no

,*,

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A review of the recent development of the nite element analysis for laminated composite plates from 1990 is presented in thispaper. The literature review is devoted to the recently developed nite elements based on the various laminated plate theories forthe free vibration and dynamics, buckling and postbuckling analysis, geometric nonlinearity and large deformation analysis, and fail-

variety of industrial areas due to their high stiness and

Using nite element method, a signicant amount ofresearch has been devoted to the analysis of vibration

1990 based on the various laminated plate theories for

The laminated plate theories are essential to provideaccurate analysis of laminated composite plates, and a vari-ety of laminated plate theories have been developed andreported in a large amount of literatures.

A review of various equivalent single layer and layer-wise laminated plate theories was presented by Reddy

* Corresponding author. Tel.: +61 2 62688169; fax: +61 2 62688276.E-mail address: y.zhang@adfa.edu.au (Y.X. Zhang).

Available online at www.sciencedirect.com

Composite Structures 88 (2strength-to-weight ratios, long fatigue life, resistance toelectrochemical corrosion, and other superior materialproperties of composites. A true understanding of theirstructural behaviour is required, such as the deections,buckling loads and modal characteristics, the through-thickness distributions of stresses and strains, the largedeection behaviour and, of extreme importance forobtaining strong, reliable multi-layered structures, the fail-ure characteristics. Finite element method is especiallyversatile and ecient for the analysis of complex struc-tural behaviour of the composite laminated structures.

the nite element analysis of composite laminated platesis presented in this paper. The nite element analysisreviewed includes the following categories: free vibrations,damping, and transient dynamic response; buckling andpostbuckling; geometric nonlinearity and largedeformation analysis; damage and failure. Some of thefuture research on composite laminated plates is alsosummarized.

2. Laminated composite plate theoriesure and damage analysis of composite laminated plates. The material nonlinearity eects and thermal eects on the buckling and post-buckling analysis, the rst-ply failure analysis and the failure and damage analysis were emphasized specially. The future research issummarised nally. 2008 Elsevier Ltd. All rights reserved.

Keywords: Laminated composite plates; Free vibration; Dynamics; Buckling; Postbuckling; Failure

1. Introduction

Composite laminates have been used increasingly in a

and dynamics, buckling and postbuckling, failure anddamage analysis and etc.

A review of the nite element models developed afterRecent developments ifor laminated c

Y.X. Zhang a

aSchool of Aerospace, Civil and Mechanical Engineering, The U

Northcott Drive, CanbebSchool of Engineering and Information Technology

Available online

Abstract0263-8223/$ - see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.compstruct.2008.02.014nite element analysismposite plates

C.H. Yang b

rsity of New South Wales, Australian Defence Force Academy,

, ACT 2600, Australia

akin University, Waurn Ponds, VIC3217, Australia

February 2008

www.elsevier.com/locate/compstruct

009) 147157

divid

(a)

siteliteratures) Higher-order shear deformation theories (HSDT)] Layer-wise lamination theory (LLT)

(b) Continuum-based 3D elasticity theory

The classical lamination theory (CLT) is based on theKirchho plate theory, it is the simplest theory amongothers, but the shear deformation eects are neglected.The rst-order shear deformation theories (FSDT) pro-vides a balance between computational eciency andaccuracy for the global structural behaviour of thin andmoderately thick laminated composite plates, but noaccurate prediction for the local eects can be obtained,for example, the interlaminar stress distribution betweenlayers, delaminations, and etc. Various higher-order sheardeformation theories have been developed to overcomethe limitations in the classical and rst-order shear defor-mation theory, and the free boundary conditions of thetransverse shear stresses on the upper and lower surfacescan usually be satised. Layer-wise lamination theoryassumes a displacement representation formula in eachlayer. It can predict accurately the interlaminar stresses,however layerwise models are computational expensivesince the number of unknown functions depends on thenumber of the layers of the laminates. The 3D contin-uum-based theory can predict the interlaminar stress of

a com3D med into the following two categories:

Equivalent single layer (ESL) theories, including Classical lamination theory (CLT) The rst-order shear deformation theory (FSDT)

(referred to as Mindlin Plate theory in someand Robbins [1]. An overall comparison of laminatedtheories based on displacement hypothesis was presentedby Liu and Li [2], including shear deformation theories,layerwise theories, generalized Zigzag theories, and theproposed globallocal double-superposition theories. Areview of theories for laminated and sandwich plateswas presented by Altenbach [3]. A review ofdisplacement and stress-based rened shear deformationtheories of isotropic and anisotropic laminated platewas given by Ghugal and Shimpi [4], in which variousequivalent single layer and layerwise theories for lami-nated plates were discussed together with their meritsand demerits. A historical review of the zig-zag theoriesfor multi-layered plates and shells up to 2003 were givenby Carrera [5]. A review of shear deformation plate andshell theories was presented by Reddy and Arciniega [6],a selective review and survey of the theories withemphasis on estimation of transverse/interlaminarstresses in laminated composites was given by Kantand Swaminathan [7], and a selective literature surveyon the free-edge eect since 1967 was given by Mittel-stedk and Becker [8].

