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Composite Structures 79 (2007) 562–570

Analysis of bolted joints in composite laminates: Strainsand bearing stiffness predictions

Marie-Laure Dano *, Elhassania Kamal, Guy Gendron

Department of Mechanical Engineering, Universite Laval, Quebec, QC, Canada G1K 7P4

Available online 3 April 2006

Abstract

A review of the investigations conducted on mechanically fastened joints is presented. A finite-element model is developed to predictthe response of pin-loaded composite plates. The model takes into account contact at the pin–hole interface, progressive damage, largedeformation theory, and a nonlinear shear stress–strain relationship. To predict progressive ply failure, four different analyses combiningHashin and the maximum stress failure criteria, and different associated degradation rules are conducted. The objectives of the study areto determine the influence of the failure criteria and the associated degradation rules on the predictions of the strains around the hole andthe bearing stiffness. Predictions are compared with experimental results. It appears that agreement between the two depends on anappropriate selection of the failure criterion and the degradation rule. Better agreement between experimental results and numerical pre-dictions is observed with the maximum stress criterion.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Composites; Bolted joints; Finite element analyses; Progressive damage model; Local strains; Bearing stiffness

1. Introduction

In structural applications such as in aircraft, spacecraftand civil engineering structures, composite componentsare often fastened to other structural members by boltedjoints [1,2]. Since bolted joints require holes to be drilledin the structure, large stress concentration tends to developaround the hole, which can severely reduce the overallstrength of the structure. Bolted joints being very oftenthe critical part of the structure, it is therefore importantto design them safely.

To exploit the full potential of composite materials instructural elements, appropriate methods for stress andfailure predictions must be developed. Several numericalmodels have already been proposed to predict bearing fail-ure of composite plates [3–15]. Most of them are reviewedin details by Camanho and Matthews [16]. Generally, the

0263-8223/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2006.02.024

* Corresponding author. Tel.: +1 418 656 2242; fax: +1 418 656 7415.E-mail address: mldano@gmc.ulaval.ca (M.-L. Dano).

proposed models use a progressive damage analysis imple-mented in a finite-element code. The following procedure isusually adopted: (1) perform a stress analysis, (2) apply aset of failure criteria to check for failure at each integrationpoint, (3) if failure has occurred, degrade the materialproperties. This procedure must be repeated for increasingload levels until properties have been degraded at so manyintegration points that the joint fails. In practice, finite-ele-ment programs often stop before the failure load is reacheddue to large deformations, which result in excessive elementdistortions. The way the stress analysis is performed, thechoice of the failure criteria and the properties degradationrules vary from authors to authors.

To perform the stress analysis, two-dimensional linearor nonlinear finite element models are mainly used. Thepin load is usually simulated either by applying a radialboundary condition to the nodes located around the platehole [3], or a cosinusoidal radial stress on the loaded side ofthe hole [4]. Although simple, these procedures are notalways accurate since the actual interaction between thepin and the hole is a true contact problem. Therefore, a

e

W

L

Dx

y

Fig. 1. Specimen configuration.

M.-L. Dano et al. / Composite Structures 79 (2007) 562–570 563

more realistic method to introduce the pin load is to modelthe pin as a rigid surface and to define master–slave contactsurfaces between the pin circumference and the nodesaround the hole. With this approach, both friction andpin clearance can be taken into account. Only few investi-gators have adopted this approach [5–7]. More recently,three-dimensional finite-element models have been pro-posed [8–10]. Although 3D models require much moreefforts to develop than their 2D counterparts, 3D modelsoffer the potential to predict more accurately failure ofbolted joints by taking into account, for example, clampingpressure and delamination effects.

