1 Introduction

A disaster chain occurs when one kind of disaster causes another, and the occurrence of these disasters has a certain sequence, which often leads to more serious damage and losses (Shi 1996; Wen 2000; Shi 2002; Delmonaco et al. 2006; Tarvainen et al. 2006; Marzocchi et al. 2009; Fan et al. 2019). Earthquake and debris flow disasters form the most frequent disaster chain, and its damage to buildings is also more serious than that from a single earthquake or debris flow disaster (Cui et al. 2008; Cui et al. 2011; Fan et al. 2019). For example, the 12 May 2008 Wenchuan Earthquake triggered a large number of debris flows and caused serious building damage under the disaster chain (Huang and Li 2009; Chen et al. 2018; Fan et al. 2019). With the acceleration of urbanization and global climate change, the number of buildings exposed to natural hazard-related disaster chains has continued to increase (Papathoma-Köhle et al. 2011; Kappes, Keiler, et al. 2012). Therefore, evaluating the physical vulnerability of buildings under a disaster chain process is not only a key step of disaster risk assessment, but also the basis for comprehensively improving regional disaster reduction capabilities.

Due to the lack of successive building damage observation data, assessing the physical vulnerability of buildings in an earthquake-debris flow disaster chain is a difficult task. Previous studies attempted to establish the joint building vulnerability curve of disaster chain based on the empirical vulnerability curves to single hazards including earthquakes (Calvi et al. 2006; Vicente et al. 2011; Suppasri et al. 2013; Karimzadeh et al. 2014) and debris flows (Fuchs et al. 2007; Papathoma-Koehle et al. 2012; Papathoma-Köhle et al. 2017; Kim et al. 2020; Luo, Fan, et al. 2020; Luo, Zhang, et al. 2020). The empirical building vulnerability indicates the cumulative probability of a specific building reaching or exceeding a certain degree of damage under a given hazard impact (Fell et al. 2005; Calvi et al. 2006; Foerster et al. 2009; Vamvatsikos et al. 2010; Quan Luna et al. 2011; Kappes, Keiler, et al. 2012). However, empirical building vulnerability models cannot provide vulnerability indicators with clear physical meaning to reflect the continuous strike process and cause and effect of multiple disasters. Some structural vulnerability indices (Kappes, Papathoma-Koehle, et al. 2012; Vega and Hidalgo 2016) have been proposed to evaluate the physical vulnerability of buildings under disaster chain strike via numerical simulations (Mahendra et al. 2011; Ribeiro et al. 2014; Chen et al. 2019). The residual capacity (REC) is the elastic spectrum acceleration of the maximum allowable capacity in the equivalent single-degree-of-freedom structure and it can be evaluated based on pushover curves. It indicats the structural capacity of buildings after a physical shock (Jalayer et al. 2011), and is widely used in physical based vulnerability assessments. However, few studies have examined building vulnerability to continuous physical destruction due to the earthquake-debris flow disaster chain, as well as examining vulnerability variations.

Beichuan County is located in an area with strong tectonic movement and highly volatile geological environment with mountain valleys, and has a high incidence of strong earthquakes and geological hazards. In the process of an earthquake-debris flow disaster chain, the earlier earthquake damages the intact buildings and then these damaged buildings suffer flow impact and debris deposit from the debris flows triggered by the earthquake (Fig. 1). Previous studies have focused more on the changes in hazard intensity and frequency caused by the triggering mechanisms of the disaster chain. The vulnerability of buildings has been assessed by dividing the continuous earthquake-debris flow disaster chain into two separate hazard events (Zhang et al. 2018; Hu et al. 2019). The physical mechanisms of building vulnerability change have not been considered, and the vulnerability of buildings is expressed only in terms of building property loss rates (Tian et al. 2022).

