Analysis of CFRP laminated plates with spatially varying non-Gaussian inhomogeneities using SFEM

Composite Structures 112 (2014) 308–326

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Composite Structures

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

Analysis of CFRP laminated plates with spatially varying non-Gaussianinhomogeneities using SFEM

http://dx.doi.org/10.1016/j.compstruct.2014.02.0250263-8223/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +91 44 2257 4055; fax: +91 44 2257 4050.E-mail addresses: [email protected] (P. Sasikumar), [email protected]

(R. Suresh), [email protected] (S. Gupta).

P. Sasikumar a, R. Suresh a, Sayan Gupta b,⇑a Vikram Sarabhai Space Centre, Trivandrum 695013, Indiab Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, India

a r t i c l e i n f o

Article history:Available online 3 March 2014

Keywords:Laminated composite plateNon-Gaussian2-D random fieldsStochastic finite element methodOptimal linear expansionFailure probability

a b s t r a c t

A stochastic finite element based methodology is developed for the uncertainty quantification and reliabilityanalysis of laminated CFRP plates with spatially varying non-Gaussian random inhomogeneities. Thematerial and strength properties in the individual laminae are modeled as 2-D non-Gaussian randomfields whose second order probabilistic characteristics are determined from test data. The 2-D randomfield discretization procedure adopted enables representing the fields through a vector of correlatedrandom variables that preserve the second order non-Gaussian characteristics of the field. Expressionsare developed that enable the construction of the stochastic stiffness matrices that are dependent on thesecorrelated random variables. Subsequently, the response analysis and failure probability estimation is car-ried out using Monte Carlo simulations. Discussions on the salient features of the proposed methodologyare highlighted based on a set of illustrative numerical examples.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Reliability analysis of structural components made up of lami-nated fiber reinforced composites require characterization of thespatially random inhomogeneities in the material properties andquantification of the propagation of these uncertainties into thestructure response. The uncertainties in the material propertiesarise due to the complexities associated with the manufacturingprocesses and processing stages and cannot be eliminated despitethe most stringent quality control. Therefore, to evolve robust andsafe designs of composite structural systems, there is a need tocharacterize these uncertainties and include their effects in anystructural analyses. Such an approach not only leads to safer de-signs, but also leads to better material utilization leading to lighterand more efficient structural systems [1,2]. The use of theories ofprobability, random variables and random processes provide a log-ical framework within which these uncertainties can be character-ized. The challenge here lies in generating adequate data fromexperimental testing such that probabilistic models for the uncer-tainties can be adopted and developing a framework within whichthe propagation of these uncertainties into the response can bequantified in a probabilistic sense.

Studies on the uncertainty characterization of the materialproperties in composite structures using probability theory havebeen carried out across different scales - the micro, meso andmacro scales. Characterization of uncertainties at the micro scaleshave been investigated in [3–7]. While these studies enable a bet-ter understanding of the origins of the uncertainties in the materialbehavior, the application of these methods for reliability analysesat the structure level poses difficulties [8]. For structural reliabilityanalyses, the meso and macro scale studies are more relevant; seefor example [9–14]. In most studies reported in the literature, theuncertainties in the ply levels are modeled as random variables [2].Subsequently, investigations on the propagation of these uncer-tainties into the response are carried out using Monte Carlosimulations [15–19].

Models that represent the uncertainties in a material propertyusing a random variable implies a homogenization of the uncer-tainties within the spatial extent of the structure. This, in turn,introduces additional epistemic uncertainties into the model andundermines the accuracy of the reliability analysis. A more accu-rate representation of the uncertainties within the spatial extentof the structural component would be to adopt random field mod-els. Random field models are more robust and can model the spa-tial fluctuations in the properties, as well as the ensemblevariations. However, such models require greater sophisticationboth in terms of testing for material characterization as well asfor response analysis. This probably explains the paucity of studies

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P. Sasikumar et al. / Composite Structures 112 (2014) 308–326 309

in the literature where the uncertainties in composites are mod-eled as random fields.

Analysis of structural systems with random field models for theuncertainties can be addressed within the framework of stochasticfinite element method (SFEM); see [20–22] for monographs onSFEM. The basic principle in traditional finite element method ison obtaining weak form solutions of the governing equations, suchthat, the error is minimized at the nodal points. This involves dis-cretization of the displacement or stress fields and representingthe same in terms of nodal variables and known interpolating func-tions. The additional complexity in SFEM is on discretization of therandom field. The challenge in SFEM lies in the development ofmethods for random field discretization which enables represent-ing the discretized random field in terms of a vector of randomvariables, such that, the probabilistic characteristics of the parentrandom field are preserved subject to error tolerance limits. As inthe case of traditional FEM, the accuracy of the SFEM approach de-pends on the random field discretization mesh size. It must benoted that the meshing for FE discretization and random field dis-cretization could be distinct from each other. The factors that areusually considered in selecting the random field mesh size arestress and strain gradients, frequency range of interest, nature ofinformation available about the random field, correlation structureand correlation length and the ability to accurately model the tailsof the distribution especially when the random field models arenon-Gaussian. A review of the available methods for SFEM discret-ization of random fields are available in [23–26].

The correlation length is usually the primary concern in choos-ing the mesh size in random field discretization. Mesh sizes aregenerally smaller for random fields having small correlationlengths. This, in turn, introduces a large number of correlated ran-dom variables into the formulation which not only increases theCPU time but also leads to numerical difficulties. Additionally,the efficiency of the field discretization scheme depends on theability to preserve the probabilistic characteristics of the parentfield. This is especially true for non-Gaussian random fields whichexhibit features such as asymmetry and long tail behavior. This hasled to the development of spectral based methods for non-Gauss-ian field discretization. More discussions on these methods willbe addressed later in this paper.

Though SFEM based methods have been used for a long time instructural engineering literature, it is only recently that attemptsare being made to extend these methods for uncertainty quantifi-cation in composite structures [27–30]. For example, the recentstudy in [31] have assumed the elements of the constitutive matrixof unidirectional composite rectangular panels as Gaussian fields.Subsequently, the authors have used the well known spectral rep-resentation methods [32] for discretizing the random field. How-ever, in real life applications, uncertainties for the primarymaterial properties are easier to measure rather than the elementsof the constitutive matrix, which are functions of multiple materialproperties. This approach was used in [33] where the authors di-rectly assume the moduli properties of a rectangular plate to beGaussian fields and treat this as the starting point of their studies.Subsequently, the constitutive matrix was constructed for the lam-inate. However, assuming Gaussian models for the material prop-erties can lead to unsatisfactory results owing to the symmetricnature of the distribution and due to the finite non-zero probabilityof the properties attaining physically impossible values. More real-istic models for the physical parameters are typically non-Gaussian[34]. The commonly used non-Gaussian models for physicalparameters in composites are lognormal, Weibull, Gamma andother forms of extreme value distributions [2,35]. Consideringnon-Gaussian random fields lead to complexities which are noteasy to handle. Thus, though the material parameters wereassumed to be band limited in [36], the distributions were

subsequently approximated as Gaussian fields for the sake of con-venience. However, most discretization methods do not preservethe non-Gaussian characteristics of the parent field. For example,the weighted integral approach used for discretizing the randomfields in composites [37] leads to Gaussian random variables irre-spective of the distribution of the random fields. It has been sug-gested in the composite literature [2,31,33] that polynomialchaos expansions could be used for non-Gaussian field discretiza-tion. However, application of this method for uncertainty quantifi-cation in composite structures is not easy and to the best of theauthors’ knowledge, such studies have not been reported in com-posites literature.

The focus of this paper is on using an alternative spectral method– the optimal linear expansion scheme (OLE)- for the discretizationof non-Gaussian random fields. The OLE method was first proposedin [38] primarily for Gaussian random fields. The applicability ofOLE in discretizing 1-D non-Gaussian random fields was discussedin greater details later in [39]. The underlying principle of this meth-od lies in representing the non-Gaussian random field as a seriesexpansion of deterministic interpolating functions and a vector ofcorrelated random variables, such that the error invoked in approx-imating the field by this series is minimized at the nodal points.Existing studies available in the literature on OLE have primarily fo-cused on 1-D random fields. In this paper, the method is extendedfor the discretization of 2-D random fields. The paper is organizedas follows. First, the problem of 2-D random field discretizationusing OLE is addressed in details. Next, the OLE based SFEM formu-lation for failure analysis of fiber laminated composite plates are de-rived. Finally, a set of numerical examples are presented that bringout the salient features of the proposed developments and illustratethe effects of the spatial uncertainties in the failure predictions of fi-ber laminated composite plates. The paper ends with a set of obser-vations and concluding remarks. The paper consist of twoappendices. Appendix A details some of the preliminary experimen-tal studies that have been undertaken by the present authors indeveloping probabilistic models for the uncertain material parame-ters. Appendix B illustrates how the proposed methodology can beused with higher order theories for the constitutive relations.

