Introduction

The evaluation of stability of jointed rock slopes has been a complex and challenging task owing to the inherent variability associated with geological material. To effectively consider the risks arising from the associated uncertainties, probabilistic-based approach has been widely adopted in a number of investigations (Vanmarcke 1980, Oka and Wu 1990, Low and Einstein 1992, Christian et al. 1994, Low 1997, Juang et al. 1998, Duncan 2000, Park and West 2001, Duzgun et al. 2003, Park et al. 2005, Low 2007, Low 2008, Ching et al. 2009, Duzgun and Bhasin 2009, Li et al. 2011, Lee et al. 2012, Park et al. 2012). However, most of these probabilistic-based stability analyses of slopes have considered the variability only in terms of strength parameters, i.e., cohesion and friction angle of intact rock and the discontinuities or the rock mass, discarding other forms of uncertainties. Moreover, most of these investigations have concentrated mainly on the determination of probability of failure with very little focus on the relative contribution of various parameters on slope stability.

However, a prominent form of uncertainty in case of jointed rock slope arises from the scattered values of the geometric parameters (such as dip and trace length) of the discontinuities. In fact, the large number of case studies has highlighted dependency of stability of slopes on geometric parameters of geological structures (Fookes and Wilson 1966; Varnes 1978; Crozier 1986; Koukis and Ziourkas 1991; Guzzetti et al. 1996, Gunther 2003; Grelle et al. 2011, Maerz et al. 2015, Youssef et al. 2014, 2015). It is believed that the consideration of this form of geometric variability in stability assessment would be of practical significance. But this form of uncertainty has seldom been considered in the stability analysis barring a few cases (Hammah et al. 2009; Brideau et al. 2012; Shamekhi and Tannant 2015).

In the present study, the influence of uncertainty in strength parameters of the intact rock and the discontinuities along with the variability in geometric parameters of two impersistent joint sets on the stability of a rock slope has been investigated. The relative contribution of various parameters and their combined effect on the stability of a jointed rock slope has been evaluated and quantified through the use of the concepts of design of experiments (DOE) and ANOVA (Analysis of Variance) approaches. Moreover, a mathematical prediction model (response surface) for the expected strength reduction factor (SRF) over the range of parameters considered in the present study has been established. It is believed that the approach highlighted in the present study will help in gaining better insight into the behavior of rock slope. Such an approach will also help in identification of parameters which should be improved for better slope performance. Therefore, this approach will be beneficial for engineers in deciding measures to be adopted for improving the stability.

Methodology

The methodology adopted in the present study is based on the application of the concepts of design of experiments (DOE) (Montgomery 2013) in a numerical framework. DOE provides a powerful tool to evaluate the cause-and-effect relation between the system response and the input parameters. For efficient implementation of the DOE approach, the factorial design has been adopted in the initial phase of the analysis using a finite element–based software package RS2.0 (Rocscience Inc., 2015). In this phase, also known as the factor screening phase, the parameters having significant influence on the behavior of considered slope, quantified in terms of SRF, are identified. In addition, the relative influence arising due to interaction among various parameters is also evaluated. Once the influencing parameters have been identified, the response surface describing the SRF of the slope are evaluated based on the central composite design model (Montgomery 2013). A step by step illustration of the adopted methodology is shown in Fig. 1. The results of the present study may be verified by following the steps provided in Fig. 1. A concise discussion of each component of the methodology is provided in subsequent sections.

Fig. 1
figure 1

Flowchart depicting the steps adopted for the present study

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Concept of factorial design

Considering the objective of the present study, one of the main challenges requires efficient management of computational burden as large combination of input parameters are involved. For this purpose, the concept of factorial design has been considered for efficient implementation of numerical simulations.

Factorial design has been widely adopted to investigate the response of systems dependent on a large number of parameters. It provides a framework within which the joint effect of several parameters on system response is evaluated by carrying out the smallest number of experiments. Thus, it is widely used in screening of important factors while conducting experiments. Following the identification of influencing parameters, more rigorous analysis may be undertaken for developing the prediction model.

