PSI - Issue 5

Khodjet-Kesba Mohamed et al. / Procedia Structural Integrity 5 (2017) 271–278 Khodjet-Kesba Mohamed et al/ Structural Integrity Procedia 00 (2017) 000 – 000

274

4

   

   

2

   2

 a

   d C d  22

   

2

2

   d C d 11

2 1

 



 

(13)

'

2 90

2 90

 C C d  2

u

t

d









c

00

02

2

2

d





a



Where,

90 1 1 E E 

(14)

C

 

00

0



0 

3 3 2    

    

(15)

C



90

02

E

E

90

0

1

(16)



 1 (3 12 8) 2    

C







22

E

60

90

  

  

3 1

1



(17)

C





11

90 G G

0

90 t l

(18)

a 

c u is the fourth order differential equation of Euler

The function  which minimizes the complementary energy '

Lagrange:

4

2

d d



 

(19)

d p d

   q

0



4

2



02 C p C C  

(20)

,

C q C 

00

11

22

22

2 4 q p  . Consequently the solutions are of the form :

Provided that

(21)

1 Ach

2 A sh

cos

sin















With

a sh a ch a a sh a a        2 sin 2 ) cos sin        a sh a ch a a sh a a        2 sin 2 ) cos sin              

2(



(22)

A



1

2(

(23)

A



1

And

cos 4 1

sin 4 1





4 1 2  q

;

;

(24)

 q 

 q 

 arctg 

2

2

p

Finally, the distribution of stress can be expressed in the form:   1 ( ) 1 0 90 90 x E E x c xx     

(25)

     1 0 0 E E x

  

0

0 E a E 90



1 ( )

(26)

x

1 

xx



c

3. Results for stress distribution

We present numerical examples for stress distribution based on analytical models (shear lag and variational approach). The results are compared with finite elements method for glass/epoxy laminate (Berthelot 1997). The material properties of the chosen composite are summarized in Table 1.