Dynamic response of multidirectional composites in hygrothermal environments
Transcript of 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
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
332 V.V.S. Rao, P.K. Sinha / Composite Structures 64 (2004) 329–338
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.
V.V.S. Rao, P.K. Sinha / Composite Structures 64 (2004) 329–338 333
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.
334 V.V.S. Rao, P.K. Sinha / Composite Structures 64 (2004) 329–338
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
4D Cell5D Cell6D Cell
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).
Moisture concentration, C (%)
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
4D Cell5D Cell6D Cell
Fig. 11. Effect of moisture on fundamental frequency of simply sup-
ported multidirectional composite plates (a=h ¼ 40).
Moisture concentration, C (%)
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
4D Cell5D Cell6D Cell
λ
Fig. 10. Effect of moisture on fundamental frequency of simply sup-
ported multidirectional composite plates (a=h ¼ 10).
V.V.S. Rao, P.K. Sinha / Composite Structures 64 (2004) 329–338 335
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
3D Cell5D Cell6D Cell
Fig. 16. Transient response of simply supported multidirectional
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
3D Cell5D Cell6D Cell
Def
lect
ion,
wx1
0-3cm
4D Cell
Fig. 17. Transient response of simply supported multidirectional
composite plates (a=h ¼ 5, C ¼ 0:5%, 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
1.6
0 50 100 150 200 250 300 350 400 450 500
3D Cell5D Cell6D Cell
Fig. 18. Transient response of simply supported multidirectional
composite plates (a=h ¼ 5, C ¼ 0:75%, T ¼ 300 K).
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
3D Cell5D Cell6D Cell
4D Cell
Fig. 19. Transient response of simply supported multidirectional
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.
336 V.V.S. Rao, P.K. Sinha / Composite Structures 64 (2004) 329–338
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
3D Cell5D Cell6D Cell
µ
Fig. 20. Transient response of simply supported multidirectional
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
3D Cell5D Cell6D Cell
Fig. 21. Transient response of simply supported multidirectional
composite plates (a=h ¼ 5, T ¼ 350 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
3D Cell5D Cell6D Cell
Fig. 22. Transient response of simply supported multidirectional
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
3D Cell5D Cell6D Cell
Fig. 23. Transient response of simply supported multidirectional
composite plates (a=h ¼ 5, T ¼ 400 K, C ¼ 0).
V.V.S. Rao, P.K. Sinha / Composite Structures 64 (2004) 329–338 337
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
338 V.V.S. Rao, P.K. Sinha / Composite Structures 64 (2004) 329–338
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.
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