# Probability-Based City-Scale Risk Assessment of Passengers Trapped in Elevators under Earthquakes

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## Abstract

**:**

## 1. Introduction

## 2. Workflow

- Step 1: Probability-based city-scale THA

- Step 2: Probability-based city-scale elevator damage assessment

- Step 3: Probability-based city-scale elevator PTE assessment

## 3. Methodologies

#### 3.1. Probability-Based City-Scale THA

- (1)
- In each realization, the first two order periods of the MDOF flexural–shear model are calculated based on the building height using empirical formulas. Then, the bending-to-shear stiffness ratio can be obtained. Consequently, the structure’s elastic bending and shear stiffness values can be calculated, respectively.
- (2)
- The design capacity of the shear and bending springs on each story is obtained by the mode-superposition response spectrum method. Subsequently, the distribution features of the yield and peak overstrength parameters, namely Ω
_{y}and Ω_{p}, respectively, are determined, based on which the Ω_{y}and Ω_{p}values are sampled stochastically in each realization. Consequently, the yield and peak points of the force–displacement skeleton curves of the shear and bending springs can be identified. - (3)
- The hysteresis parameters of the structure are determined.
- (4)
- A nonlinear THA of the MDOF flexural-shear model is performed using the central difference method to obtain the seismic response of the structure.

_{y}and Ω

_{p}are two mechanical parameters with uncertainties. For a reinforced concrete high-rise building, both (Ω

_{y}− 1) and (Ω

_{p}− 1) follow a lognormal distribution, and their logarithmic mean values can be calculated using Equations (1) and (2), respectively [17]:

^{2}under the maximum considered earthquake, respectively. In each realization, the values of Ω

_{y}and Ω

_{p}sampled stochastically based on the above lognormal distribution can be used to establish the structural model and perform the THA to predict the story-level EDPs for a specific earthquake scenario.

#### 3.2. Probability-Based City-Scale Elevator Damage Assessment

#### 3.2.1. Elevator Inventory

#### 3.2.2. Elevator Position

_{i,j,k}that the j-th elevator of building i is at the k-th story when the earthquake occurs is expressed as follows:

_{i}is the number of stories in building i.

#### 3.2.3. Damage Probability and Damage State

_{i,j}) of D1014.011 follows a lognormal distribution with a median θ

_{i,j}of 3.9 m/s

^{2}and a logarithmic standard deviation β

_{i,j}of 0.45, as expressed in Equation (4) [21,22]:

_{i,j}is the peak acceleration of the j-th elevator of building i during an earthquake, and Φ(∙) is the cumulative distribution function (CDF) of standard normal distribution. Consequently, the damage state of the j-th elevator of building i, namely DS

_{i,j}, follows Bernoulli distribution, i.e., DS

_{i,j}~ B(1, F(EDP

_{i,j})), where DS

_{i,j}= 1 implies that the elevator is damaged while DS

_{i,j}= 0 implies that the elevator can still be used.

#### 3.3. Probability-Based City-Scale PTE Assessment

#### 3.3.1. Elevator Traffic Statistics

- Step 1: Typical building selection

- Step 2: Estimation of ground-level elevator traffic statistics

_{ground,typical}, is expressed as

_{in}and TP

_{out}represent the total number of passengers entering and exiting the elevator during a certain observation period, respectively; and RT is the total number of times to record TP

_{in}and TP

_{out}values, correspondingly.

_{ground,common}, can be calculated as

_{common}and BA

_{typical}are the floorages of the target and typical buildings, respectively; and NoE

_{common}and NoE

_{typical}are the numbers of elevators in the target and typical buildings, respectively.

- Step 3: Estimation of elevator traffic statistics on all stories

_{i,j,k}and σ

_{i,j,k}are the mean and standard deviation of passengers of the j-th elevator at the k-th story of building i, respectively; and ET

_{i,j,}

_{1}and σ

_{i,j,}

_{1}are the mean and standard deviation on the ground story, respectively.

#### 3.3.2. Number of Elevator Passengers

_{i,j,k}and σ

_{i,j,k}values. Finally, the NEP can be sampled stochastically based on Equation (9).

