Information Technology Reference
In-Depth Information
FIGURE 11.22
Element 1 with local and global coordinates.
From the calculation procedure of Table 11.5, the next step is the formation of the stiffness
matrices and load vectors for each element. First, they are calculated with respect to the local
coordinates of each element. The transformation to the global coordinate systems follows.
This leads to the element matrices and load vectors which can be assembled into the system
stiffness matrix and system load vector.
As mentioned, the element matrices of the linear system (first order) have to be supple-
mented with the geometric parts of the stiffness matrices to include second order effects.
We now proceed to form the linear and geometric parts of the stiffness matrices and
vectors for the three elements of the system. The initial step of a second order anlaysis
requires the calculation of the normal forces N 0 in the elements. These are usually obtained
from a linear analysis.
Stiffness Matrices and Load Vector of Element 1
For element 1, a beam column, substitude the values of Fig. 11.22 into Eq. (11.63). For the
displacement vector arranged as v i
b ] T
=
[
w
θ
w
θ
we obtain (Table 11.3)
a
a
b
=
12
6
12
6
2109
.
38
8437
.
5
2109
.
38
8437
.
50
2
2
EI
6
4
6
2
45 000
.
0
8437
.
5
22 500
.
0
k lin =
.
.
3
12
6
12
6
2109
38
8437
50
2
2
.
6
2
6
4
Symmetric
45 000
0
(1)
From Eq. (11.64)
6
5
10
6
5
10
2
15
10
2
30
2
N 0
k geo =
6
5
10
Symmetric
2
15
2
167
.
46
111
.
64
167
.
46
111
.
64
1190
.
80
111
.
64
297
.
70
=
(2)
167
.
46
111
.
64
Symmetric
1190
.
80
Search WWH ::




Custom Search