Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 32, No. 7, pp.709-724, 1995 Copyright ~5:1995 Elsevier Science Ltd 0148-9062(95)110018-6 Printed in Great Britain. All rights reserved 0148-9062/95 $9.50 + 0.00 Pergamon Analysis and Prediction of Rockfalls Using a Mathematical Model A. AZZONIt G. LA BARBERAt A. ZANINETTI + Th& paper deals with the study of rockfalls us&g a mathematical model, codified for computer use. Called CADMA, it allows predictions to be made of fall trajectories and of the relevant parameters (energy, height of bounce, run out distance of the falling blocks)for the design of remedial works. Designed with the experience gained from several in situ tests, this model is based on rigid body mechanics, and statistically analyses a fall in a two-dimensional space. The main features of the program are presented in this paper, as well as the criteria for choosing the trajectory to be studied, and the techniques for the assessment of the most relevant parameters required for the execution of the rockfall analysis (particularly the dynamic parameters: restitution and rolling friction coefficients). Some practical aspects of the rockfall mathematical analysis are also discussed. These include the effect of topographical detail on the results and the optimal number of simulations to be carried out. The characteristics and potentials of the program were evaluated by comparing the results of in situ tests: in all cases, the program supplied generally accurate predictions in terms of fall velocity, energy, height of bounce and stopping distance. INTRODUCTION In the context of slope instability phenomena, the detachment of blocks from steep walls and their subsequent falls along slopes are particularly significant [1]. This phenomenon involves high risk in densely populated mountain areas, such as the Alps, where slopes are usually long and steep, and where housing estates and most man-made constructions are generally located at the bottom of valleys. It is particularly important in these areas to have the best possible knowledge of rockfall trajectories and energies in order to determine accurate risk zoning and construct adequate defence systems near the threatened areas. Until recently, rockfall problems, and specifically, remedial activities were mostly managed on an empirical basis, since understanding of the subject was somewhat limited. Today, computers represent an invaluable instrument in dealing with highly variable phenomena (such as rockfalls). Their development, together with valuable experience gathered through a more rational observation of the phenomenon (in particular with flSMES SpA, Via Pastrengo 9, 24068 Seriate, Bergamo,Italy. ~ENEL CRIS, Via Ornato 90/14, 20121 Milan, Italy. special/n situ and laboratory tests), has increased rockfall knowledge considerably. Such knowledge now allows us to perform more rational and repeatable analyses and gain more accurate predictions and thus more effective protective structures. MAIN APPROACHES TO THE PROBLEM Literature on the subject of rockfall analysis has been the subject of about 50 papers, written by different authors from 1963 to date. These papers may be basically divided into two groups according to the approach taken: utilizing experimental methods or computer models [2]. Experimental methods include empirical studies and physical modelling. This mainly consists of performing tests on scale models [3-10]. Because of their accurgcy, comprehensiveness and quality of results, some of these works are correctly considered as milestones in the understanding of rockfall phenomenology, and define the leading criteria for the design of protective works (particularly fences, nets and ditches). This type of methodology is undoubtedly valid, but unfortunately it is expensive and unsuitable for statistical and parametric analysis. Given the huge development of computer technology in the last 15 yr, and the 709 even if theoretically more accurate. in order to obtain realistic results that are comparable to experimental observations. defined by the rotation on a single plane of all the different vertical planes. --Block fracturing is not taken into account. 1). This approach is reasonable for obtaining conservative results. not to mention the correct calibration of the mathematical models. it is necessary: (a) to define the characteristics that the model must L I Real ~ physical system have. Careful observation of the physical phenomenon. Experimental tests Mathematicalmodelof ~ _ ~ the real physicalsystem Mathematical model analysis No I Stop ]~ Mathematical . --Blocks at the point of impact are modelled as ellipsoidal bodies rotating in a two-dimensional space around the shorter axis (rotation around the other axes is neglected) (Fig. --Rockfall trajectories are established a priori and represented as a sequence of straight segments. 2). experimental methods are still very important.: ANALYSISOF ROCKFALLS availability of powerful computers at moderate costs. Its main characteristics and principal assumptions are: --Falls analysis in a two-dimensional space. Motion kinematics are studied along a vertical plane. which can be monitored by in situ tests. Mathematical model To carry out efficient mathematical modelling of a phenomenon. This is the crucial part of modelling and it will determine the quality of the model. called CADMA. The CADMA model was developed in 1987 according to a method established by Bozzolo and Pamini at the beginning of the 1980s [17. The latter models are generally better than the former. 18]. and at the same time carried out a considerable number of in situ tests. is more expensive and in most cases unnecessary. (b) to make certain assumptions that allow less important elements to be reasonably disregarded. --the assessment of maximum run-out distances. in order to determine the areas at risk. 3). Nevertheless. heights of bounces and energies achieved during the fall. The main targets of a rockfall model are: --the assessment of velocities. since three-dimensional analysis [19]. Flow-chart showing the steps of the study procedure. is very important in this phase of the work.710 AZZONI et al. the above-mentioned limitations have been overcome using mathematical models. as they are more capable of accurately reproducing the different phases of the fall phenomena. The model is based on rigid body mechanics. respectively) [11-16]. --The fall is composed of different phases. The following paragraphs briefly describe the main characteristics of the mathematical model (techniques and assumptions) and the in situ tests. Analytical computer models can be roughly divided into two types: those considering the block either with no mass or with the mass concentrated in one point (kinematic and lumped mass methods. 