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)
