Cut slope design: limestone N48E/52

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

Read in the same way as similar exam questions: the discontinuity is N48°E/52° dipping SE = 52/138, and the natural slope is N60°E/75° dipping SE = 75/150; $\phi = 30^\circ$.

Possible failure mode

• The dip directions differ by $12^\circ$ (< 20°), so the plane is nearly parallel to the slope.
• $\psi_p = 52^\circ < \psi_f = 75^\circ$: the plane daylights in the slope face.
• $\psi_p = 52^\circ > \phi = 30^\circ$: the block can slide under gravity.

All the conditions are met, so the possible mode of failure is plane failure along the discontinuity.

Design of the cut slope

The plane is not exposed in the cut face if the cut angle does not exceed the dip of the discontinuity:

$$\psi_{cut} \leq \psi_p = 52^\circ$$

Answer: make the road cut at not more than 52° (say about 45° to 50° for a margin of safety), with the face striking parallel to the discontinuity (about N48°E to N60°E). If a steeper face is unavoidable, use rock bolts, shotcrete, drainage and benches.

(If the figures are read the other way round, with a 52° slope and a 75° joint, the joint does not daylight and the natural slope is stable; plane failure would then occur only if the cut were made steeper than 75°, so a cut of up to about 70° would be safe.)