Liquefaction - principles
Consider a box of dry sand, subjected to cycles of
shear strain. On initial loading, sand usually
compacts and then dilates. On unloading, the sand
follows a similar path to loading, but some residual
volume strain remains at zero strain. Depending on
the initial porosity, this can represent a net
compaction.
expansion
compaction
Now suppose the pores are filled with water.
For a constant-volume test, the effective stress decreases, but
the pore pressure remains constant.
For a constant-applied-load test (e.g., normal load on box is
fixed), the pore pressure increases and the effective stress
decreases.
Liquefaction occurs when the effective stress becomes zero.
Liquefaction principles (2)
Thus, pore-pressure increase is not the primary cause of
liquefaction; it is the decrease in effective stress due to lower
contact forces between particles (after re-arrangement).
Pore pressure does increase in some cases: e.g., if there is a
constant overburden (total) stress.
With a comprehensive constitutive model (such as the BSHP
model) the process of liquefaction would occur “automatically,”
with the associated pore pressure changes being handled by
FLAC’s fully coupled calculation.
However, with a simple Mohr Coulomb model, we can treat
liquefaction by simplified procedures, in which volume strains
are added to elements, as a function of cyclic strain
amplitude.
Simple models
Two simple volume-change models are built into FLAC 4.0:
Martin, Finn & Seed (1975)
Byrne (1991)
∆ε vd is the incremental volume strain
ε vd is the accumulated volume strain
γ
is the amplitude of the last cycle of strain
Shear strain
time
These relations are incorporated into a Mohr Coulomb
constitutive model, with an algorithm for detecting when a strain
“reversal” has occurred, and what was the amplitude of the last
cycle. (The model is a DLL, automatically loaded.)
Use of Martin or Byrne models
The determination of the four constants for the the Martin et al
formula is awkward, because it is necessary to simulate a cyclic
test with FLAC, and adjust the constants iteratively, to obtain a
match to the known liquefaction data (e.g., cycles to
liquefaction).
For the Byrne formula, some guidance is given in his 1991
paper, in which the two constants are related to relative
densities and SPT results:
Liquefaction tests may still be simulated with FLAC, to verify that
these empirical relations lead to a match with the data, and used
to adjust the constants if there are differences.
Simulation results
pore pressure
pore pressure
effective stress
effective stress
time
time
Martin et al formula
Byrne formula
The curves show the results of shaking table simulations, using
the two alternative relations (the actual values of the constants are
provided in the FLAC manual). The total normal stress was
constant in the simulations.
A third simple approach
Roth et al have devised a scheme
that uses a cyclic-strength curve
directly, and generates increments of
pore pressure in FLAC. Each cyclic
stress excursion (normalized to
effective vertical stress) is measured:
CSR = τ cy / σ v′
Then the number of cycles to
liquefaction is determined from the
cyclic strength curve, and used to
compute a “damage” increment:
∆D i = 0.5 / N Li
Finally, the generated pore pressure increment is derived from the
damage increment: ∆u = ∆D iσ ′
g
v
Roth approach (2)
Roth & Dawson (2000) describe this approach, which was
used in a full-scale simulation of Pleasant Valley Dam (Roth,
Bureau & Brodt, 1991), to calculate the residual deformation
that would accumulate during shaking.
Although this approach generates pore pressures directly, it is
conceptually similar to the previous two simple approaches,
which generate volume-strain increments, and use FLAC’s
coupled logic to calculate the resulting pore pressure
changes. The three schemes should give similar results when
the fluid is considerably stiffer than the soil (for volumetric
compression).
The choice of method will probably depend on the ease by
which standard test data can be used by the model.
Roth approach (3)
The scheme has been applied to several cases of dam
shaking. The following plot of excess pore pressures is
reproduced from Roth, Inel, Davis & Brodt (1993).
Full, nonlinear approach
In the previous schemes the volume-change calculation is not part of
the constitutive law (in fact, a standard Mohr Coulomb law is used).
An estimate of cyclic strain (or stress) is made by looking at
successive peaks.
With a full, nonlinear law, the volume strain evolves continuously as
part of the formulation.
stress difference
In the BSHP model mean effective stress
decreases with cycling, as shown:
Thus, volume-changes, stress changes,
pore-pressure changes, damping and
plastic flow are integrated into one FLAC
model.
mean effective stress
Choice of method
There seem to be three main approaches (if we restrict ourselves
to FLAC as the nonlinear code of choice):
1. Equivalent-linear methods, using empirical schemes to
estimate permanent displacements and pore pressure
changes after the calculation is finished.
2. Time-marching simulation with FLAC, using simple schemes
to represent damping and cyclic volume-changes (leading to
pressure and effective stress changes). Deformations and
pore pressures accumulate as the calculation proceeds.
3. Time-marching simulation with FLAC, using a comprehensive
constitutive law, such as BSHP. There is no need for
approximate damping and volume-change schemes.
Choice of method – cont.
The third approach is – potentially – the closest to reality. In one
model, all the important mechanisms are present; e.g….
Output from a
simulation of New
Exchequer Dam by
Makdisi, Wang &
Edwards (2000),
using the BSHP
model with FLAC
(Fish version).
Choice of method – cont.
However, the simplified approach (#2) uses far fewer material
parameters, and is conceptually much simpler (and therefore it’s
much easier to use common-sense checks on the results).
In both FLAC approaches, the evolution of permanent damage,
large strain, nonlinear wave effects and pore pressure build-up (&
dissipation) are simulated. In contrast, the equivalent linear
method takes drastic liberties with physics, although empirical
rules and a long history of “tuning up” allow it to make good
predictions.
Engineers should at least start using the fully nonlinear approach
in addition to the equivalent linear approach. Otherwise no
experience will be accumulated.