Resilience assessment of reinforced concrete buildings in the Indian Himalayan region under flow-type landslides

Abstract

This study presents a methodology for landslide resilience assessment of reinforced concrete (RC) buildings subjected to flow-type landslides. The methodology includes developing region-specific recovery functions that account for variations in the construction materials and practices, as well as development of fragility curves of archetype buildings using efficient intensity measures while considering the uncertainty in the input parameters related to landslide-induced forces. To this end, 2-, 3-, and 4-story archetype RC buildings, commonly observed in the landslide-prone Indian Himalayan Region, are selected, and their recovery functions are developed for various damage states based on inputs from practicing professionals. The proposed recovery functions can be used to determine the expected functional recovery quantitatively, which will be helpful for decision-makers to take action promptly to start the rescue and recovery process. Further, the effect of community preparedness levels on the landslide resilience of archetype RC buildings is examined through a detailed numerical study.

Keywords

flow-type landslide functional recovery landslide fragility recovery function reinforced concrete building resilience assessment

Introduction

The Indian Himalayan Region is a well-known hotspot for landslides, accounting for 16% of global rainfall-induced landslides (Froude and Petley 2018). These landslide events, in the form of debris flows, slides, topples, spreads, and rockfalls, cause significant damage to the built environment, especially during the monsoon season. In 2023, unprecedented rainfall triggered more than 5500 landslides in the state of Himachal Pradesh, India, resulting in the destruction of thousands of buildings and damage to roads at more than 350 locations (HPSDMA 2024). Further, due to the scarcity of flat land, people are often compelled to construct buildings adjacent to precarious slopes or near nullahs and tributaries, exposing them to debris flows and floods (Aggarwal et al., 2024). Moreover, buildings are also observed to be constructed adjacent to large hill cuts without adequate landslide mitigation measures (Kulariya et al., 2024a). These hill cuts often do not adhere to the permitted hill cut of 3.5 m outlined in Town and Country Planning guidelines (HPTCP 2022), thereby exposing the buildings to landslides. Therefore, considering such landslide scenarios and developing appropriate methodologies is essential in the quest for resilient infrastructure and communities, as targeted in global initiatives such as the Sendai Framework for Disaster Risk Reduction 2015–2030 (UNDRR 2015).

Prompt post-hazard functional recovery of buildings is a crucial aspect of community resilience. Quick functional recovery of buildings allows people to return to their homes and access other essential needs such as commerce, healthcare, education, etc. Consequently, in recent decades, there has been a growing emphasis on resilience as a key goal of national and international initiatives aimed at disaster risk reduction (e.g., NDMP 2019; UNDRR 2005, 2015). Also, considerable efforts have been made to develop methodologies for assessing the seismic resilience of buildings (e.g., Anwar et al., 2020; Burton et al., 2016; Cimellaro and Piqué 2016; Fazlalipour et al., 2023; Molina Hutt et al., 2022; Salado Castillo et al., 2022; Shamsoddini Motlagh et al., 2020; Wang et al., 2024) and other critical infrastructure such as bridges (e.g., Ashrafifar and Estekanchi 2023; Biondini et al., 2017; Dong et al., 2022; Du et al., 2019; Khan et al., 2022; Wei et al., 2023; Yang et al., 2024; Yu and Gardoni 2022), transportation networks (e.g., Mohamad El-Maissi et al., 2023; Sun et al., 2021; Xiong et al., 2023), and communities (e.g., Blagojević et al., 2023; Bruneau et al., 2003; Harati and Van de Lindt 2024; Melendez et al., 2022; Miles and Chang 2006). However, efforts addressing the resilience of buildings (particularly towards the computation of their expected functional recovery) under other natural hazards, especially landslides, one of the prominent natural hazards in the hilly regions, have been limited and are predominantly focused on landslide resilience evaluation of road networks. For example, Chen et al. (2010) examined the community resilient capacity under debris flows by simulating potential debris flow torrents in study areas in Taiwan. Aydin et al. (2018) proposed a framework for an efficient recovery process of road segments. Yousefi et al. (2022) performed a vulnerability assessment of road networks to landslides in a dry-mountainous region of Iran in order to identify those at very high risk of landslides. Zhou et al. (2024) studied the risk of road networks, situated in Sichuan province and Yunnan province, under the influence of landslides by combining landslide susceptibility with road network vulnerability. Thus, to the best of the authors’ knowledge, a methodology for assessing the landslide resilience of buildings to enhance their resilience under flow-type landslides is yet to be developed.

