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# Physical Vulnerability in Earthquake Risk Assessment

## Summary and Keywords

In the fields of earthquake engineering and seismic risk reduction the term “physical vulnerability” defines the component that translates the relationship between seismic shaking intensity, dynamic structural uake damage and loss assessment discipline in the early 1980s, which aimed at predicting the consequences of earthquake shaking for an individual building or a portfolio of buildings. In general, physical vulnerability has become one of the main key components used as model input data by agencies when developinresponse (physical damage), and cost of repair for a particular class of buildings or infrastructure facilities. The concept of physical vulnerability started with the development of the earthqg prevention and mitigation actions, code provisions, and guidelines. The same may apply to insurance and reinsurance industry in developing catastrophe models (also known as CAT models).

Since the late 1990s, a blossoming of methodologies and procedures can be observed, which range from empirical to basic and more advanced analytical, implemented for modelling and measuring physical vulnerability. These methods use approaches that differ in terms of level of complexity, calculation efforts (in evaluating the seismic demand-to-structural response and damage analysis) and modelling assumptions adopted in the development process. At this stage, one of the challenges that is often encountered is that some of these assumptions may highly affect the reliability and accuracy of the resulted physical vulnerability models in a negative way, hence introducing important uncertainties in estimating and predicting the inherent risk (i.e., estimated damage and losses).

Other challenges that are commonly encountered when developing physical vulnerability models are the paucity of exposure information and the lack of knowledge due to either technical or nontechnical problems, such as inventory data that would allow for accurate building stock modeling, or economic data that would allow for a better conversion from damage to monetary losses. Hence, these physical vulnerability models will carry different types of intrinsic uncertainties of both aleatory and epistemic character. To come up with appropriate predictions on expected damage and losses of an individual asset (e.g., a building) or a class of assets (e.g., a building typology class, a group of buildings), reliable physical vulnerability models have to be generated considering all these peculiarities and the associated intrinsic uncertainties at each stage of the development process.

# Introduction

The term physical vulnerability, which has been used in many disciplines and different contexts, defines the probability (or the potential) of a given physical component or element to be affected or damaged under a certain external excitation, e.g., a natural hazard such as an earthquake. From a seismic risk analysis perspective, the use of the physical vulnerability concept started with the development of the earthquake loss estimation (ELE) discipline in the early 1980s, which aimed at predicting the consequences of earthquake shaking for an individual structure or for a portfolio of buildings or infrastructure facilities (EERI, 1984). Nowadays, physical vulnerability has become one of the main key components used as model input data by agencies when developing disaster prevention and mitigation actions, code provisions, and guidelines. The same may apply to the insurance and reinsurance industry in developing catastrophe (CAT) models.

Within this perspective, and for a given building typology or portfolio, physical vulnerability defines the probability of suffering a certain level of physical damage. In other words, it provides a clear picture to understand which type of structure or element is more vulnerable, i.e., likely to suffer damage, and how this vulnerability is affected by the various structural and nonstructural components of a building. Furthermore, physical vulnerability has a strong impact on both monetary and social losses through the measurement of physical damage resulting from a given ground motion intensity level (FEMA, 2008).

Since the late 1990s, many methods have been introduced for quantifying physical vulnerability. The methods use different procedures and assumptions that can be based on empirical approaches (Rossetto, Ioannou, Grant, & Maqsood, 2014a), analytical (D’Ayala, Meslem, Vamvatsikos, Porter, & Rossetto, 2015), and expert judgment. One of the main challenges that are often faced in measuring seismic vulnerability is the quantification and modeling of the uncertainties (both aleatory and epistemic) that would be involved at each stage of the vulnerability model’s construction process. In many cases some of the methodologies have been implemented using simplified assumptions in order to reduce data gathering and calculation efforts for many different reasons. However, these assumptions and simplifications may greatly decrease the reliability and accuracy of the results obtained, introducing significant uncertainties into the vulnerability model construction process.

Moreover, the extensive seismic risk assessment and modeling studies that have been implemented in many different parts of the world, covering a wide range of building typologies and portfolios, have resulted in a wealth of seismic vulnerability models available in literature that can be used for future applications, such as fragility functions for North American building typology classes provided in HAZUS-MH (FEMA-NIBS, 2003), the European vulnerability database developed under SYNER-G (2011a, 2011b), or the worldwide vulnerability database compiled within the Global Earthquake Model (GEM) framework (Rossetto, Ioannou, Grant, & Maqsood, 2014; D’Ayala, & Meslem, 2013). The availability of these vulnerability models has encouraged risk analysts (e.g., academics, engineers, insurers) to use them for future applications, rather than to develop customized models (i.e., to conduct vulnerability analyses and measure physical vulnerability) that address the peculiar structural and nonstructural characteristics of the respective building stock. However, selecting vulnerability models that had been originally developed for similar building typologies in other parts of the world can be a quite challenging process in order to ensure a reliable earthquake loss assessment and modeling, considering the fact that differences in both construction techniques and structural detailing between different countries are typically significant, even when buildings are nominally designed according to similar code provisions.

Many studies have shown that the representativeness of a seismic physical vulnerability model may have a very significant effect on the outcomes of earthquake loss estimates. There is a clear understanding and agreement among the engineering and scientific communities that one should move forward using more advanced modeling strategies that are able to relax the often unrealistic assumptions and forget about the simplified assumptions used so far. Over the past ten years, several state-of-the-art frameworks and initiatives have been developed and implemented in order to build up a better strategy that would allow a reliable modeling of these uncertainties, and to maintain better seismic physical vulnerability modeling of individual structures or portfolios of buildings and infrastructure facilities.

The present article provides a comprehensive summary about the term physical vulnerability and its usage within the field of earthquake engineering and risk reduction. The article describes the development and the evolution of the different methods that are introduced in literature for measuring physical vulnerability, with an emphasis on the differences they present in terms of assumptions and approaches they incorporate and challenges that may be encountered in implementing each of these procedures.

# Concept of Physical Seismic Vulnerability

A building’s physical vulnerability to earthquakes describes its susceptibility to damage that can be caused by a given ground-motion Intensity Measure (IM). The purpose of seismic vulnerability assessment is to estimate the damage level (in a deterministic or a probabilistic way) induced to a given building typology due to a given level of ground shaking.

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Figure 1. Process for measuring physical vulnerability of a building.

Vulnerability analysis is generally conducted in three main steps: (1) definition of the building’s structural system; (2) estimation of the physical damage given the ground-motion intensity; and (3) evaluation of the overall seismic performance, i.e., the level of vulnerability, given the ground-motion intensity. The general process for identifying a building’s physical vulnerability is illustrated in Figure 1.

## Defining a Building’s Structural System

Defining the characteristics and typology of a building is a major step and represents the starting point of a physical vulnerability assessment. According to many, if not most authors dealing with this topic, the main parameter influencing physical vulnerability is the structural system, which basically is a combination of the construction materials, the load-bearing elements, and the non-load-bearing elements. Different elements can be distinguished: structural components and nonstructural components (see Figure 2).