1. --Each block falls along a trajectory not affected by those of the other blocks. and those that consider the block as a body with its own shape and volume [8. each with its own characteristics and assumptions. both for the study of the phenomenology and the assessment of the relevant physical parameters. in 1987 ISMES and ENEL CRIS started a joint research program for the study of rockfalls. They set up a mathematical model. 17-22].~ synthesisof the phenomenon " Fig. The methodology adopted for setting up the program is expressed in the flow chart (Fig. including the previously mentioned segments (Fig. METHODOLOGY FOR ROCKFALL ANALYSIS According to the above mentioned considerations. Yes . for the purpose of investigating the principal modalities of rockfalls and to determine the principal parameters involved in the model. Most programs analyse the falls in a two-dimensional space. The kinematics of the motion is studied in a vertical plane obtained by rotation into a single plane of all different vertical planes. or after an impact with the slope [Fig. Fig. local slope angle at impact. Free fall ". and therefore without any contact with the slope. X Z Fig. . C' ".Freefall . As is the case with most computer models. The following sections offer a detailed description of the characteristics and assumptions of the model. Rolling Free fall Impact '. 4(b)]." ' . rolling and sliding). For this purpose.. impact and bouncing. the model takes into consideration a large number of falls and adopts random values (chosen within a previously determined range) for each of the above-mentioned parameters. detachment area and inclination on the slope of the trajectory after detachment) requires that both description and analysis of the phenomena be statistical rather than deterministic. 2. Free falling. usually due to a sharp variation in the slope angle [Fig. 4. Different initial conditions for a free falling phase.Vm \ \ '.: 711 IDEALIZEDTOPOGRAPHIC PROFILE A TRUE TOPOGRAPHIC PROFILE 1-I' 2 ~ TRUE TRAJECTORY ~ ~ ~ ~ - I PROJECTED TRAJECTORY . 3. Model of the block at the impact. Motion takes place after a rolling or sliding phase.. and main phases of the rockfall (free falling.AZZONI et ANALYSISOF ROCKFALLS al. . \ IVy "••mpact a) ~llV b) Fig. The peculiar characteristic of free fall is that motion occurs in the air.. The natural variability of some important parameters (such as the shape of the block. the mechanical characteristics of the slope. 4(a)]. Y J. . it is important to consider the fact that the internal forces of reaction between two bodies in collision are far greater than the active external forces (e. angular momentum is not conserved. K. Likewise.. in this case. -after impact a rotation point. the phenomenon can be satisfactorily analysed by assuming the validity of conservation principles (linear and angular momentum). Relation between impellingf(f) time interval corresponding t and active F(r) forces in the to the impact. When the aerial trajectory intersects the slope. and the rotation of the block around its centre of mass.g. the impulse due to the internal reaction forces is much greater than that due to the active external forces during the same infinitesimal time interval. If we consider that the impact is partially inelastic. namely that the internal forces predominate over the external ones and since these forces act on the point P.: ANALYSIS OF ROCKFALLS CADMA analysis of free fall disregards both the effects of air friction and of aerodynamic uplift. The latter depends on the geomechanical characteristics of block and slope. The exact determination of these internal forces is very important but quite difficult to achieve.. The concept is clearly explained in Fig.. r. ..712 AZZONI ef al. * $02(Z + r’) = Gax’ Ko . the program calculates the value of c* through which the conservation principle of angular momentum is valid. The same equation shows the importance of carefully assessing angular velocity (before and after impact). The experimental analysis of the impact shows that F(t) f(t) t. it is possible to take the momentum of all forces with regards to this point as equal to zero. and thus gives: K = L.+Ar s '0 fO>dt represents the impulse of contact reactive forces. Therefore. it is possible to consider that the angular momentum at P is conserved during impact. Motion in the free falling phase is basically composed of two different movements: translation of the centre of mass. +ontact between block and slope occurs at an infinitesimal area which can be assumed as point P s fo represents the impulse of the active forces (Fig. Assumptions for the block at the impact. 5. the geometry of the slope and by the energy dissipated. it is possible to simplify the model in line with the following assumptions: to+ At F(t) dt -the block at impact has an ellipsoidal shape. the weight).. Similarly. for obtaining the correct determination of the restitution coefficient. Equation (B8) reported in Appendix B allows evaluation of the restitution coefficient of energy t *. For engineering purposes. To model the impact phase. takes place around this Because of the previous assumptions. the characteristics of motion after impact are heavily conditioned by the block’s shape. Impact and bouncing. 6. Some details of the mathematical formulation are reported in Appendix A. an impact takes place and the principal consequence is a loss in energy of variable importance. According to the previous equations. 5. As it is possible for the calculated t * to be greater than the one observed experimentally L. At Fig. that can be analytically described by a quadratic equation. 6). the impact after the free falling phase is the intersection of the parabola with the polygonal representing the slope profile. Details of the mathematical formulation can be found in Appendix B. Initial conditions are determined at the instant the block separates from the slope profile. the collision angle and the configuration of the block at impact. where: X’ Fig. the latter is considered as the upper boundary of the range of the calculated 6 *. 9). the block rolls with simultaneous slips at the points of contact [18]. The program compares the value of the normal component of velocity (Vy in the adopted X Y reference frame). or at least inaccurate. Furthermore. 7.: ANALYSIS OF ROCKFALLS Another important element in the modelling of the impact phase is the criterion used to establish whether. particularly if we consider that sliding is a phenomenon basically limited to the initial and final phases of the fall. Simplified sketches showing the influence of morphological irregularities of the block [(a) and (b)]. the model allows determination of the block's stopping point along the slope. All these data are graphically represented by histograms (Fig. to analyse rockfalls using a deterministic model. Therefore. and that it rolls on a slope with rolling friction. if Vy > Vy. Probability analysis. the block bounces or rolls and slides.AZZONI et al. The mathematical model simplifies all situations of varying complexity. b) ~03 P3 02 P ~ i X. height of the fall trajectory. by choosing within the mentioned range. (b) to make a numerical simulation using the Monte Carlo method. If Vy < Vy. mainly because of the non-linear behaviour of the materials and the morphological irregularities of the block and the slope. Given the intrinsic variability of the phenomenon. the values of all the variables at random. gives values (in terms of probability distributions) of the main parameters characterizing the falls (translational and angular velocity. C2 o (X) Fig. and at predetermined distances. where a momentum is caused by the deformability of both the wheel and the ground (Fig. 8. the block rolls (or slides). Like wheel motion. the block tends to make small jumps and slips. with a value of Vy experimentally assessed (Vy). the situation is more complex. 7).F a) - 713 a) d) Fig. it has been necessary: (a) to define all the variables within a range centred on their mean value. 8). it is impossible. and energy of the block). Interpretation of the rolling friction coefficient for a wheel. o. Rolling and sliding. On the contrary. after impact. The model analyses the rockfall trajectories graphs in the considered vertical section. which bring about the formation of more complex momenta (Fig. The same criterion is used to assess the transition from rolling to bouncing. after impact. if these dimensions are larger. sphere. the rolling movement of a falling block is connected to the momentum that occurs at the contact point between the rolling body and the slope (Appendix C). the block bounces. usually if the dimensions of the block are smaller than those of the irregularities of the slope. disc). and the slope [(c) and (d)] on antagonistic momenta. The situation is quite simple in the case of the wheel. 04 Pl = Ps i X. by considering sliding motion in equivalent rolling terms. In the case of the block rolling down the slope. This simplification is acceptable. thus allowing for sharp variations of slope angle along the fall trajectory. : . According to experimental observation. which is basically a conservative assumption. This is done by assuming that the block has a circular shape (cylinder. Detailed modelling of the phenomenon would require a very elaborate analysis. Results and the program graphs. .00 I I 64.78 °2~01" 4.63 40.33 H (m) Fig. Experimental method Experimentation on physical models allows visualization of all the aspects of a phenomenon that.67 1. 0 0 56.00 I I 72. could be difficult to . the kinetic energy of block during the falls can be easily calculated with the parameters (velocity.00 32. X . because of their specific and aleatory character. 2~oo' 3~oo .00 I m) ! .'83' 3~25' 3166' V (m/s) OBSERVATION SECTION X--35.00 48. . Although it is more directly provided by the program. 9.92 ~ 16. Graphic output of the program. frequency and mass of the falling blocks) according to equation (B7) reported in Appendix B.00 .4 0 .34" 8.34 2.00 64 -00 ?2.OO 8. e'oo '.00 16. 8 .0! f(Hz) HEIGHT OF BOUNCE .00 24.00m 66.49" 17. 0.714 AZZONI et al. 0 .00 FREQUENCY 10.92-" 15. I 4.66 3.00 Ira) ++ 4~07 --.42' 2.6oo' .00 2.: ANALYSIS OF ROCKFALLS N oo oo g g ' ".00 X VELOCITY ~6. together with a lateral moving one. (b) Throwing the rocks down the slope. --measuring the true distance between points on the trajectory. Besides these elements. The time values used for assessing the velocities are calculated by counting the number of shots taken by the camera (which works at a velocity of 24 shots/see) between two relevant positions of the block's centre of gravity. The parameters used for calibration are velocity. but through some coefficients which allow the modelling of the amount of energy dissipated during the various phases of the fall. were evaluated both through back-analysis and through the elaboration of in situ tests carried out at Strozza. This is generally considered to have an elastoplastic behaviour. and the velocity at which the block changes its movement from rolling to bouncing and vice versa). The activity can be performed with video cameras. by hand. depending on the size of the blocks. --geological and geomechanical characteristics of the falling blocks and the slope. digitizing (both for the lateral and frontal records) the shape of the falling block at different instants during the fall and evaluating the position of the centre of mass of the block. in order to evaluate the most probable falling paths and physical characteristics of the possible falling blocks [23. ---calculating translational and rotational velocities. namely the restitution coefficient and the rolling friction coefficient. The following sections describe the more relevant dynamic parameters used in computer analysis and their assessment [25. by jacks or. These activities are usually carried out in the following way: --using specific software. the following factors: --topography of the slope. the restitution coefficient expresses the amount of energy dissipated during the ground impact. frequency. The former are usually represented not directly. by which the program is able to find values of the main parameters comparable to the experimentally observed ones. In particular. One of the best is ground photogrammetry. The block's geometry is usually expressed by its volume and the ratio between its main axes. In situ tests are generally carried out using the following method: (a) Assessment of the topographical. 