Modal Analysis of Simply Supported Functionally Graded Square
Plates
by
Kevin Pendley
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
______________________________________________________
Professor Ernesto Gutierrez-Miravete, Engineering Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
May, 2014
© Copyright 2014
by
Kevin Pendley
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
NOMENCLATURE…………………………………………………………………..…vi
ACKNOWLEDGMENT ................................................................................................. vii
ABSTRACT ................................................................................................................... viii
1. Introduction…………………………………………………………………………...1
1.1 Background…………………………………………………………………...1
1.2 Description……………………………………………………………………1
2. Methodology………………………………………………………………………….5
2.1 Mori-Tanaka Method of FGM Properties………………………………….....5
2.2 Plate Modeling………………………………………………………………..6
2.3 Modal Analysis……………………………………………………………….7
3. Results and Discussions………………………………………………………………9
3.1 Isotropic Plate Comparison.…………………………………………………..9
3.2 Functionally Graded Plate Comparison (p = 2)………………………..……13
3.3 Functionally Graded Plate Comparison (p = 10)……………..……………..16
3.4 Comparison to Efraim…..………………………………………...…………20
3.5 Titanium Ceramic FGP……………………………………………………...21
4. Conclusions………………………………………………………………………….23
5. References…………………………………………………………………………...24
6. Appendices….……………………………………………………………………….25
6.1 ANSYS analysis code for isotropic plates…………………………………..25
6.2 Maple inputs – Timoshenko…………………………………….…………..34
6.3 Maple inputs – Mori-Tanaka………………………………………………..35
iii
LIST OF TABLES
Table 1: Material Properties ………………...………………………………...…………2
Table 2: Plates Studied …………………………………………………………...……...3
Table 3: Isotropic Plate Frequencies with Thickness of 0.05m……………..……………9
Table 4: Isotropic Plate Frequencies with Thickness of 0.025m………..………………10
Table 5: Functionally Graded Plate Frequencies with p = 2……………………….…...14
Table 6: Functionally Graded Plate Frequencies with p = 10………………..…………17
Table 7: Efraim and FEA Comparison………………………………………………….20
Table 8: Titanium and Ceramic Study…………………………………………………..21
iv
LIST OF FIGURES
Figure 1: Cross Sections of Functionally Graded Material…………………….……….....1
Figure 2: SHELL181 Geometry……………………………………………...…………..6
Figure 3: Plate Boundary Conditions……………………..……………………………...7
Figure 4: Isotropic Plate Mode Shapes of Steel and Aluminum……………….………..11
Figure 5: Isotropic Plate Mode Shapes of Alumina……………..………………………12
Figure 6: Isotropic Plate Mode Shapes of Zirconia..……………………………………13
Figure 7: Mode Shapes of Steel-Alumina, p = 2, H = 0.05m..………………...………..15
Figure 8: Mode Shapes of Steel-Alumina, p = 2, H = 0.025m..…………………...……16
Figure 9: Mode Shapes of Steel-Alumina, p = 10, H = 0.05m..…………………...……18
Figure 10: Mode Shapes of Aluminum-Zirconia, p = 10, H = 0.05m..………………....18
Figure 11: Mode Shapes of Aluminum-Zirconia & Steel-Alumina, p = 2, H = 0.05m...19
Figure 12: Natural Frequency of Titanium, Ceramic and Their FGPs………………….22
v
NOMENCLATURE
E
Modulus of Elasticity (Pa)
υ
Poisson’s Ratio (dimensionless)
ρ
Density (kg/m3)
D
Flexural Rigidity (Pa·m3)
a
Plate Length (m)
b
Plate Width (m)
H
Thickness (m)
z
Thickness Direction (m)
f
Frequency (Hz)
Vc
Volume Fraction, material 1, ceramic (dimensionless)
Vm
Volume Fraction, material 2, metal (dimensionless)
Vt
Volume Fraction of material 1 at top of plate (dimensionless)
Vb
Volume Fraction of material 2 at bottom of plate (dimensionless)
K
Shear Modulus (Pa)
µ
Bulk Modulus (Pa)
p
Power (dimensionless)
λ
Lamé’s first parameter (dimensionless)
vi
ACKNOWLEDGMENT
I want to thank my family and friends for their continued support in all my endeavors.
Special thanks to Professor Ernesto Gutierrez-Miravete for his invaluable guidance.
vii
ABSTRACT
This project investigated the modal response of isotropic and functionally graded plates
with the plates being simple supported using the Finite Element Method in ANSYS.
Analysis of FGPs of different power law was performed. The mode shapes were also
examined for each case. This project focused on metal/ceramic functionally graded
plates. The plates had a varying elastic modulus, Poisson ratio and density in the
thickness direction according to the power law. ANSYS was used to analyze natural
frequencies and mode shapes. The results were compared to results previously obtained
using COMSOL and to those obtained using a recently developed closed form
approximation.
viii
1. Introduction
1.1 Background
A functionally graded material (FGM) is a two phase composite characterized by a ratio
that is continuously varying from 100% of one component through to 100% of the other
component that can be defined by a function. This continuity prevents the material from
having the disadvantages of composites such as delamination due to large inter-laminar
stresses, initiation and propagation of cracks because of large plastic deformation at the
interfaces. Additionally, traditional composites (i.e. laminates) are mixtures, and they
therefore involve a compromise between the desirable properties of the component
materials. Since significant proportions of an FGM contain the pure form of each
component, the need for compromise is eliminated. The properties of both components
can be fully utilized. For example, the toughness of a metal can be mated with the
refractory properties of a ceramic, without any compromise in the toughness of the metal
side or the refractoriness of the ceramic side. They are also ideal for minimizing thermo
mechanical difference in metal-ceramic bonding.
