Issue 29

D. Addessi et al., Frattura ed Integrità Strutturale, 29 (2014) 178-195; DOI: 10.3221/IGF-ESIS.29.16

l



N

  

w

 q C d

, w n x

yz  C u

   



N

(30)

C



w

, w n w

, w n

x 

n

=1

L T     D b d

  dx

(31)

0

L



N

  

0   w =1 0  n

, w i

x w

yz

, w i   p



   dx

N

C C d

, w i

w

 x

(32)

L l

 





  

  

, w i N N  

N

N

  

  

, w n

, w i

, w n

x

xy ww

dx  C u yz 





, w n N N 



, w i N N

C

C

ww

, w i

, w n ww

, w n





x x

x

x

Eq. (30) represents the generalized section constitutive law and contains two contributions. The first term takes into account the standard deformation d associated to the cross-section rigid motion, depending on the section stiffness matrix, defined as:

T   C c  

dA

(33)

A

The second term is related to the section deformations associated to the warping. The section warping stiffness matrices are defined as follows: = = x x T x w w r w r A dA          C C P cM P 

   

   

yz

 A

= = yz w r

T yz



cM P dA

C C P w

w

r

   

  

  x w

T

x

=   C P M cM P k P C P k   = = T yz T yz yz T yz ww r w w r β r ww r β A dA            C P M cM P k P C P k   = = T xy T x yz T xy ww r w w r β r ww r β A dA            C P M cM P k P C P k = T x w T x ww r r β r ww r β A dA  

where:

T  k V V β

( ) w m  

           and          

P I

R V

r

β

w w m m  identity matrix and R a

3 w m  matrix that represents the rigid body motions of the element section,

I ( ) w m

is a

m warping points as follows:

i.e. a matrix containing the coordinates of the w

1 y z y z 2

1 1

     

      

1

2

 

R



y

z

1

w w m m

Finally, V is a matrix containing the average value and the first moments of the shape functions over the cross section:

185