Lateral earth pressure depends on soil properties, wall movement, groundwater, surcharge, geometry and drainage. This page presents simple Rankine calculations for a vertical wall retaining level, homogeneous, cohesionless backfill.
Topic guide: For background, method selection and related checks, see Lateral Earth Pressure.
The assumptions must be checked before the equations are used.
Rankine active pressure coefficient
For effective friction angle \(\phi'\):
$$ K_a = \frac{1-\sin\phi'}{1+\sin\phi'} $$An equivalent expression is:
$$ K_a = \tan^2\left(45^\circ-\frac{\phi'}{2}\right) $$Active pressure requires enough wall movement for the active condition to develop. A restrained wall may require an at-rest or soil-structure interaction assessment.
Rankine passive pressure coefficient
$$ K_p = \frac{1+\sin\phi'}{1-\sin\phi'} $$ $$ K_p = \tan^2\left(45^\circ+\frac{\phi'}{2}\right) $$Mobilising passive resistance usually requires substantially more movement than active pressure. Disturbance, excavation, erosion and future services can reduce dependable resistance. Full passive resistance should never be assumed without an explicit design decision.
At-rest coefficient
A common empirical estimate for normally consolidated soil is:
$$ K_0 = 1-\sin\phi' $$This relationship is not universal. Stress history, overconsolidation, compaction and wall construction affect at-rest pressure.
Lateral effective stress
For the simple active case:
$$ \sigma'_h(z)=K_a\gamma' z $$The resultant soil thrust over wall height \(H\) is:
$$ P_a=\frac{1}{2}K_a\gamma'H^2 $$For the triangular distribution, the resultant acts at \(H/3\) above the base.
Uniform surcharge
For uniform surcharge \(q\):
$$ \Delta\sigma'_h=K_aq $$ $$ P_q=K_aqH $$The uniform surcharge resultant acts at \(H/2\) above the base.
Hydrostatic water pressure
Where water is retained to height \(H_w\):
$$ P_w=\frac{1}{2}\gamma_wH_w^2 $$The water force is separate from effective soil pressure and acts at \(H_w/3\) above the base for a triangular hydrostatic distribution.
Drainage must not be assumed solely because a drain is shown on a drawing. Blockage, maintenance and design conditions require consideration.
Worked example
Assume:
- \(\phi'=30^\circ\)
- wall height \(H=4\) m
- effective soil unit weight \(18\) kN/m3 for this dry illustrative case
- uniform surcharge \(q=10\) kPa
- no water pressure
Soil thrust:
$$ P_a=\frac{1}{2}\left(\frac{1}{3}\right)(18)(4^2) =48.0\ \mathrm{kN/m} $$Surcharge thrust:
$$ P_q=\left(\frac{1}{3}\right)(10)(4) =13.33\ \mathrm{kN/m} $$Total horizontal thrust is 61.33 kN/m, but the two components have different points of application and must remain separate for moment calculations.
Conditions outside the simple model
Use another method or advanced analysis for:
- sloping backfill
- wall friction
- cohesion or tension cracking
- layered soils
- non-uniform surcharges
- compaction-induced pressure
- seismic loading
- partial mobilisation
- flexible or braced walls
- seepage and non-hydrostatic pore pressure
- reinforced soil systems
Calculation controls
The calculation should display:
- selected pressure state
- effective or total stress basis
- angle units
- wall and ground geometry
- groundwater and drainage assumption
- each pressure component
- force and point of application
- whether passive resistance is included
- method version and parameter source
Related resources
- Retaining Wall Stability
- Effective Stress in Soil
- Footing Pressure and Eccentricity
- Shear Strength of Soil
Authoritative reference
It does not replace a project-specific retaining-structure analysis, applicable standards or review by a suitably qualified geotechnical professional.