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
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