Composites: Part B 43 (2012) 142–149

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Composites: Part B

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Stability and failure of composite laminates with various shaped cutoutsunder combined in-plane loads

Dinesh Kumar a, S.B. Singh b,⇑a Mechanical Engineering Department, Birla Institute of Technology and Science, Pilani 333 031, Indiab Civil Engineering Department, Birla Institute of Technology and Science, Pilani 333 031, India

a r t i c l e i n f o

Article history:Received 26 March 2011Received in revised form 6 June 2011Accepted 19 September 2011Available online 25 September 2011

Keywords:A. LaminatesB. BucklingB. DelaminationCutouts

1359-8368/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compositesb.2011.09.005

⇑ Corresponding author. Tel.: +91 9414 648283 (mE-mail addresses: dineshkr@bits-pilani.ac.in (D.

ac.in (S.B. Singh).

a b s t r a c t

The objective of this paper is to study stability and failure of a composite laminate with a centrally placedcutout of various shapes (i.e., circular, square, diamond, elliptical-vertical and elliptical-horizontal) undercombined action of uni-axial compression and in-plane shear loads. The FEM formulation based on thefirst order shear deformation theory and von Karman’s assumptions has been utilized. Newton–Raphsonmethod is used to solve nonlinear algebraic equations. Failure of a lamina is predicted by the 3-D Tsai–Hill criterion whereas the onset of delamination is predicted by the interlaminar failure criterion. Theeffects of cutout shape, direction of shear load and composite lay-up on buckling and postbucklingresponses, failure loads and failure characteristics of the laminate has been discussed. An efficient utili-zation of material strength is observed in the case of laminate with circular cutout as compared to thelaminate with other shaped cutouts. In addition, it is also concluded that although the buckling strengthof the (0/90)4s laminate is lower than that of the (+45/�45/0/90)2s and (45/�45)4s laminates, but itsstrength is increased in the advanced stage of postbuckling deformation.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction cutout under combined shear and compression loads. Zhang and

Thin laminated composites find widespread applications inmechanical, aerospace, marine and automotive industries becausethey are lighter and offer versatile structural performance in com-parison to that of conventional materials. It is well known that thincomposite laminates can sustain a much higher load after bucklingunder the action of various in-plane loads, such as, uni-axial com-pression, in-plane shear and combined in-plane shear and compres-sion loads. Further, cutouts are often made in these laminatesbecause of various reasons, for instance, ports for mechanical andelectrical systems, holes for damage inspections, and cutouts toserve as doors and windows, etc. The presence of these cutoutschanges the structural stability and strength of composite laminatesdrastically. So, it is essential to have a through knowledge of re-sponse of thin composite laminates with cutouts under different ser-vice load conditions, to design composite structures with cutoutsefficiently and economically.

A considerable literatures [1–6] have been devoted to the studyof buckling and postbuckling behavior of composite panels with cut-outs subjected to either uni-axial compression or in-plane shearloads. Very limited works have been reported in the literature re-lated to the buckling and postbuckling behavior of the laminate with

ll rights reserved.

); fax: +91 1596 244183.Kumar), sbsingh@bits-pilani.

Matthews [7] carried out the study on postbuckling response ofanisotropic laminated plates without cutout under combined com-pressive and shear loading and concluded that the postbucklingbehavior of anisotropic plates is largely dependent upon the sheardirection. Kumar and Kishore [8] developed interaction curves tostudy buckling behavior of symmetric and antisymmetric angle-and cross-ply laminates without cutout under combined shear andcompression loads. Britt [9] investigated for buckling load of bi-ax-ial, shear- and combined shear and compression-loaded anisotropicpanels with centrally located elliptical cutouts. Singh and Kumar[10] studied the postbuckling behavior and progressive failure re-sponse of composite laminates without cutouts under combinedin-plane loads and constructed the load interaction diagrams forthe buckling, the first-ply failure and ultimate failure loads. Further,Iyengar and Chakraborty [11] investigated the effect of transverseshear on the stability of composite laminated plates without cutoutunder in-plane compressive and shear loading using a simple higherorder shear deformation theory and by constructing the interactioncurves. Guo et al. [12] presented numerical and experimental stud-ies on various cutout reinforcement designs and their effect on thestress concentration and buckling behavior of shear loaded lami-nated and sandwich composite panels with cutouts. Guo et al. [13]studied the performance of cutout shape and edge reinforcementsin a composite C-section beam under static shear load anddemonstrated that the cutout induced stress concentration can besignificantly by appropriate cutout shape and edge reinforcements.

