Post on 11-May-2018
Research ArticleDetermination of Slope Safety Factor withAnalytical Solution and Searching Critical Slip Surface withGenetic-Traversal Random Method
Wen-jie Niu
College of Mechanics and Engineering Department Liaoning Technical University Fuxin Liaoning 123000 China
Correspondence should be addressed to Wen-jie Niu 11862590qqcom
Received 14 September 2013 Accepted 2 February 2014 Published 17 March 2014
Academic Editors K Dincer B Podgornik and B F Yousif
Copyright copy 2014 Wen-jie Niu This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In the current practice to determine the safety factor of a slope with two-dimensional circular potential failure surface one of thesearching methods for the critical slip surface is Genetic Algorithm (GA) while the method to calculate the slope safety factor isFelleniusrsquo slicesmethodHoweverGAneeds to be validatedwithmore numeric tests while Felleniusrsquo slicesmethod is just an approx-imate method like finite element methodThis paper proposed a newmethod to determine the minimum slope safety factor whichis the determination of slope safety factor with analytical solution and searching critical slip surface withGenetic-Traversal RandomMethodTheanalytical solution ismore accurate thanFelleniusrsquo slicesmethodTheGenetic-Traversal RandomMethoduses randompick to utilize mutation A computer automatic search program is developed for the Genetic-Traversal Random Method Aftercomparison with other methods like slopew software results indicate that the Genetic-Traversal Random Search Method can givevery low safety factorwhich is about half of the othermethodsHowever the obtainedminimum safety factorwithGenetic-TraversalRandom 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 equilibriummethods of analysis when studying slope stability problemsfor example Ordinary or Felleniusrsquo method (sometimesreferred to as the Swedish circle method or the conven-tional method) Simplified Bishop method Spencerrsquosmethod Janbursquos simplified method Janbursquos rigorous methodMorgenstern-Price method or unified solution scheme[1ndash3] In order to reduce the influence of the assumptionsmade in limit equilibriummethods on the factor of safety themethods of limit analysis based on the rigid body plasticitytheory were developed by Chen (1975) Michalowski (1995)and Donald and Chen (1997) These methods based on theupper bound theorem of limit analysis are generally referredto as the upper bound methods which give an upper boundsolution to the real value of the factor of safety [4ndash6]
In the current practice searching methods for the criticalslip surface is a central issue to slope stability analysis Previ-ous research employed the Variational Calculus the dynamic
programming alternating variablemethods theMonteCarlotechnique or the genetic algorithm (GA) into slope stabilityanalysis for critical surface identification [7ndash13]
In recent years genetic algorithm search procedure hasbeen used to locate the critical slip surface of homogeneousslopes It has been found that genetic algorithm is a robustsearch technique which often gives global solution [14 15]Numerical example shows that analyzingmethod of the slopestability based on the genetic algorithm is a global optimalprocedure that can overcome the drawbacks of local optimumwidely existing in classical searchingmethods and the result issatisfactory [16] Established slope stability analysis methodscopewell withmoderately noncircular shear surfaces and thesimple genetic algorithm (SGA) has been used successfully tofind the critical slip surface [17]
In the GA the parameters in the optimization problemare translated into chromosomes with a data string (binaryor real) A range of possible solutions is obtained from thevariable space and the fitness of these solutions is com-pared with some predetermined criteria If a solution is
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 950531 13 pageshttpdxdoiorg1011552014950531
2 The Scientific World Journal
not obtained a new population is created from the original(parent) chromosomes This is achieved using ldquocrossoverrdquoand ldquomutationrdquo operations Crossover involves gene exchangefrom two random (parent) solutions to form a child (newsolution)Mutation involves the random switching of a singlevariable in a chromosome and is used to maintain popula-tion diversity as the process converges towards a solution[18]
GA includes inheritance mutation selection andcrossover [19] One of the core techniques and advantages ofGA is that mutation can consider a wide range of possiblesolutions if natural evolution continues and never ends Theother advantage is that inheritance and crossover can save allthe examples and virtues of the past age and pass them intothe next generation to save time for the best choice [18 19]
However there are many unsolved problems about GAin slope stability analysis For example how to realize GAwith hand calculationmethod How to realize GAwith auto-matic search program What is the relationship between GAand traversal random search method How to improve theFellenius method of slices concerning that the slices methodis an approximate method like finite element method Infact all these problems are mathematical problems to findthe minimum safety factor of a slope These mathematicalproblems originated from the physical equations representingthe common law of nature in slopes (eg Mohr-Columncriterion of soil 2D circular slip surface of homogeneous clayslope and safety factor definition which is the moment ofsliding resistance divided by the moment of sliding force)The physical laws of nature can be found and validated withrepeated in situ or lab experiments to measure the physicalquantities and mathematical logic to reveal the relationshipHowever mathematical problems can only be solved withlogic deduction and validated with countless numeric tests
This paper first intends to determine a cohesive soil slopesafety factor with Felleniusrsquo slices method while the 2Dcritical failure surface is searched with GA The analysis usesreal-coded methods to encode the chromosomes with thevariables of potential critical surface locations The fitness ofeach chromosome is determined using the objective functionthat the resulting safety factors should be lower enough andthe fitness of all solutions is comparedwhile the chromosomeof large safety factors shall be deleted [18] However this partis realized with hand calculation
Then a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is madeThe Genetic-Traversal Random Search Method presented inthis paper only utilizes the mutation and selection thoughtof the traditional genetic algorithm Crossover is omitteddue to the difficulty in computer program realization andcompensated with numerous random candidates due tomutation The Genetic-Traversal Random Search Methodmakes a traversal search with random method In the pro-gram random numbers for random search are generatedby computer and search boundaries are included In theprogram each slope safety factor is given by analyticalsolution rather than slices method The safety factor andfailure circle determination program developed in SilverfrostFTN95 is presented in the Appendix At last the proposed
RR
Wi
li
120572i
x = Rsin120572i
Figure 1 Felleniusrsquo method of slices
Genetic-Traversal Random Search Method is compared withother solutions such as slopew software
2 A Slope Stability Problem Example
Acohesive soil slopewith its height 25meters has a slope ratioof 1 2 The soil unit weight 120574 is 20KNm3 The soil internalfriction angle 120593 is 266 degrees and cohesion is 10 KPA Theproblem now is to give the safety factor of the slope with a 2Dcircular failure surface
3 Search the Critical Slip Surface withGA Method While Determining the SafetyFactor with Felleniusrsquo Method of Slices withHand Calculation
To solve the engineering problem in Section 2 this partwill search the critical slip surface with GA Method whiledetermining the safety factor with Felleniusrsquo method of slices
31 The Slope Safety Factor with Felleniusrsquo Method of SlicesThe potential slip surface for clay slope is two dimensionaland a part of circle In order to determine the slope safetyfactor in Figure 1 Felleniusrsquo method of slices divides the slopeinto several slices [3] Using moment equilibrium the slopesafety factor SF in Figure 1 is
SF =sum (119888119894+ 119882119894cos120572119894tan120593119894) 119897119894
119882119894sin120572119894
(1)
where 119888119894and 120593
119894are the soil slope slice cohesion and internal
friction angle119882119894is the soil slope slice self-gravity 119897
119894is the soil
slope slice slip circular arc length 120572119894is the angle between soil
slope slice slip surface tangent line and the horizontal line
32 Searching the Critical Failure Surface with GA Theassumed 2D slope failure surface is circular determined bytwo variables 119883
119888and 119883
119888119888in Figure 2 119883
119888is the abscissa of a
point on slope top surface If119883119888is determined the 2D critical
slip surface circle center with abscissa 119883119888119888must lie on the
perpendicular bisector of the straight line from the point of119883119888to the slope toe 119874 119883
119888119888is the abscissa of the critical slip
surface circle center So if 119883119888119888is given then the critical slip
surface circle center can be determined Altogether if119883119888and
119883119888119888are given 2D circular slope slip surface is determined
The Scientific World Journal 3
L
x
y
A
R
S
Search region
R
Potential 2D critical slip surface
Search region for Xc
Slope toe OXc
for Xcc
Xcc
Figure 2 2D potential failure surfaces
However in the next section all the computations accord-ing to GA are made by hand calculations If for computersimulation ldquointervalrdquo is as 119871 and 119904 as in Figure 2 and randomnumbers of computer function can help for automatic gen-eration of variables 119883
119888and 119883
119888119888in Figure 2 search regions or
boundaries for119883119888and119883
119888119888must be definedwith the definition
of the limits of119883119888and119883
119888119888 The necessity of these boundaries
is evident because computer program must avoid generatingsurfaces out of the region of primary interest [21]
The searching process for the 2D critical failure surfaceas in Figure 1 uses techniques inspired by natural evolutionsuch as inheritance mutation selection and crossover [19]In a genetic algorithm a population of strings (called chro-mosomes or the genotype of the genome) which encodecandidate solutions (called individuals creatures or pheno-types) to an optimization problem is evolved toward bettersolutions [19] In this 2D critical failure surface searchingproblem the candidate solutions are represented as 119883
119888 119883119888119888
described before The evolution starts from a population ofrandomly generated individuals as 119883
119888 119883119888119888 and happens
in generations In each generation the fitness of everyindividual in the population is evaluatedmultiple individualsare stochastically selected from the current population (basedon their fitness) and modified (recombined and possiblyrandomlymutated) to form a new populationThe evaluationstandard is that the individual 119883
119888 119883119888119888 with large safety
factor is deleted and the individual 119883119888 119883119888119888with small safety
factor is reserved The new population is then used in thenext iteration of the algorithm [19]The algorithm terminateswhen a satisfactory fitness level has been reached for thepopulation which means that it is hard to lower safety factorwith iterations
33 Searching Process with GA In the GA with hand cal-culation method the potential failure surfaces for searchare restricted that they all pass slope toe as in Figure 2 forsimplifying the search task Search process for the critical slipsurface with genetic algorithm was presented from Table 1to Table 12 In these tables the minimum safety factors aremarked with lowast in each iterationThe units for119883
119888and119883
119888119888are
meters Evaluation of individuals in Table 1 now begins The
Table 1 A population of randomly generated individuals
119883119888
119883119888119888
Safety factor51 25 272052352151 41 Safety factor is extremely large Unreasonable
lowast 51 0 13317465878lowast 60 minus10 14074551938lowast 60 25 1886105777
60 35 Safety factor is extremely large Unreasonablelowast 100 minus20 22757670756
100 21 23067661886100 70 Safety factor is extremely large Unreasonable
Table 2 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factorlowast 51 0 13317465878
60 minus10 1407455193860 25 1886105777100 minus20 22757670756
Table 3 Crossover results
119883119888
119883119888119888
Safety factorlowast 51 minus10 13799239885
51 25 272052352151 minus20 14447027254
lowast 60 0 1387959668960 minus20 14333540217100 0 2272574082100 minus10 22729514535100 25 23308411862
evaluation standard is that the individual 119883119888 119883119888119888with large
safety factor is deleted and the individual 119883119888 119883119888119888with small
safety factor is reserved Selection result will be put in Table 2Crossover of selected individuals of Table 2 will be put inTable 3 Mutation begins and results will be put in Table 4Evaluation of all previous individuals marked with lowast beginsThe evaluation standard is that the individual 119883
119888 119883119888119888 with
large safety factor is deleted and the individual 119883119888 119883119888119888with
small safety factor is reserved Selection result will be put inTable 5 Crossover of selected individuals of Table 5 beginsCrossover result will be put in Table 6 Mutation begins andthe result will be put in Table 7 Evaluation of all previousindividuals marked with lowast begins The evaluation standardis that the individual 119883
119888 119883119888119888 with large safety factor is
deleted and the individual 119883119888 119883119888119888with small safety factor is
reserved Selection result will be put in Table 8 Crossover ofselected individuals of Table 8 begins Crossover result willbe put in Table 9 Mutation begins and result will be putin Table 10 Evaluation of all previous individuals markedwith lowast begins The evaluation standard is that the individual119883119888 119883119888119888with large safety factor is deleted and the individual
119883119888 119883119888119888 with small safety factor is reserved Selection result
will be put in Table 11 The GA procedure terminates when asatisfactory fitness level has been reached for the population
4 The Scientific World Journal
Table 4 Mutation results
119883119888
119883119888119888
Safety factorlowast 50 0 13425306677
50 minus10 1400390984351 21 17627329758
lowast 55 0 13332021645lowast 55 minus10 13615830364
55 20 16493380867
Table 5 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factorlowast 51 0 13317465878
51 minus10 1379923988560 0 13879596689
lowast 50 0 13425306677lowast 55 0 13332021645
55 minus10 13615830364
Table 6 Crossover results
119883119888
119883119888119888
Safety factor51 0 1331746587851 minus10 1379923988560 0 1387959668950 0 1342530667755 0 1333202164555 minus10 13615830364
Table 7 Mutation results
119883119888
119883119888119888
Safety factorlowast 52 0 13264516064
52 minus10 13675053085lowast 53 0 13254640532
53 minus10 13611610322lowast 57 0 13503481321
57 minus10 13741239435lowast 52 11 13501275295
52 25 2515914533852 100 Safety factor is extremely large Unreasonable52 15 14194510174
which means that it is hard to lower safety factor withiterations The final result is in Table 12
34 Location of the Critical Failure Surface and Safety Factorwith GA Procedure The example was solved with foregoingGA procedure The minimum safety factor was 1325 with119883119888= 53m and 119883
119888119888= 0 The corresponding slip circle center
is at (0 688m) and the radius is 688m
Table 8 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factorlowast 51 0 13317465878lowast 50 0 13425306677lowast 55 0 13332021645lowast 52 0 13264516064lowast 53 0 13254640532
57 0 1350348132152 11 13501275295
Table 9 Crossover results
119883119888
119883119888119888
Safety factor51 11 1355689633150 11 1367290667655 11 13547892263
lowast 53 11 1348701176357 11 13696385778
Table 10 Mutation results
119883119888
119883119888119888
Safety factor54 21 16711004325
lowast 54 0 1327927158254 13 1373489757754 minus10 1359516051960 minus10 14074551938
Table 11 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factor51 0 1331746587850 0 1342530667755 0 1333202164552 0 13264516064
lowast 53 0 1325464053253 11 1348701176354 0 13279271582
Table 12 Critical failure surface and minimum safety factor
119883119888
119883119888119888
Safety factorCompleted 53 0 13254640532
4 Search the Critical Slip Surface withGenetic-Traversal Random Search MethodWhile Determining the Safety Factor withAnalytical Method
To solve the engineering problem in Section 2 this partwill search the critical failure surface with Genetic-TraversalRandom Search Method while determining the safety factorwith analytical method This part is realized with computerautomatic search program
The Scientific World Journal 5
Y
XO
A
B C
D
EF
G
H
r
m1
h
X dX
P(X Y)
Y1 = h minus Xm
Y1 = h
Y1 = 0
d120572x120572x
M(X Y2)Y2 = y + radicr2 minus (X minus x)2
Figure 3 Analytical method to determine the slope safety factor
41 Analytical Method to Determine the Slope Safety Factor inthe Above-Mentioned Slope Example in Section 2 [22] WithFelleniusrsquo method according to Zhang (1987) the analyticalsolution to give the safety factor in Figure 3 is
119896 =120574 sdot 119905119892120593 [119873] + 119888 [119871]
120574 [119879] (2)
where [119873] [119871] and [119879] were given as
119873 = [41199032minus 1199102]radic1199032 minus 1199102 + [4119903
2minus (ℎ minus 119910)
2]
times radic1199032 minus (ℎ minus 119910)2+
1
119898(21199032+ 1199092)radic1199032 minus 1199092
minus1
119898[21199032+ (119898ℎ minus 119909)
2]radic1199032 minus (119898ℎ minus 119909)
2
+119903
2
119910 arcsinradic1199032 minus 1199102
119903
minus (ℎ minus 119910) arcsinradic1199032 minus (ℎ minus 119910)
2
119903
+119909
119898arcsin 119909
119903minus
119898ℎ minus 119909
119898arcsin 119898ℎ minus 119909
119903
119879 =1
6119903[3ℎ1199032minus 1199103minus (ℎ minus 119910)
3minus
1199093
119898minus
(119898ℎ minus 119909)3
119898]
119871 = 119903[[
[
arcsinradic1199032 minus 1199102
119903+ arcsin
radic1199032 minus (ℎ minus 119910)2
119903
]]
]
(3)
where in Figure 3119875(119909 119910) is the potential failure circle center119903 is the circle radius119898 is the slope ratio ℎ is the slope height120574 is the slope soil unit weight 120593 is the soil internal frictionangle and 119888 is slope soil cohesion
42 Genetic-Traversal Random Search Method The slopestability problem example in Figure 4 is just the engineeringproblem in Section 2 Inspired by the genetic algorithmthe potential failure circle is represented with points 119860119861 and 119862 in Figure 4 The coordinates of 119860 119861 and 119862
are (119886 25) (0 119887) and (119888 0) respectively So in fact theparameters 119886 119887 and 119888 can represent the potential failurecircle In a novel Fortran program points 119860 119861 and 119862 arevaried randomly and helped with random number generatorsubprogram However points 119860 119861 and 119862 can only varyin a certain region with boundary Each group of 119886 119887 119888
gives a safety factor by (2) With random number generatorsubprogram and loop program enough groups of 119886 119887 119888 aregenerated Inspired by the genetic algorithm the relative lowsafety factor and corresponding 119886 119887 119888 are saved after eachcomparison between the old potential failure circle and thenew generated potential failure circle and helped with therandom number generator subprogram After enough timesof iterations set by the user the minimum safety factor andcorresponding 119886 119887 119888 will be determined
The safety factor and failure circle determination pro-gram developed in Silverfrost FTN95 was presented in theAppendix In fact the computer-aided genetic algorithm ofthe program presented in the Appendix only utilizes themutation and selection thought of the traditional geneticalgorithm Crossover is omitted due to the difficulty incomputer program realization and compensatedwith numer-ous random candidates due to mutation In fact geneticalgorithm (GA) is a random search method based on thebiological evolution law
43 Results of the Program in the Appendix according toGenetic-Traversal Random Search Method for the Above-Mentioned Slope Stability Problem Example After 100000times of potential failure circlesrsquo generation and selectionthe obtained minimum safety factor is 0648280 and thecorresponding 119886 119887 119888 is minus118283 327429 504410
5 Compared with Other Solutions
In order to validate the analytical solution to give safetyfactor of a specified slip surface Genetic-Traversal RandomSearchMethod to search for the critical failure surface and thecorresponding program presented in the Appendix this partwill solve the slope engineering problem in Section 2 withother methods
51 Solution of Searching the Critical Slip Surface with Fel-leniusrsquo Method While Determining the Safety Factor withFelleniusrsquo Method of Slices To solve the engineering problemin Section 2 this part will search the critical failure surfacewith Felleniusrsquo method while determining each safety factorwith Felleniusrsquo method of slices
6 The Scientific World Journal
x
y
Search region for point B
Search region for point CCirclecenter of 2D potential slipsurface
Search region for point A
O
Potential circular slip surface
A(a 25)
B(0 b)
C(c 0)
Figure 4 Potential failure circle center and radius determined and represented with points 119860 119861 and 119862
A
CB
O
1205732
1205731120572
Figure 5 Determination of 2D potential failure surfaces wheninternal friction angle 120593 = 0
Table 13 Determination of 1205731and 120573
2with slope angle 120572
Slope angle 120572 Slope ratio 1 m 1205731
1205732
60∘ 1 058 29∘ 40∘
45∘ 1 10 28∘ 37∘
33∘411015840 1 15 26∘ 35∘
26∘341015840 1 20 25∘ 35∘
18∘261015840 1 30 26∘ 35∘
14∘021015840 1 40 25∘ 36∘
11∘191015840 1 50 25∘ 39∘
In this solution safety factor is determinedwith Felleniusrsquomethod of slices as in (1) Felleniusrsquo method of searching forthe critical failure surface [20] was given as follows
If soil internal friction angle 120593 = 0 2D critical failuresurface passes through slope toe119860 and can be determined byFigure 5 and Table 13 In Figure 5 the critical failure surfacecircle center 119874 can be determined by angles 120573
1and 120573
2which
can be determined by slope angle 120572 as in Table 13 Angle 1205731
is the angle between line 119860119874 and slope surface line while 1205732
is the angle between line119874119861 and slope horizontal top surfaceline 119861119862 Point 119861 is the intersection between slope surface line119860119861 and slope horizontal top surface line 119861119862
If soil internal friction angle 120593 gt 0 2D critical failuresurface passes through slope toe and can be determined by
Figure 6 In Figure 6 point 119864 is determined by angles 1205731
and 1205732which can be determined by slope angle 120572 as in
Table 13 The critical failure surface circle center may be onthe extension line of the line 119863119864 You can try many pointson the line 119863119864 as the critical failure surface circle centercandidate like119874
1and119874
4on the line119863119864 If a point119874119909 on the
line 119863119864 is found to be the point which gives the minimumslope safety factor then draw a line 119865119866 perpendicular to theline119863119864 through the point 119874
119909 Then you can try many points
on the line 119865119866 as the critical failure surface circle centercandidate like 119874
1015840
1 11987410158402 11987410158403 and 119874
1015840
4 If a point on the line 119865119866
gives the minimum slope safety factor this point means theone that gives the final most minimum safety factor of thestudied slope
The determined minimum safety factor with Felleniusrsquomethod is 1320 while the 2D critical failure surface circlecenter is 45m 57776m in the 119909-119910 coordinate system ofFigure 2 and the radius is 57951m The corresponding 119883
119888in
Figure 2 is 52292m
52 Solution with Slopew Software To solve the engineeringproblem in Section 2 this part will determine the safety factorwith slopew software
With the Ordinarymethod theminimum safety factor tothe example slope is 1328 The corresponding critical failuresurface is presented in Figure 7 With the Bishop methodthe minimum safety factor to the example slope is 1390 Thecorresponding critical failure surface is presented in Figure 8With the Janbu method the minimum safety factor to theexample slope is 1316 The corresponding critical failuresurface is presented in Figure 9 With the Morgenstern-Pricemethod the minimum safety factor to the example slope is1389 The corresponding critical failure surface is presentedin Figure 10 With the Spencer method the minimum safetyfactor to the example slope is 1389 The correspondingcritical failure surface is presented in Figure 11 With the GLEmethod the minimum safety factor to the example slope is
The Scientific World Journal 7
A
B
F
H
H
D
G
ECircle center of 2D critical slip surface
O1
O4
1205732
1205731
Fs1
Fs4
C1 C2 C3 C4
45H
120572
O1998400
O2998400
O3998400
O4998400
Figure 6 Determination of 2D potential failure surfaces when internal friction angle 120593 gt 0 [20]
1328
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 7 Critical failure surface with the Ordinary method
1389 The corresponding critical failure surface is presentedin Figure 12With the Janbu generalizedmethod after solvingand analyzing then after selecting the critical slip surfacewith a safety factor of 1389 in the slopew software Figure 13appears and the ldquominimum factor of safetyrdquo shows that itsvalue is 1385 1385 is not identical with 1389 which is a littleweird The minimum safety factor determined by foregoingGAprocedure is 1325 After comparedwith slopew softwareforegoing GA procedure employed to search the criticalfailure surface is reasonable and applicable
53 Solution with Ansys Software To solve the engineeringproblem in Section 2 this part will determine the safety factor
1390
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 8 Critical failure surface with the Bishop method
with Ansys software [23] The slope has two layers which islayer 1 and layer 2 in Figure 14 Layer 1 is clay and Layer 2is bed rock The slope layer 1rsquos soil modulus of elasticity isassumed to be 20E7Nm2 The slope layer 1rsquos soil Poissonrsquosratio is assumed to be 03 The slope layer 1rsquos soil density isassumed to be 20408 Kgm3 The slope layer 1rsquos soil cohesionis 10000 Pa and friction angle is 266 degrees The slope layer2rsquos soil modulus of elasticity is assumed to be 32E10Nm2The slope layer 2rsquos soil Poissonrsquos ratio is assumed to be 024The slope layer 2rsquos soil density is assumed to be 2700Kgm3
The slope stability analysis problem is regarded as a plainstrain problem The left and right boundaries are restricted
8 The Scientific World Journal
1316
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 9 Critical failure surface with the Janbu method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 10 Critical failure surface with the Morgenstern-Pricemethod
horizontally The bottom boundary is restricted both hori-zontally and vertically With the drucker-prager model asthe constitutive model and with shear strength reductionmethod based on the finite element analysis the slope inFigure 14 is analyzed Assume that the real cohesion andinternal friction angle of a slope are 119888
0and 120593
0 respectively
In the shear strength reduction method when safety factoris SF the reduced cohesion and friction angle for analysis are1198880SF and 120593
0SF
The Drucker-Prager yield criterion is [24 25]
1198601198681+ radic1198692minus 119861 le 0 (4)
where 1198681= 1205901+ 1205902+ 1205903 1198692= (16)[(120590
1minus 1205902)2+ (1205902minus 1205903)2+
(1205903minus 1205901)2]
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 11 Critical failure surface with the Spencer method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 12 Critical failure surface determinedwith theGLEmethod
1385
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 13 Critical failure surface determined with the Janbugeneralized method
The Scientific World Journal 9
Layer1
Layer2
40
m80
m
80m 50m 70m
40
m105
m
25m
Figure 14 Studied region for the engineering problem in Section 2 treated with Ansys
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(a) Safety factor equals 07
Y
Z X
MXMN
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
Slope stability analysis
(b) Safety factor equals 072
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(c) Safety factor equals 073
Figure 15 No von Mises plastic strain
If we assume that the Drucker-Prager yield surfacetouches on the interior of the Mohr-Coulomb yield surfacethen the expressions [26ndash28] are
119860 =2 sin120593
radic3radic3 + sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 + sin120593
(5)
If the Drucker-Prager yield surface passes through theexternal apexes of the Mohr-Coulomb yield surface then[26 28 29]
119860 =2 sin120593
radic3radic3 minus sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 minus sin120593
(6)
where 119888 is cohesion and 120593 is internal friction angle
10 The Scientific World Journal
Y
Z X
MX
MN
Slope stability analysis0 0160E minus 04 0320E minus 04 0479E minus 04 0639E minus 04
0799E minus 05 0240E minus 04 0399E minus 04 0559E minus 04 0719E minus 04
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0719E minus 04
(a) Safety factor equals 074
Y
Z X
MX
MN
Slope stability analysis0 0627E minus 04 0125E minus 03 0251E minus 03
0314E minus 04 0941E minus 04 0157E minus 03 0219E minus 03 0282E minus 030188E minus 03
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0282E minus 03
(b) Safety factor equals 075
Y
Z X
MX
MN
Slope stability analysis0 0295E minus 03 0589E minus 03 0884E minus 03 0001178
0147E minus 03 0442E minus 03 0736E minus 03 0001031 0001325
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0001325
(c) Safety factor equals 08
Y
Z X
MX
MN
Slope stability analysis0 0926E minus 03 0001851 0002777 0003703
0463E minus 03 0001389 0002314 000324 0004166
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0004166
(d) Safety factor equals 10
Y
Z X
MX
MN
Slope stability analysis0 0006555 0013111 0019666 0026221
0003278 0009833 0016388 0022944 0029499
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9296
UNFORRFORACEL
SMX = 0029499
(e) Safety factor equals 16
Y
Z XMX
MN
Slope stability analysis0 001587 003174 0047609 0063479
0007935 0023805 0039674 0055544 0071414
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9795
UNFORRFORACEL
SMX = 0071414
(f) Safety factor equals 18
Figure 16 von Mises plastic strain occurs and develops
With the drucker-prager model as the constitutive modelto analyze the slope under only self-weight in Figure 14 theflow rule which describes the relationship between the plasticpotential function and the plastic strain could be foundin [23 30ndash34] The incremental elastic-plastic stress-strainrelationship and the corresponding elastic-plastic matrixcould be found in [23 35]
The results were presented as follows in Figures 15 16 and17
When safety factor is from 07 to 073 there is no vonMises plastic strain in slope in Figure 15 When safety factoris 074 there is local plastic strain occurring in slope inFigure 16 When safety factor is 20 von Mises plastic strainruns through from slope toe to top surface in Figure 17
According to Chen (1975) and Niu (2009) Figure 16 giveslower bound solutions of slope safety factor which are from074 to 18 And Figure 17 where vonMises plastic strain runsthrough from slope toe to top surface gives an upper bound
The Scientific World Journal 11
Y
Z X MX
MN
Slope stability analysis0 0033261 0066522 0099783 0133044
0016631 0049892 0083153 0116414 0149675
Nodal solutionStep = 1
Sub = 999999
Time = 1
EPPLEQV (avg)DMX = 6955
UNFORRFORACEL
SMX = 0149675
Figure 17 von Mises plastic strain runs through from slope toe totop surface when safety factor equals 20
solution of slope safety factor which is 20 So the true slopesafety factor is likely from 18 to 20
54 Comparisons and Discussions The obtained minimumsafety factor for the above slope stability problem examplewith Genetic-Traversal Random Search Method is so lowwhen compared with the other methods like slopew soft-ware This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This mayalso be due to the power of the computer to realize theGenetic-Traversal Random Search Method in the AppendixThe Genetic-Traversal Random Method uses random pickto utilize mutation Validation of these conclusions will beinvestigated in the future with more numeric tests
However the obtained minimum safety factor withGenetic-Traversal Random Search Method is very close tothe lower bound solutions of slope safety factor given by theAnsys software
After computation there is plastic strain in layer 2 regionin some pictures of Figures 16 and 17 This is unreasonablesince layer 2 is defined as elastic region in the analysis withAnsys This phenomenon will be investigated in the future
6 Conclusions
This paper intends to determine a cohesive soil slope safetyfactor with Felleniusrsquo method while the 2D critical failuresurface is searched with GA The 2D critical failure surfaceis represented with real-encoded chromosomes which arepotential critical surface locations variables 119883
119888and 119883
119888119888 GA
procedure for searching critical failure surface proceeds withhand calculations If for future computer automatic searchprogram with GA program code for inheritance mutationselection and crossover program code for random numbersand program code for search interval boundaries will beneeded The minimum safety factor of 1325 determined byforegoing GA procedure to search the critical slip surface isvery close to the minimum safety factor of 1320 determinedby Felleniusrsquo critical slip surface method After comparedwith slopew software the proposed foregoing GA procedure
employed to search the critical failure surface is reasonableapplicable and effective
At last a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is madewhile in the program random numbers generated by com-puter and search boundaries are included The Genetic-Traversal Random Method uses random pick to utilizemutation In the program the slope safety factor is given byanalytical solution rather than slices method Results indicatethat the new computer automatic search program can givevery low safety factor which is about half of the foregoingones This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This may alsobe due to the power of the random number generation sub-program computer operation speed and Genetic-TraversalRandom Method Further validation of the results will beinvestigated in the future However the obtained minimumsafety factor with Genetic-Traversal Random Search Methodis very close to the lower bound solutions of slope safety factorgiven by the Ansys software
Appendix
Safety factor and failure circle determination program devel-oped in Silverfrost FTN95
