....Developments in Mechanics. Vol. 7. Proceedings 01 the 13th Midwestern Mechenics Conference: 23

STABILITY AND POSTBUCKLING BEHAVIOROF HYPERELASTIC BODIES AT FINITE STRAINBY THE FINITE ELEMENT METHOD

D.W. SANDIDGE and J.T. ODENTHE UNIVERSITY OF ALABAMA AT HUNTSVillE

ABSTRAct

This paper presents an investigation of the application of the finite elementmethod to the problem of stability and postbuckling behavior of hyperelasticbodies at finite strains. A brief summary of the formulations of finiteelement models of plane-isotropic hyperelastic bodies of Oden [39] is given.The method of analysis centers around the powerful incremental loading tech-niques combined with Newton-Raphson corrections. These methods are supple-mented with other techniques to also allow for the determinstion of criticalload, the classifications of stability at critical loads and the study ofpostbuckling behavior. In conclusion, the method of solution is demonstratedby a number of representative numerical examples.

INTRODUCTION

This paper is concerned with the numerical analysis of stability and post-bUCkling behavior of highly elastic bodies at finite strain. The investiga-tion involves the application of the finite element method and the study ofthe determination of the bifurcations, limit points, and postbuckling behaviorof complex two-dimensional bodies composed of incompressible isotropicmaterials in states of plane stress.

The modern theory of stability and postbuckling behavior of elastic bodiesgrew from the celebrated work of Koiter ~l] snd was refined and generalized byThompson [2,3,4], Roorda [5J, Wempner [6), and others [7,8, 9J. A recentsurvey of the subject was contributed by Koiter and Hutchinson [10]. Despitegreat interest in this theory, it has found remarkably few applications togenuinely complex structural problems because of its complexity.

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A few attempts have been made to apply finite element methods to the analysisof postbuckling behavior. The work of Walker [11,12] dealing with rathersimple arches and shells are among the attempts that should be mentioned.More recently, Hartz and Lang [13], Conner and Morin [14], Dupuis et al. (8],and Hangai and Kawamata [16] have contributed finite-element studies ofstability and postbuckling behavior of space frames and shells of revolution.Gallagher [17] presented a simple technique for determining critical points ofstructures undergoing large deformations. Gallagher and Mau [18] also con-tributed a detailed survey of the material on finite element solution of prob-lems in elastic stability. Most of these works use variants or direct appli-cations of the Koiter-Thompson theory to assess stability at the criticalload, and all are limited to infinitesimal strains of Hookean bodies. A studyof stability of bodies at finite strain has received relatively little atten-tion; one can, however, find some works in this field by John [19] et al.[20 - 26]. Apparently all work in the stability of bodies at finite strain islimited to very simple geometries. Finite-element applications to problems ofthis type do not appear to be available, with the exception of the work byaden et al. [27 - 38]. However, the work by aden et al. concentrated ondetermining the gross behavior of hyperelastic bodies and did not deal withproblems of determining critical loads or stability at critical loads. As aresult, full automation of these methods to the analysis of completely generalnonlinear behaVior has not been possible.

The feasibility of applying methods based on a direct application of Koiter'soriginal stability theory to problems involving finite strain is not, from apractitioner's point of view, too promising. Direct application of Koiter'stheory involves the calculation of at least third and fourth variations in thepotential energy n or their equivalents. For materials with strsin energyfunctions involving, e.g., exponential or transcendental forms in the dis-placement gradients, no truncation to "lower-order terms" is possible; there-fore it is generally necessary to evaluate many thousands of terms to obtainquantitative estimates of stability. For example, in the case of a singlefinite element of 6 degrees of freedom, the fourth variation in n involvesover 1000 terma for a aingle element and for a 20-degree-of-freedom element,over 160,000 terms are required. Further', the method may fail when appliedto problema in the discrete domain (3).

To avoid such complications, it is profitable to recall that the objectives ofgeneral theories of elastic stability include the assessment of stability atcritical loads of structures, the identification of bifurcation snd limitpoints, and the prediction of the response of structures beyond criticalloads. ~lost attempts at applying the theory tacitly assume that a directaNtysis of the nonlinear problem is either impossible or, at best, impracticaland that it is possible to obtain meaningful results only through some per-turbation technique, wherein the equilibrium path is extrapolated beyondcritical points.

The present investigation adapts a qui.te different point of view. Thepremise here is that powerful methods are available for the solution oflarge systems of nonlinear equations. In fact, these methods have beenthoroughly exploited in certain of the finite-element applications in finiteelasticity, mentioned previously, and their utility for determining poat-buckling behavior of complex structures at finite strains has been demon-strated [27,32). The method described in reference [32], for example,employs an incremental loading technique with Newton-Raphson corrections,which makes it possible to traverse equilibrium paths of very complex struc-tures in a step by step fashion.

A technique to automate the nonlinear analyses and to allow for the identifi-cation of critical loads, the assessment of stability at the critical load,and the atudy of the continuation of the response beyond critical loads ialacking. The present study involves techniques which should prove useful inreaching this goal.

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FINITE ELEMENT HODEL

The derivation of the equilibrium equations of a typical flat triangular tri-angular element is given in depth by aden (27-38) and is simply stated here.

2Vob"a(6Sp +m..pu:) f~~l(6ae - ).·foe) +~t[fae(l- 2).· - 2)'·Y;.µ) - ),"'60'8)) - p~ = 0 (1)

where p~p is a generalizedAforce at node N in direction p, u" is a displacementof node M in direction p, W the strain energy per unit volumB, 11, Ia the prin-cipal invariants of the deformation tensor, and). the ratio of the deformed toundeformed thickneaa. By successive differentiation, higher-order derivativesof the total potential enerby can be obtained. For example, the followingarray, designated as the stability matrix in this paper, represents secondderivatives of the potential energy of a typical finite element:

~"'n r~Q ail [ :i ]~u~au~ = 2VobcJ>·a6"L6ep~~I1(6oe-).·foe)+a~ foe(I-2).·Yµ.µ.)+k 6oe}

+ 2Vobca(6ae + b"~u~) {-~~1[4 (k~ p)3f~ + ).·f~, p] + ~~ [f~, P(1 - 2).4

- 2).4y ) _ f .......(8().~ )3+8(kL )3y +2).·yL )+2).L 6"",)} (2)µ.µ. "" p , p µ.µ ;.µ, p ,p y;>

With the necessary equations defined for a typical element, they are appliedto each element and assembled into a connected model for the present problem.The process of assembling the elements is well-documented (e,g. [39,41) andis not elaborated on here.

STABILITY OF HODEL

In the following discussion, the stability of the discrete conservative struc-tural system is investigated using the energy criterion. Upon assembling allof the finite elements and applying boundary conditions, a discrete structuralsystem is obtained which has, e.g., n degrees of freedom. The nodal dis-placement components serve as the n generalized coordinates, denoted XI' If Adenotes a load parameter, the total potential energy TT of this system canalways be written in the form,

(3)

where U and V are the strain energy and potential energy of the external load,respectively.

BUCKLING BEHAVIOR OF THE PERFECT AND IMPERFECT STRUCTURAL SYSTEMS

We outline briefly here some essential features of stability theory needed forour analysis. Additional details can be found, for example, in [1,2,5,9,42].

The total potential energy of an imperfect system can be written as a functionn(xl, A, p), where XI is n generalized coordinates, A is the load parameter,and p is an imperfection parameter [5]. If P = 0, the energy functionn(xl, A, 0) defines perfect system. The imperfection parameter is generallyconsidered to be a function of the generalized coordinates (i.e., to representgeometric imperfections) or the load parameter A (i.e., to represent loadmisalignment). The influence of p can be assessed qualitatively by consider-ing the two general classes of critical points, limit points and bifurcationpoints, depicted in Figure I. There heavy lines indicate the response of theperfect system and the lighter lines indicate the response of the imperfectsystem; solid and dotted lines denote stable and unstable patha, respectively.

A limit point occurs when the initially stable equilibrium path emanatingfrom the origin is the unique path passing through the critical point andreaches a local maximum at the critical load AO. The equilibrium path becomesunstable at this critical load hO and the system will follow the unstable

307

path. Phyaically, the aystem exhibita a sudden change of configuration atsuch points, lurching from one stable equilibrium configuration to anotherpossibly distant stable equilibrium configuration. This phenomenon isger.erally referred to as "snapping" or "snap-through buckling". The effectsof an imperfection can increase or decrease the critical load hO for thistype buckling but the system will still exhibit a limit point type ofinstability.

