advance methods of slope-stability analysis for economical design of earth embankment

Paper No. 537

ADVANCE METHODS OF SLOPE - STABILITY ANALYSIS FORECONOMICAL DESIGN OF EARTH EMBANKMENT†

B.N.SINHA*

1. INTRODUCTION

The slope stability analysis is usually done byFellenius method also called Ordinary or Swedish Circlemethod. It does not take into consideration inter sliceforce and considers only moment equilibrium and notforce equilibrium conditions and thus provides onlymoment factor of safety and not force factor of safety.It also does not provide the closed force polygon in freebody force diagram for the individual slice within thefailure zone of earth mass. It, therefore, does not satisfythe equilibrium conditions. It provides the factor of safetyon lower side and thus results into a more conservativedesign of earth slope. The Bishop Simplified methodwhich is sometimes used takes into consideration onlyinter slice normal force, but does not take the inter sliceshear force. This also gives moment factor of safetyonly. Here also force polygon does not close.TheMorgenstern-Price method takes into consideration boththe inter slice normal and shear force and also provide

moment equilibrium and force equilibrium giving bothmoment and force factor of safety. The force polygonfor the individual slice gives closed polygon thus satisfiesthe equilibrium conditions. This paper is aimed atdiscussing the Morgenstern-Price method in detail. Acomparison of methods the merit and limitations ofdifferent methods have also been discussed. A simplecase of earth slope has been analyzed. The mathematicalequations for the equilibrium conditions and factor ofsafeties are stated. A computer software was used foranalysis and results compared. While discussing themethodology, factors like half sine functions, constantfunctions,‘λ’, the ratio of applied function with thespecified function, mobilized shear etc. concepts ofwhich are required for the analysis, have also beenexplained in the paper.

The Guidelines for the design of High EmbankmentsIRC: 75 (1979) needs updating in light of recent advancesand use of soft ware. One of the objective of the present

*General Manager, Intercontinental Consultants and Technocrats Pvt Ltd. New Delhi-110016. } E-mail: [email protected]† Written comments on this Paper are invited and will be received upto 31st December, 2007

ABSTRACT

The concept and theory involved in different methods of slope stability analysis of earth embankment have been discussed.The mathematical equations and the methodology for calculating the factor of safety of earth slope of any specified(chosen)slip circle by various methods has been given. By repeating the process for different slip circles, the minimum factor ofsafety can be calculated and critical slip circle obtained. The forces which act within a soil mass have been discussed. Theinter slice normal and shear forces which are being also considered in many methods of analysis, have been explained andmathematical equations given to calculate them for the analysis.

The specified function f (x) (including half-sine function) and applied function ratio denoted by ‘λ’ has been explained. Asimple example of earth embankment has been analyzed to illustrate the methodology. The results as obtained bymathematical calculations proceeding ab-initio have been compared with the output using a soft ware for such analysis.For the purpose of direct comparison and easy explanation the critical circles were first established by the computersoftware by various methods of analysis and to illustrate the method only these circles were analyzed through independentmathematical equations and computations using Microsoft Excel program for the iterating process. It could be seen thatwithout the use of computer for the analysis, particularly the iterating process, it would have been very cumbersome andtime consuming to do the same by manual calculations. But it is possible to do complete analysis by Excel as explained inthis paper. Graphical method can be used for marking the circle and various slices as is the normal practice for slope-stability analysis. Graphical method of analysis can be used to draw force polygon to obtain various forces and computingfactor of safety, but this paper has dealt with mathematical equations only for the analysis part. Since the main emphasis ison explaining and demonstrating the various methods, set of minimum forces (such as seismic, pore water pressure, someexternal force etc. not taken) have been considered, however, without any loss of merit for the methodology.

202 B.N. SINHA ON

discussion is to introduce advance methods of stabilityanalysis for High Embankment formed of soil/earthso that necessary modification of IRC-75 can be broughtabout.

2. DEFERENT THEORIES FOR ANALYSIS

The failure of a slope could be slippage of earthmass along a slip surface (generally circular). Coulomb(1776) considered a wedge failure in his theory ofearth pressure, Rankine (1857) considered zone offailure where each element is at the verge of failure.All the methods of slope stability analysis in practiceconsider discretization of failure zone into slices. A sliceof earth will be subjected to the forces shown in Fig.1.None of the present methods takes into considerationthe strain compatibility of the slices within the zone offailure. All methods are covered within the ambit ofLimit Equilibrium Analysis (LEA). The LEA furthercomprises of Moment Equilibrium and Force Equilibriumand provide Moment factor of safety and/or Forcefactor of safety depending on the method of analysis.

A very simple earth embankment as shown in Fig. 1has been taken for illustrating the various methods. Thesoil properties of embankment soil and foundation soil,loading etc. are given in the figure.

φ = Angle of internal frictionγ = Unit weight of soilF

m= Moment factor of safety

Ff

= Force factor of safetyx = Horizontal distance of weight of slice

from center of rotation,

The different methods of analysis consider differentforces and equilibrium conditions to arrive at factor ofsafety of earth slope. This results in different factor ofsafety for different methods of analysis. The conditionsand forces considered are illustrated in Table 1. Theearth slope do not fail in a particular/definite way andthe most appropriate method in a particular situation isdesigner’s own discretion. Further the magnitude of someof the forces (such as base normal force, inter slice shearand inter slice normal forces, factor of safety) areindeterminate as much as they are interdependent andtherefore some assumptions are made for certainfunctions to make it determinate in order to enablecomputation of factor of safety.

3. MATHEMATICAL EQUATIONS FOR FoS

3.1. Moment Equilibrium Condition

The summation of moments, about center of rotation(an axis point) of the forces acting on slices shown inFig. 1, for all slices gives following equation.

Σ W*x - Σ Sm*R = 0

substituting Sm = (c*1 + Ν* tan Φ)/ F

m, and

x= R*sin α and rearranging we get

.......... (Εq. 1)

It is a point to note that in Eq. 1 inter slice forces(normal & shear forces) are not figuring. It is becausethese are equal and opposite on interface of two slicesand sum total of all such forces is zero. The left sideforce for the first slice and the right side force of the lastslice is already zero. At all other interfaces they areequal and opposite.

The ‘mobilized shear strength (Sm)’ at slice base is

that part of the shear resistance of the soil mobilizedwhich is just enough to satisfy the equilibrium conditionsof the slice. The soil strength at the slice base is.

= ( c*l + N*tan Φ)

The ratio of shear strength of the soil at slice base bymobilized shear is the factor of safety. Thus ‘S

m’ is

Where, W = Weight of sliceR = Radius of slip circleE

L= Left side slice normal force

ER

= Right side slice normal forceX

L= Left side slice shear force

XR

= Right side slice shear forceN = Base normal forcel = Base length (along the arc)α = Base angle to horizontalS

m= Mobilized shear force

F = Factor of safety ( FoS )c = Cohesion of soil

∑ (c*l+N*tanφ)

∑ (W*sinα)Fm =

Fig. 1. Earth Embankment adopted for analysis

Live Load 24 KN/m^2

Embankment SoilC = 10 KN/m^2∅ ===== 300

= 20 KN/M^3Height of Embank-ment = 10 m.

Foundotion SoilC = 40 KN/m^2∅ ===== 50

= 20 KN/M^3Depth Considered = 10 m

HIGHLIGHTS OF THE 178TH COUNCIL MEETING 203ADVANCE METHODS OF SLOPE - STABILITY ANALYSIS FOR ECONOMICAL DESIGN OF EARTH EMBANKMENT

obtained by dividing shear strength of soil at the slicebase by factor of safety (F

m). It will be F

f in case of

force equilibrium.

3.2. Force Equilibrium Condition

Resolving forces acting on slice as shown in Fig. 1,in horizontal direction and summing them for all slices,we get the equilibrium equation as given below-

Σ( ER-E

L) + Σ S

m* cos α - Σ N* sin α = 0

substituting, Sm= (c*l + N* tan Φ) / F

f , and as the factor

Σ(ER-E

L) when summed up for all slices shall be zero,

we get in rearranging-

...... (E q. 2)

It may be seen that moment equilibrium or forceequilibrium equations contain a factor ‘N’ which dependson factor of safety and inter-slice forces and can beobtained by iterating process as stated in the followingparas.

3.3. Base Normal Force of the Slice

The normal force at the base of slice is derived atby resolving forces on the slice (Fig.-1) in verticaldirection and we get the following equation:

( XR- X

L ) + N* cos α + S

m* sin α - W = 0

substituting Sm

= (c*l + N* tan Φ F) / F and we getrearranging

------ (Eq. 3)

.

This is a non linear equation as factor of safety ‘F’is appearing in the equation. F shall be F

m (moment factor

of safety) for moment equilibrium and Ff(force factor of

safety) for force equilibrium.

When inter-slice shear forces are ignored as inBishop method the equation for ‘N’ becomes

∑ (c*l * cos α + N* tan Φ* cos α)

∑ (N*sinα)F

f =

W - (XR- X

L) -

N =

c*l* sin α F

tan Φ* sin α F

cos α +

S.No. Methods Moment Force Inter Inter Moment Force Inter

Equilibr Equilibr slice slice factor of factor slice

-ium -ium normal shear safety of Force

force force safety function

1 Culman wedge block No Yes No No No Yes No

method (no -slice)

2 Fellenius,Swedish circle Yes No No No Yes No No

or ordinary method (1936)

3 Bishop Simplified method (1955) Yes No Yes No Yes No No

4 Janbu Simplified method (1954) No Yes Yes No No Yes No

5 Spencer method (1967) Yes Yes Yes Yes Yes Yes Constant

6 Morgenstern-Price method (1965) Yes Yes Yes Yes Yes Yes Constant

Half -Sine

Clipped-

Sine

Trapezod

Specifid

7 Corps of Engineers # 1 method No Yes Yes Yes No Yes Yes

8 Corps of Engineers # 2 method No Yes Yes Yes No Yes Yes

9 Lowe-Karafiath method No Yes Yes Yes No Yes Yes

10 Sarma method (1973) Yes Yes Yes Yes Yes Yes Yes

11 Janbu Generalized method (1957) No Yes Yes Yes No Yes Yes

TABLE-1. DIFFERENT METHOD OF STABILITY ANALYSIS INDICATING EQUILIBRIUM CONDITIONS, FORCES AND FOS CONSIDERED

204 B.N. SINHA ON

-------- (Eq.4)

3.4. Inter-Slice Forces

Inter slice forces are normal and shear forces actingin the vertical faces between slices. Resolving forcesfor the slice (Fig.1) in horizontal direction we get thefollowing equation:

( ER- E

L) + S

m * cos α - N * sin α = 0

substituting, Sm = (c*l + N* tan Φ) / F and rearranging

we get

----- (Eq.5)

The left side inter slice normal force for the firstslice is zero hence right side inter slice normal forcecan be obtained provided N & F also become known.Once the inter slice normal force is known the inter sliceshear force is computed as a percentage (assumed) ofinter slice normal force. This assumption results invarious methods of slope stability analysis developed bydifferent scientists based on assumptions they made.

4. DISCUSSION ON METHODS OF ANALYSIS

4.1. Morgenstern- Price Method

A typical discertization of slices as considered inthis method is shown in Fig.2.

They proposed the empirical equation for inter sliceforce relation as given in Eq.6.

X = E* λ* f(x) (Eq. 6) Where, X = inter slice shear force

E = inter slice normal force λ = the percentage of function used

and, f(x)= inter slice force function representing thevalue of function at the location of particular

Fig. 2. Slice Discertization for Morgenstern –Price Method

N =tan Φ* sin α F

cos α +

c*l* sin α F

W -

tanΦ*cosα F )+ N (sin α-E

R = E

L - c*l* cos α

F


The value of lambda for a typical slice (say No.5)shall be ratio of ordinate f (x) against slice No 5 readfrom applied function (lower curve in Fig. 4) and dividedby the value of f (x) read from specified function (uppercurve in Fig. 4) for the same slice. In this case this ratioi.e. λ =0.239. This is to note that the λ is same for all theslices. However, the value of f (x) itself varies fromslice to slice in half sine function and hence the ratio ofslice shear force to slice normal force ( X / E ) shall alsovary from slice to slice (Eq.6). However, in a constantfunction this ratio of slice shear force to slice normalforce shall not vary as the value of f (x) for all slices is(1.0) i.e. Constant. The λ in any case is constant.

4.3. Half-sine Function

Sine curve is obtained by plotting value of angle in radian( 0 to 2π ) on the X-axis and corresponding sine of the angleon the Y-axis, is well known. The half-sine curve is from zeroto π . This can be presented in a tabular form giving values asunder:

slice in question.

Variation of f (x) with slice position is assumed tohave different shapes as shown above.

The particular shape to be adopted in a given caseis the users choice. However, the half sine function isappropriate in most of the cases, as can be seen later.

4.2. What is Lambda ( λλλλλ )

Lambda used in the Eq. 6 is the percentage of appliedfunction to specified function f (x).The specified functioncan have many shapes (refer Fig. 3). In case of half-sinefunction the shape shall be as shown in Fig. 4.

The question is how to apply to find out the value offunction f (x) for different slices when half-sine functionis chosen to apply. It is done in this way. For chosen 20slices the f (x) at the right face of the first slice shall besin π/20. For the second slice this shall be sin 2π/20 andfor 17th slice shall be sin 17 π / 20 and so on. If the no ofslices chosen is 30 the value of f (x) for first slice shallbe sin π/30, for the second slice sin 2 π/30 and so on.Thus half-sine function value for any chosen no of slicescan be obtained and used in the computation. This valueof f (x) is specified function.

4.4. Spencer Method

In this method, the constant function is adopted andis similar to what has been explained for Morgenstern-Price constant function except that in the Spencer methodthe lambda is chosen such that FoS (moment) is equalto FoS (force) all other things remains same. This isseen that for the earth slope taken for analysis both thesemethods for constant function give same FoS in case ofmoment as well as force equilibrium.

4.5. Corps of Engineers #1 Method

In this method,the resultant inter slice force isassumed to act parallel to the line joining entry pointwith exit point as shown in Fig. 5. As the inclination ofresultant is constant (parallel to the same line) the ratioof inter slice shear force to inter slice normal forceremains same for all the slices. In other words this willcompare with constant function of Morgenstern-Price.However this method considers only force equilibriumand provides only force factor of safety. Incase of Corpsof Engineers # 2 method the resultant inter slice force isassumed to be parallel to the embankment slope as shownin Fig. 6. That is it will have only inter slice normal forceand no shear force where the embankment surface atslice top is horizontal. Rest of the things remains thesame as in Corps of Engineers #1 method.

X 0 π/6 π/ 3 π/ 2 2π/ 3 5π/ 6 π

sin X 0 1/2 √3/2 1 √3/2 1/2 0 Fig. 3. Different Alternative Function

Fig. 4. Showing Specified Function and Applied Function

206 B.N. SINHA ON

4.6. Lowe-Karafiath Method

This method considers for the ratio of inter sliceshear force to inter slice normal force the average ofthe top slope of embankment and inclination of the slicebase. It only considers force equilibrium. All other thingsremains same as in Corps of Engineers method.

4.7. Janbu’s Generalized Method

This method takes the resultant inter slice force toact at 1/3 from slice base and the inter slice shear forceis obtained by taking moment of force about slice basecenter point. The inter slice normal force is obtained asin Janbu’s Simplified method and is used for obtaininginter slice shear force. It considers only force equilibriumand not moment equilibrium as also applicable for Janbu’sSimplified method .

4.8. Sarma Method

This method assumes an equation like the soil shearstrength equation for the relation of inter slice shear forceand inter slice normal force and uses the same forcomputation of these forces. It considers both momentequilibrium as well as force equilibrium.

5. COMPUTATION OF FACTOR OF SAFETY

The analysis has been done by various methodskeeping all other features such as cross-section of the

embankment, geo-technical properties of the ground soiland borrow material same, for the sole purpose of notonly understanding these methods but also provide acomparative study and help selection of most appropriatemethod for adopting in a particular case.

To start the solution we may compute Felleniusmethod factor of safety obtained by putting N=W*cos αin the general equation (Eq. 1). This will result infollowing equation

-------- (Eq.7)

Fm

can be computed by Eq. 7 as all factors are known.Substituting F

m for F and W*cos α for ‘N’ in Eq.5, E

R

for the fist slice can be obtained ( EL is zero for the first

slice). By repeating, ER for the second slice is obtained (

ER

of the first slice is EL

for the second slice). This wayE

L and E

R for all slices can be obtained.

Choosing a

function and a value for λ the inter slice shear force XL

and XR is obtained by Eq. 6. Knowing E

L, E

R, X

L and X

R

the value of ‘N’ is obtained by Eq.3. Taking this value of‘N’ factor of safety is obtained by Eq. 1. The new ‘N’and ‘F’ will provide new value for E

L, E

R, X

L and X

R by

the relevant equations and further new value of ‘N’ and‘F’. This process is repeated till a converged value of‘F’ is obtained. The converged value of ‘F’ is the requiredFoS and the E

R, E

L, X

L, X

R and ‘N’ which produced this

is required value for these factors. In case of force factorof safety the Eq.2 is used for computing FoS and therest process remains the same. This appearscumbersome but very easy to apply in a Microsoft Excelprogram as can be seen by Annexure giving computationcharts. It is seen that two/three iteration provide theconverged solution.

