Load interaction curves and postbuckling response of composite laminate cutout in-plane loading

Load interaction curves and postbuckling response of composite laminate with circular cutout under combined in-plane loadingDinesh Kumar a ,S.B.Singh b ,⇑a Mechanical Engineering Department,Birla Institute of Technology and Science,Pilani 333031,India bCivil Engineering Department,Birla Institute of Technology and Science,Pilani 333031,Indiaa r t i c l e i n f o Article history:Received 14December 2010Received in revised form 22February 2011Accepted 14March 2011Available online 17March 2011Keywords:minates B.BucklingC.Finite element analysis (FEA)Postbucklinga b s t r a c tThe objective of this paper is to study the interaction curves along with buckling and postbuckling responses of a quasi-isotropic laminate with a centrally placed circular cutout of various sizes under uni-axial compression combined with in-plane shear loads (positive and negative).The present study is carried out using finite element method.The finite element formulation is based on the first order shear deformation theory and von Karman’s assumptions to incorporate geometric nonlinearity.The resulting nonlinear algebraic equations are solved by Newton–Raphson method.The 3-D Tsai-Hill criterion is used to predict the failure of a lamina while the onset of delamination is predicted by the interlaminar failure criterion.It is observed that pure compression-buckling,-first-ply failure and -ultimate failure strengths of the quasi-isotropic laminate with and without circular cutout decreases with increasing accompanying shear load.In addition,it is also noted that for a given uni-axial compression load,a quasi-isotropic lam-inate with and without a cutout can sustain higher negative shear load in comparison to the positive shear load.Ó2011Elsevier Ltd.All rights reserved.1.IntroductionBecause of high specific stiffness and strength,composite offers versatile structural performance than the conventional materials,and hence provides unique opportunities in posite panels are used extensively in the form of thin plates which may initiate buckling under the action of in-plane compressive and shear loads.Further,the presence of cutouts in these laminates for various reasons (for instance,ports for mechanical and electri-cal systems,holes for damage inspections,and cutouts to serve as doors and windows)change the mechanical behavior of these lam-inates.An in-depth understanding of their response to different work load conditions,such as uni-axial compression,in-plane shear,and uni-axial compression combined with in-plane shear loads,is desirable to design composite structures with cutouts effectively and efficiently.As evident from the comprehensive review by Nemeth [1]in 1996,a considerable literature has been devoted to the study of buckling and postbuckling behavior of composite panels with cut-outs subjected to either uni-axial compression or in-plane shear loads.Thereafter,many investigators [2–6]also contributed in the field of buckling and/or postbuckling response of composite laminates under various edges conditions and subjected to either uni-axial compression or in-plane shear loads.As far as buckling and postbuckling behavior of the laminate with cutout under com-bined shear and compression loads is concerned,very limited works have been reported in the literature.Zhang and Matthews [7]studied the postbuckling behavior of anisotropic laminated plates without cutout under combined compressive and shear loading and concluded that the postbuckling behavior of aniso-tropic plates is largely dependent upon the shear direction.Kumar and Kishore [8]developed interaction curves to study buckling behavior of symmetric and antisymmetric angle-and cross-ply laminates without cutout under combined shear and compression loads.In addition,an approximate analysis for buckling load of bi-axial,shear-and combined shear and compression-loaded aniso-tropic panels with centrally located elliptical cutouts was carried out by Britt [9].Singh and Kumar [10]studied the postbuckling re-sponse and strength of laminates without cutouts under combined in-plane loads and constructed the load interaction diagrams for the buckling,the first-ply failure and ultimate failure loads.Fur-ther,Iyengar and Chakraborty [11]investigated the effect of trans-verse shear on the stability of composite laminated plates without cutout under in-plane compressive and shear loading using a sim-ple higher order shear deformation