1. Lecture 5: Hemispherical Projection & Slopes
D A Cameron
Rock & Soil Mechanics

2. Why are these plots needed?
to provide a simple visual reference of the various joint sets seen in rock mass exposures
to evaluate the potential for instability of engineering works in these masses
e.g. dip angle and dip direction of slopes can be compared with prevailing joint sets

3. Meridional stereographic projection net for Structural Geology

4. EXAMPLE dip direction,  = 135 dip angle,  = 50 denoted as 135/050
Plot also 000/090 and 090/000

5. N
Dip Direction  = 135°

6. Tracing paper with central drawing pin

7. Step 1
Rotate the paper until the line marking the dip direction corresponds with the equatorial position (90)

8. N 

9. Steps 2 and 3
Measure 50 ( = the dip direction, ) from the outer circle RHS and trace the great circle for the plane as shown
Measure (90 - ) or 40 from the outer circle, but this time from the LHS to locate the POLE of the great circle or plane

10. N 
40
50
90
POLE
GREAT CIRCLE

11. The Normal or Pole
The normal to a plane is an imaginary line drawn perpendicular to the plane
The downward direction of this line is plotted on the projection
The point representing the normal on the stereonet is referred to as the pole of the plane

12. Normal to the plane
The trend n and plunge n of the normal to a plane are given by:
n = w ± 180
n = 90 - w

13. Rotate back to the North position

14. N 
Add in 000/090

15. Intersections
Two planes A and B have orientations A:060/030 and B:340/075
These planes intersect on the stereonet at the point A:B
- this point represents the line of intersection of the discontinuities represented by the planes

16. N
Plane A, 060/030
Plane B, 340/075

17. Plunge of intersection line
Rotate tracing until intersection point lies on the E-W line
Read off the number of degrees from the perimeter to the intersection point
= the plunge of the intersection line

18. intersection line
The intersecting planes
plane 1
plane 2
line of intersection

19. Plunge of intersection line
Rotate tracing until intersection point lies on the E-W line
Read off the number of degrees from the perimeter to the intersection point
= the plunge of the intersection line

20. N
Plane A, 060/030
Plane B, 340/075
35

21. Dip direction of intersection line
Rotate tracing back to the datum
Mark off dip direction as indicated
The intersection point can be designated as 060/035

22. N
Plane A, 060/030
Plane B, 340/075
60

23. N
Plane A, 060/030
Plane B, 340/075

24. SLOPE STABILITY
Stereonet information can be used to indicate the likely instability
Plots of the poles of discontinuity planes
Contours to indicate high concentrations in areas of the net
prevailing discontinuities
Position of discontinuities with respect to the Great Circle for the Slope?

26. Typical Slope Instability
(a) no particular concentration of poles
- circular failure (e.g. waste rock/ fractured slate)
- similar to soil (use Bishop’s method)

27. Slope Instability
(b) single concentration of poles above cut slope - plane failure
slope
Discontinuity – strike parallel to that for the slope

28. Conditions for planar failure
The plunge of the slope > dip of the discontinuity
Discontinuity daylights on the slope face
Discontinuity has a dip angle >  for the joint
mechanically possible
Dip direction of the discontinuity and slope lie within  20

29. The last condition
Strike of discontinuity
 20
< 20°

30. Slope Instability
(c) double concentration of poles = intersecting joints - wedge failure most common
slope
discontinuities

31. Conditions for wedge failure
The plunge of the slope > dip of the Intersection line
Intersection line daylights on the slope face
Intersection line has a dip angle >  for the joints
mechanically possible

32. Dip of intersection > friction angle
intersection line
intersection line, I12
“friction circle”
Slope Great Circle
plane 2
plane 1
UNSAFE slope!

33. The Friction Circle Outer radius = 1, represents  = 0° Radius of friction circle = (90° - )/90°
= 30 (0.67)
0.75
0.5
 = 45
1.0
 = 22.5

34. The Friction Circle
The meridional plot is overlaid by the friction circle (same diameter)
The slope is safe if the intersection point, I12 is outside the friction circle () for the joint
- mechanically impossible to fail
- assumes c = 0 kPa for the joint

35. Wedges intersecting slopes
Great circle of slope surface 
intersection lines of planar discontinuities with the slope 

36. Slope Instability (d) single concentration of poles below slope
- toppling failure in hard rock
slope
discontinuity

37. Mechanics of a Planar Failure Qn 8.1, Priest (1993)
30
60
 = 27 kN/m3. Inclined tension crack.
Silt filled joint: cj = 10 kPa, j = 32

38.  U1 = 22.66 kN Area of wedge = 44.43 m2 and U2 = 101.83 kN
 W = 1200 kN
pwp at point D = 2x9.8 = 19.6 kPa
6 m
2.31 m
10 m U2 U1 W
10.38 m

39. ANSWER: FoS = 1.11 FN = Wcos30 - U1sin60 - U2 = 917.8 kN
Sliding resistance = (10.38x10 kPa) (tan32)
= kN
Sliding force =Wsin30 +U1cos60
= kN
ANSWER: FoS = 1.11

40. 2.5 m U2 U1
10.5 m

41. SUMMARY – Key Points Great Circles & poles Lines of intersection
strikes, dip directions & dip angles
Lines of intersection
Conditions for slope instability
4 potential types
Friction circle application
cohesionless joints
Analysis of planar failures