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PORE WATER PRESSURE CHANGES IN SANDS UNDER EARTHQUAKE

LOADING

T S UENG1, M C WU2, C Y LIN3 And R Y YU4

SUMMARY

During an earthquake the pore water pressure generation in a saturated sand layer leads to the
reduction of strength and even liquefaction of the soil. In this study, a model of pore water
pressure generation of sand under undrained condition was developed by adapting the previously
developed drained stress-dilatancy relation based on the particulate approach. The effects of water
modulus, system compliance, and rebound modulus on the pore water pressure changes were
discussed. This model can be used to compute the pore water pressure change in every time step
under earthquake loading. Laboratory cyclic triaxial tests, and simple shear tests were performed
for Ottawa sand, Taipei silty sand, and Mailiao sand to verify this model. It was found that the
overall trend of pore water pressure changes predicted by the model agreed very well with the test
results. According to the model the pore water pressure buildup is related to the difference
between the frictional energy and the dissipated energy for the area inside the hysteresis loop of
the shear stress-strain relation.

INTRODUCTION

Under an earthquake shaking, the pore water pressure in a saturated sandy soil will increase due to the tendency
of the volume contraction if the drainage of the water is impeded. This results in a lower effective stress and the
reduction of strength, and even the liquefaction of the sand. The phenomenon of soil liquefaction occurred
during many earthquakes in the recent years (e.g., Northridge and Kobe earthquakes) and caused severe
damages.

The liquefaction or the pore water pressure buildup induced by seismic shaking is closely related to the volume
change of the sand in drained condition. When a sand undergoes a drained cyclic shear stress action, the sand
contracts at the beginning of shearing, and as the shear strain increases, the tendency of volume reduction
reduces, eventually the volume change can become dilation. The contraction or dilation tendency depends on the
density and confining pressure of the sand. A large amount of volume reduction generally occurs at the reversal
of shear stress. When the shear strain increases, there is again volume dilation. The sand becomes denser after
every cycle of shear stress.

For a saturated sand under undrained cyclic shear stresses, the pore pressure increases at the beginning of the
shearing, whereas the increase of pore pressure gets lesser as the shear strain increases, and the pore water
pressure can even become reduced depending on the density of sand. At the reversal of the shear stress, the pore
pressure generally increases drastically. Under cyclic shear stresses, the pore pressure increases after each cycle
and the sand can reach the state of liquefaction or cyclic mobility.

1
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan, ROC Email: ueng@ce.ntu.edu.tw
2
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan, ROC
3
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan, ROC
4
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan, ROC
[Martin, Finn and Seed, 1975] based on the compatibility of volume change of soil skeleton and the pore water,
developed a relation for the pore water pressure increment for each cycle of shear stress. Other researchers, e.g.,
[Nemat-Nasser and Shokooh, 1979], [Simcock, et al., 1983], [Davis and Berrill, 1985], [Ishihara and Towhata,

1985], and [Figueroa, et al., 1994] had also established relations between the pore water pressure generation and
the dissipated energy during each cycle of shear loading based on the observations of experiment results.
However, all these relations generally consider only the pore water pressure buildup after every cycle of loading
but not the change of pore pressure during every time step within a stress cycle. The later is needed for the study
of the liquefaction problem during earthquakes, which induce irregular shear stress cycles. It was also found that
the pore pressure change at every moment could affect the behavior of sand undergoing the consecutive loading,
which in turn influences the overall pore water pressure buildup, deformation, and liquefaction of the soil during
an earthquake.

This study adopts the drained stress-dilatancy relationship proposed by [Ueng and Lee,1990] to develop a model
for the pore water pressure generation during the undrained cyclic loading process.

PORE WATER PRESSURE GENERATION UNDER UNDRAINED CONDITION

For a saturated sand under an undrained cyclic shear loading, the total volumetric strain is the sum of the
volumetric strains of solid particles and pore water. If we assume that the solid particles are incompressible, then
volumetric strain of the sand

du dσ ' n
dε v = − n ⋅ = n⋅ (1)
kw kw

where n = porosity of the sand,

du = pore water pressure change,

dσ'n = change of effective normal stress, and

kw = bulk modulus of pore water.

The volume change of a saturated sand under the undrained condition can be considered to include two parts:

(1) Volume change caused by the change of effective normal stress, dεvn,

du dσ ′
dε vn = =− n (2)
kr kr

where kr = rebound modulus of volume change of the sand structure.

(2) Volume change caused by shearing, dεvs, which can be considered by adopting the stress-dilatancy
relation developed by [Ueng and Lee, 1990], that is,
τ  dε 
= ± tan φ µ ± b vs  (3)
σ' n  dγ 

or

1  τ 
dε vs =⋅
b  σ ′n
( )
− ± tan φ µ  ⋅ dγ (4)

where τ = shear stress on the shearing plan,

1) σ'n = effective normal stress on the shearing plane,

2) φµ = basic friction angle between sand grains,
3) b = a positive value related to shearing plan orientation, friction angle, and sand fabric,
4) dεv = volumetric strain increment, and
5) dγ = shear strain increment.

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The values of tanφµ and b were found constants for every sand independent of density, method of specimen
preparation, stress path, or number of stress cycles. The ± sign in equations 3 and 4 is to indicate whether the
shear stress is in the positive or negative direction. Hence, the change of total volumetric strain is

dσ ′n 1  τ 
dεv = dεvn + dεvs = − + ⋅
b  σ ′n
( )
− ± tan φ µ  ⋅ dγ (5)
kr 
Combining equations 1 and 5, we obtain
1  τ 
dσ ′n =
 n
1
⋅ ⋅
1  b  σ ′n
(
− ± tan φ µ  ⋅ dγ )


(6)
+  

 kw kr 

The change of effective stress, or pore water pressure, for a given shear strain increment becomes,

1  τ 
∆σ ′n = ∫ dσ ′n = − ∆u =
 n
1
⋅⋅∫
1  b  σ ′n
( )
− ± tan φ µ  ⋅ dγ (7)
+  

 kw kr 
This relationship indicates that the pore water pressure can be computed by integrating the difference between
the shear stress ratio, τ/σ'n, and tanφµ against the shear strain and multiplying the result by the coefficient
composed of the soil porosity, bulk modulus of water, rebound volumetric modulus of sand structure, and the
coefficient b in the stress-dilatancy relation of the sand. Figure 1 depicts the change of effective stress during a
cycle of shear stress described by equation 7. It can be seen that when the value of tanφµ is larger than that of
τ/σ'n, the effective stress decreases. In other words, the pore water pressure increases. On the contrary, when
tanφµ is less than τ/σ'n, the effective stress increases, or the pore water pressure decreases. Equation 7 can be
rewritten in the form of energy relation:

 n

 w
k
1
k r
1
+  ∫ σ ′n ⋅ dσ ′n = ⋅
b
[∫ τ ⋅ dγ − ∫ ±(σ′ ⋅ tan φ ) ⋅ dγ ]
n µ (8)

Figure 1 also shows that the pore water pressure buildup is not directly related to the dissipated energy computed
according to the area enclosed inside the hysteresis loop of the shear stress-strain relation but the energy
pertaining to the area outside the loop between the tanφµ and the stress-strain relation. This deviates from the
concept proposed previously by others, for example, [Ishihara and Towhata, 1985] and [Figueroa, et al., 1994].

Figure 1: Shear stress ratio-strain relation and changes of effective stress under undrained loading
LABORATORY TESTS FOR MODEL VALIDATION
Undrained cyclic triaxial tests with constant confining pressure, undrained cyclic triaxial tests with constant
mean stress, and undrained cyclic simple shear tests under constant normal stress were conducted to evaluate the
proposed model. The changes of pore water pressures in the specimens were measured during the undrained
cyclic loading. CKC dynamic triaxial test apparatus was used for the cyclic triaxial test and NGI simple shear
device was used in the simple shear tests. In addition, isotropic consolidation tests, consolidated drained test,

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oedometer consolidation tests, and drained cyclic triaxial and simple shear tests were performed. Ottawa sand
(ASTM C-778), Taipei silty sand, and Mailiao sand were used in this study. The properties of these sands are
shown in Table 1. Details of these tests are given in [Wu, 1997], [Lin, 1998], and [Yu, 1999].
Table 1: Properties of the test sands
Ottawa sand Taipei silty sand Mailiao sand
Gs 2.64 2.71 2.66
D10 0.2 mm 0.075 mm 0.075 mm
D50 0.34 mm 0.160 mm 0.118 mm
Fines content 0 10% 10%
Cu 1.9 2.5 1.78
emax 0.760 1.076 1.262
emin 0.476 0.581 0.762
In the drained triaxial and simple shear tests, values of the parameters in equation 1 were determined according
to the relation between the stress ratio and the dilatancy rate on the octahedral or horizontal planes, respectively
(e.g., Fig. 2). The drained test results verify the stress-dilatancy relationship (equation 3) proposed by [Ueng and
Lee, 1990]. It was also found that there is a unique relationship between the stress ratio, τ/σ'n, and shear strain
for these test sands under different confining pressures. Figure 3 is a typical example. Therefore, the
computations of the pore water pressure generation in equation 7 can be performed without considering the
change of effective stress during the loading.
O ttaw a S a n d , 6 5%
2
N o rm a l S tress=0 .5 kg/cm
S tre ss ra tio
2
1.0 0 N o rm a l S tress=1 .0 kg/cm
2
N o rm a l S tress=1 .4 kg/cm

