PROBABILISTIC BENEFIT COST RATIOS FOR SEISMIC RETROFFITING OF BUILDINGS Mario ORDAZ1, Luis YAMIN2, Omar CARDONA3, Miguel MORA4
SUMMARY Existing infrastructure might be required to be retrofitted in order to maintain an adequate level of the seismic risk to which buildings are exposed. The decision for retrofitting involves different aspects among which the most important is the cost effectiveness of retrofitting. A methodology for evaluation of the probability distribution of the benefit-cost ratio is developed, providing a ground-breaking tool for decision-makers to analyze the net benefits of the risk mitigation measures, such as the earthquake retrofitting and the seismic code enforcement. An analytical solution for the probability distribution of the net present value of losses is presented. The results of the analytical development are verified using Monte Carlo simulation techniques. The paper describes the model and the derived tools, presenting a complete case study for three groups of public buildings in the city of Bogota, Colombia, corresponding to the educational, health and administrative sectors. The analysis permits to obtain the probability distribution for the net present value of the losses, for both the unretrofitted and the retrofitted conditions of the structures, thus allowing the determination of the probability that the net present value of savings (net present value for losses for the unretrofitted structure minus equivalent losses for the retrofitted state) is greater than the cost of retrofitting the structure at the present conditions. The methodology finally allows a rational analysis of different rehabilitations alternatives in order to have analytical parameters for final decision making.
1
Professor, Institute of Engineering, Universidad Nacional Autónoma de México (UNAM), México DF, Email:
[email protected] 2
Director - CEDERI, Universidad de Los Andes, Bogotá DC, Colombia,. Email:
[email protected] Research Engineer - CEDERI, Universidad de Los Andes, Bogotá DC, Colombia Email :
[email protected] 4 Research Engineer - CEDERI, Universidad de Los Andes, Bogotá DC, Colombia. Email :
[email protected] 3
INTRODUCTION The decision of retrofitting buildings or infrastructure can be motivated by a number of reasons. Perhaps the most common is the appearance and enforcement of new seismic design codes for buildings, which usually require higher standards. In this case, the owner of the building must, of course, comply with the new standards. But since codes establish minimum requirements, the owner could consider the possibility of attaining even higher standards, spending more money, but reducing the seismic risk to a more comfortable level. There are also cases in which, even without the compulsory upgrading dictated by a new seismic design code, the owner may wish to explore the possibility of retrofitting a structure in order to improve its performance, thus lowering its associated seismic risk, which is perhaps regarded as excessive. This is the case, for instance, of buildings of special human or historical value. From the purely economic point of view, the key issue in the decision-making process is the cost effectiveness of the retrofitting actions. In other words, how much is seismic risk reduced per dollar spent in retrofitting. In order to rationally estimate this cost effectiveness, several things must be known. First, the costs associated to several retrofitting alternatives. Second, the seismic risk associated to the present state of the structure and, also, the risk associated to the structure after several retrofitting schemes. In this paper, we will use as a measure of risk the net present value of the losses due to all future earthquakes. In the traditional approach (Referencia?), the best retrofitting alternative would be the one with highest expected savings per dollar invested, that is, the maximum expected benefit-cost ratio. However, recently (Referencia) it has been recognized that, due to its stochastic nature, the net present value of the earthquake losses is an extremely uncertain quantity, so rational decisions cannot be reached by looking only to its expected value, thus disregarding its associated uncertainty. Rather, it has been proposed (Reference) to compute the probability of having a benefit-cost ratio greater than unity, and choosing the alternative for which this probability is maximum. In this paper, we propose a method to estimate the probability distribution of the net present value of the losses due to earthquakes, and a method to compute the probability of having a benefit-cost ratio greater than unity. These methods constitute useful analytical tools to guide the decision-making process and also, as it will be presented later, to measure in economic terms the effectiveness of retrofitting actions, which is one of the main incentives for taking risk mitigation measures.
PROBABILITY DISTRIBUTION OF APPROXIMATE ANALYTICAL SOLUTION
COST-BENEFIT
RATIO-
We are interested in finding the probability distribution of a general cost-benefit ratio, Q, which is defined as follows: Q = (LU- LR)/R.
