Landslide Dynamics: The Infinite Slope Model and the Factor of Safety

Reducing a hillside to one equation

The infinite slope model is the standard first approximation for a shallow landslide: it assumes the slope is much longer and wider than the depth of the potential failure plane, so edge effects can be ignored and the problem reduces to a single slice of soil at depth z, resting on a plane tilted at angle β. Gravity pulls that slice down-slope (the driving shear stress) while friction and cohesion along the plane resist it (the shear strength). Whether the slope holds or fails comes down to comparing the two.

FS = (c' + (σn − u)·tanφ') / τ        Factor of Safety

σn = γ·z·cos²β      normal stress on the failure plane
τ  = γ·z·sinβ·cosβ  driving shear stress
u  = pore water pressure
c'  = effective cohesion, φ' = effective friction angle

FS > 1 → stable    FS = 1 → incipient failure    FS < 1 → predicted failure

Cohesion, friction, and effective stress

The resisting strength follows the Mohr-Coulomb failure criterion: a fixed cohesive component c' that holds regardless of load, plus a frictional component that scales with the normal stress pressing the failure surface together. Crucially, that frictional term depends on the effective normal stress, σn − u, not the total stress - this is Karl Terzaghi's effective stress principle, and it's the single most important idea in slope stability, because it's the lever that water pulls.

Why rain is the number one trigger

Infiltrating rainwater raises the water table inside the slope and with it the pore water pressure u. Because friction depends on σn − u, rising pore pressure directly and immediately weakens the slope in the Factor of Safety equation - nothing about the geometry or the soil has changed, only the water inside it. Two other common triggers show up as extra terms in more complete versions of the model: seismic shaking adds a pseudo-static horizontal force that increases the driving shear stress, and erosion or excavation at the toe of the slope removes the support that was resisting the driving stress in the first place.

Beyond the simple model

Real slopes complicate the picture: layered soils with different strengths, seepage running parallel to the slope (which raises pore pressure further than vertical infiltration alone), rapid drawdown after a reservoir or river level falls suddenly, and rotational or retrogressive failures where the sliding surface is curved rather than planar - a case the infinite slope model's flat-plane assumption can't capture. For those, engineers switch to limit equilibrium methods like Bishop's simplified method, which slices a curved failure surface into columns and sums the forces on each, or to full finite element models.

Frequently asked questions

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