Rock EngineeringRock EngineeringPractice & DesignPractice & Design

Lecture 5: Lecture 5: Kinematic Analysis IIKinematic Analysis IIKinematic Analysis IIKinematic Analysis II

(Underground)(Underground)

1 of 33 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

2 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Errors, omissions, comments, etc., can be forwarded to the author at: erik@eos.ubc.ca

Underground Instability MechanismsUnderground Instability Mechanisms

When considering different failure mechanisms underground, we generally di i i h b h distinguish between those that are primarily structurally-controlled and those that are stress-those that are stresscontrolled. Of course some failure modes are composites of these two conditions, and th i l th ff t others may involve the effect

of time and weathering on excavation stability.

99)

tin

et a

l.(1

99

3 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Mar

t

Underground Instability MechanismsUnderground Instability Mechanisms

1

Unstable

WedgeIn-Situ Stress 00

0)

Stress PathRelaxationRelaxation

In Situ Stress

ser

et a

l.(2

0

4 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

3 Kais

Residual StrengthResidual StrengthFor the residual strength condition any cohesion is lost once displacement For the residual strength condition, any cohesion is lost once displacement has broken the cementing action. Also, the residual friction angle is less than the peak friction angle because the shear displacement grinds the minor irregularities on the rock surface and produces a smoother, lower g pfriction surface.

5 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Discontinuity Shear Strength Discontinuity Shear Strength -- ExampleExample

The following tests were obtained in a series of direct shear tests carried out on 100 mm square specimens of granite containing clean, rough, dry joints., g , y j

Di h i Direct shear tests give normal and shear values which may be plotted directly.

6 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

plotted directly.

Discontinuity Shear Strength Discontinuity Shear Strength -- ExampleExample

Plotting the peak strength data we can

th t it t k th see that it takes the form of a bilinear strength envelope.

7 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Discontinuity Shear Strength Discontinuity Shear Strength -- ExampleExample

At higher normal stresses, however, these asperities

i

, pare sheared.

The initial slope of this envelope has an apparent

+ i

8 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

envelope has an apparent friction angle of (+i).

Discontinuity Shear Strength Discontinuity Shear Strength -- ExampleExample

= 30° basic friction angle

i = 45°i = 45°-30° = 15°

Thus…. roughness angle

+ i = 45

9 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Discontinuity Shear Strength Discontinuity Shear Strength -- ExampleExample

If we were to repeat this for the residual strength values...r

10 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

StructurallyStructurally--Controlled Instability MechanismsControlled Instability Mechanisms

Structurally-controlled instability means that blocks formed by discontinuities either fall or slide from the excavation periphery as a result of the body forces (usually gravity) enabled by the y y g y yprocess of excavation. To assess the likelihood of such failures, an analysis of the kinematic admissibility of potential wedges or planes that intersect the excavation face(s) can be performed.

97)

arri

son

(199

Hud

son

& H

a

11 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Kinematic Analysis Kinematic Analysis –– Underground Wedges Underground Wedges

The minimum requirement to define a discrete block is four non-parallel planes, which give rise to a tetrahedral block. In terms of the instability analysis, such a block can be formed by three y y ydiscontinuity planes and one plane representing the excavation periphery. On a hemispherical projection, these blocks may be identified as spherical triangles where the plane of projection

h i frepresents the excavation surface.

Given that a tetrahedral block exists, G n ,there are three kinematic possibilities to be examined: the block falls from the roof; the block slides (either along the

riso

n (1

997)

line of maximum dip of a discontinuity, or along the line of intersection of two discontinuities); or the block is stable.

dson

& H

arr

12 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Hu

Analysis of Kinematic Admissibility Analysis of Kinematic Admissibility -- FallingFallingFalling occurs when a block detaches from the roof of an excavation without sliding on any of the bounding discontinuity planes. In the case of gravitational loading, the direction of

t i ti ll d d movement is vertically downwards.

This is represented on the This is represented on the projection as a line with a dip of 90º, i.e. the centre of the projection Thus if this point 7)projection. Thus, if this point falls within the spherical triangle formed by the bounding discontinuities, falling rr

ison

(199

7

g , gis kinematically admissible.

Hud

son

& H

ar

13 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

H

Analysis of Kinematic Admissibility Analysis of Kinematic Admissibility -- SlidingSlidingKi ti th d d t l bl k lidi f th f Kinematic methods used to analyze blocks sliding from the roof, either on one discontinuity plane (planar failure) or on a line of intersection (wedge failure), generally consider the spherical triangle and whether any part of it has a dip greater than the angle triangle and whether any part of it has a dip greater than the angle of friction.

Assuming that each d l h h discontinuity plane has the same friction angle, the sliding direction will occur l n lin f m ximum dip along a line of maximum dip

(either that of a plane or a line of intersection of two planes) No other part of the planes). No other part of the spherical triangle represents a line of steeper dip than these candidates.

