Advance Methods of Slope-Stability Analysis for Earth by nuu18388

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									                       th
               The 12 International Conference of
               International Association for Computer Methods and Advances in Geomechanics (IACMAG)
               1-6 October, 2008
               Goa, India




Advance Methods of Slope-Stability Analysis                                                      for      Earth
Embankment with Seismic and Water Forces

B. N. Sinha
General Manager, Intercontinental Consultants & Technocrats Pvt. Ltd., A-8, Green Park, New Delhi, India

ABSTRACT : Slope-Stability analysis for high Embankment is essential when the highway is passing through
seismic zone-V associated with water stagnation and ground soil of low strength. The Fellenius method being
used for slope-stability analysis does not consider inter-slice forces, neither provides a closed force polygon in
a free body diagram nor satisfy equilibrium conditions. Some advanced methods such as Morgenstern-Price,
Spencer and Lowe-Krafiath take into consideration inter-slice normal and shear forces, provide closed polygon
and satisfy equilibrium conditions. These methods give more economical embankment section. Objective of
this paper is to illustrate some of these advance methods which were very cumbersome and time consuming
without computer. The application of computer has made iterating process simple and fast. Several software
program based on these advance methods have been developed and are available. They analyze many slip
surfaces and give the minimum factor of safety for the chosen embankment. They cover all kind of force
combinations and soil parameters encountered at site. Slope-stability analysis has been carried out under
three conditions, namely, normal, rapid draw down and seismic conditions. The results obtained by solving the
mathematical equations using Microsoft Excel have been compared with those computed by software.


1   Introduction
The slope-stability analysis is usually carried out by Fellenius (Fellenius,1936) method also called ordinary or
Swedish Circle method ( Terzaghi,1962.IRC-75, 1979). It ignores inter slice forces and considers only moment
equilibrium and not force equilibrium conditions. Thus, it provides only moment factor of safety and not force
factor of safety. It also does not provide closed force polygon in a free body force diagram for the individual
slices within the failure zone of earth mass. It provides the factor of safety on lower side and thus results into a
more conservative design of earth embankment. The Bishop (Bishop,1960) simplified method takes into
consideration only inter-slice normal force, but ignores the inter -slice shear force. It gives moment factor of
safety only, and the force polygon does not close. The Morgenstern-Price(Morgenstern and Price,1965)
method takes into consideration both the inter-slice normal and shear forces, provides both moment
equilibrium and force equilibrium and gives moment and force factor of safeties. This paper is aimed at
discussing the Morgenstern-Price method in detail. Corps of Engineers #1 method has also been used. It
considers both inter-slice forces but provides force factor of safety only.

 The analysis has been carried out for three conditions, namely normal, draw-down and seismic condition. The
merits and limitations of different methods have been discussed. A case study of embankment slope-stability
has been illustrated. The mathematical equations governing the equilibrium conditions and factor of safeties
are given. The results obtained using computer software (Krahn et al,1971) are compared with those obtained
by solving mathematical equations using Excel broad sheets. Several parameters like half-sine function,
constant function, the ratio of applied function with the specified function (λ), mobilized shear etc., required for
the analysis, have also been explained in the paper.

 One of the objectives of the paper is to update “ Guidelines for the Design of High Embankments [IRC-
75,1979]” in the light of recent advances(Gulati and Dutta,2005. Murthy,2001.) and use of software of stability
analysis for high embankment.


2   Different theories for slope-stability analysis
The failure of a slope could be slippage of earth mass along a slip surface (generally circular). Coulomb
(1776) considered a wedge failure in his theory of earth pressure. Rankine (1857) considered zone of failure
where each element is on the verge of motion. All the methods of slope-stability analysis in practice considers
discretization of failure zone into slices. A slice of earth will be subjected to the forces as shown in Figure 1.
None of the existing methods takes into consideration the strain compatibility of the slices within the zone of
failure. All methods are covered within the ambit of limit equilibrium analysis (LEA). Which further comprises of
moment equilibrium and force equilibrium providing Moment factor of safety and force factor of safety
respectively.




