Advances in Civil Engineering

Volume 2018, Article ID 6209137, 13 pages

https://doi.org/10.1155/2018/6209137

## Development of Empirical Fragility Curves in Earthquake Engineering considering Nonspecific Damage Information

Professor, Department of Civil Engineering, Kyungnam University, Changwon-si 51767, Republic of Korea

Correspondence should be addressed to Jung J. Kim; rk.ca.mangnuyk@mikgnuj

Received 3 September 2018; Revised 4 November 2018; Accepted 11 November 2018; Published 13 December 2018

Guest Editor: Tiago Ferreira

Copyright © 2018 Jung J. Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

As a function of fragility curves in earthquake engineering, the assessment of the probability of exceeding a specific damage state according to the magnitude of earthquake can be considered. Considering that the damage states for fragility curves are generally nested to each other, the possibility theory, a special form of the evidence theory for nested intervals, is applied to generate fragility information from seismic damage data. While the lognormal distributions are conventionally used to generate fragility curves due to their simplicity and applicability, the methodology to use the possibility theory does not require the assumption of distributions. Seismic damage data classified by four damage levels were used for a case study. The resulted possibility-based fragility information expressed by two monotone measures, “possibility” and “certainty,” are compared with the conventional fragility curves based on probability. The results showed that the conventional fragility curves provide a conservative estimation at the relatively high earthquake magnitude compared with the possibility-based fragility information.

#### 1. Introduction

In earthquake engineering, fragility curves have been used to estimate damages of infrastructures according to the magnitude of earthquake. While fragility curves can provide reasonable estimation of earthquake damages with damage levels, those might neglect the possible slight damage occurrence at the relatively low earthquake magnitude due to the nature of probability density functions consists of two parameters, expected mean value and dispersion.

Traditionally, probability theory has been used to model uncertainties in structural engineering, especially when addressing reliability for structural safety [1, 2]. However, the types of uncertainties considered in probability theory are random, chance, and likelihood, and there are limitations to model other types of uncertainties such as nonspecificity, fuzziness, and strife, using probability theory [3, 4]. Random uncertainty known as aleatory uncertainty is from inherent randomness and therefore is irreducible. However, other types of uncertainties known as epistemic uncertainties arise from lack of knowledge and therefore are reducible and subjective. Research on generalized information theory (GIT) [5, 6] showed that three types of epistemic uncertainties due to lack of knowledge and/or variability thrive when modeling complex environments [7]. While nonspecificity represents the difficulty to choose from many modeling alternatives, fuzziness represents the uncertainty due to lack of sharpness (imprecise boundaries) of the modeling parameter. Strife expresses the uncertainty due to conflict among alternatives. Given are there various types of uncertainties, appropriate modeling of uncertainty has been an interesting and challenging topic in many areas during the last few decades [8–10]. A number of theories to model uncertainties adequately have been introduced: evidence theory [11, 12], possibility theory [3, 13] and fuzzy set theory [14, 15].

In earthquake engineering, empirical fragility curves were generally presented in the form of lognormal cumulative distribution function (CDF) with respect to peak ground acceleration (PGA) representing the ground motion intensity due to earthquake [16]. To generate empirical fragility curve for a damage state, the damage reports by experts are used. The damage reports usually present the damage states in linguistic ways such as “no damage,” “slight damage,” “moderate damage,” “extensive damage,” and “collapse” for a structure experiencing earthquake of a PGA. As the fragility curve of a damage state represents the fragility of “at least” of the damage level, the evidence for a damage state includes possible higher damage states. For example, the evidence for “moderate damage” of a structure by an earthquake means that the structure is damaged at least moderately, and it might be possible for the structure to be damaged extensively or collapsed. As there exist ambiguous boundaries between the damage states, fuzzy logic and possibility theory were applied to resolve the ambiguity [17, 18].

Recently, extensive earthquake damage data are used to generate fragility curves [19–25]. Postearthquake surveys of approximately 340000 reinforced concrete structures were used to derive fragility curves for a European seismic risk assessment scenario [19]. A database of 7597 reinforced concrete buildings located in the city and the province of L’Aquila in Italy was used in order to derive fragility curves [20]. The observed damage to 9500 of low-rise residential buildings from earthquakes in South Iceland was studied by typological fragility curves [21]. Moreover, fragility curves were developed from millions of data on the basis of 665,515 building damage cases by earthquake in Nepal [22, 23]. In Italy, the postearthquake damage surveys of approximately 90,000 buildings in order to derive fragility curves were considered [24, 25]. Even with the increase of damage data to generate fragility curves, there is still an uncertainty of nonspecificity, the difficulty to choose from many modeling functions of fragility curves such as lognormal, extreme type I, extreme type II functions, and so on.

In this study, the evidence of damage state is dealt with possibility theory. It is noticeable that the fragility curves from possibility distribution representing the certainty of damage state and those are generated without any assumption of distributions. Therefore, there is no uncertainty of nonspecificity to choose functions of fragility curves.

#### 2. Possibility Information in Fragility Curves

Theories for modeling uncertainties present different types of uncertainty assignment and monotone measures. As uncertainty assignment terms, the degree of belief, probability distribution, and possibility distribution are used for evidence theory, probability theory, and possibility theory, respectively. To quantify the assigned uncertainties, monotone measures are used such as dual monotone measures of plausibility and belief, dual monotone measures of possibility and certainty, and single monotone measure of probability for evidence theory, probability theory, and possibility theory, respectively [26–28]. Considering the relationship between uncertainty assignment terms and monotone measures used for each theory, it can be known as probability theory and possibility theory are special forms of evidence theory [6]. Consider a discrete universe *D* that consists of a set of damage levels,where *d*_{N}, *d*_{S}, *d*_{M}, *d*_{E}, and *d*_{C} represent no damage, slight damage, moderate damage, extensive damage, and collapse of a structure due to a seismic force level respectively.

In evidence theory, which is also known as Dempster–Shafer theory [11, 12], the degree of belief *m* based on evidence is assigned to all countable subsets A (e.g., *Ø*, {*d*_{N}}, {*d*_{N}, *d*_{S}}, ..., {*d*_{N}, ..., *d*_{C}}) with the constraint of

Dual monotone measures, belief bel (*A*) and plausibility pl (*A*), for a subset *A* are calculated as

While belief measure represents the degree of evidence for a subset *A*, plausibility measure is defined as “Complement of the belief of the complement of a subset *A*” as

As belief measure is based on the degree of belief with its evidence, belief measure of “Complement of a subset *A*” also needs its evidence. Therefore, if there is no evidence for “Complement of a subset *A*,” one cannot determine the belief of “Complement of a subset *A*” as 1-bel (*A*). The difference between these two measures can represent our ignorance (lack of knowledge) of a subset *A* (denoted ign) as

In probability theory, probability distribution, which is equivalent to the degree of belief *m* in evidence theory, is assigned to a single variable (e.g. *d*_{N}, …, *d*_{C}) on universe *D* such aswhere *d*_{i} denotes the damage state, *d*_{N}, *d*_{S}, *d*_{M}, *d*_{E}, and *d*_{C}. Only one monotone measure, probability prob(*A*), for a subset *A* is defined asand probability measure of “Complement of a subset *A*” is defined aswith the *excluded middle axioms* [4]. Unlike evidence theory, “Complement of a subset *A*” can be determined as 1 – prob (*A*). Therefore, our lack of knowledge measured by ign (*A*) in equation (6) cannot be measured in probability theory.

In possibility theory, possibility distribution is assigned to a single damage level in possibility theory such as

The relationship between the uncertainty assignment and the degree of belief *m* in equation (10) indicates that possibility theory is a special form of evidence theory when the collective body of evidence is consonant [26–28] (see Figure 1).