Issue56

K.C. Nehar et alii, Frattura ed Integrità Strutturale, 56 (2021) 203-216; DOI: 10.3221/IGF-ESIS.56.17

1 4 1 4

     ( ) ( ) l

  

  (1 )(1 )

(4)

( , ) N l

3

2

2

     ( ) ( ) l

  

  (1 )(1 )

(5)

( , ) N l

4

1

2

In matrix form:

                          (1 )(1 ) (1 )(1 ) 1 (1 )(1 ) 4 (1 )(1 )

  i N

T



(6)





The derivatives of the interpolation functions N i are:

                   

                   

  (1 ) (1 ) 1 (1 ) 4 (1 )

  (1 ) (1 ) 1 (1 ) 4 (1 )

T

T

         i N

         i N

,

(7)

These derivatives are transformed by a function F ( x , y ) with respect to ξ and η :

    y

F y

                           

      x





F F x

x

y

    

                           [ ] F F x J x F F y y    



    



 



    







(8)







F F x

y

y

F

x

 



 

 

   

     x y

     

where J is the Jacobian operator, connecting the natural and global coordinates. The derivatives with respect to the global coordinates can be found as follows:

                        1 [ ] F F x J F F y                

(9)

The stiffness matrix is then obtained by:   [ ] [ ] [ ][ ] T V K B C B dV

(10)

where [ B ] is the strain matrix derived from the interpolation functions:

      

      



N

i

0

          i N       i N  

               i i N x N y 

x

 



N



1

i



[ ] B

     [ ] J

0

,

(11)



y

       i i N N x y

215