double precision seedreal nrndlreal NLnewsafetyfactorsafetyfactor=100000seed=50gama=20tanphi=05cohesion=10h=25m=2do 10 I=1100000a=0b=25c=50rdn=nrndl(seed)a=-20lowastrdnrdn=nrndl(seed)b=25+20lowastrdnrdn=nrndl(seed)c=50+20lowastrdnif(b==25) b=2501x=(clowastc(2lowastb)+(25+b)2-b2+alowasta(50-2lowastb))(cb+a(25-b))y=b2+(cb)lowast(x-c2)r=sqrt((x-c)lowast(x-c)+ylowasty)
12 The Scientific World Journal
AA=(4lowastrlowastr-ylowasty)lowastsqrt(rlowastr-ylowasty)BB=(4lowastrlowastr-(h-y)lowast(h-y))lowastsqrt(rlowastr-(h-y)lowast(h-y))CC=(1m)lowast(2lowastrlowastr+xlowastx)lowastsqrt(rlowastr-xlowastx)DD=(1m)lowast(2lowastrlowastr+(mlowasth-x)lowast(mlowasth-x))lowastsqrt(rlowastr-(mlowasth-x)lowast(mlowasth-x))EE=ylowastasin((sqrt(rlowastr-ylowasty))r)FF=(h-y)lowastasin((sqrt(rlowastr-(h-y)lowast(h-y)))r)GG=(xm)lowastasin(xr)-((mlowasth-x)m)lowastasin((mlowasth-x)r)N=(1(6lowastr))lowast(AA+BB+CC-DD)+(r2)lowast(EE-FF+GG)T=(1(6lowastr))lowast(3lowasthlowastrlowastr-ylowastylowasty-(h-y)lowast(h-y)lowast(h-y)-xlowastxlowastxm-(mlowasth-x)lowast(mlowasth-x)lowast(mlowasth-x)m)L=rlowast(asin((sqrt(rlowastr-ylowasty))r)+asin((sqrt(rlowastr-(h-y)lowast(h-y)))r))newsafetyfactor=(gamalowasttanphilowastN+cohesionlowastL)(gamalowastT)if(newsafetyfactorltsafetyfactor)safetyfactor=newsafetyfactorwrite(lowastlowast)safetyfactorwrite(lowastlowast)awrite(lowastlowast)bwrite(lowastlowast)c10 continueend programreal function nrndl(seed)double precision SUVseedS=655360U=20530V=138490M=seedSseed=seed-MlowastSseed=Ulowastseed+VM=seedSseed=seed-MlowastSnrndl=seedSreturnend
Conflict of Interests
The author declares that he has no financial or personalrelationships with other people or organizations that couldinappropriately influence his work The author of this paperhas chosen not to furnish the paper and its readers with infor-mation that might present a potential conflict of interests
Acknowledgment
The author acknowledges the Scientific Research StartingFunds at Liaoning Technical University (no 11-415)
References
[1] D G Fredlund and J Krahn ldquoComparison of slope stabilitymethods of analysisrdquoCanadianGeotechnical Journal vol 14 no3 pp 429ndash439 1977
[2] K S Li ldquoA unified solution scheme for slope stability analysisrdquoin Proceedings of the 5th International Symposium on LandslidesNew Zealand vol 1 pp 481ndash486 Balkema Rotterdam TheNetherlands 1992
[3] W-J NiuW-M Ye S-G Liu and H-T Yu ldquoLimit analysis of asoil slope considering saturated-unsaturated seepagerdquoRock andSoil Mechanics vol 30 no 8 pp 2477ndash2482 2009
[4] W F Chen Limit Analysis and Soil Plasticity Elsevier Amster-dam The Netherlands 1975
[5] R L Michalowski ldquoSlope stability analysis a kinematical ap-proachrdquo Geotechnique vol 45 no 2 pp 283ndash293 1995
[6] I B Donald and Z Y Chen ldquoSlope stability analysis by theupper bound approach fundamentals and methodsrdquo CanadianGeotechnical Journal vol 34 no 6 pp 853ndash862 1997
[7] J P Sun J C Li and Q Q Liu ldquoSearch for critical slip surfacein slope stability analysis by spline-based GA methodrdquo Journalof Geotechnical and Geoenvironmental Engineering vol 134 no2 pp 252ndash256 2008
[8] R Javadzadeh and E Javadzadeh ldquoLocating critical failure sur-face in rock slope stability with hybrid model based on artificialimmune system and Cellular Learning Automata (CLA-AIS)rdquoWorld Academy of Science Engineering and Technology vol 59pp 662ndash665 2011
[9] F N G Gitirana Jr and D G Fredlund ldquoAnalysis of tran-sient embankment stability using the dynamic programmingmethodrdquo in Proceedings of the 56th Canadian GeotechnicalConference 2003
[10] H T V Pham and D G Fredlund ldquoDynamic programmingmethod in slope stability computationsrdquo in Proceedings ofthe 12th Asian Regional Conference on Soil Mechanics andGeotechnica Engineering Singapore August 2003
[11] H T V Pham D G Fredlund and F N G Gitirana Jr ldquoSlopestability analysis using dynamic prograamming combined withfinite element stress analysisrdquo in Proceedings of the InternationalConference on theManagement of the Land andWater Resources(MLWR rsquo01) Hanoi Vietnam October 2001
[12] H T V Pham and D G Fredlund ldquoThe application of dynamicprogramming to slope stability analysisrdquoCanadianGeotechnicalJournal vol 40 no 4 pp 830ndash847 2003
[13] M R Taha M Khajehzadeh and A El-Shafie ldquoSlope stabilityassessment using optimization techniques an overviewrdquo Elec-tronic Journal of Geotechnical Engineering vol 15 pp 1901ndash19152010
[14] A U Rao and N Sabhahit ldquoGenetic algorithm in stabilityof non-homogeneous slopesrdquo in Proceedings of the 12th Inter-national Conference of International Association for ComputerMethods and Advances in Geomechanics (IACMAG rsquo08) GoaIndia October 2008
[15] Y-G Nie W-Q Liu J-Y Shi and W-B Zhao ldquoApplication ofaccelerating genetic algorithm for embankment slope stabilityanalysisrdquo China Journal of Highway and Transport vol 16 no4 p 16 2003
The Scientific World Journal 13
[16] Y-P Zhou S-J Li Y-X Liu H-L Sun and F Jiang ldquoSearchingcritical failure surface in slope stability analysis with geneticalgorithmrdquo Chinese Journal of Rock Mechanics and Engineeringvol 24 pp 5226ndash5230 2005
[17] P McCombie ldquoCritical failure surface location using SimpleGenetic Algorithm and multiple wedge slope stabilityrdquo in Pro-ceedings of the 1st International Symposium on ComputationalGeomechanics pp 900ndash909 Juan-les-Pins France May 2009
[18] K Gavin and J F Xue ldquoUse of a genetic algorithm to performreliability analysis of unsaturated soil slopesrdquoGeotechnique vol59 no 6 pp 545ndash549 2009
[19] Wikipedia ldquoGenetic algorithmrdquo 2012 httpenwikipediaorgwikiGenetic algorithm
[20] China University of Geosciences ldquoCohesive soil slope stabi-lity analysisrdquo httpcoursecugeducncugsoil mechanicsCOURSECHAPTER7Chap7 3 5htm
[21] J E Thomaz ldquoA general method for three dimensional slopestability analysis informational reportrdquo Tech Rep JTRP Tech-nical Reports Purdue University 1986
[22] T B Zhang ldquoEarth slope stability analysis and geotechnologyslope designrdquo Tech Rep Chengdu University of Science andTechnology Press 1987
[23] B G He Application Examples in Civil Engineering with AnsysChina Water Power Press Beijing China 2011
[24] Y Liu Q Yang and L Zhu ldquoAbutment stability analysis of archdam based on 3D nonlinear finite element methodrdquo ChineseJournal of Rock Mechanics and Engineering vol 27 supplement1 pp 3222ndash3228 2008
[25] Y R Liu ZHe B Li andQYang ldquoSlope stability analysis basedon a multigrid method using a nonlinear 3D finite elementmodelrdquo Frontiers of Structural and Civil Engineering vol 7 no1 pp 24ndash31 2013
[26] Wikipedia ldquoDruckerndashPrager yield criterionrdquo 2013 httpenwikipediaorgwikiDruckerE28093Prager yield criterion
[27] B Doran H O Koksal Z Polat and C Karakoc ldquoThe use ofldquoDrucker-Prager Criterionrdquo in the analysis of reinforced con-crete members by finite elementsrdquo Teknik Dergi vol 9 no 2pp 1617ndash1625 1998
[28] B Zoran P Verka and M Biljana ldquoMathematical modeling ofmaterially nonlinear problems in structural analyses (part Imdashtheoretical fundamentals)rdquo Facta Universitatis Architecture andCivil Engineering vol 8 no 1 pp 67ndash78 2010
[29] Q Yang X Chen and W-Y Zhou ldquoA practical 3D elasto-plastic incrementalmethod in FEMbased onD-P yield criteriardquoChinese Journal of Geotechnical Engineering vol 24 no 1 pp16ndash20 2002
[30] G D Zhang Soil Constitutive Model and Its Application inEngineering Science Press Beijing China 1995
[31] D Rakic M Zivkovic R Slavkovic and M Kojic ldquoStress inte-gration for the Drucker-Prager material model without hard-ening using the Incremental Plasticity Theoryrdquo Journal of theSerbian Society For Computational Mechanics vol 2 no 1 pp80ndash89 2008
[32] X Y Zhang Soil and Rock Plastic Mechanics China Communi-cations Press Beijing China 1993
[33] Y R Zheng Z J Shen and X N Gong The Principles of Geo-technical Plastic Mechanics China Architecture and BuildingPress Beijing China 2002
[34] S A Akers Two-dimensional finite element analysis of porousgeomaterials at multikilobar stress levels [PhD thesis] VirginiaPolytechnic Institute and State University 2001
[35] X-H Tan J-G Wang and Y Wang ldquoNonlinear finite elementanalysis of slope stabilityrdquo Rock and Soil Mechanics vol 29 no8 pp 2047ndash2050 2008
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2 The Scientific World Journal
not obtained a new population is created from the original(parent) chromosomes This is achieved using ldquocrossoverrdquoand ldquomutationrdquo operations Crossover involves gene exchangefrom two random (parent) solutions to form a child (newsolution)Mutation involves the random switching of a singlevariable in a chromosome and is used to maintain popula-tion diversity as the process converges towards a solution[18]
GA includes inheritance mutation selection andcrossover [19] One of the core techniques and advantages ofGA is that mutation can consider a wide range of possiblesolutions if natural evolution continues and never ends Theother advantage is that inheritance and crossover can save allthe examples and virtues of the past age and pass them intothe next generation to save time for the best choice [18 19]
However there are many unsolved problems about GAin slope stability analysis For example how to realize GAwith hand calculationmethod How to realize GAwith auto-matic search program What is the relationship between GAand traversal random search method How to improve theFellenius method of slices concerning that the slices methodis an approximate method like finite element method Infact all these problems are mathematical problems to findthe minimum safety factor of a slope These mathematicalproblems originated from the physical equations representingthe common law of nature in slopes (eg Mohr-Columncriterion of soil 2D circular slip surface of homogeneous clayslope and safety factor definition which is the moment ofsliding resistance divided by the moment of sliding force)The physical laws of nature can be found and validated withrepeated in situ or lab experiments to measure the physicalquantities and mathematical logic to reveal the relationshipHowever mathematical problems can only be solved withlogic deduction and validated with countless numeric tests
This paper first intends to determine a cohesive soil slopesafety factor with Felleniusrsquo slices method while the 2Dcritical failure surface is searched with GA The analysis usesreal-coded methods to encode the chromosomes with thevariables of potential critical surface locations The fitness ofeach chromosome is determined using the objective functionthat the resulting safety factors should be lower enough andthe fitness of all solutions is comparedwhile the chromosomeof large safety factors shall be deleted [18] However this partis realized with hand calculation
Then a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is madeThe Genetic-Traversal Random Search Method presented inthis paper only utilizes the mutation and selection thoughtof the traditional genetic algorithm Crossover is omitteddue to the difficulty in computer program realization andcompensated with numerous random candidates due tomutation The Genetic-Traversal Random Search Methodmakes a traversal search with random method In the pro-gram random numbers for random search are generatedby computer and search boundaries are included In theprogram each slope safety factor is given by analyticalsolution rather than slices method The safety factor andfailure circle determination program developed in SilverfrostFTN95 is presented in the Appendix At last the proposed
RR
Wi
li
120572i
x = Rsin120572i
Figure 1 Felleniusrsquo method of slices
Genetic-Traversal Random Search Method is compared withother solutions such as slopew software
2 A Slope Stability Problem Example
Acohesive soil slopewith its height 25meters has a slope ratioof 1 2 The soil unit weight 120574 is 20KNm3 The soil internalfriction angle 120593 is 266 degrees and cohesion is 10 KPA Theproblem now is to give the safety factor of the slope with a 2Dcircular failure surface
3 Search the Critical Slip Surface withGA Method While Determining the SafetyFactor with Felleniusrsquo Method of Slices withHand Calculation
To solve the engineering problem in Section 2 this partwill search the critical slip surface with GA Method whiledetermining the safety factor with Felleniusrsquo method of slices
31 The Slope Safety Factor with Felleniusrsquo Method of SlicesThe potential slip surface for clay slope is two dimensionaland a part of circle In order to determine the slope safetyfactor in Figure 1 Felleniusrsquo method of slices divides the slopeinto several slices [3] Using moment equilibrium the slopesafety factor SF in Figure 1 is
SF =sum (119888119894+ 119882119894cos120572119894tan120593119894) 119897119894
119882119894sin120572119894
(1)
where 119888119894and 120593
119894are the soil slope slice cohesion and internal
friction angle119882119894is the soil slope slice self-gravity 119897
119894is the soil
slope slice slip circular arc length 120572119894is the angle between soil
slope slice slip surface tangent line and the horizontal line
32 Searching the Critical Failure Surface with GA Theassumed 2D slope failure surface is circular determined bytwo variables 119883
119888and 119883
119888119888in Figure 2 119883
119888is the abscissa of a
point on slope top surface If119883119888is determined the 2D critical
slip surface circle center with abscissa 119883119888119888must lie on the
perpendicular bisector of the straight line from the point of119883119888to the slope toe 119874 119883
119888119888is the abscissa of the critical slip
surface circle center So if 119883119888119888is given then the critical slip
surface circle center can be determined Altogether if119883119888and
119883119888119888are given 2D circular slope slip surface is determined
The Scientific World Journal 3
L
x
y
A
R
S
Search region
R
Potential 2D critical slip surface
Search region for Xc
Slope toe OXc
for Xcc
Xcc
Figure 2 2D potential failure surfaces
However in the next section all the computations accord-ing to GA are made by hand calculations If for computersimulation ldquointervalrdquo is as 119871 and 119904 as in Figure 2 and randomnumbers of computer function can help for automatic gen-eration of variables 119883
119888and 119883
119888119888in Figure 2 search regions or
boundaries for119883119888and119883
119888119888must be definedwith the definition
of the limits of119883119888and119883
119888119888 The necessity of these boundaries
is evident because computer program must avoid generatingsurfaces out of the region of primary interest [21]
The searching process for the 2D critical failure surfaceas in Figure 1 uses techniques inspired by natural evolutionsuch as inheritance mutation selection and crossover [19]In a genetic algorithm a population of strings (called chro-mosomes or the genotype of the genome) which encodecandidate solutions (called individuals creatures or pheno-types) to an optimization problem is evolved toward bettersolutions [19] In this 2D critical failure surface searchingproblem the candidate solutions are represented as 119883
119888 119883119888119888
described before The evolution starts from a population ofrandomly generated individuals as 119883
119888 119883119888119888 and happens
in generations In each generation the fitness of everyindividual in the population is evaluatedmultiple individualsare stochastically selected from the current population (basedon their fitness) and modified (recombined and possiblyrandomlymutated) to form a new populationThe evaluationstandard is that the individual 119883
119888 119883119888119888 with large safety
factor is deleted and the individual 119883119888 119883119888119888with small safety
factor is reserved The new population is then used in thenext iteration of the algorithm [19]The algorithm terminateswhen a satisfactory fitness level has been reached for thepopulation which means that it is hard to lower safety factorwith iterations
33 Searching Process with GA In the GA with hand cal-culation method the potential failure surfaces for searchare restricted that they all pass slope toe as in Figure 2 forsimplifying the search task Search process for the critical slipsurface with genetic algorithm was presented from Table 1to Table 12 In these tables the minimum safety factors aremarked with lowast in each iterationThe units for119883
119888and119883
119888119888are
meters Evaluation of individuals in Table 1 now begins The
Table 1 A population of randomly generated individuals
119883119888
119883119888119888
Safety factor51 25 272052352151 41 Safety factor is extremely large Unreasonable
lowast 51 0 13317465878lowast 60 minus10 14074551938lowast 60 25 1886105777
60 35 Safety factor is extremely large Unreasonablelowast 100 minus20 22757670756
100 21 23067661886100 70 Safety factor is extremely large Unreasonable
Table 2 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factorlowast 51 0 13317465878
60 minus10 1407455193860 25 1886105777100 minus20 22757670756
Table 3 Crossover results
119883119888
119883119888119888
Safety factorlowast 51 minus10 13799239885
51 25 272052352151 minus20 14447027254
lowast 60 0 1387959668960 minus20 14333540217100 0 2272574082100 minus10 22729514535100 25 23308411862
evaluation standard is that the individual 119883119888 119883119888119888with large
safety factor is deleted and the individual 119883119888 119883119888119888with small
safety factor is reserved Selection result will be put in Table 2Crossover of selected individuals of Table 2 will be put inTable 3 Mutation begins and results will be put in Table 4Evaluation of all previous individuals marked with lowast beginsThe evaluation standard is that the individual 119883
119888 119883119888119888 with
large safety factor is deleted and the individual 119883119888 119883119888119888with
small safety factor is reserved Selection result will be put inTable 5 Crossover of selected individuals of Table 5 beginsCrossover result will be put in Table 6 Mutation begins andthe result will be put in Table 7 Evaluation of all previousindividuals marked with lowast begins The evaluation standardis that the individual 119883
119888 119883119888119888 with large safety factor is
deleted and the individual 119883119888 119883119888119888with small safety factor is
reserved Selection result will be put in Table 8 Crossover ofselected individuals of Table 8 begins Crossover result willbe put in Table 9 Mutation begins and result will be putin Table 10 Evaluation of all previous individuals markedwith lowast begins The evaluation standard is that the individual119883119888 119883119888119888with large safety factor is deleted and the individual
119883119888 119883119888119888 with small safety factor is reserved Selection result
will be put in Table 11 The GA procedure terminates when asatisfactory fitness level has been reached for the population
4 The Scientific World Journal
Table 4 Mutation results
119883119888
119883119888119888
Safety factorlowast 50 0 13425306677
50 minus10 1400390984351 21 17627329758
lowast 55 0 13332021645lowast 55 minus10 13615830364
55 20 16493380867
Table 5 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factorlowast 51 0 13317465878
51 minus10 1379923988560 0 13879596689
lowast 50 0 13425306677lowast 55 0 13332021645
55 minus10 13615830364
Table 6 Crossover results
119883119888
119883119888119888
Safety factor51 0 1331746587851 minus10 1379923988560 0 1387959668950 0 1342530667755 0 1333202164555 minus10 13615830364
Table 7 Mutation results
119883119888
119883119888119888
Safety factorlowast 52 0 13264516064
52 minus10 13675053085lowast 53 0 13254640532
53 minus10 13611610322lowast 57 0 13503481321
57 minus10 13741239435lowast 52 11 13501275295
52 25 2515914533852 100 Safety factor is extremely large Unreasonable52 15 14194510174
which means that it is hard to lower safety factor withiterations The final result is in Table 12
34 Location of the Critical Failure Surface and Safety Factorwith GA Procedure The example was solved with foregoingGA procedure The minimum safety factor was 1325 with119883119888= 53m and 119883
119888119888= 0 The corresponding slip circle center
is at (0 688m) and the radius is 688m
Table 8 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factorlowast 51 0 13317465878lowast 50 0 13425306677lowast 55 0 13332021645lowast 52 0 13264516064lowast 53 0 13254640532
57 0 1350348132152 11 13501275295
Table 9 Crossover results
119883119888
119883119888119888
Safety factor51 11 1355689633150 11 1367290667655 11 13547892263
lowast 53 11 1348701176357 11 13696385778
Table 10 Mutation results
119883119888
119883119888119888
Safety factor54 21 16711004325
lowast 54 0 1327927158254 13 1373489757754 minus10 1359516051960 minus10 14074551938
Table 11 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factor51 0 1331746587850 0 1342530667755 0 1333202164552 0 13264516064
lowast 53 0 1325464053253 11 1348701176354 0 13279271582
Table 12 Critical failure surface and minimum safety factor
119883119888
119883119888119888
Safety factorCompleted 53 0 13254640532
4 Search the Critical Slip Surface withGenetic-Traversal Random Search MethodWhile Determining the Safety Factor withAnalytical Method
To solve the engineering problem in Section 2 this partwill search the critical failure surface with Genetic-TraversalRandom Search Method while determining the safety factorwith analytical method This part is realized with computerautomatic search program
The Scientific World Journal 5
Y
XO
A
B C
D
EF
G
H
r
m1
h
X dX
P(X Y)
Y1 = h minus Xm
Y1 = h
Y1 = 0
d120572x120572x
M(X Y2)Y2 = y + radicr2 minus (X minus x)2
Figure 3 Analytical method to determine the slope safety factor
41 Analytical Method to Determine the Slope Safety Factor inthe Above-Mentioned Slope Example in Section 2 [22] WithFelleniusrsquo method according to Zhang (1987) the analyticalsolution to give the safety factor in Figure 3 is
119896 =120574 sdot 119905119892120593 [119873] + 119888 [119871]
120574 [119879] (2)
where [119873] [119871] and [119879] were given as
119873 = [41199032minus 1199102]radic1199032 minus 1199102 + [4119903
2minus (ℎ minus 119910)
2]
times radic1199032 minus (ℎ minus 119910)2+
1
119898(21199032+ 1199092)radic1199032 minus 1199092
minus1
119898[21199032+ (119898ℎ minus 119909)
2]radic1199032 minus (119898ℎ minus 119909)
2
+119903
2
119910 arcsinradic1199032 minus 1199102
119903
minus (ℎ minus 119910) arcsinradic1199032 minus (ℎ minus 119910)
2
119903
+119909
119898arcsin 119909
119903minus
119898ℎ minus 119909
119898arcsin 119898ℎ minus 119909
119903
119879 =1
6119903[3ℎ1199032minus 1199103minus (ℎ minus 119910)
3minus
1199093
119898minus
(119898ℎ minus 119909)3
119898]
119871 = 119903[[
[
arcsinradic1199032 minus 1199102
119903+ arcsin
radic1199032 minus (ℎ minus 119910)2
119903
]]
]
(3)
where in Figure 3119875(119909 119910) is the potential failure circle center119903 is the circle radius119898 is the slope ratio ℎ is the slope height120574 is the slope soil unit weight 120593 is the soil internal frictionangle and 119888 is slope soil cohesion
42 Genetic-Traversal Random Search Method The slopestability problem example in Figure 4 is just the engineeringproblem in Section 2 Inspired by the genetic algorithmthe potential failure circle is represented with points 119860119861 and 119862 in Figure 4 The coordinates of 119860 119861 and 119862
are (119886 25) (0 119887) and (119888 0) respectively So in fact theparameters 119886 119887 and 119888 can represent the potential failurecircle In a novel Fortran program points 119860 119861 and 119862 arevaried randomly and helped with random number generatorsubprogram However points 119860 119861 and 119862 can only varyin a certain region with boundary Each group of 119886 119887 119888
gives a safety factor by (2) With random number generatorsubprogram and loop program enough groups of 119886 119887 119888 aregenerated Inspired by the genetic algorithm the relative lowsafety factor and corresponding 119886 119887 119888 are saved after eachcomparison between the old potential failure circle and thenew generated potential failure circle and helped with therandom number generator subprogram After enough timesof iterations set by the user the minimum safety factor andcorresponding 119886 119887 119888 will be determined
The safety factor and failure circle determination pro-gram developed in Silverfrost FTN95 was presented in theAppendix In fact the computer-aided genetic algorithm ofthe program presented in the Appendix only utilizes themutation and selection thought of the traditional geneticalgorithm Crossover is omitted due to the difficulty incomputer program realization and compensatedwith numer-ous random candidates due to mutation In fact geneticalgorithm (GA) is a random search method based on thebiological evolution law
43 Results of the Program in the Appendix according toGenetic-Traversal Random Search Method for the Above-Mentioned Slope Stability Problem Example After 100000times of potential failure circlesrsquo generation and selectionthe obtained minimum safety factor is 0648280 and thecorresponding 119886 119887 119888 is minus118283 327429 504410
5 Compared with Other Solutions
In order to validate the analytical solution to give safetyfactor of a specified slip surface Genetic-Traversal RandomSearchMethod to search for the critical failure surface and thecorresponding program presented in the Appendix this partwill solve the slope engineering problem in Section 2 withother methods
51 Solution of Searching the Critical Slip Surface with Fel-leniusrsquo Method While Determining the Safety Factor withFelleniusrsquo Method of Slices To solve the engineering problemin Section 2 this part will search the critical failure surfacewith Felleniusrsquo method while determining each safety factorwith Felleniusrsquo method of slices
6 The Scientific World Journal
x
y
Search region for point B
Search region for point CCirclecenter of 2D potential slipsurface
Search region for point A
O
Potential circular slip surface
A(a 25)
B(0 b)
C(c 0)
Figure 4 Potential failure circle center and radius determined and represented with points 119860 119861 and 119862
A
CB
O
1205732
1205731120572
Figure 5 Determination of 2D potential failure surfaces wheninternal friction angle 120593 = 0
Table 13 Determination of 1205731and 120573
2with slope angle 120572
Slope angle 120572 Slope ratio 1 m 1205731
1205732
60∘ 1 058 29∘ 40∘
45∘ 1 10 28∘ 37∘
33∘411015840 1 15 26∘ 35∘
26∘341015840 1 20 25∘ 35∘
18∘261015840 1 30 26∘ 35∘
14∘021015840 1 40 25∘ 36∘
11∘191015840 1 50 25∘ 39∘
In this solution safety factor is determinedwith Felleniusrsquomethod of slices as in (1) Felleniusrsquo method of searching forthe critical failure surface [20] was given as follows
If soil internal friction angle 120593 = 0 2D critical failuresurface passes through slope toe119860 and can be determined byFigure 5 and Table 13 In Figure 5 the critical failure surfacecircle center 119874 can be determined by angles 120573
1and 120573
2which
can be determined by slope angle 120572 as in Table 13 Angle 1205731
is the angle between line 119860119874 and slope surface line while 1205732
is the angle between line119874119861 and slope horizontal top surfaceline 119861119862 Point 119861 is the intersection between slope surface line119860119861 and slope horizontal top surface line 119861119862
If soil internal friction angle 120593 gt 0 2D critical failuresurface passes through slope toe and can be determined by
Figure 6 In Figure 6 point 119864 is determined by angles 1205731
and 1205732which can be determined by slope angle 120572 as in
Table 13 The critical failure surface circle center may be onthe extension line of the line 119863119864 You can try many pointson the line 119863119864 as the critical failure surface circle centercandidate like119874
1and119874
4on the line119863119864 If a point119874119909 on the
line 119863119864 is found to be the point which gives the minimumslope safety factor then draw a line 119865119866 perpendicular to theline119863119864 through the point 119874
119909 Then you can try many points
on the line 119865119866 as the critical failure surface circle centercandidate like 119874
1015840
1 11987410158402 11987410158403 and 119874
1015840
4 If a point on the line 119865119866
gives the minimum slope safety factor this point means theone that gives the final most minimum safety factor of thestudied slope
The determined minimum safety factor with Felleniusrsquomethod is 1320 while the 2D critical failure surface circlecenter is 45m 57776m in the 119909-119910 coordinate system ofFigure 2 and the radius is 57951m The corresponding 119883
119888in
Figure 2 is 52292m
52 Solution with Slopew Software To solve the engineeringproblem in Section 2 this part will determine the safety factorwith slopew software
With the Ordinarymethod theminimum safety factor tothe example slope is 1328 The corresponding critical failuresurface is presented in Figure 7 With the Bishop methodthe minimum safety factor to the example slope is 1390 Thecorresponding critical failure surface is presented in Figure 8With the Janbu method the minimum safety factor to theexample slope is 1316 The corresponding critical failuresurface is presented in Figure 9 With the Morgenstern-Pricemethod the minimum safety factor to the example slope is1389 The corresponding critical failure surface is presentedin Figure 10 With the Spencer method the minimum safetyfactor to the example slope is 1389 The correspondingcritical failure surface is presented in Figure 11 With the GLEmethod the minimum safety factor to the example slope is
The Scientific World Journal 7
A
B
F
H
H
D
G
ECircle center of 2D critical slip surface
O1
O4
1205732
1205731
Fs1
Fs4
C1 C2 C3 C4
45H
120572
O1998400
O2998400
O3998400
O4998400
Figure 6 Determination of 2D potential failure surfaces when internal friction angle 120593 gt 0 [20]
1328
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 7 Critical failure surface with the Ordinary method
1389 The corresponding critical failure surface is presentedin Figure 12With the Janbu generalizedmethod after solvingand analyzing then after selecting the critical slip surfacewith a safety factor of 1389 in the slopew software Figure 13appears and the ldquominimum factor of safetyrdquo shows that itsvalue is 1385 1385 is not identical with 1389 which is a littleweird The minimum safety factor determined by foregoingGAprocedure is 1325 After comparedwith slopew softwareforegoing GA procedure employed to search the criticalfailure surface is reasonable and applicable
53 Solution with Ansys Software To solve the engineeringproblem in Section 2 this part will determine the safety factor
1390
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 8 Critical failure surface with the Bishop method
with Ansys software [23] The slope has two layers which islayer 1 and layer 2 in Figure 14 Layer 1 is clay and Layer 2is bed rock The slope layer 1rsquos soil modulus of elasticity isassumed to be 20E7Nm2 The slope layer 1rsquos soil Poissonrsquosratio is assumed to be 03 The slope layer 1rsquos soil density isassumed to be 20408 Kgm3 The slope layer 1rsquos soil cohesionis 10000 Pa and friction angle is 266 degrees The slope layer2rsquos soil modulus of elasticity is assumed to be 32E10Nm2The slope layer 2rsquos soil Poissonrsquos ratio is assumed to be 024The slope layer 2rsquos soil density is assumed to be 2700Kgm3
The slope stability analysis problem is regarded as a plainstrain problem The left and right boundaries are restricted
8 The Scientific World Journal
1316
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 9 Critical failure surface with the Janbu method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 10 Critical failure surface with the Morgenstern-Pricemethod
horizontally The bottom boundary is restricted both hori-zontally and vertically With the drucker-prager model asthe constitutive model and with shear strength reductionmethod based on the finite element analysis the slope inFigure 14 is analyzed Assume that the real cohesion andinternal friction angle of a slope are 119888
0and 120593
0 respectively
In the shear strength reduction method when safety factoris SF the reduced cohesion and friction angle for analysis are1198880SF and 120593
0SF
The Drucker-Prager yield criterion is [24 25]
1198601198681+ radic1198692minus 119861 le 0 (4)
where 1198681= 1205901+ 1205902+ 1205903 1198692= (16)[(120590
1minus 1205902)2+ (1205902minus 1205903)2+
(1205903minus 1205901)2]
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 11 Critical failure surface with the Spencer method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 12 Critical failure surface determinedwith theGLEmethod
1385
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 13 Critical failure surface determined with the Janbugeneralized method
The Scientific World Journal 9
Layer1
Layer2
40
m80
m
80m 50m 70m
40
m105
m
25m
Figure 14 Studied region for the engineering problem in Section 2 treated with Ansys
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(a) Safety factor equals 07
Y
Z X
MXMN
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
Slope stability analysis
(b) Safety factor equals 072
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(c) Safety factor equals 073
Figure 15 No von Mises plastic strain
If we assume that the Drucker-Prager yield surfacetouches on the interior of the Mohr-Coulomb yield surfacethen the expressions [26ndash28] are
119860 =2 sin120593
radic3radic3 + sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 + sin120593
(5)
If the Drucker-Prager yield surface passes through theexternal apexes of the Mohr-Coulomb yield surface then[26 28 29]
119860 =2 sin120593
radic3radic3 minus sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 minus sin120593
(6)
where 119888 is cohesion and 120593 is internal friction angle
10 The Scientific World Journal
Y
Z X
MX
MN
Slope stability analysis0 0160E minus 04 0320E minus 04 0479E minus 04 0639E minus 04
0799E minus 05 0240E minus 04 0399E minus 04 0559E minus 04 0719E minus 04
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0719E minus 04
(a) Safety factor equals 074
Y
Z X
MX
MN
Slope stability analysis0 0627E minus 04 0125E minus 03 0251E minus 03
0314E minus 04 0941E minus 04 0157E minus 03 0219E minus 03 0282E minus 030188E minus 03
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0282E minus 03
(b) Safety factor equals 075
Y
Z X
MX
MN
Slope stability analysis0 0295E minus 03 0589E minus 03 0884E minus 03 0001178
0147E minus 03 0442E minus 03 0736E minus 03 0001031 0001325
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0001325
(c) Safety factor equals 08
Y
Z X
MX
MN
Slope stability analysis0 0926E minus 03 0001851 0002777 0003703
0463E minus 03 0001389 0002314 000324 0004166
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0004166
(d) Safety factor equals 10
Y
Z X
MX
MN
Slope stability analysis0 0006555 0013111 0019666 0026221
0003278 0009833 0016388 0022944 0029499
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9296
UNFORRFORACEL
SMX = 0029499
(e) Safety factor equals 16
Y
Z XMX
MN
Slope stability analysis0 001587 003174 0047609 0063479
0007935 0023805 0039674 0055544 0071414
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9795
UNFORRFORACEL
SMX = 0071414
(f) Safety factor equals 18
Figure 16 von Mises plastic strain occurs and develops
With the drucker-prager model as the constitutive modelto analyze the slope under only self-weight in Figure 14 theflow rule which describes the relationship between the plasticpotential function and the plastic strain could be foundin [23 30ndash34] The incremental elastic-plastic stress-strainrelationship and the corresponding elastic-plastic matrixcould be found in [23 35]
