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IMECE2014-37254-0.pdf
DRAFT
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 approximations obtained using the
Ritz method and the finite element method. The Ritz method is
implemented with the symbolic manipulation program Maple
and ANSYS is used to perform the finite element calculations.
Reliable results are obtained using both methods.
INTRODUCTION
There are many applications for composite materials in
today's industrial markets. Composite materials have gained
popularity in the aerospace industry since they provide a high
strength to weight ratio and can be stronger than a metal in
specific configurations. However, the strength and detailed
mechanical response of composite materials can vary
depending on the orientation of the plies in the laminate. Care
must be used in composite design to account for the additional
complexity in mechanical response resulting from the intrinsic
anisotropy of composite materials. This paper investigates the
deflection of a simply supported composite plate using two
different methods. For a symmetric cross-ply laminate of 0 and
90 degree plies such as a four lamina [0/90]S, the stiffness
components A16, A26, Bij, D16, and D26 are all zero. As a result,
the maximum deflection of a symmetric cross-ply laminate is
readily solved using the method of separation of variables.
In a symmetric angle ply laminate such as [0/45]S all the
components of the bending stiffness matrix [D] are non-zero
Prof. Ernesto Gutierrez-Miravete
Rensselaer at Hartford
Hartford, CT 06120
and it is well known that the solution obtained by separation of
variables yields a large error in the computed deflection. The
Rayleigh-Ritz Method and the Finite Element Method are
approximation methods that can be used to determine the
deflection of a symmetric angle laminate. The Rayleigh-Ritz
Method is based on the principle of minimum total potential
energy for mechanical systems in static equilibrium. By
expressing the unknown deflection as a bi-linear combination
of a selected number of constant coefficients and suitably
selected basis functions, introducing this into the total energy
equation, implementing the condition for an extreme in terms
of the derivatives of the energy with respect to the values of the
coefficients, a system of simultaneous linear algebraic
equations is obtained. The solution of the system yields the
values of the coefficients in the original expansion that
minimize the total energy. In the finite element method one
subdivides the physical domain into adjoining finite elements
connected at nodal locations and introduces simple finite
element basis functions. The approximation is represented as a
linear combination of the unknown nodal values and the finite
element basis functions. When this representation is introduced
into the variational formulation of the elastic deformation
problem a system of simultaneous linear algebraic results and
the solution of the system produces the desired nodal values.
This paper presents a comparison of computed values of the
deflection of simply supported composite plates calculated
from the exact solution, with the Rayleigh-Ritz Method, and
with the ANSYS finite element program.
NOMENCLATURE
a - length of plate in x-direction (in)
b - length of plate in y-direction (in)
[A] - Extensional Stiffness Matrix (lb/in)
1
Copyright © 2014 by ASME
[B] - Coupling Stiffness Matrix (lb)
[C] - Stiffness Matrix (psi)
[D] - Bending Stiffness Matrix (lb*in)
E1 - Modulus of Elasticity – longitudinal (psi)
E2 - Modulus of Elasticity – transverse (psi)
G12, G23, G13 - Shear Moduli (psi)
M - Bending Moment Resultant (lb*in/in)
N - Force Resultant (psi/in)
Mx - Bending Moment Resultant (lb*in/in)
My - Bending Moment Resultant (lb*in/in)
Mxy - Twisting Moment Resultant (lb*in/in)
Nx - Normal Force Resultant, x-direction (psi/in)
Ny - Normal Force Resultant, y-direction (psi/in)
Nxy - Shear Force Resultant (psi/in)
P - Point Load (lb)
Q - Reduced Stiffness Matrix (psi)
Q - Transformed Reduced Stiffness (psi)
q - Applied Distributed Force (psi)
[T] - Transformation matrix m = cos(θ) n = sin(θ)
t - thickness (in)
uo – mid-thickness displacement in x-direction (in)
vo - mid-thickness displacement in y-direction (in)
wo - mid-thickness displacement in z-direction (in)
wmax - maximum deflection (in)
w - deflection (in)
x - x-direction
y - y-direction
z - z-direction
[ ]s - symmetric laminate
ε - strain (-)
γ - Engineering Shear Strain (-)
ν12, ν23, ν123- Poisson's Ratios
σ - stress (psi)
τ - shear stress (psi)
θ - ply angle
THIN PLATE THEORY
In the Kirchhoff-Love plate theory, the following
assumptions are introduced for the analysis of the small
deflection of thin plates1:



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 plane of
the plate are disregarded.
Combining the boundary conditions with the thin plate
assumptions allows the calculation of the deflection of the
plate.
MATERIAL PROPERTIES OF COMPOSITE PLY
Because of anisotropy, a composite ply has different
properties in different directions. In unidirectional fiber
reinforced composites, transverse isotropy is usually assumed.
