Lateral earth pressure is the horizontal stress exerted by soil against retaining structures such as walls, basements, and bridge abutments.
Lateral earth pressure depends on three key factors:
- Wall movement — whether the wall moves away from, towards, or remains stationary relative to the soil
- Soil properties — unit weight, shear strength (c', φ'), and drainage conditions
- Groundwater conditions — hydrostatic pressure adds to total lateral force
The Three States
| State | Wall Movement | Earth Pressure | Magnitude |
|---|---|---|---|
| At-rest ($K_0$) | No movement | Intermediate | $K_0 \gamma H$ |
| Active ($K_a$) | Wall moves away from soil | Minimum | $K_a \gamma H$ (lowest) |
| Passive ($K_p$) | Wall moves into the soil | Maximum | $K_p \gamma H$ (highest) |
Wall movement
│
Active ◄────── │ ──────► Passive
(Wall pulls away) │ (Wall pushes in)
│
At-Rest
(No movement)
At-Rest Earth Pressure ($K_0$)
Definition
The lateral stress in the ground when no lateral strain occurs — the natural "resting" pressure.
Determination
Jaky's formula (normally consolidated):
$$ K_0 = 1 - \sin\phi' $$Overconsolidated soils:
$$ K_{0(OC)} = K_{0(NC)} \times OCR^{m} $$Where $m \approx \sin\phi'$ (Mayne & Kulhawy)
| OCR | $K_0$ (for $\phi' = 30°$) |
|---|---|
| 1 (NC) | 0.5 |
| 2 | 0.66 |
| 5 | 1.03 |
| 10 | 1.46 |
At-rest pressure distribution:
$$ \sigma'_h = K_0 \sigma'_v = K_0 \gamma z \quad \text{(above water table)} $$ $$ \sigma'_h = K_0 (\gamma z - u) \quad \text{(below water table)} $$Total lateral pressure = effective lateral pressure + pore water pressure
Rankine's Earth Pressure Theory
Rankine's theory assumes:
- Smooth, vertical wall (no wall friction)
- Horizontal ground surface
- Soil is homogeneous and isotropic
- Failure occurs along a plane
Active Earth Pressure ($K_a$)
Coefficient of active earth pressure:
$$ K_a = \frac{1 - \sin\phi'}{1 + \sin\phi'} = \tan^2\left(45° - \frac{\phi'}{2}\right) $$Drained condition (c' = 0):
$$ \sigma'_a = K_a \sigma'_v = K_a \gamma' z $$Drained condition (c' > 0):
$$ \sigma'_a = K_a \sigma'_v - 2c'\sqrt{K_a} $$Undrained condition ($\phi_u = 0$, saturated clay):
$$ \sigma_a = \sigma_v - 2s_u $$Tension crack depth:
$$ z_c = \frac{2c'}{\gamma\sqrt{K_a}} $$Passive Earth Pressure ($K_p$)
Coefficient of passive earth pressure:
$$ K_p = \frac{1 + \sin\phi'}{1 - \sin\phi'} = \tan^2\left(45° + \frac{\phi'}{2}\right) $$Note: $K_p = 1/K_a$
Drained condition (c' = 0):
$$ \sigma'_p = K_p \sigma'_v $$Drained condition (c' > 0):
$$ \sigma'_p = K_p \sigma'_v + 2c'\sqrt{K_p} $$Undrained condition:
$$ \sigma_p = \sigma_v + 2s_u $$$K_a$ and $K_p$ Values
| $\phi'$ (°) | $K_a$ | $K_p$ |
|---|---|---|
| 20 | 0.49 | 2.04 |
| 25 | 0.41 | 2.46 |
| 30 | 0.33 | 3.00 |
| 32 | 0.31 | 3.25 |
| 34 | 0.28 | 3.54 |
| 36 | 0.26 | 3.85 |
| 38 | 0.24 | 4.20 |
| 40 | 0.22 | 4.60 |
| 45 | 0.17 | 5.83 |
Rankine Theory — Sloping Ground and Sloping Wall
Active — sloping ground ($\beta$ = slope angle):
$$ K_a = \cos\beta \frac{\cos\beta - \sqrt{\cos^2\beta - \cos^2\phi'}}{\cos\beta + \sqrt{\cos^2\beta - \cos^2\phi'}} $$Passive — sloping ground ($\beta$ = slope angle):
$$ K_p = \cos\beta \frac{\cos\beta + \sqrt{\cos^2\beta - \cos^2\phi'}}{\cos\beta - \sqrt{\cos^2\beta - \cos^2\phi'}} $$Coulomb's Wedge Theory
