**2. The estimation of seismic loss of structures in the PBEE framework**

The PBEE process can be expressed in terms of a four-step analysis, including [9-10]:

Hazard analysis, which results in Intensity Measures (IMs) for the facility under study,



**Table 1.** Effects from Earthquakes [8]

4 Earthquake Engineering

introduced.

through engineering strategies [4].

**1.1. The economic consequences of earthquakes** 

and "economic" effects. This is summarized in Table 1 [8].

assess the higher-order effects of an earthquake.

economic models.

for engineers, who up until then had been concerned only with risk reduction options

Seismic loss estimation is an expertise provided by earthquake engineering and the manner in which it can be employed in the processes of assessing seismic loss and managing the financial and economical risk associated with earthquakes through more beneficial retrofit methods will be discussed. The methodology provides a useful tool for comparing different engineering alternatives from a seismic-risk-point of view based on a Performance Based Earthquake Engineering (PBEE) framework [5]. Next, an outline of the regional economic models employed for the assessment of earthquakes' impact on economies will be briefly

The economic consequences of earthquakes may occur both before and after the seismic event itself [6]. However, the focus of this chapter will be on those which occur after earthquakes. The consequences and effects of earthquakes may be classified in terms of their primary or direct effects and their secondary or indirect effects. The indirect effects are sometimes referred to by economists as higher-order effects. The primary (direct) effects of an earthquake appear immediately after it as social and physical damage. The secondary (indirect) effects take into account the system-wide impact of flow losses through interindustry relationships and economic sectors. For example, where damage occurs to a bridge then its inability to serve to passing vehicles is considered a primary or direct loss, while if the flow of the row material to a manufacturing plant in another area is interrupted due to the inability of passing traffic to cross the bridge, the loss due to the business's interruption in this plant is called secondary or indirect loss. A higher-order effect is another term as an alternative to indirect or secondary effects which has been proposed by economists [7]. These potential effects of earthquakes may be categorized as: "social or human", "physical"

The term 'total impact' accordingly refers to the summation of direct (first-order effects) and indirect losses (higher-order effects). Various economic frameworks have been introduced to

With a three-sector hypothesis of an economy, it may be demonstrated in terms of a breakdown as three sectors: the primary sector as raw materials, the secondary sector as manufacturing and the tertiary sector as services. The interaction of these sectors after suffering seismic loss and the relative effects on each other requires study through proper

**2. The estimation of seismic loss of structures in the PBEE framework** 

Hazard analysis, which results in Intensity Measures (IMs) for the facility under study,

The PBEE process can be expressed in terms of a four-step analysis, including [9-10]:

Considering the results of each step as a conditional event following the previous step and all of the parameters as independent random parameters, the process can be expressed in terms of a triple integral, as shown below, which is an application of the total probability theorem [11]:

$$w(DV) = \begin{bmatrix} \iint \ \mathbf{G}[DV|DM] |d\mathbf{G}[DM|EDP] |d\mathbf{G}[EDP|IM] \end{bmatrix} \tag{1}$$

Seismic Risk of Structures and Earthquake Economic Issues 7

*Pf*=*φ* (- *β*)=1- *φ* (*β*) (4)

The Engineering Demand Parameters describe the response of the structural framing and the non-structural components and contents resulting from earthquake shaking. The parameters are calculated by structural response simulations using the IMs and corresponding earthquake motions. The ground motions should capture the important characteristics of earthquake ground motion which affect the response of the structural framing and nonstructural components and building contents. During the loss and risk estimation studies, the

The EDPs were categorized in the ATC 58 task report as either direct or processed [9]. Direct EDPs are those calculated directly by analysis or simulation and contribute to the risk assessment through the calculation of P[EDP | IM]; examples of direct EDPs include interstory drift and beam plastic rotation. Processed EDPs - for example, a damage index are derived from the values of direct EDPs and data on component or system capacities. Processed EDPs could be considered as either EDPs or as Damage Measures (DMs) and, as such, could contribute to risk assessment through P[DM | EDP]. Direct EDPs are usually introduced in codes and design regulations. For example, the 2000 NEHRP Recommended Provisions for Seismic Regulations for Buildings and Other Structures introduces the EDPs presented in Table 2 for the seismic design of reinforced concrete moment frames [12-13]:

> **Reinforced concrete moment frames** Axial force, bending moment and shear force in columns Bending moment and shear force in beams Shear force in beam-column joints Shear force and bending moments in slabs Bearing and lateral pressures beneath foundations Interstory drift (and interstory drift angle)

**Table 2.** EDPs required for the seismic design of reinforced concrete moment frames by [12-13]

summary of the available DIs is available in [14].