14 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

these candidates. Hudson & Harrison (1997)

Analysis of Kinematic Admissibility Analysis of Kinematic Admissibility -- SlidingSlidingHowever, not all lines of maximum dip on a stereonet projection will be candidates for the sliding direction. Although some planes/lines of intersection may be dipping at angles greater than th f i ti l lidi i t ki ti ll d i ibl if th li the friction angle, sliding is not kinematically admissible if the line of maximum dip is outside the spherical triangle formed by the intersecting planes (i.e. the wedge).

The spherical triangle, h ph r ca tr ang , therefore, represents the region of kinematically admissible directions of movement and any other movement and any other direction represents directions directed into the rock surrounding the block.Hudson & Harrison (1997)

15 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

g

Analysis of Kinematic Admissibility Analysis of Kinematic Admissibility -- SlidingSliding

… hence, the shaded blocks b p s nt ( ) pl n slidin above represent (a) planar sliding

along 2; and (b) wedge sliding along 31.

… of course, if the spherical triangles fall completely outside the friction circle, then the blocks are

16 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

,identified as being stable.

Geometrical Analysis of Maximum Wedge Volume Geometrical Analysis of Maximum Wedge Volume Once a series of joint sets have been identified as potentially forming tetrahedral wedges, several questions may arise as to whether they will be problematic or not:

in the case of a falling wedge, how much support will be required to hold it in place (what kind of loads on the added support can be expected, how dense will the bolting pattern have to be, etc.);p , g p , );

in the case of a sliding wedge, do the shear stresses arising due to gravitational forces exceed the shear strength along the sliding

f i id d b f i ti d ti h i (i th f surface, i.e. provided by friction and sometimes cohesion (in the form of intact rock bridges or mineralized infilling), and if so, how much support will be required to stabilize the block, how dense will the bolting pattern have to be, etc..

In both cases, the volume/weight of the maximum wedge that may form is required. This can be determined through further

t i l t ti

bolting pattern have to be, etc..

17 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

geometrical constructions.

Geometrical Analysis of Maximum Wedge Volume Geometrical Analysis of Maximum Wedge Volume

To calculate the maximum wedge volume:

1) Identify the joint planes/great circles 1) Identify the joint planes/great circles on the stereonet plot that form the wedge. In this example, the three persistent, planar discontinuity sets have dip directions/dips of: (1) 138/51 have dip directions/dips of: (1) 138/51, (2) 355/40, (3) 219/67.

Together, these joints are known to form wedges within the horizontal form wedges within the horizontal, planar roof of an excavation in sedimentary rock.

The stereonet construction is finished The stereonet construction is finished by drawing lines passing through the corners of the spherical triangle and centre of the stereonet.

Priest (1985)

18 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Priest (1985)

Maximum Wedge Volume Maximum Wedge Volume

2) On a separate sheet of paper, construct a scaled plan view, where the width of the window represents the width of the excavation. As such, the analysis will consider the largest block that could be released from the excavation roof.

In this particular example, the roof is rectangular in shape, is 6 m wide, and has it’s long axis orientated at an azimuth of 025°

025°

azimuth of 025 .

Given that the great circle representing the horizontal plane through the tunnel coincides with that f h of the stereonet projection, it is

convenient to construct the window aligned parallel to the tunnel axis.

19 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Maximum Wedge Volume Maximum Wedge Volume

3) On the scaled window, mark an arbitrary horizontal reference line and starting point. For example, about halfway along the western margin of the roof.

Inspection of the spherical triangle in the stereonet plot suggests that in the stereonet plot suggests that the corner of the face triangle formed by planes 2 and 3 will touch the western margin of the roof, and the corner formed by planes 1 and

horizontalreference line

the corner formed by planes 1 and 2 will touch the eastern margin when the largest possible tetrahedral block is considered.

As such, the arbitrary reference point can represent the corner of the face triangle formed by planes 2 and 3.

20 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Maximum Wedge Volume Maximum Wedge Volume 4) The lines associated with

planes 2 and 3 can now be added to the window construction by counting off 60°y g ffthe angles between the horizontal reference line on the stereonet plot (at 025°) and the diametral lines for planes 2 and 3 (striking at 085° and 129°, respectively).

These an les can then be These angles can then be transferred to the window construction and measured off relative to the starting point and reference line

60°

22point and reference line along the western margin of the roof.

21 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Maximum Wedge Volume Maximum Wedge Volume 4) The lines associated with

planes 2 and 3 can now be added to the window construction by counting off y g ffthe angles between the horizontal reference line on the stereonet plot (at 025°) and the diametral lines for

104°

planes 2 and 3 (striking at 085° and 129°, respectively).

These an les can then be These angles can then be transferred to the window construction and measured off relative to the starting point and reference line

104°

point and reference line along the western margin of the roof. 33

22 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Maximum Wedge Volume Maximum Wedge Volume

5) The point where the line for plane 2 intersects the eastern margin of the roof gin the window construction represents the corner of the face triangle formed by planes 1 and 2. Thus,

37°

y pthe line for plane 1 can be added by measuring the angle between the two planes on the stereonet 11pand transferring it to the window construction.

