forced nonlinear vibration of laminated composite plates with random material properties

Composite Structures 70 (2005) 334–342

www.elsevier.com/locate/compstruct

Forced nonlinear vibration of laminated composite plateswith random material properties

Amit Kumar Onkar, D. Yadav *

Department of Aerospace Engineering, Indian Institute of Technology, Kanpur 208016, India

Available online 7 October 2004

Abstract

Components in aerospace vehicles, automobiles, civil and mechanical structures that have requirements of high strength to

weight ratio are made of filamentary composite laminae. These structures are usually subjected to stochastic loads during their per-

formance. This paper deals with the nonlinear random vibration of a simply supported cross-ply laminated composite plate. The

material properties and the external excitations are treated as random process. Using basic analytical techniques, this investigation

aims at improving the accuracy of response evaluation of such plates by employing accurate models for the material properties and

external loading. The formulation uses Kirchoff–Love plate theory and Von-Karman nonlinear strain displacement relationship.

The system equation is obtained with Hamilton�s principle. Closed form solution for the variance of the response is obtained by

using first order perturbation technique. The results attempt to bring out the characteristics of the random response and its sensi-

tivity to the lamina thickness and plate aspect ratio.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Composite plates; Random material properties; Random excitation and nonlinear response

1. Introduction

The random response of a plate is an important prob-

lem for aerospace, mechanical and civil structures. Lam-

inated plates are very common components in aerospace

vehicles. Wings, skins, control surfaces and access pan-

els are just a few examples. Other vehicles like automo-

biles, ships, trains, etc. also employ laminated panels in

different locations. During typical operating conditionsstructures are constantly being subjected to random

loads like engine noise, shock waves, turbulence, gusts,

track inputs, thermal loads, wind and acoustic loads.

The response of thin plate subjected to high level of loa-

dings is nonlinear. Problems in this area can be modeled

with inherent randomness in the structural characteris-

tics, like material properties, geometry, etc., and ran-

0263-8223/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2004.08.037

* Corresponding author. Tel.: +91 11 0512 259 7951; fax: +91 11

0512 259 7561.

E-mail address: [email protected] (D. Yadav).

domness in the environment to which the structure isexposed, like loads, support conditions, etc.

Several authors have discussed the problem of dy-

namic response of composite structures to different load-

ing. However, limited published literature is available

for analysis of structure with random material proper-

ties and loading. Free vibration response has been ob-

tained by Vaicatis [1] for beams with mass and flexural

rigidity as random variables. Chen and Soroka [2] havestudied the response of a multi degree of freedom system

with random material properties to deterministic excita-

tions. The system equations have been solved by pertur-

bation technique. The second order statistics of the

system response have been investigated with variation

in the system property statistics. Shinozuka and Astill

[3] have employed a numerical technique to obtain sta-

tistical properties of eigenvalues of spring supported col-umns with the spring support, axial loading, material

and geometric properties as random. The method has

been used to investigate the accuracy of the perturbation

mailto:[email protected]

A.K. Onkar, D. Yadav / Composite Structures 70 (2005) 334–342 335

approach for evaluation of vibration and buckling

modes. Ibrahim [4] has reviewed topics pertaining to

structural dynamics with parameter uncertainties. Fuku-

naga et al. [5] have investigated the effects of scatter in

the lamina strengths, relative fiber volume fraction,

and laminate stacking sequence on the ultimate strengthof the hybrid laminates. Salim et al. [6] have obtained

the statistical response of plates considering material

properties as independent random variables. The second

order statistics for static deflection, natural frequency

and buckling load of rectangular plates have been stud-

ied using a first order perturbation technique (FOPT).

Chen et al. [7] have developed a probabilistic method

to evaluate the effect of uncertainty in geometrical andmaterial properties for truss and beam problems. FOPT

has been used to evaluate the mean and SD of displace-

ment and rotation for a rod with uncertain area, mass

and stiffness. Yadav and Verma [8] have studied the free

vibration response of thin cylindrical shell using classical

laminate theory (CLT) and have employed the FOPT

for obtaining the second order statistics of natural fre-

quencies. Gorman [9] has presented free vibration anal-ysis of thin rectangular plates with variable edge

supports using the method of superimposition. Singh

et al. [10] have studied the natural frequency of cylindri-

cal panel and composite plate with random material

properties and have obtained the second order statistics

of response.

