---
title: "Infinite slope FoS; water table rises"
subject: "Soil Mechanics"
chapter: 11
key: "semester_4/soil_mechanics:infinite-slope-cohesionless-water-table"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/infinite-slope-cohesionless-water-table
site: BCT Engineering
---

# Infinite slope FoS; water table rises

Soil Mechanics · Chapter 11: Stability of Slopes

- Asked in: 2076 Baisakh, 4 marks
- Video: [Infinite Slope Stability Example Solution (Stability of Infinite Slopes)](https://www.youtube.com/watch?v=ISThKrxuQJI) — Mahbub
- 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-cohesionless-water-table
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-11

## Question

Derive the equation of factor of safety for the infinite slope with cohesionless soil without water table. What happens if the water table rises to the surface of the slope?

## Short answer

For a cohesionless infinite slope with no water table, the factor of safety is F = tanφ / tanβ, independent of depth. When the water table rises to the surface with seepage parallel to the slope, F reduces to (γ′/γ_sat)(tanφ/tanβ), roughly half the dry value, which can trigger failure.

## Answer

#### Derivation (cohesionless soil, no water table)

Take a slice of width $b$ and depth $z$ in an infinite slope inclined at $\beta$. Soil is dry or moist with unit weight $\gamma$ and angle of friction $\phi$. Side forces cancel.

$$
W = \gamma z b, \quad N = W\cos\beta, \quad T = W\sin\beta
$$

On the failure plane of length $b/\cos\beta$:

$$
\sigma = \gamma z\cos^2\beta, \qquad \tau = \gamma z\sin\beta\cos\beta
$$

Shear strength: $\tau_f = \sigma\tan\phi = \gamma z\cos^2\beta\tan\phi$

$$
\boxed{F = \frac{\tau_f}{\tau} = \frac{\tan\phi}{\tan\beta}}
$$

The factor of safety does not depend on depth or unit weight. The slope is stable when $\beta < \phi$, and at $\beta=\phi$ ($F=1$) the slope is at its limit (angle of repose).

#### Water table rises to the surface

With seepage parallel to the slope and water at the surface, the pore pressure is $u = \gamma_w z\cos^2\beta$, so the effective normal stress falls to $\gamma' z\cos^2\beta$ while the shear stress rises to $\gamma_{sat} z\sin\beta\cos\beta$:

$$
F_{sat} = \frac{\gamma'}{\gamma_{sat}}\frac{\tan\phi}{\tan\beta}
$$

Since $\gamma'/\gamma_{sat}\approx 0.5$, the factor of safety **falls to about half**. For example, a slope with $\phi = 35^\circ$ and $\beta = 20^\circ$ has $F = 1.92$ when dry but only about 0.96 when the water reaches the surface, so the slope would fail. This is why heavy rain triggers landslides in sandy slopes.

## Check yourself

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

1. For a dry cohesionless infinite slope, what is F?
   A. tanβ/tanφ
   B. tanφ/tanβ
   C. γ z tanφ/tanβ
   D. sinφ/sinβ
   Answer: B. τf = γ z cos^2β tanφ and τ = γ z sinβ cosβ, so F = tanφ/tanβ. (Key point: Dry slope factor of safety)

2. How does the dry-slope factor of safety depend on depth z and unit weight γ?
   A. It increases with depth
   B. It decreases with depth
   C. It is proportional to γ
   D. It is independent of both
   Answer: D. z and γ cancel in the ratio of strength to shear stress. (Key point: Independence of depth)

3. At β = φ in a dry cohesionless slope, what is the situation?
   A. F = 1, the slope is at its limit (angle of repose)
   B. F = 2, the slope is safe
   C. F = 0, the slope has failed long ago
   D. F is undefined
   Answer: A. tanφ/tanβ = 1, so the slope is at limiting equilibrium, the angle of repose. (Key point: Limit condition)

4. When the water table rises to the surface with parallel seepage, what happens to the factor of safety?
   A. It doubles
   B. It is unchanged
   C. It falls to about half
   D. It rises slightly
   Answer: C. Pore pressure reduces effective stress to γ' z cos^2β, so F = (γ'/γsat) tanφ/tanβ, about half the dry value. (Key point: Water table at surface)

## 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 seepage, max angle for FoS 2](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Ainfinite-slope-seepage-max-angle) — Soil Mechanics, chapter 11
- [Infinite slope, cohesionless, steady seepage](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Ainfinite-slope-cohesionless-seepage) — 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
- [Infinite clay slope, submerged and seepage](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Ainfinite-slope-submerged-seepage-clay) — Soil Mechanics, chapter 11, asked 2×
- [Uniform sand, water table 2 m, capillary](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Auniform-sand-effective-stress-capillary) — Soil Mechanics, chapter 6

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