Introduction

In the hilly region, road connectivity plays an essential role in the community and financial activities. Landslides along road cause disturbance in traffic, loss of lives and properties. Even stabilized slopes may cause slope failure and collapse due to external loads, such as earthquakes and/or rainfall.

Built structures including roads, dams, electric towers, etc., on hilly terrains and the concentration of population on potentially unstable locations considerably increased the risk pertaining to geotechnical hazards. Landslides in the Himalayan terrain are largely controlled by frequently occurring tectonic activities, heavy rainfall, seismic activity, strength degradation due to weathering and rapid increase in anthropogenic activities like road development, hydroelectric and tunnel projects. Among these factors, earthquake has been known to be one of the major causes of devastating landslides. For example, 1999 Chamoli earthquake induced 56 landslides in the study area and also reactivated the old landslides. Therefore, the seismic parameter is important and has been included along with various other factors while carrying out landslide susceptibility assessment. Many earthquake-induced landslides have been reported due to the 1999 Chamoli earthquake [1, 2]. Damage caused by earthquake-induced landslide is often more significant than the damage related to the shaking of the earthquake itself.

It is crucial to analyze the response of a slope under the seismic condition that falls at an important location. Many methods have been developed to address this issue [3]. Jibson [4] classified these methods into three categories: pseudostatic analysis, permanent displacement analysis and stress–deformation analysis. These categories have their appropriate application for assessing seismic slope stability. Pseudostatic analysis, because of its crude characterization of the physical process, should be used only for preliminary or screening analyses. Stress–deformation analysis is best suited to large earth structures such as dams and embankments; it is too complicated and expensive for routine applications, i.e., for small structures. Permanent displacement analysis provides a valuable middle ground between these two extreme analyses. It is simple to apply and provide information about deformation, which is missing in the pseudostatic analysis.

Newmark [5] introduced a method to assess the performance of slopes during earthquakes that bridges the gap between very simplistic pseudostatic analysis and too complex stress–deformation analysis. Newmark displacement analysis has been applied in a variety of ways to slope stability problems. It has been landslides in natural slopes [6,7,8,9] and for the development of empirical, semi-empirical and predictive models for permanent displacement [10,11,12,13,14].

The major drawback of Newmark’s displacement method is that it evaluates yield acceleration (ay) using a simple function of the static factor of safety (FSStatic) and the landslide geometry. However, the seismic loading applied on the slope is dynamic and the factor of safety (FS) varies with time. So, the present paper uses an approach to find out ay, which is based on dynamic FS (FSDynamic) and depends on the level of excitation. This is the primary novelty of this paper.

Further Newmark method considers the sliding block as a rigid. However, the flexibility of the material is considered in the present study. The effects of earthquake on slope stability using nonlinear strength criterion have been investigated by many researchers [15,16,17]. These studies are based on finite element analysis using a nonlinear shear strength reduction method. However, the permanent displacements have not been computed in these studies. Calculation of earthquake-induced deformations is highly sensitive to stability analyses and thus to the assumption on failure mechanisms. Hence, finite element (FE) analysis provides advantages over other methods and the same has been used in the present study. Here, basic methodology is the same as those presented by [5].

The present study area is part of Indian Himalaya and falls in Chamoli district, Uttarakhand, India, which lies in an active seismic zone V [18]. The area witnessed a major earthquake on March 29, 1999, of magnitude (Mw = 6.8). The earthquake triggered 56 landslides within an area of 226 km2 [2]. Such vast numbers of landslides due to the earthquake indicated the need for some study for this area. Majority of the slope stability studies reported for the study area were limited to static condition only [19,20,21]. Here, in most of the studies, the effect of rainfall is not considered. A significant research gap for seismically active regions is the integration of the seismicity parameter. Sangeeta et al. [22] reported the effect of seismic parameter on landslide susceptibility zonation. The study indicated that parts of the Chamoli district fall in very susceptible class of earthquake-induced landslides.

The main focus of the present study is to evaluate the real hazard that leads to the instability of the terrains in earthquake conditions, especially in areas where there are already noticed instabilities in static conditions. This study provides a better insight into the dynamic analysis for a natural slope failure, using actual strong-motion records.

The primary aim of the present work is to check the stability of a recurring landslide for static and seismic loading assuming dry conditions. The permanent displacement (PD) of a slope using finite element (FE) analysis is evaluated for seismic conditions using linear, equivalent linear and nonlinear models. A comparison of results with those obtained using sliding block and empirical relations is carried out. The variation of PD with the peak ground acceleration (PGA) and pseudostatic factor of safety (FSpseudo) has been evaluated. The effect of nonlinearity on the landslide hazard index (LHI) for different PGA is examined. The curves to predict permanent displacement using different constitutive models have been proposed for the local site condition which would be applicable for a similar range of input data. This paper provides valuable insight into the effect of nonlinearity on the permanent displacement as discussed later. Results obtained from this study can be used as a quantitative guideline for the evaluation of seismic stability of natural slopes subjected to earthquakes with similar conditions and of the same degree of intensity.

Study Area

The present landslide with coordinates (N30º23′37″ & E79º12′4″) is located on a hill slope on the Rudraprayag-Pokhari-Gopeshwar road, 18 km from Pokhari in Chamoli district, Uttarakhand, India (Fig. 1). Figure 1 (c) also indicates the locations of other landslides and earthquakes which occurred during the period 1975–2019. In Fig. 1(c), location of the epicenter of the 1999 Chamoli earthquake is also shown. The present study area falls in very high hazard (VHH) class on earthquake-induced landslide hazard zonation map for Chamoli district, reported by [23].

Fig. 1
figure 1

Location map of the study area: (a) location of Uttarakhand in India; (b) location of Chamoli district in Uttarakhand; (c) location of landslide of present study along with other landslides and earthquakes in Chamoli district

Full size image

The study area falls under seismic Zone V [18] corresponding to zone factor 0.36 and seismic intensity IX (and above) on the MSK intensity scale. A total of 39 seismic events were recorded from 1975 to 2019 with varying moment magnitude (Mw) from 3.9 to 6.8 [24]. Landslide considered in the present study is located within 25 km vicinity from the epicenter of 1999, Chamoli earthquake of magnitude (Mw = 6.8).

The geological condition of the study area consists of Quartzite rock. On the left portion of slide above the road, a thick patch of Sericitc Quartzite is also observed. Stratigraphically the rock type belongs to Berinag unit of rocks. The river Ningol, which is passing below the site, flows parallel to the fault. It separates Quartzite of Berinag unit from Chlorite Schist of Ramgarh group, which is well exposed at the opposite side of the river.

Methodology

One of the chronic landslides has been identified from the VHH zone of Chamoli district [23] for which material and geometrical parameters were available. The selection of this landslide is primarily based on following three criteria:

  1. 1.

    This landslide is falling in VHH zone [23]

  2. 2.

    Similar kind of landslide (with respect to height and material type) in nearby areas.

  3. 3.

    Input data are available for the selected landslide.

The analyses of this representative landslide intend to estimate the probable conditions leading to failure. Therefore, this landslide has been first analyzed in the static condition to determine the likelihood of failure (based on FS) caused in the absence of earthquake shaking. Next, this is analyzed in the seismic conditions to quantify the displacement that would have occurred in earthquake shaking. Also, the simulation with FE analysis was conducted to ascertain the effect of transient excitation on permanent displacement.

As field data were not available, to validate the numerical modeling, the results of FE modeling are compared with those obtained by [25]. The results of validation are discussed in Sect. 4. The methodology of the present work follows in three stages, namely the formulation of numerical modeling, computation and analysis of outputs.

Study has four components. First, the modeling and computation techniques used in the present study are validated by comparing the results from FE analysis with those available in the literature. Second, geotechnical data and slope geometries were compiled to define the numerical modeling of slope identified in the Chamoli region. For this slope, the earthquake-induced permanent displacement of the landslide was systemically assessed using 1999 Chamoli earthquake acceleration time history for various constitutive models, namely linear, equivalent linear and nonlinear. Third, a comparison of results has been made with the sliding block method [5] and two well-established empirical models, i.e., [10 & 11] to examine the efficacy of FE modeling. Finally, the correlation of permanent displacement with landslide hazard index (LHI) has been carried out. The flowchart of this study is presented in Fig. 2. In the present study, FE based software GeoStudio [26] was used for the computation. First, values of average acceleration (aavg) and dynamic factor of safety (FSDynamic) with time are evaluated then permanent displacement is determined using double integration as discussed later.

Fig. 2
figure 2

Flowchart of the study

Full size image

Validation

It is important to validate the numerical modeling and computation technique used in the present study. For this, the results of present FE analysis are compared with those obtained by [25] for the same slope and material properties. In the validation, direction of shaking/movement of earthquake waves has been considered on the stability of slope. For this validation, the model of slope and other parameters as reported by [25] have been used. These parameters are: slope height H = 15 m, slope angle β = 45°, friction angle ϕ = 25°, cohesion c = 15 kPa and unit weight γ = 17 kN/m3. The horizontal components of the Northridge earthquake (recorded on January 17, 1994, at Moorpark station) and Kobe earthquake (recorded on January 17, 1995, at Kakogawa station) have been considered for this validation. For the analysis, linear elastic constitutive model has been adopted. For dynamic analysis, side boundary is restricted for vertical movement and ground is allowed to sway from side to side as the horizontal earthquake accelerations are applied. The bottom boundary is fixed in both x and y directions for static as well as dynamic analyses. Global element size is selected as 0.1 m. The results are given in Table 1 where permanent displacements from both studies are compared. It can be observed that the difference in results from the two studies is less than 1.3% and thus in good agreement. This validates the proposed methodology which is further used for the evaluation of permanent displacement.

Table 1 Validation: comparison of permanent displacements
Full size table

Input Data for Numerical Analysis

In this section, FE modeling of slope and input parameters used for numerical analysis has been described. Static and dynamic analyses are carried out for a chronic landslide situated in Chamoli district, Uttarakhand, India (Fig. 3a). The approximate width of the landslide is 480 m horizontally, and height is 320 m vertically above the road. The heterogeneous soil slope used for the numerical simulation is presented in Fig. 3(b). The slope consists of 2 layers, i.e., a homogeneous-intact rock mass overlain by overburden material.

Fig. 3
figure 3

Landslide site of the present study: (a) Google Earth image of landslide; (b) slope model

Full size image

Static and Pseudostatic Analysis

Table 2 shows the material properties of the slope used for the static and pseudostatic analyses. Here, rock has been considered as homogenous and intact. The analyses were performed, and it was found that the failure mechanism leads to critical slip surface in overburden. This is attributed to the fact that the shear strength of rock is much greater than that of overburden. The effect of vertical seismic coefficient (kv) is neglected [27]. The horizontal seismic coefficient (kh) considered is based on zone factor (Z). As per [18], for seismic zone V, Z is 0.36 and conservatively kh is assumed as 1/2 of Z, i.e., 0.18 [27].

Table 2 Material properties for static and pseudostatic analysis [28]
Full size table

FE Dynamic Analysis

The pseudostatic approach for stability analysis is simple and straight forward, but it cannot simulate the transient dynamic effects of earthquake shaking, because it assumes a constant unidirectional pseudostatic acceleration. However, the finite element dynamic analysis overcomes this limitation. The computational procedure for dynamic analysis is presented. The issues related to FE modeling of the slope such as meshing, boundary conditions, application of the earthquake loading are discussed in brief in the following subsections.

Meshing and Boundary Conditions

In this study, as shown in Fig. 4, the same types of two-dimensional quadrilateral, isoparametric and triangular elements are used to discretize both rock and overburden (Fig. 3). The bottom boundary is fixed in both x and y directions for static as well as for dynamic analyses. For static case, side boundary is fixed in the x-direction, while for dynamic case, side boundary is restricted for vertical movement and ground is allowed to sway from side to side as the horizontal earthquake accelerations are applied. Slope face is kept free both in x and y directions for static as well as dynamic case.

