LSSVM-based approach for refining soil failure criteria and calculating safety factor of slopes

Abstract

The failure criteria of practical soil mass are very complex, and have significant influence on the safety factor of slope stability. The Coulomb strength criterion and the power-law failure criterion are classically simplified. Each one has limited applicability owing to the noticeable difference between calculated predictions and actual results in some cases. In the work reported here, an analysis method based on the least square support vector machine (LSSVM), a machine learning model, is purposefully provided to establish a complex nonlinear failure criterion via iteration computation based on strength test data of the soil, which is of more extensive applicability to many problems of slope stability. In particular, three evaluation indexes including coefficient of determination, mean absolute percentage error, and mean square error indicate that fitting precision of the machine learning-based failure criterion is better than those of the linear Coulomb criterion and nonlinear power-law criterion. Based on the proposed LSSVM approach to determine the failure criterion, the limit equilibrium method can be used to calculate the safety factor of three-dimensional slope stability. Analysis of results of the safety factor of two three-dimensional homogeneous slopes shows that the maximum relative errors between the proposed approach and the linear failure criterion-based method and the power-law failure criterion-based method are about 12% and 7%, respectively.

Graphical abstract

Keywords

slope stability / safety factor / failure criterion / least square support vector machine

1 Introduction

‘Safety factor’ is a dimensionless number used to assess slope stability. Deterministic analysis methods based on safety factor and random analysis methods based on failure probability are two main methods for analysis of slope stability. The calculation of slope safety factor lays the groundwork for landslide disaster prevention [1].

Many methods for calculating the safety factor of slopes have been developed, which include the limit equilibrium method [1,2], the limit analysis method [3–7], and the shear strength reduction method via numerical simulation method [8,9]. Of these three methods, the limit equilibrium method is currently the most widely used analytical method in practice due to its simple conception, easy operation and extensive applicability. It can be generally further divided into the slice method and overall limit equilibrium method. Classical Fellenius’s method [10], Bishop’s method [11], Janbu’s method [12], Morgenstern and Price’s method [13], and Spencer’s method [14] belong to the slice method. The overall limit equilibrium method was first provided by Bell [15], and then Zhu and Lee [16], Zheng and Tham [17] have developed it further. In addition, with the development of computing techniques, a growing number of numerical methods have been proposed for solving practical problems in geotechnical engineering. For example, the cracking-particle method proposed by Rabczuk and Belytschko [18] is an effective tool for modeling discrete fractures in the framework of the meshless method, and this method was further simplified by Rabczuk and Belytschko [19] and Rabczuk et al. [20]. Ren et al. [21–22] developed a dual-horizon peridynamics method, which has higher computational efficiency compared with the traditional method and can serve as an extension of traditional peridynamics. Areias et al. [23–24] developed a damage and fracture algorithm based on the screened Poisson equation, obtaining remarkably consistent results as well as avoiding variable mapping within mesh adaptation algorithms.

Due to the rapid progress in machine learning techniques, using machine learning method for slope stability analysis continues to grow in popularity. Machine learning is a general term for a series of methods [25,26], including artificial neural network, support vector machine, and extreme learning machine, etc. Also, it is an important modeling technique, and has been widely used in many disciplines including finance, medicine, construction and materials technology [27–32]. Samui [33] developed a support vector machine-based method for estimating the safety factor of slope. Garg et al. [34] developed an integrated structural risk minimisation-multi-gene genetic programming method for predicting the safety factor of three-dimensional slopes. Koopialipoor et al. [35] compared the performance of imperialist competitive algorithm-artificial neural network, genetic algorithm-artificial neural network, particle swarm optimization-artificial neural network, and artificial bee colony-artificial neural network for estimating the safety factor, and found particle swarm optimization- artificial neural network was the best method of these. Ferentinou and Fakir [36] combined rock engineering systems with artificial neural networks to develop a stability analysis method for open-pit mines, and the preliminary analysis showed that the method can be used as a supplement to traditional theoretical and numerical analysis methods. Zhou et al. [37] developed the gradient boosting machine method for predicting the safety factor. Indeed, the performance of machine learning-based methods in landslide sensitivity prediction is better than traditional statistical models [38]. An agent model based on extreme learning machines to perform back analysis of slope stability has been developed by Wang et al. [39]. A recurrent neural network has been successfully used to predict pore water pressure changes in a slope in Hong Kong, China by Wei et al. [40]. Safa et al. [41] successfully used fuzzy neural networks to predict the safety factor of slopes with vegetation. Machine learning is often regarded as a black box model, and one of its greatest advantages is that it can be used to solve nonlinear problems.

