DISPLACEMENT ANALYSIS OF SUBMARINE SLOPES USING ENHANCED
NEWMARK METHOD
N. ZANGENEH and R. POPESCU
Faculty of Engineering & Applied Science, Memorial University, St. John’s,
Newfoundland, Canada A1B 3X5
Abstract
The Newmark method for predicting seismically induced displacement of slopes is
enhanced by introducing of the yield (threshold) acceleration on the real soil strength,
accounting for the effects of pore water pressure. Seismically induced pore water
pressure build-up and its dissipation after the earthquake are calculated as a function of
soil properties and ground motion characteristics, based on well-recognized practical
methods. The proposed model provides more realistic predictions of slope
displacements. It is validated based on centrifuge experimental results.
Keywords: Newmark method, submarine slopes, pore water pressure build-up and
dissipation, seismic loading.
1. Introduction
Displacement analysis is a more rational alternative to pseudo-static seismic analysis of
slope stability. Newmark (1965) introduced a limit-equilibrium-based displacement
analysis method that predicts the displacements of an infinite slope during an
earthquake based on a soil strength–dependent yield acceleration and purely kinematic
criteria. As opposed to the pseudo-static method of slope stability analysis, which
provides a factor of safety for some very short time instants during an earthquake, the
Newmark method can provide a prediction of slope performance based on the total
displacement at the end of shaking. In many applications of the Newmark method, the
yield acceleration is assumed constant during the earthquake. When applying the
method to saturated granular soils, however, due to the build-up of excess pore water
pressure (EPWP), soil strength and consequently the yield acceleration will decrease. In
addition, after the end of shaking the generated pore pressures start to dissipate with
time, which results in increasing soil strength and yield acceleration. In this paper, the
effects of EPWP build-up/dissipation are investigated, and a procedure for calculating
permanent displacements of submarine slopes subjected to seismic loads is introduced.
The method is based on the algorithm proposed by Newmark, and it uses state-ofpractice methods for estimating EPWP build-up and dissipation. The results of the
analysis are also verified using centrifuge test results.
2. Analysis Procedure
Non-cohesive soils may experience significant pore water pressure build-up due to
cyclic or earthquake loading. In the limit, it can lead to a state of zero effective stress
193
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Zangeneh and Popescu
and soil liquefaction. Therefore, in case of non-cohesive deposits, a total stress analysis
is not appropriate and may give highly over-conservative results. Instead, an effective
stress approach should be used to consider the effects of EPWP and changes in soil
shear strength.
2.1 YIELD ACCELERATION
Figure 1. Pseudo-static analysis of an infinite submarine slope.
For stability analysis of an infinite slope, the failure surface is assumed as a plane
parallel to the slope (Fig. 1). The factor of safety ( FS ) is expressed by the ratio of
available soil shear strength ( τ f ) to the shear stress developed on the failure plane ( τ ):
FS =
τf
τ
(1)
in which, soil shear strength at failure is expressed in terms of effective parameters
according to the Mohr-Coulomb failure criterion:
τ f = c ′ + (σ − u ) tan φ ′
(2)
where c ′ is the soil effective cohesion, φ ′ is the effective internal friction angle, σ is
the total stress (normal to the failure surface), and u is the total (hydrostatic + excess)
pore water pressure. Therefore, the factor of safety can be written as follows:
FS =
c ′ + (γ ′d cos 2 β − u e − kγd sin β cos β ) tan φ ′
γ ′d sin β cos β + kγd cos 2 β
(3)
γ ′ is the effective (or buoyant) unit weight of soil, d
the depth of failure plane, β is the slope angle, u e is the EPWP (in excess of
hydrostatic) generated due to earthquake, and k is the seismic coefficient defined as the
where with reference to Figure 1,
ratio between the horizontal earthquake acceleration and the gravitational acceleration
(g).
Displacement Analysis of Submarine Slopes Using Newmark Method
195
In this study, only fully saturated soils are taken into account. The buoyant (or effective)
weight of the sliding block, W ′ , is used in Equation (3) to calculate the normal
effective stress. However, because it is assumed that during a seismic event, the soil
behaviour is mostly undrained, the inertial force of the earthquake is applied to both soil
particles and pore water. Thus, the inertial force is equal to k × Wsat .
