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Slope Stability and Methods of Increasing the Factor of Safety

By Steve YaegerECI 281a

University of California, Davis

Department of Civil and Environmental Engineering

Introduction

Slope stability is one of the fundamental problems faced on a consistent basis by the

majority of practicing Geotechnical Engineers. This is why slope stability analysis is

emphasized both at the undergraduate and more so at the graduate level of Geotechnical

studies. Currently one can find and use multiple computer programs to perform anything

from integration and analysis of seepage through a section of an earth dam to an analysis

of the stability of a slope. It is in some ways it could be considered engineering by

Windows. Computers and computer programs are a very large part of current

engineering practice. So much so that it is very easy to forget the engineering principles

on which the programs are founded.

This paper serves to review and summarize some of the simpler techniques for the

analysis of slope stability as well as their applications and limitations.

This paper will also review techniques for increasing the factor of safety of unstable

slopes including the use of tiebacks, and the construction of stone columns. A case study

of the Forks of Butte project that used tiebacks as a stabilization method will also be

reviewed.

Techniques of Slope Stability Analysis

The following analysis techniques will be reviewed in this section:

1) Ordinary Method of Slices.

2) Simplified Janbu Method.

3) Simplified Bishop Method.


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All of the above methods of analysis divide a slide mass into a number of slices and

analyze the individual slices as seen in Figure 1. Dividing up the soil mass into a number

of slices allows one to accommodate differing slide mass geometries, stratified soils

within the mass and external loads.

Figure 1 Illustration of Division of Sliding Mass into Slices (Abramson et al., 1996).

F = Factor of Safety ZL = left interslice forceSa = available strength ZR= right interslice force

Sm = mobilized strength L = left interslice force angle

U = pore water force R= right interslice force angle

U = surface water force hL = height of force ZL

W = weight of slice hR= height of for ZRN = effective normal force = inclination of slice base

Q = external surcharge = inclination of slice topkv = vertical seismic coefficient h = average height of slice

kh = horiz. seismic coefficient hc = height of centroid of slice

Figure 2 Forces acting on a typical slice. (Abramson et al., 1996)


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Ordinary Method of Slices

This method assumes that the resultant of interslice forces is inclined such that it is

parallel to the base of the slice. This assumption does not satisfy interslice force

equilibrium. Please see the following equations derived from Figure 2 and the above

assumption:

)cos()cos(cos)1(sin' +++= QUkWWkUN vh

The mobilized shear strength at the base of each slice is determined using the following

equation:

F

NCSm

tan'+=

The moment equilibrium of the slices about a common center of the circular failure

surface is

=

++=n

i

vo RQUkWM1

sin]coscos)1([

=

+n

i

hRQU1

)cos](sinsin[

= =

=+n

i

n

i

chm hRWkRS1 1

0)]cos([][

If the factor of safety F is assumed to be the same for all of the slices then the following

can be derived:

=

=

+

=

=n

i

n

i

AAA

NC

F

1

321

1

)tan'(

)(cos

))(cossinsin(

sin)coscos)1((

3

2

1

R

hWkA

R

hQUA

QUkWA

ch

v

=

+=

++=


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Simplified Janbu Method

The simplified Janbu Method assumes zero interslice shear forces and does not satisfy

moment equilibrium. However, the simplified Janbu method does satisfy vertical force

equilibrium and overall horizontal force equilibrium.

The normal effective stress at the base of each slice can be determined with the following

equations:

cos

coscos)1(sincos'

QUkWSUN

vm +++=

The overall horizontal force equilibrium for the slide mass is determined from the

following:

=

= =

=+

+

+++=

n

i

n

i

n

i

hiH

F

NCQ

UWkUNF

1

1 1

0]costan'

sin[

]sinsin)'[(][

It then follows that the Factor of Safety F can be determined with the following equation:

sinsinsin

sin'

cos]tan'[

4

1

4

1

QUWkUA

NA

NC

F

h

n

i

n

i

+++=

+

+

=

=

=

The Simplified Janbu Method does not satisfy moment equilibrium for the slide mass, as

mentioned earlier. Therefore, Janbu performed more rigorous solutions and compared

the result to those found using his simplified method. He then presented the following

chart as seen in Figure 3 to correct for his over-determined solution.


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Figure 3 Janbus Correction factor for his simplified method

FJanbu=fo * Fcalcualted

Simplified Bishop Method

The Simplified Bishop assumes zero interslice shear forces, satisfies moment equilibrium

around the center of a circular failure surface and satisfies vertical force equilibrium.

The moment equilibrium of the sliding mass is given by the following:

= =

=

=

=+

+

++=

n

i

n

i

chm

n

i

n

i

vo

hRWkRS

hRQU

RQUkWM

1 1

1

1

0)]cos([][

)cos](sinsin[

sin]coscos)1([

The factor of safety, F, can be determined from the following:

=

=

+

+

=n

i

n

i

AAA

NC

F

1

765

1

tan'

)(cos

))(cossinsin(

sin)coscos)1((

7

6

5

R

hWkA

R

hQUA

QUkWA

ch

v

=

+=

++=


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The vertical effective stress at the base of the slice can be determined by summing the

vertical forces:

]tantan

1[cos

]coscoscossin

)1([1

'

Fm

QUUF

CkW

mN v

+=

++=

Bishops Simplified Method does not satisfy horizontal force equilibrium for one of the

slices and can only be applied to circular surfaces. As can be seen from the simplicity of

the above equations the method is very easy and can provide results within five percent

of those found with more rigorous methods.

