Slope Stability Analysis of Earth-Rockfill Dams Using MGA and UST

The nonlinear Unified Strength Theory (UST), which takes into account the effect of intermediate stress and nonlinear behavior on geotechnical strength, is applied in slope stability analysis of earth-rockfill dams (ERD) in this paper. The biggest drawback for general determination of slip surface is that it must presuppose the shape of slip surface and is unable to identify the critical noncircular slip surface more accurately. This paper proposes an optimal analytic model of slope stability analysis of ERD and employs modified genetic algorithm (MGA) to search for the slip surface on the basis of shear failure criteria of the nonlinear UST without prior assumption of the shape of slip surface. The application ofMGA dependent on Matlab toolbox to the slope stability analysis of ERD shows thatMGA can consequently overcome the weakness of easily falling into local optimal solutions brought by general optimal algorithms.


Introduction
The material model of geotechnical strength plays a very important role in the problems of slope stability of ERD.Many researchers [1][2][3][4] have made a large amount of research work about the effect of applied loads and groundwater on slope stability of ERD, but it is still open for further research on the slope stability of ERD because of the complexity of the soil materials.The parameters of geotechnical strength in Slices Method based on the rigid body limiting equilibrium theory are ones related to the failure criterion of linear shear strength, just like in Mohr-Coulomb failure criterion that is widely applied in geotechnical engineering.Mohr-Coulomb failure criterion is the lower limit of a linear convex function and does not consider the effect of intermediate stress on the geotechnical strength, so the numerical results trend to more safety.However, the intrinsic complexity of the mechanical property of geotechnical materials results in that there are almost few strength criteria which can accurately describe the nonlinear fractural characteristics of soils and even the same with linear failure criteria.Some researches [5][6][7] have shown that the failure of geotechnical materials with nonlinear destruction is just one special case of failure criteria under linear damage and the intermediate stress effect which can enhance the slope safety factors to some degree.The major determinants causing the dams to be broken include applied forces, seepage, and earthquake.As known to all, the strength parameters of the soil play an indispensable role in the slope stability analysis of ERD.Many experts [1][2][3][4] have done a lot of studies on the effects produced by external loads and underground water on the slope stability; however there are still many works to be done in the field of strength theory of geotechnical material.By so far, a few hundred or more of yielding and failure criteria [8][9][10][11] have been proposed, whose applications in the geotechnical engineering have to be seriously restricted because there is not a general failure formula that is appropriate for all kinds of materials and different mechanical status.In 1991 Professor Yu [11] proposed the UST on the basis of all existing strength theories, by combining the double shear strength theory with the single shear strength theory into only one formula.UST is thought as the high generalization of all kinds of existing strength theories, but UST is only asymptotic approximation for the most of the nonlinear issues.
The Slices Method, based on the rigid body limiting equilibrium theory, should suppose the shape of slip surface in advance, so the greatest weakness is that it is difficult to cast about for the slip surface quickly and accurately corresponding to the minimum safety coefficients of slip surfaces.The geometrical configuration of the slip surfaces is assumed as in circular arc for the most of the searching algorithms, but we must take an arbitrary shape of slip surfaces for heterogeneous soil materials and discrepant distribution of pore pressure in soils.In the determination of minimum safety factor and its corresponding arbitrary slip surfaces, it might be the best choice to use the optimization methods to arrive at satisfactory results when the objective function is convex and the searching domains are irregular.But for the objective function with multipeak in the complicated searching domains, the general optimization methods often get into local optimal solutions for complex geotechnical structures and heterogeneous soil layers.On the basis of biology immune principles, a novel optimization algorithm [12][13][14] is proposed for solving many optimal solutions to multimodal functions.Since the late 1980s, the slope stability analysis begins to enter the period when the numerical theory develops flourishingly and finite element method becomes one method of spatial discretization which is the most widely applied; most popular methods of slope stability analysis include Shear Strength Reduction [15,16] and Gravity Increase Method.When the Shear Strength Reduction Method is used for slope stability analysis and the mechanical states of soils are very close to limiting equilibrium, it is difficult to converge and to solve governing equations in numerical calculation.Genetic Algorithm (GA) is a global optimization algorithm, so it can overcome the shortcomings of the ordinary optimization methods that fall easily into local optimum.
How to ascertain the most critical slip surfaces faster and more accurately can be regarded as a difficult affair in the slope stability analysis of ERD, upon which this paper tries to apply the multivariable nonlinear optimal theory in slope stability analysis which is built on Matlab platform and then simulate the critical slip surfaces of ERD slope to find its minimum safety coefficient.For GA is a sort of global optimization method and can avoid simply getting into local minima, the paper will analyze slope stability by way of GA.How UST can be applied in the stability analysis of slops still makes few progresses, so this paper will try to put the linear and nonlinear UST applied to the stability analysis of ERD and compare various strength criteria to the influence of the slope stability of ERD.One of the difficulties of the slope stability analysis of ERD is how to determine the most critical slip surfaces more accurately and faster, although there have been some valuable achievements made by researchers in China, but what the most appropriate methods are still has not confirmed.Based on Matlab platform we try to apply the nonlinear optimization theory with multivariable to the stability analysis of ERD and to determine the minimum safety factor of slope stability in simulation of the critical slip surfaces of ERD.