Some authors have used Monte Carlo Simulation

(MCS) and stochastic finite element techniques to qual-ify structural response uncertainties. Raj et al. [11] have

employed MCS to analyse rectangular plates with and

without cutouts using higher order shear deformation

theory. Nakagiri et al. [12] have adopted stochastic finite

element method (SFEM) to study simply supported

graphite/epoxy plates taking fiber orientation, layer

z 0w

u

a

Fig. 1. Geometry of a lamin

thickness and layer numbers as random variables and

found that the overall stiffness of fiber reinforced com-

posite (FRP) laminated plates is largely dependent on

the fiber orientation.

The nonlinear forced vibration of circular and rectan-

gular plate with various boundary conditions has beenstudied [13,14] by applying the Galerkin or Rayleigh

Ritz method. Singh et al. [15,16] have analysed the non-

linear free and forced vibration of anti-symmetric rec-

tangular cross-ply plates by using Hamilton principle.

Lin [17] has studied nonlinear response of flat panel sub-

jected to periodic and randomly varying loading.

The nonlinear response of plates with uncertain mate-

rial properties to random excitation has not been ad-dressed fully. The present study analyses a composite

flat plate with material properties randomness. The dy-

namic external loading is also random. The formulation

uses Hamilton variational approach. Kirchoff–Love

plate theory with Von-Karman nonlinear strain dis-

placement relations have been adopted to model the sys-

tem behaviour. Perturbation approach has been

employed to develop the system equations and a closedform solution has been obtained for the response statis-

tics of flat plates with all edges simply supported.

2. Nonlinear theory of cross-ply laminated composite

plate

A rectangular composite plate of inplane dimensions�a� and �b� and constant total thickness �h�, composed of

thin orthotropic layers is considered for the present

analysis. The origin of a Cartesian coordinate system,

as shown in Fig. 1, is located in the central plane at

the left corner with x and y axes along the middle

plane and the z axis normal to this plane. The strain

0

0

y

v

x

b

ated composite plate.

336 A.K. Onkar, D. Yadav / Composite Structures 70 (2005) 334–342

components associated with Kirchoff–Love plate theory

are computed using the nonlinear Von-Karman strain-

displacement relations. Thin plate assumptions are

made in order to comply with the classical laminated

plate theory which insure that transverse shear strains

(cyz,cxz) and transverse normal strain (ez) are negligiblethroughout the plate.

Hamilton principle has been employed to develop the

system equations for the forced vibration response sta-

tistics of composite laminated plate with random mate-

rial properties having simply supported edges. In case of

anti-symmetric cross-ply all the coupling elements of

extensional stiffnesses [A], coupling stiffnesses [B] and

bending stiffnesses [D] matrices identically go to zero.By neglecting the effect of transverse shear and normal

strain, the strain energy of the plate can be expressed

as [18]:

U ¼ 1

2

Z a

0

Z b

0

A11e0x2 þ 2A12e

0xe

0y þ A22e

0y2 þ A66c

0xy2

nþ 2B11jxe

0x þ B12 jye

0x þ 2jxe

0y

� �þ 2B22jye

0y

þ 2B66jxyc0xy þ D11e

0x2 þ 2D12jxjy

þD22e0y2 þ D66jxy

2odxdy; ð1Þ

where {e0} are the membrane strains and {j} are the

curvature strains of the middle surface of the plate.

Let the plate be subjected to sinusoidal transversedynamic loading given by

Qðx; y; tÞ ¼ Q0ðtÞ sinmpxa

sinnpyb

; ð2Þ

where m and n are integers and Q0(t) is the time depend-ent forcing function distribution amplitude. This is a

generic form and series of such terms can be used to

model different loading on the plate.

