Civil Engineering Reference
In-Depth Information
where
μ
=
Poisson's ratio.
σ
µ
σ
1
i e total strain on this plane
. .
= −
(
+
σ
).
2
3
E
E
Similarly, strains on the other two planes are:
σ
µ
σ
2
−
(
+
σ
)
3
1
E
E
σ
µ
σ
3
−
(
+
σ
)
1
2
E
E
Now it can be shown that, no matter what the stresses on the faces of the cube, the volumetric strain is
equal to the sum of the strains on each face.
∆
V
V
(
σ
+ +
σ
σ
)
2
µ
σ
1
2
3
−
=
−
(
+ +
σ
σ
)
1
2
3
E
E
i.e.
∆
V
V
1 2
−
µ
σ
−
=
(
+ +
σ
σ
)
1
2
3
E
Compressibility of a material is the volumetric strain per unit pressure, i.e. for a soil skeleton,
c
=
∆
V
V
C
per unit pressure increase
1
3
Average pressure increase
=
(
σ
+ +
σ
σ
. Therefore, for a perfectly elastic soil:
)
1
2
3
3 1 2
(
−
µ σ
) (
+ +
+ +
σ
σ
)
3 1 2
(
−
µ
)
1
2
3
C
c
=
=
E
(
σ
σ
σ
)
E
1
2
3
Consider a sample of saturated soil subjected to an undrained triaxial test. The applied stress system
for this test has already been discussed (Fig.
4.16)
. The pore water pressure, u, produced during the test
will be made up of two parts corresponding to the application of the cell pressure and the deviator stress.
Let
u
a
=
pore pressure due to
σ
3
u
d
=
pore pressure due to (
σ
1
−
σ
3
).
If we consider the effects of small total pressure increments
Δ
σ
3
and
Δ
σ
1
then
Δ
σ
3
will cause a pore pres-
sure change
Δ
u
a
and
Δ
σ
1
−
Δ
σ
3
will cause a pore pressure change
Δ
u
d
.
Effect of
Δ
σ
3
When an all-around pressure is applied to a saturated soil and drainage is prevented, the proportions
of the applied stress carried by the pore water and by the soil skeleton depend upon their relative
compressibilities:
C
=
−∆
∆
σ
3
V
Compressibility of the soil C
V
