Lesson 6 ADGSE

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Lesson 6
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Rock slope stability
•Rock fall
Rock fall instability
Falls of rock blocks with a volume between 0.02m3 and about 10m3 which are
detached from a steep slope and move down along it with velocities up to
40m/s
Example of the path of a falling block
Impact kinetic energy : 100 kJ
Impact kinetic energy : 500 kJ
Rock fall analysis
Invasion area intensity
(trajectories, runout)                      (velocity, kinetic energy)
What is the dynamics of the falling block?
What are the tools available to analyse it?
What information are needed to perform the analyses?
•Rock block detachment
Source area
Shape and size of the block
Initial velocity in the detachment phase (intensity and 
direction)
•Rock block motion in the air
Trajectory in the air: position in space and velocity of the 
rock block at each instant of motion
Phases of motion
•Impact of the rock block on the slope
Position of the impact point
Energy dissipation
Rock block velocity after the impact (intensity and 
direction)
•Block motion along the slope
Rolling, sliding, rolling‐sliding, stop
The phenomenon is affected by unpredictability and 
uncertainty           the analysis has to be performed with a 
stochastic approach
Detachment 
Spread detachment areas
Trajectory in the air
When the rock block moves in the air, the friction rock‐air 
is neglected and the position of the block at any instant
follows a parabolic trajectory.
The initial conditions are:
• Starting position: P0(x0, y0)
• Starting velocity : v0(v0x, v0y)
The coordinates at a generic instant t are:
being g the gravity acceleration
0oy
2
00x
ytVtg
2
1y
xtVx
+×+××=
+×=
• The coordinates of the point in which the impact 
between the block and the slope occurs are obtained 
by the intersection of the parabola and the line 
through two points of the profile: A(x1,y1) e B(x2,y2); 
the following system has to be solved:
• The solution of the system gives the coordinates of 
the impact point  P(xp,yp) . 
( )
⎪
⎪
⎩
⎪
⎪
⎨
⎧
−
−
=
−
−
+
−
×+
−
××−=
12
1
12
1
0
0x
0
0y2
0x
2
0
xx
xx
yy
yy
y
V
xxV
V
xxg
2
1y
Block – slope interaction
The interaction between slope and rock block is
deeply influenced by:
Weight and size of the rock block
Slope topography
Mechanical characteristics of contact materials
The interaction phenomena can be:
Sliding
Rolling
Impact
Block crush
Bounce
• In this phase the block velocity components (normal and
tangential to the slope at the impact) change as a function of
the topography and the cover material characteristics
• In order to analyse the block motion, its velocity before and
after the impact has to be known
• As it is very difficult to model the very complex dynamic
phenomena that occur during the impact, , a simplify approach
is used: the impact is considered as a «closed box» and the
interest is concentrated on the energy dissipation.
• Energy dissipation is simulated by means of the restitution
coefficients:
being v’1 the block velocity after the impact and v1 the block
velocity before the impact
1
1
v
v'R =
• It is useful to consider two different restitution 
coefficients: rn ed rt, in the normal and tangential 
direction with respect to the plane of impact, respectively
• If no particular information is available on the values of 
restitution coefficients, some values available in literature 
can be used, depending on the characteristics of the cover 
material
• In any case, a rockfall analysis can not be done without a 
previous back analysis, by validating the model on the 
basis of rock fall phenomena occurred in the past. It 
means that the coefficients of restitution have to be 
changed in order to obtain a risult (runout, kinetic energy) 
very close to the real one
Rockfall analysis methods
The analysis methods are grouped into 3 categories, depending on:
spatial dimensions that allow to analyse:
•Two‐dimensional methods (2D)
•Almost three‐dimensional methods (almost 3D)
•Three‐dimensional methods (3D)
rock block simulation in the model:
•Lumped mass
•Rigid body
•Hybrid approach
Two‐dimensional models
The block trajectories are simulated in a 2D domain
The slope profile is defined in a reference plane in which one axis 
is the  progressive and the other is the altitude
The profile is defined by the operator and often it follows the 
maximum gradient line
Almost three‐dimensional models
The trajectories are simulated in a two‐dimensional domain but 
the profile is defined on the basis of GIS results. That is, two 
distinct 2D analyses are performed: the first defines the profile on 
DTM, by following the maximum dip lines; the second  simulates 
the rock block movements in a 2D domain, along the profile 
previously defined  
Three‐dimensional models
The rock block trajectories are evaluated on a 3D domain
In this case there is an interdependence  between the kinematics 
of the rock block, the location of the impact points, the bounce 
height, the impacts against vegetation (trees)
The methods provide as a result all the possible trajectories of the 
rock block, on the basis of the real topography, also the less 
predictable Source point
Maximum dip 
line
Lateral 
dispersion
Lumped mass models
The rock block is simulated as a point (the shape and size of the
block is neglected). Thence the obtained trajectories are
independent from the block mass.
Rigid body models
The rock block is simulated with a predefined shape (sphere,
cube, cylinder, ellipsoid…). All the movements of the block are
considered (rotation, for example)
Hybrid models
The rock block is simulated as a point when it moves in the air
and as a rigid body during the motion along the slope, the impact
and the bounce
In any case, the rock fall analysis requires a great number of
information:
Detachment area
Rock block volume and shape
Rock block velocity at the detachment (direction and magnitude)
Slope topography
Mechanical characteristics of the cover material along the slope
All these information are very difficult to obtain!!
therefore
Whatever the analysis method choosen for the simulation of
rock fall, the model must be calibrated on the basis of past
events. That is, a back analysismust be performed.
The goodness and truthfulness of the analysis results is strictly
dependent on the possibility to calibrate the input parameters
by simulating a real past event.
