ASME 2014 International Mechanical Engineering Congress & Exposition
IMECE2014
November 14-20, 2014, Montreal, Canada
IMECE2014-37254
A COMPARISON OF COMPUTED DEFLECTIONS OF SYMMETRIC ANGLE-PLY
LAMINATE PLATES BY THE RITZ METHOD AND THE FINITE ELEMENT METHOD
Kenneth Carroll
Sikorsky Aircraft
Rensselaer at Hartford
Stamford, CT 06901
ABSTRACT
Symmetric angle-ply laminates are characterized by full
matrices of extensional and bending stiffnesses. When a simply
supported composite plate is subjected to a lateral load, the
presence of the twist coupling stiffnesses in the governing
differential equations of equilibrium does not allow the
determination of an exact solution for the deflection and
numerical methods must be used. This paper describes a
comparison of computed results obtained using the Ritz method
and the finite element method. The symbolic manipulation
program Maple is used to implement the Ritz method and the
finite element calculations are carried out using ANSYS. The
results show that reliable results can be easily obtained using
both methods.
INTRODUCTION
There are many applications for composite materials in
today's industrial markets.
Composite materials gained
popularity from the aerospace industry with the development of
aircraft. The strength of composite materials can vary
depending on the orientation of the plies in the laminate.
Composite materials provide a high strength to weight ratio and
can be stronger than a metal in a specific configuration. This
paper analyzed the maximum deflection of a simply supported
composite plate using two different methods. A symmetric
cross-ply laminate of 0 and 90 degree plies simplifies in that
A16, A26, Bij, D16, and D26 are all zero. As a result, the
maximum deflection of a symmetric cross-ply laminate can be
solved using the method of separation of variables for an
expanded double Fourier series.
A symmetric angle ply cannot use the same method as a
symmetric cross-ply laminate because there is a fully defined
[D] matrix. The method of separation of variables cannot be
used for symmetric angle ply laminate because the Fourier
expansion will not satisfy the governing differential equation.
Prof. Ernesto Gutierrez-Miravete
Rensselaer at Hartford
Hartford, CT 06120
The Rayleigh-Ritz Method is the alternative method that can be
used for determining the deflection of a symmetric angle
laminate. The Rayleigh-Ritz Method is based on the principle
of the total potential energy. Calculating the total potential
energy with the Rayleigh-Ritz Method, when used with enough
terms in the equations, will converge to the approximate total
deflection so long as the geometric boundary conditions are
satisfied. After the exact solution of the laminated plate was
determined using the method of separation of variables and the
Rayleigh-Ritz Method, the numerical solution of the laminated
plate was solved using models in ANSYS. This paper will
compare the exact solution and numerical solution using both
methods.
NOMENCLATURE
a - length in x-direction (in)
b - length in y-direction (in)
[A] - Extensional Stiffness Matrix (lb/in)
[B] - Coupling Stiffness Matrix (lb)
C - Stiffness Matrix (psi)
[D] - Bending Stiffness Matrix (lb*in)
E - Modulus of Elasticity (psi)
ε - strain (in/in)
G - Shear Modulus (psi)
γ - Engineering Shear Strain (rad)
M - Bending Moment Resultant (lb*in/in)
N - Force Resultant (psi/in)
M+x - Bending Moment Resultant (lb*in/in)
M+xy - Twisting Moment Resultant (lb*in/in)
N+x - Normal Force Resultant (psi/in)
N+xy - Shear Force Resultant (psi/in)
ν - Poisson's Ratio
P - Point Load (lb)
Q - Reduced Stiffness Matrix (psi)
1
Copyright © 2014 by ASME
̅ - Transformed Reduced Stiffness (psi)
Q
q - Applied Distributed Force (psi)
σ - stress (psi)
[T] - Transformation matrix m = cos(θ)
n = sin(θ)
τ - shear stress (psi)
t - thickness (in)
θ - ply angle
uo - displacement in x-direction (in)
vo - displacement in y-direction (in)
wo - displacement in z-direction (in)
wmax - maximum deflection (in)
w - deflection (in)
x - x-direction
y - y-direction
z - z-direction
[ ]s - symmetric laminate
THIN PLAT THEORY
The analysis of thin plates with small deflection makes
the following three assumptions when the deflection, w, is
small in comparison to the thickness of the plate 1:
 There is no deformation in the middle plane of the
plate. This plane remains neutral during bending
 Points of the plate lying initially on a normal-to-themiddle plane of the plate remain on the normal-to-themiddle surface of the plate after bending
 The stresses in the direction transverse to the plate can
be disregarded.
These three assumptions for thin plate theory are based off of
Kirchhoff-Love Plate Theory. Thin plate theory relies on
different boundary conditions to constrain the plate. The three
assumptions that are made for thin plates with small deflections
means that the material of the plate will not be stretched. With
these three assumptions and the boundary conditions the
deflection of the plate, w, can be calculated.
MATERIAL PROPERTIES OF COMPOSITE PLY
A composite ply has material properties that are
unique in each direction. From my research in the textbook by
Hyer, the following material properties for graphite-polymer
composite plies were used:
Table 2: Material Properties of Composite Ply
Edge Length (a)
Ply Thickness
E1
E2
E3
ν12
ν23
ν13
G12
G23
G13
Applied Surface Pressure (q)
EQUATIONS FOR COMPOSITE THIN PLATE THEORY
There are a series of governing equations that are used
for determining the maximum deflection of a laminated plate.
A number of factors that need to be considered when analyzing
a laminated plate are ply material properties, ply orientation,
boundary conditions of the plate, and applied loads. A
laminated plate can be subjected to point loads, in-plane loads,
moments, and distributed applied loads2. All of the plates
analyzed for this project had a distributed applied load.
The first series of equations that are used for a
laminated plate analysis is organized into the Stiffness Matrix.
The Stiffness Matrix shows the relationship between the stress
and strain of the composite in the 1-, 2-, and 3-directions and is
organized into the following 6x6 matrix:
𝐶11
𝐶21
𝐶31
0
0
[ 0
𝐶12
𝐶22
𝐶32
0
0
0
𝜎1
σ2
σ3
τ23 =
τ13
{τ12 }
𝐶13 0
0
𝐶23 0
0
𝐶33 0
0
0 𝐶44 0
0
0 𝐶55
0
0
0
𝜀1
0
𝜀2
0
𝜀3
0
* 𝛾
23
0
𝛾
13
0
{
𝛾
12 }
𝐶 66 ]
The Plane Stress Assumption is used to simplify the
Stiffness Matrix for the laminated plate. The assumptions made
are that the stresses in the plane of the plate are much larger
than the stresses perpendicular to the plane 3. With these
assumptions we can set the σ3, τ23, and τ13 stress components
to zero. These assumptions allow the previous 6x6 Stiffness
Matrix to be reduced to a 3x3 matrix.
2
1
24 inch
0.040 inch
2.25 x 107 psi
1.75 x 106 psi
1.75 x 106 psi
0.248
0.458
0.248
6.38 x 105 psi
4.64 x 105 psi
6.38 x 105 psi
10 psi
3
Timoshenko & Woinowsky-Krieger Theory of Plates and Shells page 1
2
Hyer Stress Analysis of Fiber-Reinforced Composite Materials page 241
Hyer Stress Analysis of Fiber-Reinforced Composite Materials p 165
Copyright © 2014 by ASME
𝜎1
𝐶11
{ σ2 } = [𝐶21
τ12
0
𝐶12
𝐶22
0
This 3x3 matrix is the basis for the Reduced Stiffness Matrix:
𝜎1
𝑄11
{ σ2 } = [𝑄21
τ12
0
𝑄12
𝑄22
0
𝜎x
𝜎1
(4)
{ σ2 } = [𝑇] ∗ { σ𝑦 }
τ𝑥𝑦
τ12
𝜀1
0
0 ] * { 𝜀2 }
𝛾12
𝐶66
𝜀1
0
0 ] * { 𝜀2 }
𝛾12
𝑄66
The Transformed Reduced Stiffness Matrix relates the
stresses and strains in the x-y coordinate system for a ply
oriented at a given angle. The stress-strain relationship for a
ply at an angle, θ gives the equation:
𝜎𝑥
𝑄̅11
{ 𝜎𝑦 } = [𝑄̅12
𝜏𝑥𝑦
𝑄̅16
where:
Q11 = C11
−
2
𝐶13
𝐶33
Q22 = 𝐶22
−
Q12 = C12
2
𝐶23
𝐶33
−
𝐶13 𝐶23
𝐶33
Q66 = 𝐶66
The material properties will vary depending on the
angle of orientation. A ply that is at a 0° orientation will have
different strength properties than a ply at a 45° orientation.