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Figure 2. Defining the building structural system for vulnerability measurement.

Structural components are the main elements that contribute to the response behavior of the building, and the consequences of the response in terms of the monetary losses are connected to the repair of structural damage or the replacement of the building. Nonstructural components can be divided into two categories: those which may contribute to the response behavior of the structure (and thereby to the monetary loss connected to the damage), and those which do not contribute to the response behavior of the structure, but which are important to consider as they contribute to the reconstruction costs.

In addition to the aforementioned parameters, i.e., overall building height, level of code design, or period of construction (the age of a building is sometimes used as an indirect indicator of the seismic design level, especially in areas where seismic codes have been adopted; it can also indicate typical construction practices in a given region) can also have a strong impact on building vulnerability (FEMA-177, 1989).

The number of data points included and their level of detail required for vulnerability analysis/assessment can vary widely, mainly depending on the type of the method selected for the analysis. Accordingly, data input can be provided either qualitatively or quantitatively. Table 1 illustrates the type of assignment for the parameters governing the measurement of vulnerability with respect to the type of vulnerability assessment method.

When it comes to assessing the earthquake risk for an area or region with hundreds, often thousands of individual buildings, defining the structural system of each building with its individual structural and nonstructural characteristics would be costly and impractical, if not impossible to conduct. Usually, dividing the respective building stock into a certain number of building typology classes provides a better alternative that allows for a more manageable and hence efficient study. The classification is done based on grouping either buildings or building typologies that show comparable overall performance during earthquake shaking, that is, demonstrating similar vulnerability.

Table 1. Data input parameters governing the measurement of physical vulnerability.

Parameter (data input)

Modeling requirement for physical vulnerability assessment

Empirical-based assessment

Expert judgment-based assessment

Analytical-based assessment

Required

Desirable

Required

Desirable

Required

Desirable

Construction system

Type of load-bearing elements (structural and nonstructural)

Dimension characteristics

Height range (qualitative assignment)

Number of stories (quantitative assignment)

Dimensions of load-bearing elements (quantitative assignment)

Dimensions of non-load-bearing elements (quantitative assignment)

Mechanical characteristics of materials used in construction

Structural detailing

Geometric configuration and structural irregularity (qualitative assignment)

Level of code design and/or period of construction (qualitative assignment)

Observed damage data from past earthquakes (quantitative/qualitative assignment)

With respect to the building’s height classification, building height is often represented by number of stories or height range, the latter distinguishing between low-, mid-, and high-rise buildings that are of, e.g., 1–3, 4–6, and 7+ stories, respectively. Overall, the building classification should cover all types of conventional buildings that are available and that are representative for the target area. In doing so, local experts such as structural engineers or architects have to be consulted in order to identify the local construction typologies and to identify their major characteristics (Lang & Aldea, 2011).

Within this context, many efforts have been made over the past years aiming to define worldwide and regional building taxonomy, which have resulted in the development of many building classification schemes. For instance, the HAZUS building classification scheme (see Table 2a), which was developed by the Federal Emergency Management Agency (FEMA-NIBS, 2003), has been one of the most widely used for vulnerability assessment in the United States. This model, which has also been adopted in many other earthquake-prone regions, defines 36 model building types over 15 building classes within categories of wood, steel, concrete, masonry, or mobile homes, as shown in Table 2a.

Table 2. Examples of existing building classification schemes.

(a) HAZUS building classification scheme (FEMA-NIBS, 2003)

(b) EMS-98 building classification scheme (Grünthal, 1998)

On a global level, a very comprehensive global classification scheme for buildings, able to capture all different building types that exist around the globe, has been introduced by Brzev, Scawthorn, Charleson, Allen, Greene, Jaiswal, and Silva (2013). This worldwide classification model, which has been developed within the GEM Building Taxonomy framework, is based on data input representing a wide range of building typologies from 49 countries. Table 3 lists the most widely-used classification schemes for either global or regional construction typologies along with the different classification criteria.

Table 3. Overview of major building classification schemes (after Lang [2013] and extended herein).

Name

Regional applicability

No. of typology classes

Classification criteria

Reference

(Wall) construction material

Floor/roof system

Height range

GEM Taxonomy

Global

13 main attributes and numerous attribute values (373 in total)

Brzev, Scawthorn, Charleson, Allen, Greene, Jaiswal, & Silva (2013)

PAGER

Global

81 typologies over 9 material classes

Jaiswal & Wald (2008)

WHE

Global

45 subtypes over

www.world-housing.net

UN–Habitat

Global

20 wall type classes

UN Habitat (2007)

PSI

Global

worldwide typologies

Spence, Coburn, Sakai, & Pomonis (1991)

SYNER-G Taxonomy

Europe

32 main categories and 44 sub-categories

Hancilar & Taucer (2013)

RISK–UE

Europe

65 typologies

over 23 building classes

Lungu, Aldea, Arion, Vacareanu, Petrescu, & Cornea (2001); Milutinovic & Trendafiloski (2003)

HAZUS–MH

U.S.

36 model building types

over 15 building classes

FEMA-NIBS (2003)

ATC-13

U.S.

40 typologies

ATC-13 (ATC 1985)

RESIS–II

Central America

24 typologies

Lang, Molina, Crempien, & Erduran (2009)

CENTRAL ASIA-II

Central Asia

17 typologies over

4 material classes

Lang, Erduran, Kumar, Yasunov, & Tailiakova (2012)

HIMALAYA

India

34 typologies over 12 wall/framing types

Haldar, Singh, Lang, & Paul (2013)

SAFER

Bucharest

31 typologies over 7 wall/framing types

Lang, Molina-Palacios, Lindholm, & Balan (2012)

DACEA

Romania

8 typologies over

4 material classes

Lang & Aldea (2011)

An alternative method for classifying buildings is applied by macroseismic intensity scales such as MMI (Modified Mercalli intensity scale; Wood & Neumann, 1931, Richter, 1958), MSK (Medvedev–Sponheuer–Karnik; Medvedev, Sponheuer, & Karnik, 1965) or the European Macroseismic Scale (EMS-98; Grünthal, 1998). This alternative method categorizes buildings into vulnerability classes, as shown in Table 2b. Vulnerability classes are assigned primarily according to the main construction material and then refined according to structural characteristics, construction type (or in case of the EMS: level of earthquake resistant design). The vulnerability classes range from A to F, from the most vulnerable to the least vulnerable typologies, where the first three classes (A to C) cover adobe and stone houses, brick buildings, and reinforced-concrete constructions without any ERD, while vulnerability classes D to F address reinforced and confined masonry constructions, concrete buildings with a certain level of ERD, and steel and timber buildings. The building types are classified within a certain vulnerability class where the most likely vulnerability is marked with a circle (○); the range limits are demarcated with the (│) symbol where probable (—) and less probable (– –) ranges are identified. These ranges exist because vulnerability also depends on factors other than those previously discussed, such as quality of workmanship, state of preservation, regularity, ductility, position, interventions for strengthening, and earthquake-resistant design level. The usefulness of the EMS scale in particular relies in its feasibility to assess large building stocks in extended urban areas. The classification of buildings into vulnerability classes through a walk-down survey or based on municipal cadastral data can thereby be conducted in a cost-effective manner.

