ELSEVIER

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© 1997 Elsevier Science LtdPrinted in Great Britain. All rights reserved

0263-8223/97/$17.00

PII:S0263-8223(96)00009-3

Delamination buckling of FRP layer inlaminated wood beams

Youngchan Kim,a Julio F. Davalosb* & Ever J. Barberoc

uTechnology Research Institute, DAELIM Industrial Co., RO. Box 1117, Seoul, Koreah Department of Civil and Environmental Engineering, West Virginia University, Morgantown, WV 26506-6103, USA

CDepartment ofMechanical and Aerospace Engineering, West Virginia University, Morgantown, WV 26506-6103, USA

An analytical solution for predicting delamination buckling and growth of athin fiber reinforced-plastic (FRP) layer in laminated wood beams underbending is presented. Based on a strength-of-materials approach,displacement functions for a delaminated beam under four-point bendingare derived. Using force and displacement compatibility conditions, anexplicit form relating the applied transverse load with the delaminationbuckling load is established. An explicit form of the strain-energy releaserate is presented to study the delamination growth in beams under bending.The analytical solution is evaluated using experimental data for glued­laminated timber (glulam) beams reinforced with a thin fiber-reinforcedplastic composite on the compression face. The delamination growth inbending is shown to behave differently to that of the in-plane loading case.© 1997 Elsevier Science Ltd.

\.,

NOTATION

The notation used in this paper is listed below.

Displacement at the loading pointPrescribed rotation at the end of a dela­minated sublaminate

*Author to whom correspondence should be addressed.

AibD i

GGic

hLLd

LoM i

PQt

Wi

W2max

c3c~xp

b

Axial stiffness of segment iBeam widthBending stiffness of segment iStrain-energy release rateCritical strain-energy release rateThickness of delaminated layerBeam span lengthDelamination lengthLoading span lengthMoment acting on segment iInternal axial force acting on segment 3Applied transverse loadBeam thicknessTransverse deflection for segment iDisplacement at mid-spanStrain of a delaminated layerExperimental C3

Prescribed displacement at the end of adelaminated layer

311

INTRODUCTION

Delamination in composite laminates causesseparation of an adjoining layer from the mainlaminate structure. Delamination .growth maysignificantly influence the strength, stiffness andstability of a laminate. The initiation of delami­nation results from many sources, such asmanufacturing defects, deterioration of bondingmaterials, or local impact damage. The delami­nation behavior in laminated structures hasreceived the attention of several investigators.

One of the first analytical delaminationmodels was developed by Chai [1]. He charac­terized the delamination in homogeneous,isotropic plates using a thin-film model, andextended this approach to a general bendingcase which included the bending of a thick baselaminate. Including bending-extension coup-

312 Y Kim. J. F. Davalos, E. J~ Barbero

ling, Yin [2] derived general formulae forthin-film strips and mid-plane symmetric dela­minations in composite laminates. Kardomateasand Schmueser [3] analyzed the deformation ofdelaminated composite under axial loading witha one-dimensional beam-plate model includingtransverse shear effects. Chen [4] developed ashear deformation theory for compressive dela­mination buckling and growth with a variationalenergy principle.

The nonlinear behavior of a composite dela­minated beam under axial loading was analyzedby Sheinman and Soffer [5]. Tracy and Pardoen[6] studied the effect of delamination on theflexural stiffness of laminated beams, but theiranalytical solution did not include bending­extension coupling or delamination buckling.They tested specimens manufactured with adelamination at the mid-plane, and they con­cluded that the delamination did not degrademuch the stiffness of the laminates. However, asobserved in glulam-FRP beam tests [7], if thedelamination was placed near the top surface ofa beam, delamination buckling is likely to occur.Consequently, the stiffness of the beam wouldbe affected substantially.

