COMPDYN 2009 ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering M. Papadrakakis, N.D. Lagaros, M. Fragiadakis (eds.) Rhodes, Greece, 22–24 June 2009

A PROBABILISTIC MODEL FOR THE SEISMIC RISK OF BUILDINGS. APPLICATION TO URBAN AREAS A. Aguilar1, L. Pujades1, A. Barbat2 and N. Lantada1 1

Dept. of Geotechnical Engineering and Geosciences, Technical University of Catalonia Jordi Girona 1-3. Edificio D2. UPC Campus Nord. Barcelona. 08034 e-mail: {armando.aguilar.melendez, lluis.pujades, nieves.lantada}@upc.edu 2

Dept. of Structural Mechanics, Technical University of Catalonia Jordi Girona 1-3. Edificio C1. UPC Campus Nord. Barcelona. 08034 e-mail: [email protected]

Keywords: seismic risk, seismic vulnerability, seismic hazard, risk management, urban areas Abstract. One of the main objectives of the seismic risk management is the creation of methodologies to take decisions about the seismic risk of buildings. In order to take these decisions it is necessary to have estimations about the seismic risk of each building or group of buildings studied. In the present work a probabilistic model to compute seismic risk in buildings is developed. According to this model, the seismic risk of buildings can be estimated considering three main elements: 1) The seismic vulnerability of the buildings; 2) The seismic hazard in the place where the buildings are located; and 3) The seismic response of the buildings. It is known that each one of these three elements has important uncertainties related. For this reason, a probabilistic point of view is considered in the approach proposed to estimate seismic risk. In order to highlight this probabilistic approach, a new method to compute seismic risk in urban areas is developed. It has as starting point the LM1 RISK-UE method and it is called herein mLM1 method. Curves of physical damage states versus annual frequency of occurrence are used in this method to express the seismic risk. The main steps of the mLM1 method are: 1) Probabilistic seismic vulnerability analysis; 2) Probabilistic seismic hazard analysis; 3) Probabilistic estimation of the seismic risk. In order to highlight the application of this new method the seismic risk of 8657 buildings that are located in the Eixample district of Barcelona was estimated. Seismic risk curves were obtained for each one of the studied buildings in the Eixample district. Additionally, average seismic risk curves were obtained in order to express the risk for the whole district. According to these seismic risk curves, the annual frequency of occurrence of the moderate damage grade in the Eixample district is a value between 0.00117 and 0.00297, with a mean value of 0.00209. This also means that the moderate damage grade in the Eixample district will occur in average one time every R years; where R is a value between 337 and 857 years, with a mean value of 478 years.

A. Aguilar, L. Pujades, A. Barbat and N. Lantada

1

INTRODUCTION

The damage that is generated in some buildings during the occurrence of some earthquakes depends mainly on: a) the characteristics of the earthquake; b) the static and dynamic characteristics of the building; c) the contents of the building; d) the environment where the building is located. The earthquakes have occurred with more frequency in some regions of the world than in others. The data of the magnitude and location of the earthquakes that have occurred in the last centuries in the world are used to classify, for instance, regions of the world with low or high seismicity. At the same time, it is known that in the next years, new earthquakes will happen in different regions of the world. However, for the moment, the knowledge available is not enough to predict exactly: a) the day when will occur the next earthquakes; b) the characteristics that these new earthquakes will have. For these reasons it is necessary to do probabilistic considerations to estimate: When will the next earthquakes occur? And; which will be the characteristics of these earthquakes? [1] Nowadays, the knowledge generated by earthquake engineering is important. For example, at the present time it is known much more about the dynamic behavior of buildings than several decades ago. At the same time the information produced by earthquake engineering has contributed to the creation of new tools and materials. These tools and materials can increase the capacity of the structures of the buildings to resist the presence of dynamic loads, like those that are generated during the occurrence of ground seismic motions. Therefore, the available knowledge about earthquake engineering is enough to build structures with a high capacity to resist the effect of different kind of strong earthquakes [2]. Despite this in the world there is a significant number of buildings with low capacity to resist the effect of strong ground motions. The existence of these buildings with low seismic capacity is due to several reasons, for instance: 1) Because some of these buildings don’t need a higher seismic capacity; 2) Because some of these buildings were built before the creation of modern codes of seismic design; 3)Because some owners or other stakeholders of these buildings don’t know that their buildings have a low seismic capacity; 4) Because of the insufficient resources to do extensive studies in order to identify with high certainty, the seismic capacity of any existing building; 5) Because of the limited resources to improve the seismic capacity of an important number of these buildings. In summary, nowadays there are a lot of buildings with different levels of seismic vulnerability that can have bad behavior during the occurrence of some earthquakes. For this reason, numerous studies are being developed to create: a) new techniques to increase the seismic capacity of buildings with a reduced cost; b) innovative procedures to estimate the seismic capacity of buildings with more certainty, with a reduced cost and in less time; c) new procedures to estimate seismic risk with more certainty and a reduced cost; d) new techniques to take better decisions about the seismic risk. In this context a new probabilistic model to compute the seismic risk in buildings is proposed. This new model contributes to answer questions like this: What is the probability that a building will collapse in the next 50 years? The results of seismic risk are valuable information that allows taking important decisions about the buildings. Some of the main decisions that can be taken with the seismic risk results are shown in Figure 1.

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Seismic risk assessment

Is there seismic risk?

No

Periodic program to compute seismic risk

Yes Communicate the seismic risk computed

How to manage the risk?

