Figure 2.4 Uncertainties associated with the evaluation of Performance Point (PP).
The probability of damage to a particular typology of building will be the same under a given intensity of earthquake. This is the basic concept of constructing a DPM.
The conditional probabilities can be defined as in Eq. (2.6), where,
p
ikis the probability of reaching damage state di for ground motion intensity yk, D is the damage random variable defined on damage state vector{ } D
={ d d
0, ,...,
1d
n}
and Y is the ground motion random variable.[ | ]
ik i k
p =P D=d Y =y (2.6)
If the probability of complete damage is represented as P[C] = P[C|Y], the discrete damage probabilities of different damage states – none (N), slight (S), moderate (M), extreme (E) and collapse (C) can be expressed as: P [E] = P [C|Y] – P [E|Y], P [M] = P [E|Y] – P [M|Y], P [S] = P [M|Y] – P [S|Y], P [N] = 1 – P [C|Y], respectively. DPM can be obtained either empirically by using past earthquake data and experiences, or analytically by performing nonlinear analysis for the structure, or by judgmental and hybrid methods. DPM through empirical or expert judgement can be regarded as a direct method of vulnerability assessment where damage to building typology is directly related to ground motion intensity. Selection of any one of the methods depend upon the damage data used for their generation.
2.3.1 Damage Probability Matrix from Expert Opinion
Damage probabilities based on expert opinion was first introduced in ATC 13 (1985) that essentially derived DPM for 78 structures, 40 of which refer to buildings, by asking 58 experts (noted structural engineers, builders, etc.) to estimate the expected percentage of damage that would result to a specific structural type subjected to a given intensity. Damage was described in terms of MMI scale considering the range VI to XII.
The assessment of seismic risk and loss using ATC 13 approach was used by several
Spectral acceleration
Spectral displacement
Structural capacity curve
Seismic demand curve PP
2.3 Damage Probability Matrix (DPM)
researchers, for example, for the city of Bogotá, Colombia (Cardona and Yamin 1997) to estimate and establish damage motion relationships. The primary drawback of this method is its subjectivity, as these are based exclusively on the subjective opinion of the experts. The DPMs based on expert opinions are also difficult to calibrate or modify in order to incorporate new data or technologies. Further, it is difficult to extend ATC 13 methodology to other building types and other regions, as well as to individual building characteristics. Nevertheless, it was the first relatively thorough study on earthquake damage and loss estimation and became the standard reference for many earthquake vulnerability assessments until the mid-1990s.
2.3.2 Damage Probability Matrix from Empirical Studies
Whitman et al. (1973) was the first to have systematically compiled statistics on both structural and non-structural damage to various building typologies from damage experiences after 1971 San Fernando earthquake. This has been widely accepted and adopted by people to define the probable distribution of damage. Each value in the matrix (as indicated by “…” in Table 2.6) expresses the probability that a certain building class will experience a particular level of damage as a result of particular earthquake intensity.
The degree of damage is expressed in probabilistic terms. As a probability rule, the sum of each column must sum up to unity or 100%. The damage ratio is defined as the repair cost as a ratio of the replacement cost at the time of the earthquake. Apart from the damage data collected from the San Fernando earthquake, a questionnaire was developed to collect information on less damaged buildings. In constructing DPM each building is assigned a damage state based on damage cost and replacement cost (= building area × average building cost per unit area). Nine damage states were considered starting from 0 to 8. When the DPM is developed using observed damage of past earthquakes and applied to regions with similar characteristics to predict damage due to probable future earthquakes, a realistic indication of the expected damage is obtained and many uncertainties are inherently accounted for.
Major limitation of Whitman’s DPM lies in the unreliability of the building data collected from surveys. The building data list may contain, say for a particular typology, buildings that do not belong to it, or it may have omitted or disregarded buildings, which should have been kept in the category due to restricted specifications about it. Further,
the actual damage costs are rarely available. If the data obtained for particular ground intensity is for a lesser number of buildings, the DPM constructed is meaningless.
Table 2.6 General format for constructing Damage Probability Matrix (Whitman et al. 1973).
Damage State
Structural damage
Non-structural damage
Damage ratio (%)
Intensity of earthquake V VI VII VIII IX
0 None None 0 - 0.05 … … … … …
1 None Minor 0.05 - 0.3 … … … … …
2 None Localized 0.3 - 1.25 … … … … …
3 Not noticeable Widespread 1.25 - 3.5 … … … … …
4 Minor Substantial 3.5 - 4.5 … … … … …
5 Substantial Extensive 7.5 - 20 … … … … …
6 Major Nearly total 20 - 65 … … … … …
7 Building condemned 100 … … … … …
8 Collapse 100 … … … … …
Eleftheriadou and Karabinis (2011) aimed at developing a methodology and obtaining the DPM and vulnerability curves for the empirical seismic vulnerability assessment of typical structural types, representative of the materials, seismic codes and construction techniques used in Greece and generally in Southern Europe. The work considered seismic demand in two forms-
i. The regional macro-seismic intensity (I),
ii. The ratio (ag/ao) of the maximum PGA of a certain earthquake event to the PGA that characterizes each municipality in the Greek hazard map. Intensities and PGAs are correlated using Eq. (2.7).
ln(PGA) = 0.74I + 0.03 (2.7) Five damage state (ds) were considered (Table 2.7) and were given color codes.
Central Damage Factor was calculated for each of the damage state. Frequency of different damage states for each structural type exposed to a specific seismic demand is obtained. Next, each element of DPM is obtained simply by cumulating the frequencies from the highest to the lowest level of damage. One matrix is constructed for each structural type expressing the distribution of damage in each intensity level. DPM were