CHAPTER 4 ANALYSIS OF RECTANGULAR FGM...
Transcript of CHAPTER 4 ANALYSIS OF RECTANGULAR FGM...
42
CHAPTER 4
ANALYSIS OF RECTANGULAR FGM PLATES
4.1 INTRODUCTION
In this chapter, the newly developed EFG method is applied on the
rectangular FGM plates under various pressure loading, thermal loading and
thermo-mechanical loading. In addition a modal analysis was performed on the
rectangular FGM plates using EFG method for checking its validity under
static as well as dynamic loading conditions.
In these analyses, a plate of thickness h, is considered to be made up
of n slices, the thickness of each slice being h/n. It is shown in Fig. 4.1. The
“slices” are then treated as those made of individual homogeneous material and
are layered together and modeled similar to a composite structure. Material
properties for each “slice” are calculated according to law of variation.
4.2 RECTANGULAR PLATE UNDER THERMO-MECHANICAL
LOADING
The goal of analyzing flat plates under mechanical loading
(application of pressures, Uniformly distributed loading), thermal loading
(application of temperatures, heat flux) and thermo-mechanical loading is to
characterise the effect ‘n’ on the structural response to the loadings. In this
work Element Free Galerkein method is applied to the rectangular plate made
of FGM. A flat Aluminum-Zirconia plate with sides of 0.2 m and thickness of
43
0.01 m is exposed to various surface temperatures and various pressure
loadings. The temperature of the top surface is kept as 100°C, 200°C, 300°C,
400°C, 500°C and 600°C for various cases and the bottom temperature is
exposed to a constant temperature of 20°C. A pressure loading in terms of a
load parameter varying from 0 to 30 was also applied on the plate. The index
‘n’ showing the volume fraction of the constituents considered in this analysis
are 0 (ceramic), 0.2, 0.5, 1.0, 2.0, and ¥ (metal).
Materials properties for the bottom and top surface are listed in
Table 3.1.
4.2.1 Analysis of Rectangular Plates
A two-dimensional shell analysis with a 8´8 mesh and 9 slices is
used to solve problem. A steady state heat transfer analysis is first performed to
obtain nodal temperatures. Figure 4.2 shows the temperature profile across the
thickness for various material index (n) values.
Figure 4.1 Sliced model of FGM plate
Slice 1
Slice 2
Slice n-1
Slice n
h
44
Temperature in °CFigure 4.2 Variation of temperature across the thickness for various
‘n’ values
A summary of boundary conditions applied for analysing the
rectangular plates is listed Table 4.1.
Table 4.1 Summary of boundary conditions(BC) for rectangular plates
BC NameBC
AbbreviationWidth-wise BCs Length-wise BCs
Simple Support 1 SS1 v = w = 0; dw/dy = 0 v = w = 0; dw/dx = 0
Simple Support 2 SS2 u = w = 0; dw/dy = 0 v = w = 0; dw/dx = 0
Simple Support 3 SS3 u = w = 0; dw/dy = 0 u = v = w = 0; dw/dx = 0
Simple Support 4 SS4 u = v = w = 0 ;dw/dy = 0 u = v = w = 0; dw/dx = 0
Simple Support-
FreeSS-Free u = v = w = 0;dw/dy = 0
All displacements and
rotations are free
Clamped-Free Clamped-Freev = v = w = 0
All rotations = 0
All displacements and
rotations are free
u, v and w are the degrees of freedom for deflections in x, y and z directions respectively.
45
The FGM rectangular plate is analysed for the following loading
conditions.
a. Plate subjected to mechanical loading alone.
b. Plate subjected to only thermal loading alone.
c. Plate subjected to thermo-mechanical loading.
4.2.2 Linear and Non-linear Analysis of Rectangular Plates
In the linear analysis of rectangular plates using EFG method the
approximation in the Moving Least Square (MLS) shape function considered
will be a linear equation derived as mentioned in 3.4. Where as in non-linear
analysis the MLS shape function is a non-linear equation.
In EFG method, the non linear equations are accommodated by the
Galerkein technique by more number of scattered nodes generated with respect
to the order of the equation. Regular nodes are considered for the linear
approximations. The nodes on the boundary are considered for both the linear
and non linear analysis.
The moment caused by the loading conditions does not affect a linear
solution because the stiffness matrix is not updated after every iteration and
checked for continuity with internal forces and displacements. However, this
moment is found to have a significant impact on the non-linear solution
because the stiffness matrix is updated after every iteration.
