geometrically nonlinear nurbs isogeometric finite element analysis of laminated composite plates

Composite Structures 94 (2012) 3434–3447

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Geometrically nonlinear NURBS isogeometric finite element analysisof laminated composite plates

Hitesh Kapoor a,⇑, R.K. Kapania b,1

a Aerospace & Ocean Engineering Department, Virginia Tech, Blacksburg, VA-24060, United Statesb Aerospace & Ocean Engineering Department, 213E Randolph Hall, Virginia Tech, Blacksburg, VA-24061, United States

a r t i c l e i n f o

Article history:Available online 23 May 2012

Keywords:NURBS Isogeoemtric analysisLaminated composite plateFirst-order shear deformation theoryGeometric nonlinearityShear locking and hourglass stabilizationk-Refinement element technology

0263-8223/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compstruct.2012.04.028

⇑ Corresponding author.E-mail address: [email protected] (H. Kapoor).

1 Norris and Laura Mitchell Professor, Associate Fello

a b s t r a c t

This research present the development of geometrically nonlinear NURBS isogeometric finite elementanalysis of laminated composite plates. First-order, shear-deformable laminate composite plate theoryis utilized in deriving the governing equations using a variational formulation. Geometric nonlinearityis accounted for in Von-Karman sense. A family of NURBS elements are constructed from refinement pro-cesses and validated using various examples. k-refined NURBS elements are developed to study thinplates. Isotropic, orthotropic and laminated composite plates are studied for various boundary conditions,length to thickness ratios and ply-angles. Computed center deflection is found to be in an excellent agree-ment with the literature. For thin plate analysis, linear and k-refined quadratic NURBS element is found toremedy the shear locking problem. k-refined quadratic NURBS element provide stabilized response todistorted, coarse meshes without increasing the order of the polynomial, owing to the increased smooth-ness of solution space.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction simple spaces. The most widely used finite element, the linear tri-

Inspite of extensive use of finite element methods, the barriersbetween engineering design and analysis still exist and the way tobridge gap is to reconstitute the entire process. Idea is to use onemodel and use it as an analysis model which require a change fromclassical finite element to an anlysis procedure based CAD repre-sentation, formally known as Isogeometric analysis. There are sev-eral CAD functions which can be used for CAD representation ofanalysis module. Of most widely used CAD basis in engineering de-sign process are NURBS (non-uniform rational B-Splines) as pre-sented by Piegle and Tiller [1], Farin [2], Cohen et al. [3] andRogers [4]. NURBS represent a billion dollars CAD industry andare useful for analysis purposes because they possess useful math-ematical property of refinement through knot insertion and varia-tional deminishing property of convex hull. There are othercomputational geometry technologies that can be utilized as thebasis for isogeometric analysis such as sub-division surface by Pe-ters and Reif [5] and Warren [6] Gordon patches [7], Greogorypatch [8], S-patch [9] and A patch [10].

Finite element methods, in general, consists of variational for-mulation where trial and weight functions are defined by their ba-sis functions. These basis function are used for local representationof the spaces and the element devide the problem domain into

ll rights reserved.

w, AIAA.

angle, was formed by Courant [11] in 1943. Similarly, the bi-linearquadrilateral was proposed by Zienkiewicz and Cheung [12] andTaig [13]. Zienkiewicz et al. [14] developed a eight node serendip-ity quadrilateral element. Thin plate and shell bending analysis re-quire c1-continuous interpolation scheme due to squareintegrability of generalized second order derivatives. Reissener-Mindlin bending element requires only C0 continuity and there-fore, circumvents this problem in thin plate and bending analysis.In terms of CAD representation, CAD basis have advantage over theregular finite element due to their piecewise form and higher-or-der continuity. Besides, these functions provide computational effi-ciency, better accuracy and faster convergence. B-Splines were firstintroduced by Schoenberg [15] in order to develop piecewise poly-nomials with smoothness properties. De Boor [16] introduced astable recursion formula for evaluating B-Splines basis and theirderivatives. Gontier and Vollmer [17] used Bezier basis functionsfor nonlinear analysis of planar beams. They showed that a fewernumber of control points were required for nonlinear analysis ofstraight beams than required in finite element analysis. Hugheset al. [18] introduced the idea of isogeometric analysis usingNURBS (Non-uniform Rational B-Spline). They used NURBS to ex-actly represent the CAD geometry and then, constructed a coarsemesh for the analysis. The idea behind isogeometric analysis is tomodel the geometry exactly which also serves the basis for thesolution space i.e. invoking isoparametric concept. Similarly,Hughes et al studied the structural dynamics and wave propaga-tion [19] and fluid structure interaction [20] using isogeometric

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H. Kapoor, R.K. Kapania / Composite Structures 94 (2012) 3434–3447 3435

analysis. Cottrell et al. [21] studied vibration problem using isogeo-metric analysis. Kagan et al. [22] developed B-Spline finite elementanalysis and integrated with geometric design. Same authors [23]included adaptive refinement like hp and h-refinement techniquesin their finite element code.

1.1. Laminated composite plate theories

Advance multi-layered composite and sandwich plate/shellstructures are being increasingly used in aerospace, shipbuilding,bridge and other industries. These structures have smaller thick-ness as compared to other dimensions and therefore, are often sub-jected to large deformation behavior under external loads. Foraccurate prediction of large displacement behavior, geometric non-linear analysis is very important and developement of computa-tionally efficient geometric nonlinear finite element code forcomposite plates has been a topic of considerable interest. Theplate theories for laminated composites can be categoried intoequivalent single-layer theories (ELS) and layerwise theories.Equivalent single layer theories namely, classical, first-order andhigher-order shear doformation theories, are derived from their3D counterpart (i.e. layerwise and 3D elasticity) by making appro-priate assumptions to the state of strain/stress in the thicknessdirection, reducing 3D continuum problem to a 2D problem.

Several articles are available in literature on shear deformationtheories. Reissner–Mindlin theory [24–26] (first-order shear defor-mation theory) assumes constant state of through the thicknessstrain and requires only C0 interpolation functions to satisfy thecontinuity requirement. This theory was further extended for aniso-tropic plates by Whitney and Pagano [27]. Urthaler and Reddy [28]developed a mixed finite element for bending analysis of laminatedcomposite plates using FSDT. They treated bending moment as afield variable along with displacement and rotation. Similarly, Cenet al. [29], Kim et al. [30] and Mai-Duy et al. [31] have previouslydeveloped elements based on Mindlin-Reissner theory (FSDT).

