LIQUEFACTION ASSESSMENT BY THE ENERGY METHOD
THROUGH CENTRIFUGE MODELING
Hesham M. Dief, Associate Professor,
Civil Engineering Department,
Zagazig University, Zagazig, Egypt
J. Ludwig Figueroa, Professor
Department of Civil Engineering,
Case Western Reserve University, Cleveland, Ohio
The fundamentals of the energy method to assess the liquefaction potential of
cohesionless soils have been formulated in recent years. The procedure has been
validated through the torsional shear testing of several types of soils. An important step in
the process of development of this procedure would be to examine its validity through the
prototype-like conditions afforded by the centrifuge.
This paper discusses the results of 30 centrifuge liquefaction tests conducted at a
scale of 60g’s, on scaled pore fluid-saturated models of soil deposits with different grain
size characteristics, to assess their liquefaction potential by the energy method.
The influence of parameters such as relative density, effective confining pressure
and grain size distribution on the energy per unit volume required for liquefaction is
studied. Generalized relationships were obtained by performing regression analyses
between the energy per unit volume at the onset of liquefaction and these liquefaction
affecting parameters. These relationships are statistically compared with equations
previously developed at CWRU by Liang (1995), Kusky (1996) and Rokoff (1999). All
test results, comparisons and numerical simulations using the energy approach agree very
well in predicting whether or not liquefaction will occur, and if it does, where it will
happen within the deposit.
Introduction
Liquefaction of soils during earthquakes has received a lot of attention among the
geotechnical community and extensive research has been conducted during last three
decades to understand the mechanisms leading to it, in order to develop methods of
evaluating the potential for liquefaction. Nemat-Nasser and Shokooh (1979) introduced
the energy concept for the analysis of densification and liquefaction of cohesionless soils.
It is based on the idea that during deformation of these soils under dynamic loads part of
the energy is dissipated into the soil. This dissipated energy is represented by the area of
the hysteric shear strain-stress loop and could be determined experimentally. The
accumulated dissipated energy per unit volume up to liquefaction considers both the
amplitude of shear strain and the number of cycles, combining both the effects of stress
and strain. Compared with other methods to evaluate liquefaction potential of soils, the
energy approach is easy to deal with random loading because the amount of dissipated
energy per unit volume for liquefaction is independent of loading form. Berill et al.
(1985), Law et al. (1990), Figueroa (1990), Figueroa et al (1991, 1994) established
relationships between pore pressure development and the dissipated energy during the
dynamic motion; and to explore the use of the energy concept, in the evaluation of the
1
liquefaction potential of a soil deposit. Extensive research has been conducted at Case
Western Reserve University (CWRU) to introduce and evaluate the energy concept in
defining the liquefaction potential of soils when subjected to dynamic loads. Figueroa et
al. (1994) conducted 27 torsional shear-controlled strain liquefaction tests on Reid
Bedford sand to demonstrate the relationship between the dissipated energy per unit
volume at the onset of liquefaction and the relative density, the effective confining
pressure and the shear amplitude. Liang (1995) conducted strain-controlled torsional
triaxial experiments on hollow cylinders of sand to examine the effects of relative
density, initial confining pressure and shear strain amplitude and a numerical procedure
to simulate the seismic response of horizontal layers was also developed.
Additional torsional shear testing was conducted by Kusky (1996) and Rokoff (1999). In
order to validate the energy concept in defining the liquefaction potential of soils when
subjected to dynamic loads a number of liquefaction tests were conducted in a centrifuge
using the same soils previously tested by Liang (1995), Kusky (1996) and Rokoff (1999)
in a torsional shear device. The other objective of this study is to validate the theoretical
model developed by Liang (1995) for evaluating the liquefaction potential of a soil
deposit.
Test Equipment and Shake Table Performance
Meeting the objectives of the study requires using the shaking system installed at the 20
g-ton geotechnical centrifuge, operating at CWRU. The shake table was designed to
generate a single direction base excitation in the prototype horizontal plane that is defined
as the plane normal to the resultant direction of the centrifugal forces. Several
preliminary tests were carried out at a 60 g’s centrifuge acceleration to investigate the
performance of the shaking system. Figure 1 shows CWRU’s shake table response for
sine-wave inputs at an input amplitude of 1.0 volt. To use this facility in studying
liquefaction of soils, Dief (2000) developed a feedback correction procedure for the
shake table input signals as shown in the algorithm in Figure 2. After measuring the
frequency response function H(f), the initial input signal can be calculated by multiplying
the corrected signal x(t) by the gain factor η. The application of a gain factor during
calculation of a new signal estimate is necessary and its value depends on the dominant
frequency and average amplitude of the input signal. The results for generating the
VELACS record proved that this technique was able to simulate the desired signal with
excellent agreement in two iterations as shown in Figures 3 and 4.