Among various cases of factorial designs, the 2k designs, where k denotes the number of random parameters, have found wide application in engineering studies. In such design, each parameter is assigned two levels: low and high. Thus, for a system dependent on kfactors, 2knumber of experiments is required to be conducted. However, in the case of a large number of parameters, half factorial design may be adopted which reduces the number of experiments to half of that required in case of full factorial design (Montgomery 2013).

In the initial factor screening phase, an assumption of linear variation in system response over the range of parameters is made. Although this assumption may not be exact, it is a reasonable simplification in the initial screening phase aimed towards the identification of the influencing parameters. Once the effect of each parameter and their combined influences are known, more rigorous statistical analysis may be taken up to frame a relationship between the system response and the significant influencing parameters.

Assessment of influence of various parameters

The effects of individual parameters and their combined influences on the response of a system may be summarized in the form of main and interaction effect estimates (Montgomery 2013). For clarity of the paper, a brief discussion dealing with the evaluation of these estimates is provided below.

Suppose the system response is dependent on three factors A, B, and C each at two levels: low and high denoted by − and + respectively. In such case, a 23 factorial design may be adopted with a total of eight runs with parameter combinations as illustrated in Table 1. In any combination, the high level of a factor is defined by the corresponding lower case letter. By convention, (1) denotes the case where all the parameters are at their respective low levels. Thus, ab represents the case where factors A and B are at their corresponding high level and factor C is at low level. Similar convention may be extended for other labels. Figure 2 provides the various combinations possible in the case of 23 factorial design with an illustration of the meaning of various labels.

Table 1 Combination of parameters in 23 factorial design
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Fig. 2
figure 2

Various combinations of experiment run in a 23 factorial design

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The average effect of a factor on the response of a system is defined as the change in the response arising as a result of change in the level of that particular factor averaged over the levels of all other factors involved in the system. Thus, by definition, the average effect of A, also known as the main effect, may be evaluated using Eq. (1). The interpretation of each term of Eq. (1) is explained below.

$$ A=\frac{1}{4n}\left[a-(1)+ ab\hbox{-} b+ ac\hbox{-} c+ ab c\hbox{-} bc\right] $$
(1)

The effect of A when both B and C are at low levels is given by [a − (1)]/n. In a similar way, the effect of A when B is at low level and C is at high level is given by [ac − c]/n. [ab − b]/nrepresents the effect of A when B is at high level and C is at low level. Finally, the effect of A when both B and C are at high levels is given by [abc − bc]/n. It should be noted that the term n denotes the number of replicates of the experiment. In case of no replication, the value of n becomes 1.

In a similar way, the equations for main effects of factor B and C are given by Eqs. (2) and (3) respectively.

$$ B=\frac{1}{4n}\left[b+ ab+ bc+ ab c-(1)-a-c- ac\right] $$
(2)
$$ C=\frac{1}{4n}\left[c+ ac+ bc+ abc-(1)-a-b- ab\right] $$
(3)

For interaction between two factors say AB, Eq. (4) is used. The interaction AB is the difference between the average effects of A at two levels of B. Similarly, Eqs. (5) and (6) are used for estimating the interaction effects of BC and AC respectively.

$$ AB=\left[\frac{abc- bc-+ ab-b- ac+c-a+(1)}{4n}\right] $$
(4)
$$ BC=\frac{1}{4n}\left[(1)+a\hbox{-} b\hbox{-} ab\hbox{-} c\hbox{-} ac+ bc+ abc\right] $$
(5)
$$ AC=\left[(1)-a+b\hbox{-} ab\hbox{-} c+ ac\hbox{-} bc+ abc\right] $$
(6)

For three-factor interaction ABC, Eq. (7) is applicable which denotes the average difference between the interaction term AB at two different levels of C.

$$ ABC=\frac{\left[ abc\hbox{-} bc\hbox{-} ac+c\hbox{-} ab+b+a-(1)\right]}{4n} $$
(7)

The terms in the brackets of Eqs. (1) to (7) are also known as the contrasts in the treatment combinations (Montgomery 2013). Using the contrasts, the sum of squares for the effect estimates may be evaluated as per Eq. (8). Based on the sum of squares of various terms, the percentage contribution towards the response may be evaluated.

$$ SS=\frac{(Contrast)^2}{8n} $$
(8)

The concept discussed above for the case of three parameters may be extended for any number of parameters.

