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Analytical methods for slope stability analysis

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Figures (16)
In design, the shape of the unknown slip surface is generally assumed while the location is determined by a trial and error procedure. If the shape of the slip surface is assumed to be circular, a grid of centers can be selected, and the radius varied at each center, providing a coverage of all possible conditions (Figure 2). When the slip surface takes on a composite shape (i.e., part circular and part linear) it is still possible to use a grid of centers and varying radius to search for a critical slip surface (Fredlund, 1981). This technique can also be used to generate slip surfaces of a wedge or sliding block configuration. Although the calculations can readily be performed using a computer, the costs can be excessive. A study of the stability of a slope generally involves the following steps: i) a survey of the elevation of the ground surface on a section perpendicular to the slope, ii) the advancement of several boreholes to identify the stratigraphy and obtain undisturbed soil samples, iii) the laboratory testing of the undisturbed soil samples to obtain suitable shear strength parameters for each stratigraphic unit, and iv) the installation of piezometers to measure the pore~-water pressure. These steps provide the input data for performing a _ stability analysis. However, the location and shape of the most critical slip surface is an unknown (Figure 1). Some combination of actuating and resisting forces along a slip surface of unknown shape and location will produce the lowest factor of safety. Of course, in the case of an already failed slope, the location of the slip surface is known.
In design, the shape of the unknown slip surface is generally assumed while the location is determined by a trial and error procedure. If the shape of the slip surface is assumed to be circular, a grid of centers can be selected, and the radius varied at each center, providing a coverage of all possible conditions (Figure 2). When the slip surface takes on a composite shape (i.e., part circular and part linear) it is still possible to use a grid of centers and varying radius to search for a critical slip surface (Fredlund, 1981). This technique can also be used to generate slip surfaces of a wedge or sliding block configuration. Although the calculations can readily be performed using a computer, the costs can be excessive. A study of the stability of a slope generally involves the following steps: i) a survey of the elevation of the ground surface on a section perpendicular to the slope, ii) the advancement of several boreholes to identify the stratigraphy and obtain undisturbed soil samples, iii) the laboratory testing of the undisturbed soil samples to obtain suitable shear strength parameters for each stratigraphic unit, and iv) the installation of piezometers to measure the pore~-water pressure. These steps provide the input data for performing a _ stability analysis. However, the location and shape of the most critical slip surface is an unknown (Figure 1). Some combination of actuating and resisting forces along a slip surface of unknown shape and location will produce the lowest factor of safety. Of course, in the case of an already failed slope, the location of the slip surface is known.
DY a Stralgne L11NC Cahngene CO LNG avertayst normal stress range applicable to the slip surface under consideration. Numerous empirical nonlinear failure envelope equations have been proposed (Maksimovic, 1979). Attemps have also been made to use a bilinear shear strength envelope. The difficulty in using a bilinear or nonlinear shear strength envelope arises in the assessment of the normal stress (or force) at a particular location on the slip surface. In limit equilibrium methods, the normal force is generally computed by summing forces in a vertical or near vertical direction. The shear strength parameters appear in the derived equation. If the shear strength envelope is nonlinear, an interative approach must be adopted to compute the corresponding normal stress and shear strength parameters.
DY a Stralgne L11NC Cahngene CO LNG avertayst normal stress range applicable to the slip surface under consideration. Numerous empirical nonlinear failure envelope equations have been proposed (Maksimovic, 1979). Attemps have also been made to use a bilinear shear strength envelope. The difficulty in using a bilinear or nonlinear shear strength envelope arises in the assessment of the normal stress (or force) at a particular location on the slip surface. In limit equilibrium methods, the normal force is generally computed by summing forces in a vertical or near vertical direction. The shear strength parameters appear in the derived equation. If the shear strength envelope is nonlinear, an interative approach must be adopted to compute the corresponding normal stress and shear strength parameters.
Theoretical and quantitative comparisons have been previously made between the various limit equilibrium methods. This paper will merely attempt to summarize these findings. The general equilibrium method (i.e., GLE) (Fredlund and Krahn, 1977; Freldund, Krahn and Pufahl, 1981) is used to show the theoretical relationship between the various methods.
Theoretical and quantitative comparisons have been previously made between the various limit equilibrium methods. This paper will merely attempt to summarize these findings. The general equilibrium method (i.e., GLE) (Fredlund and Krahn, 1977; Freldund, Krahn and Pufahl, 1981) is used to show the theoretical relationship between the various methods.
\" Any Of two Orthogonal directions can be selected for the summatioi xx Of forces. Moment equilibrium is used to calculate interslice shear forces.
\" Any Of two Orthogonal directions can be selected for the summatioi xx Of forces. Moment equilibrium is used to calculate interslice shear forces.
Figure 14 shows a plot of factor of safety versus A for a similar slope with a composite slip surface (i.e., part circular and part linear). The lower portion of the slip surface has a cohesion reduced to zero and an angle of internal friction of 10 degrees. The relationship between the factors of safety remains the same for a circular slip surface. The plot also shows that the factor of safety from Bishop's Simplified method will remain close to that satisfying complete equilibrium, even for a composite slip surface. Figure 13 shows a plot of factor of safety versus A for a simple 11 metre high, 2:1 slope (Fredlund, Krahn and Pufahl, 1981). The soil properties are, as follows: total unit weight = 18.8 kN/m~; angle of internal friction = 20 degrees; cohesion = 28.75 kPa and pore-pressure ratio = 0.2. The interslice force function was assumed to be a constant. Otherwise interslice force functions can significantly affect the slope of the force equilibrium factor of safety versus A plot. The moment equilibrium factor of safety is quite insensitive to the interslice force function. As a result, any method satisfying moment equilibrium will have relatively consistent factors of safety. This does not, however, apply to the Ordinary method since it ignores all interslice forces.
Figure 14 shows a plot of factor of safety versus A for a similar slope with a composite slip surface (i.e., part circular and part linear). The lower portion of the slip surface has a cohesion reduced to zero and an angle of internal friction of 10 degrees. The relationship between the factors of safety remains the same for a circular slip surface. The plot also shows that the factor of safety from Bishop's Simplified method will remain close to that satisfying complete equilibrium, even for a composite slip surface. Figure 13 shows a plot of factor of safety versus A for a simple 11 metre high, 2:1 slope (Fredlund, Krahn and Pufahl, 1981). The soil properties are, as follows: total unit weight = 18.8 kN/m~; angle of internal friction = 20 degrees; cohesion = 28.75 kPa and pore-pressure ratio = 0.2. The interslice force function was assumed to be a constant. Otherwise interslice force functions can significantly affect the slope of the force equilibrium factor of safety versus A plot. The moment equilibrium factor of safety is quite insensitive to the interslice force function. As a result, any method satisfying moment equilibrium will have relatively consistent factors of safety. This does not, however, apply to the Ordinary method since it ignores all interslice forces.
Maksimovic (1979) used a somewhat similar iterative scheme. He used a series of progressive solutions of the force and moment equilibrium equations in an attempt to locate the intersection point between \ and the factors of safety. A A value was first selected and the force equilibrium equation was solved. The moment equilibrium equation was then solved and a new A value selected. The force and then moment equilibrium equations were repeatedly solved until the difference was less than the selected tolerance, satisfying both force and moment equilibrium (Figure 15). The initial A value was estimated and approximately three iterations were conducted for the force and moment factor of safety equations. The A value was decreased or increased depending on the relative magnitude of the moment and force equilibrium factors of safety. The intersection point was then estimated and calculations proceeded until the difference between the moment and force equilibrium factors of safety were within the required talaranca
Maksimovic (1979) used a somewhat similar iterative scheme. He used a series of progressive solutions of the force and moment equilibrium equations in an attempt to locate the intersection point between \ and the factors of safety. A A value was first selected and the force equilibrium equation was solved. The moment equilibrium equation was then solved and a new A value selected. The force and then moment equilibrium equations were repeatedly solved until the difference was less than the selected tolerance, satisfying both force and moment equilibrium (Figure 15). The initial A value was estimated and approximately three iterations were conducted for the force and moment factor of safety equations. The A value was decreased or increased depending on the relative magnitude of the moment and force equilibrium factors of safety. The intersection point was then estimated and calculations proceeded until the difference between the moment and force equilibrium factors of safety were within the required talaranca
in a negative or positive sense. This cause the normal force at the base of a slice to tend to infinity. The net result is that the normal on one or more slices may have an unreasonably large effect on the factor of safety. The suggested remedy is to ensure that the shape is such that active and passive theoretical considerations with respect to direction are not violated at the scarp and toe of the slope, respectively.
in a negative or positive sense. This cause the normal force at the base of a slice to tend to infinity. The net result is that the normal on one or more slices may have an unreasonably large effect on the factor of safety. The suggested remedy is to ensure that the shape is such that active and passive theoretical considerations with respect to direction are not violated at the scarp and toe of the slope, respectively.
FIG. 17 Typical Interslice Force Functions (From Ching and Fredlund, 1983) Nonconvergence can be encountered in any limit equilibrium method which uses an unrealistic assumption regarding’ the interslice conditions. This problem appears most often in Janbu's Generalized method (Ching, 1981). The moment equilibrium equation is used to generate the equivalent of an interslice force function based on an assumption for the line of thrust. The shape of the resulting function can be unrealistic when compared to an elastic analysis (Fan, 1983) of the mass (Figure 17). The steepness
FIG. 17 Typical Interslice Force Functions (From Ching and Fredlund, 1983) Nonconvergence can be encountered in any limit equilibrium method which uses an unrealistic assumption regarding’ the interslice conditions. This problem appears most often in Janbu's Generalized method (Ching, 1981). The moment equilibrium equation is used to generate the equivalent of an interslice force function based on an assumption for the line of thrust. The shape of the resulting function can be unrealistic when compared to an elastic analysis (Fan, 1983) of the mass (Figure 17). The steepness
}, 24 Variation of the Lambda Values with Cohesion and Angle of Internal Friction (From Fan, 1983) Equation [13] controls only the relative magnitude of the resultant interslice forces. When the interslice function is used in the GLE analysis, theaA value approached 1 for the cohesionless soil case (Figure 24). The computed lines of thrust fall near the 1/3 point and no problems have been observed with convergence (Figure 25).
}, 24 Variation of the Lambda Values with Cohesion and Angle of Internal Friction (From Fan, 1983) Equation [13] controls only the relative magnitude of the resultant interslice forces. When the interslice function is used in the GLE analysis, theaA value approached 1 for the cohesionless soil case (Figure 24). The computed lines of thrust fall near the 1/3 point and no problems have been observed with convergence (Figure 25).
QUANTITATIVE COMPARISON OF FACTORS OF SAFETY
QUANTITATIVE COMPARISON OF FACTORS OF SAFETY
Ching and FPredlund (1984) showed the "m" and "n" stability coefficients for several of the commonly used limit equilibrium methods. The results confirm that the computed factors of safety are essentially the same for those methods which satisfy complete equilibrium. Deviations in stability coefficients (i.e., m and n) are 1 to 5%. For toe circles, the differences in the m and n values obtained by the various methods are minimal. Maximum differences are approximately 10% when comparing the Ordinary and Simplified Bishop method. For deeper slip surfaces, the differences in the m and n coefficients between the Ordinary and Bishop Simplified method range from 5 to 20% (Figures 26 and 27). Janbu's Simplified method differs from the Bishop Simplified method values by 5 to 10%; Janbu's Simplified factors of safety being higher. The differences in the "m" and "n" coefficients increase with an increase in the angle of internal friction and slowe inclination.
Ching and FPredlund (1984) showed the "m" and "n" stability coefficients for several of the commonly used limit equilibrium methods. The results confirm that the computed factors of safety are essentially the same for those methods which satisfy complete equilibrium. Deviations in stability coefficients (i.e., m and n) are 1 to 5%. For toe circles, the differences in the m and n values obtained by the various methods are minimal. Maximum differences are approximately 10% when comparing the Ordinary and Simplified Bishop method. For deeper slip surfaces, the differences in the m and n coefficients between the Ordinary and Bishop Simplified method range from 5 to 20% (Figures 26 and 27). Janbu's Simplified method differs from the Bishop Simplified method values by 5 to 10%; Janbu's Simplified factors of safety being higher. The differences in the "m" and "n" coefficients increase with an increase in the angle of internal friction and slowe inclination.
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