Determination of minimum factor of safety using a genetic algorithm and limit equilibrium analysis

Transcription

1 Deterination of iniu factor of safety using a genetic algorith an liit equilibriu analysis Branon Karchewski, Dieter Stolle & Peijun Guo Departent of Civil Engineering McMaster University, Hailton, Canaa ABSTACT This paper presents the continuous paraeter genetic algorith cobine with the rigi finite eleent etho to ientify iniu factors of safety for slopes. Whereas for hoogeneous slopes, Bishop s an FEM analyses lea to siilar preictions for global iniu factor of safety an location of critical slip surface, it is shown that preictions iffer for layere systes. It is iportant that layer rigiity is inclue for layere systes. ÉSUMÉ Cet article présente le paraètre continue e l algorithe génétique cobiné avec la éthoe es éléents finis rigies pour ientifier les facteurs iniales e sécurité es pentes. Alors que pour les pentes hoogènes, la éthoe e Bishop et celle es Éléents Finis igies conuisent à es préictions siilaires pour le facteur e sécurité globale et e la position e la surface e glisseent critique; il est ontré que les préictions sont ifférentes pour un systèe hétérogène avec plusieurs couches. C est iportant que la rigiité es couches soit prise en copte pour un systèe hétérogène à plusieurs couches. 1 INTODUCTION With the availability of nuerical tools such as the finite eleent etho, the geotechnical engineer is capable of realistically solving coplex probles. Owing to the uncertainty surrouning the initial conitions, bounary conitions an istribution, as well as history-epenent properties, the quality of the preictions are ost often unerine. Siple ethos often provie solutions that are aequate for practice given that the engineer is often intereste in answering only two questions: (1) Is the structure stable?; an (2) Assuing a sufficient factor of safety, what oveents are to be expecte? These questions ay be answere by uncoupling the failure an eforation analyses. Liit equilibriu analysis presents a long-staning siple fraework for slope stability analysis. Many ethos exist an the selection of the best ay not be a trivial task. For a etaile iscussion of the relative erits of various ethos, the reaer is referre to Frelun & Krahn (1977). Iplicit in the ethoology is the use of a search algorith to ientify the surface that iniizes the factor of safety. The presence of any local inia can easily islea traitional root fining ethos. To help overcoe this proble, various algoriths have been evelope, such as: gri an raius ethos (Chen an Shao 1983); Monte Carlo rano search technique (Malkawi et al. 2001); an genetic algorith search proceures (McCobie & Wilkinson 2002; Sengupta & Upahyay 2005; Zolfaghari et. al., 2005). The objective of this paper is to present a continuous paraeter genetic algorith with ephasis on cobining it with the rigi finite eleent etho (FEM) to ientify iniu factors of safety of slopes. We begin by proviing a short overview of the FEM an then escribe a continuous genetic algorith, followe by exaples that copare FEM preictions with those obtaine by Bishop s siplifie etho. 2 BACKGOUND 2.1 igi Finite Eleent Metho Siilar to traitional liit equilibriu analyses a potential failure surface is assue, a priori. The oain is subivie into a series of rigi vertical slices as shown in Figure 1 for an assue circular slip surface that is characterize by the center of the circle ( c, c) an its raius. Figure 1. Geoetry of slope stability proble The slices are assue to be rigi, with relations efine for the interaction of eleents; i.e., between inter-slice forces an the relative oveent of ajacent slices. Siilar relations exist between the slip surface an the rigi base using non-linear springs as shown in Figure 2. The oveent of each eleent is constant an is efine in ters of horizontal an vertical isplaceent coponents. Following a finite eleent ethoology, a syste of equations is asseble consistent with virtual work for a specific slip surface. Since the interactions of