In this calculations reported in this paper, the following
material properties for graphite-polymer composite plies and
applied load were used:
Table 1: Material Properties of Composite Ply
Edge Length (a)
24 inch
Ply Thickness
0.040 inch
E1
2.25 x 107 psi
E2
1.75 x 106 psi
E3
1.75 x 106 psi
ν12
0.248
ν23
0.458
ν13
0.248
G12
6.38 x 105 psi
G23
4.64 x 105 psi
G13
6.38 x 105 psi
Applied Surface Pressure (q)
10 psi
COMPOSITE THIN PLATE THEORY
A number of factors must be considered when deriving
the governing equation for the deflection of a thin laminated
composite plate. These include individual ply material
properties, ply orientation, besides the boundary conditions and
the 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 of constant magnitude.
For thin plates, plane stress conditions are at least
approximately obtained3. With these assumption one can set
the σ3, τ23, and τ13 stress components to zero and the original
stress-strain relationships reduce to
σ
τ
0
0
=
0
*
0
where the reduced stiffness matrix components are
Q11 = C
Q22 =
−
−
Q12 = C
−
Q66 =
Boundary conditions are also used in plate theory to
constrain the plate (e.g. simply supported, clamped).
2
1
3
Timoshenko & Woinowsky-Krieger [1], page 1
2
Hyer [2], page 241
Hyer [2], page 165
Copyright © 2014 by ASME
and the Cij,s are the components of the original stiffness matrix.
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.
Determination of the transformed reduced stiffness matrix
allows accounting for the difference. 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 any angle, θ gives the
equation
=
=
=
=(
=(
N
=0
+
=0
+2
)
+ 2(
+2
+
)
−4
+
( +
)
)
−2
+(
−
+2
)
)
+ 2(
+2
+
)
−
−2
)
+(
−
+2
)
+
−2
−2
+
(
+
)
+
+
=0
+
−
⎤ ⎧
⎪
⎥ ⎪
⎥*
⎥ ⎨κ
⎥ ⎪
⎪κ
⎦ ⎩κ
⎫
⎪
⎪
⎡
⎢
N
= ⎢
⎨M ⎬ ⎢
⎪
⎪M ⎪
⎪ ⎢
M
⎩
⎭ ⎣
DEFLECTION
OF
A
SIMPLY
SUPPORTED
SYMMETRIC CROSS-PLY LAMINATE
When the laminated composite plate is symmetric with
a cross-ply orientation (specially orthotropic laminate) the
values for A16, A26, Bij, D16, and D26 are all zero. The
governing equations are:
A
+A )
+ 2(
=
1
2
Q (z − z
Q
z −z
)
=0
+A
=0
+A
+2
)
+
=
= 0, :
= 0
=−
−
=0
= 0, :
= 0
=−
−
=0
Expanding the load into a double Fourier series:
( , ) =
A =
+A )
The boundary conditions for the simply supported edges are:
⎫
⎪
⎪
⎬
⎪
⎪
⎭
+ (A
+A
(A
where
4
z −z
+
∗
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
surface4. Each segment of the ABD Matrix is taken from the
transformed reduced stiffness matrix with respect to the
thickness of the ply.
⎧
⎪
⎪
Q
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 are5:
where4:
=
=(
=(
1
3
sin
sin
Separation of variables yields the following expression for the
deflection of a plate subjected to uniform transverse load of
magnitude q will then be:
5
Hyer [2], Chapter 9
3
Hyer [2], page 584
Copyright © 2014 by ASME
=
∑
∑
, ,
, ,
(
)
Even though a truncated series must always be used in practice,
this solution will be referred to as the exact solution. For a
simply supported plate under transverse loading, the maximum
deflection will be at its center.
DEFLECTION
OF
A
SIMPLY
SUPPORTED
SYMMETRIC ANGLE PLY LAMINATE BY THE
RAYLEIGH-RITZ METHOD
The method sketched above cannot be used to
compute the deflection of a symmetric angle ply laminate.
Such a laminate has a full [D] matrix which will alter the third
governing differential equation and boundary conditions to:
+4
+ 2(
+4
)
+2
+
=
= 0, :
= 0;
=−
−
−2
=0
−2
=0
= 0, :
= 0;
=−
−
Since the deflection of symmetric angle ply laminates
cannot be calculated accurately using the method of separation
of variables an alternative method is required The RayleighRitz Method is an approximation method based on the principle
of minimum total potential energy. The total potential energy
for a symmetric angle ply laminate is given by [3]:
= ∫ ∫(
4
+2
+4
+
+4
+
−2
)
The Rayleigh-Ritz Method starts by assuming that the
deflection of the laminate plate can be expressed as:
w = ∑
∑
C sin
sin
where Cij are unknown coefficients to be determined.
The equation assumed for the deflection is next
substituted into the energy equation 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 Cij’s. There will be a total of m*n unknowns in this
algebraic equation. To determine the values of the unknown
constants, one then uses the principle of minimum total
potential energy by taking partial derivatives of V with respect
to each of the unknown Cij and equating the result to zero, i.e.
=0
This creates a m*n system of linear algebraic equations that can
then be solved standard methods to obtain the values of all the
Cij’s. An approximation to the deflection of the laminated plate
can finally be found simply by substituting into the initial
expression for w in the Rayleigh-Ritz method. All the necessary
manipulations and computations required were performed using
the symbolic manipulation program Maple [4].
DEFLECTION
OF
A
SIMPLY
SUPPORTED
SYMMETRIC ANGLE PLY LAMINATE BY THE FINITE
ELEMENT METHOD
The finite element program ANSYS [5] was used to
model 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 creation of laminate plates
consisting of layered plies.
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. It was considered that modeling
only one quarter of the plate was possible due to the symmetry
of the problem. 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.
The mesh size for the full plate was 1.0" while the mesh size
for the quarter plate was 0.75".
The boundary conditions applied to the full plate
model were based on the review of VM82 from the ANSYS
library. The following constraints were 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
4
Copyright © 2014 by ASME
Side 1
Side 4
Side 2
Side 3
.
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, an
accurate solution was found for the symmetric angle ply trials.
Ritz Method is required to calculate the deflection of the plate
when there is a full [D] matrix.
Although there is no exact solution for a symmetric
angle ply laminate, the Rayleigh-Ritz Method gives an
approximation of the deflection based on the minimization of
the total potential energy of the plate. For symmetric cross ply
laminates, the results of the Rayleigh-Ritz method converge to
the exact solution as more terms are used in the calculation. For
the symmetric angle ply laminate the results produced by both
methods are different. For the purpose of this analysis, it was
assumed that the Rayleigh-Ritz Method provides the more
accurate results for the symmetric angle ply laminate. This
assumption was tested by comparing the results against those
obtained using ANSYS. Initial tests using a quarter plate model
gave results significantly different to those obtained with the
Rayleigh-Ritz method and the exact solution. The ANSYS
model was then modified to represent 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.
Table 2 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
COMPOSITE THIN PLATE RESULTS
As indicated before, the Maple program was used to
calculate the deflection of the composite plates for both the
cross-ply laminates and the symmetric angle-ply laminates.
The equation above for the specially orthotropic laminate was
used for cross-ply laminates while the Rayleigh-Ritz Method
was used for symmetric angle ply laminates. In the case of the
Rayleigh-Ritz Method a total of 49 terms were used (i.e. M=7,
N=7). All the terms were included but he diagonal terms Cii
were always dominant.
The results of the ANSYS model for quarter plate
cross-ply laminates were in very good agreement with the exact
solution and with the results of the Rayleigh-Ritz method.
However, there was a significant difference between the
ANSYS result and the exact solution for the quarter plate
symmetric angle laminate trials. Clearly, the difference in
results is due 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. The
D16 and D26 terms are responsible for the coupling of moments
and deformations not normally associated with each other6.
The Fourier expansion used to develop the governing equations
for the cross-ply laminate cannot be applied to the symmetric
angle ply laminate because the expansion with the D16 and D26
terms will not satisfy the boundary conditions7. The Rayleigh-
6
7
Table 3 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
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 45o. 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 D16 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
Hyer [2], page 341
Jones [3], page 250
5
Copyright © 2014 by ASME
will produce a solution that is accurate with the exact solution
using the Rayleigh-Ritz Method.

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 in good agreement with the exact
solution for the symmetric cross ply laminate.
CONCLUSIONS
This project analyzed the maximum deflection for a
simply 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:
Figure 1 Nodal Displacement for Full Plate - [+/-45 0 +/-45 0]s
Laminate
Further details and additional information can be found in the
Mechanical Engineering Master’s project Report of the first
named author submitted to RPI [6].
Nodal Displacement for Quarter Plate - [+/-45 0 +/-45 0]s
Lamiante
ACKNOWLEDGMENTS
I would like to thank my fiancé and family 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. Timoshenko, S. and Woinowsky-Krieger, S. Theory of
Plates and Shells, 2nd Edition, 1959 McGraw-Hill, Inc.
2. Hyer, Michael W. Stress Analysis of Fiber-Reinforced
Composite Materials. Update Edition, 2009 DEStech
Publications, Inc.
3. Jones, Robert M. Mechanics of Composite Materials 1st
Edition, 1975 McGraw-Hill, Inc.
4. Maple 17, http://www.maplesoft.com
5. ANSYS 14.5, http://www.ansys.com
6
Copyright © 2014 by ASME
In the preparation of this report, references 7 through were also
used:
6. K. Carroll, Comparative Deflection Analysis of Aluminum
and Composite Laminate Plates using the Rayleigh-Ritz and the
Finite Element Methods, Mechanical Engineering Master’s
Project, Rensselaer Polytechnic Institute, Hartford, CT, 2013.
7. Notes from MANE 6180 Mechanics of Composite Materials
R. Naik 2013, RPI-Hartford.
8. 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.
9. Kirchoff-Love Plate Theory, Wikipedia.
10.Agarwal, Bhagwan D. and Broutman, Lawrence J. Analysis
and Performance of Fiber Composites, Second Edition 1990.
11. ANSYS Tips by Paul Dufour. http://www.ansys.belcan.com
12. Van Keuren, Kevin Structural Optimization of a Simply
Supported Orthotropic Composite Plate RPI Hartford Master's
Project December 2010.
7
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8
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