Processed EDPs are efficient parameters which could serve as a damage index during loss and risk estimation for structural systems and facilities. A Damage Index (DI), as a singlevalued damage characteristic, can be considered to be a processed EDP [10]. Traditionally, DIs have been used to express performance in terms of a value between 0 (no damage) and 1 (collapse or an ultimate state). An extension of this approach is the damage spectrum, which takes on values between 0 (no damage) and 1 (collapse) as a function of a period. A detailed

Park and Angin [15] developed one of the most widely-known damage indices. The index is a linear combination of structural displacement and hysteretic energy, as shown in the equation:

EDP with a greater correlation with damage and loss variables must be employed.

where *φ*( ) is a log-normal distribution function.

*2.1.2. Engineering demand parameters* 

The performance of a structural system or lifeline is described by comparing demand and capacity parameters. In earthquake engineering, the excitation, demand and capacity parameters are random variables. Therefore, probabilistic techniques are required in order to estimate the response of the system and provide information about the availability or failure of the facility after loading. The concept is included in the reliability design approach, which is usually employed for this purpose.

#### **2.1. Probabilistic seismic demand analysis through a reliability-based design approach**

The reliability of a structural system or lifeline may be referred to as the ability of the system or its components to perform their required functions under stated conditions for a specified period of time. Because of uncertainties in loading and capacity, the subject usually includes probabilistic methods and is often made through indices such as a safety index or the probability of the failure of the structure or lifeline.

#### *2.1.1. Reliability index and failure*

To evaluate the seismic performance of the structures, performance functions are defined. Let us assume that *z*=g(x1, x2, …,xn) is taken as a performance function. As such, failure or damage occurs when *z*<0. The probability of failure, pf, is expressed as follows:

$$\mathbf{P} \nRightarrow \mathbf{P}[\mathbf{z} \nleq 0] \tag{2}$$

Simply assume that *z*=*EDP*-*C* where *EDP* stands for Engineering Demand Parameter and *C* is the seismic capacity of the structure.

Damage or failure in a structural system or lifeline occurs when the Engineering Demand Parameter exceeds the capacity provided. For example, in a bridge structural damage may refer to the unseating of the deck, the development of a plastic hinge at the bottom of piers or damage due to the pounding of the decks to the abutments, etc.

Given that *EDP* and *C* are random parameters having the expected or mean values of µEDP and µC and standard deviation of σEDP and *σC*, the "safety index" or "reliability index", *β*, is defined as:

$$\beta = \frac{\mu\_c - \mu\_{EDP}}{\sqrt{\sigma\_c^2 + \sigma\_{EDP}^2}} \tag{3}$$

It has been observed that the random variables such as "*EDP*" or "*C*" follow normal or lognormal distribution. Accordingly, the performance function, z, also will follow the same distribution. Accordingly, probability of failure (or damage occurrence) may be expressed as a function of safety index, as follows:

Seismic Risk of Structures and Earthquake Economic Issues 7

$$P \models \phi \text{ (- }\beta\text{)} \text{=1 - }\phi \text{ ( $\beta$ )} \tag{4}$$

where *φ*( ) is a log-normal distribution function.

