6. Proposed Densification Simulation Model
6.4. Induced pore pressure due to energy dissipation
Previous studies on liquefaction (e.g. Nemat-Nasser and Shokooh 1979, Davis and Berrill 1982, Law et al. 1990, and Thevanayagam et al. 2002) have realized that liquefaction and pore pressure generation in soil is uniquely related to energy input per unit volume of soil, confining stress, and inter-grain contact density of sands and silty sands. The implication of these studies is that pore pressure profile and extent of liquefaction in a deposit can be predicted, provided that the distribution of energy dissipated in the soil surrounding the impact location can be evaluated. While most laboratory investigations concerning the liquefaction potential of sand have centered on the relationship between generated pore pressure, shear stress amplitude, and number of stress reversals to cause failure, some research has brought into question the possibility of a relationship between induced pore pressures and dissipated energy. Such a relationship offer considerable advantages in the application of experimental results to field situations.
In order to indicate the effectiveness of the energy method, an example is driven for the cases of earthquake loading; the conventional methods require the selection of an equivalent uniform shear stress and an equivalent number of loading cycles in order to model field response to actual earthquakes, by relating the pore pressure buildup to the amount of energy dissipated in the soil, then the uniform stress and equivalent cycles concepts could be completely avoided. Any sequence of variable stress reversals resulting in a certain amount of energy dissipation would be equivalent, in terms of pore pressure generated, to any other sequence resulting in the same dissipated energy.
There are many models in the literature in which the pore pressure was directly related to the amount of seismic energy dissipated in the soil, rather than the assumption of the pore pressure dependency on the shear stresses generated by ground shaking.
These models require the knowledge of the energy loss per unit volume of soil. In the following, some of the energy-based procedures developed from earthquake case histories or from experimental data for evaluation of liquefaction potential are reviewed for purpose of assessing the relation between the dissipated energy and the increase in pore pressure for potential use in pore pressure generation model.
Davis and Berrill (1982), and Trifunac (1995) have suggested that the increase in pore pressure is a simple linear function of the dissipated energy
u
c
r η w σ
= Σ
′ (6-3)
________________________________________________________________________
Where ru : Pore pressure ratio (pore pressure normalized to the effective confining stress σ’c, η : dimensionless constant relating the dynamic excess pore pressure to the
dissipated energy density, and Σw : energy loss per unit volume of soil.
Berrill and Davis (1985), in their revised model have assumed the pore pressure proportional to the square root of dissipation energy.
Law et al. (1990) have defined a nonlinear unique relation between the dissipated energy during cyclic triaxial and cyclic simple shear tests and excess pore pressure that led to liquefaction failure.
ru =β*( )WN ξ (6-4) Where 1( ) *c 2( r) *
N
h
F k F D w
W σ
= Σ
′
( )
1 c 1 * log c
F k = −χ k
( ) ( 0.7)
2 r 10 Dr
F D = θ −
WN: normalized dimensionless energy.
kc: consolidation ratio (kc = 1 for isotropic consolidation).
Dr: relative density.
F1 (kc): normalizing function to account for kc.
F2 (Dr): normalizing function to account for Dr.
χ: parameter depends on soil type and test condition, for this test series χ = 3.
θ: parameter depends on soil type and test condition, for this test series θ = −2.
By substituting
1( ) * 2( ) *
* c r
u
h
F k F D w
r
ξ
β σ
⎡ Σ ⎤
= ⎢⎣ ′ ⎥⎦ (6-5)
Fig. 6-4 shows typical test results for both cyclic triaxial and torsional simple shear tests (refer to the original paper for marks details).
0.0 0.5 1.0
0 0.02 0.04
WN Δu/σh'
Figure 6-4 Normalized excess pore pressure versus normalized dissipated energy (Law et al.
1990)
Figueroa et al. (1995), have developed a relationship based on a series of torsional shear liquefaction tests.
*
u
a b w
r c w
+ Σ
= + Σ (6-6)
Where
a,b,c: regression constants.
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Davis and Berrill (2001) have assumed a nonlinear relation between induced pore pressure and dissipated energy
1 c
w
ru e ζσ
− Σ
= − ′ (6-7) Thevanayagam et al. (2002) have developed a relation based on extensive research on liquefaction potential of granular mixes ranging from clean sands to pure silts. It has been concluded that the cyclic pore pressure ratio increases log linearly with E/EL
10
0.5* log 100 , 0.05
u
L L
E E
r = E E > (6-8) Where EL is the energy required to cause liquefaction. Appendix B summarizes some of the experimental results for EL in terms of developed contact density indices.
The model by Law et al. (1990) has been used in the analysis since it considers different values of confining stresses and relative densities. The values of β and ξ have been evaluated from the reported experimental results (law et al., 1990), and
(Thevanayagam et al. 2002) by means of regression analysis to fit the reported data (Fig.
6-5).
It should be referred here to the diminishing effect of applying more energy while the excess pore pressure has not completely dissipated, because soil lose part of its strength as the excess pore pressure develops due to surface impact and become weaker.
As a result the energy that can be imparted to the soil through further impacts decreases.
Therefore, the model was modified to be sensitive to soil status at the instant of applying further impacts.
0.3 0.5 0.7 0.9 1.1
100 1000 10000
EL (Nm/m3) (ec)eq
Model OS-00 OS-15 OS-25
Figure 6-5 Model versus experimental results