EARTH PRESSURE ON A RETAINING WALL

RANKINE ACTIVE · HYDROSTATIC PRESSURE · EN 1997

⬡ WALL GEOMETRY

Wall height H (m)

Ground level from the base z_t (m)

Water table from the base z_w (m) 0 = no water · may exceed the ground level

⬡ SOIL PARAMETERS

γ (kN/m³) above water

γ' (kN/m³) effective below water

Angle of internal friction φ (°)

Cohesion c (kPa) p = Ka·σ'v − 2c√Ka ≥ 0

⬡ SURCHARGE

INACTIVE

Surcharge q (kN/m²)

⬡ LATERAL PRESSURE DIAGRAM

⬡ CALCULATION RESULTS

Ka (Rankine)0.3333

φ30°

p la cota teren0.00 kN/m²

p la nivel apă (efic.+hidro)10.00 kN/m²

p sol activ la bază (Ka·σ'v)15.00 kN/m²

p hidrostatic la bază15.00 kN/m²

p TOTAL la bază30.00 kN/m²

Forță rezultantă E (kN/m)37.50

Punct aplicare z_R (m)0.90 m de la bază

Tool information

What this page computes

The calculator determines the active earth pressure on a retaining wall, using Rankine theory. It returns the pressure diagram over the height and its resultant — the quantities from which a retaining wall or shoring is designed.

The active earth pressure coefficient:

\(K_a = \tan^2\!\left(45° - \frac{\varphi}{2}\right)\)

and the horizontal pressure at a given depth, from the effective vertical stress \(\sigma'_v\):

\(p_a = K_a \, \sigma'_v - 2c\sqrt{K_a} \; \ge 0\)

Active pressure: why \(K_a < 1\)

Soil left free behind a wall tends to relax sideways and push against the wall. When the wall yields slightly (rotates or translates away from the soil), the soil mass reaches the active state — that of minimum thrust. Rankine relates the horizontal pressure to the vertical one through \(K_a\), which for a sand with \(\varphi = 30°\) gives \(K_a = 1/3\): horizontally it pushes a third of what it pushes vertically.

This is the usual design state for an ordinary retaining wall, which can move enough to mobilise the active thrust (lower than the at-rest thrust, \(K_0\)).

Cohesion and the tension crack

The term \(-2c\sqrt{K_a}\) shows the beneficial effect of cohesion: a cohesive soil reduces the thrust, and near the surface the active pressure can become negative (tension). Since soil carries no tension, the pressure is capped at zero over the tension depth \(z_t\) — hence the crack that appears behind walls on clayey ground.

What the calculation includes

Surcharge \(q\) at the surface — adds a constant term \(K_a \, q\) over the whole height.
Water table — below water, the submerged (effective) unit weight is used, and the water pressure is added separately, hydrostatically. Total thrust = soil thrust (from \(\sigma'_v\)) + water pressure.

Input data

\(H\) — the wall height.
\(\gamma\), \(\gamma'\) — the soil unit weight above and below water.
\(\varphi\) — the internal friction angle, from which \(K_a\) follows.
\(c\) — the cohesion (0 for cohesionless soils).
\(q\) — the surface surcharge.
Water table and tension zone positions.

Assumptions and limitations

Rankine theory: vertical wall, horizontal backfill, no wall–soil friction. For an inclined wall, sloping backfill or back friction, Coulomb theory gives different results.
It computes the active thrust; at-rest (\(K_0\), rigid wall) or passive (\(K_p\), wall pushing into the soil) are not treated.
The analysis gives the action; wall stability (overturning, sliding, bearing capacity) is checked separately.
Seismic conditions (Mononobe-Okabe dynamic thrust) are not treated.

Winkler foundation pressures — the ground pressures at the base of a wall or column
Slope stability — the global stability of the soil mass
Stress bulb — the diffusion of stress in the ground under a footing