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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 4468 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. 4469