GEO-37

Vesic rigidity index Ir

Ir = G / (c + σ′ tan φ′). Rigid if Ir > I_r,crit, otherwise a compressibility correction.

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BearingVesic Ir

Governing equation

I_r=\dfrac{G}{c+\sigma'\tan\varphi'}

$G \in \left[1, 80\right]$

where

G: Shear modulus (MPa)
c': Cohesion (kPa)
\sigma': Mean effective (kPa)
\varphi': Friction angle (°)
I_r: Rigidity index (—)

The figure, part by part

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①

Governing law — red

I_r=\dfrac{G}{c+\sigma'\tan\varphi'}

$G \in \left[1, 80\right]$

$G = 1$ $I_r = 19.03$

Along the curve

②

Domain of the variable — blue

G_{\min} = 1,\quad G_{\max} = 80

$G \in \left[1, 80\right]$

$G = 1$ $G = 1$

Along the curve

③

Response curve — green

I_r=\dfrac{G}{c+\sigma'\tan\varphi'}

$G \in \left[1, 80\right]$

$G = 1$ $I_r = 19.03$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}I_r}{\mathrm{d}G}

$G \in \left[1, 80\right]$

$G = 1.497$ $I_r^{\prime} = 19.03$

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-37 — Vesic rigidity index Ir) is the form associated with Vesic Ir. Working symbols: $G$ , $c'$ , $\sigma'$ , $\varphi'$ $\rightarrow$ $I_r$ . Vesic compared the elastic stiffness to the strength on the failure envelope. Soft soils need ξc, ξq, ξγ < 1.

Purpose

Purpose: compute

I_r

from

G

c'

\sigma'

\varphi'

in Geotechnical engineering via

I_r=\dfrac{G}{c+\sigma'\tan\varphi'}

Ir = G / (c + σ′ tan φ′). Rigid if Ir > I_r,crit, otherwise a compressibility correction. 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

G = 12.000\,\mathrm{MPa}

c' = 10.000\,\mathrm{kPa}

\sigma' = 80.000\,\mathrm{kPa}

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

, the governing relation

I_r=\dfrac{G}{c+\sigma'\tan\varphi'}

yields

I_r = 228.4\,\mathrm{—}

. A shear modulus, a strength, a dimensionless rigidity. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Shear modulus G12 MPaCohesion c'10 kPaMean effective \sigma'80 kPaFriction angle \varphi'28 °

Outputs

Rigidity index I_r228.4 —

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Narration of this film

A shear modulus, a strength, a dimensionless rigidity.

Vesic compared the elastic stiffness to the strength on the failure envelope. Soft soils need ξc, ξq, ξγ < 1.

Reading speed

Watch on YouTube