Comparative Deflection Analysis of Aluminum and Composite Laminate Plates
Using the Rayleigh-Ritz and the Finite Element Method
by
Kenneth Carroll
An Engineering Project Submitted to the
Graduate Faculty of Rensselaer Polytechnic Institute
In Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
________________________________________________
Ernesto Gutierrez-Miravete, Engineering Project Advisor
Rensselaer Polytechnic Institute
Hartford, Connecticut
December 2013
© Copyright 2013
by
Kenneth Carroll
All Rights Reserve
ii
Table of Contents
LIST OF TABLES .............................................................................................................................................. v
LIST OF FIGURES ........................................................................................................................................... vi
LIST OF SYMBOLS ........................................................................................................................................ vii
ACKNOWLEDGMENT .................................................................................................................................... ix
ABSTRACT...................................................................................................................................................... x
1. Introduction .............................................................................................................................................. 1
2. Methodology ............................................................................................................................................. 3
2.1 Thin Plate Theory ................................................................................................................................ 4
2.2 Material Properties of Aluminum ....................................................................................................... 4
2.3 Equation for Thin Plate Theory ........................................................................................................... 4
2.4 Material Properties of Composite Ply ................................................................................................. 5
2.5 Equations for Composite Thin Plate Theory ....................................................................................... 6
2.5.1 Governing Equations for a Simply Supported Symmetric Laminate ............................................ 9
2.5.2 Governing Equations for a Simply Supported Symmetric Cross-Ply Laminate .......................... 10
2.5.3 Governing Equations for a Simply Supported Symmetric Angle Laminate................................ 11
2.6 Failure Criterion ................................................................................................................................ 12
2.6.1 Maximum Stress Criterion ......................................................................................................... 13
2.6.2 Tsai-Wu Failure Criterion ........................................................................................................... 14
2.7 ANSYS Model for Aluminum Plate .................................................................................................... 14
2.8 ANSYS Model for Composite Plate.................................................................................................... 16
3. Results ..................................................................................................................................................... 19
3.1 Aluminum Thin Plate Results ............................................................................................................ 19
3.2 Composite Thin Plate Results............................................................................................................ 19
3.3 Composite Failure Criterion Results.................................................................................................. 22
3.4 Error Analysis .................................................................................................................................... 22
4. Conclusions ............................................................................................................................................. 24
REFERENCES ................................................................................................................................................ 25
APPENDIX A ................................................................................................................................................. 27
APPENDIX B ................................................................................................................................................. 32
APPENDIX C ................................................................................................................................................. 33
APPENDIX D................................................................................................................................................. 38
iii
APPENDIX E ................................................................................................................................................. 40
APPENDIX F ................................................................................................................................................. 43
APPENDIX G................................................................................................................................................. 46
APPENDIX H................................................................................................................................................. 58
iv
LIST OF TABLES
Table 1: Material Properties of Aluminum
Table 2: Material Properties of Composite Ply
Table 3: Failure Stresses for Graphite Reinforced Composite
Table 4: Composite Laminate Results
Table 5: Composite Laminate Results - Full Plate Modeled
Table 6: Maximum Stress Criterion Results
Table 7: Tsai-Wu Criterion Results
v
LIST OF FIGURES
Figure 1: Strength and stiffness of advanced composite materials
Figure 2: Fiber-reinforced material in 1-2 and x-y coordinate systems
Figure 3: SHELL63 Element
Figure 4: SHELL63 Element with Mesh (Edge Length = 0.75")
Figure 5: SHELL63 Element with Boundary Conditions and Pressure Applied
Figure 6: SHELL181 Element
Figure 7: SHELL181 Element - Full plate with Constraints and Pressure Applied
Figure 8: Nodal Displacement for Quarter Plate - [+/-45 0 +/-45 0]s Laminate
Figure 9: Nodal Displacement for Full Plate - [+/-45 0 +/-45 0]s Laminate
vi
LIST OF SYMBOLS
t - thickness (in)
σ - stress (psi)
τ - shear stress (psi)
ε - strain (in/in)
ν - Poisson's Ratio
E - Modulus of Elasticity (psi)
G - Shear Modulus (psi)
γ - Engineering Shear Strain (rad)
C - Stiffness Matrix (psi)
S - Compliance Matrix (1/psi)
Q - Reduced Stiffness Matrix (psi)
[T] - Transformation matrix
m = cos(θ)
n = sin(θ)
̅ - Transformed Reduced Stiffness (psi)
Q
θ - ply angle
M - Bending Moment Resultant (lb*in/in)
N - Force Resultant (psi/in)
P - Point Load (lb)
uo - displacement in x-direction (in)
vo - displacement in y-direction (in)
wo - displacement in z-direction (in)
εo - Extensional Strain of Reference Surface (in/in)
κox, κoy - Curvature of Reference Surface (1/in)
γoxy - Surface In-plane Shear Strain (rad)
κoxy - Reference Surface Twisting Curvature (1/in)
vii
[A] - Extensional Stiffness Matrix (lb/in)
[B] - Coupling Stiffness Matrix (lb)
[D] - Bending Stiffness Matrix (lb*in)
N+x - Normal Force Resultant (psi/in)
N+xy - Shear Force Resultant (psi/in)
M+x - Bending Moment Resultant (lb*in/in)
M+xy - Twisting Moment Resultant (lb*in/in)
Q+x - Transverse Shear Force Resultant (psi/in)
q - Applied Distributed Force (psi)
𝜎1𝐶 - compression failure stress in the 1-direction
𝜎1𝑇 - tension failure stress in the 1-direction
𝜎2𝐶 - compression failure stress in the 2-direction
𝜎2𝑇 - tension failure stress in the 2-direction
𝐹
𝜏12
- shear failure stress in the 1-2 plane
x - x-direction
y - y-direction
z - z-direction
a - length in x-direction (in)
b - length in y-direction (in)
wmax - maximum deflection (in)
w - deflection (in)
p - uniform pressure (psi)
m,n = the number of terms used in the calculation of a plate's deflection
viii
ACKNOWLEDGMENT
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.
ix
ABSTRACT
This paper analyzes the deflection of a simply supported plate that has a uniform
pressure applied to the surface. Two different analyses were conducted to compare the
deflection results of an aluminum plate with a symmetric composite plate. Results were
determined using the finite element modeling program ANSYS and compared to thin
plate theory. Different symmetric ply arrangements ranging from eight plies to 16 plies
were analyzed. Composite plate stack-ups were evaluated to determine which would
outperform the aluminum plate. The failure criterion were investigated for each
composite plate to determine the range for failure loads and which layer limited each
composite plate. The goal of this project was to analyze the response of composite
plates to find potential replacements for an aluminum plate.
x
1. 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. Composite materials are now being used in automobiles,
boats, sports equipment, protective equipment, homebuilding products, civil
engineering, and aircraft. Figure 1 compares the specific strength to the specific
modulus of a composite material. There are some composite materials that can be
stronger than a metal in a specific configuration. The greatest advantage of using
composite is that the material will have a high strength to weight ratio. While this is not
extremely important in some everyday products, materials with a high strength to weight
ratio are a staple for the aerospace industry. Aerospace companies continue to design
aircraft with lighter materials to increase range and fuel efficiency without sacrificing
strength.
