# Determination of Slope Safety Factor with Analytical Solution and Searching Critical Slip Surface with Genetic-Traversal Random Method

Authors: Wen-jie Niu
Journal: The Scientific World Journal
 (2014)

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## Abstract

In the current practice, to determine the safety factor of a slope with two-dimensional  circular potential failure surface, one of the searching methods for the critical slip surface is Genetic Algorithm (GA), while the method to calculate the slope safety factor is Fellenius’ slices method. However  GA needs to be validated with more numeric tests, while Fellenius’ slices method is just an approximate method like finite element method. This paper proposed a new method to determine the minimum slope safety factor which is the determination of slope safety factor with analytical solution and searching critical slip surface with Genetic-Traversal Random Method. The analytical solution is more accurate than Fellenius’ slices method. The Genetic-Traversal Random Method uses random pick to utilize mutation. A computer automatic search program is developed for the Genetic-Traversal Random Method. After comparison with other methods like slope/w software, results indicate that the Genetic-Traversal Random Search Method can give very low safety factor which is about half of the other methods. However the obtained minimum safety factor with Genetic-Traversal Random Search Method is very close to the lower bound solutions of slope safety factor given by the Ansys software.


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## Body

## 1. Introduction

The geotechnical engineer frequently uses limit equilibrium methods of analysis when studying slope stability problems, for example, Ordinary or Fellenius’ method (sometimes referred to as the Swedish circle method or the conventional method), Simplified Bishop method, Spencer’s method, Janbu’s simplified method, Janbu’s rigorous method, Morgenstern-Price method, or unified solution scheme [1–3]. In order to reduce the influence of the assumptions made in limit equilibrium methods on the factor of safety, the methods of limit analysis based on the rigid body plasticity theory were developed by Chen (1975), Michalowski (1995), and Donald and Chen (1997). These methods based on the upper bound theorem of limit analysis are generally referred to as the upper bound methods, which give an upper bound solution to the real value of the factor of safety [4–6].In the current practice, searching methods for the critical slip surface is a central issue to slope stability analysis. Previous research employed the Variational Calculus, the dynamic programming, alternating variable methods, the Monte Carlo technique, or the genetic algorithm (GA) into slope stability analysis for critical surface identification [7–13].In recent years, genetic algorithm search procedure has been used to locate the critical slip surface of homogeneous slopes. It has been found that genetic algorithm is a robust search technique which often gives global solution [14, 15]. Numerical example shows that analyzing method of the slope stability based on the genetic algorithm is a global optimal procedure that can overcome the drawbacks of local optimum widely existing in classical searching methods and the result is satisfactory [16]. Established slope stability analysis methods cope well with moderately noncircular shear surfaces, and the simple genetic algorithm (SGA) has been used successfully to find the critical slip surface [17].In the GA, the parameters in the optimization problem are translated into chromosomes with a data string (binary or real). A range of possible solutions is obtained from the variable space and the fitness of these solutions is compared with some predetermined criteria. If a solution is not obtained, a new population is created from the original (parent) chromosomes. This is achieved using “crossover” and “mutation” operations. Crossover involves gene exchange from two random (parent) solutions to form a child (new solution). Mutation involves the random switching of a single variable in a chromosome and is used to maintain population diversity, as the process converges towards a solution [18].GA includes inheritance, mutation, selection, and crossover [19]. One of the core techniques and advantages of GA is that mutation can consider a wide range of possible solutions if natural evolution continues and never ends. The other advantage is that inheritance and crossover can save all the examples and virtues of the past age and pass them into the next generation to save time for the best choice [18, 19].However there are many unsolved problems about GA in slope stability analysis. For example, how to realize GA with hand calculation method? How to realize GA with automatic search program? What is the relationship between GA and traversal random search method? How to improve the Fellenius method of slices concerning that the slices method is an approximate method like finite element method? In fact, all these problems are mathematical problems to find the minimum safety factor of a slope. These mathematical problems originated from the physical equations representing the common law of nature in slopes (e.g., Mohr-Column criterion of soil, 2D circular slip surface of homogeneous clay slope, and safety factor definition which is the moment of sliding resistance divided by the moment of sliding force). The physical laws of nature can be found and validated with repeated in situ or lab experiments to measure the physical quantities and mathematical logic to reveal the relationship. However mathematical problems can only be solved with logic deduction and validated with countless numeric tests.This paper first intends to determine a cohesive soil slope safety factor with Fellenius’ slices method, while the 2D critical failure surface is searched with GA. The analysis uses real-coded methods to encode the chromosomes with the variables of potential critical surface locations. The fitness of each chromosome is determined using the objective function that the resulting safety factors should be lower enough, and the fitness of all solutions is compared, while the chromosome of large safety factors shall be deleted [18]. However this part is realized with hand calculation.Then a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is made. The Genetic-Traversal Random Search Method presented in this paper only utilizes the mutation and selection thought of the traditional genetic algorithm. Crossover is omitted due to the difficulty in computer program realization and compensated with numerous random candidates due to mutation. The Genetic-Traversal Random Search Method makes a traversal search with random method. In the program, random numbers for random search are generated by computer and search boundaries are included. In the program, each slope safety factor is given by analytical solution rather than slices method. The safety factor and failure circle determination program developed in Silverfrost FTN95 is presented in the Appendix. At last, the proposed Genetic-Traversal Random Search Method is compared with other solutions such as slope/w software.


## 2. A Slope Stability Problem Example

A cohesive soil slope with its height 25 meters has a slope ratio of 1 : 2. The soil unit weightγ is 20 KN/m3. The soil internal friction angle φ is 26.6 degrees, and cohesion is 10 KPA. The problem now is to give the safety factor of the slope with a 2D circular failure surface.


## 3. Search the Critical Slip Surface with GA Method While Determining the Safety Factor with Fellenius’ Method of Slices with Hand Calculation

To solve the engineering problem in Section2, this part will search the critical slip surface with GA Method while determining the safety factor with Fellenius’ method of slices.


