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Lecture 4: Lecture 4: Ki ti A l i I Ki ti A l i I Kinematic Analysis I Kinematic Analysis I

(Slopes)(Slopes)

1 of 36 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Author’s Note:Author’s Note:The lecture slides provided here are taken from the course “Geotechnical Engineering Practice”, which is part of the 4th year Geological Engineering program at the University of British Columbia (V C d ) Th k i i d (Vancouver, Canada). The course covers rock engineering and geotechnical design methodologies, building on those already taken by the students covering Introductory Rock Mechanics and Advanced Rock Mechanics Rock Mechanics.

Although the slides have been modified in part to add context, they of course are missing the detailed narrative that accompanies any l l d h h l lecture. It is also recognized that these lectures summarize, reproduce and build on the work of others for which gratitude is extended. Where possible, efforts have been made to acknowledge th v ri us s urc s ith list f r f r nc s b in pr vid d t th the various sources, with a list of references being provided at the end of each lecture.

Errors, omissions, comments, etc., can be forwarded to the


Errors, omissions, comments, etc., can be forwarded to the author at: [email protected]

Rock Slope Rock Slope –– Continuum or Continuum or DiscontinuumDiscontinuum?? d l d k l f l ll d d In a moderately jointed rock mass, slope failure is generally dictated

directly by the presence of discontinuities, which act as planes of weakness within the rock mass. These interact to control the size and the direction of movement of the slope failure. direction of movement of the slope failure.

planar failure


wedge failure

Discontinuity MappingDiscontinuity MappingScanline mapping

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Window mapping


Discontinuity MappingDiscontinuity Mapping


Wyllie & Mah (2004)

Discontinuity Mapping Discontinuity Mapping –– Remote SensingRemote Sensing

Strouth & Eberhardt (2006)

Remote sensing techniques like LiDAR and phtogrammetry, provide a means to collect rock mass data from slopes that would otherwise be inaccessible or dangerous Discontinuity orientation persistence and spacing data can


or dangerous. Discontinuity orientation, persistence, and spacing data can be extracted from the 3-D point cloud of the scanned surface.

Stereonets Stereonets –– Pole PlotsPole PlotsPlotting dip and dip direction pole plots provide an immediate visual Plotting dip and dip direction, pole plots provide an immediate visual depiction of pole concentrations. All natural discontinuities have a certain variability in their orientation that results in scatter of the pole plots. However, by contouring the pole plot, the most highly concentrated areas of poles, representing the dominant discontinuity sets, can be identified.

It must be remembered though, that it may be difficult to distinguish which set a particular discontinuity belongs to or that in some cases a single discontinuity


a particular discontinuity belongs to or that in some cases a single discontinuity may be the controlling factor as opposed to a set of discontinuities.

Pole Plots & Modes of Slope InstabilityPole Plots & Modes of Slope InstabilityTypical pole plots for different modes of rock slope failure.


Wyllie & Mah (2004)

Discontinuity PersistenceDiscontinuity PersistencePersistence refers to the areal extent or size of a discontinuity plane Persistence refers to the areal extent or size of a discontinuity plane within a plane. Clearly, the persistence will have a major influence on the shear strength developed in the plane of the discontinuity, where the intact rock segments are referred to as ‘rock bridges’. g g

k rock bridge


increasing persistence

Discontinuity PersistenceDiscontinuity PersistenceTogether with spacing, discontinuity persistence helps to define the size of blocks that can slide from a rock face. Several procedures have been developed to calculate persistence by measuring their exposed to calculate persistence by measuring their exposed trace lengths on a specified area of the face.

Step 1: define a mapping area on the rock face scan Step 1 define a mapping area on the rock face with dimensions L1 and L2. line

Step 2: count the total number of discontinuities (N’’) of a specific set with dip in this area and

c t

t

L1

(N ) of a specific set with dip in this area, and the numbers of these either contained within (Nc) or transecting (Nt) the mapping area defined.

c

c c

t

t

L

For example, in this case:N’’ = 14Nc = 5N 4

ct t


L2 Nt = 4 Pahl (1981)

Discontinuity PersistenceDiscontinuity PersistenceStep 1: define a mapping area on the rock face with dimensions L1 and L2.

Step 2: count the total number of discontinuities

Pahl (1981)

Step 2: count the total number of discontinuities (N’’) of a specific set with dip in this area, and the numbers of these either contained within (Nc) or transecting (Nt) the mapping area defined.

c t

ct

L1c c

t t

Step 3: calculate the approximate length, l, of the discontinuities using the equations below.

