Rockfall in open pit mines: management of the pit geometry and protection measures design

International Journal of Rock Mechanics & Mining Sciences 170 (2023) 105551 1365-1609/© 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect International Journal of Rock Mechanics and Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms Rockfall in open pit mines: management of the pit geometry and protection measures design Maddalena Marchelli a,∗, Daniele Peila a, Anna Giacomini b aDepartment of Environment, Land and Infrastructure Engineering (DIATI)- Politecnico di Torino, C.so Duca degli Abruzzi 24, I–10129 Torino, Italy bPriority Research Centre for Geotechnical Science and Engineering, College of Engineering, Science and Environment, University of Newcastle, University Dr, 2308 Callaghan, NSW, Australia ARTICLE INFO Keywords: Rockfall Open pit Trajectory analysis Design charts ABSTRACT Rockfall events represent a serious hazard for people, structures, and infrastructures. The phenomenon can occur in several environments, natural and artificial. In surface mining operations, blasting, bench clean-up and scaling can induce rock blocks detachment, compromising operators’ and operations’ safety. The installation of passive mitigation measures coupled with a specific design of the pit geometry can mitigate this risk. The bench height, width, and slope angle, the number of benches, together with the material characterizing the slope, affect in different ways the kinematics of the possible detached blocks. In this paper, a large set of configurations is studied and trajectory analyses are performed to evaluate the influence of each geometrical or material variable. Results are analysed considering blocks kinematics both on the pit floor and at each bench. The findings provide a useful a tool for a preliminary design of pit geometry and mitigation measures. Design charts are proposed for different slope materials, reporting, as function of the total pit height, the blocks stopping distance, their velocity at the pit floor, their passing height and velocity at each bench, for each geometrical configuration. The obtained charts can be effectively implemented in a quantitative risk assessment procedure for mining activities. 1. Introduction Rockfall phenomena are among the most dangerous landslide events, involving the detachment and the consequent rapid movements, i.e. bouncing/rolling/sliding, along a natural or artificial slope of individual rock blocks or rocky fragments ranging from small fragments to massive boulders of tens cubic meters.1According to the intensity of the phenomena and to both exposure and characteristics of the elements at risk, serious damages can occur on infrastructures, structures, people and, in working sites, workers and machinery,2even at a large-scale. In surface mining, rockfall events can also be affected by unpredicted structural conditions ahead of mining and variations in planned slope designs (and related geometries) upon optimization of drill and blast operations, exposing workers, machinery, and site equipment to unpredicted risks. Pre-mining activities, bench cleanup and scaling, and blast activities can further enhance block detachments and rockfall phenomena.3,4In areas exposed to extreme meteorological events, natural weathering, freeze-thaw actions and intense rainfall can further contribute to an inaccurate design of blasting or scaling operations for both the intermediate and the final configurations.5,6 Rockfall risk mitigation is a fundamental requirement for safe operations, reducing the risk of unexpected work interruptions and, thus, ∗Corresponding author. E-mail address: [email protected] (M. Marchelli). substantial economic losses.7Several rockfall hazard8–10 and risk11–13 assessment methods have been proposed over the last few decades to assess the level of the hazard, to quantify the severity of the damages and to increase activities safety in resource exploitation. The installation of various mitigation solutions has significantly increased over the last decade to respond to the risks posed by rockfalls14–20: drapery meshes, waste rock and/or engineered embankments and net fences are widely adopted in surface exploitation of various rock materials. Novel monitoring procedures and technologies can also provide accurate data on event magnitude and return period, allowing detailed catalogue of events, their characteristics and possible triggering conditions, to be collected.21–25 Besides protection and mitigation works, rockfall hazard in open pit mines can be primarily controlled through bench design.7,26 The geometry of the excavation is generally defined according to a detailed geotechnical model, for which geostructural and hydrogeological features of the rock mass in the pit area have been thoroughly investigated through field and, possibly, laboratory tests. Stratification, schistosity and discontinuity sets impose limits to the height, width and slope of https://doi.org/10.1016/j.ijrmms.2023.105551 Received 13 January 2023; Received in revised form 9 June 2023; Accepted 11 July 2023 International Journal of Rock Mechanics and Mining Sciences 