#### *2.1.2. Engineering demand parameters*

6 Earthquake Engineering

**approach** 

defined as:

�(��) = ∭ �[��|��]|��[��|���]|��[���|���|��[��] (1)

The performance of a structural system or lifeline is described by comparing demand and capacity parameters. In earthquake engineering, the excitation, demand and capacity parameters are random variables. Therefore, probabilistic techniques are required in order to estimate the response of the system and provide information about the availability or failure of the facility after loading. The concept is included in the reliability design

**2.1. Probabilistic seismic demand analysis through a reliability-based design** 

The reliability of a structural system or lifeline may be referred to as the ability of the system or its components to perform their required functions under stated conditions for a specified period of time. Because of uncertainties in loading and capacity, the subject usually includes probabilistic methods and is often made through indices such as a safety index or the

To evaluate the seismic performance of the structures, performance functions are defined. Let us assume that *z*=g(x1, x2, …,xn) is taken as a performance function. As such, failure or

Pf=P[z<0] (2)

Simply assume that *z*=*EDP*-*C* where *EDP* stands for Engineering Demand Parameter and *C*

Damage or failure in a structural system or lifeline occurs when the Engineering Demand Parameter exceeds the capacity provided. For example, in a bridge structural damage may refer to the unseating of the deck, the development of a plastic hinge at the bottom of piers

Given that *EDP* and *C* are random parameters having the expected or mean values of µEDP and µC and standard deviation of σEDP and *σC*, the "safety index" or "reliability index", *β*, is

> � = ������� ��� ������ �

It has been observed that the random variables such as "*EDP*" or "*C*" follow normal or lognormal distribution. Accordingly, the performance function, z, also will follow the same distribution. Accordingly, probability of failure (or damage occurrence) may be expressed as

(3)

damage occurs when *z*<0. The probability of failure, pf, is expressed as follows:

or damage due to the pounding of the decks to the abutments, etc.

approach, which is usually employed for this purpose.

probability of the failure of the structure or lifeline.

*2.1.1. Reliability index and failure* 

is the seismic capacity of the structure.

a function of safety index, as follows:

The Engineering Demand Parameters describe the response of the structural framing and the non-structural components and contents resulting from earthquake shaking. The parameters are calculated by structural response simulations using the IMs and corresponding earthquake motions. The ground motions should capture the important characteristics of earthquake ground motion which affect the response of the structural framing and nonstructural components and building contents. During the loss and risk estimation studies, the EDP with a greater correlation with damage and loss variables must be employed.

The EDPs were categorized in the ATC 58 task report as either direct or processed [9]. Direct EDPs are those calculated directly by analysis or simulation and contribute to the risk assessment through the calculation of P[EDP | IM]; examples of direct EDPs include interstory drift and beam plastic rotation. Processed EDPs - for example, a damage index are derived from the values of direct EDPs and data on component or system capacities. Processed EDPs could be considered as either EDPs or as Damage Measures (DMs) and, as such, could contribute to risk assessment through P[DM | EDP]. Direct EDPs are usually introduced in codes and design regulations. For example, the 2000 NEHRP Recommended Provisions for Seismic Regulations for Buildings and Other Structures introduces the EDPs presented in Table 2 for the seismic design of reinforced concrete moment frames [12-13]:


**Table 2.** EDPs required for the seismic design of reinforced concrete moment frames by [12-13]

Processed EDPs are efficient parameters which could serve as a damage index during loss and risk estimation for structural systems and facilities. A Damage Index (DI), as a singlevalued damage characteristic, can be considered to be a processed EDP [10]. Traditionally, DIs have been used to express performance in terms of a value between 0 (no damage) and 1 (collapse or an ultimate state). An extension of this approach is the damage spectrum, which takes on values between 0 (no damage) and 1 (collapse) as a function of a period. A detailed summary of the available DIs is available in [14].

Park and Angin [15] developed one of the most widely-known damage indices. The index is a linear combination of structural displacement and hysteretic energy, as shown in the equation:

$$DI = \frac{u\_{\max}}{u\_{\text{c}}} + \mathcal{B}\frac{E\_h}{F\_{\text{y}}u\_{\text{c}}} \tag{5}$$

Seismic Risk of Structures and Earthquake Economic Issues 9

With an analytical approach, a numerical model of the structure is usually analysed by nonlinear dynamic analysis methods in order to calculate the *EDPs* and compare the results with the capacity to decide about the failure of the structure. The works in [21-24] are examples of analytical fragility curves for highway bridge structures by Hwang et al.

Figure 2 demonstrates the steps for computing seismic fragility in analytical approach.

