Seismic fragility analysis of coupled building‐slope systems

Buildings located on hill slopes are subjected to additional effects under seismic action compared with those on flat ground. This article presents Incremental Dynamic Analyses of coupled building‐slope systems incorporating topographic amplification, nonlinear slope movement and irregular structural configuration of hill‐side buildings. Two different stable slopes, supporting single and multiple adjacent ductile moment resisting reinforced concrete frame buildings with step‐back configuration, prevalent in Indian and Chinese hilly regions, are modelled and analyzed using the finite element software ABAQUS. A suite of 21 ground motions is incrementally scaled and used to compute the seismic response of the coupled system. Fragility curves for different damage states, derived as lognormal probability distributions in terms of three different Intensity Measures (IMs) and damage probabilities of the considered buildings, are derived and compared. The spectrum‐based IMs (e.g., spectral acceleration at fundamental period, and average of the spectral acceleration over a period range) result in lower record‐to‐record variability, compared with the peak ground acceleration as the IM. The results reveal that ignoring the effect of slope significantly underestimates the fragility for all the damage states.


NOVELTY
1. Incremental Dynamic Analyses of coupled building-slope systems 2. Consideration of topographic amplification, slope displacement, and irregular structural configuration 3. Development of fragility curves for different damage states for RC setback structures on slopes accounting for effects in 2. 4. Assessment of different IMs regarding record-to-record variability.
The governing parameters for the seismic vulnerability of buildings located on flat ground include structural type and configuration, material properties, and level of sophistication in design and detailing (commonly termed as 'design code level'). 10,11][17][18][19][20] However, such studies for buildings located on slopes are not available.In case of sloping terrain, the ground shaking causes permanent displacement or failure of slopes, [21][22][23][24] which result in increased fragility of buildings located on slopes.
6][27][28] It is reported in the past studies 29 that the irregular structural configuration alone can cause a significant increase in the collapse probability of hill-side buildings.The foundation typology and supporting ground conditions (including stratigraphy and topography) are additional important parameters, which govern the seismic vulnerability of buildings located on slope.Only a few studies are available on seismic fragility for buildings located on slopes.Most of the past studies have considered only the effect of the irregular configurations of the hill-side buildings, ignoring slope-building interaction (SBI), 26,[29][30][31][32][33][34][35] whereas others [36][37][38][39] have considered the topographic effect and nonlinear slope movement but have ignored the effect of irregular structural configuration.Fotopoulou, Pitilakis 36 used a two-step uncoupled approach to assess the vulnerability of low-rise (single storey, single bay) reinforced concrete (RC) frame buildings supported on isolated and continuous slab foundation.The buildings have been located on the crest of either cohesive or purely frictional soil slopes subjected to seismically induced slow-moving earth-slides.In this approach, the vertical settlements at the locations of column foundations have in the first step been estimated using a 2D nonlinear dynamic analysis of the slope.In the second step, the estimated maximum vertical displacements have been applied to the foundations of the building to determine the stresses in the structural system.The fragility curves have been developed considering different structural limit states defined in terms of threshold values of strains in concrete and steel.Fotopoulou, Pitilakis 37 have used the same uncoupled approach to conduct an extensive numerical analysis and have developed fragility curves for 'low-rise' (up to two storeys) RC buildings located on top of slopes.They have considered the effect of various governing parameters, viz.water level, strain softening of soil material, typology and flexibility of the foundation system, number of bays and storeys and the design code level.They have also considered different idealized slope configurations, soil and geological material properties, and distance of the structure from the slope's crest.
Fotopoulou, Pitilakis 38 have further extended the two-step uncoupled framework to consider the combined effect of ground shaking and earthquake-induced landslides on vulnerability of RC frame buildings located on the crest of slopes.They have considered the peak ground acceleration, PGA, at rock outcrop as the intensity measure (IM) to develop the fragility curves.They have studied the combined effect of ground shaking and seismically induced landslide with and without considering the interaction between the two hazards.Fatahi et al. 40 have conducted a numerical study on seismic performance of a building located close to the crest of a shallow slope, using three-dimensional finite-element simulations, and have highlighted the importance of slope-foundation-structure interaction.Wang et al. 41 have conducted numerical analysis and shaking table model test on a 'high-steep' loess slope with a building load.They have highlighted the typical failure characteristics of slope; and have recommended that a multi-story building should be located at least 0.7 times the slope height away from the crest.Some other researchers [42][43][44][45] have also considered the topography-soil-structure interaction (TSSI) effect and highlighted that the lateral response of a building located near a slope, can be significantly influenced by the TSSI.Sucasaca and Sáez 46 conducted numerical study on dynamic behavior of shear-wall buildings located on a coastal scarp.They have shown that the structure-soil-structure scenario has practically negligible additional effects with respect to soil-structure interaction.Recently, Bahuguna and Firoj 47 performed finite element analyses to evaluate the seismic response of a mid-rise building placed at various locations on slope consisting of linear and nonlinear homogenous soil strata.They found that the buildings located on the face of slope are more vulnerable to rocking failure.
The review of the available literature indicates that the past studies have not dealt with the effect of the integral (coupled) action of the building-slope system on seismic fragility of buildings located on the face of slope.The present study aims at developing a numerical approach for computation of seismic fragility curves for coupled building-slope systems consisting of single or multiple adjacent RC buildings located on the face of stable slopes.By stable slopes, it is meant slopes that could experience nonlinear movements but not collapsing due to soil instability.The considered building models are representative of the structural configurations prevailing in Indian and Chinese hilly regions and have been designed for the current Indian seismic design codes, which are at par with the other national codes, but do not consider the topographic effects.The developed approach combines all the consequences of seismic action on the building-slope system, including ground shaking, topographic effect, slope movements, SBI and irregular structural configuration of building.To this end, an extensive numerical investigation has been performed using Incremental Dynamic Analysis (IDA) 48,49 for two different idealized slope geometries with different angles of inclination and soil types but similar levels of slope stability.Models of one single and three adjacent two-storey RC buildings supported on strip foundations have been constructed on the face of the slopes and the coupled building-slope systems have been subjected to a suite of recorded ground motions.Fragility curves, in terms of three different IMs, have been developed for the coupled building-slope systems and relative suitability of different IMs is discussed.The fragility curves obtained for the coupled slope-building system are compared with their counterparts for the building structures alone, that is, without considering the effects of topography and nonlinear slope movements.

