Lecture11(Rockfall)

8/13/2019 Lecture11(Rockfall)

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Lecture 11

Rock Fall Analysis


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Rock Fall Hazard for Tuen Mun Road Widening (Tai Lam Section)

Need to identify boulder from aerial photography, site walk overand to determine the direction of boulder fall from direction ofsteepest slope (Surfer or other GIS Program)

Accurate prediction of rock falls is practically impossible


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Cross section at target 12.Grid on 1 meter spacing

Need to have very accurate slope profile


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Temporary and PermanentRock Fall Barrier


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5Does not calculate FOS from analysis


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6Comparison between risks of fatalities due to rockfalls with published and proposed acceptable risk criteria.


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Comparisons of International RiskGuidelines, GEO Report No. 80

Societal Risk Criteria

Unacceptable

ALARP

Acceptable

Unacceptable

ALARP

Acceptable


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The equations (and the physical process of a rockfall) used to simulate therockfalls are sensitive to small changes in these parameters:

Variable slope cross sectionSki-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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If, after the impact, the rock is still moving faster than the minimum velocity (V MIN ), the searchfor 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.

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 onthe 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 aline segment (a slope segment or a barrier). Once the intersection point is found, the impact iscalculated according to the coefficients of restitution.


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The equations used for the projectile calculations are listed below:

The parametric equation for a line: The parametric equation for a parabola:

segment lineof slope=

( )( )

slopetheon pointsecond ,and firsti,

rock of locationhorizontalare,

2211

121

121

Y X sY X

y x

Y Y Y y X X X x

+=+=

( )horiz.e.g.1m/suser, byinputrock of velocityinitialis,rock of positioninitialis,

21

00

00

002

00

Y X

Y

X

V V

Y X

Y t V gt y

X t V x

++=

+=

The parametric equations for the velocity of the particle:

impact before path parabolicthealong

pointanyatrock of velocityis,0

0

YB XB

Y YB

X XB

V V

gt V V

V V

+==

Equating the points of the parabola and line equations:

[ ] ( )[ ]

segment lineof slope X X Y Y

where

X X Y Y t V V t g X Y

==

=+++

12

12

011000

2

021

( )011000

2

21

24

X X Y Y c

V V b

ga

aacbb

t

X Y

+==

=

=

y x,

11 ,Y X

22 ,Y X

XBV YBV

0 X V 0Y V

Solution to quadratic equation:

00 ,Y X Beforeimpact


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The velocity just prior to impact is calculated according to

gt V V

V V

Y YB

X XB

+==

0

0

Then, these velocities are transformed into components normal and tangential to the slope according to:

( ) ( )( ) ( )

segmentlineof slopedirectionstangentialand normalthein

impact beforerock of componentvelocityis,

cossin

sincos

=

+==

TB NB

XBYBTB

XBYB NB

V V

V V V

V V V

The impact is calculated, using the coefficients of restitution, according to:

nrestitutiotangentialof tcoefficien

nrestitutionormalof tcoefficiendirectionstangentialand normalthein

impactafter rock of componentvelocityis,

==

==

T

N

TA NA

TBT TA

NB N NA

R

R

V V

V RV

V RV

y,

NBV

TBV

y x,

NAV

TAV

Energy is lost through these twocoefficients of restitution

Afterimpact


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The post-impact velocities are transformed back into horizontal and vertical components according to:

( ) ( )

( ) ( )

directionsverticaland horizontalthein

impactafter rock of componentvelocityis,cossin

cossin

YA XA

NATAYA

TA NA XA

V V V V V

V V V

=

+=

The velocity of the rock is then calculated and compared to V MIN (say 1 m/s). If it isgreater than V MIN the process starts over again, with the search for the next intersection

point. If the velocity is less than V MIN the rock can no longer be considered a particle intrajectory and is considered sliding.

The rock can begin sliding at any location along the segment and may have an initialvelocity that is directed upslope or downslope. Only the velocity component tangential tothe slope is considered in the equations.

y x, XAV

YAV

y x,


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Sliding Downslope

If the slope angle ( ) is equal to the friction angle ( ), the driving force (gravity) is equalto 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. V EXIT = V 0).

If the slope angle is greater than the friction angle, the driving force is greater than the resistingforce and the rock will slide off the downslope endpoint with an increased velocity. The speedwith which the rock leaves the slope segment is calculated by:

)(tancossin

)(tancossin

segmentlineof endpointtolocationinitialfromdistance

segmentlinethetotangentialrock,of velocityinitial

segmentlinetheof end theatrock of velocity

2

0

20

upslopeisrock of velocityinitialif k

zeroor downslopeisrock of velocityinitialif k

s

V

V

sgk V V

EXIT

EXIT

=+=

==

==

y x,VEXIT = V 0

V0

Slidingdownslope

=if

y x,

V0

>if sgk V V EXIT 220 =


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If the slope angle is less than the friction angle, the resisting force is greater than thedriving force and the rock will decrease in speed . The rock may come to a stop on thesegment, depending on the length of the segment and the initial velocity of the rock.

The length of the segment is found by setting the exit velocity (V EXIT ) to zero andrearranging:

gk V

s2

20=

If the stopping distance (s) is greater than the distance to the end of the segment (L), then therock will slide off of the end of the segment . In this case, the exit velocity is calculated usingequation:

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 sdownslope from the initial location.

sgk V V EXIT 220 =

y,

V0


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Sliding Upslope

When sliding uphill both the frictional force and the gravitational force act to decreasethe velocity of the particle . Assuming that the segment is infinitely long, the particlewill eventually come to rest. The stopping distance is calculated using equation:

gk V

s2

20=

If the stopping distance is greater than the distance to the end of the segment, the rockwill slide off of the end of the segment . In this case, the exit velocity is calculated usingequation:

sgk V V EXIT 220 =

If the stopping distance is less than the distance to the end of the segment the rock comesto rest and the simulation is stopped .

If the rock slides up and stops it is then inserted into the downslope sliding algorithm. If thesegment is steep enough to permit sliding (i.e. > ) then the rock will slide off the bottomend of the segment. If the segment is not steep enough, then the location where the rockstopped moving (after sliding uphill) is taken as the final location and the simulation is

stopped.


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No significant difference in results when angular velocity is considered


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sphereof densitysphereof massm

mr sphereaof radius

==

=

3

43

Angular Velocity

52 2mr

I Inertiaof Moment =

( ) ( )[ ]

r V

velocityangular Initial

mr I F F V m I r

V

V RV

TA A

B

TB BTA

NB N NA

==

++=

=

2

21222

( )( )

fC

s ft C

RC V

RF FunctionScaling

C r V

R RF FunctionFriction

F

N F

NB

T

F

BTB

T T

/250i i l

/20constantempirical

1.

2.1.

1

1

2

2

2

1

21

==

+

=

+

+=

Based on the concept that the coefficient ofnormal restitution is velocity dependant .

Pfeiffer, T.J., and Bowen, T.D.,(1989)Computer Simulation of Rockfalls. Bulletin ofthe Association of Engineering Geologists Vol.XXVI, No. 1, 1989 p 135-146.

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Lecture11(Rockfall)