Risk Analysis and Management: Earthquake - An example of Fragility and Load
The fully formated problem is with the solution.

The fragility of a building as function of earthquake magnitude is given as follows:
? if L  Lf Pf (L) = 1 - exp{ -C(L-Lf)2}
? if L < Lf Pf (L) = 0
where L is the magnitude of the earthquake. This fragility curve has two parameters, Lf and C. Lf is the threshold. Any earthquake with magnitude less than Lf will not damage the building. C represents the resistance of the building against earthquake. The larger value of C, the more likely the building will collapse during earthquake. Figure 1 shows Pf (L) as function of Lf and C.

Figure 1: Fragility as function of Lf and C.

According to expert opinion, the probability of earthquake in the region of interested in the next 30 years as function of earthquake magnitude can be approximated as follows:
? if L  LL p' (L) = 0
? if L < LL p' (L) = L(L-LL)2

So the load function of the earthquake is characterized by two parameters, LL and L. The probability of earthquake with magnitude greater than LL is 0. L represents the frequency of earthquakes, the greater L the more frequent the earthquakes. The values of LL and L depend on location of the building

The purpose of this exercise is to calculate the probability of damaging the building by earthquake in the next 30 years for different fragility and load curves. This probability is given by the following equation