τ  d εn 
= 0.42 + 1.45 ⋅  
σ′  dγ 
D ila ta n cy ra te
0 .0 0
-1 .0 0 0 .0 0 1 .0 0

τ  dεn 
= −0.42 − 1.45 ⋅  
σ′  dγ 

-1 .0 0

Figure: 2 Relation between shear stress ratio and dilatancy rate in simple shear test

Under undrained condition, since there is always a pore water pressure buildup, especially at the beginning of the
loading, the effective stress decreases and the sand structure tends to expand. In other words, the soil is in an
unloading condition. When there were pore pressure reductions during the cyclic loading, the sand should be
under the stage of reloading. Therefore, the rebound/reloading moduli of volume changes, kr, for these sands
obtained in the consolidation tests (e.g., Fig. 4) were used in the computations. A constant value of kr was used
in the computations, but it was found out that kr varies with σ'n, and the value will be quite different when the
effective stress approaching zero near the stage of liquefaction. Therefore, kr was considered as a function of
effective stress in the later computations.
s tre s s ra tio
T a ip e i S ilty S an d , 6 5 %
1.0 0
2
N orm a l S tre ss = 1 .4 k g /c m
2
N orm a l S tre ss = 1 .0 k g /c m
2
N orm a l S tre ss = 0 .5 k g /c m

s h e a r s tra in

0 .0 0
-0 .2 0 0 .0 0 0 .2 0

-1 .0 0
Figure 3: Stress ratio-strain relation in drained cyclic simple shear test

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 E ffe ctive C o n finin g P res su re (kP a )
1 10 100

0 .0 E + 0

4 .0 E -3

Volumetric Strain
8 .0 E -3

1 .2 E -2

L o a d in g

U n lo a d in g a n d R e lo a d in g

1 .6 E -2

Figure 4: Relation of volumetric strain versus mean effective stress for Mailiao sand

The bulk modulus of pure water is 2.15x106 kPa. For the undrained tests, the bulk modulus of water, kw, should
include the effects of the degree of saturation and the compliance of the whole testing system. Therefore, kw was
computed based on the B value obtained before the cyclic tests. They are in the order of 105 kPa which is
consistent with the values obtained by [Garga and Zhang, 1997]. The values of the parameters used in equation 7
for computations of the pore pressure changes are given in Table 2.

Table 2: Values of parameters used in model computations

Ottawa sand Taipei silty sand Mailiao sand

Test TCC TCM TCM SST SST TCM
Dr (%) 35 50 65 65 65 40
n 0.4 0.382 0.365 0.365 0.45 0.515
b 2.09 0.91 0.91 1.45 1.61 1.39
tanφµ 0.46 0.37 0.37 0.42 0.52 0.43
kr (kPa) 65000 332 σ' 401 σ' 22900 17100 369 σ'
kw (kPa) 3.95x105 2.47 x105 2.83x105 4.00 x105 3.08 x105 0.68x105

Notes: TCC = triaxial test with constant confining pressure, TCM = triaxial test with constant mean stress, SST
= simple shear test.

The values of kw were computed based on the B values in the tests.

COMPARISON OF TEST RESULTS AND MODEL COMPUTATIONS

The pore water pressure changes in these sands were computed using equation 7 according to the testing
conditions, and compared with the pore pressure measured in the undrained cyclic tests. In the cyclic triaxial test
with constant confining pressure, the mean stress of the specimens also changed along with the application of the
cyclic deviator stress. The pore pressure change due to the change of the mean stress should be deducted before
the comparison was made because equation 7 does not consider the pore pressure change induced by the mean
stress change.

Figure 5 is the typical results of pore water pressure change during tests and their comparisons with the results
obtained according to the model. The comparison shows that the trend of pore water pressure generation agrees
well between the laboratory test results and the computed results according to the model proposed in this study.
The details of pore pressure changes within each load cycle are described fairly closely by the proposed model.

