Geoscience Reference
In-Depth Information
cos
sin
R
(9.16)
res
1
sin
sin
For given friction angle
and given dilation angle
, the value of
R
res
is
independent of
, and the corresponding residual strength ratio, according to
equation (91.4), becomes
2
:
7
cos
sin
R
sin
8
9
v
5
6
res
(9.17)
2
cos
sin
R
cos
h
res
res
Note that for associative behaviour, i.e.
=
, the traditional Coulomb solution
is found from (9.16):
R
cr
= tan
, used in the
familiar passive and active earth pressure coefficients, equation (7.5). Hence, for
associative material behaviour, the residual, critical and maximum strength are
identical. An example is worked out for
, and (9.15) becomes
c
r
= 45
=
+ ½
=
=
34
=
in Fig 9.6b. The maximum,
=
62
=
.
critical and residual strength all match at
v
/
b
h
maximum strength
maximum strength
4.5
residual strength
4.0
3.54
3.5
residual strength
3.04
3.0
62º
47.5º
62º
v
b
2.5
critical strength
critical strength
h
bta
n
2.0
T
1.5
N
1.0
45º
50º
55º
60º
65º
45º
50º
55º
60º
65º
(a) test (b)
=
=
34=
(c)
= 34=
and
= 5=
Figure 9.6 Biaxial test (Teunissen)
is shown in Fig 9.6c. Here, a residual strength
lower than the maximum strength and higher than the critical strength is found for
shear plane orientations in a range of 47.5
The case for
=
34
=
and
=
5
=
=
<
< 60.0
=
, and correspondingly the
residual strength is within a range of 3.04 to 3.54.
Coulomb's associative behaviour (
) provokes unrealistic volume change.
On the other hand a non-associative behaviour (
=
) following Davis, generates
unrealistic work when the stress state moves along the failure surface. This
paradox, a consequence of incompleteness of Coulomb's and Davis' models,
causes indeterminateness. Nature will decide. According to Teunissen, any possible
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