# Earthquake Physical Risk/Loss Assessment Models and Applications: A Case Study on Content Loss Modeling Conditioned on Building Damage

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## Abstract

This paper presents a novel approach to develop content fragility conditioned on building damage for contents used in residential buildings in Turkey. The approach combines the building damage state probabilities with the content damage probabilities conditioned on building damage states to develop the content fragilities. The paper first presents the procedure and then addresses the epistemic uncertainty in building and content fragilities to show their effects on the content vulnerability. The approach also accounts for the expert opinion differences in the content replacement cost ratios (consequence functions) as part of the epistemic uncertainty. Monte Carlo sampling is used to consider the epistemic uncertainty in each model component contributing to the content vulnerability. A sample case study is presented at the end of the paper to show the implementation of the developed content fragilities by calculating the average annual loss ratio (AALR) distribution of residential content loss over the mainland Turkey.

## 10.1 Introduction

- (a)
the target exposure: a single asset or an inventory located at a specific site or a region,

- (b)
the seismic hazard: explaining the exceedance frequency of the ground-motion intensity measure (GMIM) used in defining the conditional probability of the undesirable outcome and,

- (c)
the fragility function: describing the occurrence probability of the undesirable outcome conditioned on the GMIM utilized in quantifying the seismic hazard.

When fragility functions conditioned on the ground-motion metric are combined with the consequence models, we quantify the loss (repair costs, loss of functionality). The functions measuring the loss in terms of GMIM is referred to as vulnerability functions or vulnerability models. The above terminology can be found in most of the modern seismic risk assessment text books (e.g., McGuire 2004).

*y*(

*s*) in Eq. (10.1) is the vulnerability function in terms of GMIM,

*s*, and the derivative of the seismic hazard curve

*G*(

*s*) represents the annual probability producing exactly

*s*. The negative sign accounts for the negative slope of

*G*(

*s*) at

*s*since the hazard curve slopes down to the right at all values of

*s*indicating lower exceedance frequency of higher shaking.

*DS*=

*ds*

_{i}),

*i*= 1,…,

*n; n*is the total number of damage states). The variable \({RC}^{{ds}_{i}}\) is the replacement cost corresponding to

*DS*=

*ds*

^{i}. The replacement costs are the monetary losses, representatives of different damage levels and are, therefore, called as consequence functions (or models). The vulnerability function developed by Eq. (10.2) is called as compound loss function in the HAZUS report since it accounts for all possible damage states, proportional to their occurrence probabilities, that the asset can experience during an earthquake.

In essence, the integral expression in Eq. (10.1) computes the expected annual loss of an asset by considering a range of GMIM, *s*, that are likely to occur at the site with different annual probabilities. If the consequence model used in Eq. (10.2) is dimensionless (in terms of replacement cost ratio), the resulting loss by Eq. (10.1) is called as average annual loss ratio (AALR); favored more by the insurance industry.

The accuracy of the predicted loss (in this case AAL or AALR) is confined to the reliable seismic hazard and vulnerability models hence the consistent fragility and consequence functions. This fact brings forward the modeling uncertainty (epistemic uncertainty) in these components that is addressed in a variety of scientific publications. A fairly ample review, in this respect, can be found in FEMA P-58 (ATC 2018). The lack of knowledge, insufficient data and subsequent assumptions as well as interpretations about the model behavior are the main sources of epistemic uncertainty.

This article presents a case study on modeling the uncertainty in content vulnerability functions for residential buildings in Turkey and its progressive influence on the loss computations. Since vulnerability functions are composed of fragility and consequence models (Eq. (10.2)), the progressive influence of the epistemic uncertainty is discussed by considering the interaction between these two modeling elements. The contents considered here are poorly anchored or unanchored house utensils as well as furniture and electronic equipment frequently used in the residential dwellings. Their fragility modeling presented here is conditioned on the different levels of building damage that would lead to more accountable loss predictions.

The paper starts by describing the development of content fragilities conditioned on the building damage that is followed by the development of consequence functions for different modes of content damage. The associated epistemic uncertainty in the conditional content fragilities and consequence functions are progressed to observe their influence on the content vulnerability model. To illustrate the implementation of the discussions, the last part in the paper integrates the conditional vulnerability model together with the most recent national seismic hazard maps to compute the distribution of residential building content AALR for entire Turkey.

