PSI - Issue 26

Merdaci Slimane et al. / Procedia Structural Integrity 26 (2020) 35–45 Slimane et al. / Structural Integrity Procedia 00 (2020) 000 – 000

40

6













t

t

t

(14)

b b b b x y xy x y xy N N N N M M M M M M M M , , , , , , , , = = = s s s s x y xy

Where A ij , B ij , etc. are the plate stiffness defined by

h 2 /

(

)





(

)



2

ij ij , , A B D

1 z z Q dz , ,

i j 1 2 6 , , , =

=

ij

ij

h 2 /

−

(15)

h 2 /





(

)

(

)



s s s B D H , ,

2

f z z f z f z Q dz ( ), ( ), ( )

i j 1 2 6 , , , =

=

ij

ij

ij

ij

h 2 /

−

h 2 /

(

)

  s ij A





2

(

)



g z Q dz ( )

,

i j 4 5 , , =

=

ij

h 2 /

−

6. Solutions for simply supported FG plates Rectangular plates are generally classified according to the type of support used. This paper is concerned with the exact solution for a simply supported FG plate. The following boundary conditions are imposed at the side edges:

w w y y  

s   = = = = = = = = b s v w w

(16a)

b x

N M M 0 and x 0 a , =

0

b

s

x

x

w w x x  

s   = = = = = = = = b s u w w

(16b)

b y

0 N M M 0 and y 0 b , , =

0

b

s

y

y

Substituting the expressions for δ U and δ K from Eqs.(9) and Eq.(11) into Eq.(8) , integrating the displacement gradients by parts and setting the coefficients δu, δv, δw b , and δw s zero separately. Thus, one can obtain the equilibrium equations associated with the present shear deformation theory,

x N N x y     xy N N x y    

b w w 



(17a)

s

xy

δ u

1 0 2 Ι u Ι

I

:

+ = −

−

4

x

x





(17b)

b w w 



s

y

δ v

1 0 2 Ι v Ι + = −

I

:

−

4

y

y





   

3         I

5         − I

   

(17c)

2 b M M  2 b



2 b M x

2

2

2

2



u v x y

b w w 

s w w 

   





b

s

xy

y

δ w

2

b s Ι w w I ( ) = + +

:

+

+

+ −

+

+

b

1

2

2

2

2

2

2

2

x y

 

x

y

x

y

x

y













   

5         I

6         − I

   

2 s M M  2 s

s

S





2 s M

s

2

2

2

2

S





u v x y

b w w 

s w w 

   





(17d)

b

s

xy

y

yz

x

xz + + = + + b s Ι w w I ( )

δ w

2

:

+

+

+ −

+

+

s

1

4

2

2

2

2

2

2

x y

x y

 

 

x

y

x

y

x

y













Following the Navier solution procedure, we assume the following form of solution for (u,v,w b ,w s ) that satisfies the boundary conditions given in Eq.16.

i ωt

      

      

λ x μ y λ x μ y λ x μ y λ x μ y ) sin( )cos( ) sin( ) sin(

cos( sin(

)

mn U e V e mn

s         =       u v w w

i ωt

)

  

(18)

i ωt

sin( sin(

)

bmn W e W e smn

b m 1 n 1 = =

i ωt

)