---
title: "FoS by φ = 0 analysis"
subject: "Soil Mechanics"
chapter: 11
key: "semester_4/soil_mechanics:phi-zero-factor-of-safety"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/phi-zero-factor-of-safety
site: BCT Engineering
---

# FoS by φ = 0 analysis

Soil Mechanics · Chapter 11: Stability of Slopes

- Asked in: 2079 Asoj, 3 marks; 2074 Bhadra, 4 marks (asked 2 times)
- Video: [Taylor's Stability Number | Soil Mechanics](https://www.youtube.com/watch?v=TDIw5hksRso) — APSEd
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/phi-zero-factor-of-safety
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-11

## Question

Find the factor of safety of a slope using the $\phi = 0$ analysis method. Assume necessary conditions.

## Short answer

Use the Swedish circle method: assume a circular failure arc in saturated clay with undrained strength c_u. Resisting moment is c_u * arc length * radius, driving moment is weight of sliding mass times horizontal lever arm. For typical data (c_u=30 kN/m², area 80 m², R=12 m, arc angle 80°) the factor of safety F ≈ 1.68.

## Answer

The **$\phi = 0$ analysis** is a total stress method for a **saturated clay under undrained (short-term) conditions**, where $\tau_f = c_u$ and $\phi_u = 0$. It uses the Swedish (slip) circle.

**Assumptions**

- Homogeneous saturated clay with undrained strength $c_u$ and unit weight $\gamma$.
- Failure along a circular arc of radius $R$ with centre $O$.
- Moments are taken about $O$ per metre length of slope.

**Method**

1. Draw the slope to scale and choose a trial centre $O$ and radius $R$.
2. Find the area of the sliding mass and compute its weight $W = \gamma A$; locate the centroid at horizontal distance $x$ from $O$.
3. Measure the arc length $L_a = R\theta$ (with $\theta$ in radians).
4. Moments about $O$:

$$
\text{Resisting moment} = c_u L_a R, \qquad \text{Driving moment} = Wx
$$

$$
F = \frac{c_u L_a R}{W x}
$$

5. Repeat for several trial circles; the least value of $F$ is the factor of safety of the slope.

If a tension crack of depth $z_c = \dfrac{2c_u}{\gamma}$ is considered, $L_a$ is reduced by the length in the crack, and a hydrostatic force may be added to the driving moment.

**Example (assumed data):** $c_u = 30$ kN/m², $\gamma = 18$ kN/m³, $R = 12$ m, arc angle $80^\circ$ ($1.396$ rad), area of sliding mass $= 80$ m², $x = 2.5$ m.

$$
\begin{aligned}
L_a &= 12 \times 1.396 = 16.76\ \text{m} \\
W &= 18 \times 80 = 1440\ \text{kN/m} \\
F &= \frac{30 \times 16.76 \times 12}{1440 \times 2.5} = \frac{6034}{3600} = 1.68
\end{aligned}
$$

**Answer: $F \approx 1.68$ for this trial circle**; the minimum $F$ over trial circles governs the design (should be $\geq 1.3$ to 1.5 for the short term).

## Check yourself

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

1. When is the phi = 0 analysis used?
   A. Saturated clay under undrained, short-term conditions
   B. Dry sand
   C. Long-term drained conditions
   D. Partly saturated gravel
   Answer: A. Undrained strength is c_u with phi_u = 0. (Key point: Applicability)

2. What is F for a Swedish circle in phi = 0 analysis?
   A. F = W x/(c_u L_a R)
   B. F = c_u/(gamma H)
   C. F = tan phi/tan beta
   D. F = c_u L_a R/(W x)
   Answer: D. Resisting moment c_u L_a R over driving moment W x. (Key point: Formula)

3. Which circle governs the design?
   A. The one with the largest F
   B. The average of F
   C. The trial circle giving the least F
   D. The one through the crest
   Answer: C. The minimum F is the slope's factor of safety. (Key point: Critical circle)

4. In the example, what is F?
   A. About 0.60
   B. About 1.68
   C. About 3.36
   D. About 2.5
   Answer: B. F = 30(16.76)(12)/(1440 x 2.5) = 6034/3600. (Key point: Example)

## Related in Soil Mechanics

- [Infinite slope FoS, dry cohesive soil](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Ainfinite-slope-dry-cohesive-derive) — Soil Mechanics, chapter 11
- [Infinite slope FoS; water table rises](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Ainfinite-slope-cohesionless-water-table) — Soil Mechanics, chapter 11
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- [Embankment 36°, method of slices FoS](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aembankment-method-of-slices-fos) — Soil Mechanics, chapter 11
- [Infinite dry slope, c = 10 kPa, φ = 25°](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Ainfinite-dry-slope-c10-phi25) — Soil Mechanics, chapter 11

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