The results were presented as follows in Figures 15 16 and17
When safety factor is from 07 to 073 there is no vonMises plastic strain in slope in Figure 15 When safety factoris 074 there is local plastic strain occurring in slope inFigure 16 When safety factor is 20 von Mises plastic strainruns through from slope toe to top surface in Figure 17
According to Chen (1975) and Niu (2009) Figure 16 giveslower bound solutions of slope safety factor which are from074 to 18 And Figure 17 where vonMises plastic strain runsthrough from slope toe to top surface gives an upper bound
The Scientific World Journal 11
Y
Z X MX
MN
Slope stability analysis0 0033261 0066522 0099783 0133044
0016631 0049892 0083153 0116414 0149675
Nodal solutionStep = 1
Sub = 999999
Time = 1
EPPLEQV (avg)DMX = 6955
UNFORRFORACEL
SMX = 0149675
Figure 17 von Mises plastic strain runs through from slope toe totop surface when safety factor equals 20
solution of slope safety factor which is 20 So the true slopesafety factor is likely from 18 to 20
54 Comparisons and Discussions The obtained minimumsafety factor for the above slope stability problem examplewith Genetic-Traversal Random Search Method is so lowwhen compared with the other methods like slopew soft-ware This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This mayalso be due to the power of the computer to realize theGenetic-Traversal Random Search Method in the AppendixThe Genetic-Traversal Random Method uses random pickto utilize mutation Validation of these conclusions will beinvestigated in the future with more numeric tests
However the obtained minimum safety factor withGenetic-Traversal Random Search Method is very close tothe lower bound solutions of slope safety factor given by theAnsys software
After computation there is plastic strain in layer 2 regionin some pictures of Figures 16 and 17 This is unreasonablesince layer 2 is defined as elastic region in the analysis withAnsys This phenomenon will be investigated in the future
6 Conclusions
This paper intends to determine a cohesive soil slope safetyfactor with Felleniusrsquo method while the 2D critical failuresurface is searched with GA The 2D critical failure surfaceis represented with real-encoded chromosomes which arepotential critical surface locations variables 119883
119888and 119883
119888119888 GA
procedure for searching critical failure surface proceeds withhand calculations If for future computer automatic searchprogram with GA program code for inheritance mutationselection and crossover program code for random numbersand program code for search interval boundaries will beneeded The minimum safety factor of 1325 determined byforegoing GA procedure to search the critical slip surface isvery close to the minimum safety factor of 1320 determinedby Felleniusrsquo critical slip surface method After comparedwith slopew software the proposed foregoing GA procedure
employed to search the critical failure surface is reasonableapplicable and effective
At last a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is madewhile in the program random numbers generated by com-puter and search boundaries are included The Genetic-Traversal Random Method uses random pick to utilizemutation In the program the slope safety factor is given byanalytical solution rather than slices method Results indicatethat the new computer automatic search program can givevery low safety factor which is about half of the foregoingones This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This may alsobe due to the power of the random number generation sub-program computer operation speed and Genetic-TraversalRandom Method Further validation of the results will beinvestigated in the future However the obtained minimumsafety factor with Genetic-Traversal Random Search Methodis very close to the lower bound solutions of slope safety factorgiven by the Ansys software
Appendix
Safety factor and failure circle determination program devel-oped in Silverfrost FTN95
double precision seedreal nrndlreal NLnewsafetyfactorsafetyfactor=100000seed=50gama=20tanphi=05cohesion=10h=25m=2do 10 I=1100000a=0b=25c=50rdn=nrndl(seed)a=-20lowastrdnrdn=nrndl(seed)b=25+20lowastrdnrdn=nrndl(seed)c=50+20lowastrdnif(b==25) b=2501x=(clowastc(2lowastb)+(25+b)2-b2+alowasta(50-2lowastb))(cb+a(25-b))y=b2+(cb)lowast(x-c2)r=sqrt((x-c)lowast(x-c)+ylowasty)
12 The Scientific World Journal
AA=(4lowastrlowastr-ylowasty)lowastsqrt(rlowastr-ylowasty)BB=(4lowastrlowastr-(h-y)lowast(h-y))lowastsqrt(rlowastr-(h-y)lowast(h-y))CC=(1m)lowast(2lowastrlowastr+xlowastx)lowastsqrt(rlowastr-xlowastx)DD=(1m)lowast(2lowastrlowastr+(mlowasth-x)lowast(mlowasth-x))lowastsqrt(rlowastr-(mlowasth-x)lowast(mlowasth-x))EE=ylowastasin((sqrt(rlowastr-ylowasty))r)FF=(h-y)lowastasin((sqrt(rlowastr-(h-y)lowast(h-y)))r)GG=(xm)lowastasin(xr)-((mlowasth-x)m)lowastasin((mlowasth-x)r)N=(1(6lowastr))lowast(AA+BB+CC-DD)+(r2)lowast(EE-FF+GG)T=(1(6lowastr))lowast(3lowasthlowastrlowastr-ylowastylowasty-(h-y)lowast(h-y)lowast(h-y)-xlowastxlowastxm-(mlowasth-x)lowast(mlowasth-x)lowast(mlowasth-x)m)L=rlowast(asin((sqrt(rlowastr-ylowasty))r)+asin((sqrt(rlowastr-(h-y)lowast(h-y)))r))newsafetyfactor=(gamalowasttanphilowastN+cohesionlowastL)(gamalowastT)if(newsafetyfactorltsafetyfactor)safetyfactor=newsafetyfactorwrite(lowastlowast)safetyfactorwrite(lowastlowast)awrite(lowastlowast)bwrite(lowastlowast)c10 continueend programreal function nrndl(seed)double precision SUVseedS=655360U=20530V=138490M=seedSseed=seed-MlowastSseed=Ulowastseed+VM=seedSseed=seed-MlowastSnrndl=seedSreturnend
Conflict of Interests
The author declares that he has no financial or personalrelationships with other people or organizations that couldinappropriately influence his work The author of this paperhas chosen not to furnish the paper and its readers with infor-mation that might present a potential conflict of interests
Acknowledgment
The author acknowledges the Scientific Research StartingFunds at Liaoning Technical University (no 11-415)
References
[1] D G Fredlund and J Krahn ldquoComparison of slope stabilitymethods of analysisrdquoCanadianGeotechnical Journal vol 14 no3 pp 429ndash439 1977
[2] K S Li ldquoA unified solution scheme for slope stability analysisrdquoin Proceedings of the 5th International Symposium on LandslidesNew Zealand vol 1 pp 481ndash486 Balkema Rotterdam TheNetherlands 1992
[3] W-J NiuW-M Ye S-G Liu and H-T Yu ldquoLimit analysis of asoil slope considering saturated-unsaturated seepagerdquoRock andSoil Mechanics vol 30 no 8 pp 2477ndash2482 2009
[4] W F Chen Limit Analysis and Soil Plasticity Elsevier Amster-dam The Netherlands 1975
[5] R L Michalowski ldquoSlope stability analysis a kinematical ap-proachrdquo Geotechnique vol 45 no 2 pp 283ndash293 1995
[6] I B Donald and Z Y Chen ldquoSlope stability analysis by theupper bound approach fundamentals and methodsrdquo CanadianGeotechnical Journal vol 34 no 6 pp 853ndash862 1997
[7] J P Sun J C Li and Q Q Liu ldquoSearch for critical slip surfacein slope stability analysis by spline-based GA methodrdquo Journalof Geotechnical and Geoenvironmental Engineering vol 134 no2 pp 252ndash256 2008
[8] R Javadzadeh and E Javadzadeh ldquoLocating critical failure sur-face in rock slope stability with hybrid model based on artificialimmune system and Cellular Learning Automata (CLA-AIS)rdquoWorld Academy of Science Engineering and Technology vol 59pp 662ndash665 2011
[9] F N G Gitirana Jr and D G Fredlund ldquoAnalysis of tran-sient embankment stability using the dynamic programmingmethodrdquo in Proceedings of the 56th Canadian GeotechnicalConference 2003
[10] H T V Pham and D G Fredlund ldquoDynamic programmingmethod in slope stability computationsrdquo in Proceedings ofthe 12th Asian Regional Conference on Soil Mechanics andGeotechnica Engineering Singapore August 2003
[11] H T V Pham D G Fredlund and F N G Gitirana Jr ldquoSlopestability analysis using dynamic prograamming combined withfinite element stress analysisrdquo in Proceedings of the InternationalConference on theManagement of the Land andWater Resources(MLWR rsquo01) Hanoi Vietnam October 2001
[12] H T V Pham and D G Fredlund ldquoThe application of dynamicprogramming to slope stability analysisrdquoCanadianGeotechnicalJournal vol 40 no 4 pp 830ndash847 2003
[13] M R Taha M Khajehzadeh and A El-Shafie ldquoSlope stabilityassessment using optimization techniques an overviewrdquo Elec-tronic Journal of Geotechnical Engineering vol 15 pp 1901ndash19152010
[14] A U Rao and N Sabhahit ldquoGenetic algorithm in stabilityof non-homogeneous slopesrdquo in Proceedings of the 12th Inter-national Conference of International Association for ComputerMethods and Advances in Geomechanics (IACMAG rsquo08) GoaIndia October 2008
[15] Y-G Nie W-Q Liu J-Y Shi and W-B Zhao ldquoApplication ofaccelerating genetic algorithm for embankment slope stabilityanalysisrdquo China Journal of Highway and Transport vol 16 no4 p 16 2003
The Scientific World Journal 13
[16] Y-P Zhou S-J Li Y-X Liu H-L Sun and F Jiang ldquoSearchingcritical failure surface in slope stability analysis with geneticalgorithmrdquo Chinese Journal of Rock Mechanics and Engineeringvol 24 pp 5226ndash5230 2005
[17] P McCombie ldquoCritical failure surface location using SimpleGenetic Algorithm and multiple wedge slope stabilityrdquo in Pro-ceedings of the 1st International Symposium on ComputationalGeomechanics pp 900ndash909 Juan-les-Pins France May 2009
[18] K Gavin and J F Xue ldquoUse of a genetic algorithm to performreliability analysis of unsaturated soil slopesrdquoGeotechnique vol59 no 6 pp 545ndash549 2009
[19] Wikipedia ldquoGenetic algorithmrdquo 2012 httpenwikipediaorgwikiGenetic algorithm
[20] China University of Geosciences ldquoCohesive soil slope stabi-lity analysisrdquo httpcoursecugeducncugsoil mechanicsCOURSECHAPTER7Chap7 3 5htm
[21] J E Thomaz ldquoA general method for three dimensional slopestability analysis informational reportrdquo Tech Rep JTRP Tech-nical Reports Purdue University 1986
[22] T B Zhang ldquoEarth slope stability analysis and geotechnologyslope designrdquo Tech Rep Chengdu University of Science andTechnology Press 1987
[23] B G He Application Examples in Civil Engineering with AnsysChina Water Power Press Beijing China 2011
[24] Y Liu Q Yang and L Zhu ldquoAbutment stability analysis of archdam based on 3D nonlinear finite element methodrdquo ChineseJournal of Rock Mechanics and Engineering vol 27 supplement1 pp 3222ndash3228 2008
[25] Y R Liu ZHe B Li andQYang ldquoSlope stability analysis basedon a multigrid method using a nonlinear 3D finite elementmodelrdquo Frontiers of Structural and Civil Engineering vol 7 no1 pp 24ndash31 2013
[26] Wikipedia ldquoDruckerndashPrager yield criterionrdquo 2013 httpenwikipediaorgwikiDruckerE28093Prager yield criterion
[27] B Doran H O Koksal Z Polat and C Karakoc ldquoThe use ofldquoDrucker-Prager Criterionrdquo in the analysis of reinforced con-crete members by finite elementsrdquo Teknik Dergi vol 9 no 2pp 1617ndash1625 1998
[28] B Zoran P Verka and M Biljana ldquoMathematical modeling ofmaterially nonlinear problems in structural analyses (part Imdashtheoretical fundamentals)rdquo Facta Universitatis Architecture andCivil Engineering vol 8 no 1 pp 67ndash78 2010
[29] Q Yang X Chen and W-Y Zhou ldquoA practical 3D elasto-plastic incrementalmethod in FEMbased onD-P yield criteriardquoChinese Journal of Geotechnical Engineering vol 24 no 1 pp16ndash20 2002
[30] G D Zhang Soil Constitutive Model and Its Application inEngineering Science Press Beijing China 1995
[31] D Rakic M Zivkovic R Slavkovic and M Kojic ldquoStress inte-gration for the Drucker-Prager material model without hard-ening using the Incremental Plasticity Theoryrdquo Journal of theSerbian Society For Computational Mechanics vol 2 no 1 pp80ndash89 2008
[32] X Y Zhang Soil and Rock Plastic Mechanics China Communi-cations Press Beijing China 1993
[33] Y R Zheng Z J Shen and X N Gong The Principles of Geo-technical Plastic Mechanics China Architecture and BuildingPress Beijing China 2002
[34] S A Akers Two-dimensional finite element analysis of porousgeomaterials at multikilobar stress levels [PhD thesis] VirginiaPolytechnic Institute and State University 2001
[35] X-H Tan J-G Wang and Y Wang ldquoNonlinear finite elementanalysis of slope stabilityrdquo Rock and Soil Mechanics vol 29 no8 pp 2047ndash2050 2008
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Navigation and Observation
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DistributedSensor Networks
International Journal of
The Scientific World Journal 3
L
x
y
A
R
S
Search region
R
Potential 2D critical slip surface
Search region for Xc
Slope toe OXc
for Xcc
Xcc
Figure 2 2D potential failure surfaces
However in the next section all the computations accord-ing to GA are made by hand calculations If for computersimulation ldquointervalrdquo is as 119871 and 119904 as in Figure 2 and randomnumbers of computer function can help for automatic gen-eration of variables 119883
119888and 119883
119888119888in Figure 2 search regions or
boundaries for119883119888and119883
119888119888must be definedwith the definition
of the limits of119883119888and119883
119888119888 The necessity of these boundaries
is evident because computer program must avoid generatingsurfaces out of the region of primary interest [21]
The searching process for the 2D critical failure surfaceas in Figure 1 uses techniques inspired by natural evolutionsuch as inheritance mutation selection and crossover [19]In a genetic algorithm a population of strings (called chro-mosomes or the genotype of the genome) which encodecandidate solutions (called individuals creatures or pheno-types) to an optimization problem is evolved toward bettersolutions [19] In this 2D critical failure surface searchingproblem the candidate solutions are represented as 119883
119888 119883119888119888
described before The evolution starts from a population ofrandomly generated individuals as 119883
119888 119883119888119888 and happens
in generations In each generation the fitness of everyindividual in the population is evaluatedmultiple individualsare stochastically selected from the current population (basedon their fitness) and modified (recombined and possiblyrandomlymutated) to form a new populationThe evaluationstandard is that the individual 119883
119888 119883119888119888 with large safety
factor is deleted and the individual 119883119888 119883119888119888with small safety
factor is reserved The new population is then used in thenext iteration of the algorithm [19]The algorithm terminateswhen a satisfactory fitness level has been reached for thepopulation which means that it is hard to lower safety factorwith iterations
33 Searching Process with GA In the GA with hand cal-culation method the potential failure surfaces for searchare restricted that they all pass slope toe as in Figure 2 forsimplifying the search task Search process for the critical slipsurface with genetic algorithm was presented from Table 1to Table 12 In these tables the minimum safety factors aremarked with lowast in each iterationThe units for119883
119888and119883
119888119888are
meters Evaluation of individuals in Table 1 now begins The
Table 1 A population of randomly generated individuals
119883119888
119883119888119888
Safety factor51 25 272052352151 41 Safety factor is extremely large Unreasonable
lowast 51 0 13317465878lowast 60 minus10 14074551938lowast 60 25 1886105777
60 35 Safety factor is extremely large Unreasonablelowast 100 minus20 22757670756
100 21 23067661886100 70 Safety factor is extremely large Unreasonable
Table 2 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factorlowast 51 0 13317465878
60 minus10 1407455193860 25 1886105777100 minus20 22757670756
Table 3 Crossover results
119883119888
119883119888119888
Safety factorlowast 51 minus10 13799239885
51 25 272052352151 minus20 14447027254
lowast 60 0 1387959668960 minus20 14333540217100 0 2272574082100 minus10 22729514535100 25 23308411862
evaluation standard is that the individual 119883119888 119883119888119888with large
safety factor is deleted and the individual 119883119888 119883119888119888with small
safety factor is reserved Selection result will be put in Table 2Crossover of selected individuals of Table 2 will be put inTable 3 Mutation begins and results will be put in Table 4Evaluation of all previous individuals marked with lowast beginsThe evaluation standard is that the individual 119883
119888 119883119888119888 with
large safety factor is deleted and the individual 119883119888 119883119888119888with
small safety factor is reserved Selection result will be put inTable 5 Crossover of selected individuals of Table 5 beginsCrossover result will be put in Table 6 Mutation begins andthe result will be put in Table 7 Evaluation of all previousindividuals marked with lowast begins The evaluation standardis that the individual 119883
119888 119883119888119888 with large safety factor is
deleted and the individual 119883119888 119883119888119888with small safety factor is
reserved Selection result will be put in Table 8 Crossover ofselected individuals of Table 8 begins Crossover result willbe put in Table 9 Mutation begins and result will be putin Table 10 Evaluation of all previous individuals markedwith lowast begins The evaluation standard is that the individual119883119888 119883119888119888with large safety factor is deleted and the individual
119883119888 119883119888119888 with small safety factor is reserved Selection result
will be put in Table 11 The GA procedure terminates when asatisfactory fitness level has been reached for the population
4 The Scientific World Journal
Table 4 Mutation results
119883119888
119883119888119888
Safety factorlowast 50 0 13425306677
50 minus10 1400390984351 21 17627329758
lowast 55 0 13332021645lowast 55 minus10 13615830364
55 20 16493380867
Table 5 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factorlowast 51 0 13317465878
51 minus10 1379923988560 0 13879596689
lowast 50 0 13425306677lowast 55 0 13332021645
55 minus10 13615830364
Table 6 Crossover results
119883119888
119883119888119888
Safety factor51 0 1331746587851 minus10 1379923988560 0 1387959668950 0 1342530667755 0 1333202164555 minus10 13615830364
Table 7 Mutation results
119883119888
119883119888119888
Safety factorlowast 52 0 13264516064
52 minus10 13675053085lowast 53 0 13254640532
53 minus10 13611610322lowast 57 0 13503481321
57 minus10 13741239435lowast 52 11 13501275295
52 25 2515914533852 100 Safety factor is extremely large Unreasonable52 15 14194510174
which means that it is hard to lower safety factor withiterations The final result is in Table 12
34 Location of the Critical Failure Surface and Safety Factorwith GA Procedure The example was solved with foregoingGA procedure The minimum safety factor was 1325 with119883119888= 53m and 119883
119888119888= 0 The corresponding slip circle center
is at (0 688m) and the radius is 688m
Table 8 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factorlowast 51 0 13317465878lowast 50 0 13425306677lowast 55 0 13332021645lowast 52 0 13264516064lowast 53 0 13254640532
57 0 1350348132152 11 13501275295
Table 9 Crossover results
119883119888
119883119888119888
Safety factor51 11 1355689633150 11 1367290667655 11 13547892263
lowast 53 11 1348701176357 11 13696385778
Table 10 Mutation results
119883119888
119883119888119888
Safety factor54 21 16711004325
lowast 54 0 1327927158254 13 1373489757754 minus10 1359516051960 minus10 14074551938
Table 11 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factor51 0 1331746587850 0 1342530667755 0 1333202164552 0 13264516064
lowast 53 0 1325464053253 11 1348701176354 0 13279271582
Table 12 Critical failure surface and minimum safety factor
119883119888
119883119888119888
Safety factorCompleted 53 0 13254640532
4 Search the Critical Slip Surface withGenetic-Traversal Random Search MethodWhile Determining the Safety Factor withAnalytical Method
To solve the engineering problem in Section 2 this partwill search the critical failure surface with Genetic-TraversalRandom Search Method while determining the safety factorwith analytical method This part is realized with computerautomatic search program
The Scientific World Journal 5
Y
XO
A
B C
D
EF
G
H
r
m1
h
X dX
P(X Y)
Y1 = h minus Xm
Y1 = h
Y1 = 0
d120572x120572x
M(X Y2)Y2 = y + radicr2 minus (X minus x)2
Figure 3 Analytical method to determine the slope safety factor
41 Analytical Method to Determine the Slope Safety Factor inthe Above-Mentioned Slope Example in Section 2 [22] WithFelleniusrsquo method according to Zhang (1987) the analyticalsolution to give the safety factor in Figure 3 is
119896 =120574 sdot 119905119892120593 [119873] + 119888 [119871]
120574 [119879] (2)
where [119873] [119871] and [119879] were given as
119873 = [41199032minus 1199102]radic1199032 minus 1199102 + [4119903
2minus (ℎ minus 119910)
2]
times radic1199032 minus (ℎ minus 119910)2+
1
119898(21199032+ 1199092)radic1199032 minus 1199092
minus1
119898[21199032+ (119898ℎ minus 119909)
2]radic1199032 minus (119898ℎ minus 119909)
2
+119903
2
119910 arcsinradic1199032 minus 1199102
119903
minus (ℎ minus 119910) arcsinradic1199032 minus (ℎ minus 119910)
2
119903
+119909
119898arcsin 119909
119903minus
119898ℎ minus 119909
119898arcsin 119898ℎ minus 119909
119903
119879 =1
6119903[3ℎ1199032minus 1199103minus (ℎ minus 119910)
3minus
1199093
119898minus
(119898ℎ minus 119909)3
119898]
119871 = 119903[[
[
arcsinradic1199032 minus 1199102
119903+ arcsin
radic1199032 minus (ℎ minus 119910)2
119903
]]
]
(3)
where in Figure 3119875(119909 119910) is the potential failure circle center119903 is the circle radius119898 is the slope ratio ℎ is the slope height120574 is the slope soil unit weight 120593 is the soil internal frictionangle and 119888 is slope soil cohesion
42 Genetic-Traversal Random Search Method The slopestability problem example in Figure 4 is just the engineeringproblem in Section 2 Inspired by the genetic algorithmthe potential failure circle is represented with points 119860119861 and 119862 in Figure 4 The coordinates of 119860 119861 and 119862
are (119886 25) (0 119887) and (119888 0) respectively So in fact theparameters 119886 119887 and 119888 can represent the potential failurecircle In a novel Fortran program points 119860 119861 and 119862 arevaried randomly and helped with random number generatorsubprogram However points 119860 119861 and 119862 can only varyin a certain region with boundary Each group of 119886 119887 119888
gives a safety factor by (2) With random number generatorsubprogram and loop program enough groups of 119886 119887 119888 aregenerated Inspired by the genetic algorithm the relative lowsafety factor and corresponding 119886 119887 119888 are saved after eachcomparison between the old potential failure circle and thenew generated potential failure circle and helped with therandom number generator subprogram After enough timesof iterations set by the user the minimum safety factor andcorresponding 119886 119887 119888 will be determined
The safety factor and failure circle determination pro-gram developed in Silverfrost FTN95 was presented in theAppendix In fact the computer-aided genetic algorithm ofthe program presented in the Appendix only utilizes themutation and selection thought of the traditional geneticalgorithm Crossover is omitted due to the difficulty incomputer program realization and compensatedwith numer-ous random candidates due to mutation In fact geneticalgorithm (GA) is a random search method based on thebiological evolution law
43 Results of the Program in the Appendix according toGenetic-Traversal Random Search Method for the Above-Mentioned Slope Stability Problem Example After 100000times of potential failure circlesrsquo generation and selectionthe obtained minimum safety factor is 0648280 and thecorresponding 119886 119887 119888 is minus118283 327429 504410
5 Compared with Other Solutions
In order to validate the analytical solution to give safetyfactor of a specified slip surface Genetic-Traversal RandomSearchMethod to search for the critical failure surface and thecorresponding program presented in the Appendix this partwill solve the slope engineering problem in Section 2 withother methods
51 Solution of Searching the Critical Slip Surface with Fel-leniusrsquo Method While Determining the Safety Factor withFelleniusrsquo Method of Slices To solve the engineering problemin Section 2 this part will search the critical failure surfacewith Felleniusrsquo method while determining each safety factorwith Felleniusrsquo method of slices
6 The Scientific World Journal
x
y
Search region for point B
Search region for point CCirclecenter of 2D potential slipsurface
Search region for point A
O
Potential circular slip surface
A(a 25)
B(0 b)
C(c 0)
Figure 4 Potential failure circle center and radius determined and represented with points 119860 119861 and 119862
A
CB
O
1205732
1205731120572
Figure 5 Determination of 2D potential failure surfaces wheninternal friction angle 120593 = 0
Table 13 Determination of 1205731and 120573
2with slope angle 120572
Slope angle 120572 Slope ratio 1 m 1205731
1205732
60∘ 1 058 29∘ 40∘
45∘ 1 10 28∘ 37∘
33∘411015840 1 15 26∘ 35∘
26∘341015840 1 20 25∘ 35∘
18∘261015840 1 30 26∘ 35∘
14∘021015840 1 40 25∘ 36∘
11∘191015840 1 50 25∘ 39∘
In this solution safety factor is determinedwith Felleniusrsquomethod of slices as in (1) Felleniusrsquo method of searching forthe critical failure surface [20] was given as follows
If soil internal friction angle 120593 = 0 2D critical failuresurface passes through slope toe119860 and can be determined byFigure 5 and Table 13 In Figure 5 the critical failure surfacecircle center 119874 can be determined by angles 120573
1and 120573
2which
can be determined by slope angle 120572 as in Table 13 Angle 1205731
is the angle between line 119860119874 and slope surface line while 1205732
is the angle between line119874119861 and slope horizontal top surfaceline 119861119862 Point 119861 is the intersection between slope surface line119860119861 and slope horizontal top surface line 119861119862
If soil internal friction angle 120593 gt 0 2D critical failuresurface passes through slope toe and can be determined by
Figure 6 In Figure 6 point 119864 is determined by angles 1205731
and 1205732which can be determined by slope angle 120572 as in
Table 13 The critical failure surface circle center may be onthe extension line of the line 119863119864 You can try many pointson the line 119863119864 as the critical failure surface circle centercandidate like119874
1and119874
4on the line119863119864 If a point119874119909 on the
line 119863119864 is found to be the point which gives the minimumslope safety factor then draw a line 119865119866 perpendicular to theline119863119864 through the point 119874
119909 Then you can try many points
on the line 119865119866 as the critical failure surface circle centercandidate like 119874
1015840
1 11987410158402 11987410158403 and 119874
1015840
4 If a point on the line 119865119866
gives the minimum slope safety factor this point means theone that gives the final most minimum safety factor of thestudied slope
The determined minimum safety factor with Felleniusrsquomethod is 1320 while the 2D critical failure surface circlecenter is 45m 57776m in the 119909-119910 coordinate system ofFigure 2 and the radius is 57951m The corresponding 119883
119888in
Figure 2 is 52292m
52 Solution with Slopew Software To solve the engineeringproblem in Section 2 this part will determine the safety factorwith slopew software
With the Ordinarymethod theminimum safety factor tothe example slope is 1328 The corresponding critical failuresurface is presented in Figure 7 With the Bishop methodthe minimum safety factor to the example slope is 1390 Thecorresponding critical failure surface is presented in Figure 8With the Janbu method the minimum safety factor to theexample slope is 1316 The corresponding critical failuresurface is presented in Figure 9 With the Morgenstern-Pricemethod the minimum safety factor to the example slope is1389 The corresponding critical failure surface is presentedin Figure 10 With the Spencer method the minimum safetyfactor to the example slope is 1389 The correspondingcritical failure surface is presented in Figure 11 With the GLEmethod the minimum safety factor to the example slope is
The Scientific World Journal 7
A
B
F
H
H
D
G
ECircle center of 2D critical slip surface
O1
O4
1205732
1205731
Fs1
Fs4
C1 C2 C3 C4
45H
120572
O1998400
O2998400
O3998400
O4998400
Figure 6 Determination of 2D potential failure surfaces when internal friction angle 120593 gt 0 [20]
1328
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 7 Critical failure surface with the Ordinary method
1389 The corresponding critical failure surface is presentedin Figure 12With the Janbu generalizedmethod after solvingand analyzing then after selecting the critical slip surfacewith a safety factor of 1389 in the slopew software Figure 13appears and the ldquominimum factor of safetyrdquo shows that itsvalue is 1385 1385 is not identical with 1389 which is a littleweird The minimum safety factor determined by foregoingGAprocedure is 1325 After comparedwith slopew softwareforegoing GA procedure employed to search the criticalfailure surface is reasonable and applicable
53 Solution with Ansys Software To solve the engineeringproblem in Section 2 this part will determine the safety factor
1390
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 8 Critical failure surface with the Bishop method
with Ansys software [23] The slope has two layers which islayer 1 and layer 2 in Figure 14 Layer 1 is clay and Layer 2is bed rock The slope layer 1rsquos soil modulus of elasticity isassumed to be 20E7Nm2 The slope layer 1rsquos soil Poissonrsquosratio is assumed to be 03 The slope layer 1rsquos soil density isassumed to be 20408 Kgm3 The slope layer 1rsquos soil cohesionis 10000 Pa and friction angle is 266 degrees The slope layer2rsquos soil modulus of elasticity is assumed to be 32E10Nm2The slope layer 2rsquos soil Poissonrsquos ratio is assumed to be 024The slope layer 2rsquos soil density is assumed to be 2700Kgm3
The slope stability analysis problem is regarded as a plainstrain problem The left and right boundaries are restricted
8 The Scientific World Journal
1316
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 9 Critical failure surface with the Janbu method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 10 Critical failure surface with the Morgenstern-Pricemethod
horizontally The bottom boundary is restricted both hori-zontally and vertically With the drucker-prager model asthe constitutive model and with shear strength reductionmethod based on the finite element analysis the slope inFigure 14 is analyzed Assume that the real cohesion andinternal friction angle of a slope are 119888
0and 120593
0 respectively
In the shear strength reduction method when safety factoris SF the reduced cohesion and friction angle for analysis are1198880SF and 120593
0SF
The Drucker-Prager yield criterion is [24 25]
1198601198681+ radic1198692minus 119861 le 0 (4)
where 1198681= 1205901+ 1205902+ 1205903 1198692= (16)[(120590
1minus 1205902)2+ (1205902minus 1205903)2+
(1205903minus 1205901)2]
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 11 Critical failure surface with the Spencer method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 12 Critical failure surface determinedwith theGLEmethod
1385
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 13 Critical failure surface determined with the Janbugeneralized method
The Scientific World Journal 9
Layer1
Layer2
40
m80
m
80m 50m 70m
40
m105
m
25m
Figure 14 Studied region for the engineering problem in Section 2 treated with Ansys
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(a) Safety factor equals 07
Y
Z X
MXMN
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
Slope stability analysis
(b) Safety factor equals 072
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(c) Safety factor equals 073
Figure 15 No von Mises plastic strain
If we assume that the Drucker-Prager yield surfacetouches on the interior of the Mohr-Coulomb yield surfacethen the expressions [26ndash28] are
119860 =2 sin120593
radic3radic3 + sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 + sin120593
(5)
If the Drucker-Prager yield surface passes through theexternal apexes of the Mohr-Coulomb yield surface then[26 28 29]
119860 =2 sin120593
radic3radic3 minus sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 minus sin120593
(6)
where 119888 is cohesion and 120593 is internal friction angle
10 The Scientific World Journal
Y
Z X
MX
MN
Slope stability analysis0 0160E minus 04 0320E minus 04 0479E minus 04 0639E minus 04
0799E minus 05 0240E minus 04 0399E minus 04 0559E minus 04 0719E minus 04
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0719E minus 04
(a) Safety factor equals 074
Y
Z X
MX
MN
Slope stability analysis0 0627E minus 04 0125E minus 03 0251E minus 03
0314E minus 04 0941E minus 04 0157E minus 03 0219E minus 03 0282E minus 030188E minus 03
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0282E minus 03
(b) Safety factor equals 075
Y
Z X
MX
MN
Slope stability analysis0 0295E minus 03 0589E minus 03 0884E minus 03 0001178
0147E minus 03 0442E minus 03 0736E minus 03 0001031 0001325
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0001325
(c) Safety factor equals 08
Y
Z X
MX
MN
Slope stability analysis0 0926E minus 03 0001851 0002777 0003703
0463E minus 03 0001389 0002314 000324 0004166
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0004166
(d) Safety factor equals 10
Y
Z X
MX
MN
Slope stability analysis0 0006555 0013111 0019666 0026221
0003278 0009833 0016388 0022944 0029499
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9296
UNFORRFORACEL
SMX = 0029499
(e) Safety factor equals 16
Y
Z XMX
MN
Slope stability analysis0 001587 003174 0047609 0063479
0007935 0023805 0039674 0055544 0071414
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9795
UNFORRFORACEL
SMX = 0071414
(f) Safety factor equals 18
Figure 16 von Mises plastic strain occurs and develops
With the drucker-prager model as the constitutive modelto analyze the slope under only self-weight in Figure 14 theflow rule which describes the relationship between the plasticpotential function and the plastic strain could be foundin [23 30ndash34] The incremental elastic-plastic stress-strainrelationship and the corresponding elastic-plastic matrixcould be found in [23 35]
The results were presented as follows in Figures 15 16 and17
When safety factor is from 07 to 073 there is no vonMises plastic strain in slope in Figure 15 When safety factoris 074 there is local plastic strain occurring in slope inFigure 16 When safety factor is 20 von Mises plastic strainruns through from slope toe to top surface in Figure 17
According to Chen (1975) and Niu (2009) Figure 16 giveslower bound solutions of slope safety factor which are from074 to 18 And Figure 17 where vonMises plastic strain runsthrough from slope toe to top surface gives an upper bound
The Scientific World Journal 11
Y
Z X MX
MN
Slope stability analysis0 0033261 0066522 0099783 0133044
0016631 0049892 0083153 0116414 0149675
Nodal solutionStep = 1
Sub = 999999
Time = 1
EPPLEQV (avg)DMX = 6955
UNFORRFORACEL
SMX = 0149675
Figure 17 von Mises plastic strain runs through from slope toe totop surface when safety factor equals 20