The present study has made use of the point estimate method (PEM) for the generation of samples required in the implementation of factorial design discussed above. A brief background of the original formulation of PEM along with the modification incorporated in the present study is discussed below.

Generation of samples through point estimate method

As discussed in the previous section, the factorial design approach requires the determination of response at various combinations of factors. One of the efficient ways of achieving this is with the application of point estimate method developed by Rosenblueth (1975). As per the original formulation, each random parameter is represented by two point estimates (Baecher and Christian 2003): low and high, given by Eq. (9)

$$ {\chi}^{\underline{+}}={\mu}_{\chi}\underline{+}{\sigma}_{\chi } $$
(9)

where μx is the mean and σx is the standard deviation of parameter X. For a more detailed discussion on the application of the point estimate method in the probabilistic analysis of rock slopes, the work of Ahmadabadi and Poisel (2016) may be referred. The above study highlights and summarizes advantages, limitations, and modifications to be made to the original formulation depending on the characteristic of the data considered. In the present study, the original formulation represented by Eq. (9) has been used for strength parameters. However, for geometric parameters, a modification incorporating the proposed improvement suggested by Hong (1996, 1998) is considered for the determination of point estimates as more than one skewed variables are involved. Thus, the point estimates in the present study are evaluated using Eq. (10) for skewed variables.

$$ {x}_i^{\underline{+}}={\mu}_{xi}+\left[{v}_{xi}/2\underline{+}\sqrt{n_s+{\left({v}_{xi}/ 2\right)}^2}\right]{\sigma}_{xi} $$
(10)

where μxi, vxi, and σxi represent the mean, skewness, and standard deviation of parameter xi. The term ns in the above equation represents the number of skewed parameters considered in the analysis.

Evaluation of strength reduction factor (SRF) of slope

The stability of the slope has been evaluated by determining the strength reduction factor (SRF) (Zienkiewicz et al. 1975, Naylor 1982, Donald and Giam 1988, Matsui and San 1992, Ugai and Leshchinsky 1995, Dawson et al. 1999, Griffiths and Lane 1999, Cheng et al. 2007, Wei et al. 2009, Fahimifar et al. 2012, Alemdag et al., 2014, Cai and Ugai 2000, Won et al. 2005, Wei and Cheng 2009, Griffiths et al. 2010, Yang et al. 2011, Rathod and Rao 2012, Shooshpasha and Amirdehi 2015, Zhang et al. 2015). For this, two-dimensional numerical modeling has been carried out using the finite element software RS2.0 (Rocscience Inc. 2015). This software package has been widely used in a number of numerical studies dealing with rock slope stability assessment through the Shear Strength Reduction (SSR) technique. Both persistent and impersistent joint sets along with widely adopted constitutive models for simulating rock mass behavior may be considered in this software package.

Numerical model

A schematic representation of the slope having two impersistent joint sets considered in the present study is shown in Fig. 3. The slope geometry has been adapted from the study carried out by Hammah et al. (2009). The slope has a height of 60 m and a slope angle of 60° with two impersistent joint sets J1 and J2. The spacing of the first joint set J1 is 2 m and that of the second joint set J2 is 1.5 m. In order to minimize the effect of artificial boundaries, the model dimensions have been kept as 240 m × 110 m (Wyllie and Mah 2004). The bottom boundary of the model has been fixed in all directions, whereas the lateral boundaries have been restrained in the horizontal directions. Mohr-Coulomb constitutive law has been considered for both the intact rock and the impersistent joint sets. In all the analyses, the horizontal to vertical stress ratio has been fixed as 1.0. The number of elements used in each model has been decided based on mesh sensitivity analyses. Values of various deterministic parameters are summarized in Table 2.