The work done by the external forces is

WD ¼Z a

0

Z b

0

Qðx; y; tÞw0ðx; y; tÞdxdy: ð3Þ

The kinetic energy of the plate, neglecting in-plane

inertia is given by

T ¼ 1

2

Z a

0

Z b

0

XNLi¼1

qihi

!_w20ðx; y; tÞdxdy; ð4Þ

where qi is the density of the ith lamina.

The boundary conditions for a cross-ply plate with allsides simply supported with edges free to move in their

respective inplane normal directions are

Along x ¼ 0 and x ¼ a for all y;

v ¼ 0; w ¼ 0 : Nx ¼ 0; Mx ¼ 0:

Along y ¼ 0 and y ¼ b for all x;

u ¼ 0; w ¼ 0 : Ny ¼ 0; My ¼ 0:

ð5Þ

The following sets of admissible functions satisfy the

above boundary conditions:

u0ðx; y; tÞ ¼ U 0ðtÞ cosmpxa

sinnpyb

;

v0ðx; y; tÞ ¼ V 0ðtÞ sinmpxa

cosnpyb

;

w0ðx; y; tÞ ¼ W 0ðtÞ sinmpxa

sinnpyb

;

ð6Þ

where U0(t), V0(t) and W0(t) are the maximum displace-

ments in x, y and z directions at any instant of time.Using nonlinear Von-Karman strain-displacement

relations in conjunction with Eq. (6), the strain energy,

work done by external forces and kinetic energy can

be obtained from Eqs. (1), (3) and (4) respectively.

Based on Hamilton�s principle the governing modal

equation of motion is:

XNLi¼1

qihi

!€W 0 þ L1W 0 þ L2W 2

0 þ L3W 30 ¼ Q0ðtÞ; ð7Þ

where L1, L2 and L3 are

L1 ¼ T 8 þð2T 2T 3T 6 � T 2

3T 5 � T 26T 1Þ

ðT 1T 5 � T 22Þ

;

L2 ¼ T 9 þð3T 2T 3T 7 þ 3T 2T 4T 6 � 3T 3T 4T 5 � 3T 1T 6T 7Þ

ðT 1T 5 � T 22Þ

;

L3 ¼ T 10 þð4T 2T 4T 7 � 2T 2

4T 5 � T 27T 1Þ

ðT 1T 5 � T 22Þ

:

T1,T2, . . .,T10 are coefficients depend on the plate geom-

etry, material properties and the mode shape. The

expressions for these coefficients are defined in Appen-

dix A.

The energy balance equation is obtained by multiply-

ing Eq. (7) by _W 0 and integrating with respect to time:

XNLi¼1

qihi

!_W2

0 þ L1W 20 þ

2

3L2W 3

0 þ1

2L3W 4

0

� Q0ðtÞW 0 ¼ H ¼ constant: ð8Þ

The constant H in Eq. (8) can be obtained by using

the condition that at W0 = Wmax, _W 0 ¼ 0. This gives

H ¼ L1W 2max þ

2

3L2W 3

max þ1

2L3W 4

max � QmaxW max: ð9Þ

Substituting this in Eq. (8) yields:

XNLi¼1

qihi

!_W2

0 þ L1ðW 2max � W 2

0Þ þ2

3L2ðW 3

max � W 30Þ

þ 1

2L3ðW 4

max � W 40Þ � ðQmaxW max � Q0ðtÞW 0Þ ¼ 0:

ð10Þ

The above equation at _W 0 ¼ 0 will have two real

roots, Wmax1 and Wmax2, unequal in magnitude and

opposite in sign. The plate will vibrate with different


amplitudes in the positive and negative z directions.

