Earth Pressure

Rankine Theory · Active and Passive Pressure · Retaining Walls · Basement Walls

Earth Pressure Calculator

Select Calculation:

Active Pressure Coefficient k_a

Passive Pressure Coefficient k_p

Active Earth Pressure e_a

Passive Earth Pressure e_p

At-Rest Pressure e₀

Friction Angle φ [°] Soil friction angle; 0° = non-cohesive; 30–40° = typical; max ca. 45°

Unit Weight γ [kN/m³] Typically 15–22 kN/m³ depending on soil type and moisture

Depth z [m] Depth below ground surface

Formulas & Fundamentals

Active Pressure Coefficient (Rankine):
k_a = tan²(45° − φ/2)
Pressure coefficient for horizontally exposed surface

Passive Pressure Coefficient (Rankine):
k_p = tan²(45° + φ/2)
Resistance coefficient under compression; k_p >> k_a

Active Earth Pressure:
e_a = k_a · γ · z [kN/m²]
Horizontal pressure on retaining wall (shear stress)

Passive Earth Pressure:
e_p = k_p · γ · z [kN/m²]
Earth resistance under compression (piles, beams)

At-Rest Pressure:
e₀ = k₀ · γ · z [kN/m²]
Lateral pressure without deformation; k₀ ≈ 1 − sin(φ)

Typical φ Values

Soil Type	φ	k_a	Usage
Loose sand	25–30°	0.40–0.33	simple
Dense sand	30–35°	0.33–0.27	typical
Gravel	35–40°	0.27–0.22	stable
Silt	30–35°	0.33–0.27	medium
Stiff clay	20–30°	0.49–0.33	favorable

Earth Pressure Calculation Using Rankine Theory

What is Earth Pressure?

Earth pressure is the lateral stress that a soil mass exerts on a wall or structure. Rankine Theory is a classical approximation for calculating active pressure (pushing), passive pressure (resistance), and at-rest pressure based on friction angle and depth.

e = k · γ · z
Earth pressure is proportional to the pressure coefficient, unit weight, and depth.

Active Pressure (e_a)

Soil mass push: Soil pushes against wall
k_a = tan²(45° − φ/2)

Example φ = 30°:
k_a ≈ 0.333

Passive Pressure (e_p)

Earth resistance: Soil resists penetration
k_p = tan²(45° + φ/2)

Example φ = 30°:
k_p ≈ 3.0

Applications: Retaining Walls, Basement Walls, Piles

Retaining Walls

Slope support and retention function. Active pressure (e_a) is critical. Wall must resist shear and overturning.

Basement Walls & Excavations

Underground enclosure. Active pressure on exterior, interior free.

Piles & Beams

Passive pressure (e_p) is favorable (high resistance). Important for bearing capacity and deflection.

Example: Retaining Wall in Sand

Task:

Retaining wall in dense sand φ = 32°, unit weight γ = 18 kN/m³. Wall height 6 m. Determine earth pressure distribution at 3 m depth.

Solution:

k_a = tan²(45° − 32°/2) = tan²(29°) ≈ 0.295
e_a(3 m) = 0.295 · 18 · 3 = 15.93 kN/m²
e_a(6 m) = 0.295 · 18 · 6 = 31.86 kN/m²
Total earth pressure resultant ≈ triangular distribution, Area = 0.5 · 6 · 31.86 ≈ 95.6 kN/m

Frequently Asked Questions

The passive pressure coefficient k_p >> k_a because soil must "compact" against the penetration. This requires shearing in the opposite direction and mobilizes all friction forces. In active pressure, soil deforms away, so friction helps less.

Groundwater: Pressure increases by water pressure e_w = γ_w · z. Add this to shear stress from soil friction separately.
Capillarity: Can create minor suction tension → temporary cohesion, beneficial for active pressure. Rankine without capillary/cohesion is conservative (unfavorable).

At-rest pressure e₀ is the lateral stress when the wall does not deform (elastic condition). Rule of thumb: k₀ ≈ 1 − sin(φ). For sand with φ ≈ 30–35°, k₀ ≈ 0.4–0.5, distinctly between active and passive.

Rankine is an approximation for cohesionless soils (sand, gravel). For cohesive clays with cohesion c, Rankine is conservative (overestimates active pressure). Better: Coulomb theory or textbook corrections with c-terms.

The pressure distribution is linear (linear elastic): e = k·γ·z. The resultant for a wall of height h is the triangular area:
E = 0.5 · k · γ · h²

This force acts at h/3 depth (triangle centroid). Used for stability checks (overturning, sliding).

Summary

Pressure Coefficients

k_a, k_p, k₀
from φ

Earth Pressures

e [kN/m²]
at any depth

Stability Check

Overturning, sliding
wall reinforcement

Typical Applications

Retaining Walls: Design wall thickness and reinforcement against active pressure
Basement Walls: Exterior wall pressure calculation, horizontal loads from soil
Shoring & Excavations: Sheet piles, braced cuts, bracing in deep foundations
Piles & Beams: Passive pressure resistance for bearing capacity
Slopes: Passive pressure when pushing slope (counterfort dams)
Shafts & Tunnels: Tunnel pressure distribution (simplified Rankine)

Is this page helpful?

Thank you for your feedback!

Sorry about that
How can we improve it?