Fig. 3
figure 3

A schematic representation of the slope considered

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Table 2 Deterministic parameters considered (adapted after Hammah et al. 2009)
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Random parameters considered in the study

In the present study, the variability in terms of spatial configuration of two impersistent joint sets J1 and J2 has been characterized in terms of dip angle D1 and D2, the trace length T1 and T2 respectively as shown in Fig. 3. Moreover, the stochastic nature of the strength parameters is represented by cohesion Ci and friction angle ϕi for the intact rock and friction angle ϕj for the two joint sets. Thus, there are seven random parameters which have been considered for the initial factor screening phase and have been mentioned below for brevity.

  1. 1.

    Dip angle D1 of the first joint set

  2. 2.

    Trace length T1 of the first joint set

  3. 3.

    Dip angle D2 of the second joint set

  4. 4.

    Trace length T2 of the second joint set

  5. 5.

    Cohesion of intact rock Ci

  6. 6.

    Friction angle of intact rock ϕi

  7. 7.

    Friction angle of the first and second joint set ϕj

Consideration of seven random parameters in full factorial design requires simulation of total 128 (27) numerical models. Since the number of simulation is quite large, half factorial design has been considered as has been done in many cases of design of experiments. This brings down the total number simulations to 64 in the initial factor screening phase.

Distribution type and properties for random parameters

The choice of type of statistical distribution for random parameters has been made based on the suggestions made in a number of studies. Normal and truncated normal distribution has been suggested for the friction angle of the rock discontinuities and the intact rock (Mostyn and Li 1993, Nilsen 2000, Pathak and Nilsen 2004). Cohesion for rock is also reported to follow a normal distribution (Song et al. 2011). As per Priest (1993), Fisher distribution is capable to model the dip of discontinuities with considerable accuracy. In addition, the trace length is assumed to follow lognormal distribution (Bridges 1976, Priest and Hudson 1976, Baecher et al., 1977, Baecher and Lanney, 1978, Warburton 1980, Hudson and Priest 1983, Baecher, 1983, Dershowitz and Einstein 1988, Ehlen 2002, Meyer and Einstein 2002, Gumede and Stacey 2007, Weiss 2008, Zadhesh et al. 2014). In view of the aforementioned studies, the distribution of each random parameter along with the properties considered in the investigation is presented in Table 3.

Table 3 Random parameters adopted in the present study
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SRF from numerical model

The results of the initial factor screening phase implemented with the help of numerical models based on half factorial design are discussed in terms of SRF (strength reduction factor) in the present section. Table 4 summarizes the 64 runs made with various combinations of input random parameters. The corresponding factor of safety of the slope for each run has been shown as SRF in the last column. It may be noted that the minimum SRF of the slope is 0.58 (model run 12), whereas the maximum SRF is 1.57 (model run 49).

Table 4 Model runs and corresponding SRF for the initial factor screening phase
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In the case of the lowest SRF (model run 12), the strength parameters of the intact rock and the discontinuity are minimum, whereas the dip angle and the trace length of joint set 1 have maximum value. In comparison, the case of highest SRF (model run 49) is associated with maximum point estimate for the strength parameters and minimum point estimates for the geometric parameters of joint set J1. In order to quantify the effect of each individual parameter and the combined effect or interaction among them on SRF of the slope, ANOVA (analysis of variance) has been carried out.

Assessment of effects of random parameters

Based on the results of the initial 64 numerical simulations, the main and interaction effects have been identified through the analyses of SRF utilizing the concepts discussed in the “Concept of factorial design” section. Figure 4 shows the effect of each parameter on the SRF of the considered slope.

Fig. 4
figure 4

Effect of various random parameters on the SRF of the jointed rock slope. Effect of D1 (a), effect of T1 (b), effect of D2 (c), effect of T2 (d), effect of Ci (e), effect of Fricintact (f), effect of Fricjoint (g)

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From Fig. 4a to d, it may be observed that the dip and trace length of both the first and second joint set are having significant influence on the SRF of the slope as with variation these parameters may make the slope unstable (SRF < 1). In addition, the friction angle of the joint sets also has significant impact on the stability as evident from Fig. 4g.

