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Seismic risk in Friuli-Venezia giulia: an approach
Seismic risk in Friuli-Venezia Giulia: an approach Woo Sun Yang, Dario Slejko and Dino Viezzoli
Ossercatorio Geofisico Sperimentale, P.O. Box 2011, 34016 Trieste, Italy Fulvio Gasparo
Regione Autonoma Friuli-Venezia Giulia, 34100 Trieste, Italy Seismic risk is defined in this work as the expected damage ratio from possible earthquakes, during t years. The risk associated with the earthquake having a fixed occurrence probability in t years is formulated by means of the Damage Probability Matrix. Also, an expression for the total risk from all probable earthquakes is presented. To evaluate the risks mentioned, determination of the values of the Damage Probability Matrix is carried out statistically. Friuli-Venezia Giulia (NE Italy) is chosen as study region and the above risks are evaluated for t = 100 years. Computed risk refers only to the rooms built before 1976 as no information about new buildings is available. The general picture is in accordance with previous results for the same region. Key Words: seismic risk, Damage Probability Matrix, room category.
INTRODUCTION
RISK MODEL
To develop a satisfactory strategy against damage from expected earthquakes in a region, it is necessary to know not only the nature of earthquake occurrence in that region but also its response to earthquakes. The severity of t h e expected earthquakes can, for example, be described by estimating maximum expected accelerations in fixed time periods or return periods of earthquakes of a fixed magnitude t-8. The response of a region to earthquakes depends on various factors such as population density, building types, geological conditions, etc. Several attempts to correlate building damage, appropriately defined, with these factors have been made 9't~ and the term 'seismic risk' has been introduced. Ambraseys 11 suggests defining seismic risk as:
Whitman and Cornell t2 introduced the Damage Probability Matrix (DPM) elements, P(j]I), representing the probability that an earthquake of a given intensity I causes a damage statej to a class of buildings which are assumed to respond to earthquakes with a similar degree of damage. State j is one of a set of states classified by severity of earthquake damage. In this study, a room rather than a building is chosen as the elementary unit for damage evaluation because the size of a room is generally more uniform than that of a building. With this choice, a simple criterion such as the age of the room can be used to classify rooms according to resistance to earthquake damage. Suppose we have classified all rooms in a municipality into K categories according to their strength, and that we have assigned a mean replacement cost, that is simply the economic value of a room, to each category. Furthermore, we introduced a discrete set of damage states by subdividing the whole range of damage ratio values (i.e. from 0 to l) into DS states of fixed length. The central value of the damage ratio of each damage state will be called the central damage ratio. Let us assume t h a t there a r e N k rooms of the kth category in a certain municipality, having mean replacement cost RCk, and that Pk (JlI) is the probability that a room of the kth category would be damaged to statej by an earthquake of intensity I. Then the approximate repair cost (ReCk) for all kth category rooms in the municipality due to an earthquake of intensity I can be expressed as:
(Risk) = (Seismic hazard)* (Vulnerability)* (Value)
(1)
Seismic hazard is the probability of occurrence of ground motions due to an earthquake of a magnitude capable of causing significant loss of value through damage or destruction. Vulnerability may be defined as the degree of damage inflicted by a given earthquake to man-made structures and to the ground itself, while Value may be taken either in the sense of capital value or any other quantity of interest that will adjust the right side of equation (1) to the same dimension as Risk. Risk at a site incorporates both the earthquake occurrence characteristics at that place and its response to earthquakes. Equation (1) is only a qualitative expression of Risk. The main problem is to make the formula explicit and to perform a good quantitative evaluation of Risk as a function of the relevant parameters. Communicated by G. Bonn. Accepted November 1987. Discussion closes September 1989.
/)S
ReCk= ~, Nk'Pk(j]I)'RCk'CDR j ( k = l ..... K) (2) j=l
where CDRj is the central damage ratio of the damage
(~ Computational Mechanics Publications 1989
96
Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 2
Seismic risk in Friuli-Venezia Giulia: W-S. Yang et al. state j. The total cost for repairing (TReC) all the damaged rooms in the municipality would be: K
TReC=~. ReCk = k
DS
(8)
j=l
where ~ is the Euler-Mascheroni constant and equals 0.5772157... and m is generally not an integer. If this value m (or).I"t which is close to !11)satisfies
If we define Q(I) as: DS
~ Nk'PkUII)'RCk'CDR~ Q(l)=k=, j=l t: ~. Nk'RCk
then Q(I) is the mean ratio of repair cost to total replacement cost in the municipality hit by an earthquake of intensity I, i.e. the loss ratio. If Q(l,n) denotes the sum of the loss ratios from n earthquakes of intensity 1 and P1.t(n) denotes the probability for an earthquake of intensity I occurring n times in t years, the total loss ratio in a municipality in the whole period of t years is given by:
R=E Y, 0(I,11)'P1.,(10
(5)
n=l
Since all the values of the parameters in equation (5) vary as time goes on, the defined R cannot be the actual total damage ratio which would be received during t years but just a risk weighted with earthquake occurrence probability. This R would give a general picture of possible damage severity for the next t years for a municipality in a region. In this work, the quantity Risk of equation (l) is quantified by the quantity R of equation (5) and a simple relation is assumed for Q(l,n) in each municipality of the study region, i.e.:
Q(l,n)=nQ(I)
(6)
Though equation (6) is suggested mainly for the purpose of an approximate evaluation of R, it has physical meaning. Literally, equation (6) implies that a municipality is restored to its original state after each earthquake. This assumption is not valid for all the cases but the approximation holds in most actual situations. In fact, if Q(I) is small enough equation (6) can be a reasonable approximation since it is hardly expected that the strength of a room is significantly lowered or enhanced after repairing the damage of an earthquake which caused only a minute amount of loss. A nonlinear cumulative effect of weakness should be taken into account if no restoration of damage is performed. It is not known how small Q(I) should be put the corresponding Medvedev-Sponheuer-Karnik (MSK) intensity is expected to be V or VI or less. On the other hand, if O(1) is large, the maximum of the Poisson probability of occurrence will become small (,~ l for earthquakes with I> VII for t = 100in the study area) since the exponential decrease of the annual occurrence rate of earthquakes of intensity 1,21, with respect to intensity is expected from the Gutenberg-Richter's relation ~a. Following the Poisson distribution for Pta(n) 1'~, the maximum of e - a,., (21" t)" Pl.,(n) = i1!
1 >~m~21"t
(4)
k=l
1
l)
k+l,;
ln(2t't)+?'=k~ ,
~, Nk'Pk(jII)'RCk'CDRj (3) k=l
K
lies at !11 which satisfies
(7)
(9)
and the Poisson probability decreases rapidly with increasing !1, the terms Q(I,n) with ii greater than 1 have only a small influence on the final value of R in equation (5) because of the small Pi,n) associated. In this case we can use equation (6) disregarding whether it is a valid approximation or not. It should be noted that the validity of equation (9) also depends on the time period chosen. For regions with moderate or low seismic hazard the condition of equation (9) and the rapid decrease of Pl,n) turn out to be suitable for high intensities with t--100 years. But for regions with high seismic hazard these are not true. However, for most municipalities in the study region these conditions are generally satisfied. This implies that R expressed by: co
e-a,., (21.t)~
n= 1
!1!
R=~Q(I) ~ n I
-t~Q(l)').l
(10)
l
where Q(I) is the ratio of loss expressed in equation (4), is potentially the expected total loss ratio from all possible earthquakes during the next t years. Actually, R would not give the expected ratio because not all the values of the parameters entering in equation (4) would remain the same after each earthquake. However, in this study R will be called the total risk in the next t years. Since 21 is monotonically decreasing and Q(I) is monotonically increasing with respect to I, the product Q(I)').I has a maximum in the intermediate intensity range. Then, R in equation (10) can be approximated by: 10
R=t ~ Q(I)').I
(11)
1=6
since earthquakes of intensities lower than VI do not produce damage and
l >>211> Q(ll). 211.~Q(12). :q2
(12)
This fact is checked by using the annual occurrence rates of earthquakes 6 and Q(I) in a municipality of-Friuli affected by high seismic hazard (Fig. 1). In conclusion the total risk in t years can be expressed as:
K
DS
,o E Nk'RCk'E PkUII)'CDR i R=t ~, 21k=l j=l K 1=6 ~ Nk.RCK
(13)
k=!
ASSESSMENT OF THE PARAMETERS O F THE MODEL The Damage Probability Matrix approach has been applied to the rooms classified according to age in the study region of Friuli-Venezia Giulia, located in the
Soil Dynamics aml Earthquake Engineering, 1989, Vol.8, No. 2 97
Seismic risk in Friuli-Venezia Giulia: W-S. Yanget al.
lO0~.=O(t)
least square method t7 to the following equation:
1.0
number of annual occurrences
3 total economic value of all ~. N.," R C k --k= t rooms in the ith municipality
(14)
09 08
where N~ is the number of rooms of the kth category in the ith municipality. The number of rooms of each category in each municipality is taken from the 1971 census ~s and the value of the right-hand side of equation (14) for each municipality is taken from the quoted report ~6. To obtain reasonable values for RCk, the following obvious condition is required
3.x ? ! I
07
!