Generally, the laminated plate theories can be broadly

148 Y.X. Zhang, C.H. Yang /Compoposite laminate, but the computational cost usingodels is a major concern.3. Free vibration and dampling analysis of composite

laminated plates

3.1. Computational models based on FSDT

The rst-order shear deformation theory (FSDT) hasbeen employed widely to establish nite element modelsfor free vibration analysis of the composite laminatedplates. The eects of lamination and extensionbendingcoupling, shear and twist-curvature couplings on the lowestfrequencies and corresponding mode shapes for free vibra-tion of laminated anisotropic composite plates was investi-gated using a nite element method with quadraticinterpolation functions and ve engineering degrees of free-dom (DOF) [9]. The free and forced vibration response oflaminated composite folded plate structures was predictedby a nine-node Lagrangian plate-bending nite elementwith ve engineering DOF per node that incorporatedrotary inertia [10]. A nine-node isoparametric plate-bend-ing element was used for the analysis of free undampedvibration of rectangular isotropic and ber reinforced lam-inated composite plates [11], and an eective mass lumpingscheme with rotary inertia was introduced.

The free vibration analysis of stiened laminated com-posite plates was performed using the layered (zigzag) niteelement method based on the rst-order shear deformationtheory [12]. In their work, the layers of the laminated platewere modelled using nine-node isoparametric degeneratedat shell element, and the stieners were modelled asthree-node isoparametric beam elements based on Timo-shenko beam theory. Bilinear in-plane displacement con-straints were used to maintain the inter-layer continuity,and a special lumping technique was used in deriving thelumped mass matrices.

A mixed nite element formulation with low-order dis-placement/strain interpolation for plates and shells wasused to study the eect of large spatial rotations on the geo-metric stiness for stability analysis as well as inertia oper-ators for vibrations for laminated composite plates andshells [13].

Damping analysis of composite laminated plates hasbeen carried out using the computational models developedbased on the FSDT. The eects of transverse shear defor-mation on the modal loss factors as well as the natural fre-quencies of composite laminated plates was investigatedusing a nite element method based on the shear deform-able plate theory [14]. The complex modules of an ortho-tropic lamina were employed to model damping eect. Asandwich composite beam and plate nite superelementswith viscoelastic layers were presented for vibration anddamping analysis of laminated composite beams or plates[15]. Each layer was considered as simple Timoshenkosbeam or Mindlin-Reissner plate nite element. The energydissipation in the viscoelastic layers was taken into accountwith complex modulus of elasticity theory, and the method

Structures 88 (2009) 147157of complex eigenvalues and the energy method were con-sidered for damping analysis. This nite element technique

sitewas later utilized to predict the natural frequencies and themodal damping factor of anisotropic bre-reinforced com-posite laminated plates [16].

3.2. Computational models based on HSDT

Considerable amount of free vibration analyses of vari-ous composite laminated plates has been studied using thenite element models developed based on dierent kinds ofhigher-order shear deformation theories.

A high-order quadratic isoparametric element usingboth the eight-node serendipity and the nine-nodeLagrangian shape functions was presented for free vibra-tion analysis of laminated composite plates [17]. Variousschemes for the generation of the mass matrix were dis-cussed and a comparative study of these schemes waspresented.

Based on Reddys higher-order theory, a nite elementformulation taking into account the parabolic distributionof the transverse shear deformation through the thicknessof the plate was presented for vibration analysis of lami-nated anisotropic plates [18] with dierent lay-ups and ofsandwich plates.

A four-node rectangular element with seven-degrees offreedom at each node, whose displacement model was sochosen that the parabolic distribution of transverse shearstresses and the nonlinearity of in-plane displacementsacross the thickness can be represented adequately wasdeveloped for free vibration analysis of laminated compos-ite plate structures having a constant thickness of any indi-vidual layer [19].

Latheswary et al. [20] investigated the static and freevibration analysis of moderately thick laminated compositeplates using a 4-node nite element formulation based onhigher-order shear deformation theory, and the transientanalysis of layered anisotropic plates using a shear deform-able 9-noded Lagrangian element-based on rst-ordershear deformation theory.