theory and by constructing the interaction curves.It is evident from the available literature that there is lack of studies on buckling and postbuckling response of composite1359-8368/$-see front matter Ó2011Elsevier Ltd.All rights reserved.doi:10.1016/positesb.2011.03.002⇑Corresponding author.Tel.:+919414648283(mobile);fax:+911596244183.E-mail addresses:dineshkr@bits-pilani.ac.in (D.Kumar),sbsingh@bits-pilani.ac.in (S.B.Singh).laminates with cutouts under combined in-plane compression andin-plane shear loads.The present study investigates the buckling and postbuckling failure of thin,square and symmetric quasi-iso-tropic laminate with a centrally located circular cutout of different sizes under uni-axial compression combined with in-plane shear.The interaction curves for the buckling,the first-ply failure and the ultimate failure loads are also studied.2.Present study2.1.Finite element formulation and failure modelA special-purpose computer program isent study which is based on the finite-element first order shear deformation theory with a nine element having five degrees of freedom per and h y ).Geometric non-linearity based on von tions has been incorporated.The nonlinear equations are solved by Newton–Raphson tails of FEM formulation are well [12],but for ready reference,a brief model associated with the equations governing ite plates is given in Appendix A.Failure of a lamina is predicted by tensor 3-D Tsai-Hill criterion wherein five stress directions (three in-plane stresses and two ses)were calculated at mid thickness of each element using the constitutive equations and transformation.In addition,an attempt has ent study to predict the onset of delamination two adjacent layers using interlaminar failure transverse stresses at each gauss point on interface are calculated in material directions equilibrium equations and by applying The in-plane stress variations used in each are derived from nodal values of in-plane ultimate failure of laminate,a progressive used by Singh and Kumar [14]has been progressive failure procedure,at each load step,gauss point stres-ses are used in tensor polynomial form of the Tsai-Hill failure cri-terion.If failure occurs at a gauss point in a layer of an element,a reduction in the appropriate lamina stiffness is introduced in accordance with the mode of failure.The laminate stiffness is recomputed and failure is checked again at the same load step.If no failure occurs,the process is repeated at next load step.Nomenclature{a i }nodal displacement vector for i th nodeD f a j g incremental displacements at the end of the j th iteration bin-plane dimensions of the square plate in x -and y -directionB 0,B b and B s strain–displacement matrices corresponding to in-plane axial,bending and shear strains,respectivelyd diameter of circular cutoutE 1,E 2and E 3principal Young’s moduli in fiber direction andother two transverse directions,respectivelyF external applied loads (includes in-plane loads as wellas transverse forces)G 12,G 13and G 23shear moduli associated with planes 1–2,1–3and 2–3,respectivelyh thickness of the square plate I 55Â5unit matrix K T tangent stiffness matrix M moment resultants per unit length n number of nodes in an element N stress resultants per unit lengthN i (i =1,n )shape functions of a nine noded Lagrangian element N xy applied shear load per unit width of the plateN xapplied uni-axial load per unit width applied in x -direc-tion Q transverse shear stress resultants per unit length R ,S and Tshear strengths of lamina in planes 2–3,1–3and 1–2,respectivelyu ,v and w displacements in x ,y and z directions,respectively U displacement components at a point within an element w max transverse deflection (maximum transverse deflection)X t (X c )tensile (compressive)normal strength of lamina in fiberdirection-1Y t (Y c )tensile (compressive)normal strength of lamina in thedirection transverse to the fiber direction-1Z t (Z c )tensile (compressive)normal strength of lamina in prin-cipal material direction-3,i.e.perpendicular to plane oflaminam 12,m 13and m 23Poisson’s ratios associated with planes 1–2,1–3and 2–3,respectivelyh fiber orientation with respect to x -directionh x and h y rotation of normal to the undeformed mid-plane in xz -and yz -plane,respectivelyw residual forceTable 11190 D.Kumar,S.B.Singh /Composites:Part B 42(2011)1189–1195Ultimate failure is said to have occurred when the onset of delam-ination occurs at interface of any two layers of any element orwhen the plate is no longer