If a back analysis is not possible, you can not be sure that the
results of the simulation are correct
Analisi bidimensionali
How to create the profile in a 2D analysis
In 2D analysis the profile is a succession of segments that
represent the most probable path that the rock mass will
follow during its downslope movement. It can be obtained on
the basis of:
• Past events
• Maximum dip lines between two successive contour lines
• Slope morphology
• Flow lines of a fluid that starts from the detachment area
e moves downstream
• 3D rockfall analysis
Al
m
os
t 3
D 
m
et
ho
ds
In 2D probabilistic methods the profile coordinates can be
entered as statistical variables. In this way it is possible to take
into account the mistake in the choice of the most probable
path of the rock block
Choice of the detachment area
The detection of the detachment areas is not easy. Yu can refer 
to:
•Past events, for which the detachment area is known
•Observation of the slope face in order to detect the areas with 
higher dip
•Observation of the slope face in order to detect the areas in 
which rock volumes detached in the past
In 2D probabilistic methods the detachment area can be 
simulated as a point or a line.
If it is a line, it represents a uniform distribution of source points: 
at each simulation a point of the line is extracted randomly (for 
example with Montecarlo technique)
An example: a) source point b) line source
(Rocfall)
Choice of the characteristic (design) rock volume
To date the design volume is chosen considering:
•Pastevents
•Measurements of the rock volumes that fell in the past and
stopped at the slope foot
•Structural analysis of the discontinuities that affect the slope
(orientation, spacing and persistence).
The rock block mass value is calculated on the basis of the
weight per unit volume of the rock
2D probabilistic methods allow to assign to the mass a statistical
variability
An example: design rock block 
mass enters in the analysis with a 
normal distribution  (Rocfall)
Initial velocity of the block
The choice of the initial velocity of the rock block is affected by
epistemic uncertainty.
Usually it is estimated as follows:
If the block detaches by gravity, the initial velocity is almost null
If the block mobilizes because of static or dynamic external
forces (hydraulic pressure, earthquake…), the initial velocity has
a non null value but likely lower than 1 m/s (1.5 m/s maximum)
In any case, the initial velocity is one of the parameters that has to
be changed during the validation of the model by means of back
analysis.
If the model allows to introduce the two component of the
velocity (horizontal and vertical), the direction of the velocity is
derived from the intensity of the two components
In 2D probabilistic methods the initial velocity is introduced with a
statistical variability
An example: assignment of the 
velocity component, with a 
normal distribution  (Rocfall)
Coefficienti di restituzione
Choice of restitution coefficients
The coefficient of restitution are defined as the ratio
between the block velocity before and after the impact.
In literature a large amount of values estimated for different
cover materials are available.
For a very first analysis these values can be used but then
they must be calibrated by back analysis on a past event.
The goodness of the back analysis is acceptable when the
result (in terms of trajectories, runout, bounce and total
kinetic energy) approximates to about 80% the real event.
In 2D probabilistic methods it is possible to assign a statistical distribution 
to: 
•Normal restitution coefficient
•Tangent restitution coefficient
•Friction angle at the contact block‐slope
•Superficial roughness of the slope
An example: assignment 
of restitution 
coefficients, friction 
angle and slope 
roughness values with a 
normal distribution 
(Rocfall)
Vegetation effects 
Forests are a natural protection against rock falls.
•The forest (arboreal, shrubby and herbaceous):
stops 
slows
deflects
the falling block, reducing the energy content of the phenomenon
•The tree roots can «sew» the block to the slope
But …..
• Large trees can induce the rockfall activity because their roots
can penetrate the rock mass discontinuities and act as wedges
• Movements of the trunk due to snow or wind are additional
destabilizing forces applied to the rock mass
• Some analysis methods take into account the effects of the 
vegetation in an indirect way, by changing the restitution coefficients
Restitution coefficients obtained from experiments in real size (Dorren, Berger and Putters, 2005)
Restitution coefficients obtained from experiments in real size (Rammer, Brauner, Dorren, Berger e 
Lexer, 2009)
Sito 1 (senza vegetazione) 2 (con vegetazione)
Kt Kn Kt Kn
Media 0.7 0.26 0.77 0.36
Deviaz Standard 0.12 0.09 0.1 0.13
Min 0.5 0.11 0.57 0.21
Max 0.95 0.41 0.96 0.68
Tipologia di superficie Kn Kt
Roccia dura 0.4 0.9
Roccia alterata 0.4 0.85
Detriti fini 0.3 0.75
Terreno con vegetazione  0.3 0.75
Terreno con vegetazione  (profonda) 0.3 0.8
Terreno con vegetazione  
(superficiale) 0.35 0.8
Pascolo 0.35 0.8
Pascolo (influenza dell'acqua) 0.3 0.8
Restitution coefficients suggested by Pfeiffer e Bowen (1989) and Barret (1989)
Restitution coefficients suggested by Mazzalai and Vuillermin (1995).
Restitution coefficients suggested by Piteau & Clayton (1987) and by Hoek (1987).
Choice of the number of simulations
The number of simulations must allow to obtain a good statistical 
validity of the results, that is such a number  for which with a 
higher number of simulations the results do not change anymore
Example (Rocfall)
Results of a 2D analysis
A 2D probabilistic modelling allows to obtain:
The block trajectories of 
each simulation
(an example – Rocfall)
Statistical distribution of the end points  along the slope (an example 
‐ Rocfall)
Maximum values of: kinetic energy, bounce height, velocity for each 
location along the slope (an examle – Rocfall)
Statistical trend of each parameter (kinetic energy, bounce height, 
velocity) in each specific location along the slope, with indication of 
the value corresponding to the ninety‐fifth percentile (an example: 
traslational velocity ‐ Rocfall)