Civil Engineering Reference
In-Depth Information
5.9
gEOMEtRic StiffnESS, X-Z SYStEM
The geometric stiffness matrix for the coplanar X-Z system can be
derived in a similar manner to the X-Y system performed in Section 5.8.
The primary difference is that the signs of the moments due to transla-
tion and the forces due to rotation will be the opposite of Equation 5.29.
The geometric stiffness matrix for the coplanar X-Z system is given as
Equation 5.30.
00 000
0
6
5
1
10
6
5
1
10
0
−
0
−
−
L
L
1
10
2
15
L
1
10
L
0
−
0
−
30
[
]
=
KP
m
ix
00 000
0
6
5
L
1
10
6
5
1
10
0
−
0
L
1
10
L
1
10
2
15
L
0
−
−
0
30
(5.30)
Example 5.16
Geometric stiffness
Determine the deformations at the free end of the beam by including both
the elastic and geometric stiffness contributions to the stiffness solution.
The beam is shown in Figure 5.19.
Since the
j-
end of the member is fixed, there is no need to build the
entire member stiffness matrix. The
j-
end motions will be eliminated and
Z
A=10 in
2
I=100 in
4
50kips
X
1kip
200in
E=10,000 ksi
Figure 5.19.
Example 5.16 Geometric stiffness.