Option 1. The 100 % of the seismic risk is transferred

Option 2. The 100% of the seismic risk computed is retained

Option 3. A percentage of the seismic risk is transferred, other percentage of the seismic risk is reduced and the rest of the seismic risk is retained

Option 4. A percentage of the seismic risk is transferred and the rest of the seismic risk is reduced

Option 5. A percentage of the seismic risk is retained and the rest of the seismic risk is reduced

Figure 1. Theoretical main steps of the seismic risk management

2

THE PROBABILISTIC MODEL FOR THE SEISMIC RISK

In the present work, the seismic vulnerability of a building is defined as a property of the building that indicates its weakness to resist the effects that are generated in the building by the occurrence of ground seismic motions. In other words, the seismic vulnerability of a building is a property that describes its susceptibility to suffer damage (or bad behavior) during the occurrence of ground seismic motions. For instance, a building with a low seismic vulnerability is at the same time a building that has a high capacity to resist the shaking in the building that is produced by strong ground motions. On the other hand, a building with a high seismic vulnerability is a building that has a low capacity to resist the effects that usually are produced during the occurrence of strong ground motions. Then, the seismic vulnerability of a building is a fundamental information for the seismic risk assessment. The seismic hazard, the seismic vulnerability and the earthquake damage are the three main elements that are considered in the present model to estimate the seismic risk. The way in that those three elements are taken into account to estimate the seismic risk is summarized in the equation 1. This equation is used to compute the annual probability that the damage d will be exceeded. 3

A. Aguilar, L. Pujades, A. Barbat and N. Lantada

P[ D  d ]   P[ D  d | I ,V ] '[ I ]P[V ]dVdI

(1)

where g’ [I] is the frequency of occurrence of the seismic intensity. This seismic intensity can be expressed in terms of pseudo acceleration, macroseismic intensity, etc. P [V] is the probability of occurrence of the seismic vulnerability. P [ D > d | I, V] is the probability of that the damage d will be exceeded given that a seismic intensity I, and a seismic vulnerability V have occurred. In the equation 1, the total probability theorem is applied and it is taken into account that the intensity I and the vulnerability V are independent random variables [3]. The approach that is summarized in the equation 1 allows taking into account the main uncertainties that are related to: a) the seismic vulnerability; b) the seismic hazard; c) the damage that can result in a building due the combination of a particular seismic vulnerability, and a specific seismic hazard. 3

THE MODIFIED LM1 METHOD

The probabilistic approach to compute seismic risk that is summarized in the equation 1 can be implemented in different kind of methodologies. The modified LM1 (mLM1) method is an example of the application of this probabilistic approach to compute seismic risk [3]. The main objective of the mLM1 method is to compute the seismic risk of existing buildings in urban areas. This method has as starting point the LM1 method of the RISK-UE project [5]. The seismic risk computed with the mLM1 method for each building is represented by curves of annual frequency of occurrence versus damage states. The main steps of the mLM1 method are: 1) Probabilistic seismic vulnerability analysis; 2) Probabilistic seismic hazard analysis; 3) Estimation of the seismic risk. The following sections of this paper describe in more detail each one of the main steps of the mLM1 method. 3.1

The seismic vulnerability analysis

In this step of the mLM1 method the main characteristics of the studied building and the relations of this same building with its environment are taken into account to estimate the seismic vulnerability of the building. The seismic vulnerability of the building is described through three probability density functions, which correspond to the lower, mean and upper estimation of the seismic vulnerability, respectively. Each one of these probability density functions describes the variation of a vulnerability index. The possible values of this vulnerability index are between 0 and 1. Values close to zero mean low seismic vulnerability and values close to 1 mean high seismic vulnerability. The use of three probability density functions to describe the variation of a vulnerability index allows taking into account the main uncertainties that are present in the estimation of the seismic vulnerability. The main steps in the seismic vulnerability analysis of the mLM1 method are: 1) Classify to each building into a structural type. The main structural types are shown in Table 1; 2) compute the mean vulnerability index using the equation 2 and its error related; 3) compute the most probable range of values of the vulnerability index, for each building, according to its structural type and in function of the quality and quantity of the data available to compute its seismic vulnerability; 4) estimate, for each building, the probability density functions (PDF’s) that describe the variation of the vulnerability index. These PDF’s are estimated using the mean vulnerability index and its error related that is computed in the step 2, and the most probable range of values of the vulnerability index, which is estimated in the step 3. 4

A. Aguilar, L. Pujades, A. Barbat and N. Lantada

Classification into a structural type Most of the buildings can be classified into some of the structural types that are indicated in Table 1. However, new structural types can be proposed if it is required. The structural type chosen to represent the main structural characteristics of a building has much influence in the results of seismic risk and in the confidence in these same results. For this reasons in the mLM1 method, the confidence in the main data used to compute the seismic risk is considered. For this purpose a confidence value between 0 and 10 is used to identify the certainty in the structural type assigned. Values close to 0 mean low confidence in the datum, whereas values close to 10 mean high confidence in the datum. Typology M1.1 M1.2 M1.3 M2 M3.1 M3.2 M3.3 M3.4 M4.1 M4.2 M4.3 M5 RC1.1 RC1.2 RC1.3 RC2.1 RC2.2 RC2.3 RC3.1.1 RC3.1.2 RC3.1.3 RC3.2.1 RC3.2.2 RC3.2.3 RC4 RC5 RC6 S1.1 S1.2 S1.3 S2.1 S2.2 S2.3 S3.1 S3.2 S3.3 S4 S5 W