                                                       4461
The earth embankment shown in Figure 1, has been used to illustrate the various methods of analysis. The
soil properties of embankment and foundation soil, loading etc. are shown in the figure.
The forces, equilibrium conditions and the factor of safeties considered in different methods of analysis are
illustrated in Table 1. The earth slopes do not fail in a particular / definite way and the most appropriate
method to be adopted in a particular situation is designer’s (Rodriguez et al,1988. Kaniraj,1988.) choice.

                                           x

                                   R

                                       2
                                  XL       1                    R
                                                            e
                                       W

                             EL            KW                                       AR
                                                ER
                                                     u
                                           XR

                                           Sm

                                   N



                              Figure 1. Showing slip circle and forces on a typical slice
Where W is weight of slice, kW horizontal seismic component, R radius of slip circle, E L and E R left and
right side slice normal forces, X L & X R left side and right side slice shear forces, N base normal
force, β base length, α base angle to horizontal, S m mobilized shear strength, F factor of safety ( FoS ),
 c' effective cohesion, φ ' effective angle of internal friction, γ unit weight of soil, AR water pressure,
 Fm moment equilibrium factor of safety, F f force equilibrium factor of safety, x horizontal distance of
weight of slice from center of rotation O, u pore water pressure above base of slice, e vertical distance of
center of slice and a R vertical distance of water pressure line from the center of rotation O.


3   Mathematical Equations for Factor of Safety
Several equations are derived for factor of safety in different equilibrium conditions. They are discussed below
briefly.

3.1 Moment equilibrium condition
Summation of moments of the forces acting on all the slices (Figure 1) about center of rotation (O), gives the
following relation from equilibrium condition
                                       ∑ W ∗ x − ∑ S m R + ∑ kW ∗ e − AR a R = 0
                    c' β + (N − uβ ) tan φ '
substituting, S m =                          and rearranging we get the moment factor of safety ( Fm ) given by
                             Fm
Equation 1.

                                        R}' φ nat ) β u − N( + β ' c{ ∑
                                                                        =           mF
                                       R a R A − e ∗ Wk ∑ + x ∗ W ∑
                                                                                                          (1)

In Equation 1, inter-slice forces (normal & shear forces) are not figuring. It is because these are assumed to be
equal and opposite on interface of two slices and sum total of all such forces is zero. The left side force of the
first slice and right side force of the last slice are zero.
The “mobilized shear strength (Sm)” at slice base is that part of the shear resistance of the soil mobilized which
is just enough to satisfy the moment equilibrium conditions of the slice. The soil strength at the slice base is
equal to

                                                         c' β + (N − uβ ) tan φ '
The ratio of shear strength of the soil at slice base by mobilized shear is the factor of safety. Thus ‘Sm’ is
obtained by dividing shear strength of soil at the slice base with moment factor of safety (Fm). It will be Ff in
case of force equilibrium.




                                                                    4462
3.2 Force equilibrium condition
Resolving forces acting on slice (shown in Figure 1) in horizontal direction and summing them for all slices,
we get the force equilibrium equation as below

                              ∑ (E R − E L ) + ∑ S m cos α − ∑ N sin α − ∑ kW + AR = 0

                      c' β + (N − uβ ) tan φ '
substituting, S m =                            ,and as the factor ∑(ER-EL) when summed up for all slices shall be
                               Ff
zero, rearranging we get
                                       ∑ {c'∗β cos α + (N − uβ ) tan φ ' cos α }
                                Ff =                                                                                (2)
                                              ∑ kW − AR + ∑ N sin α


It may be seen that moment equilibrium condition Equation 1 or force equilibrium condition Equation 2 contain
a parameter ‘N’ which depends on factor of safety and inter-slice forces and can be obtained by iterating
process as stated in the following paragraphs.