The studies mentioned above all show the good applicability of machine learning techniques in slope stability analysis. However, in the past, merely fitting the relationship between input variables (e.g., slope angle, slope height, soil unit weight, cohesion, and internal friction angle) and output variable (slope safety factor) through machine learning algorithms has lacked a strong basis in soil mechanics. Hence, the least square support vector machine (LSSVM) [42–43] (an important machine learning method) is used herein to reasonably determine the complex nonlinear failure criterion of soil, and it is further utilized to solve the safety factor of three-dimensional slopes based on the limit equilibrium method.

2 Calculation approach

2.1 Brief introduction to the least square support vector machine

The nonlinear fitting ability of LSSVM for small samples has attracted attention. The formula of LSSVM for regression can be expressed as [42]:

(1)f(\mathit{x})={\mathit{\omega }}^{\mathrm{T}} {\phi }_{\text{f}} (\mathit{x})+b,

where x is an input vector; ω and b are undetermined parameters; and {\phi }_{\text{f}} is a nonlinear function.

Through the Lagrange multiplier method, the parameter ω can be obtained by

(2)\mathit{\omega }={\sum }_{i=1}^{m_0}{\lambda }_{\text{l}i}{\phi }_{\text{f}} (x_i),

where λ_li is the Lagrange multiplier, and m₀ is the size of the sample.

Substituting Eq. (2) into Eq. (1), one can get

(3)f(\mathit{x})={\sum }_{i=1}^{m_0}{\lambda }_{\text{l}i}{\phi }_{\text{f}}(x_i{)}^{\text{T}}{\phi }_{\text{f}}( \mathit{x})+b .

According to the definition of the kernel function [42], Eq. (3) can be rewritten as

(4)f(\mathit{x})={\sum }_{i=1}^{m_0}{\lambda }_{\text{l}i}K(x_i,\mathit{x})+b,

where K represents the kernel function. The Radial basis function [42] is adopted here as the kernel function, which is defined as

(5)K(x_i,\mathit{x})=\exp \left\{-\Vert \mathit{x}-x_i\Vert /\phantom{-\Vert \mathit{x}-x_i\Vert 2{\delta }^2}2{\delta }^2\right\},

where δ is a kernel parameter.

Then, the unknown parameters λ_li and b can be obtained by solving the following linear equations

(6)\left[\begin{array}{cc}0& {\mathbf{1}}^{\text{T}}\\ \mathbf{1}& K(x_i,\mathit{x})+C^{-1}\mathit{I}\end{array}\right]\left[\begin{array}{c}b\\ \mathit{\lambda }\end{array}\right]=\left[\begin{array}{c}0\\ \mathit{y}\end{array}\right] ,

where \mathbf{1}={\left[1,1,\dots ,1\right]}_{m_0\times 1}, \mathit{\lambda }={\left[{\lambda }_{\text{l1}},{\lambda }_{\text{l2}},\dots ,{\lambda }_{\text{l}{m}_0}\right]}^{\text{T}}, \mathit{y}=[y_1,y_2,\dots ,y_{m_0} {]}^{\text{T}}, \mathit{I}=1_{m_0\times m_0}; y is the output vector, and C is a penalty factor. δ in Eq. (5) and C in Eq. (6) are also called hyperparameters.

2.2 Failure criterion of the soil

Failure criterion is the basis of safety factor calculation. Safety factor (F_s) can be defined as [11,44]:

(7)F_{\text{s}} =\frac{{\tau }_s}{\tau },

where τ_s and τ represent shear strength and shear stress on a potential slip surface, respectively.

The shear strength can be obtained by linear and nonlinear criteria [45]. The linear criterion is also called Coulomb strength criterion, and can be expressed as [46]:

(8){\tau }_s=c+{\sigma }_{\mathrm{n}}\tan \phi ,

where σ_n is normal stress on the slip surface; c and \phi are cohesion and internal friction angle of soil, respectively.

The power-law failure criterion is a nonlinear failure criterion [1,47–50], and it can be expressed as

(9){\tau }_s=c_0{\left(1+\frac{{\sigma }_{\mathrm{n}}}{{\sigma }_t}\right)}^{1/\phantom{1m}m},

where c₀, σ_t and m are initial cohesion, tensile strength, and nonlinear dimensionless index of soil, respectively.

