MAT-08

Mohr–Coulomb failure

τ = c + σ tan φ on the failure plane.

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FailureCoulomb 1773Mohr

Governing equation

\tau = c + \sigma\tan\varphi

$\sigma \in \left[0, 200\right]$

where

c: Cohesion (MPa)
\sigma: Normal stress (MPa)
\varphi: Friction angle (°)
\tau: Shear strength (MPa)

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

\tau = c + \sigma\tan\varphi

$\sigma \in \left[0, 200\right]$

$\sigma = 0$ $\tau = 5$

Along the curve

②

Domain of the variable — blue

\sigma_{\min} = 0,\quad \sigma_{\max} = 200

$\sigma \in \left[0, 200\right]$

$\sigma = 0$ $\sigma = 0$

Along the curve

③

Response curve — green

\tau = c + \sigma\tan\varphi

$\sigma \in \left[0, 200\right]$

$\sigma = 0$ $\tau = 5$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\tau}{\mathrm{d}\sigma}

$\sigma \in \left[0, 200\right]$

$\sigma = 1.258$ $\tau^{\prime} = 0.7$

Along the curve

Lecture brief

Historical brief

Hooke (1678), Hall–Petch grain size, Arrhenius activation and Paris fatigue-crack growth are the spine of materials selection. The sheets relate stress, microstructure and life. This sheet (MAT-08 — Mohr–Coulomb failure) is the form associated with Coulomb 1773 · Mohr. Working symbols: $c$ , $\sigma$ , $\varphi$ $\rightarrow$ $\tau$ . Coulomb's friction law on a plane, written in Mohr's stress space, is the classical failure envelope of rock, soil and concrete.

Purpose

Purpose: compute

\tau

from

c

\sigma

\varphi

in Materials via

\tau = c + \sigma\tan\varphi

τ = c + σ tan φ on the failure plane. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

c = 5.000\,\mathrm{MPa}

\sigma = 20.000\,\mathrm{MPa}

\varphi = 35.000\,\mathrm{^{\circ}}

, the governing relation

\tau = c + \sigma\tan\varphi

yields

\tau = 19.004\,\mathrm{MPa}

. 2-D, compression positive as entered (geomechanics sign). Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Cohesion c5 MPaNormal stress \sigma20 MPaFriction angle \varphi35 °

Outputs

Shear strength \tau19.004 MPa

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Narration of this film

2-D, compression positive as entered (geomechanics sign).

Coulomb's friction law on a plane, written in Mohr's stress space, is the classical failure envelope of rock, soil and concrete.

Reading speed

Watch on YouTube