Description Rubble Stone, fieldstone masonry bearing walls Simple Stone masonry bearing walls Massive stone masonry bearing walls Adobe Unreinforced masonry bearing walls with wooden slabs Unreinforced masonry bearing walls with masonry vaults Unreinforced masonry bearing walls with composite steel and masonry slabs Unreinforced masonry bearing walls with reinforced slabs Reinforced or confined masonry bearing walls – without or low earthquake resistant design (E.R.D) Reinforced or confined masonry bearing walls - moderate E.R.D. Reinforced or confined masonry bearing walls - high E.R.D. Overall strengthened masonry buildings Concrete moment frames - without or low E.R.D. Concrete moment frames - moderate E.R.D. Concrete moment frames - high E.R.D. Concrete shear walls - without or low E.R.D. Concrete shear walls - moderate E.R.D. Concrete shear walls - high E.R.D. Concrete frames with regular unreinforced masonry infill walls - without or low E.R.D. Concrete frames with regular unreinforced masonry infill walls - moderate E.R.D. Concrete frames with regular unreinforced masonry infill walls - high E.R.D. Irregular concrete frames with unreinforced masonry infill walls - without or low E.R.D. Irregular concrete frames with unreinforced masonry infill walls - moderate E.R.D. Irregular concrete frames with unreinforced masonry infill walls - high E.R.D. RC dual system (RC frames and walls) Precast concrete tilt-up walls Precast concrete frames with concrete shear walls Steel moment frames - without or low E.R.D. Steel moment frames - moderate E.R.D. Steel moment frames - high E.R.D. Steel braced frames - without or low E.R.D. Steel braced frames - moderate E.R.D. Steel braced frames - high E.R.D. Steel frames with unreinforced masonry infill walls -without or low E.R.D. Steel frames with unreinforced masonry infill walls -moderate E.R.D. Steel frames with unreinforced masonry infill walls - high E.R.D. Steel frames with cast-in-place concrete shear walls Steel and RC composite systems Wood Table 1: Building typologies (after Giovinazzi [4] and Milutinovic & Trendafiloski [5])

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A. Aguilar, L. Pujades, A. Barbat and N. Lantada

Estimation of the mean vulnerability index For each studied building, a mean vulnerability index (a value between 0 and 1) is computed using the equation 2. VI  VI*  VR  VmIn  VmEx

(2)

where V*I is a mean vulnerability index by typology (see Table 2); ΔVR is a factor that allows modifying the vulnerability index by typology, when there is enough regional knowledge or evidence that justify this change [3].

WOOD

STEEL

REINFORCED CONCRETE

MASONRY

Typology M1.1 M1.2 M1.3 M2 M3.1 M3.2 M3.3 M3.4 M4.1 M4.2 M4.3 M5 RC1.1 RC1.2 RC1.3 RC2.1 RC2.2 RC2.3 RC3.1.1 RC3.1.2 RC3.1.3 RC3.2.1 RC3.2.2 RC3.2.3 RC4 RC5 RC6 S1.1 S1.2 S1.3 S2.1 S2.2 S2.3 S3.1 S3.2 S3.3 S4 S5 W

Representative values of the vulnerability Vmin VV*I V+ Vmax 0.62 0.81 0.873 0.95 0.97 0.46 0.65 0.74 0.83 0.97 0.30 0.49 0.616 0.793 0.86 0.62 0.687 0.84 0.95 0.97 0.46 0.65 0.74 0.83 0.97 0.46 0.65 0.776 0.95 0.97 0.46 0.527 0.704 0.83 0.97 0.30 0.49 0.616 0.793 0.86 0.14 0.33 0.451 0.633 0.70 0.10 0.17 0.33 0.45 0.63 0.08 0.14 0.30 0.48 0.56 0.30 0.49 0.694 0.95 0.97 0.30 0.49 0.644 0.80 0.97 0.14 0.33 0.484 0.64 0.86 0.03 0.17 0.324 0.48 0.70 0.30 0.367 0.544 0.67 0.86 0.14 0.21 0.384 0.51 0.70 0.03 0.047 0.224 0.35 0.54 0.20 0.38 0.54 0.76 0.93 0.07 0.24 0.39 0.58 0.80 0.03 0.10 0.30 0.40 0.65 0.25 0.42 0.65 0.88 0.97 0.14 0.30 0.522 0.75 0.90 0.10 0.27 0.45 0.65 0.75 0.03 0.047 0.386 0.67 0.86 0.14 0.207 0.384 0.51 0.7 0.30 0.367 0.544 0.67 0.86 0.10 0.20 0.363 0.64 0.86 0.05 0.15 0.30 0.53 0.75 0.03 0.07 0.25 0.40 0.50 0.10 0.22 0.34 0.48 0.70 0.05 0.11 0.20 0.35 0.60 0.03 0.05 0.15 0.30 0.50 0.14 0.33 0.484 0.64 0.86 0.10 0.25 0.35 0.50 0.70 0.03 0.15 0.25 0.40 0.60 0.03 0.047 0.224 0.35 0.54 0.03 0.257 0.402 0.72 0.97 0.14

0.207

0.447

0.64

0.86

Table 2: Structural types and its corresponding representative values of vulnerability expressed in terms of a vulnerability index (after Giovinazzi [4] and Milutinovic & Trendafiloski [5])

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A. Aguilar, L. Pujades, A. Barbat and N. Lantada

The vulnerability index by structural typology, V*I , is a value that usually has an important influence in the final value of the mean vulnerability index, V I (see equation 2). For this reason, the process to select the structural typology requires an important attention. The index by structural typology represents the mean value of all the buildings that are classified into the same structural typology. Values for the vulnerability index by typology for the most representative buildings that exist in Europe were proposed by Giovinazzi [4]. The values that are proposed to be used in the mLM1 method for the mean vulnerability index by typology (V*I ) are shown in Table 2 . ΔVmIn is a factor to consider additional information to the structural typology that can modify the intrinsic vulnerability of the studied building. This factor allows taking into account specific characteristics of the studied building that usually have an important influence in the behavior of buildings during the occurrence of ground motions. The total value of this factor is equal to the sum of the scores related to each specific characteristic considered to compute the intrinsic vulnerability of the building. Then this factor allows estimating, for instance, if this building is less or more vulnerable than the mean of the buildings that are classified into the same structural type.  Examples of the characteristics that are taken into account to estimate the intrinsic vulnerability factor, ΔVmIn are shown in Table 3. 1 2 3 4 5 6 7

Characteristic of the building State of preservation Geometric irregularity in plan Irregular mass distribution in plan Vertical irregularity Superimposed floors Retrofitting intervention Foundation

Table 3: Examples of the characteristics of the building that can be taken into account to estimate ΔVmIn.

