Cut slope design: HS 110/40, B 130/20

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 (dip direction/dip)

HS 110/40; B (bedding) 130/20; J1 100/40; J2 200/50; $\phi = 32^\circ$. In dip/dip direction: HS 40/110, B 20/130, J1 40/100, J2 50/200.

Failure checks for the existing 40° face

• B: its dip direction is within 20° of the slope, but $20^\circ < \phi = 32^\circ$, so it is stable.
• J1: its dip direction differs by $10^\circ$ and its dip of $40^\circ$ equals the face angle. A steeper cut would daylight J1, and as $40^\circ > \phi$ it would slide (plane failure).
• J2: dips at $90^\circ$ to the face direction and cannot slide on its own.
• J1 x J2 intersection plunges $32.1^\circ$ towards 142°, just above $\phi$. It daylights in the face if the apparent dip of the face towards 142° is greater than $32.1^\circ$, i.e. if the cut angle is greater than $36.4^\circ$ (from $\tan\psi\cos 31.7^\circ = \tan 32.1^\circ$).
• B x J1 plunges $15.4^\circ$ and B x J2 $20.0^\circ$ (both below $\phi$): safe.

Design

Answer: the cut slope should be limited to about 35° (maximum 36°, below both the J1 dip of 40° and the wedge limit of 36.4°). Then plane, wedge and toppling failure are all kinematically avoided.