SEISMIC FRAGILITY ANALYSIS
Two procedures, namely, IDA and Multiple Stripes Analysis (MSA), are commonly used for nonlinear dynamic analysis of structures for estimation of fragility functions. 50In IDA, a suite of ground motions is iteratively scaled to achieve the level of Intensity Measure (IM) corresponding to collapse of the structure. 48,49,51On the other hand, in MSA, the dynamic analysis is performed for different levels (stripes) of IM using different suites of ground motions [52][53][54][55] consistent with the corresponding hazard level at the site.The use of hazard consistent ground motions makes the MSA a more robust and reliable method for fragility analysis.However, lack of adequate number of ground motion records, consistent with different levels of intensity, poses difficulty in practical application of this method.
On the other hand, the IDA is a practically simple approach and provides continuity in the response from elastic to collapse level.However, as the ground motion records are not consistent with all specific hazard levels, hence scaling of ground motions is used, IDA may result in some bias in the estimated capacity.In the present study, IDA method is adopted for its relative simplicity.The bias in the results has been minimized by use of an advanced Intensity Measure, namely, S a,avg which is geometric mean of the spectral acceleration ordinates over a period range 0.2T to 3T at an interval of 0.01 s, where T is the fundamental period of the structure.Further, the upper-bound scaling factors on the ground motion records are kept within 5. To control the scaling factor, selection of ground motions is made based on closeness of their response spectra with that of the Maximum Considered Earthquake (MCE) level, having an average return period of 2475 years as the buildings designed according to modern codes are expected to collapse for seismic intensities higher than MCE hazard level.
The proposed procedure for fragility analysis of coupled building-slope system is described in Figure 1.This procedure is motivated by the methods suggested by FEMA 51 and Fotopoulou and Pitilakis 38 for seismic fragility analysis of RC frame buildings located on flat ground and for buildings located near the crest of slopes, respectively.However, these studies did not consider the effect of building location on the face of slope and SBI.Therefore, in the present study, the effects of both the location of buildings and SBI have been explored in detail using IDA.

FINITE ELEMENT MODELLING
The inelastic dynamic response of the coupled building-slope systems has been simulated using the FE software package ABAQUS. 56The FE analyses have been conducted considering three stages of loading.In the first stage, the in-situ stresses in the soil and the underlying rock mass, due to self-weight, are generated for the free slope (i.e., without buildings) using a geostatic analysis.In the second stage, static analysis of the coupled building-slope system is conducted for the added gravity loads of the building.In the last stage, nonlinear time history analyses are performed on the coupled building-slope system using the Hilber-Hughes-Taylor dynamic implicit solution.Shear wave velocity, V s (m/s) 500 500

