FEA Comparison of the Deflection of Composite Laminate Polymer
Plates and Functionally Graded Polymer Plates
by
Rigels Bejleri
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
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December, 2014
(For Graduation May, 2015)
i
© Copyright 2014
by
Rigels Bejleri
All Rights Reserved
ii
CONTENTS
FEA Comparison of the Deflection of Composite Laminate Polymer Plates and
Functionally Graded Polymer Plates ............................................................................ i
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF SYMBOLS ...................................................................................................... viii
KEYWORDS .................................................................................................................... ix
ACKNOWLEDGMENT ................................................................................................... x
ABSTRACT ..................................................................................................................... xi
1. Introduction.................................................................................................................. 1
1.1
Background ........................................................................................................ 1
1.2
Problem Description........................................................................................... 3
2. Methodology ................................................................................................................ 4
2.1
Methodology Overview ..................................................................................... 4
2.2
CLP Ply Properties ............................................................................................. 4
2.3
CLP Plate Modeling ........................................................................................... 6
2.4
FGP Plate Element Properties ............................................................................ 9
2.5
FGP Plate Properties Via The Mori-Tanaka Method ....................................... 11
2.6
Functionally Graded Material Plate Modeling ................................................. 24
3. Results and Discussion .............................................................................................. 25
3.1
Laminated Plate Results and Validation of Modeling Method ........................ 25
3.2
Functionally Graded Plate Results ................................................................... 27
4. Conclusion ................................................................................................................. 33
References........................................................................................................................ 35
Appendix A...................................................................................................................... 36
Appendix B ...................................................................................................................... 39
Appendix C ...................................................................................................................... 45
iii
Appendix D...................................................................................................................... 47
Appendix E ...................................................................................................................... 52
iv
LIST OF TABLES
Table 1: Properties of AS/3501 Composite Ply................................................................. 6
Table 2: Young’s Modulus Comparison Between Two Sources..................................... 10
Table 3: Properties of FGP Plate Substrates .................................................................... 10
Table 4: Summary of FGP Plates Studied ....................................................................... 14
Table 5: Summary of FGP Plate Graphite Distribution and Deflections ........................ 29
v
LIST OF FIGURES
Figure 1: Laminated Composite Plate [4] .......................................................................... 2
Figure 2: Functionally Graded Material [5]....................................................................... 2
Figure 3: CLP Plate Layer (Ply) Arrangement .................................................................. 7
Figure 4: CLP Plate Layer (Ply) Arrangement .................................................................. 7
Figure 5: Meshed Sheet Body Representing CLP Plate .................................................... 8
Figure 6: Through Thickness Material Fractions of Plate 1 ............................................ 15
Figure 7: Through Thickness Material Fractions of Plate 2 ............................................ 15
Figure 8: Through Thickness Material Fractions of Plate 3 ............................................ 16
Figure 9: Through Thickness Material Fractions of Plate 4 ............................................ 16
Figure 10: Through Thickness Material Fractions of Plate 5 .......................................... 17
Figure 11: Through Thickness Material Fractions of Plate 6 .......................................... 17
Figure 12: Through Thickness Material Fractions of Plate 7 .......................................... 18
Figure 13: Through Thickness Material Fractions of Plate 8 .......................................... 18
Figure 14: Through Thickness Material Fractions of Plate 9 .......................................... 19
Figure 15: Through Thickness Material Fractions of Plate 10 ........................................ 19
Figure 16: Through Thickness Material Fractions of Plate 11 ........................................ 20
Figure 17: Through Thickness Material Fractions of Plate 12 ........................................ 20
Figure 18: Through Thickness Material Fractions of Plate 13 ........................................ 21
Figure 19: Through Thickness Material Fractions of Plate 14 ........................................ 21
Figure 20: Through Thickness Material Fractions of Plate 15 ........................................ 22
Figure 21: Through Thickness Material Fractions of Plate 16 ........................................ 22
Figure 22: Through Thickness Material Fractions of Plate 17 ........................................ 23
Figure 23: Through Thickness Material Fractions of Plate 18 ........................................ 23
Figure 24: Deflection of 10 Layer Cross Ply Laminated Plate ........................................ 25
Figure 25: Deflection of 20 Layer Cross Ply Laminated Plate ........................................ 26
Figure 26: Deflection of 10 Layer Angle Ply Laminated Plate ....................................... 27
Figure 27: Through Thickness Material Fractions of Plate 8, FGP Plate With Lowest
Deflection ........................................................................................................................ 30
Figure 28: Through Thickness Material Fractions of Plate 11, FGP Plate With 2nd Lowest
Deflection ........................................................................................................................ 30
vi
Figure 29: Through Thickness Material Fractions of Plate 1, FGP Plate With 3rd Lowest
Deflection ........................................................................................................................ 31
Figure 30: Through Thickness Material Fractions of Plate 5, FGP Plate With 3rd Highest
Deflection ........................................................................................................................ 31
Figure 31: Through Thickness Material Fractions of Plate 6, FGP Plate With 2nd Highest
Deflection ........................................................................................................................ 32
Figure 32: Through Thickness Material Fractions of Plate 7, FGP Plate With Highest
Deflection ........................................................................................................................ 32
vii
LIST OF SYMBOLS
Symbol Unit Meaning
E
Pa
ν
G
Elastic Modulus
Poisson’s Ratio
Pa
Q
Shear Modulus
Reduced Stiffness Matrix (Element)
tply
m
Thickness of a laminate ply
Vf
%
Volume Fraction (of fiber reinforcement)
H
m
Thickness or height of the plate
1 or r
Subscript that stands for reinforcement (FGP plates)
2 or m
Subscript that stands for matrix (FGP plates)
P
Subscript that stands for plate (FGP plate)
L
Subscript that stands for plate (FGP plate)
z
m
Variable that represents thickness in a plate (FGP plate)
K
Pa
Bulk Modulus
λ
Pa
Lamѐ first parameter
viii
KEYWORDS
Composite Laminated Polymer (CLP)
Functionally Graded Material (FGM)
Functionally Graded Polymer (FGP)
Finite Element Method (FEM)
Finite Element Analysis (FEA)
Classical Lamination Theory (CLT)
Cross Ply
Angle Ply
Mori Tanaka Method
ix
ACKNOWLEDGMENT
I would like to thank my professor and adviser, Ernesto Gutierrez-Miravete for the
guidance he provided throughout my masters project and degree. I would also like to thank
my family for their support throughout my educational career. Finally, I would like to
thank my wife Ajana for not only supporting and encouraging me throughout my masters
degree, but also for picking up my share of the chores while I was working on this project.
x
ABSTRACT
This project compares the deflection of Composite Laminate Polymer plates and
Functionally Graded Polymer plates subject to normal load, in order to determine which
configuration is stiffer. The deflection of these plates is calculated via the Finite Element
Method using the software ANSYS. The study samples were simply supported square
plates with a uniform pressure applied. Both types of composites were comprised of a set
material pair and a set volumetric ratio of the material pair.
