FACTA UNIVERSITATIS
Series: Architecture and Civil Engineering Vol. 3, No 2, 2005, pp. 121 - 136
ANALYTICAL SOLUTION FOR MULTILAYER PLATES
USING GENERAL LAYERWISE PLATE THEORY
UDC 624.011.1:624.073.8(045)=20
Đorđe Vuksanović, Marina Ćetković
University of Belgrade, Faculty of Civil Engineering, Belgrade, Serbia and Montenegro
Abstract. This paper deals with closed-form solution for static analysis of simply
supported composite plate, based on generalized laminate plate theory (GLPT). The
mathematical model assumes piece-wise linear variation of in-plane displacement
components and a constant transverse displacement through the thickness. It also
include discrete transverse shear effect into the assumed displacement field, thus
providing accurate prediction of transverse shear stresses. Namely, transverse stresses
satisfy Hook's law, 3D equilibrium equations and traction free boundary conditions.
With assumed displacement field, linear strain-displacement relation, and constitutive
equations of the lamina, equilibrium equations are derived using principle of virtual
displacements. Navier-type closed form solution of GLPT, is derived for simply
supported plate, made of orthotropic laminae, loaded by harmonic and uniform
distribution of transverse pressure. Results are compared with 3D elasticity solutions
and excellent agreement is found.
1. INTRODUCTION
In the last decades, scientists have made considerable progress in understanding
behavior of composite laminates. It is noticed that anisotropic multilayered structures
posses transverse discontinuous mechanical properties and higher transverse shear and
transverse normal stress deformability. In order to model such material behavior, two
different approaches have arise, that is equivalent single-layer theories (ESL) and
layerwise theories (LWT).
In single-layer theories one single expression is used through entire thickness to
explain the displacement field of the plate. By this, deformation of multilayer plate is
described by equivalent single layer, thus reducing 3D problem to 2D problem. In order
to include transverse shear deformation, classical (CLPT) and shear deformation theories
have been developed. Namely, CLPT based on Kirchhoff's hypothesis, ignores the effect
of transverse shear deformation. On the other hand, FSDT based on Raissner and
Received October 10, 2005
122
Đ. VUKSANOVIĆ, M. ĆETKOVIĆ
Mindlin, assume constant transverse shear stresses in the thickness direction, giving a
need for shear correction factors to adjust for unrealistic variation of the shear
strain/stress. In order to overcome the limitations of CLPT and FSDT, Higher-order
Shear Deformation Theories (HSDT) which involve higher-order terms in Taylor's
expansion of the displacements in the thickness coordinate were developed.
3,z
3,z
1,x
1,x
(2,y)
(2,y)
ISOTROPIC – SINGLE LAYER PLATE
3,z
3,z
1,x
1,x
(2,y)
(2,y)
ANISOTROPIC – MULTILAYER PLATE
Fig. 1. Displacement and shear stress distribution across the plate thickness
for isotropic and anisotropic plates
In wish of obtaining the accurate prediction of stress distribution and precisely model
kinematics of laminated composites, three-dimensional state of stresses have to be analyzed. Despite conventional 3D elasticity theory, new family of layerwise theories (LWT)
have been developed. Naimely, in LWT displacement field is defined for each layer, thus
including discrete material and discrete shear effects into the assumed displacement field.
Also, it is noticed that LWT models have some analyze advantages over the conventional
3D models. First, as LWT model allows independent in-plane and through the thickness
interpolation, the element stiffness matrix can be computed much faster. Second, even the
volume of input data is reduced, LWT are capable of achieving the same level of solution
accuracy as a conventional 3D models.
This paper deals with displacement based on layerwise theory of Reddy (1987), so
called Generalized Laminated Plate Theory (GLPT). The theory is based on piece-wise
linear variation of in-plane displacement components and constant transverse displacement through the thickness. The Navier-type closed form solution is presented for simply
supported plate loaded by harmonic and uniform distribution of transverse pressure. The
main objective is to compare results of the mentioned theory to 2D and 3D theories, and
to develop mathematical model that will be more efficient than conventional 3D model.
Analytical Solution for Multilayer Plates Using General Layerwise Plate Theory
123
2. GENERALIZED LAYERWISE PLATE THEORY
2.1 Assumptions
The following assumptions are used in the analysis of plate model:
1. Material follows Hooke's Law and each layer is made of orthotropic material.
2. Strain-displacement relation is linear, i.e. geometrical linearity.
3. Displacement and stress distributions over the z thickness plate direction is
determined by Lagrangian linear interpolation functions.
