by
Rigels Bejleri
A Thesis 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

Finite Element Study in the Deflection of Composite Laminate Plates and Functionally
Graded Material Plates ................................................................................................. i
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
GLOSSARY ..................................................................................................................... vi
KEYWORDS ................................................................................................................... vii
ACKNOWLEDGMENT ................................................................................................ viii
ABSTRACT ..................................................................................................................... ix
1.
Introduction .................................................................................................................. 1
1.1
Background ........................................................................................................ 1
1.2
Problem Description........................................................................................... 3
2.
Methodology ................................................................................................................ 4
3.
Results and Discussion .............................................................................................. 15
4.
Conclusion ................................................................................................................. 18
5.
References .................................................................................................................. 19
6.
Appendices ................................................................................................................ 21
iii

Table 1: Properties of AS/3501 Composite Ply ................................................................. 6
iv

Figure 1: Laminated Composite Plate [4] .......................................................................... 2
Figure 2: Functionally Graded Material [5] ....................................................................... 2
Figure 3: CLP Plate Layer (Ply) Arrangement .................................................................. 7
Figure 4: Meshed Sheet Body Representing CLP Plate .................................................... 8
v

vi

vii

Type the text of your acknowledgment here. viii

This project studies the deflection of Composite Laminate Plates and Functionally
Graded Material Plates in order to determine which configuration is stiffer. The deflection of these plates will be calculated via the Finite Element Method using the Finite Element
Analysis software ANSYS. The study samples will be simply supported square plates with a uniform pressure applied. Both types of composites will be comprised of a set material pair and a set volumetric ratio of the material pair. The goal of this study is to gain an understanding of the theoretical deflection differences between the two types of composites. However, this study will 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 the text of your abstract here. ix

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 in figure 1, are a type of FCP that is manufactured by stacking layers of high strength fiber reinforced polymers 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 composed of a continuously varying and smooth distribution of the two or more constituents it is made of. 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 will study the deflection capability of FGM polymer plates manufactured via 3D printing and compare it to the deflection of CLP plates
1

composed of the same two materials to see if the FGM plates could be a comparable alternative to CLPs.
Figure 1: Laminated Composite Plate [4]
Figure 2: Functionally Graded Material [5]
2

This project will utilize ANSYS to study the deflection behavior of simply supported Composite Laminate Polymer (CLP) plates and Functionally Graded Material
(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. Lastly all of the plates will be composed of the same material couple, a matrix and reinforcing material, and 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 FGM plates evaluated will have varying distributions of the matrix and reinforcement materials throughout their thicknesses. For this project, it is assumed that an unlimited range of varying material distribution can be achieved using
3D printing to manufacture the FGM plates.
3

This project builds upon the work that Kenneth Carroll and Kevin Pendley did in their Master’s Project. In “Comparative Deflection Analysis of Aluminum and Composite
Laminate Plates Using the Rayleigh-Ritz and the Finite Element Method” [6], Carroll compared the deflection of composite laminate plates with the deflection of a thin
Aluminum plate. The analysis for the composite plate was done using the finite element method (FEM) in ANSYS as well as using analytical solutions in Maple.
For this project the CLP plates will first be analyzed in ANSYS. Then the solution will be validated using Maple to solve the analytical solution. Once the modeling methodology in ANSYS is validated by using the analytical methods, the same modeling methodology will be used to analyze the FGM plates in ANSYS.
The properties of the FGM plate will be approximated using the Mori-Tanaka method referenced in “Modal Analysis of Simply Supported Functionally Graded Square
Plates” [7]. This methodology uses the material properties and the volumetric ratio of each constituent to calculate the properties of the FGM plate through its thickness.
In order to analyze the deflection of CLP plates analytically the Young’s modulus, shear modulus, and Poisson’s ratio of the composite ply are needed. These properties will be used to calculate the Reduced Stiffness Matrix which relates the laminate stress to the laminate stress. The Reduced Stiffness Matrix will be used to calculate the CLP deflection via the Classical Lamination Theory for the cross ply plate and via the Rayleigh-Ritz
Method for the angle ply plate [6].
Because the thicknesses of the CLP plates analyzed in this project is much smaller than their length and width, and the deflection of the plate is small, it is 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 see by equation 1 below [6], the 3x3 Reduced Stiffness Matrix only requires the values of the Young’s 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 E
1
, E
2
,
ν
12
, and G
12
, where x has been replaced with the subscript 1 and y with 2. The value for
ν
21
, which is also needed for the analytical solution is calculated using equation 2 [6].
𝑸
𝟏𝟏
=
𝑬
𝟏
𝟏−𝒗
𝟏𝟐 𝒗
𝟐𝟏
𝑸
𝟏𝟐
= 𝒗
𝟏𝟐
𝑬
𝟐
𝟏−𝒗
𝟏𝟐 𝒗
𝟐𝟏
= 𝒗
𝟐𝟏
𝑬
𝟏
𝟏−𝒗
𝟏𝟐 𝒗
𝟐𝟏
𝑸
𝟐𝟐
=
𝑬
𝟐
𝟏−𝒗
𝟏𝟐 𝒗
𝟐𝟏
𝑸
𝟔𝟔
= 𝑮
𝟏𝟐 𝒗
𝟐𝟏
= 𝒗
𝟏𝟐
𝑬
𝟐
𝑬
𝟏
(1)
(2)
The properties used for this project come from Appendix C of “Mechanics of
Composite Structures” by L. P. Kollár and G. S. Springer. This text provides values for
E
1
, E
2
, ν
12
, ν
23
, and G
12
. However, in order to conduct the FEM analysis in ANSYS the values of E
3
, ν
13
, G
23
, and G
13 are also required. To calculate these values we assume transverse isotropy, meaning that for the 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. Therefore E
1
>> E
2
and E
3
~ E
2
. Along the same lines ν
13
= ν
12
, and
G
13
= G
12
[8]. Lastly, by using the same assumption, the value for G
23
is calculated via equation 3 [6].
𝑮
𝟐𝟑
=
𝑬
𝟐
𝟐∗(𝟏+𝒗
𝟐𝟑
)
(3)
5