solution of slope safety factor which is 20 So the true slopesafety factor is likely from 18 to 20
54 Comparisons and Discussions The obtained minimumsafety factor for the above slope stability problem examplewith Genetic-Traversal Random Search Method is so lowwhen compared with the other methods like slopew soft-ware This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This mayalso be due to the power of the computer to realize theGenetic-Traversal Random Search Method in the AppendixThe Genetic-Traversal Random Method uses random pickto utilize mutation Validation of these conclusions will beinvestigated in the future with more numeric tests
However the obtained minimum safety factor withGenetic-Traversal Random Search Method is very close tothe lower bound solutions of slope safety factor given by theAnsys software
After computation there is plastic strain in layer 2 regionin some pictures of Figures 16 and 17 This is unreasonablesince layer 2 is defined as elastic region in the analysis withAnsys This phenomenon will be investigated in the future
6 Conclusions
This paper intends to determine a cohesive soil slope safetyfactor with Felleniusrsquo method while the 2D critical failuresurface is searched with GA The 2D critical failure surfaceis represented with real-encoded chromosomes which arepotential critical surface locations variables 119883
119888and 119883
119888119888 GA
procedure for searching critical failure surface proceeds withhand calculations If for future computer automatic searchprogram with GA program code for inheritance mutationselection and crossover program code for random numbersand program code for search interval boundaries will beneeded The minimum safety factor of 1325 determined byforegoing GA procedure to search the critical slip surface isvery close to the minimum safety factor of 1320 determinedby Felleniusrsquo critical slip surface method After comparedwith slopew software the proposed foregoing GA procedure
employed to search the critical failure surface is reasonableapplicable and effective
At last a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is madewhile in the program random numbers generated by com-puter and search boundaries are included The Genetic-Traversal Random Method uses random pick to utilizemutation In the program the slope safety factor is given byanalytical solution rather than slices method Results indicatethat the new computer automatic search program can givevery low safety factor which is about half of the foregoingones This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This may alsobe due to the power of the random number generation sub-program computer operation speed and Genetic-TraversalRandom Method Further validation of the results will beinvestigated in the future However the obtained minimumsafety factor with Genetic-Traversal Random Search Methodis very close to the lower bound solutions of slope safety factorgiven by the Ansys software
Appendix
Safety factor and failure circle determination program devel-oped in Silverfrost FTN95
double precision seedreal nrndlreal NLnewsafetyfactorsafetyfactor=100000seed=50gama=20tanphi=05cohesion=10h=25m=2do 10 I=1100000a=0b=25c=50rdn=nrndl(seed)a=-20lowastrdnrdn=nrndl(seed)b=25+20lowastrdnrdn=nrndl(seed)c=50+20lowastrdnif(b==25) b=2501x=(clowastc(2lowastb)+(25+b)2-b2+alowasta(50-2lowastb))(cb+a(25-b))y=b2+(cb)lowast(x-c2)r=sqrt((x-c)lowast(x-c)+ylowasty)
12 The Scientific World Journal
AA=(4lowastrlowastr-ylowasty)lowastsqrt(rlowastr-ylowasty)BB=(4lowastrlowastr-(h-y)lowast(h-y))lowastsqrt(rlowastr-(h-y)lowast(h-y))CC=(1m)lowast(2lowastrlowastr+xlowastx)lowastsqrt(rlowastr-xlowastx)DD=(1m)lowast(2lowastrlowastr+(mlowasth-x)lowast(mlowasth-x))lowastsqrt(rlowastr-(mlowasth-x)lowast(mlowasth-x))EE=ylowastasin((sqrt(rlowastr-ylowasty))r)FF=(h-y)lowastasin((sqrt(rlowastr-(h-y)lowast(h-y)))r)GG=(xm)lowastasin(xr)-((mlowasth-x)m)lowastasin((mlowasth-x)r)N=(1(6lowastr))lowast(AA+BB+CC-DD)+(r2)lowast(EE-FF+GG)T=(1(6lowastr))lowast(3lowasthlowastrlowastr-ylowastylowasty-(h-y)lowast(h-y)lowast(h-y)-xlowastxlowastxm-(mlowasth-x)lowast(mlowasth-x)lowast(mlowasth-x)m)L=rlowast(asin((sqrt(rlowastr-ylowasty))r)+asin((sqrt(rlowastr-(h-y)lowast(h-y)))r))newsafetyfactor=(gamalowasttanphilowastN+cohesionlowastL)(gamalowastT)if(newsafetyfactorltsafetyfactor)safetyfactor=newsafetyfactorwrite(lowastlowast)safetyfactorwrite(lowastlowast)awrite(lowastlowast)bwrite(lowastlowast)c10 continueend programreal function nrndl(seed)double precision SUVseedS=655360U=20530V=138490M=seedSseed=seed-MlowastSseed=Ulowastseed+VM=seedSseed=seed-MlowastSnrndl=seedSreturnend
Conflict of Interests
The author declares that he has no financial or personalrelationships with other people or organizations that couldinappropriately influence his work The author of this paperhas chosen not to furnish the paper and its readers with infor-mation that might present a potential conflict of interests
Acknowledgment
The author acknowledges the Scientific Research StartingFunds at Liaoning Technical University (no 11-415)
References
[1] D G Fredlund and J Krahn ldquoComparison of slope stabilitymethods of analysisrdquoCanadianGeotechnical Journal vol 14 no3 pp 429ndash439 1977
[2] K S Li ldquoA unified solution scheme for slope stability analysisrdquoin Proceedings of the 5th International Symposium on LandslidesNew Zealand vol 1 pp 481ndash486 Balkema Rotterdam TheNetherlands 1992
[3] W-J NiuW-M Ye S-G Liu and H-T Yu ldquoLimit analysis of asoil slope considering saturated-unsaturated seepagerdquoRock andSoil Mechanics vol 30 no 8 pp 2477ndash2482 2009
[4] W F Chen Limit Analysis and Soil Plasticity Elsevier Amster-dam The Netherlands 1975
[5] R L Michalowski ldquoSlope stability analysis a kinematical ap-proachrdquo Geotechnique vol 45 no 2 pp 283ndash293 1995
[6] I B Donald and Z Y Chen ldquoSlope stability analysis by theupper bound approach fundamentals and methodsrdquo CanadianGeotechnical Journal vol 34 no 6 pp 853ndash862 1997
[7] J P Sun J C Li and Q Q Liu ldquoSearch for critical slip surfacein slope stability analysis by spline-based GA methodrdquo Journalof Geotechnical and Geoenvironmental Engineering vol 134 no2 pp 252ndash256 2008
[8] R Javadzadeh and E Javadzadeh ldquoLocating critical failure sur-face in rock slope stability with hybrid model based on artificialimmune system and Cellular Learning Automata (CLA-AIS)rdquoWorld Academy of Science Engineering and Technology vol 59pp 662ndash665 2011
[9] F N G Gitirana Jr and D G Fredlund ldquoAnalysis of tran-sient embankment stability using the dynamic programmingmethodrdquo in Proceedings of the 56th Canadian GeotechnicalConference 2003
[10] H T V Pham and D G Fredlund ldquoDynamic programmingmethod in slope stability computationsrdquo in Proceedings ofthe 12th Asian Regional Conference on Soil Mechanics andGeotechnica Engineering Singapore August 2003
[11] H T V Pham D G Fredlund and F N G Gitirana Jr ldquoSlopestability analysis using dynamic prograamming combined withfinite element stress analysisrdquo in Proceedings of the InternationalConference on theManagement of the Land andWater Resources(MLWR rsquo01) Hanoi Vietnam October 2001
[12] H T V Pham and D G Fredlund ldquoThe application of dynamicprogramming to slope stability analysisrdquoCanadianGeotechnicalJournal vol 40 no 4 pp 830ndash847 2003
[13] M R Taha M Khajehzadeh and A El-Shafie ldquoSlope stabilityassessment using optimization techniques an overviewrdquo Elec-tronic Journal of Geotechnical Engineering vol 15 pp 1901ndash19152010
[14] A U Rao and N Sabhahit ldquoGenetic algorithm in stabilityof non-homogeneous slopesrdquo in Proceedings of the 12th Inter-national Conference of International Association for ComputerMethods and Advances in Geomechanics (IACMAG rsquo08) GoaIndia October 2008
[15] Y-G Nie W-Q Liu J-Y Shi and W-B Zhao ldquoApplication ofaccelerating genetic algorithm for embankment slope stabilityanalysisrdquo China Journal of Highway and Transport vol 16 no4 p 16 2003
The Scientific World Journal 13
[16] Y-P Zhou S-J Li Y-X Liu H-L Sun and F Jiang ldquoSearchingcritical failure surface in slope stability analysis with geneticalgorithmrdquo Chinese Journal of Rock Mechanics and Engineeringvol 24 pp 5226ndash5230 2005
[17] P McCombie ldquoCritical failure surface location using SimpleGenetic Algorithm and multiple wedge slope stabilityrdquo in Pro-ceedings of the 1st International Symposium on ComputationalGeomechanics pp 900ndash909 Juan-les-Pins France May 2009
[18] K Gavin and J F Xue ldquoUse of a genetic algorithm to performreliability analysis of unsaturated soil slopesrdquoGeotechnique vol59 no 6 pp 545ndash549 2009
[19] Wikipedia ldquoGenetic algorithmrdquo 2012 httpenwikipediaorgwikiGenetic algorithm
[20] China University of Geosciences ldquoCohesive soil slope stabi-lity analysisrdquo httpcoursecugeducncugsoil mechanicsCOURSECHAPTER7Chap7 3 5htm
[21] J E Thomaz ldquoA general method for three dimensional slopestability analysis informational reportrdquo Tech Rep JTRP Tech-nical Reports Purdue University 1986
[22] T B Zhang ldquoEarth slope stability analysis and geotechnologyslope designrdquo Tech Rep Chengdu University of Science andTechnology Press 1987
[23] B G He Application Examples in Civil Engineering with AnsysChina Water Power Press Beijing China 2011
[24] Y Liu Q Yang and L Zhu ldquoAbutment stability analysis of archdam based on 3D nonlinear finite element methodrdquo ChineseJournal of Rock Mechanics and Engineering vol 27 supplement1 pp 3222ndash3228 2008
[25] Y R Liu ZHe B Li andQYang ldquoSlope stability analysis basedon a multigrid method using a nonlinear 3D finite elementmodelrdquo Frontiers of Structural and Civil Engineering vol 7 no1 pp 24ndash31 2013
[26] Wikipedia ldquoDruckerndashPrager yield criterionrdquo 2013 httpenwikipediaorgwikiDruckerE28093Prager yield criterion
[27] B Doran H O Koksal Z Polat and C Karakoc ldquoThe use ofldquoDrucker-Prager Criterionrdquo in the analysis of reinforced con-crete members by finite elementsrdquo Teknik Dergi vol 9 no 2pp 1617ndash1625 1998
[28] B Zoran P Verka and M Biljana ldquoMathematical modeling ofmaterially nonlinear problems in structural analyses (part Imdashtheoretical fundamentals)rdquo Facta Universitatis Architecture andCivil Engineering vol 8 no 1 pp 67ndash78 2010
[29] Q Yang X Chen and W-Y Zhou ldquoA practical 3D elasto-plastic incrementalmethod in FEMbased onD-P yield criteriardquoChinese Journal of Geotechnical Engineering vol 24 no 1 pp16ndash20 2002
[30] G D Zhang Soil Constitutive Model and Its Application inEngineering Science Press Beijing China 1995
[31] D Rakic M Zivkovic R Slavkovic and M Kojic ldquoStress inte-gration for the Drucker-Prager material model without hard-ening using the Incremental Plasticity Theoryrdquo Journal of theSerbian Society For Computational Mechanics vol 2 no 1 pp80ndash89 2008
[32] X Y Zhang Soil and Rock Plastic Mechanics China Communi-cations Press Beijing China 1993
[33] Y R Zheng Z J Shen and X N Gong The Principles of Geo-technical Plastic Mechanics China Architecture and BuildingPress Beijing China 2002
[34] S A Akers Two-dimensional finite element analysis of porousgeomaterials at multikilobar stress levels [PhD thesis] VirginiaPolytechnic Institute and State University 2001
[35] X-H Tan J-G Wang and Y Wang ldquoNonlinear finite elementanalysis of slope stabilityrdquo Rock and Soil Mechanics vol 29 no8 pp 2047ndash2050 2008
International Journal of
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4 The Scientific World Journal
Table 4 Mutation results
119883119888
119883119888119888
Safety factorlowast 50 0 13425306677
50 minus10 1400390984351 21 17627329758
lowast 55 0 13332021645lowast 55 minus10 13615830364
55 20 16493380867
Table 5 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factorlowast 51 0 13317465878
51 minus10 1379923988560 0 13879596689
lowast 50 0 13425306677lowast 55 0 13332021645
55 minus10 13615830364
Table 6 Crossover results
119883119888
119883119888119888
Safety factor51 0 1331746587851 minus10 1379923988560 0 1387959668950 0 1342530667755 0 1333202164555 minus10 13615830364
Table 7 Mutation results
119883119888
119883119888119888
Safety factorlowast 52 0 13264516064
52 minus10 13675053085lowast 53 0 13254640532
53 minus10 13611610322lowast 57 0 13503481321
57 minus10 13741239435lowast 52 11 13501275295
52 25 2515914533852 100 Safety factor is extremely large Unreasonable52 15 14194510174
which means that it is hard to lower safety factor withiterations The final result is in Table 12
34 Location of the Critical Failure Surface and Safety Factorwith GA Procedure The example was solved with foregoingGA procedure The minimum safety factor was 1325 with119883119888= 53m and 119883
119888119888= 0 The corresponding slip circle center
is at (0 688m) and the radius is 688m
Table 8 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factorlowast 51 0 13317465878lowast 50 0 13425306677lowast 55 0 13332021645lowast 52 0 13264516064lowast 53 0 13254640532
57 0 1350348132152 11 13501275295
Table 9 Crossover results
119883119888
119883119888119888
Safety factor51 11 1355689633150 11 1367290667655 11 13547892263
lowast 53 11 1348701176357 11 13696385778
Table 10 Mutation results
119883119888
119883119888119888
Safety factor54 21 16711004325
lowast 54 0 1327927158254 13 1373489757754 minus10 1359516051960 minus10 14074551938
Table 11 Selected individuals
Selected individuals 119883119888
119883119888119888
Safety factor51 0 1331746587850 0 1342530667755 0 1333202164552 0 13264516064
lowast 53 0 1325464053253 11 1348701176354 0 13279271582
Table 12 Critical failure surface and minimum safety factor
119883119888
119883119888119888
Safety factorCompleted 53 0 13254640532
4 Search the Critical Slip Surface withGenetic-Traversal Random Search MethodWhile Determining the Safety Factor withAnalytical Method
To solve the engineering problem in Section 2 this partwill search the critical failure surface with Genetic-TraversalRandom Search Method while determining the safety factorwith analytical method This part is realized with computerautomatic search program
The Scientific World Journal 5
Y
XO
A
B C
D
EF
G
H
r
m1
h
X dX
P(X Y)
Y1 = h minus Xm
Y1 = h
Y1 = 0
d120572x120572x
M(X Y2)Y2 = y + radicr2 minus (X minus x)2
Figure 3 Analytical method to determine the slope safety factor
41 Analytical Method to Determine the Slope Safety Factor inthe Above-Mentioned Slope Example in Section 2 [22] WithFelleniusrsquo method according to Zhang (1987) the analyticalsolution to give the safety factor in Figure 3 is
119896 =120574 sdot 119905119892120593 [119873] + 119888 [119871]
120574 [119879] (2)
where [119873] [119871] and [119879] were given as
119873 = [41199032minus 1199102]radic1199032 minus 1199102 + [4119903
2minus (ℎ minus 119910)
2]
times radic1199032 minus (ℎ minus 119910)2+
1
119898(21199032+ 1199092)radic1199032 minus 1199092
minus1
119898[21199032+ (119898ℎ minus 119909)
2]radic1199032 minus (119898ℎ minus 119909)
2
+119903
2
119910 arcsinradic1199032 minus 1199102
119903
minus (ℎ minus 119910) arcsinradic1199032 minus (ℎ minus 119910)
2
119903
+119909
119898arcsin 119909
119903minus
119898ℎ minus 119909
119898arcsin 119898ℎ minus 119909
119903
119879 =1
6119903[3ℎ1199032minus 1199103minus (ℎ minus 119910)
3minus
1199093
119898minus
(119898ℎ minus 119909)3
119898]
119871 = 119903[[
[
arcsinradic1199032 minus 1199102
119903+ arcsin
radic1199032 minus (ℎ minus 119910)2
119903
]]
]
(3)
where in Figure 3119875(119909 119910) is the potential failure circle center119903 is the circle radius119898 is the slope ratio ℎ is the slope height120574 is the slope soil unit weight 120593 is the soil internal frictionangle and 119888 is slope soil cohesion
42 Genetic-Traversal Random Search Method The slopestability problem example in Figure 4 is just the engineeringproblem in Section 2 Inspired by the genetic algorithmthe potential failure circle is represented with points 119860119861 and 119862 in Figure 4 The coordinates of 119860 119861 and 119862
are (119886 25) (0 119887) and (119888 0) respectively So in fact theparameters 119886 119887 and 119888 can represent the potential failurecircle In a novel Fortran program points 119860 119861 and 119862 arevaried randomly and helped with random number generatorsubprogram However points 119860 119861 and 119862 can only varyin a certain region with boundary Each group of 119886 119887 119888
gives a safety factor by (2) With random number generatorsubprogram and loop program enough groups of 119886 119887 119888 aregenerated Inspired by the genetic algorithm the relative lowsafety factor and corresponding 119886 119887 119888 are saved after eachcomparison between the old potential failure circle and thenew generated potential failure circle and helped with therandom number generator subprogram After enough timesof iterations set by the user the minimum safety factor andcorresponding 119886 119887 119888 will be determined
The safety factor and failure circle determination pro-gram developed in Silverfrost FTN95 was presented in theAppendix In fact the computer-aided genetic algorithm ofthe program presented in the Appendix only utilizes themutation and selection thought of the traditional geneticalgorithm Crossover is omitted due to the difficulty incomputer program realization and compensatedwith numer-ous random candidates due to mutation In fact geneticalgorithm (GA) is a random search method based on thebiological evolution law
43 Results of the Program in the Appendix according toGenetic-Traversal Random Search Method for the Above-Mentioned Slope Stability Problem Example After 100000times of potential failure circlesrsquo generation and selectionthe obtained minimum safety factor is 0648280 and thecorresponding 119886 119887 119888 is minus118283 327429 504410
5 Compared with Other Solutions
In order to validate the analytical solution to give safetyfactor of a specified slip surface Genetic-Traversal RandomSearchMethod to search for the critical failure surface and thecorresponding program presented in the Appendix this partwill solve the slope engineering problem in Section 2 withother methods
51 Solution of Searching the Critical Slip Surface with Fel-leniusrsquo Method While Determining the Safety Factor withFelleniusrsquo Method of Slices To solve the engineering problemin Section 2 this part will search the critical failure surfacewith Felleniusrsquo method while determining each safety factorwith Felleniusrsquo method of slices
6 The Scientific World Journal
x
y
Search region for point B
Search region for point CCirclecenter of 2D potential slipsurface
Search region for point A
O
Potential circular slip surface
A(a 25)
B(0 b)
C(c 0)
Figure 4 Potential failure circle center and radius determined and represented with points 119860 119861 and 119862
A
CB
O
1205732
1205731120572
Figure 5 Determination of 2D potential failure surfaces wheninternal friction angle 120593 = 0
Table 13 Determination of 1205731and 120573
2with slope angle 120572
Slope angle 120572 Slope ratio 1 m 1205731
1205732
60∘ 1 058 29∘ 40∘
45∘ 1 10 28∘ 37∘
33∘411015840 1 15 26∘ 35∘
26∘341015840 1 20 25∘ 35∘
18∘261015840 1 30 26∘ 35∘
14∘021015840 1 40 25∘ 36∘
11∘191015840 1 50 25∘ 39∘
In this solution safety factor is determinedwith Felleniusrsquomethod of slices as in (1) Felleniusrsquo method of searching forthe critical failure surface [20] was given as follows
If soil internal friction angle 120593 = 0 2D critical failuresurface passes through slope toe119860 and can be determined byFigure 5 and Table 13 In Figure 5 the critical failure surfacecircle center 119874 can be determined by angles 120573
1and 120573
2which
can be determined by slope angle 120572 as in Table 13 Angle 1205731
is the angle between line 119860119874 and slope surface line while 1205732
is the angle between line119874119861 and slope horizontal top surfaceline 119861119862 Point 119861 is the intersection between slope surface line119860119861 and slope horizontal top surface line 119861119862
If soil internal friction angle 120593 gt 0 2D critical failuresurface passes through slope toe and can be determined by
Figure 6 In Figure 6 point 119864 is determined by angles 1205731
and 1205732which can be determined by slope angle 120572 as in
Table 13 The critical failure surface circle center may be onthe extension line of the line 119863119864 You can try many pointson the line 119863119864 as the critical failure surface circle centercandidate like119874
1and119874
4on the line119863119864 If a point119874119909 on the
line 119863119864 is found to be the point which gives the minimumslope safety factor then draw a line 119865119866 perpendicular to theline119863119864 through the point 119874
119909 Then you can try many points
on the line 119865119866 as the critical failure surface circle centercandidate like 119874
1015840
1 11987410158402 11987410158403 and 119874
1015840
4 If a point on the line 119865119866
gives the minimum slope safety factor this point means theone that gives the final most minimum safety factor of thestudied slope
The determined minimum safety factor with Felleniusrsquomethod is 1320 while the 2D critical failure surface circlecenter is 45m 57776m in the 119909-119910 coordinate system ofFigure 2 and the radius is 57951m The corresponding 119883
119888in
Figure 2 is 52292m
52 Solution with Slopew Software To solve the engineeringproblem in Section 2 this part will determine the safety factorwith slopew software
With the Ordinarymethod theminimum safety factor tothe example slope is 1328 The corresponding critical failuresurface is presented in Figure 7 With the Bishop methodthe minimum safety factor to the example slope is 1390 Thecorresponding critical failure surface is presented in Figure 8With the Janbu method the minimum safety factor to theexample slope is 1316 The corresponding critical failuresurface is presented in Figure 9 With the Morgenstern-Pricemethod the minimum safety factor to the example slope is1389 The corresponding critical failure surface is presentedin Figure 10 With the Spencer method the minimum safetyfactor to the example slope is 1389 The correspondingcritical failure surface is presented in Figure 11 With the GLEmethod the minimum safety factor to the example slope is
The Scientific World Journal 7
A
B
F
H
H
D
G
ECircle center of 2D critical slip surface
O1
O4
1205732
1205731
Fs1
Fs4
C1 C2 C3 C4
45H
120572
O1998400
O2998400
O3998400
O4998400
Figure 6 Determination of 2D potential failure surfaces when internal friction angle 120593 gt 0 [20]
1328
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 7 Critical failure surface with the Ordinary method
1389 The corresponding critical failure surface is presentedin Figure 12With the Janbu generalizedmethod after solvingand analyzing then after selecting the critical slip surfacewith a safety factor of 1389 in the slopew software Figure 13appears and the ldquominimum factor of safetyrdquo shows that itsvalue is 1385 1385 is not identical with 1389 which is a littleweird The minimum safety factor determined by foregoingGAprocedure is 1325 After comparedwith slopew softwareforegoing GA procedure employed to search the criticalfailure surface is reasonable and applicable
53 Solution with Ansys Software To solve the engineeringproblem in Section 2 this part will determine the safety factor
1390
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 8 Critical failure surface with the Bishop method
with Ansys software [23] The slope has two layers which islayer 1 and layer 2 in Figure 14 Layer 1 is clay and Layer 2is bed rock The slope layer 1rsquos soil modulus of elasticity isassumed to be 20E7Nm2 The slope layer 1rsquos soil Poissonrsquosratio is assumed to be 03 The slope layer 1rsquos soil density isassumed to be 20408 Kgm3 The slope layer 1rsquos soil cohesionis 10000 Pa and friction angle is 266 degrees The slope layer2rsquos soil modulus of elasticity is assumed to be 32E10Nm2The slope layer 2rsquos soil Poissonrsquos ratio is assumed to be 024The slope layer 2rsquos soil density is assumed to be 2700Kgm3
The slope stability analysis problem is regarded as a plainstrain problem The left and right boundaries are restricted
8 The Scientific World Journal
1316
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 9 Critical failure surface with the Janbu method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 10 Critical failure surface with the Morgenstern-Pricemethod
horizontally The bottom boundary is restricted both hori-zontally and vertically With the drucker-prager model asthe constitutive model and with shear strength reductionmethod based on the finite element analysis the slope inFigure 14 is analyzed Assume that the real cohesion andinternal friction angle of a slope are 119888
0and 120593
0 respectively
In the shear strength reduction method when safety factoris SF the reduced cohesion and friction angle for analysis are1198880SF and 120593
0SF
The Drucker-Prager yield criterion is [24 25]
1198601198681+ radic1198692minus 119861 le 0 (4)
where 1198681= 1205901+ 1205902+ 1205903 1198692= (16)[(120590
1minus 1205902)2+ (1205902minus 1205903)2+
(1205903minus 1205901)2]
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 11 Critical failure surface with the Spencer method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 12 Critical failure surface determinedwith theGLEmethod
1385
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 13 Critical failure surface determined with the Janbugeneralized method
The Scientific World Journal 9
Layer1
Layer2
40
m80
m
80m 50m 70m
40
m105
m
25m
Figure 14 Studied region for the engineering problem in Section 2 treated with Ansys
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(a) Safety factor equals 07
Y
Z X
MXMN
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
Slope stability analysis
(b) Safety factor equals 072
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(c) Safety factor equals 073
Figure 15 No von Mises plastic strain
If we assume that the Drucker-Prager yield surfacetouches on the interior of the Mohr-Coulomb yield surfacethen the expressions [26ndash28] are
119860 =2 sin120593
radic3radic3 + sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 + sin120593
(5)
If the Drucker-Prager yield surface passes through theexternal apexes of the Mohr-Coulomb yield surface then[26 28 29]
119860 =2 sin120593
radic3radic3 minus sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 minus sin120593
(6)
where 119888 is cohesion and 120593 is internal friction angle
10 The Scientific World Journal
Y
Z X
MX
MN
Slope stability analysis0 0160E minus 04 0320E minus 04 0479E minus 04 0639E minus 04
0799E minus 05 0240E minus 04 0399E minus 04 0559E minus 04 0719E minus 04
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0719E minus 04
(a) Safety factor equals 074
Y
Z X
MX
MN
Slope stability analysis0 0627E minus 04 0125E minus 03 0251E minus 03
0314E minus 04 0941E minus 04 0157E minus 03 0219E minus 03 0282E minus 030188E minus 03
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0282E minus 03
(b) Safety factor equals 075
Y
Z X
MX
MN
Slope stability analysis0 0295E minus 03 0589E minus 03 0884E minus 03 0001178
0147E minus 03 0442E minus 03 0736E minus 03 0001031 0001325
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0001325
(c) Safety factor equals 08
Y
Z X
MX
MN
Slope stability analysis0 0926E minus 03 0001851 0002777 0003703
0463E minus 03 0001389 0002314 000324 0004166
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0004166
(d) Safety factor equals 10
Y
Z X
MX
MN
Slope stability analysis0 0006555 0013111 0019666 0026221
0003278 0009833 0016388 0022944 0029499
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9296
UNFORRFORACEL
SMX = 0029499
(e) Safety factor equals 16
Y
Z XMX
MN
Slope stability analysis0 001587 003174 0047609 0063479
0007935 0023805 0039674 0055544 0071414
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9795
UNFORRFORACEL
SMX = 0071414
(f) Safety factor equals 18
Figure 16 von Mises plastic strain occurs and develops
With the drucker-prager model as the constitutive modelto analyze the slope under only self-weight in Figure 14 theflow rule which describes the relationship between the plasticpotential function and the plastic strain could be foundin [23 30ndash34] The incremental elastic-plastic stress-strainrelationship and the corresponding elastic-plastic matrixcould be found in [23 35]
The results were presented as follows in Figures 15 16 and17
When safety factor is from 07 to 073 there is no vonMises plastic strain in slope in Figure 15 When safety factoris 074 there is local plastic strain occurring in slope inFigure 16 When safety factor is 20 von Mises plastic strainruns through from slope toe to top surface in Figure 17
According to Chen (1975) and Niu (2009) Figure 16 giveslower bound solutions of slope safety factor which are from074 to 18 And Figure 17 where vonMises plastic strain runsthrough from slope toe to top surface gives an upper bound
The Scientific World Journal 11
Y
Z X MX
MN
Slope stability analysis0 0033261 0066522 0099783 0133044
0016631 0049892 0083153 0116414 0149675
Nodal solutionStep = 1
Sub = 999999
Time = 1
EPPLEQV (avg)DMX = 6955
UNFORRFORACEL
SMX = 0149675
Figure 17 von Mises plastic strain runs through from slope toe totop surface when safety factor equals 20
solution of slope safety factor which is 20 So the true slopesafety factor is likely from 18 to 20
54 Comparisons and Discussions The obtained minimumsafety factor for the above slope stability problem examplewith Genetic-Traversal Random Search Method is so lowwhen compared with the other methods like slopew soft-ware This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This mayalso be due to the power of the computer to realize theGenetic-Traversal Random Search Method in the AppendixThe Genetic-Traversal Random Method uses random pickto utilize mutation Validation of these conclusions will beinvestigated in the future with more numeric tests
However the obtained minimum safety factor withGenetic-Traversal Random Search Method is very close tothe lower bound solutions of slope safety factor given by theAnsys software
After computation there is plastic strain in layer 2 regionin some pictures of Figures 16 and 17 This is unreasonablesince layer 2 is defined as elastic region in the analysis withAnsys This phenomenon will be investigated in the future
6 Conclusions
This paper intends to determine a cohesive soil slope safetyfactor with Felleniusrsquo method while the 2D critical failuresurface is searched with GA The 2D critical failure surfaceis represented with real-encoded chromosomes which arepotential critical surface locations variables 119883
119888and 119883
119888119888 GA
procedure for searching critical failure surface proceeds withhand calculations If for future computer automatic searchprogram with GA program code for inheritance mutationselection and crossover program code for random numbersand program code for search interval boundaries will beneeded The minimum safety factor of 1325 determined byforegoing GA procedure to search the critical slip surface isvery close to the minimum safety factor of 1320 determinedby Felleniusrsquo critical slip surface method After comparedwith slopew software the proposed foregoing GA procedure
employed to search the critical failure surface is reasonableapplicable and effective
At last a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is madewhile in the program random numbers generated by com-puter and search boundaries are included The Genetic-Traversal Random Method uses random pick to utilizemutation In the program the slope safety factor is given byanalytical solution rather than slices method Results indicatethat the new computer automatic search program can givevery low safety factor which is about half of the foregoingones This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This may alsobe due to the power of the random number generation sub-program computer operation speed and Genetic-TraversalRandom Method Further validation of the results will beinvestigated in the future However the obtained minimumsafety factor with Genetic-Traversal Random Search Methodis very close to the lower bound solutions of slope safety factorgiven by the Ansys software
Appendix
Safety factor and failure circle determination program devel-oped in Silverfrost FTN95
double precision seedreal nrndlreal NLnewsafetyfactorsafetyfactor=100000seed=50gama=20tanphi=05cohesion=10h=25m=2do 10 I=1100000a=0b=25c=50rdn=nrndl(seed)a=-20lowastrdnrdn=nrndl(seed)b=25+20lowastrdnrdn=nrndl(seed)c=50+20lowastrdnif(b==25) b=2501x=(clowastc(2lowastb)+(25+b)2-b2+alowasta(50-2lowastb))(cb+a(25-b))y=b2+(cb)lowast(x-c2)r=sqrt((x-c)lowast(x-c)+ylowasty)
12 The Scientific World Journal
AA=(4lowastrlowastr-ylowasty)lowastsqrt(rlowastr-ylowasty)BB=(4lowastrlowastr-(h-y)lowast(h-y))lowastsqrt(rlowastr-(h-y)lowast(h-y))CC=(1m)lowast(2lowastrlowastr+xlowastx)lowastsqrt(rlowastr-xlowastx)DD=(1m)lowast(2lowastrlowastr+(mlowasth-x)lowast(mlowasth-x))lowastsqrt(rlowastr-(mlowasth-x)lowast(mlowasth-x))EE=ylowastasin((sqrt(rlowastr-ylowasty))r)FF=(h-y)lowastasin((sqrt(rlowastr-(h-y)lowast(h-y)))r)GG=(xm)lowastasin(xr)-((mlowasth-x)m)lowastasin((mlowasth-x)r)N=(1(6lowastr))lowast(AA+BB+CC-DD)+(r2)lowast(EE-FF+GG)T=(1(6lowastr))lowast(3lowasthlowastrlowastr-ylowastylowasty-(h-y)lowast(h-y)lowast(h-y)-xlowastxlowastxm-(mlowasth-x)lowast(mlowasth-x)lowast(mlowasth-x)m)L=rlowast(asin((sqrt(rlowastr-ylowasty))r)+asin((sqrt(rlowastr-(h-y)lowast(h-y)))r))newsafetyfactor=(gamalowasttanphilowastN+cohesionlowastL)(gamalowastT)if(newsafetyfactorltsafetyfactor)safetyfactor=newsafetyfactorwrite(lowastlowast)safetyfactorwrite(lowastlowast)awrite(lowastlowast)bwrite(lowastlowast)c10 continueend programreal function nrndl(seed)double precision SUVseedS=655360U=20530V=138490M=seedSseed=seed-MlowastSseed=Ulowastseed+VM=seedSseed=seed-MlowastSnrndl=seedSreturnend
Conflict of Interests
The author declares that he has no financial or personalrelationships with other people or organizations that couldinappropriately influence his work The author of this paperhas chosen not to furnish the paper and its readers with infor-mation that might present a potential conflict of interests
Acknowledgment
The author acknowledges the Scientific Research StartingFunds at Liaoning Technical University (no 11-415)
References
[1] D G Fredlund and J Krahn ldquoComparison of slope stabilitymethods of analysisrdquoCanadianGeotechnical Journal vol 14 no3 pp 429ndash439 1977
[2] K S Li ldquoA unified solution scheme for slope stability analysisrdquoin Proceedings of the 5th International Symposium on LandslidesNew Zealand vol 1 pp 481ndash486 Balkema Rotterdam TheNetherlands 1992
[3] W-J NiuW-M Ye S-G Liu and H-T Yu ldquoLimit analysis of asoil slope considering saturated-unsaturated seepagerdquoRock andSoil Mechanics vol 30 no 8 pp 2477ndash2482 2009
[4] W F Chen Limit Analysis and Soil Plasticity Elsevier Amster-dam The Netherlands 1975
[5] R L Michalowski ldquoSlope stability analysis a kinematical ap-proachrdquo Geotechnique vol 45 no 2 pp 283ndash293 1995
[6] I B Donald and Z Y Chen ldquoSlope stability analysis by theupper bound approach fundamentals and methodsrdquo CanadianGeotechnical Journal vol 34 no 6 pp 853ndash862 1997
[7] J P Sun J C Li and Q Q Liu ldquoSearch for critical slip surfacein slope stability analysis by spline-based GA methodrdquo Journalof Geotechnical and Geoenvironmental Engineering vol 134 no2 pp 252ndash256 2008
[8] R Javadzadeh and E Javadzadeh ldquoLocating critical failure sur-face in rock slope stability with hybrid model based on artificialimmune system and Cellular Learning Automata (CLA-AIS)rdquoWorld Academy of Science Engineering and Technology vol 59pp 662ndash665 2011