Fig. 1
figure 1

Schematic diagram of the continuous damage from an earthquake and a debris flow to buildings and changes in the physical vulnerability of buildings in the process of an earthquake-debris flow disaster chain

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In this study, we quantified the continuous physical vulnerability variation of buildings by adopting post-earthquake and post-debris flow building damage data and using the pushover analysis (Polese et al. 2013), aiming to illustrate the physical vulnerability of buildings due to the impact of the earthquake-debris flow disaster chain and its changes in the process. We first calculated the vulnerability of buildings before and after the earthquake. Then we reconstructed the debris flow event through the FLO-2D numerical simulation method to obtain the intensity of the debris flow,Footnote 1 and evaluated the vulnerability of earthquake-damaged buildings during the subsequent debris flow. Finally, the holistic vulnerability of buildings under the earthquake-debris flow disaster chain was estimated based on the joint probability distribution from the Bayes’ theorem.

2 Study Area

Beichuan County, situated in Sichuan Province, China, is located on a fault zone between the anterior and posterior fold belt of the Longmen Mountains, which has very active geological tectonic movement, and moderate and strong earthquakes occur frequently (der Hilst 2008; Wang 2008) (Fig. 2). The Beichuan-Yingxiu active fault passes the study area (Zhang et al. 2009). The elevation of the study area is low in the southeast and high in the northwest, with a low point of 613 m and a high point of 1,868 m, and the slope of the valleys is generally greater than 25° (Fig. 2a). Due to strong weathering, the degree of rock fragmentation in the study area is relatively high, and rocks are exposed on slopes greater than 25°. The humid climate conditions with average annual precipitation > 1340 mm (Tang et al. 2009) further increase the frequency of debris flows.

Fig. 2
figure 2

The study area. a Topography, debris flow channels, and impact area; b The county seat urban center after the Wenchuan Earthquake and before the debris flow (taken on 3 August 2008, CIGEM 2009); c The county seat urban center after the Weijia Gully debris flow (taken on 24 September 2008, CIGEM 2009); d The county seat before the Wenchuan Earthquake (May 2005, Google Earth) and e The county seat after the Weijia Gully debris flow (December 2008, Google Earth)

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According to the earthquake intensity distribution map of the 2008 Wenchuan Earthquake by the China Earthquake Administration,Footnote 2 the earthquake intensity in Beichuan was XI on the China seismic intensity scale, causing more than 90% of buildings to suffer moderate or higher damage (Earthquake Disaster Relief Expert Group 2008) (Fig. 2b). Four months after the earthquake, the 24 September Weijia Gully debris flow severely damaged the Beichuan earthquake site, causing the burial of almost all of the old town, raising the entire ground by 2−5 m and destroying many buildings that had been damaged in the Wenchuan Earthquake (Tang et al. 2009; Fan et al. 2019) (Fig. 2c). Remote sensing images before the earthquake (Fig. 2d) and after the debris flow (Fig. 2e) clearly show the damage to the buildings in the study area by the earthquake-debris flow disaster chain. The buildings damaged by the earthquake and the debris flow have been repaired and converted into earthquake museums, where much evidence of building damage during the earthquake-debris flow disaster chain process is preserved. This site provides a good opportunity for us to explore the vulnerability of buildings under the earthquake-debris flow disaster chain.

3 Data

The following subsections present the damage data of buildings under the earthquake-debris flow disaster chain, and the parameters and sources of data required for the FLO-2D model.

3.1 Building Data

Data of a total of 340 buildings in the old urban center before the earthquake, including their geographic coordinates, structure type, number of floors, building height and building area, were collected through the urban planning information system of Beichuan County.Footnote 3 These building data were examined using Google MapsFootnote 4 and a previous study (Zhang et al. 2009).

Detailed information on the 173 buildings that did not fully collapse after the earthquake and the 63 buildings that were not completely destroyed after the debris flow were obtained through remote sensing images, field survey reports, assessment reports (Earthquake Disaster Relief Expert Group 2008), reports from the Beichuan Earthquake Site Preservation Committee,Footnote 5 and photographs from a monograph. A total of 20 masonry concrete (MC) buildings was chosen to analyze their vulnerability during the earthquake-debris flow disaster chain with the following selection criteria: (1) they experienced damage from the earthquake-debris flow disaster chain and (2) the damage process of the disaster chain was representative. The damage classification scheme for earthquake-debris flow disaster chain defines five types of damage: none, slight, moderate, severe and completeFootnote 6 based on building damage characteristics.