The EMS-98 building classification concept, understandably, represents a major simplification and comes with a number of difficulties, such as the fact that building height is not addressed (this especially applies to engineered building typologies such as RC or steel, where all height ranges are involved), the fact that the concept of vulnerability classes principally allows buildings of completely different construction typologies to be assigned the same vulnerability class, leading one to expect them to demonstrate the same damage extent.

## Defining Damage to the Structural System

Estimating physical damage, on a local and global level for a building or infrastructure, given a certain ground-motion intensity level is the next major step in vulnerability assessment. As commonly used, a pragmatic simplification is adopted by categorizing building damage into discrete damage classes. The discrete damage classes are defined separately for both structural and nonstructural components of a building. Different concepts for damage classification exist, which are even based on different numbers of damage severity levels.

Table 4. Examples of existing building damage classifications that rely on qualitative-based description (after D’Ayala, Meslem, Vamvatsikos, Porter, & Rossetto [2015] and extended herein).

Table 5. Example of inter-story drift ratios associated to damage states, for RC buildings.

In general, the categorization of building damage can be either done in a qualitative descriptive manner by describing the damaging effects to the structure, or in a quantitative manner by assigning capacity thresholds (i.e., an empirical definition of damage state thresholds) to an individual structural element or to the entire building.

There are many classifications that rely on a qualitative description of the damage effects to a building, but they make use of the concept of building damage states, such as the one adopted in the earthquake loss estimation methodology developed by FEMA and NIBS, commonly known as HAZUS–MH (FEMA-NIBS, 1999, 2003). In this methodology the building damage classification, which was initially provided in the report Expected Seismic Performance of Buildings (EERI, 1994), defines four damage states: Slight, Moderate, Extensive, and Complete. Table 4 shows examples, from literature, of building damage classifications that rely on a qualitative-based description.

With respect to building damage classifications that rely on a quantitative description of the damage effects to a building, their main difference is the parameter used to differentiate between the damage thresholds, in addition to the number of damage states that are considered. This is the approach generally used in analytical-based vulnerability assessment. Table 5 shows examples of building damage classifications that are based on a quantitative description and on which element deformations are related to average inter-story drift ratios of structural damage state. Other models that use different parameters can also be found in literature: relating element to yielding and ultimate rotation/displacement limits (e.g., FEMA, 1997; CEN, 2004, 2005; Dolsek & Fajfar, 2004), or relating roof displacement to yielding and ultimate limits, i.e., on a global level (e.g., FEMA, 1997; Giovinazzi, 2005; Barbat, Moya, & Canas, 1996; Kappos, Panagopoulos, Panagiotopoulos, & Penelis, 2006; Lagomarsino & Giovinazzi, 2006).

# Methods for Physical Vulnerability Assessment

In early studies, when the term physical vulnerability was introduced, the basic principle was to express the seismic performance of a physical element (i.e., an individual building or infrastructure) to a given earthquake ground-motion level. This information was then used for developing mitigation plans and prevention actions (i.e., retrofitting solutions to improve the seismic response, etc.). Later, when the reinsurance industry also began paying greater attention to this topic in order to improve catastrophe risk models, seismic physical vulnerability became a term used to express not only the seismic performance of a structure but also to estimate the economic consequences of the physical damage in terms of monetary losses (FEMA, 2008).

## Physical Damage-to-Ground Motion Intensity Correlation

A large number of methods have been developed for quantifying physical vulnerability. These methods are differentiated through the approaches and assumptions implemented for the correlation of physical damage with a ground-motion intensity measure. They can generally be categorized into three groups (see Figure 3): empirical methods, expert judgment-based methods, and analytical methods, although in practice many efforts combine two or more of these approaches hence leading to some sort of hybrid method (Calvi, Pinho, Magenes, Bommer, Restrepo-Vélez, & Crowley, 2006; Lang, 2013; Porter, Farokhnia, Cho, Rossetto, Ioannou, Grant, et al., 2012). The results of a vulnerability assessment can be presented either in the form of discrete values or continuous curves/functions.

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Figure 3. Existing approaches for physical damage-to-ground motion intensity correlations.

Discrete forms of measured physical vulnerability were the first group to be introduced. They are based on approaches that correlate the loss,—i.e., the cost of the physical damage to buildings—, with a ground-motion intensity measure. The generation of Damage Probability Matrices (DPM), which, in discrete form, express the conditional probability of a damage state (dsi) being reached given a certain level of ground-motion intensity measure (IM), has been one of the most common vulnerability assessment methods from this category and used in different parts of the world. Table 6 illustrates an example of a DPM-based physical vulnerability model. This model, which was developed in the United States based on an empirical vulnerability analysis of damage data from the 1971 San Fernando earthquake, has nine damage states ranging from 0 for No damage to 8 for Collapse. These damage states were correlated with five intensities from V to IX.

Table 6. Format for a Damage Probability Matrix initially prepared for building damage data in the United States (taken from Whitman, Reed, & Hong, 1973). The damage percentages for each seismic intensity must sum up to 100%. Values are provided qualitatively in order to illustrate the various damage distributions.

Damage State

Structural Damage

Nonstructural Damage

Damage Ratio (%)

Seismic Intensity

V

VI

VII

VIII

IX

0

None

None

0–0.05

92

79

33

6

0

1

None

Minor

0.05–0.3

8

18

34

17

6

2

None

Localized

0.3–1.25

0

3

20

39

18

3

Not noticeable

1.25–3.5

0

0

10

11

22

4

Minor

Substantial

3.5–7.5

0

0

3

5

27

5

Substantial

Extensive

7.5–20

0

0

0

11

12

6

major

Nearly total

20–65

0

0

0

6

10

7

Building Condemned

100

0

0

0

5

4

8

Collapse

100

0

0

0

0

1

With the increasing volume of research, along with increasing availability and quality of earthquake damage and exposure data, important improvements have been achieved leading to the development of more innovative methods and procedures that allow the generation of a continuous physical damage-to-ground motion intensity relationship. This continuous correlation between physical damage with a ground-motion intensity measure is called the fragility curve. Literally, the term fragility defines the conditional probability of the seismic demand placed upon the structure exceeding its capacity for a given level of ground-motion intensity measure (IM).

$Display mathematics$
eq. (1)

For a building experiencing different damage states dsi, the conditional function expresses the probability of a damage state dsi sustained by the building, being reached or exceeded given a certain level of ground-motion intensity measure IM.

$Display mathematics$
eq. (2)

Figure 4 presents the graphical representation of a set of fragility curves for conditional probabilities of a building experiencing different damage states dsi.

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Figure 4. Example of continuous physical damage-to-measure of ground motion intensity relationship, also called seismic fragility curves.

