---
title: "Infinite slope, cohesionless, steady seepage"
subject: "Soil Mechanics"
chapter: 11
key: "semester_4/soil_mechanics:infinite-slope-cohesionless-seepage"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/infinite-slope-cohesionless-seepage
site: BCT Engineering
---

# Infinite slope, cohesionless, steady seepage

Soil Mechanics · Chapter 11: Stability of Slopes

- Asked in: 2078 Chaitra, 3 marks
- Video: [10.STABILITY OF SLOPES || INFINITE COHESIONLESS SLOPE WITH SEEPAGE | STAEDY STATE SEEPAGE | FOS |](https://www.youtube.com/watch?v=1LM-iqu5oLY) — Mk Plus Point
- 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-seepage
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-11

## Question

Analyse the infinite slope of cohesionless soil for a steady seepage condition.

## Short answer

With seepage parallel to the slope and water table at surface, effective stress is γ' z cos²β and total shear stress is γ_sat z sinβ cosβ. For c=0, F = (γ'/γ_sat)(tanφ'/tanβ). This is roughly half the dry-slope factor, and the limiting slope angle satisfies tanβ ≈ 0.5 tanφ'.

## Answer

**Conditions:** infinite slope of cohesionless soil ($c=0$), slope angle $\beta$, steady seepage parallel to the slope with the water table at the ground surface. Saturated unit weight $\gamma_{sat}$, submerged unit weight $\gamma'$.

Consider a slice of depth $z$ and width $b$. Seepage is parallel to the slope, so the equipotential lines are perpendicular to the slope and the pore pressure on the plane is

$$
u = \gamma_w z\cos^2\beta
$$

Total weight of the slice: $W = \gamma_{sat} z b$.

$$
\sigma = \gamma_{sat} z\cos^2\beta, \qquad \tau = \gamma_{sat} z\sin\beta\cos\beta
$$

Effective normal stress:

$$
\sigma' = \sigma - u = (\gamma_{sat} - \gamma_w) z\cos^2\beta = \gamma' z\cos^2\beta
$$

Shear strength $= \sigma'\tan\phi'$, so the factor of safety is

$$
F = \frac{\gamma' z\cos^2\beta\tan\phi'}{\gamma_{sat} z\sin\beta\cos\beta} = \boxed{\frac{\gamma'}{\gamma_{sat}}\,\frac{\tan\phi'}{\tan\beta}}
$$

**Remarks:**

- As $\gamma'/\gamma_{sat}\approx 0.5$, the factor of safety with seepage is about **half** the dry-slope value $\tan\phi'/\tan\beta$.
- The limiting slope angle with seepage ($F = 1$) is therefore given by $\tan\beta \approx 0.5\tan\phi'$, which is roughly half the friction angle.
- If the water table is at height $m z$ above the plane (with $0<m<1$), and the same unit weight $\gamma_{sat}$ is taken above it, $F = \left(1 - \dfrac{m\gamma_w}{\gamma_{sat}}\right)\dfrac{\tan\phi'}{\tan\beta}$.

## Check yourself

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

1. For seepage parallel to the slope with the water table at the ground surface, what is the pore pressure on the failure plane at depth z?
   A. u = γw z
   B. u = γw z cos^2β
   C. u = γw z sinβ cosβ
   D. u = γ' z cos^2β
   Answer: B. Equipotential lines are perpendicular to the slope, so the head at the plane is z cos^2β and u = γw z cos^2β. (Key point: Pore pressure on the plane)

2. What is the effective normal stress on the plane under this seepage condition?
   A. γsat z cos^2β
   B. γ' z cos^2β
   C. γ' z sinβ cosβ
   D. γw z cos^2β
   Answer: B. σ' = σ - u = (γsat - γw) z cos^2β = γ' z cos^2β. (Key point: Effective normal stress)

3. Which expression gives the factor of safety of the cohesionless infinite slope with seepage at the surface?
   A. (γsat/γ') tanφ'/tanβ
   B. tanφ'/tanβ
   C. (γ'/γsat) tanφ'/tanβ
   D. (γ'/γw) tanφ'/tanβ
   Answer: C. Shear stress uses γsat while the effective normal stress uses γ', giving F = (γ'/γsat)(tanφ'/tanβ). (Key point: Factor of safety expression)

4. Since γ'/γsat is about 0.5, how does the factor of safety with seepage compare with the dry slope?
   A. About double
   B. About the same
   C. About half
   D. About one-quarter
   Answer: C. F is multiplied by about 0.5, so the stable slope angle is roughly tanβ ≈ 0.5 tanφ'. (Key point: Effect of seepage)

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