These functionally graded materials can be designed for specific functions and
applications. Various approaches based on the bulk (particulate processing), preform
processing, layer processing and melt processing are used to fabricate the functionally
graded materials.
Figure 1. Cross Sections of Functionally Graded Material [2]
1
Figure 1 shows the gradual transition across the thickness of the material. This transition
is represented by the equation
1
𝑧 𝑝
𝑉2 = 𝑉𝑏 + (𝑉𝑡 − 𝑉𝑏 ) ∙ ( + ) where the V’s are
2
𝐻
the volume fraction of material 2 at position z through the thickness of the plate.
Functionally graded materials offer potential applications in areas where the operating
conditions are extreme. Some examples are rocket heat shields, heat exchanger tubes,
heat-engine components, thermoelectric generators, heat-engine components, plasma
facings for fusion reactors, and electrically insulating metal/ceramic joints.
1.2 Description of Problem
The purpose of this project was to perform modal analysis of functionally graded plate.
The FEA analytical tool used in this project was ANSYS. The dimensions of the FGP
were 1m x 1m and the thicknesses were 0.025m and 0.05m. The plate boundary
condition was simply supported. The natural frequency and mode shapes results were
then compared to the classical solution in [6], to previously obtained FEA COMSOL
results [1] and to results obtained using a recently developed closed form approximation
[5].
The material properties used in these modal analyses are summarized in the table 1
below.
Material
Density
Young's
(Kg/m^3)
Mod (Pa)
Poisson
Steel
7800
1.00E+11
0.3
Aluminum
2700
7.50E+09
0.33
Alumina
3690
3.00E+11
0.27
Zirconia
5700
2.00E+11
0.3
Titanium
4430
1.14E+11
0.342
Ceramic (Si3N4)
3310
3.14E+11
0.27
Steel
7800
1.00E+11
0.3
Table 1: Material Properties
2
The array of plates investigated is shown in table 2 below.
Plate Description
Materials
Steel
Aluminum
Alumina
Isotropic
Zirconia
Titanium
Ceramic (Si3N4)
Steel
Steel-Alumina
p=2
Aluminum-Zirconia
Titanium-Ceramic
p = 10
Steel-Alumina
Aluminum-Zirconia
Linear
Titanium-Ceramic
p=1
Titanium-Ceramic
p=5
Titanium-Ceramic
p=7
Titanium-Ceramic
Table 2: Plates Studied
3
2. Methodology
This project develops FEA models of simply supported plates using ANSYS. These
ANSYS models were validated by using classical results for the transverse vibrations of
isotropic plates [6] and comparing to Saunders’s FGP project [1] and the approximate
formula in [5]. The Mori-Tanaka approach was used to estimate the values of the
material properties through the thickness of the plate.
2.1 Mori-Tanaka Method of FGM Properties
The Mori-Tanaka method was employed to estimate the material properties of the
functionally graded plate. This method uses the volume fractions of each constituent
material making up the FGP throughout its thickness.
The Mori-Tanaka method is used to derive the 3 material properties for the modal
analysis. The bulk modulus for FGM is given by:
(𝐾2 −𝐾1 )∙𝑉2
𝐾𝐹𝐺𝑀 = 𝐾1 +
(1−𝑉2 )∙(𝐾2 −𝐾1 )
(1+
)
4
𝐾1 +( )∙𝜇1
3
(1)
The shear modulus for FGM is given by:
𝜇𝐹𝐺𝑀 = 𝜇1 +
(𝜇2 −𝜇1 )∙𝑉2
(1+
(1−𝑉2 )∙(𝜇2 −𝜇1 )
)
𝜇1 +𝑓1
(2)
Where:
𝑓1 =
𝜇1 ∙(9∙𝐾1 +8∙𝜇1 )
(2a)
6∙(𝐾1 +2∙𝜇1 )
The density for FGM is given by:
𝜌𝐹𝐺𝑀 = 𝜌1 ∙ 𝑉1 + 𝜌2 ∙ 𝑉2
(3)
From the above the Poisson Ratio for FGM is given by:
𝜈𝐹𝐺𝑀 =
1
(4)
𝜇
2∙(1+ 𝐹𝐺𝑀
)
𝜆
Where:
4
𝜆 = 𝐾𝐹𝐺𝑀 −
2∙𝜇𝐹𝐺𝑀
(5)
3
The modulus of Elasticity for FGM is given by:
𝐸𝐹𝐺𝑀 = 3 ∙ (1 − 2 ∙ 𝜇𝐹𝐺𝑀 ) ∙ 𝐾𝐹𝐺𝑀
(6)
In these expressions K1, µ1 and V1 represent respectively the bulk modulus, shear
modulus and volume fraction of one material and K2, µ2 and V2 represent respectively
the bulk modulus, shear modulus and volume fraction of the other material.
