Landslides, rockfalls, slumps and creeping soil are all mass wasting: rock and soil moving downhill under gravity, with no river or glacier carrying them. In Erosion and Deposition gravity was in the background; here it is the whole story, and right-triangle trigonometry tells us when a slope gives way.
Why it works
Picture a block of soil on a slope at angle $\theta$. Its weight $W$ splits into two parts: one pulling it down the slope, $W\sin\theta$, and one pressing it into the slope, $W\cos\theta$. Friction resists sliding, and it grows with the pressing force: friction $= W\cos\theta \tan\varphi$, where $\varphi$ is the material's friction angle (the steepest angle it can rest at). The factor of safety compares them: $FS = \dfrac{\text{resisting}}{\text{driving}} = \dfrac{W\cos\theta\tan\varphi}{W\sin\theta} = \dfrac{\tan\varphi}{\tan\theta}$ for dry, loose material. Above 1 the slope holds; below 1 it fails. Steepen the slope and the pull grows while the friction shrinks.
Key idea
A slope fails when the downhill pull beats the friction and strength holding it: $FS = \dfrac{\tan\varphi}{\tan\theta}$ for dry, loose material, with failure below 1. Water filling the pores lowers the friction and can cut FS roughly in half.
Worked examples
Worked example: Is it stable?
Dry gravel with a friction angle of $35^\circ$ sits on a $30^\circ$ slope. Find FS.
- $FS = \dfrac{\tan 35^\circ}{\tan 30^\circ} = \dfrac{0.700}{0.577} \approx 1.21$.
- Above 1: stable, but without much margin.
Answer: $FS \approx 1.21$: stable.
Worked example: After heavy rain
The same slope becomes soaked, with water seeping down parallel to the surface. Then $FS \approx \dfrac{\gamma_{sat} - \gamma_w}{\gamma_{sat}}\cdot\dfrac{\tan\varphi}{\tan\theta}$, with soaked soil $20\,\text{kN/m}^3$ and water $9.8\,\text{kN/m}^3$. Find FS.
- $\dfrac{20 - 9.8}{20} = 0.51$.
- $FS \approx 0.51 \times 1.21 \approx 0.62$. Below 1: the slope fails.
Answer: $FS \approx 0.62$: it fails.
Common mistake
It is natural to think water makes slopes fail mainly by adding weight. The weight adds a little, but the bigger effect is that water under pressure in the pores pushes the grains apart, so they press together less and friction drops. Check with the formula: the water term reduces the friction part, not the pull.