In contrast, Fig. 4e and f suggest that change in SRF is very minor within the range of the strength parameters of intact rock considered for the study. Thus, it may be concluded that cohesion Ci and friction angle ϕi of intact rock do not have significant influence on the SRF. This finding may also be substantiated by comparing the results of model run 1 and model run 49 (refer to Table 4) where the intact rock strength properties are different, while the geometric and strength properties of the joint sets are same. A minor variation of 0.03 (1.57–1.54) in SRF is observed.

Assessment of combined effects of random parameters

Figure 5 presents the combined effect of the first joint set dip angle with other random parameters on the mean of the SRF of the slope. It may be understood that the interaction effect will be prominent if the plots are having higher difference in gradients. Figure 5 indicates that there is a significant influence of combined effect of the following parameters with the dip angle of the first joint set (D1): trace length of the first joint set (D1T1), dip angle of the second joint set (D1D2), joint friction angle (D1ϕj). All other interaction plots have slopes which are almost parallel thereby indicating that they do not influence the SRF.

Fig. 5
figure 5

Combined effect of dip angle of the first joint set (D1) with other random parameters. Interaction between D1 and T1 (a), interaction between D1 and D2 (b), interaction between D1 and T2 (c), interaction between D1 and Ci (d), interaction between D1 and Fricintact (e), interaction between D1 and Fricjoint (f)

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Similarly, Fig. 6 represents the combined effect of the trace length of the first joint set (T1) with other random parameters. Most of the plots shown are parallel indicating very minor combined influence of various random parameters with trace length T1. However, Fig. 6a shows relatively higher difference in gradient indicating that there may be contribution of combined effect of dip angle of the first joint set with T1 (D1T1).

Fig. 6
figure 6

Combined effect of trace length of the first joint set (T1) with other random parameters. Interaction between T1 and D2 (a), interaction between T1 and T2 (b), interaction between T1 and Ci (c), interaction between T1 and Fricintact (d), interaction between T1 and Fricjoint (e)

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Similarly, combined effect of other parameters has been shown in Figs. 7, 8, 9, and 10. The gradient of the interaction plot (Fig. 7a) signifies that the SRF of the slope is dependent on the combined effect of dip angle and trace length of the second joint set (D2T2). Moreover, Fig. 7d also suggests that there may be combined effect of dip angle of the second joint set with joint friction angle (D2ϕj). Apart from the above-mentioned plots, all other interaction plots being parallel signify negligible combined effect.

Fig. 7
figure 7

Combined effect of dip angle of the second joint set (D2) with other random parameters. Interaction between D2 and T2 (a), interaction between D2 and Ci (b), interaction between D2 and Fricintact (c), interaction between D2 and Fricjoint (d)

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Fig. 8
figure 8

Combined effect of trace length of the second joint set (T2) with other random parameters. Interaction between T2 and Ci (a), interaction between T2 and Fricintact (b), interaction between T2 and Fricjoint (c)

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Fig. 9
figure 9

Combined effect of cohesion of intact rock (Ci) with random parameters. Interaction between Ci and Fricintact (a), interaction between Ci and Fricintact (b)

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Fig. 10
figure 10

Combined effect of friction angle of intact rock (ϕi) with joint friction angle (ϕj)

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Quantitative assessment of effects of random parameters

The main and combined effects of various random parameters, presented in Figs. 4, 5, 6, 7, 8, 9, and 10, may further be quantified as discussed in the “Assessment of influence of various parameters” section. Table 5 summarizes the results of ANOVA in terms of the effect estimate, percentage contribution, p value, and Fo for all the main and interaction terms considered in the present work.

Table 5 ANOVA analysis on the SRF values from the initial factor screening phase
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It may be observed that joint friction angle has maximum influence on the SRF of the slope. The percentage contribution for joint friction angle is estimated to be 35.49%. Next, higher contribution comes from dip angle and trace length of joint set J1 which characterize geometric properties. In addition, dip angle and trace length of joint set J2 also seem to have significant contribution in influencing stability of slope. Among the interaction terms, the combined effect of dip angle of both the joint sets (D1D2) has the maximum influence.

Further, the significance of each parameter is evaluated through the Fisher statistic F0 keeping the significance level as 0.05 (Montgomery 2013). Based on the F0 value, the significant parameters identified are D1, T1, D2, T2, and ϕj.

However, the quantification presented in this section requires a closer scrutiny as these results are based on the assumption of linear variation of SRF. Hence, more accurate quantitative assessment may be made by identifying curvature effects, if any, based on the influencing parameters identified at this stage. For this purpose, central composite design (CCD) is implemented subsequently by taking into consideration the important influencing parameters identified at this stage. Further, a mathematical model (response surface) capable of predicting the SRF of the considered slope has been framed.

Response surface using central composite design

The initial factor screening phase discussed in the “Assessment of effects of random parameters” section identifies the parameters affecting the stability of slope with an assumption of linear variation of SRF within the range of random parameters considered. However, the assumption of linear variation may not be accurate, and hence, a more rigorous analysis needs to be conducted to identify and account for the curvature effect, if any, on the SRF. In this regard, the central composite design (CCD), which is one of the most versatile models for evaluating the second order response surface, has been used (Box and Wilson 1951, Myers et al. 2004, Myers et al. 2009).

Background of CCD

The popularity of CCD is primarily attributed to its sequential nature which enables the partitioning of the design into two subsets. The first subset is utilized to estimate the linear and two-factor interaction estimates. If the results from the analysis of the first subset indicate the presence of significant curvature in the response, the second subset is then run to estimate the curvature effects.

A schematic representation of CCD for three parameters has been shown in Fig. 11. In the case of k factors, the design involves 2k factorial points, 2k axial point, and n′ (n′ is variable) center points with the first order variation in the model represented by the factorial points. In case of existence of significant curvature, the axial points are used for the efficient determination of the quadratic terms. The axial point is dependent on a parameter α which denotes the axial distance from the center point. Thus, the CCD facilitates in identification of nature of variation of the response as a function of various input parameters.

Fig. 11
figure 11

Design points in CCD model for the case of three parameters

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Application of CCD in the present study

In the present study, since five parameters are identified to be significant, the CCD model comprises 25 (32) factorial points, 2 × 5 (10) axial points, and center points (equal to mean value). The axial distance α is evaluated using the relation \( \alpha =\sqrt[4]{2^N} \), where N represents the number of significant parameters (five in this case). Thus, the value of α for the present study is computed to be 2.378.

The combination of the input parameters and the corresponding SRF values for the CCD analyses has been tabulated in Table 6. Here, the − and + terms represent the factorial points, whereas α corresponds to the axial points. The term 0 represents the mean values corresponding to the center points.

Table 6 SRF for the central composite design (CCD)
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Formulation of response surface

In order to develop an appropriate response surface, ANOVA has been performed on the SRF values obtained from the CCD analyses. The effect estimates, percentage contribution, and the p value of the linear and quadratic terms for all the factors are evaluated as shown in Table 7. It is observed that linear (D1) and quadratic (D1(Quad)) terms of the dip angle of the first joint set J1 have a significant contribution to the response with the quadratic term showing larger contribution. The next significant contribution is attributed to the friction angle of the joints which characterizes the resistance against instability. The contribution from linear terms T1 of joint set J1 is also comparatively significant. The other terms identified to contribute towards the response are the dip angle D2 and trace length T2 of joint set J2 and the interaction terms D1D2 which represent the block size and shape formed as a result of intersection of joint sets. In addition, the p value of the above parameters is lower than the significant value (0.05) assumed in the present case confirming the importance of these terms (Montgomery 2013).