A C0 continuous nite element model having ve- andseven-degrees of freedom per node was developed [21] forfree vibration analysis of laminated composite plates, usinga higher-order shear deformation theory to account for theparabolic variation of transverse shear stresses through thethickness and linear variation of the normal stresses.

The higher-order shear deformation theories (HST6,HST9, HST11, and HST12) and the conventional rst-order theory (FST) were employed to develop niteelement analysis methods using eight-node isoparametricelements to study the bending, free vibration and impactbehaviour of laminated composite plates [22].

Based on a higher-order shear deformation theory, a4-node, 20-DOF higher-order rectangular plate-bendingelement was developed for free vibration analysis of lami-nated composite plates [23]. The transverse displacementwas interpolated by using an optimized interpolation func-

Y.X. Zhang, C.H. Yang /Compotion while the additional rotation degrees of freedom wereapproximated by linear Lagrange interpolation. The con-sistent element mass matrix was used and a damped ele-ment was introduced to the nite element model.

A 48-degrees of freedom rectangular nite element wasformulated [24] based on a kinematics, which exactlyensured the continuity conditions for the displacements,the transverse stresses at the interfaces between the layersof a laminated structure and zero stress conditions at thetop and bottom surfaces of the plate, for static anddynamic analyses of moderately thick, multi-layered com-posite plates. Higher-order trigonometric functions wereused to dene the transverse shear deformations and thusthe shear correction factors were not required.

The free vibration analysis of multi-layered thick com-posite plates was studied by a nite element procedurebased on an accurate higher-order theory which accountedfor the realistic variation of in-plane and transverse dis-placements through the thickness [25]. The vibration andstability problems of cross-ply and angle-ply laminatedcomposite plates were investigated using general higher-order theories of laminates which took into account thecomplete eects of transverse shear and normal deforma-tions [2628].

A simple C0 higher-order facet shell element-based on ashear deformable model of higher-order theory was pre-sented for the free vibration analysis of isotropic, orthotro-pic and layered anisotropic composite and sandwichlaminates [29]. A realistic nonlinear variation of displace-ments through the shell thickness was incorporated, andshear correction coecients were eliminated.

3.3. Computational models based on layerwise theories

Compared with the computational models for the freevibration and damping analysis developed based on theFSDT and HSDT, relatively few models were developedbased on the Layerwise theories. The computational modeldeveloped based on the layerwise theories include the 18-node, three-dimensional higher-order mixed model for freevibration analysis of multi-layered thick composite plates[30], in which the continuity of the transverse stress andthe displacement elds were enforced through the thicknessof laminated composite plate, the hybrid strain-based lay-erwise shell element for free vibration of laminated com-posite plate and shell structures [31] developed based onseveral lower-order hybrid strain-based triangular shell ele-ments for the general nonlinear analysis of isotropic shellstructures, and etc.

4. Nonlinear dynamic stability and transient response of

composite laminated plates

The geometric nonlinearity or large amplitude eects onthe dynamic stability of the composite laminated plateshave been investigated. The dynamic instability of aniso-tropic laminated composite plates considering geometric

Structures 88 (2009) 147157 149nonlinearity [32], and the eect of large amplitude on thedynamic instability for a simply-supported laminated

sitecomposite plate were investigated using a C0 shear-exible,eld consistent, QUAD-9 plate element. Considering theviscoelastic properties of the material, Kim and Kim [33]studied the dynamic behavior of laminated compositeplates undergoing moderately large deection based onvon Karmans nonlinear deformation theory and Boltz-manns superposition principle. The eect of large ampli-tude on the dissipative nature as well as on the naturalfrequency of viscoelastic laminated plates was examinedusing nite element analysis and the method of multiplescales.

Based on the rst-order shear deformation theory, Ribe-iro and Petyt [34,35] studied the geometrically nonlinearvibration of thin laminated composite plates using the hier-archical nite element and the harmonic balance methods.Free and steady-state forced vibration were analysed, andthe excitations considered were harmonic plane waves atboth normal and grazing incidence. Taking into accountthe eects of the rotary inertia, transverse shear, and geo-metrical nonlinearity, a p-version, hierarchical nite ele-ment was presented for free vibration of moderately thickcomposite laminated plates [36]. The element was lateremployed to study the large amplitude, geometrically non-linear periodic vibrations of shear deformable compositelaminated plates subjected to harmonic forces appliedtransversely to the plates [37].

