chapter 4 analysis of rectangular fgm...

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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


Non

dim

ensi

onal

dis

pla

cem

ent,

w/h



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



51

Load Parameter, P

ò ò òW W W


Non d

imen

sional

dis

pla

cem

ent,

w/h



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

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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


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



55


Temperature in °C

ò ò òW W W

úûùê


Figure 4.15 Thermally induced deflection for SS3 (Linear)

56


Temperature in °C

ò ò òW W W



57


Temperature in °C

ò ò òW W W

úûùê


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=¥).

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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

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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.

chapter 4 analysis of rectangular fgm...

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