ΔVmEx is a factor to consider the extrinsic vulnerability of the studied building. This factor allows taking into account conditions of the building with its environment (extrinsic vulnerability) that usually have a significant influence in the behavior of the building during the occurrence of ground motions. The total value of this factor is equal to the sum of the scores related to each specific condition considered to compute the extrinsic vulnerability of the building. Then this factor allows estimating if this building is less or more vulnerable due to the interrelation of this building with its environment. The Table 4 shows examples of the characteristics that are taken into account to estimate the extrinsic vulnerability factor, ΔVmEx. Characteristic of the building and its environment 1 Aggregate building position (position of the building in plan with respect to the rest of the buildings in the same block) 2 Aggregate building elevation (position of the building in elevation with respect to the adjacent buildings) Table 4: Examples of the characteristics of the building with its environment that can be taken into account to estimate ΔVmEx.

For each one of the characteristic considered to compute the intrinsic and extrinsic vulnerability a confidence value between 0 and 10 is assigned. These confidence values allow identifying the quality of the data used to estimate the seismic vulnerability.

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A. Aguilar, L. Pujades, A. Barbat and N. Lantada

Estimation of the most probable range of values of the vulnerability index For each studied building, the most probable range of values of the vulnerability index is estimated. Three main elements are considered to estimate this most probable range: 1) The probable range of values of the vulnerability index delimited by V- and V+ according to the structural type of the studied building (see Table 2); 2) The exceptional range of values of the vulnerability delimited by Vmax and Vmin according to the structural type of the studied building (see Table 2); 3) The quantity and quality of the data available to compute the seismic vulnerability of the studied building. The quality of the data considered is estimated through the confidence values assigned to each one of the main data used to compute the mean vulnerability index, V I . The most probable range computed, for each building, is used to define the limits of the probability density functions that are estimated to describe the variation of a vulnerability index. Finally, these probability density functions represent the seismic vulnerability of each studied building.

Estimation of the probability density functions In the mLM1 method the probability density functions used to describe the variation of the vulnerability index are beta type. The general form of a beta probability density function (beta pdf) is shown in the equation 3.  1

f ( x;  ,  , A, B) 

1 (   )  x  A    B  A ( )(  )  B  A 

 Bx     B A

 1

A x B

(3)

The general form of a beta pdf to describe the variation of the vulnerability index VI is shown in the equation 4. f (VI ;  ,  , 0,1) 

(   )  1  1 VI  1  VI  ( )(  )

0  VI  1

(4)

The mean vulnerability index and the most probable range of the vulnerability index are used to compute the parameters α and β for the three probability density functions that represent the seismic vulnerability of each studied building. Figure 2 shows examples of the seismic vulnerability curves that can be obtained when the mLM1 method is applied. These curves correspond to two different buildings. Table 5 shows the canonical parameters α and β that correspond to each one of the cumulative density functions beta that are drawn in Figure 2. Vulnerability curves Lower Mean Upper

Building 1 α 6.12 9.37 15.53

Building 2 β 4.81 4.21 3.41

α 6.04 6.54 9.92

β 4.71 4.01 4.71

Table 5. Seismic vulnerability results for the buildings No. 1 and No.2. Values of the parameters α and β of each beta pdf that describes the variation of the vulnerability index.

The building No.1 is classified into the M34 typology (see Table 1). The seismic vulnerability curves of this building (Figure 2a) were computed using few data with low confidence. The building No.2 is also classified into the M34 typology (see Table 1). However, the seismic vulnerability curves of this building (Figure 2b) were computed using few data with high confidence.

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A. Aguilar, L. Pujades, A. Barbat and N. Lantada

1

1

lower mean upper

0.9

0.8

0.7

0.7

0.6

0.6

Probability

Probability

0.8

0.5 0.4

0.5 0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

lower mean upper

0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

0

0.1

0.2

Vulnerability index ( VI )

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vulnerability index ( VI )

(a)

(b)

Figure 2. (a) Beta cumulative distribution functions of the vulnerability index that represent the seismic vulnerability curves of the building No.1.; (b) Beta cumulative distribution functions of the vulnerability index that represent the seismic vulnerability curves of the building No.2.

3.2

The probabilistic seismic hazard analysis

The seismic hazard is one of the main elements used in the probabilistic model proposed to compute the seismic risk (see equation 1). The seismic hazard in the mLM1 method is computed through a probabilistic seismic hazard assessment and it is expressed in terms of exceedance rates of an intensity parameter (pseudo acceleration, intensity degree, etc.). There are different methodologies to compute the probabilistic seismic hazard, but important quantities of them are mainly based on the Esteva-Cornell approach [6]. In the mLM1 method is suggested apply the CRISIS2007 computer code for estimate the seismic hazard [7, 8]. This code is based on the Esteva-Cornell approach. In CRISIS2007 the results of seismic hazard are expressed in terms of exceedance rates of an intensity parameter. 3.3

The seismic risk assessment

The mLM1 method allows computing annual probability of occurrence of different damage states. The basic damage grades used in the mLM1 method are shown in Table 6 [5, 9]. Structural Damage None None Slight Moderate Heavy Very heavy

Damage grade (k) 0 1 2 3 4 5

None Negligible to slight damage Moderate damage Substantial to heavy damage Very heavy damage Destruction

Non-Structural Damage None Slight Moderate Heavy Very heavy

Table 6: Description of the damage grades used in the mLM1 method

In the mLM1 method the seismic risk is compute according to the approach that is summarized in the equation 1. The main damage function used in the mLM1 method was proposed by Giovinazzi [4]. This damage function is mainly represented by equation 5.