3.2 Debris Flow Modeling Data

Meteorological data were collected from the Tangjiashan rainfall station and hydrological data were derived from the hydrological map of Sichuan Province (Sichuan Provincial Water Resources and Power Engineering Bureau 1986). Hydrological statistics of the debris flow were gathered from the discharge hydrograph (Sichuan Provincial Water Resources and Power Engineering Bureau 1986). The digital elevation model (DEM) with a resolution of 10 m × 10 m for 2005 from the Geographic Information Monitoring Cloud PlatformFootnote 7 was used to model the debris flow. It was used in the FLO-2D model to calculate the area of the basin, the channel length of the debris flow, and the slope of the terrain. The 15 × 15 m resolution land use data from the Geographic Information Monitoring Cloud Platform for 2008Footnote 8 were used to derive the impact factor of the debris flow velocity and the inundated area since these are affected by surface roughness.

Parameters and calibration data for the debris flow simulation were obtained from field surveys, the Beichuan Earthquake Site Cultural Relics Protection Plan,Footnote 9 and previous studies (Liu and Li 2006; Tang et al. 2009; Fu et al. 2010; Jiang 2010) of disaster events in Beichuan County. The sediment concentration was obtained by the average specific gravity (1.83 g/cm3) (Deng et al. 2021). The rheological parameters were derived from previous studies by Wang et al. (2003) and Deng et al. (2021). The Manning n value (0.04−0.20) was selected according to the land use categories of the Beichuan urban area. The specific parameters that govern the debris flow behavior parameterization are shown in Table 1.

Table 1 Parameters used in the FLO-2D model
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4 Methods

The shaking from both seismic waves and debris flows damages the structure of buildings by producing lateral displacement or swing, eventually leads to building collapse and overturn (Haugen and Kaynia 2008). From this perspective, we used REC to track the successive physical structure capacity of buildings to obtain the building vulnerability under the earthquake-debris flow disaster chain process. A 3-step procedure was applied in this study to estimate the building vulnerability under the earthquake-debris flow disaster chain (Fig. 3). In Step 1, the seismic vulnerability of buildings was determined based on REC calculated by pushover analysis, and the intensity of the earthquake was derived from the Wenchuan Earthquake intensity distribution map provided by the China Earthquake Administration. In Step 2, the debris flow vulneratility of buildings was derived based on REC calculated, and the FLO-2D model was adopted to simulate the intensity of the debris flow event. Finally, in Step 3, the holistic vulnerability of buildings under the earthquake-debris flow disaster chain was estimated based on the joint probability distribution from the Bayes’ theorem.

Fig. 3
figure 3

Flowchart of the methodology for assessing building physical vulnerability in an earthquake-debris flow disaster chain

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4.1 Evaluating Seismic Vulnerability of Buildings

The residual capacity (REC) is a function of the seismic performance of the damaged building and a measure of the seismic capacity. Building seismic vulnerability can be expressed by REC reduction, indicating the lowering of seismic physical safety after an earthquake (Polese et al. 2013). In this study, pushover analysis was used to assess the residual capacity (REC) of buildings in two situations—the intact state and the damaged state—under specific seismic intensity. During the modeling process, we maintained the degradation of strength and stiffness of buildings, and the damage to the buildings was assessed by visually inspecting post-earthquake changes in its stiffness, strength, and residual displacement.

