Observation in 1-G Shaking Table Tests and Application to Prediction of Soil
Displacement due to Subsurface Liquefaction
Ikuo Towhata
Professor of Geotechnical Engineering
University of Tokyo
INTRODUCTION
The present workshop classified the author as a shaking-table investigator. This idea is not suitable, however, because the author has always been trying to combine different styles of study to achieve a single goal. The liquefaction-induced permanent displacement in subsoil has been attracting the author’s interest since 1984 when the Nihonkai-Chubu earthquake triggered liquefaction and significant lateral soil displacement in Noshiro City. Since then, the study on the causative mechanism of significant lateral displacement has been conducted by using shaking table tests as well as observation in the earthquake damage sites and laboratory shear tests. This effort then led to a development of an analytical method to predict the permanent displacement for which the knowledge of mathematics, analytical mechanics, and numerical analysis made a significant contribution. The author is recently investigating mitigative measures of permanent displacement for which shaking table tests and knowledge of structural mechanics are playing major roles. With these in mind, it is intended in this text to describe the importance of having a wide scope of study in place of focusing on a single but attractive way of research.
PERMANENT DISPLACEMENT OF LIQUEFIED SLOPE IN NOSHIRO CITY
Fig.1 illustrates the permanent displacement in Noshiro city which was discovered by Hamada by using air photographs. It is evident in this figure that the displacement is oriented towards the lower elevation. Thus, the influence of the gravity force appears much more important than that of the earthquake shaking. This finding was further confirmed by a shaking table test in the gravity field in which an axisymmetric model of embankment was shaken laterally and subsoil liquefaction induced permanent displacement in the direction of slope (Sasaki et al. 1992). Since the gravity thus played a major role in the phenomenon of lateral displacement, it was supposed that the induced lateral displacement could be assessed by a static analysis which does not require a time-consuming dynamic analysis (Towhata et al., 1992).
One of the shortcomings of the case studies in the field was that all the findings concerned the displacement at the ground surface; nothing was known about the subsurface displacement. This problem has been solved by shaking table tests in which the subsurface deformation was observed through a transparent side wall of a soil container. Fig.2 indicates an example. This is the earliest test result for which a small model of only one meter in length was shaken. The square grid of colored sand, which was introduced into model studies for the first time, clearly demonstrated the subsurface deformation of the model after shaking.

Fig.1 Permanent displacement induced by subsurface liquefaction in Noshiro City in 1983
(Sasaki et al., 1992).
Fig.2 Maximum possible deformation of liquefied model ground with small embankment at surface (Sasaki et al., 1992).
The surface embankment absorbed a sufficient amount of water and its high moisture content allowed the embankment to lose the stiffness upon shaking and became liquefied together with the water-saturated subsoil. Moreover, the shaking was continued for a very long time until the maximum possible displacement was achieved. Ultimately, the ground surface became level when it reached the final equilibrium. Recalling the fact that liquid is stable when its surface is
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level as well, the author concluded that there is a similarity between the behaviors of liquefied sand and liquid.
Another important finding was the mode of lateral displacement. Fig.2 shows that the lateral displacement increases in the vertical direction; starting from null at the bottom to the maximal near the surface. Since this mode was commonly observed in model tests on flow of liquefied slopes, it was decided to express the variation of horizontal displacement, U(x,z), in the vertical
(z) direction by a sinusoidal function;
U ( x , z )

F ( x ) sin
 z
(1)
2 H where F(x) designates the lateral displacement at the top of the liquefied layer, varying with the horizontal x coordinate. Moreover, H is the thickness of a liquefied subsoil. Thus, the amount of unknowns was drastically reduced from a two-dimensional U(x,z) to a simple function of F(x).
Fig.3 is one of the large shaking table tests on lateral flow of a liquefied slope conducted at
PWRI at Tsukuba. It is noteworthy that this large model with 6 meters in length still exhibits a similar sinusoidal mode of lateral displacement. Another important finding therein was that the surface unliquefied crust moved together with the liquefied subsoil. This means that F(x) in Eq.1 is the lateral displacement in the surface crust as well.
The continuity in the lateral displacement across the interface between the liquefied subsoil and the unliquefied surface crust accounts for the reason why the model ground in Fig.2 became level while the in-situ slope in Fig.1 did not. Although the slope in Noshiro City tried to be level, the rigidity of the unliquefied crust of the slope did not allow such a large displacement or deformation which was needed to achieve a level topography. This continuity of displacement further reduced the amount of unknowns in analysis because the liquefied soil and the crust followed the same function of F(x).
One of the shortcomings of 1-g shaking table tests is the reduced magnitude of confining pressure as compared with the extent of in-situ stress; the stress-strain-dilatancy behavior of sand is affected by the magnitude of confining pressure. Shaking table tests deal with this problem in two different ways. The one is to run a full-scale model test under the gravity field, while the other is the use of reduced density of sand. Fig.4 manifests the results of a full-scale shaking table test in which the thickness of sand was five meters and made the confining stress comparable to the in-situ level. Embedded displacement transducers indicate that the displacement increases towards the surface. Thus, the idealization of lateral displacement by
Eq.1 is reasonable.
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200 3000 2100 2700 200 1200 1600 800 200
Unit : mm
Before
After
Test Piles
Steel
PHC-1 PHC-2 PHC-3
14%
Trans-B Trans-A Trans-B
Ground surface before shaking
Ground surface after shaking
Location of markers before shaking
Location of marker after shaking
Fig.3 Lateral displacement of liquefied Fig.4 Lateral displacement of liquefied
Model slope with six meters in length. full-scale model test (Orense et al., 2001).
(Sasaki et al., 1992).
The second idea addressed to the effects of confining pressure recalls the fact that the stressstrain behavior of sand is affected not only by the stress level but also the density of sand grains.
Fig.5 illustrates a schematic idea of undrained stress-strain behavior of sand. Softening of the stress level after the peak strength affects the possibility of large deformation of sand. By using the constant-volume ring shear tests on Toyoura sand, Fig.6 was drawn which shows that a similar extent of softening (I
B
value) is attained when the relative density of sand is reduced under lower effective stress. It seems possible that the effects of reduced confining pressure are compensated for by employing a looser density of sand. Consequently, the author has been running his 1-g model tests with a reduced size at the relative density of 20 to 30 percent.
Shear
stress
Peak strength
 p
Brittleness index, IB

 p
  res
 p
Residual strength
 res
Shear strain
Fig.5 Schematic idea on softening of loose sand under undrained monotonic shear.
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Brittleness
index, I
B
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-10 0
Cons tan t volume d irect sh ear tes ts on To youra s and by W.Vargas
Brittlene ss inde x.kg
100 kPa
50 k Pa
25 k Pa
10 k Pa
10 20 30
Relative dens ity (%)
40 50 60
Fig.6 Variation of constant-volume softening with relative density of Toyoura sand
(Vargas, 1998).
Fig.7 Subsidence of river dikes due to earthqauakes
(Ministry of Construction, 1996).
ANALYTICAL SOLUTION OF MAXIMUM POSSIBLE DISPLACEMENT
Importance of Large Displacement / Deformation Formulation
The Ministry of Construction assembled the past experiences of earthquake-induced subsidence of river dikes in the past more than one hundred years. The extent of damage was variable because of different earthquake intensity and different soil conditions. Fig.7 indicates, however, that the observed subsidence was never greater than 75 percent of the original height of the dikes; at least 25 percent of the height remains even in the worst situations. The reason for this
75-percent empirical rule is interpreted as what follows.
The shaking table test in Fig.2 was conducted on, probably, a very adverse conditions in which a loose sandy deposit totally liquefied and the shaking was continued until the maximum possible deformation was attained. The development of deformation ceased when the ground surface became level. This fact implies that the deformation is not infinitely large even if sand totally loses its rigidity. In other words, the driving force of lateral displacement of liquefied sand vanishes when a liquefied slope becomes level. This situation is called the force equilibrium.
In the case of subsidence, the equilibrium of force is achieved when the weight of a dike becomes equal to the buoyancy force, or the bearing capacity, of liquefied subsoil. In the worst situation in which liquefaction occurs up to the ground surface and the state of very small rigidity
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continues for a sufficiently long time (Fig.8), the force equilibrium is achieved when the ratio of subsidence / (original height) is (unit weight of partially saturated soil in the dike) / (unit weight of water-saturated liquefied sand). Since this ratio is approximately 1.5 / 2.0, the worst subsidence of a river dike is around 75 percent of the original height. This effect of changing topography on the driving force is thus important. However, it is taken into account by analysis only when a large-displacement formulation is employed.
Model of embankment
 
