COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China
©2006 Tsinghua University Press & Springer
A Hybrid Elasticity Method for Bending and Free Vibration of
Composite Laminates
C. F. Lü, W. Q. Chen *
Department of Civil Engineering, Zhejiang University, Hangzhou, 310027 China
Email: [email protected], [email protected]
Abstract Semi-analytical solutions for bending and free vibration of composite laminated beams and plates have been
derived based on 2D/3D elasticity theory using a newly developed hybrid analysis, which perfectly combines the state
space approach (SSA) and the differential quadrature (DQ) technique. The thickness direction of laminates is selected
to be the transfer direction in SSA, and the DQ technique is employed to discretize the domain(s) normal to the transfer
domain. This enables us to transfer the original partial differential equations into a state equation consisting of
first-order ordinary differential equations. In particular, the use of DQ technique makes ease of the treatment of various
boundary conditions, which can not be considered in the conventional exact state-space approach. A transfer relation
between the state vectors at the top and bottom surfaces of the laminates using matrix theory is established, from which
the bending and free vibration problems can be solved eventually.
Comprehensive numerical results have been obtained, which converge rapidly and agree well with that in the literature
as well as finite element calculations. It is concluded that: (1) the DQ technique significantly widens the application of
SSA; and (2) the proposed method is very efficient for analyzing layered structures.
Key words: Elasticity solutions; State space approach; Differential quadrature; Laminates
INTRODUCTION
With the increasing applications of advanced composite materials in modern industry, mathematical models for
predicting mechanical behavior of composite laminates have received vast parallel developments during the past
decades. Among these, numerical and analytical models based on various simplified theories occupy a large part [1, 2].
Many such theories are suitable for global responses of thin or moderately thick laminates, such as natural frequencies
and associated vibration modes, critical buckling loads and gross deflections. In order to obtain accurate stresses and
strains at the ply level near geometric and materials discontinuities or at other required loacal regions, 3D elasticity
solutions were also presented, including analytical and numerical methods [3]. However, 3D exact solutions are always
attainable for simply supported constraints, while many existing 3D numerical methods have other shortcomings, such
as low accuracy, slow convergence or inconveniency for application.
This paper is to introduce a newly developed hybrid semi-analytical elasticity method, a combination of the state space
approach (SSA) [4] and the differential quadrature method (DQM) [5] (SS-DQM). The thickness direction of the
laminates is chosen as the transfer domain using the SSA, while the remaining domain(s) is/are discretized by the DQ
technique, thereby treating the generalized edge constaints conditions feasibly. In order to remove the numerical
instabilities always encountered in the conventional transfer matrix method (CTMM), the joint coupling matrices
(JCMs) [6] are established at the interfaces. Numerical examples are performed to validate the present method, and
effects of parameters are discussed.
THEORETICAL FORMULATIONS
Consider p -layer composite laminates with side lengths a , b and depth h as shown in Fig. 1. The orthotropic ply
orients at a positive angle φ relative to the reference x -axis. The edges are generally constrained, say simply
supported, clamped or free.
⎯ 1165 ⎯
z
b
a
φ
The p-th layer
h
The k-th layer
y
The 1st layer
zp
zk
zk −1
z1
z0
x
Figure 1: Geometry of a multi-layered composite laminates
1. Elasticity-based state equations For the angle-ply orthotropic materials in the Cartesian coordinate system, we can
derive the following state equations based on the theory of elasticity [7],
⎡0
∂
δ=⎢
∂ζ
⎣A2
A1 ⎤
δ,
0 ⎥⎦
where δT = ⎡⎣ Σζ
U V
⎡ ρ 2
⎢ − ρ (1) Ω
⎢
⎢
∂
A1 = ⎢ − sa
∂ξ
⎢
⎢
∂
⎢ − sb
∂η
⎣⎢
⎡
⎢ c10
⎢
⎢
∂
A 2 = ⎢ c4 sa
∂ξ
⎢
⎢
∂
⎢ c5 sb
η
∂
⎣⎢
(1)
− sa
∂
∂ξ
c7
0
Γηζ ⎤⎦ are the state vector, and Ai are
Γξζ
W
∂ ⎤
∂η ⎥⎥
⎥
0 ⎥,
⎥
⎥
c9 ⎥
⎦⎥
− sb
∂
c4 sa
∂ξ
−
2
2
ρ 2
2 ∂
2 ∂
c
s
c
s
Ω
−
−
1 a
6 b
ρ (1)
∂ξ 2
∂η 2
−(c3 + c6 ) sa sb
⎤
⎥
⎥
⎥
∂2
−(c3 + c6 ) sa sb
⎥.
