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deformation to the crack deformation. The structural capacity can be usually modeled by a lognormal distribution.
According to Ref. [2], on the basis of the crack feature of wall in each hysteretic skeleton curve stage and compared
with the true earthquake damage degree of buildings, the median jI R and logarithmic standard deviation C R can be
obtained, as shown in table 3. The collapse capacities of brick wall with and without constructional columns are taken
as 90% and 85% ultimate strength, respectively. The capacity for the severe damage is related to the ultimate strength.
The capacity for the initial crack damage is related to the crack strength. The capacities for the slight damage and the
moderate damage are taken as the values at 1/5 and 3/5 points between the capacities for the initial crack damage and
the severe damage, respectively. The median and logarithmic standard deviation of story capacity of the brick
buildings are obtained by composing the mean and standard deviation of the brick walls with and without
constructional columns according to their cross section ratio.
6.2 Probabilistic Response of Structures
For each earthquake load-structure system, the nonlinear seismic response analysis is carried out. The
i-th story ductility ratio is:
IT
*-^ (5)
where, U max ,i is the maximum absolute inter-story deformation of the i-th story; U c ,i is the crack
deformation of the i-th story.
The statistical analysis is utilized to determine mean, standard deviation and distribution function of
sample. The maximum structural response can be modeled by a lognormal distribution [2].
6.3 Vulnerability Analysis
The structural vulnerability with respect to a particular limit state is defined as the conditional
probability that structural response E exceeds the structural capacity R. The i-th story limit state
probability of the buildings can be written as:
P fi =P n (R,