Nomenclature

b in-plane dimensions of the square plate in x- andy- direction

c dimensions of square and diamond cutoutsd diameter of circular cutoute and f major and minor axes of the elliptical cutoutE1, E2 and E3 principal Young’s moduli in fiber direction and

other two transverse directions, respectivelyG12, G13 and G23 shear moduli associated with planes 1-2, 1-3

and 2-3, respectivelyh thickness of the square plateNxy applied shear load per unit width in x-directionNx applied uni-axial load per unit width applied in x-direc-

tionR, S and T shear strengths of lamina in planes 2-3, 1-3 and 1-2,

respectively

u, v and w displacements in x, y and z directions, respectivelywmax transverse deflection (maximum transverse deflection)X (Xt or Xc) normal strength (tensile or compressive, respectively)

of lamina in fiber direction-1Y (Yt or Yc) normal strength (tensile or compressive, respectively)

of lamina in direction transverse to the fiber direction-1Z (Zt or Zc) normal strength (tensile or compressive, respectively)

of lamina in principal material direction-3, i.e. perpen-dicular to plane of lamina

v12, v13 and v23 Poisson’s ratios associated with planes 1-2, 1-3and 2-3, respectively

h fiber orientation with respect to x-directionhx and hy rotation of normal to the undeformed mid-plane in

xz- and yz-plane, respectively

Table 1Material properties of T300/5208 (pre-peg) graphite-epoxy.

Mechanical properties Values Strength properties Values

E1 132.58 GPa Xt 1.52 GPaE2 = E3 10.80 GPa Xc 1.70 GPaG12 = G13 5.70 GPa Yt = Zt 43.80 MPaG23 3.4 GPa Yc = Zc 43.80 MPav12 = v13 0.24 R 67.60 MPav23 0.49 S = T 86.90 MPa

Table 2Details of cutout shapes and their dimensions.

Cutout shape Cutout size (expressed as ratio)a

Square c/b = 0.420Circular d/b = 0.474Diamond c/b = 0.420Elliptical-vertical e/b = 0.335

f/b = 0.670Elliptical-horizontal e/b = 0.670

f/b = 0.335

a Refer Figs. 2 and 3 for various notations.

D. Kumar, S.B. Singh / Composites: Part B 43 (2012) 142–149 143

As manifested from the available literature that very little con-cern has been shown regarding the buckling and postbuckling re-sponse and failure of composite laminates with cutouts undercombined loading (i.e., uni-axial compression combined in-planeshear loads). The aim of present investigation is to explore thebuckling and postbuckling responses of thin, square and symmet-ric laminate with a centrally located cutout of various shapes (i.e.,circular, square, diamond, elliptical-vertical and elliptical-horizon-tal) under uni-axial compression combined with in-plane shear(positive and negative) loads. The interaction curves, so called fail-ure envelopes for buckling, first-ply failure and ultimate failureloads of the laminate are also developed. In addition, the effect ofcomposite lay-up on failure characteristics of laminates with a cir-cular cutout is also examined by considering three laminate config-urations, namely, quasi-isotropic [i.e., (+45/�45/0/90)2s], angle-ply[i.e., (45/�45)4s], and cross-ply [i.e., (0/90)4s].

2. Finite element model and problem specification in thepresent study

2.1. Finite element formulation and failure model

The present study is based on finite-element formulation usingfirst order shear deformation theory and von Karman’s assumptions.The FEM formulation is based on nine-noded Lagrangian elementhaving five degrees of freedom per node (i.e., u, v, w, hx, and hy).The resulting nonlinear algebraic finite element equations aresolved by Newton–Raphson technique. The tensor polynomial formof 3-D Tsai–Hill failure criterion is used to predict failure of a lamina.Five stress components used in the criterion in material directions(three in-plane stresses and two transverse shear stresses) were cal-culated at mid thickness of each layer of individual element usingthe constitutive equations and by applying proper transformation.An attempt is also being made in the present study to predict the on-set of delamination at the interface of two adjacent layers usinginterlaminar failure criterion [14]. Three transverse stresses at eachgauss point on the corresponding interface are calculated in materialdirections using integration of equilibrium equations and by apply-ing proper transformation. The in-plane stress variations used ineach equilibrium equation are derived from nodal values of in-planestresses. To predict the ultimate failure of laminate, a progressivefailure procedure as used by Singh and Kumar [15] has been imple-mented. In this progressive failure procedure, at each load step,gauss point stresses are used in tensor polynomial form of theTsai–Hill failure criterion. If failure occurs at a gauss point in a layerof an element, a reduction in the appropriate lamina stiffness isintroduced in accordance with the mode of failure. The laminate

stiffness is recomputed and failure is checked again at the same loadstep. If no failure occurs, the process is repeated at next load step.Ultimate failure is said to have occurred when the onset of delami-nation occurs at interface of any two layers of any element or whenthe plate is no longer able to carry any further increase in load due tolarge transverse deflection.