2 slices are non-linear, an iterative schee ust be use to solve the atrix equation. This proceure satisfies both local an global force equilibriu. Siilar to traitional approaches, a coparison is ae between resisting an obilize oents to eterine the global factor of safety. An avantage of the FEM proceure is that relative oveents are eterine as part of the solution, which allow local factors of safety to be reaily eterine. The reaer is referre to Stolle & Guo (2008) for etails. The binary nuber genetic algorith search etho has receive the ost attention; see previous references. This paper escribes an alternative genetic algorith presente by Haupt an Haupt (1998) that siply stores the cobination of paraeters as a vector in ecial (continuous) for rather than in binary for. The avantages of this approach inclue: anipulation of the paraeters in crossover an utation routines are ore transparent an intuitive; an less coputation tie is spent on bookkeeping to convert the paraeters back an forth between binary an ecial notation. The general algorith involves stages: initialization, evaluation, sorting, reprouction (crossover) an utation. One perfors the last three steps iteratively for a specifie nuber of generations. The processes of crossover an utation allow the oel to ientify the global iniu even while converging on a local iniu through rano utations in the paraeters. It is iportant to use a set nuber of iterations as a stopping criterion, since a relative-error-base criterion ay cause the algorith to coplete preaturely at a local iniu Initial Population an Evaluation Figure 2. Scheatic of inter-slice an basal springs. 2.2 Genetic Algorith A genetic algorith begins with a rano saple of failure surfaces within a suitable oain, characterize by ( c, c, ); see Table 1. The selection of the oain is an iportant step as too sall of a search region ay exclue the cobination that efines the global iniu. On the other han, too large of a search region ay take too long to converge (ue to the larger nuber of possible cobinations) or lea to an unreasonable slip surface epening on the type of analysis use. Using the principle of natural selection, the critical factor of safety is eterine by systeatically ajusting the failure surface paraeters of previously generate results. At the sae tie, it allows for an appropriate level of rano searching in orer to prevent converging too quickly on local inia. This approach strikes a balance between the systeatic, resource intensive gri an raius etho an the highly ranoize Monte Carlo etho. Table 1. Paraeter ranges use for genetic algorith Paraeter 1 Miniu Maxiu c -0.25H H/ H c 1.25H 2.5H 0.2H 2.5H 1 See Figure 1: H = height of slope, = grae of slope (rise/run). After the search oain is efine, the first step in the genetic algorith is to generate an initial population of caniates for iniu global factor of safety. Chroosoes ( c, c, ) containing ranoly selecte values for each of the paraeters within their respective ranges copose this initial population. The size of the initial population is an iportant paraeter in the genetic algorith. The factor of safety F s is then evaluate for each chroosoe using FEM Sorting Once the factors of safety have been evaluate for the current population, they are sorte fro lowest to highest global factor of safety. This is in preparation for the reprouction an utation processes, where only the ost optial configurations carry their genes through to the next generation eprouction (Crossover) It is through crossover that the genetic algorith is able to converge on a particular value through the process of natural selection. The reprouction process begins by selecting a specifie upper portion of the population (typically 50%) to enter the ating pool. This represents survival of the fittest as only the ost optial proportion of the population is able to pass traits on to the next generation. The ost optial solution(s) (typically ~10-15%, but roune up to the nearest integer) are copie irectly into the next generation. This eans that the search routine always keeps the ost optial solution so that the iniu global factor of safety never increases. This is akin to the fittest ebers of the population having greater longevity an surviving for an extra

3 ating season (in aition to passing their genes on to the next generation). The algorith then assigns appropriate weights to the ebers of the ating pool accoring to the corresponing global factor of safety by the following operations: 1. Subtract the iniu factor of safety of the chroosoes not selecte for reprouction fro the factors of safety of all chroosoes selecte for reprouction. This reverses the weighting schee, as this is a iniization proble. 