I
06
Oltl 0.12
(15)
R C 1 <~ R C 2 <~ R C a
Q5
0.10
i 0.4.1
0.08
0.3
0.136
02, 100~.tO[t}
] 0.02004
0.1
V
Vl
VII
Vlll
IX
X
Xl
( MSK intensity)
Fig. 1. The maxhnum of t'2z'Q(I) with t=lO0 in the municipality Forgaria nel Friuli is placed between M S K intensity VII and VIII; the large value of 100.2 I. Q(I) at X which does not seem to decrease dramatically toward 0 at XI possibly results from the large extrapolate it, which will be discussed later
northeastern part of Italy. There are other parameters by which resistance'of rooms to earthquake damage can be better described but the proposed criterion has been chosen because of its easiness and readiness for classification. Three room categories have been adopted; namely, rooms in buildings constructed (1) before 1945, (2) between 1946 and 1960, and (3) between 1961 and 1975. Since information about rooms built after 1975, necessary for evaluating Q(I) and R, is not available, only rooms constructed before 1976 are used in all the following calculations. The model calculations have been performed using the intensity data of the 1976 earthquake (MSK intensities between VII and X ts). To determine the values of parameters entering in equation (4), a damage report on 79 relatively heavily damaged municipalities was used 16. Based on this data, we have determined the mean replacement cost RCk as well as the Damage Probability Matrix elements Pk(j[I) of equation (13).
Assessment of the replacement costs The average replacement cost R C k , of the kth category of rooms in the test region is obtained by applying the
98
The obtained average replacement costs of the three categories of rooms in the test region are listed in Table 1.
Determination of the DPM elements As a next step we must determine the average damage ratio applicable to rooms of the kth category (k = 1, 2, 3) with reference to the 1976 earthquake. The average damage ratio ADRta is defined as the ratio of the total economic loss inflicted on the rooms of the kth category to the sum of their economic values in the ith municipality, which is one of those to which MSK intensity I was assigned in 1976 ts. Generally, for the same I and k ADR's will be different between one municipality and another, and we can obtain a distribution of the ADR's over the whole interval [0, 1] for each I and k. From the distribution we can infer the probability for a room of the kth category receiving a certain degree of damage under a certain MSK intensity. The values of the average damage ratios of the three categories of rooms caused by MSK intensity from VII to X were computed for each municipality by:
3 k=l
A D R t # N ~ = e q u i v a l e n t number of destroyed rooms
ratio of lost number number value to total = of rooms + of rooms • value for the collapsed damaged damaged rooms
(16)
Where ADR, a is the average damage ratio of the kth category rooms by MSK intensity I and N~ is the number of that category rooms in the ith municipality. As for equation (14), the number of rooms of each category in each municipality is taken from the 1971 census I 8 and the value of the right side of equation (16) is taken from the damage report 16. The optimum combination of ADR's is looked for, for each municipality with the least difference between the right- and left-hand sides of equation (16) as ADR's are increased by 0.01 with the restriction 0 ~
(17)
Table 1. The replacement cost o f a room o f each room category in Friuli-Venezia Giulia. The unit Italian lira used is referred to its economic ralue in 1976 First category Second category Third category
Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 2
1.0.10 6 lira 2.5" 106 lira 11.5.10 6 lira
Seismic risk in FriulizVenezia Giulia: W-S. Yang et al.
6) CDR
ptob
0.05 O.15 0.25 0.35 0.45 0.55 0-65 0.75 0.85 0.95
O.138 0.224 0.258 0.205 0,1"15 0.045 0.013 0,002 0:000 O.O00
cumulative density function given by: 10
20
30 Nurnbe J ' ~ t municlpalit
F(y)=
Iel
e(-(x-")=/c2a~)dx
(19)
Then, the probability for ADR to lie between Yl and Y2 is given by: CDR
Pr(Yl ~
Px(x) dx 1
I0 0.05 O.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
20
30
(20)
mun|clpll;tlel
The maximum Likelihood estimates of it and tr 2 are /a =
~7= 1 ADRi
(21)
n
0.000
CDR
and 10
13} 0.05 0.15 0.25 0.35 ~.45 0.55 0.65 0.75 0.85 0,95
= (F(y2) - F(y~ ))/(F(1 ) - F(0))
,oo,0e,%,
0.260 0.372 0.261 0.090 0-015 0. O01 0.000 0.000 0.000
20
i
0.597 0.354 0.047 0.001 0.000 0.000 0.000 0.000 0.000 0.000
30 *
N um berl~o 1 mun;c;pal|lfe$
CDR Index
9
/
observed values : calculated values
z
Fig. 2. Fit of the observed distribution of the average damage ratios of M S K intensity VIII to the truncated Normal distribution; in the cohmm "prob.' the calculated probablities are given: (1) The first room category, (2) The second room category, (3) The third room category
The obtained combinations of the ADR's for all the municipalities show a root mean square of the differences of only 3.44 rooms. After grouping the obtained ADR's both by MSK intensity and by room category, the numbers of obtained ADR's falling into a 0.1 wide interval in the range between 0 and I are plotted. Thus, ADR distributions are obtained immediately for the three room categories and MSK intensities VII to X (Fig. 2). Compared to the increment 0.01 used to search the optimum ADR combinations, the 0.1 subdivision is believed to be good for the identification Of the distribution. The observed ADR distributions are fitted by a truncated Normal distribution in the interval [0, 1]. The truncated Normal distribution has been chosen to fit the data because it has the same characteristics as the ADR's distribution; in fact it is unimodal and it can reach its maximum at any point of the interval [0, 1]. Other trials with different distributions (e.g. lognormal) have been performed obtaining a similar but worse fitting. For the truncated Normal distribution the probability that the variable of interest lies between x and x + d x is given by:
e(-(x-uf /(2o)2) P x ( x ) d x = F(1)-F(O) dx
(18)
where it is the mean and a 2 is the variance and F(y) is the
a2 _ ~,,7=, ADR2 I1
It 2
(22)
respectively ~9, where the ADR~ is the average damage ratio of the involved room category in the ith municipality for the MSK intensity considered. The probability Py so obtained for the average damage ratio ADR to lie between Yl and Y2 is assumed to be equal to the probability that a room of that category would suffer a damage ratio between Yl and Y2 on the average when shaken by an earthquake of MSK intensity I. In other words, this probability is an element of DPM. Since the whole range [0, 1] of the ADR has been divided into 10 subdivisions of 0.1 width, the number ofdamage states DS in equation (3) is 10 and the central damage ratio CDR is the central value of each subdivision. All D P M elements in the MSK intensity range VII to I X determined in such a way are believed to be reasonable. Reliable elements for intensity X cannot be obtained, possibly due to lack of data (there are only two municipalities which felt intensity X during the 1976 earthquake). The elements for intensity 111 cannot be determined in this way since no data. on the 1976 earthquake damage is available for this intensity, ~For such undetermined elements, it was decided to extrapolate the values oflt and a 2 and the same procedure has been used also to obtain more realistic values associated with intensity X. If an exponential dependence ofdamage on earthquake magnitude 9 is assumed, the functional form of it is expected to be In It = a + bI
(23)
where a and b are constants determined by the least squares method. A linear relationship between magnitude and intensity is assumed for this region 2~ From equation (23) extrapolations are simply made (Fig. 3). T h e following characteristics o f a(l) are easily recognizable from the available values of a (Table 2): a(7) + a ( 9 ) - 2a(8) < 0 a(l+ l ) - a ( I ) > 0
for all the room categories for most of the cases
(24)
The continuous counterparts of these conditions are
Soil D),namics and Earthquake Engineering, 1989, Vol.8, No. 2
99
Seismic risk in Friuli-Venezia Gittlia: W-S. Yang et a l . . expressed by: I n il~ - - 1 4 . 6 4
+1.413 t
0'--0.54+0.29
O15
In I
1
d 2 a(l) - - < 0 dI 2
0.5 010 0.1
d a(I) -->0 dI
(25)
005
001
A simple relation between a a n d I satisfying the a b o v e c o n d i t i o n s is i vt
I I lrtl ~
I Ix
I at
i
i VI
INTE N$$1TY
INDEX,
9
I ~rl!
.~ o b s e r v e d
: fttted
i ~
i tl
i x
t INTENSITY
a=c + dln
vJtues
line
Fig. 3. Extrapolation o f It and a f o r M S K intensity VI and X ; it is seen (Tables 2 and 3, and Figs. I and 4) that the extrapolation o f It is not completely satisfactory
Table 2. Tl, e observed and extrapolated tt and a. The values in parentheses are extrapolated using equations (23) and (26) 9 Intensity
#
a
Category First category
VI VII VIII IX X VI VII VIII IX X
(0.0197) 0.0557 0.2376 0.5442 (1.8878)~ (0.0193) 0.0709 0.1652 0.1736 (0.2314)
Second category
Third category
(0.0073) 0.0252 O.1502 0.3925 (1.7765) ~ (0.0101) 0.0400 0. I 149 0.1071 (0.1498)
(0.0028) 0.0104 0.0648 0.1925 (0.9348) (0.0000)b 0.0178 0.0765 0.0901 (0.1279)
a Although these values are greater than 1, the DPM elementsfor these room categories and this intensity are calculated with these mean values. From this it is evident that a function p(l) with smaller radius of Curvature at high intensities should replace equation (23) b Since a negative value (-0.02) is obtained, this value is replaced by 0.000
(26)
where c a n d d are c o n s t a n t s to be determined (Fig. 3). T h e resulting fit can be considered acceptable for intensity VI while a function a(I) satisfying the conditions of e q u a t i o n (25) a n d with smaller radius of c u r v a t u r e at high intensities might be m o r e adequate. T h e D P M elements PUll) from M S K intensity VI to X o b t a i n e d by e q u a t i o n (20) a n d e x t r a p o l a t i o n s are listed in T a b l e 3.
