Bending response of FGM plate under sinusoidal Line Load

IPASJ International Journal of Mechanical Engineering (IIJME) Web Site: http://www.ipasj.org/IIJME/IIJME.htm

A Publisher for Research Motivation........ Email: [email protected] Volume 4, Issue 3, March 2016 ISSN 2321-6441

Volume 4, Issue 3, March 2016 Page 35

ABSTRACT Functionally graded materials plate made of alumina and aluminum is considered for the analysis. The governing differ equation of the plate is obtained using energy principle. Multiquadric radial basic function is applied for discreatization of the equation. The MATLAB code is developed to find out the solution. Effect of gradation index, span to thickness ratio on deflection moments and stresses are carried out under sinusoidal line load .The present results are new and may be used for validation in future. Keywords: FGM, plates, Meshless method, MQ RBF, bending, line load.

1. INTRODUCTION The increasing importance of functionally graded material (FGM) composite structures in many industries and engineering applications has resulted in the demand for more information on their behaviour. Studies involving the flexure response of FGMs structures have received the central attention for many researches in last few years. FGMs the non homogenous materials, which are commonly made of metal and ceramic, possess two main properties strength and thermal resistance. These unique characteristics make FGM more emerging composite materials in the field of engineering applications. Over the last few decades many researchers lighted on the flexural response of functionally graded materials (FGM) plates by applying different plate theories and many solving technique. Praveen et al. [1] investigated the static and dynamic responses of the common FG ceramic-metal plate considering the transverse shear deformation and rotary inertia in the Von-Karman. The finite element method was used and effect of imposed temperature field on the response of the FG plate was mentioned in detail. Ferreira [2] used meshless method with first order shear deformation theory (FSDT) for a global collocation method for Natural frequencies of FG plates. Finite element formulation for the dynamic thermoelastic responses of functionally graded cylinders and plates was introduced by Reddy [3] applying the first-order shear deformation plate theory. Tran et al. [4] introduced a formulation based on the isogeometric approach and using the HDST to examine the behavior of FGM plates. Static, dynamic and buckling behaviour of rectangular and circular plates considering different boundary conditions was examined by him Reddy[5] carried a theoretical formulation based on Navier's solutions of rectangular plates using third-order shear deformation theory (TSDT) assuming the dual-phase materials isotropic and. The RBF methods depend on the geometrical distance between two points with user-defined shape parameter. The value of this parameter not only defines the RBFs but also may make the resulting algebraic problem with good quality solutions. The numerical solution of partial differential equations (PDEs) is traditionally found by finite element methods, finite volume methods or finite difference methods. The radial based functions (RBFs) were first used by Hardy [6] for the interpolation of geographical scattered data and then Kansa [7] applied for the solution of solutions to parabolic, hyperbolic and elliptic partial differential equations. (PDEs). Fasshauer, G. E [8] carried partial differential equations by collocation (RBFs) and multiresolution methods. Ferreira [9] carried the analysis of laminated composite plates using multiquadric radial basis function. In the present paper, the flexural response of functionally graded plates using multiquadric radial basis function is presented. Functionally graded plates with simply supported boundary conditions and line load subjected to sinusoidal transverse pressure are analyzed.

2. MATHEMATICAL FORMULATION A rectangular shape plate of edge length a, b along x, y axes respectively and thickness h is the thickness along z axis whose mid plane is coinciding with x-y plane of the coordinate system is considered. The diagram of rectangular shaped functionally graded material (FGM) plate in rectangular coordinate system is shown in Figure 1.

Bending response of FGM plate under sinusoidal Line Load.

Rahul Kumar1, Dhaneshwar Mahto1, Jeeoot Singh1*

1Department of Mechanical Engineering, B.I.T., Mesra, Ranchi

*Corresponding Author




Fig 1. Geometry of rectangular FGM plate in rectangular coordinate system The homogenization technique considered in this work is the law of mixtures, which provides the following elastic properties at each material layer. The top surface of the plate is ceramic rich and the bottom surface is metal rich.