Fig. 4
figure 4

Two-dimensional model of slope used for numerical modeling

Full size image

Seismic Data

The Garhwal-Kumaun region of Western Himalayas was rocked by the Chamoli earthquake of magnitude (Mw 6.8), on March 29, 1999, at 00:35:13.4 h Indian standard time (IST) [1]. The epicenter was located at N30°24′28.8″ and E79°24′57.6″, and the focal depth has been estimated to be about 21 km. Acceleration time history of the EW component of earthquake recorded at Gopeshwar station is shown in Fig. 5 [29]. For this motion, the PGA is reported to be as 0.35 g and predominant frequency is about 3 Hz. This is used as input time history for dynamic analysis of the slope shown in Fig. 4.

Fig. 5
figure 5

Acceleration time history of EW component of 1999, Chamoli earthquake (Gopeshwar Station)

Full size image

Material Properties

The stress–strain behavior of the soil under dynamic loading is modeled using three different constitutive models, i.e., linear, equivalent linear and nonlinear. Summary of required material properties used for dynamic analysis is given in Table 3. Here, it is assumed that the rock is homogeneous and intact. Additional properties for the equivalent linear model, i.e., strength reduction and damping ratio curves, have been taken as those reported by [30] for overburden material (Fig. 6) and by [31] for rock mass (Fig. 7).

Table 3 Material properties for dynamic analysis [28]
Full size table
Fig. 6
figure 6

Overburden material: (a) modulus reduction curve; (b) damping ratio curve

Full size image
Fig. 7
figure 7

Rock mass material: (a) modulus reduction curve; (b) damping ratio curve

Full size image

First, the study is carried out for linear model, where the stress is directly proportional to the strain and the constant of proportionality is shear modulus G. The linear model is not consistent with the actual field problems because in reality, the stress–strain relationship is fairly nonlinear for earthquake shaking. Second, equivalent linear model is used where the soil stiffness G and damping ratio are modified according to computed shear strains of response. Lastly, a hyperbolic nonlinear model [27], which is widely applicable because of the simplicity of the soil required properties, is used in the analysis. In this model, shear strength of the soil varies according to the level of strain, which defines its nonlinearity (Kramer, [27]). Only maximum shear modulus (Gmax) and the shear strength parameters cohesion (c) and friction angle (ϕ) are required in this model. Values of c and ϕ used are the same as reported in Table 2. Gmax is calculated from the shear wave velocity (Vs) using the following equation:

$$G_{{\max }} = \rho v_{s}^{2}$$
(1)

Where Gmax is the shear modulus (in Pa), Vs is the shear wave velocity (in m/s), and ρ is the mass density (in kg/m3) which is equal to (γ/g) where g is acceleration due to gravity (m/s2).

Results and Discussion

The results of static and dynamic analyses are discussed in the following subsections.

Static and Pseudostatic Analysis

The stability of slope was investigated using traditional limit equilibrium (LE) analysis. The factor of safety (FS) of the slope was calculated using Bishop’s method (Eq. 2). The FS is 1.095 for the static case, indicating that the slope is marginally stable. For seismic case with kh (= 0.18) pseudostatic analysis leads to a FS equal to 0.81, which indicates that seismic loading could induce a major failure of slope causing a landslide.

$$FS = \frac{1}{\sum W\sin \alpha }\sum \left[ {\frac{cb + W\tan \phi }{{m_{\alpha } }} } \right]$$
(2a)

Where

$$m_{\alpha } = \cos \alpha \left( {1 + \frac{\tan \alpha \tan \phi }{{FS}}} \right)$$
(2b)

where W is the total weight of a slice, b is the width of each slice, and α is the angle between the tangent to the center of the base of each slice and the horizontal. As FS is coming on both sides of Eq. 1, it is solved iteratively.

Dynamic Analysis

Based on the above FE modeling, numerical simulation has been performed to ascertain the factor of safety (FS) and permanent displacement for the stability of slope. Outputs of this analysis are time histories of average acceleration (aavg), dynamic factor of safety (FSDynamic), yield acceleration (ay), velocity (v) and permanent displacement (D). The curves for these quantities are developed for three different constitutive models, i.e., linear, equivalent linear and nonlinear. Thus, the effect of nonlinearity is examined. The results are discussed in detail in the following subsections.

