GEO-07

Time factor and degree of consolidation

Tv = Cv t / Hdr² with U ≈ √(π Tv/4) for U < 0.6.

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ConsolidationTerzaghi 1925

Governing equation

T_v=\dfrac{c_v t}{H_{\mathrm{dr}}^2},\quad U\approx\sqrt{\dfrac{\pi T_v}{4}}\ (U<0.6)

where

c_v: Coefficient of consolidation (m²/yr)
t: Time (yr)
H_{\mathrm{dr}}: Drainage path (m)
T_v: Time factor (—)
U: Degree of consolidation (—)

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-07 — Time factor and degree of consolidation) is the form associated with Terzaghi 1925. Working symbols: $c_v$ , $t$ , $H_{\mathrm{dr}}$ $\rightarrow$ $T_v$ , $U$ . Terzaghi's 1-D diffusion equation ∂u/∂t = Cv ∂²u/∂z² yields a Fourier series; the early-time parabola U = √(π Tv/4) is the usual field approximation.

Purpose

Purpose: compute

T_v

U

from

c_v

t

H_{\mathrm{dr}}

in Geotechnical engineering via

T_v=\dfrac{c_v t}{H_{\mathrm{dr}}^2},\quad U\approx\sqrt{\dfrac{\pi T_v}{4}}\ (U<0.6)

Tv = Cv t / Hdr² with U ≈ √(π Tv/4) for U < 0.6. 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_v = 8.000\,\mathrm{m^{2}/yr}

t = 1.000\,\mathrm{yr}

H_{\mathrm{dr}} = 2.000\,\mathrm{m}

, the governing relation

T_v=\dfrac{c_v t}{H_{\mathrm{dr}}^2},\quad U\approx\sqrt{\dfrac{\pi T_v}{4}}\ (U<0.6)

yields

T_v = 2.0000\,\mathrm{—}

U = 0.994\,\mathrm{—}

. Double drainage: Hdr = H/2. Caps U at 1. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Coefficient of consolidation c_v8 m²/yrTime t1 yrDrainage path H_{\mathrm{dr}}2 m

Outputs

Time factor T_v2.0000 —
Degree of consolidation U0.994 —

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Narration of this film

Double drainage: Hdr = H/2. Caps U at 1.

Terzaghi's 1-D diffusion equation ∂u/∂t = Cv ∂²u/∂z² yields a Fourier series; the early-time parabola U = √(π Tv/4) is the usual field approximation.

Reading speed

Watch on YouTube