46
4.3 RECTANGULAR PLATES UNDER DISTRIBUTED PRESSURE
LOADING
The analysis of flat plates under distributed pressure loading using
Element Free Galerkein method is carried out. Analysis is performed for both
linear and non-linear form of governing equations and for various boundary
conditions stated in Table 4.1. The results are shown in Figure 4.3 to Figure
4.10.The plots show the maximum deflection (w) of the plot for various
pressure loadings. The displacement is normalized as w/h and the pressure load
is normalized as the load parameter P= (q0a4) / (Ebh
4). The load parameter for a
simple beam structure is explained in Annexure II. For each case, the results
are plotted for linear and non-linear assumptions.
It is noted that SS1 and SS-Free provide higher displacements while
SS2, SS3, and SS4 provide the lower displacements. In case of metal,
displacements with boundary conditions SS2, SS3, and SS4 are approximately
0.95. The displacements in SS1 and SS-Free boundary conditions are 0.75, i.e.,
27% lower than that of other SS2. From the results it is also found that adding
more restraints at the boundary tends to decrease the deflection in pressure
loaded plates.
Ceramic provides the lower deflections because it is the stiffer
material while Aluminum provides the higher deflections because it is the
softer material. FGM plate deflections are lower with lower ‘n’ values for a
given load with ‘ n’ = 0.2 providing the minimum deflection. Lowering ‘n’
tends to lower the maximum deflection. Of particular importance is the drastic
reduction of deflections by functionally graded materials. For all ‘n’ values,
FGMs deflections are reduced by approximately 50% over a metal plate. It is
noted from Figures 4.3 to 4.10 that a non-linear analysis provides lower
47
displacements than a linear analysis. This is due to the geometric stiffening
effect provided by a non-linear analysis.
The non-linear analysis is carried out using Element Free Galerkein
algorithm provides higher deflections than a linear analysis. Moments are
caused by the expansion of the material, which is product of the coefficient of
thermal expansion and the temperature change. A moment is created if the top
side of the plate is expanding more than the bottom side of the plate, or vice-a-
versa.
This effect of thermal moments also explains why the effect of
lowering “n” tends to lower displacements. Lowering ‘n’ creates plates that
have much more ceramic that extends deeper into the thickness. This ceramic
portion has the lowest thermal expansion coefficient; however it is exposed to
the highest temperatures. The metal found in the bottom of the plate has the
highest coefficient of thermal expansion but is exposed to the lowest
temperature. It is apparent that a lower ‘n’ value provides a combination of
a.DT that is nearly constant through-the-thickness.
It is noted that a value of n=¥ (metal provides the highest mid-plane
stress, while n=0.5 provides the lowest mid-plane stress (non-linear analysis)
and is 54% lower than a metal plate. Also, the effect of raising the ‘n’ value
tends to raise the mid-plane stress values. Also, it is noted that the linear
analysis provides lower stress values than a non-linear analysis. This is due to
the non-linear stress stiffening being negated by the non-linear thermal
moments as discussed in the deflection analysis.
48
Load Parameter, P
ò ò òW W W
êëé WF=W=W× dbdbdb kTkT rxr xxr
Non d
imen
sio
nal
dis
pla
cem
ent, w
/h
Figure 4.3 Pressure induced deflection for SS1 (Linear solution)
Figure 4.4 Pressure induced deflection for SS1 (Non-Linear solution)
49
Load Parameter, P
ò ò òW W W
êëé WF=W=W× dbdbdb kTkT rxr xxr
Non
dim
ensi
onal
dis
pla
cem
ent,
w/h
Figure 4.5 Pressure induced deflection for SS2 (Linear solution)
Figure 4.6 Pressure induced deflection for SS2 (Non-Linear solution)
50
Load Parameter, P
ò ò òW W Wêë WF=W=W× dbdbdb kk rxr xxr
Non d
imensi
onal
dis
pla
cem
ent, w
/h
Figure 4.7 Pressure induced deflection for SS3 (Linear solution)
Figure 4.8 Pressure induced deflection for SS3 (Non-Linear solution)
51
Load Parameter, P
ò ò òW W W
êëé WF=W=W× dbdbdb kTkT rxr xxr
Non d
imen
sional
dis
pla
cem
ent,
w/h
Figure 4.9 Pressure induced deflection for SS4 (Linear solution)
Figure 4.10 Pressure induced deflection for SS4 (Non-Linear solution)
52
4.3.1 Discussions
The effect of load parameter on deflection for SS1 boundary
condition for various material index ‘n’ is shown in Figure 4.3 under linear
approximation. For a given material index ‘n’, increase in load increases the
deflection linearly ( As the load is applied o the top surface, the deflections
shown are in the negative direction of z axis). For a given load parameter when
‘n’ increases from 0 to ¥, the deflection increases due to addition of metal
constituent which leads to decrease in stiffness. However the variations are
marginal. Figure 4.4 shows the results obtained from the non-linear
approximation for the same SS1 boundary conditions. The non-linear solution
captures the non-linearities better which can be seen from the curved lines. For
the boundary conditions SS2, SS3 and SS4, the linear and non linear solutions
are obtained and are plotted in Figures 4.5, 4.6, 4.7, 4.8, 4.9 and 4.10
respectively. Almost similar behaviour is observed except few changes in the
magnitude of deflection.