The expected functional recovery assessment of any component of the built environment primarily requires a thorough understanding of its vulnerability curves and recovery paths (defined through recovery functions) based on the damage incurred. Although a few studies (e.g., Kulariya and Saha 2024a, 2024b; Luo et al., 2020, 2022; Pairisi and Sabella 2017) exist that assess the landslide vulnerability of reinforced concrete (RC) buildings subjected to flow-type landslides, the recovery paths of damaged buildings, based on expert opinions, remain an area of further research. To address the research gaps mentioned above, the present study aims (i) to develop a methodology for assessing the landslide resilience of RC buildings subjected to flow-type landslides, (ii) to propose the recovery functions for archetype RC buildings commonly observed in the landslide-prone Indian Himalayan Region, (iii) to examine the expected functional recovery of considered archetype RC buildings under flow-type landslides through a detailed numerical study, and (iv) to assess the effect of community preparedness on landslide resilience of archetype RC buildings.

Methodology for landslide resilience assessment of RC buildings under flow-type landslides

A methodology for assessing the landslide resilience of RC buildings under flow-type landslides is presented herein. The procedure can be broadly divided into the following steps: (i) development of recovery functions corresponding to RC buildings with different damage states based on inputs concerning recovery time from practicing professionals, (ii) development of fragility curves for considered RC buildings under flow-type landslides, and (iii) landslide resilience assessment of considered RC buildings under flow-type landslides, as illustrated in Figure 1.

Figure 1.

Methodology for assessing the landslide resilience of RC buildings under flow-type landslides.

The first step of the methodology involves selecting archetype RC building models based on field surveys and/or existing reconnaissance surveys conducted in the study region. Likewise, typical landslide scenarios and the damage states of buildings can be identified through post-landslide reconnaissance studies or based on existing literature. Damage-state-specific recovery functions for various damage states of RC buildings can then be developed based on inputs from practicing professionals regarding recovery time.

The choice of recovery function $[F_{r e c} (t)$ ], whether linear, trigonometric, or exponential (Figure 2), largely depends on the community preparedness level and the deployability of resources after a hazard event. For instance, the linear recovery function is often used when information about preparedness, available resources, and societal response is limited (Cimellaro et al., 2005). The linear recovery function is expressed as:

F_{r e c} (t) = (1 - \frac{t - t_{0}}{t_{r}})

(1)

where

t

= time instant,

t_{0}

= time of occurrence of hazard, and

t_{r}

= total time required to repair/replace/restore the full functionality.

Figure 2.

Schematic diagram illustrating idealized recovery functions for different community preparedness levels.

A trigonometric recovery function is used when a community’s response to a hazard event is initially very slow, often due to a lack of organization and/or resources. Once the community becomes organized, possibly with help from neighboring communities, the recovery process accelerates (Chang and Shinozuka 2004). The trigonometric recovery function is expressed as:

F_{r e c} (t) = 0.5 \times {1 + \cos [\frac{π (t - t_{0})}{t_{r}}]}

(2)

On the other hand, the exponential recovery function is used when the initial response of the community to a hazard event is rapid at first due to an initial influx of resources, but the rate of recovery gradually slows over time (Kafali and Grigoriu 2005). The exponential recovery function is expressed as:

F_{r e c} (t) = \exp [- (t - t_{0}) \times \frac{\ln 200}{t_{r}}]

(3)

These general recovery functions are then tailored to specific damage states and their corresponding recovery time required to achieve 100% functional recovery of a building under consideration. This leads to the formulation of damage-state-specific recovery functions, denoted as $Q_{r} (t | {D S}_{j})$ , as shown in Figure 2, which describes the expected level of functionality at the time $t$ , given that the building is in the ${D S}_{j}$ damage state.

The second step of the methodology involves developing the landslide fragility of selected archetype RC buildings under identified landslide scenarios. The steps to develop the landslide fragility of an RC building using a suitable intensity measure (IM) can be summarized as follows:

1. Select the mean values and coefficients of variation for the various input random variables (e.g., flow density, flow depth, flow velocity) based on the laboratory and field tests of soil samples collected from active-landslide sites in the area under consideration and/or based on the historic database of landslides to account for the uncertainty in landslide-induced forces due to variations in geotechnical and flow properties. Also, select the various mean levels of the selected IM to be used for developing fragility curves.

