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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
HIGHLIGHTS OF THE 178TH COUNCIL MEETING 205ADVANCE METHODS OF SLOPE - STABILITY ANALYSIS FOR ECONOMICAL DESIGN OF EARTH EMBANKMENT
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 =
HIGHLIGHTS OF THE 178TH COUNCIL MEETING 207ADVANCE METHODS OF SLOPE - STABILITY ANALYSIS FOR ECONOMICAL DESIGN OF EARTH EMBANKMENT
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
HIGHLIGHTS OF THE 178TH COUNCIL MEETING 209ADVANCE METHODS OF SLOPE - STABILITY ANALYSIS FOR ECONOMICAL DESIGN OF EARTH EMBANKMENT
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 ).
HIGHLIGHTS OF THE 178TH COUNCIL MEETING 211ADVANCE METHODS OF SLOPE - STABILITY ANALYSIS FOR ECONOMICAL DESIGN OF EARTH EMBANKMENT
212 B.N. SINHA ON
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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
=
HIGHLIGHTS OF THE 178TH COUNCIL MEETING 213ADVANCE METHODS OF SLOPE - STABILITY ANALYSIS FOR ECONOMICAL DESIGN OF EARTH EMBANKMENT
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
=
HIGHLIGHTS OF THE 178TH COUNCIL MEETING 215ADVANCE METHODS OF SLOPE - STABILITY ANALYSIS FOR ECONOMICAL DESIGN OF EARTH EMBANKMENT
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
=
HIGHLIGHTS OF THE 178TH COUNCIL MEETING 217ADVANCE METHODS OF SLOPE - STABILITY ANALYSIS FOR ECONOMICAL DESIGN OF EARTH EMBANKMENT
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
=
HIGHLIGHTS OF THE 178TH COUNCIL MEETING 219ADVANCE METHODS OF SLOPE - STABILITY ANALYSIS FOR ECONOMICAL DESIGN OF EARTH EMBANKMENT
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 =
HIGHLIGHTS OF THE 178TH COUNCIL MEETING 221ADVANCE METHODS OF SLOPE - STABILITY ANALYSIS FOR ECONOMICAL DESIGN OF EARTH EMBANKMENT
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