Linear and Nonlinear UST
Taking a unique mechanical model as the theoretic basis, Professor Yu puts forward an UST unifying all sorts of the linear and nonlinear strength theories into one.The linear and nonlinear UST proposed by Professor Yu can be summarized as [11] follows. If (2) also if then we take following formula: in which where  1 ,  2 , and  3 are the first, second, and third principal stresses, respectively,  is the ratio of the tension with compression strength,  is the factor concerned with the normal stress effect,  is related coefficient to the intermediate principal stress,  is nonlinear coefficient reflecting failure criteria,  is cohesion, and  is the inner friction angle.When the coefficients of twins shear strength , , , and  take some specific values, UST can be converted into existing major yield and strength criteria such as Mohr-Coulomb criterion as shown in Table 1.So we might think that the linear and nonlinear UST more comprehensively reflect the strength conditions of the practical problems of geotechnical engineering.
The applications of UST in practical engineering have been in a series of researches but it is still rare in the Nonlinear criterion, 0 <  ≤ 1 Linear criterion,  = 0 slope stability of ERD.In order to reveal the influences of the nonlinearity and intermediate principal stresses on slope stability in the failure criteria of UST, we illustrate the applications of the linear and nonlinear UST in slope stability below.

Nonlinear UST Applied in Slope Stability. According to
Mohr-Coulomb strength theory, the safety coefficients along the slip surface shown in Figure 1 can be defined as where   is the shear strength of sliding surface,  is the factual shear stress,  means the arc length of sliding surface,  is number of soil slices,  is cohesion,  is inner friction angle, and  is effective stress.
To apply linear and nonlinear UST in the slope stability analysis, we need to rewrite formulas ( 2) and (4) as follows. For For where the items, related to the principal stresses at the left hand of the formulae, can be denoted by ( 1 ,  2 ,  3 ) and the items concerned with strength parameters at the right hand are written in the form (, ).Hence, the safety coefficient along a certain potential slip surface can be depicted as So we deduce following expressions. For For By means of finite element methods to calculate the stress solutions of whole ERD, then extract the principal stresses  1 ,  2 , and  3 which are substituted into the equations mentioned above and get the safety coefficients of presupposed slip surfaces.Consider one slip surface and substitution of the first, second, and third principal stresses into the formula above; then calculate the values of ∑  =1   (, )Δ  and ∑  =1   ( 1 ,  2 ,  3 )Δ  according to linear and nonlinear UST.Eventually the safety factor can be written as For application of optimal method to decide the potential slip surface of ERD, we can ascertain the minimum safety factors through comparison among the outcomes obtained above.

Analysis of Parameter Sensitivity.
In the analysis of slope stability of dams engineering, the applications of numerical simulation have become quite common and the accuracy of numerical simulation is closely related to the mathematical model and the values of the characteristic parameters of soil medium.Given the mathematical model of the slope stability, the material parameters are reasonable or do not have very great effect on the calculation results as shown in Figure 2. In the present paper we conclude that the linear and nonlinear UST associate with two main parameters, functional coefficient of intermediate principal stresses  and the nonlinear effect coefficient , in the applications of the slope stability analysis.So we will discuss the parameter sensitivity about the influence of two parameters mentioned above on the stability of slope safety as follows.There are two kinds of sensitivity analysis, that is, the single factor and multifactorial sensitivity analysis, and we will adopt the single factor sensitivity analysis in this paper.Single factor approach, by changing the value of some factor and assuming that other factors remain immovable, compares the degree of influence of the benchmark values on the factor change.
In the following examples of the slope stability analysis, we take  = 0,  = 0, and the safety factor as the benchmark values.It is illustrated in Figure 2 that the slope safety parameters change with increase of the strength parameters  and .
As Figure 2 displays, two strength parameters  and  in UST have different influences on the coefficients of slope stability of ERD.When nonlinear coefficient  develops gradually, the safety factor increases at first and then falls back slightly, but the safety coefficients increase with accretion of the parameter  concerned with the intermediate principal stresses.From Figure 2, you can also read that the nonlinear coefficient  has greater effect on the safety coefficients of slope stability of ERD more than the principal stress coefficient .

