1 Introduction

Rockfall is a significant hazardous event in the hilly regions as it has the potential to damage infrastructures and causes severe injuries or fatalities (Badger and Lowell 1992). The geotechnical setup and geodynamic behaviour of the slopes are responsible for the frequency of landslides in the Himalayan region (Kumar et al. 2018). The failure in slopes is governed by numerous factors such as geomorphological features (Hoek and Bray 1981), freeze–thaw cycle (McCarroll et al. 1998), seismic activities (Valagussa et al. 2014), rainfall/groundwater (Jaswal et al. 2020; Wei et al. 2014) and vegetation. The study conducted by Wyllie and Norrish (1996) reveals that rainstorms cause 30% of the rockfall activities, 21% due to the freeze–thaw process, and the remaining due to fractured rock, wind, etc.

Asteriou et al. (2012) describe that rockfall activity depends on numerous factors, such as slope characteristics i.e. slope height, slope angle, slope roughness etc., block characteristics i.e. block mass, block shape, block size, block strength etc. Recently, Dattola et al. (2021) investigated the effect of shape and rotation motion on the rockfall impacts based on visco-plastic theory. The higher slope height tends to increase the kinetic energy associated with the falling rock blocks. The slope angle affects the rockfall trajectory and the shear stress acting on the potential failure plane (Roth 1983). Slope excavation is one of the main factors causing an alteration in slope geometry (McColl 2015). The various types of rockfall motions, such as fall, roll and bounce, occur after separating a rock block from the detachment point (Ritchie 1963). The trajectories of falling rock blocks can be influenced by the coefficient of restitution (CoR), roughness of the slope surface, coefficient of rolling friction and the coefficient of sliding friction. Irfan and Chen (2017) developed mathematical equations to predict the rockfall trajectories.

CoR is defined as the ratio of the velocities before and after the impact of rock block on the slope’s surface. In the rockfall modelling, normal and tangential CoR (\({\text{R}}_{{\text{n}}}\) and \({\text{R}}_{{\text{t}}}\)) are being used separately to calculate the rebound velocity components in the normal and tangential direction (Wu 1985). Richards (1988) observed that the values of CoR (especially Rn) have a substantial effect on the rockfall trajectories. The value of \({\text{R}}_{{\text{n}}}\) and \({\text{R}}_{{\text{t}}}\) are used to quantify the energy dissipation during an impact in the prediction of rockfall (Buzzi et al. 2012). Asteriou (2019) demonstrates the influence of rockfall impact and rotational motion on the CoR. Slope roughness is characterized by the waviness and unevenness of the slope (ISRM 1981). The large-scale waviness contributes to resistive shear stress. These irregularities in the slope surface are responsible for variation in rockfall trajectories, as they alter the impact angle. Slope roughness plays a crucial role in the rockfall assessment. Verma et al. (2018) performed the sensitivity analysis that shows it affects the rockfall assessment outcome. Rolling friction coefficient (\(\mu_{{\text{r}}}\)) represents the slope's resistance to the angular velocity of the rock block, whereas sliding friction coefficient (\(\mu_{{\text{s}}}\)) represents the resistance offered by the slope surface to the sliding of the rock block. The rolling motion is restricted to a limiting angle of inclination. Peng (2000) investigated the characteristics of \(\mu_{{\text{r}}}\) by varying the value of \(\theta_{r}\) ranging from 1° to 5°, and observed that under the rolling conditions, coefficient of rolling friction does not play a significant role. It is actually the coefficient of sliding friction that offers a significant resistance to the movement of the rock block as compared to the coefficient of rolling friction.

