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Infinite Slope Stability Factor of Safety

Drained and undrained infinite-slope stability formulas with a worked example, assumptions and controls for engineering review.

Table of contents

The infinite-slope model is a simplified calculation for a long, uniform slope with a potential failure plane parallel to the ground surface. It can be useful for screening shallow translational failure, but it is not a substitute for a full slope-stability analysis.

Topic guide: For background, method selection and related checks, see Slope Stability.

Drained infinite-slope equation

For a potential plane at vertical depth \(z\):

$$ FS= \frac{ c'+(\gamma z\cos^2\beta-u)\tan\phi' }{ \gamma z\sin\beta\cos\beta } $$
Symbol Meaning
\(FS\) Factor of safety for the selected model
\(c'\) Effective cohesion intercept
\(\phi'\) Effective friction angle
\(\gamma\) Total unit weight of soil
\(z\) Vertical depth to the potential plane
\(\beta\) Slope angle from horizontal
\(u\) Pore water pressure acting on the plane

The depth convention must be explicit. Equations written for thickness normal to the slope are not interchangeable with a vertical-depth equation.

Dry worked example

Assume:

  • \(\beta=20^\circ\)
  • \(z=2.0\) m vertically
  • \(\gamma=18\) kN/m3
  • \(c'=5\) kPa
  • \(\phi'=28^\circ\)
  • \(u=0\) kPa
$$ FS= \frac{ 5+[18(2)\cos^2(20^\circ)]\tan(28^\circ) }{ 18(2)\sin(20^\circ)\cos(20^\circ) } $$

Using unrounded trigonometric values:

$$ FS\approx1.89 $$

This is an arithmetic example, not an acceptance assessment.

Pore-pressure ratio form

Where a defined pore-pressure ratio is used:

$$ r_u=\frac{u}{\gamma z\cos^2\beta} $$

The drained equation becomes:

$$ FS= \frac{c'}{\gamma z\sin\beta\cos\beta} + \frac{(1-r_u)\tan\phi'}{\tan\beta} $$

The meaning of \(r_u\) must be defined. Different references may use different stress bases.

Undrained total-stress case

For a total-stress model with \(\phi_u=0\):

$$ FS= \frac{s_u}{ \gamma z\sin\beta\cos\beta } $$

Where \(s_u\) is the undrained shear strength selected for the potential plane. Strength anisotropy, strain compatibility, progressive failure and construction rate may be important.

Model assumptions

The infinite-slope model generally assumes:

  • slope length is large relative to failure depth
  • uniform slope angle
  • parallel soil layers
  • a plane parallel to the surface
  • uniform parameters
  • a nominated groundwater or pore-pressure condition
  • no toe, crest or three-dimensional effects

When the model is unsuitable

Use a more complete analysis where there are:

  • deep circular or compound failure surfaces
  • layered or irregular geology
  • finite slope geometry
  • surcharge near the crest
  • excavations or retaining structures
  • tension cracks
  • seismic loading
  • rapid drawdown
  • anchors or reinforcement
  • significant three-dimensional effects

Calculation controls

A tool should:

  • state whether depth is vertical or normal to slope
  • label angles in degrees or radians
  • show driving and resisting terms
  • identify total or effective stress
  • require a pore-pressure model
  • block non-physical geometry
  • preserve unrounded values
  • avoid supplying a universal required factor of safety

Related resources

Authoritative reference

Slope design requires appropriate failure mechanisms, groundwater conditions, parameters, design criteria and professional review.