∂ξ∂η
⎥
ρ
∂2
∂2 ⎥
⎥
− (1) Ω 2 − c6 sa2 2 − c2 sb2
ρ
∂ξ
∂η 2 ⎦⎥
∂
c5 sb
∂η
∂2
∂ξ∂η
(2)
Along with the state equation, the induced variables are expressed as
Σξ = −c4 Σζ + c1 sa
∂U
∂V
∂U
∂V
∂V
∂U
, Ση = −c5 Σζ + c3 sa
, Γξη = c6 sa
.
+ c3 sb
+ c2 sb
+ c6 sb
∂ξ
∂η
∂ξ
∂η
∂ξ
∂η
(3)
In the above equations, the dimensionless variables are defined by
(σ z , τ xz , τ yz ) = C66(1) (Σζ , Γξζ , Γηζ )eiωt ,
(4)
(u , v, w) = h(U , V , W )ei ωt ,
where u , v and w are the displacements in x , y and z directions, respectively, σ x , σ y and σ z the normal stresses,
τ xz , τ yz and τ xy the shear stresses, ρ and Cij respectively the material density and elastic constants, ω the circular
frequency. ξ = x / a , η = y / b and ζ = z / h are the dimensionless coordinates, sa = h / a , sb = h / b ,
Ω = ω h ρ (1) C66(1) (the superscript denotes the 1st layer), and ci are material constants defined in Ref. [8].
2. Application of DQ procedure The plate is discretized in x and y directions with the mesh grid of N x × N y . The
DQ procedure [9] is employed to the partial derivatives about ξ and η in Eq. (1), and hence, the following state
equation at an arbitrary discrete point ( i = 1, 2, L, N x , j = 1, 2, L, N y ) is obtained,
⎯ 1166 ⎯
dΣζ ,ij
N
N
N
dU ij
ρ 2
(1)
= − sa ∑ X ik(1)Wkj + c7 Γξζ ,ij ,
= − (1) Ω Wij − sa ∑ X ik Γξζ , kj − sb ∑ Y jk(1) Γηζ ,ik ,
ρ
dζ
dζ
k =1
k =1
k =1
dVij
dζ
Ny
= − sb ∑ Y jk(1)Wik + c9 Γηζ ,ij ,
dΓξζ ,ij
dζ
dΓηζ ,ij
dζ
y
x
k =1
dWij
dζ
x
Nx
Ny
k =1
k =1
= c10 Σζ ,ij + c4 sa ∑ X ik(1)U kj + c5 sb ∑ Y jk(1)Vik ,
N
N
ρ
= c4 sa ∑ X Σζ , kj − (1) Ω 2U ij − c1 sa2 ∑ X ik(2)U kj − c6 sb2 ∑ Y jk(2)U ik − ( c3 + c6 ) sa sb ∑∑ X ik(1)Y jr(1)Vkr ,
ρ
k =1
k =1
k =1
k =1 r =1
Nx
Ny
x
x
Ny
(5)
(1)
ik
Ny
Nx N y
k =1
k =1 r =1
= c5 sb ∑ Y jk(1) Σζ ,ik − ( c3 + c6 ) sa sb ∑∑ X ik(1)Y jr(1)U kr −
N
ρ 2
2
(2)
2
(2)
V
c
s
Ω
−
i
6 a ∑ X ik Vkj − c2 sb ∑ Y jk Vik ,
ρ (1)
k =1
k =1
x
Ny
and the induced variables are
Ny
Nx
Σξ ,ij = −c4 Σζ ,ij + c1 sa ∑ X U kj + c3 sb ∑ Y jk(1)Vik ,
k =1
(1)
ik
k =1
Nx
Ny
k =1
k =1
Ση ,ij = −c5 Σζ ,ij + c3 sa ∑ X ik(1)U kj + c2 sb ∑ Y jk(1)Vik ,
Nx
Ny
k =1
k =1
(6)
Γξη ,ij = c6 sa ∑ X ik(1)Vkj + c6 sb ∑ Y jk(1)U ik .
where X ik( r ) and Y jk( r ) are the weighting coefficients for the r -th derivatives about ξ and η , respectively.
The edge conditions should be incorporated properly into Eq. (5), and summing up all the resulted equations gives
d (k )
δ (ζ ) = A ( k ) δ( k ) (ζ ) ,
dζ
(7)
where the superscript ‘ (k ) ’ denotes the k -th layer.