[9] F N G Gitirana Jr and D G Fredlund ldquoAnalysis of tran-sient embankment stability using the dynamic programmingmethodrdquo in Proceedings of the 56th Canadian GeotechnicalConference 2003
[10] H T V Pham and D G Fredlund ldquoDynamic programmingmethod in slope stability computationsrdquo in Proceedings ofthe 12th Asian Regional Conference on Soil Mechanics andGeotechnica Engineering Singapore August 2003
[11] H T V Pham D G Fredlund and F N G Gitirana Jr ldquoSlopestability analysis using dynamic prograamming combined withfinite element stress analysisrdquo in Proceedings of the InternationalConference on theManagement of the Land andWater Resources(MLWR rsquo01) Hanoi Vietnam October 2001
[12] H T V Pham and D G Fredlund ldquoThe application of dynamicprogramming to slope stability analysisrdquoCanadianGeotechnicalJournal vol 40 no 4 pp 830ndash847 2003
[13] M R Taha M Khajehzadeh and A El-Shafie ldquoSlope stabilityassessment using optimization techniques an overviewrdquo Elec-tronic Journal of Geotechnical Engineering vol 15 pp 1901ndash19152010
[14] A U Rao and N Sabhahit ldquoGenetic algorithm in stabilityof non-homogeneous slopesrdquo in Proceedings of the 12th Inter-national Conference of International Association for ComputerMethods and Advances in Geomechanics (IACMAG rsquo08) GoaIndia October 2008
[15] Y-G Nie W-Q Liu J-Y Shi and W-B Zhao ldquoApplication ofaccelerating genetic algorithm for embankment slope stabilityanalysisrdquo China Journal of Highway and Transport vol 16 no4 p 16 2003
The Scientific World Journal 13
[16] Y-P Zhou S-J Li Y-X Liu H-L Sun and F Jiang ldquoSearchingcritical failure surface in slope stability analysis with geneticalgorithmrdquo Chinese Journal of Rock Mechanics and Engineeringvol 24 pp 5226ndash5230 2005
[17] P McCombie ldquoCritical failure surface location using SimpleGenetic Algorithm and multiple wedge slope stabilityrdquo in Pro-ceedings of the 1st International Symposium on ComputationalGeomechanics pp 900ndash909 Juan-les-Pins France May 2009
[18] K Gavin and J F Xue ldquoUse of a genetic algorithm to performreliability analysis of unsaturated soil slopesrdquoGeotechnique vol59 no 6 pp 545ndash549 2009
[19] Wikipedia ldquoGenetic algorithmrdquo 2012 httpenwikipediaorgwikiGenetic algorithm
[20] China University of Geosciences ldquoCohesive soil slope stabi-lity analysisrdquo httpcoursecugeducncugsoil mechanicsCOURSECHAPTER7Chap7 3 5htm
[21] J E Thomaz ldquoA general method for three dimensional slopestability analysis informational reportrdquo Tech Rep JTRP Tech-nical Reports Purdue University 1986
[22] T B Zhang ldquoEarth slope stability analysis and geotechnologyslope designrdquo Tech Rep Chengdu University of Science andTechnology Press 1987
[23] B G He Application Examples in Civil Engineering with AnsysChina Water Power Press Beijing China 2011
[24] Y Liu Q Yang and L Zhu ldquoAbutment stability analysis of archdam based on 3D nonlinear finite element methodrdquo ChineseJournal of Rock Mechanics and Engineering vol 27 supplement1 pp 3222ndash3228 2008
[25] Y R Liu ZHe B Li andQYang ldquoSlope stability analysis basedon a multigrid method using a nonlinear 3D finite elementmodelrdquo Frontiers of Structural and Civil Engineering vol 7 no1 pp 24ndash31 2013
[26] Wikipedia ldquoDruckerndashPrager yield criterionrdquo 2013 httpenwikipediaorgwikiDruckerE28093Prager yield criterion
[27] B Doran H O Koksal Z Polat and C Karakoc ldquoThe use ofldquoDrucker-Prager Criterionrdquo in the analysis of reinforced con-crete members by finite elementsrdquo Teknik Dergi vol 9 no 2pp 1617ndash1625 1998
[28] B Zoran P Verka and M Biljana ldquoMathematical modeling ofmaterially nonlinear problems in structural analyses (part Imdashtheoretical fundamentals)rdquo Facta Universitatis Architecture andCivil Engineering vol 8 no 1 pp 67ndash78 2010
[29] Q Yang X Chen and W-Y Zhou ldquoA practical 3D elasto-plastic incrementalmethod in FEMbased onD-P yield criteriardquoChinese Journal of Geotechnical Engineering vol 24 no 1 pp16ndash20 2002
[30] G D Zhang Soil Constitutive Model and Its Application inEngineering Science Press Beijing China 1995
[31] D Rakic M Zivkovic R Slavkovic and M Kojic ldquoStress inte-gration for the Drucker-Prager material model without hard-ening using the Incremental Plasticity Theoryrdquo Journal of theSerbian Society For Computational Mechanics vol 2 no 1 pp80ndash89 2008
[32] X Y Zhang Soil and Rock Plastic Mechanics China Communi-cations Press Beijing China 1993
[33] Y R Zheng Z J Shen and X N Gong The Principles of Geo-technical Plastic Mechanics China Architecture and BuildingPress Beijing China 2002
[34] S A Akers Two-dimensional finite element analysis of porousgeomaterials at multikilobar stress levels [PhD thesis] VirginiaPolytechnic Institute and State University 2001
[35] X-H Tan J-G Wang and Y Wang ldquoNonlinear finite elementanalysis of slope stabilityrdquo Rock and Soil Mechanics vol 29 no8 pp 2047ndash2050 2008
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The Scientific World Journal 5
Y
XO
A
B C
D
EF
G
H
r
m1
h
X dX
P(X Y)
Y1 = h minus Xm
Y1 = h
Y1 = 0
d120572x120572x
M(X Y2)Y2 = y + radicr2 minus (X minus x)2
Figure 3 Analytical method to determine the slope safety factor
41 Analytical Method to Determine the Slope Safety Factor inthe Above-Mentioned Slope Example in Section 2 [22] WithFelleniusrsquo method according to Zhang (1987) the analyticalsolution to give the safety factor in Figure 3 is
119896 =120574 sdot 119905119892120593 [119873] + 119888 [119871]
120574 [119879] (2)
where [119873] [119871] and [119879] were given as
119873 = [41199032minus 1199102]radic1199032 minus 1199102 + [4119903
2minus (ℎ minus 119910)
2]
times radic1199032 minus (ℎ minus 119910)2+
1
119898(21199032+ 1199092)radic1199032 minus 1199092
minus1
119898[21199032+ (119898ℎ minus 119909)
2]radic1199032 minus (119898ℎ minus 119909)
2
+119903
2
119910 arcsinradic1199032 minus 1199102
119903
minus (ℎ minus 119910) arcsinradic1199032 minus (ℎ minus 119910)
2
119903
+119909
119898arcsin 119909
119903minus
119898ℎ minus 119909
119898arcsin 119898ℎ minus 119909
119903
119879 =1
6119903[3ℎ1199032minus 1199103minus (ℎ minus 119910)
3minus
1199093
119898minus
(119898ℎ minus 119909)3
119898]
119871 = 119903[[
[
arcsinradic1199032 minus 1199102
119903+ arcsin
radic1199032 minus (ℎ minus 119910)2
119903
]]
]
(3)
where in Figure 3119875(119909 119910) is the potential failure circle center119903 is the circle radius119898 is the slope ratio ℎ is the slope height120574 is the slope soil unit weight 120593 is the soil internal frictionangle and 119888 is slope soil cohesion
42 Genetic-Traversal Random Search Method The slopestability problem example in Figure 4 is just the engineeringproblem in Section 2 Inspired by the genetic algorithmthe potential failure circle is represented with points 119860119861 and 119862 in Figure 4 The coordinates of 119860 119861 and 119862
are (119886 25) (0 119887) and (119888 0) respectively So in fact theparameters 119886 119887 and 119888 can represent the potential failurecircle In a novel Fortran program points 119860 119861 and 119862 arevaried randomly and helped with random number generatorsubprogram However points 119860 119861 and 119862 can only varyin a certain region with boundary Each group of 119886 119887 119888
gives a safety factor by (2) With random number generatorsubprogram and loop program enough groups of 119886 119887 119888 aregenerated Inspired by the genetic algorithm the relative lowsafety factor and corresponding 119886 119887 119888 are saved after eachcomparison between the old potential failure circle and thenew generated potential failure circle and helped with therandom number generator subprogram After enough timesof iterations set by the user the minimum safety factor andcorresponding 119886 119887 119888 will be determined
The safety factor and failure circle determination pro-gram developed in Silverfrost FTN95 was presented in theAppendix In fact the computer-aided genetic algorithm ofthe program presented in the Appendix only utilizes themutation and selection thought of the traditional geneticalgorithm Crossover is omitted due to the difficulty incomputer program realization and compensatedwith numer-ous random candidates due to mutation In fact geneticalgorithm (GA) is a random search method based on thebiological evolution law
43 Results of the Program in the Appendix according toGenetic-Traversal Random Search Method for the Above-Mentioned Slope Stability Problem Example After 100000times of potential failure circlesrsquo generation and selectionthe obtained minimum safety factor is 0648280 and thecorresponding 119886 119887 119888 is minus118283 327429 504410
5 Compared with Other Solutions
In order to validate the analytical solution to give safetyfactor of a specified slip surface Genetic-Traversal RandomSearchMethod to search for the critical failure surface and thecorresponding program presented in the Appendix this partwill solve the slope engineering problem in Section 2 withother methods
51 Solution of Searching the Critical Slip Surface with Fel-leniusrsquo Method While Determining the Safety Factor withFelleniusrsquo Method of Slices To solve the engineering problemin Section 2 this part will search the critical failure surfacewith Felleniusrsquo method while determining each safety factorwith Felleniusrsquo method of slices
6 The Scientific World Journal
x
y
Search region for point B
Search region for point CCirclecenter of 2D potential slipsurface
Search region for point A
O
Potential circular slip surface
A(a 25)
B(0 b)
C(c 0)
Figure 4 Potential failure circle center and radius determined and represented with points 119860 119861 and 119862
A
CB
O
1205732
1205731120572
Figure 5 Determination of 2D potential failure surfaces wheninternal friction angle 120593 = 0
Table 13 Determination of 1205731and 120573
2with slope angle 120572
Slope angle 120572 Slope ratio 1 m 1205731
1205732
60∘ 1 058 29∘ 40∘
45∘ 1 10 28∘ 37∘
33∘411015840 1 15 26∘ 35∘
26∘341015840 1 20 25∘ 35∘
18∘261015840 1 30 26∘ 35∘
14∘021015840 1 40 25∘ 36∘
11∘191015840 1 50 25∘ 39∘
In this solution safety factor is determinedwith Felleniusrsquomethod of slices as in (1) Felleniusrsquo method of searching forthe critical failure surface [20] was given as follows
If soil internal friction angle 120593 = 0 2D critical failuresurface passes through slope toe119860 and can be determined byFigure 5 and Table 13 In Figure 5 the critical failure surfacecircle center 119874 can be determined by angles 120573
1and 120573
2which
can be determined by slope angle 120572 as in Table 13 Angle 1205731
is the angle between line 119860119874 and slope surface line while 1205732
is the angle between line119874119861 and slope horizontal top surfaceline 119861119862 Point 119861 is the intersection between slope surface line119860119861 and slope horizontal top surface line 119861119862
If soil internal friction angle 120593 gt 0 2D critical failuresurface passes through slope toe and can be determined by
Figure 6 In Figure 6 point 119864 is determined by angles 1205731
and 1205732which can be determined by slope angle 120572 as in
Table 13 The critical failure surface circle center may be onthe extension line of the line 119863119864 You can try many pointson the line 119863119864 as the critical failure surface circle centercandidate like119874
1and119874
4on the line119863119864 If a point119874119909 on the
line 119863119864 is found to be the point which gives the minimumslope safety factor then draw a line 119865119866 perpendicular to theline119863119864 through the point 119874
119909 Then you can try many points
on the line 119865119866 as the critical failure surface circle centercandidate like 119874
1015840
1 11987410158402 11987410158403 and 119874
1015840
4 If a point on the line 119865119866
gives the minimum slope safety factor this point means theone that gives the final most minimum safety factor of thestudied slope
The determined minimum safety factor with Felleniusrsquomethod is 1320 while the 2D critical failure surface circlecenter is 45m 57776m in the 119909-119910 coordinate system ofFigure 2 and the radius is 57951m The corresponding 119883
119888in
Figure 2 is 52292m
52 Solution with Slopew Software To solve the engineeringproblem in Section 2 this part will determine the safety factorwith slopew software
With the Ordinarymethod theminimum safety factor tothe example slope is 1328 The corresponding critical failuresurface is presented in Figure 7 With the Bishop methodthe minimum safety factor to the example slope is 1390 Thecorresponding critical failure surface is presented in Figure 8With the Janbu method the minimum safety factor to theexample slope is 1316 The corresponding critical failuresurface is presented in Figure 9 With the Morgenstern-Pricemethod the minimum safety factor to the example slope is1389 The corresponding critical failure surface is presentedin Figure 10 With the Spencer method the minimum safetyfactor to the example slope is 1389 The correspondingcritical failure surface is presented in Figure 11 With the GLEmethod the minimum safety factor to the example slope is
The Scientific World Journal 7
A
B
F
H
H
D
G
ECircle center of 2D critical slip surface
O1
O4
1205732
1205731
Fs1
Fs4
C1 C2 C3 C4
45H
120572
O1998400
O2998400
O3998400
O4998400
Figure 6 Determination of 2D potential failure surfaces when internal friction angle 120593 gt 0 [20]
1328
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 7 Critical failure surface with the Ordinary method
1389 The corresponding critical failure surface is presentedin Figure 12With the Janbu generalizedmethod after solvingand analyzing then after selecting the critical slip surfacewith a safety factor of 1389 in the slopew software Figure 13appears and the ldquominimum factor of safetyrdquo shows that itsvalue is 1385 1385 is not identical with 1389 which is a littleweird The minimum safety factor determined by foregoingGAprocedure is 1325 After comparedwith slopew softwareforegoing GA procedure employed to search the criticalfailure surface is reasonable and applicable
53 Solution with Ansys Software To solve the engineeringproblem in Section 2 this part will determine the safety factor
1390
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 8 Critical failure surface with the Bishop method
with Ansys software [23] The slope has two layers which islayer 1 and layer 2 in Figure 14 Layer 1 is clay and Layer 2is bed rock The slope layer 1rsquos soil modulus of elasticity isassumed to be 20E7Nm2 The slope layer 1rsquos soil Poissonrsquosratio is assumed to be 03 The slope layer 1rsquos soil density isassumed to be 20408 Kgm3 The slope layer 1rsquos soil cohesionis 10000 Pa and friction angle is 266 degrees The slope layer2rsquos soil modulus of elasticity is assumed to be 32E10Nm2The slope layer 2rsquos soil Poissonrsquos ratio is assumed to be 024The slope layer 2rsquos soil density is assumed to be 2700Kgm3
The slope stability analysis problem is regarded as a plainstrain problem The left and right boundaries are restricted
8 The Scientific World Journal
1316
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 9 Critical failure surface with the Janbu method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 10 Critical failure surface with the Morgenstern-Pricemethod
horizontally The bottom boundary is restricted both hori-zontally and vertically With the drucker-prager model asthe constitutive model and with shear strength reductionmethod based on the finite element analysis the slope inFigure 14 is analyzed Assume that the real cohesion andinternal friction angle of a slope are 119888
0and 120593
0 respectively
In the shear strength reduction method when safety factoris SF the reduced cohesion and friction angle for analysis are1198880SF and 120593
0SF
The Drucker-Prager yield criterion is [24 25]
1198601198681+ radic1198692minus 119861 le 0 (4)
where 1198681= 1205901+ 1205902+ 1205903 1198692= (16)[(120590
1minus 1205902)2+ (1205902minus 1205903)2+
(1205903minus 1205901)2]
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 11 Critical failure surface with the Spencer method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 12 Critical failure surface determinedwith theGLEmethod
1385
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 13 Critical failure surface determined with the Janbugeneralized method
The Scientific World Journal 9
Layer1
Layer2
40
m80
m
80m 50m 70m
40
m105
m
25m
Figure 14 Studied region for the engineering problem in Section 2 treated with Ansys
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(a) Safety factor equals 07
Y
Z X
MXMN
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
Slope stability analysis
(b) Safety factor equals 072
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(c) Safety factor equals 073
Figure 15 No von Mises plastic strain
If we assume that the Drucker-Prager yield surfacetouches on the interior of the Mohr-Coulomb yield surfacethen the expressions [26ndash28] are
119860 =2 sin120593
radic3radic3 + sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 + sin120593
(5)
If the Drucker-Prager yield surface passes through theexternal apexes of the Mohr-Coulomb yield surface then[26 28 29]
119860 =2 sin120593
radic3radic3 minus sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 minus sin120593
(6)
where 119888 is cohesion and 120593 is internal friction angle
10 The Scientific World Journal
Y
Z X
MX
MN
Slope stability analysis0 0160E minus 04 0320E minus 04 0479E minus 04 0639E minus 04
0799E minus 05 0240E minus 04 0399E minus 04 0559E minus 04 0719E minus 04
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0719E minus 04
(a) Safety factor equals 074
Y
Z X
MX
MN
Slope stability analysis0 0627E minus 04 0125E minus 03 0251E minus 03
0314E minus 04 0941E minus 04 0157E minus 03 0219E minus 03 0282E minus 030188E minus 03
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0282E minus 03
(b) Safety factor equals 075
Y
Z X
MX
MN
Slope stability analysis0 0295E minus 03 0589E minus 03 0884E minus 03 0001178
0147E minus 03 0442E minus 03 0736E minus 03 0001031 0001325
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0001325
(c) Safety factor equals 08
Y
Z X
MX
MN
Slope stability analysis0 0926E minus 03 0001851 0002777 0003703
0463E minus 03 0001389 0002314 000324 0004166
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0004166
(d) Safety factor equals 10
Y
Z X
MX
MN
Slope stability analysis0 0006555 0013111 0019666 0026221
0003278 0009833 0016388 0022944 0029499
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9296
UNFORRFORACEL
SMX = 0029499
(e) Safety factor equals 16
Y
Z XMX
MN
Slope stability analysis0 001587 003174 0047609 0063479
0007935 0023805 0039674 0055544 0071414
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9795
UNFORRFORACEL
SMX = 0071414
(f) Safety factor equals 18
Figure 16 von Mises plastic strain occurs and develops
With the drucker-prager model as the constitutive modelto analyze the slope under only self-weight in Figure 14 theflow rule which describes the relationship between the plasticpotential function and the plastic strain could be foundin [23 30ndash34] The incremental elastic-plastic stress-strainrelationship and the corresponding elastic-plastic matrixcould be found in [23 35]
The results were presented as follows in Figures 15 16 and17
When safety factor is from 07 to 073 there is no vonMises plastic strain in slope in Figure 15 When safety factoris 074 there is local plastic strain occurring in slope inFigure 16 When safety factor is 20 von Mises plastic strainruns through from slope toe to top surface in Figure 17
According to Chen (1975) and Niu (2009) Figure 16 giveslower bound solutions of slope safety factor which are from074 to 18 And Figure 17 where vonMises plastic strain runsthrough from slope toe to top surface gives an upper bound
The Scientific World Journal 11
Y
Z X MX
MN
Slope stability analysis0 0033261 0066522 0099783 0133044
0016631 0049892 0083153 0116414 0149675
Nodal solutionStep = 1
Sub = 999999
Time = 1
EPPLEQV (avg)DMX = 6955
UNFORRFORACEL
SMX = 0149675
Figure 17 von Mises plastic strain runs through from slope toe totop surface when safety factor equals 20
solution of slope safety factor which is 20 So the true slopesafety factor is likely from 18 to 20
54 Comparisons and Discussions The obtained minimumsafety factor for the above slope stability problem examplewith Genetic-Traversal Random Search Method is so lowwhen compared with the other methods like slopew soft-ware This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This mayalso be due to the power of the computer to realize theGenetic-Traversal Random Search Method in the AppendixThe Genetic-Traversal Random Method uses random pickto utilize mutation Validation of these conclusions will beinvestigated in the future with more numeric tests
However the obtained minimum safety factor withGenetic-Traversal Random Search Method is very close tothe lower bound solutions of slope safety factor given by theAnsys software
After computation there is plastic strain in layer 2 regionin some pictures of Figures 16 and 17 This is unreasonablesince layer 2 is defined as elastic region in the analysis withAnsys This phenomenon will be investigated in the future
6 Conclusions
This paper intends to determine a cohesive soil slope safetyfactor with Felleniusrsquo method while the 2D critical failuresurface is searched with GA The 2D critical failure surfaceis represented with real-encoded chromosomes which arepotential critical surface locations variables 119883
119888and 119883
119888119888 GA
procedure for searching critical failure surface proceeds withhand calculations If for future computer automatic searchprogram with GA program code for inheritance mutationselection and crossover program code for random numbersand program code for search interval boundaries will beneeded The minimum safety factor of 1325 determined byforegoing GA procedure to search the critical slip surface isvery close to the minimum safety factor of 1320 determinedby Felleniusrsquo critical slip surface method After comparedwith slopew software the proposed foregoing GA procedure
employed to search the critical failure surface is reasonableapplicable and effective
At last a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is madewhile in the program random numbers generated by com-puter and search boundaries are included The Genetic-Traversal Random Method uses random pick to utilizemutation In the program the slope safety factor is given byanalytical solution rather than slices method Results indicatethat the new computer automatic search program can givevery low safety factor which is about half of the foregoingones This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This may alsobe due to the power of the random number generation sub-program computer operation speed and Genetic-TraversalRandom Method Further validation of the results will beinvestigated in the future However the obtained minimumsafety factor with Genetic-Traversal Random Search Methodis very close to the lower bound solutions of slope safety factorgiven by the Ansys software
Appendix
Safety factor and failure circle determination program devel-oped in Silverfrost FTN95
double precision seedreal nrndlreal NLnewsafetyfactorsafetyfactor=100000seed=50gama=20tanphi=05cohesion=10h=25m=2do 10 I=1100000a=0b=25c=50rdn=nrndl(seed)a=-20lowastrdnrdn=nrndl(seed)b=25+20lowastrdnrdn=nrndl(seed)c=50+20lowastrdnif(b==25) b=2501x=(clowastc(2lowastb)+(25+b)2-b2+alowasta(50-2lowastb))(cb+a(25-b))y=b2+(cb)lowast(x-c2)r=sqrt((x-c)lowast(x-c)+ylowasty)
12 The Scientific World Journal
AA=(4lowastrlowastr-ylowasty)lowastsqrt(rlowastr-ylowasty)BB=(4lowastrlowastr-(h-y)lowast(h-y))lowastsqrt(rlowastr-(h-y)lowast(h-y))CC=(1m)lowast(2lowastrlowastr+xlowastx)lowastsqrt(rlowastr-xlowastx)DD=(1m)lowast(2lowastrlowastr+(mlowasth-x)lowast(mlowasth-x))lowastsqrt(rlowastr-(mlowasth-x)lowast(mlowasth-x))EE=ylowastasin((sqrt(rlowastr-ylowasty))r)FF=(h-y)lowastasin((sqrt(rlowastr-(h-y)lowast(h-y)))r)GG=(xm)lowastasin(xr)-((mlowasth-x)m)lowastasin((mlowasth-x)r)N=(1(6lowastr))lowast(AA+BB+CC-DD)+(r2)lowast(EE-FF+GG)T=(1(6lowastr))lowast(3lowasthlowastrlowastr-ylowastylowasty-(h-y)lowast(h-y)lowast(h-y)-xlowastxlowastxm-(mlowasth-x)lowast(mlowasth-x)lowast(mlowasth-x)m)L=rlowast(asin((sqrt(rlowastr-ylowasty))r)+asin((sqrt(rlowastr-(h-y)lowast(h-y)))r))newsafetyfactor=(gamalowasttanphilowastN+cohesionlowastL)(gamalowastT)if(newsafetyfactorltsafetyfactor)safetyfactor=newsafetyfactorwrite(lowastlowast)safetyfactorwrite(lowastlowast)awrite(lowastlowast)bwrite(lowastlowast)c10 continueend programreal function nrndl(seed)double precision SUVseedS=655360U=20530V=138490M=seedSseed=seed-MlowastSseed=Ulowastseed+VM=seedSseed=seed-MlowastSnrndl=seedSreturnend
Conflict of Interests
The author declares that he has no financial or personalrelationships with other people or organizations that couldinappropriately influence his work The author of this paperhas chosen not to furnish the paper and its readers with infor-mation that might present a potential conflict of interests
Acknowledgment
The author acknowledges the Scientific Research StartingFunds at Liaoning Technical University (no 11-415)
References
[1] D G Fredlund and J Krahn ldquoComparison of slope stabilitymethods of analysisrdquoCanadianGeotechnical Journal vol 14 no3 pp 429ndash439 1977
[2] K S Li ldquoA unified solution scheme for slope stability analysisrdquoin Proceedings of the 5th International Symposium on LandslidesNew Zealand vol 1 pp 481ndash486 Balkema Rotterdam TheNetherlands 1992
[3] W-J NiuW-M Ye S-G Liu and H-T Yu ldquoLimit analysis of asoil slope considering saturated-unsaturated seepagerdquoRock andSoil Mechanics vol 30 no 8 pp 2477ndash2482 2009
[4] W F Chen Limit Analysis and Soil Plasticity Elsevier Amster-dam The Netherlands 1975
[5] R L Michalowski ldquoSlope stability analysis a kinematical ap-proachrdquo Geotechnique vol 45 no 2 pp 283ndash293 1995
[6] I B Donald and Z Y Chen ldquoSlope stability analysis by theupper bound approach fundamentals and methodsrdquo CanadianGeotechnical Journal vol 34 no 6 pp 853ndash862 1997
[7] J P Sun J C Li and Q Q Liu ldquoSearch for critical slip surfacein slope stability analysis by spline-based GA methodrdquo Journalof Geotechnical and Geoenvironmental Engineering vol 134 no2 pp 252ndash256 2008
[8] R Javadzadeh and E Javadzadeh ldquoLocating critical failure sur-face in rock slope stability with hybrid model based on artificialimmune system and Cellular Learning Automata (CLA-AIS)rdquoWorld Academy of Science Engineering and Technology vol 59pp 662ndash665 2011
[9] F N G Gitirana Jr and D G Fredlund ldquoAnalysis of tran-sient embankment stability using the dynamic programmingmethodrdquo in Proceedings of the 56th Canadian GeotechnicalConference 2003
[10] H T V Pham and D G Fredlund ldquoDynamic programmingmethod in slope stability computationsrdquo in Proceedings ofthe 12th Asian Regional Conference on Soil Mechanics andGeotechnica Engineering Singapore August 2003
[11] H T V Pham D G Fredlund and F N G Gitirana Jr ldquoSlopestability analysis using dynamic prograamming combined withfinite element stress analysisrdquo in Proceedings of the InternationalConference on theManagement of the Land andWater Resources(MLWR rsquo01) Hanoi Vietnam October 2001
[12] H T V Pham and D G Fredlund ldquoThe application of dynamicprogramming to slope stability analysisrdquoCanadianGeotechnicalJournal vol 40 no 4 pp 830ndash847 2003
[13] M R Taha M Khajehzadeh and A El-Shafie ldquoSlope stabilityassessment using optimization techniques an overviewrdquo Elec-tronic Journal of Geotechnical Engineering vol 15 pp 1901ndash19152010
[14] A U Rao and N Sabhahit ldquoGenetic algorithm in stabilityof non-homogeneous slopesrdquo in Proceedings of the 12th Inter-national Conference of International Association for ComputerMethods and Advances in Geomechanics (IACMAG rsquo08) GoaIndia October 2008
[15] Y-G Nie W-Q Liu J-Y Shi and W-B Zhao ldquoApplication ofaccelerating genetic algorithm for embankment slope stabilityanalysisrdquo China Journal of Highway and Transport vol 16 no4 p 16 2003
The Scientific World Journal 13
[16] Y-P Zhou S-J Li Y-X Liu H-L Sun and F Jiang ldquoSearchingcritical failure surface in slope stability analysis with geneticalgorithmrdquo Chinese Journal of Rock Mechanics and Engineeringvol 24 pp 5226ndash5230 2005
[17] P McCombie ldquoCritical failure surface location using SimpleGenetic Algorithm and multiple wedge slope stabilityrdquo in Pro-ceedings of the 1st International Symposium on ComputationalGeomechanics pp 900ndash909 Juan-les-Pins France May 2009
[18] K Gavin and J F Xue ldquoUse of a genetic algorithm to performreliability analysis of unsaturated soil slopesrdquoGeotechnique vol59 no 6 pp 545ndash549 2009
[19] Wikipedia ldquoGenetic algorithmrdquo 2012 httpenwikipediaorgwikiGenetic algorithm
[20] China University of Geosciences ldquoCohesive soil slope stabi-lity analysisrdquo httpcoursecugeducncugsoil mechanicsCOURSECHAPTER7Chap7 3 5htm
[21] J E Thomaz ldquoA general method for three dimensional slopestability analysis informational reportrdquo Tech Rep JTRP Tech-nical Reports Purdue University 1986
[22] T B Zhang ldquoEarth slope stability analysis and geotechnologyslope designrdquo Tech Rep Chengdu University of Science andTechnology Press 1987
[23] B G He Application Examples in Civil Engineering with AnsysChina Water Power Press Beijing China 2011
[24] Y Liu Q Yang and L Zhu ldquoAbutment stability analysis of archdam based on 3D nonlinear finite element methodrdquo ChineseJournal of Rock Mechanics and Engineering vol 27 supplement1 pp 3222ndash3228 2008
[25] Y R Liu ZHe B Li andQYang ldquoSlope stability analysis basedon a multigrid method using a nonlinear 3D finite elementmodelrdquo Frontiers of Structural and Civil Engineering vol 7 no1 pp 24ndash31 2013
[26] Wikipedia ldquoDruckerndashPrager yield criterionrdquo 2013 httpenwikipediaorgwikiDruckerE28093Prager yield criterion
[27] B Doran H O Koksal Z Polat and C Karakoc ldquoThe use ofldquoDrucker-Prager Criterionrdquo in the analysis of reinforced con-crete members by finite elementsrdquo Teknik Dergi vol 9 no 2pp 1617ndash1625 1998
[28] B Zoran P Verka and M Biljana ldquoMathematical modeling ofmaterially nonlinear problems in structural analyses (part Imdashtheoretical fundamentals)rdquo Facta Universitatis Architecture andCivil Engineering vol 8 no 1 pp 67ndash78 2010
[29] Q Yang X Chen and W-Y Zhou ldquoA practical 3D elasto-plastic incrementalmethod in FEMbased onD-P yield criteriardquoChinese Journal of Geotechnical Engineering vol 24 no 1 pp16ndash20 2002
[30] G D Zhang Soil Constitutive Model and Its Application inEngineering Science Press Beijing China 1995
[31] D Rakic M Zivkovic R Slavkovic and M Kojic ldquoStress inte-gration for the Drucker-Prager material model without hard-ening using the Incremental Plasticity Theoryrdquo Journal of theSerbian Society For Computational Mechanics vol 2 no 1 pp80ndash89 2008
[32] X Y Zhang Soil and Rock Plastic Mechanics China Communi-cations Press Beijing China 1993
[33] Y R Zheng Z J Shen and X N Gong The Principles of Geo-technical Plastic Mechanics China Architecture and BuildingPress Beijing China 2002
[34] S A Akers Two-dimensional finite element analysis of porousgeomaterials at multikilobar stress levels [PhD thesis] VirginiaPolytechnic Institute and State University 2001
[35] X-H Tan J-G Wang and Y Wang ldquoNonlinear finite elementanalysis of slope stabilityrdquo Rock and Soil Mechanics vol 29 no8 pp 2047ndash2050 2008
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6 The Scientific World Journal
x
y
Search region for point B
Search region for point CCirclecenter of 2D potential slipsurface
Search region for point A
O
Potential circular slip surface
A(a 25)
B(0 b)
C(c 0)
Figure 4 Potential failure circle center and radius determined and represented with points 119860 119861 and 119862
A
CB
O
1205732
1205731120572
Figure 5 Determination of 2D potential failure surfaces wheninternal friction angle 120593 = 0
Table 13 Determination of 1205731and 120573
2with slope angle 120572
Slope angle 120572 Slope ratio 1 m 1205731
1205732
60∘ 1 058 29∘ 40∘
45∘ 1 10 28∘ 37∘
33∘411015840 1 15 26∘ 35∘
26∘341015840 1 20 25∘ 35∘
18∘261015840 1 30 26∘ 35∘
14∘021015840 1 40 25∘ 36∘
11∘191015840 1 50 25∘ 39∘
In this solution safety factor is determinedwith Felleniusrsquomethod of slices as in (1) Felleniusrsquo method of searching forthe critical failure surface [20] was given as follows
If soil internal friction angle 120593 = 0 2D critical failuresurface passes through slope toe119860 and can be determined byFigure 5 and Table 13 In Figure 5 the critical failure surfacecircle center 119874 can be determined by angles 120573
1and 120573
2which
can be determined by slope angle 120572 as in Table 13 Angle 1205731
is the angle between line 119860119874 and slope surface line while 1205732
is the angle between line119874119861 and slope horizontal top surfaceline 119861119862 Point 119861 is the intersection between slope surface line119860119861 and slope horizontal top surface line 119861119862
If soil internal friction angle 120593 gt 0 2D critical failuresurface passes through slope toe and can be determined by
Figure 6 In Figure 6 point 119864 is determined by angles 1205731
and 1205732which can be determined by slope angle 120572 as in
Table 13 The critical failure surface circle center may be onthe extension line of the line 119863119864 You can try many pointson the line 119863119864 as the critical failure surface circle centercandidate like119874
1and119874
4on the line119863119864 If a point119874119909 on the
line 119863119864 is found to be the point which gives the minimumslope safety factor then draw a line 119865119866 perpendicular to theline119863119864 through the point 119874
119909 Then you can try many points
on the line 119865119866 as the critical failure surface circle centercandidate like 119874
1015840
1 11987410158402 11987410158403 and 119874
1015840
4 If a point on the line 119865119866
gives the minimum slope safety factor this point means theone that gives the final most minimum safety factor of thestudied slope
The determined minimum safety factor with Felleniusrsquomethod is 1320 while the 2D critical failure surface circlecenter is 45m 57776m in the 119909-119910 coordinate system ofFigure 2 and the radius is 57951m The corresponding 119883
119888in
Figure 2 is 52292m
52 Solution with Slopew Software To solve the engineeringproblem in Section 2 this part will determine the safety factorwith slopew software
With the Ordinarymethod theminimum safety factor tothe example slope is 1328 The corresponding critical failuresurface is presented in Figure 7 With the Bishop methodthe minimum safety factor to the example slope is 1390 Thecorresponding critical failure surface is presented in Figure 8With the Janbu method the minimum safety factor to theexample slope is 1316 The corresponding critical failuresurface is presented in Figure 9 With the Morgenstern-Pricemethod the minimum safety factor to the example slope is1389 The corresponding critical failure surface is presentedin Figure 10 With the Spencer method the minimum safetyfactor to the example slope is 1389 The correspondingcritical failure surface is presented in Figure 11 With the GLEmethod the minimum safety factor to the example slope is
