Introduction

Probabilistic approaches are gaining popularity over the years for stability analysis of rock slopes, since they can handle the uncertainty in rock properties systematically. Uncertainties in peak strength and deformation parameters are generally considered to estimate the probability of failure and stability of the rock slopes is then defined according to the estimated probability of failure instead of a factor of safety. While uncertainty in peak strength parameters is considered in the analysis, uncertainty in post-peak strength parameters is generally ignored, which can be very important for rock slopes in an average quality rock mass because this type of rock generally shows a strength drop after the peak strength is mobilized. The effect of post-peak strength parameters becomes more important in the slope, when the rock mass suffers higher displacement than the displacement corresponding to peak strength in the presence of triggering forces like seismic forces, excavation, structural loads, as shown by some earlier researchers (Cruden and Martin 2007; Latha and Garaga 2010). The effect of post-peak strength parameters is generally neglected by assuming a rock mass as an elastic-perfectly plastic because of the limited studies and methods available to estimate post-peak strength parameters at the field scale (Crowder and Bawden 2004).

Studies on post-peak behavior of rock started with testing of intact rock samples under unconfined conditions after the development of stiff servo-controlled test machines in the 1970s (Rummel and Fairhurst 1970; Wawersik and Fairhurst 1970; Hudson et al. 1971; Wawersik and Brace 1971). Later strength and deformational behavior of intact rock under confining stress was also analyzed (Wawersik and Fairhurst 1970; Santarelli and Brown 1989; Arzua and Alejano 2013; Buzzi et al. 2014; Walton et al. 2015; Yang et al. 2012). Various stages in stress-strain behavior have been characterized and fracture patterns developed during these stages at the local and macroscopic level have been analyzed for different classes of rocks (Wawersik and Fairhurst 1970; Wawersik and Brace 1971; Tutluoglu et al. 2015). However, application of these results to a rock mass with large scale defects is not applicable, as these defects govern the peak and post-peak behavior of a rock mass (Wawersik and Brace 1971). In recent years, some studies have been carried out to investigate the effect of various parameters on the post-peak strength parameters of rock mass (Hoek and Brown 1997; Ribacchi 2000; Tiwari and Rao 2006a; Gao and Kang 2016). It was observed that the joint dip, confining stress and rock mass quality have a major influence on the post-peak strength parameters of rock mass (Reik and Zacas 1978; Yang et al. 1998; Ribacchi 2000; Kulatilake et al. 2001; Tiwari and Rao 2004; Tiwari and Rao 2006a; Tiwari and Rao 2006b; Arzua et al. 2014). Because of the influence of post-peak strength parameters on the rock structures stability, efforts have been made over the years to determine the post-peak strength parameters for the rock mass (Pan et al. 1991; Italfer 1996; Russo et al. 1998; Ribacchi 2000; Crowder and Bawden 2004). Most of these methods are not verified for different types and quality of rock masses, which makes them difficult to use. Cai et al. (2007) proposed a GSI-based method to determine peak and residual strength parameters. It was suggested to reduce the rock block volume and joint roughness to estimate GSI at the residual state and then residual strength parameters can be estimated in a similar way as peak strength parameters. This method was further verified for case studies with rock masses of different types and qualities by comparing in-situ measured and estimated strength parameters and was found to be very effective.

Many researchers have carried out the stability analysis of various natural and engineered rock slopes using deterministic approaches (Eberhardt et al. 2004; Pal et al. 2012; Pain et al. 2014; Tiwari and Latha 2016), limited efforts have been made in the application of probabilistic approaches for some real rock slopes case studies (Duzgun et al. 2003; Chowdhury and Flentje 2003; Duzgun and Bhasin 2009). Moreover, in most of these studies the importance of post-peak strength parameters was undermined by generally assuming rock mass behavior as an elastic-perfectly plastic. While some studies have been carried out over the years to investigate the effect of post-peak strength parameters on the soil slopes (Skempton 1964; Skempton 1985; Hammah et al. 2005; Maio et al. 2015; Wen and Jiang 2017), limited studies (Sjoberg 1997; Cai et al. 2007) are available on the effect of post-peak strength parameters on the rock slopes, and this is considerably an unexplored area in the field of rock mechanics.

In the current study, a GSI-based quantitative approach was used to carry out probabilistic analysis for rock slope stability analysis by considering uncertainty in peak and residual strength parameters, which can consider the strength reduction in the rock mass behavior. The method suggested by Cai et al. (2007) was included in the probabilistic framework to estimate the uncertainty in the residual strength parameters, which were later used to estimate the reliability index and the probability of failure. The approach is demonstrated using an important case study of a large Himalayan rock slope supporting the piers of the world’s highest railway bridge. The article shows the advantages of the probabilistic approach over the deterministic approach for rock slope stability analysis and the importance of including the effect of post-peak strength parameters in both the approaches.