Reddy et ale [8] developed a generalized lami­nate plate theory (GLPT), and used this theoryto account for multiple delaminations betweenlayers.· The associated finite-element. model wasdeveloped by Barbero [9] and Barbero andReddy [10]. Based on GLPT, Lee et ale [11]developed a displacement-based, one-dimen­sional finite-element model to predict criticalloads and corresponding buckling modes for amultiply delaminated composite with arbitraryboundary conditions. Sankar [12] developed anoffset-node beam finite element to model dela­minations in composite beams. The modeling ofdelaminations in a composite beam, however,was limited to the case of two sublaminates withthe same thickness, one above and one belowthe delamination plane. The offset-node ele­ment was extended to the study of dynamicdelamination propagation in composite beams[13]. Most of the existing studies focused on theresponse of a delaminated thin laminate (thin­film delamination) which does not affect thestrength and stability of a thick-base laminate.Recently, Kutlu & Chang [14] investigated thecompressive response of laminated compositeswith multiple delaminations, including the inter­action between the delamination growth andthe response of laminates.

In a hybrid system, such as glued-laminatedtimber beams (gIuIam) reinforced with fiber­reinforced plastic (FRP) strips, differentmaterials are bonded with an adhesive and,therefore, loss of face-to-face local adhesion isthe main cause of delamination at the interfacesof two distinct layers. Reinforcing glulam beamswith FRP strips on the compression and/or ten­sion side is an efficient way to increase thestiffness and strength of laminated wood beamsand to decrease the depth of the member [7].Usually, the thickness of a FRP strip is verysmall compared to that of a glulam beam. Forbeams reinforced with FRP on the compressionface, the debonding and subsequent local buck­ling of the FRP layer can result in a prematurefailure of a glulam-FRP beam. To predict theultimate strength of glulam-FRP beams rein­forced on the compression face, it is necessaryto investigate the delamination behavior of athin compression layer in a delaminated beamunder bending.

In this paper, an analytical model to predictdelamination buckling of a FRP strip in a re­inforced laminated beam under bending is pre­sented. The investigation of the critical loadsand delamination behavior are the major con­cerns of this study. In the analytical model aninitial delamination length at the interface ofthe top layer and the base laminate is assumed.Beyond the delamination length, the rest of thetop layer remains bonded to the compressionface of a relatively thick laminate. The delami­nated beam is subjected to four-point bendingand the initial delamination is symmetric aboutthe mid-span. The displacement, rotation andthe axial force acting at the delaminated layerare computed, based on the assumed displace­ment functions which are derived usingboundary conditions and compatibility condi­tions. Also, explicit expressions for criticalbuckling load and strain-energy release rate areprovided. Using the test data of a full-size lami­nated beam, the displacement and critical loadspredicted by this study are compared withexperimental values. A parametric study iscarried out to investigate the effect of laminatestiffness on the critical loads. The strain-energyrelease rate for various initial delaminationlengths are computed to find out the trend ofdelamination growth in laminated beams underbending. Finally, findings of this study aresummarized and recommendations for futureresearch are suggested.

Delamination buckling in laminated wood beams. 313

Then, we have

From the support to the loading point~ segment1 (0 <Xl <L/2 - Lj2)

(3)

(2)

(1)

The coefficient c will be determined in Section2.2 using compatibility conditions.

As the bending moment varies linearly in thisregion, the displacement function can beassumed as

The boundary condition at the support and -con­stitutive equation are

derived next, and the notation. used is given inAppendix Bo

In this paper the entire beam is involved in thederivation of the analytical solution. Considerthe beam shown in Fig. 1 subjected to four­point symmetric loading, where one half of thebeam is modeled using a shear-release at mid­span. The beam is divided into three segments:segment 1 contains no delamination; segment 2is the base laminate, or thicker lower portion ofthe delaminated region; and segment 3 is thethin upper lamina that has undergone delami­nation. The subscripts used in the derivation ofthe theory correspond to these three segments.