Modelling of slopes
In the present study two stable slopes of equal height, H = 40 m, having 20 • and 30 • angles of inclination, and consisting of stiff clay and dense sand materials, respectively, have been considered.Table 1 presents the material properties (adopted from Fotopoulou and Pitilakis 37 ) and Figure 2 shows the geometric details of the considered slopes.In the present study, the stiff clay has been considered as non-dilatant, whereas for the dense sand, a dilatancy angle of 14 • has been considered.The material properties of the slopes have been selected consistent with the slope geometry (inclination and height) to result in reasonable and close values for the static factor of safety equal to 2.3 and 2.0 for 20 • and 30 • slope, respectively, and the critical acceleration), k h = 0.36 g.Critical acceleration is the horizontal seismic coefficient at which the slope becomes unstable.
To minimize the boundary effects in the finite element models, the length of the 2D FE model has been taken at 1200 m following a sensitivity study.The model is 160 m deep on the upslope side (Figure 2).In the total 160 m depth of the model, the top 90 m consists of either stiff clay or dense sand, and the remaining depth has been considered as elastic bedrock with density, ρ = 2300 kg/m 3 ; Poisson's ratio, ν = 0.30; Young's modulus = 4.32 GPa and shear wave velocity, V s = 850 m/s.These parameters are also the same as those used by Fotopoulou and Pitilakis. 38Both soil types have been modelled using an elastoplastic constitutive model with the simple Mohr-Coulomb failure criterion, assuming non-associated flow rule. 58The Mohr-Coulomb material model is considered adequate for identification of the parameters influencing the dynamic response of slopes, [36][37][38] and the results obtained here can serve as reference for more complex soil material models.
The rock and soil mass have been discretized using four-node bilinear plane strain quadrilateral elements (CPE4R) with reduced integration and three-node linear plane strain triangle elements (CPE3) as shown in Figure 2. In order to avoid the numerical distortion of frequency content in simulation of wave propagation, mesh size smaller than 1/10 th of the smallest wavelength of interest, corresponding to 10 Hz frequency in the present study, as per ASCE 4−16 59 guidelines, has been used for rock and soil mass away from the slope.The mesh size near the slope face close to the building foundations, has been further refined to capture the displacement response of the system more accurately.Element sizes varying between 0.25 and 1.0 m have been adopted at the slope face, which increase to 5 m at the model boundaries.The suitability of the selected mesh has been confirmed by a mesh sensitivity study.To simulate the free-field conditions at the lateral boundaries, the methodology proposed by Kourkoulis et al. 60 has been used by constraining two mirror image models in lateral direction at vertical boundaries.The base of FE model has been restraint in vertical direction and the ground motions, de-convoluted at the model-base level, have been applied in horizontal direction.(De-convolution is the process 61,62 of obtaining equivalent motion at any level below the ground surface, which results in the considered motion at the ground level in free-field condition).In the present study, the recorded acceleration ground motions have been deconvoluted through a rock column of 160 m (depth of model from the slope crest) and within motion has been applied on rigid base model.Rayleigh damping has been used to account for the energy dissipation during the elastic response. 63t is noted that the dynamic response is sensitive to the assumed damping value and the corresponding frequencies used in the Rayleigh damping model. 64Rayleigh damping parameters have been estimated for soil and rock mass assuming the damping ratios of 3% and 0.5%, respectively. 38The frequencies 1.1 and 8.0 Hz have been selected for assigning the Rayleigh damping for the coupled system.This provides a frequency range that covers all the modes of the coupled system contributing to 90% modal mass participation, as well as the predominant frequency range of the ground motions evaluated from the Fourier Spectra.

Modelling of buildings
In hilly regions, the building typology is not only governed by the structural system and material of construction, but also by the arrangement of foundations on the slope.A survey of the building typologies prevalent in Indian Himalayas was conducted and a classification scheme has been presented by Surana, Singh, Lang 29 for hill-side buildings.They have identified six types of structural configurations: SC A-buildings with foundations at the same level, i.e., buildings located on a flat patch of ground, either existing naturally or made by cutting the hill slope; SC B-buildings with foundations at two different levels; SC C-'step-back' buildings with foundations at multiple levels along the hill slope; SC D -'step-back set-back' buildings, not only having the foundations at multiple levels following the natural slope, but the roof-line also following the slope; SC E-buildings with foundations located at the same level, partly on the natural ground and partly on an artificially created platform by retaining soil using a retaining wall; and SC F-buildings similar to the SC C having foundation at multiple levels, but supported on stilts.7][68] The effective moment of inertia of the beam and column sections has been considered as 35% and 70%, respectively, of the gross section inertia to account for the cracking of RC members as per IS 1893 (Part I) : 2016. 66o avoid shear failure, capacity design approach has been followed by designing the beams and columns for a shear force 1.4 times that corresponding to yielding of beams in flexure.In addition, the strength of the columns has been provided larger than 1.4 times the strength of the beams connecting at a joint 67 to ensure strong column weak beam behaviour.All the columns have been supported by strip foundations, designed using the charts presented in Raj, Singh, Shukla. 69The foundations have been embedded to an average depth of 1.5 m below the slope surface.The fundamental period of the considered buildings in fixed-base condition as computed using modal (Eigen-value) analysis has been found to be 0.82 s for both buildings.Further, the Indian code 66 applies a capping on the design period of the building.This capped design period is based on empirical expressions for estimating approximate period given in the code.Incidentally, the empirical expressions given in the Indian code are similar to those in Eurocode-8 9 , however, Eurocode-8 does not apply any capping and the use of empirical expressions is limited to estimation of approximate period of vibration in absence of analytically obtained period.On the other hand, ASCE 7−16 70 also applies a similar capping with a difference that it allows the design period to be 40−70% longer than the period obtained using empirical expressions.The Indian code, in this respect is more stringent, as it does not allow the design period to be longer than the empirical period.In the present case, the empirical period for the buildings located on the 20 • and 30 • slopes are 0.19 and 0.21 s, respectively, which results in an increase of 2.0 times in the design base shear, as compared to the numerically obtained period.This increase in design base shear results in substantial overstrength.
The beam sizes have been determined as 0.23 m × 0.35 m, so that the reinforcement at each face is in the range of 0.75% to 1.5%.This reinforcement is well within the minimum and maximum limits specified by the relevant codes.For the building on 20 • slope, the short columns, i.e., bottom storey columns on uphill side, require a size of 0.50 m × 0.50 m, due to larger share of shear attracted by them.Whereas all other columns have a size of 0.35 m × 0.35 m.In case of the building on 30 • slope, the short columns have the cross-sectional dimensions 0.60 m × 0.60 m and for all other columns the cross-sectional dimensions are 0.40 m × 0.40 m.The difference in the column sizes is due to the difference in the total building height on the two slopes.The designed buildings have also been checked for the limiting inter-storey drift permitted by the Indian codes. 66n the next step, the same RC buildings have been modelled in ABAQUS.As the computational effort required for incremental nonlinear 3D FE dynamic analysis of coupled slope-building system is formidable, 2D plane frame modelling of the building has been used.All the frames along the slope direction in the considered building, are identical, except the gravity load acting on interior frames being higher than the exterior frame.On the other hand, due to the rigid diaphragm action of the floor slabs the inertia force in the horizontal direction is shared equally by all the frames.Therefore, for the purpose of coupled nonlinear analysis, in the 2D FE model, one interior frame (Frame B in Figure 3) has been modelled with tributary loads on beams and columns and the mass distributed equally as shown in Figure 3.All the building components, namely, the beams, columns, and strip foundations, have been modeled using two-node beam elements with cubic interpolation.FEMA P-695 51 suggests targeted collapse margin ratios and collapse probabilities based on analysis of 2D models, and also provides factors to modify the estimated capacity when 3D models are used.ABAQUS provides only symmetric lumped-plastic hinges for frame elements consisting of homogeneous isotropic materials.These models are not directly applicable for RC elements having different reinforcement at opposite faces.In order to simulate the asymmetric (for reversed hogging-sagging bending) nonlinear behaviour of the RC building components, an equivalent calibrated homogeneous model has been developed. 71In this approach, an equivalent homogeneous cross-section with elastic-perfectly plastic material properties, following Mises or Hill yield surfaces and associated plastic flow rule requiring Young's modulus, Poisson's ratio and yield stress as input, has been used for beams and columns.
The developed approach enables simulation of nonlinear behaviour of each beam and column of the considered RC buildings, following the guidelines of ASCE 41-17. 72To illustrate the efficacy of the proposed calibrated modelling, nonlinear static (pushover) capacity curves of both buildings in fixed-base condition have been compared in Figure 4.In the figure, the capacity curves obtained from the calibrated models have been compared with the results obtained from a lumped plasticity model using ETABS 73 with realistic material properties for concrete and steel and distribution of reinforcement within the section.It can be observed from the figure that the capacity curves obtained from the developed calibrated models are in good agreement with the capacity curves obtained from the lumped plasticity models.The same calibrated models of buildings have been used for further study of the coupled building-slope model in ABAQUS.In the coupled building-slope model, the foundations of the buildings have been embedded in the soil.Rayleigh damping of 5% has been used for the RC building components in accordance with the IS 1893 (Part I) : 2016. 66The Rayleigh damping parameters have been estimated at periods corresponding to the fundamental mode and the mode resulting in 90% cumulative mass participation in lateral direction.

