Observations on eigenvalue buckling analysis within a finite element context

Christopher J. Earls1

INTRODUCTION There is an increasing availability of commercial finite element software systems that permit the consideration of the effects of geometric nonlinearity in structural analysis. Oftentimes these software systems will have the capability to treat stability problems through eigenvalue extraction routines applied to the global system stiffness matrix; an approach referred to, alternately, as “buckling,” “eigenvalue buckling,” or “linearized eigenvalue buckling” analysis. This type of a stability analysis is attractive from the standpoint that it is computationally inexpensive. As compared with a more general incremental analysis that traces the entire nonlinear equilibrium path of the structural system, the eigenvalue buckling approach concerns itself with only one or two points on the equilibrium path. In addition, results obtained from eigenvalue buckling analyses, when applied to stability problems exhibiting bifurcation instability, are usually quite accurate; and this accuracy is obtained without much concern on the part of the software user. However, care must be taken when applying this

1 Associate Professor, School of Civil & Environmental

Engineering, Cornell University, 220 Hollister Hall, Ithaca, New York, 14853 cje23@cornell.edu

technique to problem types exhibiting other manifestations of instability (e.g. limit point instability). In practice, situations may arise in the design office where the application of eigenvalue buckling may seem attractive for problems involving elastic beam buckling (e.g. lateral torsional buckling of a beam or planar truss) or elastic snap through buckling (e.g. lattice dome, arch, or shallow truss assembly). In the domain of stability research, too, linearized eigenvlaue buckling is oftentimes attractive as a means for identifying a “seed imperfection” for application in a more detailed incremental nonlinear finite element analysis of a beam or framework, for instance. In all of the foregoing, there are finite structural deformations prior to the onset of instability that are additive to the governing buckling mode (as compared to a bifurcation instability where the pre- and post-buckling deformations may be thought of as being orthogonal to one another). This fact creates an inconsistency with regard to assumptions made in the formulation of the linearized eigenvalue buckling procedure itself. The present paper examines the underlying assumptions within the formulation of the eigenvalue buckling method in order to highlight the problem types that most readily lend themselves to solution by this method. In addition, problems presenting responses that violate these fundamental assumptions are also examined. In this latter case, it becomes very important to understand the nature of the implementation of eigenvalue buckling in the given software system (example problems variously solved with MASTAN2, ADINA, ABAQUS, and ANSYS are included in this paper); certain implementations will make application of eigenvalue buckling, to other than bifurcation problems, extremely problematic. The present paper begins with a discussion providing background information related to the various finite element buckling formulations within a single, standardized notation to allow for a transparent comparison of underlying assumptions.

Subsequent sections separately treat bifurcation buckling and limit point instability examples. A discussion of results and concluding remarks follow. OVERVIEW OF DOMINANT FINITE ELEMENT BUCKLING ANALYSIS APPROACHES The literature adopts the term “buckling analysis” when referring to a family of finite element techniques applied to structural systems for the identification of critical load levels through the solution of an eigen-problem arising out of assumptions made relative to changes in structural stiffness, and concomitant applied loadings. While the technique is applicable to structures that exhibit critical responses arising from limit point as a well as bifurcation of equilibrium, the term buckling is nonetheless universally applied. While this may seem inconsistent, since buckling is normally associated with the condition of bifurcation in the equilibrium path only, the nomenclature is defensible nonetheless as a result of the fact that the egien-problem posed within the finite element context resembles the familiar form where the vanishing of the determinant of the stiffness matrix is associated with a certain form of the Sturm-Liouville problem [Reddy 1998, Boyce and Diprima 1986]. All formulations within the present discussion will be presented in a standardized notation (to facilitate comparison); defined subsequently. Since the analyses considered here are strictly static, time will be used to denote an equilibrium point for the subject structure within configuration space; corresponding with a certain load level:

[K0] � linear elastic stiffness matrix whose elements are independent of the current structural configuration (thus no time reference is needed as a left sub-script)

[τKσ] � initial stress matrix dependent on the state of stress at

arbitrary time, τ. This matrix is populated with terms that include

linear and quadratic dependencies on the current displacement field.