15 kN / m
3
S
H
Liquefied sand
 
20 kN / m
3
The maximum possible subsidence ;
S

15 kN / m
3 
H
20 kN / m
3
Fig.8 End of subsidence at equilibrium Fig.9 Calculation of surface uplift
of gravity and buoyancy. caused by lateral soil movement.
Closed-Form Solution of Maximum Possible Displacement
It has been the author’s consistent desire to develop an analytical method for calculation of the liquefaction-induced ground displacement. It has been thought that an analytical method works faster than nonlinear finite element method. Another advantage may be that a closed-form solution can give us an overall scope of the flow phenomenon.
A closed-form solution was intended to predict the maximum possible displacement, while the real displacement is slightly smaller than the maximum possible one and is calculated by a dynamic analysis as described in the next section. The maximum possible displacement occurs when the driving force and the resistance are in equilibrium while considering the liquefied sand as a liquid without rigidity. A constant-volume behavior which occurs under undrained condition was assumed to liquefied sand. Hence, volume change and subsidence due to consolidation is calculated separately.
The force equilibrium is equivalent with the minimum potential energy. Therefore, the liquefaction-induced displacement is predicted by finding a particular function, F(x) in Eq.1, that minimizes the overall potential energy.
As stated in the previous section, the effects of large displacement have to be taken into account in order to reproduce the variation of driving force with the changing topography. Fig.9 shows
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the calculation of the change of topography, or the change of surface elevation, with the lateral soil flow. The volume flux of lateral flow of soil is calculated by integrating Eq.1 from the bottom to the top;
Volume flux

2

FH

Due to the idealization of constant volume behavior, the difference of the volume flux in the x direction (x-derivative) gives the uplift of the ground surface,

H;


2 ( )
 dx

 as illustrated in Fig.9.
The principle of minimum potential energy is employed in the present study in which the particular function of F(x) that minimizes the potential energy is searched and detected. This procedure is identical with the energy approach of conventional finite element method except that the gravity component of the energy in the liquefied soil is calculated by firstly multiplying the weight of soil multiplied by elevation and then integrating it from the bottom to the top;
Gravity energy of liquefied soil =
H

H
0
  zdz (4)
Being similar to the practice in fluid mechanics, this way of calculating the gravity energy is reasonable because the liquefied sand is considered to be a viscous liquid. It is important that
Eq.4 is valid for a large displacement of soil, while the conventional kind of gravity energy calculation;
Change of gravity energy
H
  
0

( ) (5) cannot reproduce such a limited displacement as shown in Fig.7 and hence is not suitable.
The author’s study assembled the strain and gravity components and integrated all over the flowing ground;
Q =

0
L q x , F , dF dx

 dx
 r F , dF dx

Minimum (6) boundary in which Q is the integrated potential energy of the ground, and q is the energy at x which is integrated from x=0 to x=L. L is the length of the ground. Finally, the mathematical theory of variational principle states that the particular F(x) is obtained by solving an equivalent differential equation;
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d dx
 