2. Generate a set of random input parameters for each mean value of IM using Monte Carlo simulations, based on their assumed probability distribution.

3. Calculate the landslide-induced force using $P_{p e a k} = 0.5 k ρ g h w + α ρ v^{2} w$ for each set of input parameters and apply as a trapezoidal load on RC columns exposed to landslides, as shown in Figure 1. Herein, $h$ = flow depth, $g$ = gravitational acceleration, $k$ = static earth pressure coefficient, $v$ = flow velocity, $w$ = tributary width, $ρ$ = flow density, and $α$ = dynamic pressure coefficient.

4. Perform load-controlled nonlinear static analyses to obtain the desired response quantities [i.e., engineering demand parameters (EDPs)] at the mean levels of IM.

5. Calculate the conditional probability of attaining the ${D S}_{j}$ damage state at each $i m$ level of the selected IM as N_fail/N_total. Herein, N_fail = number of simulations causing the EDP to exceed the limit ( ${e d p}_{j}$ ) corresponding to the ${D S}_{j}$ damage state, and N_total = total number of simulations performed at each $i m$ level of the selected IM.

6. Develop the fragility curves, plotting the various levels of IM on the horizontal axis and conditional probability of attaining the ${D S}_{j}$ damage state [ $P (E D P \geq {e d p}_{j} | I M = i m)$ ] on the vertical axis.

The developed damage-state-specific recovery functions and landslide fragility curves can then be employed to evaluate the expected functional recovery [ $Q_{L F} (t | I M = {i m}_{i})$ ] of a building under consideration at a given IM level (Argyroudis et al., 2020).

Q_{L F} (t | I M = {i m}_{i}) = \sum_{j = 0}^{n_{{D S}_{j}}} Q_{r} ({t | D S}_{j}) P (E D P \geq {e d p}_{j} | I M = {i m}_{i})

(4)

Herein, $Q_{r} ({t | D S}_{j})$ = functionality of the building, that has sustained ${D S}_{j}$ damage state under a given landslide scenario, at time t after landslide event (also known as damage-state-specific recovery functions), and $n_{{D S}_{j}}$ = number of damage states selected for developing the fragility curves. In equation (4), ${D S}_{0}$ denotes that the building is fully functional and has no damage. After obtaining the $Q_{L F} (t | I M = {i m}_{i})$ , landslide resilience can be quantified in terms of the landslide resilience index ( ${L R}_{i n d e x}$ ). ${L R}_{i n d e x}$ is a quantitative measure that reflects the ability of a building to withstand and recover from landslides, accounting for various factors, including the extent of damage, recovery time, and overall building performance during such events. ${L R}_{i n d e x}$ can be calculated as (Bruneau and Reinhorn 2007):

{L R}_{i n d e x} = \frac{1}{t_{r} - t_{0}} \int_{t_{0}}^{t_{r}} Q_{L F} (t | I M = {i m}_{i}) d t

(5)

The application of the proposed methodology is further shown through a numerical study conducted for archetype RC buildings, which is discussed in the subsequent sections.

Recovery functions for archetype RC buildings

Himachal Pradesh, a mountainous state in the Indian Himalayan Region highly susceptible to flood and landslide hazards, is selected as the case study region. The existing reconnaissance studies (e.g., Aggarwal and Saha 2023; Kulariya et al., 2024a) highlight that the building stock in this region predominantly consists of 2-, 3-, and 4-story buildings. Therefore, 2-, 3-, and 4-story archetype RC buildings with different damage states are considered to propose damage-state-specific recovery functions. The archetype building models are developed based on the typical building plan, bay widths, and story heights reported in Aggarwal and Saha (2023), Kulariya et al. (2024a), and Kulariya et al. (2025). The plan and elevation views of the considered building frames are shown in Figure 3.

Figure 3.

(a) Plan and (b) elevation views of the considered RC buildings.