The volume fraction of material 2 is
1
𝑧 𝑝
𝑉2 = 𝑉𝑏 + (𝑉𝑡 − 𝑉𝑏 ) ∙ ( + )
2
𝐻
(7)
And the volume fraction of material 1 is
𝑉1 = 1 − 𝑉2
(8)
It should be noted that 𝑉1 + 𝑉2 = 1.
Here, Vb and Vt are the volume fractions of material 2 (ceramic) at the bottom and top of
the plate, respectively. p is the parameter that dictates the volume fraction profile
through the thickness (H) [4].
2.2 Plate Modeling
ANSYS was used to model the functionally graded plates. The SHELL181 element was
chosen as the element type (figure 2). SHELL181 is suitable for analyzing thin to
moderately-thick shell structures. It is a four-node element with six degrees of freedom
at each node: translations in the x, y, and z directions, and rotations about the x, y, and zaxes [3].
5
Figure 2: SHELL181 Geometry [3]
Its layered application makes it suitable for functionally graded material modeling. 10
layers were used for FGP modeling. Each layer was set to isotropic condition and the
material properties were calculated by the Mori-Tanaka method. Maple was used to
determine the material properties by calculating Young’s Modulus, Poisson Ratio and
density at the mid-span of each layer. A FE mesh consisting of 20x20 square elements
was used in all the calculations.
2.3 Modal Analysis
The analysis was carried out using modal analysis in ANSYS for the isotropic and FGM
plates. The analysis option chosen as mode extraction method was Block Lanczos. Each
edge was simply supported as shown in figure 3 below.
6
Figure 3: Plate Boundary Conditions
The isotropic plate solutions were compared to COMSOL results [1] and the classical
solution in Timoshenko given below [6].
𝜋
𝐷
𝑚2
𝑛2
𝑓𝑚,𝑛 = √ ( 2 + 2 )
2 𝜌ℎ 𝑎
𝑏
(9)
Where D is the flexural rigidity of the plate represented by [6]:
𝐷=
𝐸ℎ3
12∙(1−𝜈2 )
(10)
The functionally graded plate natural frequencies were also compared to COMSOL
results and to the Efraim frequency estimates.
7
3. Results and Discussion
3.1 Isotropic Plate Comparison
Several isotropic plates were first studied and validated against Timoshenko plate
predicted frequencies. These were also plate studied and verified against COMSOL plate
[1]. The results of these studies are shown in tables 3 and 4 below.
Material
Steel
Aluminum
Alumina
Zirconia
Mode
Frequency
Frequency
Frequency
(ANSYS)
(COMSOL)
(Theoretical)
Hz
Hz
Hz
Error %
Error %
(ANSYS)
(COMSOL)
1
169.06
166.14
170.2
-0.67
-2.39
2
419.95
413.73
425.5
-1.30
-2.77
3
419.95
413.95
425.5
-1.30
-2.71
4
663.3
651.36
680.8
-2.57
-4.32
1
79.496
77.67
80.06
-0.70
-2.99
2
197.39
192.9
200.15
-1.38
-3.62
3
197.39
192.98
200.15
-1.38
-3.58
4
311.63
301.98
320.24
-2.69
-5.70
1
421.9
412.41
424.63
-0.64
-2.88
2
1048.5
1027.44
1061.58
-1.23
-3.22
3
1048.5
1027.66
1061.58
-1.23
-3.20
4
1656.7
1612.9
1698.53
-2.46
-5.04
1
279.68
273.68
281.57
-0.67
-2.80
2
694.74
681.45
703.92
-1.30
-3.19
3
694.74
681.56
703.92
-1.30
-3.18
4
1097.3
1069.87
1126.28
-2.57
-5.01
Table 3: Isotropic Plate Frequencies with Thickness of 0.05m
The percent errors for ANSYS are less than 3% where the percent errors for COMSOL
are less than 6%. Both percent errors are considered acceptable. In addition, both the
8
ANSYS and COMSOL percent errors increase as the mode increases. These low percent
errors validate the ANSYS model.
Material
Steel
Aluminum
Alumina
Zirconia
Mode
Frequency
(ANSYS)
Hz
Frequency
(COMSOL)
Hz
Frequency
(Theoretical)
Hz
Error %
(ANSYS)
Error %
(COMSOL)
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
85.1
213.5
213.5
340.4
40.026
100.41
100.41
160.06
212.33
532.76
532.76
849.54
140.78
353.2
353.2
563.14
84.55
211.79
211.84
338.07
39.709
99.335
99.345
158.238
210.88
528.32
528.44
843.23
139.88
350.37
350.46
559.28
85.1
212.75
212.75
340.4
40.03
100.07
100.07
160.12
212.32
530.79
530.79
849.27
140.78
351.96
351.96
563.14
0.00
0.35
0.35
0.00
-0.01
0.34
0.34
-0.04
0.00
0.37
0.37
0.03
0.00
0.35
0.35
0.00
-0.65
-0.45
-0.43
-0.68
-0.80
-0.73
-0.72
-1.17
-0.68
-0.47
-0.44
-0.71
-0.64
-0.45
-0.43
-0.69
Table 4: Isotropic Plate Frequencies with Thickness of 0.025m
The natural frequency percent errors for all 4 materials are less than 2%. These low
percent errors further validate the ANSYS model. The steel, aluminum, alumina and
zirconia mode shapes are depicted in the figures below.