To overcome the uncertainties in input excitation or the developed model, usually adequate number of records and several numerical models are required so that the dispersion of the calculated data will be limited and acceptable. This is usually elaborating and increases the cost of the generation of fragility data in this approach. Probabilistic demand models are usually one of the outputs of nonlinear dynamic analysis. Probabilistic demand models establish a relationship between the intensity measure and the engineering demand

where EDP ������ is the average value of *EDP* and *a* and *b* are constants. The model has the

Assuming a log-normal distribution for fragility values, they are then estimated using the

�[��� � �|��] ��� ��(��� ������

EDP ������ � �(��)� (7)

ln (EDP ������) � ln (�) � � ln(��) (8)

�̅ � ) �����|�� � ���

�

� (9)

2001, Choi et al. 2004, Padgett et al., 2008, and Padgett et al 2008 .

**Figure 2.** Procedure for generating analytical fragility curves

parameter. Bazorro and Cornell proposed the model given below [25]:

capability to be presented as linear in a logarithmic space such that:

following equation:

*2.2.2. Analytical approach* 

where *umax* and *uc* are maximum and capacity displacement of the structure, respectively, Eh is the hysteresis energy, *Fy* is the yielding force and *β* is a constant.

See Powell and Allahabadi, Fajfar, Mehanny and Deierlein, as well as Bozorgnia and Bertero for more information about other DIs in [16-19].

#### **2.2. Seismic fragility**

The seismic fragility of a structure refers to the probability that the Engineering Demand Parameter (*EDP*) will exceed seismic capacity (*C*) upon the condition of the occurrence of a specific Intensity Measure (*IM*). In other words, seismic fragility is probability of failure, Pf, on the condition of the occurrence of a specific intensity measure, as shown below:

$$\text{Fragility} = \text{P } [\text{EDP} \% \, \text{IM}] \tag{6}$$

In a fragility curve, the horizontal axis introduces the *IM* and the vertical axis corresponds to the probability of failure, *Pf*. This curve demonstrates how the variation of intensity measure affects the probability of failure of the structure.

Statistical approach, analytical and numerical simulations, and the use of expert opinion provide methods for developing fragility curves.

#### *2.2.1. Statistical approach*

With a statistical approach, a sufficient amount of real damage-intensity data after earthquakes is employed to generate the seismic fragility data. As an example, Figure 1 demonstrates the empirical fragility curves for a concrete moment resisting frame, according to the data collected after Northridge earthquake [20].

**Figure 1.** Empirical fragility curves for a concrete moment resisting frame building class according to the data collected after the Northridge Earthquake, [20].

#### *2.2.2. Analytical approach*

8 Earthquake Engineering

**2.2. Seismic fragility** 

*2.2.1. Statistical approach* 

ܫܦ ൌ ௨ೌೣ ௨ౙ

is the hysteresis energy, *Fy* is the yielding force and *β* is a constant.

for more information about other DIs in [16-19].

affects the probability of failure of the structure.

provide methods for developing fragility curves.

to the data collected after Northridge earthquake [20].

the data collected after the Northridge Earthquake, [20].

where *umax* and *uc* are maximum and capacity displacement of the structure, respectively, Eh

See Powell and Allahabadi, Fajfar, Mehanny and Deierlein, as well as Bozorgnia and Bertero

The seismic fragility of a structure refers to the probability that the Engineering Demand Parameter (*EDP*) will exceed seismic capacity (*C*) upon the condition of the occurrence of a specific Intensity Measure (*IM*). In other words, seismic fragility is probability of failure, Pf,

Fragility=P [EDP>C|IM] (6)

In a fragility curve, the horizontal axis introduces the *IM* and the vertical axis corresponds to the probability of failure, *Pf*. This curve demonstrates how the variation of intensity measure

Statistical approach, analytical and numerical simulations, and the use of expert opinion

With a statistical approach, a sufficient amount of real damage-intensity data after earthquakes is employed to generate the seismic fragility data. As an example, Figure 1 demonstrates the empirical fragility curves for a concrete moment resisting frame, according

**Figure 1.** Empirical fragility curves for a concrete moment resisting frame building class according to

on the condition of the occurrence of a specific intensity measure, as shown below:

ா ߚ ி௨ౙ

(5)

With an analytical approach, a numerical model of the structure is usually analysed by nonlinear dynamic analysis methods in order to calculate the *EDPs* and compare the results with the capacity to decide about the failure of the structure. The works in [21-24] are examples of analytical fragility curves for highway bridge structures by Hwang et al. 2001, Choi et al. 2004, Padgett et al., 2008, and Padgett et al 2008 .