The recovery time for these buildings with four damage states, i.e., slight (hereafter denoted by “DS₁”), moderate (“DS₂”), extensive (“DS₃”), and collapse (“DS₄”) was obtained from the practicing professionals. Herein, DS₁ refers to the damage state of a building with only hairline cracks in the plastered masonry panels. DS₂ represents moderate damage in the RC building, involving flexural and shear cracks in beams, columns, and walls with cracks exceeding 2 mm and/or partial failure of infill walls. DS₃ refers to the partial collapse of lateral and gravity load-bearing elements. Severe damage, where the structure is either on the verge of collapse or at risk of total failure, is presented by DS₄ (Ghobarah 2004). It is assumed that the building functionality drops to 90% in DS₁, 70% in DS₂, 50% in DS_3, and 0% in DS₄ (Argyroudis et al., 2020; Padgett and DesRoches 2007). It is also assumed that labor and material resources are available on-site, with one mason and three helpers assigned to repair the damaged buildings, as is typical in hilly regions. The maximum and minimum recovery time required to ensure 100% functionality of any building with a given damage state is obtained from practicing professionals.

A total of 52 responses was collected from practicing professionals in the state of Himachal Pradesh, India. These responses ensure the inclusion of potential variations in construction materials and practices commonly used by these professionals. Box-and-whisker plots for the collected data are developed, as depicted in Figure 4, to identify the outliers in the dataset. The selected data, after excluding outliers, were then used to develop damage-state-specific recovery functions. The mean (μ) and COV of the dataset, after excluding outliers, are presented in Table 1. From Figure 4 and Table 1, it can be observed that the variation in recovery time obtained through field survey is greater for 2-story building as compared to its 3- and 4-story counterparts. This is because the 2-story buildings, being smaller projects, are typically constructed with locally available resources. In addition, each interviewed practitioner might have varying levels of access to skilled workforce and materials, leading to a greater variation in recovery time. However, buildings having higher stories invariably require the involvement of external workforce, materials, and machinery from outside the case study region. This results in a more uniform estimation of the repair/reconstruction time, leading to lesser variation in recovery time.

Figure 4.

Outliers in surveyed data for recovery time corresponding to archetype RC buildings with different damage states.

Table 1.

Statistics of the recovery time for archetype RC buildings subjected to flow-type landslides.

	Recovery time
Building configuration	2-Story		3-Story		4-Story
Damage state	μ (days)	COV	μ (days)	COV	μ (days)	COV
DS ₁	26	0.27	47	0.30	67	0.31
DS ₂	59	0.42	85	0.33	115	0.26
DS ₃	134	0.37	188	0.30	280	0.30
DS ₄	266	0.34	375	0.30	515	0.27

Further, landslides are one of the most commonly observed natural hazards in the case study region, which experiences heavy precipitation each year and is also a seismically active region (HPSDMA 2024). As a result, the local community is well prepared and responds quickly to recovery efforts during landslide events. Therefore, an exponential recovery of archetype RC buildings is assumed while developing the damage-state-specific recovery functions for each damage state, i.e., $Q_{r} (t | {D S}_{j})$ , which are presented in Figure 5. These functions are further used in the expected functional recovery assessment of the considered buildings which is discussed in the subsequent section.

Figure 5.

Damage-state-specific exponential recovery functions for the considered archetype RC buildings with different damage states.

Expected functional recovery assessment of archetype RC buildings under flow-type landslides

A detailed numerical study is conducted for 2-, 3-, and 4-story archetype buildings to develop their expected functional recovery curves. The structural elements of these buildings are modeled using frame element in SAP2000 (CSI 2021). The details of dead and live loads, applied confirming to IS 875 Part 1 (1987) and IS 875 Part 2 (1987), are provided in Table 2. The characteristic compressive strength of 30 MPa (M30 grade) is considered for concrete conforming to the minimum grade of concrete specified for this region as per IS 456 (2000) based upon the environmental exposure condition. The yield strength of steel is assumed to be 500 MPa (Fe500 grade). The building base is assumed to be fixed with no soil-structure interaction. The building models are analyzed to withstand live, dead, and earthquake loads as per IS 1893 Part 1 (2016), IS 456 (2000), and designed following the capacity design philosophy as per IS 13920 (2016). The details of the fundamental period of vibration of the buildings are given in Table 3. Further details of the cross-sectional sizes of structural elements and reinforcement used in the considered building models are provided in Annexure A.

Table 2.

Details of various loads considered for the archetype RC buildings.

Description	Magnitude
Live load	3.00 kN/m²
Roof live load	1.50 kN/m²
Roof treatment load	1.50 kN/m²
Slab (150 mm thick)	3.75 kN/m²
Floor finish load	1.00 kN/m²
Unit weight of masonry walls (interior wall = 110 mm thick; exterior wall = 230 mm thick; parapet wall = 230 mm thick and 1 m high)	20.0 kN/m³

Table 3.