By setting the factor of safety equal to 1, the yield acceleration coefficient at each time
instant t for downslope sliding can be obtained as follows:
k yd (t ) =
or, with
and
c ′ + [γ ′d cos 2 β − u e (t )] tan φ ′ − γ ′d sin β cos β
γd cos 2 β + γd sin β cos β tan φ ′
(4)
ru = u e / σ v′0 (the ratio between EPWP and initial effective vertical stress)
σ v′0 = γ ′d cos 2 β :
k yd (t ) =
c ′ + γ ′d cos 2 β [1 − ru (t )] tan φ ′ − γ ′d sin β cos β
γd cos 2 β + γd sin β cos β tan φ ′
(5)
The yield acceleration coefficient is defined here as k y = a y / g , where a y is the yield
acceleration and g is the acceleration of gravity. For the case of very mild slopes, it
may be worth considering also the possibility of seismically induced upslope sliding.
The yield acceleration coefficient for upslope sliding is:
k yu (t ) =
− c′ − γ ′d cos 2 β [1 − ru (t )] tan φ ′ − γ ′d sin β cos β
γd cos 2 β − γd sin β cos β tan φ ′
(6)
The sliding block downslope a bd (t ) and upslope a bu (t ) accelerations can be calculated
using the following equations:
a bd (t ) = (k (t ) − k yd (t ) )g
cos(φ ′ − β )
cos(φ ′)
(7)
a bu (t ) = (k (t ) − k yu (t ) )g
cos(φ ′ + β )
cos(φ ′)
(8)
where k (t ) ⋅ g represents the seismic acceleration time history. Finally, the slope
displacement can be computed by integrating twice the block acceleration based on the
direction of motion.
2.2 ESTIMATION OF EXCESS PORE PRESSURE BUILD-UP
According to Seed and Idriss (1982), the rate of pore pressure development in undrained
cyclic simple shear tests on most granular soils, falls within a fairly narrow range when
plotted in the normalized form shown in Figure 2.
Zangeneh and Popescu
EPWP Ratio (ru)
196
Average (α = 0.7)
(Neq/NL)
Figure 2. Rate of pore water pressure build-up in cyclic simple shear tests (Seedet al. 1975).
Curves such as those shown in Figure2 can be expressed by the following relation:
 N eq
u
2
ru = e =   arcsin 
′
σ v0  π 
 NL
where
1
 2α


(9)
N eq is the number of equivalent stress cycles applied to the sample up to a
certain moment, NL is the number of stress cycles required to produce liquefaction, and
α is called the pore pressure build-up parameter. For a real acceleration time history,
N eq , can be calculated based on a procedure
introduced by Seed (1975). By varying the value of α , Equation (9) can fit a large
the number of equivalent stress cycles,
palette of undrained pore water pressure generation curves, as shown in Fig. 3.
EPWP Ratio (ru)
α=
(Neq/NL)
Figure 3. Rate of pore pressure generation for different values of α (Seed and Idriss 1982).
The results of a typical Newmark analysis, as described before, which accounts for the
decrease of yield acceleration due to EPWP build-up is shown in Figure 4. As the
Displacement Analysis of Submarine Slopes Using Newmark Method
197
earthquake induces a gradual increase in pore pressure, the yield accelerations decrease
gradually. In this particular example, one should note that if no reduction in the yield
acceleration were considered, the permanent displacement would be much smaller, and
therefore, non-conservative. Also note that in Figure 4, two different permanent
displacements are calculated, one of which considers the possibility of upslope sliding
that is reasonable for nearly flat submarine slopes.
Displacement (without upslope)
Displacement (with upslope)
Effect of Build-up
Downslope Yield Acceleration
Earthquake Acceleration
Upslope Yield Acceleration
Figure 4. Slope displacements considering the effect of build-up.
Downslope Yield Acceleration
Indefinite
Displacement
Constant Negative Yield Acceleration
Earthquake Acceleration
Upslope Yield Acceleration
Figure 5. Slope displacement ignoring the effect of excess pore water pressure dissipation.
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2.3 ESTIMATION OF EXCESS PORE PRESSURE DISSIPATION
After the end of shaking the excess pore water pressure starts dissipating. In cases
where soil liquefies, the yield acceleration at the end of shaking is less than zero, which
results in infinite post earthquake displacements (Figure 5). In reality, however, due to
dissipation of pore pressure the soil regains part of its original shear strength and the
yield acceleration increases and becomes positive (Figure 6). This results in limiting the
displacements. Therefore, accounting for EPWP dissipation after the earthquake will
provide better prediction of post-seismic displacements.