The Scientific World Journal 7
A
B
F
H
H
D
G
ECircle center of 2D critical slip surface
O1
O4
1205732
1205731
Fs1
Fs4
C1 C2 C3 C4
45H
120572
O1998400
O2998400
O3998400
O4998400
Figure 6 Determination of 2D potential failure surfaces when internal friction angle 120593 gt 0 [20]
1328
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 7 Critical failure surface with the Ordinary method
1389 The corresponding critical failure surface is presentedin Figure 12With the Janbu generalizedmethod after solvingand analyzing then after selecting the critical slip surfacewith a safety factor of 1389 in the slopew software Figure 13appears and the ldquominimum factor of safetyrdquo shows that itsvalue is 1385 1385 is not identical with 1389 which is a littleweird The minimum safety factor determined by foregoingGAprocedure is 1325 After comparedwith slopew softwareforegoing GA procedure employed to search the criticalfailure surface is reasonable and applicable
53 Solution with Ansys Software To solve the engineeringproblem in Section 2 this part will determine the safety factor
1390
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 8 Critical failure surface with the Bishop method
with Ansys software [23] The slope has two layers which islayer 1 and layer 2 in Figure 14 Layer 1 is clay and Layer 2is bed rock The slope layer 1rsquos soil modulus of elasticity isassumed to be 20E7Nm2 The slope layer 1rsquos soil Poissonrsquosratio is assumed to be 03 The slope layer 1rsquos soil density isassumed to be 20408 Kgm3 The slope layer 1rsquos soil cohesionis 10000 Pa and friction angle is 266 degrees The slope layer2rsquos soil modulus of elasticity is assumed to be 32E10Nm2The slope layer 2rsquos soil Poissonrsquos ratio is assumed to be 024The slope layer 2rsquos soil density is assumed to be 2700Kgm3
The slope stability analysis problem is regarded as a plainstrain problem The left and right boundaries are restricted
8 The Scientific World Journal
1316
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 9 Critical failure surface with the Janbu method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 10 Critical failure surface with the Morgenstern-Pricemethod
horizontally The bottom boundary is restricted both hori-zontally and vertically With the drucker-prager model asthe constitutive model and with shear strength reductionmethod based on the finite element analysis the slope inFigure 14 is analyzed Assume that the real cohesion andinternal friction angle of a slope are 119888
0and 120593
0 respectively
In the shear strength reduction method when safety factoris SF the reduced cohesion and friction angle for analysis are1198880SF and 120593
0SF
The Drucker-Prager yield criterion is [24 25]
1198601198681+ radic1198692minus 119861 le 0 (4)
where 1198681= 1205901+ 1205902+ 1205903 1198692= (16)[(120590
1minus 1205902)2+ (1205902minus 1205903)2+
(1205903minus 1205901)2]
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 11 Critical failure surface with the Spencer method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 12 Critical failure surface determinedwith theGLEmethod
1385
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 13 Critical failure surface determined with the Janbugeneralized method
The Scientific World Journal 9
Layer1
Layer2
40
m80
m
80m 50m 70m
40
m105
m
25m
Figure 14 Studied region for the engineering problem in Section 2 treated with Ansys
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(a) Safety factor equals 07
Y
Z X
MXMN
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
Slope stability analysis
(b) Safety factor equals 072
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(c) Safety factor equals 073
Figure 15 No von Mises plastic strain
If we assume that the Drucker-Prager yield surfacetouches on the interior of the Mohr-Coulomb yield surfacethen the expressions [26ndash28] are
119860 =2 sin120593
radic3radic3 + sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 + sin120593
(5)
If the Drucker-Prager yield surface passes through theexternal apexes of the Mohr-Coulomb yield surface then[26 28 29]
119860 =2 sin120593
radic3radic3 minus sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 minus sin120593
(6)
where 119888 is cohesion and 120593 is internal friction angle
10 The Scientific World Journal
Y
Z X
MX
MN
Slope stability analysis0 0160E minus 04 0320E minus 04 0479E minus 04 0639E minus 04
0799E minus 05 0240E minus 04 0399E minus 04 0559E minus 04 0719E minus 04
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0719E minus 04
(a) Safety factor equals 074
Y
Z X
MX
MN
Slope stability analysis0 0627E minus 04 0125E minus 03 0251E minus 03
0314E minus 04 0941E minus 04 0157E minus 03 0219E minus 03 0282E minus 030188E minus 03
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0282E minus 03
(b) Safety factor equals 075
Y
Z X
MX
MN
Slope stability analysis0 0295E minus 03 0589E minus 03 0884E minus 03 0001178
0147E minus 03 0442E minus 03 0736E minus 03 0001031 0001325
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0001325
(c) Safety factor equals 08
Y
Z X
MX
MN
Slope stability analysis0 0926E minus 03 0001851 0002777 0003703
0463E minus 03 0001389 0002314 000324 0004166
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0004166
(d) Safety factor equals 10
Y
Z X
MX
MN
Slope stability analysis0 0006555 0013111 0019666 0026221
0003278 0009833 0016388 0022944 0029499
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9296
UNFORRFORACEL
SMX = 0029499
(e) Safety factor equals 16
Y
Z XMX
MN
Slope stability analysis0 001587 003174 0047609 0063479
0007935 0023805 0039674 0055544 0071414
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9795
UNFORRFORACEL
SMX = 0071414
(f) Safety factor equals 18
Figure 16 von Mises plastic strain occurs and develops
With the drucker-prager model as the constitutive modelto analyze the slope under only self-weight in Figure 14 theflow rule which describes the relationship between the plasticpotential function and the plastic strain could be foundin [23 30ndash34] The incremental elastic-plastic stress-strainrelationship and the corresponding elastic-plastic matrixcould be found in [23 35]
The results were presented as follows in Figures 15 16 and17
When safety factor is from 07 to 073 there is no vonMises plastic strain in slope in Figure 15 When safety factoris 074 there is local plastic strain occurring in slope inFigure 16 When safety factor is 20 von Mises plastic strainruns through from slope toe to top surface in Figure 17
According to Chen (1975) and Niu (2009) Figure 16 giveslower bound solutions of slope safety factor which are from074 to 18 And Figure 17 where vonMises plastic strain runsthrough from slope toe to top surface gives an upper bound
The Scientific World Journal 11
Y
Z X MX
MN
Slope stability analysis0 0033261 0066522 0099783 0133044
0016631 0049892 0083153 0116414 0149675
Nodal solutionStep = 1
Sub = 999999
Time = 1
EPPLEQV (avg)DMX = 6955
UNFORRFORACEL
SMX = 0149675
Figure 17 von Mises plastic strain runs through from slope toe totop surface when safety factor equals 20
solution of slope safety factor which is 20 So the true slopesafety factor is likely from 18 to 20
54 Comparisons and Discussions The obtained minimumsafety factor for the above slope stability problem examplewith Genetic-Traversal Random Search Method is so lowwhen compared with the other methods like slopew soft-ware This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This mayalso be due to the power of the computer to realize theGenetic-Traversal Random Search Method in the AppendixThe Genetic-Traversal Random Method uses random pickto utilize mutation Validation of these conclusions will beinvestigated in the future with more numeric tests
However the obtained minimum safety factor withGenetic-Traversal Random Search Method is very close tothe lower bound solutions of slope safety factor given by theAnsys software
After computation there is plastic strain in layer 2 regionin some pictures of Figures 16 and 17 This is unreasonablesince layer 2 is defined as elastic region in the analysis withAnsys This phenomenon will be investigated in the future
6 Conclusions
This paper intends to determine a cohesive soil slope safetyfactor with Felleniusrsquo method while the 2D critical failuresurface is searched with GA The 2D critical failure surfaceis represented with real-encoded chromosomes which arepotential critical surface locations variables 119883
119888and 119883
119888119888 GA
procedure for searching critical failure surface proceeds withhand calculations If for future computer automatic searchprogram with GA program code for inheritance mutationselection and crossover program code for random numbersand program code for search interval boundaries will beneeded The minimum safety factor of 1325 determined byforegoing GA procedure to search the critical slip surface isvery close to the minimum safety factor of 1320 determinedby Felleniusrsquo critical slip surface method After comparedwith slopew software the proposed foregoing GA procedure
employed to search the critical failure surface is reasonableapplicable and effective
At last a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is madewhile in the program random numbers generated by com-puter and search boundaries are included The Genetic-Traversal Random Method uses random pick to utilizemutation In the program the slope safety factor is given byanalytical solution rather than slices method Results indicatethat the new computer automatic search program can givevery low safety factor which is about half of the foregoingones This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This may alsobe due to the power of the random number generation sub-program computer operation speed and Genetic-TraversalRandom Method Further validation of the results will beinvestigated in the future However the obtained minimumsafety factor with Genetic-Traversal Random Search Methodis very close to the lower bound solutions of slope safety factorgiven by the Ansys software
Appendix
Safety factor and failure circle determination program devel-oped in Silverfrost FTN95
double precision seedreal nrndlreal NLnewsafetyfactorsafetyfactor=100000seed=50gama=20tanphi=05cohesion=10h=25m=2do 10 I=1100000a=0b=25c=50rdn=nrndl(seed)a=-20lowastrdnrdn=nrndl(seed)b=25+20lowastrdnrdn=nrndl(seed)c=50+20lowastrdnif(b==25) b=2501x=(clowastc(2lowastb)+(25+b)2-b2+alowasta(50-2lowastb))(cb+a(25-b))y=b2+(cb)lowast(x-c2)r=sqrt((x-c)lowast(x-c)+ylowasty)
12 The Scientific World Journal
AA=(4lowastrlowastr-ylowasty)lowastsqrt(rlowastr-ylowasty)BB=(4lowastrlowastr-(h-y)lowast(h-y))lowastsqrt(rlowastr-(h-y)lowast(h-y))CC=(1m)lowast(2lowastrlowastr+xlowastx)lowastsqrt(rlowastr-xlowastx)DD=(1m)lowast(2lowastrlowastr+(mlowasth-x)lowast(mlowasth-x))lowastsqrt(rlowastr-(mlowasth-x)lowast(mlowasth-x))EE=ylowastasin((sqrt(rlowastr-ylowasty))r)FF=(h-y)lowastasin((sqrt(rlowastr-(h-y)lowast(h-y)))r)GG=(xm)lowastasin(xr)-((mlowasth-x)m)lowastasin((mlowasth-x)r)N=(1(6lowastr))lowast(AA+BB+CC-DD)+(r2)lowast(EE-FF+GG)T=(1(6lowastr))lowast(3lowasthlowastrlowastr-ylowastylowasty-(h-y)lowast(h-y)lowast(h-y)-xlowastxlowastxm-(mlowasth-x)lowast(mlowasth-x)lowast(mlowasth-x)m)L=rlowast(asin((sqrt(rlowastr-ylowasty))r)+asin((sqrt(rlowastr-(h-y)lowast(h-y)))r))newsafetyfactor=(gamalowasttanphilowastN+cohesionlowastL)(gamalowastT)if(newsafetyfactorltsafetyfactor)safetyfactor=newsafetyfactorwrite(lowastlowast)safetyfactorwrite(lowastlowast)awrite(lowastlowast)bwrite(lowastlowast)c10 continueend programreal function nrndl(seed)double precision SUVseedS=655360U=20530V=138490M=seedSseed=seed-MlowastSseed=Ulowastseed+VM=seedSseed=seed-MlowastSnrndl=seedSreturnend
Conflict of Interests
The author declares that he has no financial or personalrelationships with other people or organizations that couldinappropriately influence his work The author of this paperhas chosen not to furnish the paper and its readers with infor-mation that might present a potential conflict of interests
Acknowledgment
The author acknowledges the Scientific Research StartingFunds at Liaoning Technical University (no 11-415)
References
[1] D G Fredlund and J Krahn ldquoComparison of slope stabilitymethods of analysisrdquoCanadianGeotechnical Journal vol 14 no3 pp 429ndash439 1977
[2] K S Li ldquoA unified solution scheme for slope stability analysisrdquoin Proceedings of the 5th International Symposium on LandslidesNew Zealand vol 1 pp 481ndash486 Balkema Rotterdam TheNetherlands 1992
[3] W-J NiuW-M Ye S-G Liu and H-T Yu ldquoLimit analysis of asoil slope considering saturated-unsaturated seepagerdquoRock andSoil Mechanics vol 30 no 8 pp 2477ndash2482 2009
[4] W F Chen Limit Analysis and Soil Plasticity Elsevier Amster-dam The Netherlands 1975
[5] R L Michalowski ldquoSlope stability analysis a kinematical ap-proachrdquo Geotechnique vol 45 no 2 pp 283ndash293 1995
[6] I B Donald and Z Y Chen ldquoSlope stability analysis by theupper bound approach fundamentals and methodsrdquo CanadianGeotechnical Journal vol 34 no 6 pp 853ndash862 1997
[7] J P Sun J C Li and Q Q Liu ldquoSearch for critical slip surfacein slope stability analysis by spline-based GA methodrdquo Journalof Geotechnical and Geoenvironmental Engineering vol 134 no2 pp 252ndash256 2008
[8] R Javadzadeh and E Javadzadeh ldquoLocating critical failure sur-face in rock slope stability with hybrid model based on artificialimmune system and Cellular Learning Automata (CLA-AIS)rdquoWorld Academy of Science Engineering and Technology vol 59pp 662ndash665 2011
[9] F N G Gitirana Jr and D G Fredlund ldquoAnalysis of tran-sient embankment stability using the dynamic programmingmethodrdquo in Proceedings of the 56th Canadian GeotechnicalConference 2003
[10] H T V Pham and D G Fredlund ldquoDynamic programmingmethod in slope stability computationsrdquo in Proceedings ofthe 12th Asian Regional Conference on Soil Mechanics andGeotechnica Engineering Singapore August 2003
[11] H T V Pham D G Fredlund and F N G Gitirana Jr ldquoSlopestability analysis using dynamic prograamming combined withfinite element stress analysisrdquo in Proceedings of the InternationalConference on theManagement of the Land andWater Resources(MLWR rsquo01) Hanoi Vietnam October 2001
[12] H T V Pham and D G Fredlund ldquoThe application of dynamicprogramming to slope stability analysisrdquoCanadianGeotechnicalJournal vol 40 no 4 pp 830ndash847 2003
[13] M R Taha M Khajehzadeh and A El-Shafie ldquoSlope stabilityassessment using optimization techniques an overviewrdquo Elec-tronic Journal of Geotechnical Engineering vol 15 pp 1901ndash19152010
[14] A U Rao and N Sabhahit ldquoGenetic algorithm in stabilityof non-homogeneous slopesrdquo in Proceedings of the 12th Inter-national Conference of International Association for ComputerMethods and Advances in Geomechanics (IACMAG rsquo08) GoaIndia October 2008
[15] Y-G Nie W-Q Liu J-Y Shi and W-B Zhao ldquoApplication ofaccelerating genetic algorithm for embankment slope stabilityanalysisrdquo China Journal of Highway and Transport vol 16 no4 p 16 2003
The Scientific World Journal 13
[16] Y-P Zhou S-J Li Y-X Liu H-L Sun and F Jiang ldquoSearchingcritical failure surface in slope stability analysis with geneticalgorithmrdquo Chinese Journal of Rock Mechanics and Engineeringvol 24 pp 5226ndash5230 2005
[17] P McCombie ldquoCritical failure surface location using SimpleGenetic Algorithm and multiple wedge slope stabilityrdquo in Pro-ceedings of the 1st International Symposium on ComputationalGeomechanics pp 900ndash909 Juan-les-Pins France May 2009
[18] K Gavin and J F Xue ldquoUse of a genetic algorithm to performreliability analysis of unsaturated soil slopesrdquoGeotechnique vol59 no 6 pp 545ndash549 2009
[19] Wikipedia ldquoGenetic algorithmrdquo 2012 httpenwikipediaorgwikiGenetic algorithm
[20] China University of Geosciences ldquoCohesive soil slope stabi-lity analysisrdquo httpcoursecugeducncugsoil mechanicsCOURSECHAPTER7Chap7 3 5htm
[21] J E Thomaz ldquoA general method for three dimensional slopestability analysis informational reportrdquo Tech Rep JTRP Tech-nical Reports Purdue University 1986
[22] T B Zhang ldquoEarth slope stability analysis and geotechnologyslope designrdquo Tech Rep Chengdu University of Science andTechnology Press 1987
[23] B G He Application Examples in Civil Engineering with AnsysChina Water Power Press Beijing China 2011
[24] Y Liu Q Yang and L Zhu ldquoAbutment stability analysis of archdam based on 3D nonlinear finite element methodrdquo ChineseJournal of Rock Mechanics and Engineering vol 27 supplement1 pp 3222ndash3228 2008
[25] Y R Liu ZHe B Li andQYang ldquoSlope stability analysis basedon a multigrid method using a nonlinear 3D finite elementmodelrdquo Frontiers of Structural and Civil Engineering vol 7 no1 pp 24ndash31 2013
[26] Wikipedia ldquoDruckerndashPrager yield criterionrdquo 2013 httpenwikipediaorgwikiDruckerE28093Prager yield criterion
[27] B Doran H O Koksal Z Polat and C Karakoc ldquoThe use ofldquoDrucker-Prager Criterionrdquo in the analysis of reinforced con-crete members by finite elementsrdquo Teknik Dergi vol 9 no 2pp 1617ndash1625 1998
[28] B Zoran P Verka and M Biljana ldquoMathematical modeling ofmaterially nonlinear problems in structural analyses (part Imdashtheoretical fundamentals)rdquo Facta Universitatis Architecture andCivil Engineering vol 8 no 1 pp 67ndash78 2010
[29] Q Yang X Chen and W-Y Zhou ldquoA practical 3D elasto-plastic incrementalmethod in FEMbased onD-P yield criteriardquoChinese Journal of Geotechnical Engineering vol 24 no 1 pp16ndash20 2002
[30] G D Zhang Soil Constitutive Model and Its Application inEngineering Science Press Beijing China 1995
[31] D Rakic M Zivkovic R Slavkovic and M Kojic ldquoStress inte-gration for the Drucker-Prager material model without hard-ening using the Incremental Plasticity Theoryrdquo Journal of theSerbian Society For Computational Mechanics vol 2 no 1 pp80ndash89 2008
[32] X Y Zhang Soil and Rock Plastic Mechanics China Communi-cations Press Beijing China 1993
[33] Y R Zheng Z J Shen and X N Gong The Principles of Geo-technical Plastic Mechanics China Architecture and BuildingPress Beijing China 2002
[34] S A Akers Two-dimensional finite element analysis of porousgeomaterials at multikilobar stress levels [PhD thesis] VirginiaPolytechnic Institute and State University 2001
[35] X-H Tan J-G Wang and Y Wang ldquoNonlinear finite elementanalysis of slope stabilityrdquo Rock and Soil Mechanics vol 29 no8 pp 2047ndash2050 2008
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International Journal of
The Scientific World Journal 7
A
B
F
H
H
D
G
ECircle center of 2D critical slip surface
O1
O4
1205732
1205731
Fs1
Fs4
C1 C2 C3 C4
45H
120572
O1998400
O2998400
O3998400
O4998400
Figure 6 Determination of 2D potential failure surfaces when internal friction angle 120593 gt 0 [20]
1328
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 7 Critical failure surface with the Ordinary method
1389 The corresponding critical failure surface is presentedin Figure 12With the Janbu generalizedmethod after solvingand analyzing then after selecting the critical slip surfacewith a safety factor of 1389 in the slopew software Figure 13appears and the ldquominimum factor of safetyrdquo shows that itsvalue is 1385 1385 is not identical with 1389 which is a littleweird The minimum safety factor determined by foregoingGAprocedure is 1325 After comparedwith slopew softwareforegoing GA procedure employed to search the criticalfailure surface is reasonable and applicable
53 Solution with Ansys Software To solve the engineeringproblem in Section 2 this part will determine the safety factor
1390
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 8 Critical failure surface with the Bishop method
with Ansys software [23] The slope has two layers which islayer 1 and layer 2 in Figure 14 Layer 1 is clay and Layer 2is bed rock The slope layer 1rsquos soil modulus of elasticity isassumed to be 20E7Nm2 The slope layer 1rsquos soil Poissonrsquosratio is assumed to be 03 The slope layer 1rsquos soil density isassumed to be 20408 Kgm3 The slope layer 1rsquos soil cohesionis 10000 Pa and friction angle is 266 degrees The slope layer2rsquos soil modulus of elasticity is assumed to be 32E10Nm2The slope layer 2rsquos soil Poissonrsquos ratio is assumed to be 024The slope layer 2rsquos soil density is assumed to be 2700Kgm3
The slope stability analysis problem is regarded as a plainstrain problem The left and right boundaries are restricted
8 The Scientific World Journal
1316
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 9 Critical failure surface with the Janbu method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 10 Critical failure surface with the Morgenstern-Pricemethod
horizontally The bottom boundary is restricted both hori-zontally and vertically With the drucker-prager model asthe constitutive model and with shear strength reductionmethod based on the finite element analysis the slope inFigure 14 is analyzed Assume that the real cohesion andinternal friction angle of a slope are 119888
0and 120593
0 respectively
In the shear strength reduction method when safety factoris SF the reduced cohesion and friction angle for analysis are1198880SF and 120593
0SF
The Drucker-Prager yield criterion is [24 25]
1198601198681+ radic1198692minus 119861 le 0 (4)
where 1198681= 1205901+ 1205902+ 1205903 1198692= (16)[(120590
1minus 1205902)2+ (1205902minus 1205903)2+
(1205903minus 1205901)2]
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 11 Critical failure surface with the Spencer method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 12 Critical failure surface determinedwith theGLEmethod
1385
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 13 Critical failure surface determined with the Janbugeneralized method
The Scientific World Journal 9
Layer1
Layer2
40
m80
m
80m 50m 70m
40
m105
m
25m
Figure 14 Studied region for the engineering problem in Section 2 treated with Ansys
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(a) Safety factor equals 07
Y
Z X
MXMN
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
Slope stability analysis
(b) Safety factor equals 072
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(c) Safety factor equals 073
Figure 15 No von Mises plastic strain
If we assume that the Drucker-Prager yield surfacetouches on the interior of the Mohr-Coulomb yield surfacethen the expressions [26ndash28] are
119860 =2 sin120593
radic3radic3 + sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 + sin120593
(5)
If the Drucker-Prager yield surface passes through theexternal apexes of the Mohr-Coulomb yield surface then[26 28 29]
119860 =2 sin120593
radic3radic3 minus sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 minus sin120593
(6)
where 119888 is cohesion and 120593 is internal friction angle
10 The Scientific World Journal
Y
Z X
MX
MN
Slope stability analysis0 0160E minus 04 0320E minus 04 0479E minus 04 0639E minus 04
0799E minus 05 0240E minus 04 0399E minus 04 0559E minus 04 0719E minus 04
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0719E minus 04
(a) Safety factor equals 074
Y
Z X
MX
MN
Slope stability analysis0 0627E minus 04 0125E minus 03 0251E minus 03
0314E minus 04 0941E minus 04 0157E minus 03 0219E minus 03 0282E minus 030188E minus 03
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0282E minus 03
(b) Safety factor equals 075
Y
Z X
MX
MN
Slope stability analysis0 0295E minus 03 0589E minus 03 0884E minus 03 0001178
0147E minus 03 0442E minus 03 0736E minus 03 0001031 0001325
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0001325
(c) Safety factor equals 08
Y
Z X
MX
MN
Slope stability analysis0 0926E minus 03 0001851 0002777 0003703
0463E minus 03 0001389 0002314 000324 0004166
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0004166
(d) Safety factor equals 10
Y
Z X
MX
MN
Slope stability analysis0 0006555 0013111 0019666 0026221
0003278 0009833 0016388 0022944 0029499
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9296
UNFORRFORACEL
SMX = 0029499
(e) Safety factor equals 16
Y
Z XMX
MN
Slope stability analysis0 001587 003174 0047609 0063479
0007935 0023805 0039674 0055544 0071414
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9795
UNFORRFORACEL
SMX = 0071414
(f) Safety factor equals 18
Figure 16 von Mises plastic strain occurs and develops
With the drucker-prager model as the constitutive modelto analyze the slope under only self-weight in Figure 14 theflow rule which describes the relationship between the plasticpotential function and the plastic strain could be foundin [23 30ndash34] The incremental elastic-plastic stress-strainrelationship and the corresponding elastic-plastic matrixcould be found in [23 35]
The results were presented as follows in Figures 15 16 and17
When safety factor is from 07 to 073 there is no vonMises plastic strain in slope in Figure 15 When safety factoris 074 there is local plastic strain occurring in slope inFigure 16 When safety factor is 20 von Mises plastic strainruns through from slope toe to top surface in Figure 17
According to Chen (1975) and Niu (2009) Figure 16 giveslower bound solutions of slope safety factor which are from074 to 18 And Figure 17 where vonMises plastic strain runsthrough from slope toe to top surface gives an upper bound
The Scientific World Journal 11
Y
Z X MX
MN
Slope stability analysis0 0033261 0066522 0099783 0133044
0016631 0049892 0083153 0116414 0149675
Nodal solutionStep = 1
Sub = 999999
Time = 1
EPPLEQV (avg)DMX = 6955
UNFORRFORACEL
SMX = 0149675
Figure 17 von Mises plastic strain runs through from slope toe totop surface when safety factor equals 20
solution of slope safety factor which is 20 So the true slopesafety factor is likely from 18 to 20
54 Comparisons and Discussions The obtained minimumsafety factor for the above slope stability problem examplewith Genetic-Traversal Random Search Method is so lowwhen compared with the other methods like slopew soft-ware This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This mayalso be due to the power of the computer to realize theGenetic-Traversal Random Search Method in the AppendixThe Genetic-Traversal Random Method uses random pickto utilize mutation Validation of these conclusions will beinvestigated in the future with more numeric tests
However the obtained minimum safety factor withGenetic-Traversal Random Search Method is very close tothe lower bound solutions of slope safety factor given by theAnsys software
After computation there is plastic strain in layer 2 regionin some pictures of Figures 16 and 17 This is unreasonablesince layer 2 is defined as elastic region in the analysis withAnsys This phenomenon will be investigated in the future
6 Conclusions
This paper intends to determine a cohesive soil slope safetyfactor with Felleniusrsquo method while the 2D critical failuresurface is searched with GA The 2D critical failure surfaceis represented with real-encoded chromosomes which arepotential critical surface locations variables 119883
119888and 119883
119888119888 GA
procedure for searching critical failure surface proceeds withhand calculations If for future computer automatic searchprogram with GA program code for inheritance mutationselection and crossover program code for random numbersand program code for search interval boundaries will beneeded The minimum safety factor of 1325 determined byforegoing GA procedure to search the critical slip surface isvery close to the minimum safety factor of 1320 determinedby Felleniusrsquo critical slip surface method After comparedwith slopew software the proposed foregoing GA procedure
employed to search the critical failure surface is reasonableapplicable and effective
At last a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is madewhile in the program random numbers generated by com-puter and search boundaries are included The Genetic-Traversal Random Method uses random pick to utilizemutation In the program the slope safety factor is given byanalytical solution rather than slices method Results indicatethat the new computer automatic search program can givevery low safety factor which is about half of the foregoingones This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This may alsobe due to the power of the random number generation sub-program computer operation speed and Genetic-TraversalRandom Method Further validation of the results will beinvestigated in the future However the obtained minimumsafety factor with Genetic-Traversal Random Search Methodis very close to the lower bound solutions of slope safety factorgiven by the Ansys software
Appendix
Safety factor and failure circle determination program devel-oped in Silverfrost FTN95
double precision seedreal nrndlreal NLnewsafetyfactorsafetyfactor=100000seed=50gama=20tanphi=05cohesion=10h=25m=2do 10 I=1100000a=0b=25c=50rdn=nrndl(seed)a=-20lowastrdnrdn=nrndl(seed)b=25+20lowastrdnrdn=nrndl(seed)c=50+20lowastrdnif(b==25) b=2501x=(clowastc(2lowastb)+(25+b)2-b2+alowasta(50-2lowastb))(cb+a(25-b))y=b2+(cb)lowast(x-c2)r=sqrt((x-c)lowast(x-c)+ylowasty)
12 The Scientific World Journal
AA=(4lowastrlowastr-ylowasty)lowastsqrt(rlowastr-ylowasty)BB=(4lowastrlowastr-(h-y)lowast(h-y))lowastsqrt(rlowastr-(h-y)lowast(h-y))CC=(1m)lowast(2lowastrlowastr+xlowastx)lowastsqrt(rlowastr-xlowastx)DD=(1m)lowast(2lowastrlowastr+(mlowasth-x)lowast(mlowasth-x))lowastsqrt(rlowastr-(mlowasth-x)lowast(mlowasth-x))EE=ylowastasin((sqrt(rlowastr-ylowasty))r)FF=(h-y)lowastasin((sqrt(rlowastr-(h-y)lowast(h-y)))r)GG=(xm)lowastasin(xr)-((mlowasth-x)m)lowastasin((mlowasth-x)r)N=(1(6lowastr))lowast(AA+BB+CC-DD)+(r2)lowast(EE-FF+GG)T=(1(6lowastr))lowast(3lowasthlowastrlowastr-ylowastylowasty-(h-y)lowast(h-y)lowast(h-y)-xlowastxlowastxm-(mlowasth-x)lowast(mlowasth-x)lowast(mlowasth-x)m)L=rlowast(asin((sqrt(rlowastr-ylowasty))r)+asin((sqrt(rlowastr-(h-y)lowast(h-y)))r))newsafetyfactor=(gamalowasttanphilowastN+cohesionlowastL)(gamalowastT)if(newsafetyfactorltsafetyfactor)safetyfactor=newsafetyfactorwrite(lowastlowast)safetyfactorwrite(lowastlowast)awrite(lowastlowast)bwrite(lowastlowast)c10 continueend programreal function nrndl(seed)double precision SUVseedS=655360U=20530V=138490M=seedSseed=seed-MlowastSseed=Ulowastseed+VM=seedSseed=seed-MlowastSnrndl=seedSreturnend
Conflict of Interests
The author declares that he has no financial or personalrelationships with other people or organizations that couldinappropriately influence his work The author of this paperhas chosen not to furnish the paper and its readers with infor-mation that might present a potential conflict of interests
Acknowledgment
The author acknowledges the Scientific Research StartingFunds at Liaoning Technical University (no 11-415)
References
[1] D G Fredlund and J Krahn ldquoComparison of slope stabilitymethods of analysisrdquoCanadianGeotechnical Journal vol 14 no3 pp 429ndash439 1977
[2] K S Li ldquoA unified solution scheme for slope stability analysisrdquoin Proceedings of the 5th International Symposium on LandslidesNew Zealand vol 1 pp 481ndash486 Balkema Rotterdam TheNetherlands 1992
[3] W-J NiuW-M Ye S-G Liu and H-T Yu ldquoLimit analysis of asoil slope considering saturated-unsaturated seepagerdquoRock andSoil Mechanics vol 30 no 8 pp 2477ndash2482 2009
[4] W F Chen Limit Analysis and Soil Plasticity Elsevier Amster-dam The Netherlands 1975
[5] R L Michalowski ldquoSlope stability analysis a kinematical ap-proachrdquo Geotechnique vol 45 no 2 pp 283ndash293 1995
[6] I B Donald and Z Y Chen ldquoSlope stability analysis by theupper bound approach fundamentals and methodsrdquo CanadianGeotechnical Journal vol 34 no 6 pp 853ndash862 1997
[7] J P Sun J C Li and Q Q Liu ldquoSearch for critical slip surfacein slope stability analysis by spline-based GA methodrdquo Journalof Geotechnical and Geoenvironmental Engineering vol 134 no2 pp 252ndash256 2008
[8] R Javadzadeh and E Javadzadeh ldquoLocating critical failure sur-face in rock slope stability with hybrid model based on artificialimmune system and Cellular Learning Automata (CLA-AIS)rdquoWorld Academy of Science Engineering and Technology vol 59pp 662ndash665 2011
[9] F N G Gitirana Jr and D G Fredlund ldquoAnalysis of tran-sient embankment stability using the dynamic programmingmethodrdquo in Proceedings of the 56th Canadian GeotechnicalConference 2003
[10] H T V Pham and D G Fredlund ldquoDynamic programmingmethod in slope stability computationsrdquo in Proceedings ofthe 12th Asian Regional Conference on Soil Mechanics andGeotechnica Engineering Singapore August 2003
[11] H T V Pham D G Fredlund and F N G Gitirana Jr ldquoSlopestability analysis using dynamic prograamming combined withfinite element stress analysisrdquo in Proceedings of the InternationalConference on theManagement of the Land andWater Resources(MLWR rsquo01) Hanoi Vietnam October 2001
[12] H T V Pham and D G Fredlund ldquoThe application of dynamicprogramming to slope stability analysisrdquoCanadianGeotechnicalJournal vol 40 no 4 pp 830ndash847 2003
[13] M R Taha M Khajehzadeh and A El-Shafie ldquoSlope stabilityassessment using optimization techniques an overviewrdquo Elec-tronic Journal of Geotechnical Engineering vol 15 pp 1901ndash19152010
[14] A U Rao and N Sabhahit ldquoGenetic algorithm in stabilityof non-homogeneous slopesrdquo in Proceedings of the 12th Inter-national Conference of International Association for ComputerMethods and Advances in Geomechanics (IACMAG rsquo08) GoaIndia October 2008
[15] Y-G Nie W-Q Liu J-Y Shi and W-B Zhao ldquoApplication ofaccelerating genetic algorithm for embankment slope stabilityanalysisrdquo China Journal of Highway and Transport vol 16 no4 p 16 2003
The Scientific World Journal 13
[16] Y-P Zhou S-J Li Y-X Liu H-L Sun and F Jiang ldquoSearchingcritical failure surface in slope stability analysis with geneticalgorithmrdquo Chinese Journal of Rock Mechanics and Engineeringvol 24 pp 5226ndash5230 2005
[17] P McCombie ldquoCritical failure surface location using SimpleGenetic Algorithm and multiple wedge slope stabilityrdquo in Pro-ceedings of the 1st International Symposium on ComputationalGeomechanics pp 900ndash909 Juan-les-Pins France May 2009
[18] K Gavin and J F Xue ldquoUse of a genetic algorithm to performreliability analysis of unsaturated soil slopesrdquoGeotechnique vol59 no 6 pp 545ndash549 2009
[19] Wikipedia ldquoGenetic algorithmrdquo 2012 httpenwikipediaorgwikiGenetic algorithm
[20] China University of Geosciences ldquoCohesive soil slope stabi-lity analysisrdquo httpcoursecugeducncugsoil mechanicsCOURSECHAPTER7Chap7 3 5htm
[21] J E Thomaz ldquoA general method for three dimensional slopestability analysis informational reportrdquo Tech Rep JTRP Tech-nical Reports Purdue University 1986
[22] T B Zhang ldquoEarth slope stability analysis and geotechnologyslope designrdquo Tech Rep Chengdu University of Science andTechnology Press 1987
[23] B G He Application Examples in Civil Engineering with AnsysChina Water Power Press Beijing China 2011
[24] Y Liu Q Yang and L Zhu ldquoAbutment stability analysis of archdam based on 3D nonlinear finite element methodrdquo ChineseJournal of Rock Mechanics and Engineering vol 27 supplement1 pp 3222ndash3228 2008
[25] Y R Liu ZHe B Li andQYang ldquoSlope stability analysis basedon a multigrid method using a nonlinear 3D finite elementmodelrdquo Frontiers of Structural and Civil Engineering vol 7 no1 pp 24ndash31 2013
[26] Wikipedia ldquoDruckerndashPrager yield criterionrdquo 2013 httpenwikipediaorgwikiDruckerE28093Prager yield criterion
[27] B Doran H O Koksal Z Polat and C Karakoc ldquoThe use ofldquoDrucker-Prager Criterionrdquo in the analysis of reinforced con-crete members by finite elementsrdquo Teknik Dergi vol 9 no 2pp 1617ndash1625 1998