To obtain FoS for Morgenstern-Price method (orany other method which considers inter slice shear force)‘λ’ is required to be decided. The best result is obtainedwhen the λ is chosen such that FoS (moment) andFoS(force) become equal. This is done by plotting FoSagainst λ for F

m and F

f (using Annexure I ,III,V and

VII by changing λ and obtaining converged FoS ) . Theintersection point provides this λ the value is then readfrom this curve and used for analysis. The curves forHalf-sine function and Constant function forMorgenstern- Price method are shown in Fig. 7 andFig. 8 .The intersection point in case of constant sinefunction gives a value of λ = 0.1814 and in case of Half-

Fig. 6. Showing Direction of Interslice Resultant force inCrops of Engineers # 2 Method.

Fig. 5. Showing Direction of Interslice Resultant force inCrops of Engineers # 1 Method

∑ (c*l+W*cosα* tan Φ)

∑ (W*sinα)Fm =


sine function the value of λ = 0.239. These valueshave been used in the computation. The SLOPE/Walso used the same value of λ and thus provided FoSfor similar input data and are comparable to computedFoS. Spencer considers only this value of λ andcomputes FoS for constant function only.

Fellenius method of computation was used forBishop critical circle and Morgenstern-Price criticalcircle. Factor of safety of F

m= 1.321 and F

m= 1.325,

respectively were worked out. The factor of safetywas obtained by solving the mathematical equations bythe use of Microsoft Excel for the iteration and resulttabulated. The FoS for corresponding situation wereobtained by SLOPE/W. The calculation was also donetaking the software out put for the value of N and FoScalculated by Eq. 1 and Eq. 2. All these results aretabulated in Table 2. It is seen that they compare quiteclose. Reference to related Annexure furnishing chartfor computation is given in this table:Fig. 8. Lambda vs. Factor of Safety

Fig. 7. Lambda vs. Factor of Safety

TABLE 2. COMPARISON OF FACTOR OF SAFETY OBTAINED INDEPENDENTLY, BY SLOPE/W AND BY CROSS CHECKING

Method Item FoS Annexure FoS AnnexureMoment Force

Morgenstern Obtained by solving 1.483 I 1.492 III-Price method mathematical equationConstant Function by Excel

SLOPE/W out put of 1.486 Give by 1.494 Given bysoft ware SLOPE/W SLOPE/W

Calculated by adopting 1.483 II 1.495 IV‘N’ from SLOPE/W out put

Morgenstern- Obtained by solving 1.493 V 1.498 VIIPrice method mathematical equationHalf-sine by Excel

Function SLOPE/W out put of 1.496 Give by 1.503 Given bysoft ware SLOPE/W SLOPE/W

Calculated by adopting 1.492 VI 1.501 VIII‘N’ from SLOPE/W out put

Bishop simplified Obtained by solving 1.509 IXmethod mathematical equation This method gives only

by Excel moment FoS an not force FoS

SLOPE/W out put 1.512 Give byof soft ware SLOPE/W

Calculated by adopting 1.511 X‘N’ from SLOPE/W out put

Fellenius Obtained for Mongenstern- 1.325 XIImethod Price This method gives only(consider Critical circle and rest same. moment FoS and not force FoS

208 B.N. SINHA ON

Method Item FoS Annexure FoS AnnexureMoment Force

ing normal baseslice force as

W* cos a) Obtained for BishopCritical 1.321 XIcircle and rest same .

Fig. 9 (a) Free body force diagram

Fig. 10 Free body force diagram

The force polygon for Morgenstern-Price methodconstant function and half-sine function were drawn andshown in Fig. 9. It is seen that the polygon gives a closedform satisfying equilibrium conditions. Similar polygon drawnfor forces in Fellenius method Fig. 10 gives an open formthus indicates lack of satisfying equilibrium conditions.

All other methods of analysis involves one of theseprocesses of calculations and fully covered by the above

Morgentern –Price Method, Half Sine Function,(Annexure -V ).

Morgentern –Price Method, Constant Function,(Annexure-I )

Fig. 9 (b) Free body force diagram

Fellenius Method for Critical Circle by Morgentern-Price ( Annexure-XII )

Fig. 11 Showing Critical Circles By Various Methods

explanation.

For various methods the computer soft ware is usedand the critical circle and factor of safety obtained. Allcritical circle by various methods have been marked onthe same sheet as shown in Fig. 11 also FoS tabulatedfor a ready comparison given in Table 3.

Chosen two slip surfaces shown in Fig. 12 havebeen analyzed through SLOPE/W and factor of safetyobtained by applying different methods. The results aretabulated in Table 4. This provides a comparison ofFoS vs. methods when all other factors are the sameincluding the slip circles. The slip surface chosen,naturally, are not the critical slip circle. When, however,the chosen circle is nearly the same as critical circle forsome of the methods, these gave nearly the same factor


TABLE 4. SHOW FACTOR OF SAFETY FOR SPECIFIED SLIP SURFACES FOR DIFFERENT METHODS

TABLE 3. SHOWS FOS FOR THE CRITICAL CIRCLE BY DIFFERENT METHODS AS OBTAINED BY SLOPE/W

Methods Morgen- Morgens- Spencer Bishop Janbu Corps Corps Lowe-

stern Price tern-Price method simplified General Of of Karafiath

method method method -ised Engineers Engineers method

Constant Half -Sine method # 1 # 2xx

function function method method

FoS Moment 1.486 1,496 1.486 1.512 No No No No

FoS force 1.494 1.503 1.494 No 1.555 1.595 1.676 1.628

Method MorgensternPrice Morgenstern Price Spencer Crops of Crops of Lowe- Bishop

Method(Constant) Method(Half-Sine) Method Engineers Engineers Karafiath Simplified

# 1 Method # 2 Method Method Method

F.O.S. 1.569 1.572 1.569 1.651 1.676 1.627 1.626

(Upper

-Curve)

F.O.S. 1.487 1.496 1.487 1.606 1.771 1.692 1.518

(Lower

-Curve)

of safety obtained by general application of the methodsas can be seen for Morgenstern-Price methods andSpencer method. It is also seen that the chosen circlesresulted in factor of safety higher than that obtained bydifferent methods for critical circles . It only proves thatcritical circle gives lowest factor of safety.