The goal of this study was to gain an understanding of the deflection differences
between the two types of composites. However, this study did not account for debits
associated with material defects such as voids, gaps, and porosity that could be present
due to the manufacturing processes used to create these composites. Therefore, it is
recommended that extensive tests be conducted to gain a full understanding of the material
capabilities of the plates studied in this project.
xi
1. Introduction
1.1 Background
Humans have taken advantage of the properties of composite materials for
thousands of years. Concrete, plywood, and cob (mud and straw) bricks are some of the
earliest examples of man-made composites that are still used today for construction
throughout the world [1]. Over the past century Fiber Reinforced Plastics (FCPs) gained
popularity in the aerospace industry as an alternative to metal because of their high
strength to weight ratio [1]. Composite Laminated Polymers (CLPs), shown schematically
in figure 1, are a type of FCP that is manufactured by stacking layers of high strength fiber
reinforced polymer in different arrangements to provide required engineering properties
[2]. The layered composition of CLPs provides flexibility in construction and allows for
the creation of a material with near metal, or in some specific aspects, better than metal
properties with a fraction of the weight. Extensive research and development in CLPs has
led to a decrease in the cost of design and manufacture of CLPs, allowing for their broader
use in automobiles, trains, electronics, sports, construction, and many other industries.
Functionally Graded Materials (FGMs) are another type of composite material.
FGMs are constituted of a continuously varying and smooth distribution of the two or
more constituents involved. FGMs provide an alternative to materials that are used in
applications that require the base material to be coated due to the severe environment that
it operates in. Inter-laminar stresses between the substrate and the coating can build up
overtime due to the adverse operating conditions and relatively abrupt transition of
material properties from the coating to the substrate. These stresses can cause coating loss,
thus exposing the substrate to the potentially corrosive environment. FGMs are a good
alternative in these applications because the smooth transition from coating material
properties to substrate properties reduces the chance for inter-laminar stress build up and
thus reduces the chance for coating loss [3].
CLPs can experience similar de-lamination due to inter-laminar stresses between
the matrix and reinforcing fibers when exposed to a challenging environment such as a
moist environment. This project studies the deflection capability of Functionally Graded
Polymer plates manufactured via 3D printing and compares it to the deflection of CLP
1
plates composed of the same two materials to determine whether the FGP plates could be
a suitable alternative to CLPs.
Figure 1: Laminated Composite Plate [4]
Figure 2: Functionally Graded Material [5]
2
1.2 Problem Description
This project utilized ANSYS FEA software to study the deflection behavior of
simply supported Composite Laminate Polymer (CLP) plates and Functionally Graded
Polymer (FGPs) plates when a uniform pressure load is applied. The goal is to determine
which type of plate, meaning CLP or FGP, has the highest stiffness. In order to isolate the
effects of the composite type on the deflection, all the plates will have the same physical
dimensions, meaning length, width and thickness. In addition, all of the plates will be
composed of the same material couple, an epoxy matrix and graphite reinforcing material.
Lastly the total volumetric ratio for the two materials will be held constant for all plates.
One of the CLP plates to be studied will be a cross ply configuration where the plies are
stacked 90 degrees relative to each other. The other CLP plate will be an angle ply with
plies stacked at 45 degrees relative to each other. Similarly, all the FGP plates evaluated
will have varying distributions of the epoxy and graphite throughout their thicknesses. For
this project, it is assumed that an unlimited range of varying material distribution can be
achieved as it is usually the case when using 3D printing to manufacture the FGP plates.
3
2. Methodology
2.1 Methodology Overview
This project was built upon the work that Kenneth Carroll and Kevin Pendley did
in their RPI Master’s Projects [6 and 7]. In [6], Carroll compared the deflection of
composite laminate plates with the deflection of a thin Aluminum plate subject to the same
loads. The analysis for the composite plate was done using the finite element method
(FEM) in ANSYS as well as with the Rayleigh-Ritz method.
For this project the Composite Laminated Polymer plates were first analyzed in
ANSYS. Then the FEA solution was compared to the analytical solution. The analytical
solution employed Maple to execute Classical Lamination Theory analysis on the CLP
plate. Once the modeling methodology in ANSYS was validated via the analytical method,
it was deemed to be acceptable to use the same modeling methodology to analyze the rest
of the CLP and Functionally Graded Polymer plates in ANSYS.
The properties of the FGP plate were approximated using the Mori-Tanaka method
referenced in [7]. This methodology uses the material properties and the volumetric ratio
of each element making up the plate to calculate the properties of the functionally graded
plate through its thickness.
2.2 CLP Ply Properties
To analyze the deflection of CLP plates analytically the elastic modulus, shear
modulus, and Poisson’s ratio of the composite ply were used to calculate the Reduced
Stiffness Matrix, which relates the laminate stress to the laminate strain. The Reduced
Stiffness Matrix was used to calculate the CLP plate deflection via the Classical
Lamination Theory for the cross ply plate [6].
Because the thicknesses of the CLP plates analyzed was much smaller than their
length and width, and the deflection of the plate was small, it was possible to take
advantage of the Thin Plate Theory assumptions to simply the Reduced Stiffness Matrix
from a 6x6 to a 3x3 matrix. These assumptions are that the middle plane of the plate does
not deform but stays neutral after bending, that straight lines that are normal to the middle
4
plane remain straight and normal to the middle plane after bending, and that stresses in the
transverse direction of the plate are low compared to in plane stresses and can therefore
be disregarded [6]. As shown in equation 1 below [6], the 3x3 Reduced Stiffness Matrix
required the values of the elastic modulus in the x and y direction, the Poisson’s ratio in
the xy direction and the shear modulus in the xy direction. Symbolically these are E1, E2,
ν12, and G12, where x has been replaced with the subscript 1 and y with 2. The value for
ν21, which is also needed for the analytical solution is calculated using equation 2 [6].