SOLUTION PROCEDURE
According to the matrix theorem, the transfer relation between the state vectors at the upper and lower surfaces of the
k -th layer is obtained as
δ1( k ) = exp ⎡⎣ (ζ k − ζ k −1 ) A ( k ) ⎤⎦ δ0( k ) = T( k ) δ0( k ) ,
(8)
where the subscripts ‘1’ and ‘0’ denote respectively the upper and lower surfaces of the k -th layer.
1. CTMM solution As usual, the global analysis of the plate can be implemented using the CTMM based on Eq. (8).
Considering the continuity conditions, we can eliminate the state vectors at all the interfaces of the lamina, and derived
the following global transfer relation,
δ1( p ) = Tδ(1)
0 ,
(9)
where T = ∏ k = p T( k ) is the global transfer matrix.
1
If the lateral surfaces of the plate subject to distributed transverse load q( x, y ) , then the boundary conditions at the top
and bottom surfaces can be expressed as
( p)
( p)
(1)
(1)
(1)
Σζ( p,1) = q t , Γξζ
,1 = Γηζ ,1 = 0 ; Σζ ,0 = q b , Γξζ ,0 = Γηζ ,0 = 0 ,
(10)
where qi = C66(1) qi , and the subscripts ‘t’ and ‘b’ denote the top and bottom surfaces, respectively. Substituting Eq. (10)
into Eq. (9) leads to the solvable equation as
⎧q t ⎫ ⎡ t12
⎪ ⎪ ⎢
⎨ 0 ⎬ = ⎢ t 52
⎪ 0 ⎪ ⎢t
⎩ ⎭ ⎣ 62
t13
t 53
t 63
(1)
t14 ⎤ ⎧ U ⎫
⎡ t11
⎪ ⎪
⎥
t 54 ⎥ ⎨ V ⎬ + ⎢⎢ t 51
t 64 ⎦⎥ ⎩⎪ W ⎭⎪0 ⎣⎢ t 61
t15
t 55
t 65
t16 ⎤ ⎧q b ⎫
⎪ ⎪
t 56 ⎥⎥ ⎨ 0 ⎬ ,
t 66 ⎦⎥ ⎩⎪ 0 ⎭⎪
⎯ 1167 ⎯
(11)
in which t ij are the partitioned matrices of the global transfer matrix T . From Eq. (11), the displacement vectors at the
bottom surface are easily solved, and all the state vectors at the interfaces can be obtained by repeated use of Eq. (8).
For free vibration, the surfaces of the plate are tractions free, leading to the following frequency equation,
t12
t13
t14
t 52
t 62
t 53
t 63
t 54 = 0 .
t 64
(12)
2. TM-JCM solution However, numerical instabilities are always encountered during the CTMM solution procedure
in the case of high aspect ratio of sa or sb , large discrete point number and high-order frequencies. Fortunately, this
can be resolved using the JCMs [6], involving which the global analysis is called the TM-JCM solution. To this end,
the continuity conditions at the interfaces and the boundary conditions at the lateral surfaces are written as
(k )
⎫⎪
⎪⎧δ
(p )
J inter ⎨ 1( k +1) ⎬ = 0 , ( k = 1, 2, L, p − 1 ), J b δ(1)
0 = f b , and J t δ1 = f t ,
⎪⎩δ0 ⎪⎭
(13)
where the joint coupling matrices are in the form of
J inter
⎡ i1
= [ I −I ] , J t = J b = ⎢⎢ 0
⎣⎢ 0
0⎤
0 ⎥⎥ ,
i 6 ⎦⎥
0 0 0 0
0 0 0 i5
0 0 0 0
(14)
in which I , i1 , i 5 and i 6 are the identity matrices with the dimensions equivalent to the length of δ , Σζ , Γξζ and
Γηζ , respectively. On the other hand, Eq. (8) is rewritten as
⎧δ(0k ) ⎫ ⎡ I ⎤ ( k )
(k )
⎨ ( k ) ⎬ = ⎢ ( k ) ⎥ δ0 = M k δ 0 , ( k = 1, 2, L, p ).
T
δ
⎦
⎩ 1 ⎭ ⎣
(15)
Assembling all the individual relations in Eqs. (13) and (15) respectively gives rise to
JΔ = f , Δ = MΔ0 ,
(16)
where
J = diag [ J b
Δ = ⎡⎣δ(1)T
b
J inter L J inter
δ1(1)T
δ(2)T
0
J t ] , M = diag ⎡⎣M1
δ1(2)T L δ(0p )T
M 2 L M p ⎤⎦ , f T = ⎡⎣f bT
T
δ(t p )T ⎤⎦ , Δ0 = ⎡⎣δ(1)T
b
0 L 0 f tT ⎤⎦
T
δ(2)T
L δ(0p )T ⎤⎦ .