(1)
where LU is the net present value of the losses due to all future earthquakes for the present structural condition, LR is the net present value of the losses due to all future earthquakes for the retrofitted structural condition, and R is the initial cost of structural retrofitting. In this paper, R is deterministic and is generally evaluated based on previous similar retrofitting projects. The value of R varies depending on the degree of intervention on the structure. For the present analysis it is considered a fixed value defined by the design level achieved after retrofitting to certain higher standards Retrofitting usually involves additional costs related to replacement or repair of nonstructural elements such as windows, ceilings, floors, old pipes and also general maintenance of the structure. Also, some retrofitting processes are usually associated to a general upgrading of the construction or even a complete renovation, including a new architectonic upgrade. In those cases, and in order to make a fare evaluation, only the direct structural retrofitting costs should be considered in the proposed analysis. Model for the loss-occurrence process We will assume that earthquake losses occur in time following a stochastic process in which both the time of occurrence and the value of the loss are uncertain. In order to completely define this process, the probability distributions of all times of occurrence and all loss values have to be known. Regarding the loss values, we will assume that losses during earthquakes are all independent and identically distributed, with probability density function p(βi), where βi is the value of the loss during the i-th earthquake. p(βi) can be obtained from a risk analysis, whose results are usually presented in terms of exceedance rates of loss values. The exceedance rate of loss βι, ν(βι), is the average number of events, per unit time, in which losses are equal o greater than βι. The inverse of the exceedance rate is the return period of the loss. ν(βι) and p(βi) are related through the following expression:
p( β i ) = −
1 dν ( β i ) ν 0 dβ i
(2)
whereν0=ν(0) is the average number of events, per unit time, that produce losses greater or equal than zero.
Regarding occurrences in time, we will assume that earthquake occurrence follows a Poisson process. Hence, the times between events are uncorrelated and exponentially distributed with parameter ν0. In view of this assumption, the time to the i-th event, ti, has the following Gamma distribution: pi (ti ) =
tii−1e −ν 0tiν 0i Γ (i )
(3)
where Γ(.) is the Gamma function. We will also make the assumption that the loss values and their times of occurrence are uncorrelated. First two statistical moments of L The present value of all the future losses due to earthquakes can be computed as: ∞
L = ∑ β i e −γti
(4)
i =1
where βi is the loss inflicted by earthquake i, which in turn takes place at time ti, and γ is the discount rate. Note that both the losses and the time of occurrence of the i-th earthquake are unknown. The sum in equation 4 extends to infinity to account for all future events. We shall now define
yi = exp(−γti )
(5)
This is an auxiliary random variable, with domain in (0,1), whose probability density function can be found to be:
p yi ( yi ) =
ν 0 / γ −1
(ν 0 / γ )i yi
(− ln( yi ))i−1 Γ (i)
(6)
or, making α=ν0/γ,
p yi ( yi ) =
α i yiα −1 (− ln( yi ))i−1 Γ (i )
(7)
It can be verified that the integral of equation 7 between 0 and 1 is unity. Furthermore, its moment-generation function is given by:
⎛ α ⎞ E ( yi ) = ⎜ ⎟ ⎝ k +α ⎠
i
k
(8)
where the operator E() stands for expected value. In particular, the expected value of yi is:
⎛ α ⎞ E ( yi ) = ⎜ ⎟ ⎝1+ α ⎠
i
(9)
We are now ready to compute the first two statistical moments of L. First, its expected value: ∞
∞
∞
i =1
i =1
i =1
E ( L) = ∑ E ( β i e −γti ) = ∑ E ( β i ) E (e −γti ) = E ( β )∑ E ( y i )
(10)
where we have replaced E(βi) with E(β) since all βi’s, which are the losses for any event, are identically distributed, and we have taken advantage of the fact that they are uncorrelated with ti. Replacing equation 9 in equation 10: ∞ ⎛ α ⎞ E ( L ) = E ( β )∑ ⎜ ⎟ i =1 ⎝ 1 + α ⎠
i
(11)
The infinite series in equation 11 is known to converge: i
⎛ α ⎞ ⎜ ⎟ =α ∑ i =1 ⎝ 1 + α ⎠ ∞
(12)
So, replacing this result and α=ν0/γ in equation 11, we find that: E ( L) = E ( β )
