PLASTICITY CORRECTION FACTORS FOR BUCKLING OF FLAT

ICAS 2002 CONGRESS

Copyright © 2002 by Delft University of Technology. Published by the 482.1International Council of the Aeronautical Sciences, with permission.

AbstractGlare is a Fibre Metal Laminate that consists ofalternate aluminium sheets and glass fibre pre-impregnated layers. For higher stress levels, thealuminium sheets will become plastic. Forrelatively thick Glare plates this also impliesthat buckling will take place non-elastically.

In this paper, a methodology is proposed tocalculate plasticity reduction factors for flatGlare plates either loaded in compression orshear. This method uses J2-deformation theoryto describe the plastic behaviour of the metallaminae in combination with a Ramberg-Osgood stress-strain relation.

A number of non-linear FEM calculationsis made on imperfect plates using J2-flow theoryfor the metal laminae. The effect of theimperfection on the buckling load is studied anda comparison is made between the reductionfactor calculated both ways.

Possible set-ups for an experimentalverification of the theory are also discussed.

Notation

�ij ��tensorial strains�ij ��tensorial stresses� ��engineering shear stress = �12� ��engineering shear strain = 2�12

E ��Young’s modulusEs ��secant modulus

Et � tangent modulusG � shear modulusGs ��shear secant modulust � plate thicknessa � plate lengthb � plate width�c ��plasticity correction factor for

compression�s ��plasticity correction factor for shear� ��elastic Poisson’s ratio�s ��elasto�plastic Poisson’s ratio�cr ��critical compressive buckling stess�y ��0.2% yield stress of aluminium�M ��Von Mises stress

[= �(�112 + �22

2 - �11�22 + 3�122)]

�cr ��critical shear buckling stress[Aij] ��laminate in-plane stiffness matrix[Dij]��laminate bending stiffness matrix[Cij] ��flexibility matrix of metal sheet[Sij] ��stiffness matrix of metal sheet

1 IntroductionGlare is a fibre metal laminate that possessesexcellent fatigue characteristic combined with arelatively low density. For these reasons,amongst others, it is a candidate material forfuselage skins in future generation Ultra-HighCapacity Aircraft (UHCA) [1]. For aircraft ofthis size, the skin will be relatively thick, so that

PLASTICITY CORRECTION FACTORS FOR BUCKLINGOF FLAT RECTANGULAR GLARE PLATES LOADED IN

COMPRESSION OR SHEART. C. Wittenberg

Faculty of Aerospace Engineering, Delft University of TechnologyKluyverweg 3, 2629 HS Delft, The Netherlands

A. de JongeFokker Aerostructures B.V.

PO Box 1, 3350 AA Papendrecht, The Netherlands

Keywords: Glare, plastic buckling, J2-deformation/flow theory, flat plates

T. C. Wittenberg and A. de Jonge

482.2

skin buckling is prone to happen in the plasticregion. It can be shown that Glare is moreefficient (in terms of weight) than aluminium inpreventing elastic plate buckling [2].

In the case of plastic buckling, it is commondesign practice to apply a “plasticity correctionfactor” to the critical buckling load calculatedwith elastic material properties. For purealuminium such factors exist for compression,and have been validated by experimentsperformed in the nineteen-forties and fifties bythe NACA [3]. The correction factor � isdefined as the ratio of the plastic buckling loadand the buckling load based on pure elasticbehaviour.

For compression loading, the followingplasticity correction factor is generally used [4]:

��

�

�

��

�

��

�

�

�

��

�

�

�

�

s

t

s

sc E

EE

E43

41

2

2

111

2 �

�� (1)

For inelastic shear buckling, Gerard [5]suggested the use of the shear secant modulus,or

GGs

s �� (2)

This approach was shown to be in goodagreement with test results of aluminium plates.

The plasticity correction factor given by Eq. (1)for an aluminium plate loaded in compressioncan be derived analytically using the so-calledJ2-deformation theory of plasticity (see, forexample, Ref. [6]). Predictions based on themore common J2-flow theory of plasticity leadto unrealistically high predictions for the plasticbifurcation-buckling load. In the past, this haslead to a lot of discussion because there is nosound physical basis for J2-deformation theory,which is essentially a non-linear elastic theory,whereas there is a good physical justification forJ2-flow theory. In the literature, this has lead toa debate as to the cause of this paradox. Someauthors argued that small initial imperfectionson the geometry would lower the buckling loadpredicted by J2-flow theory. Other people

proposed an alternative flow theory of plasticity,which can develop a corner on the yield surface.This theory leads to the same predictions asdeformation theory for bifurcation problems [7].The question whether the buckling value is dueto (imperfections on) the geometry or due to thematerial, remains up till now unresolved.However, in his famous paper [7], Hutchinson,concluded that for engineering purposes J2-deformation theory should be the preferredplasticity theory for bifurcation problems.

