State Variables for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 97

The stress-versus-strain equation in the vertical direction
for a homogeneous, isotropic, linear elastic, unsaturated soil
can be written as

The above equation can be normalized to the net vertical
stress, and the equation takes the form for the coefficient of
earth pressure at rest, K 0 :

μ E u a − u w

σ h −

u a +

σ v − u a 2 μ

E

u a − u w

K 0 =

μ −

(3.34)

ε v = − (3.31)

1 − ( 1 − μ) H σ v − u a

E H

The above equation reverts to the form common to a sat-
urated soil when matric suction goes to zero. When matric
suction is present in the soil, the horizontal stress is reduced
as the soil tends toward cracking. The reduction in horizon-
tal stress is also a function of the depth under consideration.
At shallow depths, the second part of Eq. 3.34 shows that a
small matric suction may cause the net horizontal stress to
go to zero, or tend to go negative. If the soil cannot sustain
tensile strain, cracking of the soil will occur commencing at
ground surface.
Typical K 0 values for the first loading of clay ranges
between approximately 0.3 and 0.7, depending upon Pois-
son's ratio. Let us consider the slow drying of a lacus-
trine clay deposit. For illustrative purposes, the following
properties are assumed; μ

where:

ε v = normal strain in the vertical direction,

σ v = total normal stress in the vertical direction,

σ h = total normal stress in the horizontal direction,

μ = Poisson's ratio,

E = elastic modulus with respect to a change in σ − u a ,

H = elastic modulus with respect to a change in u a − u w ,

u a = pore-air pressure, and

u w = pore-water pressure.

The stress-versus-strain equation in the horizontal direc-
tion is written as

=
1886 kg/m 3 . Figure 3.14 illustrates the relationship between
the coefficient of earth pressure at rest and matric suction
for various overburden pressures. When the soil is saturated
with zero pore-water pressure, the at-rest coefficient of earth
pressure is 0.538. The at-rest coefficient then decreases as
the matric suction of the soil increases. This behavior applies
for all depths, but the rate of reduction in the at-rest coeffi-
cient is greater at shallow depths. An at-rest coefficient K 0
of zero indicates a tendency for cracking.
The above example is a simplification which does not
take into consideration the effects of prior processes such as
wetting and drying, freezing and thawing, and loading and

= 0 . 35, E/H = 0 . 17, and ρ

E σ v +

2 u a +

σ h − u a μ u a − u w

ε h = − σ h − (3.32)

E H

The above equation applies to both horizontal directions.
For the at-rest or K 0 condition in an intact, homogeneous,
unsaturated soil mass, the strain in the horizontal directions
can be set to zero (i.e., ε h =

0). The net horizontal stress
can then be written in terms of the vertical stress:

μ σ v −

u a −

u a −

u w
(3.33)

μ

μ) E
H

σ h − u a = ( 1 −

1 −

0.6

0.538

0.5

0.4

Depth =

0.3

4.0 m
3.0 m

0.2

1.0 m 2.0 m

0.1

0

0 50 100 150 200

Matric suction ( u a − u w ), kPa

Figure 3.14 Relationship between coefficient of earth pressure at rest, K 0 , and matric suction.