Table 1.Different methods of stability analysis indicating equilibrium conditions, forces and factor of safety
considered
 S.          Methods             Moment         Force         Inter Slice   Inter Slice   Moment      Force    Inter Slice Force
 No.                            Equilibrium   Equilibrium       Normal        Shear       Factor of   Factor       Function
                                                                Forces        Forces       safety       of
                                                                                                      Safety
 1.     Culman wedge                No             Yes             No          No            No        Yes            No
        block method (no –
        slice)
 2.     Fellenius, Swedish         Yes             No              No          No           Yes        No             No
        circle or ordinary
        method (1936)
 3.     Bishop Simplified          Yes             No              Yes         No           Yes        No             No
        method (1955)
 4.     Janbu Simplified            No             Yes             Yes         No            No        Yes            No
        method (1954)
 5.     Spencer method             Yes             Yes             Yes         Yes          Yes        Yes        Constant
        (1967)
 6.     Morgenstern                Yes             Yes             Yes         Yes          Yes        Yes        Constant
        -Price method                                                                                               Half -Sine
        (1965)                                                                                                   Clipped-sine
                                                                                                                     Trapezoid
                                                                                                                  Specified
 7.     Corps of                    No             Yes             Yes         Yes           No        Yes            Yes
        Engineers # 1
        method
 8.     Corps of                    No             Yes             Yes         Yes           No        Yes           Yes
        Engineers # 2
        method
 9.     Lowe-Karafiath              No             Yes             Yes         Yes           No        Yes           Yes
        method
 10.    Sarma method               Yes             Yes             Yes         Yes          Yes        Yes           Yes
        (1973)
 11.    Janbu Generalized           No             Yes             Yes         Yes           No        Yes           Yes
        method (1957)


3.3 Base normal force of the slice
The normal force at the base of slice is derived by resolving forces on the slice (Figure 1) in vertical direction
and we get ,
                                         W − ( X R − X L ) − N cos α − S m sin α = 0

                      c' β + (N − uβ ) tan φ '
substituting S m =                             and rearranging we get ,
                                F




                                                            4463
                                                       c' β sin α uβ tan φ ' sin α
                                    W − (X R − X L ) −              +
                               N=                           F           F                                               (3)
                                                          tan φ ' sin α
                                                  cos α +
                                                                F
This is a non linear equation as factor of safety ‘F’ is appearing in the equation. F shall be Fm for moment
equilibrium and Ff for force equilibrium. Neglecting inter-slice forces and resolving forces on the slice in the
direction of ‘N’ we get
                                               N = W cos α − kW sin α
This normal base force is used to find out Fellenius factor of safety Fm by Equation 4 and this ‘F’ and ‘N’ are
used to start the iterating process.


                                             ∑ {c' β + (W cos α − kW sin α − uβ ) tan φ '}R
                                      Fm =                                                                                  (4)
                                                      ∑ W ∗ x + ∑ kW ∗ e − AR a R
3.4 Inter-slice forces
Inter slice forces are normal and shear forces acting in the vertical faces between slices. Resolving forces
acting on the slice (Figure 1) in horizontal direction we get ,
                                     (E R − E L ) + S m cos α − N sin α − kW = 0
Substituting, Sm and rearranging we get

                                       c' β cos α − uβ tan φ ' cos α     ⎛         tan φ ' cos α ⎞
                           ER = EL −                                 + N ⎜ sin α −               ⎟ + kW              (5)
                                                    F                    ⎝              F        ⎠


 The left side inter-slice normal force EL for the first slice is zero hence right side inter slice normal force ER
can be obtained provided N & F are known. Once the inter slice normal force is known, the inter slice shear
force is computed as a percentage(assumed) of inter slice normal force. This assumption results in various
methods of slope -stability analysis developed by different scientists based on their assumptions.


4   Different methods of slope-stability analysis
    The advance methods of slope-stability analysis are discussed briefly underneath


4.1 Morgenstern - Price Method
It considers inter-slice normal and shear forces. They proposed an empirical equation for inter- slice force
relation as given below
                                                    X = E ∗ λ ∗ f (x )                                               (6)
Where X is inter slice shear force, E inter slice normal force, λ the percentage of function used and, f(x) inter-
slice force function representing the value of function at the location of particular slice.


