Civil Engineering Reference
In-Depth Information
particular strain,
G
p
=
AY
p
(13.10)
G
nc
/
p
is the stiffness of normally consolidated soil at the same strain.
(Notice that Eq. (13.8) reduces to Eq. (13.10) when
n
where
A
=
1.) When soil is normally
consolidated its state is on the state boundary surface so values for
G
nc
are given by
Eq. (12.18). Values for
m
depend on the nature of the soil and on the strain. A number
of other factors, such as a rest period at constant stress and a change in the direction
of the stress path between successive loading stages, also effect soil stiffness, but the
rate of loading has virtually no effect provided that the soil is either fully drained or
fully undrained.
=
13.7 Rigidity and non-linearity
It is clear that the stress-strain behaviour of soil is highly non-linear over most of the
range of practical interest in ground engineering and this non-linear behaviour should
be taken into account in design. There are some relatively simple parameters which
can be used to describe how non-linear a soil is (Atkinson, 2000).
For materials which are linear-elastic and perfectly plastic rigidity
R
was defined in
Sec. 3.6 as
E
q
f
R
=
(13.11)
where
E
is Young's modulus and
q
f
is strength expressed as the diameter of the Mohr
circle at failure. For soils, which are highly non-linear and which have a peak and a
critical state strength, rigidity can be defined as
E
0
q
p
R
=
(13.12)
where
E
0
is Young's modulus at very small strain and
q
p
is the shear stress at the peak.
Figure 13.11(a) shows a non-linear stress-strain curve up to the peak state and
Fig. 13.11(b) is the corresponding stiffness-strain curve. At the point X the shear stress
is
q
x
, the strain is
x
and the tangent Young's modulus is
E
x
. The tangent Young's
ε
modulus is given by
d
q
d
E
t
=
(13.13)
ε
and hence
q
x
ε
x
q
x
=
d
q
=
E
t
d
ε
(13.14)
o
o
