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GEO-05

Coulomb active wedge

Coulomb Ka for a wall with friction, backfill slope and batter.

Reading speed
Earth pressureCoulomb 1776Eurocode 7

Governing equation

Ka=cos⁡2φcos⁡δ(1+sin⁡(φ+δ)sin⁡φcos⁡δ)2K_a=\dfrac{\cos^2\varphi}{\cos\delta\left(1+\sqrt{\dfrac{\sin(\varphi+\delta)\sin\varphi}{\cos\delta}}\right)^2}

φ∈[10,45]\varphi \in \left[10, 45\right]

where

\gamma
Unit weight (kN/m³)
H
Height (m)
\varphi
Friction angle (°)
\delta
Wall friction (°)
K_a
Coulomb Ka (—)
P_a
Active thrust (kN/m)

The figure, part by part

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①

Governing law — red

Ka=cos⁡2φcos⁡δ(1+sin⁡(φ+δ)sin⁡φcos⁡δ)2K_a=\dfrac{\cos^2\varphi}{\cos\delta\left(1+\sqrt{\dfrac{\sin(\varphi+\delta)\sin\varphi}{\cos\delta}}\right)^2}

φ∈[10,45]\varphi \in \left[10, 45\right]

φ=10\varphi = 10Ka=0.614K_a = 0.614

②

Domain of the variable — blue

φmin⁡=10,φmax⁡=45\varphi_{\min} = 10,\quad \varphi_{\max} = 45

φ∈[10,45]\varphi \in \left[10, 45\right]

φ=10\varphi = 10φ=10\varphi = 10

③

Response curve — green

Ka=cos⁡2φcos⁡δ(1+sin⁡(φ+δ)sin⁡φcos⁡δ)2K_a=\dfrac{\cos^2\varphi}{\cos\delta\left(1+\sqrt{\dfrac{\sin(\varphi+\delta)\sin\varphi}{\cos\delta}}\right)^2}

φ∈[10,45]\varphi \in \left[10, 45\right]

φ=10\varphi = 10Ka=0.614K_a = 0.614

④

Rate of change — amber

dKadφ\frac{\mathrm{d}K_a}{\mathrm{d}\varphi}

φ∈[10,45]\varphi \in \left[10, 45\right]

φ=10.22\varphi = 10.22Ka′=−0.022K_a^{\prime} = -0.022

Lecture brief

Historical brief

Soil mechanics became a quantitative laboratory after Karl von Terzaghi’s 1925–1943 work on effective stress, consolidation and bearing. These sheets still size shallow foundations, retaining walls, piles and drainage in SI. This sheet (GEO-05 — Coulomb active wedge) is the form associated with Coulomb 1776 · Eurocode 7. Working symbols: γ\gamma, HH, φ\varphi, δ\delta →\rightarrow KaK_a, PaP_a. Coulomb minimised the thrust of a planar wedge sliding on a plane through the heel, including wall friction δ and backfill β.

Purpose

Purpose: compute KaK_a, PaP_a from γ\gamma, HH, φ\varphi, δ\delta in Geotechnical engineering via Ka=cos⁡2φcos⁡δ(1+sin⁡(φ+δ)sin⁡φcos⁡δ)2K_a=\dfrac{\cos^2\varphi}{\cos\delta\left(1+\sqrt{\dfrac{\sin(\varphi+\delta)\sin\varphi}{\cos\delta}}\right)^2} Coulomb Ka for a wall with friction, backfill slope and batter. Use it when a real geotechnical engineering question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given γ=18.000 kN/m3\gamma = 18.000\,\mathrm{kN/m^{3}}, H=5.000 mH = 5.000\,\mathrm{m}, φ=32.000 ∘\varphi = 32.000\,\mathrm{^{\circ}}, δ=16.000 ∘\delta = 16.000\,\mathrm{^{\circ}}, the governing relation Ka=cos⁡2φcos⁡δ(1+sin⁡(φ+δ)sin⁡φcos⁡δ)2K_a=\dfrac{\cos^2\varphi}{\cos\delta\left(1+\sqrt{\dfrac{\sin(\varphi+\delta)\sin\varphi}{\cos\delta}}\right)^2} yields Ka=0.2782 —K_a = 0.2782\,\mathrm{—}, Pa=62.584 kN/mP_a = 62.584\,\mathrm{kN/m}. Pedagogical Coulomb formula for β = 0, vertical wall (α = 0). Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Coulomb Ka K_a0.2782 —
  • Active thrust P_a62.584 kN/m
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Narration of this film

Pedagogical Coulomb formula for β = 0, vertical wall (α = 0).

Coulomb minimised the thrust of a planar wedge sliding on a plane through the heel, including wall friction δ and backfill β.

Reading speed

Watch on YouTube