PSI - Issue 5

B. Boukert et al. / Procedia Structural Integrity 5 (2017) 115–122 B.Boukert/ Structural Integrity Procedia 00 (2017) 000 – 000

118

4

xx N N x y Nxy N x y Q Q        y



xy

0

 



yy

0

 

2

2

xy P P 

2





P



4 (

yy

x

xx

q  

2

)

0

 





2

2

2

x  

y

x y

 

h x 

y

3



(4)

2 M M Q x y M M Q x y Where M M P Q Q R h h                   2 0 0 4 4 : , 3 xx xy x xy yy y ij ij ij ij ij

ij

The stresses resultants are related to the strains by:

      0 1 3                      

      N M B D F P E F H                             A B E

    0 2             

    Q R D F                    A D   

(5)

,

Where :

N z

k

1 Q z z z z z 

2 3 4 6 (1, , , , , )

1 k z    

( , , A B D E F H , , ,

(6)

)

ij

ij

ij

ij

ij

ij

ij

k

2.3. The navier solution:

The development of Navier solutions of simply supported antisymmetric cross-ply laminates using the third order theory are bellow where the boundary conditions are satisfied by the following expansions.

1 1 n m         n m                   n m n m n m     1 1 1 1 1 1

u x y

mn U x

cos( ) sin( ) y  

( , )



0

v x y

V

sin( ) s( ) x co y  

( , )



mn

0

w x y

mn W x

sin( ) sin( ) y  

( , )



0

(7)

1 1

( , ) x y

mn X co x

s( ) sin( ) y  

x 



( , ) x y

Y

sin( ) s( ) x co y  

y 



mn

where

/ ,       m a

/ m b

: