# Slope Stability Analysis of Earth-Rockfill Dams UsingMGA and UST

Authors: Li Nansheng; Tang Bo; Xie Lihui
Journal: Journal of Computational Engineering
 (2015)

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## Abstract

The nonlinearUnified 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 of MGA dependent on Matlab toolbox to the slope stability analysis of ERD shows that MGA can consequently overcome the weakness of easily falling into local optimal solutions brought by general optimal algorithms.


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## Body

## 1. Introduction

The material model of geotechnical strength plays a very important role in the problems of slope stability ofERD. Many researchers [1–4] have made a large amount of research work about the effect of applied loads and groundwater on slope stability ofERD, but it is still open for further research on the slope stability ofERD 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 inMohr-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 [5–7] 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 ofERD. Many experts [1–4] 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 [8–11] 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 ProfessorYu [11] proposed theUST 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, butUST 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 [12–14] 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 ofERD, upon which this paper tries to apply the multivariable nonlinear optimal theory in slope stability analysis which is built onMatlab platform and then simulate the critical slip surfaces ofERD slope to find its minimum safety coefficient. ForGA is a sort of global optimization method and can avoid simply getting into local minima, the paper will analyze slope stability by way ofGA. HowUST can be applied in the stability analysis of slops still makes few progresses, so this paper will try to put the linear and nonlinearUST applied to the stability analysis ofERD and compare various strength criteria to the influence of the slope stability ofERD. One of the difficulties of the slope stability analysis ofERD 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 onMatlab platform we try to apply the nonlinear optimization theory with multivariable to the stability analysis ofERD and to determine the minimum safety factor of slope stability in simulation of the critical slip surfaces ofERD.


## 2. Linear and NonlinearUST

Taking a unique mechanical model as the theoretic basis, ProfessorYu puts forward anUST unifying all sorts of the linear and nonlinear strength theories into one. The linear and nonlinearUST proposed by ProfessorYu can be summarized as [11] follows.If(1)σ2≤121+βσ1+1-βσ3,then(2)1-n21+b1+βσ1-b1-βσ2-1-βσ3+n3σ1-σ22+σ2-σ32+σ3-σ12+4βσ1+σ2+σ3=A;also if(3)σ2>121+βσ1+1-βσ3,then we take following formula:(4)1-n21+βσ1+b1+βσ2-1+b1-βσ3+n3σ1-σ22+σ2-σ32+σ3-σ12+4βσ1+σ2+σ3=Ain which(5)β=31+n1+b1-α+22n1-α31+n1+b1+α+8n1-αA=1-n1+b+2/32n1+ασtα=1-sin⁡φ1+sin⁡φσt=2ccos⁡φ1+sin⁡φ,where σ1, σ2, and σ3 are the first, second, and third principal stresses, respectively, α is the ratio of the tension with compression strength, β is the factor concerned with the normal stress effect, b is related coefficient to the intermediate principal stress, n is nonlinear coefficient reflecting failure criteria, c is cohesion, and φ is the inner friction angle.When the coefficients of twins shear strengthn, b, α, and β take some specific values,UST can be converted into existing major yield and strength criteria such asMohr-Coulomb criterion as shown in Table 1. So we might think that the linear and nonlinearUST more comprehensively reflect the strength conditions of the practical problems of geotechnical engineering.Table 1
UST versus several exiting major yielding and strength criterions.
Nonlinear criterion,0<n≤1
Linear criterion,n=0
n
=
1
n
=
α
=
1 β=0
0
<
n
<
1
n
=
β
=
0 α=1
n
=
b
=
β
=
0
n
=
b
=
0 α=1
Drucker-Prager criterion
Huber-Vonmises criterion
New nonlinear strength criterion
Linear unified yielding criterion
Tresca yielding criterion
Mohr-Coulomb failure criterionThe applications ofUST in practical engineering have been in a series of researches but it is still rare in the slope stability of ERD. In order to reveal the influences of the nonlinearity and intermediate principal stresses on slope stability in the failure criteria ofUST, we illustrate the applications of the linear and nonlinearUST in slope stability below.


### 2.1. NonlinearUST Applied in Slope Stability

According toMohr-Coulomb strength theory, the safety coefficients along the slip surface shown in Figure 1 can be defined as(6)∫τfdl∫τdl=∑i=1nci+σnitan⁡φΔli∑i=1nτiΔli,where τf is the shear strength of sliding surface, τ is the factual shear stress, l means the arc length of sliding surface, n is number of soil slices, c is cohesion, φ is inner friction angle, and σ is effective stress.Figure 1
Slip surface.To apply linear and nonlinearUST in the slope stability analysis, we need to rewrite formulas (2) and (4) as follows.Forσ2≤(1/2)[(1+β)σ1+(1-β)σ3], then(7)1-n2σ1+bσ1-bσ2-σ3+n3σ1-σ22+σ2-σ32+σ3-σ12=A-43nβσ1+σ2+σ3-1-n2βσ1+bσ1+bσ2+σ3.Forσ2>(1/2)[(1+β)σ1+(1-β)σ3], then(8)1-n2σ1+bσ2-σ3-bσ3+n3σ1-σ22+σ2-σ32+σ3-σ12=A-43nβσ1+σ2+σ3-1-n2βσ1+bσ2+σ3+bσ3,where the items, related to the principal stresses at the left hand of the formulae, can be denoted by f(σ1,σ2,σ3) and the items concerned with strength parameters at the right hand are written in the form y(c,φ). Hence, the safety coefficient along a certain potential slip surface can be depicted as(9)Fs=∫lyc,φdl∫lfσ1,σ2,σ3dl=∑i=1nyic,φΔli∑i=1nfiσ1,σ2,σ3Δli.So we deduce following expressions.Forσ2≤(1/2)[(1+β)σ1+(1-β)σ3], then(10)yic,ϕ=A-43nβσ1+σ2+σ3-1-n2βσ1+bσ1+bσ2+σ3,fiσ1,σ2,σ3=1-n2σ1+bσ1-bσ2-σ3+n3σ1-σ22+σ2-σ32+σ3-σ12.Forσ2>(1/2)[(1+β)σ1+(1-β)σ3], then(11)yic,ϕ=A-43nβσ1+σ2+σ3-1-n2βσ1+bσ2+σ3+bσ3,fiσ1,σ2,σ3=1-n2σ1+bσ2-σ3-bσ3+n3σ1-σ22+σ2-σ32+σ3-σ12.By means of finite element methods to calculate the stress solutions of wholeERD, then extract the principal stresses σ1, σ2, and σ3 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 ∑i=1nyi(c,φ)Δli and ∑i=1nfi(σ1,σ2,σ3)Δli according to linear and nonlinearUST. Eventually the safety factor can be written as(12)Fs=∑i=1nyic,φΔli∑i=1nfiσ1,σ2,σ3Δli.For application of optimal method to decide the potential slip surface ofERD, we can ascertain the minimum safety factors through comparison among the outcomes obtained above.


### 2.2. 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 Figure2. In the present paper we conclude that the linear and nonlinearUST associate with two main parameters, functional coefficient of intermediate principal stresses b and the nonlinear effect coefficient n, 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.Figure 2
Strength parameters versus safety factors.In the following examples of the slope stability analysis, we taken=0, b=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 n and b.As Figure2 displays, two strength parameters n and b inUST have different influences on the coefficients of slope stability ofERD. When nonlinear coefficient n develops gradually, the safety factor increases at first and then falls back slightly, but the safety coefficients increase with accretion of the parameter b concerned with the intermediate principal stresses. From Figure 2, you can also read that the nonlinear coefficient n has greater effect on the safety coefficients of slope stability ofERD more than the principal stress coefficient b.


## 2.1. NonlinearUST Applied in Slope Stability

According toMohr-Coulomb strength theory, the safety coefficients along the slip surface shown in Figure 1 can be defined as(6)∫τfdl∫τdl=∑i=1nci+σnitan⁡φΔli∑i=1nτiΔli,where τf is the shear strength of sliding surface, τ is the factual shear stress, l means the arc length of sliding surface, n is number of soil slices, c is cohesion, φ is inner friction angle, and σ is effective stress.Figure 1
Slip surface.To apply linear and nonlinearUST in the slope stability analysis, we need to rewrite formulas (2) and (4) as follows.Forσ2≤(1/2)[(1+β)σ1+(1-β)σ3], then(7)1-n2σ1+bσ1-bσ2-σ3+n3σ1-σ22+σ2-σ32+σ3-σ12=A-43nβσ1+σ2+σ3-1-n2βσ1+bσ1+bσ2+σ3.Forσ2>(1/2)[(1+β)σ1+(1-β)σ3], then(8)1-n2σ1+bσ2-σ3-bσ3+n3σ1-σ22+σ2-σ32+σ3-σ12=A-43nβσ1+σ2+σ3-1-n2βσ1+bσ2+σ3+bσ3,where the items, related to the principal stresses at the left hand of the formulae, can be denoted by f(σ1,σ2,σ3) and the items concerned with strength parameters at the right hand are written in the form y(c,φ). Hence, the safety coefficient along a certain potential slip surface can be depicted as(9)Fs=∫lyc,φdl∫lfσ1,σ2,σ3dl=∑i=1nyic,φΔli∑i=1nfiσ1,σ2,σ3Δli.So we deduce following expressions.Forσ2≤(1/2)[(1+β)σ1+(1-β)σ3], then(10)yic,ϕ=A-43nβσ1+σ2+σ3-1-n2βσ1+bσ1+bσ2+σ3,fiσ1,σ2,σ3=1-n2σ1+bσ1-bσ2-σ3+n3σ1-σ22+σ2-σ32+σ3-σ12.Forσ2>(1/2)[(1+β)σ1+(1-β)σ3], then(11)yic,ϕ=A-43nβσ1+σ2+σ3-1-n2βσ1+bσ2+σ3+bσ3,fiσ1,σ2,σ3=1-n2σ1+bσ2-σ3-bσ3+n3σ1-σ22+σ2-σ32+σ3-σ12.By means of finite element methods to calculate the stress solutions of wholeERD, then extract the principal stresses σ1, σ2, and σ3 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 ∑i=1nyi(c,φ)Δli and ∑i=1nfi(σ1,σ2,σ3)Δli according to linear and nonlinearUST. Eventually the safety factor can be written as(12)Fs=∑i=1nyic,φΔli∑i=1nfiσ1,σ2,σ3Δli.For application of optimal method to decide the potential slip surface ofERD, we can ascertain the minimum safety factors through comparison among the outcomes obtained above.


## 2.2. 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 Figure2. In the present paper we conclude that the linear and nonlinearUST associate with two main parameters, functional coefficient of intermediate principal stresses b and the nonlinear effect coefficient n, 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.Figure 2
Strength parameters versus safety factors.In the following examples of the slope stability analysis, we taken=0, b=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 n and b.As Figure2 displays, two strength parameters n and b inUST have different influences on the coefficients of slope stability ofERD. When nonlinear coefficient n develops gradually, the safety factor increases at first and then falls back slightly, but the safety coefficients increase with accretion of the parameter b concerned with the intermediate principal stresses. From Figure 2, you can also read that the nonlinear coefficient n has greater effect on the safety coefficients of slope stability ofERD more than the principal stress coefficient b.


## 3. Stability Analysis ofERD byMGA Based onMatlab 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 ofERD 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 x and y. When the width of slices is definite, the coordinate y is only optimization variable.We choice the modifiedJanbu method [8] to calculate the safety factors of slope. Taking a general soil slice, then making force analysis is as shown in Figure 3.Figure 3
A general soil slice and applied forces.The meanings of all notations can be figured out through Figure3. If there are not external forces applied on both sides of the slices, namely, E0, En, T0, and Tn all vanish, and assuming that the number of soil slices is n, then(13)Fs=∑i=1ncib+pb+ΔTi-ubtan⁡φi1+tan2⁡αi/1+tan⁡αitan⁡φi/Fs∑i=1nΔQi+pb+ΔTitan⁡αiin which(14)tan⁡αi=yi-yi-1b,p=wib+q+Δpb,and u is the pore pressure on the slip surfaces. The optimization model of slip surface of slope stability can be concluded as the following:(15)min⁡absFs0-Fs1subject to:y0≤y0≤y0′,y1′≤y1≤h,⋮yn-1′≤yn-1≤h,y0≤yn≤yn′,-b≤yi+1-yi≤b·tan⁡αi,where Fs0 is initial estimated value of safety factor for iteration and Fs1 evaluation value by substitution of Fs0 and the shear forces between each of the soil slices into (13).