able to carry any further increase in load due to large transverse deflection.2.2.Material properties and geometric modelProperties of the material (T300/5208graphite epoxy)of eachlamina are presented in Table 1.A full square laminate of size 279mm Â279mm Â2.16mm with ply thickness 0.135mm is considered.A quasi-isotropic laminate,having stacking sequence (+45/À45/0/90)2s (i.e.,total 16layers,bottom layer being the first layer),with and without a central circular cutout of various sizes (d/b =0.16,0.32and 0.48)has been investigated.2.3.Boundary and loading conditionsIn-plane and flexural boundary conditions corresponding to simply-supported edges (i.e.,at x =0,x =b ,y =0and y =b )are de-picted in Fig.1;where,b refers to the width of the square plate.The notations for positive and negative shear loads are also shown in Fig.1.In addition to the shear load,the compression load is applied by constraining the in-plane movement in x -direction at edge x =0Table 2Convergence study.Nos.of elementsNos.of nodesNon-dimensionalized buckling load (i.e.,N xy b 2/E 2h 3)Non-dimensionalized first-ply failure load (i.e.,N xy b 2/E 2h 3)Smaller cutout aLarger cutout a Smaller cutout Larger cutout 7233615.126332.934611.049725.31789643214.911732.934610.942525.532412052814.911733.149210.942525.747014462414.911733.256410.942525.854216872014.911733.363710.942525.8542aSmaller and larger cutouts refer to the circular cutouts with d /b =0.16and 0.48,respectively.1223341011465781234456756788900111122132435466857990101116111621101114254365476698587809910012345679801021311223478911152712172241526374870659819210311451627384971608293104115112123572839461011381318236789023461151671829405173628495106117112527463193048596101491419248243901610711892031425375648697108119122232435476658798101251015202510111123465780920111101212233445577668899ElementFiberNo.Node No.xyOrientationGauss pointsD.Kumar,S.B.Singh /Composites:Part B 42(2011)1189–11951191and by applying compression load in x-direction on the edge x=b (refer Fig.1).Results for failure loads,and the corresponding deflections are presented in the following non-dimensionalized forms:Uni-axial compression in the x-direction:N x b2/E2h3;in-plane shear load:N xy b2/E2h3;maximum transverse deflection:w max/h.Here,E2is the transverse elastic modulus of a lamina;h is the thickness of the laminate;b is the width of the square plate;N x is the compression load per unit width of the plate applied in x-direction;N xy is the in-plane shear load per unit width of the plate; and,w max is the maximum transverse deflection.2.4.Convergence studyBeforefixing the number of elements in thefinite-element anal-ysis of composite laminate with a cutout,a convergence study was conducted for the laminate with a centrally located circular cutout, using72,96,120,144and168elements.Two circular cutouts with d/b=0.16and0.48(i.e.,for smaller and larger cutout sizes,respec-tively)were considered for this convergence study.The conver-gence of buckling andfirst-ply failure loads of simply-supported quasi-isotropic laminate under combined loading(i.e.,uni-axial compression combined with positive shear)with N xy=N x was checked.From Table2,it can be observed that a mesh of144ele-ments gives sufficient accurate results in terms of non-dimension-alized buckling andfirst-ply failure loads.Based on this study,a mesh of144elements is used for the laminate with circular cutout as well in the present investigation.In the case of laminate without cutout,the convergence of results has been obtained forfinite ele-ment mesh of5Â5and corroborate with the results obtained by Singh and Kumar[10].Schematic offinite element meshes along with element-and node-numbering scheme for a typical square laminate without cutout and with circular cutout are illustrated in Figs.2and3,respectively.3.Verification of resultsThe accuracy of the developed program is validated by compar-ing the results from the present study with the available results in the literature[4,11,15–16].Firstly,the validation of interaction curve for buckling load is done with that developed by Iyengar and Chakraborty[11]for(60/À60/60/À60)s laminate without cut-out under combined loading with N xy=N x.The geometry and material properties used for comparison are the same as specified in the reference.The comparative study of interaction curves ob-tained from the present study and that obtained from Ref.