Analysis and Prediction of Rockfalls Using a Methematical Model

Pergamon

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

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 rock fall 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 sub- sequent 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 some- what 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 mile- stones 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

710 AZZONI et al.: ANALYSIS OF ROCKFALLS

availability of powerful computers at moderate costs, the above-mentioned limitations have been overcome using mathematical models. Nevertheless, experimental methods are still very important, both for the study of the phenomenology and the assessment of the relevant physical parameters, not to mention the correct calibration of the mathematical models.

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, respectively) [11-16], and those that consider the block as a body with its own shape and volume [8, 17-22]. The latter models are generally better than the former, as they are more capable of accurately reproducing the different phases of the fall phenomena. Most programs analyse the falls in a two-dimensional space, since three-dimensional analysis [19], even if theoretically more accurate, is more expensive and in most cases unnecessary.

METHODOLOGY FOR ROCKFALL ANALYSIS

According to the above mentioned considerations, in 1987 ISMES and ENEL CRIS started a joint research program for the study of rockfalls. They set up a mathematical model, called CADMA, and at the same time carried out a considerable number of in situ tests, for the purpose of investigating the principal modalities of rockfalls and to determine the principal parameters involved in the model. The methodology adopted for setting up the program is expressed in the flow chart (Fig. 1). The following paragraphs briefly describe the main characteristics of the mathematical model (techniques and assumptions) and the in situ

tests.

Mathematical model

To carry out efficient mathematical modelling of a phenomenon, it is necessary:

(a) to define the characteristics that the model must

have, in order to obtain realistic results that are comparable to experimental observations,

(b) to make certain assumptions that allow less important elements to be reasonably disregarded. This is the crucial part of modelling and it will determine the quality of the model. Careful observation of the physical phenomenon, which can be monitored by in situ tests, is very important in this phase of the work.

The main targets of a rockfall model are:

-- the assessment of velocities, heights of bounces and energies achieved during the fall;

--the assessment of maximum run-out distances, in order to determine the areas at risk.

The CADMA model was developed in 1987 according to a method established by Bozzolo and Pamini at the beginning of the 1980s [17, 18]. The model is based on rigid body mechanics. Its main characteristics and principal assumptions are:

--Falls analysis in a two-dimensional space. --Rockfall trajectories are established a priori and

represented as a sequence of straight segments. Motion kinematics are studied along a vertical plane, defined by the rotation on a single plane of all the different vertical planes, including the previously mentioned segments (Fig. 2).

--The fall is composed of different phases, each with its own characteristics and assumptions.

--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. 3).

--Block fracturing is not taken into account. This approach is reasonable for obtaining conservative results.

--Each block falls along a trajectory not affected by those of the other blocks.

L Experimental tests

I Real ~ Mathematical model of ~ _ ~ Mathematical physical the real physical system model system analysis

No I

Mathematical .~ Stop ]~ synthesis of the

phenomenon "

Fig. 1. Flow-chart showing the steps of the study procedure.

Yes

AZZONI et al.: ANALYSIS OF ROCKFALLS 711

A

TRUE TOPOGRAPHIC PROFILE

IDEALIZED TOPOGRAPHIC PROFILE

1-I' 2

~ ~ I PROJECTED TRUE TRAJECTORY TRAJECTORY

~ ~ ~ - , C' " ' . ,

" , , .

Fig. 2. The kinematics of the motion is studied in a vertical plane obtained by rotation into a single plane of all different vertical planes.

Y

J ,

Z

Fig. 3. Model of the block at the impact.

X

The natural variability of some important parameters (such as the shape of the block, the mechanical characteristics of the slope, local slope angle at impact, 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. For this purpose, 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.

The following sections offer a detailed description of the characteristics and assumptions of the model, and main phases of the rockfall (free falling, impact and bouncing, rolling and sliding).

Free falling. The peculiar characteristic of free fall is that motion occurs in the air, and therefore without any contact with the slope. Motion takes place after a rolling or sliding phase, usually due to a sharp variation in the slope angle [Fig. 4(a)], or after an impact with the slope [Fig. 4(b)]. As is the case with most computer models,

Rolling Free fall

\ \

',, Free fall .

\ IVy

Impact '.Vm

",Free fall

"••mpact ~ l l V a) b)

Fig. 4. Different initial conditions for a free falling phase.

712 AZZONI ef al.: ANALYSIS OF ROCKFALLS

CADMA analysis of free fall disregards both the effects of air friction and of aerodynamic uplift.

Motion in the free falling phase is basically composed of two different movements: translation of the centre of mass, that can be analytically described by a quadratic equation; and the rotation of the block around its centre of mass. Initial conditions are determined at the instant the block separates from the slope profile. Likewise, the impact after the free falling phase is the intersection of the parabola with the polygonal representing the slope profile. Some details of the mathematical formulation are reported in Appendix A.

Impact and bouncing. When the aerial trajectory inter- sects the slope, an impact takes place and the principal consequence is a loss in energy of variable importance. To model the impact phase, 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.g. the weight). Similarly, 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. The concept is clearly explained in Fig. 5, where:

s

to + At

F(t) dt fo

represents the impulse of the active forces

s

r,,+Ar

fO>dt '0

represents the impulse of contact reactive forces.

The exact determination of these internal forces is very important but quite difficult to achieve. For engineering purposes, the phenomenon can be satisfactorily analysed by assuming the validity of conservation principles (linear and angular momentum).

The experimental analysis of the impact shows that

F(t) f(t)

t, At t

Fig. 5. Relation between impellingf(f) and active F(r) forces in the time interval corresponding to the impact.

X’ Fig. 6. Assumptions for the block at the impact.

the characteristics of motion after impact are heavily conditioned by the block’s shape, the geometry of the slope and by the energy dissipated. The latter depends on the geomechanical characteristics of block and slope, the collision angle and the configuration of the block at impact. If we consider that the impact is partially inelastic, it is possible to simplify the model in line with the following assumptions:

-the block at impact has an ellipsoidal shape, +ontact between block and slope occurs at an

infinitesimal area which can be assumed as point P

(Fig. 6), -after impact a rotation takes place around this

point.

Because of the previous assumptions, namely that the internal forces predominate over the external ones and since these forces act on the point P, it is possible to take the momentum of all forces with regards to this point as equal to zero. Therefore, it is possible to consider that the angular momentum at P is conserved during impact. Details of the mathematical formulation can be found in Appendix B.

Equation (B8) reported in Appendix B allows evaluation of the restitution coefficient of energy t *. The same equation shows the importance of carefully assessing angular velocity (before and after impact), for obtaining the correct determination of the restitution coefficient. According to the previous equations, the program calcu- lates the value of c* through which the conservation principle of angular momentum is valid.

As it is possible for the calculated t * to be greater than the one observed experimentally L;,,, the latter is considered as the upper boundary of the range of the calculated 6 *; in this case, angular momentum is not conserved, and thus gives:

K = L;,, . K, * $02(Z + r’)

= Gax ’ Ko

AZZONI e t al.: ANALYSIS OF ROCKFALLS 713

Another important element in the modelling of the impact phase is the criterion used to establish whether, after impact, the block bounces or rolls and slides.

The program compares the value of the normal component of velocity (Vy in the adopted X Y reference frame), with a value of Vy experimentally assessed (Vy). If Vy < Vy, the block rolls (or slides). On the contrary, if Vy > Vy, after impact, the block bounces. The same criterion is used to assess the transition from rolling to bouncing, thus allowing for sharp variations of slope angle along the fall trajectory.

Rolling and sliding. Like wheel motion, 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 situation is quite simple in the case of the wheel, where a momentum is caused by the deformability of both the wheel and the ground (Fig. 7). In the case of the block rolling down the slope, the situation is more complex, mainly because of the non-linear behaviour of the materials and the morphological irregularities of the block and the slope, which bring about the formation of more complex momenta (Fig. 8). Detailed modelling of the phenomenon would require a very elaborate analysis.

According to experimental observation, usually if the dimensions of the block are smaller than those of the irregularities of the slope, the block tends to make small jumps and slips; if these dimensions are larger, the block rolls with simultaneous slips at the points of contact [18]. The mathematical model sim- plifies all situations of varying complexity, by considering sliding motion in equivalent rolling terms. This is done by assuming that the block has a circu- lar shape (cylinder, sphere, disc), which is basically a conservative assumption, and that it rolls on a slope with rolling friction. This simplification is acceptable, particularly if we consider that sliding is a phenomenon basically limited to the initial and final phases of the fall.

a)

d)

Fig. 8. Simplified sketches showing the influence of morphological irregularities of the block [(a) and (b)], and the slope [(c) and (d)] on

antagonistic momenta.

Probability analysis. Given the intrinsic variability of the phenomenon, it is impossible, or at least inaccurate, to analyse rockfalls using a deterministic model. There- fore, it has been necessary:

(a) to define all the variables within a range centred on their mean value,

(b) to make a numerical simulation using the Monte Carlo method, by choosing within the mentioned range, the values of all the variables at random.

Results and the program graphs. The model analyses the rockfall trajectories graphs in the considered vertical section, and at predetermined distances, gives values (in terms of probability distributions) of the main parameters characterizing the falls (translational and angular velocity, height of the fall trajectory, and energy of the block). Furthermore, the model allows determination of the block's stopping point along the slope. All these data are graphically represented by histograms (Fig. 9).

a)

- ~ 0 3

o,F b)

P3

02 P ~

04

Pl = Ps C2

i

X, i X,

o (X) :

Fig. 7. Interpretation of the rolling friction coefficient for a wheel.

N o o

g

g ".00

! ' e'oo ' ,6oo' 2 ~ o o ' 3~oo . . . . . . . . . 40.00 48.00

X I m)

I I I I I 56.00 64.00 72 .00

oo


.OO 8.00 16.00 24 .00 32.00 40.00 48.00 66.00 64 -00 ?2 .00 X Ira)

VELOCITY FREQUENCY HEIGHT OF BOUNCE

++

~6.34" 8.63 10,92 ~ 16.92-" 15.49" 17.78 °2~01" 4.42' 2. '83' 3~25' 3166' 4~07 --.0! .67 1.34 2.00 2.66 3.33 V (m/s) f(Hz) H (m)

OBSERVATION SECTION X--35.00m

Fig. 9. Graphic output of the program.