4. The inextensibility of normal is imposed.
2.2 Displacement field
z
N
z
y
N-1
ply direction
θ
x
zk
n
k+1
h
k
hk
2
1
xk ,yk
Ω
x ,y
k-1
3
x
Ωk
k
2
1
Fig. 2. Multilayer composite plate
Consider a laminated plate (Fig. 2) composed of n orthotropic laminae. The integer k,
denotes the layer number that starts from the plate bottom. Plate middle surface
coordinates are (x, y, z), while layer coordinates are (xk, yk, zk). Plate and layer thickness
are denoted as h and hk, respectively.
The displacements components (u1, u2, u3) at a point (x,y,z) can be written as:
u1 ( x, y, z ) = u ( x, y ) + U ( x, y, z )
u 2 ( x, y , z ) = v ( x , y ) + V ( x, y , z )
(1)
u3 ( x, y , z ) = w( x, y )
where (u, v, w) are the displacements of a point (x, y, 0) on the reference plane of the
laminate, and U,V are functions which vanish on the reference plane:
U ( x , y , 0 ) = V ( x, y , 0 ) = 0
(2)
Let now reduce 3-D model to 2D format, by the following approximations:
N
U ( x, y, z ) = ∑ U I ( x, y ) ⋅ Φ I ( z )
I =1
N
(3)
V ( x , y , z ) = ∑ V ( x, y ) ⋅ Φ ( z )
I
I
I =1
I
I
where U and V are undetermined coefficients, and Φ I (z) are layerwise continuous functions of the thickness coordinate. In the view of finite element approximation, the functions Φ I (z) are the one-dimensional (linear, quadratic or cubic) Lagrange interpolation
124
Đ. VUKSANOVIĆ, M. ĆETKOVIĆ
functions of the thickness coordinates (Fig. 3), and (U I, V I) are the values of (u1, u2) at the
I-th plane.
z
N
k+1
h
z
N
1
(k)
N
1
I+1
N
1
k+1
Ψ2
Ψ1
k
h
zk
I
Φ
I
Φ
Φ
2I+1
(k)
Ψ1
k-1
b
1
Ψ2
k
I-1
k-1
1
(k)
(k)
k
2I+3
(k)
Ψ3
2I+1
Φ
2I-1
2I
I-1
1
1
1
Fig. 3. Local and global linear and quadratic Lagrangian interpolation functions
If we assume linear interpolation of in-plane displacement components through the
thickness, linear Lagrangian interpolation functions will have the form:
Φ1 ( z ) = Ψ1(1) ( z )
z1 ≤ z ≤ z 2
⎧Ψ ( I −1) ( z )
Φ I ( z) = ⎨ 2 ( I )
⎩ Ψ1 ( z )
z I −1 ≤ z ≤ z I
z I ≤ z ≤ z I +1
Φ N ( z ) = Ψ1( N ) ( z )
Ψ1( k ) = 1 −
where:
z
hk
( I = 2,3,..., N − 1)
(4)
z N −1 ≤ z ≤ z N
Ψ2( k ) =
z
hk
0 ≤ z ≤ hk
Since the displacement field of GLPT is represented as linear combination of product
of functions of in-plane coordinates and functions of thickness coordinates, independent
in-plane and through the thickness discretization of the plate may be achieved. Also, as
the thickness variation of displacement components is defined in terms of piecewise Lagrangian interpolation functions, the in-plane displacement components will be continuous through the laminate thickness.
z
y
z
z
N-1
U
n-1
I
2
1
N-1
U
I+1
I+1
U
I
x
N
U
N
n
2
U
1
U
I
2
1
u 1 (u 2 )
Fig. 4. Displacement field through the laminate thickness
Analytical Solution for Multilayer Plates Using General Layerwise Plate Theory
125
2.3 Strain-displacement relations of the laminae
The linear strain-displacement relations are given as:
ε xx =
∂u N ∂U I I
+∑
Φ
∂x I =1 ∂x
ε yy =
∂v N ∂V I I
+∑
Φ
∂y I =1 ∂y
γ xy =
∂u ∂v N ⎛ ∂U I ∂V I
+
+∑ ⎜
+
∂x
∂y ∂x I =1 ⎜⎝ ∂y
N
γ xz = ∑ U I
I =1
N
γ yz = ∑ V I
I =1
⎞ I
⎟Φ
⎟
⎠
(5)
dΦ I
dz
dΦ I
dz
From the assumed displacement and deformation field we conclude that in-plane deformation components (εxx, εyy, γxy) will be continuous through the plate thickness. while the
transverse strains (γxz, γyz) need not to be continuous.