[11]is shown in Fig.4.A good agreement can be seen from Fig.4in the results of the present study with the results in the reference.In addition,the accuracy of the program is also corroborated by mak-ing comparisons of results obtained in this investigation with that presented by Srivatsa and Krishna Murty[15],Guo[4]and Kostele-tos[16].Table3contains the details of comparison along with the validated results.The material properties used were the same as given in the respective references.From Table4,a good agreement of the results from the developed program with the results pre-sented by Srivatsa and Krishna Murty[15],Guo[4]and Kosteletos [16]can be observed.4.Results and discussionsIn this section,load interaction diagrams(i.e.,strength envel-ops)for the quasi-isotropic laminate with cutout of various sizes are discussed.The coordinates of any point on these envelops giveTable3Verification of results.S. No.References Laminate/stacking sequence BoundaryconditionsLoading conditions(i.e.,N xy/N x)Result validated In thepresentstudyIn thereference1Srivatsa andKrishnaMurty[15]279.0mmÂ279.0mmÂ2.79mm plate withcentral circular cutout with d/b=0.2/(±45)6sSimplysupportedon all edges0.0(i.e.,Uni-axialcompression alone)Non-dimensionalizedbuckling load i.e.,N x b2/E2h356.558.02Guo[4]320.0mmÂ320.0mmÂ2.0mm plate withcentral circular cutout of diameter44mm(i.e.,d/b=0.137)/(±45)4s Simplysupportedon all edges1(i.e.,Pure+veshear)Buckling load(kN)8.548.40(8.50)aClamped onall edges1(i.e.,Pure+veshear)Buckling load(kN)11.611.43Kosteletos[16]279.0mmÂ279.0mmÂ0.54mm platewithout cutout/(±45)sClamped onall edges1.0(i.e.,Compre-ssion combined with+ve shear)Non-dimensionalizedbuckling load i.e.,N x b2/E2h3(Or,N xy b2/E2h3)21.521.0w max/h at N x b2/E2h3(Or,N xy b2/E2h3)=30.02.503.201.0(i.e.,Compre-ssion combined withÀve shear)Non-dimensionalizedbuckling load i.e.,N x b2/E2h3(Or,N xy b2/E2h3)51.050.0w max/h at N x b2/E2h3(Or,N xy b2/E2h3)=30.02.463.10a The quantities inside and outside parentheses represent the critical buckling load(kN)from the FE analysis and experimental investigation,respectively. 1192 D.Kumar,S.B.Singh/Composites:Part B42(2011)1189–1195the combination of uni-axial compression and the shear loads at which the laminate would fail.Load interaction diagrams for the buckling,thefirst-ply failure and the ultimate failure loads are shown in Figs.5–7,respectively,under the combined action of uni-axial compression and in-plane shear loads.From Figs.5–7,thefirst observation that can be made is that, with the introduction of a cutout in the laminate,the buckling,first-ply failure and the ultimate failure loads of the laminate are re-duced.It can also be observed from thesefigures that the pure com-pression buckling,first-ply failure and ultimate strengths of the quasi-isotropic laminate with and without cutout of various sizes decreases with increasing accompanying shear load(positive as well as negative).An interesting effect on the combination of shear and compression that can be noted from Figs.5and6is that there is a shift in buckling andfirst-ply failure loads interaction diagrams towards the negative shear axis.Hence,for a given uni-axial com-pression load,a quasi-isotropic laminate with and without a cutout of various sizes can sustain higher negative shear load than the po-sitive shear load(i.e.,the values of N xþj N xy j for the initial buckling and thefirst-ply failure loads are more for negative shear than that for positive shear).Moreover,it can also be noticed from Figs.5and 6that under combined loading conditions,the buckling andfirst-ply failure strengths of the laminate decrease with the increase in d/b ratio of the circular cutout.Further,from Fig.7,it can be ob-served that the ultimate failure load interaction curves have more number of discontinuities than the buckling andfirst-ply failure load interaction curves due to different modes of failure.Load–deflection responses of the laminate with and without cir-cular cutout under combined loading conditions for a typical loadTable4Failure characteristics of(+45/À45/0/90)2s laminate with and without a circular cutout under combined loading with a typical load ratio N xy/N x=1.0.d/b ratio Positive shear Negative shearBL a/FPF b load/(w max/h)c FE d/mode of FPF UF e/mode of UF BL a/FPF b load/(w max/h)c FE d/mode of FPF UF e/mode of UF0.016.4/35.4/2.401/Transverse f87.2/Loss of stiffness16.5/41.0/3.395/Transverse f83.4/Loss of stiffness0.1614.9/33.3/2.51139/Transverse56.2/Loss of stiffness15.1/36.2/3.26103/Transverse66.1/Loss of stiffness0.3212.8/29.5/2.57139/Transverse61.9/Loss of stiffness13.2/33.4/3.49103/Transverse60.4/Loss