SELECTION AND SCALING OF GROUND MOTIONS
Selection and scaling of ground motions is a crucial issue in reliable assessment of seismic fragility of buildings. 51,746][77][78][79][80][81] Generally, for a fixed-base structure, 5% damped first-mode spectral acceleration, S a,T (ξ = 5%) (here, T = fundamental period of the building, and ξ = critical damping ratio) 82 and 5% damped average spectral acceleration within a period range, S a,avg (0.2T-3T, ξ = 5%) 83 are used as the Intensity Measures (IMs) for performing IDA.However, such studies for coupled building-slope, or even soil-structure, analysis are not available.Pitilakis et al. 18 and Karapetrou et al. 15 have used the PGA, recorded on rock outcrop, that is, excluding site effects, as IM for derivation of fragility curves for the non-linear SSI system, mainly due to its simplicity.On the other hand, Haghollahi and Behnamfar 20 and Tahghighi and Mohammadi 19 discussed using S a,T (ξ = 5%) as IM for fragility analysis of non-linear SSI system.They reported that the suitability of S a,T (ξ = 5%) is highly dependent on appropriate value of fundamental period, T of the coupled system.Hence, in the present study three different IMs, namely, PGA, S a,T and S a,avg have been considered for fragility analysis.In the present study, T is the fundamental period of the building considering the effect of foundation flexibility due to the slope material (soil).This is obtained by performing the free vibration analysis of the building and the slope system in which the slope is modelled as zero density elastic material.This modelling approach considers the effect of soil flexibility on the period of the building, but avoids the vibration/wave propagation within the soil.S a,avg (0.2T-3T, ξ = 5%) has been estimated by taking the geometric mean of spectral accelerations between 0.2T and 3T, at an interval of 0.01 s. 76,77 In the present study, a suite of 21 recorded acceleration time histories without velocity pulse have been selected from PEER NGA-West 2 database (https://ngawest2.berkeley.edu/).All the selected ground motions have been recorded at sites with average shear wave velocity in the upper 30 m, V s30 > 600 m/s.The moment magnitude, M w , and epicentral distance, R jb , of the selected ground motions range between 6.19 < M w < 7.62 and 0 < R jb < 10 km, respectively.Figure 5 shows the response spectra of the selected ground motions and their comparison with the design response spectrum for hard soil in IS 1893 (Part I) : 2016 66 in Seismic Zone IV having MCE with PGA = 0.36 g.The ground motions have been selected from the PEER NGA-West 2 database to have the spectral shape as close to the MCE spectrum of the design code as possible, and at the same time to result in collapse of the considered buildings with an upper-bound scaling factor of 5.It can be seen from Figure 5 that the median response spectrum of the chosen ground motion suite is close to the considered MCE spectrum.
The vertical component of ground motion can be important, especially in structures having long spans or heavy gravity loads, and in case of those located in near field. 72In case of a hilly terrain, the vertical and horizontal components of ground motion can undergo a complex interaction due to the geometry of the slope.The relative scaling of horizontal and vertical components is another challenging issue.17][18][19][20]84 The selected ground motions have been de-convoluted to a bedrock depth of 160 m with the considered properties of rock mass using DEEPSOIL software. 85The de-convoluted motions have been applied at the bedrock level (i.e. at the base of FE model).