Once stresses have been computed, the second step is tocheck for damage using failure criteria. So far, several setsof failure criteria have been proposed in the literature topredict damage in composites. Chang and Chang [17] usedthe Yamada–Sun [18] failure criterion to predict matrixcracking, fibre-matrix shear-out and fibre breakage, andHashin failure criterion [19] for fibre compressive failure.Lessard and Shokrieh [3] chose Hashin failure criterionto predict matrix failure, fibre-matrix shearing and fibrefailure modes. Kim et al. [6] also selected Hashin failure cri-terion but did not consider fibre-matrix shearing. Otherinvestigators used either maximum stress [11], Tsai–Wu[12], or Hill [13] criteria to predict failure. Dano et al. [7]studied the influence of the failure criterion on the predic-tion of the bearing response of composite plates with differ-ent stacking sequences. They observed that using themaximum stress criterion for fibre failure leads to morerealistic predictions than using Hashin criterion. This find-ing was latter verified by Tserpes et al. [9].

Finally, the last step is to degrade the elastic stiffnessproperties when failure is detected at the integration points.Among the various degradation rules that have been pro-posed in the literature, two sets of degradation rules arecommonly used [14,17,20].

Chang and Chang [14,17] developed a two-dimensionalprogressive damage model for notched laminated platessubjected to tensile loading and pin-loaded laminates.When matrix cracking failure occurs it is assumed thatthe transverse modulus E2 and Poisson’s ratio m21 werereduced to zero. In the case of the fibre-matrix shear-outor fibre breakage, both E2 and m21 were reduced to zerowhereas longitudinal modulus E1 and shear modulus G12

of the failed zone degenerate according a Weibulldistribution.

Tan [20] developed a two-dimensional progressive dam-age model for laminates with an unloaded hole subjected totensile loading. The stiffness degradation is represented byfactors reducing the material properties. It is assumed thatthe stiffness reduction associated with damage due to com-pressive load is different from the stiffness reduction associ-ated with damage due to the tensile loads. The foundationof this concept has been discussed in Tan’s earlier papers[21,22]. A parametric study shows that the effects of thestiffness degradation factors on the ultimate strength of alaminate appear to be very significant. Tan’s degradation

theory, was later extended by Camanho and Matthews[8], for a three-dimensional model. Although the model isthree-dimensional, no consideration of delamination fail-ure is made.

In spite of the progress made on progressive damagemodelling of composite laminates containing pin-boltloaded holes, the problem is far from being fullyunderstood.

Basic aspects such as failure criteria and material prop-erty degradation rules are still open for further research. Afew models [7,9,15] presented so far have compared theinfluence of failure criteria and material property degrada-tion. However, the authors [9,15] often compare only theload–bolt displacement relationship predicted by the modelwith experimental results. Therefore, there is a lack ofdetailed experimental validation of numerical finite elementmodels used to calculate stress and strain distributionsaround the hole. Thus, it would be interesting to validatefinite-element predictions on both global and local scalesby comparing not only the load–bolt displacement rela-tionship but also the strain distribution around the holewith experimental data.

The objective of this study is to investigate the effect offailure criteria and material property degradation rules onthe behaviour of pinned-joint in graphite/epoxy compositelaminates. Two sets of failure criteria and degradation ruleswere adopted in this work to examine their influence on thepredicted load–bolt displacement curves and the strainsaround the hole. The results were compared with experi-mental data from Girard et al. [23,24].

2. Problem statement

Consider a laminated composite plate of length L andwidth W with a hole diameter D, as shown in Fig. 1. Thehole is at a distance e from the free edge of the plate.The thickness of the plate, H, is equal to 4.65 mm andthe values of L, W, D and e are equal to 205 mm,95 mm, 16 mm and 80 mm. A rigid pin with a 15.8 mmdiameter is placed at the centre of the hole and a tensileload P is applied at one edge of the plate.

It has been observed experimentally that mechanicallyfastened joints under tensile loads generally fail in threebasic macroscopic failure modes, namely net-tension,shear-out and bearing, depending on the laminate lay-upand the joint geometry. To analyze the pinned-joint prob-

564 M.-L. Dano et al. / Composite Structures 79 (2007) 562–570

lem, it is desired to develop a progressive damage modelfollowing the three important steps: stress analysis, failurecriteria evaluation, and property degradation. A descrip-tion of the model will be given in the following section ofthis paper. Note that more details about the model canbe found in [7].

3. Numerical study

3.1. Finite-element modelling

The deformation behaviour of the pin-loaded joint ispredicted using a two-dimensional finite element modeldeveloped in the commercial software ABAQUS [25]. Sincethe behaviour of balanced symmetric laminate is studiedand the geometry of the problem is symmetric with respectto the y–z plane, only half the plate is modelled, as shownin Fig. 2. Symmetry boundary conditions are applied. Thenodes on the vertical centreline are restrained to movealong the vertical direction (ux = 0) only. The load (P/2)is applied to a single node of the upper edge (right end inFig. 1). A multi-point constraint forces every node on thisedge to have the same vertical displacement. To simulatethe contact between the bolt and the laminate, the pin cir-cumference is modelled as a rigid surface and the hole edgeas a deformable surface. Contact pairs are then definedbetween the two surfaces. Since in ABAQUS, shell ele-ments cannot be used to simulate in-plane contact prob-lems, the composite plate is modelled using four nodes-continuum plane stress elements (CPS4).

Fig. 2. Geometry of the finite-element model.

3.2. Progressive failure analysis

The analysis takes into account large deformation the-ory, progressive damage and the nonlinear shear behaviourof the composite. In this section, the implementation in themodel of the nonlinear shear stress–strain behaviour is dis-cussed and different failure criteria and degradation rulesused in the analyses are presented.

3.2.1. Nonlinear shear behaviour

The nonlinear shear behaviour of the composite isassumed to be of the form [26]

c12 ¼1

Go12

� �s12 þ as3

12; ð1Þ

where Go12, s12, c12 are, respectively, the initial shear modu-

lus, shear stress, shear strain, and a is a constant parameterwhich is determined experimentally. To be implemented ina finite-element program, Eq. (1) has to be expressed in adifferent form [27]. After several algebraic manipulations,it can be rewritten as,

sðiþ1Þ12 ¼ G12c

ðiþ1Þ12 ¼ ð1� dÞGo

12cðiþ1Þ12 ; ð2Þ

where i is the increment number and d, the damage param-eter, is given by

d ¼ 3aGo12ðs

ðiÞ12Þ

2 � 2aðsðiÞ12Þ3=cðiÞ12

1þ 3aGo12ðs

ðiÞ12Þ

2. ð3Þ

At the beginning of the analysis, the damage parameter iszero and the shear modulus G12 is equal to Go

12. As the loadincreases, the shear modulus is decreased linearly with re-spect to the damage parameter.

3.2.2. Failure criteria

As discussed in the introduction, several sets of failurecriteria have been proposed in the literature to predict fail-ure of composite joints. Two sets of failure criteria in com-bination with two sets of material property degradationrules were adopted in this paper for the failure analysis.The objective was to study the influence of the two failurecriteria and material properties degradation rules on thepredicted hole elongation and local strains around the hole.

The first set of failure criteria called ‘‘mixed criteria’’,combines Hashin criterion and the maximum stress crite-rion. This set is actually identical to Hashin criteria withthe difference of replacing the Hashin-type fibre tensile cri-terion by the maximum stress criterion. Table 1 specifies

Table 1Mixed criteria

Failure mode Failure criteria

Matrix tensile failure (r2 P 0) r2

Y t

� �2þ 2s2

12=Go

12þ3as412

2S2=Go12þ3aS4 P 1

Matrix compression failure (r2 6 0) r2

Y c

� �2þ 2s2

12=Go12þ3as4

12

2S2=Go12þ3aS4 P 1

Fibre tensile failure (r1 P 0) r1

X t

� �P 1

Fibre compression failure (r1 6 0) r1

X c

� �P 1

Table 2Maximum stress criteria

Failure mode Failurecriteria

Matrix tensile failure (r2 P 0) r2

Y t

� �P 1

Matrix compression failure (r2 6 0) r2

Y c

� �P 1

Fibre tensile failure (r1 P 0) r1

X t

� �P 1

Fibre compression failure (r1 6 0) r1

X c

� �P 1

Initial propertiesvalues

Load Pi

Geometric non linear analysis

Check for convergence

Stress computation

Check for failure Propertiesdegradation

Load incrementationPi = Pi-1+P

Yes

Final failure

Stop

No

Yes

No

Start

Fig. 3. Progressive damage model algorithm.