### 3.2. Slope Stability Analysis Based onMGA

TheDichotomy,Pattern Search Method, andSimplex Method are commonly adopted in slope stability analysis ofERD 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, StandardMGA 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 ofERD. Below we will detail the algorithm implementation ofMGA to slope stability ofERD.


### 3.3. Procedures of Slope Stability Analysis Based onMGA

The procedures corresponding slope stability ofERD based onMGA andUST can be summarized as follows [14].Step 1.
Define the optimal variablesy0,y1,y2,…,yn and constraint conditions in (15). In order to improve the efficiency of the optimal procedure ofMGA, the geometrical configuration of the slip surfaces should be rationalized in the optimization analysis ofERD. According to the practical features of slope stability ofERD, the slope angle of the slip slices must be subject to the following condition:(16)-45°≤αi≤80°,where αi is the angle of ith soil slice between the bottom of soil slices and horizontal coordinates. So, the last constrained condition in (15) should be modified as(17)-b≤yi+1-yi≤b·tan⁡80°.Step 2.
Build up the objective function in (13).Step 3.
Generate original population.
Producing randomlyS slip surfaces (namely S individuals) which consist of n+1y-coordinates.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 variableyi is yi′ 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-yi′)×103 sections. Binary string of digits (denoted by mi) of one variable can be obtained from the following formula:(18)2mi-1<h-yi′≤2mi.
So, individual string of digit is∑mi, 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:(19)yi=yi′+decimalsubstringi×h-yi′2mi-1,where decimal(substringi) is the decimal value of the variables yi.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 andvice versa. Consequently, the general fitness value of colony can be presented as f=∑eval(ui), which can deduce the probability of each selected individual pi=eval(ui)/∑eval(ui). Then calculate the accumulating probability of each individual Qi=∑j=1ipj, j=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 numberr(0<r<1).(2)
IfQi-1≤r≤Qi, then choose ith individual.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 Figure16.
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 whenk≤S; then generate random number rk which is a number between 0 and 1. If rk<0.3, then select the kth individual as parent generation of the crossover. Assign k=k+1. End current step.Step 9 (mutation operations).
Suppose that thejth gene of ith 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 onMGA andUST. Then repeat the steps from 6 to 9 until we obtain the optimal individuals.
We can make use ofMatlab MGA toolbox to implement the above analysis steps. Main program ofMGA and direct searching toolbox inMatlab provides an interface betweenMatlab toolbox and the external. The invoking format ofMatlab function is as follows:(20)xfval=ga@fitnessfun,nvars,options,in which x is the optimal coordinate and fval is optimal solution of objective function and both are the optimal results given by invokingMatlab function, fitnessfun stands for objective function, n  var  s denotes the number of optimization variables in objective function, and options is a parameter option of theMGA. Note that default value should be invoked without passing parameter option.


## 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 ofERD 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 x and y. When the width of slices is definite, the coordinate y is only optimization variable.We choice the modifiedJanbu method [8] to calculate the safety factors of slope. Taking a general soil slice, then making force analysis is as shown in Figure 3.Figure 3
A general soil slice and applied forces.The meanings of all notations can be figured out through Figure3. If there are not external forces applied on both sides of the slices, namely, E0, En, T0, and Tn all vanish, and assuming that the number of soil slices is n, then(13)Fs=∑i=1ncib+pb+ΔTi-ubtan⁡φi1+tan2⁡αi/1+tan⁡αitan⁡φi/Fs∑i=1nΔQi+pb+ΔTitan⁡αiin which(14)tan⁡αi=yi-yi-1b,p=wib+q+Δpb,and u is the pore pressure on the slip surfaces. The optimization model of slip surface of slope stability can be concluded as the following:(15)min⁡absFs0-Fs1subject to:y0≤y0≤y0′,y1′≤y1≤h,⋮yn-1′≤yn-1≤h,y0≤yn≤yn′,-b≤yi+1-yi≤b·tan⁡αi,where Fs0 is initial estimated value of safety factor for iteration and Fs1 evaluation value by substitution of Fs0 and the shear forces between each of the soil slices into (13).


