Issue 54

T. I. J. Brito et alii, Frattura ed Integrità Strutturale, 54 (2020) 1-20; DOI: 10.3221/IGF-ESIS.54.01

  b 

  b 



sin

cos





1

   

  b 

  b 

R

R

sin

sin

b

b

  b 

  b 

sin

cos

  B





0

  b 

  b 

    

b

R

R

sin

sin

b

b

        cos 1 sin sin b b    

        cos 1 cos sin b b    

  b 

  b 







cos

sin

0

b

b

(5)





   sin 







   cos 



         









b 

b 

b 

b 

b  R  b R

b 

b 

b 

cos

cos

sin

cos

b

b







0

  b 

  b 

R

R

R  b R

sin

sin

b

b

b

b





  b 









   cos





b 

b 

b 

b 







b 

b 

 b

sin

cos

cos

cos

sin







1

  b 

  b 

R

R

sin

sin

b

b

b

b

     cos 1 sin b  

     cos 1 cos b  









b 

b 

b 

b 











b   

b   

b 

b 

cos

sin

0

  b 

  b 

sin

sin

Then, the equilibrium relation of the structure is given by:

m

1 b    P Q

 

 

(6)

E b

being { Q E } b the expanded internal forces matrix, given as follows:

  E Q

0 0 0} T 



1 {0 0 0 node ui   wi i  Q Q Q  node i

uj Q Q Q   wj j

(7)





b

node n

node j

Therefore, the equilibrium relation can be expressed as:

      1 m T E b b b    P B M

(8)

where [ B E ] b is the expanded transformation matrix, given in a similar way of { Q E } b . Kinematics of circular arch elements

Consider again the structure depicted in Fig. 1a, the node i presents two translations parallel to the global X - and Z -axes ( u i and w i , respectively) and a rotation in the XZ plane ( θ i ). Such translations and rotation are here called generalised displacements. Then, the generalised displacements of all nodes of the structure are gathered in the generalised displacements matrix, given by:     1 1 1 ... ... ... T i i i j j j n n n u w u w u w u w      U (9) Now, the generalised deformations are defined as conjugate quantities of the generalised stress i.e.  i ,  j and δ i are conjugated to M i , M j and N i , respectively. Such quantities are gathered in the generalised deformations matrix:     T i j i b     Φ (10)

Therefore, the mechanical power of the element is expressed as:

4