Although it is more directly provided by the program, the kinetic energy of block during the falls can be easily calculated with the parameters (velocity, frequency and mass of the falling blocks) according to equation (B7) reported in Appendix B.

Experimental method

Experimentation on physical models allows visualiza- tion of all the aspects of a phenomenon that, because of their specific and aleatory character, could be difficult to


define and assess with accuracy. Rockfall experimental models enable us to define the fall modalities, assess the parameters to be used in the analyses, and to calibrate the mathematical model. In order to carry out correct analysis of in situ tests, it is crucial to determine, as carefully as possible, the following factors:

--topography of the slope, as well as the shape and dimension of the blocks,

--geological and geomechanical characteristics of the falling blocks and the slope.

These are the same elements that should be assessed when carrying out a real case analysis. In this case the work should be complemented with a geomorphological study of the slope and a geostructural assessment of the rock mass, in order to evaluate the most probable falling paths and physical characteristics of the possible falling blocks [23,24].

In situ tests are generally carried out using the following method:

(a) Assessment of the topographical, geomorphological, geological and mechanical characteristics of the slope and the blocks. 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), where greater accuracy is required, and 1 : 1000 for a long slope (hundreds of metres long). The topographical survey should be able to describe all the relevant points of the slope with adequate precision (centimetres in the former case, decimeters in the latter). The topographical survey can be done in various ways. One of the best is ground photogrammetry, which provides a remarkable accuracy (on a 50 m high slope, precision to the order of centimetres is achievable).

(b) Throwing the rocks down the slope. This activity is usually performed, depending on the size of the blocks, by hand, by jacks or, more easily when the test is carried out in a quarry, by an excavator.

(c) Recording the rockfalls. The activity can be performed with video cameras. The experimental tests are generally carried out using several fixed and moving video cameras. For short slopes of fewer than 100 m, 3-5 lateral synchronized fixed video cameras are generally used, together with a lateral moving one. These record the movement in the vertical plane parallel to the trajectory. A fixed camera is also set in front of the slope to record lateral displacement of the rock's trajectory on the slope.

(d) Elaboration and analysis of the records. These activities are usually carried out in the following way:

--using specific software, 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;

--measuring the true distance between points on the trajectory, in view of the fact that a certain amount

of distortion due to non-perpendicularity between the rockfall plane and the direction of the camera is practically unavoidable.

---calculating translational and rotational velocities, and integrating all data gathered in the analysis of all camera records for best assessment. 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;

----evaluation of the height of bounce along the whole analysis of in situ tests and assessment of experimental parameters.

ASSESSMENT OF DYNAMIC PARAMETERS AND OTHER ELEMENTS RELEVANT TO COMPUTER

ROCKFALL ANALYSIS

To predict the rockfall characteristics through computer analysis, the mechanical characteristics and geometry of the blocks and slope (topography) are needed. The former are usually represented not directly, but through some coefficients which allow the modelling of the amount of energy dissipated during the various phases of the fall. In particular, the restitution coefficient expresses the amount of energy dissipated during the ground impact. This is generally considered to have an elastoplastic behaviour. The rolling friction coefficient expresses the frictional effect of the ground on the rolling block.

The block's geometry is usually expressed by its volume and the ratio between its main axes, while the topography, once the section for study has been defined, is described in a profile which should be as detailed as possible. Besides these elements, the mathematical model utilized for this research requires the assessment of other parameters (namely, the modulus and direction of the starting velocity, and the velocity at which the block changes its movement from rolling to bouncing and vice versa). The following sections describe the more relevant dynamic parameters used in computer analysis and their assessment [25, 26].

Restitution and rolling friction coefficients

The dynamic coefficients used for the mathematical analyses, namely the restitution coefficient and the rolling friction coefficient, were evaluated both through back-analysis and through the elaboration of in situ tests carried out at Strozza, Italy. These two different approaches basically provided similar results [26].

Assessment through back-analysis. The target of back- analysis calibration is the assessment of values for the dynamic coefficients, by which the program is able to find values of the main parameters comparable to the experimentally observed ones. The parameters used for calibration are velocity, frequency, height of bounce and run-out distance. The assessment of the restitution coefficient was obtained by taking into account


Table 1. Values of the restitution and rolling friction coefficients adopted for the calibration of the mathematical model

Maximum restitution coefficient Rolling friction coefficient

Block size Rock (limestone) 0.754).90 Fine angular debris and earth, compacted 0.554).60 (gravel and cobbles, dia < 20 cm) Fine angular debris and earth, soft 0.35-0.45 Medium angular debris 0.454).50 with angular rock fragments (20-40 cm dia) Medium angular debris with scattered trees 0.40~.50 Coarse angular debris 0.554).70 with angular rock fragments (40-120 cm dia) Earth with grass and 0.504).60 some vegetation Ditch with mud <0.20 Yard (fiat surface of 0.504).65 artificially compacted ground) Road 0.75

0.3 m 3 1.2 m 3 0.40-0.45 0.40 0.504).60 0.40

0.70-0.80 0.60-0.70 0.60-0.70 0.50-0.60

0.70-1.00 0.65-1.20 0.60-0.80

0.554).65 0.454).50

0.85 0.50-0.65

0.404).45

different falls. The values represented in Table 1 cor- respond to the maximum values of the restitution coefficient assessed for different geological materials. The values of rolling friction coefficients were also evaluated through back-analysis of two different experimental tests, carried out with blocks of different shape and volume (a prismatic block, of about 1.2m 3 in volume, and a spherical one of 0.3 m3).

Since the rolling-friction coefficient in this case depends on the roughness of the slope in relation to the size of the falling block, two different values (depending on the volume of the blocks) were determined. Following some tests, it was noted that values obtained through back analysis alone, yet unconfirmed by other methods, do not always provide correct results when used on other slope types (particularly if topography is not very detailed). For this reason, another method for the assessment of the restitution coefficient was established.

Assessment by elaboration of in situ test. The ground's restitution coefficient was evaluated according to its more rigorous definition, namely the ratio between the total energy of the falling block before and after its impact on the ground. Energies were assessed by

measuring both rotational and translational block velocities, slope inclination and sizes of the block at impact. In this way it has been possible to assess that the maximum restitution coefficient (Emax) ranges on normal slopes from 0.35 to 0.95, depending on geological conditions (Fig. 10). In this figure, values larger than 1 should be disregarded, since they indicate an increase of energy (these values are related to some limitations in the analysis of video records).

Like the previous parameter, the rolling friction coefficient was assessed both through back analysis (Table 1) and the elaboration of in situ tests. 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), and basically corresponds to the slope angle at which the block moves with a steady velocity (neither accelerating nor decelerating) (Fig. 11).

Careful observation of the blocks' velocities at different positions in the fall trajectories and the measure- ment of the slope angles corresponding to the different positions of blocks, enabled us to assess rolling friction coefficients which provide a good match with those

DEBRIS

BARE ROCK

o o : o • lain

" • Reliable value • Unreliable value ]

• O ~ A

0.2 0.4 0.6 0.8 1.2 .4 1.6 1.8

Restitution coefficient

Fig. 10. Restitution coefficient values assessed for impacts on different types of ground (after [26]).


50

._. 45

40

~v 35

~ 30 E

< 25

i , • • i

. . . . . . . . . . . . . . . . . . . . . . O r . . . . . . . . . . . . i . . . . . . . . . . .

', o ; n =, ' A ' - - . . . . . . . . . . . ±__ ~ i

20 . . . . . . . . . . , ~3-o . . . . . . . . . . . . . . . . . . . . . . .

15 . . . . . . . ~ . . . . . . . . . . . ~ . . . . . . . . . . . . , . . . . . . . . . . . . ~-

0.00 O. 10 0.20 0.30 0.40

diD

0.50

Fig. 11. Assessment of rolling friction coefficient and comparison with Statham's values (after [26]). • discoidal block, accelerating; [] discoidal block, decelerating; • spheroidal block, accelerating; 0 spheroidal block, decelerating; • tabular block, accelerating; A tabular block, decelerating; • columnar block, accelerating; O columnar block, decelerating;

A' = Statham's upper boundary; A" = Statham's lower boundary; B = rolling friction coefficient by back analysis.

obtained through rockfall back-analysis. They are comparable to those provided by Bozzolo and Pamini [17, 18] and also to Statham's empirical coefficients. This method can be used to confirm the values from back- analysis or to find new values, when it is impossible (or impractical) to carry out complete in situ tests, or when the coefficients from calibration provide unreliable results.

The tests used for this assessment also revealed the important effect that the shape and dimension of the blocks have on their velocity. As a general trend, in fact, it is possible to note the progressive increase in rolling velocity with the increase both in shape parameters and dimension of the falling blocks. 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].

Starting velocity

Even if the real starting velocity generally equals zero or is slightly greater, in computer analysis, so as to "move" the block, it is sometimes necessary to provide a certain initial velocity (usually 1-3 m/sec), particularly on rough or low inclination slopes. This parameter is not so relevant to the general trajectory when the fall starts from a steep wall, since here the velocity is much more influenced by the effect of the force of gravity. In some real cases, it is useful to use a certain starting velocity, such as when only the profile of the lower part of the slope closest to the threatened area is available.

Threshold velocity between rolling and bouncing

As already mentioned, the program simulates the fall as a sequence of different phases, each of which is separately simulated and analysed. Passage from one type of motion to another depends mainly on the topography (e.g. profile irregularities produce bounces). Simulation of the different phases not only depends on topography; it also owes much to "threshold of the

normal component of the velocity". Below this threshold, the program does not allow block bouncing and therefore analyses the fall as a rolling phase. This value has been assessed through back-analysis and usually ranges from between 1 and 1.5 m/sec.

Choice of the trajectory

The choice of the rockfall trajectory to be studied is crucial when using a two-dimensional program. Choice is usually decided according to the following criteria:

- - T h e trajectory tends to follow the steeper line of the slope, and depends heavily on its topographic characteristics. Experience gathered from observation of experimental falls and case histories is an important tool for choosing reasonably representative trajectories.

- - T h e conservative strategy of considering trajectories that present the greatest risk, though less probably, should always be adopted.