𝑸𝟏𝟏 =
𝑬𝟏
𝟏−𝒗𝟏𝟐 𝒗𝟐𝟏
𝑸𝟐𝟐 =
𝑸𝟏𝟐 =
𝒗𝟏𝟐 𝑬𝟐
𝟏−𝒗𝟏𝟐 𝒗𝟐𝟏
𝑬𝟐
𝒗𝟐𝟏 𝑬𝟏
𝟏−𝒗𝟏𝟐 𝒗𝟐𝟏
(1)
𝑸𝟔𝟔 = 𝑮𝟏𝟐
𝟏−𝒗𝟏𝟐 𝒗𝟐𝟏
𝒗𝟐𝟏 = 𝒗𝟏𝟐
=
𝑬𝟐
(2)
𝑬𝟏
The properties used for this project come from Appendix C of [9]. This text
provided values for E1, E2, ν12, ν23, and G12. However, in order to conduct the FEM
analysis in ANSYS the values of E3, ν13, G23, and G13 were also required. These values
were calculated by assuming transverse isotropy. Transverse isotropy states that for a
unidirectional ply the moduli in the directions perpendicular to the fiber direction are
roughly equal and much smaller than the modulus in the fiber direction. Symbolically that
is E1 >> E2 and E3 ~ E2. Along the same lines ν13 = ν12, and G13 = G12 [8]. Lastly, by using
the same assumption, the value for G23 is calculated via equation 3 [6].
𝑮𝟐𝟑 =
𝑬𝟐
𝟐∗(𝟏+𝒗𝟐𝟑 )
(3)
A complete list of the values used for the CLP plate analysis is shown in table 1.
The properties listed in this table are for the AS/3501 (Graphite/Epoxy) combination. The
5
table also contains the thickness of a laminated ply as well as the fiber volume fraction,
which was also used in the analysis.
Table 1: Properties of AS/3501 Composite Ply
Property
Value
Comment
E1 (Pa)
138E+9
from text [9]
E2 (Pa)
8.96E+9
from text [9]
E3 (Pa)
8.96E+9
calculated
ν12
0.3
from text [9]
ν23
0.59
from text [9]
ν13
0.02
calculated
G12 (Pa)
7.1E+9
from text [9]
G23 (Pa)
2.82E+9
calculated
G13 (Pa)
7.1E+9
calculated
tply (m)
1.27E-4
from text [9]; ply thickness
Vf (%)
66
from text [9]; fiber volume fraction
2.3 CLP Plate Modeling
For this project the plates were modeled using SHELL181 element in ANSYS.
This type of element has 4 nodes with six degrees of freedom at each node and provides
the opportunity to add layers. [6]. Therefore once the material properties of a single ply
were entered, the plate was constructed in ANSYS by stacking the correct number layers,
each representing a ply, in the correct angular orientation.
The layers were laid out in a [0 0 90 90 0 0 90 90 0 0]s orientation for the cross
ply CLP plate and in a [0 0 45 45 0 0 -45 -45 0 0]s orientation for the angle ply CLP plate.
The numbers in the brackets represent the order and angle of the ply layer while the letter
“s” at the bracket end implies symmetry. Therefore, the first two layers of the cross ply
plate are oriented at 0 degrees, the next two are oriented at 90 degrees from the first two,
and so on. Layers 11 through 20 are symmetric to layers 10 through 1 respectively. Figures
6
3 and 4 show the ply arrangement for the CLP plates. In figure 3, the solid blue layers
represent the 0 degree plies while the red pattern represent the 90 degree or ±45 degree
angles plies for the CLP plates 1 and 2 respectively. Figure 4 provides a visual
representation of the fiber direction for a specific ply angle.
Figure 3: CLP Plate Layer (Ply) Arrangement
Figure 4: CLP Plate Layer (Ply) Arrangement
7
For this project all the plates will be square with constant side lengths of a = 0.2
and a height of H = 0.00254 meters, which was derived from having 20 layers with tply =
0.000127 m. These dimensions were chosen such that the plate size is manageable and can
be easily tested in a rig for the purpose of validating the results of the FEA analysis. The
sheet body plate was then meshed using quadrilateral areas and a “smart size” of 3. This
mesh size was used because it provided consistent results between the ANSYS models
and the analytical solution as it is demonstrated later in the report. Figure 5 shows a
screenshot of the meshed ANSYS model geometry.
Figure 5: Meshed Sheet Body Representing CLP Plate
Finally loads and boundary conditions were applied to the sheet body representing
the CLP plate to simulate a simply supported plate with a uniform and normal pressure
load applied to it. To achieve this loading condition all sides were constrained in the z
8
direction. Then sides 1 and 3 were prevented from rotating about the y axis and sides 2
and 4 were prevented from rotating about the x axis. To fully constrain the model side 1
was prevented from moving along the y direction and side 4 was prevented from moving
in the x direction. Lastly the pressure load of 20 kPa is applied evenly on the surface. The
ANSYS code for the 20 layer CLP plate model that is outputted to a text “log” file is
provided in Appendix A for reference.
As mentioned in section 2.1, Maple was used to perform Classical Lamination
Theory analysis on the cross ply CLP plate. This code is provided in Appendix B. The
intent of this analysis was to validate the FEA results with analytical results. To reduce
the labor involved in building a 20 layer Maple model, two ANSYS models were built for
the cross play plate. One was a 20 layer model with 0.000127m thick layers, while the
other was a 10 layer model with 0.000254m thick layer. The combination of plies into a
thicker layer for the 10 layer model was justifiable because layer orientation in the 20 layer
plate changes with every two layers. Simplifying the model to a ten layer configuration
allowed for an easier comparison using Classical Lamination Theory for a 10 layer cross
ply plate. The 10 layer CLP plate ANSYS code is provided in Appendix C. The results of
these calibration runs are explained in detail in section 3 “Results and Discussion”.
