--- title: "Principal stresses; Mohr circle at failure" subject: "Soil Mechanics" chapter: 10 key: "semester_4/soil_mechanics:principal-stresses-mohr-circle-friction" source: https://www.bctengineer.com/q/semester_4/soil_mechanics/principal-stresses-mohr-circle-friction site: BCT Engineering --- # Principal stresses; Mohr circle at failure Soil Mechanics · Chapter 10: Shear Strength of Soil - Asked in: 2078 Baisakh, 3 marks - Video: [Principal Stresses and MOHR'S CIRCLE in 12 Minutes!!](https://www.youtube.com/watch?v=aumN-Vl0OhQ) — Less Boring Lectures - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/principal-stresses-mohr-circle-friction - Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-10 ## Question Define major and minor principal stresses. What happens if the value of major principal stress increases while minor principal stress remains constant? Draw the Mohr circle of stresses at failure with Mohr-Coulomb failure line for soil having only angle of internal friction. ## Short answer Major principal stress σ1 is the largest normal stress on a plane with zero shear; minor σ3 is the smallest. If σ1 increases at constant σ3, the Mohr circle expands until it touches the failure line, causing failure. For a cohesionless soil, the failure envelope passes through the origin and sin φ = (σ1-σ3)/(σ1+σ3). ## Answer #### Major and minor principal stresses At any point in a stressed soil there are three mutually perpendicular planes on which the shear stress is zero. These are the **principal planes**, and the normal stresses on them are the **principal stresses**. - **Major principal stress $\sigma_1$:** the largest of the three normal stresses. - **Minor principal stress $\sigma_3$:** the smallest of the three normal stresses. - The third is the intermediate principal stress $\sigma_2$. #### Effect of increasing $\sigma_1$ with $\sigma_3$ constant The deviator stress $\sigma_1 - \sigma_3$ increases, so the Mohr circle grows larger (its centre moves right and its radius increases). The maximum shear stress, $\frac{\sigma_1-\sigma_3}{2}$, also increases. When the circle becomes tangent to the Mohr-Coulomb envelope, the soil fails. $\sigma_1$ cannot be increased beyond this point, because a Mohr circle cannot cut the failure envelope: the soil has already failed. #### Mohr circle at failure for a soil with only $\phi$ (cohesionless) The failure line passes through the origin, and the circle is tangent to it: ```text tau / tau = sigma tan(phi) | T/ | /.-''-. | /' '. | / phi C \ +--+------+-----+---- sigma 0 s3 s1 ``` From the right-angled triangle $OTC$: $$ \sin\phi = \frac{\sigma_1-\sigma_3}{\sigma_1+\sigma_3} $$ and the failure plane is at $45^\circ+\phi/2$ to the major principal plane. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What are major and minor principal stresses? A. The largest and smallest normal stresses, acting on planes of zero shear stress B. Shear stresses on the horizontal and vertical planes C. Total and effective stresses D. Stresses on planes of maximum shear Answer: A. Principal planes carry no shear stress; sigma1 is the largest, sigma3 the smallest. (Key point: Principal stresses) 2. What happens to the Mohr circle if sigma1 rises with sigma3 constant? A. It shrinks B. It grows larger, until it touches the envelope and the soil fails C. It stays the same D. It moves to the left only Answer: B. The deviator stress and maximum shear increase; at tangency to the envelope failure occurs. (Key point: Increasing sigma1) 3. What relation holds for the failure circle of a soil with only phi? A. sin(phi) = (s1 + s3)/(s1 - s3) B. tan(phi) = (s1 - s3)/(s1 + s3) C. sin(phi) = (s1 - s3)/(s1 + s3) D. cos(phi) = (s1 - s3)/(s1 + s3) Answer: C. In triangle OTC, radius over distance from origin to centre gives sin(phi). (Key point: Relation for cohesionless soil) ## Related in Soil Mechanics - [Derive Mohr-Coulomb principal stress relation](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Amohr-coulomb-derive-principal-stresses) — Soil Mechanics, chapter 10, asked 4× - [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 - [Three triaxial tests, Mohr circles, φ](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Amohr-circle-three-triaxial-tests-friction) — Soil Mechanics, chapter 10 ## Related in other subjects - [Principal stresses and Mohr's circle (30, 18, 75)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aprincipal-stresses-mohr-30-18-75) — Strength of Materials, chapter 4 - [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 - [Stresses on 30 degree plane; Mohr's circle](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ainclined-plane-30deg-50-30-mohr) — Strength of Materials, chapter 4, asked 3× - [Mohr's circle for a plane stress element](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Amohr-circle-plane-stress-construction) — Strength of Materials, chapter 6