ν0 γ
(13)
which is the well-known result due to Rosenblueth (1976). We will now proceed to compute the second moment of L: 2
∞ ∞ ⎛ ∞ ⎞ L2 = ⎜ ∑ β i e −γti ⎟ = ∑ β i2e −2γti + 2∑ i =1 i =1 ⎝ i=1 ⎠
∞
∑β β e
j =i +1
i
j
−γ ( ti +t j )
(14)
Since the losses due to different earthquakes are independent among them, we observe that: ∞
∞
i =1
i =1
∞
E ( L2 ) = E ( β 2 )∑ E (e −2γti ) + 2 E 2 ( β )∑
∑ E (e
−γ ( ti +t j )
)
(15)
j =i +1
Also, since we know the distribution of ti, we can determine that: E (e
− 2γti
⎛ ν0 ) = ⎜⎜ ⎝ ν 0 + 2γ
⎞ ⎟⎟ ⎠
i
(16)
and, in consequence, ∞
E ( β ) ∑ E (e 2
−2γti
i =1
⎛ ν0 ) = E ( β )∑ ⎜⎜ i =1 ⎝ ν 0 + 2γ ∞
2
⎞ ⎟⎟ ⎠
i
(17)
Again, the infinite sum in equation 17 is known to converge to:
⎛ ν0 ⎜⎜ ∑ i =1 ⎝ ν 0 + 2γ ∞
i
⎞ ν0 ⎟⎟ = ⎠ 2γ
(18)
so we find that ∞
E ( β 2 )∑ E (e −2γti ) = E ( β 2 ) i =1
ν0 2γ
(19)
In the double summation of equation 15, we observe that we need to compute − γ ( t +t ) E (e i j ) . But times ti and tj are not independent. We observe, however, that in the double summation, j > i and that tj is composed of ti plus the sum of the remaining j-i times between earthquakes, τk, which we know to be exponentially distributed. In other words, t j = ti +
j
∑τ
k =i +1
k
(20)
where al the τk’s have exponential distribution. In consequence, e
−γ (t i + t j )
j j ⎞ ⎛ = exp⎜⎜ − γ (ti + ti + ∑τ k ) ⎟⎟ = exp(− 2γti )exp(−γ ∑τ k ) k = i +1 k = i +1 ⎠ ⎝
(21)
Under these circumstances, ti is independent of all τk’s, and hence:
(
Ee
−γ (ti + t j )
) = E (exp(− 2γt ))E⎛⎜⎜ exp(−γ ∑τ ) ⎞⎟⎟ j
i
⎝
k = i +1
k
(22)
⎠
The term E (e −2γti ) was already computed (see equation 16), while, in view of the independence among the τk’s and of the fact that all of them are exponentially distributed with parameter ν0, we have that j ∞ j ⎛ ⎞ E ⎜⎜ exp(−γ ∑τ k ) ⎟⎟ = ∏ ∫ exp(−γτ k )ν 0 exp(−ν 0τ k )dτ k k = i +1 ⎝ ⎠ k = i +10
(23)
From this expression we can find that j ⎛ ⎞ ⎛ ν E ⎜⎜ exp(−γ ∑τ k ) ⎟⎟ = ⎜⎜ 0 k =i +1 ⎝ ⎠ ⎝ν 0 + γ
⎞ ⎟⎟ ⎠
j −i
(24)
Replacing equations 24 and 16 in equation 22 it results that:
(
Ee
−γ (t i + t j )
)
⎛ ν0 = ⎜⎜ ⎝ ν 0 + 2γ
i
⎞ ⎛ ν0 ⎟⎟ ⎜⎜ ⎠ ⎝ν 0 + γ
⎞ ⎟⎟ ⎠
j −i
(25)
Furthermore, ∞
∞
i =1
j =i +1
∑ ∑ E (e
−γ ( ti +t j )
∞
)=∑ i =1
⎛ ν0 ⎜⎜ ∑ j =i +1⎝ ν 0 + 2γ ∞
i
⎞ ⎛ ν0 ⎟⎟ ⎜⎜ ⎠ ⎝ν 0 + γ
⎞ ⎟⎟ ⎠
j −i
(26)
The inner sum converges, so we find that ∞
∑
∞
∑ E (e
i =1
j =i +1
∞
∞
i =1
j =i +1
−γ ( ti +t j )
∞ ⎛ ν0 ) = ∑ ⎜⎜ i =1 ⎝ ν 0 + 2γ
i
⎞ ν0 ⎟⎟ ⎠ γ
(27)
Finally:
∑ ∑ E (e
−γ ( ti +t j )
1 ⎛ν ⎞ ) = ⎜⎜ 0 ⎟⎟ 2⎝ γ ⎠
2
(28)
Replacing equations 19 and 28 in equation 15, we obtain that
⎛ν ⎞ ν E ( L ) = E ( β ) 0 + E 2 ( β )⎜⎜ 0 ⎟⎟ 2γ ⎝γ ⎠ 2
2
2
(29)
Recalling that E ( L) = E ( β )
ν0 , and that VAR(L)=E(L2)-E2(L), where VAR() stands for γ
variance, we find that VAR ( L) = E ( β 2 )
ν0 2γ
(30)
Equations 13 and 30 give the mean and the variance, respectively, of the present value of the losses due to all future earthquakes, L, in terms of the two first moments of the loss during randomly chosen earthquakes, and the parameter controlling the occurrence process with time, ν0. Let βA be the annual loss, that is, the sum of all losses accumulated during one year. It is easy to show that E (β A ) = ν 0 E (β )
(31)
VAR( β A ) = ν 0 E ( β 2 )
(32)
in view of which, E ( L) =
E (β A )
VAR( L) =
γ
VAR( β A ) 2γ
(33)
(34)
These two last equations usually present numerical advantages over equations 13 and 30 to compute E(L) and VAR(L), since to use them there is no need of an explicit knowledge of ν0. Moreover, E(βA) is the technical or pure premium, a value usually reported in risk analyses. It is interesting to note that, according to equation 33, the expected present value of the losses, from here to eternity, is the expected loss in one year divided by the discount rate. Approximate probability density function of L
In the preceding chapter we determined the first two statistical moments of L. We will proceed to investigate other features of its probability distribution. In principle, one could think that L is normally distributed, since it is computed as the sum of a potentially large number of random variables (the losses during each earthquake). However, for usual values of γ (2−5% per year), the number of terms that really contribute to L (see equation 4) is small. We have verified, through simulations, that normality is usually a poor approximation.