Details of the dynamic characteristics of the considered archetype RC buildings.

Building configuration	Fundamental period (s)
Building configuration	X-direction	Y-direction
2-Story	0.60	0.59
3-Story	0.73	0.92
4-Story	0.99	1.27

The constitutive model proposed by Mander et al. (1988) is employed to characterize the stress-strain behavior of concrete. Material nonlinearity in columns and beams is modeled using lumped plasticity modeling approach. Columns and beams are assigned with P-M2-M3 and M3 plastic hinges, respectively, following ASCE 41 (2013) guidelines. A generalized force-deformation relation for the concrete elements is illustrated in Figure 6. The backbone curve is linear up to the yield point B, followed by post-yield behavior with reduced stiffness from point B to C. The slope of the segment B-C is taken as 10% of the initial stiffness to represent strain hardening phenomena. Point C denotes the onset of concrete crushing resulting in a significant drop in strength, which in turn may cause numerical instability. Therefore, the post-capping behavior of the backbone curve beyond point C is modified to ensure a gradual softening slope, as recommended in FEMA P58 (2018). In the figure, the parameters a and b represent the deformation portions that occur after yield. The parameter c denotes the reduced resistance after the sudden reduction in strength from C to D. It is worth stating that the applicability of backbone curve properties derived as per ASCE 41 (2013) guidelines to the Indian context is demonstrated by Surana et al. (2018) through a calibration study. The length of the flexural plastic hinge ( $l_{f}$ ) is computed using the equation proposed by Paulay and Priestley (1992).

l_{f} = 0.08 L_{s} + 0.022 d_{b}

(6)

Where

L_{s}

= length of the RC element in m, and

d_{b}

= diameter of longitudinal reinforcement in m. Apart from flexural hinges, shear hinges are provided in short columns at a distance equal to the section depth (d) to model their brittle failure behavior effectively. To model the shear failure of RC columns, the parameter a is set to zero because significant plastic deformations are not expected prior to shear failure (ASCE 41 2013). In addition, the values of parameters b and c can be obtained from Tables 10–8 given in ASCE 41 (2013). Thus, the behavior of the shear hinges is modeled as force-controlled, where the segment AB in the force–deformation curve captures the initial elastic response, followed by a segment that reflects the degradation of strength and the inability to sustain gravity loads. The length of shear hinges (l_ps) is calculated as 1.5d, which is obtained based on the expression used to calculate shear crack length that is d_vcotβ' (CSA 2004). Here, the effective shear depth (d_v) is taken as 0.8d, and β′ is the angle of inclination of the shear crack, which is assumed to vary between 29° and 35° Güner (2008). Further, geometric nonlinearity is also considered by incorporating the P-delta effect into the analysis.

Figure 6.

Generalized force-deformation relation for concrete elements.

Landslide fragility assessment of RC buildings

The archetype buildings considered in this study are assumed to be exposed to uphill landslide scenarios, typically observed in the case study region, as reported by Kulariya and Saha (2024a). The input parameters required to compute the landslide-induced forces are adopted from existing literature. The value of $k$ is taken as 1 as recommended by Kwan (2012), Ashwood and Hungr (2016), and Iverson et al. (2016). The value of $α$ is taken from the calibration study performed by Kulariya and Saha (2024c). Further, based on the existing database of laboratory and field tests (Hong et al., 2015; Kang and Kim 2016; Kulariya et al., 2024b) conducted on soil samples from active-landslide sites, the uncertainty in flow density is considered through normal distribution. The mean flow density of 1950 kg/m³ is considered with a coefficient of variation (COV) of 5%. The mean flow velocities are assumed to range from 1 m/s to 10 m/s, with increments of 1 m/s and a COV of 10%. The mean flow depth is considered equal to the height of one story, i.e., 3.3 m, with an 8% COV (Parisi and Sabella 2017). The landslide-induced loads are computed for each set of random input parameters generated at each mean IM level and applied as trapezoidal loads to RC columns exposed to landslides, as discussed in the second step of the methodology. Load-controlled nonlinear static analyses are then performed using the default solution scheme “Iterative Event to Event” under the “Nonlinear Solution Control” settings. The “Maximum Total Steps” for each analysis is set to 200, and the same value is assigned to “Maximum Null Steps” to prevent premature termination due to null steps. These null steps typically occur when: (i) a frame hinge attempts to unload, (ii) an event such as yielding or unloading triggers another event, or (iii) the iteration fails to converge and a smaller step size is attempted (CSI 2021). The maximum number of Newton-Raphson and Constant Stiffness iterations per step is set to 40 and 20, respectively, with an iteration convergence tolerance of 0.0001.