Some higher-order nite element models have beendeveloped for the nite element analysis of nonlinear staticand dynamic responses of laminated composite plates, suchas the nite element analysis of geometrically nonlinear sta-tic and transiently dynamic behaviour of laminated com-posite plates [38], in which a higher-order displacementeld allowing both transverse shear and transverse normalstrains was adopted, the nite element model for the largeamplitude free vibration of the laminated composite plates[39], in which the parabolic variation of transverse shearstrains through the thickness of the laminate wasaccounted for, the continuous nite element model devel-oped based on a nonlinear higher-order shear deformationtheory for nonlinear thermal dynamic analysis of graphite/aluminium laminated composite plates [40], the C0 fourand nine-node nite elements for the transient responseof orthotropic, layered composite sandwich plates [41]developed based on a rened form of Reddys higher-ordertheory, in which parabolic variation of the transverse shearstresses was accounted for, and etc.

5. Geometric nonlinear nite element analysis of laminated

composite plates

For accurate prediction for the static structuralresponses of composite laminated plates, geometric nonlin-earity should be included in the nite element analysis.Some literatures on the geometric nonlinear nite elementanalysis of laminated composite plates existed.

150 Y.X. Zhang, C.H. Yang /CompoA procedure for the reliability analysis of laminatedcomposite plate structures with large rotations but moder-ate deformation under random static loads was presentedvia a corotational total Lagrangian nite element formula-tion which was based on the von Karman assumption andrst-order shear deformation theory [42]. An eight-node C0

membrane-plate quadrilateral nite element-based on theReissnerMindlin plate theory was presented to analysemoderately large deection, static and dynamic problemsof moderately thick laminates including buckling analysisand membrane-plate coupling eects [43]. Han et al. [44]used the hierarchical nite element method to carry outthe geometrically nonlinear analysis of laminated compos-ite rectangular plates. Based on the rst-order shear defor-mation theory and Timoshenkos laminated compositebeam functions, the current authors developed a uniedformulation of a simple displacement-based 3-node, 18-degree-of-freedom at triangular plate/shell element [45]and two simple, accurate, shear-exible displacement-based 4-node quadrilateral elements [46,47] and for linearand geometrically nonlinear analysis of thin to moderatelythick laminated composite plates. The deection and rota-tion functions of the element boundary were obtained fromTimoshenkos laminated composite beam functions.

Based on a higher-order shear deformation theory involv-ing four dependent unknowns and satisfying the vanishing oftransverse shear stresses at the top and bottom surfaces ofthe plate, geometrically nonlinear exural response charac-teristics of shear deformable unsymmetrically laminatedrectangular plates were investigated using a four-node rect-angular C1 continuous nite element having 14 degrees offreedom per node [48]. A high-order plate model whichexactly ensured both the continuity conditions for displace-ments and transverse shear stresses at the interfaces betweenlayers of a laminated structure, and the boundary conditionsat the upper and lower surfaces of the plates was used tostudy the geometrically nonlinear behaviour of multi-lay-ered plates [49], and based on this rened plate model, asix-nodeC1 conforming displacement-based triangular niteelement was developed, with the Argyris interpolation usedfor transverse displacement, the Ganev interpolation usedfor membrane displacements and transverse shear rotations,and the transverse shear strain distributions represented bycosine functions.

A three-dimensional element with two-dimensionalkinematic constraints was developed for the geometricnonlinear analysis of laminated composite plates [50] usinga total Lagrangian description and the principle of virtualdisplacements. The large deformation analysis of circularcomposite laminated plates [51] was studied using a 48-DOF four-node quadrilateral laminated composite shellnite element.

6. Buckling and postbuckling analysis of laminated composite

plates

The buckling of laminated composite plates is an impor-

Structures 88 (2009) 147157tant consideration in the design process, however the criti-cal value of load given by linear buckling analysis may not

siteaccurately represent the load-carrying capability of a plate.Although composite laminated plates generally possess lessload-carrying capacity after buckling compared to theirmetallic counterparts, the total load during the postbuck-ling of a composite laminated plate is still several times thatof the critical buckling load. In order to get the practicallimits of the load-carrying capability of the composite lam-inated plates, the postbuckling behaviour has been studiedto establish the sustained additional loads after buckling.Considerable eorts have been made for the numericalanalysis of the buckling and postbuckling analysis overthe years.

Leissa [52,53] gave a summary of the buckling and post-buckling studies of composite laminated plates up to 1986,and then he reviewed the development of buckling analysisof laminated composite plates with linear eective constitu-tive properties [54]. Later a more detailed account of theresearch on the buckling and postbukcling before 1995was presented by Noor [55].

6.1. General buckling and postbuckling analysis of composite

laminated plates

An assumed hybrid-stress nite element model togetherwith a composite multilayer element were developed tostudy the buckling of generally laminated composite plateswith arbitrary thickness and edge conditions under an in-plane stress system [56]. The equilibrium conditions withineach layer, the interlaminar traction reciprocity conditions,and the stress-free boundary conditions on the top and bot-tom surfaces of the laminate, were satised by the assumedstress eld and thus the composite shear correction factorswere not required.

