Slope Stability Analysis Using Genetic Algorithm   Sarat Kumar Das Research Scholar, Department of Civil Engineering, Indian Institute of Technology Kanpur, India e-mail: sarat@iitk.ac.in

ABSTRACT

In this paper a real-coded genetic algorithm (GA) is used to find out the factor of safety for the soil slopes using wedge method. The analysis is formulated as constrained optimization problem to solve the nonlinear equilibrium equation and finding out the factor of safety. The results are compared with the values available in literature. The sensitivity analysis of GA optimization procedure is discussed in terms of development of failure surface. In single run, the GA finds out a number of failure surfaces and the corresponding factor of safety. The shape of the failure wedge found to depend upon the slope angles with other parameters remaining constant.

Keywords: Slope stability, analysis, genetic algorithm

INTRODUCTION

The stability of slope is one of the most important problems in stability analysis of geomechnics. Out of various methods (finite element analysis, limit analysis), limit equilibrium method is widely used for its simplicity form and the results found to be close to that rigorous methods. The limit equilibrium method is taken as 2-D plane strain problem with no variation in geometry, material and surcharge in direction parallel to the crest of the slope. The problem lies in finding out the critical failure surface and its corresponding factor of safety (FOS). The above concept has given rise to consider it as an optimization problem (Basudhar, 1976; Baker, 1980).

The development of limit equilibrium as optimization is straight forward, consisting of (i) development of objective function and (ii) selection of optimization technique. Development of objective function is based on different stability analysis method for the sliding mass of the slope. The different methods in use for this are Bishop, Janbu, Spencer, Morgenstern & Price, Chen & Morgenstern, Sharma etc. (Abramson et al., 2002). The stability analysis methods basically differ from one another in the hypothesis assumed in order to satisfy the equilibrium conditions of the potential sliding mass. It has been proved that all these methods, if used respecting the basic hypothesis, gives satisfactory results.

Different sophisticated optimization techniques have been used to search for the critical slip surface, are calculus of variation, linear programming, nonlinear programming and dynamic programming. The variational technique cannot be applied to heterogeneous soil, and as the stability analysis equation is nonlinear, linear programming has not been widely accepted. Dynamic programming has the difficulty in dimensionality, so the nonlinear unconstraint optimizations like Nelder Meade, Hookes & Jeeve & Powells Conjugate direction method, steepest descent, Fletcher-Reeve (FR), Davidon,- Fletcher – Powel(DFP), Broydon-Fletcher-Goldfarb-Shanno (BFGS) have been widely used. Many practical slope problems are not convex (De Natale, 1991), there by having multiple optima. All the above optimization techniques are initial point dependant and there is a need to analyze with wide separated points. It is usually not possible to find global minimum except in special cases (De Natale, 1991).

To avoid the difficulty in finding out the global minima, evolutionary methods such as genetic algorithm is being used, which is more robust in finding out the optimal solution in many complex problems (Goldberg, 1989). Goh (1999) has used GA to find out the critical surface and the factor of safety using method of wedges. McCombie and Wilkinson (2002) used Bishop’s simplified method and Sabhahit et al. (2002) have used Janbu’s method to search for the critical surface using GA. In the above studies GA could find better solution compared to other traditional optimization tools.

With the above in view, in this study a real-coded GA has been used to find out the critical failure surface and the corresponding factor of safety for three wedge method. The real-coded GA has several advantages over binary coded GA (Deb, 2001) and three-wedge method is widely used for stability analysis of mine spoils (Huang, 1983).

METHODOLOGY

The analysis of the problem can be considered in two stages: (i) development of objective function and (ii) the application of GA in solving the objective function.

Development of objective function

In the present study, the three-wedge method for stability analysis of slopes (Huang, 1983) is used for the development of objective function. This is a force equilibrium method and development of the equations used for the analysis is described in details in Huang (1983).

Figure 1. The free-body diagram for three-wedge method

Figure 1 shows the free-body diagram showing the forces on each block. There are total six (6) unknowns (P₁, P₂, N₁, N₂, N₃ and factor of safety, F) which can be solved by six equilibrium equations, two for each block. The F can be found out by solving the nonlinear equation as shown in Eq. 1 (Huang, 1983). There are different iterative methods to solve Eq. 1. However, there are some problems in solving such equation using the iterative methods (Bhattacharya and Basudhar, 2001), which is inherent in all numerical techniques. So in the present study the Eq. 1 is solved using optimization method.

