Nonlinear Dynamic Analysis - Theory of Nonlinear Structural Analysis - page 100

#84

#79 W27x94 W27x94 W27x94

#81 #83

x 6

#12

#4 7

W36x135 W36x135 W36x135

#73

#74 #75 #76 #77 #78 x 5

#10

#9

#4 5

W36x150 W36x150 W36x150

#67 #69 #71

x 4

#4 3

W36x210 W36x210 W36x210

#62 #63 #64 #65 #66

#61 x 3

Mode Period (s) Damping

#6

1 1.000 2%

#5

#4 1

W36x210 W36x210 W36x210

2 0.362 2%

#55 #57 #58 #59 #60

#56 x 2

#4

3 0.207 2%

#3

#3 9

W36x210 W36x210 W36x210

4 0.148 2%

#49 #51 #53

x 1

5 0.114 2%

PHL#1 #25 #37

#13

6 0.093 2%

7.62 m 7.62 m 7.62 m

Figure 3.20 Six-story moment-resisting steel frame.

Example 3.9 Six-story Moment-Resisting Steel Frame
Consider the 6-story moment-resisting steel frame as shown in Figure 3.20. Assuming the
members are axially rigid, this gives a total of 30 DOFs (i.e. n = 30) and 84 PHLs (i.e. q = 84).
No mass moment of inertia is assumed at the rotational joints, and therefore static condensation
is used to eliminate the 24 rotational DOFs (i.e. r = 24), resulting in only 6 translational DOFs
for the frame (i.e. d = 6). The stiffness matrices are then formulated based on the standard
properties of each member. The 30 × 30 K matrix, 30 × 84 K 0 matrix, and 84 × 84 K 00 matrix
are first constructed by assembling the individual stiffness matrices of each member. Then
static condensation using Eq. (3.135) is applied to obtain the 6 × 6 K matrix, 6×84 K 0 matrix,
and 84 × 84 K 00 matrix. As a numerical illustration, the resulting K matrix is of the form

2
4

3
5

464300 −252200 39180 −3715 345 : 8 −29 : 4

305 : 1 −252200 445700 −266500 38160 −3550

39180 −266500 451000 −252800 36180 −3106

K =

kN = m ð3 : 156Þ

−3715 38160 −252800 384500 −190900 24440

345 : 8

−3550 36180 −190900 282900 −125000

−29 : 4 305 : 1

−3106 24440 −125000 103300