PSI - Issue 8
E. Marotta et al. / Procedia Structural Integrity 8 (2018) 43–55
48
Author name / Structural Integrity Procedia 00 (2017) 000 – 000
6
Pointing attention, as an example, to the displacement u 1 due to an applied load F 1 , the solution arises from the integral
M
1
( )
0
u
M
( ) d
( )
(11)
1
EI
F
1
The first compliance term c 11 thus requires to compute the bending moment as a function of θ
3
2
2
( ) M F y
F a
3 sin a
b
2 sin b
6 cos a
( )
( cos
cos
1
1
(12)
c
6 sin sin 2 cos a c b
d
cos 2 ) b d
cos
The integral (11) results composed by the product of several terms
2
3
2
2
a c
3 sin a
b
2 sin b
6 cos a
cos cos
cos
1
0
(13)
3 b 2
c
a
c d d
11
EI
6 sin sin 2 cos a c b
d
b d
cos 2
The resulting integral is very long, but, after the application of double-angle formulae, it results as composed by sine and cosine recursive integral terms, such those given by eq.(14) and eq.(15).
n
x
n c
sin( ) x cx dx n
n
1
cos( ) cx
x
cx dx
cos( )
(14)
c
sin x cx n n
(15)
n
n
1
x
cx dx
x
cx dx
cos( )
sin( )
c
c
Indeed, an enormous number of trivial terms occurs, all of them supply an analytical solution, but easily manageable using a symbolic solver. Repeating eq.11 for all the three degrees of freedom of the beam, it is possible to obtain, in closed form, the matrix of flexibility of the first structure shown in Figure 3.
1
u c c c
1 2 3 F F F
1
11
12
13
2 u
(16)
c c c
21
22
23
3 u c c c 31 32
33
The inversion of the flexibility matrix provides the constrained stiffness matrix in the reference of node 1
1
11 c c c c c c c c c 12 21 22
11 k k k k k k k k k 12 21 22
12
12
(17)
1
K
II
23
23
31
32
33
31
32
33
To obtain the second part of the stiffness matrix it is necessary to constrain the node 1 and apply loads at node 2.
Made with FlippingBook Digital Proposal Maker