### 3.1. The Slope Safety Factor with Fellenius’ Method of Slices

The potential slip surface for clay slope is two dimensional and a part of circle. In order to determine the slope safety factor in Figure1, Fellenius’ method of slices divides the slope into several slices [3]. Using moment equilibrium, the slope safety factor SF in Figure 1 is
(1)SF=∑(ci+Wicos⁡αitanφi)liWisinαi,
where ci and φi are the soil slope slice cohesion and internal friction angle. Wi is the soil slope slice self-gravity. li is the soil slope slice slip circular arc length. αi is the angle between soil slope slice slip surface tangent line and the horizontal line.Figure 1
Fellenius’ method of slices.


### 3.2. Searching the Critical Failure Surface with GA

The assumed 2D slope failure surface is circular determined by two variablesXc and Xcc in Figure 2. Xc is the abscissa of a point on slope top surface. If Xc is determined, the 2D critical slip surface circle center with abscissa Xcc must lie on the perpendicular bisector of the straight line from the point of Xc to the slope toe O. Xcc is the abscissa of the critical slip surface circle center. So if Xcc is given, then the critical slip surface circle center can be determined. Altogether, if Xc and Xcc are given, 2D circular slope slip surface is determined.Figure 2
2D potential failure surfaces.However, in the next section, all the computations according to GA are made by hand calculations. If, for computer simulation, “interval" is asL and s as in Figure 2 and random numbers of computer function can help for automatic generation of variables Xc and Xcc in Figure 2, search regions or boundaries for Xc and Xcc must be defined with the definition of the limits of Xc and Xcc. The necessity of these boundaries is evident because computer program must avoid generating surfaces out of the region of primary interest [21].The searching process for the 2D critical failure surface as in Figure1 uses techniques inspired by natural evolution, such as inheritance, mutation, selection, and crossover [19]. In a genetic algorithm, a population of strings (called chromosomes or the genotype of the genome), which encode candidate solutions (called individuals, creatures, or phenotypes) to an optimization problem, is evolved toward better solutions [19]. In this 2D critical failure surface searching problem, the candidate solutions are represented as {Xc,Xcc} described before. The evolution starts from a population of randomly generated individuals as {Xc,Xcc} and happens in generations. In each generation, the fitness of every individual in the population is evaluated; multiple individuals are stochastically selected from the current population (based on their fitness) and modified (recombined and possibly randomly mutated) to form a new population. The evaluation standard is that the individual {Xc,Xcc} with large safety factor is deleted and the individual {Xc,Xcc} with small safety factor is reserved. The new population is then used in the next iteration of the algorithm [19]. The algorithm terminates when a satisfactory fitness level has been reached for the population which means that it is hard to lower safety factor with iterations.


### 3.3. Searching Process with GA

In the GA, with hand calculation method, the potential failure surfaces for search are restricted that they all pass slope toe as in Figure2 for simplifying the search task. Search process for the critical slip surface with genetic algorithm was presented from Table 1 to Table 12. In these tables, the minimum safety factors are marked with * in each iteration. The units for Xc and Xcc are meters. Evaluation of individuals in Table 1 now begins. The evaluation standard is that the individual {Xc,Xcc} with large safety factor is deleted and the individual {Xc,Xcc} with small safety factor is reserved. Selection result will be put in Table 2. Crossover of selected individuals of Table 2 will be put in Table 3. Mutation begins and results will be put in Table 4. Evaluation of all previous individuals marked with * begins. The evaluation standard is that the individual {Xc,Xcc} with large safety factor is deleted and the individual {Xc,Xcc} with small safety factor is reserved. Selection result will be put in Table 5. Crossover of selected individuals of Table 5 begins. Crossover result will be put in Table 6. Mutation begins and the result will be put in Table 7. Evaluation of all previous individuals marked with * begins. The evaluation standard is that the individual {Xc,Xcc} with large safety factor is deleted and the individual {Xc,Xcc} with small safety factor is reserved. Selection result will be put in Table 8. Crossover of selected individuals of Table 8 begins. Crossover result will be put in Table 9. Mutation begins and result will be put in Table 10. Evaluation of all previous individuals marked with * begins. The evaluation standard is that the individual {Xc,Xcc} with large safety factor is deleted and the individual {Xc,Xcc} with small safety factor is reserved. Selection result will be put in Table 11. The GA procedure terminates when a satisfactory fitness level has been reached for the population which means that it is hard to lower safety factor with iterations. The final result is in Table 12.Table 1
A population of randomly generated individuals.
X
c
X
c
c
Safety factor
51
25
2.720523521
51
41
Safety factor is extremely large, Unreasonable
*
51
0
1.3317465878
*
60
−10
1.4074551938
*
60
25
1.886105777
60
35
Safety factor is extremely large, Unreasonable
*
100
−20
2.2757670756
100
21
2.3067661886
100
70
Safety factor is extremely large, UnreasonableTable 2
Selected individuals.
Selected individuals
X
c
X
c
c
Safety factor
*
51
0
1.3317465878
60
−10
1.4074551938
60
25
1.886105777
100
−20
2.2757670756Table 3
Crossover results.
X
c
X
c
c
Safety factor
*
51
−10
1.3799239885
51
25
2.720523521
51
−20
1.4447027254
*
60
0
1.3879596689
60
−20
1.4333540217
100
0
2.272574082
100
−10
2.2729514535
100
25
2.3308411862Table 4
Mutation results.
X
c
X
c
c
Safety factor
*
50
0
1.3425306677
50
−10
1.4003909843
51
21
1.7627329758
*
55
0
1.3332021645
*
55
−10
1.3615830364
55
20
1.6493380867Table 5
Selected individuals.
Selected individuals
X
c
X
c
c
Safety factor
*
51
0
1.3317465878
51
−10
1.3799239885
60
0
1.3879596689
*
50
0
1.3425306677
*
55
0
1.3332021645
55
−10
1.3615830364Table 6
Crossover results.
X
c
X
c
c
Safety factor
51
0
1.3317465878
51
−10
1.3799239885
60
0
1.3879596689
50
0
1.3425306677
55
0
1.3332021645
55
−10
1.3615830364Table 7
Mutation results.
X
c
X
c
c
Safety factor
*
52
0
1.3264516064
52
−10
1.3675053085
*
53
0
1.3254640532
53
−10
1.3611610322
*
57
0
1.3503481321
57
−10
1.3741239435
*
52
11
1.3501275295
52
25
2.5159145338
52
100
Safety factor is extremely large, Unreasonable
52
15
1.4194510174Table 8
Selected individuals.
Selected individuals
X
c
X
c
c
Safety factor
*
51
0
1.3317465878
*
50
0
1.3425306677
*
55
0
1.3332021645
*
52
0
1.3264516064
*
53
0
1.3254640532
57
0
1.3503481321
52
11
1.3501275295Table 9
Crossover results.
X
c
X
c
c
Safety factor
51
11
1.3556896331
50
11
1.3672906676
55
11
1.3547892263
*
53
11
1.3487011763
57
11
1.3696385778Table 10
Mutation results.
X
c
X
c
c
Safety factor
54
21
1.6711004325
*
54
0
1.3279271582
54
13
1.3734897577
54
−10
1.3595160519
60
−10
1.4074551938Table 11
Selected individuals.
Selected individuals
X
c
X
c
c
Safety factor
51
0
1.3317465878
50
0
1.3425306677
55
0
1.3332021645
52
0
1.3264516064
*
53
0
1.3254640532
53
11
1.3487011763
54
0
1.3279271582Table 12
Critical failure surface and minimum safety factor.
X
c
X
c
c
Safety factor
Completed
53
0
1.3254640532


