2 Proposed simulation framework

The framework of the proposed simulation on the hazards of falling objects and site selection for emergency shelters is presented in Figure 1. The framework consists of three major components: (C1) calculation of the distribution of falling objects, (C2) uncertainty analysis of the ground motion, and (C3) site selection for emergency shelters.

(C1) Calculation of the distribution of falling objects

The MCS model (Lu et al. 2014) is adopted in this study to analyze the nonlinear structural response of a building group under a given ground motion. Specifically, the time-history story drift ratios and velocities are obtained for each story. The resulting story drift ratios and the falling criteria specified in the design standards (ASCE 2010) are subsequently employed to determine the falling statuses of exterior non-structural components. If a component separates from the main structure and falls, it will undergo horizontal projectile motion from its corresponding story with an initial velocity obtained from the nonlinear THA of the building. Subsequently, the projected distances and hazard range of the falling objects from the building under a given ground motion can be determined.

(C2) Uncertainty analysis of the ground motion

The hazard ranges of falling objects subjected to different ground motions and intensities are calculated using the IDAs, from which the fragility curves of the distribution probabilities are obtained. By combining the fragility curves with the seismic hazard of the site, the distribution probabilities of the falling objects over a certain period (e.g., the design life) can be calculated.

(C3) Site selection for emergency shelters

The distribution probabilities of the falling objects of a building group are mapped to the geographic information system (GIS) platform, in which the overlapping regions of the falling objects from different buildings are considered. According to an acceptable probability level, the hazard regions of the falling objects can be determined and a rational site for an emergency shelter can subsequently be selected.

By and large, no similar framework is currently available in the literature for the distribution probabilities of falling objects. Note that the proposed framework constitutes three key problems, i.e., the uncertainty analysis, the falling criteria and the structural response analysis of a building group. The first two problems can be performed using the IDA method and the ASCE’s standard, respectively. For the third problem, traditionally the Hazus method (FEMA 2012) has been commonly used. However it is not applicable for the analysis of falling objects because only one damage state can be calculated for the whole building. On the other hand, the proposed MCS model, demonstrating several advantages over the Hazus method (detailed in Table 1), is able to determine the damage state for each story.

Table 1 Comparison between the proposed MCS model and the Hazus method

3 Simulation method

A detailed analysis method is presented in this section to elaborate upon the proposed simulation framework.

3.1 Calculation of the distribution of falling objects

(C1-1) Structural response analysis

The structural response analysis of a building group is performed using the previously developed MCS model (Lu et al. 2014). In this model, the mass of a building is assumed to be concentrated at each story, as illustrated in Figure 2. The model is capable of efficiently predicting the time-history data of each story including the story drift, the displacements and the velocities. A previous study (Xu et al. 2014) indicated that the predictions of the MCS model are as accurate as the predictions of a high-fidelity finite element (FE) model when a realistic inter-story hysteretic relationship is adopted. The MCS model was developed particularly for common buildings with regular planar and vertical layouts, as well as given building heights. The Chinese code for seismic design of buildings (GB50011, 2010) specifies the maximum heights for regular buildings. The code specified maximum heights are defined as the height limits in the proposed method. For buildings beyond the maximum heights, special designs and calculations are required. In order to determine the factors of the MCS models, Lu et al. (2014) proposed a method that requires only five macro-scale variables (i.e., structural type, story height, number of stories, construction period and story area) to obtain the inter-story hysteretic behavior based on the performance database of Hazus software. Generally, these macro-scale variables can be directly obtained from the GIS database of a city.

Fig. 2 The MCS model for a building

After determining the inter-story hysteretic parameters of each building, the nonlinear THA of a building group can be performed to obtain the structural seismic responses.

(C1-2) Determination of falling criteria

According to ASCE design standards (ASCE 2010), the falling of glazed curtain walls and exterior non-structural walls are both controlled by the relative seismic displacement D_p. Note that their respective allowable displacements differ.

The glass in glazed curtain walls, glazed storefronts and glazed partitions should satisfy the fallout relative displacement requirement of Eq. (1) or 13 mm, whichever is greater.

where Δ_fallout denotes the relative seismic displacement at which glass fallout from a curtain wall, storefront, or partition occurs; I_e represents the importance factor determined in accordance with ASCE 7–10 (ASCE 2010); and D_p is the relative seismic displacement that the component must be designed to accommodate.

