---
title: "Derive Mohr-Coulomb principal stress relation"
subject: "Soil Mechanics"
chapter: 10
key: "semester_4/soil_mechanics:mohr-coulomb-derive-principal-stresses"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/mohr-coulomb-derive-principal-stresses
site: BCT Engineering
---

# Derive Mohr-Coulomb principal stress relation

Soil Mechanics · Chapter 10: Shear Strength of Soil

- Asked in: 2078 Chaitra, 1+3 marks; 2073 Magh, 7 marks; 2074 Bhadra, 5 marks; 2076 Baisakh, 4 marks (asked 4 times)
- Video: [Mohr Coulomb's Theory of Shear Strength In Hindi | Soil Mechanics](https://www.youtube.com/watch?v=HUY8I6v6Ztk) — Chaityendra Singh
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/mohr-coulomb-derive-principal-stresses
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-10

## Question

State Mohr's failure theory and derive the Mohr-Coulomb equation, i.e. the relation between the major and minor principal stresses at failure, cohesion and angle of internal friction (draw Mohr's circle with the Mohr-Coulomb failure line).

## Short answer

Mohr's theory: failure when shear stress reaches a function of normal stress. Mohr-Coulomb: τ_f = c + σ tan φ. From Mohr circle touching failure line: sin φ = (σ1-σ3)/(σ1+σ3+2c cot φ), so σ1 = σ3 tan^2(45°+φ/2) + 2c tan(45°+φ/2); failure plane at 45°+φ/2.

## Answer

#### Mohr's failure theory

Mohr's theory states that a material fails when the shear stress on a plane reaches a limiting value that depends on the normal stress on that plane. Failure is therefore governed by a combination of normal and shear stress, not by the maximum stress alone:

$$
\tau_f = f(\sigma)
$$

The curve of $\tau_f$ against $\sigma$ is the **failure envelope**. Coulomb approximated it for soils by a straight line, giving the **Mohr-Coulomb criterion**:

$$
\tau_f = c + \sigma\tan\phi \qquad \text{(effective stress: } \tau_f = c' + \sigma'\tan\phi'\text{)}
$$

Here $c$ is cohesion and $\phi$ is the angle of internal friction. A stress state is safe if its Mohr circle lies below the line, and the soil fails when the circle touches it.

#### Derivation of the relation between $\sigma_1$ and $\sigma_3$

```text
  tau |          . failure line
      |        .   tau = c + sigma tan(phi)
      |   _.-'-._.
      | .'   |   '.
      |/  phi|  C   \
 -----+-----+-+---+---+------ sigma
      c cot(phi) O   (s3)  (s1)
```

Consider the Mohr circle at failure, touching the failure line at point $T$. Its centre $C$ is at $\frac{\sigma_1+\sigma_3}{2}$ and its radius is $\frac{\sigma_1-\sigma_3}{2}$. The failure line meets the $\sigma$-axis at $O'$, a distance $c\cot\phi$ to the left of the origin. In the right-angled triangle $O'TC$, the angle at $O'$ is $\phi$ and $CT \perp O'T$:

$$
\sin\phi = \frac{CT}{O'C} = \frac{\dfrac{\sigma_1-\sigma_3}{2}}{c\cot\phi + \dfrac{\sigma_1+\sigma_3}{2}}
$$

$$
\sigma_1 - \sigma_3 = (\sigma_1+\sigma_3)\sin\phi + 2c\cos\phi
$$

$$
\sigma_1(1-\sin\phi) = \sigma_3(1+\sin\phi) + 2c\cos\phi
$$

$$
\boxed{\sigma_1 = \sigma_3\,\frac{1+\sin\phi}{1-\sin\phi} + 2c\,\frac{\cos\phi}{1-\sin\phi}}
$$

Using $\frac{1+\sin\phi}{1-\sin\phi} = \tan^2\left(45^\circ+\frac{\phi}{2}\right)$ and $\frac{\cos\phi}{1-\sin\phi} = \tan\left(45^\circ+\frac{\phi}{2}\right)$:

$$
\sigma_1 = \sigma_3\tan^2\left(45^\circ+\frac{\phi}{2}\right) + 2c\tan\left(45^\circ+\frac{\phi}{2}\right)
$$

Equivalent form: $\sigma_3 = \sigma_1\tan^2\left(45^\circ-\frac{\phi}{2}\right) - 2c\tan\left(45^\circ-\frac{\phi}{2}\right)$.

**Failure plane angle:** in triangle $O'TC$ the angle $\angle TCO'$ is $90^\circ-\phi$. The radius $CT$ therefore makes an angle $2\theta_f = 180^\circ - (90^\circ-\phi) = 90^\circ+\phi$ with the radius to the $\sigma_1$ point, so the failure plane is inclined at $\theta_f = 45^\circ + \frac{\phi}{2}$ to the major principal plane.

For cohesionless soil ($c=0$): $\sigma_1 = \sigma_3\tan^2(45^\circ+\phi/2)$. For a saturated clay in undrained loading ($\phi_u=0$): $\sigma_1 - \sigma_3 = 2c_u$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does Mohr's failure theory say?
   A. Failure occurs when the maximum principal stress reaches a limit
   B. Failure occurs when shear stress on a plane reaches a limit that depends on the normal stress on it
   C. Failure occurs when the maximum strain reaches a limit
   D. Failure occurs only on the horizontal plane
   Answer: B. Failure is governed by a combination of normal and shear stress, tau_f = f(sigma). (Key point: Mohr's theory)

2. Which relation from the Mohr circle triangle starts the derivation?
   A. tan(phi) = (s1 - s3)/(s1 + s3)
   B. sin(phi) = [(s1 - s3)/2] / [c cot(phi) + (s1 + s3)/2]
   C. cos(phi) = [(s1 - s3)/2] / [c cot(phi) + (s1 + s3)/2]
   D. sin(phi) = (s1 + s3)/(s1 - s3)
   Answer: B. The radius is opposite the angle phi and the hypotenuse runs from the envelope intercept to the circle centre. (Key point: Geometry of derivation)

3. Which is the final Mohr-Coulomb relation?
   A. s1 = s3 tan^2(45 - phi/2) + 2c tan(45 - phi/2)
   B. s1 = s3 tan(45 + phi/2) + 2c tan^2(45 + phi/2)
   C. s1 = s3 tan^2(45 + phi/2) + 2c tan(45 + phi/2)
   D. s1 = s3 tan^2(45 + phi/2) - 2c tan(45 + phi/2)
   Answer: C. Rearranging gives (1+sin phi)/(1-sin phi) = tan^2(45+phi/2) and cos phi/(1-sin phi) = tan(45+phi/2). (Key point: Final relation)

4. At what angle to the major principal plane does failure occur?
   A. 45 - phi/2
   B. 45 degrees
   C. 90 - phi
   D. 45 + phi/2
   Answer: D. From the geometry, angle TCO' = 90 + phi, so the failure plane makes 45 + phi/2 with the major principal plane. (Key point: Failure plane angle)

## Related in Soil Mechanics

- [Principal stresses; Mohr circle at failure](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aprincipal-stresses-mohr-circle-friction) — Soil Mechanics, chapter 10
- [State Mohr-Coulomb failure criterion](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Amohr-coulomb-criterion-state) — Soil Mechanics, chapter 10
- [Limitations of Mohr-Coulomb theory](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Amohr-coulomb-limitations) — Soil Mechanics, chapter 10

## Related in other subjects

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- [Principal stresses, plane BE, Mohr's circle](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aplane-stress-40-50-30-plane) — Strength of Materials, chapter 4
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