170 (2023) 105551 2 M. Marchelli et al. the steps.5,27–29 As rockfall can originate during operations even in apparently stable zones,6a thorough understanding of rockfall kinematics in terms of trajectories, velocities, rebounds and run-out distance is indeed a priority in the decision-making processes for exploitation and for the design of mitigation measures.30 Several experimental and numerical studies31–36 have been conducted in open pit mining contexts to perform an effective design of either (i) the slope configuration, i.e. the geometric arrangement of the benches on the pit wall, to avoid rockfall propagation or (ii) the protective mitigation measures for a given geometry. Single or multiple benches slope geometries have been considered and a number of charts (or design suggestions) has been proposed. Nevertheless, these studies have considered the contribution of rockfall mitigation for a single bench geometry, only.31,32,36 A tentative of including multiple benches has been proposed,34,35 with the aim of defining the best geometry that limits (but not to completely remove) the risk. In this last case, neglecting rockfall trajectory analyses specific for a given open pit site,37–39 the literature does not provide rockfall parametric analyses evaluating for each bench and at the pit floor the kinematic parameters of the possible falling blocks during their motion. Section 1.1 provides the main results of the above mentioned studies. Hence, coupled scenarios, i.e. multiple benches with mitigation measures, are completely missing. The present works aims at providing an useful tool for the effective design, in reference to rockfall hazard, of both (i) pit slope and (ii) mitigation measures, considering single or multiple benches configurations. To achieve such goal, several parametric rockfall trajectory analyses in a hypothetical open pit are performed and analysed, considering different possible geometries and slope outcropping materials. Blocks run-out and kinematic parameters at each bench are investigated. Pit geometries and materials typical of European rock exploitation (Fig. 1(a) and (b)) and Australian coal mine environments are considered. Several analyses were conducted. First, the configurations that minimize the number of blocks arriving at the pit floor were analysed together with the parameters that mostly affect the results in terms of arrival probability, distance, and velocity. Since avoiding or minimizing the number of arriving blocks at the pit floor is not always achievable or affordable, the kinematic characteristics of falling blocks at each bench were investigated. To provide a tool for a preliminary design of both pit geometry and mitigation measures, four design charts were produced for each of the investigated slope materials, reporting the characteristic values of the kinematic parameters of the blocks at each bench and at the pit floor. These charts can also be implemented in quantitative risk assessment methods tailored for open pit, as the one proposed by Alejano et al.11 or Peila et al..13 Section 2presents the proposed methodology, the input parameters and the performed analyses, while Section 3deals with results: a deep analysis of blocks reach probability and their impacts at the pit floor are provided in Section 3.1, while blocks kinematics at each bench are addressed in Section 3.2. Section 3.3 presents the above mentioned design charts with an example of use. Finally conclusions and future perspectives are outlined in Section 4. 1.1. Previous studies As previously stated, issues related to pit design optimization against rockfall hazard have been tackled by few Authors. Even not finalized for open pit design, only, it is worth mentioning the pioneering study conducted by Ritchie,31 whose work included the rolling of hundreds of rocks off state-owned quarries and talus slopes across Washington state (US). The paths and the distances of the falling blocks were recorded and measured. The work led to the definition of an empirical method for roadway ditch design, to prevent falling rocks from reaching the travelled area of a road, proposing a set of practical design criteria further collected in a chart. For the particular open pit case, these results can be adopted for investigating blocks dynamic in a single bench geometry. Nevertheless, the chart provides the geometry Fig. 1. Sandstone quarry in Fiorenzuola, Italy (a), and aggregate quarry in Pedogna, Italy (b). Courtesy of Prof. M. Coli. of a catchment area for a 100% (total) retention of blocks, not considering variable retention levels for a cost/benefit design approach. Moreover, the proposed catchment ditch solution was not feasible in all the cases. Consequently, thanks to an extensive campaign of field tests, Pierson et al.32 investigated how slope, catchment area and rockfall properties affect the rockfall retention. The Authors developed guidelines, including design charts, for partial retention of blocks with also flat catchment area. Also in this case, the work was not specifically targeted to open pit design but, as tests were performed in such context, the results were used also for