Similarly, according to ASCE 7–10 (ASCE 2010), the threshold displacement of exterior non-structural walls should not be less than the relative seismic displacement (D_p) or 13 mm, whichever is greater.

As a critical factor of the falling criteria, D_p must not be greater than Eq. (2) (ASCE 2010).

where h_x denotes the height to which the upper connection point of the component is attached; h_y denotes the height to which the lower connection point of the component is attached; h_sx represents the story height that is used in the definition of the allowable drift; and Δ_a denotes the allowable story drift.

For glazed curtain walls and exterior non-structural walls, h_x – h_y = h_sx; thus, D_p ≤ Δ_a. The allowable story drifts of all types of structures under different risk categories are specified in ASCE 7–10 (ASCE 2010), as summarized in Table 2.

Table 2 Allowable story drift  (ASCE 2010)

Assuming that D is the story drift on a specific story of a building, based on the worst scenario, an exterior non-structural component begins to fall when D reaches the maximum allowable threshold. Thus, the falling criteria for glazed curtain walls and exterior non-structural walls can be specified by Eq. (3) and Eq. (4), respectively.

Based on Eq. (3) and Eq. (4), the falling statuses of exterior non-structural components can be determined according to the story drifts provided by the nonlinear THA. Generally, the non-structural components are attached and supported by the main structure on each story, instead of the non-structural components of the adjacent stories. As such, the non-structural components of each story are independent from each other. In this study, the effect from the adjacent stories is thus not considered in the falling criteria of the non-structural components.

(C1-3) Calculation of the projected distances

When a non-structural component falls at the i-th time step from the j-th story, where the height and velocity of the story at the moment of falling are h_j and v_i,j, respectively, the falling objects will undergo a horizontal projectile motion with an initial velocity v_i,j. Therefore, the horizontal distance from the building at which the falling objects hit the ground can be calculated using Eq. (5).

where the velocity v_i,j can be obtained from the nonlinear THA using the MCS model.

Eq. (5) indicates that the objects that fall from a higher story will have a higher speed and undergo larger displacement away from the building when they hit the ground. As a result, these falling objects are more hazardous. The hazard range of the falling objects should be determined by the maximum projected distance. If the highest story is m and the total number of time steps is n, the hazard range of the falling objects can be expressed by Eq. (6):

Due to the uncertainty of the input direction of an earthquake, the hazard range for falling objects in any direction is set to d_max. Using this calculation method, the hazard range for falling objects of each building in a building group can be calculated for any given ground motion.

3.2 Uncertainty analysis of the ground motion

Different earthquakes generate ground motions with different frequency components, durations, and amplitudes. The uncertainty of the frequency components and the durations of the ground motions can be simulated using a sufficient number of representative ground motions (Lu et al. 2013b, 2013c). Extensive work on ground motion selections has been reported by FEMA P695 (FEMA 2009), in which a typical set of ground motions including far-field and near-field records is recommended. These ground motions were selected based on the following eight principles:

(1) The earthquake magnitude is no less than 6.5.

(2) The hypocenter is located on the strike-slip fault or the thrust fault.

(3) The records are obtained from soft rock or stiff soil sites, and the shear wave velocity of the site is no less than 180 m/s.

(4) The far-field records at sites are at least 10 km from the fault rupture, whereas the near-field records at sites are within 10 km of the fault rupture.

(5) No more than two of the strongest records are taken from each earthquake.

(6) The strong motion record satisfies peak ground acceleration (PGA) exceeds 0.2 g and peak ground velocity (PGV) exceeds 15 cm/s.

(7) The ground motion measuring instruments should be located in a free-field location or on the ground floor of a small building.

(8) The frequency range of the strong motion instrument is at least 4 s.

Thus, by combining the 49 ground motions (22 far-field and 27 near-field records) proposed by FEMA P695 (FEMA 2009) with the widely used El Centro 1940 EW ground motion record, a set of 50 ground motions is considered in this study to evaluate the uncertainty in the frequency components and the durations of the ground motions.

The uncertainty of the amplitude of a ground motion is simulated using the IDAs. The previously mentioned 50 ground motions are inputted in sequence into the building group by incrementing the intensity measures (IMs).