The outline/trace of the wedge on the tunnel roof

37°

22wedge on the tunnel roof is now complete.

23 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Maximum Wedge Volume Maximum Wedge Volume

6) The next step is to add the corner edges of the wedge g gto complete the 3-D trace of the tetrahedron in the window construction box.

This can be done following a This can be done following a similar procedure by transferring the lines of intersection between the planes (i e I I I )

apex

planes (i.e. I12, I23, I13) and their measured angles from the stereonet to the window construction.

24 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Maximum Wedge Volume Maximum Wedge Volume

7) Since this construction can be completed graphically by overlaying the stereonet with the window construction or geometrically by construction, or geometrically by measuring the angles off the stereonet and transferring them onto the window construction, several checks can be made to find several checks can be made to find any errors that may have arisen.

The final step involving the finding of the location of the wedge’s apex of the location of the wedge s apex also gives a valuable check since the area of the triangle of error formed by these converging lines is a measure of any imprecision in the 85

)

apex

a measure of any imprecision in the construction.

Prie

st (1

98

25 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Maximum Wedge Volume Maximum Wedge Volume

8) The dimensions of the face triangle appearing on the excavation surface can now be scaled off directly from the construction. It’s area, Af, can be found by taking any pair of adjacent sides and their included angles:

This gives a face area of 10.1 m2.

985)

Prie

st (1

9

26 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Maximum Wedge Volume Maximum Wedge Volume

9) The areas of the three internal block surfaces can be found in a similar way from the edge lengths and appropriate internal angles:and appropriate internal angles:

985)

… geometrical properties of a

h d l bl k

Prie

st (1

9

27 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

tetrahedral block.

Maximum Wedge Volume Maximum Wedge Volume

10) To find the volume of the wedge, the wedge height and the face area are required. The face area, Af, h l d b f d h d has already been found. The wedge height, h, is given by:

which for this example problem comes to 1.47 m.

Th l V f th t t h d l The volume, V, of the tetrahedral block is then given as:

985)

resulting in a block volume of approximately 5 m3. Pr

iest

(19

28 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Maximum Wedge Volume Maximum Wedge Volume

11) Now assuming a unit weight of 25 kN/m3 for sedimentary rock, the block would have a weight of gapproximately 124 kN.

By dividing this value through by the face area, it can be seen that a support pressure of only 12 3 a support pressure of only 12.3 kN/m2, distributed over the face triangle, would be required to keep it in place.

This support pressure could, for example, be provided by rock bolts anchored beyond the block at a distance of 2 to 3 m above the excavation roof.

29 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Key Block AnalysisKey Block Analysis

The underlying axiom of block theory is that the failure of an excavation begins at the boundary with the movement of a block into the excavated space. The loss of the first block augments the space, possibly creating

t it f th f il f dditi l bl k ith ti i an opportunity for the failure of additional blocks, with continuing degradation possibly leading to massive failure.

As such the term key block identifies any

23

As such, the term key-block identifies any block that would become unstable when intersected by an excavation. The loss of a key-block does not necessarily assure

1

2subsequent block failures, but the prevention of its loss does assure stability.

Key-block theory therefore sets out to Key block theory therefore sets out to establish procedures for describing and locating key blocks and for establishing their support requirements.

30 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

Goodman & Shi (1995)

Wedge Analysis Wedge Analysis –– ComputerComputer--AidedAidedThe 3-D nature of wedge problems (i.e. size and shape of potential wedges in potential wedges in the rock mass surrounding an opening) necessitates f l l a set of relatively

tedious calculations. While these can be performed by hand performed by hand, it is far more efficient to utilise computer-based t h i techniques.

(Rocscience – Unwedge)

31 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

( g )

ComputerComputer--Aided Wedge Analysis in Design Aided Wedge Analysis in Design

The speed of computer-aided wedge analyses allow them to be employed within the design p y gmethodology as a tool directed towards "filter analysis". This is carried out during the preliminary design to determine whether or design to determine whether or not there are stability issues for a number of different problem configurations (e.g. a curving g g gtunnel, different drifts in the development of an underground mine, etc.).

32 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition

(Rocscience – Unwedge)

Lecture ReferencesLecture ReferencesGoodman, RE & Shi, GH (1995). Block theory and its application. Géotechnique 45(3): 383-423.

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

Kaiser, PK, Diederichs, MS, Martin, D, Sharpe, J & Steiner, W (2000). Underground works inhard rock tunnelling and mining. In Proceedings, GeoEng2000, Melbourne. Technomic Publishing:Lancaster pp 841 926Lancaster, pp. 841-926.

Martin, CD, Kaiser, PK & McCreath, DR (1999). Hoek-Brown parameters for predicting the depthof brittle failure around tunnels. Canadian Geotechnical Journal 36(1): 136-151.

Priest, SD (1985). Hemispherical Projection Methods in Rock Mechanics. George Allen & Unwin:, ( ) p j gLondon.

33 of 33 Erik Eberhardt – UBC Geological Engineering ISRM Edition