single bench design. Evans40 reached similar results through real tests in mine benches. Call, in Ref. 27, specifically focused on rockfall hazard in open-pit mines and in the optimization of bench and catch-ditch design from a cost perspective compatible with safety standard. Defining the excavation of a ditch as not practical and substituting the ditch with a berm, Call proposed to improve Ritchie’s criterion applied to mining, with a minimum bench width 𝑏𝑤,𝑚𝑖𝑛 equal to: 𝑏𝑤,𝑚𝑖𝑛 = 1.2+0.2𝑏ℎ(m),(1) where 𝑏ℎis the bench height (in meters). The suitability of this formula was confirmed by Ryan and Pryor,26 even though a reliability based approach to evaluate bench slopes was then proposed where a variability in geologic structure makes difficult to have a constant inter-ramp slope angle. The approach considered different issues and was not only targeted to rockfall hazard but required an amount of input data often difficult to achieve. The first study on rockfall hazard on multiple benches in open pit mines was performed by Peila et al..34 The Authors investigated, with their own 2D lumped-mass trajectory code, the specific configuration International Journal of Rock Mechanics and Mining Sciences 170 (2023) 105551 3 M. Marchelli et al. of four benches, 15 m high, 2.5 and 5 m wide. With a first attempt to insert fragmentation, the Authors analysed 9 different combinations of normal and tangential restitution coefficients and friction angle, different bench-by-bench. Despite no design charts were developed, interesting design tips were suggested: (i) reduced bench slope tends to project rock blocks at a greater distance, with a prevailing horizontal direction; (ii) for very steep bench inclination, a slight increase of bench width does not vary the results; (iii) as a preliminary rockfall mitigation measure, debris could be distributed on the bench. Starting from all the above mentioned references, Alejano et al.35 performed numerical trajectory analyses on synthetic 2D profiles, representative of hard-rock quarries, investigating the influence of the geometrical parameters in the falling block retention capacity at the pit floor. A lumped-mass approach was adopted (RocFall code), performing simulation for 2, 5 and 8 benches slopes, bench inclinations of 2:1, 3:1, and 4:1, and bench heights spanning up to 25 m, back-calculating the catch-bench widths able to retain 75%, 90%, and 95% of the blocks. A unique set of input parameters that describe rock-slope interaction was considered, with a slope roughness, i.e. a variability of the angle at the point of the impact, of 0.1 in standard deviation. This set was considered as representative of hard-rock quarry and results were organized in design charts. Nevertheless, to assess the sensitivity of results to different input data, parametric studies were performed for a series of particular cases, and the mean values for the restitution coefficients and slope angle were proved to be the most significant parameters. More recently, Ferrari et al.36 conducted rockfall sensitivity analyses on a 2D synthetic profile, representative of a rock cliff (or a single bench configuration). A lumped-mass model was adopted and RocFall code was used for the simulations. Height (from 5 m to 100 m) and slope angle (from 50◦to 85◦) were varied, together with restitution coefficients, friction angle and slope roughness. The first impact location and velocity were investigated. In a probabilistic framework, the 95𝑡ℎ and the 90𝑡ℎ percentiles for arrival and velocity, respectively, were considered as representative. Thanks to the properties of the selected trajectory model, the kinetic energy can be derived from the known mass of the falling block. Fitting the obtained data applying a linear regression through the origin, empirical equations were proposed for the preliminary calculations of the first impact position and the kinetic energy as function of the slope height, inclination and irregularity of the rock face, subdivided in classes. On the basis of this study, a qualitative rockfall hazard assessment procedure for highwall was developed.10 From a known state of activity of the highwall, i.e. describing rockfall frequency, together with the expected rockfall energy, a level of hazard could be defined. The knowledge of the first arrival is instead used for locating workers, machinery and infrastructures over the working areas at the toe of highwalls. 