PSI - Issue 13

B. Boukert et al. / Procedia Structural Integrity 13 (2018) 174–180 B.Boukert et al / Structural Integrity Procedia 00 (2018) 000–000

176

3

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   

(3)

,

where:

N z

1 Q z z z z z 

k

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

(4)

1 k z    

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

)

ij

ij

ij

ij

ij

ij

ij

k

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.         0 1 1 0 1 1 ( , ), ( , ) , .cos( ).sin( ) ( , ), ( , ) , .sin( ). s( ) x mn mn n m y mn mn n m u x y x y U X x y v x y x y V Y x co y                    

n m      

( , )

.sin( ).sin( ) y  

w x y

mn W x



(5)

0

1 1

:

/ ,       m a

/ m b

where

The mechanical transverse load q(x,y) is also expanded in double Fourrier series

0 0 4 ( , ) sin( ) sin( ) a b q x y x ab    

(6)

mn Q

y dxdy



The coefficients ( U mn ,V mn , W mn , X mn , Y mn ) of the Navier solution are obtained by solution of the system (7) : 1 1 11 12 13 14 15 2 2 T C mn mn mn T C N N U S S S S S                  

 

12 22 23 24 25 13 23 33 34 35 14 24 34 44 45 15 25 35 45 55 S S S S S S S S S S W Q S S S S S X S S S S S Y                         where N mn , M mn are given by equations 8-10:        1 , , N z k k k Ti Ti Ti mn mn mn i N M P Q      mn mn mn T mn T mn V N            

N





mn

mn

(7)

    

1

1

C

2 mn M M M M     mn

mn

2

C

mn



3

( ), T z T z z T z z dz    ( ) , ( )

(8)

1 k z k 

4 h

Ti

Ti

Ti

mn M M 

P



(9)

mn

mn

2

3

0 0   a b

4

i

i

(10)

sin( ) sin( ) x  

mn T

T

y dxdy



ab

    









(11)

( , , ), T x y z C x y z   ( , , )

1( , ), 2( , ) sin( ) sin( ) f z T f z C x y  

1 1

n m

 