Table 4Description of the analyses performed

Analysis Failure criteria Material propertiesdegradation rules

A1 Mixed ChangA2 Mixed TanA3 Maximum stress Chang

M.-L. Dano et al. / Composite Structures 79 (2007) 562–570 565

the criterion selected for each failure mode. The second setof failure criteria is simply the maximum stress criteria (seeTable 2).

3.2.3. Property degradation rules

As discussed in the introduction, two sets of propertydegradation rules have been mainly used in the literature.The first one, introduced by Chang and Chang [17], setsmost of the material properties of the failed ply to zerofor all damage modes. The second set was proposed byTan [20] and his co-workers [21,22]. A brief descriptionof these two sets is presented in Table 3. Throughout thepaper the two sets are referred to by the names ‘‘Chang’’and ‘‘Tan’’, respectively.

3.3. Numerical implementation

The progressive damage models were implemented inABAQUS by using a separate subroutine called user-defined-field [25]. This subroutine gives the ability to choosefailure criteria and degradation rules according to the spe-cific case studied. Details of the program are described bythe flow chart shown in Fig. 3. At the beginning of theanalysis, the material properties are all equal to their initialvalues and the load is increased gradually. For each loadincrement, several iterations are necessary before the analy-sis converges to an equilibrium state. At the end of eachincrement, strains, stresses, and failure criteria are com-puted. If failure does not occur, the applied load is increasedand the program returns to the initial step. If failure occurs,materials properties of the failed plies are reduced accordingto the selected properties degradation rule. The load is thenincreased and the procedure is repeated until the programstops due to excessive element distortion or the total loadP/2 has been applied.

Table 3Material properties degradation rules

Failure mode Properties degradation rules

Chang [17] Tan [20]

Matrix tensilefailure (r2 P 0)

Ed2 ¼ md

12 ¼ Gd12 ¼ 0 Ed

2 ¼ 0:2E2, Gd12 ¼ 0:2G12

Matrix compressionfailure (r2 6 0)

Ed2 ¼ md

12 ¼ Gd12 ¼ 0 Ed

2 ¼ 0:4E2, Gd12 ¼ 0:4G12

Fibre tensilefailure (r1 P 0)

Ed1 ¼ md

12 ¼ 0 Ed1 ¼ 0:07E1

Fibre compressionfailure (r1 6 0)

Ed1 ¼ md

12 ¼ 0 Ed1 ¼ 0:14E1

4. Finite-element analyses

By combining the different sets of failure criteria andmaterial properties degradation rules, four analyses werecarried out (see Table 4). Three different symmetric lay-ups were chosen for the study: cross-ply [(0/90)4]s, angle-ply [(+45/�45)4]s and quasi-isotropic [(0/±45/90)2]s. Thegeometry of the specimens is shown in Fig. 1. The materialproperties used in the analysis are presented in Table 5. Thepredictions of the four analyses are compared with experi-mental results obtained previously by Girard et al. [23].

A4 Maximum stress Tan

Table 5Mechanical properties of the pin-loaded plate

E1 (GPa) 161E2 (GPa) 8.26G12 (GPa) 4.42m12 0.32Xt (MPa) 2052Xc (MPa) 977Yt (MPa) 38.8Yc (MPa) 200S (MPa) 59.2

566 M.-L. Dano et al. / Composite Structures 79 (2007) 562–570

4.1. Prediction of the bearing stiffness

The four analyses indicated in Table 4 were performedto study the effect of the failure criteria and the propertydegradation rules. The predictions of failure strength andbearing stiffness were compared with experimental results[23]. The model predictions are summarized in Fig. 4 whichrepresents curves of the bearing stress versus the hole elon-gation, for the three different lay-ups. The bearing stresswas defined as the load divided by the hole diameter andthe thickness of the laminate rb = P/(D * H). The holeelongation is obtained by dividing the displacement ofthe specimen beside the hole by the hole diameter. The holeelongation represented on the experimental curve was eval-uated by averaging the displacement recorded by twoextensometers clamped on each side of the specimen asillustrated in Fig. 5a. Note that because of the way the

