniform column, triangularizing is shown to increase the buckling load by 20.9% over that of a circular cylinder. The results lead to isoperimetric inequalities for the buckling loads of arbitrary columns. The research reported in this paper has been sponsored by the Office of Naval Research under Contract No. (285) 46.     

Elastic Buckling with Infinitely Many Local Modes

ABSTRACT

ABSTRACT In some cases, asymptotic methods present an appealing alternative to full nonlinear analyses. In other cases, the value of an asymptotic analysis may merely be that, in a qualitative way, it can characterize the behavior of a structure. Whether an asymptotic method is applied for one or the other purpose it is of interest to attempt an estimation of its range of validity. The present paper addresses this question for an asymptotic method to predict imperfection sensitivity of elastic structures with mode interaction. The particular structure that is investigated possesses an infinity of nearly simultaneous local buckling modes. It is found that very few of these modes need to be taken into account.     

Effect of Coulomb Damping on Buckling of a Two-Rod System

This paper describes a significant influence of a slight Coulomb damping on buckling, using a simple two rods system. Coulomb damping produces equilibrium regions around the well-known stable and unstable steady states under the pitchfork bifurcation which occurs in the case without Coulomb damping. Also, the stability of the states in the equilibrium regions is examined by using the phase portrait. As a consequence, due to the slight Coulomb damping, it is theoretically clarified that the states in the equilibrium regions are locally stable, even in the neighborhood of the unstable steady states under the pitchfork bifurcation in the case without Coulomb damping, i.e., even in the neighborhood of the unstable trivial steady states in the postbuckling and the unstable nontrivial steady states under the subcritical pitchfork bifurcation. Furthermore, the experimental results are in qualitative agreement with the theoretically predicted phenomena.