2. Optimal linear expansion

The optimal linear expansion, proposed in [38], seeks to repre-sent a stationary random field gðxÞ as an expansion given by

gðxÞ � �gðxÞ ¼ g0 þXn

k¼1

SkðxÞvðxkÞ; ð1Þ

where, �gðxÞ is the discretized random field with spatial extent alongx; n denotes the number of nodal points used for the random fielddiscretization, g0 ¼< gðxÞ > denotes the mean of the process, SkðxÞare deterministic interpolating functions, xk are the nodal pointsand fvkg

nk¼1 � fvðxkÞgn

k¼1 represents a vector of random variablesdefined at the nodal points. The crux in OLE lies in estimating thedeterministic interpolating functions SkðxÞ and are determined byminimizing the variance of the error of discretization, subject tothe condition that the expectation of the discretization error is zero.Mathematically, this implies seeking SkðxÞ, such that,

< �0>2 ¼ fgðxÞ �

Xn

k¼1

SkðxÞvðxkÞg2* +; ð2Þ

is minimized where �0 is the discretization error, subject to the con-straint < �0 >¼< gðxÞ � �gðxÞ >¼ 0. Here, < � > denotes the expecta-tion operator. Expanding Eq. (2) in terms of SkðxÞ and setting@ < �0>

2=@SiðxÞ ¼ 0, it can be shown that this leads to a set ofequations

0.1

0.1

0.1

0.1

0 10 10 1

0.1

0.1

0.1

0.2

0.2

0.2

0 20 20 2

0.2

0.2

0.3

0.3

0.3

0 30 3

0.3

0.3

0.4

0.4

0 40 4

0.4

0.4 0.5

0.5

0.5

0 5

0.5

0.5

0.6

0.6

0 60 6

0.6

0.7

0.7

0 7

0.7

0.8

0 8

0.8

0 .9

0.9

s1

s 2

0 0.5 10

0.2

0.4

0.6

0.8

1

Fig. 1. Contour plots for Rggðx; yÞ; the numbers denote the correlation.

310 P. Sasikumar et al. / Composite Structures 112 (2014) 308–326

SkðxÞ ¼ C�1V; ð3Þ

where, Vi ¼< gðxÞgðxkÞ > and Cij ¼< gðxiÞgðxjÞ > is the covariancematrix of gðxÞ corresponding to the nodal points. It can be shownthat the shape functions constructed in the above manner havethe property SkðxjÞ ¼ djk, where, djk is the Kronecker-delta function.Moreover, it has been shown that the variance of the error can beexpressed as [24]

Var½gðxÞ � �gðxÞ� ¼ Var½gðxÞ� � Var½�gðxÞ�: ð4Þ

Since the error variance is always positive, it follows that the dis-cretized random field �gðxÞ always underestimates the variance incomparison to the original random field. Further, it was shown[40] that

Cov½�gðxÞ; �0ðxÞ� ¼ 0; ð5Þ

indicating that the error and the discretized random field are uncor-related. Here, Var½ � � and Cov½ � � are the the variance and covarianceoperators.

2.1. 2-D OLE representation

The equivalent optimal linear expansion representation of 2-Dstationary random fields, gðx; yÞ, is given by

gðx; yÞ � �gðx; yÞ ¼ g0 þXm

k¼1

Xn

j¼1

Skjðx; yÞvðxk; yjÞ: ð6Þ

Here, x and y denote the spatial coordinates in two orthogonaldirections along a plane, n and m represent the number of OLE nodalpoints along the two orthogonal directions, N ¼ n�m are the totalnumber of nodal points, Skjðx; yÞ are the deterministic 2-D interpo-lating functions and fvkg

Nk¼1 � fvðxk; yjÞg

m;nk;j¼1

represents a vector ofrandom variables associated with the N nodal points ðxk; yjÞ in thediscretized 2-D random field. Eq. (6) can now be rewritten as

gðx; yÞ � �gðx; yÞ ¼ g0 þXN

k¼1

Skðx; yÞvk; ð7Þ

where, the suffix k represents the k-th nodal point and Skðx; yÞ rep-resents the corresponding interpolating shape function. As in the 1-D case, the shape functions Skðx; yÞ are determined by minimizingthe variance of error of discretization, subject to the condition thatthe expectation of the discretization error is zero. Here, the varianceof the error is represented in the form of Eq. (2) but is a 2-D functionof x and y. The application of the minimization condition leads to aset of N coupled linear equations, which when written in matrixform are

C1111 C1112 : : C111n C1121 : : C11mn

C1211 C1212 : : C121n C1221 : : C12mn

: : : : : : : : :

C1n11 C1n12 : : C1n1n C1n21 : : C1nmn

C2111 C2112 : : C211n C2121 : : C21mn

: : : : : : : : :

Cmn11 Cmn12 : : Cmn1n Cmn21 : : Cmnmn

2666666666664

3777777777775

S11ðx; yÞS12ðx; yÞ

:

S1nðx; yÞS21ðx; yÞ

:

Smnðx; yÞ

2666666666664

3777777777775

¼

gðx; yÞgðx1; y1Þh igðx; yÞgðx1; y2Þh i

:

gðx; yÞgðx1; ynÞh igðx; yÞgðx2; y1Þh i

:

gðx; yÞgðxm; ynÞh i

2666666666664

3777777777775:

ð8Þ

Here, Cijkl ¼< gðxi; yjÞgðxk; ylÞ > denotes the covariance of the ran-dom field between the points ðxi; yjÞ and ðxk; ylÞ. Introducing thenotations SkðxÞ ¼ Sijðx; yÞ and VkðxÞ ¼< gðx; yÞgðxi; yjÞ >, where, xdenotes the point ðx; yÞ, Eq. (8) can be written in the compact form

½C�fSðx; yÞg ¼ fVðx; yÞg: ð9Þ

Here, ½C� is a N � N matrix and fSðx; yÞg and fVðx; yÞg denote N � 1dimensional vectors whose elements are functions of x and y.Assuming that the covariance function is non-singular, the shapefunctions fSkðx; yÞgN

k¼1 can be estimated by multiplying both sidesof Eq. (9) by C�1. Moreover, since Cijkl ¼ Cklij, it can be shown by di-rect substitution that SkðxlÞ ¼ dkl is a solution and the only solutionof Eq. (9). This implies that the k-th interpolating function takes avalue of unity at the k-th nodal point and is zero at all other nodes.An implication of this property is that at the nodal points the pdf ofthe random field is exactly preserved irrespective of the pdf of theparent random field.

To illustrate the salient features of the OLE discretization, a 2-Dstationary non-Gaussian random field having Weibull distributionof the form

pðz; k; kÞ ¼ kk

� �zk

� �k�1exp � z

k

� �k� �

; ð10Þ

is considered. Here, k ¼ 20:7 is the shape parameter and k ¼ 158:9is the scale parameter and z is the observed value at ðxj; ykÞ in a 2-Dplane. The correlation function of the 2-D random field gðx; yÞ is as-sumed to be of the form

Rggðs1; s2Þ ¼ r2 exp � s21

c1þ s2

2

c2

� �� ð11Þ

and represents the correlation between ðx; yÞ and ðxþ s1; yþ s2Þ,where s1 and s2 respectively represents the lag along the twoorthogonal directions, r2 is the variance and c1 and c2 are constants.Fig. 1 shows the contour plots for the correlation function whenr ¼ 1; c1 ¼ c2 ¼ 0:4.

The 2-D shape functions for the correlation function shown inFig. 1 are next derived. For the sake of illustration, the random fieldin the square domain is discretized into a mesh size of 2� 2, leadingto 9 nodal points. These nodes are shown as circles in Fig. 2; the so-lid grid lines indicate the FE mesh. Fig. 2 highlight that the meshingfor FE and random field discretization could be different. The 9shape functions corresponding to the OLE nodes are shown inFigs. 3–11. The property SkðxjÞ ¼ dkj is clearly discernible from thesefigures. Finer meshing would lead to a larger number of shape func-tions which have similar properties but are not shown in this paper.

The shape functions, Skðx; yÞ are derived by imposing the condi-tions that the error is zero at the OLE nodes. This is illustrated inFig. 12 where a comparison is made between the pdfs of gðx; yÞand �gðx; yÞ at an OLE node; an exact match is observed. The

0 0.5 10

0.2

0.4

0.6

0.8

1

x

y

Fig. 2. Discretization of a plane; circles: OLE nodes; solid lines: FE mesh.

00.2

0.40.6

0.81

0

0.5

1−0.5

0

0.5

1

Fig. 3. S1ðx; yÞ.

00.2

0.40.6

0.81

0

0.5

1−0.5

0

0.5

1

Fig. 4. S2ðx; yÞ.

00.2

0.40.6

0.81

0

0.5

1−0.5

0

0.5

1

Fig. 5. S3ðx; yÞ.


accuracy even at the tails of the pdf is exact as can be seen inFig. 13 where the pdfs are plotted in the log-scale. On the otherhand, the representation at locations that are not OLE nodes is ex-pected to have errors and this is clearly seen from the mismatchobserved in the corresponding pdfs shown in Fig. 14. Nevertheless,�gðx; yÞ retains the non-Gaussian characteristics even at non-nodalpoints as can be seen from the deviation from the straight linewhen the pdf is plotted in the normal probability paper; see Fig. 15.

