1981_The Relationship between Limit Equilibrium Slope Stability Methods_Fredlund

1981_The Relationship between Limit Equilibrium Slope Stability Methods_Fredlund

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Unformatted text preview: 11/17 The Relationship between Limit Equilibrium Slope Stabilitv Methods La Relation des Méthodes de Limite d’EquiIibre de Pente Stable D.G. FREDLUND .J. KRAHN D.E. PUFAHL SYNOPSIS. Dept. of Civil Engineering, University of Saskatchewan, Saskatoon, Saskatchewan, Canada Some of the methods commonly used for analyzing slopes utilizing the principles of limit equilibrium are the Ordinary or Fellenius method, the simplified Bishop method, the Corps of Engineers method, the Janbu simplified and the Janbu generalized methods, the Spencer method, and the Morgenstern-Price method. these methods have been obscure, largely because of: The similarities and differences in 1) the lack of uniformity in formulating the equations of equil~ ibrium, 2) the ambiguity concerning inter-slice forces and 3) the unknown limitations imposed by non—circular failure surfaces. Theoretical studies have shown that a common formulation of the equilibrium equations can be used for all of the methods. The factor of safety equations have been derived with respect to moment equilibrium and with respect to force equilib— rium, and all methods use either or both of these equations. In addition, the use of an inter-slice force, functional relationship (i.e., lambda, A) has been used to relate and compare the factors of safety computed for the various meth- ods. This approach has provided a clear and concise assessment of the variation in factors of safety that are computed by the different methods for typical cross—sections, pore—water pressures and soil profiles. INTRODUCTION During the past three decades, numerous methods have been proposed for performing the two—dimensional limit equili' brium method of slices (Wright, 1969). The methods most commonly used are: i) The Ordinary method. Other names given to this method are the Fellenius, Swedish Circle and Conventional method. ii) The simplified Bishop method. iii) The Spencer method. iv) The Janbu simplified and the Janbu generalized methods. v) Force equilibrium methods such as the Lowe and Karafiath method, the Corps of Engineers method, and the Taylor modified Swedish method. vi) The Morgenstern-Price method. The similarities and differences in these methods have been obscure, largely because of the lack of uniformity in formulating the factor of safety equations, the ambi- guity concerning interslice forces and the unknown limi- tations imposed by non-circular failure surfaces. Some attempts have been made to assess the quantitative dif- ferences in factors of safety obtained from the various methods (Bishop, 1955; Wright, 1969; Duncan and Wright, 1980). In general, the quantitative differences in fac- tors of safety obtained by the various methods, are not substantial with the exception of the Ordinary method which can differ by more than 60 percent from the other methods (Whitman and Bailey, 1967). Some attempts have also been made to show the relationship between the var- ious methods from a theoretical standpoint (Wright, 1969; Fredlund and Krahn, 1977; Naderi, 1977; Popescu, 1978; Janbu, 1980). The object of this paper is to present a general limit equilibrium method of slices formulation in two- dimensions and to show how each of the methods listed above is a special case of the general formulation. (Hereafter, the general limit equilibrium method of slices is simply referred to as the GLE method). The Ordinary method becomes an exception which cannot be re- lated to the general formulation since it does not sat- isfy Newtonian force principles at the interslices (Fredlund and Krahn, 1977). The paper also shows how the various methods of slices can be extended to non— circularslip surfaces. Example problems are used to demonstrate the relationship between the factors of safety. GENERAL LIMIT EQUILIBRIUM METHOD OF SLICES (GLE METHOD) The elements of statics that can be used to derive the factor of safety are the