 I  6.25VI  13.1   2.3  

 D  2.5[1  tanh 

9

(5)

A. Aguilar, L. Pujades, A. Barbat and N. Lantada

where µD is the mean damage grade, I is a macroseismic intensity and VI is a vulnerability index. The variation in the mean damage grade is represented by a binomial-equivalent beta probability density function (equation 6). f(

D  1 6

 1

(   )   D  1  ;  ,  , 0,1)    ( )(  )  6 

 D  1  1   6  

 1

;

0

D  1 6

1

(6)

0  D  5 The parameters, α and β, of this beta distribution can be computed with the equations 7 and 8, respectively.

  t (0.007  D3  0.052  D2  0.2875 D ) β=t-α

; t=8

(7) (8)

When µD=k it is possible compute the probability that the damage grade will be less or equal to some of the main damage grades k (see Table 6). Figure 3 illustrates a beta cumulative density function that was obtained through the integration of the equation 6 for µD=2 and t=8. 1 0.9 0.8

Probability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2 Damage grade (k)

3

4

5

Figure 3. Beta cumulative distribution function of the damage grade k for µD=2 and t=8

In the mLM1 method the estimation of the seismic risk of a building is based on the equation No. 1. This seismic risk is expressed in terms of the annual probability of occurrence of the damage states (see Table 6). The annual probability of occurrence for the damage states can be represented through curves of seismic risk. An example of this kind of curves is shown in Figure 4. -2

Annual probability of occurrence (1/year)

10

Lower Mean Upper

-3

10

-4

10

-5

10

-6

10

-7

10

1

2

3 Damage Grade k

4

5

Figure 4. Seismic risk curves for a hypothetic unreinforced masonry building located in Barcelona

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4

APPLICATION TO URBAN AREAS

Many works have been published in previous years on the different aspects of the seismic risk of Barcelona [10, 11, 12, 13, 14, 15, 16, 17]. In order to highlight the application of the mLM1 method the seismic risk of buildings located in the Eixample district of Barcelona is computed. The total number of buildings in the Eixample district is 8942. However, in the present work are not studied the buildings that are essential facilities or those that are considered as monumental or historical buildings. For this reason, the total number of studied buildings is 8657. 4.1

The building data

The geographical distribution of the studied buildings in the Eixample district is shown in Figure 5.

´

0

250

500

1,000 Meters

Figure 5. Location of the buildings in the Eixample district of Barcelona that are studied in the present work

According to the data, the main structural material of the 72.76 per cent of the buildings in the Eixample district is masonry, and near of 23.14 per cent of the buildings in the Eixample district is built with structures of reinforced concrete. Other basic information for each building in the Eixample district is the date of its construction. The distribution of the date of construction of the reinforced concrete buildings in the Eixample district is shown in Figure 6. Additional data was considered for each studied building in order to estimate its seismic vulnerability, its seismic hazard and its seismic risk.

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160 140

Number of buildings

120 100 80 60 40 20 0 1860

1880

1900

1920

1940

1960

1980

2000

Year of construction

Figure 6. Evolution of the construction of buildings of reinforced concrete in the Eixample district.

4.2

The seismic vulnerability analysis

The seismic vulnerability was computed, for each one of the studied buildings in the Eixample district, using the procedure that is described in the section 3.1 of this document. For comparative purpose only the intrinsic vulnerability was computed. The available data for each building were used to classify it into some of the structural typologies (see Table 1). The 8657 studied buildings were classified into some of the eight building typologies that are shown in Figure 7. S3 S5 W 2.09% 1.72% 0.29% M31 19.89% RC32 23.14% M32 0.72%

M34 4.66%

M33 47.50%

Figure 7. Distribution of the different structural typologies that were used to classify to the studied buildings in the Eixample district of Barcelona

All the data available were used to estimate the seismic vulnerability curves for each studied building. For example, Figure 8 shows the seismic vulnerability curves of the building No. 3, which is classified into the M33 structural typology (see Table 1). This figure shows that the building No. 3 has a high seismic vulnerability (low seismic resistance).

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1 lower mean upper

0.9 0.8

Probability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vulnerability index ( VI )

Figure 8. Beta cumulative distribution functions of the vulnerability index that represent the seismic vulnerability curves of the building No. 3 (M33) located in the Eixample district of Barcelona.

According to the seismic vulnerability curves in Figure 8 the probability that the vulnerability index of the building No.3 will be greater than 0.8 vary from 0.52 to 0.99 with a mean value of 0.91. Other seismic vulnerability curves that were obtained are shown in Figure 9. These curves correspond to the building No.4, which is classified into the M33 structural typology (see Table 1). This figure shows that this building has a moderate seismic vulnerability (moderate seismic resistance). 1 lower mean upper

0.9 0.8

Probability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vulnerability index ( VI )

Figure 9. Beta cumulative distribution functions of the vulnerability index that represent the seismic vulnerability curves of the building No. 4 (M33) located in the Eixample district of Barcelona.

According to the seismic vulnerability curves in figure 9 the probability that the vulnerability index of the building No.4 will be less or equal to 0.8 vary from 0.85 to 0.99 with a mean value of 0.94. Whereas, the probability that the vulnerability index of this building will be less or equal to 0.6 is between 0.29 and 0.78, with a mean value of 0.54. Figure 10 shows the seismic vulnerability curves of the building No.5, which is classified into the RC32 structural typology (see Table 1). This figure shows that the building No.5 has a high seismic vulnerability (low seismic resistance).

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A. Aguilar, L. Pujades, A. Barbat and N. Lantada

1 lower mean upper

0.9 0.8

Probability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vulnerability index ( VI )

Figure 10. Beta cumulative distribution functions of the vulnerability index that represent the seismic vulnerability curves of the building No. 5 (RC32) located in the Eixample district of Barcelona.

According to the seismic vulnerability curves in Figure 10 the probability that the vulnerability index for the building No.5 will be greater than 0.7 vary from 0.62 to 0.94 with a mean value of 0.78. Figure 11 shows the seismic vulnerability curves of the building No.6, which is classified into the S5 structural typology (see Table 1). This figure shows that the building No.6 has a low seismic vulnerability (high seismic resistance). 1 lower mean upper

0.9 0.8

Probability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vulnerability index ( VI )

Figure 11. Beta cumulative distribution functions of the vulnerability index that represent the seismic vulnerability curves of the building No. 6 (S5) located in the Eixample district of Barcelona

According to the seismic vulnerability curves in figure 11 the probability that the vulnerability index will be greater than 0.5 vary from 0.07 to 0.25 with a mean value of 0.12. Figure 12 shows the seismic vulnerability estimated for all the studied buildings in the Eixample district. According to these seismic vulnerability curves the probability that the vulnerability index of the Eixample will be greater than 0.7 vary from 0.44 to 0.92 with a mean value of 0.73.

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1 Average-lower Average-mean Average-upper

0.9 0.8

Probability

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vulnerability index ( VI )

Figure 12. Beta cumulative distribution functions of the vulnerability index that represent the average curves of seismic vulnerability of the 8657 buildings located in the Eixample District of Barcelona.

4.3

The seismic hazard

Barcelona is located in a moderate seismic zone [18]. The seismic hazard used in this work was estimated by Secanell and colleagues in 2004 [19]. Figure 13 shows the annual probability of occurrence curve of the intensity for Barcelona. -1

10

Exceedance rate (1/year)

-2

10

-3

10

-4

10

-5

10

5

6

7

8

Intensity (degree)

Figure 13. Seismic hazard curves for the city of Barcelona (mean values and one standard deviation) [11]

In order to appreciate the level of seismic hazard that exist in Barcelona it is possible identify different values of the seismic hazard curves. For instance, according to the seismic hazard curves that are shown in Figure 13 the macroseismic intensity of 7 will occur in average one time every R years. The value of R is a number of years between 667 and 2632, with a mean value of 1429. 4.4

The seismic risk estimation

Annual probabilities of occurrence for each different damage grade were estimated for each one of the 8657 buildings studied in the Eixample district of Barcelona. For example, the seismic risk results for the building No.3 (see Figure 8) are shown in Figure 14. According to these results in the building No.3 the moderate damage grade will occur in average, one time every R years. The value of R is a number of years between 193 and 510, with a mean value of 285 years. If only the mean value of R is considered, then it is possible 15

A. Aguilar, L. Pujades, A. Barbat and N. Lantada

to affirm that in the building No.3 the moderate damage grade will occur, in average, one time every 285 years. -1

Annual probability of occurrence (1/year)

10

Lower Mean Upper

-2

10

-3

10

-4

10

-5

10

-6

10

1

2

3 Damage Grade k

4

5

Figure 14. Seismic risk curves of the building No. 3 (M33) located in the Eixample district of Barcelona.

However, these same results also mean that there is a probability between 9.3 % and 22.8 % (with a mean value of 16.07%) that in the building No. 3 the moderate damage grade will occur in the next 50 years. At the same time, there is a probability between 0.4% and 1.7% (with a mean value of 0.9%) that in the building No. 3 the very heavy damage grade will occur in the next 50 years. According to the results it is possible conclude that the building No.3 is one of the buildings that have the highest seismic risk in the Eixample district of Barcelona. Other studied building is the building No. 4, whose seismic vulnerability is shown in Figure 9. The seismic risk results that are shown in Figure 15 indicate that in the building No.4 the moderate damage grade will occur in average, one time every R years. The value of R is a number of years between 1222 and 4255, with a mean value of 2049 years. If only the mean value of R is considered, then it is possible to affirm that in the building No.4 the moderate damage grade will occur, in average, one time every 2049 years. However, these same results also mean that there is a probability between 1.17 % and 4.01 % (with a mean value of 2.41%) that in the building No. 4 the moderate damage grade will occur in the next 50 years. At the same time, there is a probability between 0.01% and 0.1% (with a mean value of 0.05%) that in the building No. 4 the very heavy damage grade will occur in the next 50 years. According to the results it is possible conclude that the building No.4 is one of the masonry buildings that have the lowest seismic risk in the Eixample district of Barcelona. -2

Annual probability of occurrence (1/year)

10

Lower Mean Upper

-3

10

-4

10

-5

10

-6

10

-7

10

-8

10

1

2

3 Damage Grade k

4

5

Figure 15. Seismic risk curves of the building No. 4 (M33) located in the Eixample district of Barcelona.