Substituting W0(t) = Wmax1 sinh for the first half period,

W0(t) = �Wmax2 sinh for the next half period and

Q0(t) = Qmaxsinh, (h = xt), the nonlinear time period

for such a plate can be obtained as

T nl ¼ 2

Z p=2

0

dhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

ðqihiÞp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL1 þ 2

3L2

1þsin hþsin2h1þsin h

� �W max1 þ 1

2L3ð1þ sin2hÞW 2

max1 �Qmax

W max 1

h ir

þ 2

Z p=2

0


ðqihiÞp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL1 � 2

3L2

1þsin hþsin2h1þsin h

� �W max 2 þ 1

2L3ð1þ sin2hÞW 2

max2 þQmax

W max 2

h ir : ð11Þ

However for a square plate L2 goes to zero and Eq. (10)

at _W 0 ¼ 0 has two equal and opposite roots ±Wmax. The

nonlinear time period for a square plate is

T nl ¼2px

¼ 4

Z p=2

0


ðqihiÞp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL1 þ 1

2L3ð1þ sin2hÞW 2

max �Qmax

W max

h ir : ð12Þ

It can be put in a more compact form as

T nl ¼4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

ðqihiÞpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL1ð1þ c� dÞ

p Z p=2

0

dhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ asin2h

p : ð13Þ

where a ¼ c1þ c� d

and c ¼ L3W 2max

2L1

, d ¼ Qmax

L1W max

.

The above integral has the form of an elliptic integral,

which cannot be evaluated in terms of elementary func-

tions. An infinite series representation for it is generated

by expanding the integrands in binomial series and using

termwise integration. The final expression can be writtenas

T nl ¼2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPðqihiÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL1ð1þ c� dÞ

p 1� a

22þ a2

42� a3

82þ a4

162� � � �

� �:

ð14Þ

The above equation can be rearranged to give the un-

known Wmax in terms of the known oscillation fre-

quency x. This is a fifth order nonlinear algebraic

equation:

P 1W 5max þ P 2W 3

max þ P 3W 2max þ P 4W max þ P 5 ¼ 0; ð15Þ

where P 1 ¼50L2

3

64, P 2 ¼ L1L3 � L3x2

Pni¼1qihi, P 3 ¼

5032ðL1L3 � L3QmaxÞ, P 4 ¼ 2L2

1 � 2L1x2PNL

i¼1qihi and P 5 ¼2QmaxL1.

Hence the required nonlinear amplitude can be writ-

ten as

W max ¼ funðE11;E22;G12; m12;x and QmaxÞ: ð16Þ

3. Mean and variance of forced vibration response

First order perturbation approach has been adopted

for obtaining the response amplitude statistics with ran-

domness in material properties and loading on a flat

plate. It is assumed that all the material properties and

loading components are uncorrelated to each other. It

may be noted that in case these are not uncorrelated,it is possible to express them in terms of an uncorrelated

set by a transformation using eigenvectors of the covar-

iance matrix [19]. It is also assumed, as true in most sen-

sitive engineering application, that the dispersion of

each random quantity about its mean value is small.

Any random variable may be split up as the sum of

the mean variable and zero mean random part with gen-

erality as

L ¼ Ld þ Lr ð17Þ

where superscript �d� denotes the mean value, which is

deterministic, and �r� denotes the superimposed zero

mean random component.Using Taylor series expansion the random part of the

dependent variables can be expressed in terms of the

independent variables. The primary variables bl are as-

sumed to be the basic material properties and the ele-

ments of the applied load. As assumed, the random

part in the primary variables is small in magnitude com-

pared to their mean values, the second and higher order

terms are neglected and the expression for system oper-ator and maximum response amplitude may be put as

Lri ¼

Xsl¼1

oLdi

obdlbrl; W r

max ¼Xql¼1

oW dmax

obdlbrl; ð18Þ

where bl, l = 1,2, . . ., s are the material variables and bl,

l = s + 1, s + 2, . . ., s + s1 are the external loading varia-

bles. Thus the total number of independent basic ran-

dom variables is s + s1 = q (say).