Comparison o f the calculated Q ( I ) w i t h the observed damage ratio H a v i n g d e t e r m i n e d all q u a n t i t i e s in e q u a t i o n (4) as previously described, the values of Q(I) are calculated for the intensities observed in t h e 1976 e a r t h q u a k e a n d c o m p a r e d with the observed loss ratios (Fig. 4). I n Fig. 4 there are four r e m a r k a b l e groups of points each of which differs from other groups m a i n l y by the intensity. This discreteness is believed to c o m e from that intrinsic in the definition of intensity. If a c o n t i n u o u s variable, such as the g r o u n d acceleration rather t h a n intensity, were used a n d if c o n t i n u o u s D P M elements were d e t e r m i n e d as a f u n c t i o n of the variable, then this discreteness would p r o b a b l y disappear. T h e scattering of a m u n i c i p a l i t y with respect to the m e a n value of the g r o u p to which it belongs gives a n idea of the local v u l n e r a b i l i t y of the municipality. If the observed value is greater t h a n the m e a n value, it m a y be
Table 3. The obtained DPM elements in Friuli-Venezia Giulia. The CDR used is the central damage ratio of damage state (see text for details) CDR Intensity
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
0.000 0.000 0.205 0.124 0.000
0.000 0.000 0.115 0.197 0.000
0.000 0.000 0.045 0.228 0.000
0.000 0.000 0.013 0.190 0.008
0.000, 0.000 0.002 0.115 0.015
0.000 0.000 0.000 0.050 0.137
0.000 0.000 0.000 0.016 0.840
2. The second room category (rooms built between 1946 and 1960) V! 1.000 0.000 0.000 0.000 VII 0.958 0.042 0.000 0.000 VIII 0.260 0.372 0.261 0.1390 IX 0.003 0.033 . 0.158 0.334 X 0.000 0.000 0.000 0.000
0.000 0.000 0.015 0.314 0.000
0.000 0.000 0.001 : 0.131 0.000
0.~ 0.~ 0.~ 0.024 0.~
0.~ 0.~ 0.~ 0.002 0.~
0.~ 0.~ 0.~ 0.~ 0.~
0.~ 0.~ 0.~ 0.~ 1.~
3. The third room category (rooms built between 1961 and VI 1.000 0.000 0.000 VII 1.000 0.000 0.000 VIII 0.597 0.354 0.047 IX 0.138 0.387 0.356 X 0.000 0.000 0.000
0.000 0.000 0.000 0.010 0.000
0.000 0.000 0.000 0.000 0.006
0.~. 0.~ 0.~ 0.~ 0.041
0.~ 0.~ 0.~ 0.~ 0.162
I. The first room category (rooms built before 1945) VI 1.1300 0.000 0.000 VII 0.661 0.312 0.026 VIII 0.138 0.224 0.256 IX 0.004 0.019 0.056 X 0.000 0.000 0.000
1975) 0.000 0.000 0.001 0.108 0.000
,.0.~ 0.~ 0.~ 0.~ 0.355
0.~ 0.~ 0.~ 0.~ 0.435
According to this table, a room of the first category would receiveless damage under intensity X than one of the second category. This results from the longer tail of the distribution of the first room category whose mean is placed in the region above I (seep and a of the first two categories in Table 2). This fact suggests using a function p(I) giving smaller values at high intensities. This unsatisfactory value for MSK intensity X would not cause serious problems in estimating Qmaxand R because of the small probability of occurrence of an earthquake of this intensity in the study region
100
Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, N o . 2
Seismic risk in Friuli-Venezia Giulia: W-S. Yang et al. II*(t)
08
0.8
05
LINE 3:Q'~,I, 0.245 OcO *0.424 114 8
o o
o o
O.3 Q
o o
o o
o
o
"LINE 2: O*h~, 1.823 ('0.05.6) Oi,t-0.062 (+0Oll)
.2
0.2
,%**
0.1
.+o II o
0
"
i
I
0.1
o
o
~
LINE I:0*r Q2
0.784 Oh|
I
I
t
I
I
I
0.3
0.4
0.5
0.8
03
0.8
I
I
0.8 O h!
Fig. 4. Comparison of tile observed damage ratio Q*(I) during the 1976 earthquake ill the study region with the calculated Q(I) for the corresponding intensities: 4 groups are easily recognized. Grouping is related to the discreteness of the intensity. A relationship is drawn by 3 straight lines: Line 2 is drawn using the centres of mass of the groups and Lines I and 3 are drawn reasonably (see text for details). The far higher Q(I) of the last group than the corresponding Q*(I) indicates a not completely satisfactory extrapolation of It for intensity X. The centres of mass are denoted by open triangles said that the municipality is more vulnerable to earthquake damage than other municipalities of lower ratios. Another possible interpretation is that the scattering has been introduced by the use of a discrete quantity as intensity. In reality both factors must have an influence. The far higher Q(10) values with respect to their corresponding observed values indicate higher values of extrapolated is for intensity X than their true values. This suggests using a function with smaller radius of curvature than that of the exponential function at higher intensities. To find the relationship between the calculated Q(I) and the observed ratio of loss Q*(I), the centres of mass of the groups (groups of intensity VII, VIII, and 1X) are used for fitting with a straight line (Line 2 in Fig. 4). The line is extended by a straight line from the centre of mass of the group of intensity VII to the origin (Line 1 in Fig. 4), assuming no loss would be observed in zones where 0(I) is 0. Tentatively, the line is also extended to the higher value
of Q(I) by another straight line (Line 3 in Fig. 4) from the centre of mass of the group of intensity I X to the more reliable point ofintensity X, since the centre of mass ofthe group of intensity X has a lower 'observed' value than that of the group of intensity IX. The use of the centres of mass to draw straight lines is believed to compensate, to some extent, errors due to the scattering of values according to the second interpretation on scattering given before. The relatively low errors in the parameters of Line 2 in Fig. 4 may show the validity ofusing the centres ofmass for fitting and may indicate that good approximation is reached by using Line 2 in this range. The following relation is obtained: 0.784 Q(1),
Q*(I) = / 1.823 Q(I) "--0.062, t 0.245 Q(I) + 0.424,
for Q(I)~<0.06 for 0.06 ~
9Soil Dynamics and Earthquake Engineering, 1989, Vol.8, No. 2
(27)
101
Seismic risk in Friuli-Venezia Giulia: W-S. Yang et al. I
i
~
,.^',..
I
.-"v w " ' - ~
u
s
r
R
t
a
"'X,...--,.~"~:.k~-'~ .,,.. _ 46" 30'N
9
,,r'-,..~..I,..~...r.~..,
Province BELLUNO
/" ~...-"
..,~"
c..p"
..c
j
t""t
q~
O 46' O0'N -
.~.,,.,-.9
(3 POROENONE
t,. =
/:~ovlnce
/ ",,,
TREVISO
Province VENEZIA
k.
W "p >
AdrMtic
I 12" 30"E
Fig. 5.
I 13"09" t
I 13'30' E
Map of the maximum M S K intensity expected not to be exceeded with 37% probability in 100 years
Equation (27) is used to correct the calculated values Q(I), to obtain more reliable values of risk. Because of the little information on intensity X and possible large errors in extrapolating F and a to this intensity, this equation is believed more reliable in the range
0 <.Q(I) <~0.31
(28)
RISK MAPPING AND DISCUSSIONS When the Poisson process is assumed for the occurrence of earthquakes, their intensities with a particular re[urn period have a 37% probability of not being exceeded within a time period equal to that return period 1. Then, the intensity 1100, which is not exceeded with a 37% probability within 100 years satisfies N= 1
(29)
where N is the mean number of occurrences of earthquakes in I00 years whose intensities are I or greater. Fig. 5 shows the maximum intensity expected not to be exceeded with 37 % probability in 100 years in the Friuli-
102
eea
Venezia Giulia region obtained by an application of the Cornel114 methodology. We can see that the maximum values (I >1VIII MSK) are reached in the central part of the region with strong attenuation towards South. Similar results for the study area have already been obtained using different methodologies2'4'6. For these intensities, the risk Qm~x=Q(lm~x) is evaluated according to equations (4) and (27) for each municipality in the test region. The number of rooms of each category of each municipality is taken from the 1981 census data 21. Because the Damage Probability Matrix elements as well as the mean replacement cost have not been determined for the rooms built after the 1976 earthquake, the evaluated risk Qm~x should be interpreted as the expected loss ratio for rooms built before 1976, due to an earthquake intensity that is not exceeded with a 37% probability within 100 years. The map showing these values for the mt/nicipalities in the study area is given in Fig. 6. The general picture is that the northern part of the Friuli region is expected to suffer heavier losses than the southern part. The most hazardous municipality in this region turns out to be Forgaria nel Friuli (the black area in Fig. 6) with 35 % risk. Fig. 7 shows the distribution of the values of R, which
Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 2
Seismic risk in Friuli-Venezia Giulia: W-S. }Tang et al. I "
~
.