In structural applications, there are three discrete bifurcation points ofinterest:asymmetric, unstable symmetric, and stable symmetr~c. Bifurcationpoints occur with the intersection of distinct and continuous equilibriumpaths at the critical point. The initially stable equilibrium path emanatingfrom the origin, referred to as the fundamental equilibrium path, becomesunstable at the bifurcation; the physical system can follow this path beyondthe critical point only if restrained. The intersecting equilibrium path iareferred to as the postbuckled equilibrium path, and may be either stable orunstable.

The effects of imperfections on systems which exhibit bifurcation bucklinghave been treated in depth by others [1,S). The analysis follows the sameapproach as that for the perfect system, and relies on prior knowledge of theperfect syatem. The equilibrium paths of the imperfect system can be shownto be of hyperbolic form with the paths of the perfect system acting asasymptotes which intersect at the bifurcation point.

The asymmetric bifurcation type critical point (Figure I-b) is unstable. Thefundamental and postbuckled equilibrium paths have non-zero slopes at thepoint of intersection. Thus at loads h slightly less than the critical losdhe, two neighboring points of stationary potential energy exist: one asso-ciated with the fundamental path being stable and that of the postbuckledpath being unstable. As the system spproaches the critical point, alightperturbations will cause an exchange of equilibrium paths which reaults inviolent behavior, often leading to failure of the structure. The asymmetricbifurcation (Figure I-b) is changed by a "positive" imperfection (note thesign of the imperfection is introduced for convenience of discussion only)to a limit point type buckling at a critical load somewhat below that of theperfect structure. If the sign of the imperfection is reversed, the equilib-rium path ascends as it approaches the critical point and is stsble in thevicinity of the critical load corresponding to the perfect structure. Like-wiae, the unstable symmetric bifurcation;type criticsl point is alao unstable.The fundamentsl equilibrium path has nonzero slope at the point of intersec-tion, while the postbuckled path reaches a local maximum at the point ofinteraection. Thus at loads h, Slightly less than he, there exists threeneighboring points of atationary potential energy: one associated with thefundamental path which is stable, and two associated with the postbuckledpath, which are unstable. As in the previous case, there will be an exchangeof equilibrium paths as the load h approaches the critical load he, and thesystem will undergo large and sudden motion. In this case the system mayfollow the postbuckled path in either direction, because of the condition ofzero slope at the critical point.

When an imperfection is introduced into a perfect system which exhibits anunstable-symmetric bifurcation (Figure I.e), the resulting equilibrium pathexhibits limit point-type buckling for either positive or negative imperfec-tions. The critical load occurs somewhat below that of the perfect system.

The stable-symmetric bifurcation-type critical point is stable. The funda-mental equilibrium path has nonzero slope at the point of intersection, andthe postbuckled path curves upward on either side of this point, with zeroslope at the point of intersection. At losds A less than the critical loadA~, a single point of stationary potential energy associated with the funda-mental equilibrium path exists, and at thia point the equilibrium is stable.At loads h Slightly larger than the critical load h~, three neighboring points

308

...,

of stationary potential energy exists: one associated with the fundamentalpath which is unstable, and two associated with the postbuckled path which srestable. As the load A, is increased beyond the critical load AC

, there is anexchange of equilibrium paths, and the system will follow the stable post-buckled path.

The effect of the imperfecticn with respect to the system which exhibitsstable-symmetric bifurcation (Figure ld) is to change the equilibrium path toa stable path in the vicinity of the critical load exhibited by the perfectsystem for either a positive or negative imperfection.

A structure is said to be "imperfection sensitive" when a very small imperfec-tion results in a large deviation from the perfect system. It is well knownthst the perfect system is nonexistent in nature; therefore, the importance ofimperfections when determining buckling conditions is obvious, psrticularlywhen a small imperfection may cause buckling or large deflections well belowthe critical load of the perfect system.

NUMERICAL TECIlNIQUES

In the work presented here the nonlinear equilibrium equations are solved bythe method of incremental loading with Newton-Raphson corrections. Thefollowing discussion pertains to the relation of these methods to stabilityanalysis.

The Newton-Raphson and incremental loading techniques are appealing for thebuckling problem because, with some modifications, both allow for easy detec-tion of critical points. Difficulties arise if one wishes to project beyondthe critical point to the postbuckled path.

Both numerical schemes contain the inverse of the stability matrix, and as acriticsl point is approached this matrix becomes ill conditioned. Thus, bothmethods fsil in the vicinity of a critical point.

Certain problems have been worked successfully by rearranging the numericalscheme such that a displacement is incremented rather than the load (43). Useof incremental displscements aeema to be advantageous only in rather simplestructures With a small number of degrees-of-freedom. It is doubtful thatsuch ideas will find wide application to larger, more complex problems. Forexample, in the examples presented in this work, a number of coordInatesappesr to become critical at the same time. Further, these coordinates msyremain nearly critical at large distances from the critical point.

For a closer view of these conditions, consider the second variation of thetotal potential energy at a critical point; then

(4)

To simplify the discussion, (4) is diagonalized by the nonsingular lineartransformation, Axl ~ al 6Y,. Then n'l ,OxIAx, = X'I,6YI6Y, vanishes whereX'I' • 0 if i r j. Now ~'IJ = 0 haa nontrivial solutions only if thedeterminant of the coefficient matrix is zero:

lx, II I • Ix, nX':/li ... X, 0, I = 0 (no sum on i)

It is clesr that one or more of the diagonal terms must be zero for this con-dition to hold. The number of zero terms that appear on the diagonal of thetransformed stability matrix determines the number of critical coordinates;i.e. for each term, X ., = 0, on the diagonal, the corresponding coordinate,y., is a critical coordinate. If a single coefficient vanishes, then thecorresponding critical coordinate can be exchanged with the losd, and theaolution technique can be continued by incrementing the critical coordinate,

309

providing the stability matrix can be rearranged such that it is not ill-conditioned. If the degree of instability ia two or more, it is obvious thatthe matrix could not be rearranged in such a manner that it would be non-singular. Thus, attempts of solution by the linesrized version of either theNewton-Raphson or incremental loading techniques would fail.

It is well-known that if IT'll is positive definite, the system is stable; ifIT'I I is negative definite, negative semidefinite, or indefinite, the systemis unstable; and if IT'll is positive, semidefinite, or vanishes, the system iscritical. Thus, if all coefficients X'I) > 0, the system is stable whereasif one or more of the coefficients X,II < 0, the system is unstable, and ifone or more X'II = 0 and all other X'I I > 0, the system is critical. Toassess stability of a syatem, one must, therefore, evaluate the principalminors of the stability matrix and inspect the sign of each one. That ia, ifan even number of diagonal terma, X,II' are negative, the determinant couldbe equal to or greater than zero.

The value of the determinant of the stability matrix does provide an effectivemeans of locsting the firat critical load, in that all coordinates are assumedto be stable from the origin to the first critical point. Thus IT )AKIAKJ isassumed to be initially positive definite, becoming positive semiaefinite atthe first critical point, and one need simply search for the point st whichthe determinant vanishes. Unfortunately thia method has certain undesirableproperties when used in approximate solutions, particularly in the case offinite element problems, where the stability matrix is generally large andnarrowly banded. Because of error accumulation, the determinant will beeither extremely large and positive or extremely large and negative. Thisphenomenon is discussed in reference (44] and is not elaborated on here.

The Newton-Raphson and incremental loading techniques use a tangent projectionand, because of this, a bifurcation can be passed with small increments ofload. Unless aome check is made, tbe solution technique will continue alongthe unstable portion of the fundamental equilibrium path, projecting beyondthe critical point, and the bifurcation will not be detected. In the exampleproblems to be considered subsequently, the determinant ia evaluated for eachincrement of load. The determinant is normalized, as suggested in [44],such that only the sign of the determinant is retained, and it is used onlyto assist in locating bifurcation points.

If the solution technique paases over a bifurcation, it is signified by achange in sign of the determinant. Moreover, either of the solution tech-niques (incremental loading or Newton-Raphson) will detect a limit point, inthat both fail if the load is increased beyond the critical load.

