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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

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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

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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.

Figure 1 Shake Table Response Prediction

Figure 2 Signal Correction Algorithm

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)

D(f) = E(f)/ H (f) d (t) = F-1 D (f) Calculate new input signal

xnew (t) = xold (t) + [η . d (t)]

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0 20 40 60 80 100 120

Frequency (Hz)

(Out

put /

Inp

ut)*

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1 Volt input

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5

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0 0.05 0.1 0.15 0.2 0.25 0.3

Time (sec)

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eler

atio

n (g

)Desired acceleration 1_st Measured acceleration ----

Figure 3 First Measured and Desired Acceleration Signals

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Time (sec)

Acc

eler

atio

n (g

)

Desired acceleration 3_rd Measured acceleration -----

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.

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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 th node with

mass mj, defined by: a j = U j

..(j= 1,2,..., N). Knowing the horizontal acceleration of the

j th node and the j th mass mj, the shear stress τj in the j th 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 th node (Uj ) and the thickness of the

j t h layer (hj), the shear strain in the j t h layer, γγ j can be determined (Zeghal and

Elgamal, 1994):

j

jjj h

UU 1−−=γ (3)

53.3 cm

H Y

LVDT4 LVDT3 LVDT2

LVDT1

X

P4 P3 P2 P1

AH4 AH3 AH2 AH1

AH5

(AH) Horizontal accelerometer (LVDT) Linear variable differential transformer (P) Pore water pressure transducer

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The accumulated energy per unit volume (δW) absorbed by the specimen, until it liquefies is given by Figueroa et al. (1994):

))((2

111

1

1iii

n

iiW γγττδ −+= ++

−

=∑ ��� (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 ( −= vu pr σ , where p is excess pore pressure and −

vσ 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.

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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 stress-strain 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

00.20.40.60.8

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0 2 4 6 8 10 12 14 16Time (sec)

Exc

ess

Pore

Pre

ssur

e R

atio

P1

0

0.20.40.6

0.8

1

1.2

0 2 4 6 8 10 12 14 16Time (sec)

Exc

ess

Pore

Pre

ssur

e R

atio

P2

00.20.40.60.8

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0 2 4 6 8 10 12 14 16Time (sec)

Exc

ess

Pore

Pre

ssur

e R

atio

P3

00.20.40.60.8

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0 2 4 6 8 10 12 14 16Time (sec)

Exc

ess

Pore

Pre

ssur

e R

atio

P4

8

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.

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AH4

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AH3

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-1.5 -1 -0.5 0 0.5 1 1.5Shear Strain (%)

Shea

r St

ress

(kP

a)

(0-1.6 sec)

ru = 0.75

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0

5

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15

-1.5 -1 -0.5 0 0.5 1 1.5

Shear Strain (%)

Shea

r St

ress

(kP

a)

(10.85-11.45 sec)

ru = 1

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 ( 1=ur ). 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

Figure 10 Accumulated Energy per Unit Volume Time History at Different Depths

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0 2 4 6 8 10 12 14 16 18 20

Time (sec)

Acc

umul

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Ene

rgy/

Vol

ume

(J/m

^3)

d = 5.3 m

d = 4.6 m

d = 3.8 m

d = 3 m

d = 2.3 m

d = 1.5 m

d = 0.75 m

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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 ( wδ in J/m3) as the dependent variable and the relative density (Dr in %) and the effective

confining pressure ( ′c σ 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,

( ) rc10 D 0209.0 0124.0164.1log +′+= σδw

R2 = 0.943 (5) Reid Bedford Sand,

( ) rc10 D 0123.0 0179.0647.1log +′+= σδw R2 = 0.883 (6)

LSFD Silty Sand,

( ) rc10 D 00115.0 00448.04597.2log +′+= σδw

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.

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10000

50 55 60 65 70 75 80 85

Relative Density (%)

Ene

rgy/

Vol

ume

(J/m

3)

Centrifuge Results

Kusky's Eq.-Sinusoidal (1996)Liang's Eq.-Sinusoidal (1995)

Centrifuge Results Eq.Liang's Eq.-Random (1995)

Figure 11 wδ vs. Dr (Reid Bedford sand −cσ =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

Figure 12 wδ vs. Dr (Lower San Fernando Dam sand −cσ =31kPa)

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Relative Density (%)

Ene

rgy/

Vol

ume

(J/m

3)

Liang's Equation -Random (1995)

Centrifuge Results Equation

Centrifuge Results

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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%.