In the present paper a methodology is presentedto calculate the plasticity correction factors forbuckling of flat rectangular Glare platessubjected to compressive or shearing loads.Results are presented for two types of Glareplates and three values of the Ramberg-Osgoodstrain hardening parameter. Comparisons willalso be made with FEM calculations ofimperfect plates for which J2-flow theory isused for the aluminium layers. In these analysesthe influence of the size of the imperfection onthe plastic buckling load is investigated.

For both compression and shear loading,buckling analysis is performed using the samegeneral partial differential equation that governsequilibrium of the plate in the buckled state.This equation involves elements of the flexuralstiffness matrix of the plate, which arecalculated by using elasto-plastic moduli of thealuminium layers. The plasticity reductionfactors do not depend on the size of thespecimens, and therefore the method is suitableto generate design curves that depict theplasticity correction factor versus the bucklingstress, as commonly used in the aircraftindustry.

Although not impossible, experiments todetermine the plasticity reduction factors arevery complicated because of the difficulty toimplement totally simply supported or fullyclamped boundary conditions. It will thereforenot come as a surprise that the proposed methodhas not been experimentally verified yet. In thispaper possible set-ups for tests will bediscussed.

482.3

PLASTICITY CORRECTION FACTORS FOR FLAT RECTANGULARGLARE PLATES LOADED IN COMPRESSION OR SHEAR

2 The Glare materialGlare consists of alternate aluminium sheets andunidirectional high-strength glass fibre layerspre-impregnated with adhesive (see Ref. [1])The most common type of aluminium applied inGlare is 2024-T3 alloy. Each glass prepreg layeris composed of a certain number ofunidirectional (UD) plies, which are stackedeither unidirectionally, or, most commonly, in across-ply arrangement. The number,orientations, and the stacking sequence of theUD plies in the prepreg layer depend on theGlare grade. For example, a Glare 2 has twoUD plies in a single prepreg layer with the same0-degree orientation, while a Glare 3 has twoperpendicular UD plies. For a Glare 3 with aneven number of glass layers, the plies are placedsuch that the overall laminate is symmetric. Thethickness of each UD ply is 0.125 mm, so thatthe prepreg layers of both Glare 2 and 3 have atotal thickness of 0.25 mm. Note that the outerlayers in Glare are always aluminium, so thatthe number of glass fibre layers is always oneless.

A general Glare configuration is represented as:Glare grade-Nal/Ngl-tal

wheregrade = glare grade (currently 1, 2, 3, 4 or 5)Nal = number of aluminium layersNgl = number of glass fibre layerstal = aluminium sheet thickness

Two different Glare 3 types are considered forthe compression and shear case. These are givenin Table 1 and Table 2, respectively. Thealuminium and glass layer properties are givenin Table 3, and were taken from the MIL

handbooks [8, 9]. Note that in our analysishere, the prepreg properties of a 0/90 degreecombination are considered as one layer.Furthermore, the anisotropy of the rolledaluminium sheet is not taken into account.

3 Calculation methodFor Glare plates loaded beyond the yieldstrength, the aluminium layers become plastic,but the glass layers remain elastic. Thebehaviour of a Glare plate loaded in the plasticregion therefore essentially differs from amonolithic aluminium plate.

The proposed calculation method employsclassical laminate theory (CLT) to calculate thein-plane stiffnesses, Aij, and the out-of-planebending rigidities, Dij (according to thedefinition in Ref. [10]). Elasto-plastic moduliare used for the aluminium layers incombination with a Ramberg-Osgood (RO) fitfor the stress-strain behaviour.

Fig. 1. A typical Glare 3-3/2 laminate.

Table 1. Glare type used for the compression case.

Table 2. Glare type used for the shear case.

Table 3: Material properties used

Name Glare 3-5/4-0.4Total thickness 3.0 mmAluminium 2024-T3

Name Glare 3-7/6-0.4Total thickness 4.3 mmAluminium 2024-T3

Prepreg (0/90) 2024-T3

G4750 27218 MPa

� 0.144 0.33E11 30753 72400 MPaE22 30753 72400 MPa�y - 290 MPa


482.4

3.1 Buckling analysisFor compression, the critical buckling stress,�cr, for a simply supported, orthotropic, longplate is calculated with the following formula[2]:

� �331222112

2

22 DDDDtbcr ��

� (3)

Note that the coefficients Dij depend on thestress; suffix 1 refers to the long axes.