In practice, IM is measured in terms of a macro-seismic intensity (e.g., MMI, MSK, EMS-98, PSI) or in terms of a physical parameter, e.g., peak ground acceleration (PGA), spectral acceleration (Sa), and spectral displacement (Sd). Vulnerability curves in terms of economic loss (i.e., cost of physical damage) are then obtained by converting the fragility curves through an appropriate damage-to-loss function.

## Empirical-Based Assessment

Empirical methods that generate physical damage-to-ground motion intensity relationships incorporate assumptions and approaches that are based on field observations and statistical analysis of building damage data (i.e., the seismic performance of the building) from past earthquake events. Methods of this category are specifically suitable for poor-quality non-engineered construction whose resistance is difficult to calculate using analytical or numerical methods. In general, most of the existing empirical approaches were developed based on the use of macroseismic intensities for characterizing the earthquake shaking; examples include the Modified Mercalli Intensity (MMI) scale (Wood & Neumann, 1931), Medvedev–Sponheuer–Karnik (MSK) scale (Medvedev, Sponheuer, & Karnik, 1965), European Macroseismic Scale—EMS-98 (Grünthal, 1998), and the parameter-less scale of seismic intensity PSI (Spence, Coburn, Sakai, & Pomonis, 1991). In recent years, when more instrumental data in terms of (strong-motion) earthquake recordings became available, empirical vulnerability assessment studies based on physical parameters such as peak ground acceleration (PGA), peak ground velocity (PGV), spectral acceleration (Sa), were conducted as well.

The concept of DPM, which was initially introduced and described by Whitman, Reed, and Hong (1973), as illustrated in Table 6, and which later became the basis for ATC–13 (1985), was the first of its kind and was most widely used to represent empirical-based building vulnerability in a discrete form. Later, this concept underwent certain improvements and was implemented in many other regions of the world. The first European version of a DPM was based on Italian building damage data from the 1980 Irpinia earthquake (Braga, Dolce, & Liberatore, 1982). Within the RISK-UE framework, a research project consortium financed by the European Commission, a procedure was introduced which allowed the generation of DPMs considering the EMS-98 building vulnerability classes (Milutinovic & Trendafiloski, 2003). This DPM uses qualitative descriptions of “Few,” “Many,” and “Most” for the five damage grades (Grade 1 to Grade 5) for the levels of intensity ranging from V to XII. Table 7 shows an example of DPMs for EMS-98 vulnerability class A, containing a qualitative description of the proportion of buildings that belong to each damage grade for various levels of intensity.

Table 7. Example of damage matrices for EMS-98 building vulnerability classes (Milutinovic & Trendafiloski, 2003).

Continuous physical damage-to-ground motion intensity curves that are directly based on damage to buildings from past earthquake events were introduced slightly later than the DPMs. Most of the existing models have been mainly constructed using the parametric regression models listed in Table 8. The lognormal cumulative distribution function is most commonly used as the regression model. Its popularity can be attributed to its three properties:

• The function is constrained in the y-axis between (0, 1), which is ideal for fitting data points expressing aggregated probabilities.

• With regard to the x-axis, the values are constrained to the range (0, +∞). This agrees with the range of almost all ground-motion intensity measures.

• This distribution appears to be skewed to the left, and thus it can, at least theoretically, provide a better estimate for the smaller intensities, where the majority of the data typically lies (Ioannou & Rossetto, 2012).

Table 8. Regression models used to express empirical fragility curves (modified from Ioanna & Rossetto, 2012).

Type

$P(DS≥dsi|IM=x)$

Parameters

References

Lognormal cumulative distribution function

$Φ(ln(x)−λζ)$

Yamaguchi & Murao (2000); Shinozuka, Feng, Lee, & Naganuma (2000); Sarabandi, Pachakis, King, & Kiremidjian (2004); Rota, Penna, & Strobbia (2008); Liel & Lynch (2012)

Normal cumulative distribution function

$Φ(x−μσ)$

Yamaguchi & Yamazaki (2001)

Logistic distribution function

$11+exp(−(θ0+θ1x))$

Basöz, Kiremidjian, King, & Law (1999); O’Rourke & So (2000)

Exponential function

$1−exp(−θ0xθ1)$

Rossetto & Elnashai (2003); Amiri, Jalalian, & Amrei (2007)

The normal cumulative distribution function and the logistic distribution function have been used in cases where the intensity measure can take negative values. However, some researchers tend to use this distribution in generating fragility curves despite the fact that their intensity measure is discrete and positive (e.g., Yamaguchi & Yamazaki, 2001). Another type of distribution has also been adopted in generating empirical fragility curves, called exponential function and which is unconstrained on both x- and y-axis (e.g., Rossetto & Elnashai, 2003; Amiri, Jalalian, & Amrei, 2007). The use of a nonprobability distribution function in order to express the fragility curves may have implications in the risk assessment which requires its coupling with a hazard curve to produce the annual probability of reaching or exceeding a certain damage state.

## Expert Judgment-Based Assessment

In principle, each method used to provide building vulnerability information is based on expert opinion to some extent, since the damage predictions are based on the subjective opinion of the expert when, for instance, using the terms “few”, “many,” and “most”. In each empirical survey or analytical study, a certain number of assumptions have to be made that require the subjective opinion or decision of a (group of) expert(s), which are considered to be the best estimate for the given problem. These include questions such as: How to collect or interpret damage data (which is in turn essential to determining shaking intensity)? How to choose certain (building-related) parameters which are essential to the study’s outcome but often are not readily available, such as material parameters or reinforcement detailing? How to assign a vulnerability class to a building? How to categorize buildings with varying characteristics as in the same building class?

Such kinds of seismic vulnerability assessments were first carried out in the United States and introduced in ATC-13 (ATC, 1985). The ATC-13 report developed the DPM for 78 structural typologies, out of which 40 belonged to buildings. These DPM were developed by asking 58 experts to provide low, best, and high estimates of the damage factor (i.e., the ratio of loss to replacement cost, expressed as a percentage) for Modified Mercalli Intensities (MMI) from VI to XII for 36 different building classes. Each expert was asked to fill in a comprehensive questionnaire by utilizing his/her best knowledge. Recently, efforts have been carried out within the Global Earthquake Model (Jaiswal, Wald, Perkins, Aspinall, & Kiremidjian, 2013) trying to elicit expert opinion on uncertain quantities and providing clear definitions with respect to biases, assumptions, and expert opinions in order to improve the philosophy of the methods.

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Figure 5. (a) Photograph showing experts responding to a target question; (b) Collapse fragility estimates obtained using expert elicitation process (adapted from Jaiswal, Wald, Perkins, Aspinall, & Kiremidjian, 2013).

## Analytical-Based Assessment

The analytical methods for measuring seismic physical vulnerability may also be called purely theoretical approaches, since, in contrast to the empirical or expert judgment-based methods, they are not based on observation, but rather on the theoretical simulation of physical damage under earthquake loading. In order to analytically predict the structural damage that a building of a given capacity will produce under a given seismic impact, different methods are available in the literature covering different building typologies and locations worldwide (Meslem, D’Ayala, Ioannou, Rossetto, & Lang, 2014; D’Ayala, Meslem, Vamvatsikos, Porter, & Rossetto, 2015). The methods vary from simplified, non-numerically-based, to nonlinear static and dynamic numerically-based analyses of increasing complexity and accuracy. Table 9 shows a list of analytical-based methods that have been most commonly used for measuring physical seismic vulnerability.