6. CONCLUSIONS

(i) All methods of slope stability analysis arecovered under Limit Equilibrium Analysis method. Allmethods resort to discretization of earth mass within thefailure zone in vertical slices. Some of them considermoment equilibrium only, some consider force equilibriumonly and some consider both, but they fall within one ofthese three categories. None considers strain compatibility

for the stability analysis.

(ii) Different methods give different critical circleand different factor of safety for same situation, namely,earth fill material and ground soil properties. Even forsame slip surface different methods give different factorof safety. However, the methods, which considers sameset of forces and same factors produce similar results.For example, Morgenstern-Price and Spencer methodsfor constant function produce same factor of safety (bothconsiders inter slice normal and shear forces).

(iii) Different methods result in different criticalcircles.

(iv) The old methods (Fellenius method and Swedishmethod ) give lower factor of safety (FoS) and thereforerequires a flatter slope for the specified FoS comparedto the earth slope designed on the basis of Morgenstern-Price method which will permit a steeper slope to achievethe same FoS and will cost less.

(v) It is preferable to adopt a method which satisfiesboth moment equilibrium as well as force equilibrium.The best result is obtained when FoS for both coincidesie FoS (Moment) = FoS (Force)

Fig. 12. Showing Typical slip Circles.

210 B.N. SINHA ON

(vi) The method, which takes inter slice forces(normal and shear) into account satisfies closed forcepolygon indicating equilibrium condition of the slice in afree body force diagram.

(vii) The Morgenstern-Price method with half sincefunction, which takes into account, inter slice forces andsatisfies both moment equilibrium and force equilibriumconditions is most appropriate for design of slopes.

(viii) The mathematical equations for obtaining factorof safety with the help of Microsoft Excel can givedesired result very quickly. Convergence of iterationrequired for the analysis is achieved very fast. Henceuse of computer is preferable to conventional graphicalmethod of analysis.

(ix) The Guidelines for the design of HighEmbankments, IRC-75 (1979), needs updating to includeother improved methods of slope stability analysis byuse of soft ware. Since the conventional methodsprescribed by the IRC code are very cumbersome dueto manual calculation.

(x) GEO-SLOPE International provide a veryefficient software package for analysis of slope-stabilitycovering almost all situations. The results (out put ofpackage) tallies quite closely with the results obtainedindependently by solving mathematical equations usingMicrosoft Excel.

ACKNOWLEDGEMENT

The author likes to thank Shri K.K.Kapila, CMD,ICT for giving permission to contribute this paper to IRC.The author is indebted to the ICT for utilizing the facilitiesof the organization in bringing this paper to present shape.

The author is thankful to Dr. S.K. Majumder,Advisor, ICT for his kind help in going through this paperand giving valuable advice for improvements.

The author is thankful to Mr.Dharmendra Kumar,Ms. Jyoti Priya & Mr. Sachin Roorkiwal of the ICTfor their help in carrying out all Microsoft Excelcalculations and operating the software program forthe analysis without this the paper would not have beencompleted.

REFERENCES

1. Design Aids in Soil Mechanics and FoundationEngineering by R. Kaniraj Shenbaga.

2. The Theoretical Soil Mechanics by Terzaghi Karl.

3. GEO-SLOPE International Ltd. Canada.”An Engineeringmethodology”.

4. Geo-technical Engineering by Gulati Shashi K. & DattaManoj.

5. Principles of Soil Mechanics and Foundation Engineeringby Murthy V.N.S.

6. Soil Mechanics in Highway Engineering by RodriguezAlfonso Rico, Castillo Hermillodel and Sowers George F.

7. The Guidelines for the Design of High Embankments-IRC-75 ( 1979 ).

212 B.N. SINHA ON

Fm C

alcu

late

d by

ado

ptin

g “N

” fr

om S

lope

/w o

utpu

t for

Mor

gens

tern

pri

ce’s

Met

hod

(Con

stan

t Fun

ctio

n)

Sli

ceW

(kn)

N (k

n)αααα α

(deg

)l (

m)

c (k

n/m

^2)

φφφφ φ (d

eg)

c*l +

N*t

anφφφφ φ

w *

sin

αααα α

no.

116

0.42

914

7.78

67.9

275.

275

1030

138.

071

148.

670

230

3.96

928

9.04

56.2

243.

561

1030

202.

487

252.

664

340

3.00

937

347

.619

2.93

610

3024

4.71

229

7.69

4

450

9.05

517.

440

.098

2.74

340

515

4.98

732

7.87

8

557

4.21

570.

533

.146

2.50

540

515

0.11

231

3.96

3

647

9.01

467.

4327

.121

2.04

440

512

2.65

521

8.36

7

747

646

1.69

21.8

21.

959

405

118.

753

176.

925

846

6.03

452.

6716

.71

1.89

940

511

5.56

313

3.99

7

944

9.71

440.

6911

.734

1.85

740

511

2.83

591

.457

1042

7.46

425.

86.

847

1.83

240

511

0.53

350

.961

1139

9.52

407.

832.

011

1.81

940

510

8.44

114

.020

1236

638

6.4

-2.8

111.

821

405

106.

646

-17.

949

1332

6.89

360.

91-7

.653

1.83

540

510

4.97

6-4

3.53

3

1428

2.06

330.

41-1

2.55

21.

863

405

103.

427

-61.

299

1523

1.25

293.

47-1

7.54

61.

907

405

101.

955

-69.

715

1617

424

7.88

-22.

684

1.97

140

510

0.52

7-6

7.10

3

1713

1.24

221.

24-2

8.15

52.

159

405

105.

716

-61.

927

1887

.595

191.

9-3

4.06

2.29

840

510

8.70

9-4

9.05

9

1932

.658

143.

11-4

0.41

72.

501

405

112.

561

-21.