z=3
2.4 Constitutive equations of laminae
Laminated plate is made of laminae having a
fibers oriented at an angle θ, measured from the
material x to global x axis (Fig. 5). The stress strain
relations of the laminae is therefore defined in
material coordinate system as:
⎧ σ1 ⎫
⎪ ⎪
⎪⎪ σ 2 ⎪⎪
⎨ τ12 ⎬
⎪τ ⎪
⎪ 13 ⎪
⎩⎪τ 23 ⎭⎪
(k )
⎡C11 C12
⎢
⎢C12 C22
0
=⎢ 0
⎢
0
⎢ 0
⎢ 0
0
⎣
0
0
0
C33
0
0
0
C44
0
0
x
y
θ
2
1
Fig. 5. Global and local
coordinate system of laminae
0 ⎤
⎥
0 ⎥
0 ⎥
⎥
0 ⎥
C55 ⎥⎦
(k )
⎧ ε1 ⎫
⎪ ⎪
⎪⎪ ε 2 ⎪⎪
⎨ γ12 ⎬
⎪γ ⎪
⎪ 13 ⎪
⎩⎪γ 23 ⎭⎪
(k )
where:
σ ( k ) = {σ1 σ 2
ε ( k ) = {ε1 ε 2
)
C(k
ij
τ12
γ12
τ13
γ13
τ 23 }( k )
γ 23 }( k )
T
T
stress components of k-th laminae
in material coordinates
strain components of k-th laminae
in material coordinates
matrix of material elastic coefficients for k-th laminae, given as:
(6)
126
Đ. VUKSANOVIĆ, M. ĆETKOVIĆ
∆ = 1 − ν12 ν 21 − ν 23ν 32 − ν 31ν13 − 2ν13ν 21ν 32 ,
E1 (1 − ν 23ν 32 )
E 1 − ν 31ν13
E (ν + ν 31ν 23 )
,
, C12 = C21 = 2 21
,
C22 = 2
∆
∆
∆
C33 = G12 ,
C44 = G13 ,
C55 = G23 .
C11 =
where:
E1 , E2 , E3
Young's moduli in 1, 2 and 3 directions, respectively
ν ij
Poisson's ratio-defined as ratio of transverse strain in j-direction to axial
G12 , G23 , G13
strain in i-direction (i,j=1,2,3)
shear moduli in the 1-2, 2-3 and 1-3 planes, respectively
Along with following relations:
ν ij
Ei
=
ν ji
(i , j = 1,2 ,3)
Ej
(7)
Since all quantities should be referred to a single coordinate system, we need to establish
transformation relations among stresses and strains in global system to the corresponding
quantities in material (local) coordinate system. The constitutive matrix in global
coordinate system will than be of the form:
Q ij
(k )
(k )
= T −1Cij T
(8)
where:
⎡ cos 2 θ
sin 2 θ
2 sin θ cos θ
0
0 ⎤
⎢
⎥
2
2
− 2 sin θ cos θ
cos θ
0
0 ⎥
⎢ sin θ
T = ⎢− sin θ cos θ sin θ cos θ cos 2 θ − sin 2 θ
0
0 ⎥
⎢
⎥
0
0
0
cos θ sin θ ⎥
⎢
⎢
− sin θ cos θ⎥⎦
0
0
0
⎣
(9)
The Hooke's law for the k-th laminae in the global coordinate system is now:
⎧σ xx ⎫
⎪σ ⎪
⎪ yy ⎪
⎪ ⎪
⎨ τ xy ⎬
⎪τ ⎪
⎪ xz ⎪
⎪⎩ τ yz ⎪⎭
(k )
⎡Q11 Q12
⎢Q
⎢ 12 Q22
= ⎢Q13 Q23
⎢
0
⎢ 0
⎢ 0
0
⎣
Q13
0
Q23
0
Q33
0
0
Q44
0
Q45
0 ⎤
0 ⎥⎥
0 ⎥
⎥
Q45 ⎥
Q55 ⎥⎦
(k )
⎧ε xx ⎫
⎪ε ⎪
⎪ yy ⎪
⎪ ⎪
× ⎨γ xy ⎬
⎪γ ⎪
⎪ xz ⎪
⎪⎩γ yz ⎪⎭
(k )
(10)
where:
σ ( k ) = {σ xx
ε ( k ) = {ε xx
σ yy
ε yy
τ xy
γ xy
τ xz
γ xz
T
τ yz }( k ) stress components of k-th laminae in global coordinates
T
γ yz }( k ) strain components of k-th laminae in global coordinates
Analytical Solution for Multilayer Plates Using General Layerwise Plate Theory
127
θ4
θ3
θ2
θ1
(ε xx ,ε yy ,γ xy )
(γ xz ,γ yz )
(σ xx ,σ yy ,τ xy )
Q ij ( k )
(τ xz ,τ yz )
Fig. 6. Assumed deformation and stress field in laminated composite plate
From equation (9) and already mentioned distribution of strain field through the plate
thickness, we conclude (Fig. 6) that in-plane stresses will be discontinuous at dissimilar
material layers, leaving the possibility for transverse stresses to be continuous through the
plate thickness. This transverse stresses are the one that satisfy constitutive relations, 3D
equilibrium equations and traction free boundary conditions.
