1

Lecture 10A

Rock Slope Engineering

2

Counting circle for contouring pole concentration

3

Presentation of Structural Discontinuities

Strike or Dip

Dip or Dip Direction Identical features having the same orientation.

Traverses are used to group data units

Traverse 1 is a LINEAR traverse. For a LINEAR traverse, the Orient 1 and Orient 2 values are always in TREND/PLUNGE format.

Traverse 2 is a PLANAR traverse, corresponds to STRIKE/DIPR.

Traverse 3 is a BOREHOLE traverse

Traverse 4 is a PLANAR traverse, corresponds to DIP/DIPDIRECTION

4

The generally accepted size for the counting circle used in contour calculations is one percent of the surface area of the lower reference hemisphere.

5

Terzaghi Weighting option is to account for the sampling bias introduced by orientation data collection along Traverses.

When orientation measurements are made, a bias is introduced in favour of those features which are perpendicular to the direction of surveying.

Since the weighting function tends to infinity as alpha approaches zero, a maximum limit for this weighting must be set to prevent unreasonable results. This maximum limit corresponds to a minimum angle, which can be between 0.1° and 89.9°. However, the recommended range is limited to 5° to 25°, and the default is set to 15°. The user can change this limit with the Bias Angle option.

6

Stereonet Options

For the EQUAL ANGLE projection method, a line is drawn from the center of the top of the sphere (the zenith), to a point A on the sphere (a pole or a point on the great circle). The intersection of this line with a horizontal plane through the center of the sphere, defines the projection point B.

In the EQUAL AREA method, the bottom of the sphere rests on the projection plane. The point A on the sphere is projected down to the plane by swinging this point in an arc about the contact between the sphere and the plane, giving point B. The resultant projection is then scaled back to the size of the projection sphere.

LOWER hemisphere projection:

The plot represents the traces of planes and poles on the LOWER half of the sphere, as viewed from ABOVE.

UPPER hemisphere projection:

This displays the traces of planes and poles on the UPPER half of the sphere, when viewed from ABOVE.

EQUAL ANGLE projection preserves only the geometry of projected shapesbut not position.

EQUAL AREA projection distorts geometrical shapes such as small circles but position is correct

Slightly different position

7

In the FISHER method each pole is assigned a normal influence or FISHER distribution over the surface of the sphere, rather than a point value, as in the SCHMIDT method. The integrated zone of influence is a bell shaped distribution with a maximum height of 1, and a basal radius twice that of the counting circle radius. The influence contribution to a grid point is represented by the height of the influence zone immediately above the grid point. In this method, the total influence of an individual pole is the same as in the SCHMIDT method but its distribution of influence reflects an assumed probability of measurement error. For large quantities of data, this option will produce similar results to the familiar SCHMIDT option. The real strength of the FISHER option is in "smoothing" density plots for sparse data sets.

The SCHMIDT distribution is the classical method, in which each pole is assigned a constant influence value of 1. The integrated zone of influence is a cylinder of constant height with a radius equal to the radius of the counting circle. A counting grid is superimposed on the stereonet plane, or in the case of DIPS, on the surface of the reference sphere. Convention dictates the use of a counting circle with an area equivalent to 1% of the lower hemisphere surface. For each pole plotted, any grid point falling within a circle of arbitrary constant radius centered on this pole is incremented by the value of the pole. After the influence of all plotted poles is thus distributed, the density plotted at each grid point is calculated by dividing the pole count at that grid point by the total pole influence.

The generally accepted size for the counting circle used in contour calculations is one percent of the surface area of the lower reference hemisphere.

Pole represented by a distribution

8

9

FAILURE MODES

Assume friction angle of 35 – 40 degrees

10

Slope orientation

(Dip=45 deg. and Dip Dir.=135 deg.)

Assume a friction angle of 35 degrees

Plane representing a “slip limit”The DIP angle for this plane is derived from the PIT SLOPE ANGLE – FRICTION ANGLE = 45 – 35 = 10 degrees.The DIP DIRECTION is equal to that of the face (135 degrees)

Toppling Analysis

Add Cone to place kinematic bounds on the plot

The Trend is equal to the DIP DIRECTION of the face plus 90 degrees (135 + 90 = 225)

The 60 degree cone angle will place two limits plus / minus 30 degrees with respect to the face DIP DIRECTION as suggested by Goodman – planes must be within 30 degrees of parallel to a cut slope to topple.

Any Poles plotting within this region indicate a toppling risk.

10 deg.

30 deg.

11

Planar Sliding

A Daylight Envelope allows us to test for kinematics (ie. a rock slab must have somewhere to slide into – free space). Any pole falling within this envelope is kinematically free to slide if frictionally unstable (This case angle > 35 deg. And < 45 deg.)

Friction angle is equal to our friction estimate of 35 degrees

Select the Daylight Envelope checkbox : Pole of Dip=45 Dip Direction=135

12

Wedge Sliding: Method 1

Dealing with poles but an actual sliding surface or line, so that the friction angle (35 degrees) is taken from the EQUATOR of the stereonet, and NOT FROM THE CENTER as before. Therefore the angle we enter in the Add Cone dialog is 90 – 35 = 55 degrees.