A considerable aid in deciding which trajectory to study comes from in situ tests [26]. By defining the "dispersion" of the trajectories as the ratio between the distances separating the two extreme fall paths (i.e. the trajectory furthest to the left and to the right, when looking at the slope face) and the length of the slope, the in situ tests showed that this parameter is about 20%, unless topographical constraints (e.g. valleys) reduce this value. Furthermore, the same tests also showed that steeper slopes have smaller dispersions (Fig. 12).

C A L I B R A T I O N A N D C O M P U T E R M O D E L R E S U L T S

Effect of topography on the computer based rockfall evaluation

Topography is a key factor in evaluating the dynamic parameters, since it affects both falling blocks velocities and heights of bounce. Unfortunately, high costs and complex logistics, mean most rockfall computer analyses

718 AZZONI e t al.: ANALYSIS OF ROCKFALLS

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Distance between fall paths (m)

Fig. 12. Sketch of a typical slope used for rockfall tests (front and side view), and relation between the maximum distance separating the fall paths at the bot tom of the slope (D) and the length of the slope (L) (after [26]).

are performed on slope profiles surveyed with low detail. In view of the fact that low topographical detail corresponds to greater profile smoothness, it is important to determine the margin of the error involved in this approximation.

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, 79, 43, 25 and 16 points, respectively, 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. 13) [25]. The test enabled us to make the following main observations:

- -A progressive increase in smoothness corresponds both to an increase in calculated fall velocity and frequency of rotation, and to a decrease in height of bounce. This result is conservative in respect of fall energy, but not height of bounce, even though the experimental value still fell within the range calculated by the computer analysis. When analysing rockfalls with low detail topographical surveys, it is thus advisable to design higher fences (rather than stronger) than those hypothesized by the computer.

--Detailed topography is necessary. In particular, detail is especially important in sections where the

block bounces and rolls; where free fall is the prevailing motion type, a less accurate survey is acceptable. The most favourable topography is that surveyed at intervals similar to block dimensions, since the smaller asperities can already be satisfactorily taken into consideration by the rolling friction coefficient. When such detailed topography is not feasible, 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).

--When a special topographical survey is not available, 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, respectively.

Optimal number of simulations Dimensions of the output files and time required for

the CADMA rockfall computer analysis increase as detail increases or when, because of uncertainties about input data, parametric analyses are required. In view of this, it is important and useful to ascertain the minimum number of simulations required to provide statistically valid results [25].

To determine this, the program was run with 20, 50, 100 and 200 falls and the respective results were then

24

~ 1 6

VELOCITY (m/s) A B C D E A B C D E A B C D E A B C D E A B C D E

I I l l l

. . . . . . . . . . . . . . . . . . . . -- : . : _ _ _

. . . . . . . • • • • - - i - - ~ &

7 16 24 29 35 DISTANCE OF OBSERVATION POINT (m)

E'~-6 I - "1- O ~ 3 "1-

I BOUNCE HEIGHT (m) A B C D E A B C D E A B C D E A B C D E A B C D E

I I I

I

i

A A A A A

. . . . . . . . . . . . . . . . . . . . . . ~ i • ~ • . . . . i i ~ • • • i

Ill l # AAA~A i l 1 • •

16 24 29 35 DISTANCE OF OBSERVATION POINT (m)

4 0

3 0

20

10

. _ . 3 ¸ N

z

IJ. 0

FREQUENCY (Hz) ] A B C D E A B C D E A B C D E A B C D E A B C D E

i I i

. z~;~Z -._ - . . . . . . . . . . . . . . . . . . . . . . . . . ~ _ . . . . . . . . . . . . . . . . . . A A A A ~ . . . . i

t l t * l i i AAAAA t i t I I

_ _ l l l l I I I i # # # # # i I I I I I I[ iII I I I I

Ii--

7 16 24 29 DISTANCE OF OBSERVATION POINT (m)

- Computer max. - Computer min. • Computer mean value value value

35

* Computer modal A Exp. value value

Fig. 13. Effect o f topographical detail on the results o f the mathematical rockfall analysis: A = Slope profile with 152 points; B = Slope profile with 79 points; C = Slope profile with 43 points; D = Slope profile with 25 points; E = Slope profile with

16 points.

SLOPE A R~ck

, '~,.J Compacted '~':,~ Fine angular debris and earth

-~,;~%

.~ /k Rock

; i}k~Fine-medium angular :-'%'~ debris, loose

'; " i " : ~ / d i t c h ~ - i ; ~ ( yard

10 20 30 40 m

40

30

20

10

;':::.~ ...... SLOPE B .... "

'" ' r ,ne a n g u , a r debt, s, compoc,ed

'";~:'~'~'~.~..":~" Medium angular debris, - " ; ' k ~ loose

Coarse angular debris, l'~ :~

,'o 2'o go ;o 5'~ m Fig. 14. Topographical profiles of the test slopes (after [26]).

RMMS 32/7--(3 7 1 9


compared amongst themselves and with the experimental data. The tests showed that analyses with 100 and 200 simulations were positive and totally similar, while analyses with 20 falls were inaccurate. The results of the tests performed with 50 falls were basically similar to those with 100 simulations, but 50% faster. Thus, this number of simulations is considered advisable when dealing with detailed slope profiles or parametric analyses. In fact, for a slope about 60 m long, described with approx. 170 points, an analysis carried out with 50 simulations took about 8 min on a 386 PC and produced an output file of about 1.2 Mbytes. The same analysis, run with 100 simulations, required about 17min for calculations with a resulting output file of about 2.3

Mbytes. Using a 486 PC, the same analyses took about a third to a quarter less time.