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 100 M od e l S im u la tion 10 0 M o d e l S im u la tio n

E xp erim e ntal E x p e rim e n ta l

80 80

60 60
PW P (kPa)

P W P (kP a)
40 40

20 20

0 0

0 2000 4000 6000 8000 0 2 00 4 00 6 00 8 00

T im e (se c) T im e (se c )

(a) Mailiao sand, Dr = 40% (b) Ottawa sand, Dr = 65%

Figure 5: Comparisons of pore water pressure changes between computed and measured results

It is also found that the water bulk modulus, the system compliance, and the deformability of the specimen
boundaries have significant effects on the pore pressure changes in the undrained condition. The results show
that when kw is at least one order of magnitude higher than kr, the pore water pressure generation of sands under
cyclic shear loading would not significantly affected by the change of the modulus of pore water. Figure 6 shows
the effect on the pore water pressure changes for various values of kw.
90

kw(pure water) = 2.15x106 kPa

80
kw(GDS) = 7.5x105 kPa
Pore Water Pressure (kPa)

kw(CKC) = 3.95x105 kPa

70
kw(=kr) = 65047 kPa

60 Test Results

50

40

30

20

10

-10

0 1200 2400 3600 4800

Time (s)

Figure 6: Effect of water bulk modulus on the pore water pressure changes

The shear stress-ratio-strain relations obtained in the undrained tests should be used in the model computations
for the pore water pressure generation in sand under undrained loading. It appears that the use of the drained
stress-strain relation in the model computation does not give good results for the undrained condition in which
very little volume change is allowed.

CONCLUSIONS

In this study, a model of pore water pressure generation under the undrained cyclic shear loading was developed
based on the stress-dilatancy relation in the drained condition. According to equation 7, the change of effective
stress or the pore water pressure generation can be computed using the values of porosity, tanφµ, rebound bulk
modulus of volume change, coefficient b in the stress-dilatancy relation, and stress ratio-strain relation of the
sand obtained in the laboratory tests. The proposed model can describe the pore water pressure change for every
time step rather than that after each stress cycle during the cyclic shear loading. It is suitable for use in the
consideration of the pore water pressure generation during the irregular shaking induce by an earthquake.
Comparison between the computed pore pressure generation and the results of undrained cyclic tests for Ottawa

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sand, Taipei silty sand, and Mailiao sand showed that this model can very well describe the process and the trend
of the pore water pressure generation under cyclic loading.

ACKNOWLEDGMENT

This study is supported by the National Science Council, ROC, Grants Nos. NSC87-2621-P-002-031and
NSC88-2625-Z-002-034.

REFERENCES

Davis, R.O. and Berrill, J.B. (1985), "Energy dissipation and seismic liquefaction of sands: revised model", Soils
and Foundations, 25, 2, pp106-118.
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, 120, 9, pp1554-1569.
Garga, V.K. and Zhang, H. (1997), "Volume changes in undrained triaxial tests on sands", Canadian
Geotechnical Journal, 34, pp762-772.
Ishihara, K.L. and Towhata, I. (1985), "Shear work and pore water pressure in undrained shear", Soils and
Foundations, 25, 3, pp73-84.
Lin, C.Y. (1998), Pore Water Generation Of Sands Under Undrained Cyclic Simple Shear Loading, M.S. thesis,
Dept. of Civil Engineering, National Taiwan University.
Martin, G.R., Liam Finn, W.D., and Seed, H.B. (1975), "Fundamentals of liquefaction under cyclic loading",
Journal of Geotechnical Engineering Div., ASCE, 101, GT5, pp423-437.
Nemat-Nasser, S. and Shokooh, A. (1979), "A unified approach to densification and liquefaction of cohesionless
sand in cyclic shear", Canadian Geotechnical Journal, 16, pp659-678.
Simcock, K.J., Davis, R.O., Berrill, J.B., and Mullenger, G. (1983), "Cyclic triaxial tests with continuous
measurement of dissipated energy", Geotechnical Testing Journal, 6, 1, pp35-39.
Ueng, T.S. and Lee, C.J. (1990), "Deformation behavior of sand under shear-particulate approach", Journal of
Geotechnical Engineering, ASCE, 116, 11, pp1625-1640.
Wu, M.C. (1997), Pore Water Pressure Generation of Sands Under Cyclic Loading, M.S. thesis, Dept. of Civil
Engineering, National Taiwan University.
Yu, R.Y. (1999), Modeling of Pore Water Pressure Generation of Saturated Sands, M.S. thesis, Dept. of Civil
Engineering, National Taiwan University.

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