2.4 FGP Plate Element Properties
The average modulus of elasticity of the graphite reinforced FGP plates is lower
relative to the CLP plates. This reduction in modulus happens because the graphite in the
FGP plate is deposited in grains, thus losing some of the high strength capability derived
from a fibrous condition. Since the textbook used to obtain the laminate properties did not
have properties for graphite grains or chopped fiber, the properties were obtained from
Zoltec Companies online brochure [10]. Before the properties for chopped carbon fibers
were used for the graphite in the FGP plate, the properties for continuous tow carbon fiber
and unidirectional prepreg from Zoltec’s brochures were compared with fiber properties
and laminate ply properties from the textbook to validate that the properties used were
somewhat consistent. The modulus values for unidirectional and continuous fibers and for
unidirectional ply from both Zoltek’s brochure and the textbook matched well. This
9
closeness in the material elastic modulus properties from the two sources acts as a
validation for using materials from two different sources with the assumption that the
materials whose properties are provided are similar enough. This comparison is captured
in table 2.
Table 2: Young’s Modulus Comparison Between Two Sources
AS4,
Panex 35,
Text [9]
Zoltek [10]
Continuous Fiber E (Pa)
234E+9
242E+9
2.5
Unidirectional Ply E (Pa)
138E+9
134E+9
2.9
% Difference
The only value used from Zoltek’s brochure is the elastic modulus for the chopped
fiber. The brochure did not provide a value for Poisson’s ratio for chopped fiber, therefore
the value used for analysis was for unidirectional fibers and came from the textbook [9].
In addition the matrix or epoxy properties also came from the text book. Table 3
summarizes the values from the two different sources and the comparison between the
two.
Table 3: Properties of FGP Plate Substrates
Material
Property
Value
Source
23E+9
Zoltek [10]
Graphite,
Panex 35
Er or E1 (Pa)
Chopped Fiber
AS4
νr or ν1
Epoxy Matrix, 3501
FGP Plate
0.26
text [9]
4.4E+9
text [9]
νm or ν2
0.36
text [9]
tplate (m)
1.27E-4
text [9]
Vf (%)
66
text [9]
Em or E2 (Pa)
10
2.5 FGP Plate Properties Via The Mori-Tanaka Method
The properties of the FGP plate were estimated via the Mori-Tanaka method. This
method calculates the through thickness material properties by using the through thickness
volume fractions of each component making up the FGP plate [7]. It divides the plate into
a chosen number of isotropic layers and calculates the properties for each layer based on
the volume fraction of the constituents for that layer. In the Mori-Tanaka based equations
below the subscript “p” stands for plate, “L” for layer, “1” for material 1 which is the
graphite, and “2” for material 2, which is the epoxy. The two material properties needed
for each layer are Young’s Modulus and Poisson’s ratio. The functions representing these
properties through the FGP plate thickness are:
𝑬𝒑 = 𝟑 ∙ 𝑲𝒑 ∙ (𝟏 − 𝟐 ∙ 𝝂𝒑 )
𝝂𝒑 =
𝟏
𝑮𝒑
𝟐∙(𝟏+
𝝀𝒑
(4)
(5)
)
Gp is the equation representing the through thickness shear modulus of the FGP plate and
it is denoted by:
𝑮𝒑 = 𝑮𝟏 +
(𝑮𝟐 −𝑮𝟏 )∙𝑽𝟐
(𝟏+
(𝟏−𝑽𝟐 )∙(𝑮𝟐 −𝑮𝟏 )
)
𝑮𝟏 +𝒇𝟏
(6)
Where ƒ1 is represented by:
𝒇𝟏 =
𝑮𝟏 (𝟗 ∙ 𝑲𝟏 + 𝟖 ∙ 𝑮𝟏 )
𝟔 ∙ (𝑲𝟏 + 𝟐 ∙ 𝑮𝟏 )
λp is Lamѐ first parameter, represented by:
𝟐
𝝀𝒑 = 𝑲 𝒑 − ∙ 𝑮 𝒑
𝟑
(7)
Kp is the bulk modulus of the plate through its thickness and it is represented by the
equation:
𝑲𝒑 = 𝑲𝟏 +
(𝑲𝟐 −𝑲𝟏 )∙𝑽𝟐
(𝟏−𝑽𝟐 )∙(𝑲𝟐 −𝑲𝟏 )
(𝟏+
)
𝟒
𝑲𝟏 +( )∙𝑮𝟏
𝟑
11
(8)
V2 is the equation representing the volume fraction of material 2 through the plate
thickness and K1 and K2 are the bulk moduli of materials 1 and 2. They are represented
by:
𝑲𝒏 =
𝑬𝒏
𝟑∙(𝟏−𝟐∙𝝂𝒏 )
(9)
G1 is the shear modulus of material 1 and it represented by:
𝑮𝒏 =
𝑬𝒏
𝟐∙(𝟏−𝝂𝒏 )
(10)
Where n = 1, 2.
The FGP plate was then divided into 10 equally thick layers and the elastic
modulus and Poisson’s ratio for each layer were calculated using equations (4) and (5).
For these calculations it was assumed that the elastic modulus and Poisson’s ratio vary
linearly within the layer. This assumption is valid because the layer is very thin with
respect to the plate thickness. Then the values for E and ν were calculated for each layer
by solving the respective equations for the z value representing the middle plane of each
layer.
To explore the effects of graphite distribution on plate deflection, several plates
with different through thickness volume fractions were studied. All of these plates
maintained a total volume ratio of 66% reinforcement and 34% matrix and a total plate
thickness of 0.00254 m. These numbers were kept constant to be consistent with the
laminated plates. The volume fraction of the epoxy matrix (V2) was varied through the
plate thickness via several cosine functions to determine the distribution of graphite that
caused the least amount of deflection. This studied whether placing the highest content of
graphite on the surface, where it experiences the highest stress, or in the center, where it
experiences the lowest stress, provided the highest stiffness. The matrix distribution was
then also varied linearly and via a sine function to check whether the graphite provided
the highest stiffness when it experienced tensile stress, meaning when it was concentrated
on the opposite side the pressure was applied on, or when it experienced compressive
stress. Equation (11) through (15) represents the different variation of V2 through the plate
thickness via a cosine function that were investigated.