It is also noted that among the FGMs, increasing ‘n’ tends to raise
the stress (plots not shown) n=0.2 provided the overall lowest stress and 4%
lower than a metal plate. The highest mid-plane stress is found at n=1.0 and is
27% higher than metal’s mid-plane stress. These mid-plane stress values are a
good representation of the bend-stretch coupling found in the FGMs. Bend-
stretch coupling is found in materials that are not symmetric about their mid-
plane. That is, a material that has non-symmetric E values about it’s mid-plane
therefore as the material is stretched it undergoes non-uniform strain. This
unsymmetrical strain creates a bending moment inside the material. This
explains the metal, a homogenous material, has a very low mid-plane stress as
it exhibits no bend-stretch coupling; the material is symmetric about its mid-
plane. In addition, n=1.0 is the most unsymmetrical layup of the FGMs,
because it has a linear material distribution from ceramic on the top surface to
53
metal on the bottom surface. Consequently, it has the highest bend-stretch
coupling as evident by the high mid-plane stresses.
4.4 RECTANGULAR PLATES UNDER THERMAL LOADING
An FGM plate subjected to a temperature rise is studied. The
temperature of the top surface is kept as 100°C, 200°C, 300°C, 400°C, 500°Cand 600°C. The bottom surface was kept at the reference temperature of 20°C.
The deflections with respect to variation of constituents of FGM for
the applied temperature are plotted in Figures 4.2 to 4.11.
Temperature in °C
ò ò òW W W
êëé WF=W=W× dbdbdb kTkT rxr xxr
Figure 4.11 Thermally induced deflection for SS1 (Linear solution)
54
Temperature in °C
ò ò òW W W
úûùê
ëé WF=W=W× dbdbdb kTkT rxr xxr
Figure 4.12 Thermally induced deflection for SS1 (Non-Linear solution)
Temperature in °C
ò ò òW W W
úûùê
ëé WF=W=W× dbdbdb kTkT rxr xxr
Figure 4.13 Thermally induced deflection for SS2 (Linear solution)
55
Figure 4.14 Thermally induced deflection for SS2 (Non-Linear solution)
Temperature in °C
ò ò òW W W
úûùê
ëé WF=W=W× dbdbdb kTkT rxr xxr
Figure 4.15 Thermally induced deflection for SS3 (Linear)
56
Figure 4.16 Thermally induced deflection for SS3 (Non-Linear solution)
Temperature in °C
ò ò òW W W
êëé WF=W=W× dbdbdb kTkT rxr xxr
Figure 4.17 Thermally induced deflection for SS4 (Linear solution)
57
Figure 4.18 Thermally induced deflection for SS4 (Non-Linear solution)
Temperature in °C
ò ò òW W W
úûùê
ëé WF=W=W× dbdbdb kTkT rxr xxr
Figure 4.19 Thermally induced deflection for SS Free
(Linear solution)
58
Figure 4.20 Thermally induced deflection for SS Free
(Non-Linear solution)
4.4.1 Discussions
The effect of temperature on deflection for SS1 boundary
condition for various material index ‘n’ is shown in Figure 4.11 under linear
approximation. For a given material index ‘n’, increase in temperature
increases the deflection linearly. For a given temperature when ‘n’ increases
from 0 to ¥, the deflection increases due to addition of metal constituent which
leads to decrease in stiffness. However the variations are marginal. Figure 4.12
shows the results obtained from the non-linear approximation for the same SS1
boundary conditions. The non-linear solution captures the non-linearities better
which can be seen from the curved lines. For 600°C, for n=0, the magnitude of
deflection are 5% higher incase of non-linear solutions when compared to the
linear solutions. When n=0.5, the deflections are 10% higher and when n=¥,
the deflections are 40% because of reduction in stiffness of the plate. For the
boundary conditions SS2, SS3, SS4 and SS Free, the linear and non linear
59
solutions are obtained and are plotted in Figures 4.13, 4.14, 4.15, 4.16, 4.17,
4.18, 4.19 and 4.19 respectively. Almost similar behaviour is observed except
few changes in the magnitude of deflection.