2.2. Material properties and geometric model

Each lamina of the laminate is assumed to be made up of T300/5208(pre-peg) graphite-epoxy with material properties shown in Table 1.Size of the square laminate considered is 279 mm � 279 2.16 mmwith ply thickness 0.135 mm. A quasi-isotropic laminate, having stack-ing sequence (+45/�45/0/90)2s (i.e., total 16 layers, bottom layer beingthe first layer), with a central cutout of various shapes (i.e., circular,square, diamond, elliptical-vertical and elliptical-horizontal) is consid-ered to observe the effect of cutout shape on buckling and postbucklingresponse of the laminate under uni-axial compression combined within-plane positive and negative shear loads. The area of cutout is samefor all cutout shapes and is equivalent to the area of the square cutouthaving c/b = 0.42; where, c refers to the side of the square cutout and brefers to the width of the square plate. Detail of the cutout shapes andtheir dimensions are given in Table 2. In addition, the effect of compos-

Fig. 1. Simply-supported boundary conditions along with positive and negativeshear load directions.

Table 3Convergence study for a quasi-isotropic laminate under combined loading withNxy = Nx = N.

Nos. ofelements

Non-dimensionalized buckling

load (i.e., Nb2=E2h3)

Non-dimensionalized first-ply

failure load (i.e., Nb2=E2h3)

72 10.9425 24.030596 10.8352 23.7087120 10.8352 23.4941144 10.8352 23.2795168 10.8352 23.1723

144 D. Kumar, S.B. Singh / Composites: Part B 43 (2012) 142–149

ite lay-up [i.e., (+45/�45/0/90)2s, (0/90)4s, and (45/�45)4s] on bucklingand postbuckling responses of laminate with a typical circular cutout isalso investigated.

yy

Fig. 2. Finite element mesh for a typical sq

2.3. Boundary and loading conditions

Laminate is assumed to be simply-supported on all edges (i.e.,at x = 0, x = b, y = 0 and y = b; where, b refers to the width of thesquare plate) with in-plane and flexural boundary conditions asdepicted in Fig. 1. The directions of positive and negative shearloads are also shown in Fig. 1. In addition to the shear load (appliedon all four edges), the uni-axial compression load is applied by con-straining the in-plane movement in x-direction at edge x = 0 (referFig. 1) and by applying compression load in x-direction on the edgex = b. Results for failure loads, and the corresponding deflectionsare presented in the following non-dimensionalized forms:

Uni-axial compression in the x-direction: Nxb2=E2h3.

In-plane shear load: Nxyb2=E2h3.

Maximum transverse deflection: wmax/h.

Here, E2 is the transverse elastic modulus of a lamina; h is thethickness of the laminate; b is the width of the square plate; Nx

is the compression load per unit width of the plate applied in x-direction; Nxy is the in-plane shear load per unit width of the plate;and, wmax is the maximum transverse deflection.

2.4. Convergence study

To fix the number of elements in the finite-element analysis ofcomposite laminate with a cutout, a convergence study was con-ducted for the laminate with a centrally located square cutout of sizec/b = 0.42, using 72, 96, 120, 144 and 168 elements. The convergenceof buckling and first-ply failure loads of simply-supported quasi-iso-tropic laminate under combined loading (i.e., uni-axial compressioncombined with positive shear) with Nxy = Nx was checked. From Ta-ble 3, it can be observed that a mesh of 144 elements gives suffi-ciently accurate results in terms of non-dimensionalized bucklingand first-ply failure loads. For the sake of uniformity, finite elementmesh of 144 elements has been considered for all shaped cutouts.Schematic of finite element mesh along with element- and

Element No.

Node No.

x

Fiber Orientation

Gauss points

Element No.

Node No.

x

Fiber Orientation

Gauss pointsGauss points

uare laminate with a circular cutout.

ccc

c

x

y

ccc

cc

c

x

y

ef

ef

ef

f

e

f

e

f

e

ef

ef

ef

ef

ef

ef

ef

f

e

f

e

f

e

f

e

f

e

f

e

f

e

(a) (b)

(d)(c)Fig. 3. Meshing of square laminate with: (a) diamond, (b) square; (c) elliptical-horizontal, and (d) elliptical-vertical cutouts.

50

60

70 Present Study Reference [11]

ad, N

xb2 /E

2h3

D. Kumar, S.B. Singh / Composites: Part B 43 (2012) 142–149 145

node-numbering schemes for a typical square laminate with a circu-lar cutout is illustrated in Fig. 2. Schematics of finite element meshfor square laminates with a cutout of other shapes are shown inFig. 3. The element- and node-numbering schemes for the laminatewith non-circular cutouts follow the same pattern as in the case oflaminate with circular cutout (refer Fig. 2).

0

10

20

30

40

-200 -150 -100 -50 0 50 100 150

Inplane shear load, Nxyb2/E2h

3

Uni

-axi

al c

ompr

essi

on lo

Fig. 4. Comparison of the buckling interaction curves obtained in the present studyand Ref. [11].