2( )2

n

cz hV z

h

(1)

Where ‘n’ is exponent governing the material properties along the thickness direction known as volume fraction exponent or grading index, The volume fraction of the metal phase is obtained by

( ) 1 ( )m cV z V z (2) The material property gradation through the thickness of the plate is assumed to have the following form The displacement field at any point in the plate made up of uniform thickness is expressed as:

32

32

( , ) z 5( , ) ( , )4 3

( , ) z 5( , ) ( , )4 3

zx x

xz

y y y

zz

u x yu x y z z x yx hU

u x yU u x y z z x y

y hU

u x, y

(3)

The constitutive stress-strain relations for any FGM plate are expressed as:

xx xx11 12yy yy12 22

xy xy66

44yz yz

55zx zx

σ εQ Q 0 0 0σ εQ Q 0 0 0σ γ0 0 Q 0 0

0 0 0 Q 0σ γ0 0 0 0 Qσ γ

Where, the parameters ijQ are the stiffness coefficients and are expressed in terms of elastics constants as:

2( )2

n

c m mz hE z E E E

h

(4)

Here E denote the modulus of elasticity of FGM structure, while these parameters come with subscript m or c represent the material properties for pure metal and pure ceramic plate respectively., h is the thickness of the plate, Em and Ec are the corresponding Young’s modulus of elasticity of metal and ceramic and z is the thickness coordinate.




11 22 12 44 55 662 2, ,

1 1E EQ Q Q Q Q Q G

The governing differential equations of plate are obtained using energy equation, in mathematical form it is expressed

as:

2

1

t

t

U V dt 0 (5)

Where,, U = Strain energy

V = workdone due to transverse load

The strain energy of the plate due to internal stress resultants is expressed as:

Volume

12

xx xx yy yy xy xy yz yz xz xzU dxdydz (6)

z zArea

V u q dxdy (7)

The governing differential equations of plate are obtained using Hamilton’s principle and expressed as :

2 22yy xyxx

z2 2

0

0

M MM2 q 0

x yx y

0

0

xyxx

xy yy

ffxy fxx

x

f fxy yy f

y

NNx y

N Nx y

MMQ

x yM M

Qx y

(8)

The force and moment resultants in the plate and plate stiffness coefficients are expressed as:

h/22

ij ij i j i j i j2h/2

1 5N , M , ( , z , z. ) dz4 3

fijM z

h

(9)




3

h/2 2f fx y xz yz

h/2

z 5 z4 3hQ , Q σ , σ dz

z

(10)

2/22 2 2 2

2 2 2/2

1 5 1 5 1 5, , , , , ( ) 1, , ,z. , z. ,z.4 3 4 3 4 3

h

ij ij ij ij ij ij

h

A B D E F H Q z z z z z z z dzh h h

(11)

i, j = 1, 2, 6

2

3/2 2

/2

54 3( )

h

ij

h

z zhA Q z dz

z

(12)

i, j = 4, 5

2(, )2

nc m mij ij ijwh z hQ z Q Qere Q

h

The boundary conditions for an arbitrary edge with simply supported conditions are as follows: 0, : 0; 0; 0; 0; 00, : 0; 0; 0; 0; 0

y y z xx xx

x x z yy yy

x a u u M Ny b u u M N

3. SOLUTION METHODOLOGY

The governing differential equations (8) are expressed in terms of displacement functions. Radial basis function based formulation works on the principle of interpolation of scattered data over entire domain. A 2D rectangular domain having NB boundary nodes and ND interior nodes is shown in Figure-2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Interior Domain Nodes (NI) Boundary Nodes(NB)

Fig 2. An arbitrary two dimensional domains

The variable , , , ,x y z x yu u u can be interpolated in form of radial distance between nodes. The solution of the linear

governing differential equations (8) is assumed in terms of multiquadric radial basis function for nodes 1:N, as;




1

, , , , ( , , , , ) , ,

y yx xz

Nuu u

x y z x y j j j j j jj

u u u g X X m c

Where, N is total numbers of nodes which is equal to summation of boundary nodes NB and domain interior nodes

ND. , , jg X X m c is multiquadric radial basis function expressed as 2 2 m

g r c

, ( , , , , )y yx xzuu uj j j j j

are unknown coefficients. jX X is the radial distance between two nodes.

Where, 2 2 j j jr X X x x y y and m ,c are shape parameter. The value of 'm' and 'c' taken here is 0.5

and 1.3/(N)0.25.

4. COMPUTATION AND DISCUSSION OF RESULTS: The study here has been focused on the flexural response of simply supported square functionally graded plates under line transverse loads. A RBF based meshless code in MATLAB 2013 is developed. Several examples have been analyzed and the computed results are compared. Based on convergence study, a 15×15 node is used throughout the study. The material properties of FGMs have been taken as follows: Ceramic 151 , 0.3 c cE GPa Aluminum (Al) 70 , 0.3 m mE GPa

In order to show the accuracy and efficiency of the present solution methodology, detailed convergence studies for simply supported FGM plate (a/h=20) is carried out. The convergences of the deflection are shown in Fig. 3. It can be seen that convergence achieved is within 1 % at 15×15 nodes.