Details of the case study

General description

In this section a case study of a rock slope supporting a railway bridge was selected to demonstrate the GSI-based probabilistic approach, which can consider uncertainty in both peak and residual strength parameters. This case study was selected for the analysis since various important laboratory and in-situ test results along with various field observation data were available for the site in detail. The area under investigation covers a part of the Udhampur district of Jammu and Kashmir state in India. At a location close to Kauri and Bakkal villages the railway line crosses river Chenab at a height of 359 m where a bridge of length 1.26 km is proposed at the site where piers are situated on two rock slopes. This bridge, once constructed, will be the world’s highest railway bridge at present at a height of about 359 m. Among the 18 piers of the bridge, four piers (S10-S40) are located on the left abutment and the other 14 piers (S50-S180) on the right abutment. One of the important challenges of the project is the stability of abutment slopes during and after the construction. A photograph of the proposed bridge is given in Fig. 1. Among the two rock slopes, the right abutment slope was used for stability analysis in the present study.

Fig. 1
figure 1

Photograph of the Chenab Bridge site

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General geological details

The area of the study comes within the sub-Himalayan zone, with outcrops of unfossiliferous limestone, Sirban limestone of Hazara of presumably Permian or Permo-Carboniferous/Meso-Proterozoic age as inliers. A total of 28 drilled bore holes with a total footage of approximately 950 m along with an exploratory drift and four pits excavated along the different slope locations, were used to investigate the general geology and characteristics of general structural features at the site. Figure 2 shows the geological plan for the slope site. Petrographic examination of the rock showed that the general lithology consists primarily of a dolomitic limestone rock mass, mainly gray or dark gray in color. It was observed during borehole drilling that the rock mass at the site in general is hard in nature with slight weathering (W1), very occasionally a highly weathered zone (W3 and W4) has been recorded at places along shear zones and open joints, which is attributed to the water action along the weaker planes. The quality of the rock mass present was varying and was generally found to be blocky. However, it occurs as prominent patches of intensely jointed and fractured rock at some locations, which is referred to as fractured dolomitic limestone in this article. Figure 3 shows the schematic cross-section diagram of the slope supporting bridge with geological details, existing and excavated profiles and rock mass quality encountered during drilling of boreholes near foundation locations. Figure 4 shows the core samples extracted at different depths below the pier S50. Fresh and hard gray colored dolomite with randomly broken cores is observed throughout the depth. Percentage core recovery varied from 27% at the top surface to 99% at 50 m below the ground.

Fig. 2
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Geological map of the Chenab Bridge slope

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Fig. 3
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Details of geology and rock mass quality observed during drilling near foundation locations

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Fig. 4
figure 4

Core samples extracted below the pier S50

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Figure 5 shows the stereographic projections for the major discontinuity sets at the site. The rock mass presently has three major discontinuity sets and some random joint sets were also encountered at the site. The main discontinuities observed at the site are one foliation joint set, J1 (29°/045°) and two sub-vertical joint sets, J2 (54°/246°) and J3 (68°/162°). Quantitative description of rock discontinuities such as orientation, persistence, roughness, filling and aperture were determined in the field in accordance with the ISRM suggested methods (ISRM 1981). More details regarding the geology of the site, spacing, roughness and infilling of the discontinuities along the slope were presented by Tiwari and Latha (2016). A brief summary of the geological characteristics and orientation attributes of major joint sets along the rock slope is presented in Table 1. Figure 6a shows the typical fracture network on the cross-sectional diagram near the S50 foundation location prepared using estimated statistical parameters of discontinuities obtained from field investigations, and Fig. 6b shows the excavated drift, which shows very close spacing for J1 joint set.

Fig. 5
figure 5

Stereographic projections for slope at the Chenab Bridge site

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Table 1 Geological description and orientation attributes of discontinuities along the slope
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Fig. 6
figure 6

Details regarding discontinuity fracture network a Discontinuity fracture network near drift b Photograph of the drift showing close spacing of joint set J1

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Laboratory investigations

Different properties of the intact rock like unit weight, elastic modulus and Poisson’s ratio were estimated from laboratory tests performed on the collected samples from the site as per the guidelines (ISRM 1981). Unconfined Compressive Strength (UCS) of intact rock was estimated by carefully collecting rock samples from several boreholes located along the slope at different pier locations. The value of the intact rock Hoek-Brown parameter (mi) was taken from the literature suggested for dolomitic limestone by Marinos and Hoek (2000). Since the values of different intact rock properties were varying along the slope, a range of these properties was observed in the laboratory tests and statistical parameters of different properties are given in Table 2.

Table 2 Statistical moments, range and distribution of the intact rock properties present at the site
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Further, it was observed during the field investigations that infill was present along the joint sets J1 and J3. The infill was collected from the exposed locations inside the drift and pit and laboratory tests were conducted. Sieve analysis was carried out on the infill, and the infill material was predominantly classified as Poorly Graded Silty Sand (SP-SM) along J1 and J3 according to the Unified Soil Classification System. Direct shear tests were conducted, and the friction angle of the infill material was determined. Compaction tests were conducted to estimate the maximum dry density and optimum moisture content of the infill. Table 3 summarizes the results of laboratory tests conducted on infill material.

