GEO-14

Rankine active earth pressure

Ka = tan²(45°−φ′/2), σ′a = Ka σ′v − 2c√Ka.

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Earth pressureRankine

Governing equation

K_a=\tan^2(45^\circ-\varphi'/2),\quad \sigma'_a=K_a\sigma'_v-2c\sqrt{K_a}

where

\phi': Friction angle (°)
\sigma'_v: Vertical effective (kPa)
c': Cohesion (kPa)
K_a: Active coefficient (—)
\sigma'_a: Active stress (kPa)

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-14 — Rankine active earth pressure) is the form associated with Rankine. Working symbols: $\phi'$ , $\sigma'_v$ , $c'$ $\rightarrow$ $K_a$ , $\sigma'_a$ . A conjugate Mohr circle touching the failure envelope. Wall friction is neglected.

Purpose

Purpose: compute

K_a

\sigma'_a

from

\phi'

\sigma'_v

c'

in Geotechnical engineering via

K_a=\tan^2(45^\circ-\varphi'/2),\quad \sigma'_a=K_a\sigma'_v-2c\sqrt{K_a}

Ka = tan²(45°−φ′/2), σ′a = Ka σ′v − 2c√Ka. 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

\phi' = 30.000\,\mathrm{^{\circ}}

\sigma'_v = 50.000\,\mathrm{kPa}

c' = 5.000\,\mathrm{kPa}

, the governing relation

K_a=\tan^2(45^\circ-\varphi'/2),\quad \sigma'_a=K_a\sigma'_v-2c\sqrt{K_a}

yields

K_a = 0.333\,\mathrm{—}

\sigma'_a = 10.89\,\mathrm{kPa}

. A wall, a sliding wedge, a horizontal thrust. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Friction angle \phi'30 °Vertical effective \sigma'_v50 kPaCohesion c'5 kPa

Outputs

Active coefficient K_a0.333 —
Active stress \sigma'_a10.89 kPa

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Narration of this film

A wall, a sliding wedge, a horizontal thrust.

A conjugate Mohr circle touching the failure envelope. Wall friction is neglected.

Reading speed

Watch on YouTube