summation of forces in two di— rections and the summation of moments about a chosen point of rotation. These elements of statics, along with the failure criteria, are insufficient to make the slope stability problem determinate. Either additional elements of physics or an assumption regarding the di- rection or magnitude of some of the forces is re- quired to render the problem determinate. All methods considered in this paper use the latter procedure and make an assumption concerning the interslice forces. Theoretical studies have shown that factor of safety equations can be independently derived to satisfy moment equilibrium and force equilibrium of the slices contained above an assumed slip surface (Fredlund and Krahn, 1977). In addition, an assumed functional relationship is used to specify the direction of the interslice forces. Later in this paper, the various limit equilibrium methods of slices are viewed as special cases of the GLE formula— tron. Circular Slip Surface Figure 1 shows the forces involved in the derivation of moment and force equilibrium factor of safety equations for a circular slip surface. 11/17 CENTER OF ROTATION s-——n 7mm caucus wrm wares 1"” Fig. I Forces Acting For The Method Of Slices (Circular Slip Surface) The definition of each variable is as follows: W = the total vertical force due to the mass of a slice of width 'b' and height 'h'. P I the total normal force on the base of a slice. Sm - the shear force mobilized on the base of each slice. E - the horizontal interslice normal forces. X = the vertical interslice shear forces. R = the radius or the moment arm associated with the mobilized shear force, Sm. x = the horizontal distance from the centroid of each slice to the center of rotation. a = the perpendicular distance from the resultant external water force to the center of rotation. = the width of a slice. the resultant external water forces. a = the angle between the tangent to the center of the base of each slice and the horizontal. {>7 (I The 'L' and 'R' subscripts on the 'E‘, 'X' 'a' and 'A' variables designate the left and right sides, respective- 1y. The magnitude of the shear force mobilized at the base of a slice can be written in terms of the Mohr-Coulomb fail- ure criterion. Sm - § {c' + (an — u)tan ¢'1 [11 = effective cohesion intercept ¢' - effective angle of internal friction. o - PIE 2 - length of the failure surface at the base of each slice. F ‘ factor of safety. A moment equilibrium equation for the GLE method is des- cribed for all slices by summing moments about the center of rotation. zwx - 2st : Aa a 0 [2] The interslice shear and normal forces (i.e., X and E) do not appear directly in equation [2] since their summation over the overall slope must cancel. The mobilized shear force, 5 , is written in terms of the shear strength criterion [1]? and equation [2] can be solved for the factor of safety with respect to moment equilibrium, Fm. 410 - Z[c‘l + (P - u2)tan ¢']R Fm {fix 1 Aa [3] The force equilibrium equation for the GLE method is writ- ten by summing forces in the horizontal direction for the overall slope. 2? Sin a - 23m cos a 1 A - 0 [4] Once again the interslice forces must cancel. The mobilized shear force is again written in terms of the failure criterion [1], and equation [4) can be solved for the force equilibrium factor of safety, Ff. _ [[c'l + (P — u2)tan '] cos o F: D sin ala [5] The normal force, F, for equations [3] and [5] can be evaluated by summing forces vertically on each slice. _ _ _ c'l Sin a ul tan @' Sin a P_” (KR XL) F 4' F [6] no where: Sin o tan ¢' m - cos a + o F The factor of safety, F, in [6] can be either with res- pect to moment or force equilibrium depending upon the factor of safety equation being solved. Any one of a number of possible assumptions can be made in order to compute the interslice shear forces. The various methods of slices that are commonly used can be categorized in terms of the assumption that is made regarding the inter- slice shear forces, and whether the computed factors of safety satisfy moment or force equilibrium, or both. Non-Circular Slip Surface Figure 2 shows the