## 3.2. Slope Stability Analysis Based onMGA

TheDichotomy,Pattern Search Method, andSimplex Method are commonly adopted in slope stability analysis ofERD 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, StandardMGA 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 ofERD. Below we will detail the algorithm implementation ofMGA to slope stability ofERD.


## 3.3. Procedures of Slope Stability Analysis Based onMGA

The procedures corresponding slope stability ofERD based onMGA andUST can be summarized as follows [14].Step 1.
Define the optimal variablesy0,y1,y2,…,yn and constraint conditions in (15). In order to improve the efficiency of the optimal procedure ofMGA, the geometrical configuration of the slip surfaces should be rationalized in the optimization analysis ofERD. According to the practical features of slope stability ofERD, the slope angle of the slip slices must be subject to the following condition:(16)-45°≤αi≤80°,where αi is the angle of ith soil slice between the bottom of soil slices and horizontal coordinates. So, the last constrained condition in (15) should be modified as(17)-b≤yi+1-yi≤b·tan⁡80°.Step 2.
Build up the objective function in (13).Step 3.
Generate original population.
Producing randomlyS slip surfaces (namely S individuals) which consist of n+1y-coordinates.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 variableyi is yi′ 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-yi′)×103 sections. Binary string of digits (denoted by mi) of one variable can be obtained from the following formula:(18)2mi-1<h-yi′≤2mi.
So, individual string of digit is∑mi, 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:(19)yi=yi′+decimalsubstringi×h-yi′2mi-1,where decimal(substringi) is the decimal value of the variables yi.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 andvice versa. Consequently, the general fitness value of colony can be presented as f=∑eval(ui), which can deduce the probability of each selected individual pi=eval(ui)/∑eval(ui). Then calculate the accumulating probability of each individual Qi=∑j=1ipj, j=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 numberr(0<r<1).(2)
IfQi-1≤r≤Qi, then choose ith individual.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 Figure16.
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 whenk≤S; then generate random number rk which is a number between 0 and 1. If rk<0.3, then select the kth individual as parent generation of the crossover. Assign k=k+1. End current step.Step 9 (mutation operations).
Suppose that thejth gene of ith 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 onMGA andUST. Then repeat the steps from 6 to 9 until we obtain the optimal individuals.
We can make use ofMatlab MGA toolbox to implement the above analysis steps. Main program ofMGA and direct searching toolbox inMatlab provides an interface betweenMatlab toolbox and the external. The invoking format ofMatlab function is as follows:(20)xfval=ga@fitnessfun,nvars,options,in which x is the optimal coordinate and fval is optimal solution of objective function and both are the optimal results given by invokingMatlab function, fitnessfun stands for objective function, n  var  s denotes the number of optimization variables in objective function, and options is a parameter option of theMGA. Note that default value should be invoked without passing parameter option.