Determining the Transformed Reduced Stiffness Matrix will
allow the stiffness matrix for each ply orientation to be
combined into a single large matrix.
The Transformation Matrix is based on the
trigonometric functions sine and cosine. The matrix [T] allows
the stresses in the x-y coordinate system to correspond to the 12 coordinate system with respect to the angle of the ply. Figure
2 shows how a fiber-reinforced material in the 1-2 coordinate
system relates to the x-y coordinate system.
Figure 1: Fiber-reinforced material in 1-2 and x-y coordinate
systems4
𝑄̅12
𝑄̅22
𝑄̅26
𝜀𝑥
𝑄̅16
𝑄̅26 ] ∗ { 𝜀𝑦 }
𝛾𝑥𝑦
𝑄̅
66
where:
𝑄̅11 = 𝑄11 𝑚4 + 2(𝑄12 + 2𝑄66 )𝑛2 𝑚2 + 𝑄22 𝑛4
𝑄̅12 = (𝑄11 + 𝑄22 − 4𝑄66 )𝑛2 𝑚2 + 𝑄12 (𝑛4 + 𝑚4 )
𝑄̅16 = (𝑄11 − 𝑄12 − 2𝑄66 )𝑛𝑚3 + (𝑄12 − 𝑄22
+ 2𝑄66 )𝑛3 𝑚
𝑄̅22 = 𝑄11 𝑛4 + 2(𝑄12 + 2𝑄66 )𝑛2 𝑚2 + 𝑄22 𝑚4
𝑄̅26 = (𝑄11 − 𝑄12 − 2𝑄66 )𝑛3 𝑚
+ (𝑄12 − 𝑄22 + 2𝑄66 )𝑛𝑚3
𝑄̅66 = (𝑄11 + 𝑄22 − 2𝑄12 − 2𝑄66 )𝑛2 𝑚2 + 𝑄66 (𝑛4
+ 𝑚4 )
After developing the Transformed Reduced Stiffness
Matrix for each ply orientation, the ABD Matrix can be
determined. The ABD Matrix creates expressions for the
normal force resultants and moments acting on the laminated
plate with respect to the transformed reduced stiffness matrix
for each layer and strains and curvatures of the reference
surface5. Each segment of the ABD Matrix is taken from the
transformed reduced stiffness matrix with respect to the
thickness of the ply.
𝑁
̅ ij (zk − zk−1 )
Aij = ∑ Q
k
1
𝑘=1
𝑁
𝐵𝑖𝑗 = ∑
2
𝑘=1
𝑁
𝑚2
[𝑇]= [ 𝑛2
−𝑚𝑛
m = cosθ
4
𝑛2
𝑚2
𝑚𝑛
𝐷𝑖𝑗 =
2𝑚𝑛
−2𝑚𝑛 ]
𝑚2 − 𝑛2
2
̅ ij (zk2 − zk−1
Q
)
k
1
3
̅ ij (zk3 − zk−1
∑Q
)
k
3
𝑘=1
n = sinθ
5
Hyer, Stress Analysis of Fiber-Reinforced Composite Materials p 180
3
Hyer Stress Analysis of Fiber-Reinforced Composite Materials Chapter 9
Copyright © 2014 by ASME
𝜕 2𝑤 𝑜
𝜕 2𝑤 𝑜
𝑦 = 0, 𝑏: 𝑤 = 0 𝑀𝑦 = −𝐷12
− 𝐷22
𝜕𝑥 2
𝜕𝑦 2
=0
Nx
𝑁𝑦
N𝑥𝑦
=
M𝑥
M𝑦
{M𝑥𝑦 }
𝐴11
𝐴12
𝐴16
𝐵11
𝐵12
[𝐵16
𝐴12
𝐴22
𝐴26
𝐵12
𝐵22
𝐵26
The load can be expanded into a double Fourier series:
∞
𝑞(𝑥, 𝑦) = ∑∞
𝑚=1 ∑𝑛=1 𝑝𝑚𝑛 (sin
𝐴16
𝐴26
𝐴66
𝐵16
𝐵26
𝐵66
𝐵11
𝐵12
𝐵16
𝐷11
𝐷12
𝐷16
𝜕𝑁𝑥
𝜕𝑥
+
𝜕𝑁𝑥𝑦
𝜕𝑥
2
𝑜
𝜀𝑥
𝐵16
𝜀𝑦𝑜
𝐵26
𝑜
𝛾𝑥𝑦
𝐵66
*
𝐷16
κ𝑜𝑥
𝐷26
κ𝑜𝑦
𝐷66 ] {κ𝑜𝑥𝑦 }
𝐵12
𝐵22
𝐵26
𝐷12
𝐷22
𝐷26
𝜕𝑁𝑥𝑦
𝜕𝑦
+
𝜕𝑁𝑦
𝜕𝑦
=0
=0
GOVERNING
EQUATIONS
FOR
A
SIMPLY
SUPPORTED SYMMETRIC CROSS-PLY LAMINATE
When the laminated composite plate is symmetric with
a cross-ply orientation the values for A16, A26, Bij, D16, and
D26 are all zero. The governing equations are:
∂x2
+ A66
(A12 + A66 )
𝐷11
𝜕4 𝑤𝑜
𝜕𝑥 4
∂2 uo
∂y2
∂2 uo
∂2 vo
+ (A12 + A66 ) ∂x ∂y = 0
+ A66
∂x ∂y
+ 2(𝐷12 + 2𝐷66 )
∂2 vo
+ A22
∂x2
𝜕4 𝑤 𝑜
𝜕𝑥 2 𝜕𝑦
+
2
∂2 vo
=0
∂y2
𝜕4 𝑤 𝑜
𝐷22 4
𝜕𝑦
=𝑞
The maximum deflection of the simply supported plate will be
at its center. The boundary conditions for the simply supported
edges are:
𝑥 = 0, 𝑎: 𝑤 = 0 𝑀𝑥 = −𝐷11
𝜕 2𝑤 𝑜
𝜕 2𝑤 𝑜
−
𝐷
12
𝜕𝑥 2
𝜕𝑦 2
=0
6
sin
𝑛𝜋𝑦
𝑏
)
∞
w(x, y) = ∑∞
m=1 ∑n=1 a mn (sin
mπx
a
sin
nπy
b
)
with
𝑎𝑚𝑛
=
𝑝𝑚𝑛
𝜋4
𝑚 4
𝑚 2 𝑛 2
𝑛
𝐷11 ( ) + 2(𝐷12 + 2𝐷66 ) ( ) ( ) + 𝐷22 (
𝑎
𝑎
𝑏
𝑏
The solution for the maximum deflection of the plate will then
be:
𝑤=
∞
∑∞
𝑚=1,3,5 ∑𝑛=1,3,5
16𝑞
𝑚𝜋𝑥
𝑛𝜋𝑦
sin
sin
𝑎
𝑏
𝜋6 𝑚𝑛
𝑚 4
𝑚 2 𝑛 2
𝑛 4
𝐷11 ( ) +2(𝐷12 +2𝐷66 )( ) ( ) +𝐷22 ( )
𝑎
𝑎
𝑏
𝑏
GOVERNING
EQUATIONS
FOR
A
SIMPLY
SUPPORTED SYMMETRIC ANGLE LAMINATE
A symmetric balanced laminate
(19)cannot use the same
method as mentioned above for a cross-ply laminate. A
symmetric balanced laminate has a full [D] matrix which will
alter the third governing differential(20)
equation and boundary
conditions to:
𝜕 𝑀𝑥𝑦 𝜕 2 𝑀𝑦
𝜕 2 𝑀𝑥
+2
+
+𝑞 =0
𝜕𝑥 2
𝜕𝑥𝜕𝑦
𝜕𝑦 2
A11
𝑎
The solution for the equation to find q is:
After organizing the ABD Matrix for the symmetric
laminate, the governing equations can be organized with the
simply supported boundary conditions. The three equations
that govern the response of a laminated plate are6:
∂2 uo
𝑚𝜋𝑥
𝐷11
𝜕 4𝑤 𝑜
𝜕 4𝑤 𝑜
𝜕 4𝑤 𝑜
)
+
4𝐷
+
2(𝐷
+
2𝐷
16
12
66
𝜕𝑥 4
𝜕𝑥 3 𝜕𝑦
𝜕𝑥 2 𝜕𝑦 2
𝜕 4𝑤 𝑜
𝜕 4𝑤 𝑜
+ 4𝐷26
+
𝐷
=𝑞
22
𝜕𝑥𝜕𝑦 3
𝜕𝑦 4
𝑥 = 0, 𝑎: 𝑤 = 0 𝑀𝑥 = −𝐷11
𝜕2 𝑤 𝑜
𝜕𝑥 2
− 𝐷12
𝜕2 𝑤 𝑜
2𝐷16 𝜕𝑥𝜕𝑦 = 0𝑦 = 0, 𝑏: 𝑤 = 0 𝑀𝑦 =
𝐷22
𝜕2 𝑤 𝑜
𝜕𝑦 2
𝜕2 𝑤 𝑜
−
𝜕𝑦 2
𝜕2 𝑤 𝑜
−𝐷12 𝜕𝑥 2
−
𝜕2 𝑤 𝑜
− 2𝐷26 𝜕𝑥𝜕𝑦 = 0
Symmetric laminates cannot be solved using the
method of separation of variables because the Fourier
expansion does not satisfy the governing differential equation.
The alternative method that is required for solving for
deflection of a symmetric laminate plate is the Rayleigh-Ritz
Method. The Rayleigh-Ritz Method is based on the principle
of the total potential energy. Calculating the total potential
energy with the Rayleigh-Ritz Method, when used with enough
terms in the equations, will converge to the approximate total
deflection so long as the geometric boundary conditions are
Hyer Stress Analysis of Fiber-Reinforced Composite Materials p584
4
Copyright © 2014 by ASME
satisfied7. The total potential energy for a symmetric angle ply
laminate is given by:
𝑉=
1
∫ ∫(𝐷11
2
4𝐷66 (
𝜕2 𝑤 𝑜
𝜕𝑥𝜕𝑦
𝜕4 𝑤 𝑜
𝜕𝑥 4
+ 2𝐷12
2
) + 4𝐷16
𝜕4 𝑤 𝑜
𝜕𝑥 2 𝜕𝑦 2
𝜕2 𝑤 𝑜
𝜕𝑥 3 𝜕𝑦
+ 𝐷22
+ 4𝐷26
𝜕4 𝑤 𝑜
𝜕𝑦 4
𝜕4 𝑤 𝑜
𝜕𝑥𝜕𝑦 3
+
−
2𝑝𝑤)𝑑𝑥𝑑𝑦
The Rayleigh-Ritz Method assumes that the deflection
of the laminate plate can be expressed as:
w=
The boundary conditions applied to the full plate
model were based on the review of VM82 from the ANSYS
library and the supplemental paper Chapter 6 Shells. The full
plate model in ANSYS is shown below with the following
constraints applied:

Sides 1, 2, 3, & 4 are the simply supported edges of
the model. All four edges are constrained against translation
in the z-direction.

Side 2 and Side 4 are constrained to prevent
translation in the x-direction and rotation in the x-direction.

Side 1 and Side 3 are constrained to prevent
translation in the y-direction and rotation in the y-direction
.
mπx
nπy
∞
∑∞
m=1 ∑n=1 Cij sin a sin b
Side 1
where Cij are unknown coefficients
Equation (26) is substituted into Equation (25) and the
integration is performed. Integrating the combined equation
with respect to x and y will yield a single algebraic equation in
terms of the unknown Cijs. There will be a total of m*n
unknowns in this algebraic equation. To solve for the value of
each unknown constant, one uses the principal of total potential
energy and takes partial derivatives of the equation with respect
to each unknown. This creates a m*n system of equations with
a single unknown in each equation. The system of equations
can then be solved using matrix elimination methods with
Maple. After all of the unknowns are solved, the approximated
deflection of the laminated plate can be found using Equation
(26).
ANSYS MODEL FOR COMPOSITE PLATE
In order to model the layers of the composite plate,
the SHELL181 element was used to create the finite element
model in ANSYS. The SHELL181 element is similar to the
SHELL63 element in that they are both 4 noded elements with
six degrees of freedom at each node. The advantage to using a
SHELL181 element for a composite plate is that it allows for
the plies to layered.
Two sets of trials were run using the SHELL181
elements in ANSYS. The first set of trials modeled only one
quarter of the composite plate. Modeling only one quarter of
the plate is possible due to the symmetry of the plate. The first
set of trials included the analysis of cross-ply laminates and
symmetric angle ply laminates. The second set of trials
modeled the full plate in ANSYS for the symmetric angle ply
trials.
In both sets of trials the thickness of the composite
plies were set at .040" with a 10 psi uniform pressure applied to
the surface of the plate. The edge length for the quarter plate is
12 inches and the edge length for the full plate is 24 inches.
Due to limitations in the modeling software the mesh size for
the full plate was 1.0 while the mesh size for the quarter plate
was 0.75.
7
Side 4
Side 2
Side 3
It is interesting to note that applying the constraints
outlined by Chapter 6 Shells were not adequate to solve the full
plate in ANSYS. After additional research, the constraints
against translation in the x-direction and y-direction were
applied to the edges as previously mentioned. The translational
constraints on the edges were based on the VM82 file from the
ANSYS Verification Manual. After adding these constraints to
the edges of the full plate, a solution was found for the
symmetric angle ply trials.
COMPOSITE THIN PLATE RESULTS
Maple was used to calculate the deflection for the composite
plate trials using the governing equations for both cross-ply
laminates and symmetric angle-ply laminates. The equation for
a specially orthotropic laminate was used for cross-ply
laminates while the Rayleigh-Ritz Method was used for
symmetric ply laminates. In the case of the Rayleigh-Ritz
Method a total of 49 terms were used (i.e. M=7, N=7).
The trials in ANSYS for a quarter plate cross-ply
laminates had a high correlation with the exact solution using
the governing equations for cross-ply laminates. However,
there was a significant difference between the ANSYS result
and the exact solution for the quarter plate symmetric angle
Jones Mechanics of Composite Materials p 251
5
Copyright © 2014 by ASME
laminate trials. A contributing factor for the difference in
results can be attributed to the D16 and D26 terms from the [D]
Matrix. For a cross-ply laminate the [D] Matrix simplifies
because D16 and D26 are both equal to zero. For a symmetric
angle ply laminate both D16 and D26 are non-zero. These two
terms introduce the twisting moment resultant into the
governing equations. D16 and D26 terms responsible for the
coupling of moments and deformations not normally associated
with each other8. The Fourier expansion that is used to develop
the governing equations for the cross-ply laminate cannot be
applied because the expansion with the D16 and D26 terms will
not satisfy the boundary conditions9. The Rayleigh-Ritz
Method is required to calculate the deflection of the plate when
there is a full [D] matrix.
There is no exact solution for a symmetric angle ply
laminate. The Rayleigh-Ritz Method gives an approximation
of the deflection based on the total potential energy of the plate.
The results will converge to the exact solution as more terms
are used in the calculation. For the purpose of this analysis, it
was assumed that the Rayleigh-Ritz Method provides the more
accurate results. This assumption requires a further review of
the ANSYS model for the symmetric angle ply trials. The
model was modified to be a full plate with constraints to have
all four edges be simply supported. Running the trials using a
full plate model for the symmetric angle ply trials resulted in
much closer results to the deflections calculated using the
Rayleigh-Ritz Method. The results for the quarter plate trials
and full plate trials are provided below.
The figures below show the nodal solution for
displacement in the z-direction for both the [+/-45 0 +/-45 0]s
quarter plate model and full plate model. This laminate is
symmetric about its central axis and it is expected that the
displacement gradients would be circular. The figures show a
slightly oval pattern skewed in the direction of 45 o. It is also
interesting to note that the scale for each figure is not the same
even though the laminate stack-ups are the same. It is
concluded that the inclusion of the D 16 and D26 terns in the [D]
Matrix does not provide an accurate approximation of the
deflection using the quarter plate model in ANSYS. The full
plate model for each trial with the simply supported constraints
will produce a solution that is accurate with the exact solution
using the Rayleigh-Ritz Method.