The methods described above are very simple, however with simplicity comes

limitations. Failure is assumed to begin when the factor of safety on the failure surface is

equal to one therefore the stress-strain relationship is neglected (i.e. the methods to not

allow for strain hardening of the soil.) The methods mentioned above also assume a

constant factor of safety along the failure surface. This error in this assumption is

amplified when there are different soils along the failure plane.

Methods of Increasing the Factor of Safety

It is very well known fact that slopes fail. They fail for a variety of reasons. One may be

excavation of the toe of an embankment removing the balancing moment of the slope and

causing failure. Another could be an increase in pore pressure along the failure plane and

a corresponding decrease in vertical effective stress and a loss of strength.

There have been methods developed to help stabilize failing slopes and reinforce slopes

that could fail. This section will discuss two techniques that are used to reinforce existing

failures, installation of tiebacks and the installation of stone columns. It will be seen thatthese methods can be very effective, if utilized correctly, stabilizing existing failures and

increasing the strength of slopes on the verge of failing.


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Tiebacks

The concept of tiebacks is basically that one carries the lateral earth pressure with a tie.

The tie transfers the lateral load of the soil to a zone of soil or rock located beyond the

failure plane as can be seen in Figure 4.

Figure 4 Illustration of Tieback Mode of Stabilization

The following design guideline should be used when working with tiebacks:

1) Tiebacks are typically installed in cohesionless soils

2) Typically have a design load of 50-130 tons (for ease of installation and size of

equipment required.

3) The length if the tieback should be selected such that the anchorage zone is

beyond the original failure plane.

4) Shallow failure surfaces typically only require one row of tiebacks.

5) The inclination of the tiebacks should be between 10 and 30 degrees from the

horizontal.

6) There should be a minimum of 15 feet of overburden above the zone of

embedment.

7) All permanent tiebacks should be corrosion resistant.


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Stone Columns

Stone columns increase the factor of safety by two means:

1) Increases the average shear resistance of a soil by displacing or replacing the soil

with a series of gravel columns. (Reference Figure 5)2) Columns act as a drain to decrease the effective stress in the soil and thereby

increase the strength along the failure surface.

According to Abramson et al. (1996) stone columns should be used in soils with shear

strengths in the range of 200-1000lb/ft2. Soils in the lower range of these limits may not

provide enough lateral support and thus the soil will consume too much stone to make the

method cost effective. Soils in the higher range may not benefit from the placement of

stone columns.

Figure 5 Use of Stone Columns to Stabilize Slopes

The interested reader is encouraged to reference Abramson et al. (1996) for design

techniques.

Case Study

This case history takes us to Northern California, approximately 30 miles east of Chico,

California, to the Forks at Butte hydroelectric project. The project site was located on a

large landslide deposit (a section through the deposit along with the construction

alterations can be seen in Figure 6.)


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Figure 5 Section of Existing Site Conditions Prior to Remediation

As can be seen in Figure 5 the activities until failure included installation of a tensar wall

and excavation of the area at the top of the wall along with the installation of tiebacks to

support the new face of the slope. Slope movement was detected with the use of slope

inclinometers. All construction activity halted when slope movement was detected.

As can be seen in figure 5 the reason for movement was due to the excavation of the toe

of the slope causing an unbalance in the moment of the slope. Tiebacks were install to

maintain the face of the cut in slope however as seen in figure 5 they were useless in

increasing the factor of safety of the slope due to the fact that they did not extend beyond

the failure plain.

After considerable analysis it was determined to increase the factor of safety with a

combination of techniques. The first was using a buttress fill at the toe of the large slope

as can be seen in Figure 6. This technique was not discussed in this paper however is

pretty simple to see that by placing the buttress fill one is increasing the balancing


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moment of the slope similar to replacing what was excavated in this application. The

second was the installation of additional tiebacks.

Figure 6 Section of Site after Remediation

Five additional rows of tieback were installed above the cut slope that extended into the

bedrock underlying the site. Five additional rows of tiebacks were also placed at the very

bottom toe of the slope and were also embedded in the underlying bedrock.

Figure 7 Slope Inclinometer Data

As can be seen from the slope inclinometer data in Figure 7 the remediation activities

were successful. Upon completion the movement ceased.


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Conclusions

This paper has discussed methods of analytical analysis of the factor of safety of slopes

along with methods of increasing the factor of safety and finally a case study

demonstrating the effectiveness of one of the remediation techniques.

Currently there are several slope stability analysis computer programs on the market that

can analyze much more complex slopes and conditions then any gifted engineer could all

in a matter of seconds. It then becomes the engineers responsibility to examine the

output and determine if the program was effective in its analysis. Most of the time the

computer will be right if all of the parameters have been specified correctly. One still

needs to check.

The methods of analysis are useful for that purpose, to anticipate the solution prior to

running the program. After all this author would content that if you cant estimate the

solution prior to letting something else do it for you then you have no business trying to

solve the problem.

Methods of increasing the factor of safety were also discussed. There will always be

circumstances as illustrated in the previous case study where construction activities begin

and unbalance a slope or disturb and existing failure. Those mistakes can be costly and

life threatening. It then becomes essential to be able to fix the problem. The methods

discussed along with countless others can and should be utilized in practice to save not

only ongoing construction sites but existing structures as well.

References

Powrie, William (1997), Soil Mechanics Concepts and Applications, E&FN Spon,

London, England

Abramson, Lee W., Lee, Thomas S., Sharma, Sunhill, Boyce, Glenn M. (1996), Slope

Stability and Stabilization Methods, John Wiley & Sons, Inc., New York, New York.


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Klein, S.J., Hughes, D.K. (1992) Slope Stabilization at the Forks of Butte Project,

Stability and Performance of Slopes and Embankments II, Geotechnical Special

Publication No. 31, Session 7

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