The selection of a suitable IM is crucial for fragility assessment to accurately capture the physical characteristics of the hazard and its relationship to the structural response, measured in terms of EDP (Baker 2005; Luco and Cornell 2007; Sengar et al., 2023). A stronger correlation between the selected IM and the predicted EDP ensures a more precise fragility assessment. Herein, the overturning moment (hv) is considered as IM to develop the fragility curves due to its high efficiency and proficiency (Kulariya and Saha 2024a). Moreover, the interstory drift ratio (IDR) is taken as EDP, as damage to structural members and in-plane damage of masonry panels is associated with it. The fragility curves are developed corresponding to four damage states, i.e., DS₁, DS₂, DS_3, and DS₄. The threshold limit (edp_j) in terms of IDR corresponding to these damage states are taken as 0.4%, 1.0%, 1.8%, and 3.0%, respectively, as recommended by Ghobarah (2004). The fragility curves developed for the considered RC buildings are shown in Figure 7. For comparison purposes, the values corresponding to an hv level of 19.8 m²/s are also marked on the plots. This value represents an hv level obtained by multiplying the mean flow depth of 3.3 m, i.e., equivalent to floor height, with a mean flow velocity of 6 m/s.

Figure 7.

Fragility curves developed for the considered archetype RC buildings.

It can be observed that for the 2-story building, the conditional probability of the EDP exceeding DS₁ is 1, whereas it is 0.98 and 0.96, respectively, in the case of 3- and 4-story RC buildings. Further, the conditional probability of attaining DS₂ is 0.86 for 2-story RC building, whereas the same is observed to be 0.74 and 0.66 in the case of 3- and 4-story RC buildings, respectively. A similar trend is observed for damage states DS₃ and DS₄, therefore, not discussed herein for brevity. These results indicate that the 2-story building is more vulnerable to flow-type landslides than the 3- and 4-story buildings. This is because its columns are expected to have a lower load-carrying capacity than those in the 3- and 4-story buildings, as can be inferred from the cross-section sizes and reinforcement details provided in Annexure A. Further, all buildings are subjected to the same one-story debris height, which directly impacts about 50% of the 2-story building’s height, but only about 33% and 25% of the 3- and 4-story buildings, respectively. Therefore, as expected, the 2-story building exhibits the highest IDR, as evident from the variation shown in Figure 8. The higher IDR values indicate increased lateral deformation, leading to greater structural damage in the 2-story building. As a result, the 2-story building exhibits a higher probability of failure compared to the 3- and 4-story buildings. This observation of low-rise buildings being more vulnerable to landslides aligns with the observations made by Singh et al. (2019). The developed fragility curves are further utilized in assessing the expected functional recovery of RC buildings, as discussed in the subsequent subsection.

Figure 8.

Variation of maximum IDR (%) with respect to the height of the considered buildings under a landslide scenario at IM level, $h v$ = 19.8 m²/s.

Assessment of expected functional recovery for archetype RC buildings

The quantification of expected functional recovery comprises two parts: (a) loss model, computed in terms of fragility curves, and (b) quantifying the recovery time for a given damage scenario through damage-state-specific recovery functions. In this section, expected functional recovery curves are developed for 2-, 3-, and 4-story RC buildings subjected to flow-type landslides by using equation (4). Equation (4) can further be expanded as follows to calculate the expected functional recovery of any building under consideration:

Q_{L F} (t | I M = {i m}_{i}) = Q_{r} ({t | D S}_{0}) . P ({D S}_{0} | I M = {i m}_{i}) + Q_{r} ({t | D S}_{1}) . P ({D S}_{1} | I M = {i m}_{i}) + Q_{r} (t | {D S}_{2}) . P ({D S}_{2} | I M = {i m}_{i}) + Q_{r} ({t | D S}_{3}) . P ({D S}_{3} | I M = {i m}_{i}) + Q_{r} ({t | D S}_{4}) . P ({D S}_{4} | I M = {i m}_{i})

(7)

The values of $Q_{r} ({t | D S}_{j})$ in equation (4) required to compute $Q_{L F} (t | I M = {i m}_{i})$ can be taken from Figure 5, whereas the value of $Q_{r} ({t | D S}_{0})$ is one. The values of $P ({D S}_{j} | I M = {i m}_{i})$ for different damage states can be calculated as follows:

P ({D S}_{0} | I M = {i m}_{i}) = 1 - P (E D P \geq {e d p}_{1} | I M = {i m}_{i})

(8)

P ({D S}_{1} | I M = {i m}_{i}) = P (E D P \geq {e d p}_{1} | I M = {i m}_{i}) - P (E D P \geq {e d p}_{2} | I M = {i m}_{i})

(9)

P ({D S}_{2} | I M = {i m}_{i}) = P (E D P \geq {e d p}_{2} | I M = {i m}_{i}) - P (E D P \geq {e d p}_{3} | I M = {i m}_{i})

(10)

P ({D S}_{3} | I M = {i m}_{i}) = P (E D P \geq {e d p}_{3} | I M = {i m}_{i}) - P (E D P \geq {e d p}_{4} | I M = {i m}_{i})

(11)

P ({D S}_{4} | I M = {i m}_{i}) = P (E D P \geq {e d p}_{4} | I M = {i m}_{i})

(12)

Figure 9 presents the variation of expected functional recovery derived from equation (4) at different $I M$ levels, showing the number of recovery days required to restore a building to full functionality after a flow-type landslide event. For comparison purposes, the values of reduction in functionality corresponding to hv level of 19.8 m²/s are also marked in plots. It can be observed that for the 2-story building, at time t = 0, the functionality of the building reduced to 35.67% corresponding to hv level of 19.8 m²/s, while for the 3- and 4-story buildings, it is 62.35% and 67.78%, respectively. This is due to the fact that the 2-story building has a higher probability of attaining any damage state at a given $I M$ level, as shown in Figure 7, resulting in a higher reduction in functionality at lower hv levels.

Figure 9.

Expected functional recovery curves for the considered archetype RC buildings with different IM levels.

It is also worth noting that, as the hv level increases, the reduction in functionality of 4-story RC buildings is comparatively less than that of 2- and 3-story buildings. However, the recovery time required to restore a 4-story building to full functionality is significantly higher than that for its 2- and 3-story counterparts. For example, for 2-story building, the value of $Q_{L F} (t | I M = 19.8)$ is 35.67% at $t = 0$ and it would take approximately 79 days to reach 90% of the expected functional recovery. Whereas in the case of 4-story building, the value of $Q_{L F} (t | I M = 19.8)$ is 67.78% at $t = 0$ and it would take approximately 45 days to reach 90% of the expected functional recovery. Therefore, it can be concluded that the 2-story buildings require a longer recovery time at lower hv levels due to their higher vulnerability to flow-type landslides. However, at higher hv levels, they require a shorter recovery time compared to their 3- and 4-story counterparts owing to lesser reconstruction time. For example, for 2-story building, the value of $Q_{L F} (t | I M = 33)$ is 3.29% at $t = 0$ and it would take approximately 113 days to reach 90% of the expected functional recovery. Whereas in the case of 4-story building, the value of $Q_{L F} (t | I M = 33)$ is 30.95% at $t = 0$ and it would take approximately 151 days to reach 90% of the expected functional recovery.

Effect of community preparedness on expected functional recovery of archetype RC buildings

The expected functional recovery of a building depends on the level of community preparedness, which is represented through damage-state-specific recovery functions, as mentioned earlier. Therefore, the effect of community preparedness on landslide resilience is evaluated by comparing the expected functional recovery of considered archetype RC buildings. In this regard, the two additional recovery functions, namely, linear and trigonometric, discussed in the first step of methodology, are considered to develop damage-state-specific linear and trigonometric recovery functions. These damage-state-specific recovery functions are developed using equation (1) and (2) based on data provided in Table 1, and are presented in Figures 10 and 11, respectively. Further, these functions are subsequently used to develop the expected functional recovery curves using equation (4) for flow-type landslides.

Figure 10.

Damage-state-specific linear recovery function for the considered archetype RC buildings with different damage states.

Figure 11.

Damage-state-specific trigonometric recovery function for the considered archetype RC buildings with different damage states.

Figure 12 presents the variation of expected functional recovery computed using linear and trigonometric damage-state-specific recovery functions at different $I M$ levels. For comparison purposes, the time required to restore the building to 90% expected functionality after the landslide with hv level of 19.8 m²/s is also marked in plots. It can be observed from Figure 12(a) that the 2-story building would take 208 days to achieve 90% expected functionality when the functional recovery of the building is assumed to be linear. Whereas, for communities where the initial response to a landslide event is very slow but the recovery process accelerates once the community becomes organized possibly with help from neighboring communities (represented through trigonometric recovery function), 2-story building would take 184 days to reach the same level of functionality. In the case of well-prepared communities, the 2-story building would take 79 days to achieve 90% expected functionality, as shown in Figure 9. Similarly, the time required to achieve 90% expected functionality for 3- and 4-story buildings is significantly lower for well-prepared communities, as evident from Figures 9 and 12. This highlights the importance of selecting an appropriate recovery function for buildings, as it not only aids decision-makers in planning rescue and recovery actions but also facilitates better allocation of resources, thereby reducing downtime losses.

Figure 12.

Expected functional recovery curves developed using linear and trigonometric recovery functions for the considered archetype RC buildings with different IM levels.

Landslide resilience assessment of archetype RC buildings

The landslide resilience of the archetype RC buildings is quantitatively compared using the landslide resilience index ( ${L R}_{i n d e x}$ ), calculated using equation (5). The variation of ${L R}_{i n d e x}$ for the considered archetype RC buildings with different $I M$ levels is presented in Figure 13. The values of ${L R}_{i n d e x}$ corresponding to hv level of 19.8 m²/s are also marked for comparison purposes. It can be observed that the value of ${L R}_{i n d e x}$ at hv level of 19.8 m²/s is 0.90 for the 2-story building, while for the 3- and 4-story buildings, it is 0.94 and 0.97, respectively, in case of exponential functional recovery. Similarly, in the case of the linear and trigonometric recovery functions, the landslide resilience of the 2-story building is found to be lower than that of 3- and 4-story buildings. However, the ${L R}_{i n d e x}$ values for a given IM level are lower than those observed in the case of exponential functional recovery. For example, the value of ${L R}_{i n d e x}$ decreases from 0.90 to 0.74 when the functional recovery of the 2-story building is assumed to be linear or trigonometric. This indicates that, as expected, a building situated in a well-prepared community demonstrates higher landslide resilience.

Figure 13.

Landslide resilience index for the considered archetype RC buildings subjected to flow-type landslide with different IM levels.

It can also be observed from Figure 13 that the ${L R}_{i n d e x}$ values computed for both the linear and trigonometric functional recovery functions are identical for the considered archetype RC buildings. This is because the total area under the curve corresponding to 100% functional recovery is the same for both functions, despite the different recovery paths, ultimately resulting in identical ${L R}_{i n d e x}$ values. This observation highlights the limitation of relying solely on a single resilience metric and emphasizes the importance of visualizing recovery trajectories to gain proper insight into the post-hazard performance of buildings.

Conclusion

This study presents a methodology for assessing the landslide resilience of RC buildings subjected to flow-type landslides. The region-specific recovery functions are developed based on inputs from practicing professionals, thereby accounting for variations in construction materials and practices. The applicability of the proposed methodology is demonstrated through a detailed numerical study. In this regard, the fragility curves of archetype RC buildings typically observed in the Indian Himalayan Region are developed while considering the uncertainty in the input parameters related to landslide-induced forces. The landslide resilience assessment is then performed for buildings situated in communities with different levels of preparedness against flow-type landslides. The following conclusions are drawn from the present study:

1. The proposed recovery functions can be used to quantitatively assess the expected functional recovery of RC buildings subjected to flow-type landslides, which will be helpful for decision-makers to take action promptly to start the rescue and recovery process.

2. 2-Story RC buildings are observed to be more vulnerable to landslide damage than their 3- and 4-story counterparts.

3. 2-Story RC buildings require a longer recovery time at lower hv levels due to their higher vulnerability to flow-type landslides. However, at higher hv levels, a shorter recovery time is observed compared to their 3- and 4-story counterparts owing to lesser reconstruction time.

4. The buildings situated in a well-prepared community exhibit higher landslide resilience compared to those situated in a community that is not well-prepared for landslides.

The present study is limited to RC buildings; therefore, further studies can be carried out for other building typologies prevalent in the Indian Himalayan Region. In addition, the effect of soil-structure interaction on the landslide fragility of the RC buildings can also be investigated which has not been considered in this study.

Footnotes

ORCID iDs

Rahul Kumar

Mahipal Kulariya

Sandip Kumar Saha

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.*

Appendix

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