GEO-29

Bishop simplified slice

Fs = [c′b + (W − ub) tan φ′] / [mα W sin α], mα = cos α (1 + tan α tan φ′/F).

Reading speed

SlopesBishop 1955

Governing equation

m_\alpha=\cos\alpha\bigl(1+\tan\alpha\tan\varphi'/F\bigr),\; F=\dfrac{c'b+(W-ub)\tan\varphi'}{m_\alpha\, W\sin\alpha}

where

c': Cohesion (kPa)
\varphi': Friction angle (°)
W: Slice weight (kN/m)
b: Slice width (m)
u: Pore pressure (kPa)
\alpha: Base inclination (°)
F_0: Trial Fs (—)
F_s: Factor of safety (—)

Lecture brief

Historical brief

Soil mechanics became a quantitative laboratory after Karl von Terzaghi’s 1925–1943 work on effective stress, consolidation and bearing. These sheets still size shallow foundations, retaining walls, piles and drainage in SI. This sheet (GEO-29 — Bishop simplified slice) is the form associated with Bishop 1955. Working symbols: $c'$ , $\varphi'$ , $W$ , $b$ , $u$ , $\alpha$ , $F_0$ $\rightarrow$ $F_s$ . Circular slip, vertical interslice forces neglected, one iteration from a trial F.

Purpose

Purpose: compute

F_s

from

c'

\varphi'

W

b

u

\alpha

F_0

in Geotechnical engineering via

m_\alpha=\cos\alpha\bigl(1+\tan\alpha\tan\varphi'/F\bigr),\; F=\dfrac{c'b+(W-ub)\tan\varphi'}{m_\alpha\, W\sin\alpha}

Fs = [c′b + (W − ub) tan φ′] / [mα W sin α], mα = cos α (1 + tan α tan φ′/F). Use it when a real geotechnical engineering question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

c' = 8.000\,\mathrm{kPa}

\varphi' = 28.000\,\mathrm{^{\circ}}

W = 180.000\,\mathrm{kN/m}

b = 3.000\,\mathrm{m}

u = 20.000\,\mathrm{kPa}

\alpha = 25.000\,\mathrm{^{\circ}}

F_0 = 1.300\,\mathrm{—}

, the governing relation

m_\alpha=\cos\alpha\bigl(1+\tan\alpha\tan\varphi'/F\bigr),\; F=\dfrac{c'b+(W-ub)\tan\varphi'}{m_\alpha\, W\sin\alpha}

yields

F_s = 1.070\,\mathrm{—}

. A circular arc, one slice, a factor of safety. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Cohesion c'8 kPaFriction angle \varphi'28 °Slice weight W180 kN/mSlice width b3 mPore pressure u20 kPaBase inclination \alpha25 °Trial Fs F_01.3 —

Outputs

Factor of safety F_s1.070 —

Reading speed

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Narration of this film

A circular arc, one slice, a factor of safety.

Circular slip, vertical interslice forces neglected, one iteration from a trial F.

Reading speed

Watch on YouTube