Laboratory Testing
A total of 30 liquefaction tests were conducted on Nevada sand, Reid Bedford sand and
Lower San Fernando Dam (LSFD) Sand (which contains a significant silt content up to
28%) at 60 g’s to determine the prototype behavior in a centrifuge model. Relative
densities of 50%, 60%, 65%, 70% and 75% were considered. The model container used
in these tests is a laminar box designed to allow soil deformation in the longitudinal
direction with minimal interference in one-dimensional shear tests. Parameters such as
acceleration, displacement and pore pressure are monitored throughout the tests which
include the use of a viscosity-scaled pore fluid to ensure that the time scaling factor for
2
motion is the same as that for fluid flow. A sketch of CWRU’s laminar box and
instrumentations used for the soil model is presented in Figure 5.
90
80
(Output / Input)*100
70
60
50
40
30
1 Volt input
20
10
0
0
20
40
60
80
100
120
Frequency (Hz)
Figure 1 Shake Table Response Prediction
Measure and store
frequency response
function H(f)
Calculate initial input signal
Desired Signal: yd (t)
Desired Spectrum: Yd (f) = F yd (t)
Corrected Spectrum: X(f)= Yd (f)/ H(f)
Corrected Signal: x (t) = F-1 X(f)
xo (t) = η . x (t)
Apply Signal and
measured response
ya (t)
Calculate
Error
and
Percent Variance Error
e(t)= yd (t)- ya (t)
Correcting signal
for next input
E(f) = F e(t)
Calculate new input signal
xnew (t) = xold (t) + [η . d (t)]
Figure 2 Signal Correction Algorithm
3
D(f) = E(f)/ H (f)
d (t) = F-1 D (f)
Acceleration (g)
15
Desired acceleration
10
1_st Measured acceleration ----
5
0
-5
-10
-15
0
0.05
0.1
0.15
0.2
0.25
0.3
Time (sec)
Figure 3 First Measured and Desired Acceleration Signals
15
Desired acceleration
3_rd Measured acceleration -----
Acceleration (g)
10
5
0
-5
-10
-15
0
0.05
0.1
0.15
0.2
0.25
0.3
Time (sec)
Figure 4 Final Measured and Desired Acceleration Signals
Data Processing and Calculation
In dynamic centrifuge modeling, a procedure is developed for reconstructing the shear
stress-strain history to liquefaction at different depths, within the prototype, from the
recorded accelerations and lateral displacements of the laminar box segments as well as
for calculating the amount of dissipated energy per unit volume for each layer up to the
end of the earthquake (Dief, 2000). This dissipated energy is represented by the area of
the hysteric shear strain-stress loop (Figueroa, 1990; Figueroa et al., 1994; and Liang,
1995). The recorded horizontal accelerations and LVDT readings corresponding to
horizontal displacements can be processed and the lumped mass model may be used to
simulate the horizontal soil layers (Idriss and Seed, 1968 and Finn et al., 1977). A
horizontal soil deposit is divided into N layers and N+1 nodes. Lumped masses are
concentrated at the nodes and only have horizontal displacement.
4
53.3 cm
H
P4
AH4
LVDT4
P3
AH3
LVDT3
P2
AH2
P1
AH1
AH5
LVDT2
Y
LVDT1
X
(AH) Horizontal accelerometer
(LVDT) Linear variable differential
transformer
(P) Pore water pressure transducer
Figure 5. Laminar Box Container and Instrumentation
This lumped mass system, results in a group of equations which can be determined using
the free body diagram shown in Figure 6, where aj = acceleration of the j t h node with
..
mass mj, defined by: a j = U j (j= 1,2,..., N). Knowing the horizontal acceleration of the
j t h node and the j t h mass mj, the shear stress τj in the j t h layer can be calculated for
each node from top to bottom by using the equations of motion in the form of the central
difference method as follows (Dief, 2000):
..
mN U N = τΝ
(1)
..
mj U j = τj − τj+1
(2)
Where τj = shear stress in the j th layer
Also, knowing the horizontal displacements at the j t h node (Uj ) and the thickness of the
j t h layer (hj), the shear strain in the j t h layer, γ j can be determined (Zeghal and
Elgamal, 1994):
γ
j
=
U j −U
j −1
(3)
hj
5
The accumulated energy per unit volume (δW) absorbed by the specimen, until it
liquefies is given by Figueroa et al. (1994):
δW =
n −1
1
∑ 2 (τ
i =1
i
+ τ i +1 )(γ i +1 − γ i )
(4)
Where: n = the number of points recorded to liquefaction.
Then from equations 1, 2, 3 and 4 the accumulated energy per unit volume (δW)
absorbed by the specimen, until it liquefies can be determined (Dief, 2000).
Figure 6 Free Body Diagram of the Lumped Mass Model
Test Results and Analysis
All of the accepted tests exhibited the characteristic behavior of saturated, cohesionless
soils subjected to earthquake base excitation; therefore, the following test of Lower San
Fernando Dam sand is selected to represent the sand’s behavior. The specimen was
prepared at a relative density of 62.8% and tested at 60 g’s representing a prototype
thickness of 7.6 m. The corresponding total saturated and dry unit weights of the sand are
18.37 kN/m3 and 13.7 kN/m3 respectively (Dief, 2000). The excess pore pressure ratios
( ru = p σ v− , where p is excess pore pressure and σ v− is the initial effective vertical stress)
obtained from the records of the four pore-pressure transducers for the selected test are
shown in Figure 7. The records show the rapid build up of the pore pressure ratios of
transducer P1 during the first 2.6 seconds of base excitation.
At this stage, the soil structure loses its integrity indicating initial liquefaction. From
this point on, several decreases and increases in the excess pore pressure happen until
the end of base excitation. All layers reached final liquefaction after 6 seconds of the start
of shaking. As shown in the figure, the excess pore pressures at P1, P2, and P3 continued
without any dissipation after stopping the base excitation.
The time histories of the recorded horizontal accelerations in the soil are given in Figure
8. Results show that the soil followed the base excitation only up to approximately 6
seconds from the start of shaking followed by a decrease of the acceleration signals until
they are barely noticeable. The substantial decrease in the acceleration signals within the
upper layers indicates excellent consistency among the results. By comparing the
acceleration records with the time series of excess pore pressure ratios, an agreement
between the acceleration spikes and the instantaneous drops in pore pressure is noticed.
6
Excess Pore Pressure Ratio
1.2
1
0.8
0.6
0.4
0.2
0
P1
Excess Pore Pressure Ratio
0
2
Excess Pore Pressure Ratio
6
8
Time (sec)
10
12
1.2
1
0.8
0.6
0.4
0.2
0
14
16
P2
0
2
4
6
8
Time (sec)
10
12
14
1.2
1
0.8
0.6
0.4
0.2
0
16
P3
0
Excess Pore Pressure Ratio
4
2
4
6
8
Time (sec)
10
12
1.2
1
0.8
0.6
0.4
0.2
0
14
16
14
16
P4
0
2
4
6
8
Time (sec)
10
12
Figure 7 Excess Pore Pressure Ratio Time Histories
Characteristic hysteresis loops are generated by plotting the shear stress versus the shear
strain developed within the deposit at specific depths. Figure 9 shows the shear stressstrain relationships during two selected loading cycles. During the first cycle, (0.0- 2.6
sec.) the soil is stiff and the shearing resistance builds up without large strains. As shown
7
Acceleration (g)
0.3
0.2
0.1
0
-0.1
AH1
-0.2
-0.3
Acceleration (g)
0
5
Acceleration (g)
15
20
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
AH2
0
5
10
Time (sec)
15
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
20
AH3
0
Acceleration (g)
10
Time (sec)
5
10
Time (sec)
15
0.3
0.2
0.1
0
-0.1
20
AH4
-0.2
-0.3
0
5
10
Time (sec)
15
20
Figure 8 Horizontal Acceleration Time Histories
in Figure 7, the pore pressure generation is not high enough during these cycles (ru =
0.75) to produce liquefaction in the soil. A continuous reduction of shear strength has
been displayed after the first cycle and loops tend to become progressively flatter ending
with a clear stiffness reduction at (10.85-11.45 sec) with ru = 1, and the stress-strain
relationship is almost horizontal as shown in Figure 9.
8
15
10
10
Shear Stress (kPa)
Shear Stress (kPa)
15
5
0
-5
(0-1.6 sec)
-10
-0.5
0
0.5
Shear Strain (%)
1
-5
(10.85-11.45 sec)
-15
-1.5
-15
-1
0
-10
ru = 0.75
-1.5
5
1.5
ru = 1
-1
-0.5
0
0.5
1
1.5
Shear Strain (%)
Figure 9 Shear Stress-Strain Relationships During
Selected Cycles at a Depth of 5.3 m
The accumulated energy per unit volume (J/m3) required for liquefaction, computed from
centrifuge tests is determined at the point of initial liquefaction where the pore pressure
for the liquefied layer initially reaches the effective overburden pressure ( ru = 1 ). For all
tests, the accumulated energy per unit volume required for liquefaction is determined
using the procedure explained before in Equations 1 through 4. Figure 10 shows the
variation of the total accumulated energy per unit volume for each layer of the selected
test of Lower San Fernando Dam sand. It is observed that the major contribution to the
energy per unit volume occurs at the time of the high pore pressure build up. From Fig. 7
1400
Accumulated Energy/ Volume (J/m^3)
d = 5.3 m
1200
d = 4.6 m
1000
d = 3.8 m
800
d=3m
600
d = 2.3 m
400
d = 1.5 m
200
d = 0.75 m
0
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Figure 10 Accumulated Energy per Unit Volume Time History at Different Depths
9
and 10 it is observed that the nature of the curve of the increase of pore pressure is similar
to the one of the accumulated energy per unit volume, indicating that the energy per unit
volume buildup is related to the build up of the pore pressure as well as liquefaction.
Regression Equations
Relationships were obtained using the centrifuge test data by performing regression
analysis between the energy per unit volume dissipated in generating liquefaction ( δw in
J/m3) as the dependent variable and the relative density (Dr in %) and the effective
′
confining pressure ( σ c in kPa) as the independent variables. These equations are useful
in comparing the amount energy required for liquefaction between torsional shear and
centrifuge tests. The resulting equations were:
Nevada Sand,
′
log10 (δw) = 1.164 + 0.0124 σ c + 0.0209 D r
R2 = 0.943
(5)
R2 = 0.883
(6)
Reid Bedford Sand,
′
log10 (δw) = 1.647 + 0.0179 σ c + 0.0123 D r
LSFD Silty Sand,
′
log10 (δw) = 2.4597 + 0.00448 σ c + 0.00115 D r
R2 = 0.972
(7)
where: R2= coefficient of determination
Centrifuge test results were compared with the equations based on torsional shear test
data, developed by Liang (1995), Kusky (1996) for Reid Bedford sand and Liang (1995)
for Lower San Fernando Dam sand at a gravity level of 60 g’s as shown in Figures 11
and 12 respectively. Also, centrifuge test results are compared with the equations
specified by Rokoff (1999) for Nevada sand at a gravity level of 60 g’s as shown in
Figure 13. Centrifuge test results are averaged using the logarithmic curve fit which
provides the best approximation to the data as tested and concluded before by Liang
(1995) and Rokoff (1999).
It is noticed that the centrifuge-based equation for Reid Bedford sand is parallel with all
developed equations by Liang (1995) for both random and sinusoidal loading types as
shown in Figure 11. The curve representing Kusky’s equation converges with the curves
based on the centrifuge equation and on Liang’s equation (for random loading) as the
relative density increases, indicating a slightly different slope than the other equations. It
is seen that the shift between the curves representing the centrifuge equation and the
random loading equation developed by Liang (1995) is smaller than the shift between the
curves of Liang’s random and sinusoidal loading equations with the latter appearing on
the farther side of the curve of the centrifuge equation.
10
Energy/Volume (J/m3)
10000
Liang's Eq.-Random (1995)
Centrifuge Results Eq.
1000
Kusky's Eq.-Sinusoidal (1996)
Liang's Eq.-Sinusoidal (1995)
Centrifuge Results
100
50
55
60
65
70
75
80
85
Relative Density (%)
Figure 11 δw vs. Dr (Reid Bedford sand
σc−=34kPa)
As shown in Figure 13, the curve of centrifuge results is very close and parallel to the
torsional shear equation developed by Rokoff (1999) for Nevada sand. The equivalency
of the Nevada sand equation developed by Rokoff (1999) through sinusoidal torsional
shear testing with the curve representing the centrifuge test data supports the
1000
Energy/Volume (J/m3)
Centrifuge Results Equation
Liang's Equation -Random (1995)
Centrifuge Results
100
50
55
60
65
70
75
80
85
90
95
Relative Density (%)
Figure 12 δw vs. Dr (Lower San Fernando Dam sand
11
σc−=31kPa)
100
conclusion that the unit energy to liquefaction is independent of type of loading. Figure
12 shows the plots of the developed centrifuge equation and Liang’s equation for Lower
San Fernando Dam silty sand. As shown in the figure, the curves corresponding to the
two equations are close at their loosest state and diverge slightly at high values of relative
density. These observations confirm the conclusion reached before, indicating that the
two equations can be assumed to be equivalent up to relative densities of 75%.