A shear deformable nite element was developed for thebuckling analysis of laminated composite plates based onMindlins theory in which shear correction factors werederived from the exact expressions for orthotropic materi-als [57]. The eects of material properties, plate aspectratio, length-to-thickness ratio, number of layers and lam-ination angle on the buckling loads of symmetrically andanti-symmetrically laminated composite plates were inves-tigated. An 8-node isoparametric plate nite element with5-DOF per node was developed based on the rst-ordershear deformation theory associated with von Karmansnonlinear straindisplacement relationships to investigatethe buckling and post-buckling of moderately thick lami-nated plates subjected to uni- or bi-axial compression[58]. The eects of boundary conditions, aspect ratio, sideto thickness ratio and lay-up sequence on the bucklingand post-buckling behaviour were studied in detail.

The linear buckling analysis of multilaminated compos-ite plate-shell structures was analysed using a discrete niteelement model based on an eight-node isoparametric ele-ment with 10 degrees of freedom per node and thehigher-order theory [59]. The geometric stiness matrix

Y.X. Zhang, C.H. Yang /Compowas developed taking into consideration the eects of thehigher-order terms on the initial in-plane and transverseshear stresses. The element was then used to study thebuckling and free vibrations of multilaminated structuresof arbitrary geometry and lay-up [60].

A generalized layer-wise stochastic nite element formu-lation was developed for the buckling analysis of bothhomogeneous and laminated plates with random materialproperties [61]. The pre-buckled stresses were consideredin the derivation of geometric stiness matrix and the eectof variation in these stresses on the mean and coecient ofvariation of buckling strength was studied.

The postbuckling behaviour of laminated compositeplates under the combination of in-plane shear, compres-sion and lateral loading was investigated using an ele-ment-based Lagrangian formulation based on theassumed natural strain method for composite structures[62]. Natural coordinate-based strains, stresses and consti-tutive equations were used in the element and the element-based Lagrangian formulation was computational ecientand had the ability to avoid both membrane and shearlocking.

6.2. Eects of material nonlinearity on buckling andpostbuckling behaviour of composite laminated plates

In the literature, most stability studies of composite lam-inated plates have been limited to the geometrically nonlin-ear analysis and the research on the eect of nonlineareective constitutive material properties on compositestructural buckling and postbuckling responses has beenvery limited. The nonlinearity of in-plane shear is signi-cant for composite materials [63]. With the nonlinear com-posite constitutive properties, a few attempts have beenmade to study buckling of thin composite laminate panels[64] and postbuckling of thick-section composite laminateplates [65]. Hu [66,67] investigated the inuence of in-planeshear nonlinearity on buckling and postbuckling responsesof composite plates under uniaxial compression and bi-axial compression and of shells under end compressionand hygrostatic compression. They also investigated thenonlinear buckling of simply-supported composite platesunder uniaxial compression, and of composite laminateskew plates under uniaxial compressive loads [68].

The eect of material nonlinearity on buckling and post-buckling of bre composite laminate plates and shells sub-jected to general mechanical loading, together with theinteraction between the material and geometric nonlinear-ity was investigated [69], and it was concluded that thecomposite material nonlinearity had signicant eects onthe geometrically nonlinearity, structural buckling load,postbuckling structural stiness, and structural failuremode shape of composite laminate plates and shells.

6.3. Buckling and postbuckling analysis of composite laminated

plates under thermal eects

Structures 88 (2009) 147157 151Considerable literatures have been devoted to the buck-ling and postbuckling analyses of laminated composite

siteplates subjected to mechanical loads, while the investiga-tions on the postbuckling response of composite plates sub-jected to thermal or combined thermal and mechanicalloadings are rather limited. The thermomechanical buck-ling and postbuckling response of laminated compositeplates is clearly one of practical importance for structuresoperating at elevated temperatures and thus the under-standing of the thermal buckling and postbucklingresponse of the composite laminated plates is desirablefor the design of the composite laminates subjected to hightemperatures. Tauchert [70] presented a comprehensivereview of the studies on thermal buckling of compositelaminated plates.

Finite element method based on classical laminationtheory was applied for examining nonlinear/postbucklinganalysis of thin laminated plates subjected to uniform tem-perature distribution [71,72]. Chen and Chen investigatedthe thermal buckling behaviour of cylindrical laminatedplates subjected to a non-uniform temperature [73], thethermal buckling behaviour of composite laminated platessubjected to uniform or non-uniform temperature elds[72] and thermal postbuckling behaviour of thick compos-ite laminated plates subjected to a uniform thermal loadingwith temperature-dependent properties [74].