## 4. Numerical Examples

### 4.1. Example 1

For the comparison of the general strength theory withUST, we choose a single stageERD as an example shown in Figure 4. The intensity parameters are presumed as c=13 kP, φ=18°, ρ=19.8 kN/m3, E=2.5 mP, and μ=0.3. We adopt different strength criteria, which are provided with different strength parameters n and b inUST, to evaluate the slope safety factors and the final results shown in Table 2.Table 2
Slope safety factors evaluated by different failure criteria.
b
0
0.2
0.4
0.6
0.8
1
Nonlinear
n=α=1, β=0
0.3007
0.3007
0.3007
0.3007
0.3007
0.3007
n=0.2
1.1110
1.1170
1.1216
1.1251
1.1279
1.1302
n=0.4
1.1649
1.1668
1.1681
1.1688
1.1692
1.1694
n=0.6
1.1895
1.1910
1.1920
1.1926
1.1931
1.1932
n=0.8
1.1976
1.1989
1.2000
1.2011
1.2019
1.2026
n=1
1.1944
1.1944
1.1944
1.1944
1.1944
1.1944
Linear
n=0
0.9390
0.9700
0.9930
1.0109
1.0252
1.0368
n=β=0, α=1
0.2814
0.2885
0.2938
0.2979
0.3012
0.3039
n=0, b=1.5
1.0583Figure 4
Single stageERD.We can evaluate that the safety factor is equal to 0.9496 byUST and 0.939 byMohr-Coulomb failure criterion in this example using the formula below:(21)Fs=∫τfdl∫τdl=∑i=1nci+σnitan⁡ϕΔli∑i=1nτiΔli.The relationship of slope safety coefficientsFs via b and n is shown in Figures 5 and 6.Figure 5
Slope safety factors versus strength coefficientb.Figure 6
Slope safety factors versus strength coefficientn.By the calculation and analysis above the example, whenn=α=1, β=0 in nonlinear criterion and n=β=0, α=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 tensional-compressional strength. The nonlinear coefficient n and effect coefficient b 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 n at first, and when n is greater than 0.4 the effect of change of n to the safety factors becomes significantly less, and when n is greater than 0.8 the safety factors will reduce with increase of n instead. When n is less than 0.2, the increase of the parameter b on improvement of the safety factors is obvious, but if n is greater than 0.2, the change of the parameter b has little impact on the improvement of the safety factors.


### 4.2. Example 2

The practical working conditions of a typicalERD are demonstrated in Figure 7; The downstream water level is 1 m. Soil cohesion ofERD is 13 kP, inner friction angle 25°, effective cohesion 6 kP, effective inner friction angle 13°, natural weight-specific density 20 kN/m3, and saturated unit weight of soil 23 kN/m3. WhenERD has the different upstream water level, the stability safety factors ofERD, 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 ofERD under the conditions of different upstream water level.Figure 7
ERD (m).Figure 8
Safety factor versus upstream water level.


### 4.3. Example 3

The geotechnical structure of an inhomogeneous slope in layers is shown in Figure9. 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 theMGA andUST is taken as 2.5 m. The slope is enforced lateral acceleration 0.15 g and the recommended safety factor is 1.00. The initial population is specified manually and least safety factor is 1.346 at 30th generation inMGA. The safety factors calculated by the method proposed in this paper are 1.013; the relative error is 1.3%. The coordinates and geometrical shape of slip surface are displayed in Figure 10, respectively.Table 3
Material properties of the slope.
Number of soil layers
CohesionC (kP)
Inner friction angleφ
Weight-specific densityγ(kN/m3)
Modulus of elasticityE (mP)
Poisson’s ratioν
K
o
1
0
38
19.5
1.0
E
4
0.25
0.65
2
5.3
23
19.5
1.0
E
4
0.25
0.65
3
7.2
20
19.5
1.0
E
4
0.25
0.65Figure 9
The structure of a slope (m).Figure 10
Most dangerous slip surface.


### 4.4. Example 4

The minimum safety factors of the slop obtained fromMGA 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 andMGA; all of the results show that two strength theories have very high consistency. Note that g is gravity acceleration in Figure 14.Figure 11
Safety factors versus different slope angles.Figure 12
Safety factors versus different cohesions.Figure 13
Safety factors versus different inner friction angles.Figure 14
Safety factors versus different lateral acceleration.Figure 15Figure 16