SEISMIC FRAGILITY OF COUPLED BUILDING-SLOPE SYSTEM
To perform IDA, the selected ground motions have been incrementally scaled and applied at the base of the developed FE models.The scaling factors have been iteratively chosen to give adequate number of points on the capacity (IDA) curve before collapse and the collapse capacity is obtained as precisely as possible.Twenty-one IDA curves have been obtained for each building-slope combination, requiring more than 2600 nonlinear time-history analyses.Description and quantification of damage states, in terms of engineering demand parameter (EDP), is another important issue in development of fragility curves.In the present study, the maximum inter-storey drift ratio (IDR max , %), which is a comprehensive and most commonly used global performance parameter to quantify dynamic instability and structural damage in frame buildings, [86][87][88] has been selected as EDP.In the conventional IDA for fixed-base buildings, 48 the 'collapse' damage state can be identified from the IDA curves, because it corresponds to the value of IM beyond which a very small increment in IM leads to very large increment in the EDP.However, due to the limitation of the elasto-plastic material model used in the present study and the increasing inelastic energy dissipation in the slope material with increasing IM, the collapse damage state cannot be predicted from the shape of IDA curves (Figure 6).Therefore, the four damage states based on IDR max as proposed by Ghobarah 89 and adopted by Fotopoulou, Pitilakis 38 have been used for the derivation of fragility curves.A description of these damage states is given in Table 2.It is noted that there is no consensus on the threshold values of IDR max corresponding to different damage states, particularly about DS 4 .However, the objective of the present study is to compare the fragility of the hill-side buildings with and without considering the effect of slope.Therefore, a reasonable and consistent definition of different damage state thresholds should serve the purpose.Further, as stated earlier, the beams and columns have been designed following the capacity design method to avoid the shear failure.Therefore, the shear failure mode has not been explicitly modelled in the present study, however, the maximum shear force developed in the columns during the time history analysis, has been checked against the provided capacity to identify shear failure of columns.In the past, researchers have suggested different approaches to develop the probabilistic relationship between IM and EDP pairs derived from nonlinear dynamic analyses of structures. 90,91The prime objective of these approaches is to determine the median IM and associated record-to-record dispersion, β D corresponding to each damage state threshold.In the present study, the method of Least Square (LS) regression has been used to estimate the median and dispersion corresponding to each damage state.This method has been reported to be an efficient way to develop a strong relationship between IM and EDP using a limited number of observations. 88,92,93In this method, a power law (Equation 1) or a linear relationship between the logarithms of the IM (PGA, S a,T and S a,avg ) and the EDP (IDR max , %) (Equation 2), is fitted to the obtained data points to estimate the median value.The regression is performed in a band of EDP around the damage state under where, a and b are the coefficients obtained from regression analysis by maximizing the goodness of fit (R 2 ) between IM and EDP.It has been observed that the goodness of fit (R 2 ) for the spectrum-based IMs (S a,T and S a,avg ) is higher than that for the PGA, in all the considered cases.The median, IM i and associated dispersion parameter, β Di for each damage state, DS i , are obtained from the IM-EDP pairs bounded by the lower and upper damage state thresholds and corresponding standard deviation, β εi , as: Table 3 summarizes the median values, IM and dispersion parameter β D corresponding to different damage states and the chosen IMs.The variability in S a,T has been studied for three different definitions of the period, T: (i) fundamental period of the building in fixed-base condition ( = 0.82 s), i.e., without considering the effect of slope; (ii) fundamental period of the building with only flexibility of the soil being considered ( = 0.86 and 0.88 s for 20 • and 30 • slopes, respectively), i.e., only the stiffness of soil has been modelled but the mass of the soil is ignored; and (iii) fundamental period of the combined building-slope system ( = 1.14 s for both slopes), i.e., considering the stiffness and mass of the building as well as those of the slope material.
It has been observed (results not presented for brevity) that the variability in S a,T is minimum, in case of the second definition of the fundamental period, T. The results (IM and β D ) corresponding to this definition of the fundamental period only have been presented in Table 3.It can be observed from Table 3 that the record-to-record variability associated with damage states DS 1 and DS 2 is minimum for S a,T , whereas in case of the damage state DS 4 , the variability is minimum for S a,avg , in all of the cases.On the other hand, in case of DS 3 the record-to-record variability is minimum either for S a,T or for S a,avg .This trend can be explained by considering the fact that ductility demand is low in the case of the lower damage states DS 1 and DS 2 , whereas it is maximum in the case of DS 4 .Consequently, the period elongation due to the reduction of the effective stiffness, is maximum in DS 4 .Since, S a,avg represents a wider range of period on both sides of the elastic fundamental period, it results in the least variability in DS 4 .Further, the buildings being low-rise, the contribution of higher modes is not significant; therefore, S a,T , which depends only on the fundamental period, results in the minimum variability in case of damage states DS 1 and DS 2 .This observation is also in agreement with that in a similar study by Surana et al. 30 on fixed-base hill-side buildings.
Using the obtained IM and β D for different damage states, the fragility curves have been obtained as log-normal distribution function (Equation 5) 94 : where, P[DS ≥ DS i |IM]denotes the probability of being at or exceeding a particular damage state, DS i , for a given seismic intensity measure, IM; Φ is the normal cumulative probability distribution function; IM i is the median value of the seismic intensity measure corresponding to the i th damage state threshold; and β i is the total variability.The total variability, β i has been obtained by the combination of the three contributors, assuming them to be stochastically independent and log-normally distributed (Equation 6).
where, β DSi is the uncertainty in defining the i th damage state thresholds; β Ci is the uncertainty in the estimation of response and resistance (capacity) of the structure due to variability in structure properties (design, construction material and construction practices); and β Di is the uncertainty in demand, given by Equation 4.  The reliable estimates of uncertainty in the capacity of RC buildings pertaining to Indian codes are not available and an extensive computational effort is required to estimate the uncertainty in capacity.Hence, in the present study, the values of β DS and β C have been adopted as 0.40 and 0.30, respectively, from the previous studies 34,51,89 for all damage states.The fragility curves, developed in terms of the three IMs, are presented in Figures 7, 8.
Figure 7 illustrates the effect of slope inclination on the fragility of the considered buildings.To highlight the effect of slope, fragility analysis has also been performed considering the same RC buildings assumed to have fixed-base, i.e., without considering the effect of slope movement and topographic effects.The figures in the left column (Figure 7A,C and  E) show the fragility curves of the fixed-base buildings.It can be observed from the figures that despite some differences in the elevations of the buildings located on 20 • and 30 • slopes, the fragility curves of the two buildings are relatively close, when these are considered to have fixed-base.This is because the fundamental periods of the two buildings are close (0.82 s), and these have been designed for the same seismic base shear coefficient.On the other hand, in the case of the coupled building-slope systems, the fragility for the 30 • slope is higher than that for the 20 • slope, for all the Damage States, as observed from the figures in the right column (Figure 7B,D and F).It is observed in Figure 7A and B that some of the fragility curves for slope inclinations 20 • and 30 • intersect each other.This is due to the relatively higher record-to-record Effect of slope angle on seismic fragility curves of Building 1B, located in middle of slope: (A) fragility curves for fixed-base building with PGA as the IM; (B) fragility curves for coupled building-slope system with PGA as the IM; (C) fragility curves for fixed-base building with, Sa,T as the IM; (D) fragility curves for coupled building-slope system with, Sa,T as the IM; (E) fragility curves for fixed-base building with Sa,avg as IM; (F) fragility curves for coupled building-slope system with Sa,avg as IM.
variability in case of the 20 • slope, as compared to the 30 • slope, for which PGA is used as the intensity measure.This discrepancy is not seen in the case of other intensity measures.
Similarly, Figure 8 illustrates the effect of building location along the slope on the estimated fragility.It can be observed from the figure that the fragility of the building (3B-A), located near the crest of the slope, is higher than the same building located in the middle (3B-B) and near the toe (3B-C) of the slope.The effect of presence of adjacent buildings on the seismic fragility can be evaluated by comparing the fragility curves of building 3B-B in Figure 8 with those of building 1B in Figure 7.It can be observed that the adjacent buildings do not have any significant influence on the fragility of a building located on a reasonably stable slope for conditions similar to those considered here.The observations can be explained by considering the variation of the intensity of the ground motion at the slope surface and differential movement of the soil below the considered buildings along the slope.Figure 9 shows the variation of the ratio of peak horizontal acceleration on the slope surface, A hs , to the peak horizontal acceleration recorded at rock outcrop, A hr , for EQ-5 (Kobe, Japan (1995) earthquake with PGA = 0.60 g) along the slope.As expected, and consistent with previous studies, 21,[95][96][97][98] the acceleration of the soil mass increases towards the crest.As reported in these studies, slight de-amplification of peak horizontal acceleration near the toe is also expected.Further, the amplification and the de-amplification near the toe is higher in the case of the 30 • slope, as observed in previous studies. 96 I G U R E 9 Variation of ratio of peak horizontal acceleration on slope surface, A hs , to peak horizontal acceleration at rock outcrop, A hr , due to EQ-2 (Kobe, Japan (1995) earthquake with PGA = 0.60 g) along normalized slope height y/H, where y is height of a point on slope from its toe, and H is total height of slope.

