Lecture No.12

Consolidation of soils II:

Terzaghi’s Consolidation Equation

（土の圧密 II: テルツァギの圧密方程式）

Consolidation involves movement

of water through soil skeleton:

It takes time!

This usually matters only in clays Water

drainage

Terzaghi’s consolidation theory （テルツァギの圧密理論）

• Soil is saturated （土は飽和している）

• Darcy’s law holds （ダルシーの法則が成り立つ）

• Principle of effective stress holds （有効応力の法則が成り立つ）

• Compression expressed with mv （mvによる圧縮の表現）

• 1-D Consolidation: No lateral deformation （横方向には変形しない）

• No compression of soil particles and water （土粒子と水は圧縮しない）

■ Consolidation process （圧密過程） 1

Apply pressure p instantly

Water is far more compressible than

soil skeleton

At the first moment, all the applied

pressure is taken by water

As time passes, water is drained and

gradually p falls on soil

Before

loading

Moment of

loading

After long enough

time

Total stress s s0 (z) s0 (z) + p s0 (z) + p

Pore water pressure, u u0 (z) u0 (z) + p u0 (z)

Effective stress, s' s0'(z) s0' (z) s0' (z) + p

z

= Excess pore water pressure （過剰間隙水圧）Du

■ Consolidation process （圧密過程） 2

■ Consolidation process （圧密過程） 3

① Soil skeleton compression

zD

xD

yD

z : Compression

Volumetric strain – effective stress

relationship （体積ひずみ－有効応力関係）

s DD vv m

The volume reduction is

vzyxV DDDDD

■ Deriving consolidation equation （圧密方程式の導出） 4

② Net water flux

zD

xD

yD

ii D

i

Hydraulic gradient

Net water outflow （正味の流出体積） per unit time:

: Coefficient of permeability

（透水係数）

Water flux, q = v × Area

k

z

■ Deriving consolidation equation （圧密方程式の導出） 5

(Darcy’s law)

yxki DD

h = u /gw+ z

yxikQ DDD

Piezometric head = Pressure head + Potential head （ピエゾ水頭） （圧力水頭） （位置水頭）

■ Deriving consolidation equation （圧密方程式の導出） 6

zD

xD

yD

ii D

i

Hydraulic gradient

z

Soil skeleton compression = Net water outflow

zyxumV v DDDDD styx

z

uktQ

w

DDD

DD 1

1

g

tyxz

ukzyxum

w

v DDD

DDDDD 1

1

gs

1

1

z

u

zk

t

um

w

vg

sWhen

s is constant

0

t

s

■ Deriving consolidation equation （圧密方程式の導出） 7

Pore water pressure = Hydrostatic pressure + Excess pore water pressure （間隙水圧） （静水圧） （過剰間隙水圧）

2

2

z

u

m

k

t

u

wv

g

2

2

z

uc

t

uv

Terzaghi’s (one-dimenssional) consolidation equation （テルツァギの（一次元）圧密方程式）

zzu wg)(0 uDu

2

2

z

uzc

t

uz wv

w

D

D gg

2

2

z

uc

t

uv

D

DValid for both u and Du

■ Deriving consolidation equation （圧密方程式の導出） 8

• Can be solve with the method of separation of variables （変数分離法）

• 1 initial condition （初期条件）, 2 boundary conditions （境界条件） necessary

Example

• Boundary condition 1: Du = 0 at z = 0

• Boundary condition 2: Du = 0 at z = 2d (or, at z = d )

• Initial condition: Du = p at any z

z

p

Du = p

Apply pressure p instantly

■ Solving consolidation equation （圧密方程式を解く） 9

0/ zu

0

2d

Obtained in form of Fourier series （フーリエ級数）

D

4

12exp

2

12sin

12

14 22

0

v

n

Tn

H

zn

npu

tH

cT v

v 2

Function of z

(location)

Function of t

(time)

0

* In textbook, u is used as “excess pore water pressure” in place of Du.

∞ t

■ Solution （解） 10

2d

z

Initial condition: Du=p at Tv=0

Tv =

As Tv increases, the excess pore water pressure Du dissipates （間隙水圧が消散する）

Du/p

■ Solution （解） 11

D

4

12exp

2

12sin

12

14 22

0

v

n

Tn

H

zn

npu

2d

k ×2

Tv

H ×2

Tv

tH

cT v

v 2

20mm thick specimen

in laboratory:

1 hour for consolidation

20m thick real ground:

114 years for consolidation

Impervious （不透水性）

■ Meaning of time factor, Tv 12

Equivalent to having half thickness.

Consolidation takes 1/4 of the time for

single-side drainage case.

Du/p Single-side drainage （片面排水）

Double-side drainage （両面排水）

■ Influence of boundary conditions 13

20m

10m

10m

Impervious （不透水性）

Pervious （透水性）: Sand, gravel

20mm thick specimen

in laboratory:

1 hour for consolidation

20m thick real ground:

114 years for consolidation

20m thick real ground,

But with 10m drainage length:

29 years for consolidation

■ Example 14

In (b) ～ (e), how much time does consolidation take

when compared to (a)?

■ Further example 15

Impervious

10m

Consol. layer

（k = 1.0

×10-8 m/s）

Permeable

10m

Consol. layer

（k = 1.0

×10-8 m/s）

Permeable

10m

Consol. layer

（k = 2.0

×10-8 m/s）

Permeable

5m

Consol. layer

（k = 2.0

×10-8 m/s）Impervious

5m

Consol. layer

（k = 1.0

×10-8 m/s）

(a) (b) (c) (d) (e)

Degree of consolidation （圧密度）, Uz :

Uz = 0:

Uz = 1:

21

1),(ee

eeTzU vz

Final reduction of void ratio

Or,

Current reduction of void ratio

■ Degree of consolidation （圧密度） 16

00

0 1),(u

u

u

uuTzU vz

D

D

D

DD

Du/p

Uz depends on depth

Take the average over layer Depth

0

d

Uz(Tv=0.6)=90% 0.2H

0.8H Uz(Tv=0.6)=73%

dzTzUH

TUH

vzvz 0

),(1

)(

)( vz TU

1

2

22 2

12exp

)12(

181

n

vTn

■ Degree of consolidation （圧密度） 17

d

d

Initial Du

distribution

Ex. Properties of clay （粘土の物性）

mv = 2.0 x 10-4 (m2/kN)

k = 1.0 x 10-9 (m/s)

gw = 9.8 (kN/m3)

p

20m

Tv t (day) Uz

0.2 1822

0.5 4556

0.9 8202

Impervious

■ Calculation of degree of consolidation 18

tdm

kt

d

cT

wv

vv 22 g