2. Divie each of the (reverse) factors of safety in the ating pool by the su of the (reverse) factors of safety to obtain the iniviual weight of each. 3. A the su of the weights for lower factors of safety to obtain the cuulative weight for each iniviual chroosoe. Next, the algorith selects pairs of parent configurations in orer to fill the reaining slots in the next generation s population. If there are an o nuber of slots reaining, the progra oves the next ost optial solution after those alreay eee fittest irectly into the next generation. It then carries out the selection process by generating rano nubers between 0 an 1 an selects the first parent in the ating pool that has a cuulative weight greater than or equal to this value. This selection process, cobine with the weighting schee escribe previously, ensures that chroosoes with significantly lower factors of safety will have a greater relative probability of being selecte, while a population consisting of chroosoes with nearly equal factors of safety will have a unifor probability of being selecte The next step is to carry out the process of crossover in orer to fill the population for the next generation, which begins by generating a rano nuber between 0 an 1. If this nuber is less than a specifie probability (typically 70~90%), crossover occurs. Otherwise, the progra copies the two parents irectly into the next generation. If crossover is to occur, the progra perfors a for of interpolation on one paraeter an crossover on the reaining paraeters. Figure 3 shows a scheatic of the process. The figure uses the ter as an exaple. Since the nuber β is a ranoly generate between 0 an 1, the axiu change that the process applies to the paraeters is the ifference between the. Thus, if the paraeters are siilar then two siilar offspring will be generate an vice versa if the two paraeters are far apart. It shoul be note that there are a nuber of ways in which crossover can occur, besies what is shown in Figure 3. If were selecte for interpolation, either or woul have been ranoly selecte to be crosse; if were interpolate, or woul have been crosse; thirly, as in Figure 3, if is interpolate, either or is crosse. This reoves bias towar crossing one paraeter or the other in all cases. Finally, the progra places the two offspring into the population for the next generation. This process repeats until the population for the next generation is fille. parent parent... new1 new offspring offspring 1 2 ( ( ) ) new1 new2 Figure 3. Deterination of chroosoes via crossover Mutation The process of utation allows the genetic algorith to survey the search region for alternative inia. It prevents the algorith fro converging too quickly on a local iniu an allows the algorith an exit fro the region of a local iniu if it fins a lower value; i.e., the global iniu. If a lower iniu exists, it ay take a nuber of generations for the algorith to fin it; in fact, the closer the current iniu is to the global iniu without being in the region of the global iniu, the less probable it is to locate the global iniu. This is the ain reason why setting the nuber of generations is preferable to a convergence criterion for the genetic algorith. The nuber of utations on a given generation is n ut =3 p ut (n pop-n fit ) with n ut = total nuber of utations; p ut = probability of utation; n pop = nuber of chroosoes in population; an n fit = fittest proportion of population. Table 2 lists values use here. Table 2. Paraeters use for optiization Population (n pop) 15 Fittest Population (p fit = n fit / n pop) 0.10 Mating Pool Proportion (p at = n at / n pop) 0.50 Crossover Probability (p cross) 0.90 Mutation Probability (p ut) 0.25 Nuber of Generations (n gen) 100 Following the crossover process of each generation, a vector of (chroosoe, paraeter) aresses are selecte (excluing the fittest proportion). For each location selecte for utation, the progra selects a rano value within that paraeter s corresponing range to replace the current value. 3 NUMEICAL EAMPLES This stuy consiere various scenarios. The essence of the genetic algorith together with the FEM (an its avantages) ay be illustrate by consiering the results of two particular cases. The paraeters E, v, φ an c

4 enote elastic oulus, Poisson s ratio, friction angle an cohesion, respectively. 3.1 Case 1 Layere Ebankent Case 1 is an ebankent consisting of three strata, plus a botto layer for berock. The upper an lower strata are ientical layers of san with φ = 40, E = 1.0x10 5 kpa, an ν = The ile stratu is a layer of ixe soil with φ = 10, c = 30 kpa, E = 5x10 4 kpa, an ν = The geoetry of the slope is efine by = 5/6, H = 25 (see Figure 1), layer thicknesses of 12.5 for the san an 5 for the ixe soil an γ = 18.5 kn/ 3 for all soils. The berock lies 5 below the base of the slope. Miniu factors of safety using FEM an Bishop s etho together with the genetic algorith were obtaine for several runs, each one being a full analysis. Table 3 suarizes the statistics fro the runs, with Figure 4 presenting the iniu F s. Originally 5 runs were carrie out to confir that the algorith