Table 9. Analytical methods most commonly used for measuring seismic physical vulnerability.

Method

Implementation—Building Typology

Reference

Nonlinear Static Procedures

Capacity Spectrum Method (CSM)

Steel, Reinforced Concrete, Timber

ATC–40 (ATC 1996); FEMA-273 (FEMA 1997);

Steel, Reinforced Concrete, Timber

FEMA-440 (FEMA 2005)

Displacement Coefficient Method (DCM)

Steel, Reinforced Concrete, Timber

FEMA-356 (FEMA 2000)

Improved Displacement Coefficient Method (I-DCM)

Steel, Reinforced Concrete, Timber

FEMA-440 (FEMA 2005)

N2 Method

Steel, Reinforced Concrete

Fajfar (2002); Dolsek & Fajfar (2004); Eurocode 8 (CEN 2004)

Nonlinear Static Simplified Mechanism-based Procedures

Collapse-based Methods (CBM)

Failure Mechanism Identification and Vulnerability Evaluation (FaMIVE)

Earthen (historic buildings)

D’Ayala & Speranza (2002)

VULNU

Unreinforced Masonry

Bernardini, Gori, & Modena (1990); Cosenza, Manfredi, Polese, & Verderame (2005)

Displacement-based Methods (DBM)

Displacement-Based Earthquake Loss Assessment (DBELA)

Reinforced Concrete

Miranda (1999); Crowley, Pinho, & Bommer (2004)

Mechanical Based Procedure for the Seismic Risk Estimation (MeBaSe)

Unreinforced Masonry

Restrepo-Vélez and Magenes (2004); Restrepo-Vélez (2005); Modena, Lourenço, & Roca (2005)

Nonlinear Dynamic Procedures (NDP)

Incremental Dynamic Analysis (IDA)

For all building typologies

Shome & Cornell (1999); Vamvatsikos & Cornell (2002)

Nonlinear Static Procedures (NSP), also called Capacity Spectrum-based methods, have received the greatest attention to date, mainly because these procedures were published as various FEMA provisions (especially since some of them were established as the basis for FEMA’s HAZUS–MH methodology and implemented in many structural computer software tools), as well as earthquake building codes such as Eurocode 8 (CEN, 2004). Procedures from this category accrued from the philosophy of performance-based seismic design (PBSD), recognizing the fact that structural damage is mainly determined by lateral displacement or drifts. In this type of procedure, building vulnerability is expressed in terms of a capacity curve that represents the nonlinear behavior of the structure under lateral displacement. To identify a capacity curve, which is defined as the relationship between the base shear force and the lateral displacement of a control node of the building (Goel, 2005), a nonlinear structural analysis method such as the nonlinear quasi-static “pushover” analysis (U.S. Army, 1986; ATC, 1996; FEMA, 2000) is required. This postulates the creation of a reliable structural model of the building under consideration to which the pushover analysis can be applied (Figure 6).

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Figure 6. Principal steps of the capacity spectrum-based procedures for the calculation of seismic performance (modified from Lang, 2013).

The second component, seismic ground motion (or seismic demand), is generally represented by a response spectrum in terms of physical parameters, i.e., spectral accelerations and spectral displacements. In order to be able to correlate the response spectrum with building capacity, it needs to be converted from the (conventional) SaT domain into the domain of the capacity curve, i.e., spectral acceleration–spectral displacement domain (SaSd). The final step is to identify the target displacement (or performance point) dp. This displacement stands for the mean displacement a building typology will reach under the respective seismic demand. Hence, it represents the mean damage an individual building of this typology will experience.

In case of Nonlinear Static Simplified Mechanism-based Procedures, they have the advantage of analyzing a large number of buildings in a relatively short period of time. However, the efficiency increases only if the input parameters required for analyzing the buildings can properly capture the overall seismic behavior of the buildings.

One of these methods was developed by D’Ayala and Speranza (2003), called Failure Mechanism Identification and Vulnerability Evaluation (FaMIVE) procedure. This method uses a nonlinear pseudo-static structural analysis with a degrading pushover curve in order to estimate the performance points in a similar way to the Capacity Spectrum-based methods. It yields collapse multipliers which identify the occurrence of possible different mechanisms for a given masonry construction typology, given certain structural characteristics.

Table 10. In-plane and out-of-plane collapse mechanisms used in the simplified procedure FaMIVE.

Two types of collapse mechanisms, namely in-plane and out-of-plane are considered. The potential collapse mechanism and the corresponding capacity are determined by the geometry and boundary conditions, usually based on visual observations. The in-plane and out-of-plane collapse mechanisms are further divided into various subclasses as shown in Table 10. Each collapse mechanism is related to a damage grade recommended by the European Macroseismic Scale 1998 (EMS-98; Grünthal, 1998).

Nonlinear Dynamic Procedures (NDP) are in general limited for seismic vulnerability at an individual building level. An example of this method is the Incremental Dynamic Analysis (IDA), which was initially developed by Shome and Cornell (1999) and then later improved by Vamvatsikos and Cornell (2002, 2005). The IDA is done by subjecting a building model to nonlinear time-history analysis under a suite of ground-motion accelerograms that are scaled to increasing levels of the IM until collapse is reached (see Figure 7). The location where each IDA curve becomes flat identifies the IM level beyond which it is assumed that global collapse of the building will occur. The process should be repeated for the selected suite of ground motions. The median IDA curve is defined as 50% of all the maximum responses recorded at each level of IM, as shown in Figure 7.

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Figure 7. Procedure of seismic physical vulnerability assessment using Incremental Dynamic Analysis (IDA).

With respect to generating continuous physical damage-to-ground motion intensity relationships (i.e., fragility curves) using analytical methods, it is commonly assumed that these relationships take the form of lognormal cumulative distribution functions having a median value and logarithmic standard deviation, or dispersion. The mathematical form for such curves is:

$Display mathematics$
eq. (3)

where Φ is the standard normal cumulative distribution function; αDS|IM is the lognormal mean of the generic structural response conditioned on the ground motion intensity, IM; and β is the lognormal standard deviation of DS|IM.

The parameter lognormal standard deviation, β, describes the total uncertainty of the fragility curves, which in general should consider three primary sources of uncertainty, namely:

• the uncertainty in the demand imposed on the structure by the earthquake ground motion,

• the uncertainty associated with the structural characteristics-related parameters, and

• the uncertainty associated with the damage state threshold.

The uncertainty in the demand is introduced by the record-to-record variability, which captures the variability in the complexity of the mechanism of the seismic source, path attenuation, and site effects of the seismic event. This uncertainty is taken into account by the majority of the studies reviewed either by selecting a certain number of ground motion records and/or by scaling the records. The uncertainty in the structural characteristics-related parameters is introduced by geometrical, mechanical, structural, and modeling parameters. Typically, uncertainty in geometric parameters is accounted for by randomizing parameters such as buildings’ plan dimensions, height, and number of stories; uncertainty in structural parameters is accounted for by randomizing parameters such as bay length and column orientation; uncertainty in mechanical parameters of the construction materials is accounted for by randomizing parameters such as compressive strength and elasticity modulus of concrete, tensile strength, and elasticity modulus of steel reinforcement, hardening ratio of steel, and compressive strength of masonry infill; modeling uncertainty is typically introduced in some studies by randomizing the parameters of the hysteric models. In other cases (e.g., D’Ayala, 2005), variability in structural and geometric characteristics is accounted for by a survey of a large number of real buildings (i.e., building-to-building variability) and determining a median and standard deviation for the sample, after calculating the capacity and damage threshold for each element in the sample.

## Physical Damage-to-Economic Loss Correlation

Vulnerability functions in terms of economic loss translate the physical damage into monetary loss (i.e., estimation of repair and reconstruction costs), given a certain level of intensity measure, IM. There are alternatives that have been introduced in the literature to generate these functions, depending on the needs of the study and the availability of data/information:

### Global-Level Approach

The implementation of this approach is suited to studies of large populations of buildings. As recommended in HAZUS-MH (FEMA-NIBS, 2003), the approach consists in generating the vulnerability function by convolving building response with the cumulative cost of a given damage state (damage-to-loss functions). The transformation of the physical damage into economic loss can be conducted through the following total probability relationship:

$Display mathematics$
eq. (4)

where n is the number of damage states considered, P(dsi | IM) is the probability of a building sustaining damage state dsi given the intensity measure IM; E(C>c|dsi) is the complementary cumulative distribution of the cost (loss) given dsi; E(C>c|im is the complementary cumulative distribution of cost (or loss) given a level of intensity IM.

Many research programs have produced a compendium of empirical Damage Factor values (including material and labor costs), given a certain damage threshold, and which can be used as default values in cases when economic data (i.e., building repair or reconstruction costs for a given damage state ds) are not available (D’Ayala, Meslem, Vamvatsikos, Porter, & Rossetto, 2015). For instance, HAZUS-MH (FEMA-NIBS, 2003) has provided default values for the Damage Factor, which include material and labor costs related to 33 occupancy classifications in the United States. There is also the GEM-VEM database (Rossetto, D’Ayala, Ioannou, & Meslem, 2014) collected from different sources and regions/countries, which includes material and labor costs for structural and nonstructural components.

The example shown in Figure 8 illustrates the results of a seismic vulnerability and risk assessment that was carried out for the city of Guwahati, one of the most rapidly growing cities in India. In 1971, the city’s population was only 200,000, whereas the 2011 census revealed a population of more than 960,000 with a population density of more than 2,010 persons/km2. According to the zoning map of the Indian seismic building code IS 1893 (Part 1): 2002 (BIS, 2002), Guwahati falls into the highest seismic zone (Zone V). The vulnerability and risk assessment started by developing a customized building classification scheme for the existing building stock in the city. The classification scheme uses 12 model building types covering 6 different building classes. The building stock in Guwahati is mainly dominated by buildings of nonductile (non-engineered and low-code engineered buildings) low-rise reinforced concrete frames and confined masonry, followed by buildings of ductile (modern engineered buildings) low-rise reinforced concrete frames with unreinforced masonry infill walls. The next step was generating physical vulnerability curves, that is, fragility curves for each building typology class, as shown in Figure 8. The process of using these physical vulnerability models in order to generate physical damage distribution and the conversion in terms of monetary losses accrued as the result of the damage, is illustrated in Figure 9.

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Figure 8. Example of generated physical vulnerability (fragility) for an existing RC building in Guwahati city.

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Figure 9. Building damage distribution and the conversion in terms of economic loss for Guwahati city, for the 1897 Shillong earthquake scenario, India.

### Component-Level Approach

In the component-level approach, recommended in FEMA-P-58 (FEMA, 2012), the vulnerability functions are obtained by correlating the component level-based drifts directly to loss. In general, this approach is mostly suitable for the vulnerability analysis of single buildings, and where the majority of the economic losses are associated with nonstructural components (e.g., in case of hospitals or other highly sophisticated buildings). The implementation of such detailed analysis requires specific knowledge (e.g., the definition of the performance criteria in terms of plastic rotation values for each structural and nonstructural component), time, and monetary resources. Recently, a default source of data (average component-level performance and its associated loss/cost) for the United States building stock has been developed and provided in ATC-58 PACT/FEMA P-58 (FEMA, 2012) that can be used to implement this type of analysis.

# Current Challenges in Physical Vulnerability Assessment

Vulnerability assessment, like any other field of research, has its own special challenges and issues that are often encountered when measuring or evaluating the physical vulnerability of an individual building or a portfolio of buildings. These issues and challenges are mostly related to the level of knowledge and the level of detail of data input to be used for the assessment, as well as to the different approaches and assumptions to be adopted and implemented (e.g., the level of simplification in modeling, the analysis process to be adopted in order to reduce the calculation effort) in measuring vulnerability.

## Challenges in Choosing Between Different Methods

A number of problems can be associated with the existing empirical methods and approaches for vulnerability assessment. A large number of the existing empirical vulnerability models that were developed mainly use macroseismic intensities (e.g., MMI, MSK, EMS–98, PSI) for characterizing and representing the earthquake shaking. One of the main shortcomings when using intensities to predict earthquake damage may lie in the fact that intensity does not have any connection to the frequency (spectral) content of seismic ground motion. Hence, any damage-contributing effect that may result from agreements between the predominant frequencies of the site and the structure cannot be addressed at all. In addition, macroseismic intensity is a non-instrumental parameter primarily based on damage observations and subjective opinions (feelings, impressions, sensations) of individuals. This directly implies a certain level of uncertainty due to such subjectivity (Lang, 2013). Peak ground acceleration (PGA), which is a physical parameter, was also used in empirical studies; however, this parameter in particular shows almost no correlation to structural earthquake damage (Crowley, Pinho, & Bommer, 2004). Moreover, empirical methods rely purely on building damage observations from past earthquakes. This means that (a) generally limited data is available for lower shaking intensities (i.e., intensity I < VI) where no visible damage is produced, and (b) data for a certain test bed is typically restricted to only one or two intensity grades. Consequently, it is necessary either to use empirical data that was collected from other earthquakes and/or countries (with similar construction practice; e.g., Roca, Goula, Susagna, Chávez, González, & Reinoso, 2006) or to revert to expert opinion to supplement the database (ATC, 1985; Kappos, Stylianidis, & Pitilakis, 1998).

With respect to expert judgment-based approaches, the main shortcoming lies in the subjective opinion. Each expert, depending upon his/her knowledge and engineering judgment has his/her own opinion. In addition, the results obtained for the target area cannot be extended to other towns and cities. Another issue with the expert judgment-based methods is that they cannot be applied for the prediction of damage to regions or areas with no past earthquake experience.