The specified function giving variation of f(x) with slice position may be assumed as constant or given by
forms e.g. half-sine, clipped-sine, trapezoidal, or any other choice. One typical half-sine function has been
shown in Figure 2, discussed below.


The value of lambda for a typical slice (say No.5) shall be ratio of ordinate f (x) against slice No 5 read from
applied function (lower curve in Figure 2) and divided by the value of f(x) read from specified function (upper
curve in Figure 2)for the same slice. For slice No.5 λ =0.239. λ is the same for all the slices. However, since
the f (x) varies from slice to slice in half -sine function the ratio of slice shear force to slice normal force( X / E )
shall also vary from slice to slice. However, in a constant function value of f(x) is same( f(x)=1) for all the slices
hence this ratio shall also be constant. λ is constant in any case. When half-sine function is used for 20 slices,
then specified function f (x) at the right face of the first slice shall be sin π/20. For the second slice this shall be
sin 2π/20 and for 17th slice shall be sin17π/ 20 and so on. Thus half-sine function value for any chosen number
of slices can be obtained.




                                                         4464
                  Specified and Applied Functions-
                                                     1.2
                                                                                           Specified function
                                                     1.0
                                                     0.8




                                 f(x)
                                                     0.6
                                                     0.4                          Applied function

                                                     0.2
                                                     0.0
                                                           0   2   4   6   8      10       12        14     16   18   20
                                                                               Slice no


                               Figure 2. Specified function and applied function for chosen half-sine function
4.2 Spencer method
Spencer (Spencer,1967) method, where constant function f (x) is adopted is similar to what has been
explained for Morgenstern-Price constant function except that the lambda is chosen such that Fm is equal to Ff
. All other things remain the same. When constant function is assumed, It may be seen that both these
methods for the earth slope analysis give same factor of safety.


4.3 Corps of Engineers #1 and Corps of Engineers #2 methods
In this method, the resultant inter-slice force is assumed to act parallel to the line (dotted) joining entry point
with exit point as shown in Figure 3(a). As the inclination of resultant is constant the ratio of inter-slice shear
force to inter-slice normal force remains same for all the slices. In other words, this will compare with constant
function of Morgenstern-Price. However this method considers only force equilibrium and provides only force
factor of safety. In case of Corps of Engineers # 2 method, the resultant inter-slice force is assumed to be
parallel to the embankment slope as shown in Figure 3(b). it gives only inter-slice normal force, and shear
force will be zero, where the embankment surface at slice top is horizontal. Rest of the things remain the same
as in Corps of Engineers #1 method.




                   (a)                                                                                    (b)
Figure 3. Direction of interslice resultant force in (a) Crops of Engineers # 1 (b) Crops of Engineers # 2
methods


The other methods namely, Lowe-Krafiath, Janbu’s Generalised, Sarma etc. differ in their assumption of
relation of inter-slice normal force with shear force and equilibrium condition. Lowe-Krafiath and Janbu’s
Generalised methods consider only force equilibrium and provide only force factor of safety. Sarma method
considers both moment and force equilibrium and provides both factor of safeties. However, all these three
methods take into consideration both inter-slice normal and shear forces (Table 1)


5 Computation of factor of safety by different methods
Slope-stability analysis has been carried out by various methods for the given cross-section of the
embankment with the geo-technical properties of the fill & ground soil as indicated in Figure 1. A comparative
study is made in Table 3 to find the most appropriate method to be adopted in a given situation.




