My title https://doi.org/10.14311/APP.2022.36.0161 Acta Polytechnica CTU Proceedings 36:161–166, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague COMPARISON OF THE COLLAPSE FREQUENCY AND FAILURE PROBABILITY OF BUILDINGS Dirk Proskea,∗, Michael Schmidb a Bern University of Applied Science, School of Architecture, Wood and Civil Engineering, Pestalozzistrasse 20, 3401 Burgdorf, Switzerland b Rothpletz, Lienhard + Cie AG Projektierende Bauingenieure SIA, Blumenberggasse 50, 3013 Bern, Switzerland ∗ corresponding author: dirk.proske@bfh.ch Abstract. In a previous study, the collapse frequencies and associated mortalities of buildings were compiled based on various publications. The investigation showed a considerable scattering of collapse frequencies depending on the countries. In this paper, the collapse frequencies determined for buildings are compared with the results of probabilistic calculations. Such comparisons have already been carried out for bridges, dams, tunnels and retaining walls including the consideration of central estimators and standard deviations. In order to limit the scatter, the probabilities of failure for buildings were subdivided into the different causes of failure, and collapse frequencies were subdivided into different countries and geographical regions. Overall, the comparison shows that the probabilities of failure are on overage larger than the observed collapse frequencies. Furthermore, the comparison shows are large span, which is unusual for other types of structures. Of concern are the high observed collapse frequencies in various developing countries. Human error, lack of training, etc. are often cited as the cause. Should this correspond to the facts, the basic safety concept of modern building standards, which generally excludes human error, would only be applicable in these regions to a limited extent. However, the investigation includes some limitations, such as different safety targets in different standards and different years of constructions, which are not considered. Further work is required. Keywords: Buildings, collapse, collapse frequency, probability of failure. 1. Introduction Modern safety concepts for structures are based on probabilistic models [1, 2]. In contrast to these calcu- lation models, in which either directly or indirectly nominal failure probabilities are determined, there are the observations of building collapses. Whether and how the calculated and observed values are compara- ble and related, is the subject of scientific discussion [3, 4]. Comparison is common in other disciplines [5, 6], but is often rejected in the construction indus- try [1, 2]. Nevertheless, the calculated failure probabilities and the observed collapse frequencies of bridges [7], dams [8], tunnels [9] and supporting structures [10] were compared within the framework of a series of articles. Furthermore, the comparison was also methodologi- cally extended from central estimators to statistical uncertainty parameters [11]. Also, the mortality due to structural collapses was estimated, see for example [9, 12]. This document presents the comparison of collapse frequencies and failure probabilities for buildings. In the following section the term building is defined. 2. Definition of Buildings Structures can basically be divided into two classes: • Buildings as residential, commercial, and public buildings as well as buildings for the sports and leisure sector. • Civil engineering structures as part of the infras- tructure such as bridges, dams, retaining walls and tunnels. This paper focuses on buildings. Three definitions of buildings are given. The German Federal Statistical Office [14] defines buildings as: "...structures that generally rise sub- stantially above the earth’s surface. For technical reasons, buildings also include independently usable underground structures that can be entered by people and are suitable or intended for the protection of peo- ple, animals or property (e.g., ...., underground shop- ping centres and production facilities, underground car parks)". The Swiss Confederation [15] defines: "Buildings [are] a permanent structure, provided with a roof, firmly attached to the ground, capable of accommo- dating persons and serving residential purposes or purposes of work, education, culture, sport or any other human activity; ..." The Free State of Saxony [16] in Germany defines: "Buildings are independently usable, covered structures that can be entered by people and are suitable or in- tended to serve the protection of people, animals or 161 https://doi.org/10.14311/APP.2022.36.0161 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Dirk Proske, Michael Schmid Acta Polytechnica CTU Proceedings Figure 1. Annual frequency of building collapses based on [13]. The individual references are given in [13]. property." 