Table 3 Properties of joint infill along different joint sets
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Determination of peak and residual parameters of rock mass

Since the joint spacing in the rock mass is very small as compared to the slope dimensions as observed from Table 1 and Fig. 3, stability of the slope was analyzed using the equivalent continuum approach (Edelbro 2003; Wyllie and Mah 2004). The rock mass was modeled by substituting the original discontinuous medium by a continuum, whose constitutive law incorporates the effect of the intact material and that of the discontinuities, which eliminates the need of modeling discontinuities explicitly in the numerical modeling. Hence, for the stability analysis of the rock slope, determination of rock mass properties was carried out and is discussed in the section below.

Determination of peak GSI and residual GSI

Because of the difficulties involved in conducting the in-situ tests to determine the strength of rock mass, efforts were made over the years, to relate the strength parameters of rock mass to the rock mass classification systems like rock mass rating (RMR) or geological strength index (GSI). GSI is an established rock mass classification system, which is used along with intact rock strength properties to obtain strength parameters of the rock mass. Many researchers tried to provide a quantitative way to estimate the GSI using parameters based on block size and discontinuity conditions for better classification of rock masses for engineering purposes (Sonmez and Ulusay 1999; Cai et al. 2004). In this study, the method suggested by Cai et al. (2004) was used to estimate the peak GSI value (GSIp), i.e., GSI values used to estimate peak strength parameters. Equations required to calculate GSIpvalue for the rock mass are given below

$$ {GSI}_p\left({J}_C^p\right.,\left.{V}_b^p\right)=\frac{\ 26.5+8.79\ \mathit{\ln}\ {J}_C^p+0.9\ \mathit{\ln}\ {V}_b^p}{1+0.0151\ \mathit{\ln}\ {J}_C^p-0.0253\ \mathit{\ln}\ {V}_b^p} $$
(1)
$$ {J}_C^p=\frac{J_W^p{J}_S^p\ }{J_A^p} $$
(2)

where \( {J}_C^p \) is the peak joint condition factor; \( {V}_b^p \)is the peak block volume; \( {J}_w^p \) is the peak large-scale waviness factor in meters (varying between 1 to 10 m); \( {J}_S^p \) is the peak small-scale smoothness factor in centimeters (varying between 1 to 20 cm); \( {J}_A^p \) is the peak alteration factor.

$$ {V}_b^p={s}_1\times {s}_2\kern0.5em \times {s}_3 $$
(3)

where\( {V}_b^p \)is the peak block volume; s1 , s2 , s3 are the joint spacings for different joint sets.

A summary of the discontinuity features at the Chenab Bridge slope is given in Table 1. The average joint spacing for joint sets J1, J2 and J3 observed during field investigations were 20 cm, 60 cm and 115 cm, respectively, and the corresponding standard deviations estimated using the 3-sigma rule from the given range are 3.33 cm, 13.33 cm and 28.33 cm, respectively. Corresponding COV of the joint spacing for J1, J2 and J3 are 16.65%, 22.21% and 24.63%, respectively. It has been generally observed from the literature that joint spacing follows a negative exponential distribution or lognormal distribution (Cai 2011). For the current analysis, it was assumed that joint spacing follows a lognormal distribution. Monte-Carlo simulation was carried out to find out the statistical parameters and PDF of \( {V}_b^p \) using already known statistical parameters and PDF of joint spacing using Eq. 3. The average values for \( {J}_w^p \), \( {J}_S^p \) and \( {J}_A^p \) are 1.5, 1.5 and 4, respectively determined on the basis of discontinuities features given in Table 1 and classification provided by Cai et al. (2004). For the current analysis, a truncated normal distribution was assumed with a COV of 8% for \( {J}_w^p \),\( {J}_S^p \) and \( {J}_A^p \) as suggested by Cai (2011), giving standard deviations of 0.12, 0.12 and 0.32, respectively. The statistical parameters and PDF of \( {J}_C^p \), was estimated using already known statistical parameters and PDF of \( {J}_w^p \),\( {J}_S^p \) and \( {J}_A^p \) using Eq. 2 by Monte-Carlo simulation. Statistical parameters of GSIp were estimated using determined statistical parameters and PDF of \( {J}_C^p \) and \( {V}_b^p \) and the PDF was estimated using a curve fitting process in @Risk (Palisade 2015) using Eq. 1 by Monte-Carlo simulation. Results are summarized in Table 4.

