determination of slope safety factor with analytical ... method), simplified bishop method, ......

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


Table 4 Mutation results

119883119888

119883119888119888


50 minus10 1400390984351 21 17627329758

lowast 55 0 13332021645lowast 55 minus10 13615830364

55 20 16493380867



119883119888119888


51 minus10 1379923988560 0 13879596689

lowast 50 0 13425306677lowast 55 0 13332021645

55 minus10 13615830364


119883119888

119883119888119888

Safety factor51 0 1331746587851 minus10 1379923988560 0 1387959668950 0 1342530667755 0 1333202164555 minus10 13615830364


119883119888

119883119888119888


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



119883119888119888

Safety factorlowast 51 0 13317465878lowast 50 0 13425306677lowast 55 0 13332021645lowast 52 0 13264516064lowast 53 0 13254640532

57 0 1350348132152 11 13501275295


119883119888

119883119888119888

Safety factor51 11 1355689633150 11 1367290667655 11 13547892263

lowast 53 11 1348701176357 11 13696385778


119883119888

119883119888119888

Safety factor54 21 16711004325

lowast 54 0 1327927158254 13 1373489757754 minus10 1359516051960 minus10 14074551938



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


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


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


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


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


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


Sub = 15

Time = 1


UNFORRFORACEL


(b) Safety factor equals 072

Y

Z X

MXMN



Sub = 15

Time = 1


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


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


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0719E minus 04


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


Sub = 15

Time = 1


UNFORRFORACEL



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


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0001325


Y

Z X

MX

MN

Slope stability analysis0 0926E minus 03 0001851 0002777 0003703

0463E minus 03 0001389 0002314 000324 0004166


Sub = 15

Time = 1


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


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0029499

(e) Safety factor equals 16

Y

Z XMX

MN


0007935 0023805 0039674 0055544 0071414


Sub = 15

Time = 1


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


Y

Z X MX

MN


0016631 0049892 0083153 0116414 0149675


Sub = 999999

Time = 1


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)


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


[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


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


L

x

y

A

R

S

Search region

R



Slope toe OXc

for Xcc

Xcc



119888and 119883








119888 119883119888119888


119888 119883119888119888 and happens






119888and119883

119888119888are



119883119888

119883119888119888







119883119888119888


60 minus10 1407455193860 25 1886105777100 minus20 22757670756


119883119888

119883119888119888


51 25 272052352151 minus20 14447027254





119888 119883119888119888 with










119883119888

119883119888119888


50 minus10 1400390984351 21 17627329758


55 20 16493380867



119883119888119888


51 minus10 1379923988560 0 13879596689

lowast 50 0 13425306677lowast 55 0 13332021645

55 minus10 13615830364


119883119888

119883119888119888



119883119888

119883119888119888


52 minus10 13675053085lowast 53 0 13254640532

53 minus10 13611610322lowast 57 0 13503481321

57 minus10 13741239435lowast 52 11 13501275295








119883119888119888


57 0 1350348132152 11 13501275295


119883119888

119883119888119888

Safety factor51 11 1355689633150 11 1367290667655 11 13547892263

lowast 53 11 1348701176357 11 13696385778


119883119888

119883119888119888





119883119888119888

Safety factor51 0 1331746587850 0 1342530667755 0 1333202164552 0 13264516064

lowast 53 0 1325464053253 11 1348701176354 0 13279271582


119883119888

119883119888119888





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




119896 =120574 sdot 119905119892120593 [119873] + 119888 [119871]

120574 [119879] (2)




2]


1

119898(21199032+ 1199092)radic1199032 minus 1199092

minus1

119898[21199032+ (119898ℎ minus 119909)


2

+119903

2


119903


2

119903

+119909

119898arcsin 119909

119903minus

119898ℎ minus 119909

119898arcsin 119898ℎ minus 119909

119903

119879 =1


3minus

1199093

119898minus

(119898ℎ minus 119909)3

119898]

119871 = 119903[[

[


119903+ arcsin


119903

]]

]

(3)











x

y




O


A(a 25)

B(0 b)

C(c 0)


A

CB

O

1205732

1205731120572





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∘



1and 120573

2which








1and119874





1015840

1 11987410158402 11987410158403 and 119874

1015840




119888in

Figure 2 is 52292m




A

B

F

H

H

D

G


O1

O4

1205732

1205731

Fs1

Fs4

C1 C2 C3 C4

45H

120572

O1998400

O2998400

O3998400

O4998400


1328

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5




1390

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5





1316

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



0and 120593

0 respectively


0SF


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


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1385

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



Layer1

Layer2

40

m80

m

80m 50m 70m

40

m105

m

25m


Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL


Y

Z X

MXMN


Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL




119860 =2 sin120593

radic3radic3 + sin120593 119861 =


(5)