Average Acceleration

The initial stresses before shaking have been calculated; then, dynamic analysis is carried out. Dynamic stress (σdynamic) is obtained as a difference of total stress (σtotal) and static stress (σstatic):

$$\sigma_{dynamic } = \sigma _{total} - \sigma_{static}$$
(3)

The major difference between the sliding block method and the present method is the calculation of yield acceleration (ay). In the sliding block method, an empirical relation (discussed later) is used, while in the current method ay is evaluated as follows.

The dynamic mobilized shear (Sm) is obtained as a result of computed dynamic stress, which is integrated along the entire slip surface of the slope. It is considered as additional shear produced by seismic loading. Average acceleration (aavg) has been calculated by dividing the dynamic mobilized shear by the sliding mass (m), i.e.,

$$a_{avg} = \frac{{S_{m} }}{m}$$
(4)

The values of aavg with time for all three constitutive models are presented in Fig. 8. It can be observed that the maximum aavg recorded are 0.32 g, 0.20 g and 0.19 g for linear, equivalent linear and nonlinear cases, respectively. Result shows that the patterns of average accelerations time histories are also different in all three cases, clearly indicating the effect of nonlinearity.

Fig. 8
figure 8

Effect of nonlinearity on average acceleration

Full size image

Factor of Safety

Dynamic stress results have been used to determine the stresses at the base of each slip surface slice, and thus, a factor of safety (FS) for the slip surface is computed. The FS generally oscillates over time due to the applied dynamic forces. Thus, FS is not constant, rather varies with time. This is the major difference with traditional sliding block analysis where FS does not vary with time. Variation of FS with time is a primary novelty of the current study. Further, determining the overall slope stability during shaking is often a challenging task [4]. Figure 9 shows the variation of FS with the time for all constitutive models. The FSmin recorded is 0.69, 0.64 and 0.63 for linear, equivalent linear and nonlinear cases, respectively. Thus, it can be noted that in all cases minimum FS (FSmin) is smaller than that given by pseudostatic analysis (FS = 0.81).

Fig. 9
figure 9

Effect of nonlinearity on the factor of safety

Full size image

Further the value of FS decreases as the consideration of nonlinear behavior of soil is increased. Figures 8 and 9 have the same slip surface. The factor of safety for the critical slip surface varies during shaking (Fig. 9). The factor of safety oscillates, going above and below the static factor of safety, and in some instances, value is less than 1.0.

Yield Acceleration

The value of the yield acceleration (ay) represents the threshold for slope instability. This is defined as the value of aavg corresponding to FS equal to unity. It is the key parameter in evaluating the permanent displacement. In the present analysis, ay is strain-dependent and thus its value changes as per the constitutive model selected. However, the Newmark-based approach assumes that ay is strain independent [10]. Considering the dependency of ay on strain is a major novelty of the present research work and its effect on permanent displacement is examined.

Figure 10 is derived from Fig. 8 and Fig. 9 by eliminating time and plotting the factor of safety vs. average acceleration (aavg) for few positive values of acceleration till FS reaches less than unity. From this, the value of aavg corresponding to FS equal to unity can be found, which is yield acceleration (ay) that triggers the sliding mass. The values of ay evaluated are 0.058, 0.049 and 0.038 g for linear, equivalent linear and nonlinear cases, respectively. It shall be noted that these values are different from those obtained by the sliding block approach (as discussed later). Even for linear case, the value of ay will depend on the level of input excitation (PGA) which is not the case for traditional sliding block analysis and this is the major contribution of the present study. In Fig. 8, where the average acceleration (on positive side) exceeds ay, there will be permanent displacement. This is used to evaluate the velocity and then permanent displacement in the following subsections.

Fig. 10
figure 10

Effect of nonlinearity on yield acceleration

Full size image

Velocity

The velocity of the sliding mass is calculated by integrating the area under average acceleration time history (Fig. 8) where the acceleration exceeds ay. The velocity time history is shown in Fig. 11. The maximum velocity (Vmax) of sliding mass along the slip surface during shaking is about 0.72,0.74 and 0.76 m/s for linear, equivalent linear and nonlinear cases, respectively. The values are increasing with nonlinearity.

