---
title: "Infinite slope FoS, dry cohesive soil"
subject: "Soil Mechanics"
chapter: 11
key: "semester_4/soil_mechanics:infinite-slope-dry-cohesive-derive"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/infinite-slope-dry-cohesive-derive
site: BCT Engineering
---

# Infinite slope FoS, dry cohesive soil

Soil Mechanics · Chapter 11: Stability of Slopes

- Asked in: 2079 Asoj, 3 marks
- Video: [Derivation for Factor of Safety of Infinite Slope](https://www.youtube.com/watch?v=a5rEypKpGas) — RECKON CIVIL ACADEMY
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/infinite-slope-dry-cohesive-derive
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-11

## Question

Derive an equation for calculating the factor of safety for an infinite slope of dry cohesive soil. Assume necessary conditions.

## Short answer

For a dry c-φ soil, consider a slice of width b at depth z. Weight W = γ z b; normal stress σ = γ z cos²β, shear stress τ = γ z sinβ cosβ. Shear strength τ_f = c + σ tanφ. Factor of safety F = τ_f/τ = c/(γ z sinβ cosβ) + tanφ/tanβ.

## Answer

**Assumptions:** infinite slope of dry homogeneous $c$-$\phi$ soil, inclined at angle $\beta$ to the horizontal; failure plane parallel to the slope at vertical depth $z$; no seepage; unit weight $\gamma$.

Consider a vertical slice of horizontal width $b$, so the length of its base on the failure plane is $l = b/\cos\beta$. The forces on the two sides of the slice are equal and opposite, so they cancel.

```text
        ______ ground surface (beta)
       /|     \
      / | W    \
     /__|_____
        z   N  T  <- failure plane
```

Weight of slice: $W = \gamma z b$ per unit length.

$$
N = W\cos\beta, \qquad T = W\sin\beta
$$

Normal stress and shear stress on the plane:

$$
\sigma = \frac{N}{l} = \gamma z\cos^2\beta, \qquad \tau = \frac{T}{l} = \gamma z\sin\beta\cos\beta
$$

Shear strength (Mohr-Coulomb): $\tau_f = c + \sigma\tan\phi = c + \gamma z\cos^2\beta\tan\phi$

Factor of safety $F = \tau_f/\tau$:

$$
\boxed{F = \frac{c + \gamma z\cos^2\beta\tan\phi}{\gamma z\sin\beta\cos\beta} = \frac{c}{\gamma z\sin\beta\cos\beta} + \frac{\tan\phi}{\tan\beta}}
$$

For cohesionless soil ($c=0$), $F = \tan\phi/\tan\beta$, independent of $z$. For a given $c$, the critical (limiting) depth for $F=1$ is $z_c = \dfrac{c}{\gamma\cos^2\beta(\tan\beta - \tan\phi)}$.

## Check yourself

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

1. Why can side forces on the slice be ignored?
   A. They are zero because soil is dry
   B. They are equal and opposite and cancel
   C. They act along the base
   D. They equal W
   Answer: B. In an infinite slope the conditions are identical on both sides. (Key point: Side forces)

2. What are sigma and tau on the plane?
   A. sigma = gamma z cos^2 beta; tau = gamma z sin beta cos beta
   B. sigma = gamma z sin^2 beta; tau = gamma z cos beta
   C. sigma = gamma z; tau = gamma z tan beta
   D. sigma = gamma z cos beta; tau = gamma z sin beta
   Answer: A. N/l and T/l with l = b/cos beta and W = gamma z b. (Key point: Stresses)

3. What is the resulting F?
   A. F = tan beta/tan phi
   B. F = c/gamma z + tan phi
   C. F = c cos beta/tan phi
   D. F = c/(gamma z sin beta cos beta) + tan phi/tan beta
   Answer: D. F = tau_f / tau. (Key point: Result)

4. What does F become for c = 0?
   A. tan beta/tan phi
   B. depends on z
   C. tan phi/tan beta, independent of depth z
   D. zero
   Answer: C. The cohesion term vanishes, so F = 1 when beta = phi. (Key point: Cohesionless)

## Related in Soil Mechanics

- [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
- [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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- [10 m cut in cohesive soil, bedrock 15 m](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Acut-10m-cohesive-phi-zero-bedrock) — Soil Mechanics, chapter 11
- [FoS by φ = 0 analysis](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aphi-zero-factor-of-safety) — Soil Mechanics, chapter 11, asked 2×
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