Civil Engineering Reference
In-Depth Information
for all structures; where cross-correlations are low or non- existent, the cross -coupling terms will be
small or zero. The CQC is expressed as follows:
N
N
∑
=
∑
E
=
E E
i
ρ
(4.25.1)
j
ij
i
1
j
=
1
where
ρ
ij
is a cross- modal coeffi cient. This coeffi cient is generally expressed as a function of the modal
frequencies and damping characteristics and, for equal modal damping, i.e.
ξ
i
=
ξ
j
=
ξ
, is as follows
(Der Kiureghian, 1980 ):
2
(
)
3
2
81
ξ
+
rr
ρ
=
(4.25.2)
ij
2
(
)
+
2
2
2
(
)
1
−
r
4
ξ
r
1
+
r
where
r
=
ω
j
/
ω
i
; the coeffi cient
ρ
ij
varies between 0 and 1 for
i = j
. If the modal frequencies of the
MDOF are well separated, the off-diagonal terms tend to zero and the CQC method approaches the
SRSS.
Estimates of the total value of the response parameter
E
obtained by CQC rule may be larger or
smaller than the estimates provided by the SRSS rule (Chopra, 2002). Figure 4.33 shows the bending
moment diagrams computed from response spectral analysis for a plane frame extracted from the sample
SPEAR building in Figure 4.2. The modal combination rules discussed above, i.e. SRSS and CQC, are
utilized. The damping value
ξ
used for analyses is 5%. SRSS and CQC provide values that are in good
agreement. The modal analysis of the 3D frame shows that the frequencies of the system are not closely
spaced; the minimum difference between two frequencies is greater than 10%. However, the sample
SPEAR structure is a multi-storey building with asymmetric plan and hence the SRSS leads to reason-
able estimates of response. The differences between the values computed through the CQC and SRSS
are lower than 10%.
4.6.1.2 Response History Analysis
In contrast to the frequency-domain solutions presented in Section 4.6.1.1 (notwithstanding the special
nature of modal analysis), the response of MDOF systems to a transient signal may be calculated by
80.50
105.00
78.28
87.80
114.72
86.32
131.10
129.41
143.87
142.97
175.43
192.54
140.73
160.35
142.20
157.05
178.55
158.75
Figure 4.33
Bending moments (in kNm) computed through response spectral analyses using two different modal
combinations for three-storey frame: square root of the sum of the squares (
left
) and complete quadratic combination
(
right
)