3. Verification of results

A special-purpose computer program was developed for thepresent study. Validation of the developed program was done bycomparing the results from the present study with the results inavailable literatures [4,11,16,17]. Firstly, the validation of interac-tion curves for buckling load is done with that developed by Iyen-gar and Chakraborty [11] for (60/�60/60/�60)s laminate undercombined loading with Nxy = Nx. The geometry and material prop-erties used for comparison are the same as specified in the refer-ence. A good agreement can be seen in Fig. 4 between interactioncurves obtained from the present study and that obtained fromthe reference [11]. In addition, the accuracy of the program is alsocorroborated by making comparisons of results obtained in thisinvestigation with that presented by Srivatsa and Krishna Murty[16], Guo [4] and Kosteletos [17]. Table 4 contains the details ofcomparison along with the validated results. The material proper-ties used were the same as given in the respective references. FromTable 4, a good agreement of the results from the developed pro-gram with the results presented by Srivatsa and Krishna Murty[16], Guo [4] and Kosteletos [17] can be observed. Further, a goodcomparison of load–deflection response of (45/�45)s laminatewithout cutout, under combined loading condition (i.e., Nxy = Nx),obtained in the present study with that of Kosteletos [17] can alsobe observed in Fig. 5.

4. Results and discussions

4.1. Effect of cutout shape

In this section, the effect of cutout shape on buckling and post-buckling response of the quasi-isotropic laminate under combinedloading condition is discussed. The load–deflection response of thelaminate with various shaped cutouts under uni-axial compressioncombined with positive and negative shear loads are shown inFig. 6. The ratio of uni-axial compression load to the shear load is

Table 4Verification of results.

S.No.

Reference Laminate configuration Boundaryconditions

Loading conditions(i.e., Nxy/Nx)

Result validated In thepresentstudy

In thereference

1 Srivatsa andKrishna Murty[16]

279.0 mm � 279.0 mm � 2.79 mm (±45)6s laminatewith central circular cutout with d/b = 0.2

Simplysupported onall edges

0.0 (i.e., Uni-axialcompression alone)

Non-dimensionalizedbuckling load i.e.,

Nxb2=E2h3

56.5 58.0

2 Guo [4] 320.0 mm � 320.0 mm � 2.0 mm (±45)4s laminatewith central circular cutout of diameter 44 mm (i.e.,d/b = 0.137)

Simplysupported onall edges

1 (i.e., Pure +ve shearwith Nx = 0.0)

Buckling load (kN) 8.54 8.40(8.50)a

Clamped onall edges

1 (i.e., Pure +ve shearwith Nx = 0.0)

Buckling load (kN) 11.6 11.4

3 Kosteletos[17]

279.0 mm � 279.0 mm � 0.54 mm (±45)s laminatewithout cutout

Clamped onall edges

1.0 (i.e., Compressioncombined with +veshear)

Non-dimensionalizedbuckling load i.e.,

Nb2=E2h3

21.5 21.0

wmax/h at

Nb2=E2h3 ¼ 30:0

2.50 3.20

1.0 (i.e., Compressioncombined with �veshear)

Non-dimensionalizedbuckling load i.e.,

Nb2=E2h3

51.0 50.0

wmax/h at Nb2=E2h3 2.46 3.10

a The quantities inside and outside parentheses represent the critical buckling load (kN) from the FE analysis and experimental investigation, respectively.

0 1 2 3 40

20

40

60

80

(45/-45)sNx = Nxy = N

Inpl

ane

load

, Nb2 /E

2h3

Maximum transverse deflection, wmax /h

Present study (+ve shear) Kosteletos [17] (+ve shear) Present study (-ve shear) Kosteletos [17] (-ve shear)

Fig. 5. Comparison of the load–deflection responses obtained in the present studyand Ref. [17].

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60 Nx = Nxy = N

+ve shear-ve shear Circular Square Diamond Elliptical-vertical Elliptical-horizontal

Maximum transverse deflection, wmax /h

Inpl

ane

load

, Nb2 /E

2h3

Fig. 6. Load–deflection responses of the laminate with various shaped cutoutsunder uni-axial compression combined with positive and negative shear loads.

146 D. Kumar, S.B. Singh / Composites: Part B 43 (2012) 142–149

taken to be unity (i.e., Nxy = Nx). The corresponding details of failurecharacteristics are provided in Table 5. The bar plots of failureloads and maximum transverse deflection, associated with thefirst-ply failure load (i.e., wmax/h) for various cutout shapes is por-trayed in Fig. 7.