The i-th term in L is β i exp(−γti ) = β i yi . The probability density functions of βi and yi are given in equations 2 and 7, respectively. We have observed that the probability distribution of yi can be approximated very precisely with a Beta distribution (in fact, the Beta distribution is a limit case of the one given in equation 7). In turn, if the losses are normalized with respect to the total sum exposed, it is common to find that βi has a Beta-like distribution, which can be approximated with a Beta distribution in the range (0,1) with reasonable precision. We have, then, that every term in L is, approximately, the product of two Betadistributed, independent random variables. Surprisingly, we have observed that the product of such two variables can also be approximated with a Beta distribution. We would have in this case that L is formed with the sum of random variables that are approximately Beta distributed. Surprisingly again, simulations have shown us that the sum of k Beta distributed random variables can be approximated with another Beta distribution, but in the range (0,k). Under these circumstances, the probability density function of L would approximately be: pL ( L) =
Lc−1 (k − L) d −1 Γ (c + d ) k c+d −1Γ (c)Γ (d )
(35)
whose moment-generation function is: E ( Lm ) =
k m Γ ( c + d )Γ ( c + m ) Γ ( c )Γ ( c + d + m )
(36)
It would then suffice to compute E(L) and E(L2) with expressions 13 and 30 (or 33 and 34), and determine the values of coefficients c and d of equation 35, by solving simultaneously the equations that result from making m=1 and 2 in equation 36. It could then be found that: c=
k − E ( L)(1 + C 2 ( L)) k C 2 ( L)
d =c
k − E ( L) E ( L)
(37)
(38)
where C ( L) = VAR( L) / E ( L) is the coefficient of variation of L. Expressions 37 and 38, however, are not useful enough since we do not know k, the number of terms that effectively contribute to L. If k were known, the probability density function of L would be the following:
k −ε (1+C 2 ) −1 k C2
L pL ( L) =
k
(k − L)
k −ε (1+C 2 ) k −ε −1 ε k C2
⎛ k − ε (1 + C 2 ) k − ε (1 + C 2 ) k − ε ⎞ ⎟ + ε ⎟⎠ k C2 k C2 ⎝
Γ ⎜⎜
k −ε (1+C 2 ) k −ε (1+C 2 ) k −ε + −1 ε k C2 k C2
⎛ k − ε (1 + C 2 ) ⎞ ⎛ k − ε (1 + C 2 ) k − ε ⎟⎟Γ ⎜⎜ Γ ⎜⎜ ε k C2 k C2 ⎠ ⎝ ⎝
⎞ ⎟⎟ ⎠
(39)
where, for the sake of simplicity in the notation, we have made E(L)=ε and C(L)=C. This expression results from replacing equations 37 and 38 in equation 35. If we take the limit of pL(L) when k tends to infinity, that is, when the expected value of L is much smaller than the maximum possible value of L (which is k), we find:
lim
pL ( L) k →∞
=
L1 / C
2
−1
( )
L ) εC 2 εC 2 Γ (1 / C 2 )
exp(−
1/ C2
(40)
If we make r=1/C2 and λ=1/εC2, we recognize a Gamma distribution for L: . Lr −1 exp(−λL)λr Γ (r )
pG ( L) =
(41)
Under this distribution, the expected value and coefficient of variation of L are: E ( L) =
C ( L) =
r
=ε
(42)
1 =C r
(43)
λ
that is, the same first two moments of the original Beta distribution given in equations 35 or 39. We conclude that, for the very common situations in which E(L) is small as compared with the maximum possible value of L, it tends to have a Gamma distribution with the first two moments computed using equations 13 and 30 or 33 and 34.
Proposed approximation According to what has been presented, our approximation for the probability distribution of L consists of: 1) Compute the exact first two moments of L using equations 13 and 30 (or 33 and 34) 2) Assign L a Gamma distribution (see equation 41) with parameters r=1/C2(L) and λ=1/(Ε(L)C2(L)), where C(L) is the coefficient of variation of L. We are interested, however, in evaluating the probability of having a positive benefitcost ratio, Q, that is, Pr(Q>1). The present values of the losses in the unretrofitted state
(LU) and in the retrofitted state (LR) are random variables whose probability distributions, according to what has been shown in the preceding paragraphs, can be assumed to be of the Gamma form (see equation 41), with known parameters. Say that both the unretrofitted and the retrofitted state have been analyzed, and that we have concluded that their corresponding probability density functions are, respectively: pU ( L) =
pR ( L ) =
LrU −1 exp(−λU L)λ rU Γ(rU )
(44)
LrR −1 exp(−λR L)λrR Γ(rR )
(45)
Then, recalling the definition of Q, Pr(Q > 1) = Pr( LU − L R > R )
(46) We shall assume that LU and LR are independent (they correspond to different infinite earthquake sequences), so, for positive values of R, ∞
Pr( LU − L R > R ) =
∞
∫ ∫p 0
R ( x ) pU
( y ) dxdy
R+ y
(47)
As to the authors’ knowledge, this double integral has no analytical solution, but we have found that it can be numerically integrated using this alternative form: ∞
∫
Pr( LU − L R > R) = 1 − Gac( R + y; rU , λU ) p R ( y )dy 0
(48)
where Gac(x;r,λ) is the cumulative Gamma function, given by: x
Gac ( x; r , λ ) = ∫ 0
y r −1 exp( −λy )λr dy Γ (r )
for which various numerical approximations exist.
