Dynamic response of multidirectional composites in hygrothermal environments

Composite Structures 64 (2004) 329–338

www.elsevier.com/locate/compstruct

Dynamic response of multidirectional compositesin hygrothermal environments

V.V.S. Rao, P.K. Sinha *

Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur 721302, West Bengal, India

Abstract

The present paper deals the effects of temperature and moisture on the free vibration and transient response of multidirectional

composites. A three-dimensional finite element analysis procedure is developed using 20 noded isoparametric quadratic elements.

The analysis accounts for the degradation of composite properties due to both temperature and moisture concentration. A typical

multidirectional unit cell is assumed to consist of several unidirectional composite blocks. The transformation based relationships

are used to generate the stiffness properties of multidirectional composite plates. The numerical results for the natural frequencies

and transient response of multidirectional composites under the action of both temperature and moisture concentration are pre-

sented and discussed.

� 2003 Elsevier Ltd. All rights reserved.

Keywords: Multidirectional composites; Hygrothermal environment; Thick plates; Transient response; Free vibration

1. Introduction

Advanced composites are used extensively in aero-

space and other structural applications because of their

low density, high strength and high stiffness. Their su-

perior strength and stiffness properties are often com-

promised by the environment to which they are exposed.

Moisture and temperature may be distributed through

the volume of the structure and may induce residualstresses and extensional strains. These residual stresses

and extensional strains may also effect the gross per-

formance of the structure. In particular, the bending

characteristics, buckling loads and vibration frequencies

can be modified by the presence of moisture, tempera-

ture or both. Therefore, to utilize the full potential of

advanced composites, it will be necessary to analyze the

effects of moisture and temperature in composite struc-tural components.

The vibration characteristics of thick isotropic rect-

angular plates under an arbitrary state of initial stress

were investigated by earlier researchers [1–3]. Yang and

Shieh [4] considered the free vibration of anti-symmetric

*Corresponding author. Tel.: +91-3222-283016; fax: +91-3222-

255303/77190.

E-mail addresses: [email protected] (V.V.S. Rao),

[email protected] (P.K. Sinha).

0263-8223/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2003.09.002

cross-ply laminates in presence of a non-uniform initialstress, where the effects of transverse shear and rotary

inertia were also included. Whitney and Ashton [5] used

the classical laminate plate theory to study the effect of

hygrothermal environment on the stability, vibration

and bending behaviour of laminated composite plates.

Pipes et al. [6] presented the distribution of inplane

stresses through the thickness of symmetric laminates

subjected to moisture absorption and desorption. SaiRam and Sinha [7,8] studied the hygrothermal effects on

the bending and free vibration behaviour of laminated

composite plates. Mukherjee and Sinha [9] developed

the micro-mechanics model to obtain the thermo-me-

chanical properties for 3D multidirectional composites

and used those properties for analyzing the thermo-

structural problems. The transient analysis of isotropic,

orthotropic and layered composite plates using theNewmark’s direct integration scheme and considering

shear flexible finite element was carried out by Reddy

[10]. Meimaris and Day [11] assessed the performance of

20 noded elements for the static and dynamic response

analysis of laminated composite plates. To the best of

author’s knowledge, no results were reported on the

transient response of multidirectional composite plates

under the action of hygrothermal environment.In the present work attention is focused primarily on

investigating the effects of moisture and temperature on

mail to: [email protected]

Nomenclature

a, b dimensions of plate along x and y axes, re-spectively

fdeg element nodal displacement vector

fd ig global initial displacement vector

fF Ng non-mechanical force vector due to moisture

and temperature

h thickness of the plate

½Kr� global geometric stiffness matrix

½M � global mass matrixn number of unidirectional blocks

fPNg global non-mechanical load vector

½Tcm� transformation matrix, derived from Euler’s

angles

Vf total volume fraction of fibres in the unit cellu, v, w displacements in x, y and z axes, respectivelyq mass density

a11, a22 thermal expansion coefficient of a laminaalong and across a fibre, respectively

½a� thermal expansion matrix

b11, b22 moisture swelling coefficient of a laminaalong and across a fibre, respectively

DT change of temperatureDC change of moisture concentration

1

x

2

3

z

y

Fig. 1. Typical 1D unit cell.