Table 7 ANOVA analysis on the SRF of central composite design (CCD)
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Since quadratic term of dip angle along with linear terms associated with the dip, trace length, and joint friction angle has been identified to be significant, hence, a second order equation has been selected to frame the response surface. Equation (11) represents the response surface formulated for the present slope.

$$ {\displaystyle \begin{array}{l} SRF=1.6-0.0798{D}_1-0.315{T}_1+0.0071{D}_2+0.468{T}_2+0.0923{\phi}_j+0.000704{D}_1^2\\ {}+0.00225{T}_1^2-0.000057{D}_2^2-0.0063{T}_2^2+0.000381{\phi}_j^2-0.000657{D}_1{T}_1+0.000379{D}_1{D}_2\\ {}-0.000095{D}_1{T}_2-0.000587{D}_1{\phi}_j+0.001940{T}_1{D}_2+0.00792{T}_1{T}_2+0.00013{T}_1{\phi}_j-0.00453{D}_2{T}_2\\ {}-0.00453{D}_2{T}_2-0.00042{D}_2{\phi}_j-0.00153{T}_2{\phi}_j\end{array}} $$
(11)

To verify the effectiveness of the response surface framed on the basis of the CCD analyses, comparison of SRF calculated based on Eq. (11) has been made with the SRF of the numerical models evaluated in the initial factor screening phase (Table 4). The comparison of SRF is presented in Fig. 12 clearly highlighting the effectiveness of the equation in predicting the SRF of the considered slope.

Fig. 12
figure 12

Comparison of SRF between numerical analysis and response surface

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Variation of SRF with influencing parameters

For better insight into the variation of the behavior of slope characterized by SRF, 3D surface plots have been generated and presented in Fig. 13. The plot shows the variation in SRF as a function of two input parameters with all other parameters held constant at their mean values. From Fig. 13e to h, a noticeable curvature may be observed involving parameter D1 suggesting that the SRF decreases with increase in dip angle of up to 50°. With further increase in the dip angle D1, the SRF shows an increase. This is mainly attributed to the interaction of the dip of joint set J1 with the slope angle which is 60°. For dip angle D1 of up to 50°, joint set J1 daylights into the slope, and hence, chances of rock blocks sliding out from the slope are higher.

Fig. 13
figure 13

a–j 3D surface plots of random parameters against the SRF of the slope

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However, as the dip angle exceeds 50°, joint set J1 becomes almost parallel to the slope lowering the kinematic feasibility of the rock blocks to move out from the slope. This subsequently results in an increase of SRF which is represented by the upward curvature in Fig. 13e to h. The other prominent curvature is observed in Fig. 13a representing the combined effect of dip and trace length of the second joint set. A drop in SRF is observed for higher values of dip angle D2 and trace length T2 highlighting the influence of the second joint set in affecting the slope behavior.

Application of present methodology in explaining slope behavior

On the basis of the influencing parameters and their combined relative influence identified as discussed in the previous sections, an insight into the slope behavior can be made. The behavior seems to be dominated by the combined action of shearing along both the joint sets as schematically shown in Fig. 14 and their combined interaction. The shearing action causes sliding of rock blocks along joint set J1 coupled with detachment along joint set J2. Depending on the dip angle D1 of joint set J1, there are two possible types of behavior which may be identified from the present study.

Fig. 14
figure 14

Schematic representation of shearing action along both the joint sets

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Figure 15 provides a schematic representation of the probable behavior when the dip angle D1 is much larger than the friction angle of the joints (i.e., high point estimates). In such cases, sliding along joint set J1 is prominent which dominates the movement of the rock blocks. In such a scenario, the presence of the second joint set J2 facilitates the formation of a release surface which eventually leads to the outward movement. A close view of the discussed behavior is provided in Fig. 16 which shows the response of slope for model run 12 (refer to Table 4). A detailed discussion on this is provided below.

