Issue 58

R.N. da Cunha et alii, Frattura ed Integrità Strutturale, 58 (2021) 21-32; DOI: 10.3221/IGF-ESIS.58.02

    U



T

 1 1 ...

 i

 j

 n

u w

u w

u w

u w

...

...

(1)

i

i

j

j

n

n

1

    P



T

P P P

P P P

P P P

P P P

1 ...

...

...

(2)



 i





u

w

ui

wi

uj

wj

j

un

wn

n

1

1

where u 1 is the node 1 displacement parallel to the X-axis, w 1 is the node 1 displacement parallel to the Z-axis and θ 1 is the node 1 rotation on the XZ plane. Analogously, P un is the applied force on node n parallel to the X-axis, P wn is the applied force on node n parallel to the Z-axis and P θ n is the applied moment on node n on the XZ plane.

The structural equilibrium is given by:

      b P B M    1 m T b

(3)

b

being [ B ] b the transformation matrix and { M } b the generalised stresses of a finite element b, both given as follows:

    M



T

M M N

(4)

i

j

i

b

Frame element:

 b

 b

 

sin

cos

sin cos L L sin cos L L b b b

        

        

b





0 0 0 ...

1 ...

0 ... 0 0 0

L

L

b

b

b

 b

 b

 

sin

cos

  B

b







0 0 0 ...

0 ...

1 ... 0 0 0

(5)

b

L

L

b

b

b

b



  sin

 

0 0 0 ...

cos

0 ... cos

sin 0 ... 0 0 0

b

b

b

b

node

node i

node j

node n

1

Arch element:

node

node i

1

   b

   b



sin

cos





0 0 0 ...

1 ...

    

   b

   b

R

R

sin

sin

b

b

   b

   b

sin

cos

  B





0 0 0 ...

0 ...

   b

   b

    

b

R

R

sin

sin

b

b

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

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

   b

   b







0 0 0 ... cos

sin

0 ...

b

b





   sin







   cos



         

  

       sin b b b R     sin b b

  

           sin cos b b b b b R

cos

cos

sin

cos

b

b

b

b







...

0 ... 0 0 0

R

R

b

b





  







  



  

  

cos

cos

sin

cos

b

b

b

b

b

b

b

b

b

b







...

1 ... 0 0 0

   b

   b

R

R

R

R

sin

sin

b

b

b

b

(6)

       cos 1 sin b

       cos 1 cos b





  

  









b

b

b

b

 

 



 

 

...

cos

sin

0 ... 0 0 0

   b

   b

b

b

b

b

sin

sin

node j

node n

23