Stability Analysis of ERD by MGA
Based on Matlab Platform 3.1.Formulation of Optimal Model.The Slices Method based on the rigid body limiting equilibrium method is most widely applied in the slope stability analysis.How to determine the minimum safety coefficients and corresponding slip surfaces of ERD is the key of algorithms adopted, this problem can fall into optimization programming in mathematical terms.
If the Slices Method is adopted, the position and shape of slip surfaces are defined by two orthogonal coordinates  and .
When the width of slices is definite, the coordinate  is only optimization variable.We choice the modified Janbu method [8] to calculate the safety factors of slope.Taking a general soil slice, then making force analysis is as shown in Figure 3.
The meanings of all notations can be figured out through Figure 3.If there are not external forces applied on both sides of the slices, namely,  0 ,   ,  0 , and   all vanish, and assuming that the number of soil slices is , then in which and  is the pore pressure on the slip surfaces.The optimization model of slip surface of slope stability can be concluded as the following: where  0 is initial estimated value of safety factor for iteration and  1 evaluation value by substitution of  0 and the shear forces between each of the soil slices into (13).

Slope Stability Analysis Based on MGA.
The Dichotomy, Pattern Search Method, and Simplex Method are commonly adopted in slope stability analysis of ERD and to determine the noncircular critical slip surfaces.For the slopes with complex geometrical figuration and uneven soil layers, as well as nonconvex nonlinear constrained optimization problems, these optimization algorithms might get local optimum.However, Standard MGA is a global optimal algorithm in searching domain, which simulates the proceeding of natural evolution and can get the global best, and can easily overcome the shortcomings of general optimization algorithms to be the local minimum or maximum.So it is especially suitable to search for global minimum of the nonlinear problems of slope stability of ERD.Below we will detail the algorithm implementation of MGA to slope stability of ERD.

Procedures of Slope Stability Analysis Based on MGA. The procedures corresponding slope stability of ERD based on
MGA and UST can be summarized as follows [14].
Step 1. Define the optimal variables  0 ,  1 ,  2 , . . .,   and constraint conditions in (15).In order to improve the efficiency of the optimal procedure of MGA, the geometrical configuration of the slip surfaces should be rationalized in the optimization analysis of ERD.According to the practical features of slope stability of ERD, the slope angle of the slip slices must be subject to the following condition: where   is the angle of th soil slice between the bottom of soil slices and horizontal coordinates.So, the last constrained condition in (15) should be modified as Step 2. Build up the objective function in (13).
Step 4 (ascertaining coding method).The encoding process means transforming the variables into binary digits strings whose number of equivalent binary digits is determined by required precision.For instance, the value range of variable   is    to ℎ and precision of its value is for three decimal places, which means that the value range of each variable will be divided into (ℎ −    ) × 10 3 sections.Binary string of digits (denoted by   ) of one variable can be obtained from the following formula: So, individual string of digit is ∑   , which can be illustrated as shown in Figure 15.
Step 5 (ascertaining the decoding method).A factual value returned from binary digits string can be achieved by the following formula: where decimal (substring  ) is the decimal value of the variables   .
Step 6 (determining the fitness value of each individual).Each individual is substituted into the objective function to calculate the safety factors; if the safety factors are small, then put larger fitness value in the calculation of the safety factors and vice versa.Consequently, the general fitness value of colony can be presented as  = ∑ eval(  ), which can deduce the probability of each selected individual   = eval(  )/ ∑ eval(  ).Then calculate the accumulating probability of each individual   = ∑  =1   ,  = 1, 2, 3, . ... Step 7. Determine the operator based on the Roulette Wheel Selection.Selecting new population of a chromosome can be conducted by the next two steps: (1) Generate a stochastic number  (0 <  < 1).
Step 8 (crossover operation).Select one individual node randomly, and then produce children generation by exchanging the right sections of the node between two parent generations, as in Figure 16.
If the probability of crossover is 0.3, it indicates that only 30% of individuals on the average level have been crossed.The procedure of crossing operation is as shown below.
Continue the cross process when  ≤ ; then generate random number   which is a number between 0 and 1.If   < 0.3, then select the th individual as parent generation of the crossover.Assign  =  + 1. End current step.
Step 9 (mutation operations).Suppose that the th gene of th individual is chosen for the variation; if the digit code corresponding with the gene is 1, it will change to 0 after the variation.If probability of mutation is 0.02, namely, the 2% of the individuals above the average level must get mutated.After the completion of the variation of the population we arrive at the ultimate next generation of population.
So far, we have completed the first generation calculation of slope stability based on MGA and UST.Then repeat the steps from 6 to 9 until we obtain the optimal individuals.
We can make use of Matlab MGA toolbox to implement the above analysis steps.Main program of MGA and direct searching toolbox in Matlab provides an interface between Matlab toolbox and the external.The invoking format of Matlab function is as follows: in which  is the optimal coordinate and V is optimal solution of objective function and both are the optimal results given by invoking Matlab function,  stands for objective function,  var  denotes the number of optimization variables in objective function, and  is