5x5 7x7 9x9 11x11 13x13 15x150.00

0.01

0.02

0.03

0.04

0.05

w

Number of Nodes

Present Jeeoot Singh[10]

Fig. 3 Convergence study for deflection w of a simply supported FGM plate (a/h = 20, ‘n’=2) Table1 Effect of span to thickness ratio on deflection, stresses and Moments of a simply supported FGM Plate (n=5)

a/h

5 10 20 30 40 50 100

w 0.039282 0.0340 0.0328 0.0326 0.0325 0.0325 0.0324

xx 7.117017 3.5731 1.7924 1.1959 0.8972 0.7178 0.3590

yy 9.206115 4.6442 2.3299 1.5544 1.1661 0.9330 0.4666

xy 2.935043 1.4548 0.7262 0.4836 0.3626 0.2900 0.1449




xz 1.325961 0.3346 0.1768 0.2182 0.2327 0.2394 0.2484

xxM 0.091624 0.0461 0.0232 0.0154 0.0116 0.0093 0.0046

yyM 0.118627 0.0597 0.0299 0.0199 0.0149 0.0120 0.0060

xyM 0.039374 0.0186 0.0092 0.0062 0.0047 0.0037 0.0019 f

xxM 0.091624 0.0461 0.0232 0.0154 0.0116 0.0093 0.0046 fyyM 0.118627 0.0597 0.0299 0.0199 0.0149 0.0120 0.0060 f

xyM 0.039374 0.0186 0.0092 0.0062 0.0047 0.0037 0.0019

Table2 Effect of gradation index 'n' on deflection, stresses and Moments of a simply supported FGM Plate(a/h=5)

'n' 0 0.25 0.5 0.75 1 2 5 10 100000

w 0.0310 0.0357 0.0393 0.0420 0.0440 0.0484 0.0529 0.0565 0.0669

xx 6.0875 6.6829 7.1170 7.4309 7.6620 8.1998 9.0737 10.0401 13.1255

yy 7.8914 8.6529 9.2061 9.6044 9.8967 10.5760 11.7060 12.9736 17.0149 xy 2.5084 2.7557 2.9350 3.0642 3.1592 3.3801 3.7417 4.1419 5.4085 xz 1.0975 1.2197 1.3260 1.4140 1.4881 1.6954 1.9694 2.1252 2.3483

xxM 0.0733 0.0842 0.0916 0.0967 0.1003 0.1082 0.1189 0.1290 0.1581

yyM 0.0950 0.1091 0.1186 0.1251 0.1297 0.1399 0.1538 0.1669 0.2049

xyM 0.0316 0.0362 0.0394 0.0416 0.0431 0.0467 0.0516 0.0560 0.0681 f

xxM 0.0733 0.0842 0.0916 0.0967 0.1003 0.1082 0.1189 0.1290 0.1581 fyyM 0.0950 0.1091 0.1186 0.1251 0.1297 0.1399 0.1538 0.1669 0.2049 f

xyM 0.0316 0.0362 0.0394 0.0416 0.0431 0.0467 0.0516 0.0560 0.0681

-1 0 1 2 3 4 5 6 7 8 9 10 11

0.030

0.035

0.040

0.045

0.050

0.055

0.060

w

Effect of volume exponent 'n'

Fig 4. Effect of grading index 'n' on deflection of a square FGM plate (a/h=5)




0

2

4

6

8

10

7 8 9 10 11 12 13 14

'n'

xx

7 8 9 10 11 12 13 14yy

Fig 5. Effect of grading index 'n' on stresses of a square FGM plate (a/h=5)

0 2 4 6 8 10

0.07

0.08

0.09

0.10

0.11

0.12

0.13

Mxx

'n' Fig 6 Effect of grading index 'n' on Mxx of a square FGM plate (a/h=5)

-1 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0

0 .0 3 2

0 .0 3 3

0 .0 3 4

0 .0 3 5

0 .0 3 6

0 .0 3 7

0 .0 3 8

0 .0 3 9

0 .0 4 0

w

a /h

Fig 7 Effect of span to thickness ratio on deflection of a square FGM plate (‘n’=5)