In general, lowering “n” tends to lower the deflection in flat plates
under thermal loads with n=0.2 providing the lowest deflection. Also, non-
linear analysis must be used in a thermal FGM shell analysis, because thermal
moments play an important part in the stress stiffening of the structure. These
thermal moments are caused by the drastic difference in through-the-thickness
temperatures and by the material property variation found in an FGM. Also,
SS-Free provides displacement that is nearly identical to SS1 (which has a
“normal” boundary condition).
4.5 RECTANGULAR PLATES UNDER THERMO-MECHANICAL
LOADING
A flat Aluminum-Zirconia plate with sides a = 0.2 m and thickness
h = 0.01 m is exposed to various surface temperatures. The temperature of the
top surface is kept as 100°C, 200°C, 300°C, 400°C, 500°C and 600°C for
various cases and the bottom temperature is exposed to a constant temperature
of 20°C. Load parameter [(q0a4)/(Ebh
4)] ranges from 1 to 10 is applied. The
index ‘n’ showing the volume fraction of the constituents considered in this
analysis are 0 (ceramic), 0.2, 0.5, 1.0, 2.0, and ¥ (metal).
60
Figure 4.21 Deflection of FGM plate using EFG method for n= 0
Figure 4.22 Deflection of FGM plate using EFG method for n= 0.2
61
Figure 4.23 Deflection of FGM plate using EFG method for n= ¥
The effect of load parameter on deflection of FGM plate for simply
supported boundary condition and for various material index ‘n’ is studied
using EFG method. Figure 4.22 shows for a given material index ‘n’=0.2,
increase in load increases the deflection linearly. Similarly the increase in
temperature leads to increase in deflection. The same trend is appeared for all
the temperature inputs. Figure 4.21 shows the variation of deflection of a
ceramic plate (n=0) and figure 4.23 shows the deflection of a metal plate
(n=¥).
62
4.5.1 Comparison of prediction of EFG Method with Exact Solutions
The exact solutions are for the thermo-mechanical analysis of FGM
rectangular plates were already derived by Batra (2004) using 10X10 quadratic
rectangular nodes and using first order plate theory. These results are used for
comparing the results obtained from EFG method and FEM for validation.
The results parameters temperature, displacement and stress with
respect to thickness of the plate using EFG method are compared with exact
solutions and plotted in Figure 4.24, 4.25 and 4.26 respectively.
(a) Temperature Vs Thickness (b) Error Vs Thickness
Figure 4.24 Comparison of variations of temperature and error in
prediction by the EFG method with the exact solution
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
-10 -8 -6 -3.5 -2 0 2.5 4 6 8 10
x2 in mm
% E
rro
r
63
(a) Displacement Vs Thickness
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
4.00
5.00
-10 -6 -2 2.5 6 10
u2(L/2,x2) in mm
% E
rro
r
(b) Error in calculated Displacement Vs Thickness
Figure 4.25 Comparison of variations of displacement and error in
prediction by the EFG method with the exact solution
64
(a) Stress Vs Thickness
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
-10 0 10
Stress at (L/2,x2) in MPa
% E
rro
r
(b) Error in calculated Stress Vs Thickness
Figure 4.26 Comparison of variations of normal stress and error in
prediction the EFG method with the exact solution
65
4.5.2 Discussions
From Figures 4.24 and 4.25, it can be seen that the evolutions of the
temperature and the displacements obtained from EFG method and exact
solution are almost similar. However, the variations of the stresses were
different. This can be seen clearer in Figure 4.26, which illustrates the stress
variation through the thickness of the plate. In the graph there were still some
positions where the stress changed abruptly while it should be smooth
according to the properties of FGM. Therefore, in the cases when the stress is
not be considered but the temperature and the displacement field are necessary,
the layered model may be an acceptable choice since it does not need any
extended user defined subroutine.