Fig. 15
figure 15

Slope response dominated by sliding force along joint set J1

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Fig. 16
figure 16

Sliding of rock blocks along joint set J1 facilitated by the formation of release surface across joint set J2 (model run 12)

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Referring to Table 4, in the case of model run 24, the trace length of joint set J2 has a lower value (lower point estimate) in comparison with model run 16, while the other variables remain identical. As the resistance along the joint sets J1 and J2 is the same in both cases, the difference in SRF result can be explained on the basis that a higher trace length of joint set J2 results in larger block size subsequently leading to higher chances of formation of a release surface. Under the action of the sliding force along joint set J1, detachment of rock block occurs across joint set J2 eventually leading to unstable behavior (refer to Fig. 15). It is thus observed that the SRF in the case of model run 24 is 0.95, whereas for model run 16, it drops to 0.83. Similar reason holds true for the case of model run 24 and model run 28.

To further add to the discussion, the role of the resistive force characterized by the friction angle can be made by comparing model run 16 and model run 32 (refer to Table 4). In both cases, the kinematic feasibility of slope instability remains identical as the geometric parameters of the joint sets and the slope is the same. However, in the case of model run 32, the friction angle of the discontinuity is 16° in comparison with 24° for model run 16. The lower resistance subsequently results in a lower SRF of 0.64 in the case of model run 32 in comparison with model run 16 which has a higher resistance and subsequently a higher SRF of 0.83.

Other prominent behavior observed is associated with the case when the dip of joint set J1 is comparable with the friction angle of the discontinuities. In such cases, detachment and overturning of rock blocks across joint set J2 as shown in Fig. 17 may occur. In fact, a closer view of the results of model run 45 as shown in Fig. 18 confirms this behavior. In this particular case, block detachment and overturning lead to instability and SRF of 0.7. Also a comparison of model run 5 (refer to Table 4) with model run 45 highlights the curvature effect on SRF as highlighted in the response surface plot of D2T2 as shown in Fig. 13a. The surface plot suggests a sharp decrease in SRF as both D2 and T2 attain respective higher values. Thus, in the case of model run 5, which has a lower trace length, the SRF is 1.02 in comparison with SRF of 0.7 in the case of model run 45.

Fig. 17
figure 17

Possible mechanism of slope behavior dominated by overturning of rock blocks across joint set J2 in case of shallow dip of joint set J1

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Fig. 18
figure 18

Instability driven by the detachment and rotation of rock blocks across joint set J2 (model run 45)

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Summary and discussion

The assessment of stability of jointed rock slope has been a challenging and complex task due to the inherent variability associated with geological medium. Although probabilistic-based approaches have found wide application in slope stability investigations, a rigorous attempt to identify and quantify the relative contribution of various parameters on the integrity of the slope is few in numbers.

In this paper, the influence in variability of both the strength and geometric parameters associated with a jointed rock slope has been investigated. The random parameters considered include the cohesion of the intact rock and the friction angle of both the intact rock and the joint sets. In addition, the variability in geometric parameters characterized by the dip angle and the trace length of the discontinuities has also been taken into account. In order to optimize the process of investigating the combined effects of various parameters on the stability of the slope, concepts of design of experiments (DOE) through factorial design approach have been adopted. A brief description of each component of the adopted methodology implemented in the numerical framework has been provided.

Based on the results of the initial factor screening phase, the important factors affecting the behavior of the slope have been identified. It is observed that the performance of a slope with impersistent joints sets is majorly governed by the strength and geometric parameters of the joint sets. Moreover, it is concluded that cohesion and friction angle of intact rock have minimal effect on SRF of the slope configuration considered in the present study. Based on these findings, response surface predicting the SRF of the slope as a function of influencing parameters has been framed using the numerical results from the central composite design (CCD) approach. The response surface framed has shown good agreement with the SRF values evaluated using numerical model runs highlighting its prediction capability. Further, the nature of variation in SRF of the slope with different parameters has also been examined through surface plots.

The present investigation advances the insight into the behavior of the slope by identifying the important parameters and their combined influence. Two dominant types of slope behavior have been identified. When the dip of the first joint set exceeds the joint friction angle, the behavior is mainly governed by the sliding of rock blocks along the first joint set J1 facilitated by the formation of release surface across the second joint set J2. Another dominant behavior observed includes the sliding and overturning of rock blocks along the second joint set. This kind of behavior is observed when the dip of the first joint set becomes comparable with the joint friction angle.