[28] B Zoran P Verka and M Biljana ldquoMathematical modeling ofmaterially nonlinear problems in structural analyses (part Imdashtheoretical fundamentals)rdquo Facta Universitatis Architecture andCivil Engineering vol 8 no 1 pp 67ndash78 2010
[29] Q Yang X Chen and W-Y Zhou ldquoA practical 3D elasto-plastic incrementalmethod in FEMbased onD-P yield criteriardquoChinese Journal of Geotechnical Engineering vol 24 no 1 pp16ndash20 2002
[30] G D Zhang Soil Constitutive Model and Its Application inEngineering Science Press Beijing China 1995
[31] D Rakic M Zivkovic R Slavkovic and M Kojic ldquoStress inte-gration for the Drucker-Prager material model without hard-ening using the Incremental Plasticity Theoryrdquo Journal of theSerbian Society For Computational Mechanics vol 2 no 1 pp80ndash89 2008
[32] X Y Zhang Soil and Rock Plastic Mechanics China Communi-cations Press Beijing China 1993
[33] Y R Zheng Z J Shen and X N Gong The Principles of Geo-technical Plastic Mechanics China Architecture and BuildingPress Beijing China 2002
[34] S A Akers Two-dimensional finite element analysis of porousgeomaterials at multikilobar stress levels [PhD thesis] VirginiaPolytechnic Institute and State University 2001
[35] X-H Tan J-G Wang and Y Wang ldquoNonlinear finite elementanalysis of slope stabilityrdquo Rock and Soil Mechanics vol 29 no8 pp 2047ndash2050 2008
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 The Scientific World Journal
1316
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 9 Critical failure surface with the Janbu method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 10 Critical failure surface with the Morgenstern-Pricemethod
horizontally The bottom boundary is restricted both hori-zontally and vertically With the drucker-prager model asthe constitutive model and with shear strength reductionmethod based on the finite element analysis the slope inFigure 14 is analyzed Assume that the real cohesion andinternal friction angle of a slope are 119888
0and 120593
0 respectively
In the shear strength reduction method when safety factoris SF the reduced cohesion and friction angle for analysis are1198880SF and 120593
0SF
The Drucker-Prager yield criterion is [24 25]
1198601198681+ radic1198692minus 119861 le 0 (4)
where 1198681= 1205901+ 1205902+ 1205903 1198692= (16)[(120590
1minus 1205902)2+ (1205902minus 1205903)2+
(1205903minus 1205901)2]
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 11 Critical failure surface with the Spencer method
1389
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 12 Critical failure surface determinedwith theGLEmethod
1385
Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85
Elev
atio
n
51015202530354045505560657075
minus5
Figure 13 Critical failure surface determined with the Janbugeneralized method
The Scientific World Journal 9
Layer1
Layer2
40
m80
m
80m 50m 70m
40
m105
m
25m
Figure 14 Studied region for the engineering problem in Section 2 treated with Ansys
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(a) Safety factor equals 07
Y
Z X
MXMN
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
Slope stability analysis
(b) Safety factor equals 072
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(c) Safety factor equals 073
Figure 15 No von Mises plastic strain
If we assume that the Drucker-Prager yield surfacetouches on the interior of the Mohr-Coulomb yield surfacethen the expressions [26ndash28] are
119860 =2 sin120593
radic3radic3 + sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 + sin120593
(5)
If the Drucker-Prager yield surface passes through theexternal apexes of the Mohr-Coulomb yield surface then[26 28 29]
119860 =2 sin120593
radic3radic3 minus sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 minus sin120593
(6)
where 119888 is cohesion and 120593 is internal friction angle
10 The Scientific World Journal
Y
Z X
MX
MN
Slope stability analysis0 0160E minus 04 0320E minus 04 0479E minus 04 0639E minus 04
0799E minus 05 0240E minus 04 0399E minus 04 0559E minus 04 0719E minus 04
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0719E minus 04
(a) Safety factor equals 074
Y
Z X
MX
MN
Slope stability analysis0 0627E minus 04 0125E minus 03 0251E minus 03
0314E minus 04 0941E minus 04 0157E minus 03 0219E minus 03 0282E minus 030188E minus 03
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0282E minus 03
(b) Safety factor equals 075
Y
Z X
MX
MN
Slope stability analysis0 0295E minus 03 0589E minus 03 0884E minus 03 0001178
0147E minus 03 0442E minus 03 0736E minus 03 0001031 0001325
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0001325
(c) Safety factor equals 08
Y
Z X
MX
MN
Slope stability analysis0 0926E minus 03 0001851 0002777 0003703
0463E minus 03 0001389 0002314 000324 0004166
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0004166
(d) Safety factor equals 10
Y
Z X
MX
MN
Slope stability analysis0 0006555 0013111 0019666 0026221
0003278 0009833 0016388 0022944 0029499
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9296
UNFORRFORACEL
SMX = 0029499
(e) Safety factor equals 16
Y
Z XMX
MN
Slope stability analysis0 001587 003174 0047609 0063479
0007935 0023805 0039674 0055544 0071414
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9795
UNFORRFORACEL
SMX = 0071414
(f) Safety factor equals 18
Figure 16 von Mises plastic strain occurs and develops
With the drucker-prager model as the constitutive modelto analyze the slope under only self-weight in Figure 14 theflow rule which describes the relationship between the plasticpotential function and the plastic strain could be foundin [23 30ndash34] The incremental elastic-plastic stress-strainrelationship and the corresponding elastic-plastic matrixcould be found in [23 35]
The results were presented as follows in Figures 15 16 and17
When safety factor is from 07 to 073 there is no vonMises plastic strain in slope in Figure 15 When safety factoris 074 there is local plastic strain occurring in slope inFigure 16 When safety factor is 20 von Mises plastic strainruns through from slope toe to top surface in Figure 17
According to Chen (1975) and Niu (2009) Figure 16 giveslower bound solutions of slope safety factor which are from074 to 18 And Figure 17 where vonMises plastic strain runsthrough from slope toe to top surface gives an upper bound
The Scientific World Journal 11
Y
Z X MX
MN
Slope stability analysis0 0033261 0066522 0099783 0133044
0016631 0049892 0083153 0116414 0149675
Nodal solutionStep = 1
Sub = 999999
Time = 1
EPPLEQV (avg)DMX = 6955
UNFORRFORACEL
SMX = 0149675
Figure 17 von Mises plastic strain runs through from slope toe totop surface when safety factor equals 20
solution of slope safety factor which is 20 So the true slopesafety factor is likely from 18 to 20
54 Comparisons and Discussions The obtained minimumsafety factor for the above slope stability problem examplewith Genetic-Traversal Random Search Method is so lowwhen compared with the other methods like slopew soft-ware This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This mayalso be due to the power of the computer to realize theGenetic-Traversal Random Search Method in the AppendixThe Genetic-Traversal Random Method uses random pickto utilize mutation Validation of these conclusions will beinvestigated in the future with more numeric tests
However the obtained minimum safety factor withGenetic-Traversal Random Search Method is very close tothe lower bound solutions of slope safety factor given by theAnsys software
After computation there is plastic strain in layer 2 regionin some pictures of Figures 16 and 17 This is unreasonablesince layer 2 is defined as elastic region in the analysis withAnsys This phenomenon will be investigated in the future
6 Conclusions
This paper intends to determine a cohesive soil slope safetyfactor with Felleniusrsquo method while the 2D critical failuresurface is searched with GA The 2D critical failure surfaceis represented with real-encoded chromosomes which arepotential critical surface locations variables 119883
119888and 119883
119888119888 GA
procedure for searching critical failure surface proceeds withhand calculations If for future computer automatic searchprogram with GA program code for inheritance mutationselection and crossover program code for random numbersand program code for search interval boundaries will beneeded The minimum safety factor of 1325 determined byforegoing GA procedure to search the critical slip surface isvery close to the minimum safety factor of 1320 determinedby Felleniusrsquo critical slip surface method After comparedwith slopew software the proposed foregoing GA procedure
employed to search the critical failure surface is reasonableapplicable and effective
At last a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is madewhile in the program random numbers generated by com-puter and search boundaries are included The Genetic-Traversal Random Method uses random pick to utilizemutation In the program the slope safety factor is given byanalytical solution rather than slices method Results indicatethat the new computer automatic search program can givevery low safety factor which is about half of the foregoingones This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This may alsobe due to the power of the random number generation sub-program computer operation speed and Genetic-TraversalRandom Method Further validation of the results will beinvestigated in the future However the obtained minimumsafety factor with Genetic-Traversal Random Search Methodis very close to the lower bound solutions of slope safety factorgiven by the Ansys software
Appendix
Safety factor and failure circle determination program devel-oped in Silverfrost FTN95
double precision seedreal nrndlreal NLnewsafetyfactorsafetyfactor=100000seed=50gama=20tanphi=05cohesion=10h=25m=2do 10 I=1100000a=0b=25c=50rdn=nrndl(seed)a=-20lowastrdnrdn=nrndl(seed)b=25+20lowastrdnrdn=nrndl(seed)c=50+20lowastrdnif(b==25) b=2501x=(clowastc(2lowastb)+(25+b)2-b2+alowasta(50-2lowastb))(cb+a(25-b))y=b2+(cb)lowast(x-c2)r=sqrt((x-c)lowast(x-c)+ylowasty)
12 The Scientific World Journal
AA=(4lowastrlowastr-ylowasty)lowastsqrt(rlowastr-ylowasty)BB=(4lowastrlowastr-(h-y)lowast(h-y))lowastsqrt(rlowastr-(h-y)lowast(h-y))CC=(1m)lowast(2lowastrlowastr+xlowastx)lowastsqrt(rlowastr-xlowastx)DD=(1m)lowast(2lowastrlowastr+(mlowasth-x)lowast(mlowasth-x))lowastsqrt(rlowastr-(mlowasth-x)lowast(mlowasth-x))EE=ylowastasin((sqrt(rlowastr-ylowasty))r)FF=(h-y)lowastasin((sqrt(rlowastr-(h-y)lowast(h-y)))r)GG=(xm)lowastasin(xr)-((mlowasth-x)m)lowastasin((mlowasth-x)r)N=(1(6lowastr))lowast(AA+BB+CC-DD)+(r2)lowast(EE-FF+GG)T=(1(6lowastr))lowast(3lowasthlowastrlowastr-ylowastylowasty-(h-y)lowast(h-y)lowast(h-y)-xlowastxlowastxm-(mlowasth-x)lowast(mlowasth-x)lowast(mlowasth-x)m)L=rlowast(asin((sqrt(rlowastr-ylowasty))r)+asin((sqrt(rlowastr-(h-y)lowast(h-y)))r))newsafetyfactor=(gamalowasttanphilowastN+cohesionlowastL)(gamalowastT)if(newsafetyfactorltsafetyfactor)safetyfactor=newsafetyfactorwrite(lowastlowast)safetyfactorwrite(lowastlowast)awrite(lowastlowast)bwrite(lowastlowast)c10 continueend programreal function nrndl(seed)double precision SUVseedS=655360U=20530V=138490M=seedSseed=seed-MlowastSseed=Ulowastseed+VM=seedSseed=seed-MlowastSnrndl=seedSreturnend
Conflict of Interests
The author declares that he has no financial or personalrelationships with other people or organizations that couldinappropriately influence his work The author of this paperhas chosen not to furnish the paper and its readers with infor-mation that might present a potential conflict of interests
Acknowledgment
The author acknowledges the Scientific Research StartingFunds at Liaoning Technical University (no 11-415)
References
[1] D G Fredlund and J Krahn ldquoComparison of slope stabilitymethods of analysisrdquoCanadianGeotechnical Journal vol 14 no3 pp 429ndash439 1977
[2] K S Li ldquoA unified solution scheme for slope stability analysisrdquoin Proceedings of the 5th International Symposium on LandslidesNew Zealand vol 1 pp 481ndash486 Balkema Rotterdam TheNetherlands 1992
[3] W-J NiuW-M Ye S-G Liu and H-T Yu ldquoLimit analysis of asoil slope considering saturated-unsaturated seepagerdquoRock andSoil Mechanics vol 30 no 8 pp 2477ndash2482 2009
[4] W F Chen Limit Analysis and Soil Plasticity Elsevier Amster-dam The Netherlands 1975
[5] R L Michalowski ldquoSlope stability analysis a kinematical ap-proachrdquo Geotechnique vol 45 no 2 pp 283ndash293 1995
[6] I B Donald and Z Y Chen ldquoSlope stability analysis by theupper bound approach fundamentals and methodsrdquo CanadianGeotechnical Journal vol 34 no 6 pp 853ndash862 1997
[7] J P Sun J C Li and Q Q Liu ldquoSearch for critical slip surfacein slope stability analysis by spline-based GA methodrdquo Journalof Geotechnical and Geoenvironmental Engineering vol 134 no2 pp 252ndash256 2008
[8] R Javadzadeh and E Javadzadeh ldquoLocating critical failure sur-face in rock slope stability with hybrid model based on artificialimmune system and Cellular Learning Automata (CLA-AIS)rdquoWorld Academy of Science Engineering and Technology vol 59pp 662ndash665 2011
[9] F N G Gitirana Jr and D G Fredlund ldquoAnalysis of tran-sient embankment stability using the dynamic programmingmethodrdquo in Proceedings of the 56th Canadian GeotechnicalConference 2003
[10] H T V Pham and D G Fredlund ldquoDynamic programmingmethod in slope stability computationsrdquo in Proceedings ofthe 12th Asian Regional Conference on Soil Mechanics andGeotechnica Engineering Singapore August 2003
[11] H T V Pham D G Fredlund and F N G Gitirana Jr ldquoSlopestability analysis using dynamic prograamming combined withfinite element stress analysisrdquo in Proceedings of the InternationalConference on theManagement of the Land andWater Resources(MLWR rsquo01) Hanoi Vietnam October 2001
[12] H T V Pham and D G Fredlund ldquoThe application of dynamicprogramming to slope stability analysisrdquoCanadianGeotechnicalJournal vol 40 no 4 pp 830ndash847 2003
[13] M R Taha M Khajehzadeh and A El-Shafie ldquoSlope stabilityassessment using optimization techniques an overviewrdquo Elec-tronic Journal of Geotechnical Engineering vol 15 pp 1901ndash19152010
[14] A U Rao and N Sabhahit ldquoGenetic algorithm in stabilityof non-homogeneous slopesrdquo in Proceedings of the 12th Inter-national Conference of International Association for ComputerMethods and Advances in Geomechanics (IACMAG rsquo08) GoaIndia October 2008
[15] Y-G Nie W-Q Liu J-Y Shi and W-B Zhao ldquoApplication ofaccelerating genetic algorithm for embankment slope stabilityanalysisrdquo China Journal of Highway and Transport vol 16 no4 p 16 2003
The Scientific World Journal 13
[16] Y-P Zhou S-J Li Y-X Liu H-L Sun and F Jiang ldquoSearchingcritical failure surface in slope stability analysis with geneticalgorithmrdquo Chinese Journal of Rock Mechanics and Engineeringvol 24 pp 5226ndash5230 2005
[17] P McCombie ldquoCritical failure surface location using SimpleGenetic Algorithm and multiple wedge slope stabilityrdquo in Pro-ceedings of the 1st International Symposium on ComputationalGeomechanics pp 900ndash909 Juan-les-Pins France May 2009
[18] K Gavin and J F Xue ldquoUse of a genetic algorithm to performreliability analysis of unsaturated soil slopesrdquoGeotechnique vol59 no 6 pp 545ndash549 2009
[19] Wikipedia ldquoGenetic algorithmrdquo 2012 httpenwikipediaorgwikiGenetic algorithm
[20] China University of Geosciences ldquoCohesive soil slope stabi-lity analysisrdquo httpcoursecugeducncugsoil mechanicsCOURSECHAPTER7Chap7 3 5htm
[21] J E Thomaz ldquoA general method for three dimensional slopestability analysis informational reportrdquo Tech Rep JTRP Tech-nical Reports Purdue University 1986
[22] T B Zhang ldquoEarth slope stability analysis and geotechnologyslope designrdquo Tech Rep Chengdu University of Science andTechnology Press 1987
[23] B G He Application Examples in Civil Engineering with AnsysChina Water Power Press Beijing China 2011
[24] Y Liu Q Yang and L Zhu ldquoAbutment stability analysis of archdam based on 3D nonlinear finite element methodrdquo ChineseJournal of Rock Mechanics and Engineering vol 27 supplement1 pp 3222ndash3228 2008
[25] Y R Liu ZHe B Li andQYang ldquoSlope stability analysis basedon a multigrid method using a nonlinear 3D finite elementmodelrdquo Frontiers of Structural and Civil Engineering vol 7 no1 pp 24ndash31 2013
[26] Wikipedia ldquoDruckerndashPrager yield criterionrdquo 2013 httpenwikipediaorgwikiDruckerE28093Prager yield criterion
[27] B Doran H O Koksal Z Polat and C Karakoc ldquoThe use ofldquoDrucker-Prager Criterionrdquo in the analysis of reinforced con-crete members by finite elementsrdquo Teknik Dergi vol 9 no 2pp 1617ndash1625 1998
[28] B Zoran P Verka and M Biljana ldquoMathematical modeling ofmaterially nonlinear problems in structural analyses (part Imdashtheoretical fundamentals)rdquo Facta Universitatis Architecture andCivil Engineering vol 8 no 1 pp 67ndash78 2010
[29] Q Yang X Chen and W-Y Zhou ldquoA practical 3D elasto-plastic incrementalmethod in FEMbased onD-P yield criteriardquoChinese Journal of Geotechnical Engineering vol 24 no 1 pp16ndash20 2002
[30] G D Zhang Soil Constitutive Model and Its Application inEngineering Science Press Beijing China 1995
[31] D Rakic M Zivkovic R Slavkovic and M Kojic ldquoStress inte-gration for the Drucker-Prager material model without hard-ening using the Incremental Plasticity Theoryrdquo Journal of theSerbian Society For Computational Mechanics vol 2 no 1 pp80ndash89 2008
[32] X Y Zhang Soil and Rock Plastic Mechanics China Communi-cations Press Beijing China 1993
[33] Y R Zheng Z J Shen and X N Gong The Principles of Geo-technical Plastic Mechanics China Architecture and BuildingPress Beijing China 2002
[34] S A Akers Two-dimensional finite element analysis of porousgeomaterials at multikilobar stress levels [PhD thesis] VirginiaPolytechnic Institute and State University 2001
[35] X-H Tan J-G Wang and Y Wang ldquoNonlinear finite elementanalysis of slope stabilityrdquo Rock and Soil Mechanics vol 29 no8 pp 2047ndash2050 2008
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 9
Layer1
Layer2
40
m80
m
80m 50m 70m
40
m105
m
25m
Figure 14 Studied region for the engineering problem in Section 2 treated with Ansys
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(a) Safety factor equals 07
Y
Z X
MXMN
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
Slope stability analysis
(b) Safety factor equals 072
Y
Z X
MXMN
Slope stability analysis
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
(c) Safety factor equals 073
Figure 15 No von Mises plastic strain
If we assume that the Drucker-Prager yield surfacetouches on the interior of the Mohr-Coulomb yield surfacethen the expressions [26ndash28] are
119860 =2 sin120593
radic3radic3 + sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 + sin120593
(5)
If the Drucker-Prager yield surface passes through theexternal apexes of the Mohr-Coulomb yield surface then[26 28 29]
119860 =2 sin120593
radic3radic3 minus sin120593 119861 =
6 sdot 119888 sdot cos120593radic3radic3 minus sin120593
(6)
where 119888 is cohesion and 120593 is internal friction angle
10 The Scientific World Journal
Y
Z X
MX
MN
Slope stability analysis0 0160E minus 04 0320E minus 04 0479E minus 04 0639E minus 04
0799E minus 05 0240E minus 04 0399E minus 04 0559E minus 04 0719E minus 04
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0719E minus 04
(a) Safety factor equals 074
Y
Z X
MX
MN
Slope stability analysis0 0627E minus 04 0125E minus 03 0251E minus 03
0314E minus 04 0941E minus 04 0157E minus 03 0219E minus 03 0282E minus 030188E minus 03
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0282E minus 03
(b) Safety factor equals 075
Y
Z X
MX
MN
Slope stability analysis0 0295E minus 03 0589E minus 03 0884E minus 03 0001178
0147E minus 03 0442E minus 03 0736E minus 03 0001031 0001325
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0001325
(c) Safety factor equals 08
Y
Z X
MX
MN
Slope stability analysis0 0926E minus 03 0001851 0002777 0003703
0463E minus 03 0001389 0002314 000324 0004166
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0004166
(d) Safety factor equals 10
Y
Z X
MX
MN
Slope stability analysis0 0006555 0013111 0019666 0026221
0003278 0009833 0016388 0022944 0029499
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9296
UNFORRFORACEL
SMX = 0029499
(e) Safety factor equals 16
Y
Z XMX
MN
Slope stability analysis0 001587 003174 0047609 0063479
0007935 0023805 0039674 0055544 0071414
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9795
UNFORRFORACEL
SMX = 0071414
(f) Safety factor equals 18
Figure 16 von Mises plastic strain occurs and develops
With the drucker-prager model as the constitutive modelto analyze the slope under only self-weight in Figure 14 theflow rule which describes the relationship between the plasticpotential function and the plastic strain could be foundin [23 30ndash34] The incremental elastic-plastic stress-strainrelationship and the corresponding elastic-plastic matrixcould be found in [23 35]
The results were presented as follows in Figures 15 16 and17
When safety factor is from 07 to 073 there is no vonMises plastic strain in slope in Figure 15 When safety factoris 074 there is local plastic strain occurring in slope inFigure 16 When safety factor is 20 von Mises plastic strainruns through from slope toe to top surface in Figure 17
According to Chen (1975) and Niu (2009) Figure 16 giveslower bound solutions of slope safety factor which are from074 to 18 And Figure 17 where vonMises plastic strain runsthrough from slope toe to top surface gives an upper bound
The Scientific World Journal 11
Y
Z X MX
MN
Slope stability analysis0 0033261 0066522 0099783 0133044
0016631 0049892 0083153 0116414 0149675
Nodal solutionStep = 1
Sub = 999999
Time = 1
EPPLEQV (avg)DMX = 6955
UNFORRFORACEL
SMX = 0149675
Figure 17 von Mises plastic strain runs through from slope toe totop surface when safety factor equals 20
solution of slope safety factor which is 20 So the true slopesafety factor is likely from 18 to 20
54 Comparisons and Discussions The obtained minimumsafety factor for the above slope stability problem examplewith Genetic-Traversal Random Search Method is so lowwhen compared with the other methods like slopew soft-ware This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This mayalso be due to the power of the computer to realize theGenetic-Traversal Random Search Method in the AppendixThe Genetic-Traversal Random Method uses random pickto utilize mutation Validation of these conclusions will beinvestigated in the future with more numeric tests
However the obtained minimum safety factor withGenetic-Traversal Random Search Method is very close tothe lower bound solutions of slope safety factor given by theAnsys software
After computation there is plastic strain in layer 2 regionin some pictures of Figures 16 and 17 This is unreasonablesince layer 2 is defined as elastic region in the analysis withAnsys This phenomenon will be investigated in the future
6 Conclusions
This paper intends to determine a cohesive soil slope safetyfactor with Felleniusrsquo method while the 2D critical failuresurface is searched with GA The 2D critical failure surfaceis represented with real-encoded chromosomes which arepotential critical surface locations variables 119883
119888and 119883
119888119888 GA
procedure for searching critical failure surface proceeds withhand calculations If for future computer automatic searchprogram with GA program code for inheritance mutationselection and crossover program code for random numbersand program code for search interval boundaries will beneeded The minimum safety factor of 1325 determined byforegoing GA procedure to search the critical slip surface isvery close to the minimum safety factor of 1320 determinedby Felleniusrsquo critical slip surface method After comparedwith slopew software the proposed foregoing GA procedure
employed to search the critical failure surface is reasonableapplicable and effective
At last a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is madewhile in the program random numbers generated by com-puter and search boundaries are included The Genetic-Traversal Random Method uses random pick to utilizemutation In the program the slope safety factor is given byanalytical solution rather than slices method Results indicatethat the new computer automatic search program can givevery low safety factor which is about half of the foregoingones This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This may alsobe due to the power of the random number generation sub-program computer operation speed and Genetic-TraversalRandom Method Further validation of the results will beinvestigated in the future However the obtained minimumsafety factor with Genetic-Traversal Random Search Methodis very close to the lower bound solutions of slope safety factorgiven by the Ansys software
Appendix
Safety factor and failure circle determination program devel-oped in Silverfrost FTN95
double precision seedreal nrndlreal NLnewsafetyfactorsafetyfactor=100000seed=50gama=20tanphi=05cohesion=10h=25m=2do 10 I=1100000a=0b=25c=50rdn=nrndl(seed)a=-20lowastrdnrdn=nrndl(seed)b=25+20lowastrdnrdn=nrndl(seed)c=50+20lowastrdnif(b==25) b=2501x=(clowastc(2lowastb)+(25+b)2-b2+alowasta(50-2lowastb))(cb+a(25-b))y=b2+(cb)lowast(x-c2)r=sqrt((x-c)lowast(x-c)+ylowasty)
12 The Scientific World Journal
AA=(4lowastrlowastr-ylowasty)lowastsqrt(rlowastr-ylowasty)BB=(4lowastrlowastr-(h-y)lowast(h-y))lowastsqrt(rlowastr-(h-y)lowast(h-y))CC=(1m)lowast(2lowastrlowastr+xlowastx)lowastsqrt(rlowastr-xlowastx)DD=(1m)lowast(2lowastrlowastr+(mlowasth-x)lowast(mlowasth-x))lowastsqrt(rlowastr-(mlowasth-x)lowast(mlowasth-x))EE=ylowastasin((sqrt(rlowastr-ylowasty))r)FF=(h-y)lowastasin((sqrt(rlowastr-(h-y)lowast(h-y)))r)GG=(xm)lowastasin(xr)-((mlowasth-x)m)lowastasin((mlowasth-x)r)N=(1(6lowastr))lowast(AA+BB+CC-DD)+(r2)lowast(EE-FF+GG)T=(1(6lowastr))lowast(3lowasthlowastrlowastr-ylowastylowasty-(h-y)lowast(h-y)lowast(h-y)-xlowastxlowastxm-(mlowasth-x)lowast(mlowasth-x)lowast(mlowasth-x)m)L=rlowast(asin((sqrt(rlowastr-ylowasty))r)+asin((sqrt(rlowastr-(h-y)lowast(h-y)))r))newsafetyfactor=(gamalowasttanphilowastN+cohesionlowastL)(gamalowastT)if(newsafetyfactorltsafetyfactor)safetyfactor=newsafetyfactorwrite(lowastlowast)safetyfactorwrite(lowastlowast)awrite(lowastlowast)bwrite(lowastlowast)c10 continueend programreal function nrndl(seed)double precision SUVseedS=655360U=20530V=138490M=seedSseed=seed-MlowastSseed=Ulowastseed+VM=seedSseed=seed-MlowastSnrndl=seedSreturnend
Conflict of Interests
The author declares that he has no financial or personalrelationships with other people or organizations that couldinappropriately influence his work The author of this paperhas chosen not to furnish the paper and its readers with infor-mation that might present a potential conflict of interests
Acknowledgment
The author acknowledges the Scientific Research StartingFunds at Liaoning Technical University (no 11-415)
References
[1] D G Fredlund and J Krahn ldquoComparison of slope stabilitymethods of analysisrdquoCanadianGeotechnical Journal vol 14 no3 pp 429ndash439 1977
[2] K S Li ldquoA unified solution scheme for slope stability analysisrdquoin Proceedings of the 5th International Symposium on LandslidesNew Zealand vol 1 pp 481ndash486 Balkema Rotterdam TheNetherlands 1992
[3] W-J NiuW-M Ye S-G Liu and H-T Yu ldquoLimit analysis of asoil slope considering saturated-unsaturated seepagerdquoRock andSoil Mechanics vol 30 no 8 pp 2477ndash2482 2009
[4] W F Chen Limit Analysis and Soil Plasticity Elsevier Amster-dam The Netherlands 1975
[5] R L Michalowski ldquoSlope stability analysis a kinematical ap-proachrdquo Geotechnique vol 45 no 2 pp 283ndash293 1995
[6] I B Donald and Z Y Chen ldquoSlope stability analysis by theupper bound approach fundamentals and methodsrdquo CanadianGeotechnical Journal vol 34 no 6 pp 853ndash862 1997
[7] J P Sun J C Li and Q Q Liu ldquoSearch for critical slip surfacein slope stability analysis by spline-based GA methodrdquo Journalof Geotechnical and Geoenvironmental Engineering vol 134 no2 pp 252ndash256 2008
[8] R Javadzadeh and E Javadzadeh ldquoLocating critical failure sur-face in rock slope stability with hybrid model based on artificialimmune system and Cellular Learning Automata (CLA-AIS)rdquoWorld Academy of Science Engineering and Technology vol 59pp 662ndash665 2011
[9] F N G Gitirana Jr and D G Fredlund ldquoAnalysis of tran-sient embankment stability using the dynamic programmingmethodrdquo in Proceedings of the 56th Canadian GeotechnicalConference 2003
[10] H T V Pham and D G Fredlund ldquoDynamic programmingmethod in slope stability computationsrdquo in Proceedings ofthe 12th Asian Regional Conference on Soil Mechanics andGeotechnica Engineering Singapore August 2003
[11] H T V Pham D G Fredlund and F N G Gitirana Jr ldquoSlopestability analysis using dynamic prograamming combined withfinite element stress analysisrdquo in Proceedings of the InternationalConference on theManagement of the Land andWater Resources(MLWR rsquo01) Hanoi Vietnam October 2001
[12] H T V Pham and D G Fredlund ldquoThe application of dynamicprogramming to slope stability analysisrdquoCanadianGeotechnicalJournal vol 40 no 4 pp 830ndash847 2003
[13] M R Taha M Khajehzadeh and A El-Shafie ldquoSlope stabilityassessment using optimization techniques an overviewrdquo Elec-tronic Journal of Geotechnical Engineering vol 15 pp 1901ndash19152010
[14] A U Rao and N Sabhahit ldquoGenetic algorithm in stabilityof non-homogeneous slopesrdquo in Proceedings of the 12th Inter-national Conference of International Association for ComputerMethods and Advances in Geomechanics (IACMAG rsquo08) GoaIndia October 2008
[15] Y-G Nie W-Q Liu J-Y Shi and W-B Zhao ldquoApplication ofaccelerating genetic algorithm for embankment slope stabilityanalysisrdquo China Journal of Highway and Transport vol 16 no4 p 16 2003
The Scientific World Journal 13
[16] Y-P Zhou S-J Li Y-X Liu H-L Sun and F Jiang ldquoSearchingcritical failure surface in slope stability analysis with geneticalgorithmrdquo Chinese Journal of Rock Mechanics and Engineeringvol 24 pp 5226ndash5230 2005
[17] P McCombie ldquoCritical failure surface location using SimpleGenetic Algorithm and multiple wedge slope stabilityrdquo in Pro-ceedings of the 1st International Symposium on ComputationalGeomechanics pp 900ndash909 Juan-les-Pins France May 2009
[18] K Gavin and J F Xue ldquoUse of a genetic algorithm to performreliability analysis of unsaturated soil slopesrdquoGeotechnique vol59 no 6 pp 545ndash549 2009
[19] Wikipedia ldquoGenetic algorithmrdquo 2012 httpenwikipediaorgwikiGenetic algorithm
[20] China University of Geosciences ldquoCohesive soil slope stabi-lity analysisrdquo httpcoursecugeducncugsoil mechanicsCOURSECHAPTER7Chap7 3 5htm
[21] J E Thomaz ldquoA general method for three dimensional slopestability analysis informational reportrdquo Tech Rep JTRP Tech-nical Reports Purdue University 1986
[22] T B Zhang ldquoEarth slope stability analysis and geotechnologyslope designrdquo Tech Rep Chengdu University of Science andTechnology Press 1987
[23] B G He Application Examples in Civil Engineering with AnsysChina Water Power Press Beijing China 2011
[24] Y Liu Q Yang and L Zhu ldquoAbutment stability analysis of archdam based on 3D nonlinear finite element methodrdquo ChineseJournal of Rock Mechanics and Engineering vol 27 supplement1 pp 3222ndash3228 2008
[25] Y R Liu ZHe B Li andQYang ldquoSlope stability analysis basedon a multigrid method using a nonlinear 3D finite elementmodelrdquo Frontiers of Structural and Civil Engineering vol 7 no1 pp 24ndash31 2013
[26] Wikipedia ldquoDruckerndashPrager yield criterionrdquo 2013 httpenwikipediaorgwikiDruckerE28093Prager yield criterion
[27] B Doran H O Koksal Z Polat and C Karakoc ldquoThe use ofldquoDrucker-Prager Criterionrdquo in the analysis of reinforced con-crete members by finite elementsrdquo Teknik Dergi vol 9 no 2pp 1617ndash1625 1998
[28] B Zoran P Verka and M Biljana ldquoMathematical modeling ofmaterially nonlinear problems in structural analyses (part Imdashtheoretical fundamentals)rdquo Facta Universitatis Architecture andCivil Engineering vol 8 no 1 pp 67ndash78 2010
[29] Q Yang X Chen and W-Y Zhou ldquoA practical 3D elasto-plastic incrementalmethod in FEMbased onD-P yield criteriardquoChinese Journal of Geotechnical Engineering vol 24 no 1 pp16ndash20 2002
[30] G D Zhang Soil Constitutive Model and Its Application inEngineering Science Press Beijing China 1995
[31] D Rakic M Zivkovic R Slavkovic and M Kojic ldquoStress inte-gration for the Drucker-Prager material model without hard-ening using the Incremental Plasticity Theoryrdquo Journal of theSerbian Society For Computational Mechanics vol 2 no 1 pp80ndash89 2008
[32] X Y Zhang Soil and Rock Plastic Mechanics China Communi-cations Press Beijing China 1993
[33] Y R Zheng Z J Shen and X N Gong The Principles of Geo-technical Plastic Mechanics China Architecture and BuildingPress Beijing China 2002
[34] S A Akers Two-dimensional finite element analysis of porousgeomaterials at multikilobar stress levels [PhD thesis] VirginiaPolytechnic Institute and State University 2001
[35] X-H Tan J-G Wang and Y Wang ldquoNonlinear finite elementanalysis of slope stabilityrdquo Rock and Soil Mechanics vol 29 no8 pp 2047ndash2050 2008
International Journal of
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Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 The Scientific World Journal
Y
Z X
MX
MN
Slope stability analysis0 0160E minus 04 0320E minus 04 0479E minus 04 0639E minus 04
0799E minus 05 0240E minus 04 0399E minus 04 0559E minus 04 0719E minus 04
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0719E minus 04
(a) Safety factor equals 074
Y
Z X
MX
MN
Slope stability analysis0 0627E minus 04 0125E minus 03 0251E minus 03
0314E minus 04 0941E minus 04 0157E minus 03 0219E minus 03 0282E minus 030188E minus 03
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7537
UNFORRFORACEL
SMX = 0282E minus 03
(b) Safety factor equals 075
Y
Z X
MX
MN
Slope stability analysis0 0295E minus 03 0589E minus 03 0884E minus 03 0001178
0147E minus 03 0442E minus 03 0736E minus 03 0001031 0001325
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0001325
(c) Safety factor equals 08
Y
Z X
MX
MN
Slope stability analysis0 0926E minus 03 0001851 0002777 0003703
0463E minus 03 0001389 0002314 000324 0004166
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 7536
UNFORRFORACEL
SMX = 0004166
(d) Safety factor equals 10
Y
Z X
MX
MN
Slope stability analysis0 0006555 0013111 0019666 0026221
0003278 0009833 0016388 0022944 0029499
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9296
UNFORRFORACEL
SMX = 0029499
(e) Safety factor equals 16
Y
Z XMX
MN
Slope stability analysis0 001587 003174 0047609 0063479
0007935 0023805 0039674 0055544 0071414
Nodal solutionStep = 1
Sub = 15
Time = 1
EPPLEQV (avg)DMX = 9795
UNFORRFORACEL
SMX = 0071414
(f) Safety factor equals 18
Figure 16 von Mises plastic strain occurs and develops
With the drucker-prager model as the constitutive modelto analyze the slope under only self-weight in Figure 14 theflow rule which describes the relationship between the plasticpotential function and the plastic strain could be foundin [23 30ndash34] The incremental elastic-plastic stress-strainrelationship and the corresponding elastic-plastic matrixcould be found in [23 35]
The results were presented as follows in Figures 15 16 and17