Using Taylor series expansion, by neglecting small

quantities, one can write the random part of reduced

stiffness matrix and transformed reduced stiffness matrix

as

Qrij ¼

Xsl¼1

oQdij

obdl

!brl and Q

r

ij ¼Xsl¼1

o�Qd

ij

obdl

!brl: ð19Þ

The random part of extensional, bending-extension

coupling and bending stiffness matrices can also be ex-

pressed similarly by using Taylor series expansion keep-

ing only one term in the series and neglecting all higherorder terms as

Arij ¼

Xsl¼1

oAdij

obdl

!brl;

Brij ¼

Xsl¼1

oBdij

obdl

!brl and

Drij ¼

Xsl¼1

oDdij

obdl

!brl:

ð20Þ

The partial derivatives of the stiffness matrix elements

Adij, B

dij and Dd

ij with respect to �bdl � are required. These

can be expressed as follows:

oAdij

obdl¼XNLk¼1

o�Qd

ij

obdl

!ðhk � hk�1Þ;

oBdij

obdl¼ 1

2

XNLk¼1

o�Qd

ij

obdl

!ðh2k � h2k�1Þ;

oDdij

obdl¼ 1

3

XNLk¼1

oQd

ij

obdl

!ðh3k � h3k�1Þ:

ð21Þ

Using Eq. (16), the derivative of W dmax with respect of

bdl are

oW dmax

obdl¼

oP 5

obdl� oP 1

obdlW 5

max �oP 2

obdlW 3

max �oP 3

obdlW 2

max �oP 4

obdlW max

5P 1W 4max þ 3P 2W 2

max þ 2P 3W max þ P 4

for l 6 s ð22Þ

and

oW dmax

obdl¼

oP 5

obdl� oP 3

obdlW 2

max

5P 1W 4max þ 3P 2W 2

max þ 2P 3W max þ P 4

for l > s; ð23Þ

where partial derivative of P1, P2, P3, P4 and P5 with re-

spect to bdl can be expressed in terms of E11, E22, m12, G12

and Qmax.

Hence the total deflection response can be evaluated

as

W max ¼ W dmax þ

Xql¼1

oW dmax

obdlbrl: ð24Þ

The variance of the response is expressed as follows:

var W maxð Þ ¼ EXql¼1

oW dmax

obdlbrl

!224

35: ð25Þ

4. Results and discussion

The formulations developed in this study are used to

determine the response statistics of random vibration of

a plate that is made from graphite/epoxy orthotropic

composite material. Closed form expressions have been

developed in the previous section for the mean and

variance of maximum amplitude response along with

frequency of vibration for simply supported plates. Alllamina are assumed to have the same thickness and

the material properties are orthotropic along the mate-

rial axes. The developed expressions are validated with

available results in literature. The effects of material

property dispersion along with variations in thickness

ratio and oscillation amplitude on the response statistics

have been explored.

4.1. Validation

The validation of the present formulation is sought

by comparison of results with reported literature. How-

ever, nonlinear formulation is not available in literature

for laminated composite plate with random material

properties and random loading giving variance of the re-

sponse. Hence, comparison has been made with the

available nonlinear mean analysis only.

Table 1 presents a comparison of the non-dimensio-

nalised mean frequency for the square, two-layered

cross-ply laminate for different amplitude with results

by Singh et al. [16]. The material properties used for

the analysis are

E11 ¼ 40E22; G12 ¼ 0:5E22 and

m12 ¼ 0:25; and excitation Qmax ¼ 0:2L1:

Table 1

Comparison of the non-dimensionalised mean nonlinear frequency

(xnl/xl) for [0�/90�] laminate for aspect ration a/b = 1 with different

amplitudes

wmax/h Present work Ref. [11]

+0.4 0.8943 0.8898

�0.4 1.3408 1.3395

+0.6 1.1595 1.1486

�0.6 1.4167 1.4109

+1.0 1.6428 1.6116

�1.0 1.7581 1.7334

1 ω 2 ω 3 ω

0.15

0.2

/mea

n, w

max


The reference uses numerical integration method for

evaluation of the mean frequency whereas the present

approach gives an exact solution. A reasonably good

agreement between the two is observed.