Table 4 Statistical moments and PDF of peak rock mass properties estimated using Monte-Carlo simulations
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Cai et al. (2007) proposed to reduce the field estimated GSI (GSIp) value to residual GSI (GSIr) to obtain the residual strength parameters by reducing the block volume and joint roughness parameters. Barton et al. (1985) observed the reduction in joint roughness as displacement of joints increases towards the residual stage. They provided a curve between mobilized joint roughness condition/peak joint roughness condition vs. displacement in the joints. Based on the investigations made by Barton et al. (1985) on joint roughness degradation at residual displacement using shear tests on different rock joints, Cai et al. (2007) suggested to reduce the peak values of large scale waviness and small scale roughness of joints to the values corresponding to the residual stage. Peak values corresponds to the stage when negligible displacement occurs along the joint and residual values correspond to the higher displacement, when strength becomes constant with the displacement, as suggested by Barton et al. (1985). Furthermore, it was suggested to reduce the peak block volume of the rock to residual block volume, since block size reduces due to tensile and shear fracturing when the displacement in the rock mass increases. Data for block volume at the residual stage is collected from the tunnel case studies in Japan and numerical modeling results. Residual block volume was found to be approximately 10 cm3 from the collected data. Figure 7 shows the summary to demonstrate the effect on block volume and joint roughness after shearing in the rock mass has taken place. It can be seen that block volume and joint roughness reduces with the increasing displacement in the rock mass. In this study, a constant value of 10 cm3 is taken for residual block volume (\( {V}_b^r\Big) \) since it was suggested that residual block volume can be considered independent of the \( {V}_b^P \) for most average quality rock masses. Guidelines were provided to estimate the residual joint large-scale waviness factor (\( {J}_w^r \)), residual small-scale smoothness factor (\( {J}_S^r \)) and residual alteration factor (\( {J}_A^r \)) from the values of peak joint large-scale waviness factor (\( {J}_w^p \)), peak small-scale smoothness factor (\( {J}_S^p \)) and peak alteration factor (\( {J}_A^p \)) which were used to estimate \( {J}_C^r \) as discussed briefly below

Fig. 7
figure 7

Reduction of GSI from peak to residual value a Photograph of a Frank slide showing small block volume after failure b Reduction in block volume in slip zone (Cai et al. 2007) c Initial joint roughness before shearing (Wang et al. 2016) d Degraded joint roughness after shearing (Wang et al. 2016)

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For the reduction in joint large-scale waviness factor from \( {J}_w^p \) to \( {J}_w^r \), the following conditional equation was provided by Cai et al. (2007)

$$ {J}_w^r=1\kern0.5em \mathrm{if}\kern0.5em {J}_w^p<2;\kern0.5em \mathrm{and}\kern0.5em {J}_w^r=\frac{J_w^p}{2}\kern0.5em \mathrm{if}\kern0.5em {J}_w^p>2 $$
(4)

which was replaced by single polynomial equation using a curve fitting process that gives smooth continuous transition over the entire range of \( {J}_W^p \) instead of a conditional set of equations as given below:

$$ {J}_W^r=0.21{\left({J}_W^p\right)}^2-0.61\left({J}_W^p\right)+1.41\kern0.5em \mathrm{for}\kern0.5em 1<{J}_W^p<3 $$
(5)

Regarding reduction in the small-scale smoothness factor from \( {J}_S^p \) to \( {J}_S^r \), the following conditional equation was given

$$ {J}_S^r=0.75\kern0.5em if\kern0.5em {J}_S^p<1.5;\kern0.5em and\kern0.5em {J}_S^r=\frac{J_S^p}{2}\kern0.5em if\kern0.5em {J}_S^p>1.5 $$
(6)

which was replaced by a single polynomial equation that gives smooth continuous transition over the entire range of \( {J}_S^p \) instead of a conditional set of equations as given below:

$$ {J}_S^r=0.16{\left({J}_S^p\right)}^2-0.25\left({J}_S^p\right)+0.82\kern0.5em \mathrm{for}\kern0.5em 0.6<{J}_S^p<3 $$
(7)

No reduction is recommended for the joint alteration factor because joint alteration is unlikely to occur in a short time period and, hence,

$$ {J}_A^r={J}_A^p $$
(8)

Residual joint condition factor (\( {J}_C^r \)) can be computed using these reduced parameters as per the equation given below:

$$ {J}_C^r=\frac{J_W^r{J}_S^r\ }{J_A^r} $$
(9)

Now, GSIr can be obtained by the following equation:

$$ {GSI}_r\left({J}_C^r\right.,\left.{V}_b^r\right)=\frac{\ 26.5+8.79\ \mathit{\ln}\ {J}_C^r+0.9\ \mathit{\ln}\ {V}_b^r}{1+0.0151\ \mathit{\ln}\ {J}_C^r-0.0253\ \mathit{\ln}\ {V}_b^r} $$
(10)

Since the magnitude of \( {V}_b^r \) is considered as a constant value of 10 cm3 for average quality rock masses, Eq. (10) further reduces to Eq. (11) given below:

$$ {GSI}_r\left({J}_C^r\right.\Big)=\frac{\ 28.5+8.79\ \mathit{\ln}\ {J}_C^r}{0.9417+0.0151\ \mathit{\ln}\ {J}_C^r} $$
(11)

