A slope is stable when the resisting forces are superior or equal to the driving forces (or destabilising forces). The ratio between resisting forces and driving forces is the safety factor F. It is conventionally taken as the measure for the stability of a slope. Parameters involved in calculation of the safety factor can be summarized with figure 1 (from Besson, 2005):
With:
Sumo is the overload
The other man is the thrust force
P: weight of material (force)
Zoom corresponds to the characteristic of the slipping contact surfaces
Resisting forces:
Pn: normal component of P
Normal component of the overload (Pn)
Normal component of the thrust force (Pn)
Driving forces (or destabilising forces):
Pt: tangential component of P
U1 and U2: hydrostatic pressure due to the “Archimedes thrust”
E: flow pressure
Tangential component of the overload (Pt)
Tangential component of the thrust force (Pt)
The safety factor is also defined as the ratio between shear stress and shear strength in a loaded mass:
F =1//0
With = available shear strength and 0 = shear stress at the speed of 0
Shear strength: cohesion and effective inter-particle friction; internal resistance of a material to shear stress. The shear strength is function of temperature, confining pressure, shape, size, loading rate and amount of pores. It is an important parameter in the determination of engineering and geomorphic properties of materials.
Shear stress: two perpendicular loads or stresses applied tangential to the surface of material, producing angular deformation in the material.
A safety factor less than or equal to1 indicates that the slope is instable. Indeed, as the shear stress approaches the maximum available shear strength, the safety factor becomes 1 and failure is imminent. If the slope is destabilized by triggering factors (earthquake, rainfall, etc.), the apparent increase of the shear stress over the available shear strength is equivalent to the acceleration of the sliding mass. Three stages can be distinguished (figure 2):
- Stable, F>1,5: the margin of stability of the slope is sufficiently high to withstand all destabilising forces;
- Marginally stable, 1,0
- Actively unstable, F~1,0: destabilising forces produce acceleration of the sliding mass (continuous or intermittent movements).
Moreover, safety factor value is time-dependent (figure 2). The movement phase is split into pre-failure, failure and post-failure stages with the possibilities of occasional reactivation. All types of movements at a given stage are associated with specific controlling variables that are subdivided into predisposition and triggering factors. Slope instability responds to a combination of these factors. The predisposition factors change most times only gradually over time whereas the triggering factors are transient. Triggering factors may either increase the shear stress, decrease the shear strength of the material or both.
References:
BESSON L., 2005. Les risques naturels: de la connaissance pratique à la gestion administrative. Editions Techni. Cités, Voiron, 60 p.
VAN ASCH T., MALET J.P., VAN BEEK L., AMITRANO D., 2007. Techniques, issues and advances in numerical modelling of landslide hazard. Bulletin de la société géologique de France, n°178, p. 265-288