12
𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 + 𝑽𝒂 ∙ 𝐜𝐨𝐬 (
𝑯
𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 − 𝑽𝒂 ∙ 𝐜𝐨𝐬 (
𝑯
)
(11)
)
(12)
𝟐∙𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 + 𝑽𝒂 ∙ 𝐜𝐨𝐬 (
𝑯
𝟑∙𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 + 𝑽𝒂 ∙ 𝐜𝐨𝐬 (
𝑯
𝟒∙𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 + 𝑽𝒂 ∙ 𝐜𝐨𝐬 (
𝑯
)
(13)
)
(14)
)
(15)
Where V20 is a constant that positions the distribution curve above the y=0 axis to make
sure the material ratio is always positive. Va is the amplitude of the cosine function and
the value inside the cosine provides the frequency of the curve. H is the plate thickness
which is 0.00254 meters, and “z” is the independent variable that represents the thickness
of the plate. In order to get a plate with constant distribution of the epoxy and graphite,
V20 was set to 0.34 and Va was set to 0.
The equations describing the linear distribution of the matrix through the plate
thickness are described by equation (16) and (17). For the plates where the volume fraction
is linear the value of V20 is both a y intercept value and a contributor to the slope of the
line.
𝑽𝟐 = 𝑽𝟐𝟎 ∙ (𝟏 +
𝑽𝟐 = 𝑽𝟐𝟎 ∙ (𝟏 −
𝟐
𝑯
𝟐
𝑯
∙ 𝒛)
(16)
∙ 𝒛)
(17)
The equations describing the sine distribution of the matrix volume through the plate
thickness are described by equations (18) and (19). Similar to the cosine functions, the
sine functions have the values of V20 and Va.
𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 + 𝑽𝒂 ∙ 𝐬𝐢𝐧 (
𝑯
𝝅∙𝒛
𝑽𝟐 = 𝑽𝟐𝟎 − 𝑽𝒂 ∙ 𝐬𝐢𝐧 (
13
𝑯
)
(18)
)
(19)
Lastly, the volume fraction of the graphite through the plate thickness, or V1, is described
by equation (13).
𝑽𝟏 = 𝟏 − 𝑽𝟐
(20)
Table 4 summarizes the 18 plates analyzed in this study. Column 1 gives the plate label or
number, column 2 describes which equation was used for the volume distribution of the
matrix, and columns 3 and 4 give the values of V20 and Va used. For plates 15 and 16 the
volume distribution of the matrix is linear and therefore Va is not used. In addition, figures
6 through 23 portray the volume fraction of the graphite and epoxy for the 18 plates
studied. The Maple code used to obtain the volume fraction and the through thickness
plate properties for FGP Plate 1 is shown in Appendix D.
Table 4: Summary of FGP Plates Studied
Plate #
V2 Equation
V20
Va
Plate #
V2 Equation
V20
Va
1
(11)
0.000
0.534
10
(13)
0.340
0.050
2
(11)
0.170
0.267
11
(14)
0.430
0.430
3
(11)
0.340
0.000
12
(14)
0.280
0.280
4
(12)
0.500
0.251
13
(15)
0.340
0.340
5
(12)
0.660
0.503
14
(15)
0.340
0.340
6
(12)
0.830
0.770
15
(17)
0.340
N/A
7
(12)
0.935
0.935
16
(16)
0.340
N/A
8
(13)
0.340
0.340
17
(18)
0.340
0.340
9
(13)
0.340
0.160
18
(19)
0.340
0.340
14
Figure 6: Through Thickness Material Fractions of Plate 1
Figure 7: Through Thickness Material Fractions of Plate 2
15
Figure 8: Through Thickness Material Fractions of Plate 3
Figure 9: Through Thickness Material Fractions of Plate 4
16
Figure 10: Through Thickness Material Fractions of Plate 5
Figure 11: Through Thickness Material Fractions of Plate 6
17
Figure 12: Through Thickness Material Fractions of Plate 7
Figure 13: Through Thickness Material Fractions of Plate 8
18
Figure 14: Through Thickness Material Fractions of Plate 9
Figure 15: Through Thickness Material Fractions of Plate 10
19
Figure 16: Through Thickness Material Fractions of Plate 11
Figure 17: Through Thickness Material Fractions of Plate 12
20
Figure 18: Through Thickness Material Fractions of Plate 13
Figure 19: Through Thickness Material Fractions of Plate 14
21
Figure 20: Through Thickness Material Fractions of Plate 15
Figure 21: Through Thickness Material Fractions of Plate 16
22
Figure 22: Through Thickness Material Fractions of Plate 17
Figure 23: Through Thickness Material Fractions of Plate 18
23
2.6 Functionally Graded Material Plate Modeling
The FGP plates were modeled in ANSYS similar to the CLP plates. The element
chosen was SHELL 181. What differed between the CLP and FGP plates was that the FGP
plate was composed of 10 isotropic layers with different elastic properties. For the plates
that have volume fractions represented by the cosine function, the layers are symmetric
about the center plane. For the plates whose volume fractions are represented by linear
and sine functions there is no symmetry between the layers. The layer thickness was kept
constant to 0.000254 m, which is equal to double the layer thickness of the laminated plate.
This ensures that the total plate thickness remains constant between the two configuration
types. In addition the plate side dimensions were kept constant at a=0.2 m. The mesh size
used was smart size 3 which is the same mesh size as the CLP plates. Also the same
boundary conditions used to constrain the four sides of the plates and apply the pressure
load on the CLP plates were used to apply the boundary conditions on the FGP plates. The
ANSYS log file code used to calculate the deflection of FGP plate 1 is provided in
Appendix E for reference.
24
3. Results and Discussion
3.1 Laminated Plate Results and Validation of Modeling Method
Two types of laminated plates were studied, a cross ply and an angle ply. In
addition to providing an analysis data point for deflection, the cross ply plate was used to
validate the modeling method. First the cross ply plate was modeled in ANSYS using the
methodology described in section 2.3 of this report with 20 layers as shown by figure 3.