#### 3.3.3. Number of PTEs

_{s,i}, is expressed as

_{s,i,j}is the PTE number of the j-th elevator in building i in the s-th realization; and J

_{i}is the number of elevators in building i. Furthermore, the total number of PTEs in the target area in the s-th realization, PTE

_{s}, can be calculated as

_{s,i}and PTE

_{s}values of all realizations.

#### 3.4. Number of Realizations

## 4. Case Study

#### 4.1. Study Area

^{2}. The seismic fortification intensity at this site is eight [20]. A field investigation was conducted to obtain the building inventory data of 619 buildings. The occupancy type and number of stories of the buildings are shown in Figure 6. The campus is dominated by residential and office buildings, with high-rise buildings mainly located in the northeast and southeast regions.

#### 4.2. Elevator Inventory Data

#### 4.3. Elevator Traffic Data

_{in}, and TP

_{out}of the typical buildings were monitored every 10 min, and a piecewise linear function was applied to fit the statistical data. After repeated observations for 7 days, the time-varying mean and standard deviation of the ground-level NEP of each elevator for the typical buildings were calculated (Figure 8). It is noteworthy that data in Figure 8 were obtained during workdays (i.e., from Monday to Friday, excluding public holidays). Nevertheless, the same method can be used for weekends and holidays.

#### 4.4. Scenario-Based Assessment

^{2}obtained from the Institute of Geophysics of China Earthquake Administration was used as input (Figure 9). The same PGA was used for all buildings for simplicity as the campus area was not large.

#### 4.5. Intensity-Based Assessment

## 5. Discussions

- (a)
- For a specific urban area, a certain functional connection typically exists between different buildings, which implies that a correlation exists among the probability distributions of NEP in different buildings. It is currently challenging to identify this type of correlation at a city scale. Therefore, this study does not consider the impact of such correlation problems but assumes that the random variable of NEP is independent of each other for different buildings.
- (b)
- In addition, a linear function was used to describe the distribution pattern of elevator passengers at different stories of the same building, which is an idealized assumption. Different stories of a building may exhibit different occupancies. For example, the bottom floors of a high-rise building may be used for commercial activities, whereas the remaining floors are for residential purposes. The spatial variations in building occupancy will significantly affect the distribution pattern of elevator passengers on different stories [26].
- (c)
- Furthermore, the elevator traffic data of a specific building may be affected by the weather and season. Long-term observation of elevator traffic is crucial to the accurate prediction of city-scale PTE risks.

## 6. Conclusions

- (1)
- In the proposed method, city-scale THA is performed to simulate the seismic response of a building complex, and the Monte Carlo method is applied to generate probability-based results by considering the uncertainty of multiple factors (i.e., the mechanical properties of buildings and elevators, the elevator story position, and the spatiotemporal characteristics of elevator traffic) affecting the PTE risk level.
- (2)
- The proposed method can be used to perform both scenario- and intensity-based assessments of earthquake-induced PTE risks, thereby having the potential to provide support for applications such as virtual rescue drills and earthquake emergency plans.
- (3)
- The spatiotemporal nature of elevator traffic significantly affects the PTE risk in a building complex. In the case study of the Tsinghua University campus, the number of PTEs when an earthquake occurs during the off-peak hours of elevator traffic is approximately a quarter of that when the earthquake occurs at the morning peak time; the office building exhibits the highest PTE risk among the campus buildings; and the high-risk buildings are mainly located in the east and southeast regions of the campus under the MCE.
- (4)
- The maximum number of PTEs on the Tsinghua University campus under the MCE reaches 195, approximately five times that under the DBE. Gate 1 is identified as the optimal entrance for post-earthquake rescues under the DBE, while Gate 5 is the best choice under the MCE.
- (5)
- The fragility curves depicting the quantitative relationship between the number of PTEs in an urban area and the earthquake intensity enable a comprehensive understanding of the PTE risk in the target area under different earthquake intensities, thus promoting targeted earthquake emergency preparations.
- (6)
- The proposed method fills a gap in the research on earthquake-induced PTE risk at a city scale and provides a practical assessment workflow with acceptable labor and time costs. Additionally, compared to the data-driven solution, the workflow is a physics-driven method that has no geographical limitations.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

