Issue56

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

[ C ] is the elasticity matrix:





1

0 0

   

  

 C E [ ]



1

, in plane strains

(12)

 

2





1

    0 0 2(1 )



     

 

  1

0 0

E

   









[ ] C

1

, in plane stresses

(13)





 

  (1 ) 1 2

 (1 2 ) 

0 0

2

The volume of the element dV is given by:    .det . . dV h J d d

(14)

where det J is the determinant of the Jacobian matrix and h the thickness of the element. The surface d ξ .d η at the point ( ξ , η ) of the reference element is transformed into the surface dA at the point ( x (( ξ , η )), y (( ξ , η ))) of the natural element. Were:    det . . dA J d d (15)

The integral in the stiffness matrix on the real element is:

  [ ] [ ]. V K F dV

(16)

Therefore, becomes on the reference element:

  [ ] .[ ]. A K h F dA

(17)

Taking:

 [ ] [ ] [ ][ ] T F B C B

(18)

The integral in the stiffness matrix is then calculated numerically by Gaussian quadrature in two dimensions. In the four node element, we can use a 2 × 2 numerical integration for exact integration.

    1 1

  p q





  F d d

    , . .

  ,

(19)

w w F

i

j

i

j

  j

i

1 1

1 1

where p , q are the number of integration points in the directions ξ , η , respectively, and w i , w j are the corresponding weights. The stiffness matrix can be calculated using 2 × 2 Gauss points as:

    1 1

  e      T K h B C B dA h B C B J d d        T     . . . det . .

 A

1 1

(20)

   2 2 1 1 i j

     T B C B J w w   .det .



h

.

i

j

216