---
title: "Most critical circle, Swedish method"
subject: "Soil Mechanics"
chapter: 11
key: "semester_4/soil_mechanics:swedish-circle-critical-circle"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/swedish-circle-critical-circle
site: BCT Engineering
---

# Most critical circle, Swedish method

Soil Mechanics · Chapter 11: Stability of Slopes

- Asked in: 2078 Chaitra, 3 marks
- Video: [Stability Analysis of Finite Slope using Swedish Circle Method](https://www.youtube.com/watch?v=rgff6GUm7XM) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/swedish-circle-critical-circle
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-11

## Question

Describe the process of determining the most critical circles in the Swedish circle method.

## Short answer

Draw slope, select trial centres and radii, compute F for each circle (using F = c_u L_a R/(W x) for φ=0 or method of slices for c-φ soil). Vary R at each centre to find local minimum F. Repeat for a grid of centres, contour F values; the centre with the lowest F yields the critical circle.

## Answer

In the Swedish (slip circle) method, the failure surface is assumed to be circular. The factor of safety is calculated for many trial circles, and the **critical circle** is the one with the minimum factor of safety.

**Procedure**

1. Draw the slope section to scale, marking the toe, crest and any soil boundaries.
2. Select a trial centre $O$ (above the slope, using Fellenius' guide lines or a grid of points) and a radius $R$ so that the circle passes through the toe (or just below it).
3. Compute the factor of safety $F$ for this circle:
   - $\phi=0$ soil: $F = \dfrac{c_u L_a R}{Wx}$;
   - $c$-$\phi$ soil: divide the mass into slices and use the method of slices.
4. Keep the same centre and change $R$ to find the minimum $F$ for that centre.
5. Repeat for other centres arranged in a grid, writing the minimum $F$ at each centre.
6. Draw **contours of equal factor of safety** through the grid values. The centre of the smallest contour is the centre of the most critical circle, and its value is the minimum factor of safety.

```text
   O1*   O2*   O3*
       F=1.8 F=1.5 F=1.7    <- grid of centres
   O4*   O5*   O6*
       F=1.6 F=1.3 F=1.5
   (O5 gives lowest F: critical circle)
```

For homogeneous soil with $\phi=0$, the position of the critical centre can also be found from Fellenius' construction (direction angles at the toe and crest) or Taylor's charts.

## Check yourself

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

1. What is the critical circle?
   A. The one with the maximum F
   B. The trial circle with the minimum factor of safety
   C. The largest radius circle
   D. The one through the crest only
   Answer: B. It is the most likely failure surface. (Key point: Critical circle)

2. How is the search carried out?
   A. For each centre in a grid find the minimum F over radii, then contour the values
   B. Use one centre only
   C. Use random circles
   D. Find the maximum F at each centre
   Answer: A. Many centres are tested, each with its minimum F over R. (Key point: Procedure)

3. How is the critical centre identified?
   A. Centre of the largest contour
   B. The crest
   C. The toe
   D. Centre of the smallest contour of equal F; its value is minimum F
   Answer: D. Contours close around the lowest value. (Key point: Contours)

4. For c-phi soil how is F computed per circle?
   A. F = c_u L_a R/(W x) only
   B. Using Taylor's chart only for phi > 0
   C. By dividing the mass into slices (method of slices)
   D. Ignoring pore pressure
   Answer: C. The phi = 0 formula applies only to phi = 0 soil. (Key point: Soil type)

## Related in Soil Mechanics

- [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

- [Case study: critical factors in process control](https://www.bctengineer.com/question.md?key=semester_5%2Finstrumentation_ii%3Acs-critical-factors) — Instrumentation II, chapter 10, asked 2×
- [Critical position of UDL shorter than span](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Audl-shorter-than-span-critical-position) — Theory of Structures I, chapter 5, asked 6×
- [Contractor cash flow estimate and critical stage](https://www.bctengineer.com/question.md?key=semester_8%2Fconstruction_management%3Acontractor-cash-flow-estimate) — Construction Management, chapter 6, asked 2×
- [Shunt generator voltage build-up, critical resistance and speed](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines_i%3Ashunt-buildup-critical-define) — Electrical Machines I, chapter 3, asked 9×