The coordinate system and beam parametersare shown in Fig. 2. To simplify the problem ofinterest, we assume that ·a single symmetricdelamination is present at mid-span beforeloading. Also we assume that mode I (openingmode) fracture is the primary -mechanism. fordelamination growth, because the delaminatedsegment is in the shear-free zone, and that thethickness of a delaminated lamina is small com­pared to that of the laminated beam. Thedisplacement functions in all three segments are

DISPLACEMENT FUNCTIONS

Q .h92

®

®

L2" ... ,

Figo 10 Delaminated beam configuration.

I Segment 1

tW 1

.---~X2' x3

56 Segment 3

Segment 2

1<O~ L~o ---I~1

l

lFigo 20 Coordinate system.

314 Y Kim. J. F. Davalos, E. J. Barbero

From the loading point to the delamination tip:segment 1 (L/2-Lo/2<x1 <LI2-Ld /2)

The constitutive relation and the prescribedboundary conditions at the delamination tip are

W = (!::-. - ~) =b w' (!::-. - ~) =f}la 2 2 'lb 2 2 '

(11)

Delaminated region: segment 2 ( - L d /2 <X2 < 0)

Delaminated region: segment 3 ( -Ld /2 <X3 <0)

Assuming a constant bending moment in thisregion, the displacement function is given by

W2(X2)=hx~+ix2+ j (10)

Using the prescribed boundary conditions at thedelamination tip, we have(4)W1b(X 1) = exi+fx}+g

As the bending moment is constant in thisregion, the displacement function can be writ­ten as

(5)

Then, we obtain

The post-buckling deflection shape of a beam­column is given as (Chen & Lui [15, p. 177])

W 3(X3 ) =K 1 sin(exx3)+K2 cos(ax3 )+K3 X 3

(12)

(13)

The continuity condition of rotation at Xl =LI2-Lo/2 gives

As the deformed shape is symmetric, we haveK I =K3 = O. Using the compatibility of displace­ment b and rotation fJ at the interface betweensegments 1 and 3, eqn (12) becomes

cos(ax3)-cos f3W3(X3 ) , • _ fJ+b

ex SIn f3

where

(7) (14)

Then eqn (3) becomes

Q 3W1a(X1) =- -- Xl

6D1

The displacement functions expressed in termsof beam parameters and the prescribed bound­ary conditions are used to compute strainenergy and strain-energy release rate in a beam,as discussed next.

From the displacement continuity condition atXl = L/2 - L o/2, we have

Q(L - Lo)(L - Ld ) Q(L - Lo)3f}=- +-----

8D} 24D j (L-Ld )

(8)

(9)

CRITICAL BUCKLING LOAD ANDDELAMINATION GROWTH

Beam bending induces a strain distribution thatis linear through the thickness of the beam. Thelinear strain distribution can be represented byresultant moments for segments 1 and 2 andresultant axial forces acting at segments 2 and3, as shown in Fig. 3. From the moment equili­brium condition at the delamination tip, we canwrite

Delamination buckling in laminated wood beams 315

Segment 1

p....-.--. Segment 3

Strain Distribution Resultant Forces

Fig. 3. Resultant forces at the delamination front.

puted as follows. First, the bending energy forsegment 1 is(15)

The moment-curvature relations for M 1 .and Mzare

1 L/2-Ld /2 (d2W 1 ) 2

U =-D f -- dx1 2 1 0 dx2

(21)

(16) =

Using eqns (15) and (16), f) can be expressed as

The bending energy for segment 2 is

1 L d /2 (dZW2)2U=-D f -- dx

2 2 z0 dx2(22)

Ld .f)= - [Q(L-L )-Pt]

4D2

0(17)

The compatibility condition of axial shorteningof segments 2 and 3 is and the membrane and bending energy for seg­

ment 3 is

P 1 Lj/2 (dW2)2 P-A L d = -2 _ /2 -- dx2+tf) - - L d (18)

3 ~ dX2 A2

Then, we obtain

Using the displacement functions given in eqns(6), (8), (11) and (13), the strain energy alongthe post-buckling path for each segment is com-

Equating eqns (17) and (19), the relationbetween P and Q is established as

Pt 4D2Q= . -----(L-Lo ) Ld(L-Lo )

As the thickness of a delaminated layer is verythin compared to that of the beam, Euler'scolumn buckling formula for a fixed--fixedboundary condition can be used for Per [1].Then, substituting Per from the Euler's columnbuckling formula into eqn (20), a critical trans­verse load Qer can be· computed from eqn (20)which is valid up to the critical stage. An alter­native procedure to find Per from the totalpotential energy is described in Appendix A.Note that in eqn (23), the load in the delami­nated segment is assumed to remain constantand equal to Per. This is because of the virtuallyflat post-buckling path, similar to that of anEuler column.