3-5 lateral synchronized fixed video cameras are generally used. precision to the order of centimetres is achievable). as well as the shape and dimension of the blocks. ASSESSMENT OF DYNAMIC PARAMETERS AND OTHER ELEMENTS RELEVANT TO COMPUTER ROCKFALL ANALYSIS To predict the rockfall characteristics through computer analysis. These record the movement in the vertical plane parallel to the trajectory. (d) Elaboration and analysis of the records. and 1: 1000 for a long slope (hundreds of metres long).AZZONI et al. These are the same elements that should be assessed when carrying out a real case analysis. in view of the fact that a certain amount 715 of distortion due to non-perpendicularity between the rockfall plane and the direction of the camera is practically unavoidable. The assessment of the restitution coefficient was obtained by taking into account . For short slopes of fewer than 100 m. by an excavator. once the section for study has been defined. geological and mechanical characteristics of the slope and the blocks. the mathematical model utilized for this research requires the assessment of other parameters (namely. while the topography. In order to carry out correct analysis of in situ tests. The topographical survey should be able to describe all the relevant points of the slope with adequate precision (centimetres in the former case. The target of backanalysis calibration is the assessment of values for the dynamic coefficients. which provides a remarkable accuracy (on a 50 m high slope. Assessment through back-analysis. Restitution and rolling friction coefficients The dynamic coefficients used for the mathematical analyses. it is crucial to determine. Italy. decimeters in the latter). 26]. A fixed camera is also set in front of the slope to record lateral displacement of the rock's trajectory on the slope. the mechanical characteristics and geometry of the blocks and slope (topography) are needed. (c) Recording the rockfalls. height of bounce and run-out distance. Rockfall experimental models enable us to define the fall modalities. The experimental tests are generally carried out using several fixed and moving video cameras. The scale of the topographical survey should be as detailed as possible: usually not larger than 1:200 for a short slope (tenths of metres long). ----evaluation of the height of bounce along the whole analysis of in situ tests and assessment of experimental parameters. is described in a profile which should be as detailed as possible. and integrating all data gathered in the analysis of all camera records for best assessment. assess the parameters to be used in the analyses. as carefully as possible. The rolling friction coefficient expresses the frictional effect of the ground on the rolling block. This activity is usually performed. where greater accuracy is required.24]. The topographical survey can be done in various ways. These two different approaches basically provided similar results [26]. and to calibrate the mathematical model. In this case the work should be complemented with a geomorphological study of the slope and a geostructural assessment of the rock mass.: ANALYSISOF ROCKFALLS define and assess with accuracy. more easily when the test is carried out in a quarry. geomorphological. the modulus and direction of the starting velocity. 3 m 3 0. two different values (depending on the volume of the blocks) were determined.45 0. Values of the restitution and rolling friction coefficients adopted for the calibration of the mathematical model Maximum restitution coefficient Block size Rock (limestone) Fine angular debris and earth.90 0.35-0. do not always provide correct results when used on other slope types (particularly if topography is not very detailed). compacted (gravel and cobbles. For this reason.40 0. The ground's restitution coefficient was evaluated according to its more rigorous definition. yet unconfirmed by other methods.454).70-1.65-1.8 A 1. The values of rolling friction coefficients were also evaluated through back-analysis of two different experimental tests.554).40-0.60-0.70 0. namely the ratio between the total energy of the falling block before and after its impact on the ground.554).45 0.20 0.50-0. 11).504).2 Rolling friction coefficient 0.65 0.70 0. In this way it has been possible to assess that the maximum restitution coefficient (Emax)ranges on normal slopes from 0.60 0. Restitution coefficient values assessed for impacts on different types of ground (after [26]).4 • Unreliable value ] lain BARE ROCK 0.454).00 <0.40~.554). 10).65 0.2 . values larger than 1 should be disregarded.6 1. depending on geological conditions (Fig.65 0.50 0.85 0.6 O~ 0.50 0.60 0.404).504).754).60 0. of about 1. slope inclination and sizes of the block at impact.3 m3).8 Restitution coefficient Fig.75 0. . Energies were assessed by 0. Since the rolling-friction coefficient in this case depends on the roughness of the slope in relation to the size of the falling block. carried out with blocks of different shape and volume (a prismatic block.35 to 0. and a spherical one of 0.60-0. Like the previous parameter. The values represented in Table 1 correspond to the maximum values of the restitution coefficient assessed for different geological materials. and basically corresponds to the slope angle at which the block moves with a steady velocity (neither accelerating nor decelerating) (Fig. 10.4 1.95.70-0.716 AZZONI et al.50 0. soft Medium angular debris with angular rock fragments (20-40 cm dia) Medium angular debris with scattered trees Coarse angular debris with angular rock fragments (40-120 cm dia) Earth with grass and some vegetation Ditch with mud Yard (fiat surface of artificially compacted ground) Road different falls.40 0.50-0. Assessment by elaboration of in situ test. the rolling friction coefficient was assessed both through back analysis (Table 1) and the elaboration of in situ tests.2 m 3 0.45 measuring both rotational and translational block velocities.: ANALYSIS OF ROCKFALLS Table 1.2m 3 in volume.60-0.20 0. enabled us to assess rolling friction coefficients which provide a good match with those " • Reliable value DEBRIS oo:o • • 0. This experimental method was set up partially in line with the concepts proposed by Statham [7]: coefficient depends on the ratio between the size of the rolling block (D) and the debris (d).80 0. In this figure.80 0. another method for the assessment of the restitution coefficient was established. Following some tests. it was noted that values obtained through back analysis alone. dia < 20 cm) Fine angular debris and earth.504).60 1. since they indicate an increase of energy (these values are related to some limitations in the analysis of video records). Careful observation of the blocks' velocities at different positions in the fall trajectories and the measurement of the slope angles corresponding to the different positions of blocks.70 0. '. . . particularly on rough or low inclination slopes. when looking at the slope face) and the length of the slope. . Choice is usually decided according to the following criteria: .. . 