Vulner ability functions

When conducting comprehensive and fully probabilistic risk assessments, physical vulnerability is one of the aspects that must be represented by means of functions that relate in a continuous way the hazard intensity levels with the expected damage or what is better known as mean damage ratio (MDR). Vulnerability functions describe the loss probability moments variation as a function of the seismic demand. The loss L is defined as a random variable and then, the variation of its probability moments for different seismic demand levels are described by means of vulnerability functions. The loss probability distribution pL|S(L) is assumed as a Beta function (ATC, 1985) where then, the first two probability moments correspond to the mean (MDR) and its standard deviation

1 1 |S ( ) ( ) (1 ) ( ) ( ) a b L a b p L L L a b (4.4)

where Γ is the Gamma function and it’s a and b parameters are

2 2 1 (1 ( | )) ( | ) ( | ) c L S E L S a c L S (4.5) 1 ( | ) ( | ) E L S b a E L S (4.6)

E(L|S) is the expected loss value and c(L|S) is the loss’ coefficient of variation given a seismic demand S obtained by dividing the mean value by the standard deviation can be written as follows:

2_{( | ( ))} ( | ) ( | ( )) L s s L Sd T C L S E L Sd T (4.7)

where σL2(L|Sd(Ts)) is the variance of the loss at any spectral displacement, a value that is calculated adopting the damage probability distribution from ATC-13 (1985)

2_{( | ( )) ( ( | ( ))) (1}r1 _{( | ( )))}s1

L L Sd Ts Q E L Sd Ts E L Sd Ts (4.8)

where Q and s can be calculated as follows:

max 1_{(1 )} 1 r s M M V Q L L (4.9) 1 2 M r s r L (4.10)

Vmax is the maximum loss variance between 0 and 1, LM is the loss where the maximum variance occurs and r is a shape factor. With this, once the expected loss value and its variance are established, it is possible to estimate the probability distribution given any spectral acceleration.

Vulnerability functions, opposite to other qualitative damage scales, have all the necessary information to calculate the probability of reaching or exceeding a loss amount, given a specific seismic demand, by means of the following equation:

|

Pr( | ) _{L S}( )

l

L l S p L dL (4.11)

where l is a loss within the domain of the random variable L and S is again the sesimic demand. The damage is quantified through the MDR that is obtained as the ratio between the estimated repairing cost and the total exposed value of each element. The vulnerability function is then defined by relating the MDR and the acceleration that can be associated either to PGA for low-rise buildings or to the pseudo-spectral accelerations for medium and high-rise dwellings. For each building class, once a seismic acceleration level is known, the MDR can be obtained using the approach proposed by Ordaz et al. (1998), Miranda (1999) and Ordaz (2000):

0

( | ) 1 exp ln 0.5 i

i

E L (4.12)

where L is the loss, γ0 y γi are structural vulnerability parameters that depend on the building class and construction date, ε is the slope and E(·) is the expected value. By definition, L is the MDR, and since only direct physical losses are being assessed, takes values between 0 and 1.

Equation 4.12 can be rewritten as following to obtain the expected value of the loss as a function of the spectral displacement and account directly for structural parameters such as the fundamental period associated to each building class as well as, for example, the spectral displacement at the yielding point:

0 ( ) ( | ( ) 1 exp ln(1 ) s s y Sd T E L Sd T L Sd (4.13)

where now Sd(Ts) is the spectral displacement, Ts is the fundamental period of the associated building class, L0 corresponds to the expected loss associated an spectral displacement, Sdy is the spectral displacement at the yielding point of the structure after assuming a bilinear capacity spectrum and ε is a factor used to fit the curve to the loss levels defined by the point of ultimate capacity.

Seismic intensity is quantified in terms of spectral acceleration for any structural period and, therefore, can be converted into spectral displacements using the following expression: 2 1 2 3 2 ( ) ( ) 4 s s Ts Sd T Sa T (4.14)

Where, besides the classical conversion between spectral displacements (Sd) and spectral accelerations (Sa) (García, 1998), the ratio of maximum lateral displacement at the top of the structure to the spectral displacement considering an elastic model, the ratio of the maximum inter-story drift to the global drift of the structure and the ratio of the maximum lateral displacement assuming an inelastic model to the maximum displacement of the elastic model are considered by α1, α2 and α3, respectively.

Besides obtaining the expected damage, its dispersion is also obtained for different seismic intensity levels. Said dispersion is equal to zero for the extreme values and reaches its maximum value when MDR is equal to 50%. A hypothetical vulnerability function is shown in Figure 4.6 where the continous line corresponds to the MDR whilst the dotted line corresponds to the dispersion. It is important to bear in mind that the two probability moments have the same importance in the definition of the vulnerability and that no probabilistic seismic risk assessment can be performed if any of them is missing.

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0 1,000 2,000 3,000 4,000 5,000 MD R

Spectr al acceler ation (cm/s2)

F igure 4.6. Schematic representation of a vulnerability function

A unique physical vulnerability function is needed for each identified building class and also, the difference in the seismic performance of the dwellings is considered through the fundamental period of them. That is, each vulnerability function has associated an spectral ordinate that correspond to the fundamental elastic period of each building class for which its damage is being assessed.

As it has been already explained, all vulnerability functions take values between 0 and 100% because only direct physical damage is being quantified by means of them. Within this framework, a very important premise is that one cannot lose more of what one has which, in practical terms, can be translated as that the maximum loss would equal the total exposed value.

The seismic intensity that has been selected to connect the hazard results and their corresponding damage levels is the pseudo-spectral acceleration (Sa). Sa is merely a tool to simplify the seismic risk analysis (Baker and Cornell, 2006) since it then allows using the hazard results to calculate expected losses in a direct manner. That means that the selected GMPEs explained in the PSHA section quantify seismic hazard in terms of Sa and that the vulnerability functions use the same hazard intensity. Even if Sa is usually referred in the literature as spectral acceleration, it is noteworthy to mention that, in reality, it corresponds to the pseudo-spectral acceleration, representing then the maximum acceleration that a given ground motion can cause on a single degree of freedom oscillator with a known period and damping level (Bozzo and Barbat, 2000).

Although Sa is interpreted and understood in many cases as a unique quantity, there are several definitions for it from the hazard and the vulnerability points of view

(Baker and Cornell, 2006) and it is desirable a full compatibility when using them from different sources (i.e. hazard and vulnerability).