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(probability of being in or exceeding a specific damage state, given a value of EDP). The DMs include,
for example, descriptions of necessary repairs to structural or nonstructural components. If the
fragility functions for all relevant damage states of all relevant components are known, the DVs of
interest can be evaluated either directly or by means of cost functions that relate the damage states to
repair/replacement costs. The result of this last operation is G(DV\DM), the conditional probability
that DV exceeds a specified value, given a particular value of DM.
These steps, which form the basis of performance assessment, can be expressed in the following
equation for a desired realization of the DV, such as the MAP of the DV, A(D V), m accordance with the
total probability theorem:
This equation, which often is referred to as the framework equation for performance assessment,
suggests a generic structure for coordinating, combining and assessing the many considerations
implicit in performance-based seismic assessment. Inspection of Eq. (1) reveals that it "de-constructs"
the assessment problem into the four basic elements of hazard analysis, demand prediction, modeling
of damage states, and failure or loss estimation, by introduction of the three "intermediate variables",
DM, EDP, and IM. Then it re-couples the elements via integration over all levels of the selected
intermediate variables. This integration implies that in principle one must assess the conditional
probabilities G(EDM\IM), G(DM\EDP) and G(DV\DM) parametrically over a suitable range of DM,
EDP, and IM levels.
In the form written, the assumption is that appropriate intermittent variables (DMs and EDPs) are
chosen such the conditioning information need not be "carried forward" (e.g., given EDP, the DMs
(and DVs) are conditionally independent of /M, otherwise IM should appear after the EDP in the first
factor.) So, for example, the EDPs should be selected so that the DMs (and DVs) do not also vary with
intensity, once the EDP is specified. Similarly one should chose the intensity measures (IM) so that,
once it is given, the dynamic response (EDP) is not also further influenced by, say, magnitude or
distance (which have already been integrated into the determination of A(7MJ). This condition can
make selection of records a challenge.
Equation 1 may take on various forms, depending on the purpose and the decision variable of interest.
For instance, Miranda and Aslani (2002) use the following form to compute the expected annual loss
for component;, £"[£,], if the damage to the component can be expressed by m damage states:
] = £ J £[L y | DM = dm,] P(DM = dm, \ EDP J = edp) P(EDP J > edp \ IM = im) dv(1M \E DPd m
dIM
(D
The expected loss for the building is then computed as the sum of the expected damages for the
individual components. Clearly, expected losses do not provide a comprehensive probabilistic
performance assessment, and the assumption that component losses can be computed independently
and then summed over all components is a simplification, but Miranda's approach is the first
comprehensive implementation of the framework equation and constitutes a significant step forward in
performance assessment.