A
U
S
T
R
I
A
p.--%
~'.,~..,~,. j...1.--
46'30"N
,~
INDEX Ome= ~i 0 0,5
J
0.05 < O m l x
sO.15
015
< Om|x
__.0.25
025
< Omlx
r__ J
0 46" O0'N
/
Pro~
/
,-
)
'a
. . . . . .
9
..... Adriat~.
I 12' 30" E
~ sea
I
_~_
"\
,.-r'';~ > ~ ""
I
13'00" E
13"30' E
Fig. 6. Map of Qm~x corresponding to tile intensity which has the mean return period of 100 years or which is not exceeded with a 37 % probability within 100 years. The northern area is more risky. The most risky municipality turns out to be Forgaria nel Friuli (the black area) with 35 % risk
are obtained by using equations (13) and (27). Again, R is potentially the expected loss ratio of the rooms built before 1976 by earthquakes during the next 100 years. Fig. 7 presents a more detailed picture of the regional variation of risk than Fig. 6. The northern area of the Friuli region shows higher risk than the southern area and there is an increasing concentration of the total risk towards the centre of the northern area. Again, the most hazardous municipality is Forgaria nel Friuli with 95 % total risk. The risk map in the Pordenone province (a province of Friuli-Venezia Giulia) presented by Slejko et al. 1~ generally agrees with the present picture (the risk is defined as the number of houses destroyed or damaged by earthquakes in the Slejko et al. ~~ paper). Since severe design requirements have been adopted after the destructive earthquake of 19762, the real risk for existing rooms is certainly lower than that evaluated with the data on rooms built before 1976. Further study is required to evaluate the Damage Probability Matrix elements for all the room categories as well as their average replacement costs so that a more realistic assessment of risk could be obtained.
CONCLUSIONS The seismic risk of a municipality is quantified as the ratio of repair cost to the replacement cost of all the rooms. Borrowing on the concept of Damage Probability Matrix, we arrive at a formula expressed by equation (4). This risk is the expected damage ratio from the earthquake with a certain probability of occurrence in a fixed time period. The picture showing this value for each municipality in Friuli-Venezia Giulia is presented in Fig. 6. This figure reflects well the higher seismic activity in the northern area of this region with respect to the sourthern part. Considering the possible, even repeated, occurrences of earthquakes in the whole intensity range, we have assigned weights in terms of occurrence probability to the risk in equation (4). The weighting factor has been determined on the assumption of the Poisson process for the earthquake occurrence statistics. With further assumptions, we obtained the set of approximate 9 presented in Fig. 7. The weighted risk, which we call the total risk in a fixed time period, is potentially the sum of the loss ratios from all the possible earthquakes in that
Soil Dynamics and Earthquake Engineering, 1989, Vol.8, No. 2 103
Seismic risk in Friuli-Venezia Giulia: W-S. Yang et al. I
"X.~'~.-~.~.
A
U
S
T
R
I I 'A
Province BEL L UNO
46"30"N
':'~"~'""T'~" --/~" r
. _+..%~,
INDEX
I
"
I
.<, C-
0.20 < R S 0.40 ~
040<'R ~ 0.60"
~..,,:,..,._:_ t,...,a'-,..~ ~
,,/"
c.
/"
<
46"00"N
e,020
.......
J
0.80< R 9 0.80 O.eO< R
o u,
ooR,z, f
Province TRE VISO
VENEZIA
I._2
?..
..~ - / .
Adriatic
'T'"
I 12"30" E
~_r=~.
I 13"00" E
sea
....... \ ....
"\ ..>
%'~'--~ j r
.p
,, [ 13"30" E
Fig. 7. Map of tile total risk in 100 years. The concentric distribution of R is easily observed in the nortllern area of the region. The most risky municipality is again Forgaria nel Friuli with 95 % of tlle total risk
period. Because some restrictions, the evaluated risks, i.e., Q(I) of equation (4) and R of equation (13), only refer to the rooms built before 1976. Therefore the maps give a general picture of possible damage exposure of the old buildings, and they show where strengthening of such buildings is most needed. The highest total risk is concentrated in Forgaria nel Friuli, around which more or less concentric distribution of total risk is observed. The real risk, which should be determined by considering all rooms including those built after the 1976 earthquake, will be less than that depicted in this figure. To determine the real risk further study should be made for a homogeneous set of Damage Probability Matrix elements and a set of average replacement costs of all the room categories, taking into account the structural characteristics of the building types. However, the results obtained show agreement with a previous study in this area. In all cases determination of the DPM elements is unavoidable. Using available data, we have found the distribution of the average damage ratios by the 1976 earthquake in the test region for each room category classified by age and for each MSK intensity degree from
VII to X. These distributions, fitted by a truncated Normal model, have determined the DPM elements. For the unknown distribution of the average damage ratios of intensity VI and the unfavourable set of intensity X, simple functional relations between MSK intensity and the relevant parameters had to be arbitrarily assigned and from them the rest ofthe D P M elements were determined.
ACKNOWLEDGEMENTS Special thanks are due to Ezio Faccioli who has read the manuscript giving important suggestions and specific comments and to Peter Guidotti for checking the manuscript English. One of the present authors (W. S. Yahg) did this work with the support of the ICTP (International Centre for Theoretical Physics, Trieste, Italy) Programme for Training and Research in Italian Laboratories. The study has been undertaken, within the framework of the research unit 'Shakeability and seismic risk in northeastern Italy' (contract n. 86.00183.02) of the 'Gruppo Nazionale di Geofisica della Terra Solida'.
104 Soil Dynanlics and Eartllquake Engineering, 1989, Vol. 8, No. 2
Seismic risk in Friuli-Venezia Giulia: W-S. Yan9 et al. REFERENCES 1 2 3 4 5 6 7 8 9
10
Algermissen, S. T. and Perkins, D. M. A probabilistic estimate of maximum acceleration in rock in the contiguous United States, U.S.G.S. Open-File Report 76-416, Menlo Park, 1976 Faccioli, E. Engineering seismic risk analysis of the Friuli region, Boll. Geof. Teor. Appl., 1976, 21,173-190 Gruppo di Lavoro Scuotibilith, Carte preliminari di scuotibilit~. del territorio nazionale, Pubbl. n. 227 del P.F. Geodinamica del C.N.R., ESA, Roma, 1979 Giorgetti, F., Nieto, D. and Slejko, D. Seismic risk of the FriuliVenezia Giulia region, Publ. Inst. Geophys. Pol. Acad. Sc., 1980, A-9(135), 149-169 Mueller, S. and Mayer-Rosa, D. The new seismic hazard maps for Switzerland, Rerista Geofisica, 1980, 13, 7-19 Slejko, D. and Viezzoli, D., Scuotibilit~ del Friuli-Venezia Giulia utilizzando tecniche anisotropiche, Atti Terzo Convegno G.N.G.T.S., Roma, 1984, 829-842 Makropoulos, K. C. and Burton, P. W. Seismic hazard in Greece. I. Magnitude recurrence, Tectonophysics, 1985, 117, 205-257 Makropoulos, K. C. and Burton, P. W. Seismic hazard in .Greece. II. Ground acceleration, Tectonophysics, 1985, 117, 259294 Ambraseys, N. N. and Jackson, J. A. Earthquake hazard and vulnerability in the northeastern Mediterranean the Corinth cart hquake sequence of February-March 1981, Disasters, 1981, 5, 355-368
Il 12 13 14 15 16 17 18 19 20 21
Slejko, D., Viezzoli, D. and Gasparo, F. Esempio metodologieo del calcolo del rischio sismico, Atti Terzo Convegno G.N.G.T.S., Roma, 1984, 799-810 Ambraseys, N. N. Evaluation of seismic risk, in Seismicity and seismic risk in the offshore North Sea area, (Eds A. R. Ritsema and A. Gurpinar), Reidel Publishing Company, Dordrecht, 1983 Whitman, R. V. and Comell, C. A. Design, in Seismic risk and engineering decisions, (Eds C. Lomnitz and E. Rosenblueth), Elsevier, Amsterdam, 1976 Gutenberg, B. and Richter, C. F. Frequency of earthquakes in California, Bull. Seism. Soc. Am., 1944, 34, 185-188 Cornell, C. S. Engineering seismic risk analysis, B,dl. Seism. Soc. Am., 1968, 58, 1583-1606 Giorgetti, F. Personal communication, 1986 Regione Autonoma Friuli-Venezia Giulia, Stima dei danni provocati dal sisma del 6 maggio 1976, Relazione inedita a cura di L. Di Sopra, 1976 Lawson, C. L. and Hanson, R. J. Solving least square problems, Prentice-Hall, New York, 1974 Istituto Centrale di Statistica, 11~ Censimento generale della popolazione (24 ottobre 1971), Roma, 1972-1973 Benjamin, J. R. and Cornell, C. A. Probability, statistics, and decision for civil engineers, McGraw-Hill, New York, 1970 Peronaci, M. Intensity-magnitude relationships for Italian regions, Boll. Geof. Teor. Appl., 1982, 24, 121-128 Istituto Centrale di Statistica, 12~ Censimento generale della popolazione (24 ottobre 1981), Roma, 1983-1984
Soil Dynamics and Earthquake Engineering, 1989, Vol.8, No. 2
105
Ossercatorio Geofisico Sperimentale, P.O. Box 2011, 34016 Trieste, Italy Fulvio Gasparo
Regione Autonoma Friuli-Venezia Giulia, 34100 Trieste, Italy Seismic risk is defined in this work as the expected damage ratio from possible earthquakes, during t years. The risk associated with the earthquake having a fixed occurrence probability in t years is formulated by means of the Damage Probability Matrix. Also, an expression for the total risk from all probable earthquakes is presented. To evaluate the risks mentioned, determination of the values of the Damage Probability Matrix is carried out statistically. Friuli-Venezia Giulia (NE Italy) is chosen as study region and the above risks are evaluated for t = 100 years. Computed risk refers only to the rooms built before 1976 as no information about new buildings is available. The general picture is in accordance with previous results for the same region. Key Words: seismic risk, Damage Probability Matrix, room category.