                                                                           4465
5.1 Morgenstern-Price method
Under seismic condition , factor of safety Fm is obtained by the general Equation 4, as all parameters on the
right hand side are known. Substituting Fm for F & ( W cos α - kW sin α) for ‘N’ in Equation 5, ER 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 way EL and ER for all slices can be obtained. Choosing a function
(in the present case half-sine function) and a value for λ the inter-slice shear forces XL and XR are obtained by
Equation 6. Knowing EL, ER, XL and XR, the value of ‘N’ are obtained by Equation 3. Taking this value of ‘N’,
factor of safety Fm is obtained by Equation 1. The new ‘N’ and ‘Fm’ provides new values of EL, ER, XL and XR
by the respective equations. This process of iteration is repeated till convergence occurs. The converged
value of ‘F’ is the required FoS. In case of force factor of safety, Equation 2 is used and the rest of the process
is the same as explained above. Microsoft Excel program is adopted for computation. It is seen that 2 to 3
iteration provide the converged solution.

In case of rapid draw down, the water level comes down and follows the profile of the embankment on the
right side slope and ground. Since it is not considered to take place simultaneously with earthquake ‘k’ is taken
as zero. There is no free standing water as the water line follows the ground profile. Naturally, there would be
no water force (AR = 0 ) and N = W cos α,as ‘k’ is zero Equation 4 gets modified as below

                                                 ∑ {c' β + (W cos α − uβ ) tan φ '}R
                                            F=                                                                   (7)
                                                              ∑W ∗ x

The FoS for rapid draw down (without seismic force) is obtained on the basis of Equation 7. Taking this F & N
values, the process of obtaining ER, EL, XR, XL, N & F using respective equations( duly modified with k=0)
followed as explained above. The factor of safety is obtained when convergence occurs.

For the normal condition, without seismic force, ‘k’ is taken to be zero, and all the Equations 1 to 5 shall be
modified. The process explained above shall be repeated using respective modified equations till convergence
occurs and FoS obtained for normal condition.

To obtain FoS, ‘λ’ value is to be decided in all the cases. The best result is obtained when λ is chosen such that
Fm is equal to Ff . This is done by plotting λ against Fm and Ff by changing values of λ and computing FoS. This is
done for different conditions (normal, rapid draw down and seismic) separately (by changing λ and obtaining
converged FoS) . The intersection point (Figure 4) gives required value of λ for the particular condition. Figure 4
shows a typical curve for Morgenstern- Price method for half -sine function and normal condition. In the SLOPE/W
software also, the same value of λ is used for analysis.



                                  1.6
                                 1.58                                Fm
                                 1.56
                        F.O.S.




                                 1.54
                                 1.52
                                  1.5
                                 1.48                                  Ff
                                 1.46
                                     0.09    0.11           0.13            0.15       0.17   0.19
                                                                   Lambda


       Figure 4. Shows lambda for producing equal moment factor of safety and force factor of safety


5.2    Corps of Engineers #1 method
Like in any other method, a slip surface is chosen. The entry point and the exit point of slip circle are joined by
a straight line as shown in Figure 3(a). The resultant inter-slice force is assumed to act parallel to this line for
all the slices. The inter-slice shear force and inter-slice normal force for any slice shall be, XR = ER * tan θ
(where θ is the angle of inclination to horizontal). It will resemble to a constant function where f(x)* λ = tan θ.
This method gives only force factor of safety. Rest of the computation is same as in Morgenstern-Price
method. The Fellenius FoS and base normal is adopted to start the process of iteration as already explained
in detail in above paragraphs. After few iterations the converged FoS is obtained. The critical circle is the slip
circle corresponding to the minimum value of factor of safety.




                                                           4466
5.3        Comparison of FoS and critical circles by different methods
The computations are carried out by the above two methods under three conditions for equilibrium applicable
for the particular method. One typical Excel broad sheet for Morgenstern-Price method for half-sine function
for moment equilibrium and normal condition has been given as Table 2. Values of FoS obtained by Fellenius
method for the three conditions, are shown in Table 3. It is not possible to include all such computation
sheets for all the methods and conditions within the specified length of the paper. In a recent paper published
in Journal of the Indian Roads Congress ( Sinha, 2007) a more detailed presentation has been made.
However, these are similar. The factor of safety was also obtained using soft ware SLOPE/W (Krahn,2004)
and the results are compared in Table 3.The critical circles for normal, draw down and seismic cases for
Morgenstern-Price and Corps of Engineers # 1 methods are shown in Figure 5. The critical circles by
Morgenstern-Price method for normal and rapid draw down conditions are same but is different for seismic
condition. The critical circles by Corps of Engineers # 1 method and Morgenstern-Price method under rapid
draw down condition almost coincide.