In higher-order shear deformation theories (HSDT), Putcha andReddy [32] developed a mixed finite element approach with 9-node Lagrangian quadrilateral element, Kant and Kommineni [33]fromulated refined HSDT with 9-node quadrilateral element forlinear and nonlinear finite element analysis of laminated compos-ite and sandwich plates, Polit and Touratier [34] studied largedeflection behavior using HSDT, Reddy and Phan [35] developedspecial third order theory (STTR) and Ren and Hinton [36] devel-oped a third order plate theory for the analysis of laminated com-posite plates. Robbins and Reddy [37]’s research article provideextensive literature review on ELS and layerwise theories for lam-inated composite plates. Similarly, Carrera [38] provided extensivereview of zig-zag theories for multilayered plates and shells.

Comparing higher-order theories with FSDT, higher-order platetheories enhance the accuracy of the solution slightly but are com-putationally more expensive, especially for nonlinear analysis as itrequires C1 continuity. On the other hand, FSDT has often beenused due to its simplicity and provide best compromise betweeneconomy and accuracy in predicting the response of thin to mod-erately thick laminates [39,40].

1.2. Shear-locking and hourglass stabilization

During the last few dacades, many researchers have made signif-icant contribution to the development of efficient lower order finiteelements based on FSDT. Lower order elements suffer from shearlocking problems as thickness to span ratio becomes too small. Zie-nkiewicz et al. [41] and Hughes et al. [42] used reduced or selectiveintegration techniques to solve the shear locking problem. MacNeal[43], first, developed assumed strain method where he computedshear strain using kinematic variables at discrete collocation points

of the element other than nodes. Modified versions of this methodwere successfully proposed by various researchers such as Batheand Dvorkin [44], discrete shear elements by Batoz and Katili [45]and linked interpolation elements by Zienkewicz et al. [46].

Discrete shear gap (DSG) method improves shear lockingbehavior by invoking Kirchoff constraints on the element nodes.Echter and Bischoff [47] studied locking and unlocking of NURBSfinite element using discrete shear gap (DSG) method. Zhang andKim [48] proposed a locking-free quadrilateral plate element forgeometric nonlinear analysis of laminated composite plate. Simi-larly, Minighini et al. [49] developed a locking-free finite elementfor shear deformable orthotropic thin-walled beams. Nguyen-Xuanet al. [50] developed a smoothed finite element for plate analysis.They incorporated strain smoothing stabilization to develop a lock-ing-free plate. Cai et al. [51] developed a new shear locking-freetriangular plate element with 6 extra degrees of freedom for linearproblems. The additional degrees of freedom accounted for rota-tions caused by transverse shear deformation. Their element re-moved shear locking without extra numerical efforts such asreduced integration, assumed strain/stress or need for stabilizationof zero energy modes.

Shear locking is accompanied with hourglass instability as re-duced integration element become excessibily flexible due to rankdeficiency and require stabilization control [52–54]. Codina [55]and Lyly and Stenberg [56] developed a stable first-order sheardeformable plate finite element which was insensitive to mesh dis-tortion. Flanagan and Belytchsko [57] formulated hourglass controlfor reduced integration plate element and Tessler and Hughes [58]derived a hourglass plate element. Similarly, Kouhia developed astabilized finite element for Reissner–Mindline plates [59].

This research present the development of geometrically nonlin-ear NURBS isogeometric finite element analysis for bending analy-sis of laminated composite plate. Geometry is modeled exactly andiso-parametric finite element representation is invoked for solu-tion space. First-order, shear-deformable laminated compositeplate theory is utilized in deriving the governing equations in var-iational formulation. Geometric nonlinearity is accounted for invon-Karman sense. NURBS linear, quadratic and higher-order ele-ments are constructed from refinement processes. Isotropic, ortho-tropic and laminated composite plate are considered for theanalysis. Numerical validation is performed for different boundaryconditions, length to thickness ratios and ply-angles. Computedcenter deflection is compared with those present in the literatureand is found to be in an excellent agreement. For thin plate analy-sis, k-refined, linear and quadratic NURBS element with full inte-gration remedies the shear locking problem, eliminating the needfor reduced integration or other numerical expendices. k-refinedquadratic NURBS element is found to provide stabilized responseto mesh distortion with coarsest mesh.

2. Theoretical formulation

2.1. First-order shear deformation plate theory (FSDT)

2.1.1. Displacement and strain fieldThe origin of the material coordinate system is considered to be

the mid-plane of the laminate and the kirchhoff assumption thatthe transverse normal remain perpendicular to the mid-surfaceafter deformation is relaxed. Fig. 1 shows the coordinate systemand layer numbering of a laminated composite plate. The displace-ment field is defined as,

uðx; y; zÞ ¼ u0ðx; yÞ þ z/xðx; yÞvðx; y; zÞ ¼ v0ðx; yÞ þ z/yðx; yÞ

wðx; y; zÞ ¼ w0ðx; yÞ

ð1Þ

Fig. 1. Coordinate system and layer numbering used for a laminate plate.

3436 H. Kapoor, R.K. Kapania / Composite Structures 94 (2012) 3434–3447

where u0, v0, w0, are the displacement along the x, y and z-axis and/x, /y are the rotation of transverse normal of the mid-plane aboutthe y and x-axis respectively. Fig. 2 shows the undeformed and de-formed configuration of first order shear-deformable laminatedcomposite plate.

The strain vector is obtained in terms of Green–Lagrange strainand accounts for large transverse displacement, small strain andmoderate rotation. The strain vector corresponding to the displace-ment field is given as,

�xx ¼@u@xþ 1

2@w@x

� �2

�yy ¼@v@yþ 1

2@w@y

� �2

cyz ¼@w@yþ /y

cxz ¼@w@xþ /x

cxy ¼@u@yþ @v@xþ @w@x

@w@y

ð2Þ

Total strain vector � can be decomposed into linear and nonlinearstrain vectors.

� ¼ �l þ �nl ð3Þ

�l, linear strain vector is composed of following terms,

Fig. 2. Undeformed and deformed configuration of first order shear-deformableplate.

�lxx ¼

@u0

@xþ z

@/x

@x

�lyy ¼

@v0

@yþ z

@/y

@y

clyz ¼

@w0

@xþ /x;

clxz ¼

@w0

@yþ /y;

clxy ¼

@u0

@yþ @v0

@xþ z

@/x

@yþ@/y

@x

� �

ð4Þ

�nl, the nonlinear strain vector contains the following terms,

�nlxx ¼

12

@w0

@x

� �2

�nlyy ¼

12

@w0

@y

� �2

cnlxy ¼

@w0

@x@w0

@y

ð5Þ

Other higher order terms are considered to be negeligible.