Next, the cross ply plate was modeled assuming that the consecutive plies that were
oriented in the same direction could be combined into one layer with double the thickness,
as is shown by the color/pattern coding in figure 3. The plate dimensions, mesh size, and
boundary conditions were not changed. As shown in figures 24 and 25 both the 20 and 10
layer plates deflect 0.005139 meters. This demonstrates that combining layers with similar
orientations is an acceptable modeling methodology.
Figure 24: Deflection of 10 Layer Cross Ply Laminated Plate
25
Figure 25: Deflection of 20 Layer Cross Ply Laminated Plate
Furthermore the 10 layer plate was analyzed in Maple using the Classical
Lamination theory, as mentioned in section 2.2. The deflection obtained with the Maple
analysis was 0.005149 m which is 0.2% different than the deflection obtained in ANSYS.
This demonstrates that the layering, meshing, and boundary application method in
ANSYS is acceptable. Therefore the 10 layer modeling methodology was used to analyze
the angle ply laminate and the FGP plates.
The angle ply plate with layers at 45 degrees relative to each other was also
analyzed in ANSYS using the 10 layer method. As shown in figure 26, the angle ply plate
deflected 0.004289 m, which is less than the cross ply plate.
26
Figure 26: Deflection of 10 Layer Angle Ply Laminated Plate
3.2 Functionally Graded Plate Results
Eighteen Functionally Graded Polymer plates with different graphite distributions
through plate thickness were analyzed to determine whether FGP plate deflections can be
comparable to laminated plates. FGP plates 1 through 7 were analyzed first to understand
the effects of reinforcement material distribution on plate deflection. These 7 plates had
epoxy matrix material volume distribution represented by equations 11 and 12 which are
versions of a half of a cosine curve. The volume distribution of graphite is also represented
by a cosine curve equal to 1 minus the equation representing the volume fraction of the
epoxy. Plate 1 was 100% graphite at the surface, while the rest of the plates employed a
gradual shift of the graphite inboard toward the plate center culminating in plate 7 having
almost 0% graphite at the plate surfaces and 100% graphite in the middle plane. Plate 1
deflected by the least amount and with the gradual shift of graphite to the mid-plane and
27
away from the surfaces there was also a gradual increase in plate deflection, with plate 7
having the highest deflection.
Plates 8 through 14 further explored the effects of graphite distribution through the
plate thickness on plate deflection. The through thickness volume fractions of the epoxy
through these plates were expressed by the cos(nπ) equation, where n was 2 for plates 8
through 10, 3 for 11 and 12, and 4 for plates 13 and 14. The conclusion from the analysis
of plates 8 through 14 was the same as that of plates 1 through 7. Therefore, for a plate
that had material distribution of elements symmetric about its mid-plane, the least amount
of deflection was experienced when the distribution of reinforcing material, in this case
graphite, was biased towards the plate surfaces.
Lastly plates that had non-symmetric material distribution of elements through the
plate thickness, as represented by equations 16 through 19, were analyzed. These plates
were studied to understand whether the stiffness of the plates comes from having
reinforcing material distributed on the surface where the load is applied, on the opposite
side, or both. Biasing the graphite distribution towards the same surface that the load is
applied on produces a lower plate deflection than doing so on the opposite side. These
deflections are in the 0.0167 to 0.0178 m range, which is in the middle of the full deflection
range experienced by symmetric plates.
Table 5 summarizes the FGP plates analyzed, what equations were used to obtain
the through thickness volume fraction of the epoxy (V2), what the fraction of graphite (V1)
is at the plate surfaces and center, and the plate deflection under the applied load. Figures
27 through 29, which are also shown in section 2.5, illustrate the through thickness
volume fractions of the three plates that deflected the least. Also figures 30 through 32,
which are also repeated from section 2.5, illustrate the through thickness volume fractions
of the plates that deflected the most. The trend portrayed by table 5 and figures 27 through
32 is that plates with more reinforcing material toward the plate surfaces deflect less, while
plates with more reinforcing material concentrated toward the plate center deflect more.
28
Table 5: Summary of FGP Plate Graphite Distribution and Deflections
#
V2 Equation V1 @ z=+H/2 V1 @ z=0 V1 @ z=-H/2 Deflection (m)
1
(11)
1.000
0.466
1.000
0.013788
2
(11)
0.830
0.563
0.830
0.015257
3
(11)
0.660
0.660
0.660
0.016839
4
(12)
0.500
0.751
0.500
0.018456
5
(12)
0.340
0.843
0.340
0.020196
6
(12)
0.170
0.940
0.170
0.022204
7
(12)
0.065
1.000
0.065
0.023533
8
(13)
1.000
0.320
1.000
0.012647
9
(13)
0.820
0.500
0.820
0.014739
10
(13)
0.710
0.610
0.710
0.016152
11
(14)
0.570
0.140
0.570
0.013366
12
(14)
0.720
1.000
0.720
0.019346
13
(15)
0.320
0.320
0.320
0.018047
14
(15)
1.000
1.000
1.000
0.015223
15
(17)
1.000
0.660
0.320
0.016724
16
(16)
0.320
0.660
1.000
0.017501
17
(18)
0.320
0.660
1.000
0.017785
18
(19)
1.000
0.660
0.320
0.016870
29
Figure 27: Through Thickness Material Fractions of Plate 8, FGP Plate With Lowest Deflection
Figure 28: Through Thickness Material Fractions of Plate 11, FGP Plate With 2nd Lowest Deflection
30
Figure 29: Through Thickness Material Fractions of Plate 1, FGP Plate With 3rd Lowest Deflection
Figure 30: Through Thickness Material Fractions of Plate 5, FGP Plate With 3rd Highest Deflection
31
Figure 31: Through Thickness Material Fractions of Plate 6, FGP Plate With 2nd Highest Deflection
Figure 32: Through Thickness Material Fractions of Plate 7, FGP Plate With Highest Deflection
32
4. Conclusion
This project analyzed the deflection due to a uniform pressure load of Composite
Laminated Polymer plates and Functionally Graded Polymer plates. All of the plates
analyzed had the same dimensions and were composed of equal portions of the graphite
and epoxy. The laminated plates were composed of layers of unidirectional graphite fibers
and epoxy resin oriented in [0 0 90 90 0 0 90 90 0 0]s for the cross ply plate and [0 0 45
45 0 0 -45 -45 0 0]s for the angle ply plate. The FGP plates were composed of 3D printed
graphite and epoxy powder. The through thickness distribution of the graphite and epoxy
for the FGP plates was represented by a collection of cosine, sine, and linear functions.