2.5 Equilibrium equations
q(x,y)
x
Ω
y
t
z
s
n
Fig. 7. Geometry and load of a plate with curved boundary
The equilibrium equations are derived using the principle of virtual displacements:
0 = δU + δV
(11)
where δU is virtual strain energy, δV is virtual work done by applied forces, which are
given as:
⎧ h/2
⎫
δU = ∫ ⎨ ∫ [σ xx δε xx + σ yy δε yy + τ xy δγ xy + τ xz δγ xz + τ yz δγ yz ] dz ⎬ dx dy
(12)1
Ω ⎩− h / 2
⎭
⎡ ⎛
h⎞
h⎞
h⎞
h ⎞⎤
⎛
⎛
⎛
δV = − ∫ ⎢qb ⎜ x, y,− ⎟ δw0 ⎜ x, y,− ⎟ + qt ⎜ x, y, ⎟ δw0 ⎜ x, y, ⎟⎥ dx dy −
2⎠
2⎠
2⎠
2 ⎠⎦
⎝
⎝
⎝
⎝
Ω ⎣
h/2
⎧
⎫
∫ ⎨ ∫ [σˆ nn δun + σˆ ns δuns + σˆ nt δw] dz ⎬ ds
Γ ⎩−h / 2
⎭
(12)2
128
Đ. VUKSANOVIĆ, M. ĆETKOVIĆ
In the following text, we assume that distributed load acts in the middle surface and the
stress resultants are given as:
⎧ N xx ⎫ h / 2 ⎧σ xx ⎫
⎪
⎪
⎪ ⎪
⎨ N yy ⎬ = ∫ ⎨σ yy ⎬ dz ,
⎪ N ⎪ −h / 2 ⎪ τ ⎪
⎩ xy ⎭
⎩ xy ⎭
⎧Qx ⎫ h / 2 ⎧σ xz ⎫
⎨ ⎬ = ∫ ⎨ ⎬ dz ,
⎩Q y ⎭ −h / 2 ⎩σ yz ⎭
⎧ N I xx ⎫ h / 2 ⎧σ xx ⎫
⎪ I ⎪
⎪ ⎪ I
⎨ N yy ⎬ = ∫ ⎨σ yy ⎬ Φ dz ,
⎪ N I xy ⎪ −h / 2 ⎪ τ ⎪
⎩ xy ⎭
⎩
⎭
⎧Q I x ⎫ h / 2 ⎧σ xz ⎫ dΦ I
dz ,
⎨ I ⎬= ∫ ⎨ ⎬
⎩Q y ⎭ −h / 2 ⎩σ yz ⎭ dz
⎧ Nˆ nn ⎫
⎧σˆ nn ⎫
⎪ ˆ ⎪ h/2 ⎪ ⎪
⎨ N ns ⎬ = ∫ ⎨σˆ ns ⎬ dz .
