PSI - Issue 8

E. Marotta et al. / Procedia Structural Integrity 8 (2018) 43–55 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

49

7

R ( ɵ ) ɵ

Fig. 4. Second isostatic scheme.

The same equation as before is valid, as long as a different reference system, on node 2, is now considered (Fig. 4)

 













3

2

(18)

a  

b   

c   

d

'  

 

 

 

 

The new coefficients, identified by an apex, can be related to the previous computed ones if the spanning angle is given by    , after equating the curvature radius given by eq.s (5) and (18). The new coefficients result

2 b c d a b c d                   3 2 ' a a b b a c a ' 3 ' 3 2 '

(19)

The nomenclature for the flexibility matrix links the forces at node 2 with the displacements at the same node

2

4 u c c c u c c c        44 45

4       5        6   F F F 

46

(20)

5    

 

54

55

56

u

c c c

6   

64

65

66

accordingly, the constrained stiffness matrix is:

1



44 c c c c c c c c c 45 54 55

 

    

46

2

(21)

K

 

jj

56

 

64

65

66

Now the 2 K

jj matrix refers to the second local coordinate system. This second stiffness matrix is reconducted to the

first reference by the simple rotation

2    K T K T T

1

(22)

JJ

JJ

21

21

Where the transformation matrix, note sign opposite to usual, is given by