2.1.2. Equation of motionThe strong form of the governing equation of motion using first-

order shear deformable plate theory are,

@Nxx

@xþ @Nxy

@y

� �¼ 0

@Nxy

@xþ @Nyy

@y

� �¼ 0

@Q x

@xþ@Qy

@y

� �þ q ¼ 0

@Mxx

@xþ @Mxy

@y

� �� Q x ¼ 0

@Mxy

@xþ @Myy

@y

� �� Q y ¼ 0

ð6Þ

Where Nxx, Nyy and Nxy are the force resultants, Mxx, Mxy, Myy are themoment resultants and Qx and Qy are the shear force resultants andq corresponds to transverse load.

2.2. Variational form

Weak form of the governing equations is obtained by pre-mul-tipying the equation of motion with du0, dv0, dw0, d/x and d/y

respectively and intergrating by parts over the element domain.Substituting force, moment and shear force resultant,

0 ¼Z

Xe

@du0

@xA11

@u0

@xþ 1

2@w0

@x

� �2 !

þ A12@v0

@yþ 1

2@w0

@y

� �2 !("

þ A16@u0

@yþ @v0

@xþ @w0

@x@w0

@y

� �þ B11

@/x

@xþ B12

@/y

@y

þB16@/x

@yþ@/y

@x

� ��þ @du0

@yA16

@u0

@xþ 1

2@w0

@x

� �2 !(

þ A26@v0

@yþ 1

2@w0

@y

� �2 !

þ A66@u0

@yþ @v0

@xþ @w0

@x@w0

@y

� �

þB16@/x

@xþ B26

@/y

@yþ B66

@/x

@yþ@/y

@x

� ��dxdy�

ICe

Nndu0nds ð7Þ


0 ¼Z

Xe

@dv0

@xA12

@u0

@xþ 1

2@w0

@x

� �2 !

þ A22@v0

@yþ 1

2@w0

@y

� �2 !("

þA26@u0

@yþ @v0

@xþ @w0

@x@w0

@y

� �þ B12

/x

@xþ B22

/y

@yþ B26

/x

@yþ

/y

@x

� ��

þ @dv0

@xA16

@u0

@xþ 1

2@w0

@x

� �2 !

þ A26@v0

@yþ 1

2@w0

@y

� �2 !(

þ A66@u0

@yþ @v0

@xþ @w0

@x@w0

@y

� �þ B16

@/x

@xþ B26

@/y

@y

þB66@/x

@yþ@/y

@x

� ��dxdy�

ICe

Nsdv0sds ð8Þ

0 ¼ Ks

ZXe

@dw0

@xA55

@w0

@xþ /x

� �þ A45

@w0

@yþ /y

� ��

þ @dw0

@yA45

@w0

@xþ /x

� �þ A44

@w0

@xþ /x

� �� dxdy

þZ

Xe

@dw0

@x@w0

@xA11

@u0

@xþ 1

2@w0

@x

� �2 !(("

þ A12@v0

@yþ 1

2@w0

@y

� �2 !

þ A16@u0

@yþ @v0

@xþ @w0

@x@w0

@y

� �

þB11@/x

@xþ B12

@/y

@yþ B16

@/x

@yþ@/y

@x

� ��

þ @w0

@yA16

@u0

@xþ 1

2@w0

@x

� �2 !

þ A26@v0

@yþ 1

2@w0

@y

� �2 !(

þ A66@u0

@yþ @v0

@xþ @w0

@x@w0

@y

� �þ B16

@/x

@xþ B26

@/y

@y

þB66@/x

@yþ@/y

@x

� ��þ @dw0

@y@w0

@yA12

@u0

@xþ 1

2@w0

@x

� �2 !((

þ A22@v0

@yþ 1

2@w0

@y

� �2 !

þ A26@u0

@yþ v0

@xþ @w0

@x@w0

@y

� �

þB12/x

@xþ B22

@/y

@yþ B26

@/x

@yþ@/y

@x

� ��

þ @w0

@xA16

@u0

@xþ 1

2@w0

@x

� �2 !

þ A26@v0

@yþ 1

2@w0

@y

� �2 !(

þA66@u0

@yþ @v0

@xþ @w0

@x@w0

@y

� �þ B16

@/x

@xþ B26

@/y

@y

þB66@/x

@yþ@/y

@x

� �� dw0q

�dxdy�

ICe

Vndw0ds ð9Þ

0¼Z

Xe

@d/x

@xB11

@u0

@xþ1

2@w0

@x

� �2 !

þB12@v0

@yþ1

2@w0

@y

� �2 !("

þB16@u0

@yþ@v0

@xþ@w0

@x@w0

@y

� �þD11

@/x

@xþD12

@/y

@y

þD16@/x

@yþ@/y

@x

� ��þ@d/x

@yB16

@u0

@xþ1

2@w0

@x

� �2 !(

þB26@v0

@yþ1

2@w0

@y

� �2 !

þB66@u0

@yþ@v0

@xþ@w0

@x@w0

@y

� �

þD16@/x

@xþD26

@/y

@yþD66

@/x

@yþ@/y

@x

� ��

þKsd/x A55 /xþ@w0

@x

� �þA45 /yþ

@w0

@y

� �� dxdy�

ICe

Mnd/nds

ð10Þ

0 ¼Z

Xe

@d/y

@yB12

@v0

@yþ 1

2@w0

@y

� �2 !

þ B22@v0

@yþ 1

2@w0

@y

� �2 !("

þ B26@u0

@yþ @v0

@xþ @w0

@x@w0

@y

� �þ D12

@/x

@xþ D22

@/y

@y

þD26@/x

@yþ@/y

@x

� ��þ@d/y

@xB16

@u0

@xþ 1

2@w0

@x

� �2 !(

þ B26@v0

@yþ 1

2@w0

@y

� �2 !

þ B66@u0

@yþ @v0

@xþ @w0

@x@w0

@y

� �

þD16@/x

@xþ D26

@/y

@yþ D66

@/x

@yþ@/y

@x

� ��

þKsd/y A45 /x þ@w0

@x

� �þ A44 /y þ

@w0

@y

� �� dxdy

�I

CeMsd/sds ð11Þ

And, the secondary variables are given as,

Nn ¼ Nxxnx þ Nxyny; Ns ¼ Nxynx þ Nyyny

Mn ¼ Mxxnx þMxyny; Ms ¼ Mxynx þMyyny

Vn ¼ Q x þ Nxx@w0

@xþ Nxy

@w0

@y

� �nx þ Q y þ Nxy

@w0

@xþ Nyy

@w0

@y

� �ny

ð12Þ

where nx and ny are the directional cosines of the outward normalvector.