⎪ Qˆ ⎪ −h / 2 ⎪ σˆ ⎪
⎩ nt ⎭
⎩ n⎭
(13)
If we substitute (13) into (12),(11) we will get the following 2N + 3 partial differential
equations in 2N + 3 variables (u , v, w, U I , V I ) , ( I = 1,… N ):
δu = 0 :
N xx, x + N xy , y = 0
δv = 0 :
N xy , x + N yy , y = 0
δw = 0 :
Qx , x + Q y , y + q = 0
δU I = 0 :
N I xx, x + N I xy , y − Q I x = 0
δV I = 0 :
N I xy ,x + N I yy , y − Q I y = 0
(14)
I = 1,… N
with appropriate geometric boundary conditions:
(u , v, w, U I , V I )
(15)1
and force boundary conditions:
N
N
N
I =1
I =1
I =1
N nn + ∑ N I nn − Nˆ nn = 0 , N ns + ∑ N I ns − Nˆ ns = 0 , Qn + ∑ Q I n − Qˆ n = 0
where:
N nn = N xx nx + N xy n y
N I nn = N I xx n x + N I xy n y
N ns = N xy n x + N yy n y
N I ns = N I xy n x + N I yy n y
Qn = Qx n x + Q y n y
Q I n = Q I x nx + Q I y n y
(15)2
Analytical Solution for Multilayer Plates Using General Layerwise Plate Theory
129
2.6 Constitutive equations of laminate
The constitutive equations of laminate are given as:
N
{N 0 } = [ A]{ε 0 } + ∑ [B I ]{ε I }
I =1
N
{N I } = [B I ]{ε 0 } + ∑ [D JI ]{ε J }
(16)
J =1
where:
{N 0 } = {N xx
N yy
N xy
Qx
Q y }T force components in the middle surface
{N I } = {N xxI
N yyI
N xyI
QxI
Q yI }T force components in I-th plane
⎧ ∂u
{ε } = ⎨
⎩ ∂x
0
⎧ ∂U I
{ε } = ⎨
⎩ ∂x
I
∂v
∂y
∂u ∂v
+
∂y ∂x
∂V I
∂y
∂w
∂x
T
∂w ⎫
⎬ strain components in the middle surface
∂y ⎭
∂U I ∂V I
+
∂y
∂x
T
U
I
⎫
V ⎬ strain components in I-th plane
⎭
I
Constitutive matrixes of the laminate are given as:
n z k +1
[ A ] = [ Apq ] = ∑ ∫ [Q pq ]dz ,
(k )
p, q = 1,2,3,4,5
k =1 z k
n zk +1
I
[B] = [ B pq
] = ∑ ∫ [Q pq ] Φ I dz ,
(k )
p, q = 1,2,3
k =1 zk
n zk +1
[B] = [ B pqI ] = ∑
∫ [Q pq
(k )
]
k =1 zk
n zk +1
dΦ I
dz ,
dz
JI
[D] = [ D pq
] = ∑ ∫ [Q pq ] Φ J Φ I dz ,
(k )
p, q = 4,5
p, q = 1,2,3
k =1 zk
n z k +1
[D] = [ D pqJI ] = ∑
∫ [Q pq
k =1 z k
(k )
]
dΦ J dΦ I
dz ,
dz dz
p, q = 4,5
130
Đ. VUKSANOVIĆ, M. ĆETKOVIĆ
3. NAVIER-TYPE CLOSED FORM SOLUTION
y
z
Simply
supported edges
h
y
b
a
x
τzx
σyy
σxx
σxy
x
τ xy
Fig. 8. Geometry and boundary conditions of multilayer plate
For a rectangular (axb) simply supported cross-ply laminated plate (Fig. 8), composed
of n-layers, the following constitutive coefficients become zero:
A13 = A23 = A45 = B13 = B23 = B45 = D13 = D23 = D45 = 0
(17)
The reduced governing equations (14) become:
N
A11 u, xx + A12 v, xy + A33 ( u, yy + v, xy ) + ∑ [ B11I U ,Ixx + B12I V, xyI + B33I (U ,Iyy + V, xyI ) ] = 0
I =1
N
I
I
A12 u, xy + A22 v, yy + A33 ( u, xy + v, xx ) + ∑ [ B12I U ,Ixy + B22
V, yy
+ B33I ( U ,Ixy + V, xxI ) ] = 0
I =1
N
A44 w, xx + A55 w, yy + ∑ [ B44I U ,Ix + B55I V, yI ] + q = 0
I =1
N
∑ [ B11I u,xx + B12I v,xy + B33I ( u, yy + v,xy ) − B44I w,x ]
I =1
N
+ ∑ [ D11JI U ,Ixx + D12JI V, xyI + D33JI (U ,Iyy + V, xyI ) − D44JI U I ] = 0
I =1
N
∑ [ B12I u,xy + B22I v, yy + B33I ( u,xy + v,xx ) − B55I w,x ]
I =1
N
+ ∑[ D U
I =1
JI
12
I
, xy
+D V
JI
22
I
, yy
+ D (U
JI
33
I
, xy
J = 1,… N
(18)
+V ) − D V ] = 0
I
, yy
JI
55
I
The closed form solution that satisfy the boundary conditions
v = w = V I = N xx = N I xx = 0;
x = 0, a; I = 1,… N
u = w = U = N yy = N
y = 0, b; I = 1,… N
I
I
yy
= 0;
(19)
Analytical Solution for Multilayer Plates Using General Layerwise Plate Theory
131
and differential equations (18) can be found by assuming the following harmonic form for
the unknown variables:
∞
∞
∞
u = ∑ X mn cos αx sin βx, v = ∑ Ymn sin αx cos βy, w = ∑ Wmn sin αx sin βx,
m ,n
m ,n
∞
U = ∑R
I
m,n
I
mn
m ,n
∞
cos αx sin βx, V = ∑ S
I
I
mn
(20)1
sin αx cos βy
m,n
and applied loadings
∞
f ( x, y ) = ∑ qmn sin αx sin βy .
