Embankment basal stability analysis using shear strength reduction finiteelement method
Atsushi Nakamura

FORUM8 Co. Ltd., 2-1-1, Nakameguro GT Tower15F Kamimeguro, Meguro-ku, Tokyo, Japan

Fei Cai & Keizo Ugai

Gunma University, Kiryu, Gunma 376-8515, Japan
ABSTRACT: The stability evaluation of the slope commonly uses limit equilibrium methods, for example,
Fellenius’ method in Japan. The limit equilibrium methods have two limitations. (1) It is necessary to divide
the sliding mass into slices, and to set up some additional equations to make the problem statically determinate.
(2) It is necessary to search sliding surface with a minimum safety factor. Shear strength reduction finite
element method (SSR-FEM) can solve the limitations of the limit equilibrium methods. This paper inspects
superiority of the slope stability analysis by SSR-FEM. And, as a model case, the failure of base ground of theembankment is considered. Comparing the calculated results of SSR-FEM with limit equilibrium method, we
point out the problems inherent in limit equilibrium method for the model case when the slip surface isassumed to be circle. We propose an approach to use SSR-FEM for slope stability analysis in practical design.
1 INTRODUCTION
The stability evaluation of the slope commonly uses
limit equilibrium methods, for example, Fellenius’
method, in most design codes and standards inJapan. The limit equilibrium methods have twolimitations.(1) It is necessary to divide the sliding
mass into slices, and to set up some additional
equations to make the problem staticallydeterminate.(2) It is necessary to search sliding
surface with a minimum safety factor. Shear strengthreduction finite element method (SSR-FEM) cansolve the limitations of the limit equilibrium
methods.
SSR-FEM can obtain the safety factor and slip
surface without the analyzer to assume any
particular shape about the slip surface. SSR-FEM
begins to be used to analyze the slope stability
though it is not commonly in the design. The limit
equilibrium methods are still used in almost cases. Itis because the limit equilibrium methods are simple.
However, the limit equilibrium methods are not fully
verified for the following items.(1) Whether it iscorrect to assume that the slip surface is one circulararc.(2) How wide is the range of the grid for thecenter of slip circular arc to search the minimum
safety factor.
Then, embankment base failure of the support
ground of the fill was taken up as a model case inthis report. When the support ground consists ofclayey soil layer and bearing stratum, the thicknessof the clayey soil layer is changed, and the slip
surface are analyzed using SSR-FEM. Using theseobtained slip surfaces, we arranges the extent thatthe slip surface become a circular arc or a noncircular arc. It is the main one of objectives t thestability examination that uses the SSR-FEM fromthe stability examination by the limit equilibrium
method in this report.

2 SSR-FEM FOR STABILITY ANALYSIS
2.1 Basic concept of SSR-FEM
In the finite element method with shear strength
reduction technique(SSR-FFM),a non-associatedelasto-plastic constitutive law is adopted, where theMohr-Coulomb yield criterion is used to define theyield function.
f =σ1 − σ 22− c' cos φ '−σ1 − σ 22sin φ '(1)and the Drucker-Prager criterion to define the plastic
potential.
g = −αI 1 + J 2 − κ(2)Whereα=tanψ
9 + 12 tan ψ2,κ =3c'
9 + 12 tan 2 ψ(3)In Equation 1 and 2, c', φ ', and ψ are theeffective cohesion, friction angle, and dilatancyangle, resperctively. I1 and J2 are the first invariantof the effective stress, and the second invariants ofthe deviatoric stress, respectively. σ1 and σ3 are
the majar and minor principal effective stress,respectivety.
The global safety factor of slope in SSRFEM
identical to the one in limit equilibrium methods.
The reduced strength parameters c'F and φ'F aredefined by

c' F = c' F , φ ' F = tan −1 ( tan φ ' F )(4)In SSRFEM, firstly, the initial stresses in slope
are computed using the elastic finite element
analysis. The vector of externally nodal forcesconsists of three parts: (1) surface force; (2) bodyforce(total unit weight of soils); and (3) pore waterpressure. Secondly, stresses and strains arecalculated by the elasto-plastic finite element
analysis, where the reduced shear strength criterion.
The shear strength reduction factor F is initiallyselected to be so small, for example 0.01, that theshear strength is large enough to keep the slope inelastic stage. Stresses at some Gaussian points reachthe yielding condition with the shear strength
reduction factor F in Equation 3 increased gradually.
When the stress at anyone Gaussian point reaches
the yielding condition, the increment of the shear
strength reduction factor will make stresses at more
Gaussian points reach the yielding condition becauseof the residual force induced by the decrease in theshear strength soils.