The ANSYS model mode shapes for the steel and aluminum plates are shown in figure
4. These mode shapes are similar to the COMSOL predicted mode shapes [1].
Additionally, these mode shapes match Ferreira; et al mode shapes [8]
9
Figure 4: Isotropic Plate Mode Shapes of Steel and Aluminum
Figure 5: Isotropic Plate Mode Shapes of Alumina
10
The ANSYS model mode shapes for the alumina plates are shown in figure 5. These
mode shapes are similar to the COMSOL predicted mode shapes [1]. The 2nd and 3rd
mode shape nodal lines are approximately horizontal and vertical, respectively.
Figure 6: Isotropic Plate Mode Shapes of Zirconia
The zirconia plate ANSYS model mode shapes for above have a similar 1st, 3rd and 4th
mode shape to the COMSOL mode shapes [1]. The 2nd mode shape is different between
ANSYS and COMSOL. The 2nd mode shape for the ANSYS zirconia plate has nodal
line approximately diagonal as illustrated above where as the COMSOL zirconia plate
has nodal line approximately horizontal.
11
3.2 Functionally Graded Plate Comparison (p = 2)
Functionally graded material frequencies are bounded by the materials they are made up
of. To clarify, the FGM natural frequencies response will vary between the natural
frequencies of its isotropic constituents. This variation is governed by the volume
fraction in the thickness direction as it changes from 100% of one material to 100% of
the other material. Power law p = 2 is one example of this, which is shown in the table 4
below.
H = 0.025m
Material
Mode
H = 0.05m
Frequency
Frequency
Frequency
Frequency
(ANSYS)
(COMSOL)
(ANSYS)
(COMSOL)
Hz
Hz
Hz
Hz
H=0.025m
H=0.05m
Percent
Percent
Difference
Difference
1
114.65
113.71
227.98
222.23
0.82
2.55
Steel
2
287.77
284.47
567.39
551.71
1.15
2.80
Alumina
3
287.77
284.5
567.39
551.71
1.14
2.80
4
459.07
451.85
897.73
861.17
1.59
4.16
1
53.975
54.88
107.17
107.11
-1.66
0.06
Aluminum
2
135.38
137.2
265.95
265.16
-1.34
0.30
Zirconia
3
135.38
137.21
265.95
265.2
-1.34
0.28
4
215.83
217.72
419.71
412.47
-0.87
1.74
Table 5: Functionally Graded Plate Frequencies with p = 2
The percent difference for the 0.025m thick FGP are less than 2% where the percent
difference for the 0.05m thick FGP are less than 5%. Both percent differences are
considered acceptable. Also, the steel-alumina percent differences are showing an
increasing trend as the mode increases. The aluminum-zirconia 0.025m thick FGP shows
a decreasing trend as the mode increases. Whereas the aluminum-zirconia 0.05m thick
12
FGP shows an increasing trend as the mode increases. This suggests that the 0.025m
thick FGPs would have better agreement as the mode increases.
Figure 7: Mode Shapes of Steel-Alumina p = 2, H = 0.05m
The steel-alumina mode shapes in figure 7 are similar to the mode shapes of alumina
where the 2nd and 3rd mode shape nodal lines are approximately horizontal and vertical,
respectively. This is also in agreement with the COMSOL mode shapes.
13
Figure 8: Mode Shapes of Steel-Alumina p = 2, H = 0.025m
The ANSYS steel-alumina mode shapes in figure 8 correspond to the COMSOL mode
shapes. The COMSOL aluminum-zirconia mode shapes are same as in figure 8 above
for both 0.025m and 0.05m thick FGP. However the ANSYS aluminum-zirconia mode
shapes are somewhat different. The ANSYS aluminum-zirconia mode shapes for 0.025m
thick plate are similar to the zirconia isotropic plate mode shapes in figure 6. The
ANSYS aluminum-zirconia mode shapes for 0.05m thick plate are similar to the steelalumina plate mode shapes in figure 7.
3.3 Functionally Graded Plate Comparison (p = 10)
Power law p = 10 is another specific example of how the variation is governed by the
volume fraction as it changes from 100% of one material to 100% of the other material.
The natural frequency results are shown in the table 6 below.
14
H = 0.025m
Material
Mode
H = 0.05m
Frequency
Frequency
Frequency
Frequency
(ANSYS)
(COMSOL)
(ANSYS)
(COMSOL)
Hz
Hz
Hz
Hz
H=0.025m
H=0.05m
Percent
Percent
Difference
Difference
1
96.113
96.44
191.06
188.41
-0.34
1.40
Steel
2
241.21
241.25
475.17
467.67
-0.02
1.59
Alumina
3
241.21
241.27
475.17
467.67
-0.02
1.59
4
384.7
383.07
751.29
729.74
0.42
2.91
1
47.135
49.51
93.55
96.34
-4.91
-2.94
Aluminum
2
118.2
123.76
231.95
238.65
-4.60
-2.85
Zirconia
3
118.2
123.76
231.95
238.65
-4.60
-2.85
4
188.36
196.17
365.69
370.62
-4.06
-1.34
Table 6: Functionally Graded Plate Frequencies with p = 10
The steel-alumina percent difference for the 0.025m thick FGP are less than 1% where
the percent difference for the 0.05m thick FGP are less than 3%. The aluminum-zirconia
percent difference for the 0.025m thick FGP are less than 5% where the percent
difference for the 0.05m thick FGP are less than 3%. All these percent differences are
acceptable.