174

Ann

exur

e : I

I

∑ (c

*l+N

*tan

φφφφ φ)

∑ (w

*sin

αααα α)

Sum

2423

.664

1634

.838

=

1.4

83F

OS

=

214 B.N. SINHA ON

For

ce F

OS

Cal

cula

ting

by

Mor

gens

tern

pri

ce’s

Met

hod

(Con

stan

t fun

ctio

n)

Sli

ceW

(kn)

N (k

n)αααα α

(deg

)l (

m)

c (k

n/m

^2)

φφφφ φ (d

eg)

(c*l

+N

tan φ

)∗φ)∗

φ)∗

φ)∗

φ)∗

N *

sin

αααα α

no.

cos αααα α

116

0.42

914

7.78

67.9

275.

275

1030

51.8

8513

6.94

9

230

3.96

928

9.04

56.2

243.

561

1030

112.

572

240.

255

340

3.00

937

347

.619

2.93

610

3016

4.95

027

5.52

7

450

9.05

517.

440

.098

2.74

340

511

8.55

633

3.25

6

557

4.21

570.

533

.146

2.50

540

512

5.68

631

1.93

5

647

9.01

467.

4327

.121

2.04

440

510

9.16

821

3.08

8

747

646

1.69

21.8

21.

959

405

110.

245

171.

606

846

6.03

452.

6716

.71

1.89

940

511

0.68

413

0.15

5

944

9.71

440.

6911

.734

1.85

740

511

0.47

789

.622

1042

7.46

425.

86.

847

1.83

240

510

9.74

450

.763

1139

9.52

407.

832.

011

1.81

940

510

8.37

414

.311

1236

638

6.4

-2.8

111.

821

405

106.

517

-18.

950

1332

6.89

360.

91-7

.653

1.83

540

510

4.04

0-4

8.06

4

1428

2.06

330.

41-1

2.55

21.

863

405

100.

955

-71.

807

1523

1.25

293.

47-1

7.54

61.

907

405

97.2

12-8

8.47

3

1617

424

7.88

-22.

684

1.97

140

592

.751

-95.

595

1713

1.24

221.

24-2

8.15

52.

159

405

93.2

07-1

04.3

94

1887

.595

191.

9-3

4.06

2.29

840

590

.060

-107

.476

1932

.658

143.

11-4

0.41

72.

501

405

85.6

97-9

2.78

5

Ann

exur

e : I

V

∑

∑

∑

∑

∑ [

(c*l

+Nta

n φφφφ φ) *

cos αααα α

]

∑∑∑∑ ∑(N

*sin

αααα α)

Sum

2002

.782

1339

.926

=

1.49

5F

OS

=

216 B.N. SINHA ON

Mom

ent F

OS

Cal

cula

ting

by

Mor

gens

tern

pri

ce’s

Met

hod

Sli

ceW

(kn)

N (k

n)αααα α

(deg

)l (

m)

c (k

n/m

^2)

φφφφ φ (d

eg)

c*l +

N*t

anφφφφ φ

w *

sin

αααα α

no.

116

0.42

916

7.5

67.9

275.

275

1030

149.

456

148.

670

230

3.96

930

2.55

56.2

243.

561

1030

210.

287

252.

664

340

3.00

937

3.02

47.6

192.

936

1030

244.

723

297.

694

450

9.05

506.

9140

.098

2.74

340

515

4.06

932

7.87

8

557

4.21

547.

1933

.146

2.50

540

514

8.07

331

3.96

3

647

9.01

444.

0327

.121

2.04

440

512

0.60

821

8.36

7

747

644

0.89

21.8

21.

959

405

116.

933

176.

925

846

6.03

437.

9816

.71

1.89

940

511

4.27

813

3.99

7

944

9.71

434.

3211

.734

1.85

740

511

2.27

891

.457

1042

7.46

428.

636.

847

1.83

240

511

0.78

050

.961

1139

9.52

419.

422.

011

1.81

940

510

9.45

414

.020

1236

640

4.97

-2.8

111.

821

405

108.

270

-17.

949

1332

6.89

383.

45-7

.653

1.83

540

510

6.94

8-4

3.53

3

1428

2.06

353

-12.

552

1.86

340

510

5.40

3-6

1.29

9

1523

1.25

311.

83-1

7.54

61.

907

405

103.

562

-69.

715

1617

425

8.25

-22.

684

1.97

140

510

1.43

4-6

7.10

3

1713

1.24

220.

02-2

8.15

52.

159

405

105.

609

-61.

927

1887

.595

175.

02-3

4.06

2.29

840

510

7.23

2-4

9.05

9

1932

.658

112.

11-4

0.41

72.

501

405

109.

848

-21.

174

Ann

exur

e : V

I

Sum

2439

.247

1634

.838

∑∑∑∑ ∑ (c

*l+N

*tan

φφφφ φ)

∑

∑

∑

∑

∑ (

W*s

inαααα α

) =

1

.492

FO

S

=

218 B.N. SINHA ON

For

ce F

OS

Cal

cula

ting

by

Mor

gens

tern

pri

ce’s

Met

hod

(Hal

f-si

ne fu

ncti

on)

Sli

ceW

(kn)

N (k

n)αααα α

(deg

)l (

m)

c (k

n/m

^2)

φφφφ φ (d

eg)

(c*l

+N

tan

φ)∗

φ)∗

φ)∗

φ)∗

φ)∗

N *

sin

αααα α

no.

cos αααα α

116

0.42

916

7.50

067

.927

5.27

510

3056

.164

155.

223

230

3.96

930

2.55

056

.224

3.56

110

3011

6.90

925

1.48

5

340

3.00

937

3.02

047

.619

2.93

610

3016

4.95

727

5.54

2

450

9.05

506.

910

40.0

982.

743

405

117.

854

326.

499

557

4.21

547.

190

33.1

462.

505

405

123.

978

299.

189

647

9.01

444.

030

27.1

212.

044

405

107.

346

202.

420

747

644

0.89

021

.82

1.95

940

510

8.55

516

3.87

5

846

6.03

437.

980

16.7

11.