(19)

(20)9t2 (1 1)]-. +6P -+-

L~ A 2 A 3

9t2 (1 1)-+6 -+- P

L~ A2 A3

3t(}=--+

L d

316 Y Kim. J. F Davalos, E. J. Barbero

(24)

The strain-energy release rate (G) is definedas

a(U I+U2+U3 )G=------

aLd

and using eqns (16)-(23), the explicit expres­sion of G per unit width is

and Euler's column buckling formula. Thecritical experimental load, Q~~P = 47.2 kN, wasrecorded when buckling of the delaminatedlayer was observed during the experiment (seeFig. 5). Using LBT, the stress in the delami­nated layer at buckling can be computed usingthe known critical load Q~~P as follows

NUMERICAL EXAMPLE

PerL~[Q(L - ~o) - p ert ]2

64bD~ sin Per

(27)

Using eqns (19) and (20), P and e can be com­puted for a given Q. Then, the maximummid-span deflection w2max is given by eqn (11)as

(Jer=

(26)

Assuming that the delaminated layer behaves 'like a fixed-fixed column, the approximate dela­mination length is computed from Euler'sformula as

(25)

[Q(L - Lo ) - Pert]2

8bD2

(L-Lo )2Q2G~ ----­

8bD I

(~ Ld )

X 2-.y D3

tan Per

The accuracy of the model developed in thissection is evaluated by correlating the analyticalsolutions with experimental results, as presentednext.

The test data for glulam beams reinforced withGFRP [7] are used to validate the analyticalsolution presented previously. The beam con­figuration and average layer material propertiesare shown in Fig. 4, and the beam parametersare listed in Table 1. The delaminated length(L d ), which is unk,nown at the time of the test,is a parameter needed to carry out the compu­tations with the equations derived in Section 3.The delaminated length is estimated using lam­ination beam theory (LBT, Barbero [16])

(28)

The internal axial force Per in the delaminatedlamina and' the critical transverse load Qer arerelated by eqn (20). It is our interest to predictQer which causes local buckling of a delami­nated sublaminate. In Fig. 6, Per and Qer areplotted for various delamination lengths, wherePer is computed from eqn (AS) and also Euler'sformula, and Qer is obtained from eqn (20).

141

10.2cm _,

34cm

41.5

14.0

13.1

10.8

11.3

11.3

11.3

11.3

11.6

13.0

14.141.5

-'-

Q Q1.98 m I 1.83 m I 1.98 m

5.79m

Beam configuration

10 wood layers

::::::::::::::::::::::::::::::::::::::::::GFF.lP:l~Y$r.::::::::::::::::::: :':'::':':':':':':':':':':':':':':':'

Elastic modulus of layer (GPa)

lFigo 4L Beam configuration and elastic modulus of each layer.

Delamination buckling in laminated wood beams 317

(29)