A major part of the study was concentrated along the NH-44A highway since it is a lifeline of the Aizawl city, as this is the only link between the Aizawl city to the Lengpui Airport. The stability of rock slopes in the vicinity of Aizawl city varies from partially stable to unstable (Sardana et al. 2019a; 2019b). The preliminary study, based on Rockfall Hazard Rating System (RHRS) carried out by Verma et al. (2021), shows that the road cuts along the Lengpui-Aizawl highway are prone to rockfall and comes under moderate to higher urgency. Hoek (1999) allocated the rockfall hazard rating into three categories of urgency: high, medium and low, indicating the extent of the hazard. The slopes with RHRS scores higher than 500 require immediate attention. Based on RHRS scores, the slopes along the Lengpui-Aizawl highway are highly vulnerable and a continuous threat to the rockfall. Therefore, these slopes required a detailed rockfall assessment. Moreover, this highway has witnessed several rockfall events in the past few years (Sardana et al. 2020; Verma et al. 2019; Lallianthanga et al. 2013). The present work aims to develop a set of equations using parametric analysis to determine the rockfall output parameters: bounce height, velocity, and kinetic energy. The equations were developed using multivariate linear regression to guide predicting risk assessment concerning the bounce height, velocity, and energy accompanying the falling rock blocks. These equations serve the purpose of input parameters required by engineers associated with designing suitable mitigative measures against the adverse effect of rockfall.

2 Field Investigations

2.1 Location

The slopes under investigation are located along the Lengpui-Aizawl highway (NH-44A). The field study covers a total of 13 rock slopes from Lengpui Airport toward Aizawl city. The stretch located between the latitudes \(23^{^\circ } \;50^{\prime } \;19.85^{\prime \prime } N\)–\(23^{^\circ } \;46^{^{\prime}} \;52.95^{^{\prime\prime}} N\) and longitudes \(92^{^\circ } \;37^{\prime } \;27.01^{^{\prime\prime}} E\)–\(92^{^\circ } \;40^{^{\prime}} \;35.23^{^{\prime\prime}} E\). The study area comes under toposheet no. 83D/15, 84A/10, and 83H/4 of the Survey of India. The highway is constantly under the threat of rockfall activity at an elevation ranging from 273 to 413 m. The coordinates and elevation of the rock slopes are reported in Table 1.

Table 1 Details of road cuts along the Lengpui-Aizawl highway
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2.2 Geology

The geological investigation reveals that the area is predominantly overlain by Neogene sedimentary rocks of Tipam and Surma group formation. In Mizoram, the sedimentary column is a repetitive succession of arenaceous and argillaceous rocks comprising sandstone, shale, mudstone, silty-sandstone, silty-shale and their mixtures. Surma group covers a large portion and is subdivided into Bhuban and Bokabil formations. Bhuban formation is further divided into Upper, Middle and Lower Bhuban (Ganju 1975). The entire Aizawl city includes top and middle Bhuban dispositions of the Surma group of rocks (Kesari 2011). This group of rocks embraces alternate shale beds, siltstone, sandstone and mudstone of diverse thicknesses. Sandstones are hard, compact and stable, whereas shale beds are brittle compared to sandstone. They include bands of micaceous-felspathic and weathered sandstone (Lallianthanga et al. 2013). An arenaceous and argillaceous batch of rocks lies in relatively upper and lower ground, respectively. Reconnaissance traversing from Aizawl to Champhai resulted in identifying a Barail batch of rocks in and around the Champhai subdivision, Aizawl district, and Bhuban in the west. The geological map of the study area is shown in Fig. 1.

Fig. 1
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Geological map of Mizoram showing Aizawl and Lengpui area (after Ram and Venkataraman 1984)

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2.3 Site Characterization

Extensive site characterization includes estimating slope height, slope angle, slope geometry, slope orientation, joint orientation and discontinuity conditions and the dimension of potential falling rock blocks. Figure 2 and 3 shows the field photographs of rock slopes along this highway. Figure 4 shows the different sizes of rock blocks observed at the site. The topographical and lithological details have been provided in Table 1. Slope angle, weight of falling blocks and slope height are major parameters that affect the falling blocks' path and energy. Therefore, the histograms have been plotted, showing the values for the three factors mentioned above (Fig. 5). The height of the slope varied from 5 to 31 m; therefore, the slope height has been divided into three categories, ≤ 10 m, 11–20 m and > 20 m (Fig. 5a). Similarly, the data was divided into three categories, for slope-angle, ≤ 75°, 75–85° and > 85° (Fig. 5b); and for block-weight, ≤ 300 kg, 300–800 kg, and > 800 kg (Fig. 5c). An average height and a range of angles have been considered for 13 rock slopes for the slope height and slope-angle parameter. The block weight was estimated to consider approximately 150 fallen rock blocks observed during the site investigation.