The laminate constitutive equations relate the force and mo-ment resultant to strains in laminate coordinate system throughABD matrix.

Nxx

Nyy

Nxy

8<:

9=; ¼

A11 A12 A16

A12 A22 A26

A16 A26 A66

24

35

@u0@x þ 1

2@w0@x

� 2

@v0@y þ 1

2@w0@y

�2

@u0@y þ

@v0@x þ

@w0@x

@w0@y

�8>>><>>>:

9>>>=>>>;

þB11 B12 B16

B12 B22 B26

B16 B26 B66

24

35

@/x@x@/y

@y@/x@y þ

@phiy@x

8><>:

9>=>; ð13Þ

Mxx

Myy

Mxy

8<:

9=; ¼

B11 B12 B16

B12 B22 B26

B16 B26 B66

24

35

@u0@x þ 1

2@w0@x

� 2

@v0@y þ 1

2@w0@y

�2

@u0@y þ

@v0@x þ

@w0@x

@w0@y

�8>>><>>>:

9>>>=>>>;

þD11 D12 D16

D12 D22 D26

D16 D26 D66

24

35

@/x@x@/y

@y@/x@y þ

@/y

@x

8><>:

9>=>; ð14Þ

Q y

Q x

� �¼ kk

A44 A45

A45 A55

� � @w0@y þ /y

@w0@y þ /y

( )ð15Þ

ðAij;Bij;DijÞ ¼XN

k¼1

Qkijðzkþ1 � zkÞ;

12

Qkij z2

kþ1 � z2k

� ;13

Q kij z3

kþ1 � z3k

� � �

ðA44;A45;A55Þ ¼XN

k¼1

Q k44;Q

k45;Q

k55

�ðzkþ1 � zkÞ

ð16Þ

In ABD matrix, Aij, Bij and Dij represent extensional and shear, exten-sional-bending coupling and bending stiffness terms and Qk

ijs areplane-stress reduced stiffness terms. kk is the shear correctionfactor.


3. Geometrically nonlinear formulation

3.1. Displacement field approximation

The dependent displacement field variables are approximatedby NURBS basis as follows:

u0ðx; yÞ ¼XnCP

j¼1

ujRjðxðn;gÞ; yðn;gÞÞ; v0ðx; yÞ ¼XnCP

j¼1

v jRjðxðn;gÞ; yðn;gÞÞ

w0ðx; yÞ ¼XnCP

j¼1

wjRjðxðn;gÞ; yðn;gÞÞ

/xðx; yÞ ¼XnCP

j¼1

/xjRjðxðn;gÞ; yðn;gÞÞ;/yðx; yÞ ¼XnCP

j¼1

/yjRjðxðn;gÞ; yðn;gÞÞ

ð17Þ

where nCP are the number of control points per element and Rj are2D NURBS basis.

3.2. NURBS basis

This subsection details B-Spline and rational basis generationand describes the mapping from physical to parametric to parentdomain. A family of p and k-refined elements are constructed bymodifying the knot vector by knot-insertion. Lastly, various NURBSelements are described in details.

3.2.1. B-Spline basisA knot vector in one dimension is a set of co-ordinates in the

parametric space,N = {n1,n2, . . . . . . ,nn+p+1}, where ni is the ith knot,i is the knot index where i = 1, 2, . . ., n + p + 1, p is the order ofthe polynomial and n is the number of basis functions. The orderof the polynomial, p = 0, 1, 2, 3, . . ., refer to the constant, linear,quadratic, cubic piecewise polynomials, respectively. If more thanone knot is located at the same parametric co-ordinate, these aretermed as repeated knots. In case of the open knot vector, firstand last knots are repeated p + 1 times. Basis function formed usingthe open knot vector are interpolatory at the beginning and end ofthe parametric space interval, i.e. [n1,nn+p+1]. This distinguishes theknots from the nodes in the finite element analysis as all the nodesare interpolatory. As a starting point, B-Spline basis functions aredefined recursively using Cox-Deboor algorithm, starting withpiecewise constants (p = 0).

Npi ðnÞ ¼

1 if ni 6 n < niþ1;

0 otherwise

�ð18Þ

And, subsequently, basis functions for orders p = 1, 2, 3, . . ., are de-fined as follows:

Npi ðnÞ ¼

n� ni

niþp � niNp�1

i ðnÞ þ niþpþ1 � n

niþpþ1 � niþ1Np�1

iþ1 ðnÞ ð19Þ

3.2.2. Rational B-Spline (NURBS) basis2D NURBS basis are formed by tensor product of B-Spline basis

in n and g direction and using projective weights associated withthe control points. Rational basis are very similar to B-Spline basisand derive the continuity and support for the function from theknot vector. Also, NURBS basis form a partition of unity and arepointwise non-negetive. These properties result in a strong convexhull property. Rational basis are formed as follows:

Rp;qi;j ðn;gÞ ¼

Npi ðnÞM

qj ðgÞWi;jPCPw

i¼1

PCPui¼1Np

i ðnÞMqj ðgÞWi;j

� Rpkfk ¼ 1 : nCPg ð20Þ

NURBS surface is defined as,

Sðx; yÞ ¼XCPw

j¼1

XCPu

i¼1

Rp;qi;j ðn;gÞBi;j ð21Þ

where Npi ðnÞ and Mq

j ðgÞ are the B-Spline basis in n and g directionand Rp;q

i;j denotes a 2D NURBS basis function. CPw and CPu are thenumber of control points in n and g directions and nCP = CPu � CPwis the total number of control points per element. Wi,js are theweights associated with the control points, Bi,j and weights arethe vertical coordinates of the corresponding control points. Here,weights, Ws considered are equal to one.

3.3. NURBS elements

Various lower and higher order NURBS elements can beconstructed by utilizing refinement techniques. k-refined,9LinNURBS1,2/F, 9 control point/element (LinNURBSKR1,2 elementwith single knot insertion at location 0.25 and 0 in knot vectorrespectively), 9QuadNURBS/(F/R) element with 9 control point/element, 16CubicNURBS/(F/R) and (k-refined)16QuadNURBSKR/Fwith 16 control point/element and 25QuarticNURBS, 25Cubi-cNURBSKR/F and (k-refined)25QuadNURBSKR/F with 25 controlpoint/element are contructed. Element notation is as follows:number in the front denotes the number of control points/element,KR denotes the k-refined element and R or F denote reduced or fullintegration. For full integration, the number of quadrature pointsrequired is the smallest integer greater than or equal to N = (1/2)(p + 1), where p is the polynomial order. For k-refined elementwith full integration, number of Gauss point required is equivalentto that of p-refined element with same number of control points.Here, some of the element are described in details.

