Stochastic finite element analysis of layered composite beams with spatially varying non-Gaussian...

Acta MechDOI 10.1007/s00707-013-1009-9

P. Sasikumar · R. Suresh · Sayan Gupta

Stochastic finite element analysis of layered composite beamswith spatially varying non-Gaussian inhomogeneities

Received: 18 April 2013 / Revised: 25 September 2013© Springer-Verlag Wien 2013

Abstract This study focuses on the development of a stochastic finite element-based methodology for failureassessment of composite beams with spatially varying non-Gaussian distributed inhomogeneities. The materialproperties in the individual laminae are modeled as non-Gaussian random fields, whose probability densityfunctions and the correlations are estimated from the test data. The non-Gaussian random fields are discretizedinto a vector of correlated non-Gaussian random variables using the optimal linear expansion scheme thatpreserves the second-order non-Gaussian characteristics of the fields. Subsequently, the estimates of the failureprobability are obtained from Monte Carlo simulations carried out on the vector of correlated random variables.Issues related to the computational efficiency of the proposed framework and the variabilities in the materialproperties are discussed. Numerical examples are presented, which highlight the salient features of the proposedmethod.

1 Introduction

The response of structural systems built up with laminated fiber-reinforced composites shows significant devi-ations and scatter from the predictions obtained from theoretical and numerical analyses. The experimentallyobserved scatter in the response behavior can be attributed to the inherent randomness in the constitutiveproperties of the composite material used to build the structure. Despite the most stringent quality checks,the randomness in the constitutive properties is impossible to eliminate due to the complexities in the man-ufacturing and processing stages. It is therefore imperative that the randomness in the material properties ofcomposites should be incorporated into the analyses, for safe and reliable designs that are also economicallyoptimal. Adopting a probabilistic approach to modeling the uncertainties in the material properties enables thedevelopment of a rationale framework within which the spatially varying random fluctuations in the materialproperties can be appropriately handled. Such stochastic approaches to modeling and analysis of compositestructures offer advantages in terms of material efficiency and better utilization [1,2].

P. Sasikumar · R. SureshVikram Sarabhai Space Centre, Trivandrum 695013, IndiaE-mail: [email protected]

R. SureshE-mail: [email protected]

S. Gupta (B)Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, IndiaE-mail: [email protected].: +91-44-22574055Fax: +91-44-22574052

P. Sasikumar et al.

Uncertainties can be broadly classified as aleatory, which are intrinsic to the material and are irreducible,and epistemic, which arises due to the lack of knowledge, inadequacy of models used for analysis, etc., andare reducible. Aleatory uncertainties in composite structures are primarily at the material level [3] and arisedue to the manufacturing processes. Epistemic uncertainties arise from the methods adopted for the analysisand depend on whether the uncertainties are modeled at the micro-, meso-, or the macroscale. Studies exist inthe composite literature where uncertainty quantification has been carried out at each of these levels. Studieson the propagation of uncertainties from the constitutive level or microscale level have been carried out in[4–6]. These studies are however difficult to extend to large-scale structures where the interests are primarilyin carrying out risk assessment of structural systems. Studies in uncertainty quantification at ply levels andcomponent levels have been carried out in [7–9]. A recent review on the state-of-the-art on reliability studieson composite structures is available in [10]. The common feature in these studies, as in most studies availablein the composite literature, has been in modeling the primary uncertain parameters as random variables havingspecified probabilistic characteristics [2,3].

The drawback in using random variable models for uncertainty quantification in composite structures is thatthe spatial uncertainty variations in the material properties within the structural system cannot be incorporatedinto the analysis. Adopting random variable models for parameters that have spatial variations implies alocal homogenization of the concerned parameters and introduces additional epistemic uncertainties into theanalysis. A more appropriate approach would be to model the parameters having spatial random fluctuationsas random fields. Adopting random field models introduces significant complexities into the response analysis,especially in composite structures. This probably explains the paucity of random field models in the compositesliterature. However, the emergence of cheap computing facilities of late has made it possible to address thesecomplexities numerically.

Computational approaches to modeling systems with random fields have led to the development of thestochastic finite element method (SFEM); see [11–13] for monographs on SFEM. In SFEM, the primary focusis on the discretization of random fields and minimization of the induced error in the representation of therandom field with a vector of correlated random variables. Various random field discretization methods havebeen discussed in the structural engineering literature; see [14–17] for reviews. However, though the importanceof the application of these methods to the composites literature has been recognized [2], it is only recentlythat attempts are being made to extend these methods to problems involving composite materials. Gaussianrandom field models were fitted to test data in [18] to statistically characterize the axial, shear, and transversestrengths of composite lamina. A recent study in [19] has assumed the moduli in longitudinal, transverse,and shear as Gaussian fields with assumed correlation functions. Subsequently, the authors have used spectralrepresentation methods [20] for discretizing the random field. A similar spectral FEM approach has beenadopted in [21] where the authors directly assume the elements of the constituent matrix to be Gaussian fieldsand treat this as the starting point of their studies.

Gaussian models for the random fields for physical parameters have a finite nonzero probability of attain-ing physically impossible negative values and are hence unsuitable. More realistic models for the physicalparameters therefore need to be modeled as non-Gaussian [22]. For composite materials, the most commonmodels for the physical parameters are typically log-normal, Weibull, and other forms of extreme value distri-butions [2]. However, most random field discretization schemes available in the literature do not preserve thenon-Gaussian characteristics of the parent random fields and hence introduce additional epistemic uncertain-ties into the analysis. The application of spectral stochastic finite element methods based on polynomial chaosexpansions (PCE) allows discretization of non-Gaussian random fields, which preserve the non-Gaussian char-acteristics of the parent random field. Though this approach has been suggested in the composites literature[2,19,21], this has not been employed owing to the difficulties involved in the application of this technique tocomposites. More discussions on this issue are available later in the paper.

In this study, we use an alternative model—the optimal linear expansion (OLE)—for the discretization ofnon-Gaussian random fields having specified correlation functions. The OLE method has been proposed in[23] and discussed in greater details later in [24]. Here, the non-Gaussian random field is replaced by a vectorof correlated random variables that have the exact probability density function (pdf) as the parent randomfield at the nodal points. Moreover, the correlation structure of the parent random field is preserved among therandom variables. These characteristics limit the introduction of further epistemic uncertainties into the riskassessment calculations.

This paper is organized as follows: First, the problem considered in this paper is introduced. Next, from thelimited experimental data, parameters that show considerable scatter are selected for modeling as random fields.Statistical tests are carried out to test the suitability of fitting few well-known probability density functions, and

Stochastic finite element analysis

Fig. 1 Schematic diagram of a section of the beam; 1, 2 are the material axes, and X, Y, Z are the geometrical axes

the most appropriate models are selected for further studies carried out later in the paper. Details of the optimallinear expansion scheme are presented next, and discussions on extending the methodology to compositestructures are included. The efficacy of the proposed SFEM formalism is demonstrated by numerical examples.A comparison of the proposed SFEM formalism is examined with respect to the random variable approach.Finally, the salient features arising from this study are highlighted in the concluding section.

2 Problem definition

A composite beam made up of unidirectional carbon–epoxy HTS/M18 material is considered for the study.The beam is assumed to be made up of a stack of laminae with a [0/30/45/30/0] symmetric lay-up sequence.The thickness of each laminae is assumed to be identical and equal to 0.3 mm. Since a 5-layer sequence isconsidered, the depth of the composite beam is 1.5 mm, while the length and the breadth are taken to be 100mm and 5 mm, respectively. These dimensions were taken from an example available in [25]. A schematicdiagram of a section of the beam is shown in Fig. 1. Here, the material axes denoted by 1, 2 are shown tobe at an angle φ with respect to the geometrical axes of the beam denoted by the X, Y axes. It is to be notedthat Z is thickness direction. Three beam configurations, namely simply supported (SS), cantilever (Can), andfixed–fixed (FB) beam, are considered for the analysis. The loadings on the beams are assumed to be eitherin the form of concentrated loads (CL) or uniformly distributed loadings (UDL). The failure index (FI) of aparticular layer in the composite beam is estimated using the modified Tsai–Hill failure theory, given by

FI =[σ1

X

]2 −[σ1σ2

X2

]+

[σ2

Y

]2 +[τ12

S

]2. (1)

Here, X is the longitudinal strength, which under tension is denoted by X t and under compression is denotedby Xc; Y is the transverse strength, which under tension is denoted by Yt and under compression is denoted byYc; S is the shear strength, σ1 and σ2, respectively, which denotes the stresses developed along the material axes1 and 2; and τ12 is the corresponding shear stress. A failure in the lamina is assumed to occur when the failureindex exceeds unity. The failure of the composite beam is defined in terms of first ply failure and is deemed tooccur at the onset of failure in any laminae at any location of the composite structure. For a composite beamwith random parameters, even under deterministic loading, it is obvious that at a specified location the stressesdeveloped σ1, σ2, and τ12 and the strength parameters X, Y and S are random, implying that the failure index,FI, is a random variable with unknown pdf. The focus of this study is on the development of a SFEM formalismby which approximations for the pdf of the failure index can be developed and the failure probability of thebeam under specified loading conditions can be estimated.

3 Modeling the uncertain parameters

We first identify the parameters in a composite beam that exhibits significant scatter from the data availablefrom standardized tests on the material characterization. The conduct of the tests involves the following basicsteps:

(a) Composite laminae are taken from different manufacturing batches that use single batch of raw material(prepreg). As per MIL standards [26], statistical analyses are carried out prior to grouping of the data fromdifferent manufacturing batches.

P. Sasikumar et al.

Table 1 Details of the tests for characterizing material parameters

Test type Specimen size Gauge length Travel speed(mm/min)

ASTMstandard

Tension–longitudinal 330 × 12.5 × 0.7 150 1.5 D 3039Tension–transverse 250 × 25 × 2.5 150 1.5 D 3039Compression 145 × 10 × 1.7 15 1.0 D 3410Shear 240 × 25 × 1 150 2.0 D 3518

Table 2 Material properties of carbon–epoxy laminate

Property Notation Number ofdata points

Mean c.o.v (%)

Longitudinal elastic modulus E1 (Gpa) 83 154.9 5.9Transverse elastic modulus E2 (Gpa) 23 8.7 9.5Shear modulus G12 (Gpa) 13 4.5 8.8Ultimate longitudinal tensile strength X t (Mpa) 94 2,409 6.7Ultimate longitudinal compressive strength Xc (Mpa) 50 1,148 18.1Ultimate transverse tensile strength Yt (Mpa) 19 46 20Ultimate transverse compressive strength Yc (Mpa) 40 196 15.3Ultimate shear strength S (Mpa) 21 83 5.0

(b) Specimens with sizes complying with ASTM standards are prepared from the laminae collected in step(a). Table 1 summarizes the details of the test specimens and the testing standards. All the specimens arecured at 175 ◦C for 2 h and at vacuum pressure of 0.8 bar. The sample population of the specimens issegregated based on the parent laminae to prevent mixing of samples from different laminae batches.

(c) The specimens prepared in (b) are subsequently subjected to appropriate ASTM test standards to estimatevarious material parameters.

The details of the experimental results and their stochastic distribution modeling are explained in [27]. Asummary of the results of the experiments has been listed in Table 2. The eight parameters listed in Table 2were considered for random field modeling. In addition to the listed parameters, the test results revealed a7.5 % variability for Poisson’s ratio ν12. However, as only beamlike structures are being considered in thispaper where the effects of Poisson’s ratios are usually negligible, the Poisson’s ratio ν12 has been assumed tobe deterministic having a value equal to the mean value of 0.28. A numerical study has been carried out laterin this paper to discuss the effects of variability in Poisson’s ratio on the failure probability for the compositebeams considered in this study.

3.1 Models for the probability density function

We next consider the problem of selecting the cumulative distribution function (CDF) for the parameters listed inTable 2. The tested mechanical properties of carbon–epoxy HTS/M18 material are fitted for different probabilitydistributions that include Gaussian, log-normal, Weibull, uniform, exponential, and Gamma distributions. Testmethods, such as the Kolmogorov–Smirnov (KS) test and the Anderson-Darling (AD) test [26], have beenused to find the goodness of fit for the assumed distribution of the chosen properties. These tests are suitablefor small sample sizes, which is the case for the present study.

Figure 2 shows the best fit CDF along with the experimental data for the longitudinal elastic modulus E1;a comparison of the fit can be observed in Fig. 3 where the data and the fitted curve are shown on the Weibullpaper. It can be seen that the fitted CDF does not exactly follow the scatter observed in the experiments. Similarobservations were also reported in the literature [28]. More details about the test data and the fitted parametersare available in [27] and are not repeated here for the sake of brevity. A summary of the distributions and thenumerical values of the parameters for the various quantities are listed in Table 3.

3.2 Models for the autocorrelation function

The data required to develop models for the correlation structure for random fields g(x) require sophisticationand well-defined procedures to be adopted in the testing stage. This implies the need to cut the specimens fromthe laminate in a controlled manner and noting the spatial distances between the cut specimens. Unfortunately,


120 130 140 150 160 170 1800

0.2

0.4

0.6

0.8

1

E1 (GPa)

CD

F

WeibullData

Fig. 2 Best fit cumulative distribution function of E1

102.11

102.13

102.15

102.17

102.19

102.21

102.23

0.01

0.02

0.05

0.10

0.25

0.50

0.750.900.960.99

Data

Pro

babi

lity

Fig. 3 Weibull plot for E1

Table 3 Details of the fitted probability density functions for the material properties of HTS/M18

Property Distribution Parameters Values

E1 (Gpa) Weibull Scale, shape 158.9, 20.7E2 (Gpa) Weibull Scale, shape 9.0, 14.4G12 (Gpa) Log-normal Mean, standard deviation 1.492, 0.089X t (Mpa) Log-normal Mean, standard deviation 7.785, 0.067Xc (Mpa) Log-normal Mean, standard deviation 7.031, 0.175Yt (Mpa) Weibull Scale, shape 50.3, 5.8Yc (Mpa) Weibull Scale, shape 208.6, 7.4S (Mpa) Log-normal Mean, standard deviation 4.413, 0.05

the spatial locations of the individual specimens cut from the laminate were not recorded properly duringthe testing stage for most of the material properties. Reliable but limited test data were available only forthe samples that were used to estimate the longitudinal elastic modulus. Based on these data, an exponentialautocorrelation function [29], assumed to be of the form

Rgg(x1, x2) = e−((x2−x1)/a)2, (2)

was fitted. Here, x1 − x2 denotes the lag and a is known as the correlation constant.In the absence of reliable data for fitting the correlation functions for the material property random fields,

the correlation structure of all the random fields was assumed to be identical for the numerical examplesdiscussed later in this paper. This assumption is however not a restrictive condition on the use of the proposedmethod as has been demonstrated through a sample hypothetical case study considered later in this paper.Discussions on how the failure probability estimates change for different values of a have been presented also.

In the development of the stochastic finite element framework considered in this study, it has been assumedthat the random fields are mean square stationary processes. The following additional restrictions are necessaryto ensure the mathematical development of the stochastic finite element framework:

1. The random fields gk(x), k = 1, 2, . . ., considered in the models, are mean square bounded.

P. Sasikumar et al.

2. The fields are twice differentiable in the mean square sense. This implies that ∂4 Rgg/∂x21∂x2

2 must existfor all x1 and x2, when x1 and x2 lie within the spatial extent of the structure.

The above conditions ensure that the sample realizations of the process are sufficiently smooth such that theboundary conditions are satisfactorily described. It must be noted here that alternative models for the correlationfunction are also possible. However, the model shown in Eq. (2) is adopted since this is the most common formof the autocorrelation function that is used in structural engineering literature. Moreover, the above functionsatisfies the conditions mentioned earlier.

4 Optimal linear expansion

The method of optimal linear expansion (OLE) as a tool for random field discretization was proposed in [23]and later studied in greater details in [24]. Here, the random field g(x) is represented as an expansion

g(x) ≈ g(x) = g0 +n∑

k=1

Sk(x)χ(xk), (3)

where g is the discretized random field with the spatial extent along x , n denotes the number of nodal pointsused in the discretization, Sk(x) are the deterministic shape functions, and χk = χ(xk) are the random variablesat the kth nodal point. Here, {χk}n

k=1 represents a vector of correlated random variables. The novel feature ofOLE is that the shape functions Sk(x) do not have a predetermined form but are determined by minimizing thevariance of the error of discretization, subject to the condition that the expectation of the discretization erroris zero. Mathematically, this implies that the shape functions Sk(x) are selected such that

⟨ε2

0

⟩ =⟨{

g(x) −n∑

k=1

Sk(x)χ(xk)

}2⟩(4)

is minimized, with the constraint 〈ε0〉 = 〈g(x) − g(x)〉 = 0. Here, 〈·〉 is the expectation operator. This leadsto the set of equations

Sk(x) = C−1V, (5)

where Vi = 〈g(x)g(xk)〉 and Ci j = ⟨g(xi )g(x j )

⟩is the covariance matrix of g(x), which is assumed to

be non-singular. It can be shown that the shape functions have the property Sk(x j ) = δk j , where δk j is theKronecker’s delta function.

It is further noted that since Sk(xl) = δkl , the error of discretization at the nodes is zero. This implies thatthe first-order pdf of the parent random field and the discretized random variables at the nodes are exactlyequal. Hence, if the parent random fields are non-Gaussian, the discretized random variables capture the non-Gaussian characteristics exactly at the nodal points. To illustrate this property, we consider a random fieldhaving Weibull distribution of the form

p(x; λ, k) = k

λ

( x

λ

)k−1exp

[−

( x

λ

)k]

, (6)

where the shape parameter k = 20.7 and the scale parameter λ = 158.9. The autocorrelation function is takento be of the form in Eq. (2) with a = 0.03.

We next obtain an approximation for the Weibull distributed field with an OLE having 4 nodes. Figure 4compares the pdfs of the random field obtained from Monte Carlo simulation (MCS) —shown as a full line,and OLE when x = xk—shown as a dashed line. An exact match is observed indicating that at the nodal pointsthe discretization error is zero. A comparison of the pdf of the field when x �= xk is also shown as a dottedline in Fig. 4. As expected, the pdf is observed to have variations, especially at the tail regions. However, thenon-Gaussian features are retained as can be seen from the corresponding normal probability plot shown inFig. 5.

The number of nodal points required for the discretization of the random field is obtained by requiring thatthe global mean square error, given by

∫ xuxl

⟨ε0(x)2

⟩dx , be below a specified threshold limit. Here, xl and xu

define the lower and upper limits of the spatial domain of the problem. Figure 6 shows a plot of the globalmean square error with respect to n, the number of terms in OLE. From this figure, it can be seen that the global


100 120 140 160 18010

−6

10−5

10−4

10−3

10−2

10−1

g(xk)

Log

(pdf

) MCS OLE: x=x

k

OLE: x ≠ xk

Fig. 4 Discretized random field at nodal points and midpoints

110 120 130 140 150 160 170 180 190

0.0010.0030.010.020.050.100.25

0.50

0.750.900.950.980.99

0.9970.999

g(xk)

Pro

babi

lity

Fig. 5 Illustration of non-Gaussian features of g̃(xk) in normal probability paper: ++, probability distribution of g̃(xk) at themidpoint of the sections 1 and 2: −− corresponding Gaussian pdf

2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Number of nodes

Glo

bal m

ean

squa

re e

rror

Fig. 6 Global error

mean square error is less than 0.005 when n = 6. A similar study was carried out for the various random fielddistributions, and the global mean square error was observed to be of the same order when n = 6. Since thecomputational costs vary proportionally to the number of random variables entering the formulation, n wastaken to be 6 for all the random field representations using OLE.

The efficiency of a discretization procedure depends on the number of random variables required to representthe discretized random field, without compromising on the accuracy of the second-order characteristics of therandom field. It has been shown in the literature that the Karhunen–Loeve (KL) series expansion is the mostoptimal for Gaussian random processes [20]. Here, the basis functions are obtained by solving an integraleigenvalue problem where the correlation function constitutes the kernel. The optimality condition is satisfied

P. Sasikumar et al.

only if these eigenfunctions are available in closed form. However, for most correlation models, closed-formsolutions for the integral eigenvalue problem are not available. Instead, adopting a discrete integration rule, theintegral eigenvalue problem is converted into a matrix eigenvalue problem whose discrete eigenfunctions areused as the approximate basis functions. It was shown in [23] that this numerical procedure does not necessarilylead to optimality. Instead, the expansion OLE (EOLE) method that uses KL expansion in conjunction withOLE to derive the interpolating functions was shown to lead to more efficient discretization of the randomfields.

The KL expansion and EOLE-based approach for random field discretization are applicable for Gaussianrandom fields. For the discretization of non-Gaussian random fields, a translation process-based model [30]has been suggested in [23]. This involves transforming the problem into the Gaussian space and using EOLEto discretize the Gaussian field. Subsequently, the discretized non-Gaussian field can be obtained by theapplication of the memoryless translation processes. The crux here lies in estimating the autocovariancefunction of the process when transformed into the Gaussian space. This can be obtained by solving an integralequation given in [31]. Though analytical expressions for the bounds for the autocorrelation function for thecorresponding Gaussian field have been derived in [31] for a few commonly used autocorrelation functions, ingeneral, this requires numerical evaluation in an iterative manner [24]. However, as the discretization is carriedout in the transformed Gaussian space before transforming the problem back to the original non-Gaussianspace, it is not guaranteed that the discretization is optimal. Importantly, the numerical evaluation of thecorrelation function of the process in the transformed Gaussian space can be computationally intensive and forsome forms of correlation functions, it may not even be possible to arrive at the required correlation function.

An alternative approach used in the literature for non-Gaussian random field representation is to adopt thewidely used polynomial chaos expansions (PCE). Here, the basis functions are the polynomials of randomvariables that are mutually orthogonal. Usually, for any non-Gaussian random field representation, the basisfunctions are assumed to be Hermite polynomials, which are functions of Gaussian random variables. However,for the optimal representation of the field, depending on the non-Gaussianity, the basis functions and thedistribution of the random variables need to be appropriately selected [32,33]. This task is however not easyfor random fields with arbitrary non-Gaussian distributions.

Instead, in this study, we use the OLE expansions directly for the discretization of the non-Gaussian randomfields. The form of the shape functions are obtained from the correlation function and is derived from Eqs. (3–5), and the expansion is shown to be optimal in the sense that the variance of the error is minimized. As has beenshown earlier, this method of field discretization ensures that the second-order non-Gaussian characteristicsof the field are exactly preserved at the nodal points.

5 Stochastic finite element method

5.1 Formulation

We next focus on developing a stochastic finite element framework for composite beams, which incorporatesthe approximated random fields when optimal linear expansion is used for random field discretization. Thedevelopment of this framework is based on generalizing the available procedure for deterministic compositebeams, where the material properties in each laminae are assumed to be constant across the spatial domains.The development of the methodology is explained in this section.

Consider a multilayer laminated composite beam with uniform thickness and coordinate systems as shownin Fig. 1. We assume that the composite beam is subjected to axial and transverse loads only. The constitutiveequations for the beam can be expressed as

S = Wε̃, (7)

where the resultant matrix S is given by S = [N M]T . Here, N is the vector of membrane forces per unitlength in the global axes system, M is the vector of bending moment per unit length in the global axes system,ε̃ is the strain matrix given by ε̃ = [ε κ]T with ε being the vector of membrane strains in the global axessystem, and κ being the vector of bending strains defined in terms of mid-surface displacement and rotation.The laminate stiffness matrix W is defined as

W =[

A11 B11B11 D11

], (8)

where A11, B11, and D11 are, respectively, the first element of the extensional stiffness matrix A, bending–extension coupling matrix B and the bending stiffness matrix D, the expressions for which are given as follows:


Ai j =h/2∫

−h/2

Qi j dz, Bi j =h/2∫

−h/2

Qi j z dz and Di j =h/2∫

−h/2

Qi j z2dz. (9)

Here, the indices i, j take values 1,2,3, z is the distance from the neutral plane, and Q denotes the transformedcomposite stiffness matrix for each laminae. This is given by Q = T−1QT−t where T is the transformationmatrix and

Q =

⎡⎢⎢⎢⎢⎣

E10

1 − ν12ν21

E10ν21

1 − ν12ν210

E10ν21

1 − ν12ν21

E20

1 − ν12ν210

0 0 G120

⎤⎥⎥⎥⎥⎦

. (10)

Here, E10 , E20 , and G120 denote the mean values for the longitudinal, transverse, and in-plane shear modulus.Assuming a beam without axial load, i.e., {N} = {0}, and eliminating the membrane strain ε from Eq. (7), weget

M = Dκ, (11)

where

D = D11 − B211

A11. (12)

Note that D is scalar and models the effects of bending–stretching couplings. D is later used in formulatingthe stiffness matrix for the composite beam. When the material properties are random fields, D becomes afunction of the random fields as shown next.

We now consider E1, E2, and G12 to be random fields and assume them to be of the form

E1(x) = E10 [1 + g1(x)] ,

E2(x) = E20 [1 + g2(x)] , (13)

G12(x) = G120 [1 + g3(x)] .

Here, g1(x), g2(x), and g3(x) are stationary, zero mean, random processes having properties defined as inTable 3. Substituting these expressions in Eq. (10), it is obvious that Q can now be expressed as Q = Qd +Qs ,where Qd is as in Eq. (10) and Qs is a function of the random fields g1(x), g2(x), and g3(x). The elements ofQs can now be written as

Qs =

⎡⎢⎢⎢⎢⎢⎣

E10

∑nk=1 Sk(x)χ(xk)

1 − ν12ν21

E10

∑nk=1 Sk(x)χ(xk)ν21

1 − ν12ν210

E10

∑nk=1 Sk(x)χ(xk)ν21

1 − ν12ν21

E20

∑nk=1 Sk(x)Ψ (xk)

1 − ν12ν210

0 0 G0∑n

k=1 Sk(x)λ(xk)

⎤⎥⎥⎥⎥⎥⎦

, (14)

where in the discretized form

E1(x) = E10

[1 +

n∑k=1

Sk(x)χ(xk)

],

E2(x) = E20

[1 +

n∑k=1

Sk(x)Ψ (xk)

],

G12(x) = G120

[1 +

n∑k=1

Sk(x)λ(xk)

].

(15)

P. Sasikumar et al.

Substituting Q(x) in Eq. (9), we see that the elements of A, B, and D are functions of the spatial domain x .This implies that

D = D11(x) − B211(x)

A11(x)(16)

is a function of x and the OLE discretized random variables, collectively referred to as θ = {χ , ψ, λ}. Dcan further be written as D(x) = D0 + Ds(x), where D0 denotes the mean part and Ds(x) the randomlyfluctuating part.

5.2 Derivation of the stiffness matrix

We next develop the two-noded finite element with two degrees of freedom per node, denoted by {δ} = {w,φ},where w is the vector of transverse displacements and φ is the vector of slope at the nodal points. The internalstrain energy of the element can be determined by integrating the products of moment resultant and bendingcurvature and is given by

U = 1

2

∫

v

MT κ dv. (17)

It follows that the internal strain energy can be written in terms of stiffness and nodal displacements as

U = 1

2δT

e Keδe, (18)

where δe is the element displacement vector and Ke is the element stiffness matrix, given by

Ke =Le∫

0

b∫

0

h/2∫

−h/2

BT D B dx dy dz. (19)

Here, B is the strain displacement transfer matrix, Le is the element length, b and h are as defined in Fig. 1.When D is considered as a random field, the stiffness matrix of the composite beam can be expressed asKe = Kd

e + Kse, where the deterministic part Kd

e is given by

Kde =

Le∫

0

b∫

0

h/2∫

−h/2

BT D0 B dx dy dz, (20)

and the stochastic part Kse is given by

Kse =

Le∫

0

b∫

0

h/2∫

−h/2

BT Ds(x) B dx dy dz. (21)

It must be noted here that the above formulation allows the use of different meshing for the displacementfields and the random fields. The mesh size in the displacement field is governed by the principles of FEM,whereas the mesh size in the random field depends on the tolerance limit of the global mean square error. Itmust be remarked here that considering a large number of terms in the OLE implies the problem becomingdependent on a larger number of random variables, which increases the computational cost. Therefore, oneneeds to optimize on the number of random variables entering the formulation and an acceptable computationalcost.


5.3 Estimating the failure index

The global stiffness matrix, K, is next assembled in terms of the elemental stiffness matrices Ke. Subsequently,the nodal displacements, δ, are estimated by solving the finite element equations Kδ = P. To estimate thestress at specified nodal points, nodal strains are first estimated using standard procedures. The layer-wisestresses are obtained from the following relation:

{σ }k = [Q

]k {s} , (22)

where {σ } = {σ1 σ2 σ12} and s = {ε1 ε2 ε12} are, respectively, the stress and strain components definedalong the material axes. Note that Q here is a function of the random variables θ . Using the modified Tsai–Hill criterion given in Eq. (1), failure indices are estimated using these stresses and the material strengths. Inthe present study, the strength properties are also treated as random processes. As explained in the previoussection, we approximate the random fields for material strength properties using OLE, and the failure indicesare estimated using approximated nodal strength. Subsequently, the failure probability of the composite beamsin terms of first ply failure are estimated using Monte Carlo simulations.

6 Numerical examples and discussions

The proposed methodology is illustrated by a set of numerical examples. The choice of the numerical examplespresented here has been governed by the twin objectives of (a) validating the accuracy of the OLE-basedrandom field discretization of non-Gaussian random fields and (b) illustrating the application of the proposedOLE-based SFEM framework for reliability analysis of composite beams. To focus on the validation of theOLE-based discretization of non-Gaussian random fields, we first consider an isotropic cantilever beam withspatially randomly varying Young’s modulus of elasticity. Subsequently, a 5-layered composite beam has beenconsidered for the illustration of the reliability analysis using the OLE-based SFEM framework developed inthis paper. The results obtained by the proposed method have been compared with those obtained from MonteCarlo simulations. Some numerical case studies have been presented along with discussions to highlight theeffects of correlation lengths of the random fields for the material parameters.

6.1 Example 1: isotropic cantilever beam

The objective of this example is to examine the applicability of the OLE-based random field discretizationprocedure that forms an integral part of the SFEM framework developed in this paper. To investigate theerrors that enter the formulation when a non-Gaussian random field is represented as a series expansion ofdeterministic interpolating functions and a vector of correlated random fields, we select a problem that can besolved in the strong form and which can serve as the benchmark. Since the governing equations for compositebeams are usually written in the weak form in terms of finite elements and are not amenable for treatment inthe strong form, we therefore consider an isotropic cantilever beam subjected to uniformly distributed loadingP(x), whose field equation is given by

d2

dx2

{E(x)I

d2 y

dx2

}= P(x). (23)

Here, I is the modulus of rigidity and E(x) is the Young’s modulus of elasticity, modeled as a random fieldand is expressed as

E(x) = E0[1 + g(x)], (24)

where E0 is the mean value of E(x) and g(x) is a zero mean stationary non-Gaussian random process.Simulating sample realizations for g(x), Eq. (23) can be numerically integrated to obtain sample realizationsof the response. Here, the only source of errors arises from the step sizes being used in the numerical integrationscheme, and hence the corresponding results can serve as the benchmark against which the accuracy of theOLE discretization scheme-based response analysis can be compared.

For the numerical calculations, g(x) is assumed to be Weibull distributed with the shape and the scaleparameters being 20.7 and 158.9, respectively. The autocorrelation function for g(x) is assumed to be of theform in Eq. (2) with a = 0.03. The uniformly distributed load P(x) is taken to be 359 N/m. The beam geometryis provided in Sect. 2. As the modulus is a non-Gaussian random field, the tip displacement is a random variablewhose pdf is estimated using the following two methods:

P. Sasikumar et al.

0.018 0.019 0.02 0.021 0.022 0.023 0.024 0.025 0.026 0.0270

0.2

0.4

0.6

0.8

1

Deflection

CD

F

Method−1Method−2

Fig. 7 Example 1: cumulative distribution function of δ

0.016 0.018 0.02 0.022 0.024 0.026 0.02810−3

10−2

10−1

100

101

102

103

Deflection

pdf


Fig. 8 Example 1: probability density function of δ

(a) Method 1: The random field is discretized using OLE and the corresponding global stiffness matrix isconstructed. The stiffness matrix K can be written as K = Kd +Ks where Kd is the deterministic stiffnessmatrix and is a function of E0, whereas Ks is the stochastic component of the stiffness matrix whoseelements are obtained according to Eq. (21). Clearly, Ks is a function of the nodal non-Gaussian randomvariables θ (here, θ is a function of χ only). The numerical implementation of this method follows thesteps listed:(i) First, N realizations of the vector of correlated non-Gaussian nodal random variables θ are simulated

and the global stiffness matrix K(θ) is constructed for each realization of θ .(ii) Next, the tip deflection δ is obtained by solving the FE equation, for each realization of θ .

(iii) Finally, the pdf of the tip displacements δ is obtained by statistical processing of the samples {δi }Ni=1.

(b) Method 2: Here, the governing equation for the isotropic beam is numerically integrated using the 5-pointcentral difference scheme. The steps involved in the numerical implementation are as follows:(i) An ensemble of N realizations for the non-Gaussian random field g(x) is simulated using standard

procedures available in the literature [34].(ii) The tip displacements, δi , are obtained by direct numerical integration of the field equation, corre-

sponding to each realization of {gi (x)}Ni=1.

(iii) The pdf of the tip displacements δ is obtained by statistical processing of the samples {δi }Ni=1.

Figure 7 shows the comparison of the plots for the CDF of the tip displacement using Methods 1 and 2. Thecorresponding plots for the pdfs are shown in Fig. 8. An inspection of these two figures reveals a very closeresemblance between the two methods. The close match between the results obtained from Methods 1 and 2highlights the accuracy of the proposed OLE-based SFEM approach. With this confidence, we now apply theproposed method for the failure analysis in composite beams.

6.2 Example 2: a 5-layered laminated composite beam

We next consider 5-layered laminated composite beams having three different configurations, namely simplysupported (Model 1), cantilever (Model 2), and fixed–fixed (Model 3) beam. The geometrical dimensions ofthe beam in all three cases are assumed to be identical. For each of these three cases, we consider the effects ofboth concentrated loading (CL) as well as uniformly distributed loading (UDL). The failure index is estimated


Table 4 Summary of the beam configuration, their loadings, and the locations of stress calculations

Beam configuration Loading Case 1 Case II Location of stresscalculationL/b = 15 L/b = 20

Simply supported (a) UDL (N/m) 1,878 1,410 Mid point(Model 1) (b) CL (N)

at mid point 96 72 Mid pointCantilever (a) UDL (N/m) 479 359 Fixed end(Model 2) (b) CL (N)

at tip 24 18 Fixed endFixed–fixed (a) UDL (N/m) 2,900 2,180 Fixed ends(Model 3) (b) CL (N)

at mid point 192 144 Fixed endsUDL uniformly distributed load, CL concentrated load

at all the nodal points. First, ply failure location and the loading details are listed in Table 4. The magnitude ofthe loadings is adjusted such that the FI at the most critical locations for the deterministic case is 0.83.

We consider Model 3 with concentrated load on the middle of the beam to validate the proposed method.The material properties are modeled as random fields whose autocorrelation functions are of the form givenin Eq. (2) with correlation constant a = 0.03. The pdf of the failure index of the composite beam is estimatedusing the following two methods:

(a) Method 1: The proposed OLE-based SFEM discussed in Sect. 5 of this paper. The steps involved in thenumerical implementation of the method are as follows:

(i) Following the formulation developed in Sect. 5, the global stiffness matrix K is constructed as a func-tion of a vector of correlated random variables θ corresponding to the discretized random variables.

(ii) An ensemble of N realizations of θ is simulated.(iii) Corresponding to each realization of θ , the FE equations are solved to calculate the stresses at the

desired locations, and the failure indices are calculated according to the modified Tsai–Hill failurecriteria.

(iv) Statistical processing of the results obtained from the N realizations is carried out to estimate thefailure probability.

(b) Method 2: Here, the beam is discretized using finite elements having lengths smaller than the correlationlength of the field. Subsequently, the material properties within an element are assumed to be constantacross the length of the element. Since the material properties are assumed to be random fields, thematerial properties associated with each element are random variables. The correlation that exists betweenthe variables in adjoining elements is determined from the discretized correlation matrix, with the spatiallag defined in terms of the elemental centroidal distance. The key steps involved are as follows:

(i) The vector of correlated random variables is simulated using a standard procedure.(ii) The FE equations are now solved to obtain the stresses at the desired locations, and the failure indices

are calculated according to the Tsai–Hill failure criteria.(iii) Statistical processing of the results obtained from the N realizations is carried out to estimate the

failure probability.

It must be noted here that in Method 1 the meshing for the displacement field and the random field is differentand is governed by their individual characteristics. On the other hand, in Method 2, the FE meshing and therandom field mesh are identical. Also, as the elemental lengths are taken to be smaller than the correlation lengthof the fields, the number of finite elements is larger than what would have been required for a correspondingdeterministic beam. This increases the computational costs of the analysis. Additionally, since the materialproperties associated with each element are modeled as random variables, the number of random variablesentering the formulation is also quite large. This puts additional demands on the computational costs associatedwith the response and reliability analysis. However, the sources of errors involved in Method 2 are only dueto the choice of the element length, and as this length tends to zero, the error associated with discretization isexpected to vanish. Nevertheless, a comparison of the CDF and the pdf of the failure index, shown in Figs. 9and 10, reveals a very close agreement and is a further validation of the accuracy of the proposed SFEMframework developed in this study.

P. Sasikumar et al.

0 0.5 1 1.5 2 2.5 3 3.5 410−4

10−3

10−2

10−1

100

FILo

g (C

DF

) Method−1 Method−2

Fig. 9 Example 2: cumulative distribution function of FI

0 0.5 1 1.5 2 2.5 310−6

10−4

10−2

100

102

FI

log

(pdf

)

Method−1 Method−2

Fig. 10 Example 2: probability density function of FI

6.3 Example 3: random variable vis-a-vis random process models

Method 2 considered in the previous section assumed the material properties to have a constant value acrossthe spatial extent of the finite element and can be modeled as a random variable. However, the spatial variationsof the field are captured by the correlation characteristics between the random variables for adjacent elements.Since the elemental sizes were taken to be small, the effects of the spatial variation of the material propertieswere adequately captured. The downside of Method 2 are the associated high computational costs.

Most of the existing studies in the literature have modeled the uncertainties associated with the materialproperties as random variables. Random variable models differ from random process models in that the randomvariations along the spatial extent of the structure are neglected. Thus, only the distribution characteristics ofthe material properties are incorporated into the uncertainty model. In this section, we highlight the differencesin treating the material properties as random variables vis-a-vis as random fields.

All three types of beam models provided in Table 4 are considered for the analysis. For each of these threecases, we consider the effects of both concentrated loading (CL) and uniformly distributed loading (UDL).The failure index is estimated corresponding to all the nodal points. The location of the first ply failure andthe loading details are also listed in Table 4. The material properties considered as random fields are listed inTable 2 and their distribution properties are provided in Table 3. The following two methods of the analysisare considered.

(a) Method 1: The method proposed in this paper and is the same as Method 1 discussed in the previoussection.

(b) Method 2: The properties modeled as random fields are treated as random variables having identical pdfsas the random fields. Thus, in this method, the spatial variations in the properties are neglected and thematerial properties in all the finite elements are assumed to be the same for each realization. The numericalimplementation of this method involves the following steps:(i) The vector of random variables θ is first simulated.

(ii) The FE equations are constructed as functions of θ .Steps (i i i) and (iv) are identical to the third and fourth steps in the previous method.

The beam is discretized into 15 finite elements. The parameters listed in Table 2 are treated as random fields ineach layer. Since the beam is assumed to be made up of 5 layers, the total number of random fields entering theformulation is 30. This includes 3 moduli fields, two strength property fields (either tension or compression)


0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

Failure index

pdf


Fig. 11 Example 3: probability density function of Model 1: concentrated load

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Failure index

pdf


Fig. 12 Example 3: probability density function of Model 1: uniformly distributed load

and one shear strength field corresponding to each laminae. In Method 1, the OLE discretization of eachrandom field involves approximating the field with a vector of 6 correlated random variables. Thus, the totalnumber of random variables entering the formulation in Method 1 is 30 × 6 = 180. In Method 2, the fieldsare replaced by random variables. Thus, the number of random variables entering the formulation is just 30.

For a given beam configuration and a given loading condition, the failure index corresponding to each layerat beam locations listed in Table 4 is calculated for all N realizations. Assuming first ply failure, the failure ofa beam is assumed to occur if the failure index in any layer crosses one. The failure probability of the beam issubsequently estimated from the relative frequency definition, i.e., Pf = n f /N , where n f is the number ofsamples where failure occurs. In these studies, N is taken to be 1 × 104 samples.

It is observed that the outermost layer is the most critical in terms of first ply failure. Figure 11 illustratesthe pdf of the FI for first ply failure of the simply supported beam (Model 1) under concentrated loading. Foruniformly distributed loading, the corresponding plot for pdf is shown in Fig. 12. Figure 13 illustrates the pdffor the failure index corresponding to the cantilever beam (Model 2) under concentrated tip load. Figure 14shows the corresponding plots when the loading is UDL. Figures 15 and 16 show the pdf of the failure index forthe fixed–fixed beam conditions (Model 3) under concentrated loading and UDL, respectively. An inspectionof Figs. 11, 12, 13, 14, 15, and 16 reveals a significant difference in the probability density functions of thefailure indices when the uncertain variations are modeled as random processes and random variables. Thesevariations highlight that neglecting the spatial random fluctuations in the material properties has considerableinfluence on the reliability analysis of the structure.

A first ply failure is assumed to occur when the failure index exceeds unity. Thus, the failure probabilityis mathematically calculated as Pf = ∫ ∞

1 p(x) dx where p(x) is the pdf of the failure index. An inspectionof Figs. 11, 12, 13, 14, 15, and 16 reveals that the pdf of the failure index when random process models areconsidered is usually shifted to the right and have longer well-pronounced right tails. This implies that thecalculated failure probability would be higher when random process models are being considered. Figure 17shows a comparison of the estimated failure probabilities when Methods 1 and 2 are considered for the variouscases. As expected, it is observed that the failure probabilities are consistently higher for all cases when theuncertainties are modeled as random fields. Figure 18 summarizes the estimated failure probabilities when theL/b ratio is reduced to 15, and similar trends are observed for this case as well. It must be noted that modeling thestructural member as a beam may no longer be appropriate when the L/b ratio is reduced any further. Though

P. Sasikumar et al.

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Failure index

pdf



0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Failure index

pdf



0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

Failure index

pdf



0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

Failure index

pdf



the associated computational costs with Method 1 are about 3 times in comparison with the much simplerMethod 2, it is clear that random variable models underestimate the risk associated with laminate compositebeamlike structures. This study therefore highlights the importance of modeling the spatial fluctuations in the


Can−CL Can−UDL SS−CL SS−UDL FB−CL FB−UDL 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Type of beam

Fai

lure

pro

babi

lity


Fig. 17 Example 3: comparison of failure probabilities estimated using Methods 1 and 2; L/b ratio for beam is 20

Can−CL Can−UDL SS−CL SS−UDL FB−CL FB−UDL 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Type of beam

Fai

lure

pro

babi

lity


Fig. 18 Example 3: comparison of failure probabilities estimated using Methods 1 and 2; L/b ratio for beam is 15

Table 5 Example 4: minimum OLE nodes required to represent a random field for various values of correlation constant a

Constant a 0.005 0.008 0.01 0.02 0.03 0.04 0.05 0.1OLE nodes 27 19 15 8 6 5 4 3

material properties in uncertainty quantification and risk assessment studies, even if the computational costsare higher than the simpler models available in the literature.

6.4 Example 4: effect of correlation factor

The studies carried out in the previous section highlight the importance of modeling the spatial randomfluctuations in the material properties. Unlike random variable models, random process models incorporatethese spatial fluctuations into the model through the autocorrelation function. The correlation length definesthe distance between the two points along the spatial domain beyond which the randomness in the propertiesis assumed to have no mutual dependence. This length is governed by the correlation constant a, shown inEq. (2), and plays an important role in SFEM analysis of the study carried out in this paper.

We next carry out a numerical study where the correlation constant a was varied from 0.005 to 0.1 toinvestigate their effect on the failure probability of a 5-layered laminated composite beam. Since, in theproposed study, the random field discretization mesh is dependent on the correlation length, the size of thevector required for representing the discretized random field would also vary. For random fields with highercorrelation length, the number of discretization points would be smaller, which in turn implies the need forfewer random variables entering the formulation, for the same tolerance level for the global mean square error.Table 5 lists the number of OLE nodes required for discretizing a random field having the autocorrelationfunction of the form as in Eq. (2) with various values of constant a, when the tolerance limit for the globalmean square error is taken to be 0.005. Thus, for example, when a = 0.005, the number of random variablesrequired to discretize a random field is 27. As the formulation considers six material properties at a time to bemodeled as random fields, the total number of random variables entering the formulation is 27 × 6 × 5 = 810.In contrast, when a = 0.1, the total number of random variables in the formulation is just 3 × 6 × 5 = 90.

P. Sasikumar et al.

0.005 0.01 0.02 0.05 0.1 RV 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Correlation constant

Fai

lure

pro

babi

lity

Can−CLSS−UDL

Fig. 19 Example 4: failure probability estimate for different correlation lengths

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

Failure Index

pdf

RVRP (0.008)RP (0.03)RP (0.1)

Fig. 20 Example 4: failure probability pdf for different correlation constant for cantilever tip load

It must be noted that the number of finite elements used in the model remains the same irrespective of thecorrelation constant a.

Figure 19 summarizes the failure probability estimates of some typical beam and loading configurationswhen the correlation constant a is varied. In general, for a particular beam and load configuration, it is observedthat the failure probability decreases as the correlation length of the process increases. Figure 20 comparesthe density function for the failure index for various values of a, for the cantilever beam under tip loading.Similar curves have been observed for different beam and load configurations. It is interesting to note thatwhen the correlation length is small, the pdf is shifted to the right. However, as a increases, the pdf shifts tothe left. For higher correlation lengths, the corresponding pdf tends to the pdf obtained when random variablemodels are adopted for the material properties. This is expected as when the correlation lengths are very high, itimplies that the fluctuations within the spatial extent are small and highly dependent and thus can be adequatelymodeled as random variables instead of random fields. Also, as the pdf shifts toward left with higher valuesof correlation lengths, the failure probability is expected to decrease.

6.5 Example 5: effect of Poisson’s ratio

In the theoretical studies carried out in [35] on composite plates, the authors have observed that an assumptionof 10 % variability in the Poisson’s ratio, ν12, has contributed to about a maximum of 6 % variability in theresponse. As has been mentioned earlier, even though the testing revealed a variability of about 7.5 % in ν12, inthis study, Poisson’s ratio has been assumed to be deterministic. This is because the effects of Poisson’s ratio aremanifested usually in the lateral direction in beamlike structures and are expected to have minimal effects onthe response. The primary factors that contribute to the response are from the parameters, which are manifestedalong the longitudinal direction. To test this hypothesis, we carry out a numerical study where the effect ofvariations in the Poisson’s ratio on the failure probability indices is examined. A 5-layered laminated compositecantilever beam subjected to UDL was considered. The Poisson’s ratio, ν12, was modeled as a random fieldhaving log-normal distribution with mean and c.o.v. taken as 0.28 and 7 %, and an autocorrelation functiongiven by Eq. (2). Two cases were considered when the correlation constant a was taken to be 0.005 and 0.08,respectively. All the other material property parameters were assumed to be deterministic. The pdf of the failureindex at the nodal points was calculated using the proposed OLE-based SFEM framework and has been shownin Fig. 21 for the two cases. The pdf for the failure index when a = 0.005 and when a = 0.08 does not appear


0.828 0.83 0.832 0.834 0.836 0.8380

100

200

300

400

Failure Index

pdf

a=0.08a=0.005

Fig. 21 Effect of Poisson’s ratio variations in failure probability

to be significantly different. Moreover, an inspection of the x-axis reveals that the c.o.v. of the failure indexis less than 0.14 % for either case. This confirms the hypothesis that the effects of variations in the Poisson’sratio, ν12, are negligible in beamlike composite structures and for the sake of computational efficiency, it mayconsidered to be deterministic.

7 Concluding remarks

A stochastic finite element-based methodology has been developed for uncertainty quantification and reliabilityassessment of laminated composite beamlike structures. Following are the salient contributions arising fromthis study:

1. A framework has been developed that incorporates non-Gaussian random field models for the materialproperty uncertainties in laminated composite beamlike structures. To the best of the authors’ knowledge,studies available in the composite literature have either focussed on non-Gaussian random variable modelsfor the material properties or if the random field models have been used, the random field models havebeen limited to Gaussian fields.

2. The proposed method enables independent meshing for finite element and random field discretization. Themesh size for random field discretization is based on the correlation length, and a coarse meshing is usuallysufficient. This enables a smaller number of random variables entering the formulation. Consequently, thedemand on CPU time and computer memory are significantly less in comparison with other related methods.The development of SFEM-based approaches that can handle independent meshing for FE and randomfield discretization seems to have been addressed only in a few studies, e.g., [36,37] but appears to havenot been applied to composite structures.

3. In structures with material properties that have spatial random fluctuations with correlation lengths signifi-cantly smaller than the spatial extent of the structure, modeling the material parameters as random variablesunderestimates the probability of first ply failures. For example, in a carbon–epoxy HTS/M18 materialsystem studied in this paper, the maximum predicted failure probability when random field models areused is approximately two times in comparison with when random variable models have been used in afixed–fixed beam with uniformly distributed loading. In these situations, modeling the material propertiesas random fields is desirable as they lead to more conservative risk estimates even though the associatedcomputational costs are higher.

References

1. Carbillet, S., Richard, F., Boubakar, L.: Reliability indicator for layered composites with strongly non-linear behaviour.Compos. Sci. Technol. 69, 81–87 (2009)

2. Sriramula, S., Chryssanthopoulos, M.K.: Quantification of uncertainty modelling in stochastic analysis of FRP composites.Composites: Part A 40, 1673–1684 (2009)

3. Jeong, H.K., Shenoi, R.A.: Probabilistic strength analysis of rectangular FRP plates using Monte Carlo simulation. Comput.Struct. 76, 219–235 (2000)

4. Shiao, M.C., Chamis, C.C.: Probabilistic evaluation of fuselage-type composite structures. Probab. Eng. Mech. 14,179–187 (1999)

5. Shiao, M.C., Singhal, S.N., Chamis, C.C.: A Method for the Probabilistic Design Assessment of Composite Structures,NASA TM-106384

P. Sasikumar et al.

6. Mase, G.T., Murthy, P.L.N., Chamis, C.C.: Probabilistic Micromechanics and Macromechanics of Polymer MatrixComposites, NASA TM-103669

7. Murthy, P.L.N., Chamis, C.C.: Integrated Composite Analyzer (ICAN)—Users and Programmers Manual, NASA TP 25158. Pai Shantaram, S., Chamis, C.C.: Probabilistic Assessment of Space Trusses Subjected to Combined Mechanical and Thermal

Loads, NASA TM 1054299. Chamis, C.C., Michael, C. Shiao: IPACS—Integrated Probabilistic Assessment of Composite Structures: Code Development

and Applications, NASA, N95-2884910. Chiachio, M., Chiachio, J., Rus, G.: Reliability in composites—a selective review and survey of current develop-

ment. Composites-B 43, 902–913 (2012)11. Vanmarcke, E., Shinozuka, M., Nakagiri, S., Schueller, G.I., Grigoriu, M.: Random fields and stochastic finite

elements. Struct. Saf. 3, 143–166 (1986)12. Kleiber, M., Hien, T.D.: The Stochastic Finite Element Method: Basic Perturbation Technique and Computer Implementa-

tion. Wiley, Chichester (1992)13. Haldar, A., Mahadevan, S.: Reliability Assessment Using Stochastic Finite Element Analysis. Wiley, Canada (2000)14. Ibrahim, R.A.: Structural dynamics with parameter uncertainties. Appl. Mech. Rev. 40, 309–328 (1987)15. Manohar, C.S., Ibrahim, R.A.: Progress in structural dynamics with stochastic parameter variations 1987–1998. Appl. Mech.

Rev. 52, 177–197 (1999)16. Sudret, B., Der Kiureghian, A.: Comparison of finite element reliability methods. Probab. Eng. Mech. 17, 337–348 (2002)17. Stefanou, G.: The stochastic finite element method: past, present and future. Comput. Methods Appl. Mech. Eng. 198,

1031–1051 (2009)18. Wu, W.F., Cheng, H.C., Kang, C.K.: Random field formulation of composite laminates. Compos. Struct. 49, 87–93 (2000)19. Ngah, M.F., Young, A.: Application of the spectral stochastic finite element method for performance prediction of composite

structures. Compos. Struct. 78, 447–456 (2007)20. Ghanem, R., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)21. Nian-Zhong, C., Guedes Soares, C.: Spectral stochastic finite element analysis for laminated composite plates. Comput.

Methods Appl. Mech. Eng. 197, 4830–4839 (2008)22. Probabilistic Model Code: Technical report, Joint Committee on Structural Safety. http://www.jcss.byg.dtu.dk/Publications/

Probabilistic_Model_Code.aspx23. Li, C.-C., Der Kiureghian, A.: Optimal discretization of random fields. J. Eng. Mech. ASCE 119, 1136–1154 (1993)24. Gupta, S., Manohar, C.S.: Dynamic stiffness method for circular stochastic Timoshenko beams: response variability and

reliability analysis. J. Sound Vib. 253, 1051–1085 (2002)25. Kaw, A.K.: Mechanics of Composite Materials. 2nd edn. CRC Press/Taylor and Francis, Boca Raton (2006)26. MIL-HDBK-17-1F: Volume 1 of 5, (2002)27. Sasikumar, P., Suresh, R., Gupta, S., Vijayaghosh, P.K.: Stochastic modelling of uncertainties in layered composites. In: 4th

International Conference on Recent Advances in Composite Materials ICRACM, Goa, OP-70 (2013)28. Philippidis, T.P., Lekou, D.J., Aggelis, D.G.: Mechanical property distribution of CFRP filament wound composites. Compos.

Struct. 45, 41–50 (1999)29. Sriramula, S., Chryssanthopoulos, M.K.: An experimental characterisation of spatial variability in GFRP composite

panels. Struct. Saf. 42, 1–11 (2013)30. Grigoriu, M.: Crossings of non-Gaussian translation processes. J. Eng. Mech. ASCE 110, 610–620 (1984)31. Der Kiureghian, A., Liu, P.L.: Structural reliability under incomplete probability information. J. Eng. Mech. ASCE 112, 85–

104 (1986)32. Xiu, D., Karniadakis, G.E.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci.

Comput. 24, 6190644 (2002)33. Witteveen, J.A.S., Sarkar, S., Bijl, H.: Modeling physical uncertainties in dynamic stall induced fluid-structure interaction

of turbine blades using arbitrary polynomial chaos. Comput. Struct. 85, 866–878 (2007)34. Grigoriu, M.: Applied Non-Gaussian Processes: Examples, Theory, Simulation, Linear Random Vibration, and Matlab

Solutions. Prentice Hall, Englewood Cliffs (1995)35. Noh, H.C., Park, T.: Response variability of laminate composite plates due to spatially random material parameter. Comput.

Methods Appl. Mech. Eng. 200, 2397–2406 (2011)36. Charmpis, D.C., Papadrakakis, M.: Improving the computational efficiency in finite element analysis of shells with uncertain

properties. Comput. Methods Appl. Mech. Eng. 194, 1447–1478 (2005)37. Schenk, C.A., Schueller, G.I.: Buckling analysis of cylindrical shells with cut-outs including random boundary and geometric

imperfections. Int. J. Nonlinear Mech. 38, 1119–1132 (2003)

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Stochastic finite element analysis of layered composite beams with spatially varying non-Gaussian...

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