Property
E
1
(Pa)
E
2
(Pa)
E
3
(Pa)
ν
12
ν
23
ν
13
G
12
(Pa)
G
23
(Pa)
G
13
(Pa) t ply
(m)
V f
(%)
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 table also contains the thickness of a laminated ply as well as the fiber volume fraction, which will also be used in the analysis.
Table 1: Properties of AS/3501 Composite Ply
Value
138E+9
8.96E+9
8.96E+9
0.3
0.59
0.02
7.1E+9
2.82E+9
7.1E+9
1.27E-4
66
Comment from text [9] from text [9] calculated from text [9] from text [9] calculated from text [9] calculated calculated from text [9]; ply thickness from text [9]; fiber volume fraction
The first step in modeling the CLP plate in ANSYS is the selection of the element type. For this project the plates are modeled using SHELL181 elements. This type of element has 4 nodes with six degrees of freedom at each node. In addition SELL181 allows for layering so that a laminated plate model can be built [6]. Therefore once the element type is chosen and the material properties for the ply are entered, the next step is to add the layers that make up the plate at the correct angle.
The layer layouts are [0 0 90 90 0 0 90 90 0 0]s for the cross ply CLP plate and [0
0 45 45 0 0 -45 -45 0 0]s 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
6

symmetry. This means that one half of the layers are shown in the brackets and that the rest of them are symmetric. For example, for the first one the first two plies are oriented at 0 degrees, the next two are oriented at 90 degrees from the first two, and so on. Figure
3 shows the ply arrangement for the CLP plates. In this figure the blue layers represent the
0 degree plies while the yellow layers represent the 90 degree or 45 degree angles plies for the CLP plate 1 and 2 respectively. As shown in table 1, each layer or ply has a thickness of t ply
= 0.000127 meters and since the plates will consist of 20 layers the total plate thickness will be t plate
= 0.00254 meters. This thickness will be kept constant for all the plates analyzed in this project.
Figure 3: CLP Plate Layer (Ply) Arrangement
The shell body is then created and meshed in ANSYS. For this project all the plates will be square with constant side lengths of a = 0.2 meters. These dimensions are 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 plate sheet is then meshed using quadrilateral areas and a “smart size” of 3. Figure 4 shows a screenshot of the meshed
ANSYS model geometry. Finally loads and boundary conditions are applied to the sheet body representing the CLP plate to simulate a simply supported plate with a uniform load applied to it. First all sides are constrained in the z direction. Then sides 1 and 3 are
7

prevented from rotating about the y axis. Similarly sides 2 and 4 are prevented from rotating about the x axis. To fully constrain the model side 1 is prevented from moving along the y direction and side 4 is prevented from moving in the x direction. Lastly the pressure load of 20 kilo Pascale is applied evenly on the surface.
Figure 4: Meshed Sheet Body Representing CLP Plate
Since entering properties for 20 plies is laborious a simplification is attempted on the model. Two models were compared. One had 20 layers with a layer thickness of
0.000127m and the other had 10 layers with a layer thickness of 0.000254m. This simplification was made because layer orientation changes with every two layers which allowed me to combine layer couples with the same orientation into one thicker layer.
Simplifying the model to a ten layer configuration allows for an easier comparison using
Classical Lamination Theory (CLT) for the cross ply plate.
8

The goal of these calibrations is to be able to use a simplified 10 layer model. The way this is achieved is by using CLT is Maple to validate the 10 layer ANSYS model, and using the 10 layer ANSYS model to validate the 20 layer ANSYS model. If the deflection for these three different calculation methods are equal then the simplifications are applicable. The results of these calibration runs will be explained in detail in section 3
“Results and Discussion”.
The modulus of elasticity of the graphite reinforced FGM plates is lower relative to the CLP plate. This reduction in modulus happens because the graphite in the FGM 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. Before the properties for chopped carbon fibers were used for the graphite in the FGM 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 closeness in the material tensile 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
Continuous Fiber E (Pa)
Unidirectional Ply E (Pa)
AS4,
Text [9]
234E+9
138E+9
Panex 35,
Zoltek [10]
242E+9
134E+9
% Difference
2.5
2.9
9

The only value used from Zoltek’s brochure is Young’s 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, or source 1. 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 FGM Plate Substrates
Material Property
Graphite,
Chopped Fiber
Epoxy Matrix, 3501
Panex 35 E r
or E
1
(Pa)
AS4
ν r
or ν
1
E m or E
2
(Pa)
ν m
or ν
2
FGM Plate t plate
(m)
V f
(%)
Value
23E+9
0.26
4.4E+9
0.36
1.27E-4
66
Source
Zoltek [10] text [9] text [9] text [9] text [9] text [9]
The properties of the FGM 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 FGM plate [7]. It divides the plate into a chosen number of isentropic 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 FGM plate thickness are:
𝑬 𝒑
= 𝟑 ∙ 𝑲 𝒑
∙ (𝟏 − 𝟐 ∙ 𝝂 𝒑
) 𝝂 𝒑
=
𝟏
𝟐∙(𝟏+
𝑮𝒑 𝝀𝒑
)
(4)
(5)
10

G p
is the equation representing the through thickness shear modulus of the FGM plate and it is denoted by:
𝑮 𝒑
= 𝑮
𝟏
+
(𝟏+
(𝑮
𝟐
−𝑮
𝟏
)∙𝑽
𝟐
(𝟏−𝑽𝟐)∙(𝑮𝟐−𝑮𝟏)
𝑮𝟏+𝒇𝟏
)
Where ƒ
1
is represented by: 𝒇
𝟏
=
𝑮
𝟏
(𝟗 ∙ 𝑲
𝟏
𝟔 ∙ (𝑲
𝟏
+ 𝟖 ∙ 𝑮
𝟏
)
+ 𝟐 ∙ 𝑮
𝟏
)
λ p
is Lamѐ first parameter, represented by: 𝝀 𝒑
= 𝑲 𝒑
−
𝟐
𝟑
∙ 𝑮 𝒑
(6)
(7)
K p
is the bulk modulus of the plate through its thickness and it is represented by the equation:
𝑲 𝒑
= 𝑲
𝟏
+
(𝑲
𝟐
−𝑲
𝟏
)∙𝑽
𝟐
(𝟏+
(𝟏−𝑽𝟐)∙(𝑲𝟐−𝑲𝟏)
𝑲𝟏+(
𝟒
𝟑
)∙𝑮𝟏
)
(8)
V
2 is the equation representing the volume fraction of material 2 through the plate thickness and K
1
and K
2
are the bulk moduli of materials 1 and 2. They are represented by:
𝑲 𝒏
=
𝑬 𝒏
𝟑∙(𝟏−𝟐∙𝝂 𝒏
)
(9)
G
1
is the shear modulus of material 1 and it represented by:
𝑮 𝒏
=
𝑬 𝒏
𝟐∙(𝟏−𝝂 𝒏
)
(10)
The FGM plate is then divided into 10 equally thick layers and the tensile modulus and Poisson’s ratio for each layer is layer is calculated using equations (4) and (5). For these calculations it is assumed that the tensile 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.
11