Although no closed-form solution existsfor shear buckling, the following exact solutioncan be derived for an infinitely long, orthotropicplate [10]

2

4 322114

tbDD

cr �� (4)

which is only valid if the condition

� � 12 233122211 �� DDDD� (5)

is satisfied. The coefficient � appearing in Eq.(4) is given in Ref. [10] in tabulated form. Forthe Glare type used in this paper (Glare 3-5/4-0.4), it can be shown that 1 � � � 3, in whichrange the tabulated data can be fitted with thefollowing curve:

06.1765.476.0 2�� (6)

3.2 Ramberg-Osgood relationsThe elasto-plastic stress-strain behaviour of analuminium layer will be described by aRamberg-Osgood representation of thefollowing type:

n

y

MyM

EE ��

�

�

��

�

��

�

��

�

�

� (7)

y

E�

�

002.0� (8)

In fact, the effective strain, �, is not explicitlyused. Instead, the secant modulus, Es, and thetangent modulus, Et are used:

1

1�

��

�

�

��

�

��

� n

y

M

sEE

�

�

�

(9)

1

1�

��

�

�

��

�

��

� n

y

M

t

n

EE

�

�

�

(10)

For a state of pure shear, for which �M = ��3,the axial stress-strain curve represented by Eq.(7) can be transformed to the relation

n

y

y

GG ��

�

�

��

�

��

�

��

��

33

(11)

which now obviously represents the shearstress-strain curve. The shear secant and tangentmodulus can now conveniently be obtainedfrom this relation, analogous to Eqs. (9) and(10).

3.3 Stiffnesses of aluminium for compressionTo calculate the membrane stiffnesses, Aij, ofthe Glare plate, J2-flow theory is employed forthe (ply) stiffness moduli of the aluminiumsheets. For the bending rigidities, Dij, one theother hand, J2-deformation theory is used. Inthis way, it is guaranteed that the stresses andstrains prior to bifurcation are in accordancewith the well-verified flow theory, but thebifurcation load is determined from deformationtheory. Note that when using J2-deformationtheory for both in-plane and bending behaviour,the results change only marginally.

Essential for the behaviour of thealuminium sheets in Glare is that the loading isbiaxial, even in case where the Glare plate issubjected to uniaxial compression. This makesdefinition of the stiffness moduli a little bitmore complex. In fact, it is more appropriate towrite down the components of the compliancematrix, Cij. The inverse of this matrix is thestiffness matrix. The flexibility matrix for J2-deformation theory is defined as follows [11]:

482.5


��

�

�

��

�

�

��

�

�

��

�

�

�

��

�

�

��

�

�

12

22

11

33

2212

1211

12

22

11

2 �

�

�

�

�

�

�

�

�

�

�

�

CCCCC

(12)

with

��

��

��

�

��

�� 2

221143

21

1211

Msts

s

EEEC

�

��

��

�

�

��

�

��

�

��

�� 2

211

43

22 1111

Msts EEEC

�

�

��

�

�

��

�

��

�

��

�� 2

222

43

11 1111

Msts EEEC

�

�

� �

s

s

EC

��

�

1233 (13a, b, c, d)

where

��

�

��

��

EEEE ss

s 1121��

(14)

For J2-flow theory, the same formulas can beused with the following substitutions:

��

�

s

s EE(15)

Note that proposed moduli are correct in thefinite strain domain as well, if the strain isinterpreted as the logarithmic strain and thestress as the Kirchoff stress tensor.

3.4 Stiffnesses of aluminium for shearFor this case, there is only one stresscomponent, and, therefore, both plasticitytheories predict the same pre-bifurcation path;hence, only J2-deformation theory needs to beconsidered. The stiffness matrix can be obtainedfrom Ref. [12] for example, and is given hereexplicitly as

��

�

�

��

�

�

��

�

�

��

�

�

�

��

�

�

��

�

�

12

22

11

33

2212

1211

12

22

11

2��

�

�

�

�

�

�

�

�

�

�

SSSSS

(16)

with

22211 1 s

sESS

��

��

212 1 s

ss ES�

�

�

�

� � � � ��

��

�

��

��

�

3211

11233

sts

ts

s

s

EEEEE

S��

(17a, b, c)

Note that the shear stress does not appear in thisformula, because it is implicitly included in thesecant modulus. For J2-flow theory, thestiffnesses can again be obtained using Eq. (15).However, it can be shown that the S33-stiffnessterm is the same for flow and deformationtheory, which implies that the in-plane responseis equal for both theories. The bucklingbehaviour, on the other hand, is different, sincethe other stiffness terms (i.e. S11, S22, and S12)play a role in the calculation of the bendingstiffness matrix.

3.5 Solution schemeThe problem in establishing the correct bucklingstress is that in Eqs. (3) and (4) the coefficientsDij depend on the stress level. The equationtherefore has to be solved in an iterativemanner. The solution approach that is utilised issimilar to the incremental approach used in mostnon-linear FEM-packages. Small incrementalsteps are taken along the stress-strain path andfor each point a bifurcation check is performed.In a FEM solution, such a bifurcation wouldappear as a negative eigenvalue in the tangentstiffness matrix.

The following iterative solution procedureis used:


482.6

1. Choose a step size for the applied stress.(Typical stress level of a thousandth of yieldstress).

2. Calculate laminate membrane stiffnesses(A-matrix) using J2-flow theory incrementalmoduli for metal laminae. (Note that there isno distinction between flow and deformationtheory for pure shear loading.)