[(0/90)4]s

0

100

200

300

400

500

0 0.05 0.1 0.1

Hole elongation (%)

Bea

ring

str

ess

(MP

a)

A4 A2A3

5

A1

A2

A3

A4

Exp. [23]

A1

[(+45/-45)4]s

0

100

200

300

400

500

0 0.05 0.1 0.15

Hole elongation (%)

Bea

ring

str

ess

(MP

a)

A3. A4

A2

A1

[(0/±45/90)2]s

0

100

200

300

400

500

0 0.05 0.1 0.15

Hole elongation (%)

Bea

ring

str

ess

(MP

a)

A4A2

A3

A1

Fig. 4. Hole elongation curves.

extensometers were attached to the specimen and the fix-ture, the displacements between two referenced points,one placed on the side of the composite plate at the pinlevel and one placed on the steel fixture were measured.Therefore, the measured displacement includes the holeelongation as well as the bolt and fixture deformations.

Several interesting results can be observed from Fig. 4.First, for the three lay-ups, the slope of the curves is almostthe same for the four analyses. Therefore, bearing stiffnessprediction is only slightly affected by the failure criteria andthe material property degradation rules.

Secondly, for all analyses, the numerical predictions aremuch stiffer than the experimental results. This difference isthought to be caused by the fact that the displacement mea-sured experimentally includes the hole deformation as wellas the bolt and the fixture deformations which are nottaken into account in the model. This issue has alreadybeen raised and discussed in details by Vangrimde andBoukhili [28] who showed that the total deformation ofthe bolt and the fixture could account for up to 10% ofthe total displacement.

The third remark concerns the strength prediction. Con-sidering first analyses A1 and A3 which both use Chang’sdegradation rule but different failure criteria, one can notethat using the maximum stress criteria (A3) allows reachinga higher strength than using the mixed criteria (A1). Com-paring A2 and A4, which both use Tan’s degradation rulecombined with different failure criteria, one can observedagain that damage is predicted at an earlier stage withthe mixed failure criteria. These results verify that the con-tribution of the shear stress in the failure criteria leads tomore conservative predictions.

To examine the effect of the degradation rules on thestrength prediction, analyses A1 and A3 have to be com-pared to A2 and A4, respectively. The comparison showsthat the strength of the laminate is sensitive to the valueof the degradation factors. Tan’s degradation rule (analy-ses A2 and A4) allows the analysis to reach a higherstrength and to avoid numerical problems. The reason isthat the application of these rules imposes a lower rate ofstiffness degradation for the elements than Chang’s degra-dation rules. The stress value corresponding to the first sig-nificant slope change (analysis A2) agrees well with theexperiment results for the three lay-ups.

4.2. Prediction of the laminate stiffness and localstrains around the pin

Next, the strains predicted by the four analyses of Table4 are compared with experimental results. As explained in[23] and described in Fig. 5b, the strain distribution in thevicinity of the hole and at the centre of the specimen wasdetermined experimentally using several strain gauges onboth sides of the laminates. One pair of back-to-back straingauges was placed at the centre, on the lower half of thespecimen, to measure the stiffness of the composite lami-nate. Four pairs of back-to-back strain gauges were placed

Fig. 5. Experimental set-up [23]: (a) Bolt displacement measurement and (b) strain gauges location.

M.-L. Dano et al. / Composite Structures 79 (2007) 562–570 567

on the left and right side of the hole to detect bending in thejoint and two pairs of back-to-back strain gauges wereplaced in the bearing zone. The gauges placed just besidethe hole are called ‘‘inner gauges’’ whereas the gaugesplaced further around the hole are called ‘‘outer gauges’’.Gauges from which measurements are plotted are shownin black in Figs. 7–9.

4.2.1. Laminate stiffness

The model predictions are summarized in Fig. 6 whichrepresents the bearing stress versus the strain at the centreof the laminate for the three different lay-ups. The curvesshow a perfectly linear behaviour for the three lay-ups. Inthis case, the behaviour is not sensitive to the selected fail-ure criteria and material property degradation rulesbecause no damage occurred at the centre of the specimen.Very good correlation with experimental results isobserved.

