Lecture 11 Rock Fall Analysis
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Rock Fall Hazard for Tuen Mun Road Widening (Tai Lam Section)
Accurate prediction of rock falls is practically impossible Need to identify boulder from aerial photography, site walk over and to determine the direction of boulder fall from direction of
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Cross section at target 12. Grid on 1 meter spacing
Need to have very accurate slope profile 3
Temporary and Permanent Rock Fall Barrier
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Does not calculate FOS from analysis
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Comparison between risks of fatalities due to rockfalls with published and
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Societal Risk Criteria •Unacceptable •ALARP •Acceptable
Unacceptable
ALARP
Acceptable
Comparisons of International Risk Guidelines, GEO Report No. 80
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The equations (and the physical process of a rockfall) used to simulate the rockfalls are sensitive to small changes in these parameters: •Variable slope cross section •Ski-jump effects •Location (initial velocity of rock and location of rock) and mass of the rocks (circular vs square) •Variable from one section to another section •Materials that make up the slope (grass, bare rock) •Experimental coefficient of normal and tangential restitution •Rock breaking up not modeled
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The particle model (analysis) is a fairly crude model of the physical process of a rockfall.
•It neglects the effects that the size, shape and angular momentum of the particle have on the outcome. •The particle may be thought of as an infinitesimal circle (circular shape) with a constant mass. •Assumes that the rock has some velocity and the path the rock will take through the air is, because of the force of gravity, a parabola. •Algorithm is to find the location of intersection between a parabola (the path of the rock) and a line segment (a slope segment or a barrier). Once the intersection point is found, the impact is calculated according to the coefficients of restitution. •If, after the impact, the rock is still moving faster than the minimum velocity (VMIN), the search for next intersection point begin again. The minimum velocity defines the transition point between the projectile state and the state where the rock is moving too slowly to be considered a projectile and should instead be considered rolling, sliding or stopped.
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The equations used for the projectile calculations are listed below: The parametric equation for a parabola:
The parametric equation for a line:
x = V X 0t + X 0
x = X 1 + ( X 2 − X 1 )θ y = Y 1 + (Y 2
y =
− Y 1 )θ
gt 2 + V Y 0t + Y 0
2 X 0 , Y 0 is initial position of rock
x, y are horizontal location of rock X 1 , Y 1 is first and X 2 , Y 2 second point on the slope
V X 0 , V Y 0 is initial velocity of rock (input by user, e.g.1m/s horiz.)
θ = slope of line segment X 1 , Y 1 X 0 , Y 0
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The parametric equations for the velocity of the particle:
V X 0 V Y 0 Before impact
θ V YB
V XB
= V X 0 V YB = V Y 0 + gt V XB
V XB , V YB is velocity of rock at any point
x, y X 2 , Y 2
along the parabolic path before impact
Equating the points of the parabola and line equations:
⎡ 1 g ⎤t 2 + [V − θ V ]t + [Y − Y + θ ( X − X )] = 0 0 1 1 0 Y 0 X 0 ⎢⎣ 2 ⎥⎦ where
θ =
Y 2 − Y 1 X 2 − X 1
= slope of line segment
Solution to quadratic equation: t = a=
−b±
b
2
− 4ac
2a 1
g 2 b = V Y 0 − θ V X 0
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= V X 0 V YB = V Y 0 + gt
The velocity just prior to impact is calculated according to V XB
Then, these velocities are transformed into components normal and tangential to the slope according to:
V NB V TB
V NB
= (V YB ) cos θ − (V XB ) sin θ = (V YB )sin θ + (V XB ) cos θ
V NB , V TB is velocity component of rock before impact
V TB
in the normal and tangential directions
, y
θ = slope of line segment
The impact is calculated, using the coefficients of restitution, according to:
V NA = R N V NB After impact
x, y
V TA = RT V TB
V NA
V NA , V TA is velocity component of rock after impact
V TA
in the normal and tangential directions R N = coefficient of normal restitution RT = coefficient of tangential restitution Energy is lost through these two coefficients of restitution 11
The post-impact velocities are transformed back into horizontal and vertical components according to:
= (V NA )sin θ + (V TA ) cos θ V YA = (V TA ) sin θ − (V NA ) cos θ V XA
V XA ,V YA is velocity component of rock after impact
in the horizontal and vertical directions
V XA x, y V YA
The velocity of the rock is then calculated and compared to VMIN (say 1 m/s). If it is greater than VMIN the process starts over again, with the search for the next intersection point. If the velocity is less than VMIN the rock can no longer be considered a particle in trajectory and is considered sliding. The rock can begin sliding at any location along the segment and may have an initial velocity that is directed upslope or downslope. Only the velocity component tangential to the slope is considered in the equations.