Figure 1: Strength and stiffness of advanced composite materials
Another important advantage to using composite materials versus metallic materials is
cost. When composite technology was first being applied the cost per pound of material
was extremely high. Because of the high cost, the use of composite materials was
restricted to those companies that could afford the research and development. Most of
the early development in using composite materials was in the aerospace industry.
Since the 1960's composite materials have become easier to apply in designs. As it
becomes more common to use composite material in engineering designs, the overall
1
cost of using a composite material has decreased. This is apparent with composite
materials being used in homebuilding and sports equipment.
Further developments in the application and fabrication of composite materials will lead
to more extensive use in future designs. In the aerospace industry many recently
developed aircraft are predominantly made out of composite materials. The Boeing V22 Osprey® is made up of roughly 50% composite materials while the new Boeing 787
Dreamliner® consists of 32 short tons of carbon-fiber-reinforced polymer (CFRP)1. The
final hurdle for using composite materials in the aerospace industry is to develop the allcomposite airplane2. In order to use all composite materials, technology will need to
improve the efficiency of fabricating composite parts.
1
2
http://en.wikipedia.org/wiki/787_Dreamliner#Composite_materials - Boeing 787 Dreamliner
Jones Mechanics of Composite Materials page 25
2
2. Methodology
The deflection of a simply supported plate with a uniformly distributed load to its surface
is calculated in two different ways for this project. The first part of the analysis creates a
finite element model using ANSYS to determine the maximum deflection of the
aluminum and composite plates. ANSYS allows for many different types of analyses.
The analysis completed for this project utilized shell elements for both an aluminum
plate and different composite plate configurations. The overall size of each model in
ANSYS can be reduced due to the symmetry of the plate and the symmetry of the
boundary conditions. The symmetry of the plate requires only one quarter of the plate
to be modeled to determine the solution. The finite element model can be solved after
the boundary conditions and uniform pressure are applied.
The second part of the analysis uses thin plate theory equations solved in Maple to
determine the accuracy of the ANSYS results. To calculate the deflection of the simply
supported rectangular aluminum plate the Navier Solution outlined by Timoshenko in
Theory of Plates and Shells can be applied. The composite plate consists of layers of
plies with different material properties based on the ply angle. In cases of composite
plates different governing equations based on the Classical Lamination Theory and the
Rayleigh-Ritz Method are used to calculate the deflection of the plate.
In the case of a composite plate that is a symmetric cross-ply laminate, the case for
specially orthotropic plates can be applied. For a cross-ply laminate the D16 and D26
values are equal to zero. The case for specially orthotropic plates can only be used for
a cross-ply laminate because the [D] matrix simplifies due to the symmetry of the
laminate. In the case of a symmetric angle ply laminate there are no zero values in the
[D] matrix. The Rayleigh-Ritz Method is based on an equation for the total potential
energy of the system. Solving for the total potential energy of the system will give an
approximate result for the deflection of a composite plate.
The failure criterion for the composite plates will then be discussed. Failure criterion are
important because they calculate the limiting stresses that can be applied to a laminate.
Each type of composite fiber has failure stresses for the x-direction, y-direction, and
shear xy-direction. The failure criterion will outline the limits of the composite plates that
had a maximum deflection equal to or less than the baseline aluminum plate. The
range for failure stresses for a composite ply will be provided for the composite plates.
3
2.1 Thin Plate 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 plate3:



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-the-middle plane of the plate
remain on the normal-to-the-middle 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.
2.2 Material Properties of Aluminum
The following properties for the Aluminum plate were used:
Table 1: Material Properties of Aluminum
Modulus of Elasticity (E) 4
Thickness (h)
Poisson's Ratio (ν)
Edge Length (a)
Applied Surface Pressure (q)
10 x 106 psi
0.250 inch
0.3
24 inch
10 psi
2.3 Equation for Thin Plate Theory
For this analysis it was assumed that the thin plate is square and has a uniformly
applied surface pressure of 10 psi. The square plate will be simply supported along
each edge.
There are a series of equations that can be used for analyzing a simply supported
rectangular plate. For the square plate with a uniform load, the exact solution for the
maximum deflection of the plate is from Theory of Plates and Shells (Timoshenko &
3
4
Timoshenko & Woinowsky-Krieger Theory of Plates and Shells page 1
Young's Modulus for some common materials: http://www.engineeringtoolbox.com/young-modulus-d_417.html
4
Woinowsky-Krieger) Article 30: Alternate Solution for Simply Supported and Uniformly
Loaded Rectangular Plates. The derivation of the maximum deflection equation from
Article 30 can be found in Appendix A. The derived equation for the maximum
deflection of the plate can be expressed by:
wmax = α
where D =
(
𝑞∗𝑎4
𝐷
)
𝐸∗ℎ3
12∗(1−𝜈2 )
[1]
[2]
α is a factor that is dependent on the ratio of the edge length of the plate. Appendix B
shows Table 8 from Theory of Plates and Shells for the numerical factors for a uniformly
loaded and simply supported rectangular plate. Equation (1) is the governing equation
for calculating the maximum deflection of the aluminum plate using thin plate theory.
2.4 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)
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
Table 2 is based off of a table from Stress Analysis of Fiber-Reinforced Composite
Materials by Michael Hyer. The material properties in metric units can be found in
Appendix H.
5
2.5 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 loads5. All of the plates
analyzed for this project had a distributed applied load. The equations required for
determining the ABD of each laminated plate analysis are fully outlined in Appendix C.
The following section will provide a brief description of each equation used and the
development of the governing equations for Classical Lamination Theory and fiber
reinforce laminated plates.
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:
𝜎1
σ2
σ3
τ23 =
τ13
{τ12 }
𝐶11
𝐶21
𝐶31
0
0
[ 0
𝐶12
𝐶22
𝐶32
0
0
0
𝐶13
𝐶23
𝐶33
0
0
0
0
0
0
𝐶44
0
0
0
0
0
0
𝐶55
0
𝜀1
0
𝜀2
0
𝜀3
0
* 𝛾
23
0
𝛾13
0
𝐶 66 ] {𝛾12 }
[3]
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 plane6. 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.
𝜎1
𝐶11
{ σ2 } = [𝐶21
τ12
0
𝐶12
𝐶22
0
𝜀1
0
0 ] * { 𝜀2 }
𝛾12
𝐶66
This 3x3 matrix is the basis for the Reduced Stiffness Matrix:
5
6
Hyer Stress Analysis of Fiber-Reinforced Composite Materials page 241
Hyer Stress Analysis of Fiber-Reinforced Composite Materials p 165
6
[4]
𝜎1
𝑄11
{ σ2 } = [𝑄21
τ12
0
𝜀1
0
0 ] * { 𝜀2 }
𝛾12
𝑄66
𝑄12
𝑄22
0
[5]
where:
Q11 = C11 −
2
𝐶13
Q12 = C12 −
𝐶33
Q22 = 𝐶22 −
2
𝐶23
𝐶13 𝐶23
[6]
𝐶33
Q66 = 𝐶66
𝐶33
The relationships between the terms of the Q Matrix and the Stiffness Matrix are shown
by Equation (6). Using this method requires more calculations when organizing the
terms of the Q Matrix. A more convenient series of equations is outlined by Hyer with
each equation being in terms of the engineering constants. The alternative equations
for the 3x3 Q Matrix are7:
Q11 =
𝐸1
Q12 =
1−𝜈12 𝜈21
Q22 =
𝜈12 𝐸2
1−𝜈12 𝜈21
𝐸2
=
𝜈21 𝐸1
[7]
1−𝜈12 𝜈21
Q66 = G12
1−𝜈12 𝜈21
It is important to note that ν12 is not equal to ν21. The Poisson's Ratio in the 1-2
direction and 2-1 direction are related to one another by the composite material
properties. The three equalities for the Poisson's ratios are expressed as:
𝜈12
𝜈21
=
𝐸1
𝐸2
𝜈13
𝜈31
=
𝐸1
𝐸3
𝜈23
𝜈32
=
𝐸2
𝐸3
[8]
In the case of an isotropic material like aluminum, the equations for the 3x3 Q Matrix
simplify because the Poisson's ratio is the same in all of the principle direction. The
equations shown by (7) would then become:
Q11 = Q22 =
𝐸
1−𝜈2
Q12 =
𝐸
1−𝜈2
Q66 = G =
𝐸
[9]
2(1+𝜈)
As mentioned earlier, one of the factors that will need to be considered when analyzing
a composite plate is the ply angle. 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.
7
Hyer, Stress Analysis of Fiber-Reinforced Composite Materials p 172
7
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 1-2
coordinate system with respect to the angle of the ply. Figure 2 shows how a fiberreinforced material in the 1-2 coordinate system relates to the x-y coordinate system.
Figure 2: Fiber-reinforced material in 1-2 and x-y coordinate systems8
𝑚2
𝑛2
2𝑚𝑛
2
2
[𝑇]= [ 𝑛
𝑚
−2𝑚𝑛 ]
−𝑚𝑛 𝑚𝑛 𝑚2 − 𝑛2
m = cosθ n = sinθ
𝜎x
𝜎1
{ σ2 } = [𝑇] ∗ { σ𝑦 }
τ𝑥𝑦
τ12
[10]
[11]
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
𝑄̅12
𝑄̅22
𝑄̅26
𝜀𝑥
𝑄̅16
𝑄̅26 ] ∗ { 𝜀𝑦 }
𝛾𝑥𝑦
𝑄̅
[12]
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
8
Hyer, Stress Analysis of Fiber-Reinforced Composite Materials p 180
8
[13]
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 surface9. Each segment of the ABD Matrix is taken from the transformed
reduced stiffness matrix with respect to the thickness of the ply.
𝑁
Aij = ∑
𝑘=1
̅ ij (zk − zk−1 )
Q
k
𝑁
1
𝐵𝑖𝑗 = ∑
2
𝑘=1
𝑁
1
𝐷𝑖𝑗 = ∑
3
𝑘=1
Nx
𝑁𝑦
N𝑥𝑦
=
M𝑥
M𝑦
{M𝑥𝑦 }
𝐴11
𝐴12
𝐴16
𝐵11
𝐵12
[𝐵16
𝐴12
𝐴22
𝐴26
𝐵12
𝐵22
𝐵26
[14]
2
̅ ij (zk2 − zk−1
Q
)
k
[15]
3
̅ ij (zk3 − zk−1
Q
)
k
[16]
𝐴16
𝐴26
𝐴66
𝐵16
𝐵26
𝐵66
𝐵11
𝐵12
𝐵16
𝐷11
𝐷12
𝐷16
𝐵12
𝐵22
𝐵26
𝐷12
𝐷22
𝐷26
𝜀𝑥𝑜
𝐵16
𝜀𝑦𝑜
𝐵26
𝑜
𝛾𝑥𝑦
𝐵66
*
𝐷16
κ𝑜𝑥
𝐷26
κ𝑜𝑦
𝐷66 ] {κ𝑜𝑥𝑦 }
[17]
2.5.1 Governing Equations for a Simply Supported Symmetric Laminate
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 are10:
𝜕𝑁𝑥
𝜕𝑥
+
𝜕𝑁𝑥𝑦
𝜕𝑥
𝜕𝑁𝑥𝑦
+
𝜕𝑦
𝜕𝑁𝑦
𝜕𝑦
=0
=0
[18]
[19]
[20]
9
Hyer Stress Analysis of Fiber-Reinforced Composite Materials Chapter 9
Hyer Stress Analysis of Fiber-Reinforced Composite Materials p584
10
9
𝜕 2 𝑀𝑥𝑦 𝜕 2 𝑀𝑦
𝜕 2 𝑀𝑥
+2
+
+𝑞 =0
𝜕𝑥 2
𝜕𝑥𝜕𝑦
𝜕𝑦 2
And three partial differential equations that govern the displacement response of a fiberreinforced laminated plate are11:
𝐴11
𝜕 2 𝑢𝑜
𝜕𝑥 2
𝐵11
𝐴16
𝜕 2 𝑢𝑜
𝜕𝑥 2
+ 𝐴66
𝜕𝑥𝜕𝑦
𝜕3 𝑤 𝑜
− 3𝐵16
𝜕𝑥 3
+ (𝐴12 + 𝐴66 )
𝐵16
𝐷11
𝜕2 𝑢𝑜
+ 2𝐴16
𝜕4 𝑤 𝑜
𝜕3 𝑤 𝑜
𝜕𝑥 3
−
𝜕𝑦 2
3
𝜕 𝑤𝑜
𝜕𝑥 2 𝜕𝑦
𝜕 2 𝑢𝑜
+ 𝐴16
+ 𝐴26
𝜕𝑥𝜕𝑦
𝜕4 𝑤 𝑜
𝜕2 𝑣 𝑜
𝜕𝑥 2
𝜕 2 𝑢𝑜
𝜕𝑦 2
3
𝜕 𝑤𝑜
𝜕𝑥 2 𝜕𝑦
− 3𝐵26
𝜕3 𝑣 𝑜
𝜕𝑥 2 𝜕𝑦
− 3𝐵26
𝜕𝑥𝜕𝑦 2
𝜕𝑥𝜕𝑦 2
−
+ 4𝐷26
2
𝜕𝑥𝜕𝑦
𝜕3 𝑣 𝑜
𝜕𝑥𝜕𝑦
− 𝐵26
2
− 𝐵22
2
+ 𝐴26
𝜕2 𝑣 𝑜
𝜕𝑦 3
3
𝜕 𝑣𝑜
𝜕𝑦 3
𝜕𝑦 2
=0
+ 𝐷22
3
− 𝐵16
−
[21]
𝜕2 𝑣 𝑜
𝜕𝑦 2
=0
𝜕4 𝑤 𝑜
𝜕𝑥𝜕𝑦
3
𝜕 𝑢𝑜
𝜕2 𝑣 𝑜
+ 𝐴22
𝜕𝑥𝜕𝑦
𝜕3 𝑤 𝑜
𝐵22 3
𝜕𝑦
−
𝜕𝑥 2 𝜕𝑦
𝜕 2 𝑢𝑜
𝜕2 𝑣 𝑜
𝜕𝑥𝜕𝑦
𝜕3 𝑤 𝑜
𝐵26 3
𝜕𝑦
+ 2𝐴26
𝜕𝑥 2
𝜕3 𝑤 𝑜
𝜕4 𝑤 𝑜
− (𝐵12 + 2𝐵66 )
(𝐵12 + 2𝐵66 )
𝜕3 𝑤 𝑜
𝜕2 𝑣 𝑜
+ 𝐴66
+ 2(𝐷12 + 2𝐷66 )
𝜕𝑥 3 𝜕𝑦
𝜕 3 𝑢𝑜
3𝐵16 2
𝜕𝑥 𝜕𝑦
+ (𝐴12 + 𝐴66 )
− (𝐵12 + 2𝐵66 )
− (𝐵12 + 2𝐵66 )
+ 4𝐷16
𝜕𝑥 4
𝜕 3 𝑢𝑜
𝐵11 3
𝜕𝑥
𝜕 2 𝑢𝑜
−
[22]
𝜕4 𝑤 𝑜
𝜕𝑦 4
𝜕3 𝑣 𝑜
𝜕𝑥 3
−
−
=𝑞
[23]
2.5.2 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. This simplifies the above governing equations
to:
𝐴11
𝜕 2 𝑢𝑜
𝜕𝑥 2
+ 𝐴66
(𝐴12 + 𝐴66 )
𝐷11
𝜕4 𝑤 𝑜
𝜕𝑥 4
𝜕 2 𝑢𝑜
𝜕𝑦 2
𝜕 2 𝑢𝑜
𝜕𝑥𝜕𝑦
+ (𝐴12 + 𝐴66 )
+ 𝐴66
+ 2(𝐷12 + 2𝐷66 )
𝜕2 𝑣 𝑜
𝜕𝑥 2
+ 𝐴22
𝜕4 𝑤 𝑜
𝜕𝑥 2 𝜕𝑦
𝜕2 𝑣 𝑜
𝜕𝑥𝜕𝑦
𝜕2 𝑣 𝑜
𝜕𝑦 2
+ 𝐷22
2
=0
[24]
=0
[25]
𝜕4 𝑤 𝑜
𝜕𝑦 4
=𝑞
The maximum deflection of the simply supported plate will be at its center.
boundary conditions for the simply supported edges are:
11
Hyer Stress Analysis of Fiber-Reinforced Composite Materials p590
10
[26]
The
𝑥 = 0, 𝑎: 𝑤 = 0 𝑀𝑥 = −𝐷11
𝑦 = 0, 𝑏: 𝑤 = 0 𝑀𝑦 = −𝐷12
𝜕2 𝑤 𝑜
− 𝐷12
𝜕𝑥 2
𝜕2 𝑤 𝑜
− 𝐷22
𝜕𝑥 2
𝜕2 𝑤 𝑜
𝜕𝑦 2
𝜕2 𝑤 𝑜
𝜕𝑦 2
=0
[27]
=0
[28]
The load can be expanded into a double Fourier series:
∞
𝑞(𝑥, 𝑦) = ∑∞
𝑚=1 ∑𝑛=1 𝑝𝑚𝑛 (sin
𝑚𝜋𝑥
𝑎
sin
𝑛𝜋𝑦
𝑏
)
[29]
The solution for the equation to find q is:
∞
𝑤(𝑥, 𝑦) = ∑∞
𝑚=1 ∑𝑛=1 amn (sin
𝑚𝜋𝑥
𝑎
sin
𝑛𝜋𝑦
𝑏
)
[30]
with
𝑎𝑚𝑛 =
𝑝𝑚𝑛
𝜋4
𝑚 4
𝑎
𝑚 2 𝑛 2
𝑎
𝑏
[31]
𝑛 4
𝑏
𝐷11 ( ) +2(𝐷12 +2𝐷66 )( ) ( ) +𝐷22 ( )
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
[32]
2.5.3 Governing Equations for a Simply Supported Symmetric Angle Laminate
A symmetric balanced laminate 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 equation and boundary conditions to:
𝐷11
𝜕4 𝑤 𝑜
𝜕𝑥 4
𝜕4 𝑤 𝑜
𝜕4 𝑤𝑜
𝜕4 𝑤 𝑜
+ 4𝐷16 𝜕𝑥 3 𝜕𝑦 + 2(𝐷12 + 2𝐷66 ) 𝜕𝑥 2 𝜕𝑦 2 + 4𝐷26 𝜕𝑥𝜕𝑦 3 + 𝐷22
𝑥 = 0, 𝑎: 𝑤 = 0 𝑀𝑥 = −𝐷11
𝑦 = 0, 𝑏: 𝑤 = 0 𝑀𝑦 = −𝐷12
𝜕2 𝑤 𝑜
𝜕𝑥 2
𝜕2 𝑤 𝑜
𝜕𝑥 2
− 𝐷12
− 𝐷22
𝜕2 𝑤 𝑜
𝜕𝑦 2
𝜕2 𝑤 𝑜
𝜕𝑦 2
− 2𝐷16
− 2𝐷26
𝜕4 𝑤𝑜
𝜕𝑦 4
𝜕2 𝑤 𝑜
𝜕𝑥𝜕𝑦
𝜕2 𝑤 𝑜
𝜕𝑥𝜕𝑦
=𝑞
[33]
=0
[34]
=0
[35]
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
minimization of the total potential energy. Calculating the total potential energy with the
11
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
satisfied12. The total potential energy for a symmetric angle ply laminate is given by:
𝑉=
1
∫ ∫(𝐷11
2
𝜕4 𝑤 𝑜
𝜕𝑥 4
+ 2𝐷12
𝜕4 𝑤 𝑜
+ 𝐷22
2
𝜕𝑥 2 𝜕𝑦
𝜕4 𝑤 𝑜
4𝐷26
𝜕𝑥𝜕𝑦 3
𝜕4 𝑤 𝑜
𝜕𝑦 4
𝜕2 𝑤 𝑜
+ 4𝐷66 (
− 2𝑝𝑤)𝑑𝑥𝑑𝑦
2
𝜕2 𝑤 𝑜
) + 4𝐷16 𝜕𝑥 3𝜕𝑦 +
𝜕𝑥𝜕𝑦
[36]
The Rayleigh-Ritz Method assumes that the deflection of the laminate plate can be
expressed as:
∞
𝑤 = ∑∞
𝑚=1 ∑𝑛=1 𝐶𝑖𝑗 sin
𝑚𝜋𝑥
𝑎
sin
𝑛𝜋𝑦
𝑏
[37]
where Cij are unknown coefficients
Equation [37] is substituted into Equation [36] and the integration is performed.
Integrating equation [36] with respect to x and y will yield a single algebraic equation for
the total potential energy in terms of the pressure, p, and the unknown Cijs. The
resulting equation is differentiated with respect to each of the unknown Cijs and each
equation is made equal to zero. This creates a m*n system of coupled simultaneous
algebraic equations for the Cijs.. 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 [37].
2.6 Failure Criterion
Failure is due to a part not being able to carry an applied load. For a composite
laminate the failure needs to be a function of the direction of the applied stress relative
to the direction of the fibers13. For a composite plate that may have plies oriented at
different angles, it is important to understand how the stress components impact failure.
Failure can occur due to tension, compression, shear or a combination of all three. Two
methods that are commonly used to determine the failure loads for a ply are the
Maximum Stress Criterion and the Tsai-Wu Criterion. The failure stresses used for both
the Maximum Stress Criterion and the Tsai-Wu Criterion are shown in Table 3.
12
13
Jones Mechanics of Composite Materials p 251
Hyer Stress Analysis of Fiber-Reinforced Composite Materials p 387
12
Table 3: Failure Stress (psi) for Graphite Reinforced Composite14
𝜎1𝐶
𝜎1𝑇
𝜎2𝐶
𝜎2𝑇
𝐹
𝜏12
Graphite-Reinforced Composite (psi)
-181,297
217,557
-29,008
7,252
14,504
The values provided in Table 3 were obtained from the textbook written by Hyer. The
stresses provided are the limits for compression and tension in the 1-direction and the
2-direction. From the assumptions mentioned in thin plate theory, the stresses acting in
the 3-direction are in the transverse direction on the plate and can be neglected.
The failure criterion methods used will provide a range of stresses for the laminate plate
in the 1-direction, 2-direction, and 12-shear direction. It was assumed that the
composite laminate was subjected to biaxial forces. The stress applied to the plate in
the y-direction is equal to one half the stress that is applied in the x-direction. Not all of
the composite laminate trials had the failure criterion determined. In each case the
range for the maximum applied load that could be applied to the laminate under biaxial
stress is determined.
2.6.1 Maximum Stress Criterion
The maximum stress failure criterion can be stated as15:
A fiber-reinforced composite material in a general state of stress will fail when:
EITHER,
The maximum stress in the fiber direction equals the maximum stress in a uniaxial
specimen of the same material loaded in the fiber direction when it fails;
OR,
The maximum stress perpendicular to the fiber direction equals the maximum stress in
a uniaxial specimen of the same material loaded perpendicular to the fiber direction
when it fails;
OR
14
15
Hyer Stress Analysis of Fiber-Reinforced Composite Materials p 395
Hyer Stress Analysis of Fiber-Reinforced Composite Materials p 395
13
The maximum shear stress in the 1-2 plane equals the maximum shear stress in a
specimen of the same material loaded in shear in the 1-2 plane when it fails.
A simpler way to look at the maximum stress failure criterion is that material will not fail
as long as
𝜎1𝐶 < 𝜎1 < 𝜎1𝑇
[38]
𝜎2𝐶 < 𝜎2 < 𝜎2𝑇
[39]
𝐹
|𝜏12 | < 𝜏12
[40]
2.6.2 Tsai-Wu Failure Criterion
The Tsai-Wu Criterion is simpler than the Maximum Stress Criterion. The Tsai-Wu is
different from the maximum stress criterion in that there is only one governing equation.
2
𝐹1 𝜎1 + 𝐹2 𝜎2 + 𝐹11 𝜎12 + 𝐹22 𝜎22 + 𝐹66 𝜏12
− √𝐹11 𝐹22 𝜎1 𝜎2 = 1
[41]
where
1
1
1
1
𝐹1 = (𝜎𝑇 + 𝜎𝐶 )
1
1
2
2
1
𝐹2 = (𝜎𝑇 + 𝜎𝐶 )
1
𝐹11 = − 𝜎𝑇 𝜎𝐶
1 1
1
𝐹22 = − 𝜎𝑇 𝜎𝐶
𝐹66 = (𝜏𝐹 )
2 2
[42]
2
12
The Tsai-Wu failure criterion is considered to be a quadratic failure criteria. Solving for
the stresses of a laminate plate using the Tsai-Wu criterion will yield a positive and
negative result. This is similar to how the Maximum Stress criterion provides a range of
values where the laminate will not fail. The third difference between the Tsai-Wu
criterion and the Maximum Stress criterion is that the Tsai-Wu criterion does not
indicate the mode of failure for a ply. The mode of failure could be determined after
further calculations.
2.7 ANSYS Model for Aluminum Plate
The aluminum plate was modeled in ANSYS using a SHELL63 element. The SHELL63
element was used for the aluminum plate analysis because the element has both
bending and membrane capabilities and is widely used for a linear elastic analysis.
Each node of the element has six degrees of freedom, three translational degrees of
freedom and three rotational degrees of freedom. A representation of the SHELL63
Element from ANSYS is shown in Figure 3.
14
The SHELL63 element was used to create an area by dimensions in the active
workspace. Due to the symmetry of the aluminum plate, only a quarter of the plate has
to be created for the analysis. The 12 inch square was oriented in the x-y plane. A
thickness of 0.250 inch was entered for the plate in the z-direction. The edge length for
the generated mesh was 0.75 inch. This mesh size was used so that the number of
meshed elements for the model were evenly spaced. Figure 4 below shows the
generated plate in ANSYS with meshing.
Figure 3: SHELL63 Element16
Side 1
Side 4
Y
Z
X
Origin
Side 2
Side 3
Figure 4: SHELL63 Element with Mesh (Edge Length = 0.75")
16
SHELL63 Element diagram from ANSYS
15
The aluminum plate had a series of constraints applied in order to compute the
maximum deflection. The constraints shown for the finite element model in Figure 4 are
listed below:






The keypoint at the origin of the active workspace is where the maximum
deflection should be measured. The keypoint at the origin is constrained in Ux
and Uy directions.
Side 1 and Side 2 of the finite element model were the sides that were simply
supported.
Side 1 is constrained in the Uz direction to prevent translation in the z-direction
and in ROTy to prevent rotation in the y-direction.
Side 2 is constrained in the Uz direction to prevent translation in the z-direction
and in ROTx to prevent rotation in the x-direction.
Side 3 is constrained in ROTx to prevent rotation in the x-direction.
Side 4 is constrained in ROTy to prevent rotation in the y-direction.
These constraints create the simply supported edges along Side 1 and Side 2. The
sides are prevented from freely rotating in both the x- and y-axes. Figure 5 shows the
model in ANSYS with the applied constraints and a pressure of 10 psi acting on the
plate in the negative z-direction.
Figure 5: SHELL63 Element with Constraints and Pressure Applied
2.8 ANSYS Model for Composite Plate
The stacking of the plies to form a single composite laminate cannot be effectively
calculated using the SHELL63 element. In order to model the layers of the composite
plate, the SHELL181 element was used to create the finite element model in ANSYS.
16
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. A
representation of the SHELL181 Element from ANSYS is shown in Figure 6.
Figure 6: Shell181 Element17
Initially the composite laminate was modeled in a similar manner to the aluminum plate.
Only a quarter of the plate was modeled due to the symmetry of the plate. The length of
the edges of the quarter plate remained at 12 inches. The thickness of the composite
ply was set as 0.040". With a quarter of the plate modeled, the same mesh size of
0.75" was assigned to the model. The uniform pressure that was applied to the surface
of the composite plate was 10 psi for all composite laminate trials. Each laminate was
modeled to be symmetric about the center plane. A set of trials were run for cross-ply
laminates, where all of the plies are either at 0° or 90°. A second set of trials were run
for symmetric angle plies, where plies not at 0° or 90° were modeled. All of the
constraints for the composite laminate were the same as the constraints applied to the
aluminum plate. All results and findings for the composite laminate trials will be
discussed in the following section.
Additionally, the full plate was modeled in ANSYS for the three symmetric angle ply
trials. The length of each edge of the plate was modeled at 24 inches. Due to limits on
the number of elements that can be modeled the mesh size was changed to be 1.0.
The pressure and the ply thickness remained the same as the trials that modeled a
quarter of the plate. Due to the full plate being modeled, the constraints applied to the
edges of the model are required to be different. The following boundary conditions
based on review of VM82 from the ANSYS library and the supplemental paper Chapter
6 Shells were applied to the full plate analysis. Figure 7 show the full plate modeled in
ANSYS with the constraints applied.
The constraints for the model shown in Figure 7 are:
17
SHELL181 Element diagram from ANSYS
17



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.
Side 1
Y
Z
Side 4
X
Side 2
Side 3
Figure 7: SHELL181 Element - Full plate with Constraints and Pressure Applied
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. The
results of the models will be discussed in the next section.
18
3. Results
3.1 Aluminum Thin Plate Results
The exact solution using the equations listed in Section 2.1.2:
D=
(1𝑒7 𝑝𝑠𝑖)∗(0.25 𝑖𝑛𝑐ℎ)3
12∗(1−0.32 )
[43]
= 14,308.608 lb*inch
For b/a = 1.0, α = 0.00406
wmax = 0.00406 *
(10𝑝𝑠𝑖)∗(24 𝑖𝑛𝑐ℎ)4
14,308.608 𝑙𝑏∗𝑖𝑛𝑐ℎ
= 0.941399 inch
[44]
The maximum deflection of the simply supported aluminum plate in ANSYS was
calculated to be 0.941085".
3.2 Composite Thin Plate Results
Using the governing equations stated in Sections 2.5.1.1 and 2.5.1.2, results were
calculated using Maple. Due to the nature of the cross-ply laminate and the symmetric
angle ply laminate, the equations for both the specially orthotropic laminate and the
Rayleigh-Ritz Method were used. Please refer to Appendix G for the Maple code for
calculating the deflection for both of these methods. In the case of the Rayleigh-Ritz
Method a total of 49 terms were used (i.e. M=7, N=7).
Table 4: Composite Laminate Results - 1/4 of Plate Modeled
Laminate Stack-up
[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
Deflection - ANSYS (in)
0.7182
0.2141
0.091
0.1591
0.1452
0.1600
Deflection - Maple (in)
0.7146
0.21196
0.0895
0.1433
0.1304
0.1445
Percent Error
-0.50
-1.0
-1.7
-11.02
-11.3
-10.7
The intent of the composite laminate model in ANSYS was to model it similarly to the
aluminum plate. The trials for the cross-ply laminates had a high correlation between
the exact solution using the method outlined in Section 2.5.2 in Maple and the finite
element method in ANSYS. The close results from using both of these methods
supports the accuracy of the quarter plate finite element model for the cross-ply trials.
19
After running each trial in ANSYS and comparing it to the exact solution from Maple,
there was a significant difference for the symmetric angle ply trials. After reviewing the
results in both Maple and ANSYS, one contributing factor 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 for the composite plate. D16 and D26 are responsible for the
coupling of moments and deformations not normally associated with each other 18. 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 conditions19. The full [D] matrix requires that the Rayleigh-Ritz
Method be applied to calculate the deflection of the plate.
Due to the higher percent error for the symmetric angle ply trials, further review was
required to determine the cause. 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 if
enough terms are used in the calculation. Since the Rayleigh-Ritz Method will provide a
solution that converges to the exact solution, for the purpose of this analysis I have
assumed that the Rayleigh-Ritz Method provides the more accurate result.
By assuming that the Rayleigh-Ritz Method is calculating the more accurate solution,
the model being created in ANSYS was reviewed further. After an in-depth discussion
regarding the constraints of the model with my advisor, additional full plate trials for
symmetric angle ply laminates were modeled in ANSYS. The results from each full
plate trial compared with the calculated solution in Maple is listed in Table 5.
Table 5: Composite Laminate Results - Full Plate Modeled
Laminate Stack-up
[+/-30 0 +/-30 0]s
[+/-45 0 +/-45 0]s
[+/-60 0 +/-60 0]s
Deflection - ANSYS (in)
0.1457
0.1328
0.1469
Deflection - Maple (in)
0.1433
0.1304
0.1445
Percent Error
-1.64
-1.84
-1.66
Modeling the full plate in ANSYS for each symmetric angle ply trial produced a result
that is much closer to the calculated solution using the Rayleigh-Ritz Method. Due to
the symmetry of the plate, it was expected that the full plate model would produce the
same result as the quarter plate model. This significant difference in results can be
attributed to the change in constraints for the full plate model. The constraints for the
quarter plate model are consistent with what is outlined by Chapter 6 Shells and gives
18
19
Hyer, Stress Analysis of Fiber-Reinforced Composite Materials p 341
Jones, Mechanics of Composite Materials p 250
20
accurate results for both the aluminum plate and the cross-ply laminate plate.
The
only difference between the cross-ply laminate and the symmetric angle ply laminate
are the terms D16 and D26. Figure 8 and Figure 9 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. While this laminate is symmetric, and it is expected that the
displacement gradients would be circular, the figure shows a slightly oval pattern
skewed in the direction of 45°. Also note that the scale for the nodal solution from
ANSYS for both the quarter plate model and the full plate model are not the same. This
further supports that results for the symmetric angle ply laminate as a quarter plate
model were not producing accurate results.
Figure 8: Nodal Displacement for Quarter Plate - [+/-45 0 +/-45 0]s Laminate
Figure 9: Nodal Displacement for Full Plate - [+/-45 0 +/-45 0]s Laminate
21
It is concluded that the inclusion of these two terms 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.
3.3 Composite Failure Criterion Results
Table 5 in Appendix D has the Maximum Stress Criterion results for both cross-ply and
symmetric angle laminates. It is interesting to note what stress was required for each
ply to fail. In the cases where a 0° ply was used in a laminate, this ply orientation would
not fail due to a shear stress. This is because shear failure cannot be produced in
these layers without an applied axial load. In comparing the results on Table 5, much
larger stresses can be applied to the plies of the cross-ply composite plates before
failure.
Table 6 in Appendix D has the Tsai-Wu Criterion for both cross-ply and symmetric angle
laminates. This alternate method is a quadratic failure criterion and provides a negative
and positive value for the stress in each direction. In order to determine the range of
failure stresses in a specific ply orientation the quadratic roots from the Tsai-Wu
criterion are multiplied by the transformed stresses acting in the 1-2 coordinate system
for the ply. The values for P will provide the range that a ply will not fail under so long
as the applied stress is within the stress range.
3.4 Error Analysis
The maximum deflection using ANSYS to analyze the simply supported aluminum plate
is 0.941085". The maximum deflection using the governing equations for the simply
supported aluminum plate is 0.941399".
% Error =
𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝐷𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛−𝐴𝑁𝑆𝑌𝑆 𝐷𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛
% Error =
𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝐷𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛
(0.941399-0.941085)
0.941399"
* 100
[45]
* 100 = 0.0334%
ANSYS and Maple were used to determine the maximum deflection for a simply
supported composite plate. The first composite plate that I found deflected less than
the aluminum plate was a [0 90 0 90]s cross-ply composite laminate. The maximum
deflection using ANSYS to analyze the simply supported composite plate is 0.7182".
The maximum deflection using the governing equations in Maple for the simply
supported composite plate is 0.7146".
22
% Error =
% Error =
𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝐷𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛−𝐴𝑁𝑆𝑌𝑆 𝐷𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛
𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝐷𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛
(0.7146-0.7182)
0.7146"
* 100
* 100 = -0.5%
Additional trials for composite plates were conducted using ANSYS and Maple. The
calculated percent error is shown in Table 4 and Table 5 for all of the trials.
23
[46]
4. Conclusions
This project analyzed the maximum deflection for a simply supported aluminum plate
and composite plates. The deflection of both the aluminum plate and the composite
plates were calculated using the exact solutions in Maple. ANSYS was used to model
both an aluminum simply supported plate and composite simply supported plates. The
accuracy of the ANSYS models were verified using thin plate theory for the aluminum
plate and classical lamination theory for the composite plates. The results for the
aluminum plate are listed in section 3.1. All of the results for the composite plates
including an error calculation are listed in Table 4 and Table 5 in section 3.2. Two
tables of data are listed in Section 3.2 for the results in modeling a quarter plate and a
full plate for the symmetric angle ply laminates.
The number of plies that made up the laminated plates varied from eight plies to sixteen
plies. The laminates that were analyzed also varied from a cross-ply laminate to a
symmetric angle laminate. In both cases, the composite plate was symmetric about the
central plane. 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 D 16 &
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.
24
REFERENCES
[1] Hyer, Michael W. Stress Analysis of Fiber-Reinforced Composite Materials
Update Edition, 2009 DEStech Publications, Inc.
[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-d_417.html
Date Accessed: 9/20/2013
[9] Jones, Robert M. Mechanics of Composite Materials 1st Edition, 1975
McGraw-Hill, Inc.
[10] Boeing 787 Dreamliner®
http://en.wikipedia.org/wiki/787_Dreamliner#Composite_materials
Date Accessed: 11/20/2013
[11] Van Keuren, Kevin Structural Optimization of a Simply Supported Orthotropic
Composite Plate RPI Hartford Master's Project December 2010
[12] Chapter 6 Shells (PDF)
http://www.ewp.rpi.edu/hartford/~ernesto/F2013/EP/MaterialsforStudents/Carroll/Ch6-Shells.pdf
Date Accessed: 12/9/2013
25
[13] R. Rolfes, K. Rohwer, M. Ballerstaedt,
Efficient linear transverse normal stress analysis of layered composite plates. 1998
[14] Bending of Symmetric Angle-ply Lamianted Plates - Prof. Naik notes
[15] J.E. Ashton, An Analogy for Certain Anisotropic Plates 1969
[16] J.N. Reddy, Exact Solutions of Moderately Thick Laminates Shells 1984
[17] J.M Whitney, On the Analysis of Anisotropic Rectangular Plates 1972
[18] VM82 - ANSYS Verification Manual
[19] Laminate Plate Equations - SE 253 Mechanics of Laminated Composite Structures
Prof. Hyonny Kim
26
APPENDIX A
27
28
29
30
31
APPENDIX B
Timoshenko & Woinowsky-Krieger Theory of Plates and Shells page 120
32
APPENDIX C
Equations for Determining the ABD Matrix of a Composite Plate
33
34
35
36
37
APPENDIX D
Table 6: Maximum Stress Criterion Results
38
Table 7: Tsai-Wu Criterion Results
39
APPENDIX E
ANSYS Code for 0.25" thick Simply Supported Aluminum Plate
/BATCH
/COM,ANSYS RELEASE 10.0A1 UP20060105
20:31:34
/input,menust,tmp,'',,,,,,,,,,,,,,,,1
/GRA,POWER
/GST,ON
/PLO,INFO,3
/GRO,CURL,ON
/CPLANE,1
/REPLOT,RESIZE
WPSTYLE,,,,,,,,0
!*
/NOPR
/PMETH,OFF,0
KEYW,PR_SET,1
KEYW,PR_STRUC,1
KEYW,PR_THERM,0
KEYW,PR_FLUID,0
KEYW,PR_ELMAG,0
KEYW,MAGNOD,0
KEYW,MAGEDG,0
KEYW,MAGHFE,0
KEYW,MAGELC,0
KEYW,PR_MULTI,0
KEYW,PR_CFD,0
KEYW,LSDYNA,0
/GO
!*
/COM,
/COM,Preferences for GUI filtering have been set to display:
/COM, Structural
!*
/PREP7
!*
ET,1,SHELL63
!*
R,1,0.25, , , , , ,
RMORE, , , ,
40
10/14/2013
RMORE
RMORE, ,
!*
!*
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,1,,10e6
MPDATA,PRXY,1,,0.3
RECTNG,0,12,0,12,
/VIEW,1,1,1,1
/ANG,1
/REP,FAST
ESIZE,0.75,0,
MSHAPE,0,2D
MSHKEY,0
!*
CM,_Y,AREA
ASEL, , , ,
1
CM,_Y1,AREA
CHKMSH,'AREA'
CMSEL,S,_Y
!*
AMESH,_Y1
!*
CMDELE,_Y
CMDELE,_Y1
CMDELE,_Y2
!*
FLST,2,2,4,ORDE,2
FITEM,2,2
FITEM,2,-3
!*
/GO
DL,P51X, ,UZ,
FLST,2,2,4,ORDE,2
FITEM,2,1
FITEM,2,-2
!*
/GO
DL,P51X, ,ROTX,
41
FLST,2,2,4,ORDE,2
FITEM,2,3
FITEM,2,-4
!*
/GO
DL,P51X, ,ROTY,
FLST,2,1,3,ORDE,1
FITEM,2,1
!*
/GO
DK,P51X, , , ,0,UX,UY, , , , ,
FLST,2,1,5,ORDE,1
FITEM,2,1
/GO
!*
SFA,P51X,1,PRES,-10
FINISH
/SOL
/STATUS,SOLU
SOLVE
FINISH
/POST1
PLDISP,1
!*
/EFACET,1
PLNSOL, U,Z, 0,1.0
SAVE
FINISH
! /EXIT,ALL
42
APPENDIX F
ANSYS Code for Simply Supported [0 90 0 90]s Composite Plate
/BATCH
/COM,ANSYS RELEASE 10.0A1 UP20060105
/input,menust,tmp,'',,,,,,,,,,,,,,,,1
/GRA,POWER
/GST,ON
/PLO,INFO,3
/GRO,CURL,ON
/CPLANE,1
/REPLOT,RESIZE
WPSTYLE,,,,,,,,0
!*
/NOPR
/PMETH,OFF,0
23:27:33
KEYW,PR_SET,1
KEYW,PR_STRUC,1
KEYW,PR_THERM,0
KEYW,PR_FLUID,0
KEYW,PR_ELMAG,0
KEYW,MAGNOD,0
KEYW,MAGEDG,0
KEYW,MAGHFE,0
KEYW,MAGELC,0
KEYW,PR_MULTI,0
KEYW,PR_CFD,0
KEYW,LSDYNA,0
/GO
!*
/COM,
/COM,Preferences for GUI filtering have been set to display:
/COM, Structural
!*
/PREP7
!*
ET,1,SHELL181
!*
!*
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,1,,2.25e7
MPDATA,EY,1,,1.75e6
MPDATA,EZ,1,,1.75e6
43
10/28/2013
MPDATA,PRXY,1,,0.248
MPDATA,PRYZ,1,,0.458
MPDATA,PRXZ,1,,0.248
MPDATA,GXY,1,,6.38e5
MPDATA,GYZ,1,,4.64e5
MPDATA,GXZ,1,,6.38e5
sect,1,shell,,
secdata, .04,1,0,3
secdata, .04,1,90,3
secdata, .04,1,0,3
secdata, .04,1,90,3
secdata, .04,1,90,3
secdata, .04,1,0,3
secdata, .04,1,90,3
secdata, .04,1,0,3
secoffset,MID
seccontrol,,,, , , ,
RECTNG,0,12,0,12,
ESIZE,0.75,0,
MSHAPE,0,2D
MSHKEY,0
!*
CM,_Y,AREA
ASEL, , , ,
1
CM,_Y1,AREA
CHKMSH,'AREA'
CMSEL,S,_Y
!*
AMESH,_Y1
!*
CMDELE,_Y
CMDELE,_Y1
CMDELE,_Y2
!*
/VIEW,1,1,1,1
/ANG,1
/REP,FAST
FLST,2,2,4,ORDE,2
FITEM,2,2
FITEM,2,-3
!*
/GO
DL,P51X, ,UZ,
FLST,2,2,4,ORDE,2
FITEM,2,3
FITEM,2,-4
44
!*
/GO
DL,P51X, ,ROTY,
FLST,2,2,4,ORDE,2
FITEM,2,1
FITEM,2,-2
!*
/GO
DL,P51X, ,ROTX,
FLST,2,1,3,ORDE,1
FITEM,2,1
!*
/GO
DK,P51X, , , ,0,UX,UY, , , , ,
FLST,2,1,5,ORDE,1
FITEM,2,1
/GO
!*
SFA,P51X,1,PRES,-10
FINISH
/SOL
/STATUS,SOLU
SOLVE
FINISH
/POST1
PLDISP,1
!*
/EFACET,1
PLNSOL, U,Z, 0,1.0
SAVE
FINISH
! /EXIT,ALL
45
APPENDIX G
Maple Code for [0 90 0 90]s Laminate
For Specially Orthotropic Laminates
46
47
48
49
50
Maple Code for [+/-30 0 +/-30 0]s Laminate
For Symmetric Laminates using the Rayleigh-Ritz Method and Total Potential Energy
51
52
53
54
55
56
57
APPENDIX H
Table of Material Properties for Composite Laminates (p. 64)
from Stress Analysis of Fiber-Reinforced Composite Materials
by Michael Hyer
58