(20)2
m ,n
If we substitute (20)1, (20)2 into the system of 2N + 3equations (18) for each of the
Fourier modes (m,n), we will get the following algebraic equations in matrix form:
⎡ [k ] [k I ] ⎤ ⎧{∆1 }⎫ ⎧{f }⎫
⎬=⎨ ⎬
⎢ I T
JI ⎥ ⎨
⎣[k ] [k ]⎦ ⎩{∆ 2 }⎭ ⎩{0}⎭
(21)
for unknowns ( X mn , Ymn , Wmn , R I mn , S I mn ) ,
where:
k11 = A11α 2 + A33β 2 , k12 = ( A12 + A33 )αβ = k 21 , k 22 = A22β 2 + A33α 2 ,
k33 = A44 α 2 + A55β 2 , k13 = k31 = k 23 = k32 = 0;
k11I = B11I α 2 + B33I β 2 , k12I = ( B12I + B33I )αβ = k 21I ,
k 22I = B33I α 2 + B22I β 2 , k31I = B44I α, k32I = B55I β;
k11JI = D11JI α 2 + D33JI β 2 + D44JI , k12JI = ( D12JI + D33JI )αβ = k 21JI , k 22JI = D33JI α 2 + D22JI β 2 + D55JI
∆1 = { X mn Ymn Wmn }T , ∆ 2 = {R I mn
S I mn }T , { f } = {0 0 − qmn }T .
Once the coefficients ( X mn , Ymn ,Wmn , R j mn , S j mn ) are obtained, the in-plane stresses are
computed from the constitutive equations (6) as:
⎧σ xx ⎫
⎪ ⎪
⎨σ yy ⎬
⎪τ ⎪
⎩ xy ⎭
(k )
⎡Q11
= ⎢⎢Q12
⎢⎣ 0
Q12
Q22
0
0 ⎤
0 ⎥⎥
Q33 ⎥⎦
(k )
⎧
⎪
⎪
⎪
⎨
⎪
⎪
∂u
⎪γ xy =
∂y
⎩
⎫
∂u N ∂U I I
⎪
+∑
Φ
∂x I =1 ∂x
⎪
⎪
∂v N ∂V I I
+∑
Φ
⎬
∂y I =1 ∂y
⎪
∂v N ⎛ ∂U I ∂V I ⎞ I ⎪
⎟Φ ⎪
+
+∑⎜
+
∂x I =1 ⎜⎝ ∂y
∂x ⎟⎠ ⎭
(k )
(22)
132
Đ. VUKSANOVIĆ, M. ĆETKOVIĆ
z
y
z
z
z
N
n
h
n-1
a
k
2
2
1
1
(k)
τ 3D
h
k
h
(k)
τ const
k+1
b
x
k+1
N-1
k
τ 3D
xz
k
(yz)
Fig. 9. Shear stresses satisfying 3D equilibrium equations
Shear stresses can be computed by assuming quadratic variation of shear stresses within
each layer (Fig. 8). This require 3n equations for each of shear stresses (τ(xzk ) , τ (yzk ) ) , where n
is the number of layers. These equations can be obtained from the following conditions:
(1) satisfying traction free boundary conditions at the bottom and top surface of the plate
(2 equations):
τ (1) ( z = 0) = τ ( n ) ( z = hn )
(23)1
(2) providing the continuity of stresses along interfaces (n−1 equations):
τ ( k −1) ( z = hk −1 ) = τ ( k ) ( z = 0) ,
(23)2
(3) assuming the average shear stresses from the constitutive equations (n equations) :
1
hk
hk
∫τ
(k )
(k )
( z ) dz = τ const
.,
(23)3
0
(4) and computing the jump in (τ (xzk ,)z , τ (yzk ), z ) at each interface utilizing the first two
equations of equilibrium in terms of stresses (n−1 equations):
∂τ ( k −1) ( z = hk −1 ) ∂τ ( k ) ( z = 0) ∂τ 3( kD−1) ∂τ (3kD)