INTRODUCTION
RISK MODEL
To develop a satisfactory strategy against damage from expected earthquakes in a region, it is necessary to know not only the nature of earthquake occurrence in that region but also its response to earthquakes. The severity of t h e expected earthquakes can, for example, be described by estimating maximum expected accelerations in fixed time periods or return periods of earthquakes of a fixed magnitude t-8. The response of a region to earthquakes depends on various factors such as population density, building types, geological conditions, etc. Several attempts to correlate building damage, appropriately defined, with these factors have been made 9't~ and the term 'seismic risk' has been introduced. Ambraseys 11 suggests defining seismic risk as:
Whitman and Cornell t2 introduced the Damage Probability Matrix (DPM) elements, P(j]I), representing the probability that an earthquake of a given intensity I causes a damage statej to a class of buildings which are assumed to respond to earthquakes with a similar degree of damage. State j is one of a set of states classified by severity of earthquake damage. In this study, a room rather than a building is chosen as the elementary unit for damage evaluation because the size of a room is generally more uniform than that of a building. With this choice, a simple criterion such as the age of the room can be used to classify rooms according to resistance to earthquake damage. Suppose we have classified all rooms in a municipality into K categories according to their strength, and that we have assigned a mean replacement cost, that is simply the economic value of a room, to each category. Furthermore, we introduced a discrete set of damage states by subdividing the whole range of damage ratio values (i.e. from 0 to l) into DS states of fixed length. The central value of the damage ratio of each damage state will be called the central damage ratio. Let us assume t h a t there a r e N k rooms of the kth category in a certain municipality, having mean replacement cost RCk, and that Pk (JlI) is the probability that a room of the kth category would be damaged to statej by an earthquake of intensity I. Then the approximate repair cost (ReCk) for all kth category rooms in the municipality due to an earthquake of intensity I can be expressed as:
(Risk) = (Seismic hazard)* (Vulnerability)* (Value)
(1)
Seismic hazard is the probability of occurrence of ground motions due to an earthquake of a magnitude capable of causing significant loss of value through damage or destruction. Vulnerability may be defined as the degree of damage inflicted by a given earthquake to man-made structures and to the ground itself, while Value may be taken either in the sense of capital value or any other quantity of interest that will adjust the right side of equation (1) to the same dimension as Risk. Risk at a site incorporates both the earthquake occurrence characteristics at that place and its response to earthquakes. Equation (1) is only a qualitative expression of Risk. The main problem is to make the formula explicit and to perform a good quantitative evaluation of Risk as a function of the relevant parameters. Communicated by G. Bonn. Accepted November 1987. Discussion closes September 1989.
/)S
ReCk= ~, Nk'Pk(j]I)'RCk'CDR j ( k = l ..... K) (2) j=l
where CDRj is the central damage ratio of the damage
(~ Computational Mechanics Publications 1989
96
Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 2
Seismic risk in Friuli-Venezia Giulia: W-S. Yang et al. state j. The total cost for repairing (TReC) all the damaged rooms in the municipality would be: K
TReC=~. ReCk = k
DS
(8)
j=l
where ~ is the Euler-Mascheroni constant and equals 0.5772157... and m is generally not an integer. If this value m (or).I"t which is close to !11)satisfies
If we define Q(I) as: DS
~ Nk'PkUII)'RCk'CDR~ Q(l)=k=, j=l t: ~. Nk'RCk
then Q(I) is the mean ratio of repair cost to total replacement cost in the municipality hit by an earthquake of intensity I, i.e. the loss ratio. If Q(l,n) denotes the sum of the loss ratios from n earthquakes of intensity 1 and P1.t(n) denotes the probability for an earthquake of intensity I occurring n times in t years, the total loss ratio in a municipality in the whole period of t years is given by:
R=E Y, 0(I,11)'P1.,(10
(5)
n=l
Since all the values of the parameters in equation (5) vary as time goes on, the defined R cannot be the actual total damage ratio which would be received during t years but just a risk weighted with earthquake occurrence probability. This R would give a general picture of possible damage severity for the next t years for a municipality in a region. In this work, the quantity Risk of equation (l) is quantified by the quantity R of equation (5) and a simple relation is assumed for Q(l,n) in each municipality of the study region, i.e.:
Q(l,n)=nQ(I)
(6)
Though equation (6) is suggested mainly for the purpose of an approximate evaluation of R, it has physical meaning. Literally, equation (6) implies that a municipality is restored to its original state after each earthquake. This assumption is not valid for all the cases but the approximation holds in most actual situations. In fact, if Q(I) is small enough equation (6) can be a reasonable approximation since it is hardly expected that the strength of a room is significantly lowered or enhanced after repairing the damage of an earthquake which caused only a minute amount of loss. A nonlinear cumulative effect of weakness should be taken into account if no restoration of damage is performed. It is not known how small Q(I) should be put the corresponding Medvedev-Sponheuer-Karnik (MSK) intensity is expected to be V or VI or less. On the other hand, if O(1) is large, the maximum of the Poisson probability of occurrence will become small (,~ l for earthquakes with I> VII for t = 100in the study area) since the exponential decrease of the annual occurrence rate of earthquakes of intensity 1,21, with respect to intensity is expected from the Gutenberg-Richter's relation ~a. Following the Poisson distribution for Pta(n) 1'~, the maximum of e - a,., (21" t)" Pl.,(n) = i1!
1 >~m~21"t
(4)
k=l
1
l)
k+l,;
ln(2t't)+?'=k~ ,
~, Nk'Pk(jII)'RCk'CDRj (3) k=l
K
lies at !11 which satisfies
(7)
(9)
and the Poisson probability decreases rapidly with increasing !1, the terms Q(I,n) with ii greater than 1 have only a small influence on the final value of R in equation (5) because of the small Pi,n) associated. In this case we can use equation (6) disregarding whether it is a valid approximation or not. It should be noted that the validity of equation (9) also depends on the time period chosen. For regions with moderate or low seismic hazard the condition of equation (9) and the rapid decrease of Pl,n) turn out to be suitable for high intensities with t--100 years. But for regions with high seismic hazard these are not true. However, for most municipalities in the study region these conditions are generally satisfied. This implies that R expressed by: co
e-a,., (21.t)~
n= 1
!1!
R=~Q(I) ~ n I
-t~Q(l)').l
(10)
l
where Q(I) is the ratio of loss expressed in equation (4), is potentially the expected total loss ratio from all possible earthquakes during the next t years. Actually, R would not give the expected ratio because not all the values of the parameters entering in equation (4) would remain the same after each earthquake. However, in this study R will be called the total risk in the next t years. Since 21 is monotonically decreasing and Q(I) is monotonically increasing with respect to I, the product Q(I)').I has a maximum in the intermediate intensity range. Then, R in equation (10) can be approximated by: 10
R=t ~ Q(I)').I
(11)
1=6
since earthquakes of intensities lower than VI do not produce damage and
l >>211> Q(ll). 211.~Q(12). :q2
(12)
This fact is checked by using the annual occurrence rates of earthquakes 6 and Q(I) in a municipality of-Friuli affected by high seismic hazard (Fig. 1). In conclusion the total risk in t years can be expressed as:
K
DS
,o E Nk'RCk'E PkUII)'CDR i R=t ~, 21k=l j=l K 1=6 ~ Nk.RCK
(13)
k=!
ASSESSMENT OF THE PARAMETERS O F THE MODEL The Damage Probability Matrix approach has been applied to the rooms classified according to age in the study region of Friuli-Venezia Giulia, located in the
Soil Dynamics aml Earthquake Engineering, 1989, Vol.8, No. 2 97
Seismic risk in Friuli-Venezia Giulia: W-S. Yanget al.
lO0~.=O(t)
least square method t7 to the following equation:
1.0
number of annual occurrences
3 total economic value of all ~. N.," R C k --k= t rooms in the ith municipality
(14)
09 08
where N~ is the number of rooms of the kth category in the ith municipality. The number of rooms of each category in each municipality is taken from the 1971 census ~s and the value of the right-hand side of equation (14) for each municipality is taken from the quoted report ~6. To obtain reasonable values for RCk, the following obvious condition is required
3.x ? ! I
07
!