TA B L E 4
Probability of exceedance of different damage states at MCE for the considered RC buildings located on slope, obtained from fragility functions developed in terms of S a,T .Figure 10 shows the variation of total residual displacement, U in the two slopes due to the same EQ-6 earthquake ground motion (Northridge-01, USA (1994) with PGA = 0.60 g at rock outcrop).It can be observed that U is larger near the surface of the slopes, the movement is even more confined to the near surface zone in case of the 30 • slope, consisting of dense sand material.The figure shows that the displacement increases towards the toe of the slope.However, it should be noted that it is the differential (relative) displacement, ΔU, not the total displacement, that causes damage in the building.It has been found (not shown here for brevity) that the differential displacement in vertical direction, ΔU v is negligible in comparison to that in the horizontal ΔU h . 71Therefore, Figure 11 presents the variation of maximum differential horizontal movement, ΔU h,max below the three buildings with the intensity of shaking defined by PGA at rock outcrop.The differential displacement has been considered as the relative displacement between the outermost foundations.Interestingly, the differential displacement has a trend opposite to that of the total displacement.As a result, ΔU h,max has been observed to be the highest at the base of the building 3B-A (located near the top) in comparison with the buildings 3B-B and 3B-C for both slopes.This is in good agreement with the previous studies. 38,71Further, this trend is clearer in the 30 • slope, right from the lower intensities of shaking.The combined effect of the intensity of shaking (average PGA) and differential movement of foundations, ΔU h,max govern the extent of damage in a building, and the observed trends explain the higher fragility of the building 3B-A near the crest.
To elucidate the effect of slope on fragility of hill-side buildings, the fragility curves of the buildings considered on fixed-base, have been compared (Figure 12, reproduced from Figure 7) with those corresponding to the same building (1B) assumed to be located in the middle of the two slopes.The increase in the cumulative probability of damage exceeding a particular damage state is evident from the figure.It can be observed that by ignoring the effect of slope, one may significantly underestimate the probabilities of exceedance of damage states for both building-slope systems considered.Similarly, Table 4 presents the cumulative probability of exceeding different damage states when subjected to the MCE level ground shaking.For a fair comparison, the probabilities of exceeding different damage states have been obtained from the fragility functions developed in terms of S a,T , as these have minimum variability for most of the damage states.ASCE 7−16 70 targets a collapse probability of the designed buildings less than 10%, conditioned on the occurrence of MCE.It can be observed from the table that the considered buildings have probability of collapse almost in acceptable range when considered to have fixed-base.This observation is in good agreement with the past studies 30,99 on hill-side Comparison of fragility curves of coupled building-slope system with fragility curves of same building considered on fixed-base.Results are for Building 1B located in middle of slope.
buildings ignoring the effect of slope.However, when the buildings are located near the crest of the slopes (3B-A), the probability of collapse is unacceptably high.This increase in the collapse probability is reduced when the same buildings are located near the toe of the slope.Similar observation can be made about the other damage states (DS 2 and DS 3 ) from the values in the table.