ID | Name | ID | Name |
---|---|---|---|

1 | RSN9_BORREGO_B-ELC000.AT2 | 7 | RSN67_SFERN_ISD014.AT2 |

2 | RSN28_PARKF_C12050.AT2 | 8 | RSN76_SFERN_MA3130.AT2 |

3 | RSN40_BORREGO_A-SON033.AT2 | 9 | RSN84_SFERN_SDC000.AT2 |

4 | RSN51_SFERN_PVE065.AT2 | 10 | RSN86_SFERN_SON033.AT2 |

5 | RSN54_SFERN_BSF135.AT2 | 11 | RSN93_SFERN_WND143.AT2 |

6 | RSN55_SFERN_BVP090.AT2 |

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**Figure 8.**Time-varying traffic statistics for each elevator per 1000 m

^{2}floorage of typical buildings.

**Figure 10.**Number of PTEs on the campus when the 1679 Sanhe–Pinggu earthquake occurs at different times on a weekday.

**Figure 11.**Number of PTEs in buildings when the 1679 Sanhe–Pinggu earthquake occurs at (

**a**) 8:00, (

**b**) 12:30, (

**c**) 18:30, and (

**d**) 22:00 on a weekday.

**Figure 13.**Number of PTEs on the Tsinghua University campus under earthquakes with different intensities (earthquakes occur on weekdays): (

**a**) PGA = 0.1 g, (

**b**) PGA = 0.2 g, (

**c**) PGA = 0.3 g, and (

**d**) PGA = 0.4 g.

**Figure 14.**Number of PTEs in buildings when (

**a**) DBE- and (

**b**) MCE-level earthquakes occur at 8:00 on a weekday.

**Figure 15.**Fitted fragility curves when an earthquake occurs at 8:00, 12:30, 18:30, and 22:00 on weekdays.

Buildings Lower than 10 Stories | Buildings with 10 Stories or More | ||||
---|---|---|---|---|---|

Name | Occupancy | Stories | Name | Occupancy | Stories |

Mengminwei S&T Bldg. | Office building | 8 | Innovation Mansion | Office building | 13 |

Medical Science Bldg. | Office building | 3 | SP Bldg. B | Office building | 26 |

Bldg. 29 | Student dormitory | 6 | Zijing Student Apt. No.14 | Student dormitory | 15 |

Bldg. 21 | Student dormitory | 6 | Zijing Student Apt. No.23 | Student dormitory | 12 |

Heqingyuan No.12 | Ordinary residence | 9 | Tall Bldg. No.1 | Ordinary residence | 17 |

Heqingyuan No.1 | Ordinary residence | 7 | Lanqiying No.8 | Ordinary residence | 20 |

6th Teaching Bldg. Zone A | Teaching building | 6 | 6th Teaching Bldg. Zone B | Teaching building | 10 |

Precision Instruments Dept. | Teaching building | 4 | Liuqing Bldg. | Teaching building | 11 |

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## Share and Cite

**MDPI and ACS Style**

Gu, D.; Wang, Y.; Lu, X.; Xu, Z.
Probability-Based City-Scale Risk Assessment of Passengers Trapped in Elevators under Earthquakes. *Sustainability* **2023**, *15*, 4829.
https://doi.org/10.3390/su15064829

**AMA Style**

Gu D, Wang Y, Lu X, Xu Z.
Probability-Based City-Scale Risk Assessment of Passengers Trapped in Elevators under Earthquakes. *Sustainability*. 2023; 15(6):4829.
https://doi.org/10.3390/su15064829

**Chicago/Turabian Style**

Gu, Donglian, Yixing Wang, Xinzheng Lu, and Zhen Xu.
2023. "Probability-Based City-Scale Risk Assessment of Passengers Trapped in Elevators under Earthquakes" *Sustainability* 15, no. 6: 4829.
https://doi.org/10.3390/su15064829