The residual capacity of a specific building, \({REC}_{sa}\), was defined for each damage state \({D}_{i}\) (i = 0, 1, 2, 3, 4, and 5, where 0 represents the intact structure) as the spectral acceleration that determines the collapse of the building. The characteristics (\({T}_{eq}\),\({C}_{b}\), and \({\mu }\)) were determined by the considered damage state \({D}_{i}\). \({REC}_{sa}\) can be simply calculated by Eqs. 1 and 2 (Vidic et al. 1994):

$${REC}_{sa}={C}_{b}\bullet {\mu }, {T}_{eq}\ge {T}_{C}$$
(1)
$${REC}_{sa}={C}_{b}\bullet \left({\mu }-1\right)\bullet \frac{{T}_{eq}}{{T}_{C}}+1, {T}_{eq}<{T}_{C}$$
(2)

where \({C}_{b}\) is the base shear coefficient, measured in g (acceleration due to gravity), and this parameter is determined before an earthquake by the ratio of nonlinear strength to the mass of the equivalent single-degree-of-freedom (ESDOF) system. After the earthquake, it is calculated based on the actual ground acceleration and the equivalent mass of each building; \({\mu }\) is the displacement capacity in terms of ductility, dimensionless, and both before and after the earthquake, it is obtained by the ratio of building yield displacement to ultimate displacement; \({T}_{eq}\) (s) is the elastic period of the ESDOF system, obtained by the ratio of the product of yield displacement and the mass of the ESDOF system to nonlinear strength; and \({T}_{C}\)(s) is the characteristic period of the building. The specific calculation procedures for these parameters are provided in an online document.Footnote 10

Since the peak ground acceleration \({a}_{g}\) is used as the earthquake damage intensity parameter in the Chinese Code for Seismic Design of Buildings,Footnote 11 it can be used to evaluate the REC:

$${REC}_{{a}_{g}}={REC}_{sa}/(S\bullet \eta \bullet 2.5)({T}_{eq}/{T}_{C})$$
(3)

where \(S\) is the site correction coefficient, which is 1.15 in this study (Zhang 2012), and \(\eta \) is the damping correction coefficient, which is 1 (Zhang 2012).

Finally, vulnerability curves for individual damage states were obtained by lognormal functions (Haugen and Kaynia 2008; Polese et al. 2013)Footnote 12:

$${P}_{\mathrm{Earthquake}}=\Phi \left[\frac{1}{\beta }\bullet ln\left(\frac{{a}_{g}}{{REC}_{{a}_{g}}}\right)\right]$$
(4)

where Φ represents the standard normal cumulative distribution function (CDF), and β is a global value of dispersion.Footnote 13 Equation 4 represents the probability of reaching the collapse state under a given earthquake intensity \({a}_{g}\).

The earthquake intensity of the Wenchuan Earthquake in Beichuan County was XI (PGA is equal to 0.98 g), which was extracted directly from the Wenchuan Earthquake intensity distribution map provided by the China Earthquake Administration.

4.2 Evaluating Debris Flow Vulnerability of Buildings

Flow impact and burial are the main damage modes of debris flows to buildings (Zeng et al. 2012). The damage from flow impact is mainly caused by the dynamic pressure of debris flows, often resulting in structural damage, and the corresponding intensity parameter is the flow velocity of debris flows (Haugen and Kaynia 2008). The damage from burial mainly reflects the effect of its static pressure, and the corresponding intensity parameter is the deposition height of the debris flows. The angle between the advancing direction of a debris flow and the layout direction of buildings can also affect the degree of damage (Hu et al. 2012; Zeng et al. 2012). However, there is a lack of measured data on the angle difference within the short debris flow range (3 km), so our REC simulation adopted a general parallel distribution without considering the influence of the angle difference.

Residual capacity reduction from pushover analysis was used to assess the damage to buildings by debris flows based on building damage data and simulation of debris flow intensity. The ratio of the debris flow impact damage acceleration \({a}_{g}{\prime}\) to the residual capacity \({REC}_{{a}_{g}}{\prime}\) (Eq. 5) under pushover analysis was taken as a vulnerability index to indicate the building damage states corresponding to different debris flow energies (Haugen and Kaynia 2008; Polese et al. 2013). Therefore, we evaluated the damage to buildings by the debris flow based on the vulnerability curve obtained from the REC.

$${P}_{\mathrm{Debris flow}}=\Phi \left[\frac{1}{\beta }\bullet ln(\frac{{a}_{g}{\prime}}{{REC}_{{a}_{g}}{\prime}})\right]$$
(5)