The first observation that can be made from Fig. 6 is that exceptthe failure loads, the laminate with elliptical-vertical cutout hasthe maximum buckling and postbuckling strengths for a particularvalue of wmax/h, whereas the laminate with elliptical-horizontalcutout has the minimum strengths, irrespective of shear loaddirections. As given in Table 5, for positive shear load, the maxi-mum variations in buckling, first-ply failure and ultimate failureloads due to change in cutout shape, are within 15.4%, 11.1% and54.9%, respectively, whereas the corresponding values for negativeshear load are 8.40%, 13.4% and 53.2%, respectively. Hence, it can beobserved from Table 5 and Fig. 7 that the effect of cutout shape issignificant on the ultimate failure load because of different modesof failure. Further, as depicted in Fig. 7, the variations of failureloads and wmax/h with cutout shapes follow similar pattern for po-sitive and negative shear loads, higher values of these parametersbeing for the negative shear load for almost all cutout shapes.

Under uni-axial compression combined with positive shear load,the first-ply failure occurs at the outer edge of the laminate (i.e., atthe corner of the laminate) for all cutout shapes, except square anddiamond cutouts, wherein the first-ply failure starts at the cutoutedge near cutout corner. Under negative shear load, except in thecase of laminate with circular cutout, the critical location for first-ply failure remains cutout edge. In the case of the laminate with cir-cular cutout, the first-ply failure location remains at the outer edgeof the laminate, under negative shear load. It is important to mentionhere that the stress concentration may also occur, not only at thecutout edge, but also at the sharp corner of the laminate and hence,the first-ply failure may also occur near the corner of the laminate.This is because of the fact that rounded corners of the cutout (circu-lar or elliptical) may preclude the first-ply failure near cutout edgeand hence the first-ply failure may also start at the corner of the lam-inate. As given in Table 5, the mode of first-ply failure for the lami-nate with circular, square and elliptical-vertical cutouts remainstransverse mode of failure (i.e. matrix failure) caused by in-planenormal stresses transverse to the fiber direction, for positive as wellas negative shear load. In the case of laminate with diamond andelliptical-horizontal cutout, the modes of first-ply failure are

Table 5Failure characteristics of the quasi-isotropic laminate with various shaped cutouts under combined loading with Nxy/Nx = 1.0.

Cutout Shape Positive shear Negative shear

BLa/FPFb load/(wmax/h)c FEd/mode of FPF UFe/mode of UF BLa/FPFb load/(wmax/h)c FEd/mode of FPF UFe/mode of UF

Circular 11.1/25.9/2.60 1/Transversef 53.2/Loss of stiffness 11.8/29.1/3.66 103/Transversef 53.8/Loss of stiffnessSquare 10.7/23.3/2.42 72/Transverse 47.6/Loss of stiffness 11.7/28.0/3.52 114/Transverse 43.9/DelaminationDiamond 11.3/24.0/2.54 96/Delamination 24.0/Delamination 11.9/26.2/3.60 90/Transverse 28.3/DelaminationElliptical-vertical 11.8/26.2/2.36 139/Transverse 35.1/Delamination 11.9/26.5/3.01 54/Transverse 40.1/DelaminationElliptical-horizontal 10.0/25.0/3.12 1/Transverse 25.4/Delamination 10.9/25.2/3.85 90/Delamination 25.2/Delamination

a Buckling load.b First-ply failure.c Non-dimensionalized maximum transverse deflection at the first-ply failure.d First failed element number.e Ultimate failure load.f Transverse mode of failure refers to the failure of lamina in a direction perpendicular to the fiber direction due to in-plane stresses transverse to fiber direction.

CircularSquare

Diamond

Elliptical-vertical

Elliptical-horizontal0

10

20

30

40

50

60 +ve shear-ve shear Buckling load First-ply failure load Ultimate failure load wmax/h

Inpl

ane

load

, Nb2 /E

2h3

0

1

2

3

4

5

6Nx = Nxy = N

w max

/ h

Fig. 7. Bar plot of failure loads and the maximum transverse deflection for variouscutout shapes.

0

5

10

15

20

-30 -20 -10 0 10 20

Circular Square Diamond Elliptical-vertical Elliptical-horizontal

Inplane shear load, Nxyb2/E2h

3

Uni

-axi

al c

ompr

essi

on lo

ad, N

xb2 /E

2h3

Fig. 8. Interaction curves for buckling load of the quasi-isotropic laminate with acutout of various shapes.

0

10

20

30

40

50

-60 -45 -30 -15 0 15 30 45

Circular Square Diamond Elliptical-vertical Elliptical-horizontal

Inplane shear load, Nxyb2/E2h

3

Uni

-axi

al c

ompr

essi

on lo

ad, N

xb2 /E

2h3

Fig. 9. Interaction curves for first-ply failure load of the quasi-isotropic laminatewith a cutout of various shapes.

D. Kumar, S.B. Singh / Composites: Part B 43 (2012) 142–149 147

delamination and matrix failure, respectively, under positive shearload, and vice versa under negative shear load. In addition, it can alsobe noted from Table 5 that irrespective of shear load directions, theultimate failure is caused by the complete loss of stiffness in the caseof laminate with circular cutout. Hence, an efficient utilization ofmaterial strength is observed in the case of laminate with circularcutout as compared to the laminate with other shaped cutouts,wherein the early delamination, before the complete loss of stiff-ness, is the predominant mode of ultimate failure.