Downslope Yield Acceleration
Measured Displacement
Effect of Dissipation
Predicted Displacement
Earthquake Acceleration
Upslope Yield Acceleration
Figure 6. Slope displacement considering the effect of excess pore water pressure dissipation.
The one-dimensional consolidation theory can be applied to estimate the EPWP
dissipation rate. In a soil layer with any distribution of the initial excess pore pressure
with depth [u i ( z )] , the excess pore pressure at any time, u e (t ) at the depth d is (e.g.
Craig 1992):
n=∞
u e (t ) = ∑ [(
n =1
n 2π 2 c v t
1 2d
nπz
nπz
(
)
sin
)(sin
)
exp(
−
)]
dz
u
z
i
2d
2d
4d 2
d ∫0
(10)
d is length of longest drainage path, and c v is the coefficient of
consolidation: c v = k /( m v γ w ) where k is soil permeability , m v is the coefficient of
volume compressibility and γ w is the unit weight of water. In this study, it is assumed
that the soil layer above the failure plane liquefies during the earthquake, therefore, u i
is a linear function of z , and Equation (10) becomes:
where
n=∞
u e = ∑[γ ′ cos2 β ( −
n =1
n 2π 2Tv
nπz
4d
4d
cos nπ + 2 2 sin nπ )(sin
) exp(−
)] (11)
nπ
nπ
2d
4
Displacement Analysis of Submarine Slopes Using Newmark Method
199
3. Calibration and Validation Using Centrifuge Test Results
To calibrate and verify the analysis procedure described in the previous section, the
results of VELACS (Verification of Liquefaction Analysis by Centrifuge Studies,
Arulandan and Scott, 1993) centrifuge test for model 2 performed by RPI (Figure 7)
have been used. This test simulates an infinite submarine slope with a depth of 10 m
subject to an earthquake with maximum acceleration of about 0.2g. The soil is a
uniform sand with relative density Dr = 40%. The geomechanical soil properties were
inferred by Popescu and Prevost (1993), based on results of laboratory soil tests.
3.1 PORE PRESSURE BUILD-UP PARAMETER (α)
3.1.1 Using the acceleration at levels of each pore pressure transducer
In this section, α is back calculated for two intermediate elevations where both EPWP
and accelerations were recorded, namely (P6,AH4) and (P7,AH5) as shown in Figure 7.
In Figures 8a and 8b, values of α equal to 0.5, 0.7, 2, and 4 are shown for points P6 and
P7, respectively. The EPWP parameter α corresponding to the best curve fit is α = 4.
The value of the number of cycles to liquefaction, N L , is directly obtained from the
pore pressure records and the variation of the equivalent number of cycles of the input
motion with time.
3.1.2 Using the acceleration of the box (measured at the base of the model)
The curve-fit procedure has been repeated for the same points at the same levels but
using the centrifuge box acceleration. In this case the best curve fit can be obtained for
α = 2 to 4 (Figure 9), which is close to the values obtained in section 3.1.1. Therefore,
in real-life analyses, one could use the base (bedrock) seismic acceleration and still
obtain acceptable prediction.
Figure 7. VELACS Model #2 Configuration.
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Zangeneh and Popescu
Recorded EPWP build-up
Recorded EPWP build-up
α=4
α=4
α=2
α=2
α=0.7
α=0.7
α=0.5
α=0.5
a) P6
b) P7
Figure 8. Calibration of α using the acceleration at each level.
Recorded EPWP build-up
α=4
Recorded EPWP build-up
α=4
α=2
α=2
α=0.7
α=0.7
α=0.5
a) P6
α=0.5
b) P7
Figure 9. Calibration of a using the acceleration of the box.
3.2 CALIBRATING THE COEFFICIENT OF CONSOLIDATION
The values of EPWP after dissipation calculated using Equation (11) as well as the
values recorded in the centrifuge test at point P6 are shown in Figure 10.
Predicted using cv=5.1 m2/s
Measured
Figure 10. The predicted and measured excess pore water pressure values during the buildup and dissipation phases (at transducer P6 – see Figure 7).
Displacement Analysis of Submarine Slopes Using Newmark Method
201
The best-fit curve of post-earthquake EPWP dissipation was obtained for cv = 5.7 m/s2.