Fig. 2
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Field photographs showing road cuts (L-1–L-9) along NH-44A, Aizawl

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Fig. 3
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Field photographs showing road cuts (L-10–L-13) along NH-44A, Aizawl

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Fig. 4
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Field photographs showing falling rock blocks along NH-44A, Aizawl

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Fig. 5
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Histogram plots showing the frequency of a slope height, b slope angle and c weight of the falling rock blocks

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3 Laboratory Investigations

The restitution coefficient (CoR) plays a crucial role in the rockfall assessment. It is associated with the dissipation of energy in the course of the impact. The normal restitution coefficient (Rn) has been determined in the laboratory using a setup (Fig. 6). The experimental setup consists of a steel frame with a releaser fixed at the top and a base casing to accommodate the rock slab. The height of the releaser can be varied by adjusting the threaded rod attached to it. The supporting frame is mounted with a scale of 1 m length graduated in millimetres. The resting slab's angle can be varied by adjusting the pinion attached to the base frame (Fig. 6). A high-speed camera was used for image exposures in excess of 1/1000 or frame rates of more than 250 frames per second. It was used to record fast-moving objects as photographic images onto a storage medium. After recording, the images stored on the medium can be played back in slow-motion.

Fig. 6
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Experimental setup to determine CoR values in the laboratory

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The rock slab was placed under the sample holder and a high-speed camera was fixed parallel to the setup. The almost spherically shaped small rock sample was kept in the holder and released such that it experienced a free fall with a drop height of 0.66 m and bounced on the approximately rectangular rock slab. The test was repeated on the same rock slab with different small rock pieces. The camera captured the motion; later, the recorded video was transferred to the computer and analysed by the motion tracking software (Tracker software).

The maximum positive value of \({\text{V}}_{{\text{f}}}\) and maximum negative value of \({\text{V}}_{{\text{i}}}\) has been taken to estimate Rn as per Eq. 1.

$${\text{R}}_{{\text{n }}} = \frac{{{\text{V}}_{{\text{f}}} }}{{{\text{V}}_{{\text{i}}} }}$$
(1)

where \({\text{V}}_{{\text{f}}}\) and \({\text{V}}_{{\text{i}}}\) are the velocities of the falling rock block, after and before the bounce on the rock slab. Ten values have been plotted in Fig. 7 determined from the CoR setup, which shows 50% of the values come under less than 0.46 CoR, 20% higher than 0.5 CoR and the remaining values fall between 0.46 and 0.5.

Fig. 7
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CoR values estimated in the laboratory

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4 Rockfall Modelling

The rockfall modelling has been performed using the RocFall software package from Rocscience. The analysis in this package is based on either lumped mass or rigid body approach. In the lumped mass approach, each rock block is considered as a very small spherical particle. Therefore, the rock block's mass is considered, and the size or shape is not considered in the analysis. In contrast, the rigid body approach considers the shape and size of the rock block depending on its mass. The critical input parameters considered in the rockfall modelling are slope height, slope angle, rock block weight, source of the rockfall (i.e. seeder point), CoR and slope roughness. The major output parameters determined in the rockfall analysis are its trajectories, bounce height, velocity and energy at any point.

In this study, rockfall modelling has been performed using lumped mass formulation. The values of slope height (10–30 m), slope angle (65–85 deg) and the weight of the block (100–1000 kg) have been estimated based on extensive field investigations (Figs. 2, 3, 4). The normal restitution coefficient values have been estimated in the laboratory (Fig. 6) and the range of slope roughness has been considered from the literature (Verma et al. 2018; RocScience 2016; Srikanth 2015). Figure 8 shows rockfall modelling from the slope height of 10 m with a slope angle of 65°.