15
Figure 9: Mode Shapes of Steel-Alumina p = 10, H = 0.05m
Figure 10: Mode Shapes of Aluminum-Zirconia p = 10, H = 0.05m
16
The ANSYS steel-alumina and aluminum-zirconia mode shapes are somewhat different
to those computed by COMSOL for 0.05m thick plates as depicted figures 9 and 10,
respectively. The ANSYS and COMSOL aluminum-zirconia mode 2 and 3 mode shapes
are reversed. The may be due to interchanging vibrational modes, similar to what was
discuss by Efraim [5]
Figure 11: Mode Shapes of Aluminum Zirconia & Steel-Alumina p = 10, H = 0.025m
The ANSYS steel-alumina mode shapes in figure 11 are similar to the COMSOL mode
shapes [1]. The 2nd and 3rd modes nodal lines are diagonal.
17
3.4 Comparison to Efraim [5]
To further validate the FGP modeling, the frequencies were compared against Efraim’s
prediction. The Efraim equation for predicting frequencies is as follow.
𝑓𝐹𝐺𝑀 = 𝑓𝑀 √
𝜌𝑀 ∙𝐸𝑒𝑞
𝜌𝑒𝑞 ∙𝐸𝑀
∙ 𝑉𝑀 + 𝑓𝐶 √
𝜌𝐶 ∙𝐸𝑒𝑞
𝜌𝑒𝑞 ∙𝐸𝐶
∙ 𝑉𝐶
(11)
Where:
𝐸𝑒𝑞 = 𝐸𝑀 ∙ 𝑉𝑀 + 𝐸𝐶 ∙ 𝑉𝐶
(12)
𝜌𝑒𝑞 = 𝜌𝑀 ∙ 𝜌𝑀 + 𝜌𝐶 ∙ 𝜌𝐶
(13)
𝐻/2
𝑧
1 𝑝
𝑉𝐶 = ∫−𝐻/2 ( + ) 𝑑𝑧
𝐻
2
(14)
𝑉𝑀 = 1 − 𝑉𝐶
(15)
Equation 11 was used to compute the frequencies and the results compared to ANSYS
and COMSOL computed frequencies. The results are shown in table 7 below.
Material
Alumina
p = 10
Frequency
Efraim
Efraim
Error %
Error %
ANSYS
COMSOL
ANSYS
COMSOL
Hz
Hz
Frequency
ANSYS
COMSOL
Hz
Hz
1
191.06
188.41
188.2
189.46
1.52
-0.55
2
475.17
467.67
467.51
473.66
1.64
-1.26
3
475.17
467.67
467.51
473.66
1.64
-1.26
4
751.29
729.74
738.45
757.86
1.74
-3.71
Mode
Steel
Frequency
Frequency
Table 7: Efraim and FEA Comparison
The ANSYS error percent is less than 2% and the COMSOL error percent is less than
4%. These suggest that the Efraim predictions are in good agreement with ANSYS and
COMSOL, and the results of the three computations methods are in good agreement
with each other.
18
3.5 Titanium Ceramic FGP
Six Titanium-Ceramic isotropic and their functionally graded plate frequencies were also
studied. The results are shown in table 8 below.
Plate Frequency (Hz)
Plate 1
Mode
Plate 2
Plate 3
Plate 4
Plate 5
Titanium
Titanium
Titanium
Titanium
Ceramic
Ceramic
Ceramic
Ceramic
p=1
p=2
p=5
p=7
Ceramic
Plate 6
Titanium
1
229.36
165.88
153.36
142.56
138.93
122.38
2
575.49
416.47
384.91
357.7
348.6
306.97
3
575.49
416.47
384.91
357.7
348.6
306.97
4
917.67
664.5
613.96
570.37
555.86
489.33
Table 8: Titanium, Ceramic and Their FGP Study
The ceramic plate has a higher natural frequency than the titanium plate. The values of
the functionally graded plate frequencies vary between the frequencies of its two
constituent materials. The frequency decreases as the content of ceramic decreases. This
holds true for all the modes shown. Figure 12 is a graphical representation of table 8.
1000
Frequency, Hz
800
600
Mode 1
400
Mode 2
200
Mode 4
0
0
2
4
6
8
Plates
Figure 12: Natural frequencies of Titanium, Ceramic and Their FGPs
19
Plate1 (ceramic) has the highest natural frequencies and Plate 6 (titanium) has the lowest
natural frequencies. Other plates (FGPs) show a decreasing trend between the ceramic
and titanium plates as power law (p) increases.
20
4. Conclusions
Several modal analyses on isotropic and functionally graded plates were conducted.
ANSYS was the FEA analytical tool used to model and analyze the natural frequencies
and mode shapes of these simply supported FGPs. The results were compared to
previously obtained COMSOL FEA analysis, with Timoshenko’s classical formula for
the isotropic studies and with the recently developed closed form approximation of
Efraim for the functionally graded material studies.