Figure 1 A point

the immediately previous one. The proceduredescribed hereby can predict the factor of safety
within one loop, and can be easily implemented in acomputing code.
One of the main advantages of SSRFEM is thatthe safety factor emerges naturally from the analysis
without the user having to commit to any particularfrom of the mechanism a priori. When the slope
stability is evaluated with the effective stressmethod, the pore water pressure is usually predictedwith the finite element seepage analysis or Biot's
consolidation theory. If the same finite element
mesh is used for the seepage or consolidation
analysis and SSRFEM, the water pressure, predictedin the seepage or consolidation analysis, can bedirectly used in SSRFEM. This can simplify theslope stability analysis, and consider moreaccurately the influence of the seepage force.
2.2 Predominance of SSRFEM
This example was firstly reported by Arai and Tagyo
and concerns a layered slope (Figure 3), where alayer of low resistance is interposed between twolayers of higher shear strength. The material
properties of various layers are listed in Table 1.
Arai and Tagyo used the conjugate gradient method
to search for the critical slip surface and used
simplified Janbu's method to calculate the safety
factor, and obtained the lowest safety factor of0.405. This problem has also been examined bySridevi and Deep using the random-search technique
RST-2, by Greco using pattern search and Monte
Carlo method, and by Malkawi et al. using Monte
Carlo method (random walking). The slip surfaces
located by various investigators are of significant
difference. In this study, the problem is analyzed andTable 2 summarizes the results. Figure.4 shows theslip surface determined by SSRFEM. It is clear thatthe slip surface by SSRFEM is closer to that locatedby Greco.
Table 1 Material Properties for Example

Figure 2. Mohr's circle at A point.

The shear strength reduction factor F increases
incrementally until the global failure of the slope
reaches, which means that the finite element
calculation diverges under a physically real
convergence criterion. The lowest factor of safety ofslope lies between the shear strength reduction
factor F at which the iteration limit is reached, andFigure 3. Slope in example.

Table 3 Material Properties for Test Model

Figure 4. Slip surface by SSRFEM for example.

Table 2 Minimum Safety Factors for Example
Method for
Method forsafety factor
slip surface
Arai and Tagyo (1985)
Simplified Janbu
Conjugate gradient
Sridevi and Deep (1992)
Simplified Janbu
RST-2
Greco (1996)
Spencer
Pattern search
SpencerMonte Carlo
Malkawai et al.(2001)
SpencerMonte Carlo
Rocscience Inc. (2002)
Spencer
Simplified Janbu
Kim et Al. (2002)
Spencer
Lower-boundUpper-boundCai et Al. (2003)
SSRFEM (ψ = 0 )SSRFEM (ψ = φ )Safety factor

0.405, 0.430※
0.401, 0.423※
0.388
0.388
0.401

Random search
Random search

0.401
0.410, 0.434※

Random search
Automatic
Automatic

0.44
0.40
0.45

Automatic
Automatic

0.417
0.423

Material properties is show in Table 3. The
necessary matrix of a plasticity calculation wasdefined using Young's modulus and Poisson's ratio.
The influence of these coefficients before the failureis great, but the influence on the total safety factor isvery small (Zienkiewicz et al.1975, Griffiths andLane 1999).
Therefore, we assumed Young's modulus of
2x104kN/m2, Poisson's ratio of 0.3 regardless of thesoils. As for the embankment, φ =35.0 degrees, theadhesive strength of the soft layer assumed
35kN/m2, the adhesive strength of the bearing
stratum 100kN/m2.
3.2 Analysis result
3.2.1 h2 = 2.0m
Figure 6(a) shows an analysis result of h2=2.0m. In
SSR-FEM, the slip surface occurred in soft layerbase and was a non-circular arc. In contrast, circularslip was searched by the limit equilibrium method
located at the embankment slope. The slidingmechanics was different. As for the safety factor,
SSR-FEM gave 1.24, and the limit equilibrium
method gave 1.21.

※ corrected the safety factor calculated by simplified Janbu

3 APPLICATION OF SSR-FEM
3.1 Embankment basal failure modelThis paper compared limit equilibrium methods(Bishop’s simplified method) and SSR-FEM(ψ = φ )about embankment base failure of the support
ground of the fill. Figure 5 shows the modelembankment. The embankment was assumed 20.0min width of levee crown, 10.0m in height (h1). In twolevels of strata, the upper layer was soft, the lowerlayer was strong. The thickness (h2) of the upperlayer was changed from 2.0m, 4.0m, 6.0m, 8.0m, to10.0m, to compare the results of both analyticalmethods.
20.010.0
soft groundbearing stratum

Figure 5. Embankment basal destruction model.

Figure 6(a). Result of SSRFEM and LEM h2=2.0m

3.2.2 h2 = 4.0m
Figure 6(b) shows an analysis result of h2=4.0m. In
SSR-FEM and limit equilibrium method, the slip
surface occurs in soft layer base and become a noncircular arc. As for the safety factor, SSR-FEM gave
1.16, and limit equilibrium method gave 1.22, andsome differences occurred between the two methods.