3. Calculate incremental strains by multiplyinginverse of A-matrix with applied load vectorand calculate actual strains.

4. Calculate metal layer incremental stressesfrom the incremental strains by means of theJ2-flow theory stiffness moduli for the metallaminae and calculate actual laminaestresses.

5. Calculate D-matrix for laminate using J2-deformation theory incremental moduli.

6. Calculate buckling stress for this laminatewith Eqs. (3) or (4).

7. Check whether buckling stress is higher orlower than applied stress� if buckling stress is lower increase load

and continue at step 2,� else, if buckling load is higher, solution

is found.

In principle, the proposed algorithm (apart fromthe linear buckling analysis in step 6) is validfor biaxial loading, but in this paper only resultsfor uniaxial loading will be presented. Note thatit may easily be shown that for an aluminiumplate the proposed solution leads exactly to theclassical reduction factor of Eq. (1) if �s = 0.5 istaken.

3.6 Results for compressionFor compression, three calculations have beenmade using above given solution scheme forthree typical values of the RO-hardeningparameter n (7.5, 10, 15). The results of thesecalculations are given in Fig. 2, together with acurve for solid aluminium.

At first glance, the three curves for Glareare very similar to the curves for solidaluminium. Two major differences are striking.The first influence of plasticity occurs forbuckling stresses in the order of 0.5 times thealuminium yield stress. It is clear that this stress

is lower than for pure aluminium, because thestiffness of the glass is much lower than thestiffness of the aluminium. The actual stressesin the aluminium laminae are, therefore, higherthan the average stress in the laminate. Thesecond major difference is the behaviour forhigh buckling stresses. For aluminium, the

Fig. 2. Plasticity correction factor versus normalisedbuckling stress for compression buckling for three valuesof RO-parameter.

Fig. 3. Comparison of plasticity correction factor based onuniaxial stress-strain curve and analysis proposed in thispaper.

Fig. 4. Relative difference between solution based onuniaxal stress-strain curve and as proposed in this paper

00.10.20.30.40.50.60.70.80.9

1

0 0.5 1 1.5 2buckling stress / aluminium yield stress

plas

ticity

cor

rect

ion

fact

or

Glare (CLT)

Glare(uniaxial)

Aluminium

n = 10

0%

10%

20%

30%

40%

50%

60%

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0buckling stress / aluminium yield stress

diffe

renc

e in

redu

ctio

n fa

ctor

s

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

buckling stress / aluminium yield stress

plas

ticity

cor

rect

ion

fact

or

n=7.5n=10n=15aluminium; n=10

482.7


curve goes asymptotically to zero, whereas forGlare, the values remain finite. Of course, this isdue to the fibres that remain elastic.

An alternative calculation was made using theclassical isotropic formula for the plasticityreduction [Eq. (1)] in combination with theuniaxial stress-strain curve of Glare todetermine the tangent and secant moduli.Results of this calculation are given in Fig. 3.The curve obtained with the CLT-based analysisdescribed in this paper is below the curveobtained using the alternative solution method.The reason for this behaviour is that themembrane stiffness does not decrease as fast asthe bending stiffness for a reduced stiffness ofthe aluminium due to plasticity. This is due tothe lay-up of Glare where at the outside alwaysaluminium layers are placed. The relativedifference between the two methods isdisplayed in Fig. 4. For stress levels above 0.8times yield of aluminium the difference betweenthese methods rapidly increases. For lowerstress levels, the stiff aluminium is governingand the Glare laminate is still behavingisotropically, but for higher stress levels thecontribution of the Glass fibres becomesimportant.

It may be concluded that any simplifying modelintended to describe the plasticity reduction of afibre metal laminate over the whole range, hasto contain a parameter that expresses the metalcontribution to the bending stiffness.

As it is, the metal contribution to thebending stiffness is different for each Glareconfiguration. Curves as in Fig. 2 are therefore,in principle, only valid for one Glareconfiguration. In practice, however, thedifference between two Glare configurations ofthe same grade (Glare 2, 3 or 4) is only verysmall, and certainly much smaller than theinfluence of the RO-parameter, n.

3.7 Results for shearSimilar calculations as discussed in the previoussection were made for shear-loaded panels, andthe results are presented in Fig. 5. The curveslook very similar to the ones for compressed

plates (see Fig. 2), but, looking at the purealuminium plate, the effect of plasticity starts atlower loads. This is due to the lower value ofshear yield stress—in fact, �y = �y/�3 from theVon Mises relationship—so that inelastic effectsare more pronounced compared to thecompression case. (Note that the buckling stressis normalised with the axial yield stress in Fig.5.) Furthermore, it may be noted from Fig. 2and Fig. 5 that the difference in inelasticbuckling between Glare and pure aluminiumplates is bigger for shear buckling. This isexplained by the fact that, in shear loading, thefibres do not contribute to the shear stiffness ofthe Glare plate, which is due to the cross-plyarrangement of the fibres. This means that theshear load is almost entirely carried by thealuminium sheets, except for a smallcontribution from the resin material in theprepreg layers.

