Issue 29

S. Terravecchia et al., Frattura ed Integrità Strutturale, 29(2014) 61-73; DOI: 10.3221/IGF-ESIS.29.07

r    

r    

r    

r     



)  

 

+

(

( )

g     u g g g u g g N G N T N G N R N G N U N G N G gu f gg r gt u gr g

A

A

A

A

gu

gg

gt

gr

(14b)

1 2

g         g r g r gp    



( ) N N G N G U 0 ( )    

C

, g tc a      

C

gr

and, following the compact notation proposed by Panzeca et al. [9], the latter equations become

, u tc           a

  

   



  

  

  

         U 

uu A A A A

A C A

      T R

      0 0

ug

ut

ut

ur







)   U

(14c)

(



C

 

A A C

G a

gu

gu

gt

gr

gr

, g tc

Since some components of the vector  U and of the vector C

 U coincide, it is opportune to introduce the following

relation

C C  U H U

(15)

where the low rectangular matrix The previous relation can be rewritten

C H is a topological matrix made by

and

blocks.

I

0

2 2 

2 2 

   

   

*     

)   A C a

(

  

  

  

  

uu A A A A

A

      T R

      0 0

ut

ut

, u tc

ug

ur





)   U

 

(

( ) G

(16)



 A C

*

     

 A a

gu

gu

gr

gr



, g tc   gt

K

L

g

L

u

where

*

, g tc         , g tc  a H    

*         , u tc , u tc  a

.

(17a,b)

;

a H a

C

C

Finally, the solving system is rewritten in compact way with obvious meaning of symbols

u g    K X L L 0 

(18)

L

L load vectors due to  U e

with K symmetric flexibility matrix of the system, X vector of nodal unknowns, u L and g

G respectively, being g   L L L the total load vector. In order to evaluate the coefficients of the equation system, the numerical techniques to remove the singularities and the rigid motion strategy have to be employed. Observation about rigid motions Let the plate of Fig.2 be subjected to a) a rigid motion of translation obtained as the sum of constant displacement fields x u and y u (Fig.2a), b) a rigid motion of rotation having wideness  around any point (Fig.2b), c) a rigid motion of roto-translation obtained as a combination of a) and b). In all these cases the displacement and normal derivative of the displacement fields are entirely known both in the domain and on the boundary of the plate. Particularly - for the case of rigid motion of translation (Fig.2a) u

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