702
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
3 PARAMETER IDENTIFICATION
The calibration of the model parameters is illustrated by
considering the example of a typical kakirite sample taken from
the Sedrun section of Gotthard Base Tunnel (Anagnostou et al.
2008). The sample was subjected to a multistage consolidated
drained test (CD test) under consolidation pressures of 2, 5 and
9 MPa. The solid lines in Figures 2a and 2b show the deviatoric
stress and the volumetric strain, respectively, as a function of
the axial strain for a confining pressure of 5 MPa. The non-
linearity of the stress-strain relationship before failure is
significant. The unloading curves in Figure 2a show that
irreversible strains develop right from the start of deviatoric
loading.
The shear strength parameters of the MC model can be
determined in the usual manner, i.e. from the regression line in
the principal stress diagram. The dilatancy angle
of the MC
model can be determined from the slope of the
ε
1
and
ε
vol
curve
(Fig. 2b), taking into account that the slope, i.e.
2sin
1 sin
b
.
(16)
Three parameter sets were chosen for the MC model, which are
different with respect to the Young’s modulus and the dilatancy
angle (Table 1). Parameter set 1 assumes that the Young’s
modulus is equal to the unloading-reloading modulus. Set 2
adopts the secant modulus as Young’s modulus in order to
better map the stress curve. Set 3 is slightly different from set 1
and was chosen in order to better map the volumetric strain
behavior (Fig. 2). Poisson’s ratio, which typically is in the range
0.20-0.35, was taken equal to 0.30.
The MHS model has, as mentioned in Section 2, nine
parameters, four of which are the same as for the MC model (
,
c
f
,
f
and
f
). The reference mean stress
p
ref
is chosen as 5 MPa,
which means that the moduli
E
ur,ref
and
E
50,ref
were determined
under a radial stress of 5 MPa. The parameter
m
of the power
law that expresses the stress dependency of the moduli
(
Eqs. 3
and 5) can be determined from the slope of the (
E/E
ref
) over
(
3
*
/p
ref
*
) regression line in a log-log diagram (Fig. 3).
According to Figure 3,
m
is equal to about 0.9 or 1.9 depending
on the considered modulus (unloading-reloading or secant). In
order to consider the influence of
m
, both parameter sets will be
considered in the computations of the next Section.
4 MODEL BEHAVIOR IN TRIAXIAL DRAINED TESTS
The dashed lines in Figure 2 show the behavior of the MC
model for the three-parameter sets of Table 1. Parameter set 1
overestimates the stress before failure. Set 2 better predicts the
stress before failure, but cannot reproduce the unloading-
reloading behavior satisfactorily, of course. In addition, as
yielding occurs at a larger axial strain, the reversal in the
volumetric behavior occurs also later in the case of set 2. Set 3
was chosen in order to map the volumetric strain behavior
better. It presents of course the same problem as set 1
(overestimation of the pre-failure stress or, equivalently,
underestimation of the pre-failure strain for given axial stress).
The behavior of the MHS model (solid line with points in
Figure 2) can be easily determined by stepwise integrating the
constitutive equations in a spreadsheet. The line applies to both
parameter sets: As the radial stress is kept constant during
deviatoric loading, the moduli
E
ur
and
E
50
remain constant and,
since in the test of Figure 2 the radial stress is equal to the
reference stress (5 MPa),
E
ur
= E
ur,ref
,
E
50
=
E
50,ref
and the
parameter
m
of the power law is irrelevant. The MHS predicts a
non-linear stress-strain curve, which maps the observed
behavior better than the MC model, but slightly underestimates
the deviatoric stress close to failure, i.e. it reaches the ultimate
state more slowly than observed. The MHS model maps well
the measured peak volumetric strain, but reaches the peak value
later than observed.
One important feature of the MHS model is that it accounts
for the stress dependency of the deformation moduli. This is
why we examined the model behavior also under radial stresses
that are different than that in the test used for the parameter
determination (5 MPa). Figures 4 and 5 apply to radial stresses
of 2 and 9 MPa, respectively.
Consider the case of a lower radial stress (2 MPa, Fig 4). The
MC model greatly overestimates the stress before reaching
failure and greatly underestimates the peak volumetric strain. In
the case of a higher radial stress (9 MPa, Fig. 5), parameter set
2, which is based upon the secant modulus and maps the
observed behavior for the reference radial stress of 5 MPa well
(Figure 2), shows the greatest deviation from the test results
(both with respect to pre-failure stress and to the volumetric
strain). On the other hand, sets 1 and 3, which did not
reproduced the behavior in the reference case of 5 MPa well,
lead now to acceptable results.
The MHS model better maps the observed behavior over the
considered radial stress range, although it may also overestimate
the pre-failure stress particularly at lower radial stresses.
5 CONCLUSIONS
The MHS model has four parameters more than the widely used
MC model. The parameters have, however, a clear physical
meaning and can be determined from the same test results as the
MC model. The MHS model predicts the behavior under
different stress levels better than the MC model. This may be
significant for modeling the conditions around deep tunnels in
weak rock, where the minimum principal stress decreases
significantly in the vicinity of the opening. The effect of the
constitutive law on the response of the ground to tunnel
excavation under drained or undrained conditions is currently
under investigation.
6 ACKNOWLEDGEMENTS
The authors appreciate the financial support of the Swiss
National Science Foundation (project 200021-137888).
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