Earth Pressure (Active & Passive)

The Rankine coefficients

Earth pressure theory reduces a complicated three-dimensional soil mass to a single horizontal pressure that grows linearly with depth. The lateral pressure at depth z is the vertical overburden stress, γ·z, multiplied by an earth pressure coefficient K that captures how much of the vertical load gets turned sideways:

p(z) = K · γ · z          (effective, dry backfill)

where:
  γ = soil unit weight        (kN/m³, ~18 for sand)
  z = depth below surface     (m)
  K = earth pressure coefficient (dimensionless)

Rankine coefficients (smooth wall, level backfill):
  Active:   Ka = tan²(45° − φ/2) = (1 − sin φ)/(1 + sin φ)
  Passive:  Kp = tan²(45° + φ/2) = (1 + sin φ)/(1 − sin φ)
  At rest:  K0 = 1 − sin φ            (Jaky, normally consolidated)

Note Kp = 1/Ka for the Rankine smooth-wall case.

The whole drama lives in the angle of internal friction φ. The soil can only sustain a shear stress up to τ = σ·tan φ on any plane. When the wall yields away from the backfill, the soil is free to expand horizontally; it slides along internal failure planes and mobilizes that friction to help hold itself up, so it leans on the wall with the minimum possible force — the active state. When the wall is driven into the soil, the soil is compressed laterally; now its friction resists the compression, and the wall must overcome the soil's full passive strength — the maximum possible force.

At-rest, active and passive: one soil, three answers

The same backfill can push with three very different forces depending on whether and how the wall moves. The at-rest state is the in-situ condition before any wall movement; the active and passive states are the limiting cases reached after the soil fails internally.

	At rest (K0)	Active (Ka)	Passive (Kp)
Wall movement	None	Away from soil	Into soil
Soil deformation	None	Expands laterally	Compressed laterally
Coefficient (φ = 30°)	0.50	0.33	3.0
Coefficient (φ = 35°)	0.43	0.27	3.69
Movement needed (dense sand)	0	~0.001 H	~0.01 H
Movement needed (loose sand)	0	~0.004 H	~0.05 H
Failure-plane angle from horizontal	—	45° + φ/2 (60° at φ=30°)	45° − φ/2 (30° at φ=30°)
Typical design use	Basement, braced walls	Gravity & cantilever walls	Embedded toe, anchor blocks

Two facts from this table dominate practice. First, the active state needs only about a millimetre of yield per metre of height, so almost any real wall that is not rigidly propped reaches it — design for the active case and you are usually correct. Second, the passive state needs ten to fifty times more movement, so a designer who counts on full passive resistance is betting on deflections that may be unacceptable for the structure above.

Worked example: active thrust on a gravity wall

A 5 m high gravity wall retains dry sand with φ = 30° and unit weight γ = 18 kN/m³. The wall is free to rotate slightly about its base, so the active state applies. What is the total horizontal thrust, and where does it act?

Active coefficient:
  Ka = tan²(45° − 30°/2) = tan²(30°) = 0.333

Pressure at the base (z = H = 5 m):
  p_base = Ka · γ · H = 0.333 × 18 × 5 = 30.0 kPa

Total thrust (area of the triangular pressure block, per metre of wall):
  Pa = ½ · Ka · γ · H²
     = ½ × 0.333 × 18 × 25
     = 75 kN per metre run

Point of application (centroid of the triangle):
  acts at H/3 = 5/3 = 1.67 m above the base

Overturning moment about the toe:
  M_OT = Pa × H/3 = 75 × 1.67 = 125 kN·m per metre

That 125 kN·m overturning moment must be resisted by the wall's self-weight acting through its lever arm to the toe. A typical stability check requires the resisting moment to exceed the overturning moment by a factor of at least 1.5 to 2.0. Notice how cheap the active assumption is to the soil: had the wall been rigidly braced (at-rest, K0 = 0.5), the thrust would have been 113 kN — half again as large — and a fully passive push would have reached 675 kN, nine times the active value.

Worked example: passive resistance at the toe

The same wall embeds 1.2 m into the soil in front of the toe. How much passive resistance does that embedment provide against sliding?