2. Methodology An adequate design of catching benches in open pit mines is fundamental to prevent and/or minimize rocks detached in the upper portions of the pit slope from reaching working areas located at the pit floor, where workers and equipment are located. A balance between optimum design and production rate needs to be explored5,41 and, when necessary, the installation of protective measures should be considered. Several parametric trajectory analyses were performed within a probabilistic framework with the precise aim of evaluating (i) which geometrical configurations can provide an acceptable level of risk, i.e. blocks are unlikely to arrive on the pit floor, and (ii) which performances are required to the protective measures installed on the intermediate benches. Starting from the indefinite slope assumption, which holds for wide excavation sites, and following what proposed by Refs. 35,36, trajectory analyses on 2D synthetic profiles were performed. The heterogeneity of site conditions in term of materials was accounted thanks to a Table 1 Considered layouts for the trajectory analyses. Parameter Value bench face height 𝑏ℎ(m) 10, 20, 30, 40 bench width 𝑏𝑤(m) 3, 5, 7 bench face slope 𝛽(◦) 50◦, 60◦, 70◦, 80◦ number of benches 𝑛𝑏(–) 1, 2, 3, 4, 5, 6, 7, 8 selection of normal and tangential restitution coefficients (Section 2.2). A lumped-mass approach was selected as the most suitable to compare various profiles with different materials and geometries. In addition, the adoption of a point-mass model, in which trajectories are not affected by the block mass, allows scaling the results in terms of kinetic energy according to the mass.36 A very large number of trajectory analyses was performed to analyse all these variables: 384 different geometries with 12 different materials pairs, i.e. 4608 configurations. The need of performing such a number of analyses, recording and managing all the output quantities, underlies the choice to develop a specific trajectory code on Matlab (R2021b) environment, mimicking RocFall software (RocScience Suite).42 The code was validated by comparing the results on a sample slope with those obtained from RocFall 2019,42 as detailed in Appendix A. To encompass the uncertainties associated with the rockfall phenomenon,43 as suggested by the Eurocodes,44,45 trajectory analyses are performed in a probabilistic framework: at each bench, blocks passing height and velocity are recorded together with the stopping distance and the impact velocity at the pit floor. This approach promotes the design of protective measure. Following Eurocode 0,44 the structural works have to be designed through a partial safety factors design approach, where the effect of actions, i.e. the kinematic parameters of the blocks, are expressed through reference, or characteristic, values (determined from the probability distributions). Considering the suggestions provided by the Italian Standards for rockfall protective measures UNI 11211-4,46 widely adopted in Europe, and coherent with Eurocode 7,45 the 95𝑡ℎ percentile of the distributions can be chosen as characteristic value. Design values of the variables are obtained by multiplying the characteristic values times the partial safety factors. 2.1. Slope geometry Different layouts were considered for the 2D synthetic profiles were performed, trying to encompass the great majority of the cases. In open pit with several benches, a bench height 𝑏ℎusually ranges between 10 m and 20 m.5In particular contexts, e.g. in Australia, benches with a height spanning from 20 to 40–50 m with few steps can be observed. Referring to bench width 𝑏𝑤, the most adopted size is 7 m, although smaller values can be found in transitory or final dismantling configurations.34 Bench face angle, which is controlled by the intersecting joints and faults, typically spans from 50◦to 80–85◦.5,7 To account for a large set of configurations, the analysis was conducted considering (i) the possible number of benches ranges from 1 to 8, (ii) three bench widths (3, 5 and 7 m), (iii) four bench heights (10 to 40 m) and (iv) four bench face slopes (from 50◦to 80◦). Table 1 summarizes the studied configurations. Assuming a 2D regular profile, it should be noted that the present work does not tackle with the backbreak along the top of the bench issue, i.e. the horizontal distance between a planned and a real bench crest.47 2.2. Slope material With the leading idea to create charts for the preliminary design of both open pit geometry and mitigation/protection works, for each possible extraction site, different materials were considered. As stated in Section 1, the parameters governing the interaction between a International Journal of Rock Mechanics and Mining Sciences 170 (2023) 105551 4 M. Marchelli et al. Table 2 Values of the input parameters related to the block-open pit interaction. The subscript 𝑚accounts for the mean value of the parameters. It is worth mentioning that the maximum 𝑅𝑛and 𝑅𝑡is set to 1.0 in the Monte-Carlo sampling process. Parameter Value 𝑅𝑛,𝑚 (–) 0.3, 0.4, 0.5 𝑅𝑡,𝑚 (–) 0.35, 0.55, 0.75, 0.85 𝜙𝑚(◦) 30◦ 𝜎𝑛(–) 0.03 𝜎𝑡(–) 0.03 𝜎𝜙(◦) 1◦ Roughness (◦)±1◦ block and the terrain along which it moves are the normal restitution coefficient 𝑅𝑛, the tangential restitution coefficient 𝑅𝑡, and the friction angle 𝜙. While the first and the second stand for block energy loss during bouncing, the friction angle accounts for energy dissipation of the block during its roto-translational motion along the path. Although all these parameters are influenced by the shape and the geomechanical characteristics of both the block and the terrain, together with the velocity and orientation of the block at the impact,48 in a lumpedmass model the value of the parameters is mainly material-dependent. As often adopted, a unique friction angle 𝜙= 30◦was assumed.36 Following these considerations, hereafter, the