INGENIA
Index/Geotechnical engineering/GEO-29

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

mα=cos⁡α(1+tan⁡αtan⁡φ′/F),  F=c′b+(W−ub)tan⁡φ′mα Wsin⁡αm_\alpha=\cos\alpha\bigl(1+\tan\alpha\tan\varphi'/F\bigr),\; F=\dfrac{c'b+(W-ub)\tan\varphi'}{m_\alpha\, W\sin\alpha}

α∈[−20,55]\alpha \in \left[-20, 55\right]

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 (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

GEO-29 · bearing
00:0 / 00:08
①

Governing law — red

mα=cos⁡α(1+tan⁡αtan⁡φ′/F),  F=c′b+(W−ub)tan⁡φ′mα Wsin⁡αm_\alpha=\cos\alpha\bigl(1+\tan\alpha\tan\varphi'/F\bigr),\; F=\dfrac{c'b+(W-ub)\tan\varphi'}{m_\alpha\, W\sin\alpha}

α∈[−20,55]\alpha \in \left[-20, 55\right]

α=−20\alpha = -20Fs=−1.783F_s = -1.783

②

Domain of the variable — blue

αmin⁡=−20,αmax⁡=55\alpha_{\min} = -20,\quad \alpha_{\max} = 55

α∈[−20,55]\alpha \in \left[-20, 55\right]

α=−20\alpha = -20α=−20\alpha = -20

③

Response curve — green

mα=cos⁡α(1+tan⁡αtan⁡φ′/F),  F=c′b+(W−ub)tan⁡φ′mα Wsin⁡αm_\alpha=\cos\alpha\bigl(1+\tan\alpha\tan\varphi'/F\bigr),\; F=\dfrac{c'b+(W-ub)\tan\varphi'}{m_\alpha\, W\sin\alpha}

α∈[−20,55]\alpha \in \left[-20, 55\right]

α=−20\alpha = -20Fs=−1.783F_s = -1.783

④

Rate of change — amber

dFsdα\frac{\mathrm{d}F_s}{\mathrm{d}\alpha}

α∈[−20,55]\alpha \in \left[-20, 55\right]

α=−19.53\alpha = -19.53Fs′=−0.061F_s^{\prime} = -0.061

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′c', φ′\varphi', WW, bb, uu, α\alpha, F0F_0 →\rightarrow FsF_s. Circular slip, vertical interslice forces neglected, one iteration from a trial F.

Purpose

Purpose: compute FsF_s from c′c', φ′\varphi', WW, bb, uu, α\alpha, F0F_0 in Geotechnical engineering via mα=cos⁡α(1+tan⁡αtan⁡φ′/F),  F=c′b+(W−ub)tan⁡φ′mα Wsin⁡αm_\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 kPac' = 8.000\,\mathrm{kPa}, φ′=28.000 ∘\varphi' = 28.000\,\mathrm{^{\circ}}, W=180.000 kN/mW = 180.000\,\mathrm{kN/m}, b=3.000 mb = 3.000\,\mathrm{m}, u=20.000 kPau = 20.000\,\mathrm{kPa}, α=25.000 ∘\alpha = 25.000\,\mathrm{^{\circ}}, F0=1.300 —F_0 = 1.300\,\mathrm{—}, the governing relation mα=cos⁡α(1+tan⁡αtan⁡φ′/F),  F=c′b+(W−ub)tan⁡φ′mα Wsin⁡αm_\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 Fs=1.070 —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

Outputs

  • Factor of safety F_s1.070 —
Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

GEO-29 · bearing
00:0 / 00:08

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