4.2.2. Strains on the lateral sides of the hole

The bearing stress versus local strain curves obtainedfrom both inner and outer transversal surface gauges areshown in Fig. 7. In the figure, the curves with the highestslope correspond to the outer strain gauges. In the case

of the outer gauges, good agreement between the four anal-yses and experiments is observed. The behaviour is not sen-sitive to the failure criteria. Things are however differentfor the strains measured by the inner set of strain gauges.

For the [(0/90)4]s lay-up, analyses A3 and A4 (maximumstress failure criteria) give very similar results. The stress–strain curves are linear and on top of each other. Thecurves obtained from analyses A1 and A2 (mixed failurecriteria) are identical to A3 and A4 up to approximately180 MPa. Beyond this value, failure is detected and thecurves become irregular. Comparing with the experimentalresults, it is quite obvious that predictions from A3 and A4are much closer than predictions from A1 and A2.

For the [(+45/�45)4]s lay-up, the curves obtained fromthe four analyses exhibit a slightly nonlinear behaviour.This is induced by the assumption that the shear stress ver-sus shear strain relationship is nonlinear. Beyond 170 MPa,analyses A1 and A2 detect failure and the slope of thecurves decreases. Comparing with the experimental results,one can note again that analyses A3 and A4 predict betterthe experimental strains than analyses A1 and A2.

Finally for the [(0/±45/90)2]s lay-up, the four analysespredict similar stress–strain curves which compares wellwith the experimental results.

[(0/90)4]s

0

20

40

60

80

100

0 200 400 600 800 1000

Strain ( )

Strain ( )

Strain ( )

Stre

ss(M

Pa)

A4

A3

A2

A1

[(+45/-45)4]s

0

20

40

60

80

100

0 1000 2000 3000 4000 5000 6000

Stre

ss(M

Pa) A3, A4

A2

A1

[(0/±45/90)2]s

0

20

40

60

80

100

0 400 800 1200

Stre

ss(M

Pa)

A2, A4

A1, A3

A1

A2

A3

A4

Exp. [23]

Fig. 6. The stress versus strain curves at the centre of the laminate.

[(0/90)4]s

0

100

200

300

400

500

0 500 1000 1500 2000 2500

Strain (με)

0 2000 4000 6000 8000 1000

Strain (με)

Strain (με)

Bea

ring

str

ess

(MP

a)

A1 A2

A3 A4

Exp. [23 ]

A4A3

A2

A1

A3A4

A2

A1

[(+45/-45)4]s

0

100

200

300

400

500

Bea

ring

str

ess (

MP

a)

A3, A4

A2

A1A1

A2

A3, A4

[(0/±45/90)2]s

0

100

200

300

400

500

600

0 500 1000 1500 2000 2500 3000

Bea

ring

str

ess

(MP

a)

A4A4A2

A1, A3

A2

A1, A3

ig. 7. Stress versus local strain curves – inner and outer transversalauges.

568 M.-L. Dano et al. / Composite Structures 79 (2007) 562–570

From the results of Fig. 7, it appears that analyses A3and A4 which use the maximum stress criteria for all failuremodes are able to predict much more accurately the strainson the lateral sides of the joint. Unfortunately, the resultsdo not allow assessing which analysis between A3 andA4 is the most accurate. For the [0/±45/90]s laminate,analysis A3 predicts a bearing strength much closer thanA4. However, for the [0/90]s and the [±45]s laminates, bothanalyses overpredict the bearing strength. Additionalexperimental data would be necessary to draw a definitiveconclusion regarding the degradation rule set choice.

4.2.3. Strains in the bearing zone

Figs. 8 and 9 present the strains predicted by the analy-ses ahead of the pin for the three lay-ups. The stress–straincurves obtained from the outer bearing zone are presentedin Fig. 8. From the figure, it appears clearly that analysesA3 and A4 predict closely the experimental results for thethree lay-ups. As observed in the previous figure, analyses

Fg

A1 and A2 do not agree well with the experimental resultspast a certain bearing stress value because damage isdetected prematurely.