The FGP plates deflected two to five times more than the laminated plates. The
main reason is that the elastic modulus of the graphite used for the FGP plates was a sixth
of the value used for the laminated ply and a tenth of the elastic modulus of a fiber. The
reason for this debit is that the carbon powder loses a lot of the stiffness achieved by the
fiber form, thus resulting in higher plate deflections.
The deflection of the FGP plates depended heavily on the distribution of graphite
through the plate thickness. The plates that featured higher concentrations of graphite
along the plate surfaces deflected the least amount. These plates had a graphite distribution
that was symmetric about the mid-plane and spiked out towards the plate surfaces. These
plates deflected in the 0.013 to 0.015 m range. Plates that featured higher concentrations
of graphite in the mid-plane and lower concentrations on the plate surfaces deflected the
most. Plates that had graphite distributions that were symmetric about and spiked around
the mid-plane deflected in the 0.019 to 0.024 m range. There was a third group of plates
that deflected in the 0.016 to 0.018 m range. These were plates with an unsymmetrical
graphite distribution and plates with a symmetric graphite distribution that was evenly
spread throughout the plate thickness.
Although the FGP plates experienced higher deflections that the laminated plates,
they offer the benefit of optimizing the through thickness element distribution to minimize
plate deflection. This optimization is possible due to the flexibility in material distribution
allowed by the 3D printing process used to produce the FGP plates. This makes up for the
debit incurred from using non fibrous carbon as the reinforcing material. As seen by the
results of this study, the best performing FGP plate deflected 0.0126m, while the best
33
performing CLP plate deflected 0.0043m. While the best FGP plate deflects three times
as much as the best CLP plate, it is important to note that that the elastic modulus of the
graphite used for the FGP plates is 23 GPA, which is one tenth of the 234GPa elastic
modulus of the graphite used in the CLP plates. Therefore optimizing the graphite
distribution in the FGP plates makes up for the debit incurred from not having continuous
fiber reinforcement.
34
References
[1]
http://en.wikipedia.org/wiki/Composite_material
Date Accessed: 10/18/2014
[2]
http://en.wikipedia.org/wiki/Composite_laminates
Date Accessed: 10/18/2014
[3]
Faris Tarlochan, “Functionally Graded Material: A New Breed of Engineered
Material”, Journal of Applied Mechanical Engineering, November 2012
http://omicsgroup.org/journals/Functionally-Graded-Material-A-New-Breed-ofEngineered-Material-6832-2168-9873-1-e115.pdf
[4]
http://fugahumana.files.wordpress.com/2012/07/layup1.gif
Date Accessed: 10/18/2014
[5]
http://appliedmechanics.asmedigitalcollection.asme.org/data/Journals/JAMCAV/2
6727/015804amj2.jpeg
Date Accessed: 10/18/2014
[6]
Kenneth Carroll, “Comparative Deflection Analysis of Aluminum and Composite
Laminate Plates Using the Rayleigh-Ritz and the Finite Element Method”, RPI
Hartford Master’s Project Fall 2013
http://www.ewp.rpi.edu/hartford/~ernesto/SPR/Carroll-FinalReport.pdf
[7]
Kevin Pendley, “Modal Analysis of Simply Supported Functionally Graded Square
Plates”, RPI Hartford Master’s Project Spring 2014
http://www.ewp.rpi.edu/hartford/~ernesto/SPR/Pendley-FinalReport.pdf
[8]
Mer Arnel Manahan, “A Finite Element Study of the Deflection of Simply Supported
Composite Plates Subject to Uniform Load”, RPI Hartford Master’s Project Fall
2011
http://www.ewp.rpi.edu/hartford/~ernesto/SPR/Manahan-FinalReport.pdf
[9]
László P. Kollár and George S. Springer, “Mechanics of Composite Structures”,
Cambridge University Press 2003, Appendix C
[10] Property brochures for Panex ® 35 Continuous Tow, Chopped Fiber (Type -85), and
Prepreg Tapes from Zoltek Companies website
http://www.zoltek.com/products/panex-35/
35
Appendix A
Appendix A is the text output from the log file of the ANSYS model used to calculate the
deflection of the 20 layer CLP cross ply plate.
/PREP7
!*
ET,1,SHELL181
!*
!*
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,1,,138e9
MPDATA,EY,1,,8.96e9
MPDATA,EZ,1,,8.96e9
MPDATA,PRXY,1,,0.3
MPDATA,PRYZ,1,,0.59
MPDATA,PRXZ,1,,0.3
MPDATA,GXY,1,,7.1e9
MPDATA,GYZ,1,,2.82e9
MPDATA,GXZ,1,,7.1e9
! /REPLOT,RESIZE
sect,1,shell,,
secdata, 0.000127,1,0.0,3
secdata, 0.000127,1,0.0,3
secdata, 0.000127,1,90,3
secdata, 0.000127,1,90,3
secdata, 0.000127,1,0.0,3
secdata, 0.000127,1,0.0,3
secdata, 0.000127,1,90,3
secdata, 0.000127,1,90,3
secdata, 0.000127,1,0.0,3
secdata, 0.000127,1,0.0,3
secdata, 0.000127,1,0.0,3
secdata, 0.000127,1,0.0,3
secdata, 0.000127,1,90,3
secdata, 0.000127,1,90,3
secdata, 0.000127,1,0.0,3
secdata, 0.000127,1,0.0,3
secdata, 0.000127,1,90,3
secdata, 0.000127,1,90,3
secdata, 0.000127,1,0.0,3
secdata, 0.000127,1,0.0,3
secoffset,MID
seccontrol,,,, , , ,
RECTNG,0,0.25,0,0.25,
SMRT,6
SMRT,5
SMRT,4
36
SMRT,3
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,2,4,4,ORDE,2
FITEM,2,1
FITEM,2,-4
!*
/GO
DL,P51X, ,UZ,
FLST,2,2,4,ORDE,2
FITEM,2,2
FITEM,2,4
!*
/GO
DL,P51X, ,ROTX,
FLST,2,2,4,ORDE,2
FITEM,2,1
FITEM,2,3
!*
/GO
DL,P51X, ,ROTY,
FLST,2,1,4,ORDE,1
FITEM,2,1
!*
/GO
DL,P51X, ,UY,
FLST,2,1,4,ORDE,1
FITEM,2,4
!*
/GO
DL,P51X, ,UX,
FLST,2,1,5,ORDE,1
FITEM,2,1
/GO
!*
SFA,P51X,1,PRES,20000
FINISH
/SOL
! /STATUS,SOLU
37
SOLVE
FINISH
/POST1
!*
! /EFACET,1
! PLNSOL, U,Z, 0,1.0
38
Appendix B
Appendix B is the Maple code used to calculate the deflection of the 10 layer CLP cross
ply plate using Classical Lamination Theory.