forces that are used in the derivation of the moment and force equilibrium factor of safety equations in the GLE method for a non-circular slip sur- face. The slip surface that is shown starts and ends with a circular portion, and has a central linear portion The non-circular portion is assumed to be the result of a geological discontinuity which does not allow the slip surface to penetrate deeper. This type of non- circular surface is termed a composite slip surface (Fredlund, 1975), and it attempts to model typical ob- served modes of failure (Krahn et a1, 1979). From a theoretical standpoint there are two changes in the moment equilibrium equation. The moment arm associated with the mobilized shear force becomes a variable distance, R, and the normal force, P, has an offset arm, f. #1 norm c NTER ‘1 or WATISN 0L TENSJON CRACKS WITH WATER Fig. 2 Forces Acting For The Method Of Slices (Composite Slip Surface) The factor of safety equation for moment equilibrium be— comes, F I Z[c'l + (P - u2) tan Q']R [7] m EWx - ZPf I As The factor of safety equation for force equilibrium [5] and the equation for the normal force [6] remain unchan- ged. The center for moment equilibrium is the center of rotation for the circular portion of the slip surface. COMPARISON OF THE GLE FORMULATION AND COMMONLY USED METHODS OF SLICES Morgenstern—Price Method Horgenstern and Price (1965) solved for the factor of safety using the summation of forces tangential and norm— al to the base of a slice and the summation of moments about the center of the base of each slice. The equat— ions were written for a slice of infinitesimal thickness. The force and moment equilibrium equations were combined and a modified Newton-Raphson numerical technique was used to solve for the factor of safety satisfying force and moment equilibrium. The solution required an arbi— trary assumption regarding the direction of the result- ant of the interslice shear and normal forces. Figure 3 shows typical functional forms which can be written as x/E = Af(x) where: f(x) - a function that describes the manner in which X/E varies across the slope, and A x a constant representing the percentage (i.e., portion of the function used when solving for the factor of safety. f(x)- CONSTANT fix) - HALF— SI NE I l 12 .. v K " 3': 1 o O L x R L x ‘ f (x)-CL|PPED—SINE I fix) - TRAPEZOID 22 O L x R I f1x)- SPECIFIED L- LEFT R- RIGHT L x R Fig. 3 Typical Functional Variations For The Direction Of The lnterslice Force With Respect To The X Direction The moment and force factor of safety equations (i.e., equations [3] and [5] for the OLE method can be solved independently for a given function, f(x). by assuming various trial values of "X" (Fredlund, 1974). A best-fit regression line through the factors of safety associated with the "K" values indicates the factor of safety satis- fying both moment and force equilibrium. The factor of safety satisfying both force and moment equilibrium can 11/17 also be obtained by using a Newton-Raphson numerical sol- ver . The assumption regarding the interslice forces and the elements of statics used in the Morgenstern-Price formu— lation are the same as those used in the CLE formulation. However, there is a light difference in the way the norm— al force is applied to the base of the slice (Figure A). The Morgenstern-Price method uses integration across the slope, and this results in a linear variation of the nornr 31 force across the base of the slice. As a result, the resultant normal force, F, can have a slight offset from the center of the slice. The GLE formulation assumes that the resultant normal force acts through the center of the slice. Fredlund and Krahn (1977) used a 12,2 m high, 2:1 slope to study the difference in factor of safety and "A" be— tween the Morgenstern-Price formulation and the GLE form- ulation. The first example considered a circular slip surface passing through a homogeneous soil with an ef- fective angle of internal friction of 20 degrees and an effective cohesion intercept of 28.7 kPa. The example problem was then modified to form a composite slip sur- face by introducing a hard layer