Lateral Earth Pressure on Retaining Walls

Theory: Earth Pressure

Where the lateral pressure on a wall comes from, how the state at rest and Rankine's and Coulomb's active and passive states differ, and how cohesion, water and a surcharge enter them.

At rest, active and passive

The horizontal stress in the ground is a fraction K of the vertical effective stress. A wall that doesn't move keeps it at rest; one that yields away from the soil lets it drop to the active state, the least the soil can bear before it fails; one pushed into the soil raises it to the passive state, the most. Little movement brings the active state, about 0.1–0.4% of the height in sand; the passive state needs ten times as much or more.

\sigma'_h = K\, \sigma'_v

σ′_v: vertical effective stress, γz + q above the water table

At rest

Jaky (1944) related K₀ to the friction angle of a normally consolidated soil. Overconsolidation raises it, as Mayne and Kulhawy (1982) found, and EN 1997-1 raises it again under a backfill sloping up from the wall.

K_0 = (1 - \sin\phi')\, \mathrm{OCR}^{\sin\phi'} (1 + \sin\beta)

Used on walls that can't yield: basement walls, propped or very stiff walls

Rankine

Rankine (1857) took the soil behind a smooth vertical back to fail all through, the stress on any vertical plane following from Mohr's circle at failure. In a level backfill:

\sigma'_a = K_a \sigma'_v - 2c' \sqrt{K_a}

K_a = tan²(45° − φ′/2) · K_p = tan²(45° + φ′/2) = 1/K_a

Cohesion puts the top of the active zone in tension down to z_c, where a crack opens; the thrust is taken from the stress below it. In a backfill sloping at β, the stress on a vertical plane is parallel to the ground:

K_{a,p} = \cos\beta\, \frac{\cos\beta \mp \sqrt{\cos^2\beta - \cos^2\phi'}}{\cos\beta \pm \sqrt{\cos^2\beta - \cos^2\phi'}}

For c′ > 0 and a slope, Mazindrani and Ganjali (1997) worked out the stress, which takes the forms above in level ground

Coulomb

Coulomb (1776) looked at a wedge of soil sliding on a plane through the heel, with friction on the wall's back. The active thrust is the largest any wedge needs; the passive, the least. For a back at θ from the vertical, wall friction δ and a backfill at β:

K_a = \frac{\cos^2(\phi' - \theta)}{\cos^2\theta \cos(\delta + \theta) \left[1 + \sqrt{\dfrac{\sin(\delta + \phi') \sin(\phi' - \beta)}{\cos(\delta + \theta) \cos(\theta - \beta)}}\right]^2}

P_a = ½K_aγH², at δ to the normal of the back

K_p = \frac{\cos^2(\phi' + \theta)}{\cos^2\theta \cos(\delta - \theta) \left[1 - \sqrt{\dfrac{\sin(\phi' + \delta) \sin(\phi' + \beta)}{\cos(\delta - \theta) \cos(\theta - \beta)}}\right]^2}

P_p = ½K_pγH²

With θ = δ = β = 0 Coulomb's K_a and K_p are Rankine's. His plane wedge overestimates the passive thrust once δ passes about φ′/3, the real surface being curved; a log-spiral solution or Rankine's value is then safer.

Water and surcharge

Below the water table the coefficients act on the buoyant stress, γ′ = γ_sat − γ_w, and the full water pressure u = γ_w(z − D_w) is added: water has no K.
A uniform surcharge q adds q to σ′_v at every depth, a rectangle Kq on the pressure diagram. On Coulomb's inclined back it enters as Kq cos θ cos β / cos(θ − β).

Thrust and its line of action

P_h = \int \sigma_h \, dz

Per metre of wall; ȳ is H/3 for a triangle, H/2 for a rectangle

The horizontal thrust and its height above the base give the overturning moment on the wall; the vertical component, from the wall friction and a sloping or inclined back, helps it stand.

References

Coulomb, C. A. Essai sur une application des règles de maximis et minimis à quelques problèmes de statique relatifs à l'architecture. Mémoires de l'Académie Royale des Sciences, v. 7, p. 343–382, 1776.
Rankine, W. J. M. On the stability of loose earth. Philosophical Transactions of the Royal Society of London, v. 147, p. 9–27, 1857.
Jaky, J. The coefficient of earth pressure at rest. Journal of the Society of Hungarian Architects and Engineers, v. 78, n. 22, p. 355–358, 1944.
Mayne, P. W.; Kulhawy, F. H. K0–OCR relationships in soil. Journal of the Geotechnical Engineering Division, ASCE, v. 108, n. GT6, p. 851–872, 1982.
Mazindrani, Z. H.; Ganjali, M. H. Lateral earth pressure problem of cohesive backfill with inclined surface. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, v. 123, n. 2, p. 110–112, 1997.
EN 1997-1. Eurocode 7: Geotechnical design – Part 1: General rules. Brussels: CEN, 2004.
Das, B. M.; Sobhan, K. Principles of Geotechnical Engineering. 9th ed. Boston: Cengage, 2018.

Enter the wall — height, the back's inclination and the wall friction — and the backfill's slope, surcharge, c′, φ′ and unit weights to get the lateral pressure at rest, active and passive by Rankine and Coulomb, with each state's thrust and where it acts.

Wall

Height, HmBack's inclination, θ0°Wall friction, δ20°

θ is the back's angle from the vertical, positive when it leans away from the backfill, as on a gravity wall. δ is often taken as ⅔φ′.

Backfill

φ′30°Slope, β0°Surcharge, qkPac′kPaγ (above WT)kN/m³γ_sat (below WT)kN/m³Water table, D_wmγ_wkN/m³OCR

D_w is measured from the top of the wall; leave it blank for a dry backfill. OCR only enters K₀.

Rankine and at rest take a smooth vertical back; θ and δ only enter Coulomb.

With δ above about φ′/3, Coulomb's plane wedge overestimates the passive thrust; Rankine's is the safer figure.

Homogeneous backfill; the coefficients act on the effective vertical stress, and the water's pressure is added to it, hydrostatic below the water table.
At rest is Jaky's K₀ = 1 − sin φ′, times OCR^{sin φ′} (Mayne and Kulhawy) and 1 + sin β for a sloping backfill (EN 1997-1), acting horizontally.
Rankine takes a smooth vertical back, the stress parallel to the ground; c′ enters after Mazindrani and Ganjali, and the active tension above the crack is left out of the thrust.
Coulomb's wedges take a cohesionless backfill, the thrust at δ to the back's normal; the surcharge enters as Kq cos θ cos β / cos(θ − β).
Thrusts per metre of wall; P_v is positive down on the wall, and ȳ is the height of P_h's line of action above the base.
The 3D is an illustration: the wall's front and base are the section's gravity wall, and its weep holes (φ100 mm) show it as built but aren't designed here.