Statistical parameters and PDF of \( {J}_W^p,{J}_S^p \) and \( {J}_A^p \) were used to estimate the statistical parameters and PDF of \( {J}_W^r,{J}_S^r \) and \( {J}_A^r \) by Monte-Carlo simulation using Eqs. (5), (7) and (8). Statistical parameters and PDF of \( {J}_C^r \) was estimated using statistical parameters and PDF of \( {J}_W^r,{J}_S^r \) and \( {J}_A^r \) using Eq. (9). After obtaining statistical parameters and PDF of \( {J}_C^r \) statistical parameters and PDF of GSIr were again estimated by Monte-Carlo simulation using Eq. (11). Results regarding residual parameters for the considered slope are summarized in Table 5. It should be noted that statistical parameters of \( {J}_W^r,{J}_S^r \) can be directly estimated from the conditional equations; however, polynomial equations provide a single equation instead of conditional set of equations with smooth transition over the entire range of \( {J}_W^p\ and\ {J}_S^p \), respectively, and were found to be efficient in estimating strength parameters as discussed in the next section.

Table 5 Statistical moments and PDF of residual rock mass properties estimated using Monte-Carlo simulations
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Calculation of strength parameters of rock mass

For the current analysis, the Hoek-Brown criterion was used since the linearized Mohr-Coulomb criterion overestimates the factor of safety and reliability index, while underestimating the probability of failure. Parameters required for the Hoek-Brown strength criterion for rock mass are related to the well-accepted rock mass classification system GSI and the Hoek-Brown intact rock strength parameter as given below (Hoek et al. 2002):

$$ \frac{\kern0.5em {m}_b^{p/r}}{\kern0.5em {m}_i}={e}^{\left(\frac{GSI_{p/r}-100}{28-14D}\right)} $$
(12)
$$ {s}_b^{p/r}={e}^{\left(\frac{GSI_{p/r}-100}{9-3D}\right)} $$
(13)

where \( {m}_b^{p/r},{s}_b^{p/r} \) are peak/residual Hoek-Brown constants of rock mass, respectively, and mi is the frictional strength component of intact rock, which can be estimated using triaxial testing in the laboratory or, in the absence of laboratory results, can be taken from the literature, and D is the disturbance factor due to blast damage or stress relief. During the construction of abutment slopes, careful excavation techniques with controlled blasting are used to avoid any damage to the rock mass of the excavated profile, and; hence, this disturbance factor is considered as 0 (D = 0) (Hoek and Diederichs 2006). Strength parameters were estimated from the modified equations. To validate the above mentioned method to estimate strength parameters, a comparison is provided in Tables 10–12 in the Appendix between the estimated residual strength with the abovementioned approach and an in-situ block shear test for case studies with different types and qualities of rock masses. Statistical parameters of the rock mass strength parameters can be estimated using Monte-Carlo simulation by using Eqs. 12 and 13 using statistical parameters and PDF of mi and GSIp/r. Estimated statistical parameters and PDFs of peak and residual strength parameters are summarized in Table 4 and Table 5, respectively.

Influence of residual strength parameters on Chenab bridge slope stability

In this section, the influence of different strength parameters on the Chenab Bridge slope stability is discussed. A discussion regarding in-situ direct shear tests is included to discuss the post-peak behavior of the rock mass present at the site. Later, a sensitivity analysis was carried out to determine the influence of various rock mass strength properties on the Chenab Bridge slope stability in a quantitative way.

In-situ direct shear tests were conducted at five locations inside the drift with five different normal loads near the arch foundation location. A block of approximately 1000 mm × 1000 mm in plane and 350–400 mm in height was isolated by carrying out excavation in the drift with jack hammers. A 200 ton hydraulic jack was used to apply normal load. Subsequently, horizontal load was also applied. Horizontal load was applied till the test block was moved by about 25 mm, which corresponds to 3.6% of the strain. Figure 8 shows the typical direct shear test results for the site. It can be observed that a strength drop was observed for the tests carried out at the site which was expected as the rock mass present at the site was of average quality as can be seen from GSI values in Table 4 (Hoek and Brown 1997). The displacements corresponding to the peak strength for different tests were approximately 1–10 mm, which corresponds to 0.1%–1% strain after which a constant drop in the strength was observed. Moreover, with the limited in-situ determined peak and residual strength parameters were matching well with the estimated residual strength parameters using the GSI-based method as shown in Table 12 of the Appendix. Since the peak strength was mobilized at a very small displacement, it becomes important to take into account the effect of post-peak strength for the slope stability.

Fig. 8
figure 8

Typical results from an in-situ direct shear test conducted inside the drift at the site

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The sensitivity index (SI) was used to assess the relative importance of each random variable, i.e., peak and residual strength parameters. The SI is calculated by changing one random variable at a time over three standard deviations from the mean and determining the percent change in the material response parameter of interest (in this case it is the factor of safety) over that range (Hamby 1994; Langford and Diederichs 2013). Figure 9 shows the slope geometry used for the sensitivity and stability analysis. Factor of safety for slope was estimated by reducing the original shear strength parameter to bring the slope to the verge of failure using the finite element-based software Phase2 (Rocscience 2014). The slope was discretized, keeping the number of 3-noded triangular finite elements as 72,644. Rock mass strength was defined using the Hoek-Brown criterion for the slope. Figure 10 shows the SI values for different random variables, and it can be concluded that other than for UCS of intact rock, residual strength parameters are highly affecting the factor of safety for this rock slope. The influence of peak strength parameters is limited for the factor of safety as compared to residual strength parameters.

Fig. 9
figure 9

Geometry of slope used in the analysis (all dimensions in meters)

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Fig. 10
figure 10

Sensitivity Index value for different random variables for rock slope

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It can be seen from the above discussion that residual strength parameters have significant influence on the stability of slopes, and the behavior of the rock mass present at the site was not elastic-perfectly plastic, i.e., a strength drop was observed once the rock mass reached the displacements corresponding to peak strength. Hence, it can be concluded from this study that it is important to consider the uncertainty and strength drop in the post-peak strength parameters to properly estimate the stability of rock slopes.

Stability analysis of slope

Deterministic approach

Initially stability of the rock slope is checked using the deterministic approach, by considering rock mass properties to be constant. Details regarding geometry of the slope and numerical modeling were explained in an earlier section. Stability analysis of slope was carried out by considering strength drop and ignoring strength drop (elastic-perfectly plastic) in the analysis.

Table 6 summarizes the properties for rock mass used in the slope stability analysis for different cases. Table 7 presents the summary of results for deterministic global stability analyses of slope for both cases. It was observed that the factor of safety was overestimated by 47% when the rock mass was assumed to be elastic-perfectly plastic. Hammah et al. (2005) also observed the similar effect of post-peak strength parameters on the slope stability. A comparison between maximum displacements and shear strains is also provided. Displacement and shear strains in the table correspond to the particular factor of safety after shear strength reduction has been carried out. This shows the importance of post-peak strength parameters for rock slope stability, which is generally ignored in the analysis by assuming rock mass as elastic-perfectly plastic. Figure 11 shows the displacement contours for slope observed in deterministic analysis for both cases. Ignoring the strain softening behavior of rock can result in overestimation of post peak strength, leading to failures that are unprepared for.

Table 6 Average rock mass properties used in the deterministic analysis
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Table 7 Summary of the results for deterministic analysis
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Fig. 11
figure 11

Total displacement contours to show failure pattern for slopes a Elastic-perfectly plastic b Strength-drop

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Probabilistic approach

Probabilistic analysis of slope was also carried out by considering and ignoring strength drop in the analysis. Table 8 summarizes the rock mass properties used for the probabilistic analysis. Reliability index and probability of failure were estimated using Hong’s point estimate method since it requires a lesser number of evaluation points and can consider skewness of the distribution in the analysis. Moreover, its efficiency for the rock slopes is well established in the literature (Ahmadabadi and Poisel 2015). Hong’s PEM (Hong 1998) can be briefly summarized in following steps:

  1. (1)

    Evaluate the parameter ζi

Table 8 Rock mass properties used in the probabilistic analysis
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$$ {\zeta}_i=\sqrt{n+{\left(\frac{\ {\nu}_i}{2}\right)}^2} $$
(14)

where n = number of random variables, νi = skewness coefficient of the random variable Xi.

Skewness coefficient of the random variable Xi can be estimated as:

$$ {\nu}_i=\frac{E\left[{\left({x}_i-{\mu}_i\right)}^3\right]}{\sigma_i^3} $$
(15)

where E = expectation operator; µi = mean value of random variable; σi = Standard deviation of random variable

  1. (2)

    Evaluate the two realization points for each random variable

$$ {x}_{i,1}={\mu}_i+\left(\frac{\nu_i}{2}-{\zeta}_i\right).{\sigma}_i $$
(16)
$$ {x}_{i,2}={\mu}_i+\left(\frac{\nu_i}{2}+{\zeta}_i\right).{\sigma}_i $$
(17)

xi , 1 ,  xi , 2 = realization points for each variable.

  1. (3)

    Calculate corresponding weights for each model output to compute statistical moments of the performance functions:

$$ {p}_{i,1}=\frac{1}{2n}\left(1+\frac{\ {\nu}_i}{2\ {\zeta}_i}\right) $$
(18)
$$ {p}_{i,2}=\frac{1}{2n}\left(1-\frac{\ {\nu}_i}{2\ {\zeta}_i}\right) $$
(19)
  1. (4)

    Compute the mth statistical moment of the performance function E(Gm)

$$ E\left({G}^m\right)={\sum}_{i=1}^n{\sum}_{j=1}^2{p}_{i,j}\times {\left[G\left({\mu}_1,\dots {x}_{i,j}\dots, {\mu}_n\right)\right]}^m $$
(20)

where G is the performance function that is the factor of safety (FOS) for the present study, and it was assumed that the factor of safety is normally distributed.

  1. (5)

    Calculate the probability of failure Pf

$$ {P}_f=1-\phi \left(\frac{\mu_{FOS}-1}{\sigma_{FOS}}\right) $$
(21)

where µFOS = mean factor of safety; σFOS = standard deviation of factor of safety; ϕ( ) denotes the cumulative density function (CDF) of the standard normal distribution.

A summary of the results from probabilistic analysis is provided in Table 9. It was observed that the difference in the reliability index was approximately 52%, which reflects the amount of overestimation of the reliability index made by ignoring strength drop from the rock mass behavior. Estimation of probability of failure was done for both cases, and it was observed that probability of failure was underestimated by 99% when the rock mass was assumed as elastic-perfectly plastic. Mean factor of safety estimated for the elastic-perfectly plastic case was 2.92, while the mean factor of safety estimated by considering post-peak strength reduction was 1.57. Mean factor of safety was overestimated by 46%, when the rock mass behavior was assumed to be elastic-perfectly plastic. This shows the influence of assumptions made regarding the residual strength parameters on the reliability index and probability of failure of rock slope. It was observed that ignoring strength reduction from the post-peak behavior can lead to overestimation of the reliability index and underestimation of the probability of failure for the slope and, because of the influence of post-peak strength parameters on slope stability, uncertainty in the post-peak strength behavior should be properly considered in the stability analysis of rock slopes.

Table 9 Summary of the results from probabilistic analysis
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The reliability index and the probability of failure are the appropriate criteria of slope stability as they contain information regarding mean values and variability. However, it should be noted that the factor of safety should be used as a complementary tool to the reliability analyses and should not be ignored (Dai and Wang 1992). It was observed that the factor of safety obtained for this slope was higher than the target factor of safety 1.5 in the deterministic analysis by considering and ignoring the strength drop, which indicate that, the slope was stable. However, when the slope stability was analyzed in terms of the reliability index or the probability of failure, the expected performance level for slope was good when the rock mass behavior was assumed to be elastic-perfectly plastic,while the performance level becomes unsatisfactory when the strength drop was considered as suggested by U.S. Army Corps of Engineers (1999). This shows that it is important to express the stability of rock slopes in term of the reliability index, since it includes the uncertainties of the rock mass properties and, moreover, the uncertainty and strength-drop in post-peak strength parameters should be included as they significantly affect the reliability index of rock slopes.

Parametric study

In the probabilistic analysis carried out on the rock slope in the previous section, specific COVs of different variables, i.e., mi, UCS, \( {J}_w^p \),\( {J}_S^p \) and \( {J}_A^p \) were considered. This study is extended by varying COVs of these random variables to investigate the effect of variability of various rock properties on rock slope stability. COV of these random variables are varied between the ranges given in the literature. While many studies are available investigating the effect of variability of soil parameters on soil slope stability (Srivastava et al. 2010; Babu and Murthy 2005), limited studies have been carried out to investigate the effect of COVs of different intact rock, joint properties on the reliability index and probability of failure of rock slopes.

Influence of coefficient of variation of UCS of intact rock

Variability in UCS of intact rock results from the variability in index properties (e.g., density and porosity), petrographic characteristics (e.g., grain size, grain shape) and some knowledge-based uncertainties (Langford 2013). Influence of variability of UCS was investigated using five different values of COV (10%, 15%, 20%, 25%, 30%, 34%). Range of COV of UCS for the current study is taken as suggested by earlier researchers (Ulusay et al. 1994; Wu et al. 2000; Li et al. 2012; Langford 2013).

Figure 12 shows the variation of the reliability index and the probability of failure with COV of intact rock UCS. It was observed that as the COV of UCS increases, probability of failure increases and the reliability index decreases. Mean factor of safety was found to be 1.57 and was constant even when the COV of UCS was varied. A similar trend of probability of failure and mean factor of safety with intact rock UCS was observed by Idris et al. (2015). It can be observed that while the mean factor of safety observed for all the cases was above 1.5, which is the design factor of safety for this particular project, the performance level of the slope was observed to be at unsatisfactory level for higher COV of UCS as suggested by the U.S. Army Corps of Engineers (1999).

Fig. 12
figure 12

Variation of reliability index and probability of failure with COV of intact rock UCS

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Influence of coefficient of variation of intact rock Hoek-Brown constant (m i)

Variability in mi results from variation in mineral content, foliation and grain size and knowledge-based uncertainties (Langford 2013). Influence of COV of mi was investigated using five different values of COV (11%, 15%, 20%, 25%, 30%) by keeping COV of other random variables as a constant. Range of COV of mi used for the current study is taken from the literature (Li et al. 2012; Langford 2013).