Regarding the analytical-based methods, the main challenge remains the quantification and modeling of the uncertainties (due to simplified assumptions) that would be involved at each stage of the analysis. A recently conducted extensive literature review under the framework of developing the GEM Guide for Selecting of Existing Analytical Fragility Curves and Compilation of the Database (D’Ayala & Meslem, 2013) shows that in most vulnerability studies the examined building is typically simulated in terms of a 2D symmetrical model with deterministic geometrical properties, reducing the ability of the model to capture the real behavior of the building and the variability in the structural characteristics. A more realistic 3D model was only used in few studies (e.g., Rossetto & Elnashai, 2003). Moreover, few publications simulated the infill walls in reinforced concrete (RC) buildings, while Dymiotis, Kappos, and Chryssanthopoulos (1999, 2001) highlighted the different performance of the RC building due to vertical irregularities, e.g., an existing soft story.

On the other hand, the challenge with the implementation of nonlinear dynamic-based methods is that they involve intense calculations and require detailed mathematical models of multi degree of freedom (MDoF) systems. In fact, this technique of analysis is considered to be quite impractical for everyday use. Depending on the level of complexity and material type of the building, the length of time required to perform a computation process might be significant.

Basically, the choice between the different existing methods for physical vulnerability quantification must consider a number of aspects. This includes the purpose of the assessment (at both the individual building level and the building stock level), the size of the urban center, the prevalent building typologies within it, and the availability of the required data input (i.e., the quality and level of details) in order to accurately define the typology class and to develop a consistent model that would best represent the real behavior of the individual building or building stock selected, and thereby better quantify the uncertainty (Figure 10).

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Figure 10. Calculation efforts and uncertainties associated to various methods of evaluating physical vulnerability

Within the framework of the Global Earthquake Model (GEM), Porter, Farokhnia, Cho, Rossetto, Ioannou, Grant, et al. (2012) studied methods from different categories which led to the development of guideline documents that would assist analysts in ensuring the consistency between the purpose of the type of analysis (approach/method), the mathematical modeling, and the type and quality of data input to be used (Rossetto, D’Ayala, Ioannou, & Meslem, 2014; D’Ayala, Meslem, Vamvatsikos, Porter, & Rossetto, 2015; D’Ayala & Meslem, 2013; Jaiswal, Aspinall, Perkins, Wald, & Porter, 2012).

## Challenges in Selecting Existing Vulnerability Models Database from the Literature

The world has witnessed many earthquake disaster events causing significant property damage and economic losses, as well as social losses. This, in turn, has pushed governments from different earthquake-prone countries to implement many research programs aimed at developing prevention and mitigation actions, or in refining code provisions and guidelines. Further,insurance and reinsurance companies spent significant investments in developing or refining earthquake catastrophe models. These activities have resulted in a wealth of seismic vulnerability models covering a wide range of building typologies and portfolios. These are made available in the literature, such as the U.S. vulnerability database provided in HAZUS-MH (FEMA-NIBS, 2003), the European vulnerability database by SYNER-G (2011a, 2011b), and the worldwide vulnerability database compiled within the framework of the Global Earthquake Model (GEM) (Rossetto, D’Ayala, Ioannou, & Meslem, 2014; D’Ayala & Meslem, 2013).

The availability of these vulnerability models has encouraged risk analysts (e.g., academics, engineers) to use them for future applications, rather than to develop customized models (i.e., conduct vulnerability analysis and measure physical vulnerability) that addresses the peculiar structural and nonstructural characteristics of the respective building stock. The reasons for this are either a desire to reduce the calculation efforts, especially when studies are conducted for large portions of the building stock, a lack of available resources, or a lack of information that does not allow for a detailed survey and data acquisition (Meslem, Lang, & Molina, 2015).

However, the main challenge in using these predefined physical vulnerability models, is how to identify suitable ones in order to ensure a reliable earthquake loss assessment. In general, these existing models have been derived using a variety of approaches, assumptions, and methodologies that employ diverse structural modeling and analysis techniques (as discussed in the previous section). A range of sampling methods has also been applied to parameters of the structural models and the seismic demand in order to account for uncertainties and intrinsic differences observable in the building stock and its response to seismic loading. It can therefore be difficult to appraise existing physical vulnerability models, even when derived for the same structural typology class.

An application example is presented in Figure 11, showing the strength of the physical vulnerability representativeness on the risk assessment outcomes (i.e., damage and economic loss). The selected test bed is Santiago de Cuba, Cuba’s second largest city and the capital of Santiago de Cuba province. According to a census conducted in 2012, the city’s population was estimated to be more than 400,000. According to the zoning map of the Cuban National Bureau of Standards (NC46, 2013), the city of Santiago de Cuba is situated in the country’s highest seismic zone (Zone 5).

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Figure 11. Earthquake risk maps for the city of Santiago de Cuba (Cuba), using customized and collected physical vulnerability models. The selected scenario is a M 7.5 earthquake located at 40 km distance to the city center.

The influence of physical vulnerability models has been investigated by comparing risk results obtained using customized physical vulnerability models with those obtained using a set of physical vulnerability models (i.e., fragility curves) provided by HAZUS-MH (FEMA-NIBS, 2003). Note that the customized physical vulnerability models are regional level-based models, while the ones from HAZUS-MH are country level-based models.

The comparison of the customized models with the collected ones from HAZUS-MH shows a remarkable bias (with a factor of 2.5), leading to a significant difference of more than 100% in predicting the seismic performance of the buildings, and hence, earthquake damage and economic loss estimates. Typically, differences in construction techniques and detailing between different countries are significant, even when buildings are nominally designed according to similar code provisions. This result clearly indicates that physical vulnerability models may have a very significant effect on the earthquake loss estimates. Hence, special care should be given when selecting the existing vulnerability models that are available from literature, in order to ensure a reliable earthquake loss assessment.

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Figure 12. Physical vulnerability models selection framework considering size and regional factor and their uncertainties.

Recently, the different parameters influencing the development of physical vulnerability models have been investigated (D’Ayala & Meslem, 2013; Rossetto, D’Ayala, Ioannou, & Meslem, 2014) resulting in the development of a Relevance Ranking System that can assist analysts in selecting a vulnerability model appropriate for their application scope. In general, the rating system considers three main attributes:

• Size and regional attribute: to consider the level of the assessment, i.e., on an individual building/facility level, city level, country level, or global level (see Figure 12);

• Defining structural system and classification modeling: to consider the representativeness of the structural system, ensuring an appropriate definition of the building classification; and

• Method of analysis and generating process used in correlating physical damage with ground motion intensity.

Special care should be given to these factors when selecting the existing vulnerability models that are available from the literature, in order to ensure a reliable earthquake loss assessment.

# Conclusions

It is the main purpose of the present overview to provide specialist and nonspecialist readers with comprehensive information that would help the reader gain a clear understanding of the term “building physical vulnerability” and its use in the field of earthquake engineering and disaster risk management. In earthquake risk assessment, the term “physical vulnerability” is basically defined as a component that describes the susceptibility of a given building type or infrastructure to experience damage (and its associated economic loss) caused by a given level of ground shaking.