Fig. 8
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Rockfall modelling showing the slope geometry, slope material, seeder point and trajectories of the falling rock blocks

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A total of 243 simulations have been performed by varying the three sets of values for these five input parameters. Three-level values of the aforesaid five parameters were varied, usually referred to as low, intermediate and high values, as given in Table 2.

Table 2 Parameters with three levels of value (low, intermediate, and high) used in rockfall modelling and parametric study
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5 Parametric Analysis

A parametric analysis has been performed to observe the effects of rockfall input parameters on the rockfall output parameters. Further, multivariate linear regression has been used to develop the generalised equations to predict the rockfall outcome. One of the five input parameters was varied while keeping the other four constant to observe its influence on the output parameters. The procedure was repeated for each input parameter. A factorial design of experiments was used in the parametric study (Sinha 2013; Montgomery et al. 2011). In the factorial experiment with k factors (here, k = 5), each factor has three levels of value (provided in Table 2), will need \(3^{{\text{k}}}\) number of runs (i.e. \(3^{5} = 243\)). This is recognized as \(3^{{\text{k}}}\) factorial design and read as three-level ‘k’ factor design.

5.1 Factorial Plots for Bounce Height, Velocity and Kinetic

The factorial plots were plotted using the Minitab software package to observe the main effects among the rockfall input parameters for their different values on the rockfall output parameters. Figure 9 shows the main effect of slope height, slope angle, block weight, CoR and slope roughness on the bounce height, velocity and kinetic energy.

Fig. 9
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Main effects plot for a bounce height, b velocity, and c kinetic energy associated with the falling rock blocks

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The factorial plots reveal that, for bounce height, the maximum variation is shown by slope-angle (Fig. 9a); therefore, slope-angle will influence the bounce height higher than the other four rockfall input parameters. Also, no variation was observed in block weight; hence, it will not affect the bounce height in any manner. The influence of slope roughness will be higher than the slope height and CoR, and lower than the slope angle. In the factorial velocity plots (Fig. 9b), slope height shows higher variation followed by slope angle, while the remaining three parameters were constant. Therefore, slope height is the major influencing input for the velocity of the rockfalls. For the factorial plot of kinetic energy (Fig. 9c), the major variation was observed by block weight and slope height. The slope angle also shows a minor variation. Therefore, these three inputs will affect the kinetic energy associated with the falling rock blocks.

5.2 Multivariate Linear Regression for Bounce Height, Velocity and Kinetic Energy

Multivariate linear regression analysis is used to develop the linear equations that can predict the bounce height, velocity and kinetic energy of rockfalls. The data used in the regression analysis has been generated through the 243 rockfall simulations. A linear form of regression was taken for analysis, as given in Eq. 2.

$${\text{Output}} =\, {\text{a}}_{1} + {\text{a}}_{2} \times {\text{W}} + {\text{a}}_{3} \times {\text{H}}_{{\text{s}}} + {\text{a}}_{4} \times \theta + {\text{a}}_{5} \times {\text{CoR}} + {\text{a}}_{6} \times {\text{R}}_{{\text{s}}}$$
(2)

where, a1 through a7 are constants. The output parameters are Bounce height, Kinetic energy and Velocity. The linear regression analysis aims to find out the respective constants. Minitab software package was used to carry out regression analysis.

5.2.1 Observation of the Results for Bounce Height, Velocity and Kinetic Energy

The regression analysis was performed using ‘Analysis of Variance’ (ANOVA) (Table 3). The ‘P-value’ in the last column of the ANOVA section of Tables 3, 4, 5 indicated the significance of the model. It represents the goodness of fit; the lower the value, the better the fit. The P-value must be lesser than 0.05 for the model to fit into the data. For all three outputs, bounce height, velocity and kinetic energy, its value is 0.00 (i.e. less than 0.05); therefore, the model is a good fit.

Table 3 Result of regression analysis for bounce height versus parameters
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Table 4 Result of regression analysis for velocity versus parameters
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Table 5 Result of regression analysis for kinetic energy versus parameters
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Similarly, for the coefficients section in Tables 3, 4, 5, P-value represents the significance of the variable coefficients. In case if P-value is higher than 0.05, it indicates that the coefficient estimate is not consistent as it would have too much variation or dispersion. The individual coefficients column gives the respective constants of the regression equation. Here, in bounce height, the ‘P-value’ of all the parameters is 0.000, except the ‘block weight’. Therefore, the model is a good fit for the data, whereas the block weight parameter has wide variations and is not a good fit for the data. R-sq = 74.31% value indicates that one can predict bounce height’s value with 74.31% confidence. The value of R-sq (adj) = 74.79% suggests that this model can be used to explain 74.79% of the present data. In the velocity, the ‘P-value’ for slope height and slope angle was less than 0.05, indicating the model is a good fit for the data. However, for the remaining three parameters, their values are higher than 0.05 (i.e.1.000, 0.980, and 0.992), indicating that these variables' coefficient estimates are less reliable than slope height and slope angle. In the kinetic energy, block weight, slope height and slope angle have P-values of 0.000 or 0.001, indicating that the model is a good fit. However, for the remaining two input parameters, its value was 0.992 and 1.000 (i.e. higher than 0.05). Hence, the coefficients for these two variables can be considered less reliable.

Figure 10 shows the residual plots of the regression analysis for rockfall output parameters. The two major plots are the normal probability plot and histogram plot. The purpose of a normal probability plot is to identify outliers of the data, whereas a histogram plot represents the distribution of the data. The outliers of data and distribution of the data for bounce height, velocity and kinetic energy have been shown in Fig. 10a–c respectively.

Fig. 10
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Residual plot of the regression analysis for a bounce height, b velocity c kinetic energy associated with the falling rock blocks

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5.2.2 Regression Equations

According to the regression analysis results in Tables 3 through 5, the coefficients of equations are substituted in Eq. 2 to derive the equations of bounce height, velocity and kinetic energy, respectively, for the rockfalls. These equations are given below (Eqs. 3 through 5).

$${\text{Bounce height}} = - 17.87 + 0.11 \times {\text{H}} + 0.18 \times \theta + 7.76 \times {\text{CoR}} + 0.52 \times {\text{R}}_{{\text{s}}} \;\left( {{\text{R}}^{2} = 75.31} \right)$$
(3)
$${\text{Velocity}} = 3.15 + 0.49 \times {\text{H}} + 0.07 \times \theta \;\left( {{\text{R}}^{2} = 99.17} \right)$$
(4)
$${\text{Kinetic energy}} = - 147.4 + 0.17 \times {\text{W}} + 4.65 \times {\text{H}} + 0.73 \times \theta \;\left( {{\text{R}}^{2} = 88.53} \right)$$
(5)

5.3 Statistical Validation of the Regression Equations

The validation has been performed by plotting the predicted values estimated from the developed equations versus values of output variables i.e. bounce height, velocity and kinetic energy observed through rockfall simulations and measuring its correlation. Plots of the correlation of the predicted versus observed values are given in Fig. 11a through 11c for bounce height, velocity and energy, respectively. The correlation coefficient \(\left( {{\text{R}}^{2} } \right)\) of 0.78, 0.98 and 0.92 for bounce height, velocity, and energy have been observed, respectively, indicating the confidence level with which these equations can predict the output parameters for rockfall at specified locations.

Fig. 11
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Correlation of a bounce height b velocity c kinetic energy associated with the falling rock blocks

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6 Design Sample of Rockfall Barrier and Other Mitigative Measures

Vogel et al. (2009) explained selecting appropriate protective measures based on falling rock blocks’ energy. The commonly used protective measures worldwide are benched slopes, ditches, rockfall barriers, gabion walls, nets, rock sheds, earth dams and reinforced dams (Fig. 12).

Fig. 12
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Selection of various protection structures based on their energy absorbing capacity (after Vogel et al. 2009)

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A rockfall barrier is one of the suitable preventive measures for rockfall hazards. The crucial properties to design a rockfall barrier are capacity (energy), height, inclination and location. The barrier's design can be achieved by equations proposed by (Peila and Ronco 2009).

According to the energy design equation, the energy absorbed by the barrier must be greater than the energy computed by the simulation program (Eq. 6).

$${\text{E}}_{{\text{d}}} - \frac{{{\text{E}}_{{{\text{ETA}}}} }}{{\gamma_{{\text{E}}} }} < 0$$
(6)
$${\text{h}}_{{\text{d}}} > \left( {{\text{h}}_{{{\text{b95}}}} \times \gamma_{{{\text{Tr}}}} \times \gamma_{{{\text{Dp}}}} + {\text{f}}_{{\text{b}}} } \right)$$
(7)

where \({\text{E}}_{{{\text{ETA}}}}\) is the energy of barrier certified by European Technical Approval (ETA), \({\text{E}}_{{\text{d}}}\) is design energy for a barrier that can be computed through simulation, \({\text{h}}_{{\text{d}}}\) is the design height of the barrier, \({\text{h}}_{{{\text{b95}}}}\) is 95 percentile of the height of the trajectories of falling rock blocks over the slope (i.e. bounce height), \({\text{f}}_{{\text{b}}}\) is half of the average size of the falling rock block, \(\gamma_{{\text{E}}}\), \(\gamma_{{{\text{Tr}}}}\) and \(\gamma_{{{\text{Dp}}}}\) are the safety factors concerning energy computation, trajectory computation and slope discretization quality, respectively.

The rockfall assessment conducted by Verma et al. (2018), using Rocfall software package, reveals that kinetic energy was estimated to be 58.2 kJ. Subsequently, using Eq. 6, the minimum barrier capacity was 75.7 kJ. Similarly, the bounce height was 1.4 m and using Eq. 7. The minimum barrier height was 2.1 m. Therefore, a rockfall barrier of a minimum of 2.5 m in height and 100 kJ capacity was proposed to arrest all the falling blocks (Fig. 13a). Considering the same inputs parameters, the bounce height and kinetic energy calculated through Eqs. 3 and 5 were 1.5 m and 50.5 kJ, respectively. Further, using Eqs. 6 and 7, the minimum barrier height and capacity were 2.2 m and 65.6 kJ, respectively (Fig. 13b). The calculated values show a 5–13% variation between the present study and Verma et al. (2018). However, the overall proposed height and barrier capacity remain the same, i.e. 100 kJ of barrier capacity with a barrier height of 2.5 m.

Fig. 13
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Histogram plots showing a comparison of the present study with Verma et al. (2018) for a kinetic energy, and b bounce height

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The magnitude of energy and the bounce height can be estimated through Eqs. 3 and 5 instead of computing through the rockfall simulation package. Similarly, for selecting other energy-based protection structures suggested by Vogel et al. 2009, the energy magnitude can be estimated using Eq. 5.

7 Conclusion

The present study includes (i) the determination of input parameters through an extensive field investigation of 13 rock slopes carried out at the Lengpui-Aizawl highway, (ii) the testing carried out in the laboratory on collected rock samples from the site and (iii) a total of 243 simulations of rockfall by varying three level of values of five input parameters.

  • The parametric analysis helps to identify the influence of each input parameter on the outcome of the rockfall. The results reveal that the slope-angle, slope-height and block-weight are three parameters that affect the bounce height, velocity and kinetic energy, respectively, compared to the rest of the input parameters.

  • A set of generalised equations was developed and further validated using multivariate linear regression analysis to predict the bounce height, velocity, and kinetic energy associated with the rockfall activity.

  • The assessment of rockfall using software packages is costly and time-consuming. Similar results (variation approximately up to 13%) were observed through the developed equations compared to the outcome of rockfall modelling. Further, these generalised equations will assist the field engineers in selecting and designing a suitable mitigative measure against rockfall.

  • The study is limited to the rock slopes of NH-44A. The work can be further expanded by including more case studies from other parts of the country.