Coulomb's theory accounts for:
- Wall friction ($\delta$)
- Wall inclination ($\alpha$)
- Sloping backfill ($\beta$)
- Any soil type (c, φ)
Coulomb Active Pressure
$$ K_a = \frac{\sin^2(\alpha + \phi')}{\sin^2\alpha \sin(\alpha - \delta) \left[1 + \sqrt{\frac{\sin(\phi' + \delta)\sin(\phi' - \beta)}{\sin(\alpha - \delta)\sin(\alpha + \beta)}}\right]^2} $$Coulomb Passive Pressure
$$ K_p = \frac{\sin^2(\alpha - \phi')}{\sin^2\alpha \sin(\alpha + \delta) \left[1 - \sqrt{\frac{\sin(\phi' + \delta)\sin(\phi' + \beta)}{\sin(\alpha + \delta)\sin(\alpha + \beta)}}\right]^2} $$Wall Friction Angle ($\delta$)
| Wall Surface Material | $\delta$ |
|---|---|
| Smooth concrete | $\frac{1}{3}\phi'$ to $\frac{1}{2}\phi'$ |
| Rough concrete | $\frac{1}{2}\phi'$ to $\frac{2}{3}\phi'$ |
| Cast-in-place concrete | $\frac{2}{3}\phi'$ |
| Steel sheet pile | 20° (maximum) |
Effect of Wall Friction
| Condition | Active $P_a$ | Passive $P_p$ |
|---|---|---|
| Smooth wall ($\delta = 0$) | Higher | Lower |
| Rough wall ($\delta > 0$) | Lower | Higher |
| For $\phi' = 30°$, $\alpha = 90°$ | ||
| $\delta = 0$ | $K_a = 0.33$ | $K_p = 3.00$ |
| $\delta = 20°$ | $K_a = 0.30$ | $K_p = 4.98$ |
| $\delta = 30°$ | $K_a = 0.28$ | $K_p = 8.91$ |
Lateral Earth Pressure Distributions
Basic Triangular Distribution (Homogeneous Soil)
Active: Passive:
┌──┐ ┌──┐
│ │ │ │
│ │ KaγH │ │ KpγH
│ │ │ │
│ │◄───── │ │─────►
H │ │ │ │
│ │ │ │
│ │ │ │
│ │▼ Pa = ½KaγH² │ │▲ Pp = ½KpγH²
└──┘ └──┘
Resultant force location: $H/3$ from the base (for uniform soil, no surcharge)
Soil with Cohesion + Surcharge
Active pressure at depth z:
σ'_a(z) = K_a(q + γz) - 2c'√K_a
Tension zone: where σ'_a < 0 (tension cracks)
Total force: Pa = ½K_aγH² + K_aqH - 2c'H√K_a
The tension zone (typically the top 1–2 m for clay) is usually ignored in design — the soil above the tension crack depth cannot sustain tension.
Layered Soils
For layered soil profiles, compute the lateral pressure increment for each layer using the appropriate $K_a$ or $K_p$ value:
| Layer | Depth | $\gamma$ | $\phi'$ | $K_a$ |
|---|---|---|---|---|
| 1: Sand fill | 0–2 m | 18 kN/m³ | 32° | 0.31 |
| 2: Clay | 2–5 m | 17 kN/m³ | — | $s_u$ = 40 kPa |
Total force: Sum of trapezoidal or triangular areas from each layer.
Surcharge Loads
| Load Type | Lateral Pressure |
|---|---|
| Uniform surcharge ($q$) | $\Delta\sigma_h = K_a q$ (constant with depth) |
| Line load ($Q$ per m) | Varies with depth per Boussinesq distribution |
| Strip load ($q \times B$) | Varies — use elastic solution charts |
Effect of Groundwater
Hydrostatic Pressure
Water pressure is always perpendicular to the wall surface and acts in addition to effective earth pressure:
$$ u = \gamma_w h_w $$Where $h_w$ = height of water against the wall.
Total Lateral Force (with groundwater)
h₁ dry ─┬─────────────
│ σ'_h = K_a γ z
│
h₂ ──────┼─── Water table
│ σ'_h = K_a (γ_z) for z > h₁
│ u = γ_w(z - h₁)
│
Total at base:
σ_h = K_a(γh₁ + γ'h₂) + γ_w h₂
Key point: Water pressure typically accounts for 40–60% of total lateral force below the water table — proper drainage is critical for economical retaining wall design.