10 0. .. Experience gathered from observation of experimental falls and case histories is an important tool for choosing reasonably representative trajectories. Threshold velocity between rolling and bouncing As already mentioned. . . . . . ~v 35 ~ . .30 0. since here the velocity is much more influenced by the effect of the force of gravity. . [] discoidal block. the same tests also showed that steeper slopes have smaller dispersions (Fig. .. . . . . . mean most rockfall computer analyses . . Simulation of the different phases not only depends on topography. and depends heavily on its topographic characteristics. This parameter is not so relevant to the general trajectory when the fall starts from a steep wall. the in situ tests showed that this parameter is about 20%. so as to " m o v e " the block. • columnar block. . . They are comparable to those provided by Bozzolo and Pamini [17. . . ~3-o . ' i A'-- 30 E < 25 20 15 0. . or when the coefficients from calibration provide unreliable results. . . 45 .. . . . . ... . high costs and complex logistics. unless topographical constraints (e. . .g. . . Passage from one type of motion to another depends mainly on the topography (e. . . . . profile irregularities produce bounces).40 0. . n i i • . O columnar block. In some real cases. • discoidal block. This value has been assessed through back-analysis and usually ranges from between 1 and 1.: ANALYSISOF ROCKFALLS 717 50 . The tests used for this assessment also revealed the important effect that the shape and dimension of the blocks have on their velocity. Unfortunately. . As a general trend. accelerating. 40 . Starting velocity Even if the real starting velocity generally equals zero or is slightly greater. . the program does not allow block bouncing and therefore analyses the fall as a rolling phase.. . . in computer analysis.5 m/sec. ~ O r o.. . Assessment of rolling friction coefficientand comparison with Statham's values (after [26]). such as when only the profile of the lower part of the slope closest to the threatened area is available. . . . A" = Statham's lower boundary. . . By defining the "dispersion" of the trajectories as the ratio between the distances separating the two extreme fall paths (i. . . In a relatively small number of cases the tests also showed that low-shape coefficient blocks (specially tabular and discoidal types) behave like spheroidal ones when their velocities are large enough for them to roll around their minimum axis ("wheel-like" movement) [23]. . . . 11. . . .. . .AZZONI et al.. .. . Furthermore. each of which is separately simulated and analysed.T h e conservative strategy of considering trajectories that present the greatest risk. ... . . .. . . Below this threshold.. ±__ • . . . • spheroidal block. . . it also owes much to "threshold of the normal component of the velocity". decelerating. . the trajectory furthest to the left and to the right. . should always be adopted. . 18] and also to Statham's empirical coefficients. . ~. ... .00 . . . This method can be used to confirm the values from backanalysis or to find new values. • tabular block.50 diD Fig. . 0 spheroidal block. it is useful to use a certain starting velocity. =. accelerating. B = rolling friction coefficientby back analysis. accelerating. .T h e trajectory tends to follow the steeper line of the slope.. CALIBRATION AND COMPUTER MODEL RESULTS Effect of topography on the computer based rockfall evaluation Topography is a key factor in evaluating the dynamic parameters. A considerable aid in deciding which trajectory to study comes from in situ tests [26]. ~. . decelerating. when it is impossible (or impractical) to carry out complete in situ tests. .. . i . .. .. it is possible to note the progressive increase in rolling velocity with the increase both in shape parameters and dimension of the falling blocks.e. since it affects both falling blocks velocities and heights of bounce. . it is sometimes necessary to provide a certain initial velocity (usually 1-3 m/sec). . . . . . .g. A tabular block. accelerating. A' = Statham's upper boundary. . in fact. obtained through rockfall back-analysis. ~O.. . the program simulates the fall as a sequence of different phases. . . valleys) reduce this value._. . . decelerating. though less probably. decelerating.20 0. . . Choice of the trajectory The choice of the rockfall trajectory to be studied is crucial when using a two-dimensional program.. 12). . ~/ D q 10000 i "~ V ® . L ~ . ..-.. J-J-L~JJ i d r .. where free fall is the prevailing motion type.. ... the program was run with 20.'-. F .... . .. --When a special topographical survey is not available.. .. /.i . .r . . even though the experimental value still fell within the range calculated by the computer analysis. . .. The test enabled us to make the following main observations: .. a less accurate survey is acceptable. . . . .. . . To determine this. .I .- i ~ ..A progressive increase in smoothness corresponds both to an increase in calculated fall velocity and frequency of rotation. :I:=.::" ... it is important to determine the margin of the error involved in this approximation. .: ANALYSIS OF ROCKFALLS Slope top Slope top Rock f o i l / / . and to a decrease in height of bounce. 