Energy/Volume (J/m3)
10000
1000
Rokoff's Equation (1999)
Centrifuge Results Equation
Centrifuge Results
100
45
50
55
60
65
70
75
80
Relative Density (%)
Figure 13 δw vs. Dr (Nevada Sand
σc−=34 kPa)
A rational procedure introduced by Liang (1995) to decide whether or not liquefaction of
a soil deposit is imminent can be formulated by comparing the calculated unit energy
from the time series record of a design earthquake with the resistance to liquefaction in
terms of energy, based on in situ soil properties. The dissipated energy per unit volume
during the earthquake can be computed using this numerical procedure to calculate the
seismic response of horizontal soil layers to give shear stresses and shear strain histories
for each layer (Figueroa et al., 1998). The in situ resistance to liquefaction in terms of
energy can be determined by applying the pre-described Equation 5 for Nevada sand and
Equation 6 for Reid Bedford sand and Equation 7 for Lower San Fernando Dam sand.
Figure 14 shows a comparison between the energy per unit volume dissipated into the
soil layers obtained from centrifuge test results and the numerical procedure developed
by Liang (1995), corresponding to a selected test of Nevada sand of a relative density of
58.5% representing a prototype thickness of 7.6 m (with a total saturated and dry unit
weights of 19.7 kN/m3 and 15.85 kN/m3 respectively) as well as the numerical results of
the energy required for liquefaction obtained by applying Equation 5 (Dief, 2000).
According to the experimental results it is seen that liquefaction is initiated in the middle
12
layers of the deposit at a depth of about 4.5 m from the soil surface, where the dissipated
energy exceeds the resistance. Numerical results show that liquefaction is imminent at
depths of 5.5 to 6.5 m and extended to a depth of 4.5 m from the soil surface after about
10 seconds from start of shaking. Centrifuge test results show a reasonable supported
agreement with the results of the numerical procedure developed by Liang (1995),
confirming the accuracy of the energy method for evaluating the liquefaction potential of
a soil deposit.
0
1
Eq u a t i o n 5
2
Depth (m)
3
t = 8 s e c ( EXP )
t =1 2 s e c ( EXP )
4
t = 5 s e c ( EXP )
t =1 2 s e c ( N U M )
5
t = 5 sec (NUM)
t =8 s e c ( N U M )
6
Dissipa t e d En e r g y ( Ex p e r i m e n t a l )
7
Dissipa t e d En e r g y ( N u m e r i c a l)
R e s i s t a n c e En e r g y
8
0
200
400
600
Accumulated Energy/Volume (J/m^3)
800
1000
Figure 14 Determination of the Liquefaction Potential of a Soil Deposit of Nevada
Sand Using the Energy Method
Summary and Conclusions
The use of the energy concept to define the liquefaction potential of soils when subjected
to dynamic loading has been examined and validated through a series of centrifuge tests.
A total of 30 liquefaction tests at selected relative densities were conducted on specimens
of Nevada sand, Lower San Fernando Dam sand and Reid Bedford sand. Parameters such
as accelerations, displacements and pore pressure were monitored through the tests. The
amount of the dissipated energy per unit volume in a soil deposit in centrifuge modeling
can be determined by using the simplified procedure developed herein. The values of the
energy per unit volume at the onset of liquefaction in each of the thirty individual cases
are estimated, and the influence of the relative density and the confining pressure on the
unit energy level required for liquefaction was examined. It is observed that at the same
confining pressure, finer soils need lower energy per unit volume to reach liquefaction.
Centrifuge test results show that the total energy required for liquefaction increases as the
13
relative density increases confirming the results of torsional shear tests conducted at
CWRU. It is noticed that the energy per unit volume increase is related to the pore
pressure development, with the major contribution to the energy per unit volume
occurring at the time of the higher pore pressure build up. Centrifuge test results
equations showed a consistent trend and close agreement with measured data provided by
the regression equation developed by Liang (1995), Kusky (1996) and Rokoff (1999) to
estimate the resistance of a soil deposit to liquefaction. The theoretical predictions of
liquefaction using the numerical procedure developed by Liang (1995) were compared
with actual observations during centrifuge testing. A rational procedure to decide whether
or not liquefaction is imminent can then be formulated by comparing the calculated unit
energy from the time series record of a design earthquake with the resistance to
liquefaction in terms of energy, based on in situ soil properties, and if it does where and
when it will happen within the deposit.
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15