Issue 52

N. Hebbar et alii, Frattura ed Integrità Strutturale, 52 (2020) 230-246; DOI: 10.3221/IGF-ESIS.52.18

where: (z) is the density and I 0 , I 1 , J 1 , I 2 , J 2 and K 2 are the coefficients of inertia as defined below:

h

h

h

/2

/2

/2







0 I b 

( ) , z dz I b z z dz J b f z z dz      ( ) , ( ) ( ) ,

1

1







h

h

h

/2

/2

/2

(22)

h

h

h

/2

/2

/2







2

2

( ) , I b z z dz J b zf z z dz K b f z z dz       ( ) ( ) , ( ) ( )

2

2

2







h

h

h

/2

/2

/2

A NALYTICAL SOLUTION he motion equations admit Navier's solutions for simply supported beams. The variables u 0 , w 0 and  ϕ  can be written assuming the following variations:

T

i t 

    

     

cos sin cos w xe xe     m m u xe

0         0 u w 

 

   

i t 

(23)



m

1,3,5

i t 

m

with: i = 1  and α = m π /L The transverse load q is also expressed by the double series of Fourier sine as follows:



4 sin q

m   

0



q

x

1,3,5 (24) Substitute the expressions u 0 , w 0 and  ϕ of Eqn. (23) in the equation of motion (20). The analytical solution is given in the following form: m 

0 q

 

    



    

11 12 13 m m m u m m m w    12 22 23

11 12 13 12 22 23 13 23 33 s s s s s s s s s

    

    

    

0      4          m m             m m       0    

2 



(25)

m m m

13 23 33

In which: S 11 = A 11 α 2 , S 12 = -B 11 α 3 , S 13 =

2 a K A'

2 H s 11 α 2 + 2

a K A'D 11 α

a K B

a K A'D

a K A' A

2 , S 22 =

s 11 α 2 , S 23 = -

s 11 α 2 , S 33 =

s 55 (26)

2 a K A'

m 11 = I 0 , m 12 =- I 1 α , m 13 = a K A' J 1 α

2 , m 22 = I 2 α 2 + I 0 , m 23 = - J 2 α 2 , m 33 =

2 K 2 α 2

N UMERICAL RESULTS AND DISCUSSION

I

n this section, various numerical examples are presented to verify the accuracy of the theory presented for the purpose of predicting bending, buckling, and vibration responses of a simply supported FG beam. The properties of the materials change through the thickness of the beam according to a power-law. The lower surface of the beam is rich in aluminum and the upper surface of the beam is rich in alumina. For convenience, the following dimensionless form is used:

3

3

2

2

100 w E h

100 w E h

12

h

h

P

0 N a

 x



2 h

2 L

, ( , 0, ) 2 2  x L h

L



(27)

(0, 0,

)  

, ( , 0, 0) w

, (0, 0, 0) 

,

,

u

N

m

m

x

m













xz

cr

4

4

3

0 q L

0 q L

h E

0 q L

0 q L

m E h

m

237