The following observation emerged from this project:

Overall ANSYS and COMSOL FEA analysis tool provide excellent prediction of
the natural frequencies and mode shapes. COMSOL directly uses the FGM
equations Young’s modulus, Poisson ratio and density. ANSYS uses a layer
method where each layer has its input of Young’s modulus, Poisson ratio and
density. Both FEA analysis tools are suitable for further studies of FGM.

The metal-ceramic functionally graded plates suggest that the frequencies are
strongly influenced by the metal component whereas the mode shapes are
strongly influenced by the ceramic component.
21
5. References
[1] Wesley L Saunders, Modal Analysis of Rectangular Simply-Supported Functionally
Graded Plates, Rensselaer Polytechnic Institute (2011)
[2] http://www2.haut.edu.cn/cllx2/Article/UploadFiles/200805/2008052123142867.jpg
[3] ANSYS Element Reference. ANSYS R15.0 Academic Help Viewer
[4] Senthil S. Vel, R.C. Batra, Three-dimensional exact solution for the vibration of
functionally graded rectangular plates, Journal of Sound and Vibration 272 (2004), pages
703-730
[5] Elia Efraim, Accurate formula for determination of natural frequencies of FGM
plates basing on frequencies of isotropic plates, Engineering Procedia 10 (2011), pages
242-247
[6] S. Timoshenko, D.H. Young, W. Weaver Jr., Vibration Problems in Engineering.
Forth Edition. John Wiley & Sons, 1974, pages 481-502
[7] T. Mori, K. Tanaka, Average stress in matrix and average elastic energy of materials
with misfitting inclusions, Acta Metallurgica 21 (1973), pages 571-574
[8] A.J.M. Ferreira, C.M.C. Roque, E. Carrera, M. Cinefra, Analysis of thick isotropic
and cross-ply laminated plates by radial basis functions and a Unified formulation,
Journal of Sound and Vibration 330 (2011), pages 771-787
22
6. Appendices
6.1 ANSYS analysis code for isotropic and FG plates
/BATCH
! /COM,ANSYS RELEASE 15.0
04/02/2014
/input,menust,tmp,''
! /GRA,POWER
! /GST,ON
! /PLO,INFO,3
! /GRO,CURL,ON
! /CPLANE,1
! /REPLOT,RESIZE
WPSTYLE,,,,,,,,0
RESUME
! /COM,ANSYS RELEASE 15.0
04/02/2014
/PREP7
BLC4,0,0,1,1
! /REPLOT,RESIZE
!*
ET,1,SHELL181
!*
!*
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,1,,1e11
MPDATA,PRXY,1,,0.3
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,DENS,1,,7800
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,2,,1e11
MPDATA,PRXY,2,,0.3
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,DENS,2,,7800
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,3,,1e11
MPDATA,PRXY,3,,0.3
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,DENS,3,,7800
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,4,,1e11
MPDATA,PRXY,4,,0.3
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,DENS,4,,7800
UP20131014
23:11:38
UP20131014
23:11:51
23
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,5,,1e11
MPDATA,PRXY,5,,0.3
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,DENS,5,,7800
sect,1,shell,,
secdata, .005,1,0.0,3
secdata, .005,2,0.0,3
secdata, .005,3,0.0,3
secdata, .005,4,0.0,3
secdata, .005,5,0.0,3
secoffset,MID
seccontrol,,,, , , ,
FLST,5,4,4,ORDE,2
FITEM,5,1
FITEM,5,-4
CM,_Y,LINE
LSEL, , , ,P51X
CM,_Y1,LINE
CMSEL,,_Y
!*
LESIZE,_Y1, , ,20, , , , ,1
!*
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
!*
FINISH
/SOL
!*
ANTYPE,2
!*
!*
MODOPT,LANB,12
EQSLV,SPAR
MXPAND,0, , ,0
LUMPM,0
PSTRES,0
!*
MODOPT,LANB,12,1,5000, ,OFF
! LPLOT
24
FLST,2,4,4,ORDE,2
FITEM,2,1
FITEM,2,-4
!*
