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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
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