C O N C L U S I O N

A mathematical model was used to analyse and predict rockfall trajectories on two different slopes, where in situ tests had been performed. The analyses were carried out using dynamic coefficients values specifically determined for this program. A list of suggested values, as evaluated by an elaboration of in situ tests and through back-analysis of the tests and real rockfalls, is presented in Table 1. These values,

24.0

E 16.0

0 o 8.0 .._1 m

0.0

9.0

E 6.0

T (3 ~'~ 3.0 "1"

0.0

9.0

T v

>" 6.0 O Z UJ

O 3.0 UJ rY LL

0.0

VELOCITY (m/s)

7.0 16.0 24.0 29.0 35.0

DISTANCE OF OBSERVATION POINT (m)

BOUNCE HEIGHT (m)

?

7.0 16.0 24.0 29.0 35.0


FREQUENCY (Hz)

I I I I

7.0 16.0 24.0 29.0 35.0


-- Computer av. max. • Computer av. 4. Computer av. & min. value mean value modal value

• Computer max. O Av. experimental O Max. experimental value value value

Fig. 15. Comparison between the computer analysis results and the experimental data, for slope A.


even if correct overall (for a rolling block of up to 1 m3), should be used by taking into account all the observations and criteria discussed in the previous paragraphs.

With regard to the computer analysis results, a comparison between experimental and calculated values of height of bounce, velocity and frequency of rotation for two different slopes at Strozza quarry (Fig. 14) are shown in Figs 15 and 16. 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, average value of the maximum, minimum, mean and modal values; and the overall maximum calculated value). These values were assessed at specific observation points placed at critical positions on the slopes.

From these results it is possible to draw the following conclusions:

Translational and rotational velocity and energy. The program is generally able to make correct (or at least acceptable) predictions of these parameters. In particular, the experimental velocity generally falls within

the range of the predicted values and is always described satisfactorily by the mean and the modal values.

Height of bounce. If the topographical input is good, the program is generally able to find correct results for this parameter. In the sections beneath the steep rock slopes, it tends to slightly underestimate the values, though in this case a possible inaccuracy in the experimental values due to over-estimation should be taken into account.

Run-out distance. The program provides acceptable results for this parameter. In particular, 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 simple to run and provides clear and easily read graphical outputs, such as tables with all the numbers generated by the calculations, slope profiles with fall trajectories, as well as histograms of velocities, frequency of rotation, height of bounce and stopping distances. Now that calibration has been completed, the program is currently undergoing slight changes,

20.0

VELOCITY (m/s)

15.0 E

---- 10.0 ¢D o ._1 LU > 5.0

0.0

....... IIIIIIIIIIZIIIIIIIIIIil ..............

2.0

40.0 54.0

DISTANCE OF OBSERVATION POINTS (m)

BOUNCE HEIGHT (m)

1.6

E " " 1.2 I - "1-

-~ 0.8 I.U "I"

0.4

0.0

c i , L

40.0 54.0

DISTANCE OF OBSERVATION POINTS (m)

• = Computer av. max. & • Computer av. modal • Computer av. mean min. value value value

• Computer max. O Av. experimental O Max. experimental value value value

Fig. 16. Comparison between the computer analysis results and the experimental data, for slope B.


VELOCITY (m/s) BOUNCE HEIGHT (m)

25

20 v

15

o, lo

W 5 >

I I I I

7 16 24 29 35


~ 6

0 7 16 24 29 35


- Computer • Computer • Computer o Experimental max & min. mean value modal value value value

Fig. 17. Comparison between the results of the computer back-analysis and the experimental data for roekfall No. 26.

to f u r t h e r i m p r o v e o u t p u t d a t a a n d r e n d e r i t m o r e

u s e r - f r i e n d l y f o r al l r o c k fal l spec ia l i s t s .

Accepted for publication 15 January 1995.

R E F E R E N C E S

1. Spang R. M. Protection against rockfalls---stepchild in the design of rock slopes. Proceedings of 6th Int. Congress on Rock Mechanics, Montreal, Canada, pp. 551-557 (1987).

2. Richards L. R. Rockfall protection: a review of current analytical and design methods. Secondo Ciclo di Conferenze di Meccanica e Ingegneria delle Rocce, MIR, Politecnico di Torino, pp. 11.1-11.13 (1986).

3. Ritchie R. M. Evaluation of rockfalls and its control. Highways Res. Record. 17, 14-28 (1963).

4. Camponuovo G. F. ISMES experience on the model of St. Martino. Proc. Meet. Rockfall Dynamics Protective Works Effectiveness 90, 25-39 (1977).

5. Broili L. Relations between scree slope morphometry and dynamics of accumulation processes. Proc. Meet. Roekfall Dynamics Protective Works Effectiveness 90, 11-24 (1977).

6. Habib P. Notes sur le robondissement des blocs rocheux. Proc. Meet. Rockfall Dynamics Protective Works Effectiveness 90, 123-125 (1977).

7. Statham I. A simple dynamic model of rockfall: some theoretical principles and field experiments. International Colloquium on Physical and Geomechanical Models, pp. 237-258 (1979).

8. Falcetta J. L. Etude cynematique et dynamique de chute de blocs rocheux. Th~se, INSA, Lyon (1985).

9. Chan Y. C., Chan C. F. and Au W. C. Design of a boulder fence in Hong Kong. Conference on Rock Engineering in an Ubran Environment, Inst. Min. Metall., Hong Kong, pp. 87-96 (1986).

10. Mak N. and Blomfield D. Rock trap design for pre-splitting slopes. Conference on Rock Engineering in an Urban Evironment, Inst. Min. Metall., Hong Kong, pp. 263-270 (1986).

11. Piteau D. R. Computer rockfall model. Commun. Proc. Meet. Rockfall Dynamics Protective Works Effectiveness 90, 123-125 (1977).

12. Hacar B., Bollo F. and Hacar R. Bodies falling down on different slopes. Dynamic studies. Proc. 9th Int. Conf. Soil Mech. Found Engng 2, 91-95 (1977).

13. Azimi C., Desvarreux P., Giraud A. and Martin Cocher J. Methode de calcul de la dynamique des chutes de blocks. Application ~ l'Otude du versant de la montagne de la Pale (Vercors), Bull. liaison Labo P. et Ch. 122, 93-102 (1982).

14. Hock E. A program in Basic for the analysis of rockfalls from slopes. Unpublished notes (1987).

15. Hungr O. and Evans S. G. Engineering evaluation of fragmental rockfall hazard. Proceedings of the 5th International Symposium on Landslides, Lausanne, pp. 685~590 (1988).

16. Paronuzzi P. Probabilistic approach for design optimisation of rockfall protective barriers. Quarterly J. Engng Geol. 22, 135-146 (1989).

17. Bozzolo D. and Pamini R. Modello matematico per lo studio della caduta dei massi. Laboratorio di Fisica Terrestre-ICTS, Lugano-Trevano (1982).

18. Bozzolo D. and Pamini R. Simulation of Rock Falls down a valley side. Acta Mech. 63, 113-130 (1986).

19. Descoeudres F. and Zimmermann T. Three-dimensional dynamic calculation of rockfalls. Proceedings o f the 6th International Congress on Rock Mechanics, Montreal, pp. 337-342 (1987).

20. Rochet L. Application des modeles numeriques de propagation a l'etude des eboulements rocheux. Bull liaison Labo P. et Ch. 1501151, 84-95 (1987).

21. Spang R. M. and Rautenstrauch R. W. Empirical and mathematical approaches to rockfall protection and their practical application. Proceedings of the 5tb International Symposium on Landslides, Lausanne, pp. 1237-1243 (1988).

22. Pfeiffer T. J. and Bowen T. D. Computer simulation of rockfalls. Bull. Ass. Engng Geol. XXVl, 135-146 (1989).

23. Azzoni A., Drigo E., Giani G. P., Rossi P. P. and Zaninetti A. In situ observation of rockfall analysis. Proceedings of the 6th International Symposium on Landslides, Christchurch, pp. 307-314 (1992).

24. Giani G. P. Rock Slope Stability Analysis, p. 361. Balkema, Rotterdam (1992).

25. Azzoni A. Methods for predicting rockfalls. M.Sc. disser- tation, Imperial College of Science, Technology and Medicine, Department of Engineering Geology, London (1993).

26. Azzoni A. and de Freitas M. H. Prediction of rockfall trajectories with the aid of in situ tests. Rock Mech. Rock Engng. Submitted.

APPENDIX A

In the assumed OXY reference frame (Fig. AI), the components of the acceleration are:

a.,.(t) = 0 ax(t) = - g . (A1)

The initial conditions are as follows, denoting the components of the initial velocity V(to) at time t o as:

VAt0) = V0x V,,(to) = Vow. (A2)

and the co-ordinates of the initial position of the center of mass at time t o by:

X(to) = XA Y(to) = YA + ho" (A3)

After integrating equations (A1) over time:

x ( t ) = vo,." (t - to) + XA Y(t)= - ~ ' g ' ( t -- to)2 + Voy" (t - - to )+(YA+ho) (A4)


Y

X

Fig. AI. Definition of the free falling problem in an assumed OXY reference frame.

obtaining (t - to) from the first equation of (A4) and substituting it into the second one, the following equation is obtained:

1 g V0, Y(t) = - - 2 " V ~ ' [ X ( t ) -- XA] 2 + I:~x'[X(t) -- XA] + (YA + h0)

(AS)

which corresponds to the equation of a parabola.

A P P E N D I X B

Applying the principle of the conservation of the angular momentum over the infinitesimal time interval, before and after the impact (Fig. B1), the following relation can be written:

1 . ~ o o + V o ~ ' d , - V o ~ . . d ~ = l . c o + V ~ . d , . - L . d ~ (a l )

where: dy = }Io- Yp and d x = X o - Xp

I = moment of inertia of the block about the centre of mass co 0, co = angular velocities before and after the impact

V0x, Iz~ = x components of velocity before and after the impact Voy; V,, = y componcnts of velocity before and after the impact.

Assuming that a rotational motion about the contact point P takes place after the impact (Fig. 6), the velocity of the center of mass can be obtained as follows:

V = to x r = to x PG. (B2)

Since t o = O - i + O . j - o ~ . k and P G = ( X G - X p ) . i + ( Y o - Y p ) . j + O . k

x PG = /" f V = to 0 0 - c o ~

o ( x o - x p ) ( r o - Y~)

=¢n~ ' (Yo - Y v ) ' i - c o z . ( X o - - X v ) " j. (83)

Y

x

Fig. BI. Configuration of the block before and after the impact.

Then:

V x ~ (d)z • d r

V,, = - co z . de ( B 4 )

assuming co = coz:

V=co "dy . i - co 'd~'j = V,'i+ V,.'j. (B5)

Since Yo > YP is always thc casc, then V~ is always > 0. As for dx, it could be less, equal or greater than 0, depending on the

position of the center of mass G with respect to the contact point P (Fig. B2). Three different possibilities can occur:

(a) X G > Xp ~ d, > 0 ~ V y < 0 [Fig. B2(a)] (b) X G = Xp = d, = 0 ~ V x = 0 [Fig. B2(b)] (c) X o < Xp ~ d~ < 0 ~ V,. > 0 [Fig. B2(c)].

Obviously if Vv<~ 0 bounces can not occur. In this case, the possibility of a second impact has been introduced. In this way the block assumes a symmetric position with rcspcct to the previous onc, and thus Vy bccomcs positive.

Substituting equations (84) into the right hand side of the cquation (Bl), the following equation can be obtained:

1 coo+Vo~'dy-Vo,.'d~ ~o = " " ( 8 6 )

I + d2~ + d~

~ ~ ~ ~ , CASEa):Xa'>X"

0

~ ' , ~ o ( ~ ) ~ 7, CASE b):x,~:x~

( ~ ) ~ ~0 (~ CASE ¢): X~ < X e

o

e e_ II I Fig. 82. Different possibilities for the block at the impact.


The components of the velocity after the impact can be determined by substituting the value of co, calculated with the previous equation, into the equation (B4). The total kinetic energy for the unit mass after the impact can be calculated by the following equation:

K = ~. ( I" co2 + V~. + V,2,) = ~" co2. ( I + d~ + d~) = ½" co2. ( I + r2).

(B7)

Therefore, it is possible to evaluate a coefficient of restitution of energy with the following relation:

K Q~ co2 co "Q0 E * ( / + r 2) = ( B 8 )

K o 2 ' K o - ( l + r 2) 2 . K o 2 - K o

within (0~<E* < l) where:

Qo = I " coo + Vo.,= " ~, - Voy " d,.

K0 = total kinetic energy before impact.

A P P E N D I X C

The dynamic equilibrium equations of the rigid body, in the assumed reference frame (Fig. CI), are as follows:

f 0 = N - m ' g ' c o s ~ t m • XG = m - g . s ina - T ( C 1 )

d28 I . ~ - ~ = T . R - N . a .

The 3rcl equation can be rewritten as:

I . - ~ = T . R - N . ~

T _ I -~.i'~+N.~. From the first equation N = m .g .cos ~t then:

I T = - ~ " ~G + m " g " c°s °~ R

Substituting this equation into the second one of (C1):

~ G = m ( R ) I ' g " sin a - cos e • . (C2)

m + ~ Defining

m A = - -

I m + ~

and #r = ~/R = tan So is defined as the Rolling Friction coefficient. Equation (C2) can be rewritten as follows:

-~o = A • g . cos ~t • (tan ~t - tan 4~a)- (C2a)

The integration of differential equation (C2a) gives:

)/'~ (t) = A .g .cos a - ( t a n ct - tan Sd)" t + gG (t0) (C2b)

X ~

Fig. C1. Definition of the rolling problem in the assumed O X ' Y ' reference frame.

1 X o ( t ) = ~" A "g ' cos ~ ' (tan ct - tan eke)" t 2 + Yr,(to)" t + XG(tO).

(C2c)

Equation (C2a) shows that three different situations can be possible:

X~ = 0 when tan ~d = tan ~ ~ uniform rolling motion with constant velocity

.~.~ < 0 when tan 4~d > tan a ~ uniformly decelerated rolling motion X~ > 0 when tan 4~d < tan ~ = uniformly accelerated rolling motion.

Obtaining t from equation (C2b):

XG(t) -- ,~'o(t0) t = A ' g ' cos at' (tan ~t -- tan ~bd)' (C3)

Substituting equation (C3) in to (C2c), the velocity of the block during the rolling or sliding motion can be determined with the following equation:

)~o(t) = x / 2 ' A ' g 'cos =- (tan ~ - tan ~bd). [XG(t) -- X~(t0)] + ~ ( t 0 ) .

(C4)

From equation (C4), the rolling friction coefficient can be determined a s :

[?(~ (t) -- )f~ (t o )] #, = tan q~a = tan ~ 2. A .g .cos ~ "[Xo(t)-- X~(t0)]" (C5)

Documents

Analysis and Prediction of Rockfalls Using a Methematical Model