Figure 2 demonstrates the steps for computing seismic fragility in analytical approach.

**Figure 2.** Procedure for generating analytical fragility curves

To overcome the uncertainties in input excitation or the developed model, usually adequate number of records and several numerical models are required so that the dispersion of the calculated data will be limited and acceptable. This is usually elaborating and increases the cost of the generation of fragility data in this approach. Probabilistic demand models are usually one of the outputs of nonlinear dynamic analysis. Probabilistic demand models establish a relationship between the intensity measure and the engineering demand parameter. Bazorro and Cornell proposed the model given below [25]:

$$\overrightarrow{\rm EDP} = a(IM)^b \tag{7}$$

where EDP ������ is the average value of *EDP* and *a* and *b* are constants. The model has the capability to be presented as linear in a logarithmic space such that:

$$
\ln(\text{EDP}) = \ln(a) + b\ln(IM) \tag{8}
$$

Assuming a log-normal distribution for fragility values, they are then estimated using the following equation:

$$P\left[EDP > C\left|IM\right] = \phi\left[\frac{\ln(^{EDP}/\_{\mathcal{C}})}{\sqrt{\beta\_{EDP|IM}^2 + \beta\_{\mathcal{C}}^2}}\right] \tag{9}$$

The parameter *β* introduces the dispersion in the resulting data from any calculations. An example of analytical fragility curves for highway bridges is shown in Figure 3.

**Figure 3.** Fragility curve for the 602-11 bridge for 4 damage states [21]

#### *2.2.3. Expert opinion approach*

Given a lack of sufficient statistical or analytical data, expert opinion provides a valuable source for estimating the probability of the failure of typical or specific buildings for a range of seismic intensity values. The number of experts, their proficiency and the quality of questionnaires, including the questions, their adequacy and coverage, can affect the uncertainty of the approach and its results.

#### **2.3. Seismic risk**

The expected risk of a project, assuming that the intensity measure as the seismic hazard parameter is deterministic, is calculated by equation 10, below:

$$R \text{=} P \times \text{L} \tag{10}$$

Seismic Risk of Structures and Earthquake Economic Issues 11

in Figure 4, is usually collected through questionnaires, statistical data from post-earthquake observations or else calculated through numerical simulations. ATC 13 provides an example of the collection of earthquakes' structural and human damage and loss data for California [26].

A summary of calculations required for estimating the risk of a project under a specific

0 200 400 600 800 1000 1200 1400 1600

**PGA (gal)**

An Event tree diagram is a useful tool for estimation of the probability of occurrence of damage and corresponding loss in a specific project due to a certain seismic event. The procedure requires information about seismic intensity, probable damage modes, seismic

As an example, suppose that partial seismic damage, structural collapse, partial fire and extended fire are considered to be the loss-generating consequents of an earthquake for a building. Figures 5 and 6 are the event tree diagrams, which demonstrate the procedure followed to calculate the corresponding risk for the seismic intensity of two levels of PGA=300gal and 500gal. To select the probability of the occurrence of each damage mode, (i.e., the probability of the exceedance of damage states) the fragility curves can be utilized. Each node is allocated to a damage mode. The probability of the incidence or non-incidence of each damage mode is mentioned respectively on the vertical or horizontal branch immediately after each node. The probability of the coincidence of the events at the same root is calculated by multiplying the probability of incidence of the events on the same root.

Figure 7.a demonstrates the distribution of risk values for different damage modes. In addition, it can be seen how increasing seismic intensity increased the risk of the project. Figure 7.b shows the distribution of the probability of the occurrence of different loss values

fragility values and the vulnerability and loss function of the facility under study.

seismic intensity level may be illustrated by an "event tree" diagram.

The final total risk, R, is then calculated as the summation of all Ris.