(1)

Where

r_u = pore pressure parameter, W₃ = weight of the 3^rd wedge, C_s is horizontal seismic acceleration coefficient and P₁, P₂, N₁, N₂, N₃ are as shown in Fig. 1.

The optimization method may be described as finding out the minimum factor safety which satisfies the Eq. 1 and in mathematical programming form it can be written as:

Min F:
Subjected to

(2)

The variables (design vectors) are l₁, l₂, l₃, N₁, N₂, N₃ and F and the application dependent input parameters are slope angle (b), cohesion (c_i), height of slope (H), angle of internal friction (f) pore pressure parameter r_u) and seismic acceleration coefficient C_s

In order to ascertain that the shape and location of the slip surface are physically reasonable and kinematically compatible, the following constraints need to be imposed on the choice of design variable. As per physically condition it is found that the direction of the Ti should be positive (Huang, 1983), and the kinematical conditions are applied for the geometry of the failure surface.

(3)

(4)

b3 q1

(5)

Ti³0.0; i = 1, 2, 3

(6)

Genetic algorithm

The GA is a random search algorithm based on the concept of natural selection inherent in natural genetics, presents a robust method for search for the optimum solution to the complex problems. The algorithms are mathematically simple yet powerful in their search for improvement after each generation (Goldberg, 1989). The artificial survival of better solution in GA search technique is achieved with genetic operators: selection, crossover and mutation, borrowed from natural genetics. The major difference between GA and the other classical optimization search techniques is that the GA works with a population of possible solutions; whereas the classical optimization techniques work with a single solution. Another difference is that the GA uses probabilistic transition rules instead of deterministic rules. The GA that employs binary strings to represent the variables (chromosomes) is called binary-coded GA. The binary-coded GA consists of three basic operators, selection, crossover or mating, and mutation, which are discussed as follow. In the selection procedure, the chromosomes compete for survival in a tournament selection, where the chromosomes with high fitness values enter the mating population and the remaining ones die off. The selection probability (Ps) determines the number of chromosomes to take part in tournament selection process. The selected chromosomes form an intermediate population known as the mating population, on which crossover and mutation operator is applied. The selected chromosomes are randomly assigned a mating partner from within the mating population. Then, a random crossover location is selected in any two parent chromosomes and the genetic information is exchanged between the two mating parent chromosomes with a certain mating probability (P_c), giving birth to a child (new variable) or the next generation. In binary-coded GA, mutation is achieved by replacing 0 with 1 or vice versa in the binary strings, with a probability of P_m. This process of selection, crossover, and mutation is repeated for many generations (iterations) with the objective of reaching the global optimal solution. The flow chart of the general solution procedure of GA is depicted in Figure 2.

Figure 2. Flow chart for working principles of genetic algorithm

In the present analysis, a real- coded GA has been used, in which there is no need of coding and decoding the design variables. The real- coded GA with simulated binary crossovers (SBX), polynomial mutations and a tournament selection type of selection procedure have been used, details of which are available in Deb (2001). The GA was implemented using pseudo code available as freeware at

http://www.iitk.ac.in/mech/research_labs.htm.

The GA has an inherent limitation of not being able to handle the equality constraints. The equality constraints need to be converted to inequality constraint using a dummy variable e (0.001 is considered for the present study) in the GA formulation and the revised constraint is written as in Eq.7-11. So the total objective function and the corresponding constraints can be written as

(7)

Subject to:

(8)

(9)

b3 q1

(10)

Ti³0.0; i = 1, 2, 3

(11)

The common method of handling the constraints is by penalty function method. However, in the present study the following method (Deb, 2001) is used for constraint handling.

• The method uses tournament selection as the selection operator and two solutions are compared at a time.
• Any feasible solution is preferred to any infeasible solution;
• Among two feasible solutions the one having better objective function is preferred and among two infeasible solutions, the one having smaller constraint violation is preferred.

Thus, at any iteration, the infeasible solutions are not computed for objective function if some feasible solutions are present, which helps in reducing the computational effort.

For the present study the GA parameters of the crossover probability (P_c) varied from 0.58 to 0.78 and the mutation probability (P_m) 0.21 to 0.41.

