Civil Engineering Reference
In-Depth Information
Figure 10.8
Power law peak strength envelope for shear tests.
φ
c
. The only smooth envelope which can meet these requirements is
the curved line OAC shown in Fig. 10.8(a).
A convenient equation for this line is a power law
τ
c
=
σ
c
tan
where
τ
p
=
σ
p
a
(10.11)
or
τ
p
=
σ
p
+
log
log
a
b
log
(10.12)
so the curved peak strength envelope is linear if the stresses are plotted with logarithmic
axes, as shown in Fig. 10.8(b). If
b
=
1 then Eq. (10.11) is linear and the parameter
a
φ
c
in Eq. (9.2). The parameter
b
describes how curved the peak
envelope is: the smaller the value of
b
the greater the curvature. Both parameters
a
and
b
are state dependent parameters and they depend on the voids ratio or water content.
To take account of voids ratio Eq. (10.11) can be normalized with respect to the
critical stress
is equivalent to tan
σ
c
. The power law for the peak failure envelope becomes
A
σ
p
σ
c
B
τ
p
σ
c
=
(10.13)
φ
c
and Eq. (10.13)
Since the peak envelope ends at the critical state point,
A
=
tan
becomes
σ
p
σ
c
B
τ
p
σ
c
=
φ
c
tan
(10.14)
or
log
τ
p
σ
p
B
log
σ
p
σ
c
φ
c
)
=
log(tan
+
(10.15)