25 and 16 points. . : : = :'' ~ : : " . ' I & . Optimal number of simulations Dimensions of the output files and time required for the CADMA rockfall computer analysis increase as detail increases or when. . .1 . .1 . . . . . --Detailed topography is necessary.. The most favourable topography is that surveyed at intervals similar to block dimensions. When such detailed topography is not feasible. I [ . I ...:. . " I 7 7 . i i I ~ ~ i -*- "1 f7 p i 1 ' r "I ['1 1 I I I I I J .. I I I I I I • 25-30* ~ ~ ... .. ..1 - 50-60* [E i t i i i ..t --'1 I . \ / I I Limits of r o c k / ( .i .r .. . it is thus advisable to design higher fences (rather than stronger) than those hypothesized by the computer. . . are performed on slope profiles surveyed with low detail. In view of the fact that low topographical detail corresponds to greater profile smoothness. . . .. ... . 50.. but not height of bounce. 100 a.. . In particular.] I'l I " .r i i • i 'l-- i i i i i i i i i i i i i i i i i i i i ~ I I I I I I I I i i i i i i i i t i I i / i 100 6 0 ° ~ _ ~ r r l i i i > i i 10 "t . I --=--:-=aZ_---.... ..r r i i I i i i I I Slone annie .i : : [ -... . .718 A Z Z O N I et al. .. .. I J _ _1_ _1_ ~ J _ l _ l _ ! T. . . .. .-_ i '1" I I ) I I . . .~ 7 --III . it is important and useful to ascertain the minimum number of simulations required to provide statistically valid results [25].. . Sketch of a typical slope used for rockfall tests (front and side view). .. I -- 1l .. . . I . I. and by comparing the analysis results among themselves and with those obtained from an experimental test performed with a block of the same characteristics (Fig. . ..T -..... .. 1 7 7 " . respectively. . . .. . detail is especially important in sections where the block bounces and rolls. . . I c~!:::::r::~::c:.. . ... . .7 . . . . 12.: : 3 : 2 _ . because of uncertainties about input data... An assessment of the relevancy of topographical detail for the computer results has been carried out by performing the same analysis on the same slope surveyed with 152. I I I I I . .. .. In view of this. :::::c::~::: I I I . ..r -i- . . .. ..--•l . . . I I r d .F F i . i i T~_I_Iq r ...~ i i -.. . parametric analyses are required. 0 I i I i i i i i -F_l_ i i .. 100 and 200 falls and the respective results were then . the maps used for drawing slope profiles should not be scaled at lower detail that 1:200 and 1:1000 for short (tenths of metres) and long (hundreds of metres) slopes.fell/. This result is conservative in respect of fall energy. ..r T i i i 7 ~ . respectively.. 13) [25]. . and relation between the maximum distance separating the fall paths at the bottom of the slope (D) and the length of the slope (L) (after [26]). . .~: czt~' ====================== I-- . . .~ -. .~ . .T .i ... . When analysing rockfalls with low detail topographical surveys.. . F i 1 ~ I -. .[ 7 . i i . . .. I 1000 o I .[ 3 = r J__II__L_LJ_LL i I '-.. . . 7 . . LLI I. T--i- I l ' I i i r- - 7- . . . " ='k t o " t' • I- .:z~-:.. there should be at least surveys made of all the relevant points of the slope (corresponding to significant changes in slope angle) and some points taken at reasonable intervals (not greater than 20-50 m). ~. I I ... .. I . .~-_-c~_:ccc -~ . . . 79. \ I oreo / .. 43. I t . . . . since the smaller asperities can already be satisfactorily taken into consideration by the rolling friction coefficient.- T777 T--I-q-F~777 r-r 7-rrr 1000 Distance between fall paths (m) Fig. . .. .. . . . .. value value Fig.. Topographical profiles of the test slopes (after [26])...: _ _ _ • •••--i-- ~ & 16 24 29 DISTANCE OF OBSERVATION POINT (m) ABCDE ABCDE 35 IBOUNCE HEIGHT (m) ABCDE ABCDE ABCDE I I I I E'~-6 I"1O i AAAAA ~3 . ..~:'~'~'~. i}k~Fine-medium angular :-'%'~ debris. . ":~" Rock Medium angular debris. E = Slope profile with 16 points.. . .Computer max. 13..~/k '". . .. RMMS 32/7--(3 debt.~. . " i " : ~ / d i t c h ~-i. loose '. .. .ne '" .'k~ loose 10 Coarse angular debris. 20 20 ... . B = Slope profile with 79 points...._ _.... value .. . SLOPE A SLOPE B . . . AAAAA i __llll II I[ i z~.~. "1- ~i•~• . . i i t*l II I I I I I - A AAA~ I I Ii-0 7 16 24 29 DISTANCE OF OBSERVATION POINT (m) .. . s..~ ( yard 10 10 20 30 40 .... . ii~• • •i AAA~A l # Ill il 1•• 16 24 29 DISTANCE OF OBSERVATION POINT (m) 35 ] FREQUENCY (Hz) ABCDE ABCDE ABCDE ABCDE ABCDE .." R~ck 40 40 '~':...... .. . . ..~ 30 ' r. Effect o f topographical detail on the results o f the mathematical rockfall analysis: A = Slope profile with 152 points. 7 --:. .'~...o 5'~ m .. . c o m p o c .Computer min. ..... .VELOCITY (m/s) ABCDE 24 ABCDE ABCDE ABCDE ABCDE I I l l l ~16 .....':::. .. value 35 • Computer mean value * Computer modal A Exp..ar 719 go l'~: ~ . .~% .. .. . ._. I I I i tit ##### iII -. D = Slope profile with 25 points. .. . . . .... .'o m 2'o Fig. . ..J Compacted Fineangular debrisand earth 30 -~. .. .~Z tl . . . . ..".... 14. ~ . e d angu.. . C = Slope profile with 43 points. .. z i IJ.3 ¸ i N I . modal value • O Av.0 35. VELOCITY (m/s) 24.. The analyses were carried out using dynamic coefficients values specifically determined for this program. mean value 4. The tests showed that analyses with 100 and 200 simulations were positive and totally similar.0 T (3 ~'~ 3. & min. for slope A. value Fig. experimental value O Max.720 AZZONI et al.0 DISTANCE OF OBSERVATION POINT (m) FREQUENCY (Hz) 9.0 DISTANCE OF OBSERVATION POINT (m) -.0 . A list of suggested values.0 29.0 T v >" 6. 170 points. 15. These values.0 0 o 8.0 7. run with 100 simulations.0 29. is presented in Table 1.0 I 29. where in situ tests had been performed. The results of the tests performed with 50 falls were basically similar to those with 100 simulations.0 UJ rY LL 0.: ANALYSIS OF ROCKFALLS compared amongst themselves and with the experimental data.Computer av. an analysis carried out with 50 simulations took about 8 min on a 386 PC and produced an output file of about 1.0 "1" 0. Thus.0 7. In fact.0 16.0 16. Using a 486 PC.0 O Z UJ O 3. this number of simulations is considered advisable when dealing with detailed slope profiles or parametric analyses.2 Mbytes. while analyses with 20 falls were inaccurate. CONCLUSION A mathematical model was used to analyse and predict rockfall trajectories on two different slopes. Comparison between the computer analysis results and the experimental data.0 24. max.0 DISTANCE OF OBSERVATION POINT (m) BOUNCE HEIGHT (m) 9._1 m 0.0 I 7. value • Computer av.0 35.3 Mbytes.0 ? E 6. described with approx. The same analysis.0 24. .0 I 16. for a slope about 60 m long. the same analyses took about a third to a quarter less time. experimental value Computer max.0 E 16. Computer av.0 35. required about 17min for calculations with a resulting output file of about 2.0 I 24. but 50% faster. as evaluated by an elaboration of in situ tests and through back-analysis of the tests and real rockfalls. . mean value O Max. From these results it is possible to draw the following conclusions: Translational and rotational velocity and energy.0 ¢D o .0 40.. These values were assessed at specific observation points placed at critical positions on the slopes. max. The program provides acceptable results for this parameter.0 L 54. Now that calibration has been completed. 