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10000

45 50 55 60 65 70 75 80

Relative Density (%)

Ene

rgy/

Vol

ume

(J/m

3)

Rokoff's Equation (1999)

Centrifuge Results Equation

Centrifuge Results

Figure 13 wδ vs. Dr (Nevada Sand −cσ =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

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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

2

3

4

5

6

7

8

0 200 400 600 800 1000Accumulated Energy/Volume (J/m^3)

Dep

th (

m)

t =1 2 s e c ( EXP )

Eq u a t ion 5

Dissipa t e d En e r g y ( Ex p e r i m e n t a l )

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

t = 8 s e c ( EXP )

t = 5 s e c ( EXP )

t = 1 2 s e c ( N U M )

t = 5 sec (NUM)t = 8 s e c ( N U M )

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

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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.

References Berrill, J.B. and Davis, R.O. (1985). "Energy Dissipation and Seismic Liquefaction of Sands: Revised Model," Soils and Foundations, Vol. 25, No. 2, pp. 106-118. Dief, H.M. (2000). “Evaluating the Liquefaction Potential of Soils by the Energy Method in the Centrifuge” Ph.D. Dissertation, Department of Civil Engineering, Case Western Reserve University, Cleveland, OH. Figueroa, J. L. (1990). "A Method for Evaluating Soil Liquefaction by Energy Principles," Proceedings, Fourth U.S. National Conference on Earthquake Engineering, Palm Springs, CA , May. Figueroa, J. L. and Dahisaria, M. N. (1991). "An Energy Approach in Defining Soil Liquefaction," Proceedings, Second International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, University of Missouri-Rolla, March. Figueroa, J.L., Saada, A.S., Liang, L. and Dahisaria, M.N. (1994). "Evaluation Of Soil Liquefaction By Energy Principles," Journal of Geotechnical Engineering, ASCE, Vol. 120, No. 9, September, 1554-69. Figueroa, J.L., Saada, A.S., Rokoff, M.D. and Liang L.(1998) “Influence of Grain-Size Characteristics in Determining the Liquefaction Potential of a Soil Deposit by the Energy Method”, Proceedings of the International Workshop on the Physics and Mechanics of Soil Liquefaction, Baltimore/, Maryland/, USA/. 10-11 September, 237-245. Finn, W. D. L., Lee, K. W. and Martin, G. R. (1977). "An Effective Stress Model for Liquefaction," Journal of the Geotechnical Engineering Division, ASCE, Vol. 103, No. GT6, June, pp. 517-533. Idriss, I.M. and Seed, H.B. (1968). “Seismic Response of Horizontal Soil Layers,” Journal of Soil Mechanics and Foundations Division, ASCE, Vol. 94, SM4, 1003 - 1031

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Kusky, P.J., (1996). “Influence of Loading Rate on the Unit Energy Required for Liquefaction”, M.S.Thesis, Department of Civil Engineering, Case Western Reserve University, Cleveland, Ohio. Law, K. T., Cao, Y. L. and He, G. N. (1990). "An Energy Approach for Assessing Seismic Liquefaction Potential," Canadian Geotechnical Journal, Vol. 27, pp. 320-329. Liang, L. (1995). "Development of an Energy Method for Evaluating the Liquefaction Potential of a Soil Deposit" Ph.D. Dissertation, Department of Civil Engineering, Case Western Reserve University, Cleveland, Ohio. Nemat-Nasser, S. and Shokooh, A. (1979). "A Unified Approach to Densification and Liquefaction of Cohesionless Sand in Cyclic Shearing," Canadian Geotechnical Journal, Vol. 16, pp. 659-678. Rokoff, M.D. (1999). "The Influence of Grain-Size Characteristics in Determining the Liquefaction Potential of a Soil Deposit by the Energy Method". M.S. Thesis, Department of Civil Engineering, Case Western Reserve University, Cleveland, Ohio. Zeghal, M. and ElGamal, A-W. (1994). Analysis of Site Liquefaction Using Earthquake Records. Journal of Geotechnical Engineering, ASCE, Vol. 120, No. 6, June, 996-1017.