PSI - Issue 26

Merazi Mohamed et al. / Procedia Structural Integrity 26 (2020) 129–138 Merazi et al / Structural Integrity Procedia 00 (2020) 000 – 000

134

6

The stress resultants of a the plate can be related to the total strains by

s

    

    

N

B D H 0 D D A 0 B s s

    

    

k k      

    

s xz s yz

s xz s yz

   

   

   

   

s 44 0 A A 0

   

   

S S

 

s b

s

s b

=

(16)

=

M M

,

s 55

s

where



 b t xy



 s t xy

  t x y xy N N , N , N = ,

b y b x b M M ,M ,M = 0 , 0 , 0 } , = { , , } , = { , , } ,

s s x s M M ,M ,M = y

(17a)

= {

(17b)

Where ij A , ij D , etc., are the plate stiffness, defined by { 11 11 1 1 1 1 1 1 12 12 1 2 1 2 1 2 and

66 66 6 6 6 6 6 6 } = ∫ 11 (1, 2 , ( ), ( ), 2 ( )) ℎ 2 − − ℎ 2 −

{ 1 1− 2 }

(18a)

(

) (

) s 11

s 22

s 22

s 22

s 11

s 11

11 22 22 A , D , B , D , H A , D , B , D , H = 11

(18b)

h d

2  − − −





2

= = s A A Q g z s

dz

( ) ns

,

(18c)

ns

44

55

55

h d

2

3. Analytical solution

Rectangular plates are generally classified in accordance with the type of support used. We are here concerned with the exact solution for a simply supported FG plate. The following boundary conditions are imposed at the side edges: 0 = = = = = = = = 0 at x= -a/2, a/2 (19a) 0 = = = = = = = = 0 at y= -b/2, b/2 (19b) The equations of motion admit the Navier solutions for simply supported plates. The displacement variables 0 u , 0 v , b w , s w can be written as product of arbitrary parameters and known trigonometric functions by assuming the following variations

V U

        y) x) sin( sin( y) x) sin( sin( y) x) cos( sin( y) x) sin( cos(

v u

      

      

      

      

0 0

mn mn

  =  = m 1n 1

(20)

=

W

w w

s b

smn bmn

W

n / b  =  .

mn U , mn V , bmn W , and

smn W are arbitrary parameters to be determined, and

m / a  =  and