Cut slope design: N41W/56 hillslope

Method (lower-hemisphere stereonet, equal-angle): plot the great circle of the slope face and of each discontinuity, draw the friction circle of radius $\phi$, and mark intersections of great circles. Then apply the kinematic tests below.

Data and assumptions

Hill slope N41°W/56° dipping south-west = 56/229; discontinuity N50°W/37° dipping SW = 37/220; $\phi = 28^\circ$ (both are assumed to dip towards the same side, which is the critical case).

Possible failure mode

• The dip directions differ by only $9^\circ$ (< 20°).
• Joint dip $37^\circ <$ slope $56^\circ$, so the joint daylights in the natural slope.
• Joint dip $37^\circ > \phi = 28^\circ$.

All the plane-failure conditions are satisfied, so the possible mode is plane failure along the discontinuity.

Design of the cut slope

Plane failure is avoided if the cut slope does not expose the plane, i.e. the cut angle is not steeper than the dip of the discontinuity:

$$\psi_{cut} \leq \psi_p = 37^\circ$$

On the stereonet, the great circle of the cut slope must then lie outside (nearer the primitive than) the great circle of the joint, so the joint's dip vector no longer falls between the slope great circle and the friction circle.

Answer: cut the slope at about 35° (not more than 37°), keeping the road face along the existing strike (about N41°W). If a steeper cut (up to 56°) is needed for the road, support it with rock bolts or shotcrete and drain the slope, because the joint dip (37°) exceeds the friction angle (28°) and sliding will occur on any cut steeper than 37°.