3.3.1. Quadratic NURBS (9QuadNURBS/(F/R))A tensor product of knot vector {�1 �1 �1 1 1 1} in n and g

direction results in a bi-quadratic NURBS element in the parent do-main. ni denote the ith knot value in the knot vector. The elementconsists of 9 control points. The shape function in n direction canbe written as follows:

M21ðnÞ ¼

n4 � n

n4 � n2

!n4 � n

n4 � n3

!

M22ðnÞ ¼

n� n2

n4 � n2

!n4 � n

n4 � n3

!þ n5 � n

n5 � n3

!n� n3

n4 � n3

!n3 6 n < n4

M23ðnÞ ¼

n� n3

n5 � n3

!n� n

n4 � n3

!

ð22Þ

Similarly, the shape functions in g direction are obtained. The prod-uct rule yields the following bi-quadratic shape functions,

R21ðn; gÞ ¼ M2

1ðnÞN21ðgÞ R2

2ðn; gÞ ¼ M21ðnÞN

22ðgÞ R2


23ðgÞ

R24ðn; gÞ ¼ M2

2ðnÞN21ðgÞ R2


22ðgÞ R2


23ðgÞ

R27ðn; gÞ ¼ M2

3ðnÞN21ðgÞ R2


22ðgÞ R2


23ðgÞð23Þ

Fig. 3 shows NURBS basis in n and g directions alond side the parentelement.

3.3.2. k-refined quadratic NURBS I (16QuadNURBSKR/F)A tensor product of knot vector {�1 �1 �1 0 1 1 1} in n and g

directions result in a bi-quadratic NURBS element in parent do-main. The knot vector is divided into two intervals and shape func-tions have unique value over each interval. ni denote the ith knotvalue in the knot vector. The shape functions in n direction overeach interval are derived as follows:

Fig. 3. 9 control point quadratic NURBS element with basis functions in eachdirection.

Fig. 4. k-refined quadratic NURBS element with basis functions in each direction.


M21ðnÞ ¼

n4 � n

n4 � n2

!n4 � n

n4 � n3

!

M22ðnÞ ¼

n� n2

n4 � n2

!n4 � n

n4 � n3

!þ n5 � n

n5 � n3

!n� n3

n4 � n3

!n3 6 n < n4

M23ðnÞ ¼

n� n3

n5 � n3

!n� n

n4 � n3

!

M24ðnÞ ¼ 0

ð24Þ

M21ðnÞ ¼ 0

M22ðnÞ ¼

n5 � n

n5 � n3

!n5 � n

n5 � n4

!

M23ðnÞ ¼

n� n3

n5 � n3

!n� n

n4 � n3

!þ n� n3

n5 � n3

!n5 � n

n5 � n3

!n4 6 n < n5

M24ðnÞ ¼

n� n4

n6 � n4

!n4 � n

n5 � n4

!

ð25Þ

Similarly, the shape functions in g direction are obtained. The prod-uct rule yields the 16 bi-quadratic shape functions, R2

k , requiring4 � 4 Guass points. In k-refinement process, the order of the exist-ing knot vector is elevated first and then, a knot is inserted insteadof elevating the order after knot insertion. This results in q � 1 con-tinous derivatives instead of p � 1 continuous derivatives, where pand q are the order of the knot vector before and after order eleva-tion respectively. Fig. 4 shows k-refined, quadratic NURBS elementin the parent domain and NURBS basis in each direction.

3.3.3. k-refined quadratic NURBS II (25QuadNURBSKR/F)A tensor product of knot vector {�1 �1 �1 �0.5 0.5 1 1 1} in n

and g direction results in a k-refined, bi-quadratic NURBS elementin parent domain. The knot vector is divided into three knot inter-vals and shape functions have unique value over each interval. ni

denote the ith knot value in the knot vector. The shape functionsin n direction over each interval are derived as follows:

M21ðnÞ ¼

n4 � n

n4 � n2

!n4 � n

n4 � n3

!

M22ðnÞ ¼

n� n2

n4 � n2

!n4 � n

n4 � n3

!þ n5 � n

n5 � n3

!n� n3

n4 � n3

!n3 6 n < n4

M23ðnÞ ¼

n� n3

n5 � n3

!n� n3

n4 � n3

!

M24ðnÞ ¼ 0

M25ðnÞ ¼ 0

ð26Þ

M21ðnÞ ¼ 0

M22ðnÞ ¼

n5 � n

n5 � n3

!n5 � n

n5 � n4

!

M23ðnÞ ¼

n� n3

n5 � n3

!n5 � n

n5 � n4

!þ n6 � n

n6 � n4

!n� n4

n5 � n4

!n4 6 n < n5

M24ðnÞ ¼

n� n4

n6 � n4

!n� n4

n5 � n4

!

M25ðnÞ ¼ 0

ð27Þ

M21ðnÞ ¼ 0

M22ðnÞ ¼ 0

M23ðnÞ ¼

n6 � n

n6 � n4

!n6 � n

n6 � n5

!

M24ðnÞ ¼

n7 � n

n7 � n5

!n� n5

n6 � n5

!n5 6 n < n6

M25ðnÞ ¼

n� n5

n7 � n5

!n� n5

n6 � n5

!ð28Þ

The element consists of 25 control functions and require 5 � 5Guass integration points (full). The derivatives are q � 1 continuousinstead of p � 1 continuous. Fig. 5 shows the NURBS element andthe basis in n and g directions.

Fig. 5. k-refined quadratic NURBS element with basis in each direction.


3.3.4. Quartic NURBS (25QuarticNURBS/F)A tensor product of knot vector, {�1 �1 �1 �1 �1 1 1 1 1 1} in n

and g directions result in a bi-quartic NURBS element in the parentdomain. Quartic NURBS element consists of 25 control points andrequires 5 � 5 Gauss points. It is equivalent to 25QuadNURBSKR/F element in terms of number of control points/element and Guassintegration points. The shape functions in explicit form are not pro-vided here. Fig. 6 shows the NURBS element and basis in n and gdirections.