The ratio of actual burial depth (obtained from the field investigation) to building height was used as the index of debris flow burial vulnerability (Eq. 6) (Totschnig et al. 2011) to establish a debris flow burial vulnerability assessment model for earthquake-damaged buildings:

$$I=\frac{h}{H}$$
(6)

where I is the index of debris flow burial vulnerability, h is the actual burial depth of the building (m) and H is the building height (m). A classification scheme for debris flow burial of damaged buildings defines five damage classes: none, slight, moderate, severe and complete (Table 2) (Hu et al. 2012; Zhang et al. 2018).

Table 2 Debris flow burial damage degree of damaged buildings (the floor height is 3 m)
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The FLO-2D model (O’Brien et al. 1993; Muir et al. 2006; O’Brien and Garcia 2009; Quan Luna et al. 2011) was used to reconstruct the debris flow event to quantify the intensity of the debris flow, and field observations were taken into account and used for calibration and validation. The modeling of the Weijia Gully debris flow has two parts. The first part simulated the rainfall in the area to calculate the discharge hydrograph and the influence of rainfall intensity on the flow. The second part was the intensity of the debris flow simulation,Footnote 14 including the debris flow velocity and debris flow deposition height.

4.3 Evaluating Physical Vulnerability of Buildings Under a Disaster Chain

The shaking from seismic waves and debris flows damages the structure of buildings and leads to building collapse and overturn (Haugen and Kaynia 2008). Bayes’ theorem describes the probability of occurrence of an event related to other events, and is adopted to model compound hazards as it provides the joint probabilities of two hazard continuously occurring (Liu et al. 2018; Tilloy et al. 2019). We evaluated the joint vulnerability of buildings under a disaster chain by subtracting the probability of not being damaged after succeesive hazard impact (Eq. 7)

$${P}_{Deb|Ear} ={P}_{Ear} +(1-{P}_{Ear})\times {P}_{Deb}$$
(7)

where \({P}_{Deb|Ear}\) is the vulnerability of the building after the impact of an earthquake-debris flow chain, \({P}_{Ear}\) is the vulnerability of the building to earthquake and \({P}_{Deb}\) is the vulnerability of the damaged building to debris flow impact. All three vulnerability, \({P}_{Deb|Ear}\), \({P}_{Ear}\), and \({P}_{Deb}\) can be defined as the REC of post-disaster buildings.

5 Results and Discussion

This section depicts the REC of buildings continuously hit by earthquakes and debris flows from the pushover analysis, and then presents the physical damage probability of buildings or vnlnerability under the earthquake-debris flow disaster chain process. The actual building damage in the case of the Wenchuan Earthquake and its triggered debris flows was used to check the vulnerability evaluation results. Finally, we demonstrates the joint probability distribution of building physical damage when encountering continuous disaster strikes, and presents its physical vulnerability curves.

5.1 Seismic Vulnerability of Buildings

Applying the methodology described in Sect. 4.1, the building residual capacity for the intact and damaged states was estimated under different seismic intensities. Table 3 summarizes the results for \({REC}_{{a}_{g}}\), together with the values of \({C}_{b}(\mathrm{g})\) and \({\mu }\). Our results show that the average REC of the studied buildings is 0.11862. The highest REC is 0.21665 for a slightly damaged building, and the lowest REC is 0.04725 for a collapsed building. The REC reduction of buildings under earthquake strike was sourced from the base shear coefficient (\({C}_{b}\)) and ductility (\({\mu }).\) When the earthquake intensity is 0.2 g (PGA of the Chinese Code for Seismic Design of BuildingsFootnote 15), \({C}_{b}\) and \({\mu }\) are normal, indicating that the building has better residual capacity at this damage level. In contrast, when the earthquake intensity is 0.98 g (PGA of Beichuan County in the Wenchuan Earthquake), \({C}_{b}\) and \({\mu }\) decrease by about 50%, indicating that the building contains less residual capacity at this damage level.