The sum of the foregoing two stiffness matrices is typically what is referred to as the “tangent stiffness matrix” associated with a specific equilibrium point in configuration space. Some readers may be more familiar with the notion of the tangent stiffness being associated with a Taylor series expansion of the internal force vector about the current configuration during the solution, while others may recognize it as emanating from the stationarity of the total potential functional whose internal energy term includes the influence of finite strains. While other options exist for the population of the tangent stiffness matrix [Wood and Schrefler 1978][Holzer et al. 1990][Chang and Chen 1986] the former definition has emerged as the most dominant to date.

0T � equilibrium configuration associated with the trivial case of no external actions

tT � equilibrium configuration associated with an intermediate loading condition occurring between the unloaded condition and the unstable condition

�t � denotes an incremental motion through configuration space, along the equilibrium path

t + �tT � equilibrium configuration associated with an intermediate loading condition, occurring between the unloaded condition and the unstable condition; that is arbitrarily close to configuration tT

Tcr � equilibrium configuration at incipient instability

In the present discussion it will be helpful to define two applied loading conditions that are used to reckon an assumed characteristic change in the system stiffness. In general, the applied load will be referred to as “P.”

{Pbaseline} � the loading condition used to bring the structure to a point in configuration space denoted by tT.

{Pcharacteristic} � the loading condition resulting in the structure assuming configuration t+�tT

{Pcr} � the critical load associated with the equilibrium configuration at incipient instability, Tcr

The structural state may be characterized using tangent stiffness measures defined according to the following.

[Kbaseline] � the instantaneous stiffness of the structure arrived at by retaining only the terms in a Taylor series expansion of the load – deflection response of the structure about the point in configuration space corresponding to the applied loading {Pbaseline}.

[Kcharacteristic] � the instantaneous stiffness of the structure arrived at by retaining only the terms in a Taylor series expansion of the load – deflection response of the structure about the point in configuration space corresponding to the applied loading {Pcharacteristic}.

Classical Formulation The initial treatment of the finite element buckling analysis appeared in the literature prior to the formal naming of the finite element method [Gallagher et al. 1967]; this earliest reference identified the approach as being based on the “discrete element procedure.” In light of the foregoing, and based on a survey of the literature, it appears that in the most commonly held definition of the classical formulation for finite element buckling analysis, the following problem is solved [Cook et al. 2002, Holzer et al. 1990, Chang and Chen 1986, Brendel and Ramm 1980]

[ ] [ ]( ) 0det 00=+ σλ KK TT t

(1)

It is frequently assumed that the equilibrium point at Time tT is very close to the initial configuration at time 0T, but this is not a requirement in the literature. The subsequent buckling load is computed as

{ } { }cr baselineP Pλ= (2)

Secant Formulation The present discussion adopts the name “secant formulation” to describe the variation of the finite element buckling problem that is referred to variously as the “secant formulation” [Bathe and Dvorkin 1983, ADINA 2006] and the “linear and nonlinear analysis” [Holzer et al. 1990]. This problem is posed as

[ ] [ ] [ ]( )( ) 0det =−+ baselinesticcharacteribaseline KKK λ (3)

The subsequent buckling load is computed as

{ } { } { } { }( )cr baseline characteristic baselineP P P Pλ= + − (4)

EXAMPLE PROBLEMS In the following examples, comparisons of results from the two approaches to finite element buckling analyses are considered (i.e. those involving software packages employing buckling approaches characterized by equations (1) and (2), and (3) and (4), respectively). In some instances the results from closed form solutions are also presented. In addition, some cases also include results from manual implementation of the finite element buckling approaches as encapsulated in equations (1) and (2). We begin by distinguishing between bifurcation and limit point instability in a formalized way. Consider the classical form of the finite element statement of the incremental equilibrium equations:

( ){ } { }0 0 t t tt tT T T TK K u Rσ +∆+∆

� � � �+ ∆ =� � � � (5)

Where {∆u} are the incremental nodal displacements and {R} is the residual vector representing the imbalance between the internal forces at time t, and the desired load level associated with some set of external forces. We may use the standard form

of the eigenvalue problem to compute eigenvalues, ωi, and

eigenvectors, φi, for the tangent stiffness matrix according to

( ){ } { }0 0 tT T i i iK Kσ φ ω φ� � � �+ =� � � � (6)

We may thus form a spectral representation of the tangent stiffness matrix according to

[ ] { }{ }T

T i i ii

K ω φ φ=� (7)

These same eigenvectors may also be used to define a projection operator that takes the original displacement and load vectors and projects them onto a new vector space spanned by

the eigenvectors, φi, such that

{ } { } { } { }; t tT i i t t i i

i i

u Pα φ ρ φ+∆ +∆∆ = =� � (8)

Where {P} is the externally applied load vector on the structure

and ρi � {φi}T{P}. The transformations embodied in equations (7)

and (8) effectively diagonalize equation (5) and result in the following transformation (Pecknold et al. 1985)

i i iω α λρ= (9)

In an elastic structure, the first critical point occurs when the equilibrium equations become singular, or in other words, when the tangent stiffness matrix is no longer positive definite. Since we have well ordered eigenpairs as a result of this being one case of the Sturm-Liouville problem, we recognize that loss of

positive definiteness of the stiffness occurs when ω1 = 0; at

which point we then have λρ1 = 0. Based on this fact, we can recognize a distinction between bifurcation and limit point

instability (Pecknold et al. 1985). If ρ1 � 0, then we see that the eigenvector is not orthogonal to the externally applied loading

vector {P} and thus λ will have to be zero in order for λρ1 = 0 to be true. In this case we have a limit point instability (this point will become more clear as we continue the discussion). Conversely,

ρ1 = 0, when the loading is orthogonal to the eigenmode and we have buckling occurring. We may thus summarize as follows:

Limit point instability: { } { }1 10, and 0;T

Pω φ= ≠ (10)

Bifurcation instability: { } { }1 10, and 0T

Pω φ= = (11)

Equations (10) and (11) may be interpreted according to the following. In the neighborhood of limit points on the equilibrium path in configuration space, there is no increasing load and the eigenvector is not orthogonal to the external load vector. Conversely, if an increase in loading is possible in the neighborhood of the critical point, and the eigenvector is orthogonal to the load vector, then bifurcation instability is present. Bifurcation instability The following three cases are classical examples of bifurcations instability in the sense of equation (11). Pre-buckling deformations are small and thus the underlying assumptions, in this regard, present in the linearized eigenvalue buckling approach are valid. Flat rectangular plate under edge loading Consider the case of the flat plate under edge loading depicted in Figure 1.

Figure 1. Plate buckling example

For this problem, the closed form solution is easily developed and the critical edge stress is identified as

( )

2

2

212 1

cr

Ek

b

t

π ησ

ν

=� �

− � �

(12)

Substitution of the problem parameters described in Figure 1 yields a critical stress of 1.114 ksi. As expected, this is in close agreement with finite element buckling results presented in Table 1 for both finite element buckling approaches under discussion.

fre

e

pinY

rollerY

sim

ple

X

X, u1

Y, u2

Boundary Conditions:Free? u1 u2 u3 ur1 ur2 ur3

rollerY Y Y N Y N NpinY Y N N Y N NsimpleX N Y N N Y Nfree Y Y Y Y Y N

b

a

a = 50”b = 10”t = 0.1”E = 29,000 ksiν= 0.3

since a>b and b>>tthen

k = 0.425� = 1

σcr

Table 1. FEM buckling results - plate

Euler Column Another classical example of bifurcation instability, where both finite element buckling analysis methods work quite well, is the case of the Euler column depicted in Figure 2.