It is also important to observe from Fig. 6 that under combinedloading condition, the shear load direction has significant effect onpostbuckling stiffness (given by the slope of load versus deflectioncurve at a particular value of maximum transverse deflection) ofthe laminate. Under positive shear load, the laminate has greaterpostbuckling stiffness than that under negative shear load.

The effect of cutout shape on buckling and failure loads (corre-sponding to first-ply and ultimate failure) under combined uni-axialcompression and in-plane shear loads is examined by developinginteraction diagrams (failure envelopes) as shown in Figs. 8–10.The co-ordinates of any point on interaction diagrams for buckling,first-ply failure and ultimate failure loads correspond to the com-bined state of uni-axial compression and in-plane shear loads. FromFigs. 8–10, it is observed that the interaction diagrams of the lami-nate with various shaped cutouts for buckling, first-ply failure andultimate failure loads, respectively, are not symmetric aboutNxy = 0 line. It can also be noted that irrespective of cutout shapes,the buckling, first-ply failure and ultimate failure loads of the lami-nate with a cutout subjected to uni-axial compression will decrease

if an in-plane positive or negative shear is also applied in combina-tion with uni-axial compression. From Fig. 8, it is also worth men-tioning that for a given uni-axial compression load, a quasi-isotropic laminate with a cutout of various shapes can sustain highernegative shear load than the positive shear load, before the platebuckles [i.e., the values of (Nx + |Nxy|) for the initial buckling of the

0

20

40

60

80

-100 -75 -50 -25 0 25 50 75 100

Circular Square Diamond Elliptical-vertical Elliptical-horizontal

Inplane shear load, Nxyb2/E2h

3

Uni

-axi

al c

ompr

essi

on lo

ad, N

xb2 /E

2h3

Fig. 10. Interaction curves for ultimate failure load of the quasi-isotropic laminatewith a cutout of various shapes.

148 D. Kumar, S.B. Singh / Composites: Part B 43 (2012) 142–149

plate is more for negative shear than that for positive shear]. For highvalues of in-plane shear (positive and negative) load combined withlow values of uni-axial compression load, the effect of orientation ofan elliptical cutout (i.e., elliptical-vertical and elliptical-horizontal)on the buckling strength of the laminate is not significant; and, thelaminate with a diamond cutout gives the highest value of(Nx + |Nxy|) with respect to the buckling (Fig. 8) of the laminate.Moreover, in the middle range (i.e., low in-plane shear load com-bined with high uni-axial compression), the laminate with an ellip-tical-vertical cutout gives the highest value of (Nx + |Nxy|). On the

Table 6Failure characteristics of the laminate of different lay-ups with a circular cutout under co

Composite lay-ups Positive shear

BLa/FPFb load/(wmax/h)c FEd/Mode of FPF UFe/Mode of U

(45/�45/0/90)2s 11.1/25.9/2.60 1/Transversef 53.2/Loss of st(0/90)4s 6.7/25.1/2.88 139/Transverse 41.3/Loss of st(45/�45)4s 11.8/20.0/2.77 72/Transverse 38.4/Loss of st

a Buckling load.b First-ply failure.c Non-dimensionalized maximum transverse deflection at the first-ply failure.d First failed element number.e Ultimate failure load.f Transverse mode of failure refers to the failure of lamina in a direction perpendicul

0 2 4 6 80

10

20

30

40

50

60

+ve shear-ve shear (45/-45/0/90)2s (0/90)4s (45/-45)4s

Nx = Nxy = N

Inpl

ane

load

, Nb2 /E

2h3

Maximum transverse deflection, wmax /h

Fig. 11. Effect of composite lay-up on load–deflection response of the laminatewith a typical circular cutout under combined shear and compressive loads.

other hand, the laminate with an elliptical-horizontal cutout givesthe lowest value of (Nx + |Nxy|) corresponding to buckling of theplate, for almost all combinations of in-plane shear and uni-axialcompression load. It is also worth mentioning that the failure envel-ops corresponding to the buckling of the laminate with variousshaped cutouts intersect at some unique points, meaning therebythat for these combinations of loadings the buckling strengths oflaminates will be same for different shapes of the cutout. In addition,from Figs. 9 and 10, it can be observed that the effect of cutout shapeon the first-ply and ultimate failure loads is most significant whenhigh uni-axial compression load is combined with low in-planeshear load. As shown in Fig. 9, the effect of cutout shape on thefirst-ply failure load of the laminate is more substantial under highnegative shear than that under high positive shear. Further, fromFig. 10, it can be observed that the ultimate failure load interactioncurves have more number of discontinuities than the buckling andfirst-ply failure load interaction curves due to different modes offailure (i.e., loss of stiffness and delamination). It can be seen thatthe laminate with a circular cutout has the highest value of ultimatefailure load under combined shear and compressive loading,whereas the laminate with either a diamond or an elliptical-hori-zontal cutout is found to have the lowest value of ultimate failureload.

