Civil Engineering Reference
In-Depth Information
2
1.2
Geo nonlinear
Geo linear
Geo nonlinear
Geo linear
1.5
0.8
1
0.4
0.5
0
0
-0.5
-0.4
-1
-0.8
∙∙
4
(
∙∙
6
(
y
t
)
-1.5
y
t
)
-2
-1.2
0
5
10
15
20
0
5
10
15
20
Time (s)
Time (s)
1.5
1
Geo nonlinear
Geo linear
Geo nonlinear
Geo linear
0.8
1
0.6
0.4
0.5
0.2
0
0
-0.2
-0.5
-0.4
-0.6
-1
∙∙
3
(
∙∙
1
(
y
t
)
y
t
)
-0.8
-1.5
-1
0
5
10
15
20
0
5
10
15
20
Time (s)
Time (s)
Figure 7.30
Global acceleration responses of the 6-story frame with geometric nonlinearity.
Figures 7.31 and 7.32 show the moment and plastic rotation responses, respectively, of
selected plastic hinges. These selected plastic hinges include the base of the exterior column
(PHL #1), base of the interior column (PHL #13), exterior hinge of the beam on the second
floor (PHL #49), and interior hinge of the beam on the third floor (PHL #56). In addition, the
same responses but for the case without considering any geometric nonlinearity (see Example
3.9) are presented in the figures for comparisons. Finally, the corresponding hysteresis loops
are plotted in Figure 7.33.
Example 7.9 Ten-story Moment-Resisting Steel Frame
Consider the 10-story moment-resisting steel frame as shown in Figure 7.34. Assuming the
members are axially rigid, this gives a total of 50 DOFs (i.e.
n
= 50) and 140 PHLs (i.e.
q
=
140). Also assume that updating of geometric nonlinearity due to changes in axial forces in
columns is ignored. A uniform gravity load of 21.89 kN/m is applied on all the beams, and
this gives the axial compressive forces in the columns that are used to calculate the
i
th member
stiffness matrices
K
i
,
K
0
i
, and
K
0
i
along with the standard properties of each member. Finally,
the 50 × 50
K
matrix, 50 × 140
K
0
matrix, and 140 × 140
K
00
matrix are constructed by assem-
bling the individual stiffness matrices of each member.
No mass moment of inertia is assumed at the rotational joints, and therefore static con-
densation is used to eliminate the 40 rotational DOFs (i.e.
r
= 40), resulting in only