4.6 MODAL ANALYSIS OF RECTANGULAR FGM PLATES
USING EFG METHOD
The EFG method is also applied for the modal analysis of
rectangular FGM plates for the study of vibration characteristics and to check
the suitability for dynamic problems. For the modal analysis, a sinusoidal
distribution of the transverse load is considered and it is defined as
0( , ) cos sin2
aq x y q x y
a b
p pæ ö æ ö= - -ç ÷ ç ÷è ø è ø (4.9)
where a and b are the length of the two sides in x- and y-axis, h is the total
thickness of the plate. The deflection is presented in the following non-
dimensional form
66
4
0
3
0aq
hEww
c= (For load in sinusoidal distribution)
where w0 is the maximum deflection in the particular mode.
)1(12
124
0
3
0 g-=aq
hEww
c (For uniform load)
(4.10)
For the analysis, the plate is discretised by (12´12) uniformly
distributed nodes and (6´6) quadrilateral background cells are used for the
purpose of integration in this different power ‘n’.
The cantilever plate is initially subjected to uniformly distributed load of
100 N/m2 and the centerline deflection is computed. The FGM plate is studied
again under a thermal gradient through its thickness direction. The temperature
of the top ceramic-rich surface is fixed at 600°C and that of the bottom metal-
rich surface is kept constant at 20°C.
4.6.1 Response of rectangular plates using EFG Method
The first modes of the FGM plate are calculated using EFG method
as a function of the volume fraction power law exponent ‘n’. The preceding
plate model is still considered herein. Both the cantilever and simply supported
boundary conditions are considered. The natural frequencies of the plate
corresponding to two different boundary conditions are given in Tables 4.2 to
4.7 respectively. It can be seen that the natural frequencies decrease gradually
as the exponent n gets larger, which agree well with the solutions obtained
using FEM. The prediction of EFG are better than FEM.
67
Table 4.2 Natural frequencies of a square Al/ZrO2 functionally graded
thick plate - +
c c(V = 0,V =1)
Schemen=1.0 (constant) a/h=5(constant)
a/h=5 a/h=10 a/h=20 n=2.0 n=3.0 n=5.0
MTm (Batra 2004) 5.4806 5.9609 6.1076 5.4923 5.5285 5.5632
EFG Method 5.7001 6.2133 6.3989 5.6681 5.6988 5.7356
% error 4.01 4.23 4.77 3.20 3.08 3.10
Table 4.3 Non-dimensionalised center deflections of the FGM plate
(a/h=10, n=2.0)
FEM EFG METHOD
Elements )10(2-´
h
w%
error
Regular
nodes )10(2-´
h
w%
error
Irregular
nodes )10(2-´
h
w%
error
5´5 2.4407 47.15 5´5 4.2661 7.63 25 4.2646 7.66
7´7 3.3059 28.42 7´7 4.3824 5.11 49 4.4150 4.40
9´9 3.37753 18.25 9´9 4.4960 2.65 81 4.5273 1.97
11´11 4.0410 12.50 11´11 4.5387 1.72 121 4.5668 1.12
13´13 4.2018 9.02 13´13 4.5636 1.18 169 4.5866 0.69
15´15 4.3050 6.78 15´15 4.5785 0.86 225 4.5981 0.44
17´17 4.3748 5.27 17´17 4.5882 0.65 289 4.6056 0.27
21´21 4.4599 3.43 21´21 4.5997 0.40 441 4.6146 0.08
68
Table 4.4 Natural frequencies of the FGM plate (a/h=10, n=2.0)
(w = frequency in a particular mode / max. frequency)
FEM EFG Method
Elements w %
error
Regular
nodesw %
error
Irregular
nodesw % error
5´5 6.9138 44.78 5´5 5.0200 5.12 25 5.0599 5.95
7´7 5.7747 20.92 7´7 4.8627 1.82 49 4.8599 1.77
9´9 5.3503 12.03 9´9 4.8237 1.01 81 4.8166 0.86
11´11 5.1476 7.79 11´11 4.8072 0.66 121 4.8011 0.53
13´13 5.0355 5.44 13´13 4.7981 0.47 169 4.7928 0.36
15´15 4.9672 4.01 15´15 4.7924 0.35 225 4.7838 0.25
17´17 4.9226 3.08 17´17 4.7886 0.27 289 4.7838 0.17
* The percentage errors are calculated with respect to the result w =4.7756 using 10´10
quadratic rectangular 8-node finite elements of first-order theory (Exact solutions).
