Mohr's Circle and Mohr-Coulomb Failure

Theory: Mohr's Circle

How the stresses at a point of soil change with the plane, how Mohr's circle shows them, and how the Mohr-Coulomb envelope tells whether the soil fails and on which plane.

Stresses at a point

In plane strain, three values set the stresses at a point: the normal stresses σ_x and σ_y on the vertical and horizontal faces of an element and the shear τ_xy between them. From them follow the stresses on any plane through the point, turned θ from the x-plane:

\sigma_\theta = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2} \cos 2\theta - \tau_{xy} \sin 2\theta

Compression positive; θ counterclockwise from the x-plane

\tau_\theta = \frac{\sigma_x - \sigma_y}{2} \sin 2\theta + \tau_{xy} \cos 2\theta

Positive when it turns the element counterclockwise

Since soils are nearly always compressed, soil mechanics takes compression as positive, the opposite of structural mechanics.

Mohr's circle

Mohr showed in 1882 that the pairs (σ_θ, τ_θ) of all planes fall on a circle of center s and radius t:

s = \frac{\sigma_x + \sigma_y}{2}

s: mean stress in the plane · t: largest shear

The vertical and horizontal faces are the points X (σ_x, τ_xy) and Y (σ_y, −τ_xy), at the ends of a diameter. Turning the plane θ turns its point 2θ around the circle, the same way with this page's signs.

Principal stresses

\sigma_1 = s + t

σ₁, σ₃: major and minor principal stresses · θ₁: angle of the plane σ₁ acts on

The principal planes carry no shear and are square to each other. The largest shear, equal to t, acts at 45° to them.

Mohr-Coulomb criterion

Coulomb proposed in 1776 that a soil's shear strength rises with the normal stress on the failure plane. In effective stresses:

\tau_f = c' + \sigma' \tan\phi'

c′: effective cohesion · φ′: effective friction angle

Mohr joined his circle to that line: the soil fails when the circle touches the envelope, and no stress state can cross it. At failure:

\sigma'_1 = \sigma'_3 \tan^2\left(45^\circ + \frac{\phi'}{2}\right) + 2c' \tan\left(45^\circ + \frac{\phi'}{2}\right)

The same condition written with the principal stresses

This page compares the circle's radius with that of the circle of the same center tangent to the envelope, R_f = (s + c′ cot φ′) sin φ′. The ratio t / R_f is the stress level: 100% is failure.

Failure plane

The point of tangency is 90° + φ′ around the circle from σ₁'s point, so the failure plane makes 45° + φ′/2 with the plane σ₁ acts on — on either side, in a pair of conjugate planes. Before failure, these are the critical planes, whose points come closest to the envelope.

The failure plane isn't the plane of largest shear: there, the normal stress is higher, and so is the strength.

Limitations

The criterion holds in effective stresses; with total stresses, use the undrained parameters.
Plane strain leaves out the intermediate principal stress σ₂, which the Mohr-Coulomb criterion ignores too.
The real envelope is usually curved; the line holds over the range of stresses c′ and φ′ were fitted to.
At low or tensile stresses the apparent cohesion may not exist, and the soil may crack before touching the envelope.

References

Coulomb, C. A. Essai sur une application des règles de maximis et minimis à quelques problèmes de statique, relatifs à l'architecture. Mémoires de l'Académie Royale des Sciences, v. 7, 1776.
Mohr, O. Über die Darstellung des Spannungszustandes und des Deformationszustandes eines Körperelementes. Civilingenieur, v. 28, 1882.
Lambe, T. W.; Whitman, R. V. Soil Mechanics. New York: Wiley, 1969.
Das, B. M.; Sobhan, K. Principles of Geotechnical Engineering. 9th ed. Boston: Cengage, 2018.

Enter the effective stresses at a point of soil in plane strain — σ′_x, σ′_y and τ_xy — and the strength parameters c′ and φ′ to see Mohr's circle, the principal stresses, the Mohr-Coulomb failure envelope and the stresses on any plane, drawn on the soil element.

Mohr's circle

σ₁: 212.1 kPa
σ₃: 67.9 kPa
Plane of σ₁: -73.2°
τ_max: 72.1 kPa
Center, s: 140.0 kPa

Below the envelope

Stress level: 92%
Radius at failure: 78.7 kPa
Critical planes: -13.2° · 46.8°

Plane angle, θ0.0°

The element turns θ counterclockwise from the x-plane (dashed). Arrows pointing at the faces are compression; pointing away, tension.

σ′ on the plane: 80.0 kPa
τ on the plane: 40.0 kPa
Strength, c′ + σ′ tan φ′: 56.2 kPa
Factor of safety on the plane: 1.40

Stresses at the point

σ′_xkPaσ′_ykPaτ_xykPa

Compression positive. τ_xy acts on the vertical faces and is positive when it turns them counterclockwise (up on the right face).

Strength

Effective cohesion, c′kPaEffective friction angle, φ′30°

Envelope τ = c′ + σ′ tan φ′, in effective stresses.

Compression is positive; a shear is positive when the pair on opposite faces turns the element counterclockwise, and angles run counterclockwise from the x-plane (the vertical face). So a plane turned θ shows on the circle turned 2θ, the same way, with τ up. The stress level is the circle's radius over that of the circle with the same center tangent to the envelope.