CONCLUSION
Results of an extensive numerical investigation, using more than 2600 Incremental Dynamic Analyses (IDA) of two different idealized slope geometries with different soil properties, different RC buildings and locations on the slope, were presented.The buildings have been supported by strip foundations on the face of slopes.The analysis framework used in the study has considered all the consequences of seismic action on the building-slope system, including ground shaking, topographic effect, ground movements, and slope-building interaction.Fragility curves, in terms of three different IMs [PGA, S a,T (ξ = 5%) and S a,avg (0.2T-3T, ξ = 5%)] have been developed for the coupled building-slope systems.To illustrate the effect of slope, the fragility curves of the coupled building-slope systems have been compared with the corresponding buildings assumed to have fixed-base.It has been found that in all the considered cases the spectrum-based IMs (S a,T and S a,avg ) result in lower record-torecord variability (β D ) than to PGA for all the damage states.In case of lower damage states DS 1 and DS 2 , the spectral acceleration, S a,T corresponding to the fundamental period of building, including the effect of the flexibility of the slope material, results in the lowest record-to-record variability.This can be attributed to two facts: (i) in case of the low-rise buildings considered in the present study, the contributions of higher modes are not significant, and (ii) the period elongation due to inelasticity is small for lower damage states because of the low ductility demand at these damage states.As a result, the relevant intensity of ground shaking is better represented by spectral acceleration at the fundamental period.On the other hand, as the period elongates significantly due to high ductility demand in case of the collapse damage state (DS 4 ), the variability is minimum for S a,avg , which takes into consideration a range of period around the fundamental period.In case of damage state DS 3 , the record-to-record variability is minimum either for S a,T or for S a,avg .
The results have illustrated that ignoring the effect of slope-building interaction underestimates the fragility of the considered buildings for all the damage states.The degree of underestimation increases with the slope inclination.The ductile moment resisting frame buildings, having fixed-base, and designed using Indian codes, have shown acceptable (≈10%) collapse probabilities at MCE, however, when the same buildings are located on (reasonably stable) slopes, they have shown unacceptable (up to 36%) collapse probabilities.
For lower damage states (DS 1 and DS 2 ), there is only small variation in the fragility of the RC buildings located on the two different slopes, despite some variation in the building elevations due to change in slope inclination.This can be attributed to the design of the two buildings for identical seismic forces and relatively similar critical yield accelerations for the two slopes.However, for the higher damage states (DS 3 and DS 4 ), the fragility of the buildings located on the 30 • slope is slightly higher than those on 20 • slope.Further, it has also been observed that the fragility of the building located near the crest is higher than the same building located in the middle and near the toe of the slope.It has been observed that the collapse probability of the building near the crest is increased by 35% and 110% in case of 20 • and 30 • slopes, respectively, compared with the same buildings located near the toe of the slope.The highest collapse probability (36%) is for the building located near the crest of the steeper (30 • ) slope.
The present study has been conducted using simplified non-degrading hysteretic models for both slope material and building components.Consideration of degradation in hysteretic models is expected to affect the computed collapse probabilities.However, the relative influence of slope is expected to remain about the same.In the building-slope system, the three-dimensional effects due to geometry of the slope and structural configuration of building may also be important and need further investigation.Further, a more detailed study considering different structural configurations and wider range of slope geometry and material properties, will be useful in developing generic fragility functions for their application in seismic risk studies in hilly regions.However, the observations regarding amplification of ground motion along the slope, comparison with fixed-base (or on flat ground) buildings, and interaction between adjacent buildings, are expected to be valid for other configurations.Additional supporting information on calibration procedure, selected ground motions, least square regression, and uncertainty parameters can be found in the Supporting Information section at the end of this paper.