4.2. Effect of composite lay-up

In this section, the effect of composite lay-up on load deflectionresponse and failure characteristics of a simply-supported lami-nate with a typical circular cutout is investigated under combinedin-plane shear (positive and negative) and uni-axial compressionloads for unity load ratio (i.e., Nxy = Nx). The load deflection re-sponse of the laminate for various lay-ups [(+45/�45/0/90)2s, (0/90)4s and (45/�45)4s] is shown in Fig. 11 and corresponding detailsof failure characteristics are presented in Table 6.

From Table 6, it can be noted that irrespective of shear loaddirections, the angle-ply [i.e., (45/�45)4s] and the cross-ply [i.e.,(0/90)4s] laminates have maximum and minimum bucklingstrengths, respectively; whereas, the strengths corresponding tofirst-ply failure are maximum and minimum for the quasi-isotropic[i.e., (+45/�45/0/90)2s] and the angle-ply laminates, respectively.As far as the ultimate failure is concerned, the quasi-isotropic lam-inate has the maximum strength, for positive as well as negativeshear load, whereas the angle-ply and cross-ply laminates haveminimum strengths for positive and negative shear loads, respec-tively. It is also important to observe from Table 6 that the buck-ling, first-ply failure and ultimate failure strengths of quasi-isotropic and angle-ply laminate are more under negative shearload than that under positive shear load, whereas reverse is truefor cross-ply laminate. Moreover, there is no significant effect ofcomposite lay-ups on the maximum transverse deflection corre-sponding to first-ply failure load, for positive as well as negative

mbined loading with Nxy/Nx = 1.0.

Negative shear

F BLa/FPFb load/ (wmax/h)c FEd/Mode of FPF UFe/Mode of UF

iffness 11.8/29.1/3.66 103/Transversef 53.8/Loss of stiffnessiffness 5.8/25.0/3.84 103/Transverse 30.8/Loss of stiffnessiffness 13.1/24.3/4.04 114/Transverse 39.6/Loss of stiffness

ar to the fiber direction due to in-plane stresses transverse to fiber direction.

D. Kumar, S.B. Singh / Composites: Part B 43 (2012) 142–149 149

shear load direction. It is also worth mentioning that irrespectiveof shear load directions, the first-ply failure occurs at the outeredge of the laminate for (+45/�45/0/90)2s and (0/90)4s laminationsequences, whereas the cutout edge is the critical location for first-ply failure in the case of (45/�45)4s laminate. The mode of first-plyfailure remains transverse mode of failure (i.e. matrix failure)caused by in-plane normal stresses transverse to the fiber direc-tion, irrespective of composite lay-ups and directions of shear load.Further, the ultimate failure is caused by complete loss of stiffnessin all cases.

From Fig. 11, it can also be observed that irrespective of com-posite lay-ups, the laminate under positive shear load have morepostbuckling stiffness (given by the slope of the load deflectionplot at a particular value of maximum transverse deflection) thanthat under negative shear load. Further, it is to be noted thatalthough the buckling load of the (0/90)4s laminate is lower thanthat of the other two laminates, its strength is increased in the ad-vanced stage of the postbuckling deformation and becomes morethan that of (45/�45)4s laminate.

5. Concluding remarks

Based on the results of the present investigation, followingimportant conclusions can be drawn:

� For unit load ratio (i.e., Nxy/Nx), except the values of first-ply andultimate failure loads, the quasi-isotropic laminate with ellipti-cal-vertical cutout has the maximum buckling load and post-buckling strengths for a particular value of wmax/h, whereas thelaminate with elliptical-horizontal cutout has the minimumstrengths; the maximum effect of the cutout shape is observedon the ultimate failure load, irrespective of shear load directions.� For combined loading with equal compression and shear, the

angle-ply [i.e., (45/�45)4s] and the cross-ply [i.e., (0/90)4s] lam-inates with a circular cutout have maximum and minimumbuckling strengths, respectively, whereas the strengths corre-sponding to first-ply failure are maximum and minimum forthe quasi-isotropic [i.e., (+45/�45/0/90)2s] and the angle-ply[i.e., (45/�45)4s] laminates with a circular cutout, respectively.� The quasi-isotropic laminate subjected to combined loading has

more buckling and failure loads under negative shear load thanthat under positive shear load, for all cutout shapes and thesame is true for an angle-ply laminate with a circular cutout;whereas, the opposite is true for a cross-ply laminate with a cir-cular cutout.� The first-ply failure occurs either at the outer edge of the lami-

