Assessing seismic risks for new and existing buildings using performance-based earthquake engineering (PBEE) methodology - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 311

1

0.9

0.8

Return period = 36 yrs
Return period = 72 yrs
Return period = 475 yrs
Return period = 975 yrs

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0

1 2 3

Repair cost, $C

4 5 6 7

10 6

×

12.4 Example of distribution of repair cost.

cumulative distribution function shown in Fig. 12.4, then multiplying it by
the slope of the hazard curve at the corresponding ground motion intensity
measure level, and fi nally integrating the resulting curves across the inten-
sity measure interval considered in the seismic hazard analysis. Repeating
this process for all repair cost values produces a loss curve that represents
the annual rate of the total repair cost ( TC ) exceeding a threshold value.
Figure 12.5 shows a sample loss curve obtained from the sample building
performance assessment shown in Fig. 12.4. Note that the procedure pre-
sented here is identical to the process presented in Eq. [12.4] where the
conditional probability, G ( TC | im ) (complement of the CDF generated
using the procedure presented in Step 1 through Step 5), is multiplied by
the derivative of the seismic hazard curve and integrated through all seismic
intensity measure levels.
Lastly, following the derivation presented in Der Kiureghian (2005), valid
for a non-negative random variable X , the expected cumulative value of
X is

∞ ∞

∑

∫

() =

∫

()

EX x x ⎣ ⎦ = d λ λ xx d

[12.14]

0 0

where the last expression is obtained from integration by parts. Thus, the
area under the loss curve represents the mean cumulative annual total
repair cost for all earthquake events occurring in one year. Information