3. Definition of Collapse and Failure After the definition of the term building, the terms collapse and failure should be defined. Usually, a collapse is a sequence of failures of struc- tural elements leading to the destruction of the build- ing (non-repairable). The sequence is often described as chain reaction and progressive collapse. Collapse frequencies are computed based on observed and doc- umented collapses of buildings. Another definition considers the building collapse as the entire structure or parts come down. In contrast, failure is the computational exceedance of a single ultimate limit state equation, which may or may not yield to a non-repairable damage. This ex- ceedance of the ultimate load state equation is related to one internal force (moment, shear force, axial force, buckling etc.) and one location at a structural ele- ment. Probabilistic failure computations are related to numerical models incorporating random variables as well as individual limit state functions. The results of individual probabilistic computations for one struc- tural element and one limit state can be combined to consider the entire structures. 4. Collapse Frequency of Buildings The collapse frequency is computed as follows: F = nC nB (1) The stock of buildings nB , the number of collapses nC and the collapse frequencies of buildings F were investigated and visualized in [13] based on a large number of publications. This reference also includes the investigation of the mortality due to building col- lapses. However, the study does not include collapses from earthquakes and floods. Figure 1 gives an overview of references related to the collapse frequency of buildings and Figure 2 shows the mortalities based individual references. The individual references are given in [13]. 5. Probability of Failure of Buildings While only a relatively small number of publications can be used to determine collapse frequencies, there is an almost unmanageable number of publications with calculations of the probability of failure of build- ings. In the context of this study, 22 publications were selected, however finally only 21 are used. The publications refer to a wide range of actions, but they are not yet a representative selection of the building stock. The discussion of the assumptions and limita- tions of each of the used publications is beyond the scope of this article. Even further, the study should be extended using more references and distinctions of the building stock. Table 1 and Figure 3 visualises the probabilities of failure of the used references. Furthermore, Figure 3 includes the observed collapse frequencies from Fig- ure 1 in grey colour. Additionally, the references of the collapse frequencies have been excluded. 6. Discussion Based on Table 1 and Figure 3, it can be seen, that the probabilities of failure show an extraordinary scatter. The values range 10−19 to 10−1 per year. The scatter 162 vol. 36/2022 Collapse Frequency and Failure Probability of Buildings Figure 2. Mortality due to building collapses based on [13]. The individual references are given in [13]. Number Reference Action Range of Probability of Failure 1 [17] Structural Analysis 3.17 × 10−5 to 9.68 × 10−4, 6.21 × 10−3 to 2.12 × 10−1 2 [18] Snow 3.17 × 10−5 to 1.39 × 10−2 3 [19] Earthquake 1 × 10−5 to 1.05 × 10−2 4 [20] Earthquake 9.87 × 10−10 to 1.07 × 10−2 5 [21] Structural Analysis 3.47 × 10−3 to 6.87 × 10−4 6 [22] Wind 6 × 10−4 to 8 × 10−4 7 [23] Earthquake 0.310 × 10−2/0.363 × 10−2 8 [24] Human Failure 0.3 × 10−3 to 1.3 × 10−3 9 [25] Structural Analysis 8.275 × 10−6 to 1.285 × 10−5 10 [26] Human Failure 1.35 × 10−2 to 8.63 × 10−2 11 [27] Foundation 1 × 10−6 to 1.3 × 10−4 12 [28] Foundation 4.5 × 10−6/4.21 × 10−4/1.86 × 10−8/2.47 × 10−8/1.70 × 10−6 13 [29] Wind 1 × 10−5 14 [30] Wind, Typhoon 1.7 × 10−4/1.69 × 10−3 15 [31] Earthquake 3.4 × 10−4 and 2.8 × 10−3 16 [32] Wind 1.0 × 10−6 to 5.58 × 10−5 17 [33] Foundation 2.72 × 10−8/1.22 × 10−9/1.61 × 10−11/1.89 × 10−16 18 [34] Fire 3.8 × 10−7/7 × 10−10 19 [35] Human Failure 6.4 × 10−6 20 [36] Fire 3.3 × 10−9/3.3 × 10−8 21 [37] Structural Analysis 1 × 10−5 Table 1. References and related Probabilities of Failure. is significantly larger than the scatter of the collapse frequencies. Furthermore, the probabilities of failure are on aver- age clearly above the average of the collapse frequen- cies. Of course, the collapse frequencies in Figure 1 and 3 do not consider large-scale accidental actions such as earthquakes or floods, which are considered in the probabilities of failure. Whereas the median of the probabilities of failure without accidental actions is in the range of 1.06×10−5, the median of all results is in the range of 6 × 10−4 per year. The collapse frequency on a worldwide scale without earthquakes and floods was 3.3 × 10−6 per year [13]. Own estimations considering earthquakes and floods yields to a factor 3 to 20 for this collapse frequency, either based on Table 2 and 3 or on the average number of fatalities due to earthquake and the number of collapsed buildings [39, 40]. This would yield to a 163 Dirk Proske, Michael Schmid Acta Polytechnica CTU Proceedings Figure 3. Computed probabilities of failure for buildings based on various references given in Table 1 and related to actions. Year Country Action Number of buildings destroyed 1970 Bangladesh Flood 400,000 1995 Kobe, Japan Earthquake 46,000 - 100,000 2001 El Salvador Earthquake 200,000 2001 Peru Earthquake 20,000 2004 Southeast Asia Tsunami 300,000 2010 Haiti Earthquake 250,000 2011 Japan Earthquake and Tsunami 45,000 - 130,000 destroyed 190,000 - 240,000 damaged Table 2. Number of destroyed houses during large-scale accidental actions. Year Damaged houses Destroyed houses Collapse Frequency Ratio to 3.3 × 10−6 2015 660,000 85,000 6.5 × 10−5 19.81 2016 380,000 90,000 6.9 × 10−5 20.98 2017 560,000 115,000 8.8 × 10−5 26.81 2018 450,000 95,000 7.3 × 10−5 22.14 2019 340,000 90,000 6.9 × 10−5 20.98 2020 170,000 30,000 2.3 × 10−5 6.99 Table 3. Number of damaged and destroyed houses during earthquakes per year worldwide (Earthquake Impact Database). The total stock of buildings has been assumed with 1.3 billion, which is slightly below 1.5 used in [38]. 164 vol. 36/2022 Collapse Frequency and Failure Probability of Buildings collapse frequency in range from 1 × 10−5 to 7 × 10−4. Therefore, the ratio of median probabilities of fail- ure to median collapse frequencies is 3.2 ((1.06 × 10−5)/(3.3 × 10−6)) without considering earthquakes and floods. For the consideration of earthquakes and floods this ratio would be in the range just below 1 to well over 20 with an average value in the upper range ((6 × 10−4)/(1 × 10−5 to 7 × 10−4). The large span of the difference between the calcula- tions and the observations stays in contrast to all other types of structures, such as bridges, dams, retaining structures. For those the values are in the range of 2. On the other hand, buildings form the largest number among the structures. Therefore, subgroups have probably to be formed to increase the quality of this comparison, especially with focus on earthquakes. Even further, we have not considered the different target values in different standards, such as ASCE and Eurocode, the different consequence classes, and the different construction years of the buildings and the relevant standards. It is well known that newer standards yield to better structural behaviour under extreme loads such as earthquakes. These distinctions can be investigated in future works. On the other hand, this study shows that the com- puted probabilities of failure in average are conserva- tive. 7. Conclusion The sample size of the publications used for the inves- tigation of the failure probabilities and the collapse frequencies is roughly comparable. However, the dif- ferences between the average calculated failure prob- abilities and the observed collapse frequencies shows an large span, which is in contrast to other types of structures. 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