Authors use the term material to indicate a couple of 𝑅𝑛and 𝑅𝑡values. Considering the ranges suggested in the literature for rocky outcrops, 𝑅𝑛generally spans from 0.30 to 0.50 and 𝑅𝑡from 0.65 to 0.9,16,33,49–54 taking specifically into account simulations performed with lumped-mass 2D model.42,55 For specific cases, Refs. 16,36 found that 𝑅𝑡can reach 0.35. To account for a large set of materials, the analysis was conducted considering that (i) the normal restitution coefficient 𝑅𝑛ranges from 0.3 to 0.5, (ii) the tangential restitution coefficient 𝑅𝑡ranges from 0.35 to 0.85 and (iii) a unique friction angle 𝜙= 30◦is assumed. As further detailed in the following section, in a probabilistic framework these represent the mean values of material parameters identified with 𝑚 subscript. For a simplified and conservative assumption, and for having an overview of general validity, an homogeneous material was assumed for the whole profile, neglecting thus the presence of debris on the benches or particular situations that can be developed in future studies. 2.3. Trajectory analyses As in a probabilistic approach, the performed trajectory analyses consist in a set of throws, i.e. simulations, from the individuated source area, in which the input parameters related to the material are randomly selected in a predefined range. Random sampling approaches should be based on theoretical knowledge for all the probabilistic parameters. In the case of the restitution coefficients, Gaussian and uniform distribution represent the most adopted distributions, even though currently there is no evidence that the values in a range have equal probabilities or conform to other types of probability distribution.43,56,57 In the present work, a Gaussian distribution was assigned, as already performed by Alejano et al.,35 Ferrari et al.,36 and Frattini et al..58 The mean values correspond to 𝑅𝑛,𝑚,𝑅𝑡,𝑚, and 𝜙𝑚, while the standard deviations are 𝜎𝑛,𝜎𝑡, and 𝜎𝜙, respectively. Table 2 reports the values adopted in the simulations, resulting in 3 × 4 = 12 combinations of 𝑅𝑛,𝑚, and 𝑅𝑡,𝑚. Since it was intended to represent widely different materials, a narrow Gaussian distribution is assumed, i.e. low values of 𝜎𝑛and 𝜎𝑡, following the findings of Chau et al.,52 who experienced standard deviations from 1% to 15%, but generally less than 5%, and of Asteriou et al..48 To account for a possible slight variability in the local surface angle of segments of the slope, a slope roughness was considered, i.e. allowing a variation of ±1◦to the angle at the point of the impact. Fig. 2. Scheme of a considered synthetic profile used for the trajectory analyses. The sketch reports a 8 steps configurations. Blue cross represents the source area, while red and the purple dashed lines the vertical and the horizontal collectors, respectively. As the number of simulations for each configuration should be sufficiently large to guarantee the statistical significance of the results, 6000 throws were performed in each of the considered configurations. Even though rockfalls may start anywhere along the slope, their occurrence is more common from the upper part of the benches.35 Hence, the detachments of blocks from the top of the pit was considered, with a non-null, but low, initial velocity (i.e. 1 m/s) to mimic the trigger conditions related to external agents. A number of collectors, or notphysical sections in which the kinematic parameters of the blocks are recorded, were considered according to a potential suitable location of mitigation measures. In particular, 1 to 8 vertical sections on the steps represent the possible locations for the installation of passive measures, while a horizontal section at the pit bottom was considered to record the run-out distances of the blocks, together with their velocity at the impact (Fig. 2). Under lumped-mass assumption, blocks velocity was considered to implicitly account for kinetic energy, as, known the mass 𝑚𝑏of the possible impacting blocks, energy can be derived as 𝐸= 1 2𝑚𝑏𝑣2. Thus, in the vertical sections, the height of the trajectory, ℎ, and the velocity of the blocks, 𝑣𝑣, were recorded. In the horizontal section, the run-out distance, 𝑑, and the impact velocity, 𝑣ℎ, were measured. 3. Results and discussion This section details the results of the propagation analyses. First, the probability of reach and blocks kinematics at the pit floor were investigated, individuating the geometrical parameters and the materials, i.e. the couples 𝑅𝑛,𝑚 and 𝑅𝑡,𝑚, which minimize the numbers of arriving trajectories and the distance, together with the velocity. The velocity of the falling blocks impacting on the pit floor is indeed fundamental to evaluate the kinetic energy and, thus, the degree of damages on workers and infrastructures. Second, since a specific geometry cannot be implemented, protective measures can be required to prevent blocks impacting on the floor, or on the working benches. For this reason, the knowledge of passing height and velocity at each bench was evaluated and trends analysed. Following Eurocode 0 design principles, the 95𝑡ℎ percentiles of the distributions of the output quantities are considered as the characteristic values of the output variables. The results, grouped for each material, were presented, for each layout, in terms of: International Journal of Rock Mechanics and Mining Sciences 170 (2023) 105551 5 M. Marchelli et al. Fig. 3. Values of 𝑑95 for different couples of 𝑅𝑛,𝑚 and 𝑅𝑡,𝑚, grouped for different values of 𝛽. Details of the plots are reported in the text. •run-out distance of the blocks on the pit floor, 𝑑95; •velocity of the blocks when impacting on the horizontal pit floor, 𝑣ℎ,95; •passing height of the blocks recorded on each vertical section, ℎ95; •velocity of the blocks recorded on each vertical section, 𝑣𝑣,95. 