0
(17)
Using Eq. (16), we get
JMΔ0 = f ,
(18)
from which the state vectors at the lower surfaces of all the lamina can be obtained. For free vibration, the frequency
equation is
JM = 0 .
(19)
NUMERICAL EXAMPLES
In order to illustrate the efficiency of SS-DQM for static and vibration analysis of laminated beams and plates, several
numerical examples are performed. The grid points employed for the DQ procedure in the numerical calculations are
designated according to the Chebyshev-Gauss-Lobatto grid spacing pattern [10],
ξi = 1/ 2 − cos[(i − 1)π /( N x − 1)] / 2 , ( i = 1, 2, L , N x ),
η j = 1/ 2 − cos[( j − 1)π /( N y − 1)] / 2 , ( j = 1, 2, L , N x ).
(20)
1. Convergence study Firstly, free vibration of an isotropic homogeneous SS beam is considered to validate the
convergence of SS-DQM. The aspect ratio is sa = 1/10 and the Poisson’s ratio is ν = 0.3 . It should be pointed out that,
⎯ 1168 ⎯
for laminated beams, the solution procedure is similar to that of plates. The difference lies in that the constitutive
equations for the plane stress problems are obtained by setting the stress components in one direction to be zero, say
σ y = τ xy = τ yz = 0 , and hence, the dimensions of the governing equations are reduced. For simplicity, the relative
formulations are eliminated.
Table 1 listed the lowest six frequencies ω = ( ρ Aω 2 l 4 EI )
14
with different grid point number N x , as well as the exact
elasticity solutions ω0n [11]. It is seen that the present results converge fast with increasing grid point number, and
identical to the exact solutions.
Table 1 Comparisons of the first six frequencies of an isotropic SS beam with the exact elasticity solutions
ω01 = 3.1164264
ω02 = 6.0959843
ω03 = 8.8558660
ω04 = 11.373498
ω05 = 13.662388
ω06 = 15.749609
(3.1416)
(6.2832)
(9.4248)
(12.5664)
(15.7087)
(18.8496)
Nx
Results
Nx
Results
Nx
Results
Nx
Results
Nx
Results
Nx
Results
5
3.1093288
6
6.0167404
8
8.8890550
9
11.499652
12
13.681423
13
15.802314
6
3.1160663
7
6.0985250
9
8.8520906
11
11.378756
14
13.662446
15
15.750743
7
3.1164387
9
6.0961172
11
8.8557620
13
11.373439
15
13.662405
16
15.749516
8
3.1164306
12
6.0959844
12
8.8558454
14
11.373508
17
13.662385
18
15.749596
9
3.1164264
13
6.0959844
14
8.8558667
15
11.373503
18
13.662387
19
15.749605
10
3.1164264
14
6.0959843
15
8.8558660
16
11.373498
19
13.662388
20
15.749609
Table 2 Comparisons of natural frequencies (Hz) with the experimental results for a CF beam ( sa = 1/ 60 )
φ = 15o
φ = 30o
Mode
1
2
3
4
5
Mode
1
2
3
4
5
6
Semi.-15
82.17
512.32
1422.73
2755.34
4487.15
Semi.-15
52.63
328.95
917.32
1787.00
2931.51
4343.23
Exp. [12]
82.5
511.3
1423.4
2783.6
4364.6
Exp. [12]
52.7
331.8
924.7
1766.9
2984
4432.4
Re (%)
0.27
0.20
0.05
1.02
2.81
Re (%)
0.17
0.85
0.80
1.14
1.76
2.01
Note: Re = |SS-DQM-Exp.|/Exp. × 100%.
Table 3 Comparisons of the present results with exact solutions for a 0o/90o/0o SSSS square plate
sa
σ x (a / 2,
σ y (a / 2,
τ xz (0,
τ yz (a / 2,
b / 2, h / 2)
b / 2, h)
b / 2, h / 3)
b / 2, h / 2)
0, h / 2)
τ xy (0, 0, h)
Results
Semi.−5
2.015387
0.802045
−0.558987
0.252788
0.218126
−0.050483
CTMM
Semi. −7
2.005930
0.800848
0.255943
0.217183
0.25
TM-JCM
Semi. −9
---
---
−0.556261
---
---
---
−0.051069
---
Semi. −5
2.020343
0.804023
−0.558415
0.253380
0.214756
−0.049804
Semi. −7
2.005954
0.800855
−0.556268
0.255946
0.217221
−0.051077
Semi. −9
Exact
2.005911
0.800840
−0.556256
0.255902
0.217180
−0.051061
2.005911
0.800840
−0.556256
0.255902
0.217181
−0.051061
Semi. −5
--5.130080
0.801
1.439165
−0.0556
−0.745960
0.256
0.162386
0.2172
0.256375
−0.0511
−0.083804
Semi. −7
5.095495
1.435999
−0.742368
0.163987
0.259162
−0.085939
Pagano [12]
0.5
w(a / 2,
Solution
procedure
TM-JCM
Semi. −9
5.095384
1.435984
−0.742352
0.163958
0.259115
−0.085911
Exact
5.095383
1.435983
−0.742351
0.163958
0.259116
−0.085911
Pagano [12]
---
1.436
−0.742
0.164
0.2591
−0.0859
Secondly, natural frequencies of a single-layered angle ply CF beam made of graphite-epoxy and tested by Abarcar and
Cunniff [12] are computed using SS-DQM. Numerical and experimental results are tabulated in Table 2, from which, it
is seen that the two series of results agree very well.