I
06
Oltl 0.12
(15)
R C 1 <~ R C 2 <~ R C a
Q5
0.10
i 0.4.1
0.08
0.3
0.136
02, 100~.tO[t}
] 0.02004
0.1
V
Vl
VII
Vlll
IX
X
Xl
( MSK intensity)
Fig. 1. The maxhnum of t'2z'Q(I) with t=lO0 in the municipality Forgaria nel Friuli is placed between M S K intensity VII and VIII; the large value of 100.2 I. Q(I) at X which does not seem to decrease dramatically toward 0 at XI possibly results from the large extrapolate it, which will be discussed later
northeastern part of Italy. There are other parameters by which resistance'of rooms to earthquake damage can be better described but the proposed criterion has been chosen because of its easiness and readiness for classification. Three room categories have been adopted; namely, rooms in buildings constructed (1) before 1945, (2) between 1946 and 1960, and (3) between 1961 and 1975. Since information about rooms built after 1975, necessary for evaluating Q(I) and R, is not available, only rooms constructed before 1976 are used in all the following calculations. The model calculations have been performed using the intensity data of the 1976 earthquake (MSK intensities between VII and X ts). To determine the values of parameters entering in equation (4), a damage report on 79 relatively heavily damaged municipalities was used 16. Based on this data, we have determined the mean replacement cost RCk as well as the Damage Probability Matrix elements Pk(j[I) of equation (13).
Assessment of the replacement costs The average replacement cost R C k , of the kth category of rooms in the test region is obtained by applying the
98
The obtained average replacement costs of the three categories of rooms in the test region are listed in Table 1.
Determination of the DPM elements As a next step we must determine the average damage ratio applicable to rooms of the kth category (k = 1, 2, 3) with reference to the 1976 earthquake. The average damage ratio ADRta is defined as the ratio of the total economic loss inflicted on the rooms of the kth category to the sum of their economic values in the ith municipality, which is one of those to which MSK intensity I was assigned in 1976 ts. Generally, for the same I and k ADR's will be different between one municipality and another, and we can obtain a distribution of the ADR's over the whole interval [0, 1] for each I and k. From the distribution we can infer the probability for a room of the kth category receiving a certain degree of damage under a certain MSK intensity. The values of the average damage ratios of the three categories of rooms caused by MSK intensity from VII to X were computed for each municipality by:
3 k=l
A D R t # N ~ = e q u i v a l e n t number of destroyed rooms
ratio of lost number number value to total = of rooms + of rooms • value for the collapsed damaged damaged rooms
(16)
Where ADR, a is the average damage ratio of the kth category rooms by MSK intensity I and N~ is the number of that category rooms in the ith municipality. As for equation (14), the number of rooms of each category in each municipality is taken from the 1971 census I 8 and the value of the right side of equation (16) is taken from the damage report 16. The optimum combination of ADR's is looked for, for each municipality with the least difference between the right- and left-hand sides of equation (16) as ADR's are increased by 0.01 with the restriction 0 ~
(17)
Table 1. The replacement cost o f a room o f each room category in Friuli-Venezia Giulia. The unit Italian lira used is referred to its economic ralue in 1976 First category Second category Third category
Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 2
1.0.10 6 lira 2.5" 106 lira 11.5.10 6 lira
Seismic risk in FriulizVenezia Giulia: W-S. Yang et al.
6) CDR
ptob
0.05 O.15 0.25 0.35 0.45 0.55 0-65 0.75 0.85 0.95
O.138 0.224 0.258 0.205 0,1"15 0.045 0.013 0,002 0:000 O.O00
cumulative density function given by: 10
20
30 Nurnbe J ' ~ t municlpalit
F(y)=
Iel
e(-(x-")=/c2a~)dx
(19)
Then, the probability for ADR to lie between Yl and Y2 is given by: CDR
Pr(Yl ~
Px(x) dx 1
I0 0.05 O.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
20
30
(20)
mun|clpll;tlel
The maximum Likelihood estimates of it and tr 2 are /a =
~7= 1 ADRi
(21)
n
0.000
CDR
and 10
13} 0.05 0.15 0.25 0.35 ~.45 0.55 0.65 0.75 0.85 0,95
= (F(y2) - F(y~ ))/(F(1 ) - F(0))
,oo,0e,%,
0.260 0.372 0.261 0.090 0-015 0. O01 0.000 0.000 0.000
20
i
0.597 0.354 0.047 0.001 0.000 0.000 0.000 0.000 0.000 0.000
30 *
N um berl~o 1 mun;c;pal|lfe$
CDR Index
9
/
observed values : calculated values
z
Fig. 2. Fit of the observed distribution of the average damage ratios of M S K intensity VIII to the truncated Normal distribution; in the cohmm "prob.' the calculated probablities are given: (1) The first room category, (2) The second room category, (3) The third room category
The obtained combinations of the ADR's for all the municipalities show a root mean square of the differences of only 3.44 rooms. After grouping the obtained ADR's both by MSK intensity and by room category, the numbers of obtained ADR's falling into a 0.1 wide interval in the range between 0 and I are plotted. Thus, ADR distributions are obtained immediately for the three room categories and MSK intensities VII to X (Fig. 2). Compared to the increment 0.01 used to search the optimum ADR combinations, the 0.1 subdivision is believed to be good for the identification Of the distribution. The observed ADR distributions are fitted by a truncated Normal distribution in the interval [0, 1]. The truncated Normal distribution has been chosen to fit the data because it has the same characteristics as the ADR's distribution; in fact it is unimodal and it can reach its maximum at any point of the interval [0, 1]. Other trials with different distributions (e.g. lognormal) have been performed obtaining a similar but worse fitting. For the truncated Normal distribution the probability that the variable of interest lies between x and x + d x is given by:
e(-(x-uf /(2o)2) P x ( x ) d x = F(1)-F(O) dx
(18)
where it is the mean and a 2 is the variance and F(y) is the
a2 _ ~,,7=, ADR2 I1
It 2
(22)
respectively ~9, where the ADR~ is the average damage ratio of the involved room category in the ith municipality for the MSK intensity considered. The probability Py so obtained for the average damage ratio ADR to lie between Yl and Y2 is assumed to be equal to the probability that a room of that category would suffer a damage ratio between Yl and Y2 on the average when shaken by an earthquake of MSK intensity I. In other words, this probability is an element of DPM. Since the whole range [0, 1] of the ADR has been divided into 10 subdivisions of 0.1 width, the number ofdamage states DS in equation (3) is 10 and the central damage ratio CDR is the central value of each subdivision. All D P M elements in the MSK intensity range VII to I X determined in such a way are believed to be reasonable. Reliable elements for intensity X cannot be obtained, possibly due to lack of data (there are only two municipalities which felt intensity X during the 1976 earthquake). The elements for intensity 111 cannot be determined in this way since no data. on the 1976 earthquake damage is available for this intensity, ~For such undetermined elements, it was decided to extrapolate the values oflt and a 2 and the same procedure has been used also to obtain more realistic values associated with intensity X. If an exponential dependence ofdamage on earthquake magnitude 9 is assumed, the functional form of it is expected to be In It = a + bI
(23)
where a and b are constants determined by the least squares method. A linear relationship between magnitude and intensity is assumed for this region 2~ From equation (23) extrapolations are simply made (Fig. 3). T h e following characteristics o f a(l) are easily recognizable from the available values of a (Table 2): a(7) + a ( 9 ) - 2a(8) < 0 a(l+ l ) - a ( I ) > 0
for all the room categories for most of the cases
(24)
The continuous counterparts of these conditions are
Soil D),namics and Earthquake Engineering, 1989, Vol.8, No. 2
99
Seismic risk in Friuli-Venezia Gittlia: W-S. Yang et a l . . expressed by: I n il~ - - 1 4 . 6 4
+1.413 t
0'--0.54+0.29
O15
In I
1
d 2 a(l) - - < 0 dI 2
0.5 010 0.1
d a(I) -->0 dI
(25)
005
001
A simple relation between a a n d I satisfying the a b o v e c o n d i t i o n s is i vt
I I lrtl ~
I Ix
I at
i
i VI
INTE N$$1TY
INDEX,
9
I ~rl!
.~ o b s e r v e d
: fttted
i ~
i tl
i x
t INTENSITY
a=c + dln
vJtues
line
Fig. 3. Extrapolation o f It and a f o r M S K intensity VI and X ; it is seen (Tables 2 and 3, and Figs. I and 4) that the extrapolation o f It is not completely satisfactory
Table 2. Tl, e observed and extrapolated tt and a. The values in parentheses are extrapolated using equations (23) and (26) 9 Intensity
#
a
Category First category
VI VII VIII IX X VI VII VIII IX X
(0.0197) 0.0557 0.2376 0.5442 (1.8878)~ (0.0193) 0.0709 0.1652 0.1736 (0.2314)
Second category
Third category
(0.0073) 0.0252 O.1502 0.3925 (1.7765) ~ (0.0101) 0.0400 0. I 149 0.1071 (0.1498)
(0.0028) 0.0104 0.0648 0.1925 (0.9348) (0.0000)b 0.0178 0.0765 0.0901 (0.1279)
a Although these values are greater than 1, the DPM elementsfor these room categories and this intensity are calculated with these mean values. From this it is evident that a function p(l) with smaller radius of Curvature at high intensities should replace equation (23) b Since a negative value (-0.02) is obtained, this value is replaced by 0.000
(26)
where c a n d d are c o n s t a n t s to be determined (Fig. 3). T h e resulting fit can be considered acceptable for intensity VI while a function a(I) satisfying the conditions of e q u a t i o n (25) a n d with smaller radius of c u r v a t u r e at high intensities might be m o r e adequate. T h e D P M elements PUll) from M S K intensity VI to X o b t a i n e d by e q u a t i o n (20) a n d e x t r a p o l a t i o n s are listed in T a b l e 3.
