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record, the intensity of the ground motion is increased step
by step until it reaches the threshold intensity IMcritical at
which the structure is collapsed by the ground motion. This
IMcritical is defined as the collapse resistant capacity (denoted
as CRC) of that structure subjected to that ground motion.
The value of CRC for a given structure signifies the maxi-
mum intensity level IMcritical of ground motion that the
structure is able to resist. CRC has the same unit as the cor-
responding IM. Because of the record-to-record uncertainty
of different ground motions, CRC is also a random variable.
Hence, the collapse fragility curve also represents the cu-
mulative distribution function of CRC.
From the above discussion, the collapse fragility curve
can be understood from the perspective of the conditional
probability of collapse at a given intensity level (Figure
1(a)) and it can also be understood from the perspective of
the cumulative probability distribution of CRC (Figure 1(b)).
For example, point A(IM*, P(collapse|IM*)) on the curve in
Figure 1(a) represents the conditional collapse probability at
a given intensity equal to IM*, while the same point
A(CRC*, P(CRC<CRC*)) in Figure 1(b) represents the
probability that the CRC of the structure is not greater than
CRC*.
3 Methodology for assessment of the collapse
risk
A structure’s collapse resistant capacity can be expressed in
the form of the conditional collapse probability or the
probability distribution of CRC by conducting a collapse
fragility analysis. Earthquake engineering addresses how to
assess and control the risk of structural collapse, which re-
quires consideration of both the ground motion demand and
the collapse resistant capacity. The ground motion demand
is determined via probabilistic seismic hazard analysis [13]
in terms of the probability, denoted by P(IM), that a given
building site experiences an earthquake of a given intensity
during a given period of time. By integrating the ground
motion demand with the collapse resistant capacity, the
structural collapse risk is expressed by the total probability
of earthquake-induced collapse in a period of Y years [12]
and is calculated as follows: where P(collapse in Y years) is the total probability of col-
lapse during Y years, that is, the risk of earthquake-induced
collapse; P(collapse|IM) is the conditional collapse proba-
bility of the structure subjected to earthquakes of a given
intensity level, obtained through collapse fragility analysis;
and P(IM) is the probability density that the structure site is
hit by earthquakes of a given intensity level during Y years,
obtained through a seismic hazard analysis.
Considering that structural collapse occurs when the
ground motion demand exceeds the structural collapse re-
sistant capacity, the calculation of the risk of earth-
quake-induced collapse in eq. (1) can be expressed in an-
other way, namely where P(CRC) denotes the probability density function of
CRC, obtained by differentiating the cumulative probability
function of CRC (note that CRC and IM share the same
physical meaning, as explained in the previous section) and
P(IM>CRC) represents the probability that earthquakes with
intensity levels higher than CRC hit the building site (i.e.,
exceedance probability corresponding to the intensity level
of CRC), obtained by implementing integration operation to
P(IM).
Eqs. (1) and (2) are equivalent in terms of the calculation
of the collapse risk, as shown in the appendix.
4 Example
4.1 Structural layout
The reinforced concrete frame structure shown in Figure 2