METHOD OF SOLtrl'ION

Consider now the case of bifurcation buckling. Difficulties arise in theexchange of stability paths. As noted in previous diacussion, it is simpleto traverse an equilibrium path, except in the immediate vicinity of acritical point, and here the problem is not too great if a unique equilibriumpath passes through the critical point. Thus, by introducing an imperfection,the bifurcation can be eliminated.

Symmetric bifurcations occur in systems with complete symmetry of load andgeometric shape of the structure. The symmetric bifurcation can generally bedistinguished from the asymmetric bifurcation by inspection. By introducingan imperfection and increasing the load incrementally to the critical value ofthe imperfect system, the type bifurcation can be determined and informationon the postbuckling behavior can be obtained. In the case of symmetric bifur-cation, if the imperfect system exhibits a critical load larger than that ofthe perfect system, it is of the stable symmetric type and the postbuckledpath is atable. If the critical load of the imperfect system is less thanthat of the perfect system, it is of the unstable symmetr;c type bifurcation

310

and the poatbuckled path is unstable. Symmetric behavior can be assured byreversing the imperfection and observing that the same critical load is ob-tained for both imperfect systems.

The asymmetric point of bifurcation is always unstable, but the path of the im-perfect system can be either stable or unstable. Assuming a positive imperfec-tion yields a critical load larger than that of the perfect system, a negativeimperfection would yield a critical load less than that of the perfect system.

In the work presented, only load perturbations are used to produce the modifiedsystem. Because the material is highly elastic, slight loads selectively ap-plied can cause noticeable geometric change, producing essentially the Sameeffect ss geometric imperfections. With a finite element formulstion of a sys-tem avsilsble, virtually any conceivable imperfection can be introduced intothe system With ease.

NUMERICAL EXAMPLES

The follOWing examples involve neo-Hookean material (Cl ; 24 psi, Ca; 0).bodies are 1/8 inch thick in the ~ direction and restrained from out ofbuckling, 1.e., buckling is restricted to the Xl - ~ plane.

COLUMN TYPE BUCKLING

Theplane

EXAMPLE I. This example considers a 2- by 8-inch rectangular body (Figure 2)subjected to axial load in the ~ direction. The Kg; 0 surfsce ts fixed, tl1'e~ ; L surface is reatrained from rotations (where L denotes the height of thebody in the ~ direction) and deformation in the Xl direction, but can undergorigid body motion in the Xl - ~ plane. These end conditions are as if theends were bonded to steel plates.

The finite element approximation consists of 64 elements connected at 45 nodeawhich resulta in 72 unknown nodal displacements. The resulting 72 nonlinearequilibrium equationa, which are sixth-degree polynomials, were solved by themethod of incremental loading with Newton-Raphson corrections. The load wasapplied in increments of AP - 0.4 pound. The determinant of the stabilitymatrix changed from s positive to negative value at the seventh increment ofload (P-2.8 pounds) indicating that a bifurcation point occurred within theincrement. The incremental solution was restarted at a load of P=2.4 pounds,with load increments ~-O.05 pound, for which case the Newton-Rsphson tech-nique failed to converge at a load of 2.65 pounds. Hence, the critical loadpe was iaolated to the range 2.60 < pc < 2.65 pounds.

Because no experimental results were available, the critical load was comparedwith the standard Euler buckling load, as a guide, which is pe = rfEI/L'" - 2.35pounds, where the moduluS of elaaticity was approximated by E = 6C1 and themoment of inertia I and length L were evaluated in the deformed atate near thecritical load, neglecting the change in thickness.

Before buckling, the vertical center line remained vertical, and the verticaldisplacements were linear. The vertical displacement at buckling was 0.51inch or 6.4 percent of the original height, with s maximum change in width of0.07 inch or 3.5 percent of the original width. Figure 3 shows the column atstate of zero load, the deformed shape at the critical load, and the post-buckled shape. After the first critical load Was determined, a small horizon-tal load (p - 0.15 pound) was applied at the top (x",= L) of the column to givethe syatem an initial deformation. While holding the horizontal losd constant,the vertical load was again increased fran zero to the critical by the incre-mental technique. For this case the equilibrium path of the imperfect systemwas stable to a load of 3.60 pounds where limit-point type buckling was indi-cated. Holding the vertical load constant at 3.0 pounds, the horizontal losdwas incremented to zero to project to the postbuckled path of the perfectsystem. The motion followed by the structure in removal of the horizontal load

311

is shown by line segment I-J in Figure 4.