of stiffness0.4811.1/25.9/2.601/Transverse53.2/Loss of stiffness11.8/29.1/3.66103/Transverse53.8/Loss of stiffnessa Buckling load.b First-ply failure.c Non-dimensionalized maximum transverse deflection at thefirst-ply failure.d First failed element number.e Ultimate failure.f Transverse mode of failure refers to the failure of lamina in a direction perpendicular to thefiber direction due to in-plane stresses transverse tofiber direction.D.Kumar,S.B.Singh/Composites:Part B42(2011)1189–11951193ratio (i.e.,N xy /N x )equal to 1.0are shown in Fig.8for positive and negative shear loads.The corresponding details of the failure char-acteristics in terms of buckling,first-ply failure and ultimate failure loads,location and maximum transverse deflection associated with first-ply failure,and the modes of first-ply and ultimate failures are provided in Table 4.It can be noted from Fig.8and Table 4that,except in the case of laminate with circular cutout with d /b =0.32under positive shear load,the buckling and postbuckling strengths of the laminate decrease with the increase in cutout size.Further,the maximum transverse deflection corresponding to first-ply failure load remains,more or less,constant as d/b is varied,for positive as well as negative shear load.It is also important to note from Table 4that,irrespective of d/b ratio and shear load direction,the first-ply failure starts at the corners of the laminate with its mode of failure as matrix failure which is caused by in-plane nor-mal stresses transverse to the fiber direction.In addition,for all cases,there is no delamination observed before the laminate losses its stiffness completely and hence,the ultimate failure is caused by complete loss of the stiffness.For N xy /N x equal to 1.0,the variations in load–deflection re-sponses with the shear load directions are shown in Fig.9for the laminate with a circular cutout of sizes (i.e.,d/b ratio)0.0and 0.32.It can be seen from Fig.9that,for d/b ratios 0.0and 0.32,the buckling and postbuckling responses of the laminate are al-most same for positive and negative shear loads in the initial stage of postbuckling (i.e.,for w max /h <1.4),but as the maximum trans-verse deflection increases,the difference in load–deflection re-sponses becomes large.Further,it can also be observed that at higher values of w max /h (i.e.,>1.4),the postbuckling stiffness (given by the slope of load versus deflection curve at a particular value of maximum transverse deflection)of the laminate under positive shear load is more than that under negative shear load.Furthermore,it is worth mentioning that from the detailed study of failure characteristics of the laminate with and without circular cutout of various sizes and subjected to combined loading conditions,it is observed that,irrespective of N xy /N x ratio and shear load direction,the mode of first-ply failure remains matrix failure.Further,it is also noted that for almost all N xy /N x and d/b ratios,the first-ply failure occurs near the diagonal corner of the laminate,ex-cept in the case of laminate with circular cutout of d/b =0.16and subjected to negative shear with higher N xy /N x ratios (i.e.,from À1to À5.6),wherein it starts at the cutout edges.In addition,it is also found that,irrespective of load ratio,direction of shear load and cutout size,the ultimate failure of the laminate is caused by the complete loss of its stiffness.5.ConclusionsBased on the results and discussions on the present investiga-tion,following important conclusions can be drawn:Under combined loading conditions,the introduction of a cut-out at the center of the laminate reduces the buckling,first-ply failure and the ultimate failure loads of the laminate,irre-spective of shear load direction and load ratio (i.e.,N xy /N x ). Compression buckling,first-ply failure and ultimate strengths of the quasi-isotropic laminate with and without cutout of var-ious sizes decreases with increasing accompanying shear load (for positive as well as negative shear loads).The values of ðN x þj N xy jÞfor the initial buckling and the first-ply failure loads are more for negative shear as compared to the positive shear.1194 D.Kumar,S.B.Singh /Composites:Part B 42(2011)1189–1195Under combined loading conditions,the buckling andfirst-ply failure strengths of the laminate decrease with the increase in d/b ratio of the circular cutout.Irrespective of N xy/N x ratio and shear load direction,the mode of first-ply failure remains matrix failure for the