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= a(lM) b
(6)
A typical result of an EDP hazard curve is shown in Figure 5.
' * Incremental Dynamic Analysis
AVERAGE DRIFT HAZARD CURVE-T^l.8 sec.
N=9, ysfl 10, §=0 05, Peak oriented model, 6*0 060, BH. K,, Si, LMSR
1 •
J# ft fl1 •
V
\ | — Hazard Curve Obtained from Numerical Integration
1 N xJ _i_ '
f
EDP (e.g., max. interstory dnft)
Figure 4. Incremental Analyses and Their Use in
Probabilistic Seismic Demand Analysis
A nnrvi •
0 0.005 0.01 0.015 0.02 O.I
Average of Maximum Story Drifts, 9
Figure 5. Example of a Hazard Curve for
an Engineering Demand Parameter
A special problem arises when the limit state of collapse is of interest. Collapse is caused by
detenoration in strength, and for this reason response prediction has to resort to the use of deteriorating
systems. Models for such systems are available, see Fig 6 (Ibarra et al., 2002). The effect of
detenoration on the seismic response is illustrated in Fig. 7, which shows IDA curves for a 9-story
frame without and with deterioration. The implication of the IDA turning horizontal is that an
infinitesimal increase in ground motion intensity causes a very large increase in the maximum story
drift. This implies dynamic instability (global collapse), which cannot be evaluated by means of
nondeterioratmg systems.
Pinching Hysteretic Model, Hahl-Column 1,P-A=0,
a=Q.10,ac,p=-024,ic=fl.5,Yit=100,Y,=50,Y«=30,YB=40,5c=2.35y
MAX. STORY DUCTILITY vs. NORM. STRENGTH
N=», T,=0.9, §=O.OS, K,, S,, BH, 9=0.015, Peak-Oriented Model. LP89svl
s 10 is
Maximum Story Ductility Over the Height, fi^,.
Figure 6. Comparison Between Exp. Results and
Analytical Predictions for a Deteriorating System
Figure 7. IDA for a MDOF Structure Without
and With Deteriorating Elements
The use of deteriorating systems permits a direct evaluation of the probability of collapse in
accordance with Eq. (1), without the help of an intermittent DM, Collapse statistics can be performed
on the last point of the ID As, see Fig. 8, which provides statistical information in terms of a suitable
intensity measure. In Fig. 8, S a (Tj) is used as the IM (j is the base shear coefficient V/W).
Alternatively, collapse fragility curves of the type shown in Fig. 9 can be developed, which permits an
evaluation of the probability of collapse for different deterioration properties (Ibarra et al., 2002).