Table 3 Representative parameter and corresponding residual capacity (REC) values of masonry concrete (MC) buildings in intact, post-earthquake, and post-debris flow states
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The damage distribution was estimated using a beta-distribution. By calculating the probability density function, we derived the vulnerability curve according to the REC values of the 20 buildings before and after the earthquake (Fig. 4). For the slightly damaged building, the change of the vulnerability curve from that of the intact building is not very significant, indicating that the building has better residual capacity at this damage level. For the moderately damaged building, the probability of collapse when the ground acceleration is 0.2 g is 71%—compared to the 31% probability of the intact building, the probability of collapse of the moderately damaged building increases by 40%. For the structure with severe damage and collapse damage, the probability of collapse rises by 46% and 57%, respectively, from 31% for intact buildings when the ground acceleration is 0.2 g.

Fig. 4
figure 4

Vulnerability curves of buildings pre-earthquake (intact structure) and post-earthquake (four damage levels of earthquake-damaged structures)

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After the Wenchuan Earthquake, nearly 51% of the 340 buildings in the old urban center (Fig. 5a) collapsed directly and 173 buildings were not completely destroyed (Fig. 5b). The collapsed buildings were mainly concentrated in places where the fault passes, and the remaining buildings suffered moderate or above damage. Of the 20 MC buildings we studied, 70% had a pre-earthquake vulnerability between 0.3 and 0.4 at an earthquake intensity of 0.2 g and only 30% were below 0.3. The maximum vulnerability value of MC buildings before the earthquake was 0.39. After the Wenchuan Earthquake, the damage to buildings led to a general increase in their vulnerability values, which were all between 0.45 and 1, with 50% of them between 0.8 and 0.9, and the highest was 0.99.

Fig. 5
figure 5

Vulnerability of the 20 masonry concrete (MC) buildings and the spatial distribution of the buildings in the study area before and after the earthquake-debris flow disaster chain damage process. a Pre-earthquake; b Post-earthquake; c Post-debris flow

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5.2 Debris Flow Vulnerability of Buildings

Earthquake-damaged buildings were again damaged by the debris flow, causing the structural properties of the buildings to continue to change, and each node of the building was further damaged, causing the building to become more tilted (Fig. 6). Our simulation results show that the earthquake caused on average 55% loss in REC values of our studied buildings, and the subsequent debris flow caused an average loss of 26% of the REC values (Table 3). These REC losses were mainly sourced from the the tilting of the building and the connection failure among building elements (Fig. 6). As a result, the REC curve shifts to the left side, indicating that the same external impact may lead to greater building damage, which is the most obvious expression of the increased vulnerability of the buildings (Polese et al. 2013). In the field of seismic engineering, this is demonstrated through the reduction of the foundation shear coefficient (\({C}_{b}\)) and ductility (\({\mu }\)).

Fig. 6
figure 6

Illustration of the physical vulnerability of buildings to the earthquake-debris flow chain. The upper part is the change of the residual capacity of the building with the development of the earthquake-debris flow chain, and the lower part is the corresponding simple mechanical model; a–c represent the residual capacity (REC) curve of the intact building, the damaged building after the earthquake, and the damaged building after the debris flow, respectively and the corresponding simple pushover model. The x-axis represents the hazard intensity, and the y-axis represents the probability density of REC

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The REC values were evaluated to reprensent the physical vulnerability of buildings under earthquake-debris flow chain strike (Fig. 6). The simulation results show that vulnerability of buildings continued to rise as the earthquake-debris flow disaster chain continued to damage the buildings. At the intensity of 0.2 g, the vulnerability of earthquake-damaged buildings to debris flow impact increases from 79% after the earthquake to 92% (Fig. 7). The probability of building collapse increased by 48% before and after the earthquake, while it increased by only 13% before and after the flow impact of the debris flow. This indicates that in the earthquake-debris flow disaster chain, building damage is dominated by earthquakes, which also coincides with the decreasing energy release of each hazard in the chain. When the hazard intensity is 0.2 g, the vulnerability of buildings to debris flow burial is 28% (Fig. 7). This indicates that post-earthquake buildings are often semi-open due to earthquake damage, and the static pressure damage suffered is less than that of intact buildings (which is 31%, see Fig. 4).