nate or at the cutout edge, irrespective of shear load directions,cutout shapes and composite lay-ups. The mode of first-ply fail-ure for the quasi-isotropic, angle-ply and cross-ply laminateswith circular cutout, and the quasi-isotropic laminate withsquare and elliptical-vertical cutouts is matrix failure, for posi-tive as well as negative shear load, whereas in the case ofquasi-isotropic laminate with diamond and elliptical-horizontalcutout, the modes of first-ply failure are either delamination ormatrix failure depending upon the direction of the shear load.� The quasi-isotropic laminate with a circular cutout has the

maximum ultimate strength as compared to the cross-ply andangle-ply laminates with circular cutouts, for combined loadwith equal level of compression and shear loads; however, thepostbuckling reserve strength, defined as ratio of the first-plyfailure or ultimate failure load to the buckling load, is highestfor the cross-ply laminate.

� In the case of (+45/�45/0/90)2s, (45/�45)4s and (0/90)4s lami-nates with a circular cutout, the ultimate failure is caused bylarge transverse deflection; whereas, delamination is the pre-dominant cause of ultimate failure for the quasi-isotropic lami-nate with non-circular cutouts, irrespective of shear loaddirections and hence, an efficient utilization of material strengthis observed in the case of laminates with circular cutouts.� Irrespective of the cutout shape and lamination sequence, the

shear load direction has significant effect on postbuckling stiff-ness of the laminate with cutout under combined in-planeloads; the greater postbuckling stiffness being under positiveshear load than that under negative shear load.� The effect of orientation of an elliptical cutout on the buckling

strength of the quasi-isotropic laminate is not significant forhigh values of in-plane shear (positive and negative) load com-bined with low values of uni-axial compression load; further,there are unique values (i.e., corresponding to the point of inter-action of different failure envelopes) of combined loading forwhich the buckling strength of the laminate will be same fordifferent shapes of the cutout.

Acknowledgements

The present work is the part of CSIR project (No. 22(0442)/07/EMR-II) and K.K. Birla Academy project. The financial support byCSIR, New Delhi and K.K. Birla Academy to execute the project ishighly appreciated.

References

[1] Nemeth MP. Buckling and postbuckling behavior of laminated compositeplates with a cutout. NASA technical paper 3587, July 1996.

[2] Jha PN, Kumar A. Response and failure of square laminates under combinedloads. Compos Struct 2002:55(3):p. 337–45.

[3] Jain P, Kumar A. Postbuckling response of square laminates with a centralcircular/elliptical cutout. Compos Struct 2004;65:179–85.

[4] Guo SJ. Stress concentration and buckling behaviour of shear loaded compositepanels with reinforced cutouts. Compos Struct 2007;80(1):1–9.

[5] Kumar D, Singh SB. Effects of boundary conditions on buckling andpostbuckling responses of composite laminate with various shaped cutouts.Compos Struct 2010;92:769–79.

[6] Kumar D, Singh SB. Postbuckling strengths of composite laminate with variousshaped cutouts under in-plane shear. Compos Struct 2010;92:2966–78.

[7] Zhang Y, Matthews FL. Postbuckling behavior of anisotropic laminated platesunder pure shear and shear combined with compressive loading. AIAA1984;22(2):281–6.

[8] Kumar A, Kishore BNR. Buckling of antisymmtric angle- and cross-plyrectangular plates under shear and compression. Int J Mech Sci1991;33(1):31–9.

[9] Britt VO. Shear and compression buckling analysis for anisotropic panels withelliptical cutouts. AIAA 1994;32(11):2293–9.

[10] Singh SB, Kumar A. Postbuckling response and strength of laminates undercombined in-plane loads. Compos Sci Technol 1999;59:727–36.

[11] Iyengar NGR, Chakraborty A. Study of interaction curves for compositelaminate subjected to in-plane uniaxial and shear loadings. Compos Struct2004;64(3–4):307–15.

[12] Guo S, Zhou L, Cheung CW. Cutout reinforcements for shear loaded laminateand sandwich composite panels. Int J Mech Mater Des 2008;4:157–71.

[13] Guo S, Morishima R, Zhang X, Mills A. Cutout shape and reinforcement designfor composite C-section beams under shear load. Compos Struct2009;88:179–87.

[14] Singh SB, Kumar D. Postbuckling response and failure of symmetric laminatedplates with rectangular cutouts under uniaxial compression. Struct Eng Mech2008;29(4):455–67.

[15] Singh SB, Kumar A. Postbuckling response and failure of symmetric laminatesunder in-plane shear. Compos Sci Technol 1998;58:1949–60.

[16] Srivatsa KS, Krishna Murty AV. Stability of laminated composite plates withcut-outs. Comput Struct 1992;43:273–9.

[17] Kosteletos S. Postbuckling response of laminated plates under shear loads.Compos Struct 1992;20:137–45.