(49)
CASE STUDY
Portfolio for analysis The evaluation of the cost-benefit ratio is carried out for a group of public buildings in three public sectors: educational, health and administrative buildings, for the city of Bogotá, Colombia. For these buildings, detailed structural and economic information is readily available [1]. For the purpose of this analysis, a data base was built using different sources of information [1] [2] [3]. Table 1 summarizes the information available for the three groups of buildings, discriminating replacement values in direct, contents and business interruption, BI, values.
Table 1 Summary of information available for analysis S ec tor
No B uilding s
R eplac ement % P artic ipation value + c ontents % P artic ipation
Education Health Administrative Total
691 39 94 824
[x 10 US D]
[x10 US D]
83.9% 4.7% 11.4% 100.0%
$ 250,114 $ 25,542 $ 49,489 $ 325,146
R eplac ement value 3
3
[units ]
76.9% 7.9% 15.2% 100.0%
$ 146,902 $ 17,676 $ 31,352 $ 195,930
R eplac ement c ontent 3
[x10 US D]
$ 103,212 $ 7,867 $ 18,137 $ 129,216
R eplac ement BI 3
[x10 US D]
$ 164,530 $ 35,351 $ 35,114 $ 234,996
For each group of buildings a complete set of information was gathered from different sources. First of all, information was collected from official cadastral data bases. In addition, some technical visits were performed in order to complete information related to special characteristics for important buildings. Finally, some correlations and proxies were used (ERN, 2006-2007) in order to complete all required information. In summary, each building from the data base includes the following specific parameters: location, replacement value, contents value, business interruption reference value, number of stories, type of structural system, approximate year of construction, presence or not of previous damages, present condition, and specific information related to nonsymmetrical geometry, presence of short columns, corner buildings, possibility of interaction with adjacent buildings and other critical structural characteristics.
Risk Analysis Method Risk analysis is performed using software RN-COL (ERN, 2006-2007). The general analysis in order to determine cost-benefit ratios consists of the following steps: (a) A preliminary risk analysis is performed for the three groups of buildings in order to select the most critical seismic scenario. An earthquake with magnitude Ms=7.3 originating in the Eastern Cordillera Frontal Fault, at 60 km from Bogotá turns out to be the most critical event for the portfolio. (b) For the most critical event, a risk analysis is performed in order to classify buildings from the three groups according to the level of damage expected.
(c) Buildings of high vulnerability (unreinforced masonry, adobe wall constructions, moment resisting frames with more than 5 stories and more than 10 years old) and those observed to present excessive damage (mean damage ratio greater than 10%) for the most critical event, are subjected to a hypothetical retrofitting strategy. Actual retrofitting cost estimates are made based on actual budgets from retrofitting projects recently implemented in public buildings in Colombia. (d) A new risk analysis is performed for the hypothetical rehabilitated data base, for buildings of each sector. Retrofitting is considered by means of a change in the type of structural system and/or a change in the vulnerability curve. Expected damage ratio for the most critical earthquake scenario and expected annual losses are estimated for the three portfolios considered. The analysis is performed for each group of buildings in an individual manner and also for a unique portfolio of public buildings, constructed by joining together the three data bases. Risk associated to buildings and contents are computed using software RS-COL. Business interruption costs are included in the analysis through an approximate correlation between direct and indirect costs related to the mean damage ratio of the building structure itself and the estimated time for recovery.
Estimation of Retrofitting Costs In order to have a reasonable estimation of retrofitting costs, we considered a group of retrofitting projects directly related to similar buildings as those considered in the present analysis. Table 2 presents a summary of unit cost per area of construction of different types of buildings (adapted from reference [2]). Only structural retrofitting is considered in the previous figures. No architectonic or functional retrofitting is considered.
Table 2 Retrofitting cost estimates for different construction types Building Structure Moment resisting reinforce concrete frames for office buildings (5 to 10 stories)
Reinforced conrete moment resisting frames with infilled unreinforced masonry walls (5 to 10 stories) Partially confined masonry walls (1 to 3 stories) Reinforced conrete moment resisting frames with infilled unreinforced masonry walls with partial or incomplete diaphragm configuration (2 and 6 stories)
Retrofitting strategy Reinforced concrete share walls Stell bracings Colums reinforcement and foundation retrofitting Reinforced concrete walls and reinforced concrete elements confining masonry walls Steel plates as reinforcement of walls Reinforced concrete walls and reinforced concrete elements confining masonry walls + slab reinforcement
Range of retrofitting cost $ 91 USD/m2 $ 136 USD/m2 $ 68 USD/m2
$ 114 USD/m2
$ 182 USD/m2
$ 227 USD/m2
$ 45 USD/m2
$ 91 USD/m2
$ 23 USD/m2
$ 45 USD/m2
$ 409 USD/m2
$ 591 USD/m2
Model Application Table 3 summarizes the retrofitting strategy adopted and the costs involved.