Fig. 5. Plasticity correction factor versus normalisedbuckling stress for shear buckling for three values of RO-parameter.

Fig. 6. Comparison of plasticity reduction factor based on“uniaxial” shear stress-strain curve and CLT-basedanalysis.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.25 0.5 0.75 1

buckling stress / aluminium yield stress

plas

ticity

cor

rect

ion

fact

or n=7.5n=10n=15aluminium; n=10

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1buckling stress / aluminium yield stress

plas

ticity

cor

rect

ion

fact

or uniaxial (secant)

CLT (secant)

CLT (deformation)

n = 10


482.8

As an alternative, one could directly apply the“uniaxial” shear stress-strain curve of the Glarematerial to calculate the correction factor fromEq. (2) (and thereby neglecting the bendingeffects). The shear stress-strain curve of thealuminium layers is then represented byEq.(11), from which the response of the entirelaminate can be calculated when combined withthe (elastic) glass properties. Results of thisapproach are presented in Fig. 6 for a ROparameter of n = 10, along with the solutionfrom a CLT-based procedure where the shearsecant modulus is used for the individualaluminium layers [13].

Contrary to the compression case, we seethat the curve for the “uniaxial” approach differsonly slightly from the curve for which the effectof bending is included (i.e. the CLT-basedprocedure) over the entire range. This isexplained by the fact that the latter procedureexplicitly calculates the correction factor as theratio of the inelastic and elastic buckling stress.When in the plastic region, the stiffness moduli(i.e. E and G) of the aluminium sheets changeaccording to the same ratio, (Gs/G)al [i.e. of thealuminium layers, calculated from Eq. (11)], sothat the material remains more or less isotropic.Examining the bending stiffnesses (D11, D22,D12, and D33) of the bulk material in the plasticregion, it shows that they all change inapproximately equal ratios as well, especially inthe early/moderately-plastic range where thealuminium is still dominating. It so turns outthat the relative change in bending stiffness issimilar to the relative change in secant modulusof the entire laminate, (Gs/G)lam. Therefore,when applying Eq. (4) and assuming that the“4�-term” remains constant, a division of theplastic and elastic buckling stress gives a valuenot very different from (Gs/G)lam.

It should be noted that the procedure ofRef. [13] uses the instantaneous value ofPoisson’s ratio given by Eq. (14). When usingthe elastic value of Poisson’s ratio at all stresslevels, the results of the CLT-based procedureand the “uniaxial G/Gs” method almost overlap.In reality, the change in bending stiffnessincreases over the change in secant moduluswhen advancing in the plastic region, but the

“4�-term” in Eq. (4) actually decreases;therefore the laminated plate procedureproduces a similar plasticity correction factor asEq. (2).

Also shown in Fig. 6 is a curve corresponding tothe J2-deformation theory approach proposed inthe present work, and it can be seen that thismethod is more conservative than thoseemploying the shear secant modulus. In thehighly plastic region the results tend to convergeto the same value for the correction factor,which should theoretically be the case since atthese stress-levels the response is predominantlygoverned by the elastic fibre layers.

4 FEM analysisTo compare the plastic buckling results from theJ2-deformation theory, a number of calculationswere made with imperfect plates on the basis ofJ2-flow theory. For this purpose the STAGS[14] FEM code, which utilises animplementation of J2-flow theory according tothe White-Besseling model, was used. Since themagnitude of the initial imperfection greatlyaffects the plate response, this particularphenomenon has been investigated as well.

4.1 Analysis procedureThe first step performed was always a linearbifurcation buckling analysis, in order to obtainthe elastic critical load. This load was then usedin the non-linear analysis to normalise theapplied increments along the equilibrium path.The subsequent non-linear analyses can bedivided into the following steps:

1. The critical bifurcation buckling load of theperfect plate is calculated following theelasto-plastic pre-buckling path. (Thisessentially yields bifurcation results on thebasis of J2-flow theory). The main purposeis to determine the inelastic buckling mode,which may be different from the elastic one.

2. The buckling mode corresponding to theelasto-plastic critical load is imposed uponthe plate geometry, serving as an initialimperfection. Non-linear plate response is

482.9


then calculated for different amplitudes ofthe imperfection.

The buckling analyses represented by Eqs. (3)and (4) are theoretically only valid for infinitelylong plates. The plates used in the FEMcalculations were taken with an aspect ratio ofa/b = 5, which can be considered “sufficientlylong”; for longer plates, the buckling load isvirtually independent of the length. Moreover,the length effect is minimised by the fact thatthe plasticity correction factor is merelycalculated as a ratio of two buckling loads—theelastic and inelastic one—which both containthe length effect. The FEM calculations weremade with a RO-parameter of n = 10.