2.2. Number of nodal points

The number of shape functions required for the discretization ofthe random field depend on the number of nodal points, N, used inthe 2-D discretization. To investigate the effects of selecting differ-ent mesh sizes for the random field discretization, a parametricstudy is carried out where the error Eðx; yÞ ¼ gðx; yÞ � �gðx; yÞ iscomputed for different mesh sizes ranging from 2� 2 to 6� 6.For example, Fig. 16 shows Eðx; yÞ when a 2� 2 meshing is used.Fig. 17 shows the corresponding contour plots with the numbersindicating the error levels. The red1 crosses in Fig. 17 shows the no-

1 For interpretation of color in Fig. 17, the reader is referred to the web version othis article.

f

dal points. It can be seen that the zero error contours pass throughthe nodal points. Figs. 18,19 show the error plots when the mesh sizeis 4� 4 and Figs. 20,21 are the corresponding figures when the meshsize is 6� 6. An inspection of these figures reveal that the percentageof the area in the x� y domain such that �� < Eðx; yÞ < � increases asthe meshing becomes finer. Here, � is a tolerance value. This indi-cates that a finer meshing leads to a better representation of the ran-dom field. However, finer meshing leads to larger values of N andincreases the computational costs. In the limiting case when themesh size is infinitely small, gðx; yÞ and �gðx; yÞ coincide withN !1. There is therefore a need to optimize on N.

A criterion for selecting N, used in this study, is to impose thecondition that the global mean square error is below a specifiedthreshold value �. Mathematically, this condition is represented as

00.2

0.40.6

0.81

0

0.5

1−0.5

0

0.5

1

Fig. 6. S4ðx; yÞ.

00.2

0.40.6

0.81

0

0.5

1−0.2

0

0.2

0.4

0.6

0.8

1

Fig. 7. S5ðx; yÞ.

00.2

0.40.6

0.81

0

0.5

1−0.5

0

0.5

1

Fig. 8. S6ðx; yÞ.

00.2

0.40.6

0.81

0

0.5

1−0.5

0

0.5

1

Fig. 9. S7ðx; yÞ.


Z xu

xl

Z yu

yl

�0ðx; yÞ2D E

dx dy 6 �: ð12Þ

Here, ðxl; ylÞ and ðxu; yuÞ respectively define the lower and upperlimits of the spatial domain of the problem. For the auto-correlationfunction in Eq. (11), since the coefficients c1 and c2 are equal, it im-plies that the correlation lengths along x and y are equal. The num-ber of nodal points along either x or y directions can be selectedbased on selecting a tolerance limit for the global mean square erroralong either of the axes. This leads to the number of nodal points mand n along the x and y axes and helps in defining the mesh grid.Fig. 22 shows a plot of the global mean square error with respectto N ¼ m� n the number of terms in OLE. It is seen that the globalmean square error is less than 0:0174 when n ¼ 16. The globalmean square error can also be expressed as a 2-D function of m

and n. The corresponding contour plot for the 2-D function, shownin Fig. 23, reveals its symmetric nature owing to the correlationlengths being equal along x and y directions.

However, in the more general case, the correlation lengthsalong x and y need not be equal and hence m and n need not beequal. For example, if c1 ¼ 0:4 and c2 ¼ 0:2 in Eq. (11), the correla-tion length along x axis is smaller and correspondingly requires fi-ner meshing along x-axis. The contour plots for the global meansquare error now are shown in Fig. 24 and are observed to be nolonger symmetric. Fig. 25 shows the dependence of the globalmean square error as a function of N ¼ m� n. It can be seen thatin this case, fewer nodal points are required when � is consideredto be the same as in the case when c1 ¼ c2 ¼ 0:4. However, in thenumerical studies considered later in this paper, the autocorrela-tion function is assumed to be of the form as in Eq. (11) with

00.2

0.40.6

0.81

0

0.5

1−0.5

0

0.5

1

Fig. 10. S8ðx; yÞ.

00.2

0.40.6

0.81

0

0.5

1−0.5

0

0.5

1

Fig. 11. S9ðx; yÞ.

100 120 140 160 1800

0.01

0.02

0.03

0.04

0.05

g &g

pdf

gg

Fig. 12. Comparison of pdf of �gðx; yÞ and gðx; yÞ at an OLE node.

100 120 140 160 18010−8

10−6

10−4

10−2

100

g &g

Log

(pdf

)

gg

Fig. 13. Comparison of pdf of �gðx; yÞ and gðx; yÞ in the log scale, at an OLE node.

100 120 140 160 18010−8

10−6

10−4

10−2

100

g

Log

(pdf

)

OLE Node pointGauss point

Fig. 14. Comparison of pdf of �gðx; yÞ and gðx; yÞ when ðx; yÞ is at Gauss point.


c1 ¼ c2 ¼ 0:4. Consequently, the mesh size is selected such thatm ¼ n ¼ 4 and N ¼ 16.

2.3. Discussions

The discussion in the preceding section highlights that the effi-ciency of random field discretization depends on selecting ascheme that requires the least number of random variables forthe discretized random field representation, and preserving theprobabilistic characteristics of the parent random field. Here, onemust mention that the probabilistic descriptors are usually limitedto second order characteristics as stronger forms of representation,in terms of multi-dimensional joint pdfs, are not feasible even for1-D random fields. In this context, it has been shown in the litera-ture that the Karhunen–Loeve (KL) series expansion is the most

optimal series representation for a Gaussian field for the same er-ror tolerance [32]. Here, the eigenfunctions of the correlation func-tion of the random field constitute the basis for the seriesrepresentation.

Unfortunately, the optimality condition is satisfied only if theeigenfunctions are obtained in closed form. For most correlationmodels, closed form solutions for the integral eigenvalue problemassociated with KL expansions are not available. Adopting

100 110 120 130 140 150 160 170 180

0.001

0.010.05

0.25

0.75

0.950.99

0.999

g

Prob

abilit

y

Fig. 15. Pdf of �gðxk; ykÞ in normal probability paper.

00.2

0.40.6

0.81

0

0.5

1

−0.05

0

0.05

xy

Erro

r

Fig. 16. Error plot for 2 � 2 random field discretization.

−0.02−0.01 −0.0

1

−0.01

0

0

0

0

0

0

00

0.01 0.01

0.01 0.01

0.01

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0.05

0.05

0.05

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fig. 17. Contour error plot for 2 � 2 random field discretization: + OLE nodal points.

00.2

0.40.6

0.81

0

0.5

1−0.05

0

0.05

xy

Erro

r


−0.03

−0.02

−0.02

−0.02

−0.01

−0.01

−0.01

−0.0

1

−0.01

−0.01

−0.01−0

.01

0 0

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0.04

0.04

0.04

0.04

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


00.2

0.40.6

0.81

0

0.5

1−2

0

2

4x 10−3

xy

Erro

r



numerical techniques for such problems involves discretizationthat converts the integral eigenvalue problem into a matrix eigen-value problem, whose eigenvectors serve as the approximation forthe eigenfunctions. It was shown in [38] that this numerical proce-dure does not necessarily lead to optimality. In fact, the authorsshowed that the expansion OLE (EOLE) method that uses KL expan-sions in conjunction with OLE to derive the basis functions, to be amore efficient random field discretization scheme.

The application of the EOLE based approach is relativelystraightforward when the random fields to be discretized are

Gaussian. For discretizing non-Gaussian fields, a translation pro-cess based model [41] has been suggested in [38]. This involvestransforming the problem into the Gaussian space and using EOLEto discretize the Gaussian field. Subsequently, the discretized non-Gaussian field can be obtained by retransforming the problem intothe non-Gaussian space. Though this approach is simple in princi-ple, the difficulty lies in estimating the corresponding correlationfunction of the transformed Gaussian process. The correlationfunction for the transformed Gaussian field can be obtained bysolving a two dimensional integral equation [42]. Though

−0.001

−0.001

−0.0

01−0

.001

−0.0005

−0.0005

−0.0005

−0.0005

−0.0005

−0.0005

−0.0005

−0.0005

0

0

0

0

00

00

0

0

0 0

0

00

0

0

0

0

0.00050.0005

0.0005 0.0005

0.00050.0005

0.00

050.

0005

0.00050.0005

0.0005

0.0005

0.0005

0.00050.00050.0005

0.0005

0.0005

0.0010.001

0.0010.001

0.0010.001

0.00

10.

001

0.001

0.00

1

0.0010.001

0.00

1

0.001

0.0010.001

0.0015

0.0015

0.0015

0.0015

0.0015

0.0015

0.00

15

0.0015

0.0015

0.0015

0.0015

0.0015

0.002

0.002

0.002

0.002

0.0

02

0.002

0.0

02

0.002

0.00

2

0.002

0.002

0.002

0.0025

0.0025

0.0025

0.0025

0.00

3

0.003

0.003

0.003

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


5 10 15 20 250

0.1

0.2

0.3

0.4

OLE nodes

Mea

n sq

uare

erro

r

Fig. 22. Global mean square error; c1 ¼ c2 ¼ 0:4.

2 3 4 52

3

4

5

m

n

0.050.05

0.05 0.05

0.10.1

0.1 0.1

0.150.15

0.15 0.15

0.20.2

0.2

0.2 0.20.250.30.35

Fig. 23. Contour plots for global mean square error; c1 ¼ c2 ¼ 0:4.

2 3 4 52

3

4

5

m

n

0.050.05

0.05

0.10.1

0.10.1

0.150.15

0.150.15

0.20.2

0.20.2

0.250.25

0.25

0.30.3

0.3

0.350.35

0.35

0.40.4

0.4

0.450.5

Fig. 24. Contour plots for global mean square error; c1 ¼ 0:4; c2 ¼ 0:2.

