Retaining and trench safety
Active earth pressure on a retaining wall (Rankine)
- where
- active earth pressure coefficient
- angle of shearing resistance of the soil (°)
- total active thrust per metre run (kN/m)
- soil unit weight (kN/m³)
- retained height (m)
- uniform surcharge (kPa)
- overturning moment at the base per metre (kN·m/m)
Explain this equation
This gives the sideways thrust of soil on a small retaining wall, and the overturning moment at its base, using Rankine’s active earth pressure.
φ is the soil’s angle of shearing resistance and Ka the active pressure coefficient, the fraction of vertical stress that pushes sideways. γ is the unit weight of soil, H the retained height and q any uniform load on the ground behind. Pa is the total thrust per metre of wall and M the moment at the base.
Soil pressure grows with depth, making a triangle with total ½Ka·γ·H² acting a third of the way up. A surcharge adds an even pressure Ka·q over the full height, acting halfway up. Moment is each force times its height above the base.
Sense check: for φ = 30°, Ka is 1/3, so a 1.5 m wall in 18 kN/m³ soil gets about 6.75 kN per metre; doubling the height quadruples the thrust.
Watch out: water behind an undrained wall can more than double the load. These are unfactored values, not a wall design.
Characteristic lateral earth thrust on a vertical wall retaining level, drained, cohesionless soil, from Rankine’s theory.
Enter your values to see the result.
Good to know
Ka = (1 − sin φ′)/(1 + sin φ′): about 0.33 for φ′ = 30° and 0.27 for φ′ = 35°. The triangular soil pressure gives a thrust ½·Ka·γ·H² acting at H/3 above the base; a uniform surcharge q (for example 10 kPa for light traffic or stored materials) adds a rectangular block Ka·q·H acting at H/2. Typical γ for soil is 17–20 kN/m³. These are characteristic values: design to Eurocode 7 applies partial factors and checks sliding, overturning and bearing.
Common mistake
Ignoring water. If the backfill is not drained, water pressure (9.81 kN/m³ × depth) acts on top of the soil pressure and can more than double the thrust. Weep holes and free-draining backfill matter.
Limits
Vertical smooth wall, level backfill, dry or drained cohesionless soil, wall free to move enough to mobilise active pressure. Clay, sloping backfill, propped walls and walls over about 1 m that retain people, roads or buildings need an engineer.
Sources
- Craig’s Soil Mechanics (Knappett & Craig), Rankine’s theory of earth pressure: Ka = (1 − sin φ′)/(1 + sin φ′), Pa = ½·Ka·γ·H²
- BS EN 1997-1 Eurocode 7: Geotechnical design (retaining structures need design to EC7 with partial factors)
Results are calculations from the published methods and figures shown, and depend on the data you enter. Follow BS 8500, BS EN 206, BS EN 752, Building Regulations Part H, manufacturers’ data and your engineer’s design. Structural concrete and foundations need a competent designer.