## 4.1. Example 1

For the comparison of the general strength theory withUST, we choose a single stageERD as an example shown in Figure 4. The intensity parameters are presumed as c=13 kP, φ=18°, ρ=19.8 kN/m3, E=2.5 mP, and μ=0.3. We adopt different strength criteria, which are provided with different strength parameters n and b inUST, to evaluate the slope safety factors and the final results shown in Table 2.Table 2
Slope safety factors evaluated by different failure criteria.
b
0
0.2
0.4
0.6
0.8
1
Nonlinear
n=α=1, β=0
0.3007
0.3007
0.3007
0.3007
0.3007
0.3007
n=0.2
1.1110
1.1170
1.1216
1.1251
1.1279
1.1302
n=0.4
1.1649
1.1668
1.1681
1.1688
1.1692
1.1694
n=0.6
1.1895
1.1910
1.1920
1.1926
1.1931
1.1932
n=0.8
1.1976
1.1989
1.2000
1.2011
1.2019
1.2026
n=1
1.1944
1.1944
1.1944
1.1944
1.1944
1.1944
Linear
n=0
0.9390
0.9700
0.9930
1.0109
1.0252
1.0368
n=β=0, α=1
0.2814
0.2885
0.2938
0.2979
0.3012
0.3039
n=0, b=1.5
1.0583Figure 4
Single stageERD.We can evaluate that the safety factor is equal to 0.9496 byUST and 0.939 byMohr-Coulomb failure criterion in this example using the formula below:(21)Fs=∫τfdl∫τdl=∑i=1nci+σnitan⁡ϕΔli∑i=1nτiΔli.The relationship of slope safety coefficientsFs via b and n is shown in Figures 5 and 6.Figure 5
Slope safety factors versus strength coefficientb.Figure 6
Slope safety factors versus strength coefficientn.By the calculation and analysis above the example, whenn=α=1, β=0 in nonlinear criterion and n=β=0, α=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 tensional-compressional strength. The nonlinear coefficient n and effect coefficient b 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 n at first, and when n is greater than 0.4 the effect of change of n to the safety factors becomes significantly less, and when n is greater than 0.8 the safety factors will reduce with increase of n instead. When n is less than 0.2, the increase of the parameter b on improvement of the safety factors is obvious, but if n is greater than 0.2, the change of the parameter b has little impact on the improvement of the safety factors.


## 4.2. Example 2

The practical working conditions of a typicalERD are demonstrated in Figure 7; The downstream water level is 1 m. Soil cohesion ofERD is 13 kP, inner friction angle 25°, effective cohesion 6 kP, effective inner friction angle 13°, natural weight-specific density 20 kN/m3, and saturated unit weight of soil 23 kN/m3. WhenERD has the different upstream water level, the stability safety factors ofERD, 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 ofERD under the conditions of different upstream water level.Figure 7
ERD (m).Figure 8
Safety factor versus upstream water level.


## 4.3. Example 3

The geotechnical structure of an inhomogeneous slope in layers is shown in Figure9. 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 theMGA andUST is taken as 2.5 m. The slope is enforced lateral acceleration 0.15 g and the recommended safety factor is 1.00. The initial population is specified manually and least safety factor is 1.346 at 30th generation inMGA. The safety factors calculated by the method proposed in this paper are 1.013; the relative error is 1.3%. The coordinates and geometrical shape of slip surface are displayed in Figure 10, respectively.Table 3
Material properties of the slope.
Number of soil layers
CohesionC (kP)
Inner friction angleφ
Weight-specific densityγ(kN/m3)
Modulus of elasticityE (mP)
Poisson’s ratioν
K
o
1
0
38
19.5
1.0
E
4
0.25
0.65
2
5.3
23
19.5
1.0
E
4
0.25
0.65
3
7.2
20
19.5
1.0
E
4
0.25
0.65Figure 9
The structure of a slope (m).Figure 10
Most dangerous slip surface.


## 4.4. Example 4

The minimum safety factors of the slop obtained fromMGA 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 andMGA; all of the results show that two strength theories have very high consistency. Note that g is gravity acceleration in Figure 14.Figure 11
Safety factors versus different slope angles.Figure 12
Safety factors versus different cohesions.Figure 13
Safety factors versus different inner friction angles.Figure 14
Safety factors versus different lateral acceleration.Figure 15Figure 16


## 5. Conclusions

The numerical results indicate that the nonlinear coefficientn and intermediate principal stress effect coefficient b inUST have effect on the slope stability factors at some degree. In general, nonlinear coefficient n has greater effect on the slope stability factors more than the principal stress effect coefficient b. In the geotechnical engineering, the determination of linear and nonlinear parameters inUST 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 byMGA as that by Strength Reduction Method, butMGA is a highly efficient, simple, and good performance of fault-tolerance algorithm. Compared with the general optimization methods,MGA andUST used in the stability analysis ofERD 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 whichMGA 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.



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*Source: 895142-2015-06-10.xml*