Table 3. Comparative Statement of various FoS by various methods and under different conditions.

              Moment factor of safety (Fm)                 Force factor of safety (Ff)
              Morgenstern-Price method       Fellenius     Morgenstern-Price method       Corps of Engineers #1
              Half-sine function             method        Half-sine function             method
 Condition    Computed         Soft-ware     Computed      Computed, Soft-ware            Computed Soft-ware            Remarks
              using Excel      SLOPE/W       using         using         SLOPE/W          using        SLOPE/W
                               out put       Excel         Excel         out put          Excel        out put
 Normal       1.553            1.5569        1.302         1.553         1.558            1.735        1.741            Computed
                                                                                                                        FoS tallies with
 Rapid        1.333           1.3362         1.139         1.340         1.334            1.428           1.427         SLOPE/W
 Draw                                                                                                                   out put.
 Down                                                                                                                   Fellenius gives
 Seismic      1.127           1.1236         o.980         1.126         1.133            1.136           1.136         less FoS for
                                                                                                                        similar
                                                                                                                        situation



                                                                                 LE G E N D
                                                                       M ethod of analysis                    Sym bol
                                                             M orgenstern-Price ( half- sine function )
                                                             norm al and rapid draw dow n
                                                             M orgenstern-Price ( half -sine function )
                                                             seism ic condition
                                                            C orps of Engineers # 1, norm al
                                                            Corps of Engineers # 1, rapid draw dow n
                                                            Corps of Engineers # 1,seism ic



                               Figure 5. Critical slip circles by different methods


6     Summary and conclusions
Different theories of slope-stability analysis of embankment are discussed. Mathematical equations for
moment and force factor of safeties have been given under different conditions. FoS are computed using
Microsoft excel program and also by use of software SLOPE/W. Table 2 illustrates typical computation of
moment factor of safety (Fm ) by Morgenstern-Price method using half-sine function and constant λ value of
0.1555 corresponding to Fm= Ff (Figure 4). Value of factor of safeties computed by different methods under
different conditions are given in Table 3. Based on the results , following broad conclusions can be drawn.
      The old methods (Fellenius method and Swedish Circle method ) give lower factor of safety (FoS) and
      therefore requires a flatter slope for the specified FoS compared to the earth slope obtained on the basis
      of Morgenstern-Price method which takes inter-slice forces (normal and shear) into account and satisfies
      closed force polygon indicating equilibrium condition of the slice in a free body force diagram.
      The Morgenstern-Price method with half-sine function, which takes into account inter-lice forces and
      satisfies both moment equilibrium and force equilibrium conditions, is most appropriate for the design of
      slope-stability for a value of λ corresponding to Fm= Ff.




                                                         4467
                                             Table 2. Moment FoS Calculating by Morgenstern- Price Method for Half-sine function and Normal condition
                                                                                                                                                          N=(W1+W2) -
                                                                                 f(x)=                                                 ER=EL-                                        Σ{c'*β+
Slice Surface                                     c' (kn                                                                         (c'β*cosα− β φ'∗ α)/F       (XR-XL) -    (AR*aR)/
                                                                                           XR=λ*      XL=λ*
               W1 (kn) N (kn) α (deg) β (m)                φ     λ       u      (sin x π                        F        EL                                                           (N-uβ)    Σw*sinα
                                                                                                                                            *
                                                                                                                                            u   a
                                                                                                                                                n
                                                                                                                                                t   o
                                                                                                                                                    cs




no.(x) load W2
                                                           '




                                                                                           f(x)*ER   f(x)*EL                          +N*( sinα-        (c'*β*sinα)/F+uβ     R
                                                  /m^2)
                                                                                  /20)                                              tanφ'*cosα/F)      (tanφ*sinα)/F)/(co
                                                                                                                                                           '




                                                                                                                                                                                      tanφ'}
                                                                                                                                                        sα+tanφ'sinα/F)
 1     71.696   200.69   296.49   64.45    6.67    10      30   0.16     0       0.16       4.904       0      1.557     0             201.62              298.808         ……..      237.873    245.757
 2     40.366   231.42   286.48   53.82    2.74    10      30   0.16   11.437    0.31      17.616      4.904   1.557   201.62          366.59              285.778         ……..      174.694    219.363
 3     40.366   295.37   345.20   47.99    2.41    10      30   0.16   30.859    0.45      38.512    17.616    1.557   366.59          545.52              342.764         ……..      180.436    249.461
 4     57.743   512.5    613.13   41.756   3.10    40      5    0.16   49.959    0.59      79.999    38.512    1.557   545.52          875.25              614.671         ……..      164.025    379.759
 5     57.743   596.85   667.17   35.059   2.82    40      5    0.16   67.868    0.71      129.444   79.999    1.557   875.25         1177.24              669.537        ……..       154.531    376.011
 6     57.743   663.3    709.92   28.881   2.64    40      5    0.16   81.975    0.81      180.711   129.444   1.557   1177.24        1436.47              712.144         ……..      148.743    348.258
 7     57.743   715.18   747.92   23.056   2.51    40      5    0.16   92.99     0.89      227.697   180.711   1.557   1436.47        1643.41              749.160        ……..       145.462    302.700
 8       0      631.38   607.16   17.838   2.10    40      5    0.16   100.86    0.95      259.821   227.697   1.557   1643.41        1756.86              605.036         ……..      118.639    193.409
 9       0      613.53   594.41   13.137   2.05    40      5    0.16   106.29    0.99      279.523   259.821   1.557   1756.86        1819.98              592.561         ……..      115.073    139.443
 10      0      588.82   580.08   8.5258   2.02    40      5    0.16   110.1     1.00      285.300   279.523   1.557   1819.98        1834.72              578.768         ……..      112.181     87.295
 11      0      557.57   563.00   3.9696   2.01    40      5    0.16   112.19    0.99      276.969   285.300   1.557   1834.72        1803.35              562.371         ……..      109.785     38.599
 12      0      519.95   541.88 -0.56145 2.00      40      5    0.16   112.77    0.95      255.682   276.969   1.557   1803.35        1728.87              541.941         ……..      107.694     -5.095
 13      0      476.00   515.29   -5.096   2.01    40      5    0.16   111.8     0.89      223.821   255.682   1.557   1728.87        1615.43              515.939         ……..      105.776    -42.281
 14      0      435.46   492.05 -9.6631    2.03    40      5    0.16   109.26    0.81      184.485   223.821   1.557   1615.43        1466.47              493.103         ……..      104.823    -73.094
 15      0      398.00   471.42 -14.293    2.06    40      5    0.16   105.1     0.71      141.273   184.485   1.557   1466.47        1284.83              472.476         ……..      104.841    -98.258
 16      0      353.81   441.80 -19.022    2.12    40      5    0.16   99.242    0.59      98.449    141.273   1.557   1284.83        1077.12              442.796         ……..      104.920    -115.318
 17      0      302.02   401.53 -23.891    2.19    40      5    0.16   91.541    0.45      60.201    98.449    1.557   1077.12         852.76              402.079         ……..      105.129    -122.317
 18      0      280.24   403.36 -29.281    2.58    40      5    0.16   81.13     0.31      28.264    60.201    1.557   852.76          588.19              401.095         ……..      120.111    -137.063
 19      0      216.5    348.65 -35.314    2.76    40      5    0.16   67.225    0.16       7.818    28.264    1.557   588.19          321.39              346.970         ……..      124.527    -125.149
 20      0      136.12   269.38 -41.846    3.02    40      5    0.16   49.69     0.00       0.000      7.818   1.557   321.39           78.90              268.652        51.205     131.200    -90.810
                                                                                                                                                               SUM        51.205     2670.463   1770.670
                                                                                                                                          F O S =Σ{c'*β+(N-uβ)tanφ'}/{Σw*sinα−(AR*a)/R} =1.553
                                                                                                                                                                                           R