## 10.2 Development of Content Fragilities Conditioned on Building Damage

### 10.2.1 Review of Some Benchmark Documents

- (a)
regular building construction quality,

- (b)
structures on firm soil,

- (c)
merely ground shaking without damage aggravation due to collateral hazard (e.g., fire, fault rupture and inundation) and,

- (d)
the content is at the ground level and unanchored.

- (a)
structural damage states,

- (b)
building-type sensitive content replacement value and,

- (c)
the probability of building being in non-structural acceleration sensitive damage state.

FEMA P-58 (ATC 2018) apriori assumes that the contents are sensitive to peak floor acceleration and velocity, and provides building-type dependent content fragilities as well as normative content quantitates to predict the content loss under ground shaking. The content loss assessment tools provided in the FEMA P-58 and HAZUS documents are more comprehensive than the one in ATC-13 but it seems that they are more suitable to assess a specific single asset as long as the content loss is of concern. Naturally, all three documents establish their methodologies considering the structural typologies and construction quality in the United States. In fact, ATC-13 (1985) was prepared for earthquake loss in California.

### 10.2.2 Theoretical Background

*j*= 1,…,

*n*

_{str}, as well as the likelihood of building being not damaged, \({DS}^{Str}={ds}_{nd}^{str}\).

*i*th state content damage probability conditioned on the

*j*th state building damage and \(Pr\left({DS}^{Cnt}={ds}_{i}^{Cnt}|{DS}^{Str}={ds}_{nd}^{Str}\right)\) is the

*i*th state content damage probability conditioned on undamaged building state. The terms \(Pr\left({DS}^{Str}={ds}_{j}^{Str}\right)\) and \(Pr\left({DS}^{Str}={ds}_{nd}^{Str}\right)\) refer to the

*j*th damage state and no damage state probabilities of the building, respectively.

Figure 10.4 indicates that the content damage probabilities are assumed to be represented by three different damage states (slight, light and moderate) when the building sheltering the content does not suffer any structural damage (i.e., when only a limited nonstructural damage is observed in the architectural and mechanical/electrical building components). Note that the approach primarily rates the slight content damage, which is followed by the light and moderate damages associated with very small probabilities. When the building suffers from moderate structural damage (Fig. 10.5), the content damage probabilities are represented by all five states and the proposed approach apriori prefers moderate content damage and then rates the occurrence probability of heavy content damage more than the other three damage states (slight, light and very heavy). Note that the likelihood of very heavy content damage is more than the slight content damage prorating the existence of fragile content in the residential buildings. Upon very severe structural damage (Fig. 10.6), the approach almost exclusively favors very have content damage, practically advocating its full replacement. As a final remark, the sampling should be tailored such that given building damage state, the assigned content damage probabilities should sum up to unity at every sampling.

### 10.2.3 Case Studies on Developed Content Fragilities

- a.
uncertainty in the damage state threshold by different studies,

- b.
variability in the modeling aspects of the buildings and,

- c.
variability in building response due to intricate nature of earthquake ground-motion records.

For this reason, the Monte Carlo sampling results in bands of probabilities given a specific MMI value. The upper and lower end of the bands, hence the damage probabilities, overlap each other due to excessive variability in the fragilities at each damage state.

Figure 10.8 shows the content fragilities for the same residential building class after implementing Eq. (10.3). The conditional content probabilities conditioned on different building damage states and the building damage probabilities are populated as described in the above paragraphs. They are presented in Figs. 10.4, 10.5, 10.6 and 10.7. To this end, the resulting content fragilities account for the model uncertainty due to building response and content damageability associated with differences in the residential equipment, their locations, placements and etc. That’s why the overlapping of damage probabilities at different content damage states are increased with respect to those presented for the buildings (Fig. 10.7).

The comparisons of content fragilities provided in this study and those given in the ATC-13 document are in agreement to a limited extent. There are major differences in terms of the uncertainty in content damageability predictions by each approach. ATC-13 defines a unique content damage probability at each damage state given a specific MMI value. The content fragilities developed in this study yield a range of damage probability at each content damage state by considering the progression of the epistemic uncertainty in building response as well as content damageability. As depicted by Figs. 10.9 and 10.10, the slight content damage is similarly predicted by the two studies but the ATC-13 probabilities for the rest of the damage states seem to be closer to the lower bound fragility predictions of this study. Hence, upon the use of these two different fragility sets in a probabilistic risk assessment study, one may obtain completely different loss pictures. This fact brings forward the importance of how epistemic uncertainties are handled in a given methodology as well as the country-based differences in loss assessment.