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A. Aguilar, L. Pujades, A. Barbat and N. Lantada

The building No. 5 is other studied case, whose seismic vulnerability is shown in Figure 10. The seismic risk results that are shown in Figure 16 indicate that in the building No.5 the moderate damage grade will occur in average, one time every R years. The value of R is a number of years between 212 and 486, with a mean value of 285 years. If only the mean value of R is considered, then it is possible to affirm that in the building No.5 the moderate damage grade will occur, in average, one time every 285 years. However, these same results also mean that there is a probability between 9.8 % and 20.98 % (with a mean value of 16.07%) that in the building No. 5 the moderate damage grade will occur in the next 50 years. At the same time, there is a probability between 0.53% and 1.57% (with a mean value of 1.1%) that in the building No. 5 the very heavy damage grade will occur in the next 50 years. According to the results it is possible conclude that the building No.5 is one of the reinforced concrete buildings that have the highest seismic risk in the Eixample district of Barcelona. -1

Annual probability of occurrence (1/year)

10

Lower Mean Upper

-2

10

-3

10

-4

10

-5

10

-6

10

1

2

3 Damage Grade k

4

5

Figure 16. Seismic risk curves of the building No. 5 (RC32) located in the Eixample district of Barcelona.

Other studied case is the building No. 6, whose seismic vulnerability is shown in Figure 11. The seismic risk results that are shown in Figure 17 indicate that there is a probability between 0.26 % and 0.69 % (with a mean value of 0.37%) that in the building No. 6 the moderate damage grade will occur in the next 50 years. At the same time, there is a probability between 0.0014% and 0.0079% (with a mean value of 0.0022%) that in the building No. 6 the very heavy damage grade will occur in the next 50 years. According to the results it is possible conclude that the building No.6 is one of the buildings that have the lowest seismic risk in the Eixample district of Barcelona. -3

Annual probability of occurrence (1/year)

10

Lower Mean Upper

-4

10

-5

10

-6

10

-7

10

-8

10

-9

10

1

2

3 Damage Grade k

4

5

Figure 17. Seismic risk curves of the building No. 6 (S5) located in the Eixample district of Barcelona

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A. Aguilar, L. Pujades, A. Barbat and N. Lantada

In order to represent results for the complete Eixample district average values considering the seismic risk results for each building were computed (Figure 18). According to these results it is possible to conclude that in the Eixample district the moderate damage grade will occur in average, one time every R years. The value of R is a number of years between 337 and 857 years, with a mean value of 478 years. However, these same results also indicate that there is a probability between 5.67 % and 13.78 % (with a mean value of 9.92%) that in the 8657 buildings located in the Eixample district the moderate damage grade will occur in the next 50 years. At the same time, there is a probability between 1.34% and 4.27% (with a mean value of 2.7%) that in the 8657 buildings located in the Eixample district the substantial damage grade will occur in the next 50 years. -2

Annual probability of occurrence (1/year)

10

Average-lower Average-mean Average-upper -3

10

-4

10

-5

10

-6

10

1

2

3 Damage Grade k

4

5

Figure 18. Curves with average values of the annual probability of occurrence of the damage grade k, for the Eixample district.

It is important to notice that there are some similarities between the results of seismic risk obtained in the present work and the results of seismic risk obtained by previous studies. For example, in the Lantada study [20] seismic scenarios were computed for Barcelona. In this same study two methods were used to compute the seismic risk: 1) the vulnerability index method (VIM) of the RISK-UE project; 2) the capacity spectrum method (CSM) of the RISKUE project. According to the results of the Lantada study the expected mean damage grade in the Eixample district is equal to: a) 2.03 for a return period of 475 years, when the VIM method is used; b) 1.90 for a return period of 475 years, when the CSM method is used. These last two results have some similarities with the seismic risk computed in the present work, where, for instance, the moderate damage grade (k=2) has a mean return period of 478 years (see Figure 18). However, it is important to notice that this comparison is partially valid because there are difference in the data used in this study and the data used in previous studies. On the other hand it is also important to consider that one of the main advantages of the mLM1 is that the results are range of values of seismic risk and not unique values as the results obtained in the previous studies that are cited above. In the present work the uncertainty of one standard deviation in the seismic hazard results was considered to compute the seismic risk. However, for comparative purposes Table 7 shows the annual probability of occurrence of the moderate damage grade for two cases: a) when the uncertainty of one standard deviation in the results of the seismic hazard is considered to compute the seismic risk; b) when only the mean value in the results of the seismic hazard is considered to compute the seismic risk.

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According to the results that are shown in Table 7 it is possible to conclude that for the studied buildings in the Eixample district of Barcelona, when one standard deviation in the results of the seismic hazard is considered, then the annual probability of occurrence of the moderate damage grade is 10 per cent greater than the case where only the mean values of the seismic hazard results are considered. Seismic risk

Average-lower Average-mean Average-upper

Annual probability of occurrence of the moderate damage grade 2 Case a) Uncertainty in the seismic Case b) Uncertainty in the hazard results is not considered seismic hazard results is considered (one standard deviation) 0.001049 0.001167 0.001891 0.002090 0.002698 0.002966

Table 7: Annual probability of occurrence of the moderate damage grade in the Eixample district.