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 first plate that was looked at had an even constant mixture of the components. Then the volume fraction of 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 graphite on the surface, where it experiences the highest stress, or on the center, where it experiences the lowest stress, provided the highest stiffness. The matrix distribution was then also varied linearly to check whether the graphite provided the highest stiffness when it experienced tensile stress, meaning when it is on the opposite side the pressure is applied on, or when it experienced compressive stress. Equation (11) through (15) represents the variation of V
2 through the plate thickness via a cosine function.
𝑽
𝟐
= 𝑽𝟐𝟎 + 𝑽 𝒂
∙ 𝐜𝐨𝐬 ( 𝝅∙𝒛
𝑯
)
𝑽
𝟐
= 𝑽𝟐𝟎 − 𝑽 𝒂
∙ 𝐜𝐨𝐬 ( 𝝅∙𝒛
𝑯
)
𝑽
𝟐
= 𝑽𝟐𝟎 + 𝑽 𝒂
∙ 𝐜𝐨𝐬 (
𝟐∙𝝅∙𝒛
𝑯
)
𝑽
𝟐
= 𝑽𝟐𝟎 + 𝑽 𝒂
∙ 𝐜𝐨𝐬 (
𝟑∙𝝅∙𝒛
𝑯
)
𝑽
𝟐
= 𝑽𝟐𝟎 + 𝑽 𝒂
∙ 𝐜𝐨𝐬 (
𝟒∙𝝅∙𝒛
𝑯
)
(11)
(12)
(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. V a
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 describes the location of the thickness of the plate. In order to get a plate with constant distribution of the matrix and reinforcement V20 is set to 0.34 and V a
is set to 0.
12

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.
𝑽
𝟐
𝟐
= 𝑽𝟐𝟎 ∙ (𝟏 +
𝑯
∙ 𝒛)
𝑽
𝟐
𝟐
= 𝑽𝟐𝟎 ∙ (𝟏 −
𝑯
∙ 𝒛)
(16)
(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 V a.
𝑽
𝟐
= 𝑽𝟐𝟎 + 𝑽 𝒂
∙ 𝐬𝐢𝐧 ( 𝝅∙𝒛
𝑯
)
𝑽
𝟐
= 𝑽𝟐𝟎 − 𝑽 𝒂
∙ 𝐬𝐢𝐧 ( 𝝅∙𝒛
𝑯
)
(18)
(19)
Lastly, the volume fraction of the graphite through the plate thickness, or V1, is described by equation (13).
𝑽
𝟏
= 𝟏 − 𝑽
𝟐
(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 V a
used. For plates 15 and 16 the volume distribution of the matrix is linear and therefore V a
is not used.
13

Table 4: Summary of Plates Studied
7
8
5
6
9
Plate # V
2
Equation V20 Va
1 (11) 0.000 0.534
2
3
4
(11)
(11)
(12)
0.170 0.267
0.340 0.000
0.500 0.251
(12)
(12)
(12)
(13)
(13)
0.660 0.503
0.830 0.770
0.935 0.935
0.340 0.340
0.340 0.160
Plate # V
2
Equation V20 Va
10 (13) 0.340 0.050
11
12
13
(14)
(14)
(15)
0.430 0.430
0.280 0.280
0.340 0.340
14
15
16
17
18
(15)
(17)
(16)
(18)
(19)
0.340 0.340
0.340 N/A
0.340 N/A
0.340 0.340
0.340 0.340
The FGM plates were modeled in ANSYS similar to the CLP plates. The element chosen was SHELL 181. What differed between the CLP and FGM plates is that the FGM plate is composed of 10 isentropic layers that differ from each other. 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 vary linearly there is no symmetry between the layers. The layer thickness is kept constant to 0.000254 m, which is equal to 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 FGM plates.
14

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 coding in figure 3. The plate dimensions, mesh size, and boundary conditions were not changed. As shown in figures 5 and 6 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 5: Deflection of 10 Layer Cross Ply Laminated Plate
15

Figure 6: 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 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 FGM plates. The full maple code of the Classical Lamination Theory is shown in Appendix A.
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 7, the angle ply plate deflected 0.004289 m, less than the cross ply plate.
16

Figure 7: Deflection of 10 Layer Angle Ply Laminated Plate
17

18

[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-of-
Engineered-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/
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[11] http://en.wikipedia.org/wiki/Functionally_graded_material
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