With the system now preaumably on the postbuckled path of the perfect system,the vertical load was increased from 3.0 pounds to the second criticsl load of3.75 pounds < pC < 3.80 pounds were limit-point type buckling was experienced.The first two increasing load increments from point J (Figure 4 indicates thatthe system has not returned to a point of complete relative minimum of thetotsl potential energy. This action is attributed to the weak condition ofthe system. The postbuckled path of the perfect system from point K to thesecond critical point at point L does sppear to be of correct form in that itfollows nesrly parallel to the imperfect system.

EXAMPLE 2. This example, like example 1, also considers the column type buck-ling problem. The rectangular sheet is 2 by 10 inches with the additionalboundary condition that the top (~-L) of the column is restricted fromlateral movement (Figure 5). The finite element model consists of 80 elementsconnected at 55 nodes, which results in 91 unknown nodal displscements. Thefirst critical load was determined as described in example 1 and wss found tobe 10.25 pounds < pO < 10.30 pounds. This was compared with the standardEuler buckling load ss a guide which yielded 10.8 pounds.

In this example, very large displacements were experienced before buckling (21percent of the original height) and, as a result, the buckling behavior wasquite violent; i.e., the system abruptly shaped to a grosaly deformed position.Figure 6 shows the column at a state of zero load, the deformed shape near thecritical state, snd the postbuckled shape.

Having determined the first critical load, a horizontal load of p = 0.1 poundwas introduced at the center node of the structure (xl'~) B (0,L/2), and theload wss applied incrementally to determine the critical load of the imperfectsystem. The critical load of the imperfect system was found to be 8.95 pounds< pC < 9.00 pounds. A plot (Figure 7) of the center horizontal deflectionversus the vertical load ahows the deflection to increase rapidly with loadbeyond a load of 8 pounds approaching tangency to the horizontal plane at thecritical load. Initial attempts to project past this critical point were un-successful, and, the additional computational time needed to extend the solu-tion was regarded as prohibitive. This analysis seems to incidate that thepostbuckled behavior of the perfect system is initially unstable, because theimperfect aystem reaches a critical point well below that of the perfectsystem for "small" imperfections. In view of this, it is conjectured that thecritical point of the imperfect system is a limit point, which indicates thatthe perfect system exhibits an unstable symmetric bifurcation.

FRAME TYPE BUCKLING

EXAMPLE 3. This example considers the buckling behaVior of the knee-bentstructure depicted in Figure 8. The frame has pinned supports, and the load isapplied vertically at the intersection of the center lines of the horizontaland vertical members. The finite element model consists of 72 elements con-nected at 57 nodes with 110 unknown displacements. For frames, such as this,constructed of stiff metallic materials, it is well known that the instabilityis of the asymmetric bifurcation type. However, auch is not the cale with thehighly flexible materials considered here. The equilibrium path is nonlinearfrom the origin. The structure undergoes very large deformation before reach-ing the first critical load, with the vertical and horizontal members curvinginward and outwsrd, respectively. (Figure 9). With the load applied alongthe center line of the vertical member, the critical load was found to be 2.115pounds < p. < 2.120 pounds. Figure 10 shows a plot of horizontal deflectionof the center of the vertical member Veraue load, which shows that the struc-ture is very weak ss it approaches the critical load, i.e., the slope of thecurve is nesr zero for a relative long distance.

Although this problem is not of the bifurcation type, it was found to exhibit

312

characteristics similar to the asymmetric bifurcation problem. Initial curva-tures of the members are produced by applying a small couple p at the intersec-tion of the vertical and horizontal members. With the couple applied in acounter-clockwise direction, the load resistance of the structure was notablyreduced. However, when the couple was applied in a clockwise direction, theload carrying capability was increased. This is illustrated in Figure 10.

EXAMPLE 4. The structure investigated in this example is identical to that ofexample 3. Now, however, the load is applied along a line of symmetry, asshown in Figure II.