3.4. Geometric nonlinear stiffness matrix

Substituting the displacement field approximation into theweak form, we obtain element stiffness matrix and load vector.

Fig. 6. Quartic NURBS element with basis functions in each direction.

The size of a sub-matrix, K11, in element stiffness matrix, is(nCP) � (nCP) and the size of element load vector, F, is(nCP � dof,1). nCP stands for control points per element and dofare the No. of degrees of freedom/control point. a and b are the de-grees of freedom per control point.

½Kab� ¼

½K11� ½K12� ½K13� ½K14� ½K15�

½K21� ½K22� ½K23� ½K24� ½K25�

½K31� ½K32� ½K33� ½K34� ½K35�

½K41� ½K43� ½K43� ½K44� ½K45�

½K51� ½K52� ½K53� ½K54� ½K55�

2666666664

3777777775

ð29Þ

Fa¼3i ¼

ZXe

qRpi dxdy ð30Þ

From the element stiffness matrix, the expression for K11, a sub-ma-trix of element stiffness matrix, K, is as follows:

K11ij ðx;yÞ¼

ZXe

A11@Rp

i

@x

@Rpj

@xþA66

@Rpi

@y

@Rpj

@yþA16

@Rpi

@x

@Rpj

@yþ@Rp

i

@y

@Rpj

@x

!" #dxdy

ð31Þ

The equation of motion can be written in the matrix form as,

½K�fDg ¼ fFg ð32Þ

The solution of nonlinear equations is obtained by Newton–Raph-son iterative process as follows:

fRgðfDgÞ ¼ ½KTðfDgÞ�fDg � fFg ¼ 0 ð33Þ

Tangent stiffness matrix is obtained as,

KT;abij ¼ @Ra

i

@Dbj

KT;abij ¼ @

@Dbj

X5

c¼1

XnCP

k¼1

Kacik Dc

k � Fai

! ð34Þ

For example, a sub-matrix in element tangent stiffness matrix is,

KT;33ij ¼

X5

c¼1

XnCP

k¼1

@K3cik

@wjDc

k þ K33ij ð35Þ

The incremental displacement vector is given as,

fdDg ¼ �½KTðfDgrÞ��1fRgr ð36Þ

And, total displacement vector is obtained as,

fDgrþ1 ¼ fDgr þ fdDg ð37Þ

KT and R are the tangent stiffness and residual load vector. At thebeginning of iteration, r = 0 and nonlinear stiffness terms reducesto zero. The iterative process is continued till the convergence is ob-tained within the error tolerence of 10�3. The error norm used forconvergence is as follows:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNI¼1 Drþ1

I � DrI

2PNI¼1 Drþ1

I

2vuut < 10�3 ð38Þ

3.5. Numerical integration

In Isogeometric finite element analysis, there are 2 notions ofmeshes, one in physical domain i.e. a patch and other in parametricdomain i.e. a knot vector. For instance, in general, plate geometrycan be represented by a single patch and refinement is performed

Fig. 7. Mapping between physical and parent domain: a framework for isogeo-metric finite element analysis.


at subdomain level i.e. in parametric domain. For example, a tensorproduct of knot vectors in n and g directions, i.e. {0 0 0.5 1 1} � {0 00.5 1 1} along with associated control points, forms a 2 � 2 platemesh in parametric and physical domain. Fig. 7 shows the sche-matic of isogeometric framework. It shows the physical mesh (a),parametric domain with control net (b) and index space (c).Shaded region represent a unique knot interval (an element) inthe knot vector and parent NURBS element (d).

In order to perform numerical integration, Guass quadrature isemployed. Integration is performed by mapping from physical (x,y)to parametric (n,g) to parent domain ðn; gÞ and is performed overthe index space. Index space is the space of tensor product of knotvectors in n and g directions. The mapping between physical andparametric domain is performed as follows:

x

y

� �¼XnCP

k¼1

R1kðn;gÞ

Bxk

Byk

� �@Rp

k@x

@Rpk

@y

8<:

9=; ¼ ½J�1�

@Rpkðn;gÞ@n

@Rpkðn;gÞ@g

8<:

9=;

½J� ¼@x@n

@y@n

@x@g

@y@g

" #ð39Þ

Bxk and Byk are control point coordinates and Jxnyg is the mappingfrom physical to parametric domain. Mapping from parametric toparent domain is the standard mapping as is done in finite element.

Table 1Comparison of various NURBS elements, including k-refined with analytical solution for c

Load, �p Levy’s analytical MXFEM/Reddy 9

0 0 0 017.8 0.237 0.2392 038.3 0.471 0.4738 063.4 0.695 0.6965 095 0.912 0.9087 0

134.9 1.1210 1.1130 1184 1.323 1.308 1245 1.521 1.501 1318 1.714 1.688 1402 1.902 1.866 1

4. Numerical testing

Isotropic, orthotropic and laminated composite plates are stud-ied here. Geometric nonlinearity is accounted for in von-Karmansense including membrane-bending coupling. Several, thin andmoderately thick, isotropic, orthotropic and laminated compositeplate examples are studied for validation purposes. Differentboundary conditions, plate to thickness ratios and ply angles areconsidered. Due to bi-axial symmetry, only a quadrant of the plateis modeled and center deflections are computed and validated withthe literature. The computed center deflection is normalized as�w ¼ w=h. Three different boundary conditions are considered forthe analysis.

SS1 : v0¼w0¼/y¼0; at x¼ a=2;

u0¼w0¼/x¼0; at y¼b=2SS3 : u0¼v0¼w0¼0 at x¼ a=2; y¼ b=2

Clamped : u0¼v0¼w0¼/x¼/y¼0; at x¼ a=2 and y¼ b=2symmetry B:C:; x¼0; u0¼/x¼0

y¼0; v0¼/y¼0

4.1. Clamped isotropic plate under uniform loading

In this example, a thin a/h = 100, clamped, isotropic square plateunder uniform loading is analyzed. The plate length and thicknessare a = 300 in. and h = 3 in. and the material properties areE1 = 30E6 psi and m = 0.316. The center deflection of a quater plateis computed and is validated with Reddy [28]’s mixed finite ele-ment solution and Levy’s analytical [61] and Urthaler. Levy’s solu-tion is considered as a benchmark solution for validatinggeometrically nonlinear analysis of thin plates. For most of the ele-ments tetsed, solution converged well with the analytical and Red-dy’s mixed finite element solution. Urthaler and Reddy [28]required 4 � 4 mesh of 9 node quadratic element with 7 dof/nodeto compute the center deflection while 9QuadNURBS/R required2 � 2 mesh with 5 dof/control point. From the integration pointof view, quadratic and cubic NURBS, with reduced integration pro-duced equivalent center deflection. Higher-order p and k-refinedNURBS element performed equally well with full integration. Sim-ilarly, k-refined quadratic NURBS elements (25QuadNURBSKR and16QuadNURBSKR) with full integration produced the equivalentcenter deflection as its counterpart 9QuadNURBS/R element withreduced integration. Table 1 and Fig. 8 give the load vs deflectiondata and the curve, respectively.