at a depth of 1.52 m be- low the toe of the slope. A thin, soft layer with an ef- fective angle of internal friction of 10 degrees and zero effective cohesion was located immediately above the hard layer. Two pore pressure coefficients were used for each case. The computed factors of safety and "A" values are presented in Table 1. The Morgenstern-Price method was solved using the University of Alberta computer program (Krahn et a1, 1971) and the OLE formulation was solved using the SLOPE computer program (Fredlund, 1974). MORGEN STERN - PRICE FORMULATION C / ¢ GENERAL LIMIT / EQUILIBRIUM FORMULATION Fig. h Point of Application of Normal Force For Morgenstern-Price Formulation and General Formulation 411 11/17 Table I. Comparison of Factors of Safety and "A" Using the Morgenstern-Price Method and the GLE Method __..._________..._____..___________.__.__________________ Shape of Side Force Morgenstern—Price GLE Failure r Function Method Method Surface u * * F X F X Circular 0.0 Constant 2.085 0.257 2.076 0.254 Circular 0.0 Half Sine 2.085 0.314 2.076 0.318 Circular 0.25 Constant 1.772 0.351 1.765 0.244 Circular 0.25 Half Sine 1.770 0.434 1.764 0.304 Composite 0.0 Constant 1.394 0.182 1.378 0.159 Composite 0.0 Half Sine 1.386 0.218 1.370 0.187 Composite 0.25 Constant 1.137 0.334 1.124 0.116 Composite 0.25 Half Sine 1.117 0.4A1 1.118 0.130 * The tolerance is 0.001 The average difference in the factors of safety obtained from the Morgenstern-Price method and the GLE method was negligible (i.e., less than 0.01). The average difference in the "A" values was in the order of 0.1 with the Morgenstern—Price formulation being higher. The differ- ence in "A" value is attributed to the procedure used to handle the normal force at the base of the slice. The results indicate that there is a small difference in the computations due to the manner in which this force is handled. The Spencer Method The Spencer method assumes a constant relationship be- tween the magnitude of the interslice shear and normal forces (Spencer, 1967). x/E = tan 6 [9] where: 6 - angle of the resultant interslice force from the horizontal. Equation [9] is the same as equation [8} if the inter- slice force function, f(x), is equal to 1; then "X" is equal to tan 8. Spencer (1967) summed forces perpendicu- lar to the interslice forces to derive the normal force equation. However, the same equation can be derived by sunning forces in a vertical and horizontal direction (i.e., equation [6]. Spencer (1967) derived two factor of safety equations; one satisfying force equilibrium. These equations are essentially the same as those proposed in the GLE form— ulation when the interslice force function, f(x), is as- sumed to be a constant. Two example problems were selected to demonstrate the re— lationship between the Spencer method and the GLE formu- lation (Figure 5). Example No. 1 considers a circular slip surface while example No. 2 is forced into a composite mode by a bed- rock layer. The examples ate similar to those described previously in this paper (Fredlund and Krahn, 1977) with the exception that a 3.05 m tension crack zone with no water is assumed and the pore pressure coefficient is 0.2. The results for example No. 1 (Figure 6) show that the Spencer method and the OLE formulation give the same results. Likewise, the results for example No. 2 (Fig— ure 7) show agreement between the Spencer method and the GLE formulation. The interslice force directions on Figures 6 and 7 are presented in terms of tan 9 (i.e., A). 