Course:GEOBrefGuide/Geomorphology/factor of safety

The basic principle

A factor of safety is simply a ratio that indicates how close a slope is to the threshold for a landslide to occur. A ratio of 1 indicates that the slope is on the threshold for failing: as the ratio increases above 1, the slope moves farther and farther from the threshold, and the slope becomes progressively more stable (hence the term factor of safety). We typically conceive of failures being driven by the downslope component of gravity acting parallel to the failure plane, which is called the shear force. The forces resisting failure are referred to as the material strength. Therefore, the most fundamental expression of the factor of safety is:

• $FS={\frac {Material\,Strength}{Shear\,Force}}$

We typically attribute material strength to cohesion along the failure plane, which is related to the sedimentological properties of the soil (in large part, the clay content), and to friction along the failure plane, generated by the component of gravity acting perpendicular to the failure plane. So a more explicit equation for the factor of safety is:

• $FS={\frac {F_{cohesion}+F_{friction}}{F_{shear}}}$

In order to produce an equation that we can use to make predictions about the relative stability of a given slope, we need to make some further assumptions about the geometry and hydrology of the slope.

The factor of safety equation

Here we consider a stability analysis for an infinitely long, straight, inclined hillslope, where groundwater is flowing parallel to the slope (the resulting stability equation is often referred to as the infinite slope model). The basis for assessing slope stability is the force balance between the driving forces (F_driving) tending to move the material down the slope and the resisting forces (F_resisting) tending to keep the material in place. The force balance is expressed using the factor-of-safety (FS). In this discussion of slope stability, we will make some assumptions to make our lives easier, but you should realize that the concept of the factor-of-safety can be applied in a more complex and physically realistic ways.

Driving force

The driving force is the component of gravity acting on the soil mass, parallel to the potential failure plane. In mountainous environments, where soils are relatively shallow, the potential failure plane is usually assumed to be the bedrock surface lying beneath a thin layer of soil, the slope of which can be estimated from topographic maps. If we consider a rectangular block of soil resting on bedrock, then the total force of gravity is:

• $F_{g}=gM=g\rho _{b}V=g\rho _{b}(D_{x}D_{y})d$

That is, the force of gravity is equal to the acceleration of gravity (g = 9.81 m/s²) times the mass of the block (M). The mass of the block is equal to the volume (V) of the block times its “bulk” density (ρ_b ~ 1900 kg/m³ for a wet soil), and the volume is equal to the width (D_x) times the length (D_y) times the depth (d) of the rectangle. Our first assumption is that our rectangle will have unit width and length, so D_x = D_y = 1, and those variables and drop of the equation. Really, we are expressing the force as a force per unit area, which we call a stress. The units for measuring a force are Newtons (N), and for a stress they are Pascals (Pa = N/m²). So the total stress due to gravity is:

• $\sigma _{g}=g\rho _{b}d=\gamma _{b}d$

where γ_b is the bulk unit weight of the soil (~ 18.6 kN/m³ for wet soil), d is the thickness of the soil perpendicular to the potential failure plane. If the failure plane is flat, then all of the gravity force is perpendicular to it and there is no component parallel to it and hence no driving force. Once we incline the plane, the component of the gravity force acting parallel to the surface increases continuously. The component of σ_g acting parallel to a failure plane, inclined at angle β, is called the shear stress (τ):

• $\tau =\sigma _{g}\sin {\beta }=\gamma _{b}d\sin {\beta }$

Resisting force

The resisting force includes two components:

• the friction produced by the σ_g acting normal to the failure plane, and
• the attractions between the soil particles that bind them together, called cohesion.

In a mountainous environment, the soils tend to be coarse-grained and cohesionless, but slopes are frequently forested and the roots of the trees produce a kind of cohesion that binds the soil together, adding to the resisting forces. The friction force is the product the normal force and the coefficient of friction. The normal force, expressed as a stress (σ), is given by:

• $\sigma =\sigma _{g}\cos {\beta }=\gamma _{b}d\cos {\beta }$

The coefficient of friction is usually expressed using a friction angle (φ ≈ 35^o for coarse grained soils), so that the coefficient is expressed as tan(φ). The pressure of water in the pore spaces acts in the opposite direction to σ, and reduces the amount of friction at the potential failure plane. The pore pressure is related to the height of the water table above the failure plane, measured perpendicular to the failure plane, and the density of water (ρ). The pore pressure, u, (also measured in Pa) is given by:

• $u=\gamma md\cos {\beta }$

The term γ is the unit weight of water (≈ 9.81 kN/m³), and m is the ratio of the height of the water table (measured perpendicular to the failure plane) to the total soil depth, d. The term m varies from 0 for a completely dry soil to 1 for a completely saturated one. The frictional stress (σ_f) is thus:

• $\sigma _{f}=(\sigma -u)\tan {\phi }=d\cos {\beta }(\gamma _{b}-m\gamma )\tan {\phi }$

The factor-of-safety

If a soil has no cohesion of any sort, then the factor-of-safety is given by the ratio of the frictional stress and the shear stress. If the factor-of-safety drops below 1.0, a landslide can be expected to occur. If it is above 1.0, the slope should be stable. The equation is:

• $FS={\frac {d\cos {\beta }(\gamma _{b}-m\gamma )\tan {\phi }}{\gamma _{b}d\sin {\beta }}}$

which can also be written

• $FS=\left(1-m{\frac {\gamma }{\gamma _{b}}}\right){\frac {\tan {\phi }}{\tan {\beta }}}$

If the soil has measurable cohesion (C, measured in kN/m³), then the cohesive component is added to the frictional stress, σ_f, and the equation becomes:

• $FS={\frac {C+d\cos {\beta }(\gamma _{b}-m\gamma )\tan {\phi }}{\gamma _{b}d\sin {\beta }}}$

These last two equations are the basis for the 'infinite slope model for slope stability analysis. While simple, they are surprisingly powerful.