Table 3 Seismic retrofitting strategy and costs S ec tor No. B uilding s No. R etrofitted B uilding s R eplac ement direc t value R eplac ement c ontents v alue B I value T otal replac ement+B I value S truc tural retroffitng c os t % of direc t value
Education 691 222 $ 146,902 $ 103,212 $ 164,530 $ 414,644 $ 34,065 23.2%
[units ] [units ] 3
[x10 US D] 3
[x10 US D] 3
[x10 US D] 3
[x10 US D] 3
[x10 US D] [% ]
Health 39 24 $ 17,676 $ 7,867 $ 35,351 $ 60,894 $ 6,113 34.6%
Administrative 94 73 $ 31,352 $ 18,137 $ 35,114 $ 84,604 $ 21,639 69.0%
Total 824 319 $ 195,930 $ 129,216 $ 234,996 $ 560,142 $ 61,818 31.6%
Table 4 summarizes the expected damages for the most critical scenario before and after the retrofitting strategy. In addition, expected annual losses are estimated again for unretrofitted and retrofitted structures.
Table 4 Expected damages and losses S ec tor Direct Unretroffited + Contents E x pec ted + BI Damag e (c ritic al Direct s c enario) R etroffited + Contents + BI Direct Unretroffited + Contents + BI E x pec ted Annual Direct L os s R etroffited + Contents + BI
Education 3
[x10 USD]
$ 23,734 $ 40,410 $ 82,318 $ 2,632 $ 3,556 $ 4,370 $ 334 $ 569 $ 1,085 $ 30 $ 41 $ 46
Health 3
[x10 USD]
[%]
16.2% 16.2% 19.9% 1.8% 1.4% 1.1% 2.3‰ 2.3‰ 2.6‰ 0.2‰ 0.2‰ 0.1‰
$ 2,595 $ 3,750 $ 11,293 $ 1,611 $ 1,969 $ 3,845 $ 71 $ 103 $ 275 $ 45 $ 55 $ 77
Administrative 3
[x10 USD]
[%]
14.7% 14.7% 18.5% 9.1% 7.7% 4.9% 4.0‰ 4.0‰ 4.5‰ 2.5‰ 2.1‰ 1.0‰
$ 5,211 $ 8,226 $ 17,001 $ 1,631 $ 2,102 $ 2,716 $ 137 $ 216 $ 405 $ 37 $ 47 $ 54
[%]
16.6% 16.6% 20.1% 5.2% 4.2% 3.2% 4.4‰ 4.4‰ 4.8‰ 1.2‰ 1.0‰ 0.6‰
The exceedance rates of loss for βι , ν(βι), which correspond to the average number of events per unit time for which losses are equal o greater than βι, are presented in Figure 1, for the three groups of buildings and for both conditions of analysis, unretrofitted and retrofitted. As noted before, the inverse of the exceedance rate is the return period of the loss. Also, using equation 1, probability density functions for losses are calculated and presented in Figure 2 for the different conditions of analysis.
Figure 1 Exceedance rate of loss Education
Health 1E+00
Eexceedance rate ν
Retroffited Unretroffited
1E-01 1E-02 1E-03 1E-04
1E+00
Retrof ited
Exceedance rate ν
1E+00
Exceedance rate ν
Administrative Unretrofited
1E-01 1E-02 1E-03 1E-04 1E-05
$0
$2
$4
$6
$8
$ 10
Retrofitted Unretrofitted
1E-01 1E-02 1E-03 1E-04
$0
$2
$4
$6
$8
Millions
$ 10
$0
$2
$4
$6
$8
Millions
PML - β [USD]
$ 10 Millions
PML - β [USD]
Loss - β [USD]
Figure 2 Probability density functions Education
Health p( βi) = −
1.E-02
1 dν ( β i ) dβ i
νo
p(βi) = −
1.E-03
Retroffited Unretroffited
1.E-04
1.E-01
1.E-02
p (βi)
p (βi)
1.E-03
1.E-05
Administrative
1.E-01
1 dν ( β i ) ν o dβ i
Unretrofited
1.E-05
1.E-04 1.E-05
1.E-06
1.E-06
1.E-07
1.E-07
1.E-07
1.E-08 $0
$2
$4
$6
$8
$ 10
1.E-08
$0
$1
$2
$3
$4
$5
Millions
Loss - β [ USD]
Retrofitted
1.E-03
1.E-06
1.E-08
1 dν ( β i ) dβ i
νo
Unretrofitted
Retrofited
1.E-04
p(βi) = −
1.E-02
p (βi)
1.E-01
Loss - β [ USD]
$6
$7
$8
Millions
$0
$2
$4
$6
Loss - β [ USD]
$8
$ 10 Millions
Monte Carlo Simulation Using Monte Carlo simulation, a group of hypothetical seismic events were generated, using the probability density functions of losses given in Figure 2. The resulting empirical loss distribution is shown in Figure 3. A total of 100 stochastic events where generated for each condition of analysis
Figure 3 Cumulative density function and random βi events CDF de β curve
1.000
P (ββi) Retrof ited
0.3
Γ Retrof ited
0.3
0.2
P(β>βi) Unretrof ited
0.2
Γ Unretrof ited
0.2
Γ Retrof ited
0.1
Γ Unretrof ited
0 $0
$ 10
$ 20
$ 30
Loss β [USD]