woul converge to a consistent iniu F s. One observes fro the figure that F s is relatively constant when using Bishop s etho for slope stability, but the FEM preictions show sensitivity to the (rano) see value. Aitional runs were therefore coplete to further investigate sensitivities. The sensitivity that was originally note was confire. More iportant is the observation that the iniu value was achievable only after 65 runs. In other wors, there is no guarantee that a genetic algorith will converge quickly. It epens on the sensitivity of the evaluation function to input. Table 3 clearly shows that there is consierable variation in critical failure surface copare to changes in iniu F s for this particular proble. The FEM stability proceure preicte a lower global iniu. Table 3. Case 1: Genetic algorith statistics for 85 runs. c () c () () F s Bishop Average SD Min Max FEM Average SD Min Max Figure 4. Miniu factor of safety as function of run. The critical surface for run 5 (iniu F s 0.84) is shown in Figure 5, along with the variation of local F s along the failure surface as eterine fro the sliing law; see Stolle an Guo (2008). An exaination of the local factors of safety along the slip surface reveals that F s is highest near the base of the failure surface an steaily ecreases as one oves up the surface. This is expecte as the level of confineent also ecreases as one oves up the surface. Although the ixe soil is the key for the failure oe given that the F s is slightly greater than 1 in the san, the transfer in loa ue to the instability of the ixe soil causes the local factors of safety to further ecrease in the san when loa is reistribute. Bishop s proceure was not capable of transferring loa an therefore the search for the iniu tene to focus on echaniss that were restricte to the san. This siple exaple eonstrates the iportance of accooating loa reistribution in stability analysis. y() 0.85 F s 0.75 x() Figure 5. Preicte failure surface (FEM) for run 5. The iniu factor of safety corresponing to the FEM analysis was foun to be approxiately 0.61.

5 y () Figure 6 shows the critical surface an copares it with that preicte when using the Bishop algorith (soli line for FEM, ashe line for Bishop s etho). The surface in Figure 6 is not uch ifferent fro that of Figure 5. proceures, for FEM, greater relative oveent between slices is observe at the crest, which is consistent with the local F s, being saller; i.e., large oveents being consistent with greater yieling F s = 1.02 San F s = 0.61 Mixe San x () -5 Figure 6. Failure surface for global iniu, Case Case 2 Narrow, Weak Layer Case 2 correspons to a slope siilar to that use by Frelun & Krahn (1977) for stuying the various classical ethos of slope stability analysis. The slope consists of a ense, ixe soil lying over a berock base. At the interface of these two layers, there is a very thin layer of weak frictional san. The upper layer is characterize by γ = kn/ 3, φ = 20, c = 29 kpa, E = 7.5x10 4 kpa, ν = 0.4, an has a thickness of The thin ile layer is weak frictional san having γ = kn/ 3, φ = 10, E = 5.0x10 4 kpa, ν = 0.45, an is 0.5 thick. The slope has a grae = 0.5 with H = 12. Since their paper only consiers classical ethos of analysis, it oes not provie values for elastic oulus or Poisson s ratio, so these values were assue. The ephasis with this exaple is the effect of the nuber of slices use on the global factor of safety. Figure 7 copares the preictions fro the Bishop s an FEM siulations. It is clear that the iniu factor of safety for each etho levels off above 10 slices. There is a slight increase for Bishop s etho, but this ay be ue to slight increase nuerical errors when the slices becoe thinner. One observes that Bishop s etho preicts lower F s, which ay inicate that the etho is conservative ue to the siplifying assuptions. One ust, however, be careful with aking conclusions an generalizing as there are there are special cases (such as Case 1) where FEM preicts lower values as it is capable of accooating failure oes resulting fro stiffness-epenent loa transfer, which Bishop s etho ignores. Figure 8 copares the critical surfaces preicte by each stability analysis proceure for the 10 slice analyses. The corresponing local safety factors (FEM) are also shown. Both preict siilar critical surfaces, although the iniu factor of safety for each is ifferent. With the FEM one gets a better picture of the way the slope oves. While there is a rotation for both Figure 7. Coparison of factors of safety fro Bishop s an FEM analyses. Figure 8. Coparison of failure surfaces fro Bishop s an FEM analyses.