/GO
DL,P51X, ,UZ,0.0
FLST,2,1,4,ORDE,1
FITEM,2,4
!*
/GO
DL,P51X, ,ROTX,0
FLST,2,1,4,ORDE,1
FITEM,2,2
!*
/GO
DL,P51X, ,ROTX,0
FLST,2,1,4,ORDE,1
FITEM,2,3
!*
/GO
DL,P51X, ,ROTY,0
FLST,2,1,4,ORDE,1
FITEM,2,1
!*
/GO
DL,P51X, ,ROTY,0
! /STATUS,SOLU
SOLVE
FINISH
/POST1
SET,LIST,999
SET,,, ,,, ,1
!*
! /EFACET,1
! PLNSOL, U,SUM, 0,1.0
!*
! /SHRINK,0
! /ESHAPE,1.0
! /EFACET,1
! /RATIO,1,1,1
/CFORMAT,32,0
! /REPLOT
!*
! /VIEW,1,1,1,1
! /ANG,1
! /REP,FAST
! /DIST,1,1.08222638492,1
! /REP,FAST
! /DIST,1,1.08222638492,1
! /REP,FAST
! /DIST,1,0.924021086472,1
! /REP,FAST
! /DIST,1,0.924021086472,1
! /REP,FAST
25
! /DIST,1,0.924021086472,1
! /REP,FAST
! /DIST,1,0.924021086472,1
! /REP,FAST
! /DIST,1,0.924021086472,1
! /REP,FAST
! /DIST,1,0.924021086472,1
! /REP,FAST
! /DIST,1,0.924021086472,1
! /REP,FAST
! /DIST,1,0.924021086472,1
! /REP,FAST
! /DIST,1,0.924021086472,1
! /REP,FAST
! /DIST,1,0.924021086472,1
! /REP,FAST
! /DIST,1,0.924021086472,1
! /REP,FAST
! /DIST,1,0.924021086472,1
! /REP,FAST
! /DIST,1,0.924021086472,1
! /REP,FAST
!*
! /SHRINK,0
! /ESHAPE,0.0
! /EFACET,1
! /RATIO,1,1,1
/CFORMAT,32,0
! /REPLOT
!*
!*
! /SHRINK,0
! /ESHAPE,0.0
! /EFACET,1
! /RATIO,1,1,1
/CFORMAT,32,0
! /REPLOT
!*
! EPLOT
!*
! /SHRINK,0
! /ESHAPE,1.0
! /EFACET,1
! /RATIO,1,1,1
/CFORMAT,32,0
! /REPLOT
!*
!*
! /SHRINK,0
! /ESHAPE,0.0
! /EFACET,1
! /RATIO,1,1,1
/CFORMAT,32,0
! /REPLOT
26
!*
! SAVE, file,db,
! /COM,ANSYS RELEASE 15.0
04/03/2014
! /VIEW,1,,,1
! /ANG,1
! /REP,FAST
! /DIST,1,1.08222638492,1
! /REP,FAST
! /DIST,1,1.08222638492,1
! /REP,FAST
! /DIST,1,1.08222638492,1
! /REP,FAST
! /DIST,1,1.08222638492,1
! /REP,FAST
! /DIST,1,1.08222638492,1
! /REP,FAST
! /DIST,1,1.08222638492,1
! /REP,FAST
! /REPLOT,RESIZE
!*
!*
FINISH
/FILNAME,Ti_Cer_10_layer,0
!*
/PREP7
!*
!*
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,6,,1e11
MPDATA,PRXY,6,,.3
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,DENS,6,,7800
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDE,DENS,6
MPDATA,DENS,6,,7800
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,7,,1e11
MPDATA,PRXY,7,,.3
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,DENS,7,,7800
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDE,DENS,7
MPDATA,DENS,7,,7800
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,8,,1e11
MPDATA,PRXY,8,,.3
UP20131014
27
22:05:01
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,DENS,8,,7800
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,9,,1e11
MPDATA,PRXY,9,,0.3
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,DENS,9,,7800
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,10,,1e11
MPDATA,PRXY,10,,.3
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,DENS,10,,7800
sect,1,shell,,
secdata, 0.0025,1,0,3
secdata, 0.0025,2,0,3
secdata, 0.0025,3,0,3
secdata, 0.0025,4,0,3
secdata, .0025,5,0,3
secdata, 0.0025,6,0.0,3
secdata, 0.0025,7,0.0,3
secdata, 0.0025,8,0.0,3
secdata, 0.0025,9,0.0,3
secdata, 0.0025,10,0.0,3
secoffset,MID
seccontrol,0,0,0, 0, 1, 1, 1
! DLLIS, ALL
! SAVE, Ti_Cer_10_layer,db,
FINISH
/SOL
!*
! /STATUS,SOLU
SOLVE
FINISH
/POST1
SET,LIST,999
SET,,, ,,, ,1
!*
! /EFACET,1
! PLNSOL, U,SUM, 0,1.0
! SAVE, Ti_Cer_10_layer,db,
! SAVE, Ti_Cer_10_layer,db,
! /COM,ANSYS RELEASE 15.0
UP20131014
04/05/2014
! DLLIS, ALL
/POST1
SET,LIST,999
! MPLIST,ALL,,,EVLT
FINISH
/PREP7
28
11:34:57
!*
MP,EX,1,1.14E+11
MP,nuXY,1,0.342
MP,EX,2,1.14E+11
MP,nuXY,2,0.342
MP,EX,3,1.14E+11
MP,nuXY,3,0.342