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      Different methods give different critical circle and different factor of safeties for the same situation,
      namely, earth fill material and ground soil properties. Even for same slip surface different methods give
      different factor of safety.
      Microsoft Excel can also be used for finding FoS quickly by solving the equations (1) to(7). Convergence
      is achieved very fast. Use of computer is preferable to conventional graphical method of analysis.
      SLOPE/W by GEO-SLOPE International provide a very efficient software package for analysis of slope-
      stability covering almost all situations. The results (out put of package) tallies quite closely with the
      results obtained independently by solving mathematical equations using Microsoft Excel for all the
      conditions.
      The Guidelines for the design of High Embankments, IRC-75 (1979), needs updating to include other
      advance methods of slope-stability analysis by use of soft ware. The conventional methods prescribed
      by the IRC code are cumbersome due to manual calculation.


7 Acknowledgements
The author likes to thank Shri K.K.Kapila, CMD, ICT Pvt. Ltd., New Delhi for giving all help in contributing this
paper to IACMAG. The author is indebted to the ICT for utilizing the facilities of the organization in bringing this
paper to the present shape. The author is thankful to Dr.S.K. Majumder, Advisor, ICT for his kind help in going
through this paper and giving valuable advice for improvements. The author is thankful to Mr. Sachin Roorkiwal
and Ms. Jyoti Priya of the ICT for their help in carrying out all Microsoft Excel calculations and operating the
software program for the analysis.


8 References
Bishop, A. W.and Morgenstern,N.,1960.Stability coefficients for earth slopes. Geotechnique, Vol. 10,No.4, pp. 164-169.
Fellenius, W.,1936. Calculation of the Stability of Earth Dams. Proceedings of the Second Congress of Large Dams,Vol. 4,
     pp. 445-463.
Gulati, Shashi K. & Datta Manoj. 2005. Geo-technical Engineering. Tata McGraw Hill, New Delhi.
IRC-75 (1979). The Guidelines for the Design of High Embankments.
Kaniraj, Shenbaga. R, 1988. Design Aids in Soil Mechanics and Foundation Engineering. Tata McGraw Hill,
     New Delhi.
Krahn, J., Price, V.E., and Morgenstern, N.R. 1971.Slope Stability Computer Program for Morgenstern-Price Method of
    Analysis. User’s Manual No.14, University of Alberta, Edm,onton, Canada.
Krahn, John. 2004. Stability Modeling with SLOPE/W. GEO-SLOPE/W International Ltd. Calgary, Alberta, Canada.
Morgenster, N.R. and Price, V.E., 1965. The Analysis of the Stability of General Slip Surfaces. Geotechnique, Vol. 15,
    pp. 79-93.
Murthy, V.N.S. 2001 “Principles of Soil Mechanics and Foundation Engineering”. UBS Publishers, New Delhi, Fifth edition.
Rodriguez, Alfonso Rico., Castillo, Hermillo del., and Sowers, George F.. 1988 Soil Mechanics in Highway Engineering.
    Trans Tech Publications, Germany.
Sinha, B.N. 2007. Advance Methods of Slope-Stability Analysis for Economical Design of Earth Embankment. IRC Journal
    Volume 68-3, pp. 201-222.
Spencer, E. 1967. A Method of Analysis of Embankments assuming Parallel Inter-slice Forces. Geotechnique, Vol. 17(1),
    pp. 11-26.
Terzaghi, Karl. 1962. The Theoretical Soil Mechanics John     Wiley and Sons, New York, Tenth edition.




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