3.1. Rockfall hazard on pit floor Fig. 3 displays the run-out distance of the blocks on the pit floor, 𝑑95, for the four extreme value pairs of 𝑅𝑛,𝑚 and 𝑅𝑡,𝑚, i.e. with minimum and maximum values of both parameters. Each material condition is reported in the row of the plot; the graphs are grouped for 𝛽values (columns of the plot). In the 𝑥-axis, the total height of the pit 𝐻𝑡𝑜𝑡 is indicated. Different colours and line types are used to indicate 𝑏ℎand 𝑏𝑤 values, respectively, as displayed in the legend. The number of benches is thus univocally determined according to 𝐻𝑡𝑜𝑡 and 𝑏ℎ. Coloured bullets indicate geometrical configurations for which blocks arrive only in the case of a single bench (𝑛𝑏=1). It could be noticed that for very low values of 𝑅𝑡,𝑚 (first and third rows), the blocks tend not reach the pit floor (in particular with multiple benches) and the range of 𝑑95 values remains below 7 or 10 m, increasing 𝑅𝑛,𝑚. Despite being in the same range and with the similar trend, the bench height 𝑏ℎhas the effect of slightly increasing 𝑑95; a similar effect is noted if the bench width 𝑏𝑤reduces. The number of benches 𝑛𝑏does not influence the output. It is worth mentioning that for a bench inclination 𝛽= 80◦, blocks arrive on the floor if 𝑏ℎ= 40 m and 𝑏𝑤= 3 m, only. Neglecting 𝑛𝑏= 1, for which unavoidably blocks arrive on the floor, the linear fit the results leads to almost a constant value 𝑑95 ∼ 4 m (Fig. 3, dotted line). For high values of 𝑅𝑡,𝑚, regardless of 𝑅𝑛,𝑚 (second and fourth rows), for almost all the geometrical configurations the blocks reach the pit floor, with distances that are positively affected by 𝐻𝑡𝑜𝑡 or, as a consequence, by the total number of benches 𝑛𝑏. For a given 𝛽, a fitting is with a straight line through the origin (Fig. 3, dotted line), irrespective of 𝑏ℎand 𝑏𝑤, is possible and highlighted. The slope of the linear fit is similar for 𝛽= 50–60◦, while it decreases for higher 𝛽, especially for 𝛽= 80◦. The value of the slope has also a great variability according to 𝑅𝑛,𝑚. It is interesting to observe that for 𝑅𝑛,𝑚 = 0.5and 𝛽= 50◦–60◦the slope of the fit line is almost equal to one, i.e. 𝑑95 tends to 𝐻𝑡𝑜𝑡, while it reduces to almost 0.3𝐻𝑡𝑜𝑡 for 𝛽= 80◦. All these aspects can be explained in terms of trajectories. In case of low tangential restitution coefficient, the falling blocks, once reaching each bench tread, dissipate a great amount of their kinetic energy and continue their motion to the following bench with a trajectory pattern quite similar to the one with which they started their initial fall (Fig. 4.a). In this case, the influence of the number of benches is negligible, as in all the benches similar trajectories occur, even though some blocks stop on some intermediate benches. The higher the bench, the higher the acquired velocity at the impact and, thus, the outgoing velocity and the final value of 𝑑95. Increasing 𝑏𝑤, some of the trajectories stop on the first bench. On the contrary, for high 𝑅𝑡,𝑚, once impacting a bench, the velocity vector of the blocks acquires a tangential component higher than the normal one, and the resulting parabolic motion cannot directly reach the following bench, but the one beyond (Fig. 4.b). The increment in 𝛽causes the parabolic motion to prevail on the sliding/rolling one, resulting in an almost vertical impact on bench tread and, thus, in trajectories similar to those for low 𝑅𝑡,𝑚 values, but with less energy dissipation. The study reveals that low 𝑅𝑡,𝑚 values and 𝛽= 80◦allow obtaining zero (or negligible) block probability of reach for several configurations in terms of 𝑏ℎand 𝑏𝑤. It should be noticed that the high 𝑑95 values obtained in this study are the result that the pit floor is completely free of obstacles and debris. Nevertheless, the knowledge of 𝑑95 rather than recording the arrival (or not) of the block on the pit floor can provide important information on the level of hazard. To better analyse the influence of the parameters on both arrival probabilities on the pit floor and 𝑑95 values, all the results related to the 4608 configurations are statistically organized and summarized into International Journal of Rock Mechanics and Mining Sciences 170 (2023) 105551 6 M. Marchelli et al. Fig. 4. Results of the trajectory analyses for 𝑏ℎ= 20 m, 𝑏𝑤= 3 m, 𝑛𝑏= 8,𝛽= 60◦, for (a) 𝑅𝑛,𝑚 = 0.30 &𝑅𝑡,𝑚 = 0.35 and (b) 𝑅𝑛,𝑚 = 0.30 &𝑅𝑡,𝑚 = 0.85. Red stars indicate blocks stopping points. Table 3 Maximum values for 𝑑95 and 𝑣ℎ,95 recorded at the pit floor for all the configurations. 