## 10.3 Content Consequence Model

## 10.4 Vulnerability Model and Country-Wide Content AALR

The maps suggest an AALR interval ranging between 4 × 10^{–3} and 6 × 10^{–3} for the most seismic prone settlements in Turkey (e.g., Istanbul, Izmir, Canakkale, Erzincan, Aydin, Denizli, etc.). This value suggests a yearly basis pure premium of €40 to €60 Euros for residential equipment of €10,000 worth in such cities. The AALR values go down to as much as 1 × 10^{–3} (i.e., a yearly basis pure premium of €10 for residential equipment of €10,000 worth) in the least seismic regions in the country such as the large portion of the south Eastern Turkey and the central Anatolia. Needless to say the presented numbers are valid for the residential content in the mid-rise, high-code, MRF buildings. They would be scaled up and down depending on the building type, construction period and height as partially discussed in Fig. 10.9.

## 10.5 Summary and Conclusions

This study proposes a procedure to develop content fragilities conditioned on building damage for loss and risk modeling that can be used in computing metrics relevant to insurance and reinsurance. The conditional content fragilities can account for the epistemic uncertainty in assessing the earthquake induced building damage states as well as the different likelihoods of content damage under different modalities of building damage. These uncertainties are handled via Monte Carlo sampling that enables the risk expert to trace forward or backward the progression of model uncertainty and its effects on the computed loss and risk. The proposed procedure is analytical, and its systematic utilization can result in country-specific vulnerability and risk models. This feature makes the procedure appealing because the current well-organized and state-of-the-art tools in this field seem to be tailored for the construction quality and building classification in the US practice (e.g., ATC-13 1985; FEMA 2003). The systematic efforts for improving this procedure should involve calibrations through comparisons with other approaches as well as sensitivity analyses to understand the behavior of critical components contributing the most to loss and risk assessment results.

## Notes

### Acknowledgements

This study is done under the financial supports provided by Turkish Catastrophe Insurance Pool and Turkish Insurance Association via Turkish Earthquake Foundation. The author benefitted significantly from the fruitful discussions with Prof. Mustafa Erdik and Prof. Ufuk Yazgan while establishing the presented content fragility model.

## References

- Akkar S, Eroğlu Azak T, Çan T, Çeken U, Demircioğlu Tümsa MB, Duman TY, Erdik ÖM, Ergintav S, Kadirioğlu FT, Kalafat D, Kale Ö, Kartal RF, Kekovalı K, Kılıç T, Özalp S, Altuncu Poyraz S, Şeşetyan K, Tekin S, Yakut A, Yılmaz MT, Yücemen MS, Zülfikar Ö (2018) Evolution of seismic hazard maps in Turkey. Bull Earthq Eng 16:3197–3228CrossRefGoogle Scholar
- Applied Technology Council, ATC (1985) ATC-13 Earthquake damage evaluation data for California. Prepared by C. Rojahn and RL Sharpe, funded by Federal Emergency Management AgencyGoogle Scholar
- Applied Technology Council, ATC (2018) FEMA P58 Seismic performance assessment of buildings. Funded by Federal Emergency Management AgencyGoogle Scholar
- Federal Emergency Management Agency, FEMA (2003) Multi-hazard loss estimation methodology HAZUS
^{®MH}MR4 Technical ManualGoogle Scholar - McGuire R (2004) Seismic hazard and risk analysis. Earthquake Engineering Research Institute, Oakland, CAGoogle Scholar
- Porter K (2019) A beginner’s guide to fragility, vulnerability, and risk. University of Colorado Boulder, p 119. https://spot.colorado.edu/~porterka/Porter-beginners-guide.pdf
- Turkiye Deprem Vakfi, TDV (2018) Revision of earthquake premiums in accordance with the revised national seismic hazard maps of Turkey (in Turkish)Google Scholar
- Wald JW, Quitariano V, Heaton TH, Kanamori H (1999) Relationships between peak ground acceleration, peak ground velocity, and modified Mercalli intensity in California. Earthq Spect 15:557–564CrossRefGoogle Scholar

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