where Rw =Re = 1.0 corresponds to the testbeam parameters given in Table 1 and Fig. 4.As seen in Fig. 7, the response of Qer is linearlyproportional to the value of the multipliers. Asexpected, increasing the elastic modulus ofwood layers affect Qer more significantly thanincreasing the modulus of the top compositelayer. The corresponding internal axial force Peris shown in Fig. 8. Unlike the curves for Qer, thevalue of Per is not affected by the bending stiff­ness of segment 2, and it depends only on thestiffness of segment 3. Therefore, only thematerial properties of the delaminated sublami­nate affect· the stress level in the delaminatedlayer. As in the case of Fig. 6, the valueobtained by using eqn (AS) is almost identicalto that computed with Euler's formula. Toinvestigate the effect of delamination length onbeam stiffness, the response of a delaminatedbeam is compared with that of a perfect beam,and the effect of wood properties is alsoexplored for different values of Rw • As shown inFig. 9, the degradation of the delaminatedbeam stiffness is not affected by the delamina­tion length. However, the delamination lengthcan significantly influence the state of stress inthe delaminated sublaminate, and the ultimatestrength and failure mode of the beam. Afterthe buckling of a delaminated sublaminateoccurs, it is of interest to investigate the growthof the delamination length; i.e. whether or notthe delamination will grow or be arrested asfunction of the applied load. For this purpose,the strain-energy release rate given in eqn (25)is plotted for various initial delaminationlengths in Figo 10. To interpret the physicalmeaning of the curves, a critical strain-energy

E:omposite = ReEeomposite

this difference is that the material properties ofthe wood layers vary along the length of thebeam. Compared with test results, the criticalload Qer predicted by this study is within 2.6%of the experimental value, and the predicted Peris within 1% of the experimental Per. Theexperimental Per is computed using the experi~

mental strain at the top surface measured withstrain gages.

To investigate the effect of the beam stiffnesson Qer and Per the elastic moduli of the GFRPlayer and wood layers are modified using multi­pliers as follows

12

45

8

30

o Euler's formula

This study

- Per: Eq. (A.5)-_ .. Ocr: Eq. (19)

L =5.79, L o = 1.83A z =428.8,A3 = 20.1D] = 7.96, D z = 7 03, D 3 = 5.87

h =0.48, t =33.9b = 10.15

4

15

Ld(cm)

Figo 60 Critical loads vs delamination length.

O-F------......-------~----___1

o

75

Z~

t;50 Per

0 '.....

....

a.'G O~r'.....

"-.... ...

25 .....

Length (m)Axial stiffness (MN)Bending stiffness

(GNomZ)

Thickness· (cm)Width (cm)

Delamination buckling

Table 10 Beam parameters

Displacement (em)

Figo 50 Experimental load-displacement curve of theglulam-GFRP beam.

120-----------------.

ZOO~

g~..9 40

100-.-----------------,

Euler's formula and the results of eqn (AS) pro­vide nearly identical values for Per. However,Euler's formula can not by itself provide anyinformation on the magnitude of the transverseload (Q). Therefore, eqn (20) must be used.The experimental load-displacement path ofthe. glulam-GFRP beam is given in Fig. 5, andthe experimental values and analytical solutionsare compared in Table 2. The mid-span maxi­mum displacement relative to the point loads isunderpredicted by 7.5%. One of the .reasons for

318 Y Kim. J. F. Davalos, E. J. Barbero

Table 2. Comparison of critical load, force and displacement

Per (kN)

Test This study Test This study Test l . Euler eqn (AS)

8.37 7.74 47.2 48.5 61.9 61.5 61.4

6.3~------------------,

/release rate (G1c) is assumed as 87.6 N/m. Thisvalue of G1c .is representative of graphite-epoxyT300/976 (see [14]). When the transverse loadreaches a critical value (Qcr = 48.5 kN for theexample considered in this study), the delami­nation becomes unstable, and it grows from theinitial delamination length (L d = 0.16 m) to astable condition (L d = 0.3 m). Increasing theload by a factor of 10% of Qcr the delaminationlength decreases slightly, which means that nofurther delamination growth occurs; i.e. thedelamination growth is arrested.

E 5.6Z~.<;

.~

E

~o 4.9

This point correspondsto example

............. Perfect beam

- Delaminated beam

65 --------------------.

58.5-ro----------------......252117

R w=O.6.....................................................................................................................................................................................................................

4.213

Ld (em)

Fig. 9. Effect of delamination on beam stiffness (atQ =22.2 kN).