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Appendix C
Appendix C is the text output from the log file of the ANSYS model used to calculate the
deflection of the 10 layer CLP cross ply plate. This code, with modified layer properties,
was also used to analyze the deflection of the angle ply CLP plate.
/PREP7
!*
ET,1,SHELL181
!*
!*
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,1,,138e9
MPDATA,EY,1,,8.96e9
MPDATA,EZ,1,,8.96e9
MPDATA,PRXY,1,,0.3
MPDATA,PRYZ,1,,0.59
MPDATA,PRXZ,1,,0.3
MPDATA,GXY,1,,7.1e9
MPDATA,GYZ,1,,2.82e9
MPDATA,GXZ,1,,7.1e9
! /REPLOT,RESIZE
sect,1,shell,,
secdata, 0.000254,1,0.0,3
secdata, 0.000254,1,90,3
secdata, 0.000254,1,0.0,3
secdata, 0.000254,1,90,3
secdata, 0.000254,1,0.0,3
secdata, 0.000254,1,0.0,3
secdata, 0.000254,1,90,3
secdata, 0.000254,1,0.0,3
secdata, 0.000254,1,90,3
secdata, 0.000254,1,0.0,3
secoffset,MID
seccontrol,,,, , , ,
RECTNG,0,0.25,0,0.25,
SMRT,6
SMRT,5
SMRT,4
SMRT,3
MSHAPE,0,2D
MSHKEY,0
!*
CM,_Y,AREA
ASEL, , , ,
1
CM,_Y1,AREA
CHKMSH,'AREA'
CMSEL,S,_Y
45
!*
AMESH,_Y1
!*
CMDELE,_Y
CMDELE,_Y1
CMDELE,_Y2
!*
FLST,2,4,4,ORDE,2
FITEM,2,1
FITEM,2,-4
!*
/GO
DL,P51X, ,UZ,
FLST,2,2,4,ORDE,2
FITEM,2,2
FITEM,2,4
!*
/GO
DL,P51X, ,ROTX,
FLST,2,2,4,ORDE,2
FITEM,2,1
FITEM,2,3
!*
/GO
DL,P51X, ,ROTY,
FLST,2,1,4,ORDE,1
FITEM,2,1
!*
/GO
DL,P51X, ,UY,
FLST,2,1,4,ORDE,1
FITEM,2,4
!*
/GO
DL,P51X, ,UX,
FLST,2,1,5,ORDE,1
FITEM,2,1
/GO
!*
SFA,P51X,1,PRES,20000
FINISH
/SOL
! /STATUS,SOLU
SOLVE
FINISH
/POST1
!*
! /EFACET,1
! PLNSOL, U,Z, 0,1.0
46
Appendix D
Appendix D is the Maple code used to calculate the volume fraction and elastic properties
of FGP plate 1. This same code was used for FGP plates 2-18 with the modification of the
appropriate equation representing the through thickness volume fraction of the epoxy
matrix, or V2.
FGP Plate 1 Properties
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Appendix E
Appendix E is the text output from the log file of the ANSYS model used to calculate the
deflection of FGP plate 1. This code, modified with layer properties outputted from the
Mori-Tanaka Maple code, was also used to analyze the deflection of the rest of the FGP
plates.
/PREP7
!*
ET,1,SHELL181
!*
!*
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,1,,20.58e9
MPDATA,PRXY,1,,0.265
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,2,,16.61e9
MPDATA,PRXY,2,,0.275
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,3,,13.75e9
MPDATA,PRXY,3,,0.285
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,4,,11.91e9
MPDATA,PRXY,4,,0.293
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,5,,11.01e9
MPDATA,PRXY,5,,0.298
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,6,,11.01e9
MPDATA,PRXY,6,,0.298
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,7,,11.91e9
MPDATA,PRXY,7,,0.293
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,8,,13.75e6
MPDATA,PRXY,8,,0.285
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,9,,16.61e9
MPDATA,PRXY,9,,0.275
52
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,10,,20.58e9
MPDATA,PRXY,10,,0.265
sect,1,shell,,
secdata, .000254,1,0.0,3
secdata, .000254,2,0.0,3
secdata, .000254,3,0.0,3
secdata, .000254,4,0.0,3
secdata, .000254,5,0.0,3
secdata, .000254,6,0.0,3
secdata, .000254,7,0.0,3
secdata, .000254,8,0.0,3
secdata, .000254,9,0.0,3
secdata, .000254,10,0.0,3
secoffset,MID
seccontrol,,,, , , ,
RECTNG,0,0.25,0,0.25,
SMRT,6
SMRT,5
SMRT,4
SMRT,3
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,2,4,4,ORDE,2
FITEM,2,1
FITEM,2,-4
!*
/GO
DL,P51X, ,UZ,
FLST,2,2,4,ORDE,2
FITEM,2,2
FITEM,2,4
!*
/GO
DL,P51X, ,ROTX,
FLST,2,2,4,ORDE,2
FITEM,2,1
FITEM,2,3
53
!*
/GO
DL,P51X, ,ROTY,
FLST,2,1,4,ORDE,1
FITEM,2,1
!*
/GO
DL,P51X, ,UY,
FLST,2,1,4,ORDE,1
FITEM,2,4
!*
/GO
DL,P51X, ,UX,
FLST,2,1,5,ORDE,1
FITEM,2,1
/GO
!*
SFA,P51X,1,PRES,20000
FINISH
/SOL
! /STATUS,SOLU
SOLVE
FINISH
/POST1
!*
54