- 0 .4

- 0 .2

0 .0

0 .2

0 .4

-2 -1 0 1 2

x x

z/h

'n '= 0 'n '= 0 .5 'n '= 1 'n '= 1 .5 'n '= 2 'n '= 5

Fig 8 Effect of grading index 'n' on normalized stress xx of simply supported square FGM plate along the thickness

- 0 . 4

- 0 . 2

0 . 0

0 . 2

0 . 4

- 3 - 2 - 1 0 1 2y y

z/h

' n '= 0 ' n '= 0 . 5 ' n '= 1 ' n '= 1 . 5 ' n '= 2 ' n '= 5

Fig 9 Effect of grading index 'n' on normalized stress yy of simply supported square FGM plate along the

thickness

- 0 . 4

- 0 . 2

0 . 0

0 . 2

0 . 4

- 0 . 5 - 0 . 4 - 0 . 3 - 0 . 2 - 0 . 1 0 . 0 0 . 1

4 1 0x y

z/h

' n ' = 0 ' n ' = 0 . 5 ' n ' = 1 ' n ' = 1 . 5 ' n ' = 2 ' n ' = 5

Fig 10: Effect of grading index 'n' on normalized stress xy of simply supported square FGM plate along the thickness The result obtained for deflection, stresses and moments due to different span ratio for simply supported FGM plates with gradation index 5 is shown in table 1 and table 2 shows the effect of gradation index ’n’ for a thick simply supported FGM plates. It is observed from Fig 4, Fig 5 and Fig 6 that the effect of grading index is more prominent




when the value of n is less than 2 for deflection stresses and moments respectively. Fig 7 shows the variation in deflection become almost negligible as plates become thinner (i.e a/h > 30). However it is more prominent for thick plate. Fig 8, 9 and10 represent the through thickness variation of stresses for different values of gradation index ‘n’.

5. CONCLUSION Bending response of functionally graded material plate (FGM) is presented using shear deformation theory. The effect of span to thickness ratio decreases for a/h ≥ 30. The effect of gradation index 'n' is prominent for lesser values of 'n' and decreases as 'n' increases. The present results can be used for validation purpose. Present solution mythology is good for obtaining the result and the concentrated load. The same can be extended for other types of concentrated load like sinusoidal varying line load, point load, patch load etc.

REFERENCES [1] Praveen, G.N. and J.N. Reddy,. “Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal

plates,” Int. J. Solids Struct, 35(33), 4457-4476, 1998. [2] Ferreira, A.J.M., R.C. Batra, C.M.C. Roque, L.F. Qian, R.M.N. Jorge, 2006. “Natural frequencies of functionally

graded plates by a meshless method,” Comp Struct, 75, 593-600. [3] Reddy JN, Chin CD. “Thermomechanical analysis of functionally graded cylinders and plates,” J Therm Stresses

1998, 21:593–626. [4] Tran LV, Ferreira AJM, Nguyen-Xuan H. “Isometric analysis of functionally graded plate using higher-order

shear deformation theory,” Composites:Part B 51;368-383, 2013. [5] Reddy JN. “ Analysis of functionally graded plates,” International Journal for Numerical Methods in

Engineering;47, 663-684, 2000. [6] Hardy, R. L., “Multiquadric equations of topography and other irregular surfaces,” Geophys. Res. 176, 1905–1915

(1971). [7] Kansa, E. J., “Multiquadrics- a scattered data approximation scheme with applications to computational fluid

dynamics. ii: Solutions to parabolic, hyperbolic and elliptic partial differential equations,” Comput. Math. Appl. 19(8/9), 147–161,1990.

[8] Fasshauer, G. E., “Solving partial differential equations by collocation with radial basis functions,” Surface fitting and multiresolution methods,” Vol. 2 Proceedings of the 3rd International Conference on curves and surfaces 2, 131–138 (1997).

[9] Ferreira AJM, “A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates,” Composite Structures, 59: 385–392, 2003.

[10] Jeeoot Singh “Some studies on Linear and Nonlinear analysis of rectangular plates using RBF based meshfree method,” PHD thesis, page 82, 2012.

AUTHOR

Jeeoot Singh, graduated in Mechanical Engineering from MNREC, Allahabad in 1999, M.Tech from IIT Delhi in 2001 and PhD from MNNIT Allahabad in 2012. Presently working as Associate Professor in BIT Mesra, Ranchi and carrying out his research in field of Computational Mechanics and Meshfree Methods.

Documents

Bending response of FGM plate under sinusoidal Line Load