330 V.V.S. Rao, P.K. Sinha / Composite Structures 64 (2004) 329–338

the natural frequency and transient response of multi-

directional composite plates using 20 noded isopara-

metric multidirectional composite finite element, based

on a multidirectional micro-mechanics model. The ap-

propriate 3D finite element procedure is developed to

include the strain energy due to the initial stresses. The

finite element formulation accounts for the hygrother-

mal strains and reduced lamina properties at elevatedtemperature and moisture concentration. The natural

frequencies are evaluated using the inverse iteration

scheme for multidirectional composite models subjected

to uniform temperature and moisture concentration.

The Newmark’s average acceleration method [12] is

employed to integrate the dynamic equations resulting

from the finite element approximation. The fundamental

frequencies and transient response solutions are pre-sented for various multidirectional composite plates

subjected to different temperature and moisture con-

centration.

Total volume of the unit cell

2. Elastic rigidity matrix of multidirectional composite

The generalized Hooke’s law for a 3D orthotropic

material is defined as [13]

ri ¼ Qijej; i; j ¼ 1; 2; 3; 4; 5; 6 ð1Þ

where ri are the stress components, Qij is the rigidity

matrix, and ej are the engineering strains. The 3D mul-

tidirectional material model is represented by a unit cellcontaining �n’ number of unidirectional blocks. Eachunidirectional block consists of a set of fibres and ma-

trix. Each block again can be randomly oriented within

a unit cell. A typical 1D unit cell model is shown in Fig.

1. The cells are designated according to the number of

unidirectional blocks they contain. The volume of fibres

is distributed equally in all the multidirectional com-

posite models. The multidirectional composite models

3D, 4D, 5D and 6D cells are shown in Figs. 2–5. The

orientation of the unidirectional block is measured fromthe Euler’s angles [14] as shown in Fig. 6. The elastic

rigidity matrix for a multidirectional cell containing

�n’ unidirectional composite blocks can be expressed as[9]

½Q� ¼Xni¼1

Wf i½T3rm�Ti ½Q�i½T3rm� ð2Þ

where [T3rm] is the 3D transformation matrix, Wfi is theweight factor for the ith block, defined as

Wfi ¼ Vfi=Vf ð3Þ

Vf i ¼Volume of fibres in the ith direction block ð4Þ

F5

F3

F4

F2F1

z

y

x

Fig. 4. Typical 5D unit Cell with F3, F4, F5 fibre blocks oriented di-agonally.

F3

F2

F1

z

y

x

Fig. 2. Typical 3D unit cell with fibre block F3 in the x-plane ath ¼ 45�.

z

y

x

F2

F1

F4 F3

θ

Fig. 3. Typical 4D unit cell with fibre blocks F3 and F4 lying on the x-plane oriented at h ¼ 45� to the z-axis.

F6

F5

F3

F4

F2F1

z

y

x

Fig. 5. Typical 6D unit cell with F3, F4, F5, F6 fibre blocks orienteddiagonally.

x

z

y

1

3

2

φ ψ

β

Fig. 6. Euler’s angles.