899

405

109.

453

125.

931

944

9.71

434.

320

11.7

341.

857

405

109.

932

88.3

27

1042

7.46

428.

630

6.84

71.

832

405

109.

990

51.1

01

1139

9.52

419.

420

2.01

11.

819

405

109.

387

14.7

18

1236

640

4.97

0-2

.811

1.82

140

510

8.14

0-1

9.86

0

1332

6.89

383.

450

-7.6

531.

835

405

105.

995

-51.

065

1428

2.06

353.

000

-12.

552

1.86

340

510

2.88

4-7

6.71

6

1523

1.25

311.

830

-17.

546

1.90

740

598

.743

-94.

008

1617

425

8.25

0-2

2.68

41.

971

405

93.5

88-9

9.59

4

1713

1.24

220.

020

-28.

155

2.15

940

593

.113

-103

.818

1887

.595

175.

020

-34.

062.

298

405

88.8

37-9

8.02

2

1932

.658

112.

110

-40.

417

2.50

140

583

.633

-72.

686

Ann

exur

e : V

III

∑

∑

∑

∑

∑ [

(c*l

+Nta

n φφφφ φ) *

cos αααα α

]

∑

∑

∑

∑

∑(N

*sin

αααα α)

Sum

2009

.458

1338

.542

=

1.50

1F

OS

=


FO

S C

alcu

lati

ng b

y B

isho

p’s M

etho

d

Sli

ceW

(kn)

αααα α (d

eg)

l (m

)F

cφφφφ φ

(deg

)E

R=E

L-c

*l*c

osa/

F+

E Lc*

l +N

*tan

φφφφ φw

* s

inαααα α

N=[

W-{

(C*l

*Sin

αααα α)/

no.

(kn/

m^

2) N

*(si

na-(

tanf

*cas

a)/F

) F

}] /

[(si

nαααα α

* ta

nφφφφ φ)

/F

+cos

αααα α]

114

8.84

569

.03

5.18

41.

5090

1030

117.

811

0.00

014

6.10

813

8.98

716

3.27

7

228

2.87

558

.038

3.50

01.

5090

1030

304.

592

117.

811

212.

946

239.

991

308.

206

337

6.48

549

.96

2.87

91.

5090

1030

493.

100

304.

592

251.

949

288.

235

386.

522

445

3.05

343

.076

2.55

91.

5090

405

781.

915

493.

100

148.

570

309.

421

528.

183

551

1.55

336

.877

2.33

71.

5090

405

1047

.046

781.

915

143.

201

306.

983

568.

309

655

7.75

31.1

522.

184

1.50

9040

512

76.1

7810

47.0

4613

9.49

028

8.53

059

5.85

4

757

1.51

25.5

782.

218

1.50

9040

514

46.6

7612

76.1

7814

0.26

924

6.74

358

9.11

2

856

5.14

20.0

732.

130

1.50

9040

515

57.9

5014

46.6

7613

4.98

119

3.96

656

9.00

2

955

0.2

14.7

582.

068

1.50

9040

516

13.4

5315

57.9

5013

0.50

514

0.15

654

6.18

8

1052

7.43

9.56

92.

028

1.50

9040

516

17.2

3516

13.4

5312

6.67

787

.677

520.

720

1149

7.26

4.46

12.

006

1.50

9040

515

74.0

5916

17.2

3512

3.31

938

.677

492.

395

1245

9.95

-0.6

124

2.00

01.

5090

405

1489

.405

1574

.059

120.

317

-4.9

1646

0.82

9

1341

5.55

-5.6

92.

010

1.50

9040

513

69.6

7314

89.4

0511

7.61

5-4

1.20

042

5.37

4

1436

3.96

-10.

814

2.03

61.

5090

405

1222

.474

1369

.673

115.

133

-68.

287

385.

114

1530

4.89

-16.

028

2.08

11.

5090

405

1057

.062

1222

.474

112.

873

-84.

182

338.

710

1623

7.83

-21.

384

2.14

81.

5090

405

885.

093

1057

.062

110.

781

-86.

717

284.

160

1716

4.02

-26.

626

1.99

31.

5090

405

729.

752

885.

093

98.6

27-7

3.50

821

6.24

4

1812

8.59

-31.

832.

096

1.50

9040

557

1.38

072

9.75

210

0.70

6-6

7.81

819

2.77

8

1984

.999

-37.

291

2.23

91.

5090

405

420.

456

571.

380

103.

476

-51.

498

159.

062

2031

.416

-43.

222.

445

1.50

9040

529

3.24

542

0.45

610

7.42

4-2

1.51

411

0.00

9

Ann

exur

e : I

X

Sum

2684

.968

1

779.

726

FO

S

=

∑

∑

∑

∑

∑ (

c*l+

N*t

anφφφφ φ)

∑

∑

∑

∑

∑ (W

*sin

αααα α)=

1.

509

220 B.N. SINHA ON

=

1.51

1

FO

S C

alcu

lati

ng b

y B

isho

p’s M

etho

d

Sli

ceW

(kn)

N (k

n)αααα α

(deg

)l (

m)

c (k

n/m

^2)

φφφφ φ (d

eg)

w *

sin

αααα αc*

l +N

*tan

φφφφ φ

no.

114

8.84

516

3.52

69.0

35.

184

1030

138.

987

146.

248

228

2.87

530

8.46

58.0

383.

500

1030

239.

991

213.

092

337

6.48

538

6.78

49.9

62.

879

1030

288.

235

252.

098

445

3.13

352

8.43

43.0

762.

559

405

309.

475

148.

592

551

1.55

356

8.42

36.8

772.

337

405

306.

983

143.

210

655

8.60

364

1.76

331

.152

2.18

440

528

8.97

114

3.50

7

757

1.51

589.

225

.578

2.21

840

524

6.74

314

0.27

6

856

5.14

569.

0620

.073

2.13

040

519

3.96

613

4.98

6

955

0.2

546.