5 10 15 20 250

0.2

0.4

0.6

OLE nodes

Mea

n sq

uare

erro

r

Fig. 25. Global mean square error; c1 ¼ 0:4; c2 ¼ 0:2.


analytical expressions for the bounds of the auto-correlationfunction for the transformed Gaussian field are provided for afew common forms of the correlation function for the parentnon-Gaussian field, in general, the integral equation can be solvednumerically in an iterative manner [39] which can be computa-tionally intensive. Moreover, for some forms of the correlationfunction in the non-Gaussian space, a corresponding correlationfunction in the transformed Gaussian space may not exist. Most

importantly, application of EOLE for non-Gaussian space does notguarantee that the discretization is most optimal when the prob-lem is transformed back to the non-Gaussian space.

An alternative spectral approach used in the literature for non-Gaussian random field discretization is to adopt the polynomialchaos expansions (PCE). Here, the basis functions are polynomialsof random variables. The optimality condition is ensured when,depending on the distribution of the parent random field, the formof the polynomials are selected from the Askey scheme [43]. Thistask however is not easy for arbitrary non-Gaussian random fields.Extending the PCE expansions for 2-D random fields involves fur-ther complexities that require assumptions on 2D correlation func-tion modeling as well. Moreover, adopting PCE representation forthe non-Gaussian random fields makes the associated SFEM devel-opments significantly more cumbersome and involved and is out-side the scope of this paper. Instead, in this study, the OLEexpansions are used directly for the discretization of the non-Gaussian field. The shape functions are derived from the correla-tion function of the parent non-Gaussian random field. The OLErepresentation for the non-Gaussian random fields are subse-quently used for the SFEM development. This is discussed in thefollowing section.

3. Stochastic finite element method

This section focuses on the development of an OLE based sto-chastic finite element framework for composite plates. The formu-lation presented in this section is based on generalizing the


procedure available for deterministic composite plates. First, the FEequations are developed for the laminated composite plate wherethe laminae are characterized by spatial random inhomogeneities.Subsequently, the procedure for failure analysis have beendeveloped.

3.1. Formulation

Consider a multi layer laminated composite plate with uniformthickness and coordinate systems as shown in Fig. 26. Here, theplate dimensions are expressed as a� b, while the thickness is h.It is assumed that the composite plate is subjected to uniformlydistributed transverse loads only. The geometrical axes of the plateare denoted by the x� y� z axes system.

The constitutive equations for the plate can be expressed as

NMV

264

375 ¼

A B 0B D 00 0 H

264

375

em

j

es

264

375; ð13Þ

where, N;M and V are the vector of membrane forces per unit length,vector of bending moment per unit length and vector of shear forcesper unit length in the global axes system respectively. The mem-

brane strain is expressed as em ¼ u; x; v; x; u; y þ v; x� T , the shear

strain is given by es ¼ w; x � /x; w; y � /y

� T and the curvature/

bending strain is represented as j ¼ /x; x; /y; y; /x; y þ /y; x

� T .Here, u;v and w represent the deflections respectively along x; yand z axes, /x and /y are respectively the rotations along x and y axesand, x and y denote derivatives with respect to x and y axisrespectively.

The stiffness matrix in Eq. (B.1) depends on the extensionalstiffness matrix A, bending-extension coupling matrix B, the bend-ing stiffness matrix D and the shear stiffness matrix H. The ele-ments of these matrices are expressed as

Aij ¼Z h=2

�h=2Q mij

dz; Hkl ¼Z h=2

�h=2Q skl

dz;

Bij ¼Z h=2

�h=2Q mij

z dz; Dij ¼Z h=2

�h=2Q mij

z2 dz;

ð14Þ

where, i; j ¼ 1;2;3, and k; l ¼ 1;2; z is distance from neutral plane.Here, Q m and Q s are the stiffness matrices for a lamina whose ele-ments are dependent on the material properties E1; E2;G12;G13;G23;

m12; m21 and the direction cosines of the individual lamina axes withrespect to the global structure axes [44].

Fig. 26. Schematic diagram of a section of a laminated composite plate; x; y; z arethe geometrical axes, 1;2 are material axis, h ply orientation.

The derivations of the FE equations in the traditional frame-works consider the material properties to be deterministic. Exist-ing studies that take into account the effect of uncertainties inthe material properties treat these parameters as random vari-ables; see [7,13–15]. The development of the corresponding FEequations lead to stiffness matrices which are functions of theserandom variables. However, a random variable approach to model-ing the material properties implies that the randomness is consid-ered across sample realizations only; the random variations in thematerial properties along the spatial extent are ignored.

For the development of a finite element framework throughwhich these random spatial inhomogeneities are incorporated intothe analysis, the following material properties are assumed to be2� D random fields and are expressed as follows:

E1ðx; yÞ ¼ E10 1þ g1ðx; yÞ½ �;E2ðx; yÞ ¼ E20 1þ g2ðx; yÞ½ �;G12ðx; yÞ ¼ G120 1þ g3ðx; yÞ½ �;G13ðx; yÞ ¼ G130 1þ g4ðx; yÞ½ �;G23ðx; yÞ ¼ G230 1þ g5ðx; yÞ½ �;m12ðx; yÞ ¼ m120 1þ g6ðx; yÞ½ �:

ð15Þ

Here, the property parameters with subscript zero indicates themean value (the deterministic value of the parameter) about whichthe random fluctuations occur, and fgiðx; yÞg

6i¼1 are assumed to be a

vector of stationary, mutually independent 2-D random fields. Theassumption of mutual independence has been considered for thesake of simplicity and is not a restrictive assumption. The probabi-listic characteristics of these random fields can be established onlyafter exhaustive experimental testing. Based on some limited test-ing, the marginal pdf of the above parameters have been deter-mined for certain grades of laminated composites. These aredetailed in Appendix A. For estimating the correlation propertiesof these random fields, more detailed testing needs to be carriedout and is currently in progress. For the sake of simplicity, the cor-relation functions for these random fields have been arbitrarily se-lected to be of the form as in Eq. (11).

Next, the random fields are discretized using the OLE methoddiscussed in the previous section. Following Eq. (7), each of the2-D random fields can be represented as follows:

E1ðx; yÞ ¼ E10 1þXN

k¼1

Skðx; yÞvk

" #;

E2ðx; yÞ ¼ E20 1þXN

k¼1

Skðx; yÞWk

" #;

G12ðx; yÞ ¼ G120 1þXN

k¼1

Skðx; yÞkk

" #;


k¼1

Skðx; yÞuk

" #;


k¼1

Skðx; yÞUk

" #;

m12ðx; yÞ ¼ m120 1þXN

k¼1

Skðx; yÞak

" #: ð16Þ

Note that in the general case, the random field discretization meshcould be different for each of the random fields. This would meanthat the shape functions and the number of OLE nodes for each ofthe random fields could be different. However, this does not affectthe formulation that is being presented in this section.

Here, h ¼ ½v;W; k;u;U;a� represents a vector of correlated ran-dom variables having distributions corresponding to their parentrandom fields. Substituting Eq. (16) in Eq. (14), it can be seen that


the elements of composite stiffness matrices can be expressed asnon-linear functions of the interpolating shape functionsfSkðx; yÞg and the random variables h. As a result, the compositestiffness matrices �Q mðhÞ and �Q sðhÞ and in turn the elements ofthe matrices AðhÞ;BðhÞ;DðhÞ and HðhÞ are also functions of theserandom variables and x and y. These matrices henceforth are writ-ten as functions of h to denote their dependence on the discretizedrandom fields.The derivation of the stochastic stiffness matrix isdescribed in the following section.

3.2. Derivation of the stiffness matrix

We next develop the eight noded finite element with each nodehaving five degrees of freedom df g ¼ u;v;w;/x;/y

�, where, u;v

and w are the displacement vectors in x; y; z directions respectivelyand /x;/y are respectively the rotations about x; y axes. The inter-nal strain energy U of the element is obtained as volume integralsof the form

U ¼ 12

Zv

NTðhÞ em dv þZ

vMTðhÞ j dv þ

Zv

VTðhÞ es dv� �

; ð17Þ

where, the superscript T denotes matrix transpose. Substituting theexpressions for NðhÞ;VðhÞ and MðhÞ from Eq. (B.1), U can beexpressed in terms of the elemental stiffness matrix KeðhÞ as

U ¼ 12

dTe KeðhÞde ð18Þ

where, de is the elemental displacement vector and

KeðhÞ¼Z a

0

Z b

0

Z h=2

�h=2BT

m AðhÞBmþBTb DðhÞBbþBT

s HðhÞBs�

dxdydz: ð19Þ

Here, Bm;Bb and Bs are the strain displacement transfer matricescorresponding to membrane, bending and shear strains respec-tively and are independent of material properties and matricesAðhÞ;BðhÞ;DðhÞ and HðhÞ are as defined in Eq. (14). It is clear thatthese matrices are functions of the material random fieldsexpressed as in Eq. (16). Since the shape functions in the OLErepresentation of the random fields are usually available numeri-cally, the integrals in Eq. (19) need to be evaluated numerically.In this study, a Gauss quadrature integration scheme has beenadopted to numerically determine the element stochastic stiffnessmatrix KeðhÞ. The formulation for the derivation of KeðhÞ allowsthe use of different meshing for the displacement and randomfields. The mesh size in the displacement field is governed bythe principles of FEM, whereas the criteria that determine themesh size in the random field discretization have already beendiscussed in the previous section. The use of higher order sheardeformation theories into the SFEM formalism has been discussedin Appendix B.

