SUMMARY, CONCLUSIONS AND FUTURE SCOPE
CHAPTER 3 NUMERICAL MODELLING
3.2. WORKING PRINCIPLES OF VARIOUS MODULES OF GEOSTUDIO AND RSLOPE2D
3.2.1. Slope/W for Slope Stability Modeling
This section briefs the theoretical basis utilised in the development of Slope/W module. Slope/W basically solves for two factor of safety equations considering static equilibrium; one satisfying the force equilibrium (Ff) and the other satisfying the moment equilibrium (Fm), following the basic conventions of slope stability analysis. The commonly used methods of slices (for example, Bishop’s Simplified method, Janbu’s Simplified method, Spencer method, Morgenstern-Price method) are special cases of the General Limit Equilibrium (GLE) solution which allows a range of assumptions related to the interrelationship between the interslice shear forces and normal forces. Limit equilibrium types of analyses for assessing the stability of earth slopes have been in use in geotechnical engineering for many decades. The idea of discretizing a potential sliding mass into vertical slices was introduced early in the 20th century and is consequently the oldest numerical
analysis technique in geotechnical engineering. Many different solution techniques for the method of slices have been developed over the years. Basically, all of them are very similar. The differences between the methods depends on the equations of statics that are satisfied, the interslice forces that are considered in the equilibrium solution, and the assumed relationship between the interslice shear and normal forces.
The Ordinary, or Fellenius (1936), method was the first method developed. The method ignored all interslice forces and satisfied only moment equilibrium. Adopting these simplified assumptions made it possible to compute a factor of safety using hand calculations, which was important since there were no computers available. Later, Bishop (1955) devised a scheme that included interslice normal forces but ignored the interslice shear forces. Again, Bishop’s Simplified method satisfies only moment equilibrium. Of interest and significance with this method is the fact that by including the normal interslice forces, the factor of safety equation became nonlinear, and an iterative procedure was required to calculate the factor of safety. The Janbu’s Simplified method (1954) is similar to the Bishop’s Simplified method in that it includes the normal interslice forces and ignores the interslice shear forces. The difference between the Bishop’s Simplified and Janbu’s Simplified methods is that the Janbu’s Simplified method satisfies only horizontal force equilibrium, as opposed to moment equilibrium. Later, computers made it possible to more readily handle the iterative procedures inherent in the limit equilibrium method, and this lead to mathematically more rigorous formulations which include all interslice forces and satisfy all equations of statics. Two such methods are the Morgenstern-Price (M-P) (1965) and Spencer (1967) methods. Spencer only considered a constant X/E ratio for all slices in the General Limit Equilibrium (GLE) equation (Eqn. 1), which infers that the ratio of shear to normal is a constant
between all slices. The M-P method can utilize any general appropriate function and hence considered in this study.
The interslice shear forces in the GLE method are governed by the equation proposed by Morgenstern and Price (1965), which is expressed as
( )
X =Ef x (3.1)
where, f(x) is a function which provides the relation between the interslice normal force (E) and shear force (X), and λ is the reduction factor.
Slope/W can accommodate a wide range of different interslice force functions, namely Constant, Half-sine, Clipped-sine, Trapezoidal and Data point fully specified. The Spencer method uses a constant interslice function which infers that the ratio of shear to normal is a constant between all slices. One does not need to select the function; it is fixed to be a constant function in the software when the Spencer method is selected. Only the Morgenstern-Price allows for user-specified interslice functions. Some of the functions available are the constant, half-sine, clipped-sine, trapezoidal and data-point specified. The most commonly used functions are the constant and half- sine functions. A Morgenstern-Price analysis with a constant function is the same as a Spencer analysis. Slope/W by default uses the half-sine function for the M-P method. The half-sine function tends to concentrate the interslice shear forces towards the middle of the sliding mass and diminishes the interslice shear in the crest and toe areas. Defaulting to the half-sine function for these methods is based primarily on experience and intuition and not on any theoretical considerations. Other functions can be selected if deemed necessary. However, in the present study an extensive analysis has not been conducted to study the influence of different functions on the
slope stability assessment and the Slope/W default Half Sine function is utilised. The GLE formulation estimates Fm and Ff for a range of lambda (λ) values. A plot similar to Fig. 3.1 can be drawn with these computed Fm and Ff values with lambda (λ). The Morgenstern-Price (M-P) FoS is the factor of safety corresponding to the point where the two curves intersects in Fig. 3.1.
However, it is to be noted that since the method is purely based on the principles of statics, and there is no mention of displacement or deformations, it is not always possible to obtain realistic stress distributions from this method.
Fig. 3.1 A typical plot of factor of safety versus lambda (λ) (adopted from Slope/W v2018)
As an alternative to the limit-equilibrium (LE) stability analysis, the theory of the Finite Element Stress method is applied. This method computes the stability factor of a slope based on the stress state in the soil obtained from a finite element stress analysis. A FoS is defined as that factor by
which the shear strength of the soil must be reduced in order to bring the soil mass into a state of limiting equilibrium along a selected slip surface. For an effective stress analysis, the shear strength is defined as
' ( n ) tan '
s c= + −u (3.2)
where, s is effective shear strength of the soil, c' is effective cohesion, ' is effective angle of internal friction, σn is normal stress on shear plane, and u is pore-water pressure.
For a total stress analysis, the strength parameters are defined in terms of total stresses and pore- water pressures are not taken into consideration. The stability analysis involves passing a slip surface through the earth mass and dividing the inscribed portion into vertical slices. The slip surface may be circular, composite (i.e., combination of circular and linear portions) or consist of any shape defined by a series of straight lines (i.e., fully specified slip surface). Slope/W conducts a slope stability analysis as described above to estimate stress-based FoS against failure.
Slope/W accommodates a comprehensive algorithm for conducting stability analysis within a probabilistic framework. Most of the input parameters can be considered as random variable characterising the same by assigning a probability distribution function (pdf), and a Monte Carlo Simulation (MCS) scheme is then adopted to produce a pdf of the resulting FoS. Once the pdf of the FoS are known, the probability of failure and reliability index of the earth structure can be computed. Although most natural material parameters may vary statistically in a Normal distribution manner, for general purposes, Slope/W includes several probability density functions (pdf) namely Normal, Lognormal, Uniform, Triangular and Generalized Spline function for
accounting the special circumstances. The present study utilises the Lognormal pdf for characterising various input random variables.