The Scientific World Journal

The Scientific World Journal / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 950531 | 13 pages | https://doi.org/10.1155/2014/950531

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

Academic Editor: B. F. Yousif
Received14 Sep 2013
Accepted02 Feb 2014
Published17 Mar 2014

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.

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 [13]. 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 [46].

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 [713].

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 weight is 20 KN/m3. The soil internal friction angle 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 Section 2, 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 Figure 1, Fellenius’ method of slices divides the slope into several slices [3]. Using moment equilibrium, the slope safety factor SF in Figure 1 is where and are the soil slope slice cohesion and internal friction angle. is the soil slope slice self-gravity. is the soil slope slice slip circular arc length. is the angle between soil slope slice slip surface tangent line and the horizontal line.

3.2. Searching the Critical Failure Surface with GA

The assumed 2D slope failure surface is circular determined by two variables and in Figure 2. is the abscissa of a point on slope top surface. If is determined, the 2D critical slip surface circle center with abscissa must lie on the perpendicular bisector of the straight line from the point of to the slope toe . is the abscissa of the critical slip surface circle center. So if is given, then the critical slip surface circle center can be determined. Altogether, if and are given, 2D circular slope slip surface is determined.

However, in the next section, all the computations according to GA are made by hand calculations. If, for computer simulation, “interval" is as and as in Figure 2 and random numbers of computer function can help for automatic generation of variables and in Figure 2, search regions or boundaries for and must be defined with the definition of the limits of and . 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 Figure 1 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 described before. The evolution starts from a population of randomly generated individuals as 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 with large safety factor is deleted and the individual 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 Figure 2 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 and are meters. Evaluation of individuals in Table 1 now begins. The evaluation standard is that the individual with large safety factor is deleted and the individual 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 with large safety factor is deleted and the individual 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 with large safety factor is deleted and the individual 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 with large safety factor is deleted and the individual 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.


Safety factor

51252.720523521
5141Safety factor is extremely large, Unreasonable
*5101.3317465878
*60−101.4074551938
*60251.886105777
6035Safety factor is extremely large, Unreasonable
*100−202.2757670756
100212.3067661886
10070Safety factor is extremely large, Unreasonable


Selected individuals Safety factor

*5101.3317465878
60−101.4074551938
60251.886105777
100−202.2757670756


Safety factor

*51−101.3799239885
51252.720523521
51−201.4447027254
*6001.3879596689
60−201.4333540217
10002.272574082
100−102.2729514535
100252.3308411862


Safety factor

*5001.3425306677
50−101.4003909843
51211.7627329758
*5501.3332021645
*55−101.3615830364
55201.6493380867


Selected individuals Safety factor

*5101.3317465878
51−101.3799239885
6001.3879596689
*5001.3425306677
*5501.3332021645
55−101.3615830364


Safety factor

5101.3317465878
51−101.3799239885
6001.3879596689
5001.3425306677
5501.3332021645
55−101.3615830364


Safety factor

*5201.3264516064
52−101.3675053085
*5301.3254640532
53−101.3611610322
*5701.3503481321
57−101.3741239435
*52111.3501275295
52252.5159145338
52100Safety factor is extremely large, Unreasonable
52151.4194510174


Selected individuals Safety factor

*5101.3317465878
*5001.3425306677
*5501.3332021645
*5201.3264516064
*5301.3254640532
5701.3503481321
52111.3501275295


Safety factor

51111.3556896331
50111.3672906676
55111.3547892263
*53111.3487011763
57111.3696385778


Safety factor

54211.6711004325
*5401.3279271582
54131.3734897577
54−101.3595160519
60−101.4074551938


Selected individuals Safety factor

5101.3317465878
5001.3425306677
5501.3332021645
5201.3264516064
*5301.3254640532
53111.3487011763
5401.3279271582


Safety factor

Completed5301.3254640532

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 with  m and . 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 Section 2, 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 Section 2 [22]

With Fellenius’ method, according to Zhang (1987), the analytical solution to give the safety factor in Figure 3 is where , , and were given as where, in Figure 3, is the potential failure circle center, is the circle radius, is the slope ratio, is the slope height, is the slope soil unit weight, is the soil internal friction angle, and is slope soil cohesion.

4.2. Genetic-Traversal Random Search Method

The slope stability problem example in Figure 4 is just the engineering problem in Section 2. Inspired by the genetic algorithm, the potential failure circle is represented with points , , and in Figure 4. The coordinates of , , and are , , and , respectively. So, in fact, the parameters , , and can represent the potential failure circle. In a novel Fortran program, points , , and are varied randomly and helped with random number generator subprogram. However, points , , and can only vary in a certain region with boundary. Each group of gives a safety factor by (2). With random number generator subprogram and loop program, enough groups of are generated. Inspired by the genetic algorithm, the relative low safety factor and corresponding 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 will be determined.

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 is .

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 Section 2 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 Section 2, 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 angle , 2D critical failure surface passes through slope toe and can be determined by Figure 5 and Table 13. In Figure 5, the critical failure surface circle center can be determined by angles and which can be determined by slope angle as in Table 13. Angle is the angle between line and slope surface line, while is the angle between line and slope horizontal top surface line . Point is the intersection between slope surface line and slope horizontal top surface line .


Slope angle Slope ratio 1 : m

60°1 : 0.5829°40°
45°1 : 1.028°37°
33° 1 : 1.526°35°
26° 1 : 2.025°35°
18° 1 : 3.026°35°
14° 1 : 4.025°36°
11° 1 : 5.025°39°

If soil internal friction angle , 2D critical failure surface passes through slope toe and can be determined by Figure 6. In Figure 6, point is determined by angles and which can be determined by slope angle as in Table 13. The critical failure surface circle center may be on the extension line of the line . You can try many points on the line as the critical failure surface circle center candidate like and on the line . If a point on the line is found to be the point which gives the minimum slope safety factor, then draw a line perpendicular to the line through the point . Then you can try many points on the line as the critical failure surface circle center candidate like , , , and . If a point on the line gives the minimum slope safety factor, this point means the one that gives the final most minimum safety factor of the studied slope.

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 the coordinate system of Figure 2 and the radius is 57.951 m. The corresponding in Figure 2 is 52.292 m.

5.2. Solution with Slope/w Software

To solve the engineering problem in Section 2, 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 Figure 7. 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.

5.3. Solution with Ansys Software

To solve the engineering problem in Section 2, 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.

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 Figure 14 is analyzed. Assume that the real cohesion and internal friction angle of a slope are and , respectively. In the shear strength reduction method, when safety factor is SF, the reduced cohesion and friction angle for analysis are and .

The Drucker-Prager yield criterion is [24, 25] where .

If we assume that the Drucker-Prager yield surface touches on the interior of the Mohr-Coulomb yield surface, then the expressions [2628] are

If the Drucker-Prager yield surface passes through the external apexes of the Mohr-Coulomb yield surface, then [26, 28, 29] where is cohesion and is internal friction angle.

With the drucker-prager model as the constitutive model to analyze the slope under only self-weight in Figure 14, the flow rule which describes the relationship between the plastic potential function and the plastic strain could be found in [23, 3034]. 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 Figures 15, 16, and 17.

When safety factor is from 0.7 to 0.73, there is no von Mises plastic strain in slope in Figure 15. 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), Figure 16 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 Figures 16 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 variables and . 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.

Appendix

Safety factor and failure circle determination program developed in Silverfrost FTN95double precision seedreal nrndlreal N,L,newsafetyfactorsafetyfactor=100000seed=5.0gama=20tanphi=0.5cohesion=10h=25m=2do 10 I=1,100000a=0b=25c=50rdn=nrndl(seed)a=-20*rdnrdn=nrndl(seed)b=25+20*rdnrdn=nrndl(seed)c=50+20*rdnif(b==25) b=25.01x=(c*c/(2*b)+(25+b)/2-b/2+a*a/(50-2*b))/(c/b+a/(25-b))y=b/2+(c/b)*(x-c/2)r=sqrt((x-c)*(x-c)+y*y)AA=(4*r*r-y*y)*sqrt(r*r-y*y)BB=(4*r*r-(h-y)*(h-y))*sqrt(r*r-(h-y)*(h-y))CC=(1/m)*(2*r*r+x*x)*sqrt(r*r-x*x)DD=(1/m)*(2*r*r+(m*h-x)*(m*h-x))*sqrt(r*r-(m*h-x)*(m*h-x))EE=y*asin((sqrt(r*r-y*y))/r)FF=(h-y)*asin((sqrt(r*r-(h-y)*(h-y)))/r)GG=(x/m)*asin(x/r)-((m*h-x)/m)*asin((m*h-x)/r)N=(1/(6*r))*(AA+BB+CC-DD)+(r/2)*(EE-FF+GG)T=(1/(6*r))*(3*h*r*r-y*y*y-(h-y)*(h-y)*(h-y)-x*x*x/m-(m*h-x)*(m*h-x)*(m*h-x)/m)L=r*(asin((sqrt(r*r-y*y))/r)+asin((sqrt(r*r-(h-y)*(h-y)))/r))newsafetyfactor=(gama*tanphi*N+cohesion*L)/(gama*T)if(newsafetyfactor<safetyfactor) safetyfactor=newsafetyfactorwrite(*,*)safetyfactorwrite(*,*)awrite(*,*)bwrite(*,*)c10 continueend programreal function nrndl(seed)double precision S,U,V,seedS=65536.0U=2053.0V=13849.0M=seed/Sseed=seed-M*Sseed=U*seed+VM=seed/Sseed=seed-M*Snrndl=seed/Sreturnend

Conflict of Interests

The author declares that he has no financial or personal relationships with other people or organizations that could inappropriately influence his work. The author of this paper has chosen not to furnish the paper and its readers with information that might present a potential conflict of interests.

Acknowledgment

The author acknowledges the Scientific Research Starting Funds at Liaoning Technical University (no. 11-415).

References

  1. D. G. Fredlund and J. Krahn, “Comparison of slope stability methods of analysis,” Canadian Geotechnical Journal, vol. 14, no. 3, pp. 429–439, 1977. View at: Google Scholar
  2. K. S. Li, “A unified solution scheme for slope stability analysis,” in Proceedings of the 5th International Symposium on Landslides, New Zealand, vol. 1, pp. 481–486, Balkema, Rotterdam, The Netherlands, 1992. View at: Google Scholar
  3. W.-J. Niu, W.-M. Ye, S.-G. Liu, and H.-T. Yu, “Limit analysis of a soil slope considering saturated-unsaturated seepage,” Rock and Soil Mechanics, vol. 30, no. 8, pp. 2477–2482, 2009. View at: Google Scholar
  4. W. F. Chen, Limit Analysis and Soil Plasticity, Elsevier, Amsterdam, The Netherlands, 1975.
  5. R. L. Michalowski, “Slope stability analysis: a kinematical approach,” Geotechnique, vol. 45, no. 2, pp. 283–293, 1995. View at: Google Scholar
  6. I. B. Donald and Z. Y. Chen, “Slope stability analysis by the upper bound approach: fundamentals and methods,” Canadian Geotechnical Journal, vol. 34, no. 6, pp. 853–862, 1997. View at: Google Scholar
  7. J. P. Sun, J. C. Li, and Q. Q. Liu, “Search for critical slip surface in slope stability analysis by spline-based GA method,” Journal of Geotechnical and Geoenvironmental Engineering, vol. 134, no. 2, pp. 252–256, 2008. View at: Publisher Site | Google Scholar
  8. R. Javadzadeh and E. Javadzadeh, “Locating critical failure surface in rock slope stability with hybrid model based on artificial immune system and Cellular Learning Automata (CLA-AIS),” World Academy of Science, Engineering and Technology, vol. 59, pp. 662–665, 2011. View at: Google Scholar
  9. F. N. G. Gitirana Jr. and D. G. Fredlund, “Analysis of transient embankment stability using the dynamic programming method,” in Proceedings of the 56th Canadian Geotechnical Conference, 2003. View at: Google Scholar
  10. H. T. V. Pham and D. G. Fredlund, “Dynamic programming method in slope stability computations,” in Proceedings of the 12th Asian Regional Conference on Soil Mechanics and Geotechnica Engineering, Singapore, August 2003. View at: Google Scholar
  11. H. T. V. Pham, D. G. Fredlund, and F. N. G. Gitirana Jr., “Slope stability analysis using dynamic prograamming combined with finite element stress analysis,” in Proceedings of the International Conference on the Management of the Land and Water Resources (MLWR '01), Hanoi, Vietnam, October 2001. View at: Google Scholar
  12. H. T. V. Pham and D. G. Fredlund, “The application of dynamic programming to slope stability analysis,” Canadian Geotechnical Journal, vol. 40, no. 4, pp. 830–847, 2003. View at: Publisher Site | Google Scholar
  13. M. R. Taha, M. Khajehzadeh, and A. El-Shafie, “Slope stability assessment using optimization techniques: an overview,” Electronic Journal of Geotechnical Engineering, vol. 15, pp. 1901–1915, 2010. View at: Google Scholar
  14. A. U. Rao and N. Sabhahit, “Genetic algorithm in stability of non-homogeneous slopes,” in Proceedings of the 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG '08), Goa, India, October 2008. View at: Google Scholar
  15. Y.-G. Nie, W.-Q. Liu, J.-Y. Shi, and W.-B. Zhao, “Application of accelerating genetic algorithm for embankment slope stability analysis,” China Journal of Highway and Transport, vol. 16, no. 4, p. 16, 2003. View at: Google Scholar
  16. Y.-P. Zhou, S.-J. Li, Y.-X. Liu, H.-L. Sun, and F. Jiang, “Searching critical failure surface in slope stability analysis with genetic algorithm,” Chinese Journal of Rock Mechanics and Engineering, vol. 24, pp. 5226–5230, 2005. View at: Google Scholar