Figure 2. Euler column buckling example

For this problem, too, the closed form solution is easily obtained (considering the first mode only):

2

2cr

EIP

L

π= (13)

Y, u2

Z, u3

Boundary Conditions:Free? u1 u2 u3 ur1 ur2 ur3

roller N Y N Y N Npin N N N Y N N Pcr

10”

1”

1”

b = 1”L = 10”I = 0.0833

E = 29,000 ksi

Linearized Eigenvalue Buckling Results:

(10 ADINA MITC4 four node shell elementsacross width)

Pbaseline σσσσcr Method0.001 k/in 1.222 ksi Classical

0.01 k/in 1.217 ksi Classical0.04 k/in 1.217 ksi Classical0.05 k/in 1.217 ksi Classical

0.12 k/in NA Classical

0.001 k/in NA Secant

0.01 k/in 1.217 ksi Secant0.04 k/in 1.217 ksi Secant0.05 k/in 1.217 ksi Secant

0.12 k/in NA Secant

which yields Pcr = 238 kips in the case of the proportions described in Figure 2. This compares quite favorably with the finite element buckling results presented in Table 2.

Table 2 FEM buckling results - column

Cylindrical shell loaded at one end While it is encouraging to note the agreement between the two dominant methods of finite element buckling analysis and the related closed form solutions in the previous two cases, this type of result is not universally true for all bifurcation type problems. Consider the case of a silicon nitride cylindrical shell pinned at one end and longitudinally loaded along the free edge at the other end as depicted in Figure 3.

Linearized Eigenvalue Buckling Results:

(5 ADINA Hermitian beam elements)

Pbaseline Pcr Method

200k 235 Classical

100k 235.8 Classical

10k 236.5 Classical

1k 236.6 Classical200k 235 Secant

100k 235.8 Secant

10k 236.5 Secant

1k 236.6 Secant

Figure 3. Cylindrical shell buckling example

While it is that good results are obtained for the classical buckling responses that characterize the problems described in Figures 1 and 2, irrespective of baseline load level, etc., the same cannot be said for the case depicted in Figure 3. In this case, the ADINA finite element buckling analyses were quite sensitive to the baseline load level and thus in some cases a result was obtained, while in other cases, the ADINA eigenvalue extraction routine was unable to converge to a reasonable eigenpair (i.e. one satisfying the Sturm sequence requirement). When convergent eigenpairs were obtained, the agreement with the much more computationally demanding incremental nonlinear solution (LDC solution) was encouraging. However, the lack of robustness, in terms of solution dependence on Pbaseline, was anything but encouraging. It is also interesting to note that ABAQUS was simply unable to achieve a convergent solution for this problem; a fact may help to put the AINDA results into context. It is noted here that in the case of the ADINA two-point buckling formulation (known as the secant formulation in the ADINA

Linearized Eigenvalue Buckling Results:(MITC4 shells) LDC solution = 984.9k

Pbaseline Pcr Method

0.7539k failure Classical

3.77k 975.6k Classical37.7k failure Classical

75.4k 975.6 Classical0.7539k failure Secant3.77k 974.55 Secant37.7k failure Secant75.4k failure Secant

literature) it is not possible to know what the load level of Pcharacteristic is since the information contained in the output file and portal file are incomplete in this regard. This is considered to be a critical shortcoming in ADINA in terms of reliance on the so-called secant formulation for finite element buckling analysis. As will be seen subsequently, it is not possible to employ the secant method intelligently when using ADINA since insufficient information, regarding the underlying solution process, is available to the user, and thus critical judgment related to the specification of Pbaseline, or even simply the interpretation of results, is severely compromised by this lack of information. Limit point instability The remaining cases for discussion all exhibit limit point (snap-through) instability and thus pre-buckling deformations tend to be finite. Strictly speaking, this may be viewed as a violation of the underlying assumptions used in the formulations of equations (1) and (3). Thus we might reasonably conclude that the finite element buckling analyses are technically not applicable to such instances. However, the fact remains that engineers do employ this method to cases that are not strictly in consonance with the underlying assumptions of the formulation; and thus it is important to consider this class of problems within the present discussion. In addition, it is not possible to escape the fact that, under certain circumstances, very good answers are obtained when comparing finite element buckling results with the results of more exact methods of analysis. Toggle frame Consider the toggle frame structure depicted in Figure 4. The model created to treat this case is constructed using 5 Hermitian beam elements per side of the framework.

Figure 4. Toggle frame buckling example

An incremental nonlinear finite element solution (LDC) for this problem yields a critical load of 68 kips; in addition to allowing a trace of the equilibrium path to be developed (Figure 5).

Figure 5. Equilibrium path for toggle frame example

The limit point instability response of the toggle frame case is clearly visible in the depiction of the equilibrium path, as it is not possible to resist increasing loads in the neighborhood of 0.5” in vertical displacement without developing a large motion (a jump of �2.75” in vertical deflection). We may now examine the predictive capabilities of the two finite element buckling approaches as applied to such limit point problems.

Pcr

X, u1

Y, u2 1”

b = 1”Span = 20”Height = 2”

E = 29,000 ksi

1”

Boundary Conditions:Free? u1 u2 u3

pin N N Nbrace Y Y N

20”

2”LDC Pcr = 68 kips

-50.00

0.00

50.00

100.00

150.00

200.00

250.00

300.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50

Equilibrium Trace

Vertical deflection (in.)

Ap

pli

ed

lo

ad

(k

ips)

In the subsequent discussion, it will be useful to refer to a class of diagrams known as eigenvalue plots [Brendel and Ramm 1980, Holzer et al. 1990]. Such a plot depicts a graph of Pcritical versus Pbaseline (both normalized by dividing by the exact critical load). An example of such a plot appears in Figure 6; as related to the toggle frame example described in Figures 4 and 5.

Figure 6. Eigenvalue plot – toggle frame Pcritical versus Pbaseline (normalized by exact critical load)

The depiction of the eigenvlaue plot in Figure 6 is useful to consider in the case of limit point instabilities since it highlights the dependence of the finite element buckling solution on the selection of a reasonable base loading, Pbaseline. The predicted critical loading from the finite element buckling solution is arrived at by multiplying the eigenvalues by the base loads, as described in equations (2) and (4), respectively, for the classical and secant approaches. The results presented in Figure 6 are consistent with what is expected, from the standpoint that the approximate

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1

Base Load

Classical

SecantPbase

λλλλsecantPbase

finite element buckling loads improve in accuracy as the magnitude of the base load increases. This is the case since at higher loads, the softening response of the initial loading portion of the equilibrium curve (Figure 5) is manifest, and thus, to some degree, reasonably described by the stiffness change terms present in equations (1) and (3). We may now use the eigenvalue plot in our subsequent discussions. For the case of the toggle frame structure from Figure 4, we may study the finite element buckling predictions contained in Figure 6, and also in Table 3 below.

Table 3 FEM buckling results – toggle frame

From these results, it could be argued that the classical method of finite element buckling analysis seems most appropriate since it has the greatest consistency in results (i.e. the difference between the classical Pcr and the exact value of 68 kips is uniformly smaller than what is obtained using the secant method.) Unfortunately, this will turn out to be highly problem specific. Truss Arch Considering now the similar case of a truss arch, we may study instances involving various height-to-span ratios. In all of the

Linearized Eigenvalue Buckling Results:(5 ADINA Hermitian beam elements / side)

Pbaseline Pcr Method70k NA Classical60k 74.79 Classical50k 79.89 Classical10k 92.23 Classical1k 94.38 Classical70k NA Secant60k 62.34 Secant

50k 55.44 Secant10k 36.29 Secant1k 33.57 Secant

following examples, the span length is held constant at 20” and the height is varied from 2” to 25” (see Figure 7.)