RESULTS & DISCUSSION

Three field problems (Problem I, Problem II and Problem III) as discussed in Jade and Shankar (1995) were considered for the present study. The three dry slopes with its geometry, soil parameter and the corresponding critical factor safety using Bishops simplified method is shown in Table 1. Unlike previous GA studies, in this study the fitness is considered as the objective function itself. As tournament selection is used, so the minimum value is the winner. The variation of the fitness function (objective function) with generation number is shown in Figure 3. It can be seen that the best fitness (minimum objective function) is almost constant after 100 generation. However, there is sufficient diversity in the results as observed from average fitness and worst fitness. The results of the present study are compared with the results of Jade and Shankar (1995). As it can be seen from Table 1 that for all the three cases the factor of safety obtained by the present study is less than that obtained by Bishop’s simplified method. Though, it can be pointed out that the Bishop’s simplified method assumes circular slips surface, where as the present study assumes the wedge failure. So it was observed that though the factor of safety is similar the slip surface was different and the above observations have been observed by other studies in slope stability analysis.

Figure 4, 5 and 6 shows the different failure surface along with the critical surface for the Problem I, II and III respectively. Unlike traditional optimization problem, working with GA has the advantages of storing a number of solutions close to the optimum value, along with the optimum one. This will help in finding the variability of the FOS over the slope (Baker and Leshchinsky, 2001). It can be observed that for the Problem I and II, the critical failure surface is almost single wedge. This may be due to the fact that both the problems may be considered as steep slope b = 78⁰ and 65⁰ respectively). The critical surface in Problem III (b = 45^o) shows a two wedge failure surface. To study the effect of slope angle on the type of failure wedge, Problem II was solved for different slope angle and the results are shown in Figure 7. It can be seen that when b = 65^o the critical failure surface is single wedge and as the slope angle reduces the failure wedge changes to two wedges and subsequently to three wedges. This may help in choosing the type of wedge analysis depending upon the slope angle.

Figure 3. Variation of the fitness of the objective function with generation number

Table 1. Comparisons of results of present study with Jade and Shankar (1995)

	H (m)	b	c (kN/m2)	f	g (kN/m3)	ru	Cs	F1	F2
Problem I	8.3	780	7.5	280	19.04	0.0	0.0	0.92	0.663
Problem II	3.8	650	7.5	280	19.04	0.0	0.0	1.30	1.178
Problem III	11.6	450	15.0	220	19.10	0.0	0.0	1.15	1.13
	1 Jade and Shankar (1995); 2 Present study

Figure 4. Different failure surfaces along with the critical failure surface for Problem I

Figure 5. Different failure surfaces along with the critical failure surface for Problem II

Figure 6. Different failure surfaces along with the critical failure surface for Problem III

Figure 7. Variation of type of critical failure surface with different slope angle for Problem II

Sensitivity Analysis

Like NLP optimization methods, population based algorithms first optimizes the parameter which is the most sensitive to the objective function (Marseguerra et al. 2003). Marseguerra et al. (2003) have shown that for GA, the sensitivity of the parameter can be correlated with square of coefficient of variation. The coefficient of variation is defined as the ratio of the standard deviation to the mean of the population. The parameter which is the most sensitive should quickly converge to low value of square of coefficient of variation with generation. The behavior of square of coefficient of variation with generation is shown in Figure 8. It can be seen that the GA first optimize the material parameter FOS. The sensitivities of the geometric terms are in the order as, q₃, l₃, q₂, l₂, q₁, and l₁. It can be seen that GA tries to first minimize the angle and then the corresponding length of wedge. Malkawi et al. (2001) through Monte Carlo simulation observed that the minimization of factor of safety is achieved by gradual fixing of failure segments.

Figure 8. Variation of type of square of coefficient variation with generation number

CONCLUSIONS

In this paper, application of genetic algorithm in analyzing soil slopes using three-wedge method was discussed. The factor of safety obtained by the present study for three cases was less than that obtained by Bishop’s simplified method. Unlike traditional optimization problem, working with GA has the advantages of storing a number of solutions close to the optimum value, along with the optimum one. This will help in finding the variability of the FOS over the slope. The number of failure wedge at critical condition found to depend upon the slope angle. The sensitivities of the geometric terms in terms of minimizing the FOS are in the order as, l₃, q₃, q₁, l₁, q₂, and l₂.

REFERENCES

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