5

CONCLUSIONS

The probabilistic model that is developed in this work can be successfully applied in different methodologies to compute the seismic risk of buildings. The mLM1 method is a useful methodology to compute the seismic risk based on a probabilistic approach. The seismic vulnerability curves that can be estimated with the mLM1 method allow capturing important uncertainties related to the seismic vulnerability of buildings. These vulnerability curves also show the uncertainty related to the results of the seismic vulnerability. In order to compute the seismic risk the mLM1 method allows considering the uncertainty related to the seismic hazard results. Damage curves that can be obtained with the mLM1 method contain useful information, for instance, these curves describe numerous damage scenarios with its respective return period. Other important advantage of this method is that the seismic risk results are expressed in terms of range of values. Then, they are not expressed in terms of a unique value. The seismic risk results that are obtained with the application of the mLM1 method, show the inherent uncertainty to the seismic risk. This kind of seismic risk results contributes to improve the way in that the seismic risk is represented and communicated, and it also allows improving the process of the seismic risk management. Thus, the seismic risk results that can be obtained with the mLM1 method constitute a qualitative improvement with respect to the previous results that were obtained to express the seismic risk in urban areas. It has been observed that there are buildings in the Eixample district of Barcelona that have a significant seismic risk. The mean values of the seismic risk results obtained with the mLM1 method for the Eixample district have some similarities with the seismic risk scenarios obtained for the same district with other methodologies. The seismic results obtained for the Eixample district allows taking relevant decisions in order to manage the seismic risk of the buildings located in this site. ACKNOWLEDGEMENTS

This work has been partially supported by the Spanish Ministry of Education and Science and with FEDER funds (project: CGL-2005-04541-C03-02/BTE) and by the project “Contribuciones sismológicas, geofísicas y de ingeniería a la predicción y prevención del riesgo sísmico” (CGL2008-00869/BTE). The first author acknowledges the big support of the Uni-

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versity of Veracruz in México, the CONACYT (National Council for Research and Technology of Mexico) and the Technical University of Catalonia.

REFERENCES

[1] McGuire, R. K., Seismic Hazard and Risk Analysis, Earthquake Engineering Research Institute, Oakland, CA, 2004. [2] Bozorgnia, Y. & Bertero, V. eds. Earthquake Engineering from Engineering Seismology to Performance-based Engineering, 1st Edition. CRC press, 2004. [3] A. Aguilar, L. Pujades, A. Barbat & M. Ordaz, Probabilistic assessment of seismic risk in urban areas. Proceedings of the 14th World Conference on Earthquake Engineering, Paper No. 09_01_0158, Beijing, China, October 12-17, 2008. [4] S. Giovinazzi, The vulnerability assessment and the damage scenario in seismic risk analysis. Doctoral thesis, Technical University of Braunschweig, and University of Florence, 2005. [5] Z. Milutinovic & G. Trendafiloski, WP4: Vulnerability of current buildings. RISK-UE. An advanced approach to earthquake risk scenarios with applications to different European towns, Contract: EVK4-CT-2000-00014, 2003. [6] L. Esteva, Regionalización sísmica de México para fines de ingeniería. Institute of Engineering Series-246, UNAM, Mexico, 1970. [7] M. Ordaz, A. Aguilar, J. Arboleda, CRISIS2007. Program for Computing Seismic Hazard. Institute of Engineering UNAM, Mexico, 2007. [8] E. Faccioli, V. Pessina, WP2: Basis of a handbook of earthquake ground motions scenarios. An advanced approach to earthquake risk scenarios with applications to different European towns, Contract: EVK4-CT-2000-00014, 2003. [9] G. Grüntal, European Macroseismic Scale 1998. Centre Européen de Géodynamique et de Séismologie, Luxemburg, 1998. [10] A. H. Barbat, F. Yépez Moya and J. A. Canas, “Damage scenarios simulation for risk assessment in urban zones”, Earthquake Spectra, 2:3, 371-394, 1996. [11] F. Yépez, A. H. Barbat y J. A. Canas, “Simulación de escenarios del daño sísmico en zonas urbanas”, Revista internacional de métodos numéricos para cálculo y diseño en ingeniería, 12:3, 331-358,1996. [12] A. H. Barbat, U. Mena y F. Yépez, “Evaluación probabilista del riesgo sísmico en zonas urbanas”, Revista internacional de métodos numéricos para cálculo y diseño en ingeniería, 14:2, 247-268, 1998. [13] A. H. Barbat, L. G. Pujades and N. Lantada, “Performance of buildings under earthquakes in Barcelona, Spain”, Computer-Aided Civil and Infrastructure Engineering, 21, 573-593, 2006. [14] M. L. Carreño, O. D. Cardona and A. H. Barbat, “Urban seismic risk evaluation: A holistic approach”, Natural Hazards, 40, 137-172, 2007.

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[15] A. H. Barbat, L. G. Pujades and N. Lantada, “Seismic damage evaluation in urban areas using the capacity spectrum method: application to Barcelona”, Soil Dynamics and Earthquake Engineering, 28, 851–865, 2008. [16] N. Lantada, L. G. Pujades and A. H. Barbat, “Vulnerability index and capacity spectrum based methods for urban seismic risk evaluation. A comparison”, Natural Hazards, 2009 (in press). [17] A. H. Barbat, M. L. Carreño, L. G. Pujades, N. Lantada, O. D. Cardona and M. C. Marulanda, “Seismic vulnerability and risk evaluation methods for urban areas. A review with application to a pilot area”, Structure and Infrastructure Engineering, 2009 (in press). [18] J. Cid, S. Figueras, J. Fleta, X. Goula, T. Susagna & C. Amieiro, Zonación sísmica de la ciudad de Barcelona. Proceedings 1er Congreso Nacional de Ingeniería Sísmica. Murcia, Spain, 1999. [19] R. Secanell, X. Goula, T. Susagna, J. Fleta, & A. Roca, Seismic hazard zonation of Catalonia, Spain, integrating random uncertainties. Journal of Seismology. 8:, 25-40, 2004. [20] N. Lantada, Evaluación del riesgo sísmico mediante métodos avanzados y técnicas GIS. Aplicación a la ciudad de Barcelona. Doctoral thesis, Universitat Politécnica de Catalunya, 2007.

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