PSI - Issue 8

Francesco Penta et al. / Procedia Structural Integrity 8 (2018) 399–409 Francesco Penta et al./ Structural Integrity Procedia 00 (2017) 000 – 000

409 11

Appendix A - Pratt girder stiffness sub-matrices The (5x5) blocks forming the leading diagonal of the Pratt unit cell stiffness matrix are given by

              

  

  

2

2

2

3

sin    

cos sin

sin    

cos sin

sin cos     

cos sin

c 

1 2

d

d

d

d

d

d

d

d

d

d

d

d

d

2

12

6

1

2











2

2

2

2

2

2

2

l

l

l

l

l

l

l

d 

sin

d 

d 

d 

d

d

d

d

d

d

d

   

   

  

  

2

3

2 sin cos

cos     sin

cos   

 

c 

1

1 4

1 2

2

4

d

d

d

d

d

d

d

d

d

6

cos        3 sin 12

6











  d 

d

d

d

d

t

2 s l

l

l

l

in

2

d

d

d

d

  

  

2 sin cos      cos

2

sin cos     

cos sin

cos    

1 2





d

d

d

d

d

d

d

d

d

d

c

d  

d

d

12

6

24

12







x

x

2

2

2

2

2

l

l

l

l

l

l

l

d 

d

d

d

d

c

d

d

  

  

  

  

sin

cos   

d d  

2

12 6 c 

d

d

6

3 sin 12  

  







d

d

t

l

l

l

d

c

d

sin

cos  

d d  





2  

d

d

6

3 sin

6





d

d

l

l

d

d

         

 

  

sin  

sin  

d

d

d

d



6

6

 

l

l

d

d





2

2     cos d d



3 sin 12  

3 sin

cos       d d t

,

 

  

12 6 c 

d

d

d

d

6

 



l

l

l

c

d

d





8 4 12  

4

  

d 





c

d

t







4

4 8 4 



    

d 

d

c

t

, x l r  and where

For the

with

2 E A l l c c t

2 E A l l d d t

c c c E I l

E I l

t t E I l

,

,

,

and

.

c 



d 



t 



c 



d 



d d d

t

c

d

xx  affected by the  , the upper sign is for x l  , the lower for x r  . Furthermore, for the upper

components of

and lower out-diagonal blocks it results: 2

              

2

2

3

sin    

cos sin

sin    

cos sin

sin cos     

cos sin

c 

1 2

d

d

d

d

d

d

d

d

d

d

d

d

d

2

12

6

12













2

2

2

2

2

2

2

l

l

l

l

l

l

l

d 

sin

d 

d 

d 

d

d

d

d

d

d

d

2

3

2 sin cos      cos

sin    

cos sin

c 

1 2

1 4

1 2

2

4

d

d

d

d

d

d

d

d

d

6

cos      3 sin

6





 





d

d

d

d

l

l

l

l

2

d 

d

d

d

d

2 sin cos      cos

2

sin cos     

cos sin

cos    

1 2

T

 





d

d

d

d

d

d

d

d

d

d

24    c d

d

d

12

6

12







l

r

r

l

2

2

2

2

2

l

l

l

l

l

l

l

d 

d

d

d

d

c

d

d

sin  

cos   

2  

d

d

12 6 c

d

d

6

3 sin



 

d

d

l

l

l

d

c

d

sin  

cos  

2  

d

d

d

d

6

3 sin

6



d

d

l

l

d

d

sin  

sin  

.         

d

d

d

d

6

6





l

l

d

d

2

2

3         d   sin 3 sin cos cos d  d d

12 6 c 

d

d

d

d

6

l

l

l

c

d

d

4 2 

2

 

d 

c

d

2

4 2 



d 

 

c

d