GEO-22

Hansen bearing capacity

qult = c Nc sc dc ic + q Nq sq dq iq + ½ γ B Nγ sγ dγ iγ. Inclination cuts the Nγ term hardest.

Reading speed

BearingHansen 1970

Governing equation

q_{ult}=c N_c s_c d_c i_c+q N_q s_q d_q i_q+\tfrac12\gamma B N_\gamma s_\gamma d_\gamma i_\gamma

$i_c \in \left[0.3, 1\right]$

where

c': Cohesion (kPa)
N_c: Nc (—)
N_q: Nq (—)
N_\gamma: Nγ (—)
q: Overburden (kPa)
\gamma: Unit weight (kN/m³)
B: Width (m)
i_c: Inclination ic (—)
i_q: Inclination iq (—)
i_\gamma: Inclination iγ (—)
q_{ult}: Ultimate bearing (kPa)

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.

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①

Governing law — red

q_{ult}=c N_c s_c d_c i_c+q N_q s_q d_q i_q+\tfrac12\gamma B N_\gamma s_\gamma d_\gamma i_\gamma

$i_c \in \left[0.3, 1\right]$

$i_c = 0.3$ $q_{ult} = 606.2$

Along the curve

②

Domain of the variable — blue

i_c_{\min} = 0.3,\quad i_c_{\max} = 1

$i_c \in \left[0.3, 1\right]$

$i_c = 0.3$ $i_c = 0.3$

Along the curve

③

Response curve — green

q_{ult}=c N_c s_c d_c i_c+q N_q s_q d_q i_q+\tfrac12\gamma B N_\gamma s_\gamma d_\gamma i_\gamma

$i_c \in \left[0.3, 1\right]$

$i_c = 0.3$ $q_{ult} = 606.2$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}q_{ult}}{\mathrm{d}i_c}

$i_c \in \left[0.3, 1\right]$

$i_c = 0.304$ $q_{ult}^{\prime} = 363$

Along the curve

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-22 — Hansen bearing capacity) is the form associated with Hansen 1970. Working symbols: $c'$ , $N_c$ , $N_q$ , $N_\gamma$ , $q$ , $\gamma$ , $B$ , $i_c$ , $i_q$ , $i_\gamma$ $\rightarrow$ $q_{ult}$ . Hansen added load inclination, base tilt and ground slope to Brinch Hansen's general formula.

Purpose

Purpose: compute

q_{ult}

from

c'

N_c

N_q

N_\gamma

q

\gamma

B

i_c

i_q

i_\gamma

in Geotechnical engineering via

q_{ult}=c N_c s_c d_c i_c+q N_q s_q d_q i_q+\tfrac12\gamma B N_\gamma s_\gamma d_\gamma i_\gamma

qult = c Nc sc dc ic + q Nq sq dq iq + ½ γ B Nγ sγ dγ iγ. Inclination cuts the Nγ term hardest. 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' = 12.000\,\mathrm{kPa}

N_c = 25.000\,\mathrm{—}

N_q = 12.000\,\mathrm{—}

N_\gamma = 10.000\,\mathrm{—}

q = 30.000\,\mathrm{kPa}

\gamma = 18.000\,\mathrm{kN/m^{3}}

B = 2.500\,\mathrm{m}

i_c = 0.850\,\mathrm{—}

i_q = 0.800\,\mathrm{—}

i_\gamma = 0.700\,\mathrm{—}

, the governing relation

q_{ult}=c N_c s_c d_c i_c+q N_q s_q d_q i_q+\tfrac12\gamma B N_\gamma s_\gamma d_\gamma i_\gamma

yields

q_{ult} = 805.9\,\mathrm{kPa}

. A tilted load on a strip, three reduced capacity bars. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Cohesion c'12 kPaNc N_c25 —Nq N_q12 —Nγ N_\gamma10 —Overburden q30 kPaUnit weight \gamma18 kN/m³Width B2.5 mInclination ic i_c0.85 —Inclination iq i_q0.8 —Inclination iγ i_\gamma0.7 —

Outputs

Ultimate bearing q_{ult}805.9 kPa

Reading speed

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Narration of this film

A tilted load on a strip, three reduced capacity bars.

Hansen added load inclination, base tilt and ground slope to Brinch Hansen's general formula.

Reading speed

Watch on YouTube