A note on calculating stresses on slopes in geomorphology

Two common problems occur when calculating the stresses on a slope (or in a slope). One is confusion over the use of sines, tangents or cosines, the other is doubt over the appropriate depth to use in the calculation. Unfortunately the literature is not much help since it is often in error. The main problem comes from the fact that at small slope angles, it is possible to be very sloppy and still get the right answer.

So here are some slope equations, with notes.

Shear stress under a uniform slab lying at an angle on a slope (or the normal and shear stresses within a slope, at a depth, parallel to the surface);

, (1)

where d is the perpendicular to slope thickness of the slab or layer, and a is the slope angle from the horizontal. If the thickness is given as a vertical (to gravity) thickness and is given (say) the symbol h, then the equation becomes;

, (2)

where the cosine is introduced to correct the depth.

For angles of less than about 10⁰, the cosine is very close to 1 and the sine is close to the tangent. In addition, the tangent is convenient since it is just “rise over run” or the commonly referred to slope. As a result you will often see (1) or (2) written with a tangent, and even more common is (2) without the cosine.