𝑅𝑛,𝑚,𝑅𝑡,𝑚 𝑑95,𝑚𝑎𝑥 Geometry of 𝑑95,𝑚𝑎𝑥 𝑣ℎ,95,𝑚𝑎𝑥 Geometry of 𝑣ℎ,95,𝑚𝑎𝑥 (–) (m) 𝑏ℎ𝑏𝑤𝛽 𝑛𝑏(m/s) 𝑏ℎ𝑏𝑤𝛽 𝑛𝑏 0.3 , 0.35 7.8 40 7 60◦3 26.5 40 3 80◦2 0.3 , 0.55 13.0 40 5 60◦6 29.5 40 3 70◦2 0.3 , 0.75 93.0 40 3 50◦8 51.1 40 3 60◦8 0.3 , 0.85 165.4 40 5 60◦8 65.8 40 3 60◦8 0.4 , 0.35 9.3 40 3 60◦1 26.5 40 3 80◦2 0.4 , 0.55 17.8 40 3 50◦4 30.2 40 3 70◦2 0.4 , 0.75 148.3 40 3 60◦8 58.5 40 3 70◦7 0.4 , 0.85 243.7 40 3 50◦8 66.7 40 3 60◦8 0.5 , 0.35 12.0 40 5 60◦1 26.5 40 5 80◦2 0.5 , 0.55 40.4 40 3 70◦8 42.9 40 3 70◦5 0.5 , 0.75 212.6 40 5 60◦8 65.4 40 3 70◦8 0.5 , 0.85 336.6 40 5 50◦8 67.6 40 5 60◦8 pivot plots. Fig. 5 displays the percentage of arrivals on the pit floor with respect to the blocks detached, i.e. the arrival or reach probability, subdivided for 𝛽and further subdivided in the 𝑥-axis according to 𝑅𝑡,𝑚, which is found to be the relevant parameter. The pivot plots report the data also for the remaining four parameters, i.e. 𝑅𝑛,𝑚,𝑏ℎ,𝑏𝑤, and 𝑛𝑏(Fig. 5.a to d, respectively). In each case, the configurations of the slope are grouped according to the parameters of the 𝑥-axis, the mean percentage value is considered. For a given tangential restitution coefficient, the graph of Fig. 5.a reveals that the increase of 𝑅𝑛,𝑚 has the effect of increasing the number of arrivals. This trend is less evident if 𝑅𝑡,𝑚 increases, as the majority of the blocks arrive on the pit floor. The graph highlights an interesting situation: for 𝑅𝑡,𝑚 ≥0.55, the increase of 𝑅𝑛,𝑚 involves higher percentages of arrival, while the case 𝑅𝑡,𝑚 = 0.35, for all 𝑅𝑛,𝑚, displays a higher percentage of arrival than for 𝑅𝑡,𝑚 = 0.55. This trend is more evident in several 𝑅𝑛,𝑚 and 𝛽configurations: for 𝛽= 50◦for 𝑅𝑛,𝑚 ≥0.4and for 𝛽= 60◦for 𝑅𝑛,𝑚 ≤0.4. Analysing in detail the cases with 𝑅𝑡,𝑚 = 0.35, it results that there are few configurations in which the blocks reach the pit floor, whereas in such scenarios, almost the totality of the throws arrives. Besides, since in the pivot plots the mean value is reported, the resulting figure is low. In 𝑅𝑡,𝑚 = 0.55 it results that there are more configurations in which the blocks reach the pit floor, but with a lower arrival probability. That is why, in general, the bars in the pivot plots of Fig. 5 are shorter or equal to the ones related to 𝑅𝑡,𝑚 = 0.35. Analysing the trajectories, for 𝑏ℎ= 40 m, 𝑏𝑤≤5m, and 50◦≥𝛽≤60◦, in case on 𝑅𝑡,𝑚 = 0.55, a significant percentage of blocks stops at the second bench (see also Fig. 5.b and c): after the first step, bouncing becomes the prevailing motion, instead of sliding as in 𝑅𝑡,𝑚 = 0.35, and, thus, energy dissipates more rapidly. Both 𝑏ℎand 𝑏𝑤(Fig. 5.b and .c) influence the results: the decrease of 𝑏ℎand the increase of 𝑏𝑤has the effect of reducing the number of arrivals. Finally, neglecting 𝑛𝑏= 1, for which blocks unavoidably arrive on the pit floor, negligible influence can be observed changing 𝑛𝑏(Fig. 5.d), except for 𝑅𝑡,𝑚 = 0.55 and 𝑛𝑏= 2, for which, as observed before, some configurations involve a significant percentage of stopping blocks on the second bench. In the four subplots, the influence of 𝛽is negligible except for 𝛽= 80◦(for which a reduced number of arrivals is displayed), while for 𝛽= 50◦–70◦the results are comparable. With a similar approach, Fig. 6 reports the pivot charts of the mean (bars) and the maximum (points) values of 𝑑95. The 𝑦-axis is in logarithmic scale for sake of readability. As in the previous case, the 𝑥-axis relates to 𝑅𝑡,𝑚 and the pivot plots divide the data also for the remaining four parameters, i.e. 𝑅𝑛,𝑚,𝑏ℎ,𝑏𝑤, and 𝑛𝑏. As observed in Fig. 3, and particularly in the fitting lines equations, for 𝑅𝑡,𝑚 = 0.35, the influence of 𝑅𝑛,𝑚 is almost negligible. Observing Fig. 6.a, the increase of 𝑅𝑛,𝑚 results in an increase of 𝑑95. A similar conclusion can be drawn observing Fig. 6.d where the results are grouped with respect to 𝑛𝑏. A positive correlation can be observed for 𝑏ℎ(Fig. 6.b), while the effect of 𝑏𝑤on the values of 𝑑95 is negligible (Fig. 6.c). The bench angle does not affect the results, except for 𝛽= 80◦, as observed in the previous analyses. Table 3 reports, for each material, the maximum values of 𝑑95 together with the correspondent slope geometrical configurations. Confirming the previous observations, all the maximum values are reached with 𝑏ℎequal to 40 m, the great majority with 𝑏𝑤equal to 3 m and 𝛽∼ 50◦–60◦. No evident trends are observable for 𝑛𝑏, for 𝑅𝑡,𝑚 ≤0.55. Fig. 7 displays the velocity of the blocks when impacting on the horizontal pit floor 𝑣ℎ,95 for the four extreme value pairs of 𝑅𝑛,𝑚 and 𝑅𝑡,𝑚, i.e. with minimum and maximum values of both parameters. Each material condition is reported in the row of the plot. The graphs are grouped for 𝛽values (columns of the plot), and on the 𝑥-axis the total height of the pit 𝐻𝑡𝑜𝑡 is indicated. It is worth mentioning that the velocity refers to the first impact of the block on the pit floor. For International Journal of Rock Mechanics and Mining Sciences 170 (2023) 105551 7 M. Marchelli et al. Fig. 5. Pivot charts of the number of arrival at the pit floor grouped according to the value for the tangential restitution coefficient 𝑅𝑡,𝑚 and an additional variable (𝑅𝑛,𝑚 ,𝑏ℎ,𝑏𝑤, and 𝑛𝑏), as explained in the text and further divided by 𝛽. 