Major planes representing the mean concentrations for each set are used

Plane intersections are shown as black dots on this plot

13

ALTERNATIVE METHOD FOR CHECKING KINEMATIC ADMISSIBILITY OF WEDGES: Method 2

The method of using great circles to define the lines of intersections of planes is tedious for large data sets

A simpler technique is to use the ‘pole’ to the line of intersection rather than the line of intersection itself

The ‘pole’ to the line of intersection (PAB ) is the dip/direction of the great circle that passes through the poles of the two planes (PA , PB ) – the line of intersection (IAB ) being normal to this great circle.

14

The trend and plunge of the line of intersection between two planes can be calculated from the orientation of the two planes

mmll and βαβα ,,If the trend and plunge of the normals to planes 1 and 2 are

respectively, then the direction cosines for the two normals are:

where x, y, z represent the right hand coordinate system north (x), east (y) and down (z), then

lz

lly

llx

I

II

β

βαβα

sin

cossincoscos

=

==

If it is expressed in terms of Cartesian coordinate,

mz

mmy

mmx

m

mm

β

βαβα

sin

cossincoscos

=

==

The Cartesian components of the line of intersection are given by the vector product as follows:

xyyxz

zxxzy

yzzyx

mImIi

mImIi

mImIi

−=

−=

−=

Priest SD (1993). Discontinuity analysis for rock engineering. Chapman & Hall, London

Direction Cosines of Plane L

Direction Cosinesof Plane M

15

The trend and plunge of the line of intersection can be calculated as follows:11 βα and

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

++=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+=

+⎟⎟⎠

⎞⎜⎜⎝

⎛=

222221

1

arcsinarctan

arctan

zyx

z

yx

z

x

y

iii

i

ii

i

Qii

β

α

Laurie Richards (2004) AN ALTERNATIVE METHOD FOR CHECKING THE KINEMATIC ADMISSIBILITY OF WEDGES

If the trend/plunge data for the lines of intersection are used as though these were the dip direction/dip of planes, then the program will plot the wedge poles.

Wedge poles formed by combinations of all planes

16Ref.:2001 Rocscience Inc.

PLANAR FAILURE

17

Flow Chart:

1. Solve for N

(eq. [1])

2. Solve for B

(eq. [2])

3. Solve for L

(eq. [6])

4. Solve for M

(eq. [7])

5. Solve for C

(eq. [8])

6. Solve for A

(eq. [9])

7. Solve for W

(eq. [10])

βsinHN =

{ } { }HHHNB ,cot,cos ββ ==( )

( )ψααψβ

tancossintancot1

−−

=HL

ψβα

coscotcos HLM −

=

{ }αα sin,cos LLC =

xyyx CBCBA −=21

γ⋅= AW

18

19

20

21

Flow Chart:

1. Solve for N

(eq. [1])

2. Solve for B

(eq. [2])

3. Solve for C

(eq. [13])

4. Solve for M

(eq. [14])

5. Solve for Q

(eq. [19])

6. Solve for L

(eq. [17])

7. Solve for D

(eq. [16])

8. Solve for A

(eq. [20])

9. Solve for W

(eq. [10])

βsinHN =

{ } { }HHHNB ,cot,cos ββ ==

{ }ψtan,TTBC +=

ψcosTM =

{ }αα sin,cos LLD =

αθ

coscosQCL x −=

θαθα

coscotsincot

−

−= xy CC

Q

( )( ) ( )( )xxyyyyxxxyyx BCBDBCBDDBDBA −−−−−+−=21

21

γ⋅= AW

22

23

24

25

26

27

28

29

S from last page

30

Release surfaces required to allow planar sliding to occur.

31

Stability of surface wedges in rock slopes

Hoek, E. and Bray, J.W. Rock Slope Engineering, Revised 3rd edition, The Institution of Mining and Metallurgy, London, 1981, pp 341 - 351.

32

33Duncan Wyllie (1992) Foundations on Rock. Chapman and Hall.

Stability of fractured rock mass

•Non circular

•Controlled by sets of planar discontinuities

•Method of slices not applicable to non- vertical joints

•Use Sarma Method (Sarma, 1979)

•Slices can be triangular or quadrilateral

•Closed form solution to calculate the critical horizontal acceleration factor Kc required to induce limit equilibrium

•FOS is calculated by reducing the values of tan φ

and cohesion to tan φ/FOS and cohension/FOS until Kc =0

•Useful in simulating earthquake load by applying an external load on each slice (e.g., a horizontal force of 0.05Weight can be assumed to be equivalent to an acceleration of 5% of gravity)

Sarma, S. K. (1979) Stability analysis of embankments and slopes. J. Geotech. Eng. Div. ASCE, vol. 105, GT12, pp. 1511- 1524

Satisfy Summation of Fx and Fy only, not moment equilibrium

34

Kc =0 critical horizontal acceleration factor