Passive coefficient:
  Kp = tan²(45° + 30°/2) = tan²(60°) = 3.0

Passive thrust over the 1.2 m embedded depth d:
  Pp = ½ · Kp · γ · d²
     = ½ × 3.0 × 18 × (1.2)²
     = 38.9 kN per metre run

This acts at d/3 = 0.4 m above the base of embedment.

Per metre of wall the toe embedment offers nearly 39 kN of passive push-back against sliding — comparable to half the active driving thrust — from only 1.2 m of buried depth. But because reaching full passive needs about 1% of the embedded height in movement (roughly 12 mm here), most codes apply a reduction factor or use a mobilized fraction of Kp. Many designers ignore passive resistance entirely as a conservative simplification, especially if the soil in front of the wall could be excavated for a future trench or service.

Rankine vs Coulomb: when wall friction matters

Rankine's elegant closed form assumes a smooth, vertical wall back and a horizontal backfill — no friction develops between soil and wall, and the resultant thrust is purely horizontal. Real walls have rough backs, batter, and sloping fills. Coulomb's wedge theory generalizes the problem by analysing a trial failure wedge and including the wall friction angle δ, the wall-back inclination, and the backfill slope.

• Active case: wall friction tilts the resultant downward and slightly reduces the active thrust. Coulomb-active is safe and widely used; for δ = 0 it collapses back to Rankine.
• Passive case: wall friction can dramatically inflate the predicted Kp, and the planar-wedge assumption overestimates passive resistance when δ is large. For critical passive checks engineers use curved or log-spiral failure surfaces (Caquot–Kérisel charts) instead of the simple wedge.
• Sloping backfill (angle β): Rankine extends to Ka = cos β · (cos β − √(cos²β − cos²φ)) / (cos β + √(cos²β − cos²φ)), and the thrust acts parallel to the slope rather than horizontally.
• Earthquakes: the Mononobe–Okabe method adds a horizontal seismic coefficient to Coulomb's wedge, raising the active thrust and shifting its point of application upward toward mid-height.

Water and cohesion change everything

• Water table behind the wall. Below the water table use the buoyant unit weight γ' ≈ γ − 9.81 kN/m³ for the soil pressure, then add the full hydrostatic water pressure as a separate triangular block with K = 1. Water has no shear strength, so it cannot enter the active or passive states — it simply pushes. A blocked drain that lets the water table rise to the surface can more than double the total thrust; poor drainage is the single most common cause of retaining-wall failure.
• Cohesive backfill (c > 0). The active pressure becomes p = Ka·γ·z − 2c√Ka and the passive becomes p = Kp·γ·z + 2c√Kp. The active formula predicts negative (tensile) pressure near the surface down to a critical depth z_c = 2c/(γ√Ka), creating a tension crack. Designers conservatively ignore that tension zone because cracks fill with rainwater and apply hydrostatic pressure instead.
• Compaction stress. Mechanically compacting the backfill in lifts locks in horizontal stress well above the active value near the top of the wall, sometimes approaching K0 or higher. Lightweight compaction near the wall and a designed drainage layer mitigate this.
• Surcharge. A uniform surcharge q on the surface adds a rectangular pressure block Ka·q over the full height, whose resultant acts at H/2, raising the effective point of application of the combined thrust.

How retaining walls fail

• Overturning. The active thrust, acting at H/3, rotates the wall about its toe. Resisted by self-weight; checked with a factor of safety of 1.5–2.0 on moments. Under-estimating the thrust (ignoring surcharge or water) is the usual culprit.
• Sliding. The horizontal thrust exceeds the friction available at the base, μ·N, plus any mobilized passive resistance at the toe. A shear key or a wider base increases base friction; relying on passive toe resistance that needs large movement is risky.
• Bearing failure. The overturning moment shifts the resultant toward the toe, concentrating pressure there until it exceeds the soil's bearing capacity. The heel can lift off and the wall punches into the foundation.
• Global (slope) instability. A deep failure surface passes beneath the entire wall and footing through weak soil — a slope-stability problem the wall design alone cannot prevent.
• Hydrostatic build-up. Clogged weep holes or filter fabric let pore water accumulate; the added water pressure (K = 1) overwhelms a wall sized for drained active pressure. Almost always preventable with drainage.
• Excessive deflection. A flexible sheet-pile or anchored wall may not fail structurally yet deflect enough to damage adjacent structures or utilities — a serviceability limit that can govern over strength.