GEO-38

Rankine tension-crack depth

zc = 2c /(γ √Ka). Active pressure is zero down to zc; a crack may open.

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Earth pressureRankine tension crack

Governing equation

z_c=\dfrac{2c}{\gamma\sqrt{K_a}}

$c' \in \left[1, 60\right]$

where

c': Cohesion (kPa)
\gamma: Unit weight (kN/m³)
K_a: Active Ka (—)
z_c: Crack depth (m)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

z_c=\dfrac{2c}{\gamma\sqrt{K_a}}

$c' \in \left[1, 60\right]$

$c' = 1$ $z_c = 0.193$

Along the curve

②

Domain of the variable — blue

c'_{\min} = 1,\quad c'_{\max} = 60

$c' \in \left[1, 60\right]$

$c' = 1$ $c' = 1$

Along the curve

③

Response curve — green

z_c=\dfrac{2c}{\gamma\sqrt{K_a}}

$c' \in \left[1, 60\right]$

$c' = 1$ $z_c = 0.193$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}z_c}{\mathrm{d}c'}

$c' \in \left[1, 60\right]$

$c' = 1.371$ $z_c^{\prime} = 0.193$

Along the curve

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-38 — Rankine tension-crack depth) is the form associated with Rankine tension crack. Working symbols: $c'$ , $\gamma$ , $K_a$ $\rightarrow$ $z_c$ . σ′a = Ka σ′v − 2c √Ka vanishes at zc. For φu = 0, Ka = 1 and zc = 2c/γ.

Purpose

Purpose: compute

z_c

from

c'

\gamma

K_a

in Geotechnical engineering via

z_c=\dfrac{2c}{\gamma\sqrt{K_a}}

zc = 2c /(γ √Ka). Active pressure is zero down to zc; a crack may open. 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

c' = 15.000\,\mathrm{kPa}

\gamma = 18.000\,\mathrm{kN/m^{3}}

K_a = 0.330\,\mathrm{—}

, the governing relation

z_c=\dfrac{2c}{\gamma\sqrt{K_a}}

yields

z_c = 2.90\,\mathrm{m}

. A wall, a dry crack, a shortened thrust diagram. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Cohesion c'15 kPaUnit weight \gamma18 kN/m³Active Ka K_a0.33 —

Outputs

Crack depth z_c2.90 m

Reading speed

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Narration of this film

A wall, a dry crack, a shortened thrust diagram.

σ′a = Ka σ′v − 2c √Ka vanishes at zc. For φu = 0, Ka = 1 and zc = 2c/γ.

Reading speed

Watch on YouTube