Drainage Effects
| Drainage Condition | Lateral Pressure |
|---|---|
| Fully drained (weep holes, drainage blanket) | No hydrostatic component |
| Partially drained | Reduced hydrostatic component |
| No drainage | Full hydrostatic + effective earth pressure |
| Hydrostatic only (no soil — submerged wall) | $P = \frac{1}{2}\gamma_w H^2$ |
Earth Pressure due to Compaction
Compaction near retaining walls induces residual lateral stresses that can exceed active pressure:
$$ \sigma'_h(\text{compaction}) = \frac{\gamma r}{(1 - \sin\phi')} \sqrt{\frac{2r}{3}} $$Where $r$ = roller drum radius and $\gamma$ = soil unit weight.
Practical approach: For design, the compaction-induced lateral pressure is typically taken as at least $K_0$ or the active pressure, whichever is higher. Many codes specify a minimum equivalent fluid pressure.
Special Cases
Seismic Earth Pressure (Mononobe-Okabe)
Pseudo-static analysis for earthquake conditions:
$$ K_{AE} = \frac{\sin^2(\alpha + \phi' - \theta)}{\cos\theta \sin^2\alpha \sin(\alpha - \theta - \delta)\left[1 + \sqrt{\frac{\sin(\phi' + \delta)\sin(\phi' - \beta - \theta)}{\sin(\alpha - \delta - \theta)\sin(\alpha + \beta)}}\right]^2} $$Where $\theta = \tan^{-1}\left(\frac{k_h}{1 - k_v}\right)$, with $k_h$ = horizontal seismic coefficient.
Total seismic lateral force:
$$ P_{AE} = P_A + \Delta P_{AE} $$The seismic increment increases total force by approximately 15–40% depending on seismicity.
Braced Excavations (Trapezoidal Pressure)
For braced cuts, the apparent earth pressure distribution differs from the triangular Rankine distribution:
Soft to medium clay: $\sigma_h = K_a \gamma H$ to $0.3\gamma H$ (Terzaghi & Peck)
Stiff clay: $\sigma_h = 0.2\gamma H$ to $0.4\gamma H$
Sand: $\sigma_h = 0.65K_a \gamma H$ to $0.75K_a\gamma H$
Negative Wall Friction (Downdrag)
If the wall settles relative to the soil (e.g., pile in consolidating soil), negative skin friction adds a downward load. This typically increases lateral earth pressure and must be considered for piles adjacent to fill or in settling ground.
Design Checklist
For Retaining Wall Design
| Item | Check |
|---|---|
| Determine wall type (gravity, cantilever, anchored, etc.) | — |
| Drainage provision (weep holes, drainage blanket, filter) | — |
| Select appropriate earth pressure theory | Rankine or Coulomb |
| Select wall friction angle ($\delta$) | — |
| Apply appropriate factor of safety | — |
| Check water table effects | — |
| Consider surcharge loads | — |
| Check seismic conditions | — |
| Check compaction effects | — |
| Verify sliding, overturning, bearing, and global stability | — |
Typical Factors of Safety (AS 4678)
| Failure Mode | FS Required |
|---|---|
| Sliding | 1.5 |
| Overturning | 2.0 |
| Bearing capacity | 2.5 |
| Overall stability | 1.3–1.5 |
Australian Standards
| Standard | Title | Application |
|---|---|---|
| AS 4678 | Earth Retaining Structures | Design of retaining walls |
| AS 4678-2002 | (replaced by 2023 in some states) | Active/passive pressure, drainage, seismic |
| AS 5100.3 | Bridge Design — Foundations and Substructures | Abutment earth pressure |
| AS 1170.4 | Structural Design Actions — Earthquake | Seismic coefficient $k_h$ |
| TfNSW QA Specification B80 | Earth Retaining Structures | NSW Roads and Maritime |
| AS 2159 | Piling | Lateral load on piles |
Worked Example: Gravity Retaining Wall
Given:
- Wall height $H = 5.0$ m
- Backfill: granular, $\gamma = 18$ kN/m³, $\phi' = 32°$
- Wall friction: $\delta = 16°$ ($\frac{1}{2}\phi'$)
- No water table, no surcharge
- Vertical wall, horizontal backfill
Step 1 — Active pressure coefficient (Coulomb):
$K_a = 0.307$ (approximately, for $\phi' = 32°$, $\delta = 16°$, vertical wall)Step 2 — Total active force:
$P_a = \frac{1}{2} \times 0.307 \times 18 \times 5.0^2 = 69.1$ kN/mStep 3 — Horizontal and vertical components:
$P_{ah} = P_a \cos(\delta) = 69.1 \times \cos(16°) = 66.4$ kN/m $P_{av} = P_a \sin(\delta) = 69.1 \times \sin(16°) = 19.0$ kN/mStep 4 — Point of application:
$H/3 = 1.67$ m above the base