 q dF dx 






F

F

0
(7) which finally becomes dF dx

C
4 x
2
C
1 x
2

C
5 x

C
2 x


A
C
3
1
(8)
The solution of this differential equation is easily obtained by elementary mathematics. It is noteworthy that the required input data is the thickness of the surface unliquefied layer, which resists compressional deformation of the ground, its elastic modulus, E, and the thickness of liquefied subsoil together with the overall topography. This amount of data requirement is much smaller than other method of analysis which requires undisturbed soil sampling and precise laboratory tests. For details, however, refer to Towhata et al. (1999).
Fig.10 reveals a calculated example of the maximum possible distortion of a dike of the
Shiribeshi-Toshibetsu River. This site was shaken by the 1993 Hokkaido-Nansei-oki earthquake and its subsoil liquefied. The left and right parts of the dike were analyzed separately and the subsidence of 1.91m was obtained. The extent of subsidence is smaller than the observed 2.67m.
7000
6000
Toyoura s and of 40% re lative de nsity.
Is otropically consolidate d : 100 kPa.
Radial e ffe ctive stre ss = 30kPa.
5000
4000
3000
Rate of deviatoric
strain (1/sec) :
2000
1000
0.0 003
0.0 006
0.0 009
0
0 0.01
0.02
0.03
0.04
0.05
0.06
Deviatoric strain
Fig.10 Calculated maximum possible distortion of Shiribeshi-Toshibetsu river dike.
Fig.11 Viscosity of sand at 30kPa effective stress (Nishimura et al., 2001).
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DYNAMIC ANALYSIS IN TIME DOMAIN
A dynamic analysis was developed so that the increase of lateral displacement with time could be calculated. The details of the mathematical aspects are presented in Towhata et al. (1999). The present text describes three interesting issues in the dynamic analysis.
Rate-Dependent Behavior of Liquefied Sand
Nishimura et al. (2001) attempted to measure the rate-dependent behavior of sand at low effective stress levels. After reducing the effective stress from the isotropic consolidation of 100 kPa down to 10, 20, or 30 kPa, an axial load was applied quickly and the delayed strain response was monitored. The viscosity coefficient was then assessed by using the given stress and the rate of deviatoric strain. The specimens remained drained in the meantime with the reduced effective stress in order to avoid the increase of effective stress due to dilatancy if tested undrained. Fig.11 indicates the viscosity thus obtained under 30 kPa; the viscosity ranging from 2000 to 4000 kPa in this figure depending upon the magnitude of deviatoric strain. Finally, the data from different effective stress levels were compared with each other and it was found that the stress level does not affect the obtained viscosity significantly.
Continuation Time of Flow Failure
In the dynamic analysis, it is very important to determine when the flow of soil starts and when it ends. The tentative practice assumes that the lateral flow of liquefied soil starts at the onset of the maximum acceleration. This seems reasonable because the maximum acceleration determines the thickness of liquefied subsoil and the extent of flow phenomenon.
In contrast, the end of flow significantly affects the duration time of flow and consequently the calculated magnitude of permanent displacement. According to the past shaking table tests, the flow of soil was terminated at either the dissipation of excess pore water pressure or the end of shaking. Normally, the duration of real earthquake motion is shorter that the time needed for insitu dissipation of excess pore water pressure. Hence, the authors (Okada et al., 1999) analyzed real earthquake motions to get an idea of the duration of strong phase of shaking. It was therein decided that “strong earthquake” means the amplitude of acceleration greater than 50 gal in order to avoid overestimation of duration time due to weak phase of shaking. Table 1 shows the results.
Rate of Propagation of Flow Failure
One of the important problems induced by lateral flow of liquefied subsoil is the ground deformation and damage to embedded lifelines caused by quay wall displacement. Those pipes which are situated within the area of large lateral displacement are prone to breakage. In order to assess the risk of this kind, the author has been attempting to evaluate the range of lateral displacement induced by the quay wall displacement.
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Table 1 Upperbound of duration time of strong phase of earthquake shaking (Okada et al., 1999).
JMA earthquake magnitude Duration of strong motion (seconds).
6 10
7 20
7.5 50
8 more than 60
Towhata et al. (1999) analytically derived the rate of propagation of lateral displacement in the horizontal direction;
V f

2 g

 Eh

4

H 2

2




H

2 P
 (9) in which H is the thickness of liquefied sand, P the overburden pressure of the surface unliquefied crust, h the thickness of crust, and E the elastic modulus of the crust. E>0 is used until the active state of stress is reached, beyond which E=0 is appropriate because soil cannot sustain a tensile state of stress anymore. This rate of propagation is multiplied by the duration time of flow in Table 1, and the range of influence is obtained; see the next section for example.
DEVELOPMENT OF THREE-DIMENSIONAL DYNAMIC ANALYSIS WITH LARGE-
DISPLACEMENT FORMULATION
The most recent version of analysis is conducted in a three-dimensional manner with again a large-displacement formulation as well. One important development from the previous version lies in the mode of lateral displacement;
( , , , )

( , , ) sin
π z
2 H

π z
(10)
H in which “f” stands for the amplitude of sinusoidal function which is similar to Eq.1 but varies with time. In contrast, “j” represents an additional mode of lateral displacement which is frequently observed under sinking embankment and foundation; see Photo 1. This new mode is characterized by the null displacement at both top and bottom, while taking the maximal in the middle. By thus combining two modes of displacement, the analysis has obtained a more universal use.
Fig.12 is indicative of the calculated lateral ground displacement at the surface of Kobe Port
Island in which the northern part developed a significant liquefaction and lateral displacement of quay walls. A new numerical technique to take into account the behavior of a gravity-type caisson quay wall resting on liquefied subsoil was developed. Due to the relatively small
9

magnitude of the earthquake, only 10 seconds of flow was assumed. The overall feature of the three dimensional analysis is evident.
2200
2000
1800
1600
10.0[m]
1400
1200
1000
800
Photo 1 Deformation of liquefied subsoil
induced by subsidence of embankment
model.
600
400
200
0 400 800 1200 1600 x[m]
2000 2400 2800
Fig.12 Calculated lateral displacement at the surface of Kobe Port Island.
For the aforementioned assessment of the area of influence, the use of Eq.9 (rate of propagation of quay-wall displacement) multiplied by the duration time of flow was examined against the numerical method (Fig.13). Apparently, the former method is simpler. Solid squares indicates the lateral displacement obtained by the numerical study, while open squares shows the simple calculation of the size of the affected area. The consistency of the two different methods seems acceptable. Finally, the results of simple calculation was examined against the case histories during past major earthquakes. Fig.14 compares the calculation with the data by Yasuda et al.
(1997). In spite of the simple nature of Eq.9, the agreement is reasonable.
CONCLUDING REMARKS
The long history of studies on the liquefaction-induced ground displacement was summarized briefly in the present text. The major issues of the study are summarized in what follows.
1) The gravity force plays the chief role in the development of lateral displacement, while seismic shaking triggers the development of high excess pore water pressure and liquefaction.
2) The change of topography due to lateral soil displacement controls the extent of the soil displacement. Therefore, a reasonable prediction of soil displacement needs to take into account this fact by employing a large-displacement formulation.
3) Liquefied subsoil behaves like a viscous liquid while strong shaking is going on. The viscosity coefficient was assessed by triaxial compression tests.
4) The amount of calculation was drastically reduced by using the modes of displacement which are observed in shaking table tests.
5) The principle of minimum potential energy as well as the variational principle of mathematics helps derive a closed-form solution of the maximum possible soil displacement.
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6) The input data which is required to carry out the present analysis consists of such limited ones as topography, thickness of liquefiable soil, and elastic modulus of surface unliquefied crust. This is a big advantage over other methods which need undisturbed soil sampling and elaborate laboratory shear tests which are normally time-consuming and expensive.
0.000
2.000
0.000
2.000
0.000
1.500
0.000
: Computed Result
: V f
×Elapsed Time
10[s]
9[s]
8[s]
7[s]
Measured data by Yasuda et al.(1997)+ADEP(2000)
Kawashima's duration time of shaking.
Calculation with E=1960 kPa
400
Data from Akita Harbor
300
1.500
0.000
6[s]
1.250
0.000
5[s]
200
1.000
0.000
4[s]
Data from Niigata
0.700
0.000
3[s]
100
0.500
0.000
2[s]
Data from Kobe
0.350
0.000
0.125
0 50 150 200
1[s]
0
0 100 200 300
Affected area (m)
400
100
X[m]
Caisson
Fig.13 Development of lateral displacement Fig.14 Comparison of calculated size of between three-dimensional analysis earthquakes. and simple formula.
ACKNOWLEDGMENT
The author expresses sincere his gratitudes to the Association for Development of Earthquaek
Prediction which has continuously supported the study since 1984 until the present time.
LIST OF REFERENCES
Nishimura,S., Honda,T., and Towhata,I. (2001): "Torsion Shear Tests on Rate-Dependent
Behavior of Liquefied Sand," CD-ROM Proceedings, 10th International Conference on Soil
Dynamics and Earthquake Engineering, Philadelphia.
Okada,S., Orense,R., Kasahara,Y., and Towhata,I. (1999): “ Prediction of Liquefaction-induced deformations of River Embankments,” Proc. Second Int. Conf. Earthquake Geotechnical
Engineering, Lisboa , Vol.2, pp.543-548.
Orense,R., Yasuda,S., Morimoto,I., and Takagi,M. (2001): “Full-scale shaking table tests on the behavior of piles during earthquakes,” Proc. 2nd Workshop on Safety and stability of
Infrastructure against Environmental Impacts, pp. 159-168, Manila.
Sasaki,Y., Towhata,I., Tokida,K., Yamada,K., Matsumoto,H., Tamari,Y., and Saya,S. (1992):
"Mechanism of permanent displacement of ground caused by seismic liquefaction", Soils and
Foundations, Vol.32, No.3, pp.79-96.
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Towhata,I., Sasaki,Y., Tokida,K., Matsumoto,H., Tamari,Y. and Yamada,K. (1992): "Prediction of Permanent Displacement of Liquefied Ground by Means of Minimum Energy Principle",
Soils and Foundations, Vol.32, No.3, pp.97-116.
Towhata,I., Orense,R.P. and Toyota,Hirofumi (1999): "Mathematical principles in prediction of lateral ground displacement induced by seismic liquefaction," Soils and Foundations, Vol.39,
No.2, pp.1-19.
Vargas-Monge,W. (1998): “Ring shear tests on large deformation of sand,” Ph.D.Thesis,
University of Tokyo.
Yasuda,S., Ishihara,K., Harada,K., and Nomura,H. (1997): “Factors affecting the area of lateral flow in the ground behind quaywalls,” Proc. 2nd Conf. on Earthquake Disasters in Hanshin-
Awaji area, JSCE., pp.113-120.
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