2314

.758

2.06

840

514

0.15

613

0.50

9

1052

7.43

520.

749.

569

2.02

840

587

.677

126.

679

1149

7.26

492.

414.

461

2.00

640

538

.677

123.

320

1245

9.95

460.

83-0

.612

42.

000

405

-4.9

1612

0.31

7

1341

5.55

425.

36-5

.69

2.01

040

5-4

1.20

011

7.61

4

1436

3.96

385.

09-1

0.81

42.

036

405

-68.

287

115.

131

1530

4.89

338.

67-1

6.02

82.

081

405

-84.

182

112.

870

1623

7.83

284.

11-2

1.38

42.

148

405

-86.

717

110.

776

1716

4.02

216.

18-2

6.62

61.

993

405

-73.

508

98.6

21

1812

8.59

192.

61-3

1.83

2.09

640

5-6

7.81

810

0.69

1

1984

.999

158.

97-3

7.29

12.

239

405

-51.

498

103.

468

2031

.416

109.

88-4

3.22

2.44

540

5-2

1.51

410

7.41

3

Ann

exur

e : X

∑

∑

∑

∑

∑

(c*

l+N

*tan

φφφφ φ)

∑

∑

∑

∑

∑ (W

*sin

αααα α)

Sum

1780

.222

2689

.421

FO

S =


FO

S C

alcu

lati

ng b

y F

elle

nius

Met

hod

,for

Bis

hop’

s cri

tica

l cir

cle

Sli

ceW

(kn)

αααα α (d

eg)

l (m

)c

(kn/

m^

2)φφφφ φ

(deg

)w

*cos

α∗α∗α∗

α∗ α∗

w *

sin

αααα α

no.

tan φ

+φ+φ+φ+ φ+cl

114

8.84

569

.03

5.18

410

3082

.595

138.

987

228

2.87

558

.038

3.50

010

3012

1.45

623

9.99

1

337

6.48

549

.96

2.87

910

3016

8.62

528

8.23

5

445

3.05

343

.076

2.55

040

513

0.95

330

9.42

1

551

1.55

336

.877

2.33

740

512

9.28

130

6.98

3

655

7.75

31.1

522.

184

405

129.

120

288.

530

757

1.51

25.5

782.

218

405

133.

829

246.

743

856

5.14

20.0

732.

130

405

131.

640

193.

966

955

0.2

14.7

582.

068

405

129.

277

140.

156

1052

7.43

9.56

92.

028

405

126.

631

87.6

77

1149

7.26

4.46

12.

006

405

123.

616

38.6

77

1245

9.95

-0.6

124

2.00

040

512

0.24

3-4

.916

1341

5.55

-5.6

92.

010

405

116.

573

-41.

200

1436

3.96

-10.

814

2.03

640

511

2.72

3-6

8.28

7

1530

4.89

-16.

028

2.08

140

510

8.87

3-8

4.18

2

1623

7.83

-21.

384

2.14

840

510

5.29

0-8

6.71

7

1716

4.02

-26.

626

1.99

140

592

.475

-73.

508

1812

8.59

-31.

832.

096

405

93.3

98-6

7.81

8

1984

.999

-37.

291

2.23

940

595

.476

-51.

498

2031

.416

- 43.

222.

445

405

99.8

03-2

1.51

4

Ann

exur

e : X

I

Sum

2351

.876

1779

.726

∑

∑

∑

∑

∑ (

w*c

osα∗

α∗

α∗

α∗

α∗

tan

φ+φ+φ+φ+ φ+cl

)

∑

∑

∑

∑

∑ (w

*sin

αααα α)=

1.

321

FO

S

=

222 B.N. SINHA ON

FO

S C

alcu

lati

ng b

y F

elle

nius

Met

hod

, for

Mor

gens

tern

Pri

ce’s

cri

tica

l cir

cle

Sli

ceW

(kn)

αααα α (d

eg)

l (m

)c

(kn/

m^

2)φφφφ φ

(deg

)w

*cos

α∗α∗α∗

α∗ α∗

w *

sin

αααα α

no.

tan φ

+φ+φ+φ+ φ+cl

116

0.42

967

.927

5.27

510

3087

.557

148.

670

230

3.96

956

.224

3.56

110

3013

3.17

725

2.66

4

340

3.00

947

.619

2.93

610

3018

6.19

829

7.69

4

450

9.05

40.0

982.

743

405

143.

788

327.

878

557

4.21

33.1

462.

505

405

142.

262

313.

963

647

9.01

27.1

212.

044

405

119.

060

218.

367

747

621

.82

1.95

940

511

7.02

117

6.92

5

846

6.03

16.7

11.

899

405

115.

011

133.

997

944

9.71

11.7

341.

857

405

112.

802

91.4

57

1042

7.46

6.84

71.

832

405

110.

411

50.9

61

1139

9.52

2.01

11.

819

405

107.

692

14.0

20

1236

6-2

.811

1.82

140

510

4.82

2-1

7.94

9

1332

6.89

-7.6

531.

835

405

101.

744

-43.

533

1428

2.06

-12.

552

1.86

340

598

.607

-61.

299

1523

1.25

-17.

546

1.90

740

595

.570

-69.

715

1617

4-2

2.68

41.

971

405

92.8

85-6

7.10

3

1713

1.24

-28.

155

2.15

940

596

.483

-61.

927

1887

.595

-34.

062.

298

405

98.2

69-4

9.05

9

1932

.658

- 40.

417

2.50

140

510

2.21

5-2

1.17

4

Ann

exur

e : X

II

Sum

2165

.576

1634

.838

∑

∑

∑

∑

∑ (

w*c

osα∗

α∗

α∗

α∗

α∗

tan

φ+φ+φ+φ+ φ+cl

)

∑

∑

∑

∑

∑

(w*s

inαααα α)

=

1.32

5F

OS

=

B.N. SINHA ON

ADVANCE METHODS OF SLOPE - STABILITY ANLYSIS FOR ECONOMICAL DESIGN OF EARTH EMBANKMENT

222