Since its introduction, this component has received much attention by research engineers and insurance analysts and has resulted in the development of a large number of methods, which can be divided into three main categories: empirical, analytical, and expert judgment/opinion.

The quantification and modeling of the uncertainties that are involved at each stage of a vulnerability analysis has always been, and still remains, one of the main challenges encountered when measuring physical vulnerability in seismic risk assessment. These uncertainties can derive from the definition of the structural capacity-related characteristics of the building; the uncertainty in estimating the ground-motion intensity for a given event; the uncertainty in estimating physical damage given the ground-motion intensity for a given event; and finally the uncertainty in estimating the economic loss given damage to the building. Each of these elements could be made more precise with additional efforts and resources to improve the quality and quantity of data input.

The new development of high-performance computer systems and advanced programs (e.g., with nonlinear analysis capabilities), and the increasing volume of research, together with the increasing availability of earthquake damage and exposure data, is resulting in the development of more innovative procedures/approaches and important improvements in the reliability and robustness of physical vulnerability modeling. However, it is important to keep in mind that any known value assigned to a given parameter when computing physical vulnerability has an uncertainty associated with it. Uncertainties are inevitable in any practical study of vulnerability assessment and should be expressed and quantified. However, reducing or better quantifying the uncertainty associated with one of the parameters when computing physical vulnerability does not necessarily mean improving the overall reliability and robustness of the results.

# Acknowledgments

This present work is part of international collaborative research projects carried out by NORSAR in collaboration with local governmental organizations and research institutions from different earthquake-prone countries. The authors acknowledge financial support from the Royal Norwegian Ministry of Foreign Affairs. We further appreciate the comments of two anonymous reviewers that allowed us to greatly improve the present manuscript.

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# Overlapping between two intervals: reasoning / algorithm to find the set of disjoint and overlapping intervals

Consider the positive integers {1, 2, 3, 4, ...} and the corresponding Integer Number Line.

Suppose we have four integer numbers, A, B, C and D.

For example:

_________________A___________________________B__________________ ___________________________C__________D_________________________

or

_________________A___________________________B__________________ ___________________________C____________________________D_______

Consider the notation AB as the set of all consecutive numbers A, ...., B, and the interval CD as the set of all consecutive numbers C, ..., D.

I would like to know a simple yet abstract and generalizaed resoning (mathematical demonstration would be really apreciated) that we can apply if we want to find all intervals consisting of the "disjoint" numbers and the "overlapping" numbers between both set of numbers, AB and CD.

Example 1: If AB = {2, 3, 4, 5, 6, 7, 8, 9} and CD = {4, 5, 6, 7}, we have the following subsets for each set AB and CD:

• for AB: {2, 3}, {4, 5, 6, 7} and {8, 9}
• for CD: {4, 5, 6, 7}

Example 2: If AB = {2, 3, 4, 5, 6, 7, 8, 9} and CD = {4, 5, 6, 7, 8, 9, 10, 11, 12}, we have the following subsets for each set AB and CD:

• for AB: {2, 3}, {4, 5, 6, 7, 8, 9}
• for CD: {4, 5, 6, 7, 8, 9} and {10, 11, 12}

But, which one would be a great and yet simple reasoning behind disjoint and overlapping sets of numbers between two given set of integer numbers?

Thanks very much for any clue!!

For two integers $$A$$ and $$B$$ where $$A\le B$$, let $$[A,B)$$ denote the half-closed half-open interval that consists of the numbers $$A, A+1, \cdots, B-1$$. In particular, if $$A, then there are $$B-A$$ numbers in $$[A,B)$$. If $$A=B$$, then there is no numbers in $$[A,B)$$. For example,

• $$[0,1)$$ means the number 0.
• Both $$[0,0)$$ and $$[3, 3)$$ mean no numbers.
• $$[5,11)$$ means the numbers $$5,6,7,8,9,10.$$
• To denote the numbers $$-2, -1, 0, 1, 2$$, we can use $$[-2, 3)$$.

A nice property of half-closed half-open interval: If $$L\le M \le R$$, then $$[L,R) = [L,M) \sqcup [M,R)$$.

Proof: I am afraid it is obvious by definition. Done.

Suppose we are given $$[A, B)$$ and $$[C, D)$$, where $$A\lt B$$ and $$C\lt D$$. Then $$[A,B)$$ overlaps $$[C,D)$$ if and only $$\max(A,C)\lt \min(B,D)$$.

Proof:

"$$\Rightarrow$$": Let $$n$$ be a number in both $$[A,B)$$ and $$[C,D)$$, i.e., $$A\le n\lt B$$ and $$C\le n \lt D$$. Then $$\max(A,C)\le n\lt \min(B,D)$$.

"$$\Leftarrow$$": $$A\le \max(A,C)\lt \min(B,D)\le B$$, i.e., $$\max(A,C)\in [A,B)$$. Similarly, $$\max(A,C)\in [C,D)$$. So $$[A,B)$$ overlaps $$[C,D)$$.

Done.

Suppose $$[A,C)$$ and $$[B,D)$$ overlap. Then we have the following set decompositions.

• $$[A,B) = [A, \max(A,C))\sqcup[\max(A,C), \min(B,D)]\sqcup[\min(B,D), B)$$
• $$[C,D) = [C, \max(A,C))\sqcup[\max(A,C), \min(B,D)]\sqcup[\min(B,D), D)$$

Proof:

We have $$A\le \max(A,C) \le \min(B,D)\le B$$. So,

\begin{align} [A,B) &= [A, \max(A,C))\sqcup[\max(A,C),B)\\ &= [A, \max(A,C))\sqcup[\max(A,C), \min(B,D)]\sqcup[\min(B,D), B). \end{align}

Example 1: If $$I = \{2, 3, 4, 5, 6, 7, 8, 9\}$$ and $$J = \{4, 5, 6, 7\}$$, then $$I=[2,10)$$ and $$J=[4,8)$$. We have the following set decompositions.

• $$[2,10) = [2, 4)\sqcup [4, 8) \sqcup [8,10)$$.
• $$[4,8) = [4, 4)\sqcup [4, 8) \sqcup [8,8) = [4,8)$$.

Example 2: If $$I = {2, 3, 4, 5, 6, 7, 8, 9}$$ and $$J = {4, 5, 6, 7, 8, 9, 10, 11, 12}$$, then $$I=[2,10)$$ and $$J=[4,13)$$. We have the following set decompositions.

• $$[2,10) = [2, 4)\sqcup [4, 10) \sqcup [10,10)= [2, 4)\sqcup [4, 10)$$.
• $$[4,13) = [4, 4)\sqcup [4, 10) \sqcup [10,13)= [4, 10) \sqcup [10,13)$$.

The above examples should indicate clearly enough the algorithm that decomposes two overlapping intervals into disjoint intervals.

• John, thank you so much!! Mar 13 '20 at 13:21
• @strajano, welcome! Mar 13 '20 at 16:57

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