4.2. Second order amplitude statistics

The material used for the graphite/epoxy compositeplate and the excitation amplitude used for this analysis

is same as employed for generating Table 1. All the four

material properties and excitation are considered as the

basic random variables for the analysis. These are se-

quenced as follows:

b1 ¼ E11; b2 ¼ E22; b3 ¼ m12; b4 ¼ G12; b5 ¼ Qmax:

ω ω 1

2ω 3

0.2

0

0.05

0.1

0 0.05 0.1 0.15 0.2 0.25

sd/mean, all random inputs

sd

Fig. 2. Influence of SD of all basic random inputs changing simulta-

neously on coefficient of variation of amplitude for [0�/90�/90�/0�]laminate with b/a = 1 and b/h = 100.

4.2.1. Maximum mean amplitude

Table 2 presents the non-dimensionalised maximum

mean amplitude for different mean frequencies for sym-

metric and anti-symmetric stacking sequences. It is ob-

served that the mean amplitude ratio increases with

increase in mean oscillation frequency. It is also ob-served that the anti-symmetric laminates have higher

mean amplitude compared to the symmetric laminate.

4.2.2. Variance of amplitude

Influence of the scattering in the material properties

and loading on the mean amplitude has been obtained

by allowing the coefficient of variation to change from

0% to 20% for laminated cross-ply plates. The variations

Table 2

Non-dimensionalised maximum mean amplitude (wmax/h) for square

plates with different frequencies and stacking sequences

Frequency (x) (rad/s) (wmax/h)

Stacking sequence

[0�/90�/90�/0�]Stacking sequence

[0�/90�/0�/90�]

1.00 · 10�3 0.2287 0.2337

1.25 · 10�3 0.2935 0.3108

1.50 · 10�3 0.4070 0.4458

1.75 · 10�3 0.5838 0.6389

2.00 · 10�3 0.7900 0.8390

of non-dimensionalised amplitude with dispersion in all

the basic inputs changing simultaneously are presented

in Figs. 2 and 3 for three different excitation frequen-

cies x1 = 1.0 · 10�3, x2 = 1.5 · 10�3, and x3 = 2.0 ·10�3 rad/s. Two cases, square symmetric and anti-sym-

metric four layered cross ply with b/h = 100, have beenexamined. It is observed that as the excitation frequency

increases the amplitude sensitivity increases for all cases.

The sensitivity of the square anti-symmetric cross ply is

marginally greater than the symmetric case.

Fig. 4(a)–(d) and Fig. 5(a)–(d) present the influence of

only one material property random at a time on dis-

placement response dispersion for square symmetric

and anti-symmetric cross ply respectively with b/h = 100. The amplitude variations are most affected by

change in E11 and least affected by dispersion in m12. Itis also seen that sensitivity of the oscillation amplitude

to plate frequency increases with increased variations

in E11, E22, G12 and m12.

max

0

0.05

0.1

0.15

0 0.05 0.1 0.15 0.2 0.25sd/mean, all random inputs

sd/

mea

n, w

Fig. 3. Influence of SD of all basic random inputs changing simulta-

neously on coefficient of variation of amplitude for [0�/90�/0�/90�]laminate with b/a = 1 and b/h = 100.

Fig. 4. Influence of SD of basic material properties on coefficient of variation of amplitude for [0�/90�/90�/0�] laminate with b/a = 1.0 and b/h = 100.

(a) Only E11 varying, (b) only E22 varying, (c) only G12 varying, (d) only m12 varying.

Fig. 5. Influence of SD of basic material properties on coefficient of variation of amplitude for [0�/90�/0�/90�] laminate with b/a = 1.0 and b/h = 100.

(a) Only E11 varying, (b) only E22 varying, (c) only G12 varying, (d) only m12 varying.