Figure 7. Truss arch buckling example

Case with 2” height The toggle frame is a particularly useful example, since it affords the opportunity to easily obtain results using “hand” calculations. In the case of the truss arch from Figure 7, with a height of 2”, an energy formulation involving the stationarity of the total potential yields a critical load of 85.4 kips. This agrees well with the results from a nonlinear incremental finite element solution using ADINA (85.8 kips). These results can be considered as exact when comparing the results from various methods obtained within the computational framework of various software packages. In the case of the truss arch with 2” height, finite element buckling results were obtained using MASTAN2, ANSYS and ADINA. In the case of the first two pieces of software, only a very small axial force is considered in the formulation of the tangent stiffness matrix used in the classical approach from Equation (1). These results, along with a “hand” calculation meant to parallel the classical formulation, as presented in Equations (1) and (2), appear in Table 4 and Figure 8 below.

Pcr

X, u1

Y, u2 1”

b = 1”Span = 20”

Height = 2”

E = 29,000 ksi

1”

Boundary Conditions:Free? u1 u2 u3

pin N N Nbrace Y Y N

20”

height

Figure 8. Eigenvalue plot – truss arch (ht. = 2”) Pcritical versus Pbaseline (normalized by exact critical load)

Table 4 FEM buckling results – truss arch

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

Base LoadADINA - classicalADINA - secantHand CalculationMASTAN2ANSYS

Linearized Eigenvalue Buckling Results:(1 truss element per side)

Pbaseline Pcr Method100k NA Classical

85k 108.63 Classical

80k 151.44 Classical70k 204.89 Classical50k 287 Classical

10k 423.5 Classical1k 451.9 Classical

100k NA Classical (hand)85k 270 Classical (hand)80k 303 Classical (hand)

70k 335 Classical (hand)50k 375 Classical (hand)

10k 423.5 Classical (hand)1k 451.9 Classical (hand)100k NA Secant

85k 92.99 Secant

80k 104.08 Secant70k 115.29 Secant50k 129.4 Secant

10k 147.9 Secant1k 151.3 Secant

From Figure 8 it is clear that MASTAN2, ADINA, ANSYS, and the hand calculation all agree reasonably well for small values of Pbaseline. However, as Pbaseline grows, MASTAN2 and ANSYS remain constant in their predictions since their implementation of the classical method does not admit the possibility of a varying Pbaseline. In addition, while the hand calculations and ADINA both permit a variation in Pbaseline, the agreement is less favorable than at low values of Pbaseline. This may be as a result of subtle difference in the way the classical formulation is implemented in ADINA; but as an unfortunate by product of a lack of inclusion (within the .out files of ADINA) of intermediate values in the solution process, it very difficult to test any theories aimed at understanding the nature of the differences. Case with 17” height In the case of the 17” high truss arch, we see a difference in the trending of the response observed in the eigenvalue plots appearing in Figure 9.

Figure 9. Eigenvalue plot – truss arch (ht. = 17”) Pcritical versus Pbaseline (normalized by exact critical load)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.2 0.4 0.6 0.8 1

Base Load

ADINA - classicalADINA - secant

Hand CalculationMASTAN2

While the MASTAN2 and hand calculations agree well with each other at low loads, and produce a reasonable estimate for the critical load, the same is not true for the ADINA results. The ADINA secant results are clearly diverging from the correct solution (13,000 kips – obtained from a incremental nonlinear finite element analysis) while the ADINA classical result begin at a point inexplicably far away from the other two classical implementations. Case with 25” height Adding a small amount to the truss arch height allows for the illustration of several important points within the context of finite element buckling analysis. This first point to consider is that since the truss arch is so deep, the classical nonlinear snap-through equilibrium path actually bifurcates into a sway mode prior to the attainment of the limit load. Indeed when tracing the nonlinear equilibrium path in ADINA using an incremental nonlinear finite element solution approach, a negative eigenvalue appears in the global system stiffness matrix at a load of 12,313 kips while the limit point is only attained at a load level of 19,488 kips. Indeed, this bifurcation point, corresponding to the sway mode, can be readily observed if a tiny (fraction of an inch) imperfection in the horizontal position of the loaded joint is used as an initial imperfection in the structure. The truss-arch response for this case appears in Figure 10.