Next, % error in center displacement w.r.t Levy’s solution innonlinear analysis is plotted for 9LinNURBS2 and quadratic NURBSelements. % error is greater in the case of 9QuadNURBS/F as com-pared with 9QuadNURBS/R due to slenderness of the plate. LinearNURBS element, 9LinNURBS2, perform well due to increased

lamped, isotropic plate under uniform loading.

QuadNURBS/R 16QuadNURBSKR 25QuadNURBSKR

0 0.2393 0.2354 0.2328.4741 0.4678 0.4635.6975 0.6951 0.6854.9103 0.9036 0.8985.1148 1.1092 1.1045.3101 1.3059 1.302.5024 1.4942 1.4969.6885 1.6758 1.6856.8657 1.8662 1.8652

Fig. 8. Load vs. deflection curve for a clamped, isotropic plate under increasinguniform load.

Fig. 9. % error in displacement w.r.t Levy’s analytical solution for various NURBSelements for 2 � 2 mesh in nonlinear analysis.

Fig. 10. Shear locking test for various NURBS elements for 2 � 2 mesh.

Fig. 11. Shear locking test for various NURBS elements for 3 � 3 mesh.

Table 2Center deflection (F) vs. load values of a simply supported (SS3) isotropic square plate un

Load, P FE/Reddy 9QuadNurbs 9Q

0 0 0 06.25 0.278 0.2735 0.2

12.5 0.4619 0.4575 0.425 0.6902 0.6874 0.650 0.957 0.9563 0.975 1.133 1.1337 1.1

100 1.2686 1.2702 1.2125 1.3809 1.382 1.3150 1.4774 1.4802 1.4175 1.5629 1.5661 1.5200 1.6399 1.6437 1.6


smoothness of basis function in solution space. Fig. 9 shows the %error in nonlinear deflection for increasing loads.

4.2. Shear locking test in thin plates

Shear Locking is the over constraining of element stiffness ma-trix as the side to thickness ratio become large, resulting in under-estimation of displacement. Shear locking test in linear analysiscontext is performed with k-refined linear and quadratic NURBSelement for various a/h ratios.

A square plate with length, L = 10 and material properties asE = 10.92 and m = 0.3 with SS1 simply-supported boundary condi-tion under uniform loading, for various a/h ratios is analyzed. Per-centage error in center displacement w.r.t reference solution iscomputed to study the shear locking effects. Figs. 10 and 11 showshear locking free behavior for increasing a/h ratios for 2 � 2 and3 � 3 meshes. In case of k-refined linear NURBS element, additionof a knot in the knot vector provide increased smoothness andseems to remove numerical ill-conditioning effect. QuadraticNURBS element also performs excellently with full and reducedintegration.

4.3. Simply supported isotropic plate under uniform loading

A square plate with length L = 10 in., thickness h = 1 in andmaterial properties E1 = 7.8E6 psi and m = 0.3 with simply-sup-ported boundary conditions, SS1 and SS3, under uniform load isconsidered here. The computed center deflection is compared withReddy and Putcha [32]’s solution. Tables 2–4 show the centerdeflection data for various NURBS elements under increased load.9QuadraticNURBS/R element (4 elements, 25 control point, 125 de-grees of freedom) is in an excellent agreement with 4 � 4 mesh (16elements, 81 nodes and 405 dof) 9QuadraticReddy/(F/R) finite ele-ment analysis. Also, the center deflection values computed using p

der uniform loading.

uadNurbsHR 25QuarticNurbs 25QuadNurbsKR

0 0752 0.2778 0.280259 0.4617 0.4645879 0.6899 0.6925563 0.9566 0.958632 1.1326 1.1342679 1.2682 1.2694803 1.3803 1.3813771 1.4769 1.4776627 1.5624 1.5629399 1.6395 1.6398

Table 3Center deflection (R) of a simply supported (SS3) isotropic square plate under uniform loading.

Load, P FE/Reddy 9QuadNurbs 9QuadNurbsHR 25QuarticNurbs 16QuadNurbsKR

0 0 0 0 0 06.25 0.279 0.2784 0.278 0.278 0.2811

12.5 0.463 0.4626 0.4618 0.4619 0.468125 0.6911 0.691 0.6897 0.6901 0.700750 0.9575 0.9579 0.9561 0.9567 0.972875 1.1333 1.1339 1.1319 1.1327 1.1524

100 1.2688 1.2696 1.2674 1.2683 1.2907125 1.3809 1.3817 1.3794 1.3804 1.4051150 1.4774 1.4783 1.476 1.477 1.5036175 1.5628 1.5638 1.5614 1.5625 1.5909200 1.6398 1.6408 1.6385 1.6396 1.6695

Table 4Center deflection of a simply supported (SS1/F) isotropic square plate under uniform loading.

Load, P FE/Reddy 9QuadNurbs 9QuadNurbsHR 25QuarticNurbs 16QuadNurbsKR

0 0 0 0 0 06.25 0.2812 0.284 0.2836 0.2842 0.2854

12.5 0.5185 0.5244 0.5229 0.524 0.525625 0.8672 0.879 0.8725 0.8765 0.877850 1.3147 1.3341 1.3267 1.3296 1.329475 1.6237 1.6467 1.6385 1.6433 1.6411

100 1.8679 1.8918 1.8849 1.8921 1.8876125 2.0746 2.0967 2.0925 2.1028 2.0957150 2.2549 2.2744 2.274 2.2881 2.2779175 2.4168 2.4322 2.4364 2.455 2.4414200 2.5645 2.5747 2.5841 2.608 2.5903

Fig. 12. Load vs. deflection curve for a simply supported (SS3/F), isotropic plateunder increasing uniform load.

Fig. 13. Load vs. deflection curve for a simply supported (SS3/R), isotropic plateunder increasing uniform load.

Fig. 14. Load vs. deflection curve for a simply supported (SS1/F), isotropic plateunder increasing uniform load.