𝑅𝑡,𝑚 = 0.35,𝑣ℎ,95 spans from 10 to 20 m/s for all the configurations for which the blocks arrive. For 𝑅𝑡,𝑚 = 0.85, values up to 60 m/s are observed increasing the number of benches. Referring to the velocity, a trend similar to that of Fig. 3 related to the distance is observed, except for the influence of bench angle 𝛽, which seems to be negligible. While blocks reach lower distances on the pit floor for decreasing 𝛽, the right tail of the distribution of the velocity, i.e. 𝑣ℎ,95, does not change as it reflects the velocity in the first impact on the floor. This behaviour International Journal of Rock Mechanics and Mining Sciences 170 (2023) 105551 8 M. Marchelli et al. Fig. 6. Pivot charts of 𝑑95 at the pit floor: bar display the mean values, while points the maximum values, grouped according to the value for the tangential restitution coefficient 𝑅𝑡,𝑚 as explained in the text and further divided by 𝛽. confirms the fact that the slope angle affects the way the blocks are projected out of the face, while the falling height, thus the potential energy of the boulder, affects the velocity, i.e. the kinetic energy. The linear fitting of the plots allows to appreciate an almost horizontal trend of 𝑣ℎ,95 for 𝑅𝑡,𝑚 = 0.35, regardless of 𝑅𝑛,𝑚, while a slope equal to 0.15 and an intercept at around 13 m/s is found for International Journal of Rock Mechanics and Mining Sciences 170 (2023) 105551 9 M. Marchelli et al. Fig. 7. Values of 𝑣ℎ,95 for different couples of 𝑅𝑛,𝑚 and 𝑅𝑡,𝑚, grouped for different values of 𝛽. 𝑅𝑡,𝑚 = 0.85, independently from the bench angle. Table 3, reports the maximum values of 𝑣ℎ,95 for each materials couples. The worst (in terms of velocity) configuration presupposes a bench height of 40 m and a bench width of 3 m. Different number of benches are noted, as there are configurations for which blocks do not arrive on the pit floor. Distance and velocity can be fitted with a line having equation 𝑋=𝑚𝐻𝑡𝑜𝑡 +𝑐, (2) where 𝑋is the observed parameter, 𝑚is the slope of the fitting line and 𝑐is its intercept, i.e. the value at 𝐻𝑡𝑜𝑡 = 0.Table 4 reports the values of 𝑚and 𝑐for all the studied configurations. Some consideration can be done: •for 𝑅𝑡,𝑚 ≤0.55, the equations 𝑑95 ∼ 0.1𝐻𝑡𝑜𝑡 + 4.45,(3) and 𝑣ℎ,95 ∼ 0.2𝐻𝑡𝑜𝑡 + 18.67,(4) allow a rough and conservative estimation of 𝑑95 and 𝑣ℎ,95, even though for 𝛽= 80◦slightly different results can be displayed. It should be noted that these equations do not allow to define whether the blocks arrive, or not, on the pit floor. •for 𝑅𝑡,𝑚 >0.55,𝑚and 𝑐varies according to 𝑅𝑛,𝑚 and, for a given 𝑅𝑛,𝑚, the case of 𝛽= 80◦shows different values, generally with a lower slope 𝑚and higher intercept 𝑐. 3.2. Mitigation measures With the leading idea that controlling the geometry is not always possible or affordable, the kinematic parameters of the blocks along the bench edges were recorded in terms of characteristic values, i.e. ℎ95 and 𝑣𝑣,95.Fig. 8 displays ℎ95 and 𝑣𝑣,95, grouped with respect to the bench slope 𝛽. The way the results are presented is similar to what proposed for pit floor in Section 3.1. For sake of simplicity, two extreme cases are reported, only. The first two rows of Fig. 8 refer to 𝑅𝑛,𝑚 = 0.3, 𝑅𝑡,𝑚 = 0.35, the last two to 𝑅𝑛,𝑚 = 0.5, 𝑅𝑡,𝑚 = 0.85. The results display that 𝑅𝑡,𝑚 is the most influencing parameter. For low 𝑅𝑡,𝑚, as observed in Fig. 4, the trajectories are similar for each bench. For this reason, a specific range of values can be found, almost irrespective of the number of benches 𝑛𝑏, except in those cases in which the blocks stop at the first bench. For a given 𝛽, the increase of 𝑏𝑤and the decrease of 𝑏ℎpromote the stop of the block at the first bench. This trend is further enhanced for large bench angles. Saw-tooth like curves refer to the fact that the position of the vertical recording section is fixed irrespective of the position of the bouncing of the bench. The obtained range of values of trajectory height ℎ95 spans from 0 m, i.e. pure translational motion, up to 3 m high. The range for block velocity 𝑣𝑣,95 is between 2 m/s and 8.5 m/s. Despite the observed sawtooth behaviours, an angular coefficient equal to 0 (horizontal line) for the linear fitting equation of ℎ95 is generally observed for 𝑅𝑡,𝑚 = 0.35, while for 𝑣𝑣,95 its average value is around 0.01. On the contrary, for 𝑅𝑡,𝑚 = 0.85, the values are affected by the height and the number of benches. The influence of 𝛽is negligible, for both height and velocity, ss noted for the velocity at the pit floor. 3.3. Design charts In order to provide a useful tool for the designers, the obtained results are organized in charts: for each material, four separate charts are realized for 𝑑95,𝑣ℎ,95,ℎ95 and 𝑣𝑣,95, displaying the output values for each geometrical configuration. As for a given 𝐻𝑡𝑜𝑡, different 𝑏ℎ values are possible by varying 𝑛𝑏, a different marker is adopted: circle, triangle, diamond and square for 10, 20, 30, 40 m high benches, respectively. The number of steps should be calculated according to 𝑏ℎ. On the 𝑥-axis, 𝛽values are provided, subdivided each for the possible 𝑏𝑤. According to the marker shape and position, a geometrical