4. Failure probability estimation

For estimating the failure probability of the composite plate, theglobal stiffness matrix KðhÞ is first assembled in terms of the ele-mental stiffness matrices KeðhÞ. Subsequently, the nodal displace-ments d are estimated by solving the finite element equationsKðhÞd ¼ P. Clearly, the response will also be uncertain even if theload is assumed to be deterministic. The layer wise stresses are ob-tained from the following relation

rf gk ¼ Q ðhÞh ik

ef g; ð20Þ

where, rf g ¼ r1 r2 r12f g and ef g ¼ e1 e2 e12f g are respectively thestress and strain components defined along the material axes and

the superscript k indicates the k-th layer. The stresses are computedat the Gauss quadrature points used for the numerical evaluation ofthe integrals in Eq. (19). The failure index is next calculated for allthe Gauss points in all the elements, using the modified Tsai-Hillfailure criterion, given by

FI ¼ r1

X

h i2� r1r2

X2

� �þ r2

Y

h i2þ s12

S

h i2: ð21Þ

Here, X;Y and S are the longitudinal, transverse and shear strengthrespectively. r1;r2 and s12 respectively denote the induced plystresses. Clearly, even under deterministic loading the stressesdeveloped are functions of h and are uncertain. Moreover, for com-posite plates with material random inhomogeneities, the strengthparameters X;Y and S in Eq. (21) are also expected to be randomfields. Thus, the failure indices calculated at the Gauss points inall elements are random variables whose pdf need to be estimatedfor reliability calculations.

As a first step, the strength parameters need to be modeled as2-D random fields with specified pdf and auto-correlation struc-tures. These random fields are subsequently discretized using the2-D OLE discretization scheme discussed earlier in Section 2. Thisintroduces additional vectors of correlated random variablesC ¼ ½C1;C2;C3�, where Ci are the vectors of correlated random vari-ables corresponding to the three strength fields. Thus, the failureindex is now a function of the random variables h and C, i.e.,FI � FIðh;CÞ.

A failure in the lamina is assumed to occur when the failure indexexceeds unity. The failure of the composite plate is defined in termsof first ply failure. Now, at a specified location corresponding to theq-th Gauss point, sq ¼ ðxq; yqÞ, the ply failure can originate at any ofthe k laminae. Correspondingly, at sq, for a given realization of C, FIcorresponding to each laminae can be computed. Defining themaximum failure index at sq as Fq ¼max ½FIj�kj¼1, a failure at sq

occurs when Fq P 1. Since C are random variables, Fq is a randomvariable. Therefore, the failure probability at sq, denoted by Pfq , isgiven by

Pfq ¼ P½Fq > 1� ¼Z 1

1pqðsÞds: ð22Þ

Here, pqð�Þ is the pdf of Fq at sq. Estimates of pqð�Þ are numericallyobtained through the following steps:

(a) Discretize the 2-D material property (moduli, Poisson’s ratioand strength) random fields using the proposed OLE frame-work. The 2-D shape functions are obtained as matrices bynumerically solving Eq. (9).

(b) Ensembles of the vectors of correlated random variables atthe OLE nodes, (h; C), are simulated.

(c) KeðhÞ and KðhÞ are numerically evaluated corresponding toeach sample realization of h.

(d) The displacement, stresses and the failure indices are com-puted at all elements at its Gauss points.

(e) Steps (c)-(d) are repeated for all the sample realizations thathave been simulated in step (b).

(f) Finally, pqð�Þ corresponding to all the Gauss points are esti-mated by statistically processing.

Further illustration of the proposed framework is presentedthrough a set of numerical examples discussed in the next section.

5. Numerical examples and discussions

A square composite plate of dimension a ¼ 1, made up ofunidirectional carbon epoxy materials, is considered for the study.


The a=h for the plate is assumed to be 0:01. The plate is assumed tobe made up of a stack of three laminae of equal thickness. A 6� 6FE mesh has been used to discretize the plate. The elements ofKeðhÞ are numerically evaluated using selective integration. TheFI are estimated at all 3� 3 Gauss points of the 6� 6 elements.

Three different carbon epoxy materials, namely, HTS/M18,UMS/M18 and M55J/M18 have been considered in this study.Based on the testing, the properties that have been identified asshowing significant variabilities have been modeled as randomfields. Table A.1 in Appendix A list the properties of the probabilitydensity functions that have been fitted for these random fields.Two plate configurations, namely, simply supported (SS) andclamped–clamped (CC) conditions in all edges are considered forthe analysis. The following three different layup sequences areconsidered to capture the effects of spatial inhomogeneities onthe failure probability:

Case 1: A lay up sequence of ½0=90=0� is considered. Thissequence does not have either shear extension coupling or bendtwist coupling. This case is referred as C1 for CC plate and S1 forSS plate. Case 2: A symmetric angle ply laminate with sequence of [30/�30/30] is considered. Due to symmetry, bending-extensioncoupling matrix is zero here. This case is referred as C2 for CCplate and S2 for SS plate. Case 3: An unsymmetric generic laminate with lay-up sequence

of [0/30/60] is considered so that all coupling terms are non-zero. This case is referred as C3 for CC plate and S3 for SS plate.

The loadings on the plate are assumed to be uniformly distrib-uted loadings (UDL). The magnitude of the deterministic loadingshave been selected such that the FI at the most critical locationsin all cases is 0:75. This is to ensure an identical safety marginfor all cases, such that, a comparative study can be carried out toassess the effects of spatial inhomogeneities on the failure proba-bilities. Fig. 27 provides a summary of the details of the loadingsfor the various cases.

5.1. Methods for analysis

The following three methods of analysis are considered.

(a) Method 1 (Random variable approach): The spatial varia-tions in the material and strength parameters are neglectedand only the ensemble variations are considered. Thus, theseparameters are modeled as mutually independent randomvariables. As a result, only FE discretization is required.The construction of the FE matrices are identical as for

S1 C1 S2 C2 S3 C30

2

4

6

8

10x 104

Type of Plate

Load

in N

/m2

HTS/M18UMS/M18M55J/M18

Fig. 27. Summary of the loadings for the various composite plates.

deterministic studies. The number of random variablesentering the formulation is M ¼ P � k, where P is the numberof parameters considered random in each lamina and k indi-cates the number of layers. The numerical implementationof this method involves the following steps:

(i) S realizations of the vector of M dimensional randomvariables ½h;C� are simulated.

(ii) The FE equations are constructed as a function of hi.(iii) Corresponding to each realization of hi, the FE equations

are solved deterministically and the failure indices arecalculated at all the Gauss points using the respectivestrength parameters Ci.

(iv) Statistical processing of the results obtained from the Srealizations are carried out to estimate the failure prob-ability at specified locations sq.

(b) Method 2 (Random process approach): The spatial randomfluctuations in the material and strength properties areincorporated into the analysis by modeling them as 2-D ran-dom fields. The proposed OLE based SFEM framework is usedto formulate the global stiffness matrices. The steps involvedin the numerical implementation of the method are:

(i) An ensemble of S realizations of h are simulated, whereh has the same meaning as in Section 3.1.

(ii) Following the formulation developed in Sections 2 and3, the global stiffness matrix KðhÞ is constructednumerically.

Steps ðiiiÞ and ðivÞ are identical to Method 1. The number of ran-dom variables entering the formulation is M ¼ P � N � k, where,P is the number of random fields in each laminae, N is the numberof OLE nodes for each random field and k indicates the number oflayers. The FE and the random field discretization meshes are takento be different.

(c) Method 3: As in Method 2, the spatial random fluctuations inthe material and strength properties are incorporated intothe analysis by modeling them as 2-D random fields. How-ever, no separate random field meshing is considered.Instead, once the FE meshing is carried out, the strengthand material property random fields within an element areassumed to be spatially uniform and is taken to be the repre-sentative value as at the elemental centroid. Thus, the fieldsare now represented through a vector of correlated randomvariables which capture the pdf and the correlation charac-teristics exactly. However, the spatial variations within eachelement are neglected. Moreover, the FE formulation followsthe well accepted deterministic approach and does not needto adopt the formulation mentioned in this paper. As the FEmeshing is usually much finer than the OLE random fieldmeshing, the number of random variables entering the for-mulation is larger than in Method 2. Thus, if there aren N FE elements, the total number of random variablesentering the formulation is M ¼ P � n� k. The steps involvedin the numerical implementation of this method are:

(i) The FE meshing is carried out and the composite plateis discretized into n elements.

(ii) An ensemble of S realizations of M dimensional h aresimulated.

(iii) The elemental stiffness matrices are constructed in thesame way as the deterministic case. Here, Ke are func-tions of P � k random variables.

(iv) n distinct Ke elements are constructed and the globalstiffness matrix K is assembled.

The next two steps are identical to steps ðiiiÞ and ðivÞ of Method 1.The difference between Methods 1 and 3 are that in Method 1, aparticular property has the same value in all the finite elementalmatrices, while in Method 3, these values vary across the finiteelements.


The composite plate is discretized into 36 finite elements. Theparameters listed in Table A.1 are treated as random fields in eachlayer. Since the plate is assumed to be made up to 3 layers, the totalnumber of random fields entering the formulation is 7� 3 ¼ 21.This includes 3 independent moduli fields, one Poisson’s ratio field,two strength property fields (either tension or compression) andone shear strength field corresponding to each laminae. In Method1, where the spatial variations are neglected and the material prop-erties are modeled as random variables, the number of randomvariables entering the formulation is just 21. In Method 2, theOLE discretization of each random field involves approximatingthe field with a vector of correlated random variables. In the ab-sence of any measurement data, the correlation structure of therandom fields are assumed to be identical. This assumption hasbeen considered for the sake of simplicity and is not a restrictiveassumption. Assuming the random field meshing to be 4� 4, thenumber of OLE nodes for each random field is 16. Thus, the totalnumber of random variables entering the formulation in Method2 is 21� 16 ¼ 336. In Method 3, since the mesh size for finite ele-ment discretization and random field discretization are the same,each field is approximated by 36 correlated random variables.Thus, the total number of random variables entering the formula-tion is 21� 36 ¼ 756.

5.2. Validation study

The focus of this example is to validate the proposed OLE basedSFEM framework (Method 2). The proposed method is used tocompute the pdf of the failure index of the first ply failure of theplate, when the boundary conditions are assumed to beclamped–clamped. The composite plate is analyzed for all the threedifferent lay-up sequences. In all three cases, 336 number ofrandom variables are used to estimate FI. Using Monte Carlo sim-ulations, S ¼ 5000 realizations of the vector of the discretized ran-dom variables were simulated and the failure index was computedat the Gauss points. The pdf of Fq; pqð�Þ, for the three cases, C1;C2and C3, have been shown in Fig. 28. These predictions were com-pared with the corresponding Method 3 predictions. In Method

0 1 2 310−4

10−2

100

102

Failure Index

log

(pdf

)

Method−2Method−3

0 110−4

10−2

100

102

Failu

log

(pdf

)

Fig. 28. Comparison of pqð�Þ using Methods

Table 1Validation of proposed method: comparison of the estimates with literature.

Input c.o.v SSFEM [33] MCS [33]

Mean (�10�2 m) Std deviation (�10�3 m) Mean (�10�2 m)

0.10 1.648 0.974 1.6490.15 1.656 1.481 1.6570.20 1.668 2.033 1.6730.25 1.685 2.632 1.688

3, there is no separate random field discretization. The FE meshingwas selected such that there are 36 elements and the random fieldis represented by the values at the centroidal locations of these 36finite elements. Hence, in this method, there are no additionalapproximations due to random field discretization. In this case,the FI is dependent on 756 number of random variables.S ¼ 5000 realizations of these vectors of correlated random vari-ables are simulated and the pdf, pqð�Þ are calculated. Fig. 28 reveala very close match between Methods 2 and 3, even at the tails ofthe distribution. This serves to validate the proposed OLE basedSFEM framework.

To further validate the results obtained by the proposed meth-od, a comparison is carried out with available results in the litera-ture and those obtained by the proposed method. Here, we take thelaminated composite plate example considered in [33], where themean and the standard deviation of the deflection is estimated. Theparameters E1;G12;G23 and G13 are modeled as Gaussian randomfields. The c.o.v. of these fields are varied from 0:1 to 0:25.Estimates of the same quantities are also calculated using the pro-posed method and a comparison of the estimates are provided inTable 1. It is observed that the estimates are in very goodagreement with literature results.

5.3. Random variable vis-a-vis random process modeling

This study investigates the necessity of adopting random pro-cess models for the material properties. Here, pqð�Þ at specifiedlocations of composite plates of identical dimensions but withthree different ply sequences are computed using Methods 1 and2, for both types of boundary conditions – simply supported andclamped–clamped cases. Figs. 29 and 30 show the comparison ofthe plots of pqð�Þ obtained using Methods 1 and 2 when the com-posite material used is HTS/M18.

An inspection of Figs. 29 and 30 show that there is a significantdifference in the estimated pqð�Þ of the composite plate when thematerial property uncertainties are modeled as random fields orrandom variables. The predictions obtained from random variablemodels show higher peaks and flatter tails. This implies that

2 3

re Index


0 1 2 310−4

10−2

100

102

Failure Index

log

(pdf

)


2 and 3; material property HTS/M18.

Method-2

Std deviation (�10�3 m) Mean (�10�2 m) Std deviation (�10�3 m)

1.051 1.649 0.9951.619 1.661 1.562.217 1.671 2.272.931 1.687 3.05

0 1 2 30

0.5

1

1.5

2

Failure Index

pdf


0 1 2 30

0.5

1

1.5

2

Failure Index

pdf


0 1 2 30

0.5

1

1.5

2

Failure Index

pdf


Fig. 29. Comparison of pqð�Þ using Methods 1 and 2; material HTS/M18.

0 1 2 30

0.5

1

1.5

2

Failure Index

pdf


0 1 2 30

0.5

1

1.5

2

Failure Index

pdf


0 1 2 30

0.5

1

1.5

Failure Index

pdf


Fig. 30. Comparison of pqð�Þ using Methods 1 and 2; material HTS/M18.

C1 C2 C3 S1 S2 S30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Type of Plate

Failu

re p

roba

bilit

y


Fig. 32. Comparison of failure probability for UMS/M18.

0.1Method−1


random variable models (a) overestimates the probability associ-ated with the FI near the modal value but underestimates FI valueswhich are high, and (b) underestimates the modal value for FI. Onthe other hand, the pdf predictions obtained by Method 2 showmuch longer right tails. More importantly, the changes in pdf ofFI with changes in the lay up sequence are observed to be marginalin the random variable model when compared to the random fieldmodel. This seems to imply that random field models are moresensitive to the variations in the composite structure materialand geometric properties. A physical explanation could be for thisobservation is that random variable models imply a homogeniza-tion of the variations in the properties along the spatial extentand hence the effects of changes in the properties are notadequately captured.

The importance of random field models for the uncertainvariations in the material properties become clearer when failureprobabilities are estimated. Figs. 31–33 show a comparison of thefailure probability estimates for first ply failures in a composite

C1 C2 C3 S1 S2 S30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Type of Plate

Failu

re p

roba

bilit

y


Fig. 31. Comparison of failure probability for HTS/M18.

C1 C2 C3 S1 S2 S30

0.02

0.04

0.06

0.08

Type of Plate

Failu

re p

roba

bilit

y Method−2

Fig. 33. Comparison of failure probability for M55J/M18.

plate with identical dimensions and identical safety margins butfor different lay-up sequences. Three separate materials havebeen considered, namely, HTS/M18, UMS/M18 and M55J/M18.


An inspection of Figs. 31–33 clearly show that irrespective ofthe lay up sequence, boundary conditions or the grade of the com-posite used, Method 1 underestimates the failure probability incomparison to Method 2. This implies that random variable mod-eling of the uncertainties when the correlation length of the ran-dom fields is small leads to overestimating the safety of thestructure and hence such designs are less conservative.

Fig. 34 provides a comparison of the failure probability esti-mated using Method 2 for all six case studies conducted for eachof the three materials considered in this study. It must be remarkedthat consistently, the failure probability for plates made up of HTS/M18 is the highest and for M55J/M18 to be the lowest even thoughthe geometrical dimensions of the plates were identical for allthree materials. Here, one must note that the deterministic load-ings considered were such that the design margins were the samein all the plates (when deterministic analysis is carried out) irre-spective of the grade of the material used in constructing the com-posite plate. An explanation for the variations in the failureprobability estimates can be understood from Table A.2 wherethe coefficient of variation (c.o.v.) of the material properties, forthe different composite grades, are listed. It is seen that stiffnessand strength properties show the greatest variations for HTS/M18 and the least for M55J/M18. The differences observed inFig. 34 could be due to the greater variabilities observed in thematerial and strength properties in HTS/M18. One can thereforesummarize that the use of grade materials that show greater vari-abilities lead to higher variabilities in its response (as expected)and in turn, implies higher probabilities of failure as well.

5.4. Failure locations on the plate

Estimates of pqð�Þ at all the elemental Gauss points, calculatedbased on the studies carried out so far, can now be used to identify

C1 C2 C3 S1 S2 S30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Type of Plate

Failu

re p

roba

bilit

y

HTS/M18UMS/M18M55J/M18

Fig. 34. Comparison of failure probability of different material systems usingMethod2.

x

y

0 0.5 1

1

0.5

0.2

0.4

0.6

1

0.5

0

0

0.1

0.2

y

Failu

re p

roba

bilit

y

Fig. 35. Composite plate C

the most likely failure locations on the plate. Assuming that a fail-ure occurs due to first ply failure, the absolute failure probability ateach Gauss point can be calculated using Eq. (22) and is denoted byPfq . To identify the most likely failure locations, the relative nor-malized frequency of failures corresponding to each Gauss point,are calculated. In mathematical terms, this is the conditional fail-ure pdf at any location q, given that failure has occurred and is rep-resented as Pf jF>1.

The contour plots for the estimated FI at all elements at itsGauss points have a similar resemblance when the material andstrength properties are modeled as deterministic and when theseare modeled as random variables only. For case C1 and for compos-ite grade HTS/M18, these contour plots are shown in Fig. 35(a).Here, one can see that the locations with darker shades have higherFI with the maximum values being 0.75 located at the edges. Eventhough the structure is geometrically symmetric along all axes andhave symmetric loading, the distribution of the FI is not symmetricbecause of the lay-up sequences which induce asymmetry in thecomposite material property.

The composite plate is next analyzed when the material prop-erties are modeled as 2-D random fields using Method 2 andpqðsÞ is computed at all Gauss points. The 3-D pdf of the abso-lute failure probability associated with each location is shownin Fig. 35(b). This plot gives an indication of the weak pointsassociated with the composite plate C1. Fig. 35(c) shows thecontour plots for the conditional pdf Pf jF>1 and clearly identifiesnot only the most likely failure locations, but also the associatedprobability that the failure will occur at these locations. A com-parison of Figs. 35(a) and (c) show that the most likely failurepoints occur in the same region where highest values of FI areobtained from the deterministic analysis. When the boundaryconditions are changed from clamped to simply supported con-dition, Figs. 36 show the distributions of FI for the deterministiccase and the corresponding most likely first ply failure points.Similar studies have been carried out for the lay up sequencescorresponding to Cases 2 and 3. These results are shown InFigs. 37–40.

Based on the studies carried out and an inspection of theFigs. 35–40 the following observations can be made:

The most likely failure points in the composite plates with ran-dom spatial inhomogeneities occur at regions which matchwith the regions with the highest FI when the plates are ana-lyzed deterministically. This is true for all lay-up sequencesand boundary conditions. For clamped–clamped conditions, the probabilistic analysis

reveals that the most likely failure regions are at the edgesand is concentrated in very small zones. The locations of thesefailure zones are dependent on the lay-up sequence; seeFigs. 35(c), 37(c) and 39(c).

0.51

x x

y

0 0.5 1

1

0.5

0.05

0.1

0.15

0.2

1; material HTS/M18.

x

y

0 0.5 1

1

0.5

0.2

0.4

0.6

0.51

1

0.5

0

0

0.1

0.2

xy

Failu

re p

roba

bilit

y

x

y

0 0.5 1

1

0.5

0

0.02

0.04

0.06

0.08

Fig. 36. Composite plate S1; material HTS/M18.

x

y

0 0.5 1

1

0.5

0.2

0.4

0.6

0.51

1

0.5

0

0

0.1

0.2

xy

Failu

re p

roba

bilit

y

x

y

0 0.5 1

1

0.5

0

0.1

0.2

0.3

0.4

Fig. 37. Composite plate C2; material HTS/M18.

0.05

0.1

0.15

0.2

x

y

0 0.5 1

1

0.5

0.2

0.4

0.6

0.51

1

0.5

0

0

0.1

0.2

xy

Failu

re p

roba

bilit

y

x

y

0 0.5 1

1

0.5


0

0.05

0.1

0.15

x

y

0 0.5 1

1

0.5

0.2

0.4

0.6

0.51

1

0.5

0

0

0.1

0.2

xy

Failu

re p

roba

bilit

y

x

y

0 0.5 1

1

0.5

Fig. 39. Composite plate C3; material HTS/M18.


For simply supported plates, deterministic analyses reveals largezones of FI above 0.5; see Figs. 36(a), 38(a) and 40(a). However,the probabilistic analyses for the corresponding plates reveals

failure zones which are much smaller; see Figs. 36(c), 38(c) and40(c). This indicates that most likely failure regions is not neces-sarily at maximum FI observed through deterministic analysis.

0

0.05

0.1

x

y

0 0.5 1

1

0.5

0.2

0.4

0.6

0.51

1

0.5

0

0

0.1

0.2

xy

Failu

re p

roba

bilit

y

x

y

0 0.5 1

1

0.5



Importantly, in the case of simply supported plates, the proba-bilistic analysis reveals certain locations in the plate where thelikelihood of first ply failures to be comparable to the regions ofmost likely failures. However, for clamped–clamped plates, thisis not observed and the locations of first ply failures are concen-trated in very small failure regions.

5.5. Failure probability and correlation length

Next, a parametric study is carried out to investigate the effectof the correlation lengths of the random fields on the failure prob-ability of the HTS/M18 simply supported (S3) and clamped–clamped (C2) composite plates. The correlation lengths of the2-D random fields are dependent on the constants c1 and c2 consid-ered in Eq. (11). As has been mentioned earlier, in the absence ofrelevant test data, all the 2-D random fields have been assumedto have identical correlation structure and is of the form as in Eq.(11). Moreover, c1 and c2 are assumed to be equal implying sym-metry in the correlation lengths along the global X and Y directions.Here, the correlation parameters c1 ¼ c2 ¼ c are changed and thecorresponding first ply failures are computed using the proposedOLE based SFEM. As c is changed, the size of the random field meshis also changed such that the global error is kept below the thresh-old value 0:02. Thus, when c ¼ 2:5, a 2� 2 mesh was consideredfor random field discretization. On the other hand, when c ¼ 0:2,the corresponding mesh size was taken to be 5� 5. This has impli-cations on the number of random variables entering the formula-tion and in turn, the CPU time expended in the analysis. Fig. 41shows the estimated pqð�Þ at a specified location for various valuesof c. The corresponding pdf when random variable models areadopted are also shown in the same figure. It is observed that as

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

Failure Index

pdf

RP−0.2RP−0.6RP−2.5 RV

Fig. 41. Estimates of pqð�Þ for S3 HTS/M18 composite plate for various correlationlengths.

the correlation length decreases, there is a rightwards shift of thepdf and they have longer tails. It must be noted that smaller corre-lation lengths require finer meshing and in turn, larger number ofrandom variables in the formulation. Thus, the random variablemodel can be assumed to be a case where the correlation lengthof the random fields is very large. Thus, the pdf obtained for therandom variable case can be viewed as a limiting case.

A comparison of the estimated first ply failure probabilities, as afunction of c, is shown in Fig. 42. It is observed that as c decreases, i.e.,as the correlation length is smaller, the composite plate under iden-tical deterministic loading has a higher failure probability. Onceagain, the limiting case is observed to be the random variable modelwhich can be considered for random fields with large correlationlengths. As c is increased from 0:2 to infinity (the random variablemodel), the failure probability decreases from 0.55 to 0.3, a decreaseof almost 50%. This gives an indication of the error in reliability pre-dictions when random variable models are used for probabilisticanalysis where random field modeling is more appropriate.

5.6. Computational costs

The computational costs associated with the analysis is directlyproportional to the number of random variables entering the for-mulation. In this respect, it is quite obvious that Method 3, whichhas the highest number of random variables in the formulation, de-mands the highest computational costs. Table 2 lists the computa-tional costs for the various case studies that have been analyzed,expressed as a percentage of the computational cost when Method3 is used. In all the three methods, the sample size S was taken tobe 5000. The plates were analyzed in a personal computer with adual core 2 GB processor. Though Method 1 demands the least

RP−0.2 RP−0.4 RP−0.6 RP−1.0 RP−1.5 RP−2.5 RV0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Correlation constant

Failu

re p

roba

bilit

y

C2

S3

Fig. 42. Comparison of first ply failure probability for various correlation lengths.

Table 2Summary of computational costs expressed as a percentage of the computationalcosts in Method 3.

Model (%) C1 C2 C3 S1 S2 S3

Method 1 12 12.1 11.8 11.7 11.6 12Method 2 46 37.8 35.2 45.4 45 48

0 0

0 0

00

00

0.005 0.005

0.005 0.005

0.00

5

0.005

0.0050.005

0.005

0.005

0.005

0.010.01

0.010.01

0.01

0.010.01

0.01 0.01

0.010.01

0.01

0.015

0.0150

.015

0.01

5

0.0150.015

0.015

0.015

0.0

15

0.015

0.01

5

0.01

5

0.015

0.015

0.02

0.02

0.02

0.02

0.02

0.02

0.02

0.02

0.025

0.025

0.025

0.025

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1



computational costs, usually a 10% of the computational costs ofMethod 3, it does not take into account the spatial random fluctu-ations of the material and strength properties. Consequently, ashas been shown in the studies carried out in the previous section,the method underestimates the failure probabilities. On the otherhand, Table 2 shows that the savings in the computational costsassociated with the proposed OLE based SFEM framework is morethan 50%. The costs associated with Method 2 could increase forcorrelations with smaller correlation lengths. However, moreimportantly, depending on the structure and loading, a finer meshwould increase the computational costs associated with Method 3significantly irrespective of the correlation structure in the randomfields. This inconvenience is avoided in the proposed method.

5.7. Non-uniform FE discretization

In all the numerical examples considered so far, the correlationlength of the fields along the two orthogonal axes were assumed tobe the same, leading to identical random field meshing along bothaxes. However, if the correlation lengths are different along thetwo axes, the number of OLE nodes along the two axes would bedifferent. However, this poses no further difficulties in the imple-mentation of the proposed method. As an illustration, the failureprobability estimates for the S3 case are computed for differentcorrelation lengths along the two orthogonal directions of the plateand are shown in Table 3. The 2-D auto-correlation function is as-sumed to be of the same form as in Eq. (11) but with different val-ues for the constants c1 and c2. Here, nx and ny respectively denote

Table 3Estimates of Pf for case S3 for non-uniform random field mesh size.

c1 c2 nx ny N Pf

0.2 0.2 5 5 25 0.5460.2 0.4 5 4 20 0.5400.2 0.6 5 3 15 0.5230.4 0.4 4 4 16 0.5360.4 0.6 4 3 12 0.5120.6 0.6 3 3 9 0.484

−0.03

−0.03

−0.03

−0.03

−0.02

−0.02

−0.0

2

−0.0

2

−0.0

2

−0.02

−0.01

−0.01

−0.0

1

−0.0

1

−0.0

1

−0.01

0

0

0

0

0

0

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.01

0.02

0.02

0.02

0.02

0.02

0.02

0.02

0.02

0.02

0.020.02

0.0

2

0.03

0.03

0.03

0.03

0.03

0.03

0.03

0.03

0.03

0.030.03

0.03

0.04

0.04

0.04

0.04

0.04

0.04

0.04

0.04

0.04

0.040.04

0.04

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.06

0.06

0.06

0.06

0.06

0.06

0.06

0.06

0.07

0.07

0.07

0.07

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


the number of OLE nodes along x and y directions and N ¼ nx � ny

is the total number of OLE nodes. It is observed that the failureprobability estimates are higher for smaller values of constantsc1 and c2, which corroborates the findings in Section 5.5. Addition-ally, for the case c1 ¼ c2 ¼ 0:6 which can be solved using a 3� 3uniform mesh, a finer mesh size was considered along one of theaxes. Here, the mesh size considered was 5� 3. The failure proba-bility estimate was found to be 0.472 for the non-uniform mesh.The slight difference of 1:2% can be attributed to statistical fluctu-ations. Figs. 43 and 44 show the error plots for the non-uniformmeshing of the random fields for the case when c1 ¼ 0:4 andc2 ¼ 0:6. As can be clearly observed, the nodal points lie on the zeroerror contours even when non-uniform meshing is adopted.