Figs. 6 and 7 show amplitude sensitivity to dispersion

in external excitation for square symmetric and anti-

symmetric cross-ply respectively for b/h = 100. The load-

ing adopted for generating the result is small in magni-

max

ω 3ω 2ω 1

max

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 0.05 0.1 0.15 0.2 0.25

sd/mean, Q

sd/

mea

n, w

Fig. 6. Influence of SD of excitation �Q0� on coefficient of variation of

amplitude for [0�/90�/90�/0�] laminate with b/a = 1.0 and b/h = 100.

max

ω ω ω

1 2 3

max

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 0.05 0.1 0.15 0.2 0.25 sd/mean, Q

sd/

mea

n, w

Fig. 7. Influence of SD of excitation �Q0� on coefficient of variation of

amplitude for [0�/90�/0�/90�] laminate with b/a = 1.0 and b/h = 100.

sd/mean, all random inputs, 5%

10%

15%

20%

Wmax/h

0

0.05

0.1

0.15

0.2

0.2 0.3 0.4 0.5 0.6 0.7 0.8

sd/m

ean,

Wm

ax

Fig. 9. Influence of mean amplitude on coefficient of variation of

amplitude for [0�/90�/90�/0�] laminate with all random inputs changing

simultaneously.


tude. Even then it is observed that variation of excita-

tion has a dominant effect on the deflection scattering

as compared to E22, G12 and m12. It can also be seen that

sd/mean, all random inputs, 5%

10%

15%

20%

ω x 10–3 (rad/s)

0

0.05

0.1

0.15

0.2

1 1.2 1.4 1.6 1.8 2

sd/m

ean,

Wm

ax

Fig. 8. Influence of frequency on coefficient of variation of amplitude

for [0�/90�/90�/0�] laminate for all random inputs changing

simultaneously.

the variation in external loading has stronger effect on

symmetric cross-ply compared to anti-symmetric cross-

ply laminates.Influence of frequency on coefficient of variation of

amplitude with a constant SD/mean for all basic inputs

changing simultaneously for square symmetric cross-ply

with b/h = 100 is shown in Fig. 8. It is found that the

amplitude scatter increases nonlinearly with increase in

excitation frequency and also increases with increase in

variation of basic inputs.

The interrelation between (wmax/h) on coefficient ofvariation of amplitude with a constant SD/mean for

all basic inputs changing simultaneously for square

symmetric cross-ply with b/h = 100 is shown in Fig.

9. This also shows that the amplitude scatter increases

nonlinearly with increase in response amplitude and

also increases with increase in variation of basic

inputs.

5. Conclusions

A general formulation for the nonlinear random

vibration analysis of composite laminated plates mode-

led by Kirchhoff–Love plate theory has been presented.

It has been observed that the dispersions in the response

amplitude show linear increase with SD of the materialproperties in the range studied whereas nonlinear behav-

iour in amplitude has been seen with the oscillation

amplitude and frequency. Dispersion of E11 has domi-

nant effect on the scattering of amplitude as compared

to the material properties E22, G12 and m12. Variationin the external load causes larger variations in the deflec-

tion response compared to the effect of E22, G12 and m12.It is seen that E11 has more effect on response statisticscompared to Qmax. This may be due to the small value

of mean excitation used for the analysis.


Appendix A

T 1¼mpa

� �2A11þ

npb

� �2A66;

T 2¼mpa

� � npb

� �ðA12þA66Þ; T 3¼� mp

a

� �3B11;

T 4¼� 4Smn

9mnp2

mpa

� �3A11þ

mpa

� � npb

� �2ðA12�A66Þ

� �;

T 5¼mpa

� �2A66þ

npb

� �2A22; T 6¼� np

b

� �3B22;

T 7¼� 4Smn

9mnp2

npb

� �3A22þ

npb

� � mpa

� �2ðA12�A66Þ

� �;

T 8¼mpa

� �4D11þ2

mpa

� �2 npb

� �2ðD12þ2D66Þþ

npb

� �4D22;

T 9¼4Smn

3mnp2

mpa

� �4B11þ

npb

� �4B22

� �;

T 10¼9

32

mpa

� �4A11þ

npb

� �4A22

� �

þ 1

16

mpa

� �2 npb

� �2ðA12þ2A66Þ;

where Smn = (1 � (�1)m)(1 � (�1)n).

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