Fig. 7
figure 7

Vulnerability curves of buildings in an earthquake-debris flow disaster chain. The red, yellow and brown lines represent the vulnerability curves of earthquake-damaged buildings, flow impact-damaged buildings, and flow burial-damaged buildings, respectively

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The 173 buildings that did not completely collapse in the Wenchuan Earthquake (Fig. 5b) were further damaged by the Weijia Gully debris flow four months after the earthquake, resulting in nearly 64% of the buildings being completely destroyed (impacted or buried), but 63 buildings still survived (Fig. 5c). The FLO-2D simulation result shows that the average velocity of the Weijia Gully debris flow is 12.5 m/s in the study area, and the maximum flow velocity is 28.9 m/s, which occurs in transportation area (upper-left corner of Fig. 8a). The velocity of the debris flow gradually decreases (to 0−10.8 m/s) toward the sides of roads and passages where buildings are concentrated. The velocity of the debris flow in the streets and channels (8.8−28.9 m/s) is significantly higher than that on both sides (0−8.7 m/s) of these features due to the narrow pipe effect on the debris flow (Fig. 8a). Debris flow deposition are concentrated in areas where buildings are concentrated and oriented perpendicular to the direction of debris flow movement and in low-lying riverbank areas (Fig. 8b). The maximum depth of 10.7 m is found around Buildings #2 and #3 (Fig. 8b). By comparing the actual heights of Building #7 (Fig. 8c and d) before and after the debris flow damage, it can be seen that the building was buried by the debris flow up to 1.5 floors (4.5 m height). The simulation result, 3.3−4.8 m, is consistent with the observed heights.

Fig. 8
figure 8

Debris flow simulation results: a Maximum flow velocity and b debris flow deposition; c Actual height of Building #7 before debris flow damage; d Actual height of Building #7 after debris flow damage; e Statistics of the simulated maximum flow velocity; f Statistics of the simulated debris flow deposition. MC masonry concrete

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Statistical analysis result shows that 53% (91) of the damaged buildings suffer debris flow impact velocities exceeding 9 m/s that can cause complete damage (Fig. 8e), and 35% (60) of the buildings are buried deeper than 3 m (Fig. 8f). The vulnerability evaluation result shows that all 20 MC buildings we studied are impacted or buried by the debris flow and their vulnerability after the debris flow impact is basically between 0.9 and 1. Only Building #4 has a vulnerability value of 0.8 due to the topography of the location. Generally, buildings damaged by the earthquake-debris flow disaster chain are relatively better preserved (approximately 30%) in areas where earthquake damage level is less than or equal to severe damage, and the debris flow velocity is less than 6 m/s. In contrast, other buildings are completely destroyed and became ruins.

5.3 Earthquake-Debris Flow Disaster Chain Vulnerability of buildings

A disaster chain consists of a series of hazards that have a certain triggering mechanism and sequence of occurrence and are causally related (Shi 2002; Kappes, Keiler, et al. 2012; Fan et al. 2019). Its impact on hazard-affected objects is often presented as continuous impact and destruction (Shi 2002; Delmonaco et al. 2006; Kappes 2011; Fan et al. 2019). The earthquake and its triggered debris flow strike buildings’ physical structures continuously, leading to partial or total functional failure (Cui et al. 2008; Cui et al. 2011; Fan et al. 2019). The seismic waves bring structural damage by generating vertical and horizontal vibrations in buildings, and then the compromised buildings after absorbing the brunt of seismic shaking exhibit lower structural stability and resillience to the shock of the following debris flow (Kappes, Keiler, et al. 2012; Luo, Fan, et al. 2020; Luo, Zhang, et al. 2020). In this study, we tracked the physical destruction of buildings based on the REC from the pushover analysis, and evaluated building vulnerability under this disaster chain process. Our results show that the building damage started by the Wenchuan Earthquake, contributing to 51% buildings’ total functional failure and the rest with partial functional failure. Thereinto, the vulnerability of buildings with slight damage (REC > 83%) by the earthquake increases by about 13% under a debris flow with average velocity of 9 m/s in our studied case.