A C K N O W L E D G E M E N T S
The work presented in this article was partially supported by the Ministry of Education, India (MoE) and Department of Science and Technology, Government of India, under the Scheme Impacting Research, Innovation and Technology IMPRINT II.The support received is gratefully acknowledged.

C O N F L I C T O F I N T E R E S T S TAT E M E N T
The authors declare that they have no conflict of interest.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data can be made available on request to the corresponding or first author if considered reasonable.

F
I G U R E 2 2D FE model in ABAQUS showing single and multiple adjacent two-storey buildings located on face of slope (1B indicates location of single building, 3B-A, 3B-B and 3B-C indicate locations of three adjacent buildings).

F I G U R E 3
Plan and elevations of buildings: (A) Plan; (B) Elevation on 20 • slope; (C) Elevation on 30 • slope.regions of China.25,26,65Surana et al.29 have reported that SC C is the most common building typology in this region, and the same has been chosen in the present study.In the present study, two storeys tall single, denoted by '1B' in Figure2, and multiple buildings, (adjacent three buildings denoted by '3B-A', '3B-B' and '3B-C', in Figure2, are considered.Note that the number of storeys is counted above the top-most foundation level.The buildings having irregular step-back configuration to suit the slope geometry, have been considered to be founded on the face of the slopes.The plan and elevations of the buildings are shown in Figure3.The adjacent buildings have been considered with a 5 m clear horizontal gap.The buildings have been designed in accordance with the relevant Indian standards, 66-68 using the design response spectrum of IS 1893 (Part I) : 2016 66 corresponding to Soil Type I in Seismic Zone IV, and Response Reduction Factor, R = 5, corresponding to special ductile RC moment resisting frame (SMRF) buildings.The design has been performed assuming the buildings to have fixed-base, i.e., assigning fixed supports at the foundations which are located at different elevations due to slope.In this modelling, the irregular geometry of the buildings due to foundations placed at different levels is considered, but slope movement and topographical effects are ignored.The design has been based on a linear dynamic analysis of 3D models considering the irregular distribution of stiffness and mass in the buildings.The beams, columns and strip foundations have been considered to consist of M30 grade concrete, with unit weight = 25 kN/m 3 ; Poisson's ratio = 0.20; Young's modulus, E conc = 27 GPa, and nominal cube compressive strength = 30 MPa.Fe500 grade steel with unit weight = 78.5 kN/m 3 , Poisson's ratio = 0.30, Young's modulus, E steel = 200 GPa and nominal yield strength = 500 MPa has been used.

F I G U R E 4
Comparison of capacity curves obtained from nonlinear static analyses of RC buildings located on: (A) 20 • slope and (B) 30 • slope.

F I G U R E 5
Comparison of response spectra of selected ground motions with design response spectrum of IS 1893 (Part I) : 201666 in Seismic Zone IV, having PGA = 0.36 g at MCE.

F I G U R E 8
Comparison of seismic fragility curves of three buildings in different locations on slope (Building 3B-A is located towards crest of slope, Building 3B-B is located in middle of slope, and Building 3B-C is locate towards toe of slope, with 5 m clear horizontal gap between the three buildings).

F I G U R E 1 0 1
Variation of total residual displacement, U (m) due to EQ-6 (Northridge-01, USA (1994) earthquake with PGA = 0.60 g at rock outcrop for: (A) 20 • slope and (B) 30 • slope.Variation of maximum differential horizontal movement, ΔU h,max between outermost foundations of the three buildings at different locations on: (A) 20 • slope and (B) 30 • slope; with increasing intensity (PGA at rock outcrop) of ground motions. 57

F I G U R E 1
Framework for seismic fragility analysis of buildings located on the face of a slope.DS i , the i th Damage State; EDP, Engineering Demand Parameter; IM, Intensity Measure.

6
88namic capacity (IDA) curves for buildings '1B' located in middle of slope.Description and quantification of damage states adopted in present study for ductile moment resistance frame (MRF) buildings (taken from Ghobarah89).This band is usually taken from one damage state lower to one damage state higher than the considered damage state.Details of this 'piece-wise' regression are available in Cǎrǎuşu, Vulpe93and Gehl, Douglas, Seyedi.88 Fragility functions parameters for different damage states of the considered building-slope systems.
TA B L E 3