Comparison o f the calculated Q ( I ) w i t h the observed damage ratio H a v i n g d e t e r m i n e d all q u a n t i t i e s in e q u a t i o n (4) as previously described, the values of Q(I) are calculated for the intensities observed in t h e 1976 e a r t h q u a k e a n d c o m p a r e d with the observed loss ratios (Fig. 4). I n Fig. 4 there are four r e m a r k a b l e groups of points each of which differs from other groups m a i n l y by the intensity. This discreteness is believed to c o m e from that intrinsic in the definition of intensity. If a c o n t i n u o u s variable, such as the g r o u n d acceleration rather t h a n intensity, were used a n d if c o n t i n u o u s D P M elements were d e t e r m i n e d as a f u n c t i o n of the variable, then this discreteness would p r o b a b l y disappear. T h e scattering of a m u n i c i p a l i t y with respect to the m e a n value of the g r o u p to which it belongs gives a n idea of the local v u l n e r a b i l i t y of the municipality. If the observed value is greater t h a n the m e a n value, it m a y be
Table 3. The obtained DPM elements in Friuli-Venezia Giulia. The CDR used is the central damage ratio of damage state (see text for details) CDR Intensity
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
0.000 0.000 0.205 0.124 0.000
0.000 0.000 0.115 0.197 0.000
0.000 0.000 0.045 0.228 0.000
0.000 0.000 0.013 0.190 0.008
0.000, 0.000 0.002 0.115 0.015
0.000 0.000 0.000 0.050 0.137
0.000 0.000 0.000 0.016 0.840
2. The second room category (rooms built between 1946 and 1960) V! 1.000 0.000 0.000 0.000 VII 0.958 0.042 0.000 0.000 VIII 0.260 0.372 0.261 0.1390 IX 0.003 0.033 . 0.158 0.334 X 0.000 0.000 0.000 0.000
0.000 0.000 0.015 0.314 0.000
0.000 0.000 0.001 : 0.131 0.000
0.~ 0.~ 0.~ 0.024 0.~
0.~ 0.~ 0.~ 0.002 0.~
0.~ 0.~ 0.~ 0.~ 0.~
0.~ 0.~ 0.~ 0.~ 1.~
3. The third room category (rooms built between 1961 and VI 1.000 0.000 0.000 VII 1.000 0.000 0.000 VIII 0.597 0.354 0.047 IX 0.138 0.387 0.356 X 0.000 0.000 0.000
0.000 0.000 0.000 0.010 0.000
0.000 0.000 0.000 0.000 0.006
0.~. 0.~ 0.~ 0.~ 0.041
0.~ 0.~ 0.~ 0.~ 0.162
I. The first room category (rooms built before 1945) VI 1.1300 0.000 0.000 VII 0.661 0.312 0.026 VIII 0.138 0.224 0.256 IX 0.004 0.019 0.056 X 0.000 0.000 0.000
1975) 0.000 0.000 0.001 0.108 0.000
,.0.~ 0.~ 0.~ 0.~ 0.355
0.~ 0.~ 0.~ 0.~ 0.435
According to this table, a room of the first category would receiveless damage under intensity X than one of the second category. This results from the longer tail of the distribution of the first room category whose mean is placed in the region above I (seep and a of the first two categories in Table 2). This fact suggests using a function p(I) giving smaller values at high intensities. This unsatisfactory value for MSK intensity X would not cause serious problems in estimating Qmaxand R because of the small probability of occurrence of an earthquake of this intensity in the study region
100
Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, N o . 2
Seismic risk in Friuli-Venezia Giulia: W-S. Yang et al. II*(t)
08
0.8
05
LINE 3:Q'~,I, 0.245 OcO *0.424 114 8
o o
o o
O.3 Q
o o
o o
o
o
"LINE 2: O*h~, 1.823 ('0.05.6) Oi,t-0.062 (+0Oll)
.2
0.2
,%**
0.1
.+o II o
0
"
i
I
0.1
o
o
~
LINE I:0*r Q2
0.784 Oh|
I
I
t
I
I
I
0.3
0.4
0.5
0.8
03
0.8
I
I
0.8 O h!
Fig. 4. Comparison of tile observed damage ratio Q*(I) during the 1976 earthquake ill the study region with the calculated Q(I) for the corresponding intensities: 4 groups are easily recognized. Grouping is related to the discreteness of the intensity. A relationship is drawn by 3 straight lines: Line 2 is drawn using the centres of mass of the groups and Lines I and 3 are drawn reasonably (see text for details). The far higher Q(I) of the last group than the corresponding Q*(I) indicates a not completely satisfactory extrapolation of It for intensity X. The centres of mass are denoted by open triangles said that the municipality is more vulnerable to earthquake damage than other municipalities of lower ratios. Another possible interpretation is that the scattering has been introduced by the use of a discrete quantity as intensity. In reality both factors must have an influence. The far higher Q(10) values with respect to their corresponding observed values indicate higher values of extrapolated is for intensity X than their true values. This suggests using a function with smaller radius of curvature than that of the exponential function at higher intensities. To find the relationship between the calculated Q(I) and the observed ratio of loss Q*(I), the centres of mass of the groups (groups of intensity VII, VIII, and 1X) are used for fitting with a straight line (Line 2 in Fig. 4). The line is extended by a straight line from the centre of mass of the group of intensity VII to the origin (Line 1 in Fig. 4), assuming no loss would be observed in zones where 0(I) is 0. Tentatively, the line is also extended to the higher value
of Q(I) by another straight line (Line 3 in Fig. 4) from the centre of mass of the group of intensity I X to the more reliable point ofintensity X, since the centre of mass ofthe group of intensity X has a lower 'observed' value than that of the group of intensity IX. The use of the centres of mass to draw straight lines is believed to compensate, to some extent, errors due to the scattering of values according to the second interpretation on scattering given before. The relatively low errors in the parameters of Line 2 in Fig. 4 may show the validity ofusing the centres ofmass for fitting and may indicate that good approximation is reached by using Line 2 in this range. The following relation is obtained: 0.784 Q(1),
Q*(I) = / 1.823 Q(I) "--0.062, t 0.245 Q(I) + 0.424,
for Q(I)~<0.06 for 0.06 ~
9Soil Dynamics and Earthquake Engineering, 1989, Vol.8, No. 2
(27)
101
Seismic risk in Friuli-Venezia Giulia: W-S. Yang et al. I
i
~
,.^',..
I
.-"v w " ' - ~
u
s
r
R
t
a
"'X,...--,.~"~:.k~-'~ .,,.. _ 46" 30'N
9
,,r'-,..~..I,..~...r.~..,
Province BELLUNO
/" ~...-"
..,~"
c..p"
..c
j
t""t
q~
O 46' O0'N -
.~.,,.,-.9
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t,. =
/:~ovlnce
/ ",,,
TREVISO
Province VENEZIA
k.
W "p >
AdrMtic
I 12" 30"E
Fig. 5.
I 13"09" t
I 13'30' E
Map of the maximum M S K intensity expected not to be exceeded with 37% probability in 100 years
Equation (27) is used to correct the calculated values Q(I), to obtain more reliable values of risk. Because of the little information on intensity X and possible large errors in extrapolating F and a to this intensity, this equation is believed more reliable in the range
0 <.Q(I) <~0.31
(28)
RISK MAPPING AND DISCUSSIONS When the Poisson process is assumed for the occurrence of earthquakes, their intensities with a particular re[urn period have a 37% probability of not being exceeded within a time period equal to that return period 1. Then, the intensity 1100, which is not exceeded with a 37% probability within 100 years satisfies N= 1
(29)
where N is the mean number of occurrences of earthquakes in I00 years whose intensities are I or greater. Fig. 5 shows the maximum intensity expected not to be exceeded with 37 % probability in 100 years in the Friuli-
102
eea
Venezia Giulia region obtained by an application of the Cornel114 methodology. We can see that the maximum values (I >1VIII MSK) are reached in the central part of the region with strong attenuation towards South. Similar results for the study area have already been obtained using different methodologies2'4'6. For these intensities, the risk Qm~x=Q(lm~x) is evaluated according to equations (4) and (27) for each municipality in the test region. The number of rooms of each category of each municipality is taken from the 1981 census data 21. Because the Damage Probability Matrix elements as well as the mean replacement cost have not been determined for the rooms built after the 1976 earthquake, the evaluated risk Qm~x should be interpreted as the expected loss ratio for rooms built before 1976, due to an earthquake intensity that is not exceeded with a 37% probability within 100 years. The map showing these values for the mt/nicipalities in the study area is given in Fig. 6. The general picture is that the northern part of the Friuli region is expected to suffer heavier losses than the southern part. The most hazardous municipality in this region turns out to be Forgaria nel Friuli (the black area in Fig. 6) with 35 % risk. Fig. 7 shows the distribution of the values of R, which
Soil Dynamics and Earthquake Engineering, 1989, Vol. 8, No. 2
Seismic risk in Friuli-Venezia Giulia: W-S. }Tang et al. I "
~
.
A
U
S
T
R
I
A
p.--%
~'.,~..,~,. j...1.--
46'30"N
,~
INDEX Ome= ~i 0 0,5
J
0.05 < O m l x
sO.15
015
< Om|x
__.0.25
025
< Omlx
r__ J
0 46" O0'N
/
Pro~
/
,-
)
'a
. . . . . .
9
..... Adriat~.
I 12' 30" E
~ sea
I
_~_
"\
,.-r'';~ > ~ ""
I
13'00" E
13"30' E
Fig. 6. Map of Qm~x corresponding to tile intensity which has the mean return period of 100 years or which is not exceeded with a 37 % probability within 100 years. The northern area is more risky. The most risky municipality turns out to be Forgaria nel Friuli (the black area) with 35 % risk
are obtained by using equations (13) and (27). Again, R is potentially the expected loss ratio of the rooms built before 1976 by earthquakes during the next 100 years. Fig. 7 presents a more detailed picture of the regional variation of risk than Fig. 6. The northern area of the Friuli region shows higher risk than the southern area and there is an increasing concentration of the total risk towards the centre of the northern area. Again, the most hazardous municipality is Forgaria nel Friuli with 95 % total risk. The risk map in the Pordenone province (a province of Friuli-Venezia Giulia) presented by Slejko et al. 1~ generally agrees with the present picture (the risk is defined as the number of houses destroyed or damaged by earthquakes in the Slejko et al. ~~ paper). Since severe design requirements have been adopted after the destructive earthquake of 19762, the real risk for existing rooms is certainly lower than that evaluated with the data on rooms built before 1976. Further study is required to evaluate the Damage Probability Matrix elements for all the room categories as well as their average replacement costs so that a more realistic assessment of risk could be obtained.
CONCLUSIONS The seismic risk of a municipality is quantified as the ratio of repair cost to the replacement cost of all the rooms. Borrowing on the concept of Damage Probability Matrix, we arrive at a formula expressed by equation (4). This risk is the expected damage ratio from the earthquake with a certain probability of occurrence in a fixed time period. The picture showing this value for each municipality in Friuli-Venezia Giulia is presented in Fig. 6. This figure reflects well the higher seismic activity in the northern area of this region with respect to the sourthern part. Considering the possible, even repeated, occurrences of earthquakes in the whole intensity range, we have assigned weights in terms of occurrence probability to the risk in equation (4). The weighting factor has been determined on the assumption of the Poisson process for the earthquake occurrence statistics. With further assumptions, we obtained the set of approximate 9 presented in Fig. 7. The weighted risk, which we call the total risk in a fixed time period, is potentially the sum of the loss ratios from all the possible earthquakes in that
Soil Dynamics and Earthquake Engineering, 1989, Vol.8, No. 2 103
Seismic risk in Friuli-Venezia Giulia: W-S. Yang et al. I
"X.~'~.-~.~.
A
U
S
T
R
I I 'A
Province BEL L UNO
46"30"N
':'~"~'""T'~" --/~" r
. _+..%~,
INDEX
I
"
I
.<, C-
0.20 < R S 0.40 ~
040<'R ~ 0.60"
~..,,:,..,._:_ t,...,a'-,..~ ~
,,/"
c.
/"
<
46"00"N
e,020
.......
J
0.80< R 9 0.80 O.eO< R
o u,
ooR,z, f
Province TRE VISO
VENEZIA
I._2
?..
..~ - / .
Adriatic
'T'"
I 12"30" E
~_r=~.
I 13"00" E
sea
....... \ ....
"\ ..>
%'~'--~ j r
.p
,, [ 13"30" E
Fig. 7. Map of tile total risk in 100 years. The concentric distribution of R is easily observed in the nortllern area of the region. The most risky municipality is again Forgaria nel Friuli with 95 % of tlle total risk
period. Because some restrictions, the evaluated risks, i.e., Q(I) of equation (4) and R of equation (13), only refer to the rooms built before 1976. Therefore the maps give a general picture of possible damage exposure of the old buildings, and they show where strengthening of such buildings is most needed. The highest total risk is concentrated in Forgaria nel Friuli, around which more or less concentric distribution of total risk is observed. The real risk, which should be determined by considering all rooms including those built after the 1976 earthquake, will be less than that depicted in this figure. To determine the real risk further study should be made for a homogeneous set of Damage Probability Matrix elements and a set of average replacement costs of all the room categories, taking into account the structural characteristics of the building types. However, the results obtained show agreement with a previous study in this area. In all cases determination of the DPM elements is unavoidable. Using available data, we have found the distribution of the average damage ratios by the 1976 earthquake in the test region for each room category classified by age and for each MSK intensity degree from
VII to X. These distributions, fitted by a truncated Normal model, have determined the DPM elements. For the unknown distribution of the average damage ratios of intensity VI and the unfavourable set of intensity X, simple functional relations between MSK intensity and the relevant parameters had to be arbitrarily assigned and from them the rest ofthe D P M elements were determined.
ACKNOWLEDGEMENTS Special thanks are due to Ezio Faccioli who has read the manuscript giving important suggestions and specific comments and to Peter Guidotti for checking the manuscript English. One of the present authors (W. S. Yahg) did this work with the support of the ICTP (International Centre for Theoretical Physics, Trieste, Italy) Programme for Training and Research in Italian Laboratories. The study has been undertaken, within the framework of the research unit 'Shakeability and seismic risk in northeastern Italy' (contract n. 86.00183.02) of the 'Gruppo Nazionale di Geofisica della Terra Solida'.
104 Soil Dynanlics and Eartllquake Engineering, 1989, Vol. 8, No. 2
Seismic risk in Friuli-Venezia Giulia: W-S. Yan9 et al. REFERENCES 1 2 3 4 5 6 7 8 9
10
Algermissen, S. T. and Perkins, D. M. A probabilistic estimate of maximum acceleration in rock in the contiguous United States, U.S.G.S. Open-File Report 76-416, Menlo Park, 1976 Faccioli, E. Engineering seismic risk analysis of the Friuli region, Boll. Geof. Teor. Appl., 1976, 21,173-190 Gruppo di Lavoro Scuotibilith, Carte preliminari di scuotibilit~. del territorio nazionale, Pubbl. n. 227 del P.F. Geodinamica del C.N.R., ESA, Roma, 1979 Giorgetti, F., Nieto, D. and Slejko, D. Seismic risk of the FriuliVenezia Giulia region, Publ. Inst. Geophys. Pol. Acad. Sc., 1980, A-9(135), 149-169 Mueller, S. and Mayer-Rosa, D. The new seismic hazard maps for Switzerland, Rerista Geofisica, 1980, 13, 7-19 Slejko, D. and Viezzoli, D., Scuotibilit~ del Friuli-Venezia Giulia utilizzando tecniche anisotropiche, Atti Terzo Convegno G.N.G.T.S., Roma, 1984, 829-842 Makropoulos, K. C. and Burton, P. W. Seismic hazard in Greece. I. Magnitude recurrence, Tectonophysics, 1985, 117, 205-257 Makropoulos, K. C. and Burton, P. W. Seismic hazard in .Greece. II. Ground acceleration, Tectonophysics, 1985, 117, 259294 Ambraseys, N. N. and Jackson, J. A. Earthquake hazard and vulnerability in the northeastern Mediterranean the Corinth cart hquake sequence of February-March 1981, Disasters, 1981, 5, 355-368
Il 12 13 14 15 16 17 18 19 20 21
Slejko, D., Viezzoli, D. and Gasparo, F. Esempio metodologieo del calcolo del rischio sismico, Atti Terzo Convegno G.N.G.T.S., Roma, 1984, 799-810 Ambraseys, N. N. Evaluation of seismic risk, in Seismicity and seismic risk in the offshore North Sea area, (Eds A. R. Ritsema and A. Gurpinar), Reidel Publishing Company, Dordrecht, 1983 Whitman, R. V. and Comell, C. A. Design, in Seismic risk and engineering decisions, (Eds C. Lomnitz and E. Rosenblueth), Elsevier, Amsterdam, 1976 Gutenberg, B. and Richter, C. F. Frequency of earthquakes in California, Bull. Seism. Soc. Am., 1944, 34, 185-188 Cornell, C. S. Engineering seismic risk analysis, B,dl. Seism. Soc. Am., 1968, 58, 1583-1606 Giorgetti, F. Personal communication, 1986 Regione Autonoma Friuli-Venezia Giulia, Stima dei danni provocati dal sisma del 6 maggio 1976, Relazione inedita a cura di L. Di Sopra, 1976 Lawson, C. L. and Hanson, R. J. Solving least square problems, Prentice-Hall, New York, 1974 Istituto Centrale di Statistica, 11~ Censimento generale della popolazione (24 ottobre 1971), Roma, 1972-1973 Benjamin, J. R. and Cornell, C. A. Probability, statistics, and decision for civil engineers, McGraw-Hill, New York, 1970 Peronaci, M. Intensity-magnitude relationships for Italian regions, Boll. Geof. Teor. Appl., 1982, 24, 121-128 Istituto Centrale di Statistica, 12~ Censimento generale della popolazione (24 ottobre 1981), Roma, 1983-1984
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