V.V.S. Rao, P.K. Sinha / Composite Structures 64 (2004) 329–338 331

3. Finite element formulation

A 20 noded hexahedral isoparametric quadratic ele-

ment with three degrees of freedom at each node is as-

sumed for the present analysis. The displacements are

expressed in terms of the nodal values using the element

shape functions [15], as defined by

uvw

8<:

9=; ¼

X20i¼1

Niui ð5Þ

The element stiffness matrix for a 20 noded hexahedral

element is given as

½Ke� ¼ZV½B�T½Q�½B�dxdy dz ð6Þ

where ½Q� is the rigidity matrix as defined in Eq. (2).In a similar manner the element mass matrix [Me] is

calculated as


½Me� ¼ZV½N �T½q�½N �dxdy dz ð7Þ

The element level nodal load vector due to non-me-

chanical forces is given by

fPNe g ¼ZVBTfF Ngdv ð8Þ

The element level nodal load vector due to external

transverse load is obtained as

fPeg ¼ZS½N �Tfqgdxdy ð9Þ

The integrations in Eqs. (6)–(8) are carried out using the

3 · 3 Gauss quadrature method. The element stiffnessmatrices and element load vectors so formed are as-

sembled with respect to the common global coordinates,and the resulting equilibrium condition becomes

½K�fd ig ¼ fPg þ fPNg ð10Þ

The solution to the initial displacements fd ig is obtainedfrom the equilibrium condition. The initial stress values

are computed using the nodal displacements to enhance

the accuracy of the stress values, the stress smoothing

method as suggested by Hinton et al. [16] is used. The

initial stresses are evaluated using the following relation

fr0g ¼ ½Q�½B�fd ig ½Q�ðf�agDT þ f�bgDCÞ ð11Þ

where f�ag and f�bg are defined in Eq. (12).The off-axis coefficient of thermal expansion (CTE)

can be related to the on-axis CTE through the relation

½�a� ¼XNi¼1

wi½Tcm�Ti ½a�½Tcm�i ð12Þ

where

½a� ¼a11 0 0

0 a22 00 0 a33

24

35 and ½Tcm� ¼

l1 m1 n1l2 m2 n2l3 m3 n3

24

35

In vector notation, the global coefficient of thermal ex-

pansion for the multidirectional composite plate can beexpressed as

f�ag ¼ ½ax; ay ; az; ayz; azx; axy �T

¼ ½�a11; �a22; �a33; 2�a23; 2�a13; 2�a12�T ð13Þ

The moisture coefficient f�bg is computed in a similarway as f�ag.The strain energy due to initial stresses fr0g is defined

as [17]

Ur ¼ 12

ZVfdgT

S 0 00 S 0

0 0 S

24

35fdgdv ð14Þ

where fdg ¼ ½u;x; u;y ; u;z; v;x; v;y ; v;z;w;x;w;y ;w;z�T and

S ¼rx0 sxy0 szx0sxy0 ry0 syz0szx0 syz0 rz0

24

35

And also

fdg ¼ ½G�fdeg ð15Þ

where ½G� is obtained from the shape functions [N ] byappropriate differentiation and ordering of terms. Thus,

the strain energy is

Ur ¼ fdegT½Ker�fdeg2

ð16Þ

where ½Ker� is the element initial stress stiffness matrix,defined as

½Ker� ¼ZV½G�T

S 0 0

0 S 0

0 0 S

24

35½G�dv ð17Þ

3.1. Solution procedure

The finite element analysis of free vibration and

transient response is carried out in two phases. The first

part of the solution is to obtain the initial stress resul-tants induced by the external static load as well as by

moisture and temperature environments. The initial

displacements fd ig are found from the equilibrium

condition as given by

½K�fd ig ¼ fPg þ fPNg ð18Þ

From the initial displacements, fd ig the initial stressstiffness matrix is computed using the above mentioned

procedure and Eq. (17).The second part of the solution involves the deter-

mination of natural frequencies and transient response

using the following equations, respectively

j½K� þ ½Kr� x2½M �j ¼ 0 ð19Þ

½K þ Kr�fdg þ ½M �f€dg ¼ fRg ð20Þ

3.2. Finite element code

The computer programme to implement the present

finite element analysis procedure is developed in C lan-guage. The finite element code is capable to handle a 3D

multidirectional composite structure subjected to static

and dynamic loadings and having arbitrary boundary

conditions. The FE code is thus generalized to the solve

bending, free vibration and transient response problems,

including the hygrothermal effects. It can also analyze

thick laminated composite plates.


