Short note: Factor of safety

Factor of safety (FS) is the ratio of the forces (or shear strength) resisting failure of a slope to the forces (or shear stress) driving it.

$$FS = \frac{\text{Resisting force (shear strength)}}{\text{Driving force (shear stress)}}$$

• $FS > 1$: stable; $FS = 1$: limiting equilibrium; $FS < 1$: failure.
• Usual design values are 1.3 to 1.5 for permanent slopes, and 1.1 to 1.2 for temporary cuts (higher when failure would be disastrous).
• For a planar rock slide on a plane dipping at $\psi_p$, with block weight $W$, cohesion $c$, area $A$ of the sliding plane and friction angle $\phi$:

$$FS = \frac{cA + W\cos\psi_p\tan\phi}{W\sin\psi_p}$$

Example: $W = 1000$ kN, $\psi_p = 30^\circ$, $\phi = 35^\circ$, $c = 10$ kPa, $A = 20$ m$^2$. Resisting $= 200 + 606.4 = 806.4$ kN; driving $= 500$ kN. Answer: $FS = 1.61$ (stable).