Benefit cost ratio
$ 40
$ 50
M illions
P(β>βi) Unretrofited Γ Retrofited Γ Unretrofited
0
0
$ 60
P(β>βi) Retrofited
0.1
0.1 $0
$1
$2
$3
$4
$5
Loss β [USD]
$6
$7
$8 M illions
$0
$2
$4
$6
$8
Loss β [USD]
$ 10
$ 12
$ 14 M illions
The expected value and variance of the net present values of the losses, E(L) and VAR(L) respectively, were calculated using equations 33 and 34 and numerical values from Table 4 Then, the Gamma distribution parameters for unretrofitted and retrofitted losses were calculated with equation 42 and 43 and the expected value of the benefit-cost ratio defined in equation 1 was calculated using the following expression: E (Q) =
E ( LU ) − E ( L R ) R
(52)
Finally, the probability of having Q>1 or LU-LR>R was calculated using equation 48 and 49. Table 5 summarizes such results for the different conditions of analysis.
Table 5 Expected value and probability of Q Building Building+Contents Building+Contents+BI Unretrofitted Retrofitted Unretrofitted Retrofitted Unretrofitted Retrofitted 24,808 2,251 41,968 3,040 80,571 3,372 259,761,297 3,047,933 752,705,117 5,701,575 3,315,317,234 8,166,923 2.37 1.66 2.34 1.62 1.96 1.39 10,471 1,354 17,935 1,876 41,148 2,422 0.30 0.51 1.01 47.3% 45.8% 42.9% 5,327 3,337 7,697 4,078 20,523 5,754 5,776,099 2,579,307 12,061,608 3,948,861 102,657,790 12,497,116 4.91 4.32 4.91 4.21 4.10 2.65 1,084 773 1,567 968 5,002 2,172 0.15 0.26 1.08 94.7% 90.3% 67.9% 10,225 2,721 16,140 3,507 30,161 4,005 22,359,324 2,522,761 55,712,443 4,380,307 228,655,822 7,157,018 4.68 2.94 4.68 2.81 3.98 2.24 2,187 927 3,452 1,249 7,581 1,787 0.15 0.26 0.54 68.7% 64.2% 54.8%
E(L) VAR(L ) λ r E(Q) Pr( Q>E(Q) ) E(L) VAR(L )
Education Cost-benefint ratio Health
λ r E(Q) Pr( Q>E(Q) ) E(L) VAR(L ) λ r E(Q) Pr( Q>E(Q) )
Cost-benefint ratio Administrative Cost-benefint ratio
Interpretation of results Figure 5 presents the cumulative probability distribution of the benefit-cost ratio for the different conditions of analysis.
Figure 5 Cost benefit probability distribution Education
Health
1.000
1.000
0.900
0.900
0.900
0.800
Pr(L U-L R>R)
0.700 0.600 0.500 0.400
Pr(Q>1)=0.44
0.300
0.800
0.700
0.700
Pr(Q>1)=0.73
0.600
Pr(L U-LR>R)
0.800
Pr(L U-LR>R)
Administrative
1.000
0.500 0.400 0.300 0.200
0.200
0.100
0.100
0.000 0.0
0.5
1.0
1.5
0.500 0.400 0.300 0.200 0.100
0.0
0.000
0.600
2.0
0.5
1.0
1.5
2.0
Pr(Q>1)=0.12 0.000 0.0
Q=(L U -L R )/R
0.5
Q=(L U -L R )/R Building+Contents+Profit
Building+Contents
1.0
1.5
2.0
Q=(L U -L R )/R Building
Building+Contents+BI
Building+Contents
Building
Building+Contents+BI
Building+Contents
Building
From Figure 5 it can be concluded that the probability of having a benefit-cost ratio greater than one, (Q>1 or Lu-LR>R), would be (44%, 73%, 12%) for the three sectors analyzed, and considering direct, content and business interruption costs. On the other hand if we only consider direct costs, the equivalent probabilities would be (0.14%, 0.5%, 0%)
REFERENCES ERN-Colombia, (2007) Diseño de la Estrategia de Aseguramiento para la Protección Financiera de las Edificaciones Públicas Distritales Establecidas en la Ciudad de Bogotá D.C. en el Caso de Desastres Naturales, Secretaría Distrital de Hacienda, Bogotá ERN-Colombia, (2006) [1] Economic Loss Estimation of Different Seismic Risk Scenarios for Public and Private Buildings in Bogotá and Economic Analysis of Residual Risk to the City, Bacno Mundial Fondo Japones, Ministerios de Ambiente Vivienda y Desarrollo Territorial y Fondo Fianaciero para Proyectos de Desarrollo, Bogotá Arámbula, S., Ordaz, M., Yamin, L.E. and Cardona, O.D. (2001) Evaluación de Pérdidas por Sismo en Colombia: Aplicación a la Industria Aseguradora, VII Seminario Internacional y