6 4 CONCLUDING EMAKS Only a few results have been shown in this paper. Nevertheless, observations presente in this section take into account the experiences fro the entire stuy. This stuy akes the following conclusions: For hoogeneous slopes, Bishop s etho an FEM analysis prouce coparable results both in ters of the nuerical value of the global iniu factor of safety an the location of the critical slip surface. For slopes with ultiple strata, it is iportant to consier whether the ifferent layers have significantly ifferent values for the stiffness ters, E an ν. If the values iffer significantly, classical approaches such as Bishop s etho ay not give accurate nuerical estiates of the global iniu factor of safety. This is likely ue to how the soil ass transits lateral stresses, inucing shear flow along the interface between two strata. For slopes with ultiple strata, Bishop s etho an FEM analysis prouce coparable results for the location of the critical slip surface. It ay be ost coputationally efficient to use Bishop s etho analysis to arrive at the critical slip surface, but finally evaluate the actual factor of safety using FEM. Nevertheless, converging to a global iniu requires ore generations for a ultilayer slope. For cohesionless an ixe aterial slopes, the ( c, c, ) paraeters that efine the critical surface ay vary consierably for a given F s. This iplies that such slopes ay be less sensitive to variations in the acrostructure of the slope (e.g. inclusions such as boulers) as there is a wie range of equally critical failure surfaces. For slopes of cohesive aterial with φ = 0, the critical surface tens to be ore unique, with inial variation in the ( c, c, ) paraeters. This eans that if there is a region of localize strength along the critical surface (e.g. a shear key), stability increases significantly. The reverse is true for regions of localize weakness along the critical failure surface. The nuber of slices use in the analysis has an ipact on preictions. With too few (< 10), the analysis overestiates the iniu factor of safety as an accurate representation of the failure surface geoetry is lacking. On the other han, too any slices ay prove coputationally inefficient an introuce an unesirable level of nuerical error into the coputation. It is therefore optial to use a balance nuber of slices in the range [20, 30] to get an accurate solution while keeping the coputational expense in a satisfactory range. The following was observe when coparing preictions by binary nuber an continuous paraeter algoriths at the beginning of the stuy: The binary nuber etho tens to be ore powerful at rano searching for a given set of genetic algorith paraeters. This gives the binary nuber etho an avantage in situations where there are any local inia because it is ore likely to iscover the region of the global iniu. The binary nuber etho is therefore best suite to situations where a iniu nuber of runs are necessary ue to either tie or coputational constraints since it is ore likely to fin the global iniu in fewer runs. The continuous paraeter etho, while slightly less effective at conucting a rano search of the iniu F s, it oes ten to run faster ue to less bookkeeping require in converting to an fro binary nubers uring each iteration. In ost situations, the ability to o ore runs in less tie is avantageous because having ore output provies a better iea of how unique the critical surface is an how uch influence it has on overall slope stability. Although the proceure has been teste on various slopes, aitional testing is require to check robustness of the ethoology an influence of interlayer stiffness on the iniu factor of safety, as well as fully explore the liitations of the rigi finite eleent proceure. ACKNOWLEDGEMENTS The authors acknowlege the contribution of NSEC for funing this research. EFEENCES Chen, Z. an Shao, C Evaluation of iniu factor of safety in slope stability analysis, Canaian Geotechnical Journal 25: Frelun, D. an Krahn, J Coparison of slope stability ethos of analysis, Canaian Geotechnical Journal 14: Haupt,. an Haupt, S Practical Genetic Algoriths, John Wiley an Sons, Inc., Toronto. Malkawi, A. H., Hassan, W. an Sara, S An efficient search etho for fining the critical circular slip surface using the Monte Carlo technique, Canaian Geotechnical Journal, 38: McCobie, P. an Wilkinson, P The use of the siple genetic algorith in fining the critical factor of safety in slope stability analysis. Coputers an Geotechnics, 29: Sengupta, A. an Upahyay, A An evolutionary algorith for locating the critical failure surface in a soil slope, The Electronic Journal of Geotechnical Engineering, 10 (F). Stolle, D. an Guo, P Liit equilibriu slope stability analysis using rigi finite eleents, Canaian Geotechnical Journal, 45: Zolfaghari, A., Heath, A. an McCobie, P Siple genetic algorith search for critical non-circular failure surface in slope stability analysis. Coputers an Geotechnics, 32: