Content uploaded by Tanmoy Mukhopadhyay

Author content

All content in this area was uploaded by Tanmoy Mukhopadhyay on Oct 08, 2017

Content may be subject to copyright.

Original Article

Probabilistic

characterisation for

dynamics and stability

of laminated soft core

sandwich plates

SDey

1

, T Mukhopadhyay

2

, S Naskar

3

,

TK Dey

4

, HD Chalak

5

and S Adhikari

2

Abstract

This paper presents a generic multivariate adaptive regression splines-based approach

for dynamics and stability analysis of sandwich plates with random system parameters.

The propagation of uncertainty in such structures has significant computational chal-

lenges due to inherent structural complexity and high dimensional space of input par-

ameters. The theoretical formulation is developed based on a refined C

0

stochastic finite

element model and higher-order zigzag theory in conjunction with multivariate adaptive

regression splines. A cubical function is considered for the in-plane parameters as a

combination of a linear zigzag function with different slopes at each layer over the entire

thickness while a quadratic function is assumed for the out-of-plane parameters of the

core and constant in the face sheets. Both individual and combined stochastic effect of

skew angle, layer-wise thickness, and material properties (both core and laminate) of

sandwich plates are considered in this study. The present approach introduces the

multivariate adaptive regression splines-based surrogates for sandwich plates to achieve

computational efficiency compared to direct Monte Carlo simulation. Statistical analyses

are carried out to illustrate the results of the first three stochastic natural frequencies

and buckling load.

Journal of Sandwich Structures and Materials

0(00) 1–32

!The Author(s) 2017

Reprints and permissions:

sagepub.co.uk/journalsPermissions.nav

DOI: 10.1177/1099636217694229

journals.sagepub.com/home/jsm

1

Mechanical Engineering Department, National Institute of Technology Silchar, Silchar, India

2

College of Engineering, Swansea University, Swansea, UK

3

School of Engineering, University of Aberdeen, Aberdeen, UK

4

Department of Civil Engineering, National Institute of Technical Teachers’ Training & Research (NITTTR)

Kolkata, Kolkata, India

5

Civil Engineering Department, National Institute of Technology Kurukshetra, Kurukshetra, India

Corresponding author:

Tanmoy Mukhopadhyay, College of Engineering, Swansea University, Swansea SA1 8EN, UK.

Email: 800712@swansea.ac.uk; www.tmukhopadhyay.com

Keywords

Uncertainty quantification, sandwich plates, composites, stochastic natural frequency,

stochastic buckling load, multivariate adaptive regression splines

Introduction

The application of sandwich structures has gained immense popularity in advanced

engineering applications, especially in aerospace, marine, civil, and mechanical

structures that require superior performances and outstanding properties such as

lightweight, high stiﬀness, high structural eﬃciency, and durability. The construc-

tion of sandwich panels consisting of thin face sheets of high strength material

separated by a relatively thick and low density material oﬀers excellent mechanical

properties such as high strength-to-weight ratio and high stiﬀness-to-weight ratios.

The characteristic features of such structures are aﬀected by their layered construc-

tion and variations in properties through their thickness, and therefore it is import-

ant to predict their overall response in a realistic manner considering all these

features. The eﬀect of shear deformation plays a vital role in the structural analysis

of sandwich and composite constructions because of their low shear modulus

compared to extensional rigidity with a large variation in material properties

between the core and the face layers. Moreover, due to their special type of con-

struction and behavior, sandwich structures possess high statistical variations in

the material and geometric properties. These inherent uncertainties should be prop-

erly taken into account in the analysis in order to have more realistic and safe

design. This cannot be mapped by the conventional deterministic analysis. In fact,

accurate predictions of the vibration response of these structures become more

challenging to the engineers in the presence of inherent scatter in stochastic input

parameters consisting of both material and geometric properties. Stochastic natural

frequencies of such sandwich structures consist of overall mode and localised ones

or through the thickness that the classical deterministic theories lack to detect. Due

to the dependency of a large number of parameters in complex production and

fabrication processes, the system properties are inevitably random in nature result-

ing in uncertainty in the response of the sandwich plate. Therefore, there is a need

for an eﬃcient and accurate computational technique which takes into account the

eﬀects of parameter uncertainty on the structural response. In the deterministic

analysis of structures, the variations in the system parameters are neglected and the

mean values of the system parameters are used in the analysis. But the variations in

the system parameters should not be ignored for accurate and realistic studies that

require a probabilistic description in which the response statistics can be adequately

achieved by considering the material and geometric properties to be stochastic in

nature.

Many review articles [1–4] are published on deterministic analysis of sandwich

composite plates. Several investigators [5–11] studied on deterministic bending,

2Journal of Sandwich Structures and Materials 0(00)

buckling and free vibration analysis of skew composite and sandwich plates and

thereby optimising such structures. Recently, a study has been published on ana-

lytical development for free vibration analysis of sandwich panels with randomly

irregular honeycomb cores [12]. Free vibration response of laminated skew sand-

wich plates is investigated by Garg et al. [13] using C

0

isoparametric ﬁnite element

model based on HSDT. The vibration behaviour of imperfect sandwich plates with

in-plane partial edge load is presented by Chakrabarti and Sheikh [14] in a deter-

ministic framework, while free vibration analyses of sandwich plates subjected to

thermo-mechanical loads is studied by Shariyat [15] using a generalised global–

local higher order theory. Many literatures [16–20] are found which investigate on

dynamic and stability of soft core sandwich plates by analytical or ﬁnite element

method. Radial basis function is used by Roque et al. [21], Ferrera et al. [22–24]

and Rodrigues et al. [25,26] to analyse bending, buckling and free vibration

characteristics of composite sandwich plates. The mesh-free moving Kriging inter-

polation method is presented by Bui et al. [27] for analysis of natural frequencies

of laminated composite plates while Yang et al. [28] studied the vibration and

damping analysis of thick sandwich cylindrical shells with a viscoelastic core.

There is plenty of literature found which presents buckling and free vibration

analyses of sandwich plates using Rayleigh-Ritz method [29–32]. A failure analysis

of laminated sandwich shells has been carried out by Kumar et al. [33]. Several

recent reports investigate bending and buckling analysis of sandwich plates with

functionally graded material [34,35]. Researchers reported their results using

Galerkin method [36], quadrature method [37], state–space method [38,39],

Levy’s method [40,41], Navier’s method [42] or exact solutions method [43]

for buckling and free vibration analyses of laminated sandwich plates. Recently,

an analytical approach has been presented to obtain equivalent elastic properties

of spatially irregular honeycomb core [44–46], which is often used as the core of

sandwich panels. Such equivalent elastic properties of irregular honeycomb core

can be an attractive solution for stochastic analyses of honeycomb panels with

honeycomb core.

Most of the investigations carried out so far concerning the analysis of sandwich

composite panels are deterministic in nature that lacks to cater the necessary

insight on the behaviour of diﬀerent structural responses generated from inherent

statistical variations of stochastic material and geometric parameters. The studies

on the stochastic analysis of sandwich plates with transversely ﬂexible core are very

scarce in literature [47]. In general, Monte Carlo simulation (MCS) is commonly

used for stochastic response analysis. But the traditional MCS-based stochastic

analysis approach is very expensive because it requires thousands of ﬁnite element

simulations to be carried out to capture the random nature of parametric uncer-

tainty. Hence, reduced order modelling (ROM) techniques are used to reduce the

computational time and cost. In past, reduced order computational models are

found to be employed in stochastic analysis of structures and materials [48–50]

and some of them are speciﬁcally applied in laminated composite plates and shells

including the eﬀect of noise [51–65]. In the present study, we propose a multivariate

Dey et al. 3

adaptive regression splines (MARS)-based eﬃcient uncertainty quantiﬁcation algo-

rithm for composite sandwich structures. In this approach, the expensive ﬁnite

element model for sandwich composite structures is eﬀectively replaced by the

computationally eﬃcient MARS model making the overall process of uncertainty

quantiﬁcation much more cost-eﬀective. Compared to other reduced order model-

ling techniques, the use of MARS for engineering design applications is relatively

new. Sudjianto et al. [66] used MARS model to emulate a conceptually intensive

complex automotive shock tower model in fatigue life durability analysis

while Wang et al. [67] compare MARS to linear, second-order, and higher-order

regression models for a ﬁve variable automobile structural analysis. Friedman [68]

integrated MARS procedure to approximate behaviour of performance variables

in a simple alternating current series circuit. Literature suggests that the major

advantages of using the MARS-based reduced order modelling appears to be

the accuracy and signiﬁcant reduction in computational cost associated with con-

structing the surrogate model compared to other conventional emulators such as

Kriging [69].

To the best of the authors’ knowledge, no work is reported in scientiﬁc literature

on the study of dynamics and stability for laminated sandwich skewed plates with

random geometric and material properties based on eﬃcient MARS approach.

MARS constructs the input/output relation from a set of coeﬃcients and basis

functions that are entirely driven from the regression data. The algorithm allows

partitioning of the input space into regions, each with its own regression equation.

This makes MARS particularly suitable for problems with high dimensional input

parameter space. As ﬁnite element models of sandwich structures normally have

large number of stochastic input parameters, MARS has the potential to be an

eﬃcient mapping route for the inputs and responses of such structures. In the

present study, a stochastic analysis for free vibration and buckling of laminated

sandwich skewed plates is carried out by solving the random eigenvalue problem

through an improved higher-order zigzag theory in conjunction with MARS fol-

lowing a bottom-up random variable framework. Characterisation of probabilistic

distributions for natural frequencies and buckling load of sandwich plates is ﬁrst

attempted in this study. Subsequently, relative individual eﬀect of diﬀerent stochas-

tic input parameters towards natural frequencies and buckling is discussed. The

uncertain geometric and material properties are considered along with the eﬀect of

the transverse normal deformation of the core. The in-plane displacement ﬁelds are

assumed as a combination of a linear zigzag model with diﬀerent slopes in each

layer and a cubically varying function over the entire thickness. The out-of-plane

displacement is assumed to be quadratic within the core and constant throughout

the faces. The sandwich plate model is implemented with a stochastic C

0

ﬁnite

element formulation developed for this purpose. The proposed computationally

eﬃcient MARS-based approach for uncertainty quantiﬁcation of sandwich com-

posite plates is validated with direct Monte Carlo simulation. The numerical results

are presented for both individual and combined layer-wise variation of the sto-

chastic input parameters.

4Journal of Sandwich Structures and Materials 0(00)

Theoretical formulation

Consider a laminated soft core sandwich plate (Figure 1) with thickness ‘t’ and

skew angle ‘’ (as shown in Figure 2), consisting of ‘n’ number of thin lamina, the

stress–strain relationship considering plane strain condition of an orthotropic layer

or lamina (say k-th layer) having any ﬁber orientation angle ‘’ with respect to

structural axes system (X–Y–Z) can be expressed as: fð~

!Þg ¼ fQkð~

!Þgf"ð~

!Þg

xxð~

!Þ

yyð~

!Þ

zzð~

!Þ

xyð~

!Þ

xzð~

!Þ

yzð~

!Þ

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

¼

Q11 ð~

!Þ

Q12ð~

!Þ

Q13ð~

!Þ

Q14ð~

!Þ00

Q21ð~

!Þ

Q22ð~

!Þ

Q23ð~

!Þ

Q24ð~

!Þ00

Q31ð~

!Þ

Q32ð~

!Þ

Q33ð~

!Þ

Q34ð~

!Þ00

Q41ð~

!Þ

Q42ð~

!Þ

Q43ð~

!Þ

Q44ð~

!Þ00

0000

Q55ð~

!Þ

Q56ð~

!Þ

0000

Q65ð~

!Þ

Q66ð~

!Þ

2

6

6

6

6

6

6

6

6

4

3

7

7

7

7

7

7

7

7

5

"xð~

!Þ

"yð~

!Þ

"zð~

!Þ

xy ð~

!Þ

xzð~

!Þ

yz ð~

!Þ

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

9

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

;

ð1Þ

where fð~

!Þg,f"ð~

!Þg and fQkð~

!Þg are random stress vector, random strain vector

and random transformed rigidity matrix of k-th lamina, respectively. Here the

symbol ~

!indicates the stochasticity of respective input parameters. Figure 2(a)

represents the in-plane displacement ﬁeld. The in-plane displacement parameters

are expressed as

Uxð~

!Þ¼uoþzxþX

nu1

i¼1

ðzzu

iÞð ~

!Þðzzu

iÞi

xu

Figure 1. Simply supported soft core sandwich plate.

Dey et al. 5

þX

nl1

j¼1

ðzzl

jÞð ~

!Þðzþzl

jiÞj

xl þxz2þxz3ð2Þ

Vxð~

!Þ¼voþzyþX

nu1

i¼1

ðzzu

iÞð ~

!Þðzzu

iÞi

yu

þX

nl1

j¼1

ðzzl

jÞð ~

!Þðzþzl

jiÞj

yl þyz2þyz3

ð3Þ

where uoand voare the in-plane displacements of any point in the X-axis and

Y0-axis on the mid-surface, xand yare the rotations of the normal to the

middle plane about the Y-axis and X-axis respectively, nuand nlare the number

of upper and lower layers, respectively while x,y,x, and yare the higher-order

unknown co-eﬃcient, i

xu,i

yu,j

xl, and j

yl are the slopes of i-th and j-th layer

corresponding to upper and lower layers, respectively, and ðzzu

iÞand

ðzþzl

jiÞare the unit step functions. The general lamination scheme, governing

equations and displacement conﬁguration are considered as per Pandit et al. [70].

The transverse displacements are assumed to vary quadratically through the core

thickness and constant over the face sheets and it may be expressed as

Wð~

!Þ¼ zðzþtlÞ

tuðtuþtlÞwuð~

!ÞþðtlþzÞðtuzÞ

tutl

woð~

!Þþ zðtuzÞ

tlðtuþtlÞwlð~

!Þðfor coreÞ

ð4Þ

Wð~

!Þ¼wuð~

!Þðfor upper face layersÞð5Þ

Wð~

!Þ¼wlð~

!Þðfor lower face layersÞð6Þ

Figure 2. (a) General lamination and displacement configuration. (b) Skewed laminate

geometry.

6Journal of Sandwich Structures and Materials 0(00)

where wuð~

!Þ,woð~

!Þand wlð~

!Þare the values of the transverse displacement at the

top layer, middle layer and bottom layer of the core, respectively. Utilising the

conditions of zero transverse shear stress at the top and bottom surfaces of the

plate and imposing the conditions of the transverse shear stress continuity at the

interfaces between the layers along with the conditions, u¼u

u

and v¼v

u

at the top

and u¼u

l

and v¼v

l

at the bottom of the plate, the generalised displacement vector

fgfor the present plate model can be expressed as

fg¼fuovowoxyuuvuwuulvlwlgTand yl¼y0

lcos ½ð~

!Þ ð7Þ

where ð~

!Þdenotes the random skew angle (Figure 2). For the skewed plates, the

elements on the inclined edges may not be parallel to the global axes (xgygzg).

To determine the elemental stiﬀness matrix at skew edges, it becomes necessary to

use edge displacements (uo,vo,wo,x,y,uu,vu,wu,ul,vland wl) in local coord-

inates (x0y0z0) (Figure 2). It is thus required to transform the element matrices

corresponding to global axes to local axes with respect to which the elemental

stiﬀness matrix can be conveniently determined. The relation between the global

and local degrees of freedom of a node on the skew edge can be obtained through

the simple transformation rules and the same can be expressed as

fLð~

!ÞgT¼½Tnð~

!Þ fgTð8Þ

A nine-noded isoparametric element is used for ﬁnite element formulation con-

sidering 11 degrees of freedom, where fLð~

!Þg and ½Tnð~

!Þ are the displacement vector

in the localised coordinate system and node transformation matrix, respectively.

Using the node transformation matrix, the elemental transformation matrix

½Teleð~

!Þ can be determined, which is used to transfer the elemental stiﬀness matrix

of the skew edge elements from the global axes to local axis. The node transformation

matrix ½Tnð~

!Þ and the elemental transformation matrix ½Teleð~

!Þ are expressed as

½Tnð~

!Þ ¼

mn000000000

nm000000000

00100000000

000mn000000

000nm000000

00000mn00 0 0

00000nm00 0 0

00000001000

00000000mn0

00000000nm0

00000000001

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

and ð9Þ

Dey et al. 7

½Teleð~

!Þ ¼

½Tnð~

!Þ 0000000000

Tnð~

!Þ 000000000

Tnð~

!Þ 00000000

Tnð~

!Þ 0000000

Tnð~

!Þ 000000

Tnð~

!Þ 00000

Sym:Tnð~

!Þ 0000

Tnð~

!Þ 000

Tnð~

!Þ 00

Tnð~

!Þ 0

Tnð~

!Þ

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

ð10Þ

8Journal of Sandwich Structures and Materials 0(00)

where m¼sin ð~

!Þand n¼cos ð~

!Þ, wherein ð~

!Þis the random ﬁbre orientation

angle.

Using linear strain–displacement relation, the strain ﬁeld "ð~

!Þ

may be

expressed in terms of unknowns (for the structural deformation) as

"ð~

!Þ

¼@Uð~

!Þ

@x

@Vð~

!Þ

@y

@Wð~

!Þ

@z

@Uð~

!Þ

@x

þ@Vð~

!Þ

@y

@Uð~

!Þ

@zþ@Wð~

!Þ

@x

@Vð~

!Þ

@zþ@Wð~

!Þ

@x

i:e:"ð~

!Þ

¼Hð~

!Þ½"ð~

!Þ

ð11Þ

where

"fg¼½u0v0w0xyuuvuwuulvlwlð@u0=@xÞð@u0=@yÞð@v0=@xÞð@v0=@yÞð@w0=@xÞð@w0=@yÞ

ð@x=@xÞð@x=@yÞð@y=@xÞð@y=@yÞð@uu=@xÞð@uu=@yÞð@vu=@xÞð@vu=@yÞð@wu=@xÞ

ð@wu=@yÞð@ul=@xÞð@ul=@yÞð@vl=@xÞð@vl=@yÞð@wl=@xÞð@wl=@yÞ

and the elements of [H] are functions of z and unit step functions. In the present

problem, a nine-node quadratic element with 11 ﬁeld variables (u

0,

v

0

,w

0,

x

,

y

,u

u

,

v

u

,w

u

,u

l

,v

l

and w

l

) per node is employed. Using ﬁnite element method, the general-

ised displacement vector fð~

!Þg at any point may be expressed as

ð~

!Þ

¼X

n

i¼1

Nið~

!Þið~

!Þð12Þ

where fg¼ u0v0w0xyuuvuwuulvlwl

Tas deﬁned earlier,

i

is the displacement

vector corresponding to node i,N

i

is the shape function associated with the node

iand nis the number of nodes per element. With the help of equation (12), the

strain vector {"} that appeared in equation (11) may be expressed in terms of

unknowns (for the structural deformation) as

"ð~

!Þ

¼½Bð~

!Þ ð~

!Þ

ð13Þ

where [B] is the strain–displacement matrix in the Cartesian coordinate system.

From Hamilton’s principle [71], the dynamic equilibrium equation for free

vibration analysis can be expressed as [72]

½Kð~

!Þ f

g¼ 2½Mð~

!Þ f

gð14Þ

where ½ð~

!Þ is the stochastic free vibration frequencies for diﬀerent modes and the

global mass matrix ½Mð~

!Þ may be formed by assembling a typical element mass

Dey et al. 9

matrix as shown below

½Mð~

!Þ ¼ X

nuþnl

i¼1ZZZ ið~

!Þ½NT½PT½N½Pdxdydz¼ZZ ½NT½Rð~

!Þ½Ndxdyð15Þ

where ið~

!Þis the random mass density of the i-th layer, matrix [P] is of order

311 and contains z terms and some constant quantities, matrix [N] is the shape

function matrix and the matrix ½Rð~

!Þ can be expressed as

½Rð~

!Þ ¼ X

nuþnl

i¼1

ið~

!Þ½PT½Pdzð16Þ

A numerical code is developed to implement the above-mentioned operations

involved in the proposed ﬁnite element model to determine the vibration response

of laminated skew composite sandwich plates. The skyline technique is used to

store the global stiﬀness matrix in a single array. Simultaneous iteration technique

of Corr and Jennings [73] is used in free vibration analysis. In the present study, a

nine-noded isoparametric element with 11 degrees of freedom at each node is con-

sidered for ﬁnite element formulation. The elemental potential energy can be

expressed as [17]

e¼UsUext ð17Þ

where Usand Uext are the strain energy and the energy due to external in-plane

load, respectively.

Ye¼1

2ZZ fg

TBð~

!Þ½

TDð~

!Þ½Bð~

!Þ½fgdxdy

1

2ZZ fg

TBð~

!Þ½

TGð~

!Þ½Bð~

!Þ½fgdxdy

¼1

2fg

TKeð~

!Þ½fg

1

2lfg

TKGð~

!Þ½fg

ð18Þ

where Keð~

!Þ½¼

RBð~

!Þ½

TDð~

!Þ½Bð~

!Þ½dxand KGð~

!Þ½¼

RBð~

!Þ½

TGð~

!Þ½Bð~

!Þ½dx

Here Bð~

!Þ½is the random strain–displacement matrix while Keð~

!Þ½and KGð~

!Þ½

are the stochastic elastic stiﬀness matrix and geometric stiﬀness matrix, respect-

ively. The equilibrium equation can be obtained by minimising O

e

as given in

equation (18) with respect to {}as

Keð~

!Þ½fg¼lð~

!ÞKGð~

!Þ½fg ð19Þ

10 Journal of Sandwich Structures and Materials 0(00)

where ð~

!Þis a stochastic buckling load factor. The skyline technique has been used

to store the global stiﬀness matrix in a single array and simultaneous iteration

technique is used for solving the stochastic buckling equation (19).

Formulation of multivariate adaptive regression

splines (MARS)

MARS [66] provides an eﬃcient mathematical relationship between input param-

eters and output feature of interest for a system under investigation based on few

algorithmically chosen samples. MARS is a nonparametric regression procedure

that makes no assumption about the underlying functional relationship between

the dependent and independent variables. MARS algorithm adaptively selects a set

of basis functions for approximating the response function through a forward and

backward iterative approach. The MARS model can be expressed as

Y¼X

n

k¼1

kHf

kðxiÞð20Þ

with Hf

kðx1,x2,x3...xnÞ¼1, for k¼1

where kand Hf

kðxiÞare the coeﬃcient of the expansion and the basis functions,

respectively. Thus, the ﬁrst term of equation (20) becomes

1

, which is basically an

intercept parameter. The basis function can be represented as

Hf

kðxiÞ¼Y

ik

i¼1zi,kðxjði,kÞti,kÞq

tr ð21Þ

where ikis the number of factors (interaction order) in the k-th basis function,

zi,k1, xjði,kÞis the j-th variable, 1 j(i,k)n, and ti,kis a knot location on each of

the corresponding variables. qis the order of splines. The approximation function

Yis composed of basis functions associated with ksub-regions. Each multivariate

spline basis function Hf

kðxiÞis the product of univariate spline basis functions zi,k,

which is either order one or cubic, depending on the degree of continuity of

the approximation. The notation ‘tr’ means the function is a truncated power

function.

½zi,kðxjði,kÞti,kÞk

tr ¼½zi,kðxjði,kÞti,kÞqfor ½zi,kðxjði,kÞti,kÞ 50ð22Þ

½zi,kðxjði,kÞti,kÞq

tr ¼0 Otherwise ð23Þ

Here each function is considered as piecewise linear with a trained knot ‘tr’ at

each xði,kÞ. By allowing the basis function to bend at the knots, MARS can model

functions that diﬀer in behaviour over the domain of each variable. This is applied

Dey et al. 11

to interaction terms as well. The interactions are no longer treated as global across

the entire range of predictors but between the sub-regions of every basis function

generated. Depending on ﬁtment, the maximum number of knots to be considered,

the minimum number of observations between knots, and the highest order of

interaction terms are determined. The screening of automated variables occur as

a result of using a modiﬁcation of the generalised cross-validation (GCV) model ﬁt

criterion, developed by Craven and Wahba [74]. MARS ﬁnds the location and

number of the needed spline basis functions in a forward or backward stepwise

fashion. It starts by over-ﬁtting a spline function through each knot, and then by

removing the knots that least contribute to the overall ﬁt of the model as deter-

mined by the modiﬁed GCV criterion, often completely removing the most insig-

niﬁcant variables. The equation depicting the lack-of-ﬁt (Lf) criterion used by

MARS is

LfðY~

kÞ¼Gcvð~

kÞ¼

1

nPn

i¼1½YiY~

kðxiÞ2

1~

cð~

kÞ

n

hi

2ð24Þ

where ~

cð~

kÞ¼cð~

kÞþM:~

k

Here ‘n’ denotes the number of sample observations, ~

cð~

kÞis the number of

linearly independent basis functions, ~

kis the number of knots selected in the for-

ward process, and ‘M’ is a cost for basis-function optimisation as well as a smooth-

ing parameter for the procedure. Larger values of ‘M’ result in fewer knots and

smoother function estimates. The best MARS approximation is the one with the

highest GCV value. Thus, MARS is also compared with parametric and nonpara-

metric approximation routines in terms of its accuracy, eﬃciency, robustness,

model transparency, and simplicity and it is found suitable methodologies because

it is more interpretable than most recursive partitioning, neural and adaptive stra-

tegies wherein it distinguishes well between actual and noise variables. Moreover,

the MARS are reported [75] to work satisfactorily in terms of computational cost

irrespective of dimension (low–medium–high) and noise.

Random input representation

The layer-wise random input parameters such as ply-orientation angle, skew angle,

thickness and material properties (e.g. mass density, elastic modulus, Poisson’s

ratio) of both core and face sheet are considered for sandwich plates. It is assumed

that the random uniform distribution of input parameters exists within a certain

band of tolerance with their deterministic values. The individual and combined

cases wherein the input variables considered in both soft core and each layer of

face sheet of sandwich are as follows

(Case-a) Variation of ply-orientation angle only:ð~

!Þ¼ ½f123...i...lgð ~

!Þ

12 Journal of Sandwich Structures and Materials 0(00)

(Case-b) Variation of thickness only:ttot ð~

!Þ¼ ½ftcg,ftfsð1Þtfsð2Þ...tsðlÞgð ~

!Þ

(Case-c) Variation of mass density only:ð~

!Þ¼ ½ffsð1Þcð2Þgð ~

!Þ

(Case-d) Variation of skew angle only:’ð~

!Þ

(Case-e) Variation of material properties

Pð~

!Þ¼½Exðfs,cÞ,Eyðfs,cÞ,Ezðfs,cÞ,G12ðfs,cÞ,G13ðfs,cÞ,G23ðfs,cÞ,12ðfs,cÞ,21ðfs,cÞ,

...13ðfs,cÞ,23ðfs,cÞ,32ðfs,cÞ,ðfs,cÞð ~

!Þ

(Case-f) Combined variation of ply orientation angle, thickness, mass density, skew

angle, elastic moduli, shear moduli, Poisson ratios and mass density for both

core and face sheet (total 63 numbers of random input variables):

Cð~

!Þ¼½ð~

!Þ,ttot ð~

!Þ,ð~

!Þ,’ð~

!Þ,Pð~

!Þ

where ,t,and ’are the ply orientation angle, thickness, mass density and skew

angle, respectively. The subscripts cand fs are used to indicate core and face sheet,

respectively. ‘l’ denotes the number of layer in the laminate, where i¼1, 2,...,l. Six

diﬀerent cases are considered for the analysis: layer-wise stochasticity in ply orien-

tation angle (ð~

!Þ), combined eﬀect for thickness of core and face sheet (ttot ð~

!Þ),

combined eﬀect for mass density of core and face sheet (ð~

!Þ), skew angle (’ð~

!Þ),

combined eﬀect for material properties of core and face sheet (Pð~

!Þ) and combined

variation of all parameters (Cð~

!Þ). In the present study, 5variation for ply

orientation angle and skew angle and 10% variation in material properties are

considered from their respective deterministic values unless mentioned otherwise

for some analyses. Figure 3 presents the ﬂowchart of proposed stochastic frequency

analysis using MARS model for the laminated soft core sandwich structure.

Results and discussion

The present study considers a sandwich composite plate with soft core (both upper

and lower as 0) and two facesheets with four-layered cross-ply (90/0/90/0)

laminate covering the core in both top and bottom side. A nine-noded isopara-

metric plate bending element is considered for ﬁnite element formulation. For the

analysis, the dimensions and boundary conditions considered for the sandwich

composite plate are as follows: length (L)¼1 m, width (b)¼0.5 m and thickness

(t)¼L/10, with simply supported boundary conditions (unless otherwise men-

tioned). The considered material properties of the sandwich plate are provided in

Table 1.

The present MARS model is employed to ﬁnd a predictive and representative

surrogate model relating each natural frequency to a number of stochastic input

variables. The MARS-based surrogate models are used to determine the ﬁrst three

natural frequencies corresponding to a set of input variables, instead of time-

consuming and expensive ﬁnite element analysis. The probability density function

Dey et al. 13

is plotted as the benchmark of bottom line results. The variation of geometric and

material properties is considered to ﬂuctuate within the range of lower and upper

limit (tolerance limit) as 10% with respective mean values while for ply orienta-

tion angles and skew angles as within 5ﬂuctuation (as per industry standard)

Start

Construct MARS models using the design points

Identification and statistical description of stochastic input parameters

Probabilistic characterization, statistical

analysis and interpretation of results

FE formulation to evaluate natural frequencies and buckling load

Selection of design points based on Latin hypercube sampling

FEM

Code

Input Output

MCS using MARS model

Mapping natural frequencies and buckling load using FEM

Figure 3. Flowchart of stochastic analysis using MARS model.

Table 1. Material properties for core and face sheet of sandwich plate.

Material properties Core Face sheet

E

1

0.5776 GPa 276 GPa

E

2

and E

3

0.5776 GPa 6.9 GPa

G

12

and G

13

0.1079 GPa 6.9 GPa

G

23

0.2221 GPa 6.9 GPa

n

12

and n

13

0.25 0.25

n

21

and n

31

0.00625 0.00625

n

23

and n

32

0.25 0.25

1000 kg/m

3

681.8 kg/m

3

14 Journal of Sandwich Structures and Materials 0(00)

with respect to their deterministic mean values. A layer-wise random variable

approach is employed for generating the set of random input variables which are

considered for surrogate-based numerical ﬁnite element iteration to obtain the

respective set of random output parameters accordingly. The transverse shear

stresses vanish only at the top and bottom surfaces of the laminate irrespective

of the considered boundary conditions, e.g. for clamped boundary condition, all

the kinematic variables vanish at clamped edges. Results are presented for stochas-

tic natural frequencies and stochastic buckling load for the sandwich plate.

Stochastic natural frequency analysis

Mesh convergence and validation of the ﬁnite element model for the sandwich plate

is conducted ﬁrst considering a deterministic analysis. The optimum mesh size is

ﬁnalised on the basis of a mesh convergence study as presented in Figure 4, wherein

a mesh size of (14 14) is found to be adequate. The non-dimensional natural

frequencies ($¼100 !Lﬃﬃﬃﬃﬃ

c

E2f

q, where cis the density of the core layer) for the

ﬁrst two modes based on the present model are obtained for various skew angles

and are tabulated in Table 2 along with the previous results obtained by Wang

et al. [76] and Kulkarni and Kapuria [77]. Table 3 presents the results for non-

dimensional natural frequencies of a four-layered clamped symmetric (0/90/90/

0) laminated composite plate obtained from present analysis for various aspect

ratios with respect to the previous analyses reported by Kulkarni and Kapuria [76]

and Khandelwal et al. [78]. The results corroborate good agreement of the deter-

ministic natural frequencies obtained using the present ﬁnite elememt model with

respect to previous works. The validation of the MARS model as a surrogate of the

actual ﬁnite elemet model is presented using scatter plots and probability density

function plots (refer to Figures 5 and 6). The low deviation of points from the

diagonal line in the scatter plot (Figure 5) corroborates the high accuracy of pre-

diction capability of the MARS model with respect to ﬁnite element model for all

the random input parameter sets (combined eﬀect of 63 numbers of random input

parameters). The probability density function plots presented in Figure 6 show a

negligible deviation between MARS model and original MCS model indicating

validity and high level of precision for the present surrogate-based approach fur-

ther. It is noteworthy that the proposed MARS-based approach requires 256 num-

bers of original ﬁnite element simulations for the layer-wise individual variation of

stochastic input parameters, while due to increment in number of input variables,

512 ﬁnite element simulations are found to be adequate for combined random

variation of input parameters. Here, although the same sample size as in direct

MCS (10,000 samples) is considered for characterising the probability distributions

of natural frequencies, the number of actual ﬁnite element simulations is much less

compared to direct MCS approach. Hence, the computational time and eﬀort

expressed in terms of expensive ﬁnite element simulations is reduced signiﬁcantly

compared to full scale direct MCS. This provides an eﬃcient aﬀordable way for

simulating the uncertainties in natural frequency. The optimum number of ﬁnite

Dey et al. 15

element simulations (i.e. the number of design points in Latin hypercube sampling)

required to construct the MARS models is decided based on a convergence study as

presented in Table 4.

In the present analysis, all the layer-wise individual cases of stochasticity are

studied as described in the Random input representation section. It is, however,

noticed that skew angle, mass density and transverse shear modulus are the three

most sensitive factors for ﬁrst three stochastic natural frequencies (refer to

Figure 4. Mesh convergence study of finite element analysis with different mesh sizes with

respect to fundamental and second natural frequencies of sandwich skewed plates

FNF: first natural frequency; SNF: second natural frequency.

Table 2. Non-dimensional natural frequencies of a four-layered (0/90/0/90) anti-symmetric

composite plate.

Skew angle Mode Present analysis Wang et al. [76]

Kulkarni and

Kapuria [77]

301 1.8889 1.9410 1.9209

2 3.4827 2.9063 3.5353

451 2.5806 2.6652 2.6391

2 3.7516 3.2716 4.1810

16 Journal of Sandwich Structures and Materials 0(00)

Figure 7) by analysing the relative coeﬃcient of variations [64]. Relative combined

eﬀect of the other parameters are (logitudinal and transverse elastic modulus, ply

orientation angle, thickness, longtudinal shear modulus and Poisson ratio) also

shown in Figure 7 for the ﬁrst three natural frequencies. As the eﬀect of other

parameters has neglegible sensitivity on stochastic natural frequencies,

Table 3. Non-dimensional natural frequencies of a four-layered clamped symmetric (0/90/

90/0) laminated composite plate.

Aspect ratio Mode Present analysis

Kulkarni and

Kapuria [77]

Khandelwal

et al. [78]

10 1 18.0843 18.2744 17.9550

2 28.9441 28.9047 28.9674

20 1 23.4534 24.1130 23.9339

2 37.0587 36.7473 37.0614

Figure 5. Scatter plot of finite element (FE) model with respect to MARS model for

(a) fundamental natural frequency (FNF), (b) second natural frequency (SNF) and (c) third

natural frequency (TNF) of simply supported sandwich skewed plates considering combined

variation (total 63 numbers of random input variables) for ð~

!Þ¼45.

Dey et al. 17

representative results are furnished for stochastic eﬀect of two most eﬀective par-

ameters (skew angle and mass density) for analysis of individual cases.

Probability distributions for ﬁrst three stochastic natural frequencies of a simply

supported composite sandwich plate due to only variation in skew angles are fur-

nished in Figure 8. As the skew angle increases, the mean of stochastic natural

frequencies is also found to increase, while probability distributions corresponding

to diﬀerent skew angles vary consideranly. Figure 9 presents the stochastic ﬁrst

three natural frequencies of a simply supported sandwich composite skewed plate

(for skew angle ð~

!Þ¼45) due to only variation of mass density (layer-wise) with

diﬀerent degree of stochasticity. As the percentage of stochasticity in mass density

increases, the response bounds are found to increase accordingly, while the mean

does not change for diﬀerent percentage of variation in mass density. The eﬀect of

combined stochasticity in all input parameters (referred as Cð~

!Þin the Random

input representation section) is also analysed for diﬀerent skew angles in addition

to individual eﬀect of the input parameters for stochastic natural frequnencies of

sandwich plates. In Figure 10, the stochastic ﬁrst three natural frequencies are

presented for simply supported sandwich composite plates with diﬀerent skew

angles considering combined variation of input parameters Cð~

!Þ(total 63 numbers

random input variables), wherein a general trend is noticed that the mean and

response bounds increase with the increase in skew angle. Response bounds of

the ﬁrst three natural frequencies due to combined variation are noticed to be

Figure 6. Probability density function for MCS as well as MARS model for the first three nat-

ural frequencies of simply supported compsoite sandwich skewed plates considering combined

variation (a total of 63 random input variables) for ð~

!Þ¼45.

18 Journal of Sandwich Structures and Materials 0(00)

Table 4. Convergence study of first three modes due to individual and combined variation of inputs for simply supported sandwich plates.

Individual

variation Value

f

1

f

2

f

3

MCS

(10,000)

MARS (Sample run)

MCS

(10,000)

MARS (Sample run)

MCS

(10,000)

MARS (Sample run)

64 128 256 64 128 256 64 128 256

ð~

!ÞMax 39.6407 39.6614 39.6527 39.6493 54.6218 54.7123 54.6531 54.6338 67.8206 67.9845 67.9214 67.8584

Min 38.9549 38.9420 38.9456 38.9485 53.8200 53.8064 53.8119 53.8168 67.4350 67.4220 67.4307 67.4321

Mean 39.3039 39.3115 39.3101 39.3092 54.2480 54.2311 54.2342 54.2381 67.6402 67.6998 67.6831 67.6587

SD 0.1186 0.1194 0.1195 0.1196 0.1458 0.1473 0.1468 0.1461 0.0668 0.0735 0.0698 0.0681

tsð~

!ÞMax 39.6858 39.7564 39.7313 39.7257 54.7669 54.8214 54.7917 54.7764 68.4852 68.6154 68.5978 68.5432

Min 38.8943 38.8021 38.8264 38.8409 53.7134 53.3116 53.3982 53.4918 66.8631 66.8167 66.8227 66.8497

Mean 39.3044 39.3164 39.3114 39.3081 54.2644 54.4951 54.3718 54.3083 67.6574 67.6831 67.7952 67.7098

SD 0.1159 0.1173 0.1176 0.1179 0.1527 0.1584 0.1562 0.1552 0.2391 0.2487 0.2423 0.2416

ð~

!ÞMax 40.3206 41.1121 40.9821 40.5127 55.6577 56.1064 55.9561 55.7473 69.3890 69.9983 69.8134 69.4835

Min 38.4011 38.3942 38.3964 38.9873 53.0081 52.6876 52.7942 52.9264 66.0856 65.8421 65.9226 65.9928

Mean 39.3299 39.6734 39.5154 39.4212 54.2901 54.5876 54.5083 54.4221 67.6840 67.8674 67.8050 67.7213

SD 0.4905 0.6154 0.5584 0.5129 0.6772 0.7954 0.7054 0.6997 0.8444 0.9533 0.8624 0.8517

’ð~

!ÞMax 41.7226 42.2134 42.0219 41.8687 56.7749 57.1259 56.9641 56.7963 70.1767 70.9897 70.6245 70.3516

Min 37.2399 36.8276 36.9893 37.0867 52.1238 51.7383 51.9767 52.1013 65.5452 64.9984 65.1137 65.4194

Mean 39.3663 39.6124 39.5483 39.4468 54.3251 54.6682 54.5437 54.4198 67.7213 67.9457 67.8438 67.7438

SD 1.2740 1.3130 1.3030 1.2991 1.3218 1.3356 1.3286 1.3264 1.3164 1.3552 1.3487 1.3258

Combined

variation

MCS

(10,000)

MARS (Sample run)

MCS

(10,000)

MARS (Sample run)

MCS

(10,000)

MARS (Sample run)

128 256 512 128 256 512 128 256 512

Cð~

!ÞMax 46.59067 47.00219 46.9832 46.9265 62.80511 63.51472 63.2164 63.05806 77.35152 78.02273 77.9516 77.8462

Min 33.0163 33.50135 33.4134 33.3372 46.86064 47.0558 46.9671 46.9493 59.25218 59.32715 59.2971 59.2832

Mean 39.43564 39.39828 39.4002 39.4138 54.4066 54.36105 54.3883 54.3921 67.8260 67.77762 67.7884 67.7935

SD 2.5081 2.4872 2.4889 2.4992 2.8085 2.7907 2.7921 2.7944 3.0515 3.0439 3.0476 3.0497

Dey et al. 19

higher than individual variation of input parameters in all cases. The stochastic ﬁrst

three natural frequencies of sandwich composite skewed plates with diﬀerent

boundary conditions (C-Clamped, S-Simply supported, F-Free) are shown in

Figure 11 considering combined variation of input parameters to investigate the

Figure 8. Stochastic first three natural frequencies (rad/s) of simply supported composite

sandwich plates due to only variation of skew angles.

Figure 7. Sensitivity for first three natural modes for simply supported sandwich plates.

20 Journal of Sandwich Structures and Materials 0(00)

Figure 9. Stochastic first three natural frequencies (rad/s) of simply supported sandwich

composite skewed plates for ð~

!Þ¼45due to only variation of mass density with different

degree of stochasticity.

Figure 10. Stochastic first three natural frequencies (rad/s) of simply supported sandwich

composite plates for different skew angles considering combined variation of input parameters

(a total of 63 random input variables).

Dey et al. 21

eﬀect of boundary conditions. The probability distributions are found to vary

signiﬁcantly depending on the considered boundary condition. Both mean and

standard deviation of CCCC boundary condition are found to be highest for

combined variation of all input parameters.

Stochastic buckling load analysis

Mesh convergence and validation of the ﬁnite elemnt model for deterministic buck-

ling load is presented in Figure 12. The convegence study on ﬁnite element mesh

size is conducted to obtain the optimum mesh size. In the present study, the results

of buckling load corresponding to diﬀerent mesh sizes are found to be convergent

as depicted in Figure 12, wherein the mesh convergence study is carried out to

compare the critical bi-axial buckling load for laminated sandwich plates with

diﬀerent boundary conditions such as CCCC, SCSC and SSSS (where S – simply

supported, C – clamped, indicating boundary condition of four sides). As the

computational iteration time increases with the increase of mesh size, a (14 14)

optimal mesh size is considered in the present study. The present buckling load are

also validated with the results obtained by Liew and Huang [79]. The results cor-

roborate good agreement of the buckling load obtained using the present ﬁnite

elememt model with respect to previous works of Liew and Huang irrespective of

imposed bounary conditions. Further, the MARS model that is employed to

achieve computational eﬃciency is validated with traditional Monte Carlo simu-

lation (MCS). Representative results are furnished for combined variation of all

Figure 11. Stochastic first three natural frequencies (rad/s) of sandwich composite skewed

plates for ð~

!Þ¼45with different boundary conditions considering combined variation of

input parameters (a total of 63 random input variables) (C: clamped; S: simply supported;

F: free).

22 Journal of Sandwich Structures and Materials 0(00)

input parameters (512 samples) using probability density function plots and scatter

plot as shown in Figure 13. The ﬁgures indicate high degree of precision and

accuracy of the present MARS model with respect to original ﬁnite element

model. The results for buckling load are presented hereafter (Figure 13 to 19) as

a ratio of stochastic buckling load and deterministic buckling load to provide a

clear and direct interpretation for stochasticity in diﬀerent input parameters.

The eﬀects of variation of core thickness and face sheet thickness on stochastic

normalised buckling load of sandwich plates are shown in Figures 14 and 15,

respectively. It is found that as the percentage of variation of both core and face

sheet thickness increases, the response bound of stochastic buckling load also

increases, while the mean does not vary. The sparsity of stochastic normalised

buckling load due to variation of core thickness is observed to be signiﬁcantly

higher than that of the same due to variation of face sheet thickness. The eﬀect

of variation of all core material properties on stochastic buckling load of sandwich

plates is furnished in Figure 16, while Figure 17 presents the eﬀect of ply

Figure 12. Mesh convergence study and validation for comparison of non-dimensionalised

critical bi-axial buckling load [

¼ðl2Þ=ðh2ETf Þwhere ,l,hand ETf are the buckling load

factor, depth of the plate and transverse modulus of elasticity of face layer, respectively] for

laminated sandwich plates with different boundary conditions.

Figure 13. Probability density function and Scatter plot for buckling load of sandwich plates

considering combined variation of all input parameters (Cð~

!Þ).

Dey et al. 23

Figure 14. Effect of variation of core thickness on normalised buckling load of sandwich

plates with SCSC (S – simply supported, C – clamped).

Figure 15. Effect of variation of face sheet thickness on normalised buckling load of sandwich

plates with SCSC (S – simply supported, C – clamped).

Figure 16. Effect of variation in material properties of core on normalised buckling load of

sandwich plates with SCSC (S – simply supported, C – clamped).

24 Journal of Sandwich Structures and Materials 0(00)

orientation angle of face sheet on stochastic normalised buckling load of sandwich

plates. Besides variation of core and face sheet thickness (Figures 14 and 15), the

mean value for stochastic buckling load remains unaltered with diﬀerent degrees of

stochasticity in core material properties, while the standard deviation increases

with increase in degree of stochasticity. In contrast, both mean and standard devi-

ation of stochastic buckling load are found to increase with increasing degree of

stochasticity in ply orientation angle. The variation in buckling load due to sto-

chasticity of core material properties (Figure 16) is found to be higher than the

other three individual cases (Figures 14,15 and 17). However, the maximum vari-

ation in normalised buckling load is observed in case of combined stochasticity of

core and face sheet thickness, ply-orientation angle of face sheet and material

properties (Figure 18). The eﬀect of diﬀerent boundary conditions (CCCC,

Figure 17. Effect of variation of ply orientation angle of face sheet on normalised buckling

load of sandwich plates with SCSC (S – simply supported, C – clamped).

Figure 18. Effect of combined variation of stochastic input parameters (core and face sheet

thickness, ply-orientation angle of face sheet and material properties) on normalised buckling

load of sandwich plates.

Dey et al. 25

CFCF, SCSC and SSSS; where S – simply supported, C – clamped and F – ﬁxed

end condition) on normalised stochastic buckling load of sandwich plates is pre-

sented in Figure 19. Even though the response bounds for diﬀerent boundary

conditions for normalised buckling load does not vary, the probability distribu-

tions for buckling loads in actual values will vary signiﬁcantly depending on their

deterministic values. The coeﬃcient of variation corresponding to diﬀerent degrees

of stochasticity for diﬀerent cases considered in this study is plotted in Figure 20.

From the ﬁgure it is evident that the eﬀect on buckling load due to stochastic

variation of diﬀerent input parameters in a decreasing order is: combined variation

of all stochastic input parameters, core material properties, core thickness, ply

orientation angle and face sheet thickness. The slope of the curves for diﬀerent

parameters corresponding to diﬀerent degrees of stochasticity provides a clear

interpretation about their relative sensitivity towards buckling load.

Figure 19. Effect of boundary condition on normalised buckling load of sandwich plates.

Figure 20. Coefficient of variation on buckling load with respect to degree of stochasticity of

input parameters for simply supported sandwich plates.

26 Journal of Sandwich Structures and Materials 0(00)

Conclusions

This article illustrates the layer-wise propagation of uncertainties in sandwich

skewed plates in an eﬃcient surrogate based bottom-up framework. The probabil-

ity distributions of ﬁrst three natural frequencies and buckling load are analysed

considering both individual and combined stochasticity in input parameters.

A multivariate adaptive regression splines (MARS)-based approach is developed

in conjunction with ﬁnite element modelling to map the variation of ﬁrst three

natural frequencies and buckling load caused due to uncertain input parameters,

wherein it is found that the number of ﬁnite element simulations is exorbitently

reduced compared to direct Monte Carlo simulation without compromising the

accuracy of results. The computational expense is reduced by (1/78) times (indi-

vidual eﬀect of stochasticity) and (1/39) times (combined eﬀect of stochasticity) of

direct Monte Carlo simulation. The skew angle is found to be most sensitive to the

frequencies corresponding to the ﬁrst three modes. The mass density and transverse

shear modulus are other two eﬀective factors for the ﬁrst three natural frequencies

among the considered stochastic input parameters, respectively. The combined

eﬀect of the material properties of soft-core has the most sensitivity for buckling

load, followed by core thickness, ply orientation angle and face sheet thickness,

respectively.

Novelty of the present study includes probabilistic characterisation of natural

frequencies and buckling load for laminated sandwich plates following an eﬃ-

cient MARS-based uncertainty propagation algorithm. The numerical results

presented in this article shows that stochasticity in diﬀerent material and geo-

metric properties of laminated sandwich plates has considerable inﬂuence on the

dynamics and stability of the structure. Thus, it is of prime importance to incorp-

orate the eﬀect of stochasticity in subsequent analyses, design and control of such

structures. The proposed MARS-based uncertainty quantiﬁcation algorithm can

be extended further to explore other stochastic systems in future course of

research.

Declaration of conflicting interests

The author(s) declared no potential conﬂicts of interest with respect to the research, author-

ship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following ﬁnancial support for the research, author-

ship, and/or publication of this article: TM acknowledges the ﬁnancial support from

Swansea University through the award of Zienkiewicz Scholarship during the period of

this work. SA acknowledges the ﬁnancial support from The Royal Society of London

through the Wolfson Research Merit award.

References

1. Noor AK, Burton WS and Bert CW. Computational models for sandwich panels and

shells. Appl Mech Rev 1996; 49: 155–199.

Dey et al. 27

2. Bert CW. Literature review: research on dynamic behavior of composite and sandwich

plates – V: Part II. Shock Vib Digest 1991; 23: 9–21.

3. Mallikarjuna and Kant T. A critical review and some results of recently developed

refined theories of fiber-reinforced laminated composites and sandwiches. Compos

Struct 1993; 23: 293–312.

4. Altenbach H. Theories for laminated and sandwich plates. Mech Compos Mater 1998;

34: 243–252.

5. Alibeigloo A and Alizadeh M. Static and free vibration analyses of functionally graded

sandwich plates using state space differential quadrature method. Eur J Mech A/Solid

2015; 54: 252–266.

6. Carrera E and Brischetto S. A survey with numerical assessment of classical and refined

theories for the analysis of sandwich plates. Appl Mech Rev 2009; 62: 1–17.

7. Mukhopadhyay T, Dey TK, Chowdhury R, et al. Optimum design of FRP bridge deck:

an efficient RS-HDMR based approach. Struct Multidiscipl Optimiz 2015; 52: 459–477.

8. Dey TK, Mukhopadhyay T, Chakrabarti A, et al. Efficient lightweight design of FRP

bridge deck. Proc Inst Civil Eng Struct Build 2015; 168: 697–707.

9. Singh A, Panda S and Chakraborty D. A design of laminated composite plates using graded

orthotropic fiber-reinforced composite plies. Compos Part B: Eng 2015; 79: 476–493.

10. Mukhopadhyay T, Dey TK, Dey S, et al. Optimization of fiber reinforced polymer web

core bridge deck – A hybrid approach. Structural Engineering International 2015; 25:

173–183.

11. Mukhopadhyay T. A multivariate adaptive regression splines based damage identifica-

tion methodology for web core composite bridges including the effect of noise. Journal

of Sandwich Structures & Materials (In Press). DOI: 10.1177/1099636216682533.

12. Mukhopadhyay T and Adhikari S. Free vibration analysis of sandwich panels with

randomly irregular honeycomb core. J Eng Mech 142: 06016008. DOI: 10.1061/

(ASCE)EM.1943-7889.0001153.

13. Garg AK, Khare RK and Kant T. Free vibration of skew fiber-reinforced composite

and sandwich laminates using a shear deformable finite element model. J of Sandw

Struct and Mater 2006; 8: 33–53.

14. Chakrabarti A and Sheikh AH. Vibration of laminated faced sandwich plate by a new

refined element. ASCE J Aerosp Eng 2004; 17: 123–134.

15. Shariyat M. A generalized global–local high-order theory for bending and vibration

analyses of sandwich plates subjected to thermo-mechanical loads. Int J Mech Sci

2010; 52: 495–514.

16. Elmalich D and Rabinovitch O. A high-order finite element for dynamic analysis of

soft-core sandwich plates. J Sandwich Struct Mater 2012; 14: 525–555.

17. Chalak HD, Chakrabarti A, Sheikh AH, et al. Stability analysis of laminated soft core

sandwichplates using higher order zig-zag plate theory. Mech Adv Mater Struct 2015; 22:

897–907.

18. Singh SK and Chakrabarti A. Static vibration and buckling analysis of skew composite

and sandwich plates under thermo mechanical loading. Int J Appl Mech Eng 2013; 18:

887–898.

19. Aguib S, Nour A, Zahloul H, et al. Dynamic behavior analysis of a magnetorheological

elastomer sandwich plate. Int J Mech Sci 2014; 87: 118–136.

20. Zhen W, Wanji C and Xiaohui R. An accurate higher-order theory and C0 finite elem-

ent for free vibration analysis of laminated composite and sandwich plates. Compos

Struct 2010; 92: 1299–1307.

28 Journal of Sandwich Structures and Materials 0(00)

21. Roque CMC, Ferreira AJM and Jorge RMN. Free vibration analysis of composite and

sandwich plates by a trigonometric layerwise deformation theory and radial basis func-

tions. J Sandwich Struct Mater 2006; 8: 497–515.

22. Ferreira AJM, Roque CMC, Jorge RMN, et al. Static deformations and vibration

analysis of composite and sandwich plates using a layerwise theory and multiquadrics

discretizations. Eng Anal Bound Elem 2005; 29: 1104–1114.

23. Ferreira AJM, Fasshauer GE, Batra RC, et al. Static deformations and vibration ana-

lysis of composite and sandwich plates using a layerwise theory and RBF-PS discret-

izations with optimal shape parameter. Compos Struct 2008; 86: 328–343.

24. Ferreira AJM, Roque CMC, Carrera E, et al. Two higher order Zig-Zag theories for the

accurate analysis of bending, vibration and buckling response of laminated plates by

radial basis functions collocation and a unified formulation. J Compos Mater 2011; 45:

2523–2536.

25. Rodrigues JD, Roque CMC, Ferreira AJM, et al. Radial basis functions-differential

quadrature collocation and a unified formulation for bending, vibration and buckling

analysis of laminated plates, according to Murakami’s Zig-Zag theory. Comput Struct

2012; 90–91: 107–115.

26. Rodrigues JD, Roque CMC, Ferreira AJM, et al. Radial basis functions–finite differ-

ences collocation and a Unified Formulation for bending, vibration and buckling ana-

lysis of laminated plates, according to Murakami’s zig-zag theory. Compos Struct 2011;

93: 1613–1620.

27. Bui TQ, Nguyen MN and Zhang C. An efficient mesh-free method for vibration analysis

of laminated composite plates. Comput Mech 2011; 48: 175–193.

28. Yang C, Jin G, Liu Z, et al. Vibration and damping analysis of thick sandwich cylin-

drical shells with a viscoelastic core under arbitrary boundary conditions. Int J Mech Sci

2015; 92: 162–177.

29. Narita Y. Closure to Discussion of combinations for the free-vibration behavior of

anisotropic rectangular plates under general edge conditions. J Appl Mech 2001; 68: 685.

30. Carrera E, Fazzolari FA and Demasi L. Vibration analysis of anisotropic simply sup-

ported plates by using variable kinematic and Rayleigh–Ritz method. J Vib Acoust 2011;

133: 1–16.

31. Watkins RJ and Barton O. Characterizing the vibration of an elastically point supported

rectangular plate using eigensensitivity analysis. Thin Wall Struct 2009; 48: 327–333.

32. Iurlaro L, Gherlone M, Sciuva MD, et al. Assessment of the refined zigzag theory for

bending, vibration, and buckling of sandwich plates: a comparative study of different

theories. Compos Struct 2013; 106: 777–792.

33. Kumar A, Chakrabarti A, Bhargava P, et al. Probabilistic failure analysis of laminated

sandwich shells based on higher order zigzag theory. J Sandwich Struct Mater 2015; 17:

546–561.

34. Taibi FZ, Benyoucef S, Tounsi A, et al. A simple shear deformation theory for thermo-

mechanical behaviour of functionally graded sandwich plates on elastic foundations.

J Sandwich Struct Mater 2015; 17: 99–129.

35. Gulshan Taj MNA, Chakrabarti A and Talha M. Bending analysis of functionally

graded skew sandwich plates with through-the thickness displacement variations.

J Sandwich Struct Mater 2014; 16: 210–248.

36. Liu J, Cheng YS, Li RF, et al. A semi-analytical method for bending, buckling, and free

vibration analyses of sandwich panels with square-honeycomb cores. Int J Struct Stab

Dynam 2010; 10: 127–151.

Dey et al. 29

37. Zhang QJ and Sainsbury MG. The Galerkin element method applied to the vibration of

rectangular damped sandwich plates. Comput Struct 2000; 74: 717–730.

38. Makhecha DP, Ganapathi M and Patel BP. Vibration and damping analysis of lami-

nated/sandwich composite plates using higher-order theory. J Reinf Plast Compos 2002;

21: 559–575.

39. Xiang Y and Wei GW. Exact solutions for buckling and vibration of stepped rectangu-

lar Mindlin plates. Int J. Solids Struct 2004; 41: 279–294.

40. Aydogdu M and Ece MC. Buckling and vibration of non-ideal simply supported rect-

angular isotropic plates. Mech Res Commun 2006; 33: 532–540.

41. Dehkordi MB, Khalili SMR and Carrera E. Non-linear transient dynamic analysis of

sandwich plate with composite face-sheets embedded with shape memory alloy wires

and flexible core-based on the mixed LW (layer-wise)/ESL (equivalent single layer)

models. Compos Part B: Eng 2016; 87: 59–74.

42. Xiang S, Jin YX, Bi ZY, et al. A n-th order shear deformation theory for free vibration

of functionally graded and composite sandwich plates. Compos Struct 2011; 93:

2826–2832.

43. Douville MA and Le Grognec P. Exact analytical solutions for the local and global

buckling of sandwich beam-columns under various loadings. Int J Solids Struct 2013; 50:

2597–2609.

44. Mukhopadhyay T and Adhikari S. Equivalent in-plane elastic properties of irregular

honeycombs: an analytical approach. Int J Solids Struct 2016; 91: 169–184.

45. Mukhopadhyay T and Adhikari S. Effective in-plane elastic properties of auxetic honey-

combs with spatial irregularity. Mech Mater 2016; 95: 204–222.

46. Mukhopadhyay T and Adhikari S. Stochastic mechanics of metamaterials. Composite

Structures 2017; 162: 85–97.

47. Ying ZG, Ni YQ and Ye SQ. Stochastic micro-vibration suppression of a sandwich

plate using a magneto-rheological visco-elastomer core. Smart Mater Struct 2013; 23:

025019.

48. Batou A and Soize C. Stochastic modeling and identification of an uncertain compu-

tational dynamical model with random fields properties and model uncertainties.

Archive Appl Mech 2013; 83: 831–848.

49. Mahata A, Mukhopadhyay T and Adhikari S. A polynomial chaos expansion based

molecular dynamics study for probabilistic strength analysis of nano-twinned copper.

Mater Res Express 2016; 3: 036501.

50. Mukhopadhyay T, Mahata A, Dey S, et al. Probabilistic analysis and design of HCP

nanowires: an efficient surrogate based molecular dynamics simulation approach.

Journal of Materials Science & Technology 2016; 32: 1345–1351.

51. Mukhopadhyay T, Chakraborty S, Dey S, et al. A critical assessment of Kriging model

variants for high-fidelity uncertainty quantification in dynamics of composite shells.

Archive Computat Meth Eng, DOI: 10.1007/s11831-016-9178-z.

52. Dey S, Mukhopadhyay T and Adhikari S. Stochastic free vibration analysis of angle-ply

composite plates – A RS-HDMR approach. Compos Struct 2015; 122: 526–536.

53. Chakraborty D. Artificial neural network based delamination prediction in laminated

composites. Mater Des 2005; 26: 1–7.

54. Dey S, Mukhopadhyay T, Spickenheuer A, et al. Bottom up surrogate based approach

for stochastic frequency response analysis of laminated composite plates. Compos Struct

2016; 140: 712–727.

30 Journal of Sandwich Structures and Materials 0(00)

55. Dey S, Mukhopadhyay T, Sahu SK, et al. Thermal uncertainty quantification in fre-

quency responses of laminated composite plates. Compos Part B: Eng 2015; 80: 186–197.

56. Mukhopadhyay T, Naskar S, Dey S, et al. On quantifying the effect of noise in surro-

gate based stochastic free vibration analysis of laminated composite shallow shells.

Compos Struct 2016; 140: 798–805.

57. Dey S, Mukhopadhyay T, Haddad Khodaparast H, et al. Rotational and ply-level

uncertainty in response of composite shallow conical shells. Compos Struct 2015; 131:

594–605.

58. Naskar S, Mukhopadhyay T, Sriramula S, et al. Stochastic natural frequency analysis of

damaged thin-walled laminated composite beams with uncertainty in micromechanical

properties. Composite Structures 2017; 160: 312–334.

59. Dey S, Mukhopadhyay T and Adhikari S. Metamodel based high-fidelity stochastic

analysis of composite laminates: A concise review with critical comparative assessment.

Composite Structures 2017. DOI: http://dx.doi.org/10.1016/j.compstruct.2017.01.061.

60. Dey S, Mukhopadhyay T, Sahu SK, et al. Effect of cutout on stochastic natural fre-

quency of composite curved panels. Composites Part B: Engineering 2016; 105: 188–202.

61. Dey S, Mukhopadhyay T, Spickenheuer A, et al. Uncertainty quantification in natural

frequency of composite plates – An Artificial neural network based approach. Advanced

Composites Letters 2016; 25: 43–48.

62. Dey S, Mukhopadhyay T, Khodaparast HH, et al. Fuzzy uncertainty propagation in

composites using Gram-Schmidt polynomial chaos expansion. Applied Mathematical

Modelling 2016; 40: 4412–4428.

63. Dey S, Naskar S, Mukhopadhyay T, et al. Uncertain natural frequency analysis of

composite plates including effect of noise – A polynomial neural network approach.

Composite Structures 2016; 143: 130–142.

64. Dey S, Mukhopadhyay T, Khodaparast HH, et al. Stochastic natural frequency of

composite conical shells. Acta Mechanica 2015; 226: 2537–2553.

65. Dey S, Mukhopadhyay T and Adhikari S. Stochastic free vibration analyses of compo-

site doubly curved shells – A Kriging model approach. Composites Part B: Engineering

2015; 70: 99–112.

66. Sudjianto A, Juneja L, Agrawal A, et al. Computer aided reliability and robustness

assessment. Int J Reliab Qual Safe 1998; 5: 181–193.

67. Wang X, Liu Y and Antonsson EK. Fitting functions to data in high dimensional design

spaces. Advances in design automation (held in Las Vegas, NV), Paper No. DETC99/

DAC-8622. ASME, 1999.

68. Friedman JH. Multivariate adaptive regression splines. Ann Stat 1991; 19: 1–67.

69. Jin R, Chenand W and Simpson TW. Comparative studies of metamodelling techniques

under multiple modelling criteria. Struct Multidisc Optim 2001; 23: 1.

70. Pandit MK, Sheikh AH and Singh BN. An improved higher order zigzag theory for the

static analysis of laminated sandwich plate with soft core. Finite Element Anal Des 2008;

44: 602–610.

71. Meirovitch L. Dynamics and control of structures. New York, NY: John Wiley & Sons,

1992.

72. Chakrabarti A and Sheikh A. Vibration of laminate-faced sandwich plate by a new

refined element. J Aerosp Eng 2004; 17: 123–134.

73. Corr RB and Jennings A. A simultaneous iteration algorithm for symmetric eigenvalue

problems. Int J Numer Meth Eng 1976; 10: 647–663.

Dey et al. 31

74. Craven P and Wahba G. Smoothing noisy data with spline functions. Numer Math 1979;

31: 377–403.

75. Crino S and Brown DE. Global optimization with multivariate adaptive regression

splines. IEEE Transact Syst Man Cybernet Part B: Cybernet 2007; 37: 333–340.

76. Wang CM, Ang KK and Yang L. Free vibration of skew sandwich plates with lami-

nated facings. J Sound Vibrat 2000; 235: 317–340.

77. Kulkarni SD and Kapuria S. Free vibration analysis of composite and sandwich plates

using an improved discrete Kirchoff quadrilateral element based on third order zigzag

theory. Comput Mech 2008; 42: 803–824.

78. Khandelwal RP, Chakrabarti A and Bhargava P. Vibration and buckling analysis of

laminated sandwich plate having soft core. Int J Struct Stabil Dynam 2013; 13: 20–31.

79. Liew KM and Huang YQ. Bending and buckling of thick symmetrical rectangular

laminates using the moving least squares differential quadrature method. Int J Mech

Sci 2003; 45: 95–114.

32 Journal of Sandwich Structures and Materials 0(00)