The probability of the maximum projected distance of the falling objects exceeding d₀ is defined as P(d_max≥d₀), where d₀ is a given distance in the horizontal plane from the perimeter of the building. By conducting the IDAs and a seismic hazard analysis, the total probabilities P(d_max≥d₀) over the design life of a building group can be obtained. The detailed analysis procedure is elaborated upon in the followings:

(C2-1) IDA of buildings

The number of ground motions employed in the IDA is defined as N_total. In this study, N_total = 50. These ground motion records are individually assigned to the buildings to perform the corresponding nonlinear THAs. To remain consistent with current seismic design codes, the highly accepted PGA is employed as the IM of the ground motions. If the hazard range d_max of a building, which is derived from the nonlinear THA of a ground motion at a given IM level, exceeds d₀, this ground motion is recorded as a “dangerous ground motion” at the corresponding IM level. The total number of “dangerous ground motions” among the selected N_total ground motions is defined as . Thus, P(d_max≥d₀) at this IM level can be calculated as follows:

By incrementing IM from 0 to , the fragility curve of P(d_max≥d₀) is obtained. To explain the variation in the probabilities of falling objects under different IMs, a lognormal distribution function is employed in this study to fit the fragility curve of P(d_max≥d₀). A numerical example, which is presented in Section 4, reveals that the fitted curve passes the Kolmogorov–Smirnov test at a 5% significance level. This result confirms that P(d_max≥d₀) for different IMs follows a lognormal distribution.

(C2-2) Seismic hazard analysis

The seismic hazard analysis provides the probability of a specific building site experiencing an earthquake of a given IM during a certain period of time (Y years), which is denoted as P(IM). Generally, the exceedance probability over a certain design period can be obtained by function fitting method according to the design code and earthquake data for a given site.

(C2-3) Calculation of total probability

Based on the above  and P(IM), the total probability of P(d_max≥d₀) over a period of Y years can be calculated using Eq. (8).

where P(d_max≥d₀ in Y years) denotes the total probability of the maximum projected distance of the falling objects exceeding d₀ in Y years; P(d_max≥d₀ |IM) is the conditional probability of P(d_max≥d₀) for a given IM, which is obtained from the fragility curve of P(d_max≥d₀); and P(IM) represents the probability density that a structure site is hit by an earthquake of a given IM in Y years, which is obtained from a seismic hazard analysis.

The distribution probabilities of falling objects from a building can be obtained by gradually increasing the value of d₀, which can be used to demonstrate the risk from the hazards of falling objects. In doing so, the risk of these hazards for an entire building group can be obtained, and a suitable and safe site for emergency shelters can then be selected.

3.3 Site selection for emergency shelters

A rational prediction of the hazard region of falling objects for a building group is critical for the site selection of emergency shelters. To determine the hazard region, the overlapping of objects that fall from different buildings should be considered. On the GIS platform, the target area in which a building group is located is divided into a number of grids. For each grid, the falling objects from different buildings are assumed to be independent. Therefore, the total probability of the falling objects in a grid is equivalent to the sum of the probabilities of individual buildings.

Further, the acceptable probability level of falling hazards should also be determined. An emergency shelter should be located a considerable distance from buildings with a high collapse risk. In ASCE 7–10 (ASCE 2010), the acceptable earthquake-induced collapse risk of a building is 1% in 50 years. Accordingly, the exceedance probability of 1% in 50 years is also adopted in this study as the acceptable probability level of falling objects. The risk of falling objects should be less than or equal to the collapse risk of the buildings. Given that the falling objects are important factors to be considered in the site selection of emergency shelters, all grids in the GIS platform in which the probability of falling objects exceeds 1% over 50 years are considered to form the hazard regions, which are unacceptable for emergency shelter sites.

4 Case study

The case study is related to a residential community area. This area encompasses a total of 19 reinforced-concrete buildings with an average of 20 stories and heights of approximately 60 m. According to the Chinese code for seismic design of buildings (GB50011, 2010), the maximum height considered for these residential buildings is 100 m, therefore the buildings having 60.5m in height can be simulated by the proposed method. Light hollow brick-infill walls are the primary form of exterior non-structural components of these buildings. Due to limited stiffness of the light walls, their falling does not significantly change the dynamic characteristics of the building. Therefore the impact of non-structural walls to the building's dynamic characteristics is not considered in this study. According to ASCE 7–10 (ASCE 2010), the risk category for these buildings is Ⅱ and the allowable story drift ratio of the masonry infill walls is 0.020.