6. Concluding remarks

A stochastic finite element based methodology is developed forfailure probability estimation in laminated composite plates. Thematerial properties in each laminae are assumed to have randomspatial inhomogeneities and are modeled as two dimensionalnon-Gaussian random fields. A stochastic finite element basedframework has been developed that discretizes the random fieldusing the optimal linear expansion scheme. In this context, the no-vel features discussed in this study can be summarized as follows:

(1) The material properties are modeled as non-Gaussian ran-dom fields at the ply levels. To the best of our knowledge,studies in the composite literature considering random fieldmodeling are few [31,33,36,37] and these have consideredonly Gaussian models. The non-Gaussian characteristics ofthe random uncertainties in the material properties is wellestablished in the literature and has been borne out by ourstudies as well [46].

(2) Extension of the spectral methods discussed in [32] alongthe lines of [31,33] for treatment of the non-Gaussian inho-mogeneities in 2-d planes is not easy and fraught with com-plexities. In this study, we adopt a different spectralapproach for the 2-d non-Gaussian random field discretiza-tion, namely, the optimal linear expansions. This methodhas advantages over the PCE approach, which have beenhighlighted in Section 2.

(3) The proposed OLE based scheme for random field discretiza-tion preserves the second order non-Gaussian characteris-tics of the parent random field.

(4) The use of OLE also enables easy construction of the globalstructure matrices, which are now functions of random vari-ables. This formulation is also new.

Table A.2Mean and coefficient of variations of the material properties.

Material HTS/M18 UMS/M18 M55J/M18

Property Mean c.o.v Mean c.o.v Mean c.o.v

m12 0.281 7.5 0.25 8.7 0.31 3.1E1 (Gpa) 154.9 5.9 253.1 2.6 346.1 5.7E2 (Gpa) 8.7 9.5 6.4 7.8 6.5 3.1G12 (Gpa) 4.5 8.8 4.2 5.8 4.6 3.2Xt (Mpa) 2409 6.7 2102.8 7.2 1890.9 10.9Xc (Mpa) 1148 18.1 826.9 10.4 587 5.2Yt (Mpa) 46 20 28.1 11.1 21.1 5.5Yc (Mpa) 196 15.3 174.3 3.9 106 2S (Mpa) 83 5 60.3 2.9 53.9 1


(5) The proposed OLE based SFEM framework allows the use ofdifferent mesh sizes for finite element and random field dis-cretizations. The FE discretization mesh size is governed bythe structure geometry and the loading, while the randomfield discretization is governed by the correlation structureof the 2-D random fields. The use of different mesh sizesenables significant savings in the computational costs.

(6) Random variable models for materials with spatially varyingrandom uncertainties with correlation lengths significantlysmaller than the spatial dimensions of the composite plateresult in significant underestimation of the failures. Thishighlights the importance of random field modeling in com-posite plate analyses.

(7) The proposed SFEM framework enables identifying the mostlikely failure regions in the composite plate under specifiedloadings. The analysis reveals that the most likely failureregions is not necessarily at maximum FI observed throughdeterministic analysis or a random variable approach tomodeling the uncertainties.

(8) Unlike in the existing literature where the failure probabilityestimates are computed using approximate methods such asFORM/SORM which only lead to reliability indices andnotional estimates of the structure reliability, the use ofMCS not only leads to estimates of the absolute failure prob-abilities, but simultaneously enables identification of themost probable failure regions. The computational advanta-ges associated with FORM/SORM over MCS diminishes asthe number of random variables entering the formulationincreases. This is particularly true if one is interested in esti-mating the most probable failure domains which wouldrequire a large number of FORM/SORM analyses correspond-ing to each limit state.

Appendix A. Modeling the uncertain parameters

Developing models for the uncertainties in the material andstrength properties associated with laminated composites requireextensive testing and measurements. This is carried out on threecarbon epoxy materials, namely, HTS/M18 [46], UMS/M18 andM55J/M18. Based on the limited tests, some well known probabil-ity distribution functions were tested for the goodness of fit forthese properties. The acceptable models and their parameters arelisted in Table A.1. Here, the distributions are referred to by a letterfollowed by two numbers in braces, the key for which are as

Table A.1Distribution parameters for carbon/epoxy material systems.

Property HTS/M18 UMS/M18 M55J/M18

m12 LN (�1.271,0.078)

W (0.254277,15.44203)

G (1170.749,0.000268)

E1 (Gpa) W (158.9, 20.7) W (256.1161,47.5331)

W (355.0401,19.25807)

E2 (Gpa) W (9.0, 14.4) G (173.5372,0.036686)

U (6.209, 6.935)

G12

(Gpa)LN (1.492,0.089)

W (18.69539,4.282489)

LN (1.519087,0.032468)

Xt (Mpa) LN (7.785,0.067)

W (2172.303,16.09276)

W (1981.832,10.19817)

Xc (Mpa) LN (7.031,0.175)

LN (6.712217,0.108411)

W (600.4284,23.99974)

Yt (Mpa) W (50.3, 5.8) LN (3.331139,0.114088)

W (21.58399,20.27171)

Yc (Mpa) W (208.6, 7.4) W (177.3584,30.63628)

W (106.961,60.61156)

S (Mpa) LN (4.413,0.050)

W (61.11555,37.11639)

U (52.48, 55.06)

follows: Uða; bÞ: Uniform (lower limit, upper limit), Wða; bÞWeibull(Scale, Shape), LNða; bÞ: Log-normal (Mean, standard deviation) andGða; bÞ ¼ Gamma (Shape, Scale). Table A.2 lists the mean valuesand the associated coefficient of variations for the parametersbeing considered for the three grades of composites.

Studies on estimating the correlation structure requires moresophisticated testing which is currently being pursued. The out ofplane shear modulus is considered 0.8 times of the in-plane shearmodulus for the study. It is expected that details of these testingwill be published at a later stage once the studies are completed.

Appendix B. Using higher oder shear deformation theory

The developments presented in this paper can be easily ex-tended when higher order shear deformation theories are used.Thus, for example, using the third order shear deformation theoryproposed in [45], the constitutive equations for the plate can be ex-pressed as

NMPQR

26666664

37777775¼

A B E 0 0B D F 0 0E F H 0 00 0 0 As Ds

0 0 0 Ds Fs

26666664

37777775

e0

e1

e2

m0

m1

26666664

37777775; ðB:1Þ

where,

e0 ¼ u0; x; v0; y; u0; y þ v0; x� T

;

e1 ¼ /x; x; /y; y; /x; y þ /y; x

� T;

e2 ¼ �4

3h2 /x; x þw0;yy; /y; y þw0;yy; /x; y þ /y; x þ 2w0;xy� T

;

m0 ¼ /y þw0;y;/x þw0;x� T

;

m1 ¼ �4

h2 /y þw0;y;/x þw0;x� T

:

ðB:2Þ

u;v and w represent the deflections in the x; y and z directionsrespectively, given by

uðx; y; zÞ ¼ u0ðx; yÞ þ z/xðx; yÞ �4

3h2 z3 /x þw0;xð Þ;

vðx; y; zÞ ¼ v0ðx; yÞ þ z/yðx; yÞ �4

3h2 z3 /y þw0;x�

;

wðx; y; zÞ ¼ w0ðx; yÞ;

ðB:3Þ

and /x and /y are the rotations. Thus, Eq. (14) can now be written as

Aij;Bij;Dij; Eij; Fij;Hij�

¼Z h=2

�h=2Q mij

1; z; z2; z3; z4; z6� dz; ðB:4Þ

Askl;Dskl

; Fskl

� ¼Z h=2

�h=2Q skl

1; z2; z4� dz; ðB:5Þ


where, i; j ¼ 1;2;3, and k; l ¼ 1;2; z is distance from neutral plane.Here, Q m and Q s are the stiffness matrices for a lamina whoseelements are dependent on the material propertiesE1; E2;G12;G13;G23; m12; m21 and the direction cosines of the individuallamina axes with respect to the global structure axes. Eq. (17) canbe now rewritten as

U ¼ 12

Zv

NTðhÞ e0 dv þZ

vMTðhÞ e1 dv þ

Zv

PTðhÞ e2 dv�

þZ

vQ TðhÞ m0 dv þ

Zv

RTðhÞ m1 dv�: ðB:6Þ

The above derivation needs to be suitably developed for otherapproaches such as NURBS based isogeometric analysis for compos-ite materials [47,48].

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Analysis of CFRP laminated plates with spatially varying non-Gaussian inhomogeneities using SFEM

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