3.2.4 h2 = 8.0m
Figure 6(d) shows an analysis result of h2=6.0m. In
SSR-FEM and limit equilibrium method, the slip
surface occurs in soft layer base. As for the safety
factor is both 1.10.

Figure 6(d). Result of SSR-FEM and LEM (h2=8.0m)

Figure 6(b). Result of SSRFEM and LEM h2=4.0m

3.2.3 h2 = 6.0m
Figure 6(c) shows an analysis result of h2=6.0m. In
SSR-FEM and limit equilibrium method, the slip
surface occurs in soft layer base. As for the safetyfactor, SSR-FEM gave 1.16, and LEM gave 1.22.

Figure 6(c). Result of SSR-FEM and LEM (h2=6.0m)

3.2.5 h2 = 10.0m
Figure 6(e) shows an analysis result of h2=10.0m. In
SSR-FEM and limit equilibrium method, the slip
surface occurs in soft layer base. As for the safety
factor is both 1.08.

Figure 6(e). Result of SSR-FEM and LEM (h2=10.0m)

3.3 The comparison of the analysis result
Table 4 shows the results of both methods. Figure 7
shows the results between the safety factor and theratio of thickness (h2) of the soft layer andembankment height (h1=10.0m).

Table 5 shows the result that measured the lengthof the straight line of the non-circular arc in SSRFEM.
Table 5. Length of the straight line every layer thickness

Table 4. An analysis result table every layer thickness

According to the list, the length of the straightline gets longer so that the thickness of the soft layer
increases, but the ratio for the layer thickness
understands a thing becoming small. In other words
the glide plane suffers from the shape that is near toan arc so that a soft layer is thick.

1.300
1.250

Fs

1.200
1.150

SSR
LEM

1.100
1.050
1.000
0.950
0.200

0.400

0.600
h2/h1

0.800

1.000

Figure7. A figure of comparison

Figure shows that both analysis results consistent
with each other when the thickness of the soft layer
is as above 6.0m (h2/h1≧0.6).
However, in the situation that a soft layer is thin,a difference occurs in SSRFEM and limit
equilibrium method.
When a soft layer is very thin h2=2.0m(h2/h1=0.2), in SSR-FEM, bottom failure occurs. On
the other hand, embankment slope failure occurs bylimit equilibrium method. The calculated failure
mechanics was different with each other.
Slip surface in the straight line occurs to go along
the soft layer lower part from a result of SSR-FEM.
In other words this case becomes the non-circulararc.
The limit equilibrium method assumes that it iscircular arc sliding. As a result, it is thought that asafety factor searched the lowest circular arc sliding.
However, the limit equilibrium method was not able
to search the smallest safety factor and the sliding
shape, because a safety factor of SSR-FEM is 1.21,
and a safety factor of the limit equilibrium method is1.24.
In h2=4.0m, embankment base failure occurs
both in SSRFEM and limit equilibrium method. The
slip surface occurs in soft layer base and become anon-circular arc. As for the safety factor, the latter
was 1.16 former 1.22. Among both, somedifferences occur. In h2=6.0m are regarded as noncircular arc about the sliding shape in SSR-FEM
about 8.0, 10.0m as follows. But, the ratio that thetangent holds shrinks as a soft layer thickens.
Therefore, for a safety factor, is approximatelyequal.

Figure 8. A non-arc by limit equilibrium method slips.

Figure 8 is a result of non-arc sliding by the limit
equilibrium method that considered a straight line
calculated by SSR-FEM about layer thickness 4.0m.
It was 1.17, and the safety factor almost equal to thatof SSR-FEM.
4 CONCLUSION
By this report, we compared an analysis result ofSSR-FEM with the limit equilibrium method. With
that in mind, we wanted to grasp such soil layer
condition that sliding surface became the non-arc
from the analysis result of SSR-FEM.
It is necessary to give the center of slip circular
act by the judgment of the designer after having
supposed sliding shape to be an arc by the limit
equilibrium method. Therefore, the limit equilibrium
method cannot cope with a non-arc even if theminimum safety factor was able to search thecircular act.
SSR-FEM does not have to assume anexamination condition, in a given condition, there is
the big advantage that it is slippery, and can searchshape to become the minimum safety factor.
When the thickness of the layer was thin, SSRFEM could search a non-arc as sliding line
associated with the minimum safety factor, and, inthe example, the difference with the limit
equilibrium method became great.

By this report, SSR-FEM analyses were
performed to find the parameter which wasascertained whether the sliding shape was noncircular. The soft layer thickness was changed and aseries of analysis was carried out.
Although the series of analyses was performed, agood parameter was not found to indicate the layer
thickness of the soft layer under which non-arc
sliding should occur. A future search should beperformed.
At present, we advise that the sliding shape wascalculated using SSR-FEM, the safety factor of thecalculated slip surface was evaluated using limit
equilibrium method wanted by design codes andstandards.
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