For an initial delamination length of ·0.25 mthe flat zone of the curve moves down, but thecurve is still above the assumed G1c value, whichindicates an unstable delamination growth inthe beam leading to a stable condition at thedelamination length of around 0.48 m. How­ever, in this case, Qcr decreases by 60% to avalue of 18.3 kN, when compared to the pre­vious case for L d =0.16 m. For a furtherincrease of delamination length to 0.38 m, the.flat zone of the curve moves below the assumedG1c value and, therefore, there is no delamina­tion growth.

Compared to the initial case (L d = 0.16 m),Qcr in this case decreases by 80% to a value of8.1 kN. From this observation, we may inferthat delamination growth in a thin layer on thecompression face of a beam is arrested, while inlaminates under axial load the delaminationgrowth can grow indefinitely. Additional experi­mental data are needed to corroborate thisobservation. Usually delamination buckling con­tributes to premature failure of glulam-GFRPbeams reinforced on the compression facebecause the reinforcement ceases to be effect­ive. After delamination, the wood laminatealone must carry any additional load. Testresults of 5.79-m long beams, similar to the one

1.21.0

- This study (Eq. (A.S»

-_.. Euler's formula

0.8

_11- _ •• __ •• __ •• __ •• __ le __ •• __ .... _ .... _ le __ ' , __ "1 __ '. __ 11 .. - •• __ •• __ •• _- ••

_ •• __ •• __ ~. __ II __ " • __ •• __ •• __ •• __ 11 __ le .... _ • __ •• __ •• __ II __ 11 "" _ •• -_ ••

26.0+--------.----------------f0.6

52.0

45.5

Z~

b0 39.0

32.5

Rw

Fig. 7. Effect of elastic moduli on Qer'

This point corresponds toexample

Fig. 8. Effect of.elastic moduli on Per.

354---------T-------r--------t0.6 0.8 1.0 1.2

-..-_ ..__ ..__ ..-_ .._- .. __ ..__ .. __ .._- ..__ .. -_ ..-_ .. __ .. __ .. __ .. __ .. __ ..

~- 55o..b

Del(lmination buckling in laminated wood beams 319

755025L~ (em)

Fig. 10. Strain-energy release rate curves.

- Ocr....u, ...... 1.1 ft Ocr

/'==~Initial Ld= 0.16 m Initial Ld lD 0.25 m

Per'" 61.5 KN Per'" 23.2 KN

Oer""' 48.5 KN Q cr"'" 18.3 KN

V Initial L d= 0.38 mperID 10.3 KN

~ / Q crc:a8.1 KN

G Ic '" 87.6 /\.. \ /I

oo

350

1750

1400

(!:J 700

.-. 1050

-E-Z'-'"

used in this example, indicated a significantdecrease in the expected ultimate load dueprimarily to delamination of the top GFRPlayer [7].

detect the onset of delamination buckling andmeasure delamination growth.

REFERENCES

CONCLUSION

Most of the existing studies on delamination inlaminated structures deal with an axial loading,irrespective of whether a thin-film model or ageneral bending model is used. However, theexisting models can not be directly applied to atransverse load case in which the major concernis to find a critical transverse load. The mainfindings of this study are: (1) the internal axialforce induced from a transverse load is close toEuler's column buckling load for a fixed-fixedboundary condition; (2) Euler's formula alonecannot be used to predict the critical transverseload (Qcr); (3) an explicit form to relate thetransverse load (Q) with the internal axial force(P) is established; (4) the critical load (Qcr) canbe accurately predicted as corroborated byexperiment; and (5) a simulated delaminationphenomenon indicates an unstable delamina­tion growth after buckling of the delaminatedsublaminate, followed by arrested delaminationgrowth. This response is different to the axialloading case, for which unbounded delamina­tion growth is generally predicted. To furtherverify the present model, it is desirable to con­duct well-controlled tests of laminated beamswith predetermined embedded delaminationlengths and instrumented with transducers to

1. Chai, H., The growth of impact damage in compres­sively load laminates. Ph.D. dissertation, CaliforniaInstitute of Technology, Pasadena, CA, 1982.