The equivalent single layer rst-order shear deformationtheories have been employed widely for the thermal buck-ling and postbuckling analysis of composite laminatedplates. A mixed formulation with the fundamentalunknowns consisting of the generalized displacementsand the stress resultants of the plate was used to analysethe thermomechanical buckling of composite plates sub-jected to combined thermal and axial loadings [75], ther-momechanical buckling and postbuckling responses ofat unstiened composite panels subjected to combinedtemperature change and applied edge displacement [76],and the buckling and postbuckling responses of at,unstiened composite panels subjected to various combina-tions of mechanical and thermal loads [77]. A 9-node shear-exible isoparametric quadrilateral nite element was usedto study the buckling behaviour of laminated compositeplates subjected to a uniform temperature eld [78], andthe inuence of boundary conditions, ply orientation, andplate geometries on the critical buckling temperature wasexamined. Prabhu and Dhanaraj [79] also employed a 9-node Lagrangian isoparametric element for the thermalbuckling analysis of symmetric cross-ply, symmetricangle-ply and quasi-isotropic laminates subjected to uni-form temperature distribution. Thermal buckling and post-buckling behaviour of shear deformable laminatedcomposite plates was investigated by employing a four-node rectangular C1-continuous nite element by Singhet al. [48]. A nonlinear nite element formulation of aC0-continuity element [80,43] based on the rst-order sheardeformation theory was used to study the postbucklingbehaviour of laminated plates induced by a uniform/non-

152 Y.X. Zhang, C.H. Yang /Compouniform temperature eld [81]. The nonlinearity due tomoderately large deformation of the plate was includedin the formulation and the inuences of various parameterssuch as number of layers, ply-angle, aspect and thicknessratios and boundary conditions on the thermal postbuck-ling behaviour of laminates subjected to arbitrary temper-ature distribution were investigated.

The thermal buckling and postbuckling analysis hasbeen carried out to skew composite laminates and sand-wich plates. Two shear deformable nite element modelsbased on rst-order shear deformation theory and thehigher-order shear deformation theory, respectively, wereemployed to study the elastic buckling of both thin andthick skew bre-reinforced composite and sandwich plateswith various skew angles, lamination parameters andboundary conditions subjected to thermal loads [82]. Thebuckling and postbuckling analysis of shear deformablecomposite skew plates subjected to combined uniaxial com-pression and uniform temperature rise was performed [83].Thermal buckling response of laminated composite squareand skew plates was studied using a three-node plate ele-ment developed based on the rst-order shear deformationtheory [84], thermal buckling temperatures including thecritical one and mode shapes were numerically investigatedand the element showed excellent performance in the mod-erately thick to very thin plates.

A 3-node triangular facet nite element which accountsfor transverse shear deformation was used to examine thebending, buckling, and postbuckling behaviours of lami-nated composite plates under thermally-induced loadsbased on a natural thermoelastic theory with a linearthrough the thickness temperature variation [85]. Thematerial properties were assumed independent of tempera-ture, and the natural mode method was used. Thermalbuckling behaviour of composite laminated plate subjectedto a uniform temperature eld was investigated by consid-ering the temperature-dependent elastic and thermal prop-erties [86], and it was concluded that the inuence oftemperature-dependent properties on the thermal bucklingbehaviour was signicant.

The higher-order shear deformation theories have alsobeen employed for buckling analysis of laminated compos-ite plates. Based on a 9-node Lagrangian isoparametric ele-ment and two rened higher-order theories, two discretenite element models, with the eect of transverse normaldeformation included in one and neglected in the other,were developed for the thermal buckling analysis of com-posite laminated and sandwich plates [87]. The geometricstiness matrices were developed with the considerationof the eects of the higher-order terms on the initial in-plane and transverse shear stresses. Singha et al. [88] inves-tigated the thermal postbuckling behaviour of graphite/epoxy multi-layered rectangular plates with various bound-ary conditions considering the temperature-dependentthermal and elastic properties of the material. A 4-nodelock-free rectangular composite plate nite element having6-DOF per node based on a bi-cubic representation of the

Structures 88 (2009) 147157transverse displacement eld was employed to investigatethe post-buckling behaviour of rectangular laminated

siteplates subjected to thermal loads [89], and the eects ofboundary conditions, aspect ratio, number of layers andlay-up sequence on the post-buckling behaviour were stud-ied in detail.