4. Numerical results and discussion

In the present analysis, free and forced vibration

problems of multidirectional composites subjected to anexternal transverse static load and uniform distribution

of moisture and temperature through the volume of the

plates are analyzed. The finite element analysis code as

reported in the previous section is used for the purpose.

The boundary conditions used in the present investiga-

tion are shown in Fig. 7.

The material properties of a graphite/epoxy lamina

[18] at different moisture concentration and temperatureare listed in Tables 1–3.

4.1. Free vibration of multidirectional composite plates

The present results are first tested with the published

ones for accuracy and convergence. As an example

problem, a simply supported cross-ply laminated

[0=90=90=0] square plate (a=h ¼ 5) is considered for freevibration analysis. A finite element mesh size of 8 · 8 · 4is used for the full plate. Fundamental frequencies for

different E11=E22 ratios are evaluated using the inverseiteration method and the frequencies are tabulated in

Table 4. The present non-dimensional fundamental

frequencies, k ¼ xa2ðq=E22hÞ0:5 are found to agree withthe published results [19].

Non-dimensional fundamental frequencies, k areobtained for a cross-ply [0=90=90=0] laminated plate(a=h ¼ 10) with simply supported and clamped bound-ary conditions. The results generated at different tem-

perature and moisture concentration levels are

compared with those of Sai Ram and Sinha [8] as shown

in Figs. 8 and 9. The present results are found to match

well.

Non-dimensional fundamental frequencies are thengenerated for simply supported multidirectional com-

posite plates under the action of hygrothermal envi-

ronments. Multidirectional composite square plates

v=w

=0

u=w=0

y

xa

b

i) Simply supported

Fig. 7. Boundary

(a=h ¼ 10, 40) are considered for the analysis. The ma-terial properties listed in Tables 1 and 2 are used to

compute the hygro-thermo-elastic properties for multi-

directional composite plates. In the present analysis, thereference temperature T0 ¼ 300 K and the reference

moisture concentration C0 ¼ 0% are assumed. The non-dimensional fundamental frequencies, k are computedfor the variations of temperature, T (300–400 K) andmoisture concentration, C (0–1%). The effects of uni-

form temperature and uniform moisture concentration

on the non-dimensional fundamental frequencies, k ofmultidirectional composite models are plotted in Figs.10–13. It is observed that the fundamental frequencies

reduce with the increase of moisture and temperature

levels. For the a=h ratio to be 40, the composite plateswith 4D and 5D cells buckle, when moisture concen-

tration, C is more than 0.75%. The 6D cell model be-

comes instable at a lower moisture concentration,

C ¼ 0:5%.Non-dimensional fundamental frequencies, k are

evaluated, when a simply supported plate is exposed to

both temperature T ¼ 366 K and moisture C ¼ 0:75%.The material properties listed in Table 3 are used. The

results are computed for different a=h ratios and aretabulated in Table 5. All multidirectional composite

models become instable at a=h ¼ 40. The results showthat the thin models are more susceptible to environ-

mental effects. A 3D multidirectional composite exhibitsbetter performance.

4.2. Transient analysis of multidirectional composites

Transient dynamic solutions are obtained using the

implicit Newmark’s constant average acceleration

method. Initially, the finite element code is tested forisotropic materials. Transient solutions are generated

for a square simply supported isotropic plate subjected

to a uniform pulse load. The results are plotted in

u=v=

w=0

u=v=w=0

y

xa

b

ii) Clamped supported

conditions.

Table 4

Non-dimensionalized fundamental frequencies, k ¼ xa2

h

ffiffiffiffiffiqE22

qof a simply supported cross-ply [0=90=90=0] composite plate (a=b ¼ 1, a=h ¼ 5)

E11E22

3D elasticity [20] HSDT [19] Present

3 6.6815 6.5597 6.5778

10 8.2103 8.2718 8.2791

20 9.5603 9.5263 9.5033

30 10.272 10.272 10.2132

40 10.752 10.787 10.6916

Table 2

Elastic moduli of graphite/epoxy lamina at different temperatures; G13 ¼ G12, G23 ¼ 0:5G12, m12 ¼ m13 ¼ m23 ¼ 0:3, a11 ¼ 0:3� 106/K,a22 ¼ 28:1� 106/KElastic moduli (GPa) Temperature, T (K)

300 325 350 375 400 425

E11 128 128.1 129 130.6 131.8 131.1

E22 9.4 8.69 7.84 7.12 6.71 6.61

G12 6.28 5.88 5.33 5.07 4.66 4.60

Table 1

Elastic moduli of graphite/epoxy lamina at different moisture concentrations; G13 ¼ G12, G23 ¼ 0:5G12, m12 ¼ m13 ¼ m23 ¼ 0:3, b11 ¼ 0 and b22 ¼ 0:44Elastic moduli (GPa) Moisture concentration, C (%)

0.0 0.25 0.5 0.75 1.0 1.25

E11 128 128 128 128 128 128

E22 9.4 7.26 6.12 5.56 5.43 5.43

G12 6.28 5.39 5.0 4.0 3.65 3.5

Table 3

Elastic moduli of graphite/epoxy lamina at temperatures T ¼ 366 K, moisture concentration, C ¼ 0:75%; a11 ¼ 0:3� 106/K, a22 ¼ 28:1� 106/K,b11 ¼ 0 and b22 ¼ 0:44E11 (GPa) E22 (GPa) G12 (GPa) G13 (GPa) G23 (GPa) m12 ¼ m13 ¼ m23

127 5.94 4.71 4.71 2.355 0.3

Non

-dim

ensi

onal

fund

amen

tal f

requ

ency

,

Moisture concentration, C (%)

SS, a/b=1, a/h=10

CC, a/b=1, a/h=10

0

5

10

15

20

25

30

35

40

45

50

0 0.2 0.4 0.6 0.8 1 1.2 1.4

PresentRef. [8]λ

Fig. 8. Effect of moisture on fundamental frequency of [0=90=90=0]

laminate, SS¼ simply supported, CC¼ clamped.

Non

-dim

ensi

onal

fund

amen

tal f

requ

ency

,

Temperature, T (K)

SS, a/b=1, a/h=10

CC, a/b=1,a/h=10

0

5

10

15

20

25

30

35

40

45

50

300 320 340 360 380 400 420

PresentRef. [8]λ

Fig. 9. Effect of temperature on fundamental frequency of [0=90=90=0]

laminate, SS¼ simply supported, CC¼ clamped.


Temperature, T (K)

Non

-dim

ensi

onal

fund

amen

tal f

requ

ency

,

3D Cell

8

8.5

9

9.5

10

10.5

11

11.5

12

300 320 340 360 380 400

4D Cell5D Cell6D Cell

λ

Fig. 12. Effect of temperature on fundamental frequency of simply

supported multidirectional composite plates (a=h ¼ 10).

Temperature, T (K)

3D Cell

0

2

4

6

8

10

12

300 320 340 360 380 400


Non

-dim

ensi

onal

fund

amen

tal f

requ

ency

, λ

Fig. 13. Effect of temperature on fundamental frequency of simply

supported multidirectional composite plates (a=h ¼ 40).


Non

-dim

ensi

onal

fund

amen

tal f

requ

ency

, λ

3D Cell

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1


Fig. 11. Effect of moisture on fundamental frequency of simply sup-

ported multidirectional composite plates (a=h ¼ 40).


Non

-dim

ensi

onal

fund

amen

tal f

requ

ency

,

3D Cell

8

8.5

9

9.5

10

10.5

11

0 0.2 0.4 0.6 0.8 1


λ

Fig. 10. Effect of moisture on fundamental frequency of simply sup-

ported multidirectional composite plates (a=h ¼ 10).


Fig. 14 and the results are found to be in good agree-

ment with those of Reddy [10].

A simply supported square multidirectional com-

posite plate of side, 250 mm and thickness, 50 mm is

considered for the transient analysis. Transient solutions

are generated for a uniform pulse load, q0 ¼ 10 N/cm2.

The full plate model is considered with the optimum

finite element mesh of 8 · 8 · 2. The time step, Dt ¼ 5 lsis chosen for the Newmark’s integration scheme. The

Table 5

Non-dimensionalized fundamental frequencies, k ¼ xa2

h

ffiffiffiffiffiqE22

qfor multidirectio

perature, T ¼ 366 Kah 3D cell 4D cell

10 9.1336 8.3768

20 8.5309 7.2079

30 6.4002 3.8184

40 – –

transverse central deflections are computed for different

multidirectional composite models subjected to various

levels of moisture concentrations and temperature.

These are plotted in Figs. 15–23. It is observed that the

transverse central deflections of multidirectional com-

posites increase with the increase of moisture concen-

tration and temperature. The period of oscillation also

increases with the rise of temperature and moistureconcentration levels. It is observed from the results that

nal composite models at moisture concentration C ¼ 0:75% and tem-

5D cell 6D cell

8.3768 8.4337

7.2079 7.0660

3.8184 3.0880

– –

Fig. 15. Transient response of simply supported multidirectional

composite plates (a=h ¼ 5, C ¼ 0, T ¼ 300 K).

Time, t (µs)

4D Cell

Def

lect

ion,

wx1

0-3cm

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 50 100 150 200 250 300 350 400 450 500



composite plates (a=h ¼ 5, C ¼ 0:25%, T ¼ 300 K).

Time, t (µs)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 50 100 150 200 250 300 350 400 450 500


Def

lect

ion,

wx1

0-3cm

4D Cell



Time, t (µs)

4D Cell

Def

lect

ion,

wx1

0-3cm

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 50 100 150 200 250 300 350 400 450 500




Def

lect

ion,

wx1

0-3cm

Time, t (µs)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 50 100 150 200 250 300 350 400 450 500


4D Cell


composite plates (a=h ¼ 5, C ¼ 1%, T ¼ 300 K).

Time, t ( s)

Def

lect

ion,

wx1

0-3cm

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 50 100 150 200 250 300 350 400

PresentReddy

µ

Fig. 14. Transient response of simply supported isotropic plate.


4D Cell

Def

lect

ion,

wx1

0-3cm

Time, t ( s)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 50 100 150 200 250 300 350 400 450 500


µ


composite plates (a=h ¼ 5, T ¼ 325 K, C ¼ 0).

Time, t (µs)

4D Cell

Def

lect

ion,

wx1

0-3cm

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 50 100 150 200 250 300 350 400 450 500




Time, t (µs)

4D Cell

Def

lect

ion,

wx1

0-3cm

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 50 100 150 200 250 300 350 400 450 500



composite plates (T ¼ 375 K, C ¼ 0).

Time, t (µs)

4D Cell

Def

lect

ion,

wx1

0-3cm

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 50 100 150 200 250 300 350 400 450 500





the transient response behaviour of all multidirectional

models are same up to approximately t ¼ 75 ls and thevariations in transient behaviour are noticed, when

t > 75 ls, due to different fibre orientations in the

model. The lowest transverse central deflections are

noticed in the 6D cell model, as the balanced diagonalfibres make the structure stiffer. The 4D cell model ex-

hibits the highest transverse central deflection.

5. Conclusions

3D finite element analysis procedures for free vibra-

tion and transient response problems are developed for

multidirectional composite plates under the hygrother-

mal environments. The stiffness properties for multidi-

rectional composites are computed based on the

transformation principles. Results show that the present

finite element analysis procedure using 20 noded iso-

parametric elements is capable of predicting the natural

frequencies of anisotropic laminated composite plates

subjected to moisture and temperature environments.

The fundamental frequencies of multidirectional

composite plates are evaluated under the effect of uni-form moisture and temperature. The free vibration

analysis is carried out for multidirectional composite

plates having different aspect ratios subjected to both

temperature and moisture. Thin (a=h ¼ 40) multidirec-tional composite plates with fibres aligned along more

than three directions, i.e., 4D, 5D, 6D cells, become

unstable at higher moisture concentration levels. Tran-

sient solutions are carried out for simply supportedmultidirectional composite plates subjected to a uniform

pulse load. An increase in transverse central deflections


and period of oscillations is observed for all multidi-

rectional composite models with the increase of mois-

ture concentration and temperature levels. The

variations in transient response are noticed after t ¼ 75ls due to different fibre orientations. Results indicatethat a multidirectional composite with 6D cell model

configuration exhibits higher stiffness under the action

of a uniform pulse load. It can be remarked that the

diagonal fibres make the composite much stiffer.

References

[1] Herrmann G, Armenakas AE. Vibrations and stability of plates

under initial stress. Trans Am Soc Civil Eng 1962;127:458–91.

[2] Brunelle EJ, Robertson SR. Initially stressed mindlin plates.

AIAA J 1974;12:1036–45.

[3] Brunelle EJ, Robertson SR. Vibrations of an initially stressed

thick plate. J Sound Vibrat 1976;45:406–16.

[4] Yang IH, Shieh JA. Vibrations of initially stressed thick,

rectangular, orthotropic plates. J Sound Vibrat 1987;119(3):545–

58.

[5] Whitney JM, Ashton JE. Effect of environment on the elastic

response of layered composite plates. AIAA J 1971;9:1708–13.

[6] Pipes RB, Vinson JR, Chou TW. On the hygrothermal response

of laminated composite systems. J Compos Mater 1976;10:129–

48.

[7] Sai Ram KS, Sinha PK. Hygrothermal effects on the bending

characteristics of laminated composite plates. Comput Struct

1991;40(4):1009–15.

[8] Sai Ram KS, Sinha PK. Hygrothermal effects on the free vibration

of laminated composite plates. J Sound Vibrat 1992;158(1):133–

48.

[9] Mukherjee N, Sinha PK. Three-dimensional Thermo structural

analysis of multidirectional fibrous composite plates. Compos

Struct 1994;28(3):333–46.

[10] Reddy JN. Dynamic (transient) analysis of layered anisotropic

composite material plates. Int J Numer Methods Eng 1983;

19:237–55.

[11] Meimaris C, Day JD. Dynamic response of laminated anisotropic

plates. Comput Struct 1995;55(2):269–78.

[12] Bathe KJ. Finite element procedures in engineering analysis.

Englewood Cliffs, NJ: Prentice-Hall; 1982.

[13] Agarwal BD, Broutman LJ. Analysis and performance of fibre

composites. New York: John Wiley & Sons; 1980.

[14] Whitney JM, McCoullough RL. In: Introduction to anisotropic

elasticity, micro mechanical materials modeling, vol. 2. Lancan-

ster, Basel: Technomic Publishing; 1990.

[15] Zienkiewicz OC. The finite element method. New Delhi: McGraw-

Hill Book Company; 1989.

[16] Hinton E, Scott FC, Ricketts RE. Local least squares stress

smoothing for parabolic isoparametric elements. Int J Numer

Methods Eng 1975;9:235–56.

[17] Cook RD. Concepts and applications of finite element analysis.

New York: John Wiley; 1981.

[18] Tsai SW, Hahn HT. Introduction to composite materials.

Westport, CT: Technomic; 1980.

[19] Phan ND, Reddy JN. Analysis of laminated composite plates

using a higher order shear deformation theory. Int J Numer

Methods Eng 1985;21:2201–19.

[20] Noor AK. Free vibrations of multilayered composite plates.

AIAA J 1972;11(7):1038–9.

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Dynamic response of multidirectional composites in hygrothermal environments