**Figure 4.** Seismic loss data

0

0.2

0.4

0.6

**Loss %**

0.8

1

1.2

*3.3.1. Event tree diagram* 

where P is the probability of the occurrence of damage and L indicates the corresponding loss. The equation shows that any factor which alters either the probability or the value of the resulted loss affects the related risk. Diverse damage modes and associated loss values, Li (i=1 to a number of probable damage modes), with a different probability of occurrence, Pi, may be envisaged for a structure. The probable risk of the system, R, can be estimated as a summation of the loss of each damage mode:

$$\mathbf{R} \mathbf{=} \boldsymbol{\Sigma} \mathbf{P} \mathbf{\times} \mathbf{L} \tag{11}$$

Loss functions are usually defined as the replacement cost - corresponding to each damage state - versus seismic intensity. The loss associated with each damage mode, presented schematically in Figure 4, is usually collected through questionnaires, statistical data from post-earthquake observations or else calculated through numerical simulations. ATC 13 provides an example of the collection of earthquakes' structural and human damage and loss data for California [26].

**Figure 4.** Seismic loss data

10 Earthquake Engineering

The parameter *β* introduces the dispersion in the resulting data from any calculations. An

Given a lack of sufficient statistical or analytical data, expert opinion provides a valuable source for estimating the probability of the failure of typical or specific buildings for a range of seismic intensity values. The number of experts, their proficiency and the quality of questionnaires, including the questions, their adequacy and coverage, can affect the

The expected risk of a project, assuming that the intensity measure as the seismic hazard

where P is the probability of the occurrence of damage and L indicates the corresponding loss. The equation shows that any factor which alters either the probability or the value of the resulted loss affects the related risk. Diverse damage modes and associated loss values, Li (i=1 to a number of probable damage modes), with a different probability of occurrence, Pi, may be envisaged for a structure. The probable risk of the system, R, can be estimated as

R=∑PiLi (11)

Loss functions are usually defined as the replacement cost - corresponding to each damage state - versus seismic intensity. The loss associated with each damage mode, presented schematically

*L* (10)

*R=P*

example of analytical fragility curves for highway bridges is shown in Figure 3.

**Figure 3.** Fragility curve for the 602-11 bridge for 4 damage states [21]

parameter is deterministic, is calculated by equation 10, below:

*2.2.3. Expert opinion approach* 

**2.3. Seismic risk** 

uncertainty of the approach and its results.

a summation of the loss of each damage mode:

A summary of calculations required for estimating the risk of a project under a specific seismic intensity level may be illustrated by an "event tree" diagram.

#### *3.3.1. Event tree diagram*

An Event tree diagram is a useful tool for estimation of the probability of occurrence of damage and corresponding loss in a specific project due to a certain seismic event. The procedure requires information about seismic intensity, probable damage modes, seismic fragility values and the vulnerability and loss function of the facility under study.

As an example, suppose that partial seismic damage, structural collapse, partial fire and extended fire are considered to be the loss-generating consequents of an earthquake for a building. Figures 5 and 6 are the event tree diagrams, which demonstrate the procedure followed to calculate the corresponding risk for the seismic intensity of two levels of PGA=300gal and 500gal. To select the probability of the occurrence of each damage mode, (i.e., the probability of the exceedance of damage states) the fragility curves can be utilized. Each node is allocated to a damage mode. The probability of the incidence or non-incidence of each damage mode is mentioned respectively on the vertical or horizontal branch immediately after each node. The probability of the coincidence of the events at the same root is calculated by multiplying the probability of incidence of the events on the same root. The final total risk, R, is then calculated as the summation of all Ris.

Figure 7.a demonstrates the distribution of risk values for different damage modes. In addition, it can be seen how increasing seismic intensity increased the risk of the project. Figure 7.b shows the distribution of the probability of the occurrence of different loss values

and how an increase of seismic intensity from 300gal to 500gal affects it in this structure. As mentioned, the calculations in an event tree diagram are performed for a special level of hazard. The curves present valuable probabilistic data about the points on the seismic loss curve. A seismic loss curve may be developed according to the information from event trees for a range of probable seismic intensities of the site. Figure 8 shows a schematic curve for the seismic loss of a project. The curve is generated by integrating the seismic risk values for each damage mode. It provides helpful data for understanding the contents and elements of the probable loss for each level of earthquake hazard.