Nodal Displacement for Full Plate - [+/-45 0 +/-45 0]s
Laminate
Composite Laminate Results - 1/4 of Plate Modeled
LAMINATE
STACK-UP
DEFLECTION
ANSYS (IN)
DEFLECTION
MAPLE (IN)
%
ERROR
[0 90 0 90]S
[0 90 0 90 0 90]S
[0 90 0 90 0 90 0 90]S
[+/-30 0 +/-30 0]S
[+/-45 0 +/-45 0]S
[+/-60 0 +/-60 0]S
0.7182
0.2141
0.091
0.1591
0.1452
0.1600
0.7146
0.21196
0.0895
0.1433
0.1304
0.1445
-0.50
-1.0
-1.7
-11.02
-11.3
-10.7
Nodal Displacement for Quarter Plate - [+/-45 0 +/-45 0]s
Lamiante
Composite Laminate Results - Full Plate Modeled
LAMINATE
DEFLEC DEFLECTION
%
STACK-UP
TION
MAPLE (IN)
ERROR
ANSYS
(IN)
[+/-30 0 +/-30 0]S
0.1457
0.1433
-1.64
[+/-45 0 +/-45 0]S
0.1328
0.1304
-1.84
[+/-60 0 +/-60 0]S
0.1469
0.1445
-1.66
8
9
Hyer, Stress Analysis of Fiber-Reinforced Composite Materials p 341
Jones, Mechanics of Composite Materials p 250
6
Copyright © 2014 by ASME
CONCLUSIONS
This project analyzed the maximum deflection for a
simply supported aluminum plate and composite plates. The
deflection of the composite plates were calculated using the
exact solutions in Maple. ANSYS was used to model
composite simply supported plates. The number of plies that
made up the laminated plates varied from eight plies to sixteen
plies and varied in orientation from a cross-ply laminate to a
symmetric angle laminate. The orientation of the composite
fibers had a significant effect on the maximum deflection of the
laminated plate. The following conclusions were made:

The composite plate that had the smallest deflection
was the 16 ply [0 90 0 90 0 90 0 90]s laminate.

The thinnest plate that had the smallest deflection is
the 12 ply [+/-45 0 +/-45 0]s laminate

The higher percent error for the symmetric angle ply
laminates in Table 4 can be attributed to multiple factors
including the introduction of the terms D16 & D26 and
inconsistent constraints along the ANSYS model edges. The
full plate model produces more accurate results as shown in
Table 5.

The quarter plate model for symmetric angle
composite plates was not consistent with a full plate model.
It was found that the full plate model, when constrained
against translation and rotational displacements along the
edges, calculated the more accurate deflection of the
symmetric angle ply plates.

All of the symmetric angle ply laminates are
symmetric about the center plane of the composite plate.
However the nodal solution contour plot produced by
ANSYS for the symmetric angle ply laminates shows oval
shaped displacement gradients skewed at the angle of the
angle plies.

Using a total of 49 terms for the Rayleigh-Ritz Method
does provide a solution that is convergent to the exact
solution.
2. Timoshenko, S. and Woinowsky-Krieger, S. Theory of
Plates and Shells, 2nd Edition, 1959 McGraw-Hill, Inc.
3. Notes from MANE 6180 Mechanics of Composite Materials
R. Naik 2013
4. Manahan, Mer Arnel A Finite Element Study of the
Deflection of Simply Supported Composite Plates Subject to
Uniform Load. RPI Hartford Master's Project December 2011
5. Kirchoff-Love Plate Theory Wikipedia
http://en.wikipedia.org/wiki/Kirchhoff%E2%80%93Love_plate
_theory. Date Accessed: 9/20/2013
6.Agarwal, Bhagwan D. and Broutman, Lawrence J. Analysis
and Performance of Fiber Composites, Second Edition 1990
7. ANSYS Tips by Paul Dufour. http://www.ansys.belcan.com
Date Accessed: 10/15/2013
8.
Young's
Modulus
for
common
materials
http://www.engineeringtoolbox.com/young-modulus-_417.html
Date Accessed: 9/20/2013
9. Jones, Robert M. Mechanics of Composite Materials 1st
Edition, 1975 McGraw-Hill, Inc.
10. Van Keuren, Kevin Structural Optimization of a Simply
Supported Orthotropic Composite Plate RPI Hartford Master's
Project December 2010
11. Chapter 6 Shells (PDF)
http://www.ewp.rpi.edu/hartford/~ernesto/F2013/EP/Materialsf
orStudents/Carroll/Ch6-Shells.pdf
Date Accessed: 12/9/2013
ACKNOWLEDGMENTS
I would like to thank my family and fiancé for supporting me in
my academic career. It has been a long journey, but with their
support I have gotten to my goal. A special thanks to Prof. Ken
Brown and Prof. Rajiv Naik. The courses in Finite Element
Analysis and Mechanics of Composite Materials were the most
interesting classes I took at RPI Hartford. I will use all that I
learned in these classes throughout my career. I also would like
to thank my advisor Prof. Ernesto Gutierrez-Miravete for all of
his guidance during the completion of my degree.
REFERENCES
1. Hyer, Michael W.
Composite Materials.
Publications, Inc.
Stress Analysis of Fiber-Reinforced
Update Edition, 2009 DEStech
7
Copyright © 2014 by ASME
8
Copyright © 2014 by ASME