119860 =2 sin120593



(6)



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0001325


Y

Z X

MX

MN


0463E minus 03 0001389 0002314 000324 0004166


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0004166


Y

Z X

MX

MN


0003278 0009833 0016388 0022944 0029499


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0029499


Y

Z XMX

MN


0007935 0023805 0039674 0055544 0071414


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0071414








Y

Z X MX

MN


0016631 0049892 0083153 0116414 0149675


Sub = 999999

Time = 1


UNFORRFORACEL

SMX = 0149675






6 Conclusions


119888and 119883

119888119888 GA




Appendix







Acknowledgment


References






































L

x

y

A

R

S

Search region

R



Slope toe OXc

for Xcc

Xcc



119888and 119883








119888 119883119888119888


119888 119883119888119888 and happens






119888and119883

119888119888are



119883119888

119883119888119888







119883119888119888


60 minus10 1407455193860 25 1886105777100 minus20 22757670756


119883119888

119883119888119888


51 25 272052352151 minus20 14447027254





119888 119883119888119888 with










119883119888

119883119888119888


50 minus10 1400390984351 21 17627329758


55 20 16493380867



119883119888119888


51 minus10 1379923988560 0 13879596689

lowast 50 0 13425306677lowast 55 0 13332021645

55 minus10 13615830364


119883119888

119883119888119888



119883119888

119883119888119888


52 minus10 13675053085lowast 53 0 13254640532

53 minus10 13611610322lowast 57 0 13503481321

57 minus10 13741239435lowast 52 11 13501275295








119883119888119888


57 0 1350348132152 11 13501275295


119883119888

119883119888119888

Safety factor51 11 1355689633150 11 1367290667655 11 13547892263

lowast 53 11 1348701176357 11 13696385778


119883119888

119883119888119888





119883119888119888

Safety factor51 0 1331746587850 0 1342530667755 0 1333202164552 0 13264516064

lowast 53 0 1325464053253 11 1348701176354 0 13279271582


119883119888

119883119888119888





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




119896 =120574 sdot 119905119892120593 [119873] + 119888 [119871]

120574 [119879] (2)




2]


1

119898(21199032+ 1199092)radic1199032 minus 1199092

minus1

119898[21199032+ (119898ℎ minus 119909)


2

+119903

2


119903


2

119903

+119909

119898arcsin 119909

119903minus

119898ℎ minus 119909

119898arcsin 119898ℎ minus 119909

119903

119879 =1


3minus

1199093

119898minus

(119898ℎ minus 119909)3

119898]

119871 = 119903[[

[


119903+ arcsin


119903

]]

]

(3)











x

y




O


A(a 25)

B(0 b)

C(c 0)


A

CB

O

1205732

1205731120572





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∘



1and 120573

2which








1and119874





1015840

1 11987410158402 11987410158403 and 119874

1015840




119888in

Figure 2 is 52292m




A

B

F

H

H

D

G


O1

O4

1205732

1205731

Fs1

Fs4

C1 C2 C3 C4

45H

120572

O1998400

O2998400

O3998400

O4998400


1328

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5




1390

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5





1316

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



0and 120593

0 respectively


0SF


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


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1385

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



Layer1

Layer2

40

m80

m

80m 50m 70m

40

m105

m

25m


Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL


Y

Z X

MXMN


Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL




119860 =2 sin120593

radic3radic3 + sin120593 119861 =


(5)


119860 =2 sin120593



(6)



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0001325


Y

Z X

MX

MN


0463E minus 03 0001389 0002314 000324 0004166


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0004166


Y

Z X

MX

MN


0003278 0009833 0016388 0022944 0029499


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0029499


Y

Z XMX

MN


0007935 0023805 0039674 0055544 0071414


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0071414








Y

Z X MX

MN


0016631 0049892 0083153 0116414 0149675


Sub = 999999

Time = 1


UNFORRFORACEL

SMX = 0149675






6 Conclusions


119888and 119883

119888119888 GA




Appendix







Acknowledgment


References







































119883119888

119883119888119888


50 minus10 1400390984351 21 17627329758


55 20 16493380867



119883119888119888


51 minus10 1379923988560 0 13879596689

lowast 50 0 13425306677lowast 55 0 13332021645

55 minus10 13615830364


119883119888

119883119888119888



119883119888

119883119888119888


52 minus10 13675053085lowast 53 0 13254640532

53 minus10 13611610322lowast 57 0 13503481321

57 minus10 13741239435lowast 52 11 13501275295








119883119888119888


57 0 1350348132152 11 13501275295


119883119888

119883119888119888

Safety factor51 11 1355689633150 11 1367290667655 11 13547892263

lowast 53 11 1348701176357 11 13696385778


119883119888

119883119888119888





119883119888119888

Safety factor51 0 1331746587850 0 1342530667755 0 1333202164552 0 13264516064

lowast 53 0 1325464053253 11 1348701176354 0 13279271582


119883119888

119883119888119888





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




119896 =120574 sdot 119905119892120593 [119873] + 119888 [119871]