After the study of the interlaminar stresses and displace-ments in cross-ply laminated composite and sandwichplates subjected to mechanical/thermal loading based onthe global higher-order theory [28,90,91], Matsunaga ana-lysed thermal buckling problems of cross-ply laminatedcomposite and sandwich plates [92], and angle-ply multi-layered composite and sandwich plates [93] based on theglobal higher-order theory with the power series expan-sions of continuous displacement components. Several setsof truncated Mth-order approximate theories were appliedto solve the eigenvalue problems of simply supported lam-inated composite and sandwich plates.

The three-dimensional layerwise analysis has made acontribution to obtain accurate prediction of the free vibra-tion and buckling of thermally stressed mutilayered angle-ply composite plates [94], thermal buckling and sensitivelyderivatives of temperatures sensitive multi-layered angle-ply plates [95], thermal buckling of multi-layered aniso-tropic plates [96], and the response of angle-ply laminatedcomposite and sandwich plates [97]. Both in-plane and nor-mal displacements were assumed to be C0 continuous in thecontinuity conditions at the interface between layers in thethree-dimensional layerwise theory. The number ofunknowns was dependent on the number of layers in a lam-inate, thus the three-dimensional layerwise analysis areoften computationally intractable, especially for laminatedplates with a large number of layers.

7. Failure analysis

Under normal operating conditions, local failures suchas matrix cracks, bre breakage, bre matrix debondingand inter-layer delamination, may be developed in the lam-inated composite structures, and the failure may cause per-manent loss of integrity within the laminate and result inloss of stiness and strength of the material. Prediction ofthe failure process, the initiation and growth of the dam-ages, and the maximum loads that the structures can with-stand before failure occurs is essential for assessing theperformance of composite laminated plates and for devel-oping reliable and safe design. In particular, the rst-plyfailure analysis of laminated composite plates has beenactively investigated in recent years, and the mechanicalbehaviour and the rst-ply failure load of laminated com-posite plates subjected to in-plane loading conditions, suchas tension, compression, shear, and out-of-plane loadingsuch as transverse loads have been studied. Compared withthe failure analysis of composite laminates subjected to in-plane loading, the failure analysis of composite laminatessubjected to out-of-plane loading seems more complicateddue to material and geometric nonlinearities that come into

Y.X. Zhang, C.H. Yang /Compoplay when the loads are increased beyond the rst-ply fail-ure. The dierent laminated plate theories, such as theCLT, FSDT, HSDT and layer-wise theories have beenemployed for failure analysis.

Chang and Lessard [98] studied the damage in laminatedcomposites containing an open hole, subjected to compres-sive loading, and the in-plane response of the laminatesfrom initial loading to nal collapse was studied consider-ing the geometrically nonlinearity. Sahid and Chang [99]developed a progressive failure model for predicting theaccumulated damage and the eects of such damage onthe in-plane response of laminated composites subjectedto tensile and shear loads. Based on the classical laminatedplate theory, Sleight and Knight [100] studied the damageof composite plates subjected to shear and compressiveloading. The postbuckling behaviour and progressive fail-ure response of thin, symmetric laminates under uniaxialcompression and uniaxial compression combined with in-plane shear loads was studied based on the rst-order sheardeformation theory and geometric nonlinearity [101], andthe 3D TsaiHill criterion was used to predict failure of alamina and the maximum stress criterion was used to pre-dict onset of delamination at the interface of two adjacentlayers.

Based on rst-order shear deformation theory and sev-eral phenomenological failure criteria, a nite elementmodel has been developed to nd linear and nonlinearrst-ply failure loads of composite laminates subjected toin-plane and transverse loads [102], and failure analysisand the rst-ply failure load in both linear and the geomet-rically nonlinear stage of thin and thick plates under a uni-formly distributed transverse load was studied [103]. First-ply failure of laminated composite plates was analysedusing the nite element method developed based on theReissnerMindlin plate theory that accounted for moder-ate rotation [104], and failure loads were obtained for dif-ferent laminate thickness, stacking sequences and aspectratios and dierent failure criteria. Kam and Lin [105]developed a stochastic nite element method for the reli-ability analysis of linear laminated composite plates sub-jected to transverse loads, and procedures for thereliability analysis of laminated composite plate structuressubjected to large deections under random static loadswas also presented [42]. The rst-ply failure probabilitiesof linear and nonlinear centrally loaded laminated compos-ite plates including the geometric nonlinear eects wereexamined [106,107], and an 8-node element of the serendip-ity family and 9-node Lagrangian elements with dierentnumerical integration rules were used to study the nonlin-ear deection and rst-ply failure load of thin laminatedcomposite plates subject to transverse loading based onseveral phenomenological failure criteria [108]. The rst-ply failure load, progression of damage and ultimate col-lapse load in the nonlinear deformation regime of lami-nated composite plates subjected to uniform transversepressure was studied with the large strain and large rota-tion included in the geometric nonlinearity analysis [109].