−
=
−
∂z
∂z
∂z
∂z
(23)4
4. NUMERICAL EXAMPLES
We considered cross-ply laminated plate, in which each lamina is made of material:
E1 / E2 = 25, G12 / E2 = G13 / E2 = 0.5, G23 / E2 = 0.2, ν12 = ν13 = ν 23 = 0.25
The results are presented in the following normalized form:
2
E 100 h 3
⎛h⎞ 1
⎛h⎞ 1
w = 2 4 w , ( σ xx , σ yy , τ xy ) = ⎜ ⎟
(σ xx , σ yy , τ xy ) , ( τ xz , τ yz ) = ⎜ ⎟ (τ xz , τ yz )
a
q
q0 a
⎝ ⎠ 0
⎝ a ⎠ q0
where h is plate thickness, a,b are plate dimensions in x,y directions, respectively.
Analytical Solution for Multilayer Plates Using General Layerwise Plate Theory
133
The transverse deflection and stresses are computed at the following locations:
h⎞
⎛a b ⎞
⎛a b h⎞
⎛a b h⎞
⎛
w = w ⎜ , ,0 ⎟ , σ xx = σ xx ⎜ , , ± ⎟ , σ yy = σ yy ⎜ , , ± ⎟ , τ xy = σ xx ⎜ 0, 0, ± ⎟
2
2
2
2
2
2
2
4
2⎠
⎠
⎠
⎠
⎝
⎝
⎝
⎝
⎞
⎛a
⎛ b ⎞
τ xz = τ xz ⎜ , 0, 0 ⎟ , τ yz = τ yz ⎜ 0, , 0 ⎟
⎠
⎝2
⎝ 2 ⎠
Results are obtained using program CS_GLPT coded in MATLAB.
4.1 Example I: Cross-ply plate loaded by bi-harmonic distribution of transverse pressure
A cross-ply plate is loaded by bi-harmonic distribution of transverse pressure:
q ( x, y ) = q 0 sin
π
π
x sin y
a
b
shown on Fig. 10.
Fig. 10. Example I: bi-harmonic distribution of transverse pressure
Square, symmetrically laminated plate made of nine layers 00/900/00/900/00/900/00/900/00
is analyzed. Results are shown in Table 1. and Fig. 11.
Table 1. Example I: Comparison between 3D and GLPT
a/h
Solution
3Д
4
[1]
[2]
GLPT
3Д[1]
10
10
0
[2]
GLPT
3Д[1]
GLPT[2]
w
σ xx
σ yy
τ xy
τ xz
τ yz
─
1.797
1
─
0.657
6
─
0.433
5
0.684
0.671
8
0.551
0.552
1
0.539
0.538
8
0.628
0.618
7
0.477
0.471
7
0.431
0.419
6
0.0337
0.0346
6
0.0235
0.0237
5
0.0213
0.0213
1
0.2134
0.223
0.232
5
0.226
0.261
1
0.259
0.275
9
0.223
0.247
0.2118
1
0.219
0.2015
134
Đ. VUKSANOVIĆ, M. ĆETKOVIĆ
⎛a b
⎝2 2
⎞
⎠
σ xx ⎜ , , z ⎟
⎛a
⎝2
⎛a b
⎝2 2
σ yy ⎜ ,
⎛
⎝
⎞
⎠
b
2
τ yz ⎜ 0 , ,
τ xz ⎜ ,0 , z ⎟
Fig. 11. Normalized stresses for square plate (a/h = 10) made of nine layers , loaded with
bi-harmonic transverse pressure
4.2 Example II: Cross-ply plate loaded by transverse distribution of constant pressure
A cross-ply plate is loaded transverse distribution of constant pressure:
q ( x, y ) = q 0
shown on Fig. 12.