Example 1.
For the comparison of the general strength theory with UST, we choose a single stage ERD as an example shown in Figure 4.The intensity parameters are presumed as  = 13 kP,  = 18 ∘ ,  = 19.8kN/m 3 ,  = 2.5 mP, and  = 0.3.We adopt different strength criteria, which are provided with different strength parameters  and  in UST, to evaluate the slope safety factors and the final results shown in Table 2.
We can evaluate that the safety factor is equal to 0.9496 by UST and 0.939 by Mohr-Coulomb failure criterion in this example using the formula below: The relationship of slope safety coefficients   via  and  is shown in Figures 5 and 6.
By the calculation and analysis above the example, when  =  = 1,  = 0 in nonlinear criterion and  =  = 0,  = 1 in linear criterion, for the reason of no consideration of normal stress effect, so the results are not reasonable and not applicable to the materials with large tensionalcompressional strength.The nonlinear coefficient  and effect coefficient  relative to failure criteria and intermediate principal stresses, respectively, may improve the slope stability safety at some degrees, but the former to the improvement of safety factors is more obvious than the latter.From Figures 5 and 6 we can see that the increase of safety factor gets faster with the increase of parameter  at first, and when  is greater than 0.4 the effect of change of  to the safety factors becomes significantly less, and when  is greater than 0.8 the safety factors will reduce with increase of  instead.
When  is less than 0.2, the increase of the parameter  on improvement of the safety factors is obvious, but if  is greater than 0.2, the change of the parameter  has little impact on the improvement of the safety factors.

Example 3.
The geotechnical structure of an inhomogeneous slope in layers is shown in Figure 9.The material properties of the slope are listed in Table 3.The width of soil slices in the determination of the most dangerous slip surfaces by the MGA and UST is taken as 2.5 m.The slope is enforced lateral acceleration 0.15 g and the recommended safety factor  strength theories have very high consistency.Note that g is gravity acceleration in Figure 14.

Conclusions
The numerical results indicate that the nonlinear coefficient  and intermediate principal stress effect coefficient  in UST have effect on the slope stability factors at some degree.In general, nonlinear coefficient  has greater effect on the slope stability factors more than the principal stress effect coefficient .In the geotechnical engineering, the determination of linear and nonlinear parameters in UST must rely on the unique characteristics of soil-rock; then geotechnical experimental research plays more effective role in the quantification of strength parameters of geotechnical material, so as to save the construction cost of the dams project.
By the above examples we can easily know that although the minimum safety factors are almost the same values deduced by MGA as that by Strength Reduction Method, but MGA is a highly efficient, simple, and good performance of fault-tolerance algorithm.Compared with the general optimization methods, MGA and UST used in the stability analysis of ERD possess the following specialties:  (1) The searching process does not operate on objective functions and optimal variables, instead of encoding the individuals on parameter set, of which MGA enables performing operation on structural objects (sets, series, matrix, trees, figures, chains, and charts).(2) Searching processes from a set of optimal iterative solutions to another, use the way of handling more individuals in one group at the same time, thus reducing the possibility of falling into local optimal solution and being easy for parallel processing.(3) The adoption of probabilistic transition rules confirms the searching direction, not the certainty searching rules.(4) There is no specific restriction to the search space such as connectivity and convexity, only using adaptive information, and there is no need of derivatives and other auxiliary information.

Figure 3 :
Figure 3: A general soil slice and applied forces.

4. 2 .
Example 2. The practical working conditions of a typical ERD are demonstrated in Figure 7; The downstream water level is 1 m.Soil cohesion of ERD is 13 kP, inner friction angle 25 ∘ , effective cohesion 6 kP, effective inner friction angle 13 ∘ , natural weight-specific density 20 kN/m 3 , and saturated unit weight of soil 23 kN/m 3 .When ERD has the different upstream water level, the stability safety factors of ERD, calculated by the Strength Reduction Method and method presented in this paper, are shown in Figure 8.The results indicate that two strength theories almost give the same factors of slope stability of ERD under the conditions of different upstream water level.

Table 1 :
UST versus several exiting major yielding and strength criterions.

Table 2 :
Slope safety factors evaluated by different failure criteria.

Table 3 :
Material properties of the slope.
4.4.Example 4. The minimum safety factors of the slop obtained from MGA and Strength Reduction Method can be compared in detail as follows.Figures 11-14 depict the safety factors change with slope angles, cohesions, inner friction angles, and lateral acceleration, respectively, by Strength Reduction Method and MGA; all of the results show that two