Background Paper on the Conversion of Damage Probability Matrices into Fragility Curves

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1 Robert Reitherman December 12, 1986 Written for: Panel on Earthquake Loss Estimation National Research Council Background Paper on the Conversion of Damage Probability Matrices into Fragility Curves INTRODUCTION This background paper describes a method for converting a damage probability matrix into a set of fragility curves. Damage probability matrices are described in the work of Whitman (Whitman et al., 1973), and the application of fragility curves to the subject of urban scale earthquake loss estimation is discussed by Kircher and McCann (Kircher and McCann, 1984). The Applied Technology Study, ( ATC-13 ), Earthquake Damage Evaluation Data for California, (Rojahn et al., 1985) is the source of the damage probability matrix converted here. It uses the Whitman scale mentioned above. The Jack Benjamin and Associates study, ( the JBA study ) Development of Fragility Curves for Sixteen Types of Structures Common to Cities of the Mississippi Valley Region, (Kircher and McCann, 1984), is the source of the example fragility curve, and it uses a different damage scale as noted in Table 1. STEPS IN THE CONVERSION PROCESS Step 1. Relate the damage scales used in the ATC-13 and JBA studies. Table 1 shows the two damage scales and gives their original sources. Various interpretations are possible, but the main difference of opinion on how to convert these two scales is probably with regard to ATC-13 damage levels 4, 5, 6, and 7. One could argue that ATC-13 levels 4 and 5 combined ( significant localized damage of many components warranting repair and extensive damage requiring major repairs ) are equivalent to JBA damage state 3 ( general yielding large deep cracks in walls load carrying capacity of the structures is partially reduced.) ATC-13 damage level 6 ( major widespread damage that may result in the facility being razed, demolished, or repaired ) would then correspond to JBA state 4 ( ultimate yielding of some main elements approximately 50% of the main structural elements fail and partially collapse). ATC-13 damage level 7 ( total destruction of the majority of the facility and a 100% damage factor) corresponds to JBA state 5 ( a large part of whole of the building collapses clearing the site and reconstruction. )

2 One could also argue that ATC-13 damage levels 6 and 7 should be combined and made equivalent to the JBA state 5. ATC-13 level 6 may be heavy enough damage to be equivalent to JBA 5 s a large part of whole of the building collapses. In this case, ATC-13 level 5 corresponds to JBA state 4 and ATC-13 level 4 corresponds to JBA state 3. The lower levels correspond as before. This latter alternative was selected, partly because it is the preferred alignment of Charles Kircher, who was one of the developers of the JBA construction class used in this example (Kircher, 1986). This correspondence of the two scales is the one shown in Table 1. Kircher s suggested ATC-13/JBA construction class correspondence is also used later in this analysis, with the aim of greater consistency. The goal is to match up what the ATC-13 analysts had in their minds when using the seven damage levels as compared with what was in the minds of the JBA analysts in using levels or states. The number of ATC-13 analysts involved makes a retrieval of their interpretations of the seven-level scale impractical, while Kircher was one of two individuals responsible for the use of the six level scale and so his version of the meaning of each of the six levels, in terms of the Whitman seven-level scale used in the ATC-13 study, has been used here. This issue may seem trivial, but it makes a significant difference in the plotting of the ATC-13 fragility curves for the highest two damage states. This issue is very commonly faced in the field of earthquake damage estimation, because original damage data and studies projecting earthquake damage have used a variety of scales. There is also the perennial problem of the lack of definition in the Modified Mercalli Intensity Scale, or another intensity scale, which typically comprises the abscissa on a damage estimation graph. Different estimators may have different levels of motion in mind when they rate damage vis-à-vis one of this Roman numerals. The JBA study used peak ground acceleration as the ground motion parameter, then converted this to MMI, to produce the MMI abscissa shown in the fragility graphs here. Step 2. Compare equivalent ATC-13 and JBA construction classes The JBA construction class all bearing wall buildings is most equivalent to the ATC-13 class 75, unreinforced masonry buildings with bearing walls only and without frames, of one to three stories in height. This is because in the study area (Memphis, Tennessee) for which the JBA classes were calibrated, most of the low-rise non-wood frame bearing wall buildings were of unreinforced masonry, and those that were not were still relatively non-resistant to earthquakes. The masonry buildings were on average lower in earthquake resistance than typical California unreinforced masonry construction, while the non-masonry bearing wall buildings were somewhat higher in earthquake resistance, so that the overall class of all bearing wall buildings is comparable to the ATC-13 unreinforced masonry (Kircher, 1986).