MP,EX,4,1.14E+11
MP,nuXY,4,0.342
MP,EX,5,1.14E+11
MP,nuXY,5,0.342
MP,EX,6,1.14E+11
MP,nuXY,6,0.342
MP,EX,7,1.14E+11
MP,nuXY,7,0.342
MP,EX,8,1.14E+11
MP,nuXY,8,0.342
MP,EX,9,1.14E+11
MP,nuXY,9,0.342
MP,EX,10,1.14E+11
MP,nuXY,10,0.342
MP,dens,1,4430
MP,dens,2,4430
MP,dens,3,4430
MP,dens,4,4430
MP,dens,5,4430
MP,dens,6,4430
MP,dens,7,4430
MP,dens,8,4430
MP,dens,9,4430
MP,dens,10,4430
FINISH
/SOL
! /STATUS,SOLU
SOLVE
! /REPLOT,RESIZE
FINISH
/POST1
SET,LIST,999
SET,,, ,,, ,1
!*
! /EFACET,1
! PLNSOL, U,SUM, 0,1.0
SET,LIST,999
SET,,, ,,, ,2
!*
! /EFACET,1
! PLNSOL, U,SUM, 0,1.0
SET,LIST,999
SET,,, ,,, ,3
!*
! /EFACET,1
! PLNSOL, U,SUM, 0,1.0
SET,LIST,999
29
SET,,, ,,, ,4
!*
! /EFACET,1
! PLNSOL, U,SUM, 0,1.0
! SAVE, Ti_Cer_10_layer,db,
! SAVE, Ti_Cer_10_layer,db,
! SAVE, Ti_Cer_10_layer,db,
FINISH
/PREP7
!*
MP,EX,1,1.19415E+11
MP,nuXY,1,0.3877
MP,EX,2,1.3145E+11
MP,nuXY,2,0.332312
MP,EX,3,1.44696E+11
MP,nuXY,3,0.325824
MP,EX,4,1.59353E+11
MP,nuXY,4,0.319262
MP,EX,5,1.75664E+11
MP,nuXY,5,0.312575
MP,EX,6,1.93932E+11
MP,nuXY,6,0.305701
MP,EX,7,2.14541E+11
MP,nuXY,7,0.298568
MP,EX,8,2.37976E+11
MP,nuXY,8,0.291084
MP,EX,9,2.6487E+11
MP,nuXY,9,0.283135
MP,EX,10,2.96058E+11
MP,nuXY,10,0.274572
MP,dens,1,4374
MP,dens,2,4262
MP,dens,3,4150
MP,dens,4,4038
MP,dens,5,3926
MP,dens,6,3814
MP,dens,7,3702
MP,dens,8,3590
MP,dens,9,3478
MP,dens,10,3366
FINISH
/SOL
! /STATUS,SOLU
SOLVE
FINISH
/POST1
SET,LIST,999
! SAVE, Ti_Cer_10_layer,db,
! SAVE, Ti_Cer_10_layer,db,
!*
!*
FINISH
/FILNAME,Ti_10_layer,0
!*
30
! SAVE, Ti_10_layer,db,
! MP,EX,1,1.14E+11
! MP,nuXY,1,0.342
! MP,EX,2,1.14E+11
! MP,nuXY,2,0.342
! MP,EX,3,1.14E+11
! MP,nuXY,3,0.342
! MP,EX,4,1.14E+11
! MP,nuXY,4,0.342
! MP,EX,5,1.14E+11
! MP,nuXY,5,0.342
! MP,EX,6,1.14E+11
! MP,nuXY,6,0.342
! MP,EX,7,1.14E+11
! MP,nuXY,7,0.342
! MP,EX,8,1.14E+11
! MP,nuXY,8,0.342
! MP,EX,9,1.14E+11
! MP,nuXY,9,0.342
! MP,EX,10,1.14E+11
! MP,nuXY,10,0.342
/PREP7
!*
MP,EX,1,1.14E+11
MP,nuXY,1,0.342
MP,EX,2,1.14E+11
MP,nuXY,2,0.342
MP,EX,3,1.14E+11
MP,nuXY,3,0.342
MP,EX,4,1.14E+11
MP,nuXY,4,0.342
MP,EX,5,1.14E+11
MP,nuXY,5,0.342
MP,EX,6,1.14E+11
MP,nuXY,6,0.342
MP,EX,7,1.14E+11
MP,nuXY,7,0.342
MP,EX,8,1.14E+11
MP,nuXY,8,0.342
MP,EX,9,1.14E+11
MP,nuXY,9,0.342
MP,EX,10,1.14E+11
MP,nuXY,10,0.342
MP,dens,1,4430
MP,dens,2,4430
MP,dens,3,4430
MP,dens,4,4430
MP,dens,5,4430
MP,dens,6,4430
MP,dens,7,4430
MP,dens,8,4430
MP,dens,9,4430
MP,dens,10,4430
! SAVE, Ti_10_layer,db,
31
FINISH
/SOL
! /STATUS,SOLU
SOLVE
FINISH
/POST1
SET,LIST,999
! SAVE, Ti_10_layer,db,
FINISH
! /EXIT,MODEL
! /COM,ANSYS RELEASE 15.0
04/20/2014
! DLIST, ALL
/PREP7
! /REPLOT,RESIZE
FINISH
/SOL
! /STATUS,SOLU
SOLVE
FINISH
/POST1
SET,LIST
SET,LIST,999
SET,,, ,,, ,1
!*
! /EFACET,1
! PLNSOL, U,SUM, 0,1.0
SET,LIST,999
SET,,, ,,, ,2
!*
! /EFACET,1
! PLNSOL, U,SUM, 0,1.0
SET,LIST,999
SET,,, ,,, ,3
!*
! /EFACET,1
! PLNSOL, U,SUM, 0,1.0
SET,LIST,999
SET,,, ,,, ,4
!*
! /EFACET,1
! PLNSOL, U,SUM, 0,1.0
FINISH
UP20131014
32
13:50:33
6.2 Maple inputs - Timoshenko
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>
>
>
>
>
6.3 Maple inputs – Mori-Tanaka
>
>
>
>
>
>
>
33
>
>
>
>
>
>
34
>
>
>
35
>
>
36
>
>
37
6.4 Maple inputs - Efraim
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>
>
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>
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38
>
>
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>
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39