Fig. 11
figure 11

Effect of nonlinearity on velocity

Full size image

Permanent Displacement

The amount of permanent displacement of the slope during seismic loading is estimated by integrating velocity time history. Figure 12 shows the cumulative parallel movement of sliding mass during the shaking which initiates permanent displacement of about 77, 92 and 98 cm for linear, equivalent linear and nonlinear cases, respectively, clearly indicating the effect of nonlinearity on these values.

Fig. 12
figure 12

Effect of nonlinearity on permanent displacement

Full size image

It shall be noted that in Fig. 12, the permanent displacement (PD) keeps increasing due to shaking and reported as a cumulative value. This is well-defined method proposed by [5]. The maximum PD reported in Table 4 is considering whole time history. Here, maximum displacement for highest PGA (0.35 g) for nonlinear case is 0.98 m which is just 0.3% of the slope height (320 m) and within acceptable limit of 1%.

Table 4 Effects of nonlinearity on various parameters
Full size table

Summary of Results of FE Analysis

Table 4 shows the effect of nonlinearity on different parameters. It can be observed that minimum FS and yield acceleration decrease due to nonlinearity, while maximum velocity and maximum permanent displacement increase from linear to nonlinear cases. However, in general, the difference is more between linear to equivalent linear cases, particularly in values of permanent displacements. Values of aavg for equivalent and nonlinear are comparatively smaller than the linear case, but the areas which exceed the yield acceleration are quite higher for equivalent and nonlinear cases. This leads to higher permanent displacements (after double integration) for nonlinear cases. Here, nonlinear model better represents the field condition.

Comparison of Numerical Modeling with other Methods

The results obtained from linear finite element dynamic analysis are compared with those obtained using sliding block method [5] and two empirical displacements-based predictive models [10, 11] to check the reliability of proposed FE analysis. For this comparison, the acceleration time history of 1999, Chamoli earthquake (Fig. 5) is used.

Newmark [5] assumed that the dynamic behavior of a soil mass could be regarded as a rigid block sliding on an inclined plane. The sliding would initiate when the inertial forces were large enough to overcome the shear resistance between the block and the base, while the movement would stop when the inertia forces were reversed. Next, ay, commonly referred to as the “yield” or “critical” acceleration, is determined by increasing horizontal earthquake acceleration in a pseudostatic analysis until the factor of safety approaches unity. The yield acceleration (ay) can be established using the following relation

$$a_{y} = \left( {FS - 1} \right)g \times \sin \beta$$
(5)

where FS is a static factor of safety determined using LE analysis and \(\beta\) is an angle of sliding surface from the horizontal direction. In this study, \(\beta\) is calculated as 25°. So, the value of yield acceleration in the sliding block approach is 0.04 g.

When the input acceleration [a(t)] exceeds critical acceleration (ay), the permanent displacement (D) generates and is calculated by double integration of the part of acceleration time history which exceeds critical acceleration, i.e., where [a(t) > ay]. This is expressed by the following equation

$$D = \iint {(a\left( t \right) - a_{y} )dtdt}$$
(6)

where a(t) is the acceleration at time t.

There are several simplified sliding block empirical equations available based on rigid [10, 11]-, coupled [14]- and decoupled [32]-based models to calculate permanent displacement of earth dams, embankments, natural slopes, shallow and deep slope failures. In these models, different parameters are used, e.g., the initial fundamental period (TD), acceleration response spectrum (Sa), residual strength of earthquake (IE), mean period of earthquake acceleration (Tm), PGA, arias intensity (Ia).

In the present study, the effect of PGA on permanent displacement has been evaluated on natural slope. Therefore, two simplified sliding block models [10, 11] were chosen to compare with the present study. Both empirical equations had used PGA to evaluate permanent displacement applicable for natural slopes.

Ambraseys and Menu [10] were the first to propose a regression equation to estimate the permanent displacement (D) as a function of the yield acceleration ratio (the ratio of yield acceleration ay to maximum acceleration amax) based on analysis of 50 strong-motion records from 11 earthquakes, they concluded that the following equation best characterizes the results of their study:

$$\log D = 0.90 + \log \left[ {\left( {1 - \frac{{a_{y} }}{{a_{\max } }}} \right)^{2.53} \left( {\frac{{a_{y} }}{{a_{\max } }}} \right)^{ - 1.09} } \right]$$
(7)

where D is in cm and the standard deviation (σ) of this model is 0.30.

The other empirical equation used for comparison is a simplified rigid block model proposed by [11] using displacements calculated from over 2,000 acceleration time histories, which predicts slope displacement as a function of ratio of yield acceleration and PGA.

$$\ln D = 5.52 - 4.43\left( {\frac{{a_{y} }}{PGA}} \right) - 20.93\left( {\frac{{a_{y} }}{PGA}} \right)^{2} + 42.61\left( {\frac{{a_{y} }}{PGA}} \right)^{3} - 28.74\left( {\frac{{a_{y} }}{PGA}} \right)^{4} + 0.72\ln \left( {PGA} \right)$$
(8)

where D is in cm and both PGA and ay are in g. The regression model given in Eq. 7 has a standard deviation of 1.13. Empirical Eqs. (6 and 7) are valid for natural slopes.

For evaluating the effect of PGA, the time history of the 1999 Chamoli earthquake has been scaled down so that PGA varies from 0.05 g to 0.35 g. The results are presented in Table 5.

Table 5 Permanent displacements evaluated using FE analysis and empirical equations
Full size table

To validate the accuracy of the proposed model, a comparison is made between values of permanent displacements obtained by the numerical modeling for a linear case with sliding block method and empirical equations in Fig. 13. It can be observed that empirical relation proposed by [10] over predicted the permanent displacement at most of the PGA except at higher PGA, i.e., 0.3 g and 0.35 g. Thus, relative errors are higher at most of the PGA. The probable reason for this could be that authors considered a higher PGA dataset to develop empirical relations. On the other side, values of D by [11] are in good agreement with those obtained from the present study at lower PGA up to 0.2 g and under predicted at higher PGA. However, the relative error is less than 25%. The results of FE modeling are also compared with the sliding block method which under predicted permanent displacements as compared to [11] and that to present study.

Fig. 13
figure 13

Comparison of permanent displacements

Full size image

From Table 5 and Fig. 13, it can be observed that the displacements calculated by the FE method up to 0.15 g are closer to the sliding block method. Beyond 0.15 g, FE analysis overpredicts the displacement; however, still, results are comparable to that shown by empirical relation [10].

Thus, some difference was found by comparing the presented model with those reported in the literature. In order to develop these empirical relations, researchers typically have performed analytical sliding block analysis. This assumes that the sliding block was a rigid block and thus, the internal deformation was ignored. However, this is not completely in line with the FE modeling which gives more realistic displacement results. Vahedifard and Meehan [33] indicated that almost all predicted displacements are smaller than those observed during seismic events. The tendency of the models for under prediction of earthquake-induced displacements is an issue of concern for practicing engineers, especially in high-risk areas. So, it is important to emphasize that the presented FE numerical approach gives better and more reliable results for local site conditions as values are within the range of that suggested by the literature. Moreover, the present approach employs a dynamic analysis to find the yield acceleration.

Effect of Nonlinearity on Permanent Displacement

Effect of nonlinearity on permanent displacement has been examined by carrying out FE analysis using three constitutive models, i.e., linear, equivalent linear and nonlinear. This has been examined considering the variation of PD with PGA and pseudostatic FS.

Variation of Permanent Displacement with PGA

Table 6 shows the comparative results for all three constitutive models based on FSmin, ay and D for different PGA. It can be observed that as the nonlinearity increases, values of FSmin and ay decrease, while permanent displacement increases for a particular value of PGA. Thus, for a particular value of PGA, permanent displacement calculated from the linear model shows the lowest value whereas the nonlinear model gives the highest value. The variation of permanent displacement with PGA is not linear. It shows an exponential increase at the higher end of PGA. However, it is acknowledged that this conclusion is based only on one input ground motion used in the current study. This should be verified for different ground motions with varying dominant frequencies.

Table 6 Results for linear, equivalent linear and nonlinear cases
Full size table

Figure 14 shows the variation of permanent displacement with PGA for 3 constitutive models. For all three models, permanent displacement shows a polynomial variation with PGA. However, at all PGA, the value of D is higher for the equivalent linear model than that for the linear model which becomes highest for nonlinear models. Thus, the effect of nonlinearity is clearly visible.

Fig. 14
figure 14

Variation of permanent displacement with PGA for linear, equivalent linear and nonlinear soil models

Full size image

Variation of Permanent Displacement with Pseudostatic FS

FE numerical modeling is a rigorous approach that requires various input parameters and consumes a significant amount of time. Therefore, to overcome this limitation predictive curves for permanent displacement based on the pseudostatic factor of safety (FSpseudo) are proposed. Values of permanent displacements with FSpseudo for linear, equivalent linear and nonlinear are given in Table 6, and variations are presented in Fig. 15. These curves can be used to predict the permanent displacement corresponding to a particular FSpseudo for a selected material model. Though, it is acknowledged that the result will hold for the particular slope discussed here or a similar slope.

Fig. 15
figure 15

Variation of permanent displacement with FSpseudo for linear, equivalent linear and nonlinear soil models

Full size image

Correlation of Permanent Displacement with Hazard Class

Interpreting permanent displacements is very important. The significance of the Newmark displacements must be judged in terms of the probable effect on the potential landslide mass. Predicted displacements do not necessarily correspond directly to measurable slope movements in the field; instead, permanent displacements provide an index to correlate with field performance [34, 35].

Table 7 presents hazard categories for the representative landslide based on [36]. Further, the permanent displacement results for different PGA (in bracket) are categorized in different hazard classes for all three constitutive models. These results are based on numerical modeling and field investigation has not been involved. Results show that in all three cases, displacements at PGA 0.3 and 0.35 g are falling in very high hazard class. Further, it can be observed from Table 7 that the hazard class shifts toward a higher range when constitutive model changes from linear to equivalent linear and then to nonlinear for a particular value of PGA, thus clearly indicating the effect of nonlinearity. Hence, it can be concluded that it is important to consider the material nonlinearity in the analysis, while correlating with the hazard class.

Table 7 Comparison of hazard indices with the literature
Full size table

Summary and Conclusions

In this study, finite element dynamic analyses are carried out for a chronic landslide situated in Himalaya located at Chamoli district, Uttarakhand, India. This landslide has been analyzed in static and dynamic conditions to determine the likelihood of failure caused due to the 1999, Chamoli earthquake. For finite element (FE) dynamic analysis, the factor of safety (FS) is considered to vary with time and yield acceleration (ay) depends on the level of excitation. The validation of numerical modeling and computation techniques is carried out by comparing the results of present FE analysis with those available in the literature. The difference in results from the two studies is less than 1.3% and thus found in good agreement.

Three different constitutive models, namely linear, equivalent linear and nonlinear, have been considered to examine the effect of nonlinearity on the permanent displacement (PD). Numerically calculated displacements results are compared with well-established sliding block and empirical models. Based on results, curves to predict permanent displacement for different constitutive models have been proposed for the local site condition which would be applicable for a similar range of input data. The following conclusions can be drawn based on this study:

  1. i.

    A significant effect of nonlinearity was observed in the FS, yield acceleration, velocity and permanent displacement. The minimum FS and yield acceleration decrease, while maximum velocity and maximum permanent displacement increase due to nonlinearity. Hence, it is important to consider material nonlinearity.

  2. ii.

    Permanent displacements computed from FE numerical modeling are in a comparable range of that computed using empirical equations.

  3. iii.

    The study shows that effect of nonlinearity on permanent displacement increases with PGA.

  4. iv.

    The study also proposed predictive curves for permanent displacement based on pseudostatic factor of safety (FSpseudo) which could be used to evaluate the PD for a particular FSpseudo in a similar situation

  5. v.

    Finally, the study examines the variation in the value of the hazard class index based on linear, equivalent linear and nonlinear constitutive models. It is inferred that the nonlinear model leads to a higher hazard class for the same PGA.

The impact and the consequences of presented landslide in this paper are on a local scale, but the possible consequences due to seismic potential of the region are at a broader scale. This problem of instability might cause significantly more massive landslides during the earthquake. Therefore, the study will be beneficial to predict earthquake-induced slope displacements in the future scenario. The presented model can be used as a reference for performing on-site slope stability evaluations for road and tunnel construction projects in the region.