A value of cv = 5.1 m/s2 was calculated using a value k = 3.3 × 10 −3 m / s for soil
permeability (Popescu & Prevost 1993) and computing the coefficient of volume
compressibility as m v = 1 / B . B is the low strain bulk modulus of the soil that is a
function of the average effective confining stress during the dissipation phase.
3.3 SLOPE DISPLACEMENTS CONSIDERING DISSIPATION
Finally, by applying all the previously described procedures the permanent displacement
of the slope is calculated and shown in Figure 11. The predicted value is satisfactorily
close to the measured value.
Downslope Yield Acceleration
Measured Displacement
Predicted Displacement
Earthquake Acceleration
Upslope Yield Acceleration
Figure 11. The predicted and measured permanent displacements considering EPWP buildup and dissipation effects (at transducer P6 – see Figure 7).
4. Model Limitations and Further Research
This paper presents a study in progress that needs further addressing of a few aspects:
(1) the acceleration applied at the base of the block was that recorded in the experiment
at that location (or level), and (2) the number of cycles to liquefaction was estimated
from the measurements using the time at which ru reached one in the centrifuge model.
These two elements are unknown when making a prediction of slope displacements.
They were used in the current study to check the proposed method for calculating
excess pore water pressure build-up and dissipation. In the next phase, for a given
seismic motion the number of cycles to liquefaction will be estimated based on
information on soil strength and maximum seismic acceleration, and using a method
compatible with the current guidelines for liquefaction strength assessment (Youd et al.
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Zangeneh and Popescu
2000) using in-situ tests results (e.g. CPT).
The Newmark model assumes that slope displacements are concentrated in a narrow
band, i.e. below a moving rigid block. This is a good assumption for layered soils
having a weaker layer sandwitched between more resistant soils. In homogeneous soils,
such as that in the VELACS model #2, the downslope displacements are distributed
with depth, with maximum values at the soil surface. For such situation, the rigid block
assumption is a limitation of the Newmark model. This can be mitigated by either: (1)
considering the flexibility of the moving block (e.g. Rathje and Bray, 2000) and
accounting for continuous softening, or (2) considering a "stack of rigid blocks".
5. Summary and Conclusions
The original Newmark model has been enhanced by applying state-of-practice methods
of estimating the EPWP build-up during seismic events and dissipation after the
earthquake to obtain more realistic predictions of permanent slope displacements. The
results have been calibrated, and validated based on centrifuge test results. The results
show that the proposed procedure is promising, especially for risk assessment, involving
a large number of analyses and requiring a reliable and time effective algorithm. The
method has some limitations that will be mitigated as discussed in section 4.
6. References
Craig, R.F. 1992. Soil Mechanics, Chapman & Hall, London
Newmark, N.M., 1965. Effects of earthquakes on dams and embankments. 5th. Rankine Lecture,
Géotechnique, 15(2): 137-160.
Popescu, R. and Prevost, J.H. 1993. Centrifuge validation of a numerical model for dynamic soil liquefaction.
Soil dynamics and Earthquake Engineering, 12: 73-90.
Rathje, E. M., and Bray, J. D. 2000. Nonlinear Coupled Seismic Sliding Analysis of Earth Structure. Journal
of Geotechnical and Geoenvironmental Engineering, 126(11): 1002-1014.
Seed, H.B. and Idriss, I.M. 1982. On the importance of dissipation effects in evaluating pore pressure changes
due to cyclic loading. Soil Mechanics - Transient and Cyclic Loads, eds., Pande, N., Zienkiewics,
O.C.: 53-70.
Seed, H. B., Idriss, I. M., Makdisi, F., and Banerjee, N. 1975. Representation of irregular stress-time history
by equivalent uniform stress series in liquefaction analyses. Report No. EERC 75-29, Earthquake
Engineering Research Center, University of California, Berkeley.
Youd, T.L., Idriss, I.M., Andrus, R.D., Arango, I., Castro, G., Christian, J.T., Dobry, R., Finn, W.D.L., Harder
, L.F., Hynes, M.E., Ishihara, K., Koester, J.P., Liao, S.S.C., Marcuson, W.F., Martin, G.R., Mitchell,
J.K.,
Moriwaki, Y., Power, M.S., Robertson, P.K., Seed, R.B., and Stokoe, K.H. 2000, Liquefaction Resistance of
Soils: Summary report from the 1996 NCEER and 1998 NCEER/NSF workshops on Evaluation of
liquefaction resistance of soils, Journal of Geotechnical and Geoenvironmental Engineering, Vol.
127(10): 817-833.