### 3.4. Location of the Critical Failure Surface and Safety Factor with GA Procedure

The example was solved with foregoing GA procedure. The minimum safety factor was 1.325 withXc=53 m and Xcc=0. The corresponding slip circle center is at (0, 68.8 m) and the radius is 68.8 m.


## 3.1. The Slope Safety Factor with Fellenius’ Method of Slices

The potential slip surface for clay slope is two dimensional and a part of circle. In order to determine the slope safety factor in Figure1, Fellenius’ method of slices divides the slope into several slices [3]. Using moment equilibrium, the slope safety factor SF in Figure 1 is
(1)SF=∑(ci+Wicos⁡αitanφi)liWisinαi,
where ci and φi are the soil slope slice cohesion and internal friction angle. Wi is the soil slope slice self-gravity. li is the soil slope slice slip circular arc length. αi is the angle between soil slope slice slip surface tangent line and the horizontal line.Figure 1
Fellenius’ method of slices.


## 3.2. Searching the Critical Failure Surface with GA

The assumed 2D slope failure surface is circular determined by two variablesXc and Xcc in Figure 2. Xc is the abscissa of a point on slope top surface. If Xc is determined, the 2D critical slip surface circle center with abscissa Xcc must lie on the perpendicular bisector of the straight line from the point of Xc to the slope toe O. Xcc is the abscissa of the critical slip surface circle center. So if Xcc is given, then the critical slip surface circle center can be determined. Altogether, if Xc and Xcc are given, 2D circular slope slip surface is determined.Figure 2
2D potential failure surfaces.However, in the next section, all the computations according to GA are made by hand calculations. If, for computer simulation, “interval" is asL and s as in Figure 2 and random numbers of computer function can help for automatic generation of variables Xc and Xcc in Figure 2, search regions or boundaries for Xc and Xcc must be defined with the definition of the limits of Xc and Xcc. The necessity of these boundaries is evident because computer program must avoid generating surfaces out of the region of primary interest [21].The searching process for the 2D critical failure surface as in Figure1 uses techniques inspired by natural evolution, such as inheritance, mutation, selection, and crossover [19]. In a genetic algorithm, a population of strings (called chromosomes or the genotype of the genome), which encode candidate solutions (called individuals, creatures, or phenotypes) to an optimization problem, is evolved toward better solutions [19]. In this 2D critical failure surface searching problem, the candidate solutions are represented as {Xc,Xcc} described before. The evolution starts from a population of randomly generated individuals as {Xc,Xcc} and happens in generations. In each generation, the fitness of every individual in the population is evaluated; multiple individuals are stochastically selected from the current population (based on their fitness) and modified (recombined and possibly randomly mutated) to form a new population. The evaluation standard is that the individual {Xc,Xcc} with large safety factor is deleted and the individual {Xc,Xcc} with small safety factor is reserved. The new population is then used in the next iteration of the algorithm [19]. The algorithm terminates when a satisfactory fitness level has been reached for the population which means that it is hard to lower safety factor with iterations.


## 3.3. Searching Process with GA

In the GA, with hand calculation method, the potential failure surfaces for search are restricted that they all pass slope toe as in Figure2 for simplifying the search task. Search process for the critical slip surface with genetic algorithm was presented from Table 1 to Table 12. In these tables, the minimum safety factors are marked with * in each iteration. The units for Xc and Xcc are meters. Evaluation of individuals in Table 1 now begins. The evaluation standard is that the individual {Xc,Xcc} with large safety factor is deleted and the individual {Xc,Xcc} with small safety factor is reserved. Selection result will be put in Table 2. Crossover of selected individuals of Table 2 will be put in Table 3. Mutation begins and results will be put in Table 4. Evaluation of all previous individuals marked with * begins. The evaluation standard is that the individual {Xc,Xcc} with large safety factor is deleted and the individual {Xc,Xcc} with small safety factor is reserved. Selection result will be put in Table 5. Crossover of selected individuals of Table 5 begins. Crossover result will be put in Table 6. Mutation begins and the result will be put in Table 7. Evaluation of all previous individuals marked with * begins. The evaluation standard is that the individual {Xc,Xcc} with large safety factor is deleted and the individual {Xc,Xcc} with small safety factor is reserved. Selection result will be put in Table 8. Crossover of selected individuals of Table 8 begins. Crossover result will be put in Table 9. Mutation begins and result will be put in Table 10. Evaluation of all previous individuals marked with * begins. The evaluation standard is that the individual {Xc,Xcc} with large safety factor is deleted and the individual {Xc,Xcc} with small safety factor is reserved. Selection result will be put in Table 11. The GA procedure terminates when a satisfactory fitness level has been reached for the population which means that it is hard to lower safety factor with iterations. The final result is in Table 12.Table 1
A population of randomly generated individuals.
X
c
X
c
c
Safety factor
51
25
2.720523521
51
41
Safety factor is extremely large, Unreasonable
*
51
0
1.3317465878
*
60
−10
1.4074551938
*
60
25
1.886105777
60
35
Safety factor is extremely large, Unreasonable
*
100
−20
2.2757670756
100
21
2.3067661886
100
70
Safety factor is extremely large, UnreasonableTable 2
Selected individuals.
Selected individuals
X
c
X
c
c
Safety factor
*
51
0
1.3317465878
60
−10
1.4074551938
60
25
1.886105777
100
−20
2.2757670756Table 3
Crossover results.
X
c
X
c
c
Safety factor
*
51
−10
1.3799239885
51
25
2.720523521
51
−20
1.4447027254
*
60
0
1.3879596689
60
−20
1.4333540217
100
0
2.272574082
100
−10
2.2729514535
100
25
2.3308411862Table 4
Mutation results.
X
c
X
c
c
Safety factor
*
50
0
1.3425306677
50
−10
1.4003909843
51
21
1.7627329758
*
55
0
1.3332021645
*
55
−10
1.3615830364
55
20
1.6493380867Table 5
Selected individuals.
Selected individuals
X
c
X
c
c
Safety factor
*
51
0
1.3317465878
51
−10
1.3799239885
60
0
1.3879596689
*
50
0
1.3425306677
*
55
0
1.3332021645
55
−10
1.3615830364Table 6
Crossover results.
X
c
X
c
c
Safety factor
51
0
1.3317465878
51
−10
1.3799239885
60
0
1.3879596689
50
0
1.3425306677
55
0
1.3332021645
55
−10
1.3615830364Table 7
Mutation results.
X
c
X
c
c
Safety factor
*
52
0
1.3264516064
52
−10
1.3675053085
*
53
0
1.3254640532
53
−10
1.3611610322
*
57
0
1.3503481321
57
−10
1.3741239435
*
52
11
1.3501275295
52
25
2.5159145338
52
100
Safety factor is extremely large, Unreasonable
52
15
1.4194510174Table 8
Selected individuals.
Selected individuals
X
c
X
c
c
Safety factor
*
51
0
1.3317465878
*
50
0
1.3425306677
*
55
0
1.3332021645
*
52
0
1.3264516064
*
53
0
1.3254640532
57
0
1.3503481321
52
11
1.3501275295Table 9
Crossover results.
X
c
X
c
c
Safety factor
51
11
1.3556896331
50
11
1.3672906676
55
11
1.3547892263
*
53
11
1.3487011763
57
11
1.3696385778Table 10
Mutation results.
X
c
X
c
c
Safety factor
54
21
1.6711004325
*
54
0
1.3279271582
54
13
1.3734897577
54
−10
1.3595160519
60
−10
1.4074551938Table 11
Selected individuals.
Selected individuals
X
c
X
c
c
Safety factor
51
0
1.3317465878
50
0
1.3425306677
55
0
1.3332021645
52
0
1.3264516064
*
53
0
1.3254640532
53
11
1.3487011763
54
0
1.3279271582Table 12
Critical failure surface and minimum safety factor.
X
c
X
c
c
Safety factor
Completed
53
0
1.3254640532


## 3.4. Location of the Critical Failure Surface and Safety Factor with GA Procedure

The example was solved with foregoing GA procedure. The minimum safety factor was 1.325 withXc=53 m and Xcc=0. The corresponding slip circle center is at (0, 68.8 m) and the radius is 68.8 m.


## 4. Search the Critical Slip Surface with Genetic-Traversal Random Search Method While Determining the Safety Factor with Analytical Method

To solve the engineering problem in Section2, this part will search the critical failure surface with Genetic-Traversal Random Search Method while determining the safety factor with analytical method. This part is realized with computer automatic search program.


### 4.1. Analytical Method to Determine the Slope Safety Factor in the Above-Mentioned Slope Example in Section2 [22]

With Fellenius’ method, according to Zhang (1987), the analytical solution to give the safety factor in Figure3 is
(2)k=γ·tgφ[N]+c[L]γ[T],
where [N], [L], and [T] were given as
(3)N={[4r2-y2]r2-y2+[4r2-(h-y)2]d×r2-(h-y)2+1m(2r2+x2)r2-x2f-1m[2r2+(mh-x)2]r2-(mh-x)2}+r2{yarcsinr2-y2rddfidd-(h-y)arcsinr2-(h-y)2rddfidd+xmarcsinxr-mh-xmarcsinmh-xrr2-(h-y)2r},T=16r[3hr2-y3-(h-y)3-x3m-(mh-x)3m],L=r[arcsinr2-y2r+arcsinr2-(h-y)2r],
where, in Figure 3, P(x,y) is the potential failure circle center, r is the circle radius, m is the slope ratio, h is the slope height, γ is the slope soil unit weight, φ is the soil internal friction angle, and c is slope soil cohesion.Figure 3
Analytical method to determine the slope safety factor.


### 4.2. Genetic-Traversal Random Search Method

The slope stability problem example in Figure4 is just the engineering problem in Section 2. Inspired by the genetic algorithm, the potential failure circle is represented with points A, B, and C in Figure 4. The coordinates of A, B, and C are (a,25), (0,b), and (c,0), respectively. So, in fact, the parameters a, b, and c can represent the potential failure circle. In a novel Fortran program, points A, B, and C are varied randomly and helped with random number generator subprogram. However, points A, B, and C can only vary in a certain region with boundary. Each group of {a,b,c} gives a safety factor by (2). With random number generator subprogram and loop program, enough groups of {a,b,c} are generated. Inspired by the genetic algorithm, the relative low safety factor and corresponding {a,b,c} are saved after each comparison between the old potential failure circle and the new generated potential failure circle and helped with the random number generator subprogram. After enough times of iterations set by the user, the minimum safety factor and corresponding {a,b,c} will be determined.Figure 4
Potential failure circle center and radius determined and represented with pointsA, B, and C.The safety factor and failure circle determination program developed in Silverfrost FTN95 was presented in the Appendix. In fact, the computer-aided genetic algorithm of the program presented in the Appendix only utilizes the mutation and selection thought of the traditional genetic algorithm. Crossover is omitted due to the difficulty in computer program realization and compensated with numerous random candidates due to mutation. In fact, genetic algorithm (GA) is a random search method based on the biological evolution law.


### 4.3. Results of the Program in the Appendix according to Genetic-Traversal Random Search Method for the Above-Mentioned Slope Stability Problem Example

After 100,000 times of potential failure circles’ generation and selection, the obtained minimum safety factor is 0.648280, and the corresponding{a,b,c} is {-11.8283,32.7429,50.4410}.


## 4.1. Analytical Method to Determine the Slope Safety Factor in the Above-Mentioned Slope Example in Section2 [22]

With Fellenius’ method, according to Zhang (1987), the analytical solution to give the safety factor in Figure3 is
(2)k=γ·tgφ[N]+c[L]γ[T],
where [N], [L], and [T] were given as
(3)N={[4r2-y2]r2-y2+[4r2-(h-y)2]d×r2-(h-y)2+1m(2r2+x2)r2-x2f-1m[2r2+(mh-x)2]r2-(mh-x)2}+r2{yarcsinr2-y2rddfidd-(h-y)arcsinr2-(h-y)2rddfidd+xmarcsinxr-mh-xmarcsinmh-xrr2-(h-y)2r},T=16r[3hr2-y3-(h-y)3-x3m-(mh-x)3m],L=r[arcsinr2-y2r+arcsinr2-(h-y)2r],
where, in Figure 3, P(x,y) is the potential failure circle center, r is the circle radius, m is the slope ratio, h is the slope height, γ is the slope soil unit weight, φ is the soil internal friction angle, and c is slope soil cohesion.Figure 3
Analytical method to determine the slope safety factor.


## 4.2. Genetic-Traversal Random Search Method

The slope stability problem example in Figure4 is just the engineering problem in Section 2. Inspired by the genetic algorithm, the potential failure circle is represented with points A, B, and C in Figure 4. The coordinates of A, B, and C are (a,25), (0,b), and (c,0), respectively. So, in fact, the parameters a, b, and c can represent the potential failure circle. In a novel Fortran program, points A, B, and C are varied randomly and helped with random number generator subprogram. However, points A, B, and C can only vary in a certain region with boundary. Each group of {a,b,c} gives a safety factor by (2). With random number generator subprogram and loop program, enough groups of {a,b,c} are generated. Inspired by the genetic algorithm, the relative low safety factor and corresponding {a,b,c} are saved after each comparison between the old potential failure circle and the new generated potential failure circle and helped with the random number generator subprogram. After enough times of iterations set by the user, the minimum safety factor and corresponding {a,b,c} will be determined.Figure 4
Potential failure circle center and radius determined and represented with pointsA, B, and C.The safety factor and failure circle determination program developed in Silverfrost FTN95 was presented in the Appendix. In fact, the computer-aided genetic algorithm of the program presented in the Appendix only utilizes the mutation and selection thought of the traditional genetic algorithm. Crossover is omitted due to the difficulty in computer program realization and compensated with numerous random candidates due to mutation. In fact, genetic algorithm (GA) is a random search method based on the biological evolution law.


## 4.3. Results of the Program in the Appendix according to Genetic-Traversal Random Search Method for the Above-Mentioned Slope Stability Problem Example

After 100,000 times of potential failure circles’ generation and selection, the obtained minimum safety factor is 0.648280, and the corresponding{a,b,c} is {-11.8283,32.7429,50.4410}.


## 5. Compared with Other Solutions

In order to validate the analytical solution to give safety factor of a specified slip surface, Genetic-Traversal Random Search Method to search for the critical failure surface and the corresponding program presented in the Appendix, this part will solve the slope engineering problem in Section2 with other methods.


### 5.1. Solution of Searching the Critical Slip Surface with Fellenius’ Method While Determining the Safety Factor with Fellenius’ Method of Slices