Figure 13 shows the variation of reliability index and probability of failure with COV of mi. It was observed that the probability of failure was increasing and the reliability index was decreasing for the slope with the increasing COV of mi, however, reliability index and probability of failure are less sensitive to the variability of mi. In addition to this, the mean factor of safety obtained was 1.57 and was constant with the increasing COV of mi. While the mean factor of safety is marginally above the design factor of safety of 1.5, probability of failure and reliability index were showing an unsatisfactory performance level of slope. Li et al. (2012) carried out a study to investigate the influence of COV of strength parameters (mi,UCS) on the probability of failure (Pf) of rock slopes. They concluded that the Pf is sensitive to strength parameters; however, they have investigated the effect by simultaneously varying COVs of both parameters making it difficult to analyze the individual influence of COV of mi. This study shows that mi has little significance on the probability of failure and the reliability index as compared to UCS of intact rock.

Fig. 13
figure 13

Variation of reliability index and probability of failure with COV of mi

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Influence of coefficient of variation of peak joint roughness and alteration parameters

The uncertainty in roughness parameters can result from both spatial variability of the joint surface condition and human errors in field mapping (Cai 2011). While Cai (2011) has assumed a COV of 8% for \( {J}_w^p \),\( {J}_S^p \) and \( {J}_A^p \), range of COV for these parameters cannot be found from the literature. Influence of COV of these parameters was investigated by assuming four values of COV (8%, 15%, 20%, 25%) by keeping COV of other parameters as constant.

Figure 14 shows the variation of reliability index and probability of failure with COV of joint roughness and joint alternation factors. It was observed that probability of failure was increasing and the reliability index was decreasing for the slope; however, the influence was limited with the increasing COV of \( {J}_w^p \),\( {J}_S^p \) and \( {J}_A^p \). In addition to this, the mean factor of safety obtained was 1.58 and was constant with the increasing COV of these factors.

Fig. 14
figure 14

Variation of reliability index and probability of failure with COV of joint condition parameters

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The method demonstrated here suffers from some limitations. The value of the drop modulus was considered to be infinity, i.e., the strength drop from peak to residual strength was considered to be instant. While the effect of the deformation modulus remains limited on the factor of safety, the effect of drop modulus is still unknown. The effect of the drop modulus can be included in the future in the current method by considering the reduction of GSIp to GSIr with the softening parameters like plastic strains or plastic work; however, this can be done by the back analysis of a large number of well documented case histories, which is absent from the present literature. Hence, the reliability index and factor of safety estimated using current method are slightly conservative.

Slope monitoring

Borehole inclinometers and total stations were placed at the foundation locations to estimate the displacements of slopes during excavation and placement of foundation loads over the years. For displacement estimation, a probabilistic stress analysis using Hong’s PEM was carried out to estimate the statistical parameters of displacements, and the PDFs of displacements was assumed as normal (Cai 2011). Probabilistic analysis was carried out since deterministic analysis for displacement estimation could be misleading owing to the variability in rock mass properties. Estimated displacements and in-situ displacements near slope foundation locations S50 and S60 measured after laying the foundations are compared. Figure 15 shows the results of comparison of estimated displacements with the in-situ measured displacements. It was observed that the measured in-situ displacements were in good agreement with the range of displacements estimated from probabilistic analysis. The displacements in the slope were found to be very small and negligible, and; hence, it can be concluded that slopes are stable.

Fig. 15
figure 15

Comparison between estimated displacement probability distributions and measured displacements over the years for slopes a S50 b S60

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Summary and conclusions

This paper illustrates a method for assessing reliability of rock slope, considering the uncertainty in peak and residual strength parameters, utilizing quantitative risk assessment using the GSI system. The method was applied to a large Himalayan rock slope supporting the piers of a bridge, and its reliability analysis was carried out. For this case, some in-situ direct shear tests were conducted, and it was observed that rock mass present at the site was showing a strain softening behavior. Hence, this case study was used to investigate the effect of post-peak strength parameters on reliability of rock slope. It was observed that the factor of safety and the reliability index are very sensitive to residual strength parameters, and; hence, uncertainty and strength drop in the post-peak strength parameters should be properly considered.

A parametric study was carried out to investigate the effect of COV of the UCS of intact rock, Hoek-Brown constant for intact rock (mi) and roughness and alteration parameters of discontinuities on the reliability index and probability of failure of slope. It was observed that, for this particular rock slope COV of intact rock, UCS has a significant impact on the probability of failure and reliability index. Effect of mi and roughness and alteration parameters on the probability of failure and reliability index of this slope was observed to be limited. However, with the increasing COV of these parameters, the reliability index was decreasing and probability of failure was increasing. This study brought out the fact that the factor of safety alone is an inappropriate indicator of slope stability, and it should be complemented with the reliability index to assess the slope stability accurately. The residual strength parameters should be included properly in the analysis, since after a certain displacement, the strength along the slip surface is controlled by post-peak strength parameters. The method can be used while carrying out staged analysis, i.e., to simulate excavation stages and, along with seismic forces, to find out the probability of rock slides.