120574 [119879] (2)




2]


1

119898(21199032+ 1199092)radic1199032 minus 1199092

minus1

119898[21199032+ (119898ℎ minus 119909)


2

+119903

2


119903


2

119903

+119909

119898arcsin 119909

119903minus

119898ℎ minus 119909

119898arcsin 119898ℎ minus 119909

119903

119879 =1


3minus

1199093

119898minus

(119898ℎ minus 119909)3

119898]

119871 = 119903[[

[


119903+ arcsin


119903

]]

]

(3)











x

y




O


A(a 25)

B(0 b)

C(c 0)


A

CB

O

1205732

1205731120572





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∘



1and 120573

2which








1and119874





1015840

1 11987410158402 11987410158403 and 119874

1015840




119888in

Figure 2 is 52292m




A

B

F

H

H

D

G


O1

O4

1205732

1205731

Fs1

Fs4

C1 C2 C3 C4

45H

120572

O1998400

O2998400

O3998400

O4998400


1328

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5




1390

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5





1316

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



0and 120593

0 respectively


0SF


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


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1385

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



Layer1

Layer2

40

m80

m

80m 50m 70m

40

m105

m

25m


Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL


Y

Z X

MXMN


Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL




119860 =2 sin120593

radic3radic3 + sin120593 119861 =


(5)


119860 =2 sin120593



(6)



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0001325


Y

Z X

MX

MN


0463E minus 03 0001389 0002314 000324 0004166


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0004166


Y

Z X

MX

MN


0003278 0009833 0016388 0022944 0029499


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0029499


Y

Z XMX

MN


0007935 0023805 0039674 0055544 0071414


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0071414








Y

Z X MX

MN


0016631 0049892 0083153 0116414 0149675


Sub = 999999

Time = 1


UNFORRFORACEL

SMX = 0149675






6 Conclusions


119888and 119883

119888119888 GA




Appendix







Acknowledgment


References






































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




119896 =120574 sdot 119905119892120593 [119873] + 119888 [119871]

120574 [119879] (2)




2]


1

119898(21199032+ 1199092)radic1199032 minus 1199092

minus1

119898[21199032+ (119898ℎ minus 119909)


2

+119903

2


119903


2

119903

+119909

119898arcsin 119909

119903minus

119898ℎ minus 119909

119898arcsin 119898ℎ minus 119909

119903

119879 =1


3minus

1199093

119898minus

(119898ℎ minus 119909)3

119898]

119871 = 119903[[

[


119903+ arcsin


119903

]]

]

(3)











x

y




O


A(a 25)

B(0 b)

C(c 0)


A

CB

O

1205732

1205731120572





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∘



1and 120573

2which








1and119874





1015840

1 11987410158402 11987410158403 and 119874

1015840




119888in

Figure 2 is 52292m




A

B

F

H

H

D

G


O1

O4

1205732

1205731

Fs1

Fs4

C1 C2 C3 C4

45H

120572

O1998400

O2998400

O3998400

O4998400


1328

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5




1390

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5





1316

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



0and 120593

0 respectively


0SF


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


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1385

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



Layer1

Layer2

40

m80

m

80m 50m 70m

40

m105

m

25m


Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL


Y

Z X

MXMN


Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL




119860 =2 sin120593

radic3radic3 + sin120593 119861 =


(5)


119860 =2 sin120593



(6)



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0001325


Y

Z X

MX

MN


0463E minus 03 0001389 0002314 000324 0004166


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0004166


Y

Z X

MX

MN


0003278 0009833 0016388 0022944 0029499


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0029499


Y

Z XMX

MN


0007935 0023805 0039674 0055544 0071414


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0071414








Y

Z X MX

MN


0016631 0049892 0083153 0116414 0149675


Sub = 999999

Time = 1


UNFORRFORACEL

SMX = 0149675






6 Conclusions


119888and 119883

119888119888 GA




Appendix







Acknowledgment


References






































x

y




O


A(a 25)

B(0 b)

C(c 0)


A

CB

O

1205732

1205731120572





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∘



1and 120573

2which








1and119874





1015840

1 11987410158402 11987410158403 and 119874

1015840




119888in

Figure 2 is 52292m




A

B

F

H

H

D

G


O1

O4

1205732

1205731

Fs1

Fs4

C1 C2 C3 C4

45H

120572

O1998400

O2998400

O3998400

O4998400


1328

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5




1390

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5





1316

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



0and 120593

0 respectively


0SF


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


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1385

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



Layer1

Layer2

40

m80

m

80m 50m 70m

40

m105

m

25m


Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL


Y

Z X

MXMN


Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL




119860 =2 sin120593

radic3radic3 + sin120593 119861 =


(5)