To solve the engineering problem in Section2, this part will search the critical failure surface with Fellenius’ method while determining each safety factor with Fellenius’ method of slices.In this solution, safety factor is determined with Fellenius’ method of slices as in (1). Fellenius’ method of searching for the critical failure surface [20] was given as follows.If soil internal friction angleφ=0, 2D critical failure surface passes through slope toe A and can be determined by Figure 5 and Table 13. In Figure 5, the critical failure surface circle center O can be determined by angles β1 and β2 which can be determined by slope angle α as in Table 13. Angle β1 is the angle between line AO and slope surface line, while β2 is the angle between line OB and slope horizontal top surface line BC. Point B is the intersection between slope surface line AB and slope horizontal top surface line BC.Table 13
Determination ofβ1 and β2 with slope angle α.
Slope angleα
Slope ratio 1 : m
β
1
β
2
60°
1 : 0.58
29°
40°
45°
1 : 1.0
28°
37°
33°41′
1 : 1.5
26°
35°
26°34′
1 : 2.0
25°
35°
18°26′
1 : 3.0
26°
35°
14°02′
1 : 4.0
25°
36°
11°19′
1 : 5.0
25°
39°Figure 5
Determination of 2D potential failure surfaces when internal friction angleφ=0.If soil internal friction angleφ>0, 2D critical failure surface passes through slope toe and can be determined by Figure 6. In Figure 6, point E is determined by angles β1 and β2 which can be determined by slope angle α as in Table 13. The critical failure surface circle center may be on the extension line of the line DE. You can try many points on the line DE as the critical failure surface circle center candidate like O1 and O4 on the line DE. If a point Ox on the line DE is found to be the point which gives the minimum slope safety factor, then draw a line FG perpendicular to the line DE through the point Ox. Then you can try many points on the line FG as the critical failure surface circle center candidate like O1′, O2′, O3′, and O4′. If a point on the line FG gives the minimum slope safety factor, this point means the one that gives the final most minimum safety factor of the studied slope.Figure 6
Determination of 2D potential failure surfaces when internal friction angleφ>0 [20].The determined minimum safety factor with Fellenius’ method is 1.320, while the 2D critical failure surface circle center is 4.5 m, 57.776 m, in thex-y coordinate system of Figure 2 and the radius is 57.951 m. The corresponding Xc in Figure 2 is 52.292 m.


### 5.2. Solution with Slope/w Software

To solve the engineering problem in Section2, this part will determine the safety factor with slope/w software.With the Ordinary method, the minimum safety factor to the example slope is 1.328. The corresponding critical failure surface is presented in Figure7. With the Bishop method, the minimum safety factor to the example slope is 1.390. The corresponding critical failure surface is presented in Figure 8. With the Janbu method, the minimum safety factor to the example slope is 1.316. The corresponding critical failure surface is presented in Figure 9. With the Morgenstern-Price method, the minimum safety factor to the example slope is 1.389. The corresponding critical failure surface is presented in Figure 10. With the Spencer method, the minimum safety factor to the example slope is 1.389. The corresponding critical failure surface is presented in Figure 11. With the GLE method, the minimum safety factor to the example slope is 1.389. The corresponding critical failure surface is presented in Figure 12. With the Janbu generalized method, after solving and analyzing, then after selecting the critical slip surface with a safety factor of 1.389 in the slope/w software, Figure 13 appears and the “minimum factor of safety” shows that its value is 1.385. 1.385 is not identical with 1.389 which is a little weird. The minimum safety factor determined by foregoing GA procedure is 1.325. After compared with slope/w software, foregoing GA procedure employed to search the critical failure surface is reasonable and applicable.Figure 7
Critical failure surface with the Ordinary method.Figure 8
Critical failure surface with the Bishop method.Figure 9
Critical failure surface with the Janbu method.Figure 10
Critical failure surface with the Morgenstern-Price method.Figure 11
Critical failure surface with the Spencer method.Figure 12
Critical failure surface determined with the GLE method.Figure 13
Critical failure surface determined with the Janbu generalized method.


### 5.3. Solution with Ansys Software

To solve the engineering problem in Section2, this part will determine the safety factor with Ansys software [23]. The slope has two layers which is layer 1 and layer 2 in Figure 14. Layer 1 is clay and Layer 2 is bed rock. The slope layer 1’s soil modulus of elasticity is assumed to be 2.0E7 N/m2. The slope layer 1’s soil Poisson’s ratio is assumed to be 0.3. The slope layer 1’s soil density is assumed to be 2040.8 Kg/m3. The slope layer 1’s soil cohesion is 10000 Pa and friction angle is 26.6 degrees. The slope layer 2’s soil modulus of elasticity is assumed to be 3.2E10 N/m2. The slope layer 2’s soil Poisson’s ratio is assumed to be 0.24. The slope layer 2’s soil density is assumed to be 2700 Kg/m3.Figure 14
Studied region for the engineering problem in Section2 treated with Ansys.The slope stability analysis problem is regarded as a plain strain problem. The left and right boundaries are restricted horizontally. The bottom boundary is restricted both horizontally and vertically. With the drucker-prager model, as the constitutive model, and with shear strength reduction method based on the finite element analysis, the slope in Figure14 is analyzed. Assume that the real cohesion and internal friction angle of a slope are c0 and φ0, respectively. In the shear strength reduction method, when safety factor is SF, the reduced cohesion and friction angle for analysis are c0/SF and φ0/SF.The Drucker-Prager yield criterion is [24, 25]
(4)AI1+J2-B≤0,
where I1=σ1+σ2+σ3,J2=(1/6)[(σ1-σ2)2+(σ2-σ3)2+(σ3-σ1)2].If we assume that the Drucker-Prager yield surface touches on the interior of the Mohr-Coulomb yield surface, then the expressions [26–28] are
(5)A=2sinφ33+sinφ,B=6·c·cos⁡φ33+sinφ.If the Drucker-Prager yield surface passes through the external apexes of the Mohr-Coulomb yield surface, then [26, 28, 29]
(6)A=2sinφ33-sinφ,B=6·c·cos⁡φ33-sinφ,
where c is cohesion and φ is internal friction angle.With the drucker-prager model as the constitutive model to analyze the slope under only self-weight in Figure14, the flow rule which describes the relationship between the plastic potential function and the plastic strain could be found in [23, 30–34]. The incremental elastic-plastic stress-strain relationship and the corresponding elastic-plastic matrix could be found in [23, 35].The results were presented as follows in Figures15, 16, and 17.No von Mises plastic strain.
(a)
Safety factor equals 0.7
(b)
Safety factor equals 0.72
(c)
Safety factor equals 0.73von Mises plastic strain occurs and develops.
(a)
Safety factor equals 0.74
(b)
Safety factor equals 0.75
(c)
Safety factor equals 0.8
(d)
Safety factor equals 1.0
(e)
Safety factor equals 1.6
(f)
Safety factor equals 1.8Figure 17
von Mises plastic strain runs through from slope toe to top surface when safety factor equals 2.0.When safety factor is from 0.7 to 0.73, there is no von Mises plastic strain in slope in Figure15. When safety factor is 0.74, there is local plastic strain occurring in slope in Figure 16. When safety factor is 2.0, von Mises plastic strain runs through from slope toe to top surface in Figure 17.According to Chen (1975) and Niu (2009), Figure16 gives lower bound solutions of slope safety factor which are from 0.74 to 1.8. And Figure 17 where von Mises plastic strain runs through from slope toe to top surface gives an upper bound solution of slope safety factor which is 2.0. So the true slope safety factor is likely from 1.8 to 2.0.


### 5.4. Comparisons and Discussions

The obtained minimum safety factor for the above slope stability problem example with Genetic-Traversal Random Search Method is so low when compared with the other methods like slope/w software. This may be due to the fact that the analytical solution is more accurate than Fellenius’ slices method. This may also be due to the power of the computer to realize the Genetic-Traversal Random Search Method in the Appendix. The Genetic-Traversal Random Method uses random pick to utilize mutation. Validation of these conclusions will be investigated in the future with more numeric tests.However the obtained minimum safety factor with Genetic-Traversal Random Search Method is very close to the lower bound solutions of slope safety factor given by the Ansys software.After computation, there is plastic strain in layer 2 region in some pictures of Figures16 and 17. This is unreasonable since layer 2 is defined as elastic region in the analysis with Ansys. This phenomenon will be investigated in the future.


## 5.1. Solution of Searching the Critical Slip Surface with Fellenius’ Method While Determining the Safety Factor with Fellenius’ Method of Slices

To solve the engineering problem in Section2, this part will search the critical failure surface with Fellenius’ method while determining each safety factor with Fellenius’ method of slices.In this solution, safety factor is determined with Fellenius’ method of slices as in (1). Fellenius’ method of searching for the critical failure surface [20] was given as follows.If soil internal friction angleφ=0, 2D critical failure surface passes through slope toe A and can be determined by Figure 5 and Table 13. In Figure 5, the critical failure surface circle center O can be determined by angles β1 and β2 which can be determined by slope angle α as in Table 13. Angle β1 is the angle between line AO and slope surface line, while β2 is the angle between line OB and slope horizontal top surface line BC. Point B is the intersection between slope surface line AB and slope horizontal top surface line BC.Table 13
Determination ofβ1 and β2 with slope angle α.
Slope angleα
Slope ratio 1 : m
β
1
β
2
60°
1 : 0.58
29°
40°
45°
1 : 1.0
28°
37°
33°41′
1 : 1.5
26°
35°
26°34′
1 : 2.0
25°
35°
18°26′
1 : 3.0
26°
35°
14°02′
1 : 4.0
25°
36°
11°19′
1 : 5.0
25°
39°Figure 5
Determination of 2D potential failure surfaces when internal friction angleφ=0.If soil internal friction angleφ>0, 2D critical failure surface passes through slope toe and can be determined by Figure 6. In Figure 6, point E is determined by angles β1 and β2 which can be determined by slope angle α as in Table 13. The critical failure surface circle center may be on the extension line of the line DE. You can try many points on the line DE as the critical failure surface circle center candidate like O1 and O4 on the line DE. If a point Ox on the line DE is found to be the point which gives the minimum slope safety factor, then draw a line FG perpendicular to the line DE through the point Ox. Then you can try many points on the line FG as the critical failure surface circle center candidate like O1′, O2′, O3′, and O4′. If a point on the line FG gives the minimum slope safety factor, this point means the one that gives the final most minimum safety factor of the studied slope.Figure 6
Determination of 2D potential failure surfaces when internal friction angleφ>0 [20].The determined minimum safety factor with Fellenius’ method is 1.320, while the 2D critical failure surface circle center is 4.5 m, 57.776 m, in thex-y coordinate system of Figure 2 and the radius is 57.951 m. The corresponding Xc in Figure 2 is 52.292 m.


## 5.2. Solution with Slope/w Software

To solve the engineering problem in Section2, this part will determine the safety factor with slope/w software.With the Ordinary method, the minimum safety factor to the example slope is 1.328. The corresponding critical failure surface is presented in Figure7. With the Bishop method, the minimum safety factor to the example slope is 1.390. The corresponding critical failure surface is presented in Figure 8. With the Janbu method, the minimum safety factor to the example slope is 1.316. The corresponding critical failure surface is presented in Figure 9. With the Morgenstern-Price method, the minimum safety factor to the example slope is 1.389. The corresponding critical failure surface is presented in Figure 10. With the Spencer method, the minimum safety factor to the example slope is 1.389. The corresponding critical failure surface is presented in Figure 11. With the GLE method, the minimum safety factor to the example slope is 1.389. The corresponding critical failure surface is presented in Figure 12. With the Janbu generalized method, after solving and analyzing, then after selecting the critical slip surface with a safety factor of 1.389 in the slope/w software, Figure 13 appears and the “minimum factor of safety” shows that its value is 1.385. 1.385 is not identical with 1.389 which is a little weird. The minimum safety factor determined by foregoing GA procedure is 1.325. After compared with slope/w software, foregoing GA procedure employed to search the critical failure surface is reasonable and applicable.Figure 7
Critical failure surface with the Ordinary method.Figure 8
Critical failure surface with the Bishop method.Figure 9
Critical failure surface with the Janbu method.Figure 10
Critical failure surface with the Morgenstern-Price method.Figure 11
Critical failure surface with the Spencer method.Figure 12
Critical failure surface determined with the GLE method.Figure 13
Critical failure surface determined with the Janbu generalized method.