Primer Congreso Nacional de Ingeniería Sísmica, Bogotá. Cardona O.D., Hurtado J.E (2000) Holistic seismic risk estimation of a metropolitan center, in Proceedings of 12th World Conference of Earthquake Engineering, Auckland, New Zealand. Cardona, O.D. Ordaz, M.G., Arámbula, S., Yamín, L.E., Mahul, O. and Ghesquiere, F. (2006) “Detailed earthquake loss estimation model for comprehensive risk management”, Proceedings of First European Conference on Earthquake Engineering and Seismology, European Association of Earthquake Engineering EAEE, Geneva. Carreño M.L., Cardona O.D., Barbat A.H. (2007) Urban Seismic Risk Evaluation: A Holistic Approach, Natural Hazards vol 40 num 1, pp 137:172, DOI 10.1007/s11069-006-0008-8. CEDERI, (2005) Earthquake Loss and Risk Scenarios for Bogota City, Dirección de prevención y atención de emergencies, Bogotá. CEDERI, (2004) Strategy for Transfer, Retention and Seismic Risk Mitigation for Community Essential Buildings, Departamento Nacional de Planeación (DNP), Agencia Colombiana Cooperación Internacional (ACCI), Bogota. EMI (2006) Megacity Indicators System MIS: Implementation in Metro Manila, 3cd program of Earthquake and Megacities Inititative EMI, Pacific Disaster Center, ProVention Consortium. ERN-Colombia, (2007) Diseño de la Estrategia de Aseguramiento para la Protección Financiera de las Edificaciones Públicas Distritales Establecidas en la Ciudad de Bogotá D.C. en el Caso de Desastres Naturales, Secretaría Distrital de Hacienda, Bogotá ERN-Colombia, (2006) [1] Economic Loss Estimation of Different Seismic Risk Scenarios for Public and Private Buildings in Bogotá and Economic Analysis of Residual Risk to the City, Bacno Mundial Fondo Japones, Ministerios de Ambiente Vivienda y Desarrollo Territorial y Fondo Fianaciero para Proyectos de Desarrollo, Bogotá ERN-Colombia, (2005a) Definición de la Responsabilidad del Estado, su Exposición ante Desastres Naturales y Diseño de Mecanismos para la Cobertura de los Riesgos Residuales del Estado, Reports prepared for Departamento Nacional de Planeación (DNP), Agencia Colombiana Cooperación Internacional (ACCI) and the World Bank, Bogotá. ERN-Manizales, (2005b) Diseño de Esquemas de Transferencia de Riesgo para la Protección Financiera de Edificaciones Públicas y Privadas en Manizales en el Caso de Desastres por Eventos Naturales, Reports prepared for DNP, ACCI and the World Bank, Bogotá. Esteva, L. (1970) Regionalización sísmica para fines de ingeniería; serie azul, Instituto de Ingeniería, UNAM. Miranda, E. (1999) Approximate seismic lateral deformation demands on multistory buildings, Journal of Structural Engineering, Vol 125 No 4, 417-425. Ordaz, M; Miranda, E; Reinoso, E and Pérez-Rocha, L.E. (1998) Seismic Loss Estimation Model for México City, Universidad Nacional Autónoma de México, México DF.
Ordaz, M. (2000) Metodología para la Evaluación del Riesgo Sísmico Enfocada a la Gerencia de Seguros por Terremoto, Universidad Nacional Autónoma de México, México DF. CEDERI (2005) Estrategia para Transferencia, Retención, Mitigación del Riesgo Sísmico en Edificaciones Indispensables y de Atención a la Comunidad del Distrito Capital, Universidad de Los Andes, Reports prepared for DNP, ACCI and the World Bank, Bogotá. Rosenblueth, E., (1976) Optimum Design for Infrequent Disturbances, Journal of the Structural Division, ASCE, Vol. 102, No. 9, pp. 1807-1825. Smyth, A.W., G. Altay, G. Deodatis, M. Erdick, G. Franco, P. Gülkan, H. Kunreuther, H. Lu, E. Mete, N. Seeber, and Ö Yüzügüllü, (2004) Probabilistic Benefit-Cost Analysis for Earthquake Damage Mitigation: Evaluating Measures for Apartment Houses in Turkey, EERI Earthquake Spectra, 20, February, 171-203.