  17. P. McCombie, “Critical failure surface location using Simple Genetic Algorithm and multiple wedge slope stability,” in Proceedings of the 1st International Symposium on Computational Geomechanics, pp. 900–909, Juan-les-Pins, France, May 2009. View at: Google Scholar
  18. K. Gavin and J. F. Xue, “Use of a genetic algorithm to perform reliability analysis of unsaturated soil slopes,” Geotechnique, vol. 59, no. 6, pp. 545–549, 2009. View at: Publisher Site | Google Scholar
  19. Wikipedia, “Genetic algorithm,” 2012, http://en.wikipedia.org/wiki/Genetic_algorithm. View at: Google Scholar
  20. China University of Geosciences, “Cohesive soil slope stability analysis,” http://course.cug.edu.cn/cug/soil_mechanics/COURSE/CHAPTER7/Chap7_3_5.htm. View at: Google Scholar
  21. J. E. Thomaz, “A general method for three dimensional slope stability analysis: informational report,” Tech. Rep., JTRP Technical Reports, Purdue University, 1986. View at: Google Scholar
  22. T. B. Zhang, “Earth slope stability analysis and geotechnology slope design,” Tech. Rep., Chengdu University of Science and Technology Press, 1987. View at: Google Scholar
  23. B. G. He, Application Examples in Civil Engineering with Ansys, China Water Power Press, Beijing, China, 2011.
  24. Y. Liu, Q. Yang, and L. Zhu, “Abutment stability analysis of arch dam based on 3D nonlinear finite element method,” Chinese Journal of Rock Mechanics and Engineering, vol. 27, supplement 1, pp. 3222–3228, 2008. View at: Google Scholar
  25. Y. R. Liu, Z. He, B. Li, and Q. Yang, “Slope stability analysis based on a multigrid method using a nonlinear 3D finite element model,” Frontiers of Structural and Civil Engineering, vol. 7, no. 1, pp. 24–31, 2013. View at: Google Scholar
  26. Wikipedia, “Drucker–Prager yield criterion,” 2013, http://en.wikipedia.org/wiki/Drucker%E2%80%93Prager_yield_criterion. View at: Google Scholar
  27. B. Doran, H. O. Koksal, Z. Polat, and C. Karakoc, “The use of “Drucker-Prager Criterion” in the analysis of reinforced concrete members by finite elements,” Teknik Dergi, vol. 9, no. 2, pp. 1617–1625, 1998. View at: Google Scholar
  28. B. Zoran, P. Verka, and M. Biljana, “Mathematical modeling of materially nonlinear problems in structural analyses (part I—theoretical fundamentals),” Facta Universitatis. Architecture and Civil Engineering, vol. 8, no. 1, pp. 67–78, 2010. View at: Publisher Site | Google Scholar
  29. Q. Yang, X. Chen, and W.-Y. Zhou, “A practical 3D elasto-plastic incremental method in FEM based on D-P yield criteria,” Chinese Journal of Geotechnical Engineering, vol. 24, no. 1, pp. 16–20, 2002. View at: Google Scholar
  30. G. D. Zhang, Soil Constitutive Model and Its Application in Engineering, Science Press, Beijing, China, 1995.
  31. D. Rakić, M. Živković, R. Slavković, and M. Kojić, “Stress integration for the Drucker-Prager material model without hardening using the Incremental Plasticity Theory,” Journal of the Serbian Society For Computational Mechanics, vol. 2, no. 1, pp. 80–89, 2008. View at: Google Scholar
  32. X. Y. Zhang, Soil and Rock Plastic Mechanics, China Communications Press, Beijing, China, 1993.
  33. Y. R. Zheng, Z. J. Shen, and X. N. Gong, The Principles of Geotechnical Plastic Mechanics, China Architecture and Building Press, Beijing, China, 2002.
  34. S. A. Akers, Two-dimensional finite element analysis of porous geomaterials at multikilobar stress levels [PhD thesis], Virginia Polytechnic Institute and State University, 2001.
  35. X.-H. Tan, J.-G. Wang, and Y. Wang, “Nonlinear finite element analysis of slope stability,” Rock and Soil Mechanics, vol. 29, no. 8, pp. 2047–2050, 2008. View at: Google Scholar

Copyright © 2014 Wen-jie Niu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

17184 Views | 3616 Downloads | 3 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.