3 Step 3. Sum the cumulative probability of reaching or exceeding each damage state for a given Modified Mercalli Intensity Table 2 presents the ATC-13 damage probability matrix for low rise unreinforced masonry bearing wall buildings. These data are manipulated as follows. For a given modified Mercalli Intensity, sum up the percentages that correspond to JBA damage state 1 or greater, then 2 or greater, and so on, or in other words, each column is cumulatively added. The sum of the values in any MMI column is 100%. This results in Table 3. Step 4. Plot the results of Table 3 on the JBA fragility graph This results in Figure 1. Each curve shows the probability of reaching or exceeding a particular damage state. The very steep curves for the low damage states approximate a step function, implying that the analysts thought that there was virtually no chance of experiencing at least minor damage at less than a certain intensity, such as VI, while after some slightly higher intensity, such as VII, there was a very high chance that at least minor damage would be experienced. It is also true that because of the fact that the origin is at the left hand side of the graph, the curves that indicate high probabilities of reaching or exceeding a damage state at a low intensity must steeply rise, even if one thought that there was a smoothly declining probability of damage at very low intensities. However, since very little of engineering significance occurs at intensities less than MMI VI or 0.1 PGA, this is not a significant problem in this case One can also interpret the probabilities in a damage probability matrix, or the probability ordinate on a fragility graph, as indicating the percentage of buildings in a population that would experience damage at least as severe as the indicated damage state. This would be the way fragility curves could be applied to large numbers of buildings in the loss estimation process, and also allows comparison of such curves with historical loss data that indicate the percentages of a collection of buildings that were damaged to varying degrees. COMPARISON OF JBA AND ATC-13 FRAGILITY CURVES Figure 2 again plots the JBA and ATC-13 fragility curves, with a separate graph used for each damage state simply for clarity. For the lower damage levels, the two sets of curves are very similar. For damage state 4 & 5, the median values are similar, but the curves diverge significantly above and below this common meeting ground at the median. The steeper slopes of the ATC-13 curves for damage level 4 and 5 imply less variability than do the flatter slopes of the corresponding JBA curves. The ATC-13 expert opinion, a collective opinion produced by reiteration of the process of having

4 individual experts fill out questionnaires after comparing their earlier estimates with the other experts, in effect makes the statement that the higher damage states for unreinforced masonry are easier to predict, as compared with the flatter slopes of the JBA curves that state that there is more variability. This is expectable, questions of validity aside, since the ATC-13 collective opinion approach was aimed at reducing the spread in the individual experts opinions. A secondary reason is that the construction class dealt with in the JBA study was more broadly defined. Figure 3 shows two hypothetical cases illustrating this point. If, for a given damage state such as collapse and for a given construction class, one thought that the distribution of buildings across intensities would graph as a wide or flatly sloped bell shaped curve, this would mean one thought there was large variability. This is shown in Figure 3-A, which shows the bell shaped curve arbitrarily symmetrical to illustrate a generic case. This graph states that one can t rule out the collapse of a few buildings at about intensity VIII, and one also thinks that a few would be uncollapsed at intensity XII. There is a wide range in between where most of the buildings will be found, and the cumulative total of buildings with collapse gradually increases with increasing ground motion. When translated into a fragility curve, this would state that the probability of reaching this state would only gradually increase, and thus there would be a more flatly sloped fragility curve as compared with the next example. If, on the other hand, one thought that the distribution of buildings in this class, for the collapse state of damage, would graph as a steep bell shaped curve of narrow range, then this would translate into a fragility curve of steeper slope. This would be the case of Figure 3-B. As one moves from left to right across the curve, there are very few buildings found until, at a narrow range of intensity, almost all the buildings are concentrated. This example states the variability is low, and the fragility curve will graph as a curve that suddenly, steeply, rises where it hits the narrow intensity range where the cumulative total of buildings quickly increases. The question of the variability in the performance of buildings is one of the recurring issues in the field of earthquake damage estimation. One problem concerns how to determine how much variability there is, which is a technical matter. The other concerns the issue of communicating to the people who will use an earthquake loss estimation study the amount of variability inherent in the study s loss estimates.

5 REFERENCES Kircher, Charles and Martin McCann, 1984, Appendix A: Development of Seismic Fragility Curves for Sixteen Types of Structures Common to Cities of the MISSISSIPPI Valley Region, Jack Benjamin and Associates, Inc.; unpublished paper written to be part of a FEMA published study by Allen and Hoshall, Inc., 1985, An Assessment of Damage and Casualties for Six Cities in the Central United States Resulting From Earthquakes in the New Madrid Seismic Zone. Kircher, Charles, 1986, personal communication. Rojahn, Chris, et al., 1985, Earthquake Damage Evaluation Data for California, Applied Technology Council, a study prepared by FEMA. Whitman, R. V., et al., 1973, Earthquake Damage Probability Matrices, Proceedings of the Fifth World Conference on Earthquake Engineering, International Association for Earthquake Engineering, Rome, Italy.

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