16.10. the experimental velocity generally falls within the range of the predicted values and is always described satisfactorily b y the mean and the modal values. & min. as well as histograms of velocities.AZZONI et al. 14) are shown in Figs 15 and 16.. In the sections beneath the steep rock slopes.. mean and modal values... the stopping effect of a ditch full of muddy water was also simulated correctly in view of the fact that this ditch stopped over 80% of the falling blocks. . the program is generally able to find correct results for this parameter. The program is simple to run and provides clear and easily read graphical outputs. frequency of rotation. average value of the maximum.0 54.0 E ---.8 I. 40. it tends to slightly underestimate the values. In particular. the program is currently undergoing slight changes.0 15.. slope profiles with fall trajectories.0 .. If the topographical input is good.6 E " " 1..4 c i 0. though in this case a possible inaccuracy in the experimental values due to over-estimation should be taken into account. height of bounce and stopping distances. V E L O C I T Y (m/s) 20. should be used by taking into account all the observations and criteria discussed in the previous paragraphs.. and the overall maximum calculated value).. 5.0 0. a comparison between experimental and calculated values of height of bounce. value 721 Computer av..0 DISTANCE OF OBSERVATION POINTS (m) BOUNCE HEIGHT (m) 2._1 LU > .0 DISTANCE OF OBSERVATION POINTS (m) • = Computer av. such as tables with all the numbers generated by the calculations.. The program is generally able to make correct (or at least acceptable) predictions of these parameters.U "I" 0. experimental value Computer max.2 I"1- -~ 0.. Run-out distance... minimum..0 1. The diagrams also highlight the relation between experimental values (maximum and average values of 15 falls for each slope) and the values provided by the computer analysis (for each slope.. With regard to the computer analysis results. experimental value Fig. value • • O Av. velocity and frequency of rotation for two different slopes at Strozza quarry (Fig. Comparison between the computer analysis results and the experimental data.: ANALYSIS OF ROCKFALLS even if correct overall (for a rolling block of up to 1 m3). for slope B. In particular.. IIIIIIIIIIZIIIIIIIIIIil . modal value • Computer av. Height of bounce. Mak N. 19. Montreal. Ritchie R. 20. J. Hong Kong.. Meet. 263-270 (1986). Acta Mech. it m o r e Accepted for publication 15 January 1995... Probabilistic approach for design optimisation of rockfall protective barriers. F. 17. and Hacar R. L. 16. 122. et Ch. Habib P. Highways Res.. Simulation of Rock Falls down a valley side. and Zimmermann T. Pfeiffer T. International Colloquium on Physical and Geomechanical Models. Soil Mech. Bodies falling down on different slopes. Chan Y.(to) = Vow. 17. 123-125 (1977). Proc. Bozzolo D. 685~590 (1988).722 AZZONI et al. P. Min.. Laboratorio di Fisica Terrestre-ICTS.to) + XA Y(t)= . Quarterly J. Azzoni A.(t) = 0 ax(t) = . Relations between scree slope morphometry and dynamics of accumulation processes. Descoeudres F. Azzoni A.~ ' g '(t -. XXVl. 87-96 (1986). (A1) The initial conditions are as follows.: ANALYSIS OF ROCKFALLS B O U N C E H E I G H T (m) V E L O C I T Y (m/s) 25 ~ 20 v 15 o. Ass. pp. Technology and Medicine. Richards L. Design of a boulder fence in Hong Kong. Broili L. dissertation. Martino. Rotterdam (1992). Canada. Department of Engineering Geology. Comparison between the results of the computer back-analysis and the experimental data for roekfall No. M. Application ~ l'Otude du versant de la montagne de la Pale (Vercors). 237-258 (1979).. Meet. Lausanne. 24. Meet. Rock Engng. Evaluation of rockfalls and its control. Bozzolo D. Desvarreux P. and Evans S. 337-342 (1987).. Th~se. 9th Int. 3. pp. Proceedings of 6th Int. 2. 11. 551-557 (1987). C. to further improve output data and render u s e r . 26. Hungr O. Rock Slope Stability Analysis. ISMES experience on the model of St. pp.. Hong Kong. Methods for predicting rockfalls. A program in Basic for the analysis of rockfalls from slopes. Modello matematico per lo studio della caduta dei massi.Computer max & min. M. 93-102 (1982). 84-95 (1987). 12.. Meet. Inst. Congress on Rock Mechanics. Proceedings of the 5th International Symposium on Landslides. 14-28 (1963). Engineering evaluation of fragmental rockfall hazard. Giani G. 26. D. Camponuovo G. liaison Labo P. the components of the acceleration are: a.1-11. Metall. pp. MIR. Roekfall Dynamics Protective Works Effectiveness 90. pp. Balkema.t o ) + ( Y A + h o ) (A4) .. et Ch. 23. Rochet L. R. and Martin Cocher J.. lo W 5 > I 7 I 16 I 24 0 I 29 35 DISTANCE OF OBSERVATION POINT (m) . G. Dynamic studies. Lugano-Trevano (1982). Rockfall Dynamics Protective Works Effectiveness 90. Imperial College of Science. 91-95 (1977). and Pamini R. 13.f r i e n d l y f o r all r o c k fall s p e c i a l i s t s . Record.to)2 + Voy"(t . Unpublished notes (1987). 6 • Computer mean value 7 16 24 29 35 DISTANCE OF OBSERVATION POINT (m) • Computer modal value o Experimental value value Fig. Etude cynematique et dynamique de chute de blocs rocheux. In situ observation of rockfall analysis. 22. Drigo E. 25-39 (1977). Bollo F. Proc. pp. 10. 1501151. 17. Conference on Rock Engineering in an Ubran Environment. Giani G. and Bowen T.g . Azzoni A.Sc. Lausanne. Bull. Application des modeles numeriques de propagation a l'etude des eboulements rocheux. Proc. Metall. Rossi P. Giraud A. Politecnico di Torino. pp. A simple dynamic model of rockfall: some theoretical principles and field experiments. Empirical and mathematical approaches to rockfall protection and their practical application. 307-314 (1992). Engng Geol. and de Freitas M. Proc. R. 113-130 (1986). REFERENCES 1. 7. Conf. 22. Commun. Secondo Ciclo di Conferenze di Meccanica e Ingegneria delle Rocce. pp. 11. Min. 63. and Au W. Methode de calcul de la dynamique des chutes de blocks. 1237-1243 (1988). Conference on Rock Engineering in an Urban Evironment. 135-146 (1989). and Blomfield D." (t . Chan C. M. 6. C. Statham I. P. and Rautenstrauch R. Spang R. Bull liaison Labo P. 25. Proceedings o f the 6th International Congress on Rock Mechanics. Lyon (1985). INSA. 5. Inst. 9. Hock E. (A2) and the co-ordinates of the initial position of the center of mass at time to by: X(to) = XA Y(to) = YA + ho" (A3) After integrating equations (A1) over time: x ( t ) = vo. M. 123-125 (1977). and Pamini R. Spang R. Falcetta J. Azimi C. P. Rock trap design for pre-splitting slopes. Engng Geol. London (1993). Found Engng 2. Hacar B. 8.. Notes sur le robondissement des blocs rocheux. p. Proc. Three-dimensional dynamic calculation of rockfalls. APPENDIX A In the assumed OXY reference frame (Fig.. Computer rockfall model. Rockfall Dynamics Protective Works Effectiveness 90. 14. Submitted. Protection against rockfalls---stepchild in the design of rock slopes. Rockfall protection: a review of current analytical and design methods. Proceedings of the 6th International Symposium on Landslides. Paronuzzi P. Rockfall Dynamics Protective Works Effectiveness 90. 18. Proceedings of the 5tb International Symposium on Landslides. pp. Montreal. 135-146 (1989). H. W. AI). 11-24 (1977). and Zaninetti A.. 361. denoting the components of the initial velocity V(to) at time to as: VAt0) = V0x V. 4. F. Piteau D. Bull. Christchurch. Prediction of rockfall trajectories with the aid of in situ tests. 21.13 (1986). Rock Mech. 15. Computer simulation of rockfalls. B2(a)] (b) XG = Xp = d.~:x~ CASE ¢): X~ < X e ~0 ( ~ o x Fig.-L. (B4) V = c o " d y .'d~ " " I + d2~+ d~ (86) (B2) and P G = ( X G . depending on the position of the center of mass G with respect to the contact point P (Fig.c o z . B2)..'i+ V. .co+V~. k j+O. 6). then V~ is always > 0. Obviously if Vv<~0 bounces can not occur. Substitutingequations (84) into the right hand side of the cquation (Bl).o ~ .. CASEa):Xa'>X" 0 o = ¢ n ~ ' ( Y o . Assuming that a rotational motion about the contact point P takes place after the impact (Fig.~o Y ( ~ ) ~ (~)~ 7.2 " V ~ ' [ X ( t ) -. > 0 ~ V y < 0 [Fig.. Three different possibilities can occur: (a) XG > Xp ~ d.d~=l. V.: ANALYSIS OF ROCKFALLS 723 Y X Fig. Y(t) = . Since t o = O . AI.~oo+Vo~'d. B1). e e_ II I Fig.. Configuration of the block before and after the impact.i + O . i . the velocity of the center of mass can be obtained as follows: V = to x r = to x PG. obtaining (t .X p ) .X v ) " j..Xp I = moment of inertia of the block about the centre of mass co0. Definition of the free falling problem in an assumed O X Y reference frame.. the possibilityof a second impact has been introduced. which corresponds to the equation of a parabola.c o 'd~'j = V. = 0 ~ Vx = 0 [Fig.> 0 [Fig. co = angular velocities before and after the impact V0x. Different possibilities for the block at the impact.. In this way the block assumes a symmetric position with rcspcct to the previous onc.to) from the first equation of (A4) and substituting it into the second one. = - B Applying the principle of the conservation of the angular momentum over the infinitesimal time interval.Y~) ~ ~ -co~ ~ . equal or greater than 0. Iz~ = x components of velocity before and after the impact Voy. it could be less. As for dx.. de assuming co = coz: (B5) Since Yo > YP is always thc casc. before and after the impact (Fig. = y componcnts of velocity before and after the impact. (83) ~'. i + ( Y o .AZZONI et al.XA] + (YA + h0) (AS) V. CASEb):x.-Vo~. the following relation can be written: 1.d.XA]2 + I:~x'[X(t) -.xp) f0 ( r o .Y p ) . and thus Vy bccomcs positive. In this case.Y v ) ' i . B2(b)] (c) X o < Xp ~ d~ < 0 ~ V. the following equation can be obtained: ~o = 1 coo+Vo~'dy-Vo.'j.Yp and dx = X o .d~ (al) where: dy = }Io. ( X o . APPENDIX coz . the following equation is obtained: Then: V x ~ (d)z • d r 1 g V0. j .k ~ V = to x PG = 0/" (xo . B2(c)]. BI. 82.. Definition of the rolling problem in the assumed O X ' Y ' reference frame.) = ~" co2. (B7) Therefore. R-N.R-N.cos e • R ) .( l + r 2) 2 . are as follows: f 0=N-m'g'cos~t m • XG = m .T d28 I.~ " ~G + m " g " c°s °~ R Obtaining t from equation (C2b): Substituting this equation into the second one of (C1): ~G= m I ' g " ( sin a .cos ~ "[Xo(t)-.(tan ~ .)f~ (t o)] 2.cos a . and #r = ~ / R = tan So is defined as the Rolling Friction coefficient.~'o(t0) t = A ' g ' cos at' (tan ~t -.2.X~(t0)] + ~ ( t 0 ) . (I + d~ + d~) = ½" co2.~-~= (C1) X~ Fig. . g .~ Equation (C2a) shows that three different situations can be possible: -~. Equation (C2) can be rewritten as follows: -~o = A • g .tan eke)" t 2 + Yr. C1.a. APPENDIX C The dynamic equilibrium equations of the rigid body.tan 4~a)- (C2a) (C4) From equation (C4). sina .. 1 X o ( t ) = ~" A "g 'cos ~ ' (tan ct .tan Sd)" t + gG (t0) (C2b) #.(to)" t + XG(tO). CI).=" ~. m+~ m - m+~ (C3) Substituting equation (C3) in to (C2c).: ANALYSIS OF ROCKFALLS The components of the velocity after the impact can be determined by substituting the value of co.g .( t a n ct . From the first equation N = m .tan ~bd)' (C2) I )~o(t) = x / 2 ' A ' g 'cos =. (I + r2). it is possible to evaluate a coefficient of restitution of energy with the following relation: K Q~ co2 co "Q0 E* ( / + r 2) = (B8) K o 2 ' K o . A . calculated with the previous equation.~. the rolling friction coefficient can be determined as: The integration of differential equation (C2a) gives: )/'~ (t) = A .i'~+N.g . + V. the velocity of the block during the rolling or sliding motion can be determined with the following equation: Defining A = - XG(t) -.~ < 0 when tan 4~d> tan a ~ uniformly decelerated rolling motion X~ > 0 when tan 4~d < tan ~ = uniformly accelerated rolling motion. (I" co2 + V~. The total kinetic energy for the unit mass after the impact can be calculated by the following equation: K = ~. K o 2-K o within (0~<E* < l) where: Qo = I " coo + Vo.tan ~bd). T _I X~ = 0 when tan ~d = tan ~ ~ uniform rolling motion with constant velocity .724 A Z Z O N I et al. cos ~t • (tan ~t . K0 = total kinetic energy before impact. = tan q~a = tan ~ [?(~ (t) -... The 3rcl equation can be rewritten as: (C2c) I. in the assumed reference frame (Fig.~. T. [XG(t) -.cos ~t then: I T = .-~=T. into the equation (B4).Voy " d.g .X~(t0)]" (C5) .
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