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.”)
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).
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
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.
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.