L2

c

Again, for this case:

If L1 = 15 m, L2 = 5 m and = 35°, then H’ = 4.95 m and m = -0.07. From this the average length/persistence of the discontinuity set l = 4 3 m


From this, the average length/persistence of the discontinuity set l = 4.3 m.

Discontinuity SpacingDiscontinuity Spacing

Spacing is a key parameter in that it controls the block size distribution related to a potentially unstable mass (i.e. failure of a massive block or unravelling-type failure).


Discontinuity RoughnessDiscontinuity RoughnessFrom the practical point of view of quantifying joint roughness, only one technique has received some degree of universality – the some degree of universality – the Joint Roughness Coefficient (JRC). This method involves comparing discontinuity surface

l profiles to standard roughness curves assigned numerical values.

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Dilatancy and Shear StrengthDilatancy and Shear Strength h f l d f In the case of sliding of an

unconstrained block of rock from a slope, dilatancy will accompany shearing of all but the smoothest shearing of all but the smoothest discontinuity surfaces. If a rock block is free to dilate, then the second-order asperities will have a di i i h d ff t h t th diminished effect on shear strength.

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By increasing the normal force across a shear surface by adding tensioned rock bolts, dilation can be limited and interlocking along &

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be limited and interlocking along the sliding surface maintained, allowing the second-order asperities to contribute to the shear strength

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shear strength.

Mechanical Properties of DiscontinuitiesMechanical Properties of Discontinuities

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Discontinuity Data Discontinuity Data –– Probability DistributionsProbability DistributionsDiscontinuity properties can vary over a wide range, even for those belonging to the same set. The distribution of a property can be described by means of probability distributionsof probability distributions.

A normal distribution is applicable where a particular property’s mean value is the most commonly occurring. This is usually the case for dip and dip direction This is usually the case for dip and dip direction.

A negative exponential distribution is applicable A negative exponential distribution is applicable for properties of discontinuities, such as spacing and persistence, which are randomly distributed.

Negative exponential function:


Wyllie & Mah (2004)

Discontinuity Data Discontinuity Data -- Probability DistributionsProbability Distributions

N i Negative exponential function:

From this the probability that a given value

Wyllie & Mah (2004)

From this, the probability that a given value will be less than dimension x is given by:

For example, for a discontinuity set with a mean spacing of 2 m, the b b l h h ll b l h probabilities that the spacing will be less than:

1 m


5 m

StructurallyStructurally--Controlled Instability MechanismsControlled Instability MechanismsStructurally-controlled instability means that blocks formed by discontinuities may be free to slide from a newly excavated slope face under a set of body forces (usually gravity). To assess the lik lih d f h f il l i f th ki m ti dmi ibilitlikelihood of such failures, an analysis of the kinematic admissibilityof potential wedges or planes that intersect the excavation face(s) can be performed.


Kinematic Analysis Kinematic Analysis –– Planar Rock Slope FailurePlanar Rock Slope FailureTo consider the kinematic admissibility of plane instability fiveTo consider the kinematic admissibility of plane instability, fivenecessary but simple geometrical criteria must be met: (i) The plane on which sliding occurs

ik ll l h must strike near parallel to the slope face (within approx. ±20°).

(ii) Release surfaces (that provide n li ibl sist nc t slidin ) must negligible resistance to sliding) must be present to define the lateral slide boundaries.

(iii) The sliding plane must “daylight” in (iii) The sliding plane must daylight in the slope face.

(iv) The dip of the sliding plane must be greater than the angle of frictiongreater than the angle of friction.

(v) The upper end of the sliding surface either intersects the upper slope, or terminates in a tension crack.


Wyllie & Mah (2004)

Kinematic Analysis Kinematic Analysis –– Rock Slope Wedge FailureRock Slope Wedge FailureSimilar to planar failures several conditions relating to the line of Similar to planar failures, several conditions relating to the line of intersection must be met for wedge failure to be kinematically admissible :

(i) The dip of the slope must exceed the dip of the line of intersectionof the two wedge forming di ti it ldiscontinuity planes.

(ii) The line of intersection must “daylight” on the slope face.

(iii) The dip of the line of intersectionmust be such that the strength of the two planes are reached.

( ) Th d f h l f (iv) The upper end of the line of intersection either intersects the upper slope, or terminates in a tension crack.


tension crack.Wyllie & Mah (2004)

Kinematic Analysis Kinematic Analysis –– Daylight EnvelopesDaylight EnvelopesDaylight Envelope: Zone within which all poles belong to planes that daylight, and are therefore potentially unstable.

slopeslopefaces

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daylightenvelopes Li

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Kinematic Analysis Kinematic Analysis –– Friction ConesFriction ConesFriction Cone: Zone within which all poles belong to planes that dip at angles less than the friction angle, and are therefore stable.