When safety factor is from 07 to 073 there is no vonMises plastic strain in slope in Figure 15 When safety factoris 074 there is local plastic strain occurring in slope inFigure 16 When safety factor is 20 von Mises plastic strainruns through from slope toe to top surface in Figure 17
According to Chen (1975) and Niu (2009) Figure 16 giveslower bound solutions of slope safety factor which are from074 to 18 And Figure 17 where vonMises plastic strain runsthrough from slope toe to top surface gives an upper bound
The Scientific World Journal 11
Y
Z X MX
MN
Slope stability analysis0 0033261 0066522 0099783 0133044
0016631 0049892 0083153 0116414 0149675
Nodal solutionStep = 1
Sub = 999999
Time = 1
EPPLEQV (avg)DMX = 6955
UNFORRFORACEL
SMX = 0149675
Figure 17 von Mises plastic strain runs through from slope toe totop surface when safety factor equals 20
solution of slope safety factor which is 20 So the true slopesafety factor is likely from 18 to 20
54 Comparisons and Discussions The obtained minimumsafety factor for the above slope stability problem examplewith Genetic-Traversal Random Search Method is so lowwhen compared with the other methods like slopew soft-ware This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This mayalso be due to the power of the computer to realize theGenetic-Traversal Random Search Method in the AppendixThe Genetic-Traversal Random Method uses random pickto utilize mutation Validation of these conclusions will beinvestigated in the future with more numeric tests
However the obtained minimum safety factor withGenetic-Traversal Random Search Method is very close tothe lower bound solutions of slope safety factor given by theAnsys software
After computation there is plastic strain in layer 2 regionin some pictures of Figures 16 and 17 This is unreasonablesince layer 2 is defined as elastic region in the analysis withAnsys This phenomenon will be investigated in the future
6 Conclusions
This paper intends to determine a cohesive soil slope safetyfactor with Felleniusrsquo method while the 2D critical failuresurface is searched with GA The 2D critical failure surfaceis represented with real-encoded chromosomes which arepotential critical surface locations variables 119883
119888and 119883
119888119888 GA
procedure for searching critical failure surface proceeds withhand calculations If for future computer automatic searchprogram with GA program code for inheritance mutationselection and crossover program code for random numbersand program code for search interval boundaries will beneeded The minimum safety factor of 1325 determined byforegoing GA procedure to search the critical slip surface isvery close to the minimum safety factor of 1320 determinedby Felleniusrsquo critical slip surface method After comparedwith slopew software the proposed foregoing GA procedure
employed to search the critical failure surface is reasonableapplicable and effective
At last a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is madewhile in the program random numbers generated by com-puter and search boundaries are included The Genetic-Traversal Random Method uses random pick to utilizemutation In the program the slope safety factor is given byanalytical solution rather than slices method Results indicatethat the new computer automatic search program can givevery low safety factor which is about half of the foregoingones This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This may alsobe due to the power of the random number generation sub-program computer operation speed and Genetic-TraversalRandom Method Further validation of the results will beinvestigated in the future However the obtained minimumsafety factor with Genetic-Traversal Random Search Methodis very close to the lower bound solutions of slope safety factorgiven by the Ansys software
Appendix
Safety factor and failure circle determination program devel-oped in Silverfrost FTN95
double precision seedreal nrndlreal NLnewsafetyfactorsafetyfactor=100000seed=50gama=20tanphi=05cohesion=10h=25m=2do 10 I=1100000a=0b=25c=50rdn=nrndl(seed)a=-20lowastrdnrdn=nrndl(seed)b=25+20lowastrdnrdn=nrndl(seed)c=50+20lowastrdnif(b==25) b=2501x=(clowastc(2lowastb)+(25+b)2-b2+alowasta(50-2lowastb))(cb+a(25-b))y=b2+(cb)lowast(x-c2)r=sqrt((x-c)lowast(x-c)+ylowasty)
12 The Scientific World Journal
AA=(4lowastrlowastr-ylowasty)lowastsqrt(rlowastr-ylowasty)BB=(4lowastrlowastr-(h-y)lowast(h-y))lowastsqrt(rlowastr-(h-y)lowast(h-y))CC=(1m)lowast(2lowastrlowastr+xlowastx)lowastsqrt(rlowastr-xlowastx)DD=(1m)lowast(2lowastrlowastr+(mlowasth-x)lowast(mlowasth-x))lowastsqrt(rlowastr-(mlowasth-x)lowast(mlowasth-x))EE=ylowastasin((sqrt(rlowastr-ylowasty))r)FF=(h-y)lowastasin((sqrt(rlowastr-(h-y)lowast(h-y)))r)GG=(xm)lowastasin(xr)-((mlowasth-x)m)lowastasin((mlowasth-x)r)N=(1(6lowastr))lowast(AA+BB+CC-DD)+(r2)lowast(EE-FF+GG)T=(1(6lowastr))lowast(3lowasthlowastrlowastr-ylowastylowasty-(h-y)lowast(h-y)lowast(h-y)-xlowastxlowastxm-(mlowasth-x)lowast(mlowasth-x)lowast(mlowasth-x)m)L=rlowast(asin((sqrt(rlowastr-ylowasty))r)+asin((sqrt(rlowastr-(h-y)lowast(h-y)))r))newsafetyfactor=(gamalowasttanphilowastN+cohesionlowastL)(gamalowastT)if(newsafetyfactorltsafetyfactor)safetyfactor=newsafetyfactorwrite(lowastlowast)safetyfactorwrite(lowastlowast)awrite(lowastlowast)bwrite(lowastlowast)c10 continueend programreal function nrndl(seed)double precision SUVseedS=655360U=20530V=138490M=seedSseed=seed-MlowastSseed=Ulowastseed+VM=seedSseed=seed-MlowastSnrndl=seedSreturnend
Conflict of Interests
The author declares that he has no financial or personalrelationships with other people or organizations that couldinappropriately influence his work The author of this paperhas chosen not to furnish the paper and its readers with infor-mation that might present a potential conflict of interests
Acknowledgment
The author acknowledges the Scientific Research StartingFunds at Liaoning Technical University (no 11-415)
References
[1] D G Fredlund and J Krahn ldquoComparison of slope stabilitymethods of analysisrdquoCanadianGeotechnical Journal vol 14 no3 pp 429ndash439 1977
[2] K S Li ldquoA unified solution scheme for slope stability analysisrdquoin Proceedings of the 5th International Symposium on LandslidesNew Zealand vol 1 pp 481ndash486 Balkema Rotterdam TheNetherlands 1992
[3] W-J NiuW-M Ye S-G Liu and H-T Yu ldquoLimit analysis of asoil slope considering saturated-unsaturated seepagerdquoRock andSoil Mechanics vol 30 no 8 pp 2477ndash2482 2009
[4] W F Chen Limit Analysis and Soil Plasticity Elsevier Amster-dam The Netherlands 1975
[5] R L Michalowski ldquoSlope stability analysis a kinematical ap-proachrdquo Geotechnique vol 45 no 2 pp 283ndash293 1995
[6] I B Donald and Z Y Chen ldquoSlope stability analysis by theupper bound approach fundamentals and methodsrdquo CanadianGeotechnical Journal vol 34 no 6 pp 853ndash862 1997
[7] J P Sun J C Li and Q Q Liu ldquoSearch for critical slip surfacein slope stability analysis by spline-based GA methodrdquo Journalof Geotechnical and Geoenvironmental Engineering vol 134 no2 pp 252ndash256 2008
[8] R Javadzadeh and E Javadzadeh ldquoLocating critical failure sur-face in rock slope stability with hybrid model based on artificialimmune system and Cellular Learning Automata (CLA-AIS)rdquoWorld Academy of Science Engineering and Technology vol 59pp 662ndash665 2011
[9] F N G Gitirana Jr and D G Fredlund ldquoAnalysis of tran-sient embankment stability using the dynamic programmingmethodrdquo in Proceedings of the 56th Canadian GeotechnicalConference 2003
[10] H T V Pham and D G Fredlund ldquoDynamic programmingmethod in slope stability computationsrdquo in Proceedings ofthe 12th Asian Regional Conference on Soil Mechanics andGeotechnica Engineering Singapore August 2003
[11] H T V Pham D G Fredlund and F N G Gitirana Jr ldquoSlopestability analysis using dynamic prograamming combined withfinite element stress analysisrdquo in Proceedings of the InternationalConference on theManagement of the Land andWater Resources(MLWR rsquo01) Hanoi Vietnam October 2001
[12] H T V Pham and D G Fredlund ldquoThe application of dynamicprogramming to slope stability analysisrdquoCanadianGeotechnicalJournal vol 40 no 4 pp 830ndash847 2003
[13] M R Taha M Khajehzadeh and A El-Shafie ldquoSlope stabilityassessment using optimization techniques an overviewrdquo Elec-tronic Journal of Geotechnical Engineering vol 15 pp 1901ndash19152010
[14] A U Rao and N Sabhahit ldquoGenetic algorithm in stabilityof non-homogeneous slopesrdquo in Proceedings of the 12th Inter-national Conference of International Association for ComputerMethods and Advances in Geomechanics (IACMAG rsquo08) GoaIndia October 2008
[15] Y-G Nie W-Q Liu J-Y Shi and W-B Zhao ldquoApplication ofaccelerating genetic algorithm for embankment slope stabilityanalysisrdquo China Journal of Highway and Transport vol 16 no4 p 16 2003
The Scientific World Journal 13
[16] Y-P Zhou S-J Li Y-X Liu H-L Sun and F Jiang ldquoSearchingcritical failure surface in slope stability analysis with geneticalgorithmrdquo Chinese Journal of Rock Mechanics and Engineeringvol 24 pp 5226ndash5230 2005
[17] P McCombie ldquoCritical failure surface location using SimpleGenetic Algorithm and multiple wedge slope stabilityrdquo in Pro-ceedings of the 1st International Symposium on ComputationalGeomechanics pp 900ndash909 Juan-les-Pins France May 2009
[18] K Gavin and J F Xue ldquoUse of a genetic algorithm to performreliability analysis of unsaturated soil slopesrdquoGeotechnique vol59 no 6 pp 545ndash549 2009
[19] Wikipedia ldquoGenetic algorithmrdquo 2012 httpenwikipediaorgwikiGenetic algorithm
[20] China University of Geosciences ldquoCohesive soil slope stabi-lity analysisrdquo httpcoursecugeducncugsoil mechanicsCOURSECHAPTER7Chap7 3 5htm
[21] J E Thomaz ldquoA general method for three dimensional slopestability analysis informational reportrdquo Tech Rep JTRP Tech-nical Reports Purdue University 1986
[22] T B Zhang ldquoEarth slope stability analysis and geotechnologyslope designrdquo Tech Rep Chengdu University of Science andTechnology Press 1987
[23] B G He Application Examples in Civil Engineering with AnsysChina Water Power Press Beijing China 2011
[24] Y Liu Q Yang and L Zhu ldquoAbutment stability analysis of archdam based on 3D nonlinear finite element methodrdquo ChineseJournal of Rock Mechanics and Engineering vol 27 supplement1 pp 3222ndash3228 2008
[25] Y R Liu ZHe B Li andQYang ldquoSlope stability analysis basedon a multigrid method using a nonlinear 3D finite elementmodelrdquo Frontiers of Structural and Civil Engineering vol 7 no1 pp 24ndash31 2013
[26] Wikipedia ldquoDruckerndashPrager yield criterionrdquo 2013 httpenwikipediaorgwikiDruckerE28093Prager yield criterion
[27] B Doran H O Koksal Z Polat and C Karakoc ldquoThe use ofldquoDrucker-Prager Criterionrdquo in the analysis of reinforced con-crete members by finite elementsrdquo Teknik Dergi vol 9 no 2pp 1617ndash1625 1998
[28] B Zoran P Verka and M Biljana ldquoMathematical modeling ofmaterially nonlinear problems in structural analyses (part Imdashtheoretical fundamentals)rdquo Facta Universitatis Architecture andCivil Engineering vol 8 no 1 pp 67ndash78 2010
[29] Q Yang X Chen and W-Y Zhou ldquoA practical 3D elasto-plastic incrementalmethod in FEMbased onD-P yield criteriardquoChinese Journal of Geotechnical Engineering vol 24 no 1 pp16ndash20 2002
[30] G D Zhang Soil Constitutive Model and Its Application inEngineering Science Press Beijing China 1995
[31] D Rakic M Zivkovic R Slavkovic and M Kojic ldquoStress inte-gration for the Drucker-Prager material model without hard-ening using the Incremental Plasticity Theoryrdquo Journal of theSerbian Society For Computational Mechanics vol 2 no 1 pp80ndash89 2008
[32] X Y Zhang Soil and Rock Plastic Mechanics China Communi-cations Press Beijing China 1993
[33] Y R Zheng Z J Shen and X N Gong The Principles of Geo-technical Plastic Mechanics China Architecture and BuildingPress Beijing China 2002
[34] S A Akers Two-dimensional finite element analysis of porousgeomaterials at multikilobar stress levels [PhD thesis] VirginiaPolytechnic Institute and State University 2001
[35] X-H Tan J-G Wang and Y Wang ldquoNonlinear finite elementanalysis of slope stabilityrdquo Rock and Soil Mechanics vol 29 no8 pp 2047ndash2050 2008
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 11
Y
Z X MX
MN
Slope stability analysis0 0033261 0066522 0099783 0133044
0016631 0049892 0083153 0116414 0149675
Nodal solutionStep = 1
Sub = 999999
Time = 1
EPPLEQV (avg)DMX = 6955
UNFORRFORACEL
SMX = 0149675
Figure 17 von Mises plastic strain runs through from slope toe totop surface when safety factor equals 20
solution of slope safety factor which is 20 So the true slopesafety factor is likely from 18 to 20
54 Comparisons and Discussions The obtained minimumsafety factor for the above slope stability problem examplewith Genetic-Traversal Random Search Method is so lowwhen compared with the other methods like slopew soft-ware This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This mayalso be due to the power of the computer to realize theGenetic-Traversal Random Search Method in the AppendixThe Genetic-Traversal Random Method uses random pickto utilize mutation Validation of these conclusions will beinvestigated in the future with more numeric tests
However the obtained minimum safety factor withGenetic-Traversal Random Search Method is very close tothe lower bound solutions of slope safety factor given by theAnsys software
After computation there is plastic strain in layer 2 regionin some pictures of Figures 16 and 17 This is unreasonablesince layer 2 is defined as elastic region in the analysis withAnsys This phenomenon will be investigated in the future
6 Conclusions
This paper intends to determine a cohesive soil slope safetyfactor with Felleniusrsquo method while the 2D critical failuresurface is searched with GA The 2D critical failure surfaceis represented with real-encoded chromosomes which arepotential critical surface locations variables 119883
119888and 119883
119888119888 GA
procedure for searching critical failure surface proceeds withhand calculations If for future computer automatic searchprogram with GA program code for inheritance mutationselection and crossover program code for random numbersand program code for search interval boundaries will beneeded The minimum safety factor of 1325 determined byforegoing GA procedure to search the critical slip surface isvery close to the minimum safety factor of 1320 determinedby Felleniusrsquo critical slip surface method After comparedwith slopew software the proposed foregoing GA procedure
employed to search the critical failure surface is reasonableapplicable and effective
At last a computer automatic search program (Genetic-Traversal Random Search Method) inspired by GA is madewhile in the program random numbers generated by com-puter and search boundaries are included The Genetic-Traversal Random Method uses random pick to utilizemutation In the program the slope safety factor is given byanalytical solution rather than slices method Results indicatethat the new computer automatic search program can givevery low safety factor which is about half of the foregoingones This may be due to the fact that the analytical solutionis more accurate than Felleniusrsquo slices method This may alsobe due to the power of the random number generation sub-program computer operation speed and Genetic-TraversalRandom Method Further validation of the results will beinvestigated in the future However the obtained minimumsafety factor with Genetic-Traversal Random Search Methodis very close to the lower bound solutions of slope safety factorgiven by the Ansys software
Appendix
Safety factor and failure circle determination program devel-oped in Silverfrost FTN95
double precision seedreal nrndlreal NLnewsafetyfactorsafetyfactor=100000seed=50gama=20tanphi=05cohesion=10h=25m=2do 10 I=1100000a=0b=25c=50rdn=nrndl(seed)a=-20lowastrdnrdn=nrndl(seed)b=25+20lowastrdnrdn=nrndl(seed)c=50+20lowastrdnif(b==25) b=2501x=(clowastc(2lowastb)+(25+b)2-b2+alowasta(50-2lowastb))(cb+a(25-b))y=b2+(cb)lowast(x-c2)r=sqrt((x-c)lowast(x-c)+ylowasty)
12 The Scientific World Journal
AA=(4lowastrlowastr-ylowasty)lowastsqrt(rlowastr-ylowasty)BB=(4lowastrlowastr-(h-y)lowast(h-y))lowastsqrt(rlowastr-(h-y)lowast(h-y))CC=(1m)lowast(2lowastrlowastr+xlowastx)lowastsqrt(rlowastr-xlowastx)DD=(1m)lowast(2lowastrlowastr+(mlowasth-x)lowast(mlowasth-x))lowastsqrt(rlowastr-(mlowasth-x)lowast(mlowasth-x))EE=ylowastasin((sqrt(rlowastr-ylowasty))r)FF=(h-y)lowastasin((sqrt(rlowastr-(h-y)lowast(h-y)))r)GG=(xm)lowastasin(xr)-((mlowasth-x)m)lowastasin((mlowasth-x)r)N=(1(6lowastr))lowast(AA+BB+CC-DD)+(r2)lowast(EE-FF+GG)T=(1(6lowastr))lowast(3lowasthlowastrlowastr-ylowastylowasty-(h-y)lowast(h-y)lowast(h-y)-xlowastxlowastxm-(mlowasth-x)lowast(mlowasth-x)lowast(mlowasth-x)m)L=rlowast(asin((sqrt(rlowastr-ylowasty))r)+asin((sqrt(rlowastr-(h-y)lowast(h-y)))r))newsafetyfactor=(gamalowasttanphilowastN+cohesionlowastL)(gamalowastT)if(newsafetyfactorltsafetyfactor)safetyfactor=newsafetyfactorwrite(lowastlowast)safetyfactorwrite(lowastlowast)awrite(lowastlowast)bwrite(lowastlowast)c10 continueend programreal function nrndl(seed)double precision SUVseedS=655360U=20530V=138490M=seedSseed=seed-MlowastSseed=Ulowastseed+VM=seedSseed=seed-MlowastSnrndl=seedSreturnend
Conflict of Interests
The author declares that he has no financial or personalrelationships with other people or organizations that couldinappropriately influence his work The author of this paperhas chosen not to furnish the paper and its readers with infor-mation that might present a potential conflict of interests
Acknowledgment
The author acknowledges the Scientific Research StartingFunds at Liaoning Technical University (no 11-415)
References
[1] D G Fredlund and J Krahn ldquoComparison of slope stabilitymethods of analysisrdquoCanadianGeotechnical Journal vol 14 no3 pp 429ndash439 1977
[2] K S Li ldquoA unified solution scheme for slope stability analysisrdquoin Proceedings of the 5th International Symposium on LandslidesNew Zealand vol 1 pp 481ndash486 Balkema Rotterdam TheNetherlands 1992
[3] W-J NiuW-M Ye S-G Liu and H-T Yu ldquoLimit analysis of asoil slope considering saturated-unsaturated seepagerdquoRock andSoil Mechanics vol 30 no 8 pp 2477ndash2482 2009
[4] W F Chen Limit Analysis and Soil Plasticity Elsevier Amster-dam The Netherlands 1975
[5] R L Michalowski ldquoSlope stability analysis a kinematical ap-proachrdquo Geotechnique vol 45 no 2 pp 283ndash293 1995
[6] I B Donald and Z Y Chen ldquoSlope stability analysis by theupper bound approach fundamentals and methodsrdquo CanadianGeotechnical Journal vol 34 no 6 pp 853ndash862 1997
[7] J P Sun J C Li and Q Q Liu ldquoSearch for critical slip surfacein slope stability analysis by spline-based GA methodrdquo Journalof Geotechnical and Geoenvironmental Engineering vol 134 no2 pp 252ndash256 2008
[8] R Javadzadeh and E Javadzadeh ldquoLocating critical failure sur-face in rock slope stability with hybrid model based on artificialimmune system and Cellular Learning Automata (CLA-AIS)rdquoWorld Academy of Science Engineering and Technology vol 59pp 662ndash665 2011
[9] F N G Gitirana Jr and D G Fredlund ldquoAnalysis of tran-sient embankment stability using the dynamic programmingmethodrdquo in Proceedings of the 56th Canadian GeotechnicalConference 2003
[10] H T V Pham and D G Fredlund ldquoDynamic programmingmethod in slope stability computationsrdquo in Proceedings ofthe 12th Asian Regional Conference on Soil Mechanics andGeotechnica Engineering Singapore August 2003
[11] H T V Pham D G Fredlund and F N G Gitirana Jr ldquoSlopestability analysis using dynamic prograamming combined withfinite element stress analysisrdquo in Proceedings of the InternationalConference on theManagement of the Land andWater Resources(MLWR rsquo01) Hanoi Vietnam October 2001
[12] H T V Pham and D G Fredlund ldquoThe application of dynamicprogramming to slope stability analysisrdquoCanadianGeotechnicalJournal vol 40 no 4 pp 830ndash847 2003
[13] M R Taha M Khajehzadeh and A El-Shafie ldquoSlope stabilityassessment using optimization techniques an overviewrdquo Elec-tronic Journal of Geotechnical Engineering vol 15 pp 1901ndash19152010
[14] A U Rao and N Sabhahit ldquoGenetic algorithm in stabilityof non-homogeneous slopesrdquo in Proceedings of the 12th Inter-national Conference of International Association for ComputerMethods and Advances in Geomechanics (IACMAG rsquo08) GoaIndia October 2008
[15] Y-G Nie W-Q Liu J-Y Shi and W-B Zhao ldquoApplication ofaccelerating genetic algorithm for embankment slope stabilityanalysisrdquo China Journal of Highway and Transport vol 16 no4 p 16 2003
The Scientific World Journal 13
[16] Y-P Zhou S-J Li Y-X Liu H-L Sun and F Jiang ldquoSearchingcritical failure surface in slope stability analysis with geneticalgorithmrdquo Chinese Journal of Rock Mechanics and Engineeringvol 24 pp 5226ndash5230 2005
[17] P McCombie ldquoCritical failure surface location using SimpleGenetic Algorithm and multiple wedge slope stabilityrdquo in Pro-ceedings of the 1st International Symposium on ComputationalGeomechanics pp 900ndash909 Juan-les-Pins France May 2009
[18] K Gavin and J F Xue ldquoUse of a genetic algorithm to performreliability analysis of unsaturated soil slopesrdquoGeotechnique vol59 no 6 pp 545ndash549 2009
[19] Wikipedia ldquoGenetic algorithmrdquo 2012 httpenwikipediaorgwikiGenetic algorithm
[20] China University of Geosciences ldquoCohesive soil slope stabi-lity analysisrdquo httpcoursecugeducncugsoil mechanicsCOURSECHAPTER7Chap7 3 5htm
[21] J E Thomaz ldquoA general method for three dimensional slopestability analysis informational reportrdquo Tech Rep JTRP Tech-nical Reports Purdue University 1986
[22] T B Zhang ldquoEarth slope stability analysis and geotechnologyslope designrdquo Tech Rep Chengdu University of Science andTechnology Press 1987
[23] B G He Application Examples in Civil Engineering with AnsysChina Water Power Press Beijing China 2011
[24] Y Liu Q Yang and L Zhu ldquoAbutment stability analysis of archdam based on 3D nonlinear finite element methodrdquo ChineseJournal of Rock Mechanics and Engineering vol 27 supplement1 pp 3222ndash3228 2008
[25] Y R Liu ZHe B Li andQYang ldquoSlope stability analysis basedon a multigrid method using a nonlinear 3D finite elementmodelrdquo Frontiers of Structural and Civil Engineering vol 7 no1 pp 24ndash31 2013
[26] Wikipedia ldquoDruckerndashPrager yield criterionrdquo 2013 httpenwikipediaorgwikiDruckerE28093Prager yield criterion
[27] B Doran H O Koksal Z Polat and C Karakoc ldquoThe use ofldquoDrucker-Prager Criterionrdquo in the analysis of reinforced con-crete members by finite elementsrdquo Teknik Dergi vol 9 no 2pp 1617ndash1625 1998
[28] B Zoran P Verka and M Biljana ldquoMathematical modeling ofmaterially nonlinear problems in structural analyses (part Imdashtheoretical fundamentals)rdquo Facta Universitatis Architecture andCivil Engineering vol 8 no 1 pp 67ndash78 2010
[29] Q Yang X Chen and W-Y Zhou ldquoA practical 3D elasto-plastic incrementalmethod in FEMbased onD-P yield criteriardquoChinese Journal of Geotechnical Engineering vol 24 no 1 pp16ndash20 2002
[30] G D Zhang Soil Constitutive Model and Its Application inEngineering Science Press Beijing China 1995
[31] D Rakic M Zivkovic R Slavkovic and M Kojic ldquoStress inte-gration for the Drucker-Prager material model without hard-ening using the Incremental Plasticity Theoryrdquo Journal of theSerbian Society For Computational Mechanics vol 2 no 1 pp80ndash89 2008
[32] X Y Zhang Soil and Rock Plastic Mechanics China Communi-cations Press Beijing China 1993
[33] Y R Zheng Z J Shen and X N Gong The Principles of Geo-technical Plastic Mechanics China Architecture and BuildingPress Beijing China 2002
[34] S A Akers Two-dimensional finite element analysis of porousgeomaterials at multikilobar stress levels [PhD thesis] VirginiaPolytechnic Institute and State University 2001
[35] X-H Tan J-G Wang and Y Wang ldquoNonlinear finite elementanalysis of slope stabilityrdquo Rock and Soil Mechanics vol 29 no8 pp 2047ndash2050 2008
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
12 The Scientific World Journal
AA=(4lowastrlowastr-ylowasty)lowastsqrt(rlowastr-ylowasty)BB=(4lowastrlowastr-(h-y)lowast(h-y))lowastsqrt(rlowastr-(h-y)lowast(h-y))CC=(1m)lowast(2lowastrlowastr+xlowastx)lowastsqrt(rlowastr-xlowastx)DD=(1m)lowast(2lowastrlowastr+(mlowasth-x)lowast(mlowasth-x))lowastsqrt(rlowastr-(mlowasth-x)lowast(mlowasth-x))EE=ylowastasin((sqrt(rlowastr-ylowasty))r)FF=(h-y)lowastasin((sqrt(rlowastr-(h-y)lowast(h-y)))r)GG=(xm)lowastasin(xr)-((mlowasth-x)m)lowastasin((mlowasth-x)r)N=(1(6lowastr))lowast(AA+BB+CC-DD)+(r2)lowast(EE-FF+GG)T=(1(6lowastr))lowast(3lowasthlowastrlowastr-ylowastylowasty-(h-y)lowast(h-y)lowast(h-y)-xlowastxlowastxm-(mlowasth-x)lowast(mlowasth-x)lowast(mlowasth-x)m)L=rlowast(asin((sqrt(rlowastr-ylowasty))r)+asin((sqrt(rlowastr-(h-y)lowast(h-y)))r))newsafetyfactor=(gamalowasttanphilowastN+cohesionlowastL)(gamalowastT)if(newsafetyfactorltsafetyfactor)safetyfactor=newsafetyfactorwrite(lowastlowast)safetyfactorwrite(lowastlowast)awrite(lowastlowast)bwrite(lowastlowast)c10 continueend programreal function nrndl(seed)double precision SUVseedS=655360U=20530V=138490M=seedSseed=seed-MlowastSseed=Ulowastseed+VM=seedSseed=seed-MlowastSnrndl=seedSreturnend
Conflict of Interests
The author declares that he has no financial or personalrelationships with other people or organizations that couldinappropriately influence his work The author of this paperhas chosen not to furnish the paper and its readers with infor-mation that might present a potential conflict of interests
Acknowledgment
The author acknowledges the Scientific Research StartingFunds at Liaoning Technical University (no 11-415)
References
[1] D G Fredlund and J Krahn ldquoComparison of slope stabilitymethods of analysisrdquoCanadianGeotechnical Journal vol 14 no3 pp 429ndash439 1977
[2] K S Li ldquoA unified solution scheme for slope stability analysisrdquoin Proceedings of the 5th International Symposium on LandslidesNew Zealand vol 1 pp 481ndash486 Balkema Rotterdam TheNetherlands 1992
[3] W-J NiuW-M Ye S-G Liu and H-T Yu ldquoLimit analysis of asoil slope considering saturated-unsaturated seepagerdquoRock andSoil Mechanics vol 30 no 8 pp 2477ndash2482 2009
[4] W F Chen Limit Analysis and Soil Plasticity Elsevier Amster-dam The Netherlands 1975
[5] R L Michalowski ldquoSlope stability analysis a kinematical ap-proachrdquo Geotechnique vol 45 no 2 pp 283ndash293 1995
[6] I B Donald and Z Y Chen ldquoSlope stability analysis by theupper bound approach fundamentals and methodsrdquo CanadianGeotechnical Journal vol 34 no 6 pp 853ndash862 1997
[7] J P Sun J C Li and Q Q Liu ldquoSearch for critical slip surfacein slope stability analysis by spline-based GA methodrdquo Journalof Geotechnical and Geoenvironmental Engineering vol 134 no2 pp 252ndash256 2008
[8] R Javadzadeh and E Javadzadeh ldquoLocating critical failure sur-face in rock slope stability with hybrid model based on artificialimmune system and Cellular Learning Automata (CLA-AIS)rdquoWorld Academy of Science Engineering and Technology vol 59pp 662ndash665 2011
[9] F N G Gitirana Jr and D G Fredlund ldquoAnalysis of tran-sient embankment stability using the dynamic programmingmethodrdquo in Proceedings of the 56th Canadian GeotechnicalConference 2003
[10] H T V Pham and D G Fredlund ldquoDynamic programmingmethod in slope stability computationsrdquo in Proceedings ofthe 12th Asian Regional Conference on Soil Mechanics andGeotechnica Engineering Singapore August 2003
[11] H T V Pham D G Fredlund and F N G Gitirana Jr ldquoSlopestability analysis using dynamic prograamming combined withfinite element stress analysisrdquo in Proceedings of the InternationalConference on theManagement of the Land andWater Resources(MLWR rsquo01) Hanoi Vietnam October 2001
[12] H T V Pham and D G Fredlund ldquoThe application of dynamicprogramming to slope stability analysisrdquoCanadianGeotechnicalJournal vol 40 no 4 pp 830ndash847 2003
[13] M R Taha M Khajehzadeh and A El-Shafie ldquoSlope stabilityassessment using optimization techniques an overviewrdquo Elec-tronic Journal of Geotechnical Engineering vol 15 pp 1901ndash19152010
[14] A U Rao and N Sabhahit ldquoGenetic algorithm in stabilityof non-homogeneous slopesrdquo in Proceedings of the 12th Inter-national Conference of International Association for ComputerMethods and Advances in Geomechanics (IACMAG rsquo08) GoaIndia October 2008
[15] Y-G Nie W-Q Liu J-Y Shi and W-B Zhao ldquoApplication ofaccelerating genetic algorithm for embankment slope stabilityanalysisrdquo China Journal of Highway and Transport vol 16 no4 p 16 2003
The Scientific World Journal 13
[16] Y-P Zhou S-J Li Y-X Liu H-L Sun and F Jiang ldquoSearchingcritical failure surface in slope stability analysis with geneticalgorithmrdquo Chinese Journal of Rock Mechanics and Engineeringvol 24 pp 5226ndash5230 2005
[17] P McCombie ldquoCritical failure surface location using SimpleGenetic Algorithm and multiple wedge slope stabilityrdquo in Pro-ceedings of the 1st International Symposium on ComputationalGeomechanics pp 900ndash909 Juan-les-Pins France May 2009
[18] K Gavin and J F Xue ldquoUse of a genetic algorithm to performreliability analysis of unsaturated soil slopesrdquoGeotechnique vol59 no 6 pp 545ndash549 2009
[19] Wikipedia ldquoGenetic algorithmrdquo 2012 httpenwikipediaorgwikiGenetic algorithm
[20] China University of Geosciences ldquoCohesive soil slope stabi-lity analysisrdquo httpcoursecugeducncugsoil mechanicsCOURSECHAPTER7Chap7 3 5htm
[21] J E Thomaz ldquoA general method for three dimensional slopestability analysis informational reportrdquo Tech Rep JTRP Tech-nical Reports Purdue University 1986
[22] T B Zhang ldquoEarth slope stability analysis and geotechnologyslope designrdquo Tech Rep Chengdu University of Science andTechnology Press 1987
[23] B G He Application Examples in Civil Engineering with AnsysChina Water Power Press Beijing China 2011
[24] Y Liu Q Yang and L Zhu ldquoAbutment stability analysis of archdam based on 3D nonlinear finite element methodrdquo ChineseJournal of Rock Mechanics and Engineering vol 27 supplement1 pp 3222ndash3228 2008
[25] Y R Liu ZHe B Li andQYang ldquoSlope stability analysis basedon a multigrid method using a nonlinear 3D finite elementmodelrdquo Frontiers of Structural and Civil Engineering vol 7 no1 pp 24ndash31 2013
[26] Wikipedia ldquoDruckerndashPrager yield criterionrdquo 2013 httpenwikipediaorgwikiDruckerE28093Prager yield criterion
[27] B Doran H O Koksal Z Polat and C Karakoc ldquoThe use ofldquoDrucker-Prager Criterionrdquo in the analysis of reinforced con-crete members by finite elementsrdquo Teknik Dergi vol 9 no 2pp 1617ndash1625 1998
[28] B Zoran P Verka and M Biljana ldquoMathematical modeling ofmaterially nonlinear problems in structural analyses (part Imdashtheoretical fundamentals)rdquo Facta Universitatis Architecture andCivil Engineering vol 8 no 1 pp 67ndash78 2010
[29] Q Yang X Chen and W-Y Zhou ldquoA practical 3D elasto-plastic incrementalmethod in FEMbased onD-P yield criteriardquoChinese Journal of Geotechnical Engineering vol 24 no 1 pp16ndash20 2002
[30] G D Zhang Soil Constitutive Model and Its Application inEngineering Science Press Beijing China 1995
[31] D Rakic M Zivkovic R Slavkovic and M Kojic ldquoStress inte-gration for the Drucker-Prager material model without hard-ening using the Incremental Plasticity Theoryrdquo Journal of theSerbian Society For Computational Mechanics vol 2 no 1 pp80ndash89 2008
[32] X Y Zhang Soil and Rock Plastic Mechanics China Communi-cations Press Beijing China 1993
[33] Y R Zheng Z J Shen and X N Gong The Principles of Geo-technical Plastic Mechanics China Architecture and BuildingPress Beijing China 2002
[34] S A Akers Two-dimensional finite element analysis of porousgeomaterials at multikilobar stress levels [PhD thesis] VirginiaPolytechnic Institute and State University 2001
[35] X-H Tan J-G Wang and Y Wang ldquoNonlinear finite elementanalysis of slope stabilityrdquo Rock and Soil Mechanics vol 29 no8 pp 2047ndash2050 2008
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 13
[16] Y-P Zhou S-J Li Y-X Liu H-L Sun and F Jiang ldquoSearchingcritical failure surface in slope stability analysis with geneticalgorithmrdquo Chinese Journal of Rock Mechanics and Engineeringvol 24 pp 5226ndash5230 2005
[17] P McCombie ldquoCritical failure surface location using SimpleGenetic Algorithm and multiple wedge slope stabilityrdquo in Pro-ceedings of the 1st International Symposium on ComputationalGeomechanics pp 900ndash909 Juan-les-Pins France May 2009
[18] K Gavin and J F Xue ldquoUse of a genetic algorithm to performreliability analysis of unsaturated soil slopesrdquoGeotechnique vol59 no 6 pp 545ndash549 2009
[19] Wikipedia ldquoGenetic algorithmrdquo 2012 httpenwikipediaorgwikiGenetic algorithm
[20] China University of Geosciences ldquoCohesive soil slope stabi-lity analysisrdquo httpcoursecugeducncugsoil mechanicsCOURSECHAPTER7Chap7 3 5htm
[21] J E Thomaz ldquoA general method for three dimensional slopestability analysis informational reportrdquo Tech Rep JTRP Tech-nical Reports Purdue University 1986
[22] T B Zhang ldquoEarth slope stability analysis and geotechnologyslope designrdquo Tech Rep Chengdu University of Science andTechnology Press 1987
[23] B G He Application Examples in Civil Engineering with AnsysChina Water Power Press Beijing China 2011
[24] Y Liu Q Yang and L Zhu ldquoAbutment stability analysis of archdam based on 3D nonlinear finite element methodrdquo ChineseJournal of Rock Mechanics and Engineering vol 27 supplement1 pp 3222ndash3228 2008
[25] Y R Liu ZHe B Li andQYang ldquoSlope stability analysis basedon a multigrid method using a nonlinear 3D finite elementmodelrdquo Frontiers of Structural and Civil Engineering vol 7 no1 pp 24ndash31 2013
[26] Wikipedia ldquoDruckerndashPrager yield criterionrdquo 2013 httpenwikipediaorgwikiDruckerE28093Prager yield criterion
[27] B Doran H O Koksal Z Polat and C Karakoc ldquoThe use ofldquoDrucker-Prager Criterionrdquo in the analysis of reinforced con-crete members by finite elementsrdquo Teknik Dergi vol 9 no 2pp 1617ndash1625 1998
[28] B Zoran P Verka and M Biljana ldquoMathematical modeling ofmaterially nonlinear problems in structural analyses (part Imdashtheoretical fundamentals)rdquo Facta Universitatis Architecture andCivil Engineering vol 8 no 1 pp 67ndash78 2010
[29] Q Yang X Chen and W-Y Zhou ldquoA practical 3D elasto-plastic incrementalmethod in FEMbased onD-P yield criteriardquoChinese Journal of Geotechnical Engineering vol 24 no 1 pp16ndash20 2002
[30] G D Zhang Soil Constitutive Model and Its Application inEngineering Science Press Beijing China 1995
[31] D Rakic M Zivkovic R Slavkovic and M Kojic ldquoStress inte-gration for the Drucker-Prager material model without hard-ening using the Incremental Plasticity Theoryrdquo Journal of theSerbian Society For Computational Mechanics vol 2 no 1 pp80ndash89 2008
[32] X Y Zhang Soil and Rock Plastic Mechanics China Communi-cations Press Beijing China 1993
[33] Y R Zheng Z J Shen and X N Gong The Principles of Geo-technical Plastic Mechanics China Architecture and BuildingPress Beijing China 2002
[34] S A Akers Two-dimensional finite element analysis of porousgeomaterials at multikilobar stress levels [PhD thesis] VirginiaPolytechnic Institute and State University 2001
[35] X-H Tan J-G Wang and Y Wang ldquoNonlinear finite elementanalysis of slope stabilityrdquo Rock and Soil Mechanics vol 29 no8 pp 2047ndash2050 2008
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of