Table 4.5 Variation of the natural frequencies (Hz) with the volume
fraction exponent n for a cantilever FGM plate
Mode
No.
n=0 n=0.2 n=1 n=5
FEM EFG FEM EFG FEM EFG FEM EFG
1 36.36 36.42 34.623 34.682 31.447 31.510 29.640 29.699
2 89.15 90.02 84.881 85.733 77.094 77.943 72.654 73.398
3 225.88 228.81 214.08 217.94 195.35 198.13 184.06 186.65
4 286.13 288.28 272.43 274.50 247.44 249.36 233.17 234.88
5 327.42 333.59 311.76 317.72 283.16 288.78 266.78 271.80
6 573.09 589.61 545.72 561.59 495.66 510.34 466.85 479.89
69
Table 4.6 Variation of the natural frequencies (Hz) with the volume
fraction exponent n for a simply supported FGM plate
Mode
No.
n=0 n=0.2 n=1 n=5
FEM EFG FEM EFG FEM EFG FEM EFG
1 206.50 207.13 196.61 197.18 178.57 179.00 168.31 168.59
2 522.10 528.18 497.13 502.91 451.53 456.64 425.43 429.87
3 522.10 528.18 493.13 502.91 154.53 456.64 425.43 429.87
4 835.81 844.98 795.92 804.56 722.93 730.42 680.75 687.10
5 1070.9 1087.0 1010.8 1035.1 926.80 940.06 872.13 884.51
6 1070.9 1087.0 1010.8 1035.1 926.80 940.06 872.13 884.51
4.7 CHAPTER SUMMARY
4.7.1 Summary of Thermo-Mechanical Analysis of Rectangular FGM
Plates
The results of mechanical loading have revealed that SS1 and SS-
Free boundary conditions provide the higher displacements, whereas SS2, SS3,
and SS4 provide the lower displacements. The displacements for metal plate
with boundary conditions SS2, SS3, and SS4 are found to be equal and have a
dimensionless deflection of 0.95 and the same for SS1 and SS-Free boundary
conditions are 0.75 or 27% lower. For flat plates with thermal loading alone,
this trend is opposite where SS1 has 35% lower deflections than SS2. Adding
more restraints at the boundary is found to decrease the deflection in pressure
loaded plates. Ceramic provides the lower deflection because of higher stiffer,
while aluminum provides the higher deflection because of its soft nature. FGM
plate deflections fall in the middle with index ‘n’=0.2 providing the minimum
deflection. Lowering the index ‘n’ tends to lower the center deflection. For
70
values of index ‘n’ varying from 0.1 to 15, the deflections of FGM plates are
reduced approximately 50% over a pure metal plate. The trend of lowering
index ‘n’ reduces the deflection and follows the same trend in thermally loaded
plates. In addition, n=0.2 provides the lowest deflection on both thermally
loaded plates and pressure loaded plates.
The results thermo-mechanical analyses shows that EFG method
yields very less percentage of error i.e., -2.5 to +5 for deflection and -3.2 to
+2.5 for maximum stress compared to the exact solutions derived by
Batra(2004).
4.7.2 Summary of Modal Analysis of Rectangular FGM Plates
From the results of modal analysis, the FEM results an error of
47.15% to 3.43% eror for increase in number of elements from 25 to 441 for
non-dimensionalised central deflection. The EFG method with structured nodes
provides an error of 7.63% to 0.4% and the EFG method with irregular nodes
provides an error of 7.66% to 0.08% for the same number of elements for
deflection.
For the frequency calculated from FEM and EFG produces 44.78%
to 3.08% error for increase in number of elements from 25 to 441 for non-
dimensionalised natural frequency. The EFG method with structured nodes
provides an error of 5.12% to 0.27% and the EFG method with irregular nodes
provides an error of 5.95% to 0.17% for the same number of elements.
Similarly for different values of material index ‘n’ the mode shape
frequency is almost same when calculated using Fem and EFG method. This
shows the validity of the newly developed EFG method for dynamic problems.