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Pole Plots Pole Plots -- Kinematic AdmissibilityKinematic Admissibility

friction

Having determined from the daylight envelope whether block failure is kinematically friction

coneblock failure is kinematically permissible, a check is then made to see if the dip angle of the failure surface (or li f i i ) i

slopeface

line of intersection) is steeper than the with the friction angle.

daylightenvelope Thus, for poles that plot

inside the daylight envelope, y g p ,but outside the friction circle, translational sliding is possible.


Pole Plots Pole Plots -- Kinematic AdmissibilityKinematic Admissibility

< f

> f

Wyllie & Mah (2004)


y

Wedge Failure Wedge Failure –– Direction of SlidingDirection of Sliding

Scenario #1: If the dip directions of the two planes lie outside the included angle between i(trend of the line of intersection) and f (dip direction of face) the wedge will slide on both direction of face), the wedge will slide on both planes.

Example scenario #2: If the dip directions of one 2004

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Example scenario #2: If the dip directions of one plane (e.g. Plane A) lies within the included angle between i (trend of the line of intersection) and f (dip direction of face), the wedge will slide on only that plane lli

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only that plane.

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Case History: Rock Slope StabilizationCase History: Rock Slope Stabilizationfe

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A rock slope with a

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history of block failures is to be stabilized through anchoring. To carry out the design, a y gback analysis of earlier block failures is first performed to obtain joint shear strength


j gproperties.

Case History: Rock Slope StabilizationCase History: Rock Slope Stabilizationel

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Case History: Rock Slope StabilizationCase History: Rock Slope Stabilization

Assume: Water in tension crack@ 50% the tension crack height &water along discontinuitywater along discontinuity.



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Given:• Unstable Rock Slope

40 ft t ll• 40 ft tall• About 55 degrees• Joint Set Dips 38 degrees

’ + i 38 40 degrees

55 deg slope

• + i ~ 38 - 40 degrees

From previous back analysis From previous back analysis of failed block below bridge abutment.



1. “Worst case” tension crack distance is 8.6 ft for a “dry” condition

9.9’condition.

2. Assume 50% saturation for tension crack.

55 deg slope3. Estimate “super bolt” tension

given desired bolt inclination.

4. Distribute “super bolt” tension over slope face based on available bolts.

5. Make sure and “bolt” all unstable blocks.



Results:

1. 22 kips tension/ft required at 5 d d d l f F 5 deg downward angle for F = 1.5

2. Slope face length is equal to:

9.86’

55 deg slopeV

U



Recommendations:

1. 8 rows of bolts (40/5 = 8)

2. Try to bolt every block

3. Grout length determined by contractor

55 deg slope

contractor

4. Rule of thumb, grout length; UCS/30 < 200 psi adhesion

l f 5. Contractor responsible for testing or rock bolts

6. Engineer responsible to “sign ff” off” on Contractors tests



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ComputerComputer--Aided Planar AnalysisAided Planar Analysis

(R cscience R cPlane)


(Rocscience – RocPlane)

Lecture ReferencesLecture ReferencesBarton, NR & Choubey, V (1977). The shear strength of rock joints in theory and practice. RockMechanics 10: 1–54.

Hoek, E, Kaiser, PK & Bawden, WF (1995). Support of Underground Excavations in Hard Rock.Balkema: Rotterdam.

Hudson, JA & Harrison, JP (1997). Engineering Rock Mechanics – An Introduction to the Principles .Elsevier Science: Oxford.

Lisle, RJ (2004). Calculation of the daylight envelope for plane failure of rock slopes. Géotechnique54: 279 28054: 279-280.

Pahl, PJ (1981). Estimating the mean length of discontinuity traces. International Journal of RockMechanics & Mining Sciences & Geomechanics Abstracts 18: 221-228.

Strouth, A & Eberhardt, E (2006). The use of LiDAR to overcome rock slope hazard data collection, , ( ) pchallenges at Afternoon Creek, Washington. In 41st U.S. Symposium on Rock Mechanics: 50 Years ofRock Mechanics, Golden. American Rock Mechanics Association, CD: 06-993.

Wyllie, DC & Mah, CW (2004). Rock Slope Engineering (4th edition). Spon Press: London.

W lli DC & N i h NI (1996) R k t th ti d th i M t I L d lidWyllie, DC & Norrish, NI (1996). Rock strength properties and their Measurement. In Landslides:Investigation and Mitigation – Special Report 247. National Academy Press: Washington, D.C., pp. 372-390.


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