Figure 10. Equilibrium path for toggle frame example (load versus vertical displacement)

Considering now the eigenvalue plot depicted in Figure 11, we observe several interesting points:

Figure 11. Eigenvalue plot – truss arch (ht. = 25”) Pcritical versus Pbaseline (normalized by exact critical load)

-20000

-15000

-10000

-5000

0

5000

10000

15000

20000

0 5 10 15 20 25 30 35

LDC-equilibrium trace

LDC - imperfection trace

1st Negative eigenvaluein LDC and bucklingsolution result

Asymmetrical mode

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Base Load

ADINA - classicalADINA - secant

Hand CalculationMASTAN2

1) The finite element buckling solutions are tending to converge to a load level consistent with the equilibrium bifurcation associated with the sway mode (i.e. not the limit load) 2) In the three earlier limit point stability analyses employing finite element buckling, the ADINA classical method was always predicting loads greater than those predicted using the secant formulation. For the case of height = 25”, this trend has been reversed. 3) The results from MASTAN2 and the hand calculations agree less well with the bifurcation load for this particular truss arch geometry. In contrast, ADINA does quite a bit better in this regard than was the case when the truss arch height was only moderately different (at 17”). CONCLUSIONS Finite element buckling analysis results should be interpreted with great care. The results of ostensibly identical formulations (as described in theory manuals, etc.) within various software packages, frequently lead to estimates of critical loads that vary significantly for identical structural configurations. It seems reasonable to avoid using such finite element buckling approaches for all but the simplest cases of bifurcation buckling; but even then care must be taken when considering the validity of the results. Perhaps, given the ever increasing speed and core memory size of desktop computers, stability analyses should be undertaken within the context of incremental nonlinear finite element solutions carried out using commercial software such as ABAQUS and ADINA. However, even then, the results should be interpreted with care, and by an analyst who is well versed in the theoretic foundations of the method.

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th Ed., John Wiley &

Sons. Brendel, B., Ramm, E. (1980) “Linear and nonlinear stability analysis of cylindrical shells,” Computers and Structures, Vol. 12, Pergamon Press, Great Britain, pp.549-558. Chang, S.C., Chen, J.J. (1986) “Effectiveness of linear bifurcation analysis for predicting the nonlinear stability limits of structures,” International Journal for Numerical Methods in Engineering, Vol. 23, John Wiley & Sons, Ltd., pp. 831-846. Cook, R.D., Malkus, D.S., Plesha, M.E., Witt, R.J. (2002) Concepts and Applications of Finite Element Analysis, 4

th Ed.,

John Wiley & Sons. Gallagher, R.H., Gellatly, R.A., Padlog, J., Mallett, R.H. (1967) “A discrete element procedure for thin-shell instability analysis,” AIAA Journal, 5(1), pp.138-145. Holzer, S.M., Davalos, J.F., Huang, C.Y. (1990) “A review of finite element stability investigations of spatial wood structures,” Bulletin of the International Association for Shell and Spatial Structures, 31(2), pp. 161-171 Pecknold, D.A., Ghaboussi, J., Healey, T.J. (1985) “Snap-through and bifurcation in a simple structure,” Journal of

Engineering Mechanics, American Society of Civil Engineers, Reston, Virginia, pp.909-922. Reddy, B.D. (1991) Introductory Functional Analysis with Applications to boundary Value Problems and Finite Elements, Springer, Berlin, Germany. Wood, R.D., Schrefler, B. (1978) “Geometrically non-linear analysis – A correlation of finite element notations,” International Journal for Numercial Methods in Engineering, Vol. 12, John Wiley & Sons, Ltd., pp. 635-642