Seismic Risk of Structures and Earthquake Economic Issues 13

**Figure 7.** a) Distribution of seismic risk values vs. damage, b) Probability of occurrence vs. probable loss

(a) (b)

**0 0.1 0.2 0.3 0.4 0.5**

**Probability of occurrence** 

**(%)**

Ri(PGA=300) Ri(PGA=500)

The total probable loss calculated by event trees provides valuable information for

If the uncertainties in the seismic hazard assessment of a specific site could be avoided, a deterministic approach could provide an easy and rational method for this purpose. However, the nature of a seismic event is such that it usually involves various uncertainty sources, such as the location of the source, the faulting mechanism and the magnitude of the event, etc. The probabilistic seismic hazard analysis offers a useful tool for the assessment of

In an active area source, k, with a similar seismicity all across it, the seismicity data gives the maximum magnitude of muk and a minimum of mlk and the frequency of the occurrence of

**3. The employment of seismic hazard analysis for the assessment of** 

estimating the annual probable loss of facilities, as shown in the next part.

**Figure 8.** Seismic loss curve

**Probability of exceedance**

**Ri(%)**

0

100

**ND PF CF PD PD+PF PD+CF CO Damage**

200

300

400

500

600

700

800

**Acceleration (gal)**

900

1000

1100

}R2=P2L2

**Probable loss**

**0 15 25 40 65 75 100 Li (%)**

}R2=P2L2

1200

1300

1400

1500

P(PGA=300gal) P(PGA=500gal)

annual norms of seismic loss and risk. [27]

**3.1. Probabilistic seismic hazard analysis** 

**seismic risk** 


ND: No Damage, F: Partial Fire, CF: Complete Fire, PD: Partial Damage, CO: Collapse



#### **Figure 6.** Event Tree, PGA=500gal

The information provided by an event tree simply increases the awareness of engineers and stakeholders about the importance and influence of each damage mode on the seismic risk of the project and demonstrates the distribution of probable loss among them.

**Figure 7.** a) Distribution of seismic risk values vs. damage, b) Probability of occurrence vs. probable loss

**Figure 8.** Seismic loss curve

and how an increase of seismic intensity from 300gal to 500gal affects it in this structure. As mentioned, the calculations in an event tree diagram are performed for a special level of hazard. The curves present valuable probabilistic data about the points on the seismic loss curve. A seismic loss curve may be developed according to the information from event trees for a range of probable seismic intensities of the site. Figure 8 shows a schematic curve for the seismic loss of a project. The curve is generated by integrating the seismic risk values for each damage mode. It provides helpful data for understanding the contents and elements of

0.95 0.8 0.9 ND 0.684 0% 0%

Fire Extended Fire Pi Li Ri

0.9 PD 0.171 25% 4.28%

Fire Extended Fire Pi Li Ri

0.8 PD 0.2560 25% 6.40%

 0.9 PF 0.0684 15% 1.03% 0.1 CF 0.0076 65% 0.49%

 0.9 PD+PF 0.0171 40% 0.68% 0.1 PD+CF 0.0019 75% 0.14%

 0.8 PF 0.0768 15% 1.15% 0.2 CF 0.0192 65% 1.25%

 0.8 PD+PF 0.0512 40% 2.05% 0.2 PD+CF 0.0128 75% 0.96%

CO 0.2000 100% 20

∑Ri=31.81%

CO 0.05 100% 5.0%

∑Ri=11.62%

=Pi Li

=Pi Li

the probable loss for each level of earthquake hazard.

0.05 0.2 0.1

0.2 0.4 0.2

0.2

0.1

Partial Seismic

Damage Collapse

**Figure 5.** Event Tree, PGA=300gal

Damage Collapse

Partial Seismic

**Figure 6.** Event Tree, PGA=500gal

ND: No Damage, F: Partial Fire, CF: Complete Fire, PD: Partial Damage, CO: Collapse

The information provided by an event tree simply increases the awareness of engineers and stakeholders about the importance and influence of each damage mode on the seismic risk

0.8 0.6 0.8 ND 0.3840 0% 0

of the project and demonstrates the distribution of probable loss among them.

The total probable loss calculated by event trees provides valuable information for estimating the annual probable loss of facilities, as shown in the next part.