As an example, the FEM analysis of aGlare plate with a length of a = 675 mm andwidth of b = 125 mm will be described in detail,for both the compression (with Glare 3-7/6-0.4)and shear (with Glare 3-5/4-0.4) case. The platewas modelled with 50 elements in the longdirection and 10 elements in the transversedirection. The STAGS 410 shell element wasused here. All four edges were provided withsimple-support conditions (i.e. rotationallyunrestrained), while the in-plane displacementswere unrestrained. Note that the in-planeconditions at the edges are unimportant in abifurcation analysis, but do have a greatinfluence on the post-buckling behaviour of theplate.

4.2 Compression loadingThe linear buckling analysis gave a criticalcompressive stress of �cr = 253.8 MPa, whereasthe analytical solution of Eq. (3) yields a valueof �cr = 252.5 MPa, from which it is evidentthat the FEM result is very accurate. The criticalbuckling mode has five half-waves (see Fig. 7a),which is also according to theory. Thebifurcation analysis using the elasto-plastic pre-buckling path gave a buckling load at 89% ofthe elastic solution. This means that theplasticity correction factor according to J2-flowtheory is �c = 0.89. The buckling pattern nowhas six half-waves—as can be seen in Fig. 7b—which is the result of a “change in anisotropy”during plastic deformation. The value of the

correction factor for the calculation procedurediscussed in this paper (i.e. based ondeformation theory) is �c = 0.83, which is about7% lower than the value for flow theory.

The inelastic buckling pattern of six half-waves was used as an initial deviation fromflatness of the plate, w0, for the subsequent non-linear analyses. Different amplitudes of thisimperfection were considered; results for valuesof w0/t = 0.0001, 0.001, 0.01, and 0.1 are plottedin Fig. 8, where the applied load is normalisedwith the elastic buckling load. The results show

Fig. 7. Elastic and inelastic buckling modes of theexample plate loaded in compression.

Fig. 8. Non-linear response curves for differentimperfection magnitudes of the example plate loaded incompression.

Fig. 9. Comparison of deformation theory and flow theoryapplied to compressed plates with different imperfectionsizes.

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8maximum normalised deflection, (w-w0)/t

norm

alis

ed lo

ad

w0/t = 0.0001w0/t = 0.001w0/t = 0.01w0/t = 0.1

0.00

0.20

0.40

0.60

0.80

1.00

0 200 400 600 800elastic buckling stress [MPa]

plas

ticity

cor

rect

ion

fact

or

Deformationtheoryw0/t = 0.001

w0/t = 0.01

(a) elastic (b) inelastic


482.10

that the imperfect plates exhibit limit pointbuckling behaviour (i.e. a maximum in the load-deflection curve), and that the magnitude of thislimit point progressively decreases for growinginitial imperfections. As the amplitude of theimperfection approaches zero, the value of thelimit point moves towards the bifurcation loadof the perfect plate for flow theory (previouslycalculated as 0.89). Taking the limit point as thecritical buckling load, the correction factor fordeformation theory (�c = 0.83) lies between theresults for imperfection sizes w0/t = 0.001 (�c =0.84) and 0.01 (�c = 0.81).

Results for Glare 3-7/6-0.4 plates ofdifferent widths are shown in Fig. 9 for twosmall imperfection sizes, w0/t = 0.001 and 0.01.It can be seen that the buckling loads forimperfections of w0/t = 0.01 agree generallywell with the critical bifurcation loads of thedeformation theory, but advancing in the plasticregion, the deviation of the two differentanalysis methods increase.

4.3 Shear loadingThe linear shear buckling stress of the Glare 3-5/4-0.4 example plate—with a = 675 mm and b= 125 mm—calculated with STAGS was foundto be �cr = 181.4 MPa. For a very long plate (i.e.a � �, b = 125 mm), Eq. (4) yields a result of�cr = 173.0 MPa, which is about 5% differentfrom the FEM result. This discrepancy is higherthan for the compression case, due to the highercomplexity (e.g. skewedness) of the shearbuckling deformation pattern, which is shown inFig. 10. To reach the same accuracy as forcompression, the shear model requires a (much)finer mesh density, which dramaticallyincreases the computing time, in particular for anon-linear analysis. However, since theplasticity correction is calculated as the ratio oftwo buckling loads (which containapproximately the same error), this inaccuracyof 5% is irrelevant. The elasto-plasticbifurcation load on the basis of flow theory wascalculated as 0.81 times the elastic one, so that�s = 0.81. This value is considerably higher than�s = 0.69, which is the result from the procedurebased on deformation theory discussed in this

paper. The plastic buckling mode is the same asfor the elastic case (shown in Fig. 10).