Structures 88 (2009) 147157 153The rst-ply failure of laminated panels under transverseloading was analysed using an eight-node isoparametric

sitequadratic shell element [110], and various failure criteriawere studied to predict the load of various plates and shellshaving varying lamination schemes. The rst-play failureof thin laminated composite plates under combined trans-verse load with uni-axial compression and transverse loadwith in-plane shear was studied [111]. An 8-node isopara-metric plate-bending element was used to model the pro-gressive failure of laminated composite plates undertransverse static loading in linear and elastic range[112,113]. After the failure of the weakest ply, the stinesswas reduced by either bre failure or matrix failure. Thestiness of failed lamina was then totally discarded andother existing lamina was considered to remain unchangedafter the weakest ply failure.

Fewer nite element models have been developed forfailure analysis of composite laminated plate based onHSDT, and one example is the 7-DOF nite element modelincluding three displacements, two rotations of normalabout the plate mid-plane, and two warps of the normal,which was developed to determine the rst-ply failureand the last-ply failure of laminated composite plates sub-jected to both in-plane and sinusoidal transverse loads by aprogressive stiness reduction technique under conditionsof complex loading [114].

Some progressive failure analysis of composite lami-nates based on the 3D layerwise plate theories have beencarried out. For example, Reddy and Reddy [115] usedgeneralized layerwise plate theory and a progressive failuremodel to determine rst-ply and ultimate failure loads of athree-point bending specimen with geometric nonlinearity.The failure mechanism and ultimate failure loads of thecross-ply and quasi-isotropic laminates for dierent stack-ing sequences with the same thickness subjected to axialextension was conducted [116] based on the generalizedlayerwise plate theory (GLPT) in order to consider thelocal eect near the free edges. A 3D layer-wise mixed niteelement model [117] was employed for the computation ofstress and strain components for the rst-ply failure analy-ses of composite laminated plates [118], and the maximumstress, the maximum strain, TsaiHill, TsaiWu and Ho-man failure theories were used for the failure analysis. Therst-ply failure of moderately thick laminated compositeplates [107] was studied using a nite element formulationbased on the layerwise linear displacement theory, in whicha laminated composite element was divided into a numberof mathematical layer groups and displacements wereassumed to vary linearly in each layer group.

Viscoelastic behaviour of composite materials inuencesthe failure behaviour, particularly when nonlinear geomet-rical eects are important. The failure behaviour ofcomposite laminates in the presence of large displacementsand creep was modelled including the material behaviourof thermal, hygroscopic and viscoelastic eects [119]. Incre-mental damage is determined and used to calculate a mod-ied stiness matrix, and the procedure can be used to

154 Y.X. Zhang, C.H. Yang /Compoanalyse buckling, creep buckling and creep bucklingincluding damage.8. Summary and future research

The recent advances of the nite element analysis ofcomposite laminated plates based on various laminationtheories, with the focus on the free vibration and dynamics,buckling and postbuckling analysis, geometric nonlinearityand large deformation analysis and failure and damageanalysis of composite laminated plates, are reviewed in thispaper. The development of buckling and postbucklinganalysis under material nonlinearity and thermal eectsare emphasised and in the failure analysis, the concentra-tion is especially on the advances of the rst-ply failureanalysis.

Based on the authors investigation, it has been foundthat the research on the following aspects of the compositelaminated plates is relatively limited and may attract moreinterests in the future research.

Material nonlinearity eects on structural behaviour ofcomposite laminates.

Failure and damage analysis under viscoelastic eectssuch as thermal and creep eects.

Failure and damage analysis under cyclic loading. Micromechanical approach for damage analysis. Analysis of the damage evolution in compositelaminates.

Multiscale modelling of crack initiation, propagation,and overall structural failure.

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Y.X. Zhang, C.H. Yang /Composite Structures 88 (2009) 147157 157

Recent developments in finite element analysis for laminated composite platesIntroductionLaminated composite plate theoriesFree vibration and dampling analysis of composite laminated platesComputational models based on FSDTComputational models based on HSDTComputational models based on layerwise theories

Nonlinear dynamic stability and transient response of composite laminated platesGeometric nonlinear finite element analysis of laminated composite platesBuckling and postbuckling analysis of laminated composite platesGeneral buckling and postbuckling analysis of composite laminated platesEffects of material nonlinearity on buckling and postbuckling behaviour of composite laminated platesBuckling and postbuckling analysis of composite laminated plates under thermal effects

Failure analysisSummary and future researchReferences