Finally, Fig. 9 presents the stress–strain curves obtainedfrom the inner bearing zone for the three lay-ups. For the[(0/90)4]s laminate, strain measurements turned out to beunsatisfactory. A large discrepancy could be observedbetween the strains obtained from the two back-to-backstrain gauges. It is thought that the bolt was not loadingthe hole uniformly through the specimen thickness. There-fore, the average value represented in the graphic cannot becompared with the predictions. For the [(+45/�45)4]s andthe [(0/±45/90)2]s laminates, it appears that the four anal-yses predict a stiffer stress–strain behaviour. The discrep-ancy may be explained by some experimental errors(gauge location and alignment) which, given the very highstress concentration near the hole, could lead to significantdifferences in the measured strains. Also, along the hole

[(0/90)4]s

0

100

200

300

400

500

Bea

ring

str

ess

(MP

a)

A1 A2

A3 A4

Exp. [23 ]

A2

A1

A3A4

[(+45/-45)4]s

0

100

200

300

400

500

Bea

ring

str

ess

(MP

a)

A3, A4

A2

A1

[(0/±45/90)2]s

0

100

200

300

400

500

-3500 -3000 -2500 -2000 -1500 -1000 -500 0

Bea

ring

str

ess

(MP

a)

A4A2

A1, A3

-5000 -4000 -3000 -4000 -1000 0

-6000 -4000-5000 -3000 -1000-2000 0

Fig. 8. Stress versus local strain curves – outer longitudinal gauges.

[(0/90)4]s

0

100

200

300

400

500

-6000 -5000 -4000 -3000 -2000 -1000 0Strain (με)

Strain (με)

Bea

ring

str

ess

(MP

a)

A1 A2

A3 A4

Exp. [23]

A3, A4

A2

A1

[(+45/-45)4]s

0

100

200

300

400

500

600

-40000 -30000 -20000 -10000

Bea

ring

str

ess

(MP

a)

A3, A4 A2

A1

[(0/±45/90)2]s

0

100

200

300

400

500

Bea

ring

str

ess

(MP

a)

A3A4

A2

A1

0

-6000 -5000 -4000 -3000 -2000 -1000 0Strain (με)

Fig. 9. Stress versus local strain curves – inner longitudinal gauges.

M.-L. Dano et al. / Composite Structures 79 (2007) 562–570 569

edge, the presence of interlaminar stresses may have aninfluence on the strains very close to the hole. However,these interlaminar stresses were not accounted in the model.It would be very interesting to investigate if a three-dimen-sional model is able to capture better the strain behaviour ofthe joint just ahead of the pin where the compressive stressesare the highest. This work is currently being conducted andwill be presented in a forthcoming paper.

5. Conclusions

A two-dimensional finite-element progressive failuremodel of pin-loaded composite plates was developed toinvestigate the effects of failure criteria and material prop-erties degradation rules on the bearing behaviour. A totalof four analyses using a combination of different failure cri-teria and degradation rules were performed. By comparingthe predictions of the different analyses with experimental

results, the accuracy of each analysis was assessed andthe effects on the predictions of the failure criteria andthe material properties degradation rules were studied.

From the theoretical and experimental results presented,it can be concluded that:

• The predicted hole elongation was smaller than the onemeasured. This is probably due to the fact that themodel did not account for the bolt and the fixturedeformations.

• The bearing strength is sensitive to the value of the deg-radation factors selected for the degradation rules. UsingTan’s degradation rule allows the analysis to reach ahigher strength and to avoid numerical problems.

• Damage is detected prematurely when the shear stressterm is included in the failure criteria. Comparisons withexperimental results show that using the maximumstress criterion for all failure mode leads to more accu-rate predictions.

570 M.-L. Dano et al. / Composite Structures 79 (2007) 562–570

• Correlation was generally very good between the strainspredicted using the maximum stress criteria and thestrains measured around the hole. However, the strainmeasured right beside the hole in the bearing zone couldnot be predicted well. In the bearing zone close to thehole edge, the stress concentration is very high and inter-laminar stresses must be present. Using a 3D modelwould probably result in more accurate predictions.

References

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