The load was applied in increments of 6A;O.2 pound, and the bifurcation pointwas passed in the increment from 1.4 to 1.6 pounds. This is shown graphicallyin Figure 12. The incremental solution was restarted at a load of p~ 1.4pounds with the load increment 6PgO.02 pound. In this particular caae, theincremental solution did actually pick up the postbuckled psth at a load ofpsl.SO pounds. Geometric deformations were rapid from this point (Figure 12),and the critical load pC was isolated to the interval 1.SIS pounds < pC < 1.520pounds. Successive plots of the deformed structure with increaaing load (Fig-ure 13) are interesting in that they show the step-wise transition to a buckledmode. It is noted that one of the members reverses curvature, which is typicalof this type structure.

As in example 3 a small couple was applied at the vertex of the structure andthe load was increased to the critical. The critical load of the imperfectsystem was found to be 1.00 < p' < 1.05 pound. The critical load of the per-turbed system occurs well below that of the perfect system, which indicatesthat the perfect system exhibits unstable symmetric bifurcation. This, ofcourse, is typical of this type structure.

ACKNOWLEDGEMENT: The support of this work by the U.S. Air Force Office ofScientific Research under Contract F44620-69-C-0124 is gratefully acknowledged.

A

(a) LIMIT POINT lbl ASYMMETRIC BIFURCATION

Fig. Is-c. Limit point and bifurcation point type critical points.

313

"':';'",," I, , I

," ~ ," , ,"'z' - ' -- ' ..-";.",,.. ~"...."- ,--- "~- ~, F ~~,

leI UNITAaU-IY ... ETRICWUIlCA TION

r8 In.

2in.

Fig. 2a,b. Finite-element model, example 1.

P • 0

(dl STABLE·SYMMETRICBIFURCATION

P

rL

b

p > pC

314

Fig. 3a-c. Deformed snd undeformed geometry, example 1.

4

I ~I"'" Lx(72)

Lx(711

3

~pe

rr:'"''c<

x(72)

g 2

1.0 2.0 3.0

DISPLACEMENT tin.'4.0

Fig. 4. Load-displacement curves, example 1.

x2

I10 in.

L ' "1-.2---1.n.b

Fig. 5a,b. Finite-element model,example 2.

P • 0

P < pCb

Fig. 6a-c. Deformed and undeformedgeometry, example 2.

315

:J I I ~

9L , x{4S1 I f'r () ~o.I In.,

n / I 17~ I P I L.....I......I-X1I 1-1--1I I I~ ~

~ 8

Q

~ I J 0-1 +\x(451oJ 5

4

3

2

Fig. 8. Finite-element model,example 3.

0.5

OIIPLAC:UlENT IInJ1.0

Fig. 7. Losd-displacement curve, example 2.

P • 0 p ~ pC

Fig. 9a-c. Deformed and undeformed geometry,example 3.

316

·.(261--'-.0.5 .4 .0.3 .0.2

p

P'F-3.0~

.(25)g!r

- 0

Fig. 10. Load-displacement curves, ex~ple 3.

r5 in.

l~_.,f-l-l

in.

Fig. 11. Finite-element model, example 4.

317

3.0 .J2Pr-x(251

x(261

ocl:o..J

2.0

1.0

0.1 0.2 0.3 0.4 0.5

DISPLACEMENT (in.1

0.6 0.7

318

Fig. 12. Load-displacement curves, example 4.

p • pC

Fig. 13. Buckling behavior, example 4.

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[IJ

[2]

[3J

[4]

[6)

[7J

[a]

[9]

(10]

ell]

(12)

[13]

(14J

[15]

[16]

[17]

[18]

[19J

[20]

[21]

[22]

L23]

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J. M. T. Thompson and A. C. Walker: The Branching Analysis of Perfect andImperfect Discrete Structural Systems. J. Mech. Phys. Solids, Vol.17 (1969).

J. Roorda: On the Buckling of Symmetric Structural Systems with First andSecond Order Imperfections. Int. J. Solids Struct., Vol. 4, p. 1137-1148 (1968).

G. A. Wempner: The Energy Criteria for Stability of StrucLures. U.S. AirForce AFOSR-TR-72-0875, UARI Research Report No. 119, The Universityof Alabama in Huntsville, Alabama (1972).

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