x, y
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Sliding Downslo pe
If the slope angle ( θ ) is equal to the friction angle (φ ), the driving force (gravity) is equal to the resisting force (friction) and the rock will slide off the downslope end of the segment, with a velocity equal to the initial velocity (i.e. VEXIT = V0). V0 if θ = φ
VEXIT = V0
Sliding downslope
x, y
If the slope angle is greater than the friction angle, the driving force is greater than the resisting force and the rock will slide off the downslope endpoint with an increased velocity. The speed with which the rock leaves the slope segment is calculated by:
V EXIT = V 0
2
− 2sgk
V0
V EXIT = velocity of rock at the end of the line segment
if θ > φ = initial velocity of rock, tangential to the line segment s = distance from initial location to endpoint of line segment k = + sin θ − cos θ tan φ (if initial velocity of rock is downslope or zero) k = − sin θ − cos θ tan φ (if initial velocity of rock is upslope)
V EXIT = V 0
V 0
x, y
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2
− 2sgk
If the slope angle is less than the friction angle, the resisting force is greater than the driving force and the rock will decrease in speed . The rock may come to a stop on the segment, depending on the length of the segment and the initial velocity of the rock. V0
if θ < φ
s=
V 0
2
, y V EXIT = 0, stop
2 gk L
The length of the segment is found by setting the exit velocity (VEXIT) to zero and rearranging: 2
s=
V 0
2 gk
If the stopping distance (s) is greater than the distance to the end of the segment (L), then the rock will slide off of the end of the segment. In this case, the exit velocity is calculated using equation:
V EXIT = V 0
2
− 2sgk
V EXIT = V 0
2
− 2sgk
Slide to next segment
If the stopping distance is less than the distance to the end of the segment then the rock will stop on the segment and the simulation is stopped . The location where the rock stops is a distance of s downslope from the initial location. 14
Slidin g Upslop e
When sliding uphill both the frictional force and the gravitational force act to decrease the velocity of the particle. Assuming that the segment is infinitely long, the particle will eventually come to rest. The stopping distance is calculated using equation:
s=
V 0
2
2 gk
If the stopping distance is greater than the distance to the end of the segment, the rock will slide off of the end of the segment. In this case, the exit velocity is calculated using equation:
V EXIT = V 0
2
− 2sgk
If the stopping distance is less than the distance to the end of the segment the rock comes to rest and the simulation is stopped . If the rock slides up and stops it is then inserted into the downslope sliding algorithm. If the segment is steep enough to permit sliding (i.e. θ > φ) then the rock will slide off the bottom end of the segment. If the segment is not steep enough, then the location where the rock stopped moving (after sliding uphill) is taken as the final location and the simulation is stopped. 15
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No significant difference in results when angular velocity is considered
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Angular Velocity radius of a sphere r = 3
3m 4πγ
m = mass of sphere
γ = density of sphere
Moment of Inertia I =
2mr 2 5
V NA = R N V NB V TA
=
[
+ m(V TB )2 ]F 1F 2 I + mr 2
r I (ω B ) 2
2
ω B = Initial angular velocity ω A
=
V TA r
Friction Function F 1 = RT +
Scaling Function F 2
C F 1
=
(1 − RT ) ⎛ (V TB − ω B .r )2 ⎞ ⎜⎜ ⎟⎟ + 1.2 C F 1 ⎝ ⎠ RT 2
⎛ V NB ⎞ ⎜⎜ ⎟⎟ + 1 . C R ⎝ F 2 N ⎠
= empirical constant 20 ft / s
Based on the concept that the coefficient of normal restitution is velocity dependant. Pfeiffer, T.J., and Bowen, T.D.,(1989) Computer Simulation of Rockfalls. Bulletin of the Association of Engineering Geologists Vol.
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