Table 5Center deflection of a clamped, cross-ply (0/90/90/0), laminated composite squareplate under uniform loading.

Load, P Experimental 9QuadNURBS/R 9LinNURBS/F FE/Reddy

0 0 0 0 00.4 0.78 0.6573 0.70962 0.65040.8 1.2 1.0134 1.09567 1.00391.2 1.48 1.2512 1.3578 1.24061.6 1.74 1.4328 1.5607 1.42162 1.87 1.58175 1.7284 1.5701


and k-refined NURBS element converges to the required solution.The load vs deflection curve for SS3 boundary condition with full

and reduced integration and for SS1 boundary condition with fullintegration are shown in Figs. 12–14 respectively. It is observedthat SS3 boundary constraint result in lower deflection than SS1boundary condition.

Fig. 15. Load vs. deflection curve for clamped, cross-ply (0/90/90/0) laminatedcomposite plate under uniform loading.

Fig. 16. Load vs. deflection curve for 6 layer cross-ply (0/90) laminated compositeplate under uniform loading.

Fig. 17. Load vs. deflection curve for 2 layer cross-ply (0/90) laminated compositeplate under uniform loading.

Fig. 18. Load vs. deflection curve for angle ply (45/�45) laminated composite plateunder uniform loading for a/h = 40.

Fig. 19. Load vs. deflection curve for angle ply (45/�45) laminated composite plateunder uniform loading, a/h = 10.

Fig. 20. Load vs. deflection curve for orthotropic plate under uniform loading andSS1 boundary condition.


4.4. Square, symmetric cross-ply (0/90/90/0) laminated compositeplate under uniform loading

A square, symmetric cross-ply, (0/90/90/0), laminated compos-ite plate subjected to uniform load is considered. Dimensions of theplate are a = 12 in., thickness, t = 0.096 in. Material properties areE1 = 1.8282e6 psi, E2 = 1.8315e6 psi, G12 = G13 = G23 = 0.3125e6 psiand m = 0.2395. The center deflection computed using p and k-re-fined NURBS elements compare well with those in the literature.Center deflection computed using NURBS quadratic element is inexcellent agreement with Reddy’s finite element solution [60].

Also, the center deflection is found to be closer to the experimentalcurve than Reddy and Putcha [32] and Zhang [48]’s deflection re-sponse. Table 5 and Fig. 15 shows load vs deflection data and curverespectively.

4.5. Effect of number of layers and thickness on laminated compositeplate

This example studies the effect of number of layers on centerdeflection in a 2 and 6 layer cross-ply, composite laminates and ef-fect of thickness on 2 layer, angle ply (45/�45), laminate. Thedimension of the plate considered are a = 12 in, a/h = 10, 40 and

Fig. 22. Physical meshes with different level of mesh distortion for a clamped,isotropic, square plate.

Fig. 23. Mesh distortion sensitivity test using 9QuadNURBS/R element for 2 � 2mesh.

Fig. 24. Mesh distortion sensitivity test using 25QuadNURBSKR/F element for 2 � 2mesh.

Fig. 25. Mesh distortion sensitivity test using 25QuarticNURBS/F element for 2 � 2mesh.

Fig. 26. % error (center displacement) w.r.t structured mesh in Mesh distortionsensitivity test.

Fig. 21. Load vs. deflection curve for orthotropic plate under uniform loading andSS3 boundary condition.


the material properties are E1 = 40e6 psi, E2 = 1e6 psi,G12 = G13 = 0.6e6, G23 = 0.5e6 psi and m = 0.25. Figs. 16 and 17 showsthe deflection curve for 6 and 2 layer (0/90) cross-ply composite

plates. Increasing number of layers reduce the bending deflectionin a cross-ply laminate. Figs. 18 and 19 show the effect of thicknesson nonlinear deflection on a 2-layer, (45/�45) angle ply, laminatedcomposite plate. The nonlinear effects are reduced as the thicknessis increased, resulting in straightening of the deflection curve to-wards the linear solution.

4.6. Orthotropic square plate under uniform loading

An orthotropic plate with dimension of a = 12 in., thickness ofh = 0.138 and material properties, E1 = 3e6 psi, E2 = 1.28e6 psi,G12 = G13 = G23 = 0.37e6 psi and m = 0.32 is considered. SS1 and SS3boundary conditions are used in the analysis. The results are


compared with the Agyris [62] and Zhang [48] results. Figs. 20 and21 show the deflection vs load curve for SS1 and SS3 boundary con-ditions respectively. Center deflection is in excellent agreementwith the experimental data given by Agyris [62]. Some of theobservations are as follows; k-refined quadratic NURBS elementproduce excellent results as compared to the quadratic NURBS ele-ment with full integration. SS1 boundary condition provide lessercontraint on deflection than SS3 boundary condition.

4.7. Clamped, isotropic square plate with diffferent level of meshdistortion

A thin a/h = 100, clampled, isotropic square plate under uniformloading is analyzed here. The plate length, thickness and materialproperties are a = 300 in., h = 3 in. and E1 = 30E6 psi, m = 0.316respectively. Analysis is performed with a coarsest, 2 � 2, unstruc-tured physical mesh for different levels of mesh distortion. Fig. 22shows the unstructured meshes. 9QuadNURBS/R, 25Quad-NURBSKR/F and 25QuadNURBS/F elements are considered for theanalysis. Figs. 23–25 show the comparison of nonlinear deflectionresponse for the various NURBS elements. 9QuadNURBS/R elementexhibit unstable response due to under-integrated stiffness matrix.However, k-refined quadratic NURBS element i.e. without increas-ing the order of the polynomial and higher-order quartic NURBSelement with full integration produce a stabilized nonlinear deflec-tion response. Fig. 26 shows the % error in delfection in nonlinearanalysis.

5. Conclusions

Geometrically nonlinear NURBS isogeometric finite elementanalysis of laminated composite plate is presented here. Plategeometry is modeled using linear NURBS basis with associatedcontrol net. Subsequent refinements are done at the parametric le-vel. p and k-refinement techniques are explored in element formu-lation and various lower and higher-order elements areconstructed. Numerical results are presented for thin to moder-ately thick plates for various length to thickness ratios, ply-anglesand boundary conditions. The computed center deflection is foundto be in an excellent agreement with the literature and requiredfewer degrees of freedom/control point when compared with reg-ular finite element analysis. For thin plates analysis, k-refined, lin-ear and quadratic NURBS elements remedied the shear lockingproblem. In addition, k-refined quadratic NURBS element producedstable response in nonlinear analysis.

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