GEO-21

Vesic bearing capacity

qult = c Nc sc dc + q Nq sq dq + ½ γ B Nγ sγ dγ. Shape and depth factors on Terzaghi's three terms.

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BearingVesic 1975

Governing equation

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

$B \in \left[0.4, 8\right]$

where

c': Cohesion (kPa)
N_c: Nc (—)
N_q: Nq (—)
N_\gamma: Nγ (—)
q: Overburden q (kPa)
\gamma: Unit weight (kN/m³)
B: Width (m)
s_c: Shape sc (—)
s_q: Shape sq (—)
s_\gamma: Shape sγ (—)
d_c: Depth dc (—)
d_q: Depth dq (—)
d_\gamma: Depth dγ (—)
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+q N_q s_q d_q+\tfrac12\gamma B N_\gamma s_\gamma d_\gamma

$B \in \left[0.4, 8\right]$

$B = 0.4$ $q_{ult} = 1366.2$

Along the curve

②

Domain of the variable — blue

B_{\min} = 0.4,\quad B_{\max} = 8

$B \in \left[0.4, 8\right]$

$B = 0.4$ $B = 0.4$

Along the curve

③

Response curve — green

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

$B \in \left[0.4, 8\right]$

$B = 0.4$ $q_{ult} = 1366.2$

Along the curve

④

Rate of change — amber

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

$B \in \left[0.4, 8\right]$

$B = 0.448$ $q_{ult}^{\prime} = 118.8$

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-21 — Vesic bearing capacity) is the form associated with Vesic 1975. Working symbols: $c'$ , $N_c$ , $N_q$ , $N_\gamma$ , $q$ , $\gamma$ , $B$ , $s_c$ , $s_q$ , $s_\gamma$ , $d_c$ , $d_q$ , $d_\gamma$ $\rightarrow$ $q_{ult}$ . Vesic recast Prandtl–Reissner with rigidity Ir and shape sc, sq, sγ plus depth dc, dq, dγ.

Purpose

Purpose: compute

q_{ult}

from

c'

N_c

N_q

N_\gamma

q

\gamma

B

s_c

s_q

s_\gamma

d_c

d_q

d_\gamma

in Geotechnical engineering via

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

qult = c Nc sc dc + q Nq sq dq + ½ γ B Nγ sγ dγ. Shape and depth factors on Terzaghi's three terms. 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' = 15.000\,\mathrm{kPa}

N_c = 30.000\,\mathrm{—}

N_q = 18.000\,\mathrm{—}

N_\gamma = 15.000\,\mathrm{—}

q = 27.000\,\mathrm{kPa}

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

B = 2.000\,\mathrm{m}

s_c = 1.200\,\mathrm{—}

s_q = 1.200\,\mathrm{—}

s_\gamma = 0.800\,\mathrm{—}

d_c = 1.200\,\mathrm{—}

d_q = 1.150\,\mathrm{—}

d_\gamma = 1.100\,\mathrm{—}

, the governing relation

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

yields

q_{ult} = 1556.3\,\mathrm{kPa}

. A footing, three capacity bars scaled by shape and embedment. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Cohesion c'15 kPaNc N_c30 —Nq N_q18 —Nγ N_\gamma15 —Overburden q q27 kPaUnit weight \gamma18 kN/m³Width B2 mShape sc s_c1.2 —Shape sq s_q1.2 —Shape sγ s_\gamma0.8 —Depth dc d_c1.2 —Depth dq d_q1.15 —Depth dγ d_\gamma1.1 —

Outputs

Ultimate bearing q_{ult}1556.3 kPa

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Narration of this film

A footing, three capacity bars scaled by shape and embedment.

Vesic recast Prandtl–Reissner with rigidity Ir and shape sc, sq, sγ plus depth dc, dq, dγ.

Reading speed

Watch on YouTube