119860 =2 sin120593



(6)



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0001325


Y

Z X

MX

MN


0463E minus 03 0001389 0002314 000324 0004166


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0004166


Y

Z X

MX

MN


0003278 0009833 0016388 0022944 0029499


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0029499


Y

Z XMX

MN


0007935 0023805 0039674 0055544 0071414


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0071414








Y

Z X MX

MN


0016631 0049892 0083153 0116414 0149675


Sub = 999999

Time = 1


UNFORRFORACEL

SMX = 0149675






6 Conclusions


119888and 119883

119888119888 GA




Appendix







Acknowledgment


References






































A

B

F

H

H

D

G


O1

O4

1205732

1205731

Fs1

Fs4

C1 C2 C3 C4

45H

120572

O1998400

O2998400

O3998400

O4998400


1328

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5




1390

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5





1316

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



0and 120593

0 respectively


0SF


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


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1385

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



Layer1

Layer2

40

m80

m

80m 50m 70m

40

m105

m

25m


Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL


Y

Z X

MXMN


Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL




119860 =2 sin120593

radic3radic3 + sin120593 119861 =


(5)


119860 =2 sin120593



(6)



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0001325


Y

Z X

MX

MN


0463E minus 03 0001389 0002314 000324 0004166


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0004166


Y

Z X

MX

MN


0003278 0009833 0016388 0022944 0029499


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0029499


Y

Z XMX

MN


0007935 0023805 0039674 0055544 0071414


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0071414








Y

Z X MX

MN


0016631 0049892 0083153 0116414 0149675


Sub = 999999

Time = 1


UNFORRFORACEL

SMX = 0149675






6 Conclusions


119888and 119883

119888119888 GA




Appendix







Acknowledgment


References






































1316

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



0and 120593

0 respectively


0SF


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


1389

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5


1385

Distance0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Elev

atio

n

51015202530354045505560657075

minus5



Layer1

Layer2

40

m80

m

80m 50m 70m

40

m105

m

25m


Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL


Y

Z X

MXMN


Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL




119860 =2 sin120593

radic3radic3 + sin120593 119861 =


(5)


119860 =2 sin120593



(6)



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0001325


Y

Z X

MX

MN


0463E minus 03 0001389 0002314 000324 0004166


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0004166


Y

Z X

MX

MN


0003278 0009833 0016388 0022944 0029499


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0029499


Y

Z XMX

MN


0007935 0023805 0039674 0055544 0071414


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0071414








Y

Z X MX

MN


0016631 0049892 0083153 0116414 0149675


Sub = 999999

Time = 1


UNFORRFORACEL

SMX = 0149675






6 Conclusions


119888and 119883

119888119888 GA




Appendix







Acknowledgment


References






































Layer1

Layer2

40

m80

m

80m 50m 70m

40

m105

m

25m


Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL


Y

Z X

MXMN


Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MXMN



Sub = 15

Time = 1


UNFORRFORACEL




119860 =2 sin120593

radic3radic3 + sin120593 119861 =


(5)


119860 =2 sin120593



(6)



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0001325


Y

Z X

MX

MN


0463E minus 03 0001389 0002314 000324 0004166


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0004166


Y

Z X

MX

MN


0003278 0009833 0016388 0022944 0029499


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0029499


Y

Z XMX

MN


0007935 0023805 0039674 0055544 0071414


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0071414








Y

Z X MX

MN


0016631 0049892 0083153 0116414 0149675


Sub = 999999

Time = 1


UNFORRFORACEL

SMX = 0149675






6 Conclusions


119888and 119883

119888119888 GA




Appendix







Acknowledgment


References






































Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL



Y

Z X

MX

MN




Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0001325


Y

Z X

MX

MN


0463E minus 03 0001389 0002314 000324 0004166


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0004166


Y

Z X

MX

MN


0003278 0009833 0016388 0022944 0029499


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0029499


Y

Z XMX

MN


0007935 0023805 0039674 0055544 0071414


Sub = 15

Time = 1


UNFORRFORACEL

SMX = 0071414








Y

Z X MX

MN


0016631 0049892 0083153 0116414 0149675


Sub = 999999

Time = 1


UNFORRFORACEL

SMX = 0149675






6 Conclusions


119888and 119883

119888119888 GA




Appendix







Acknowledgment


References






































Y

Z X MX

MN


0016631 0049892 0083153 0116414 0149675


Sub = 999999

Time = 1


UNFORRFORACEL

SMX = 0149675






6 Conclusions


119888and 119883

119888119888 GA




Appendix







Acknowledgment


References









































Acknowledgment


References





































determination of slope safety factor with analytical ... method), simplified bishop method, ......

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