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

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

Meridional stereographic projection net for Structural Geology

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

N Dip Direction  = 135°

Tracing paper with central drawing pin 

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

N 

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

N  40 50 90 POLE GREAT CIRCLE

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

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

Rotate back to the North position 

N  Add in 000/090

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

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

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

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

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

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

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

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

 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

ANSWER: FoS = 1.11 FN = Wcos30 - U1sin60 - U2 = 917.8 kN Sliding resistance = (10.38x10 kPa) + 917.8(tan32) = 677.3 kN Sliding force =Wsin30 +U1cos60 = 611.3 kN ANSWER: FoS = 1.11

2.5 m U2 U1 10.5 m

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