## 5.3. Solution with Ansys Software

To solve the engineering problem in Section2, this part will determine the safety factor with Ansys software [23]. The slope has two layers which is layer 1 and layer 2 in Figure 14. Layer 1 is clay and Layer 2 is bed rock. The slope layer 1’s soil modulus of elasticity is assumed to be 2.0E7 N/m2. The slope layer 1’s soil Poisson’s ratio is assumed to be 0.3. The slope layer 1’s soil density is assumed to be 2040.8 Kg/m3. The slope layer 1’s soil cohesion is 10000 Pa and friction angle is 26.6 degrees. The slope layer 2’s soil modulus of elasticity is assumed to be 3.2E10 N/m2. The slope layer 2’s soil Poisson’s ratio is assumed to be 0.24. The slope layer 2’s soil density is assumed to be 2700 Kg/m3.Figure 14
Studied region for the engineering problem in Section2 treated with Ansys.The slope stability analysis problem is regarded as a plain strain problem. The left and right boundaries are restricted horizontally. The bottom boundary is restricted both horizontally and vertically. With the drucker-prager model, as the constitutive model, and with shear strength reduction method based on the finite element analysis, the slope in Figure14 is analyzed. Assume that the real cohesion and internal friction angle of a slope are c0 and φ0, respectively. In the shear strength reduction method, when safety factor is SF, the reduced cohesion and friction angle for analysis are c0/SF and φ0/SF.The Drucker-Prager yield criterion is [24, 25]
(4)AI1+J2-B≤0,
where I1=σ1+σ2+σ3,J2=(1/6)[(σ1-σ2)2+(σ2-σ3)2+(σ3-σ1)2].If we assume that the Drucker-Prager yield surface touches on the interior of the Mohr-Coulomb yield surface, then the expressions [26–28] are
(5)A=2sinφ33+sinφ,B=6·c·cos⁡φ33+sinφ.If the Drucker-Prager yield surface passes through the external apexes of the Mohr-Coulomb yield surface, then [26, 28, 29]
(6)A=2sinφ33-sinφ,B=6·c·cos⁡φ33-sinφ,
where c is cohesion and φ is internal friction angle.With the drucker-prager model as the constitutive model to analyze the slope under only self-weight in Figure14, the flow rule which describes the relationship between the plastic potential function and the plastic strain could be found in [23, 30–34]. The incremental elastic-plastic stress-strain relationship and the corresponding elastic-plastic matrix could be found in [23, 35].The results were presented as follows in Figures15, 16, and 17.No von Mises plastic strain.
(a)
Safety factor equals 0.7
(b)
Safety factor equals 0.72
(c)
Safety factor equals 0.73von Mises plastic strain occurs and develops.
(a)
Safety factor equals 0.74
(b)
Safety factor equals 0.75
(c)
Safety factor equals 0.8
(d)
Safety factor equals 1.0
(e)
Safety factor equals 1.6
(f)
Safety factor equals 1.8Figure 17
von Mises plastic strain runs through from slope toe to top surface when safety factor equals 2.0.When safety factor is from 0.7 to 0.73, there is no von Mises plastic strain in slope in Figure15. When safety factor is 0.74, there is local plastic strain occurring in slope in Figure 16. When safety factor is 2.0, von Mises plastic strain runs through from slope toe to top surface in Figure 17.According to Chen (1975) and Niu (2009), Figure16 gives lower bound solutions of slope safety factor which are from 0.74 to 1.8. And Figure 17 where von Mises plastic strain runs through from slope toe to top surface gives an upper bound solution of slope safety factor which is 2.0. So the true slope safety factor is likely from 1.8 to 2.0.


## 5.4. Comparisons and Discussions

The obtained minimum safety factor for the above slope stability problem example with Genetic-Traversal Random Search Method is so low when compared with the other methods like slope/w software. This may be due to the fact that the analytical solution is more accurate than Fellenius’ slices method. This may also be due to the power of the computer to realize the Genetic-Traversal Random Search Method in the Appendix. The Genetic-Traversal Random Method uses random pick to utilize mutation. Validation of these conclusions will be investigated in the future with more numeric tests.However the obtained minimum safety factor with Genetic-Traversal Random Search Method is very close to the lower bound solutions of slope safety factor given by the Ansys software.After computation, there is plastic strain in layer 2 region in some pictures of Figures16 and 17. This is unreasonable since layer 2 is defined as elastic region in the analysis with Ansys. This phenomenon will be investigated in the future.


## 6. Conclusions

This paper intends to determine a cohesive soil slope safety factor with Fellenius’ method, while the 2D critical failure surface is searched with GA. The 2D critical failure surface is represented with real-encoded chromosomes which are potential critical surface locations variablesXc and Xcc. GA procedure for searching critical failure surface proceeds with hand calculations. If for future computer automatic search program with GA, program code for inheritance, mutation, selection and crossover, program code for random numbers, and program code for search interval, boundaries will be needed. The minimum safety factor of 1.325 determined by foregoing GA procedure to search the critical slip surface is very close to the minimum safety factor of 1.320 determined by Fellenius’ critical slip surface method. After compared with slope/w software, the proposed foregoing GA procedure employed to search the critical failure surface is reasonable, applicable, and effective.At last, a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is made, while in the program random numbers generated by computer and search boundaries are included. The Genetic-Traversal Random Method uses random pick to utilize mutation. In the program, the slope safety factor is given by analytical solution rather than slices method. Results indicate that the new computer automatic search program can give very low safety factor which is about half of the foregoing ones. This may be due to the fact that the analytical solution is more accurate than Fellenius’ slices method. This may also be due to the power of the random number generation subprogram, computer operation speed, and Genetic-Traversal Random Method. Further validation of the results will be investigated in the future. However the obtained minimum safety factor with Genetic-Traversal Random Search Method is very close to the lower bound solutions of slope safety factor given by the Ansys software.



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*Source: 950531-2014-03-17.xml*