Analytical Solution for Multilayer Plates Using General Layerwise Plate Theory
135
Fig. 12. Example II: transverse distribution of constant pressure
Square, symmetrically laminated plate made of three layers 00/900/00 is analyzed.
Results are shown in Table 2.
Table 2. Example II: Comparison between GLPT and 3D, 2D
a/h
Solution
s
w
σxx
σ yy
τxy
τ xz
τ yz
GLPT[2]
3.079
0
3.044
4
2.334
4
0.658
8
1.156
4
1.154
1
0.954
6
0.658
8
0.671
3
0.671
3
0.661
8
0.658
8
1.115
7
1.117
3
0.667
6
0.804
0
0.872
1
0.870
8
0.773
3
0.804
0
0.808
3
0.808
3
0.803
7
0.804
0
0.780
4
0.1094
0.4552
0.5214
0.4435
0.4956
0.6466
9
0.7211
6
0.5312
5
0.3842
6
0.6305
0.4087
0.6279
0.4017
0.7060
0
0.7211
6
0.4150
7
0.3842
6
0.7201
0.3854
0.7201
0.3852
0.7209
6
0.7211
6
0.3845
7
0.3842
6
CC[3]
4
FSDT
CLPT
GLPT[2]
CC[3]
10
FSDT
CLPT
GLPT[2]
10
0
CC[3]
FSDT
CLPT
[1]
─
─
─
0.359
7
─
─
─
0.193
5
─
─
─
0.0973
4
0.0692
3
0.0418
9
0.0611
7
0.0597
4
0.0494
3
0.0418
9
0.0428
5
0.0428
5
0.0420
2
0.0418
9
Pagano N.J.
CS_GLPT - MATLAB program
[3]
E. Carrera, A. Ciuffreda
CLPT - Classical Laminate Plate Theory
FSDT - First-order Shear Deformation Theory
[2]
136
Đ. VUKSANOVIĆ, M. ĆETKOVIĆ
5. CONCLUSION
The proposed mathematical model can be used for analysis of thick, as well as thin
plates. It is shown that differences among 2D and 3D theories are greater for transverse shear
stresses, than for in-plane stresses. Also, differences among theories vanish by increasing
a/h. Finally, as GLPT model is capable to achieve the same solution accuracy as
conventional 3D elasticity model, it can be used as test model of approximation models.
REFERENCES
1. Reddy, J. N., Mechanics Of Laminated Composite Plates, CRC Press, 1996
2. Reddy, J. N., A plate bending element based on a generalized laminated plate theory, International
Journal For Numerical Methods In Engineering, vol. 28, 2275-2292, 1989
3. Chaudhuri, R. A., An Approximate Semi-Analytical Method For Prediction Of Interlaminar Shear Stress
In Arbitrarily Laminated Thick Plate, Comp. Struct., 25, 627-636, 1987
4. Carrera, E., Ciuffreda, A., A Unified Formulation To Asses Theories Of Multilayered Plates For Various
Bending Problems, Composite Structures, in press.
ANALITIČKO REŠENJE VIŠESLOJNIH PLOČA
ZASNOVANO NA OPŠTOJ LAMINATNOJ TEORIJI PLOČA
Đorđe Vuksanović, Marina Ćetković
U ovom radu prikazano je anlitičko rešenje statičke anlize slobodno oslonjene kompzitne
ploče, zasnovano na opštoj laminatnij teoriji ploča (GLPT). Matematički model pretpostavlja deo
po deo linearnu raspodelu komponenata pomeranja u1,u2 i konstantnu promenu komponente
pomeranja u3 po debljini ploče. Pomenuti model u obzir uzima i uticaj smičuće deforamcije unutar
svakog sloja, čime se dobija realna procena smicanja po debljini ploče. Naime, smičući naponi
zadovoljavaju Hook-ov zakon, 3D uslove ravnoteže i granične uslove po naponima. Koristeći
pretpostavljeno polje pomeranja, linearne veze deformacija i pomeranja i konstitutivne jednačine
lamine, primenom principa virtualnih pomeranja izvedeni su uslovi ravnoteže. Navier-ovo rešenje
GLPT prikazano je za slobodno oslonjenu ploču, sačinjenu od ortotropnih lamina, opterećenu biharmonijskom i konstantnom raspodelom poprečnog opterećenja. Dobijeno je izvrsno slaganje sa
3D rešenjem elastične teorije.