2. Yin, W.L., The effects of laminated structure on dela­mination buckling and growth. I. Composite Mater.1958,22, (6),502-517.

3. Kardomateas, G.A. and Schmueser, D.W., Bucklingand postbuckling of delaminated composites undercompressive loads including transverse shear effects.AIAA In/. 1988,26, (3),337-343.

4. Chen, H.P., Shear deformation theory for compressivedelamination buckling and growth. AIAA Inl. 1991,29, (5), 813-819.

5. Sheinman, I. and Soffer, M., Post-buckling analysis ofcomposite delaminated beam. Int. 1. Solids Struct.1991,27, (5),639-646.

6. Tracy, J.J. and Pardoen, G.e., Effect of delaminationon the flexural stiffness of composite laminates. Thin­Walled Struct. 1988,6, 371-383.

7. Kim, Y., A layer-wise theory for linear and failureanalysis of laminated composite beams. PhD disserta­tion, West Virginia University, Morgantown, WV,1995.

8. Reddy, J.N., Barbero, E.J. and Teply, J.L., A platebending element based on a generalized laminateplate theory. Int. J. Numer. Meth. Engng 1989, 28,2275-2292.

9. Barbero, E. J., On a generalized laminate theory withapplication to bending, vibration, and delaminationbuckling in composite laminates. Ph.D. dissertation,Virginia Polytechnic Institute and State University,Blacksburg, VA, 1989.

10. Barbero, E.J. and Reddy, J.N., Modelling of delami­nation in composites laminates using a layer-wiseplate theory. Int. I. Solid Struct. 1991, 28, (3),373-388.

11. Lee, J., Gurdal, Z. and Griffin, O.H. Jr., Layer-wiseapproach for the bifurcation problem in laminated

320 Y Kim. J. F. Davalos, E. J. Barbero

Q2(L - Lo)2(L+2Lo- 3Ld )

48D 1

Then, the critical buckling load (bifurcationpoint) can be found as (see Timoshenko &Gere [17] and Ziegler [18], eqn (1.36), p. 11)

composites with delaminations. AIAA In/. 1993, 31,(2), 331-338.

12. Sankar, B.V., A finite element for.modeling delamina­tions in composite beams. Comput. Struct. 1991, 38,(2), 239-246.

13. Sankar, B.V. andHu, S., Dynamic delaminationpropagation in composite beams. I. Composite Mater.1991, 25, (11), 1414-1426.

14. Kutlu, Z. and Chang, F.K., Modeling compressionfailure of laminated composites containing multiplethrough-the-width delaminations. 1. Composite Mater.1992, 26, (3), 350-387.

15. Chen, W. F. and Lui, E. M., Structural Stability. Else­vier, New York, 1987.

16. Barbero, E.J., Pultruded structural shapes-From theconstituents to the structural behavior. SAMPE In/.1991, 27, (1), 25-30.

17. Timoshenko, S. P. and Gere, J. M., Theory of ElasiicStability, 2nd edn. McGraw-Hill, New York, 1961.

18. Ziegler, H., Principles of Structural Stability. BlaisdellPublishing, Massachusetts, 1968.

APPENDIX A

an-=0oQ

(A3)

(A4)

The total potential energy can be written as

II= U 1+U2+U3 -QL\ (AI)

where Ll, the displacement at a loading pointcomputed using eqn (8), is

~= O(L-Lo ) + Q(L-Lo)2(L+2Lo -3Ld )

2 241)1

(A2)

The total potential energy along the post-buck­ling path can be expressed in terms of a singleparameter (Q) as

1 Ld[Q(L - La) - Pert]2ll=- .

16 D2

1 PerL~[Q(L-Lo)-Pert]2+ - ---------

64 D~[1-cos(2Per)]

or

3 PerL~[Q(L-Lo)-Pert]

8 D~ sin2 Per(L - Lo)

(A5)

At the bifurcation point, the load Q in eqn (20)can be substituted into eqn (AS). Then, Per isfound by an iterative numerical method usingeqn (A5).