In previous assessments of the vulnerability of buildings in disaster chains, vulnerability was viewed as to independent hazards (Luo, Fan, et al. 2020; Luo, Zhang, et al. 2020) and was often calculated based on the original building, which is structurally intact and has the highest resistance to hazards. Such method fails to consider the dynamic changes of vulnerability during a disaster chain process (additional factors that contribute to building vulnerability may have occurred), while using inappropriate assessment methods will affect the accuracy and variability of vulnerability. For comparison, this study also generated the statistics-based vulnerability curves that were calculated by summing the independent vulnerability values of intact buildings under earthquake and debris flow respectively, as used in previous disaster chain studies (Kappes, Keiler, et al. 2012; Julià and Ferreira 2021) (Fig. 9b). Our results show that the mechanical-based method can avoid probability overflow that exist in the vulnerability summation method. For example, the holistic vulnerability of buildings to earthquake-debris flow disaster chain is 57% at hazards intensity 0.2 g (Fig. 9a), while the vulnerability to earthquake and debris flow (flow impact) is 39% and 90%, respectively, according to the statistics-based assessment of building vulnerability to independent earthquake (AQSIQ 2001; Earthquake Disaster Relief Expert Group 2008) and debris flow (flow impact) (Hu et al. 2012; Zeng 2014; Papathoma-Köhle et al. 2017) hazards at an intensity of 0.2 g (Fig. 9b).

Fig. 9
figure 9

Differences between a the mechanical-based method for assessing the physical vulnerability of buildings in the earthquake-debris flow disaster chain and b statistics-based simple summation method for earthquake and debris flow vulnerability

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This phenomenon exists not only in earthquake or geological disaster chains but also in other disaster chains. Therefore, during the assessment of building vulnerability under a disaster chain process, it should not only consider the relationships between the various hazard processes in the disaster chains (Granger et al. 1999; Kappes, Keiler, et al. 2012; Marzocchi et al. 2012) but also assess the vulnerability of buildings based on their physical damage processes. There are also other parameters that affect the vulnerability of buildings, such as the size of building openings, location, site conditions, and angle of debris flow impact in relation to the orientation of the buildings, which are not represented in this study but play a role in actual situations and need to be considered to improve the assessment of vulnerability to disaster chains.

6 Conclusion

This study selected 20 representative buildings in Beichuan County that had been damaged by the earthquake-debris flow disaster chain. Based on the mechanical-based physical vulnerability assessment model for buildings, the study obtained the vulnerability curve of buildings to the earthquake-debris flow disaster chain. According to the specific sequence of events in the earthquake-debris flow disaster chain, the vulnerability of buildings to earthquake is 79% and the vulnerability of damaged buildings to debris flow impact and burial are 92% and 28% respectively. The holistic vulnerability of buildings under the earthquake-debris flow disaster chain is 57%. The physical vulnerability of buildings under the disaster chain process is not equal to the summation of the vulnerabilities to independent hazards estimated by statistical models. Our calculations show that there is a significant difference in the results based on the method of structural mechanical assessment of buildings and using traditional statistical methods that do not take into account changes in the structural properties of buildings between hazard processes, which causes a probability overflow problem in the representation of the holistic vulnerability of a disaster chain.

The mechanical-based physical vulnerability model for assessing the vulnerability of buildings under continuous damage of the earthquake-debris flow disaster chain can reflect the sequential damage process of buildings under the actual disaster chain, and the specific change of vulnerability. Therefore, in order to better serve disaster management and ensure the accuracy of risk assessment and prediction of disaster losses, it is necessary to conduct an overall vulnerability assessment of the areas with frequent disaster chains and dense exposure. This method can also be applied to the assessment of vulnerability in other disaster chains by quantitatively and continuously describing and evaluating the vulnerability changes of exposed elements in the process of the disaster chains.