412 °(56£,2741 18 E 5 12 E P: 1.92 Mq/m3 I a ¢= 20° 5,1 6 c‘=2e.75 kPa (u: 0.2 6 12 1e 24 30 36 42 DISTANCE (m) o (36.6.2141 P=1.92 Mq/m3 A '8 ¢‘=10° E _____ 6:0 5 12 P=1.92 Mg/m3 E 49: 20" a c'=28.75ch1 d 6 ru=o_2 BEDROCK [2 18 24 30 36 42 WSTANCE [NU Fig. 5 Example Problems with Circular And Composite Slip Surfaces 220 215 210 205 ',_ 200 E < 195 U) 3 1.90 K .9 '35 SIMPLIFIED g BISHOP u 1.80 175 1.70 +SPENCER METHOD 0 GENERAL LIMIT EQUILIBRIUM 155 1111) - CONSTANT ISO JANBU'S SIMPLIFIED (WITHOUT CORRECTION FACTOR) O Q! 02 Q3 Q4 Q5 03 I Fig. 6 Comparison of Factors 0f Safety For Example No. 1 (Circular Slip Surface) l25 [20 [15 110 SIMFHJFIED '// mSHOP FACTOR OF'SAFETY JANBU'S SIMPLIFIED (WITHOUT CORRECTION FACTOR) +SPENCER 0 GENERAL LIMIT EQUILIBRIUM f(x)-CONSTANT I00 0 0,1 (12 0.3 0.4 0.5 0.6 I Fig. 7 Comparison Of Factors Of Safety For Example No. 2 (Composite Slip Surface) Simplified Bishop Method The simplified Bishop method neglects the interslice shear forces (Bishop, 1955). The normal force equation is the same as equation [6] with the interslice shear forces set to zero. The factor of safety equation is derived by taking moments about the center of rotation. In other words, the simplified Bishop method corresponds to the moment equilibrium factor of safety equation [3] when "X" is equal to zero or the Spencer moment equilib- rium equation when 9 is equal to zero (See Figure 6 and 7). The simplified Bishop method bears the same rela- tionship to the GLE formulation (or the Spencer equat- ions) regardless of whether the slip surface is circular or composite. In general, the difference between the simplified Bishop factor of safety and the factor of safety satisfying both force and moment equilibrium, de— creases as a particular slip surface has an increasing planar portion. Janbu Simplified Method In the derivation of the Janbu simplified method the interslice shear forces are assumed to be zero (Janbu et a1, 1956). The normal force equation is the same as equation [6] with the interslice shear forces set to zero The factor of safety is computed from the horizontal force equilibrium equation (i.e., equation [5]). Then an empirical correction factor is multiplied by the com- puted factor of safety in an attempt to account for the effect of the interslice shear forces. ‘The empirical correction factor is related to the shear strength prop— erties and the shape of the slip surface. Moment equili— brium is not satisfied. The uncorrected factors of safe- ty correspond to the force equilibrium factor of safety, [5], when "A" equals zero. For examples No. 1 and No. 2, the uncorrected factors of safety for the Janbu simpli— fied method are 1.609 and 1.005, respectively, (See Fig— ures 6 and 7). The empirical correction factor generally 11/17 increases the factor of safety by up to approximately 10 percent. Janbu Generalized Method The Janbu generalized method includes the effect of inter- slice forces by making an assumption regarding the point at which the interslice forces act (i.e., the line of thrust; Janbu, 1954; Janbu et al, 1956). The normal force equation is derived from the summation of vertical forces equation [6]. The factor of safety equation is derived from the horizon— tal force equilibrium equation (equation [5]). In order to solve for the factor of safety, the interslice shear forces are computed from the summation of the moments about the center of the base of each slice. XR = ER tan at — (ER - EL) tR/b [10] where: at = angle between the line of thrust on the right side of a slice and the horizontal. tR = vertical distance from the base of the slice to the line of thrust on the right side of the slice. The horizontal interslice forces required for equation [10], are obtained by summing forces in the horizontal direction on each slice. EL — ER Sm cos a ~ P sxn a [11] Once the Janbu generalized factor of safety equation, [5), has been solved, it is possible to plot the computed in- terslice shear and normal forces and determine a corres- ponding side force function. It was not possible to ob- tain a side force function for examples No. 1 and No. 2 since convergence difficulty was encountered because of the geometry that was arbitrarily chosen in these two cases. Example No. 3 (Figure 8) is used to demonstrate the rela— tionship between the Janbu generalized method and the GLE formulation. The factor of safety by the Janbu general- ized method is 1.19. Px 208 Mq/m3 dxs7s7hPo 49-245- so 995 P: 205 Mm! c'- 9.59 “’0 so 4’" “' ELEVATION (m) 30 SLIP SURFACE .J_ _L_ _L J 1.