Finally, consider a symmetric laminated (0o/90o/0o) SSSS square plate subjected to sinusoidal distributed load
q( x, y ) = q0 sin(nπ x / a) cos(mπ y / b) at the top surface. Numerical results for deflection and stresses are obtained
using SS-DQM and compared to the exact ones [11] and that from Pagano [13], see Table 3. All the variables are
⎯ 1169 ⎯
non-dimensionalized following Pagano [13]. It is seen from Table 3 that SS-DQM converges very well and the results
agree perfectly with exact solutions. Note that, for the cases of sa = 0.25 (9 × 9) and sa = 0.5 , the CTMM calculation
deteriorates so that the results are absolutely wrong. In contrast, TM-JCM delivers highly stable and accurate results
even for the strongly thick plate.
2. Effect of aspect ratio Consider the bending of a symmetric laminated (0o/90o/0o) CC beam subjected to uniformly
distributed load q0 on its top surface. Material properties are: EL / ET = 25 , GLT = 0.5 ET , GTT = 0.2 ET and
ν LT = ν TT = 0.25 , where the subscript ‘L’ and ‘T’ denote along and perpendicular to the fiber direction, respectively.
The through-thickness distributions of displacements and stresses for different aspect ratios sa are depicted in Fig. 2,
where U = U (0.15, ζ ) , W = W (0.5, ζ ) , Γ = Γ(0.15, ζ ) and Σξ = Σξ (0.5, ζ ) .
Figure 2: Through-thickness distributions of displacements and stresses of CC beams (0o/90o/0o) with various sa (N = 9)
It is seen that with the increasing of sa , the distribution curves becomes increasingly non-linear. Fig. 2(a) and Fig. 2(d)
show that, when sa increases from 0.05 to 0.5, the axial displacement at the upper interface shifts from negative to
positive value, while the axial normal stress shifts from tensile to compressive stress. Converse phenomenon is
observed at the lower interface. The transverse displacements achieve maximum at the loaded surface (Fig. 2(b)). The
higher the aspect ratio sa , the more significant the deviation of W at the top and bottom surfaces. When sa = 0.05
and sa = 0.1 , maximal shear stress occurs at the mid-surface; it shifts to the surface layer and approaches increasingly
to the beam surface when sa ≥ 0.2 (see Fig. 2(c)).
3. Effect of ply angle Influences of the ply angle φ on the natural frequencies of symmetric laminated SSSS and
CCCC square plates are investigated. The aspect ratio is sa = sb = 0.1 , and material properties are: EL / ET = 40 ,
GLT = 0.6 ET , GTT = 0.5ET and ν LT = ν TT = 0.25 . Variations of the fundamental frequency parameter ω =
ω a 2 ρ / ET h 2 versus φ are plotted in Fig. 3. It is seen that frequencies of the five-layer plates are more sensitive to
the variation of φ than that of the three-layer plates. Frequency of SSSS plate attains maximum when φ = 45o , while
for CCCC plate, φ = 90o .
⎯ 1170 ⎯
Figure 3: Variation of frequencies versus the ply angle φ for symmetric laminated plates
CONCLUSIONS
A hybrid semi-analytical elasticity method (SS-DQM) is proposed for the static and free vibration of multi-layered
beams and plates generally supported at the ends/edges. Introduction of JCMs removes the numerical instabilities
frequently encountered in the CTMM computation. Numerical examples indicate that the present method can deliver
highly converged and accurate results, which can serve as benchmarks for future numerical analyses.
Acknowledgements
This work was supported by the National Science Foundation of China (No. 10432030) and the Program for New
Century Excellent Talents in University.
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⎯ 1171 ⎯