Seismic risk models for aging and deteriorating buildings and civil infrastructure - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 387

v 0

Y ∼ G ( x )

( n ) ( x )

D s ( t ) ∼ G

d G (x)

V(t)

k*

Failure region

X 1 X 2 T n

t Time

15.2 Description of the probability of failure for systems abandoned
after fi rst intervention.

(Fig. 15.2). Note that failure or serviceability assessments can be computed
by exchanging s* and k* in equation 15.7 (see Fig. 15.1).
The distribution G in equation 15.7 describes the probability of having a
certain damage level as a result of a shock (e.g., loss of structural capacity
after an earthquake). This distribution is obtained after studying the dynamic
interaction between the earthquake demand and the structural response.
Commonly G is derived from the so-called fragility curves, which describe
the probability that the system reaches a certain damage level in terms of
a demand parameter. Several approaches to compute these curves have
been proposed in the literature but their discussion is beyond the scope of
this chapter (Hazus, 1997). Finally, it is important to stress that although the
assumption of iid shocks might seem strong, it relaxes assumptions made
by other models in which the structural damage, as a result of successive
shocks, does not accumulate with time (Rackwitz, 2000).

15.3.4 Systems abandoned after failure
Instantaneous failure rate

This section considers the case of a structure exposed to successive shocks
until the remaining life falls below a prescribed threshold value, i.e., s* .
Once this happens, the structure is abandoned; this is, the system is not
reconstructed or intervened in any way. The instantaneous failure probabil-