— .I O 30 60 90 I20 I50 ISO 210 DISTANCE lm) Fig. 8 Example No. 3 To Demonstrate The Janbu General- ized Method The ratio of the computed interslice shear and nor— mal force was plotted versus the distance along the slip surface (Figure 9). The resulting plot was used as a side force function, f(x), in the OLE equations with the "A" value set to 1. The factor of safety from the GLE (Force) equation, {5], yielded the same factor of safety 413 11/17 as that obtained by the Janbu generalized method. In other words, the summation of forces on each slice can be viewed as a means of obtaining a particular type of side force function. In this way, moment equilibrium is im~ Dlicitlv satisfied. 20 L5 10 1(1) Th __+ _._.,L _1. I L— 30 60 90 IZO I50 I80 DISTANCE ( m 1 Fig. 9 Side Force Function For Example No. 3 Using The Janbu Generalized Method Force Equilibrium Methods — Lowe and Karafiath Method The Lowe and Karafiath method computes the factor of safe- ty from a force equilibrium equation. The direction of the resultant of each interslice force is assumed to be equal to the average of the surface and slip surface slopes. The computed side force functions for examples No. 1 and 2 (Figure 5) are shown in Figures 10 and 11, respectively. These side force functions were then used in the GLE formulation with "X" set to 1. 24 EXAMPLE N01 5 ELEVATION (m) K) 6 I2 I8 24 30 36 42 48 DISTANCE (m) L0 11“ DISTANCE (m) Fig. 10 Side Force Function Using Lowe and Karafiath Method (Example No. I) 414 N b EXAMPLENQZ E ELEVATION (m) E 6 12 I8 24 30 36 42 48 DISTANCE ("U DISTANCE (m) Fig. 11 Side Force Function Using Lowe And Karafiath Method (Example No. 2) Table 2 shows the force and moment equilibrium factors of safety from the GLE equations when using the Love and Karafiath assumption regarding the interslice force dir- actions. Table II. Factors of Safety Using the Lowe and Karafiath Interslice Force Assumption Example GLE Method No. F F f m I 1.880 1.791 2 1.096 1.068 The force equilibrium factor of safety, [5], corresponds to the Love and Karafiath factor of safety. These fac- tors of safety can be compared with those obtained by other methods (See Figure 6 and 7). The Lowe and Kara- fiath assumption regarding the interslice forces can be interpreted as a special side force function applied to the CLE formulation. — Corps of Engineers Method The Corps of Engineers method (1970) computes the factor of safety from the force equilibrium equation [5]. The direction of the resultant interslice force is assumed to be equal to the average surface slope. This appears to be interpreted as either equal to the average slope be— tween the extreme extrance and exit of the failure sur— face, (Case 1) or the changing slope of the ground sur- face (Case 2). A side force function is computed for Case I and 2, (Figure 12). 24 EXAMPLENQZ ELEVAWON (m) 4 BEDROCK *rauunrnm 6 I2 |8 24 30 36 42 4B NSTANCE (m) LO 05 I'llIllllllllliiiillllllllllllllll mSTANCE (m) (CASEZ Fig. 12 Side Force Function Using the Corps 0f Engineers Method (Example No. 1 and 2) The side force functions, f(x), are the same in both ex- amples No. 1 and 2. Table 3 summarizes the factors of safety obtained using the OLE equations along with the Corps of Engineers interslice force assumptions. The "A" value is set equal to 1. Table III. Factors of Safety Using the Corps of Engineers Interslice Force Assumptions Example Case GLE Method No. No. F F f m 1 1 1.893 1.810 2 2.000 1.801 2 1 1.102 1.068 2 1.100 1.050 The force equilibrium factors of safety (GLE method) cor- respond to the Corps of Engineers factor of safety. Since the function, f(x), is a constant, it can be set to 1 and "A" can be used as before. The factors of safety for the Case 1 assumption can be seen to correspond to a "K" value of 0.364 in Figure 6 and 7. Again, the Corps of Engineers assumption regarding the direction of the re- sultant interslice forces can be interpreted as a special side force function applied to the GLE formulation. The Corps of Engineers factors of safety (as well as the Lowe and Karafiath factors of safety) lie along a force equilibrium factor of safety line. The magnitude of the factor of safety may either be higher or lower than the factor of safety satisfying both force and moment equilib- rium. It should be noted that the force equilibrium factor of safety is more highly influenced by the side force assumption than the moment equilibrium factor of safety. 11/17 SUMMARY The factor of safety equations for all methods of slices considered can be written in the same form if the moment and/0r force equilibrium are explicity satisfied. This applies for both circular and composite (non-circular) slip surfaces. The normal force at the base of each slice can be solved using the same equation for all meth- ods, with the exception of the Ordinary method. The type of side force function assumed or computed, results in a variation in the normal force at the base of a slice. This also accounts for the difference in factors of safety between various methods when force or moment equilibrium are considered independently. The analytical aspects of slope stability analysis can be viewed in terms of one factor of safety equation satisfying overall moment equi- librium and another satisfying overall force equilibrium for various "X/E” values. Each of the methods of slices coneideredbecomes a special case of the proposed GLE fonr ulation. LIST OF REFERENCES Bishop, A. W. (1955), "The Use of the Slip Circle in the Stability Analysis of Slopes", Geotechnique, 5, pp. 7—17. Duncan, J. M. and Wright, S. G. (1980), "The Accuracy of Equilibrium Methods of Slope Stability Analysis", Proceed— ings of the International Symposium on Landslides, New Delhi, Vol. 1, pp. 247—255. Fredlund, D. G. (1974), "Slope Stability Analysis", User's Manual CD-é, Dept. of Civil Engineering, University of Saskatchewan, Saskatoon, Canada. Fredlund, D. G. (1975), "A Comprehensive and Flexible Slope Stability Program", Presented at the Roads and Transportation Association of Canada Meeting, Calgary, Alberta, Canada. Fredlund, D. G. and Krahn, J. (1977), "Comparison of Slope Stability Methods of Analysis", Canadian Geotechni- cal Journal, Vol. 1h, pp. 429-A39. Janbu, N. (1980), "Critical Evaluation of the Approaches to Stability Analysis of Landslides and Other Mass Move- ments, Free. International Symposium on Landslides, New Delhi, Vol. 2, pp. 109-128. Janbu, N. (1954), "Application of Composite Slip Surfaces for Stability Analysis", Proceedings of the European Con- ference on Stability of Earth Slopes, Stockholm, Vol. 3 pp. 43—49. Janbu, N.; Bjerrum, L. and Kjaernsli, B. (1956), "Stabil— itetsberegning for fyllinger skjaeringer og naturlige skraninger", Norwegian Geotechnical Publication No. 16, Oslo, Norway. Krahn, J.: Johnson, R. P.; Fredlund, D. G. and Clifton, A. W. (1979), "A Highway Cut Failure in Cretaceous Sedi— ments at Maymont, Saskatchewan", Canadian Geotechnical Journal, Vol. 16, No. 4, pp. 703-715. 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, Edmonton, Alberta. Lowe, J., III and Karafiath, L. (1960), "Stability of Earth Dams Upon Drawdown”, Proceedings First Pan-American Conference on Soil Mechanics and Foundation, Mexico City, Vol. 2, pp. 537-552. 415 11/17 Morgenstern, N. R. and Price, V. E. (1965), "The Analysis of the Stability of General Slip Surfaces", Geotechnique, 15, pp. 70-93. Naderi, F. (1977), "Examination of the 'SLOPE' Computer Program", M.Sc. Thesis, University of Saskatchewan, Sask— atoon, Canada. Popescu, M. (1978), "Analiza Comparative a Metodelor de Calcul al Slabilitatic Taluzurilor”, Hidrotehnica, 23, A, pp. 76—79. Spencer, E. (1967), "A Method of Analysis of the Stability of Embankments Assuming Parallel Interslice Forces", Geo- technique, 17, pp. 11—26. U.S. Army Corps of Engineers (1970), "Engineering and Design, Stability of Earth and Rock—Fill Dams", Depart— ment of the Army, Corps of Engineers, Engineer Manual, EM1110-2-1902, April. Whitman, R. V. and Bailey, W. A. (1967), “Use of Computers for Slope Stability Analysis", ASCE Journal of the Soil Mechanics and Foundation Division, 93(SM4). Wright, S. (1969), "A Study of Slope Stability and the Undrained Shear Strength of Clay Shales", Ph.D. Thesis, University of California, Berkeley, Calfornia. 416 ...
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1981_The Relationship between Limit Equilibrium Slope Stability Methods_Fredlund

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