Analogous to the compressed plate (seeFig. 8), Fig. 11 presents load-deflection curvesfor the shear case for several amplitudes of themode shape shown in Fig. 10, which is used asan initial imperfection in the non-linearanalyses. The trends are the same as for thecompression case, but it can be noted from thehigher deflections that the plate loaded in shearwas analysed further into the post-bucklingregion. It should be noted that in all cases—

Fig. 10. Critical buckling mode of the example plateloaded in shear (same for elastic and inelastic case).

Fig. 11. Non-linear response curves for differentimperfection magnitudes of the example plate loaded inshear.

Fig. 12. Comparison of deformation theory and flowtheory applied to plates loaded in shear with differentimperfection sizes.

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4

maximum normalised deflection, (w-w0)/t

norm

alis

ed lo

ad

w0/t = 0.0001w0/t = 0.001w0/t = 0.01w0/t = 0.1

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800elastic buckling stress [MPa]

plas

ticity

cor

rect

ion

fact

or

Deformationtheoryw0/t = 0.001

w0/t = 0.01

Secant modulus

482.11


including the compressed plates—the non-linearanalysis was terminated by STAGS due toexcessive plastic strains, which resulted in asingular stiffness matrix. In this case, the resultfor deformation theory (bifurcation buckling,perfect plate), �s = 0.69, is bounded by the flowtheory results (limit point buckling, imperfectplate) for imperfection sizes w0/t = 0.01 (�s =0.74) and 0.1 (�s = 0.64). Employing the shearsecant modulus method of Ref. [13] (whichcorresponds to the “CLT-curve” in Fig. 6) aresult of �s = 0.73 is found, which is in betteragreement with the flow theory predictions ofw0/t = 0.01 and 0.001 (�s = 0.76).

Fig. 12 presents a comparison of bucklingloads calculated with deformation theoryanalysis and STAGS results on the basis of flowtheory, for Glare 3-5/4-0.4 plates. Also plottedis the curve according to the shear secantmodulus method of Ref. [13]. It shows that inthe early plastic region (�s = 0.80-1, say) thereis a good agreement of the deformation theoryand flow theory for small imperfections (w0/t =0.001-0.01). For higher plastic bucklingstresses, though, the shear secant modulusmethod correlates better with the STAGS resultsfor imperfections of w0/t = 0.01, but thedifference tends to increase when advancing inthe plastic region. However, for designpurposes, use of the deformation theory seemsfavourable since it is the most conservativeapproach. It should be noted that in Fig. 12 thecorrection factors obtained from the FEManalyses are plotted against the analyticalsolution of the elastic shear buckling stress. Inthis way, the 5% error in elastic critical load iseliminated.

5 DiscussionThe inelastic buckling behaviour of flat,rectangular Glare plates looks very similar forcompression and shear loading, but theplasticity effect is more pronounced for thelatter case. This is caused by two effects: forone, the shear yield stress of aluminium is lowerthan the direct yield stress and, secondly, thefibres are not stressed when the Glare plate isloaded in pure shear, so that virtually all the

load is carried by the aluminium layers.Obviously, to achieve a better performance inshear, the fibres should be aligned at �45-degrees, in which case the fibres are fullyutilised.

The results from the CLT-based procedureproposed in the present paper were alsocompared to the “classical” approach (i.e. forfully metallic plates) employing the uniaxialstress-strain curve of the Glare plate. Forcompressed plates, the results from bothmethods agree closely for buckling stresses upto 80% of the aluminium yield stress. This canbe explained by the fact that the bendingstiffness decreases more rapidly than themembrane stiffness when the aluminiumbecomes plastic, which is due to the stackingsequence (aluminium is at the outside of thelaminate). In the advanced-plastic region, wherethe aluminium has become weak, the glass fibrelayers govern the laminate response.

For shear, the “classical” reduction factorof Eq. (2), which utilises the shear stress-straincurve of the laminated plate, gives results thatare almost equal to those from a procedurewhere the bending is explicitly included (Ref.[13]). The reason for this is that the latterprocedure explicitly calculates the plasticitycorrection factor as the ratio of two bucklingstresses. This ratio is mainly determined by thechanges in plate bending stiffnesses, which aresimilar to the change in shear modulus, Gs/G, ofthe plate. Application of the deformation theoryin conjunction with CLT gives the mostconservative plastic shear buckling stresses, andis therefore—while no experimental data isavailable for Glare plates yet—recommendedfor design purposes.

Principally, the results presented in thispaper are only valid for the specific Glare typesconsidered, since the contributions of the metaland fibre parts to the laminate stiffnesses aredifferent for each Glare configuration. Inpractice, however, the difference between Glareconfigurations of the same grade (i.e. 2, 3, or 4)is relatively limited; on the other hand, theinfluence of the strain hardening parameter, n, isvery pronounced indeed.


482.12

Comparison of STAGS FEM calculations ofplates containing initial imperfections and usingJ2-flow theory, with bifurcation buckling resultsof perfect plates on the basis of J2-deformationtheory, revealed that similar results areobtaineable for relatively small imperfections(typically 0.1 to 1 percent of the platethickness). For both the compression and shearcase the limit point was taken as the inelasticbuckling load of the imperfect plates.