quasi-isotropic laminate with and without a circular cutout.Under combined loading with positive shear load,thefirst-ply failure occurs near the diagonal corner of the laminate,for all N xy/N x and d/b ratios;but,in the case of laminate under uni-axial compression combined with negative shear load with higher N xy/N x ratios,thefirst-ply failure takes place at the cut-out edges for the laminate with smaller size circular cutout(i.e.,d/b=0.16),whereas for other cutout sizes and load ratios,it starts at the diagonal corners.For N xy/N x equal to unity and at higher values of w max/h(i.e., >1.4),the laminate with a circular cutout of sizes d/b=0.0and0.32have more postbuckling stiffness under positive shear loadthan that under negative shear load.Irrespective of load ratio,direction of shear load and cutout size, the ultimate failure of the laminate is caused by the complete loss of its stiffness and there is no delamination observed before the laminate losses its stiffness completely. AcknowledgementsThe present work is the part of CSIR Project(No.22(0442)/07/ EMR-II)and K.K.Birla Academy project.Thefinancial support by CSIR,New Delhi and K.K.Birla Academy to execute the project is highly appreciated.Appendix AThefinite element formulation for laminated composite plates based on thefirst order shear deformation theory is briefly de-scribed here.The formulation is based on the virtual work equation for a continuum in the total Lagrangian coordinate system under the assumption of small strains.The displacement within an element is interpolated by an expression of the formf U g¼½u;v;w;h x;h y T¼X Ti¼1½N i I5 f a i g;where{U}is the value of displacement components at a point with-in an element,n the number of nodes in an element,N i the shapefunctions of a nine noded Lagrangian element,for i=1,n,I5the5Â5unit matrix.f a i g¼½u0i;v0i;w0i;h xi;h yi T is the nodal displace-ment vector for i th node.From the principle of virtual work and the total Lagrangian for-mulation,the element nonlinear equilibrium equation is derivedas:w f a g¼ZA½B0 T f N gþ½B b T f M gþ½B s T f Q g dAÀF¼0where w f a g is the residual force which is a function of displacementvector{a},[B0],[B b]and[B s]the strain–displacement matrices cor-responding to in-plane axial,bending and shear strains,respec-tively,{N}the stress resultants per unit length,{M}the momentresultants per unit length,{Q}the transverse shear stress resultantsper unit length,F is the external applied loads(includes in-planeloads as well as transverse forces).The Newton–Raphson method is used to solve these nonlinearalgebraic equations using a combined incremental and iterative procedure.If for an initial estimate of{a j}(i.e.,for j th iteration), the residual forces w f a gj–0,then an improved solution f a jþ1g is ob-tained by equating to zero the linearized Taylor’s series expansionof w f a gjþ1in the neighborhood of{a j}asw f a gjþ1ffiw f a gjþK T D f a j g¼0where D f a j g is the incremental displacement vector and K T is the tangent stiffness matrix evaluated at{a j}and is given by:K T¼@w f a gj@½aThe improved solution is then found as:f a jþ1g¼f a j gþD f a j g:To improve on the numerical stability and convergence of the solu-tion,the load is applied in small increments.The iterative solution is checked for convergence using the following criterion:w T wF F"#1=2Â1006bwhere b is sufficiently small number,i.e.,0.001%.The integration of expressions for w{a}and K T is carried out using the Gaussian quadrature.A selective integration scheme is adopted with a3Â3integration rule to evaluate integrals of the functions of the membrane and the bending behavior and a2Â2 integration rule is used for the transverse shear component.References[1]Nemeth MP.Buckling and postbuckling behavior of laminated compositeplates with a cutout.NASA Technical Paper3587;July1996.[2]Jha PN,Kumar A.Response and failure of square laminates under combinedput Struct2002;55:337–45.[3]Jain P,Kumar A.Postbuckling response of square laminates with a centralcircular/elliptical put Struct2004;65:179–85.[4]Guo SJ.Stress concentration and buckling behaviour of shear loaded compositepanels with reinforced put Struct2007;80(1):1–9.[5]Guo S,Morishima R,Zhang X,Mills A.Cutout shape and reinforcement designfor composite C-section beams under shear pos Struct 2009;88(2):179–87.[6]Kumar D,Singh SB.Postbuckling strengths of composite laminate with variousshaped cutouts under in-plane pos Struct2010;92:2966–78. 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