Take a typical 20-story building with a height of 60.5 m as an example. The MCS model of this building is created using five variables (i.e., structural type, story height, number of stories, construction period and story area) (Lu et al 2014), and the geometric nonlinearity has been considered in this MCS model using Shi et al.’s (2014) method. The seismic responses of this building are simulated using the MCS model, and the fragility curves of P(d_max≥d₀) for different values of PGAs are calculated. Figure 3 illustrates the fragility curve of the probabilities P(d_max≥d₀) against different PGAs when d₀ is equal to 10 m (i.e., P(d_max≥10 m)), which is fitted by a lognormal distribution (Zareian and Krawinkler 2007). P(d_max≥10 m) is nearly zero when the PGA is less than 0.3g, which implies that the non-structural components will neither fall nor obtain a sufficient velocity to be displaced a distance of 10 m when PGA is less than 0.3g. Note that when a PGA of 1.0 g is attained, P(d_max≥10 m) is 100%.

According to the seismic hazard map of the site, the design PGAs with exceedance probabilities of 63%, 10% and 2% over 50 years are 0.07 g, 0.20 g and 0.41 g, respectively. Adopting the above exceedance probabilities and considering the upper and lower limits of P(PGA) (i.e., P(PGA ≥ 0) = 100% and P(PGA ≥ + ∞)= 0%), Ma and Xie (2002) proposed Eq. (9) to produce a seismic hazard curve, which is shown in Figure 4.

By integrating the fragility curve (Figure 3) and the seismic hazard curve (Figure 4), the total probability of P(d_max≥10 m) over 50 years can be calculated to be 6.8%. By changing d₀, the distribution of the total probabilities of falling objects over 50 years, P(d₀), can be obtained, as illustrated in Figure 5. P(d₀) decreases as the projected distance d₀ increases, which agrees well with real-world observations during earthquakes. In particular, the probability P(d₀) approaches 0 when d₀ exceeds 15 m, which indicates that the maximum projected distance of falling objects for this building over a period of 50 years will not exceed 15 m.

From the analysis of the distribution probabilities of falling objects for all 19 buildings in the selected community area, the entire distribution probabilities can be obtained, as shown in Figure 6. The red contours denote larger distribution probabilities for falling objects. The overlapping regions of hazard due to falling objects from different buildings are considered. As a result, the regions between adjacent buildings are indicated by the darker contours. In terms of the computational cost using a desktop computer (Intel Core2 Q8200, 2.33GHz, 4GB memory), the calculation of the distribution of falling objects takes approximately 10 s for 19 buildings, under each ground motion and with a specific PGA. However, when considering the uncertainty of different ground motions with different PGAs, the total calculation time is much longer to about 1.4 h.

All hazard regions identified for this community area are shown in Figure 7. The hazard regions encompass a large area of the useable spaces between each building; thus, the available space for emergency shelters is limited. Figure 7 also shows the originally allocated site for the emergency shelter in this community area. Certain portions of the original site overlap with the hazard regions, which is considered to be unsafe and may result in injuries or casualties caused by falling objects. An alternative site location outside the hazard regions is thus selected as a safe area for an emergency shelter. This is also shown in this figure. Note that the total area of the alternative emergency shelter is identical to that of the original shelter, and both are in close proximity to roads and are convenient for evacuation purposes. To this end, the proposed simulation of hazards due to the falling objects has demonstrated a useful application for the site selection of emergency shelters.

5 Conclusions

This study proposes a novel simulation method for earthquake-induced hazards caused by falling exterior non-structural components, which has rarely been investigated in the existing literature. The hazard ranges of falling objects in a building group are predicted with a nonlinear THA using the previously developed MCS model. The distribution probabilities of falling objects during the design life of a building is calculated via IDAs and a seismic hazard analysis. A residential community area is selected as a case study to evaluate the distribution probabilities of falling objects, which in turn helps to select a suitable and safe site for an emergency shelter. Additional applications of the proposed simulation method, such as the selection of the excavation paths, will be conducted in the future. This study is expected to provide a useful reference for emergency and disaster management. In the future study, the impact of non-structural walls to the building's dynamic characteristics will be further investigated.

Acknowledgements

The authors are grateful for the financial support received from the National Key Technology R&D Program (No. 2015BAK14B02), the National Natural Science Foundation of China (No. 51578320, 51308321), and National Non-profit Institute Research Grant of IGP-CEA (Grant No: DQJB14C01).

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