The logical next step in the investigation is tocorrelate the results obtained in this paper withexperimental data, in order to verify theaccuracy of the developed calculationprocedure. A practical problem, however, is thedifficulty to implement exactly simply-supported or clamped edge conditions of theplate, which is needed for comparison with thetheoretical solutions.

For aluminium plates loaded incompression, the test method employed in Ref.[3] seems a practical one. The specimens areactually long, extruded tubes of square cross-section, where the walls are the actual platesbeing tested. Because these “plates” have all thesame dimensions and material properties, theybuckle at (approximately) the same point,thereby simulating nearly simply-supportedconditions at the long edges. (Note that forsufficiently long plates the conditions at theshort edges are not so relevant.) If this methodis to be used for Glare plates, a specialmanufacturing process needs to be devised sothat the four “plates” have exactly equalproperties, since extrusion is obviously notpossible for Glare.

For shear loading, a three-rail shear testset-up as applied by Gerard [5] can also be usedfor Glare plates. In this case, the long edges areclamped. (Again, the individual “plates” arelong so that the influence of the short-edgeconditions is limited). Alternatively, a pictureframe may be used to apply a pure shear load toa Glare plate, in which case all the plate edgesare clamped. When installing torsionally rigidstiffeners, the test plate can be divided intosmaller ones where, again, all edges are

clamped or nearly clamped. In this manner, thesame test fixture can be used to obtain variousvalues of buckling loads or, in other words, arange of plasticity correction factors.

Finally, it should be noted thatexperimental data does exist for plastic bucklingof stiffened Glare panels loaded in shear, andthat good correlation was obtained with FEMsimulations using STAGS [15]. Noting this fact,the results for imperfect plates using flow theorymay tentatively be considered for verification-purposes as the “next best thing” to actual testresults.

AcknowledgementThe research presented in this paper wassponsored by the Dutch Technology Foundation(STW).

References[1] Vlot, A and Gunnink, JW (eds.). Fibre Metal

Laminates – an introduction. Kluwer AcademicPublishers, Dordrecht (NL), 2001.

[2] Verolme, K. The development of a design tool forFiber Metal Laminate compression panels. Ph.D.thesis, Faculty of Aerospace Engineering, DelftUniversity of Technology, 1995.

[3] Pride, RA and Heimerl, GJ. Plastic Buckling ofSimply Supported Compressed Plates. NACA TN1817, National Committee for Aeronautics andAstronautics, April 1949.

[4] Gerard, G and Becker, H. Handbook of StructuralStability – Part I, Buckling of Flat Plates. NACA TN3781, National Committee for Aeronautics andAstronautics, July 1957.

[5] Gerard, G. Critical Shear Stress of Plates above theProportional Limit. Journal of Applied Mechanics,Vol. 15, No. 1, pp. 7-12, 1948.

[6] Shrivastava, SC. Inelastic buckling of platesincluding shear effects. Int. J. Solids Structures, Vol.15, pp. 567-575, 1979.

[7] Hutchinson, JW. Plastic Buckling. Advances inApplied Mechanics, Vol. 14, pp. 67-144, 1974.

[8] Anonymous. Metallic Materials and Elements forAerospace Vehicle Structures. MIL-HDBK-5H, U. S.Government Printing Office, Washington, D.C.,December 1998.

[9] Anonymous. Polymer Matrix Composites – Volume2. Material Properties. MIL-HDBK-17-2E, U. S.Government Printing Office, Washington, D.C.,January 1997.

[10] Ashton, JE and Whitney, JM. Theory of LaminatedPlates. Technomic Publishing Co., Inc., Stamford

482.13


(Conn., USA), 1970.[11] Peek, R and De Jonge, A. A plasticity theory to

predict wrinkling of imperfect pipes. Paper presentedat Offshore Mechanics and Arctic EngineeringConference 1998 (OMAE 1998).

[12] Needleman, A and Tvergaard, V. Necking ofbiaxially stretched elastic-plastic circular plates. J.Mech. Phys. Solids, Vol. 25, pp. 159-183, 1977.

[13] Wittenberg, TC and Van Baten, TJ. Plastic BucklingAnalysis of Flat Rectangular FML Plates Loaded inShear. In Proceedings of the 32nd InternationalSAMPE Technical Conference, Boston (MA., USA),pp. 673-686, November 2000.

[14] Rankin, CC, Brogan, FA, Loden, WA, and Cabiness,HD. STAGS Users Manual – Version 3.0, LockheedMartin Missiles & Space Co., Inc., December 1997.

[15] Wittenberg, TC, Van Baten, TJ and De Boer, A.Design of FML shear panels for Ultra-High CapacityAircraft. Aircraft Design, No. 4, pp. 99-113, 2001.

Documents

PLASTICITY CORRECTION FACTORS FOR BUCKLING OF FLAT