The above two criteria have been widely used in practice. However, they may not fully and reasonably describe the relationship between shear strength and normal stress [51,52]. As a matter of fact, it can be seen from Eq. (7) that the accuracy of prediction of the safety factor depends on the accuracy of τ_s. If a failure criterion in better agreement with the test data can be established, the calculated safety factor will be more precise. Thus, one can carry it out via a machine learning method. Herein the LSSVM is adopted to construct the relationship between normal stress and shear strength on the slope slip surface. As a widely used machine learning method for solving classification and regression problems [53–55], LSSVM can express the failure criterion of soils as follows:

(10){\tau }_{\text{s}} =LSSV M({\sigma }_{\text{n}} ).

2.3 Slope safety factor based on the LSSVM failure criterion

Let the three-dimensional rectangular coordinate system of the slope surface and slip surface, respectively, be {(x,y,l)} and {(x,y,z)} (where x represents horizontal coordinate in the direction of slope length, y represents horizontal coordinate in the direction of slope width, z represents vertical coordinate in the direction of slope height, and l represents vertical coordinate of slop surface) (Fig.1). The following assumptions/simplifications are made:

1) the motion of sliding body is assumed to a rigid body motion;

2) the sliding body is assumed to be symmetrical about the xoz plane;

3) the potential tensile stress in the sliding body is not considered.

Assumption 1) is the premise of most analytical methods of slope safety factor [13,14,16,17]; assumption 2) is to simplify this problem, because when the sliding body is symmetrical about the xoz plane, only three equations (see below) are needed to strictly describe the static equilibrium conditions of the sliding body; assumption 3) is also widely used in slope stability analyses [3–5].

According to Fig.1, the equilibrium equations of a three-dimensional homogeneous slope can be written as

\begin{array}{}\text{(11a)}& \iint \left(-{\sigma }_{\text{n}}{{\mathit{\text{z}}}^{\prime }}_x+\tau \frac{\omega }{{\omega }_1}\right)\text{d}x\text{d}y=\iint K_{\text{c}}w\text{d}x\text{d}y,\end{array}

\begin{array}{}\text{(11b)}& \iint \left({\sigma }_{\text{n}}+{{\mathit{\text{z}}}^{\prime }}_x\tau \frac{\omega }{{\omega }_1}\right)\text{d}x\text{d}y=\iint w\text{d}x\text{d}y,\end{array}

\begin{array}{rl}& \iint \left({\sigma }_{\text{n}}+{{\mathit{\text{z}}}^{\prime }}_x\tau \frac{\omega }{{\omega }_1}\right)x\text{d}x\text{d}y-\iint wx\text{d}x\text{d}y\\ +& \iint K_{\text{c}}w{\mathit{\text{z}}}_{\text{c}}\text{d}x\text{d}y=\iint \left(-{\sigma }_{\text{n}}{{\mathit{\text{z}}}^{\prime }}_x+\tau \frac{\omega }{{\omega }_1}\right)\mathit{\text{z}}\text{d}x\text{d}y\end{array}

where

\begin{array}{}\text{(12a)}& \omega =\sqrt{1+{({{\mathit{\text{z}}}^{\prime }}_x)}^2+{({{\mathit{\text{z}}}^{\prime }}_y)}^2},\end{array}

\begin{array}{}\text{(12b)}& {\omega }_1=\sqrt{1+{({{\mathit{\text{z}}}^{\prime }}_x)}^2},\end{array}

where w represents self-weight of soil, which is a function of unit weight (γ_g), i.e., w=\iint {\gamma }_{\text{g}}(l-\mathit{\text{z}})\text{d}x\text{d}y; K_c represents the horizontal seismic coefficient; the superscript ‘'’ is the derivative symbol.

The normal stress can be expressed as [15]:

(13){\sigma }_{\text{n}} =w\left({\lambda }_{1}x+{\lambda }_2\right),

where λ₁ and λ₂ are undetermined parameters.

Combining the linear failure criterion with Eqs. (11)–(13), the analytical solutions of F_s, λ₁ and λ₂ can be easily obtained. However, when nonlinear functions (including the power-law failure criterion of Eq. (9) and LSSVM-based failure criterion of Eq. (10)) are used to describe the failure criterion of the soil, the analytical solutions cannot be obtained. By using exhaustive or optimization algorithms, the analytical solutions of F_s, λ₁ and λ₂ satisfying Eq. (11) can be obtained based on the nonlinear failure criterion, but it is time-consuming and inefficient. Referring to the tangential principle used in the limit analysis method with a nonlinear failure criterion [50], the slope safety factor can be solved by transforming the LSSVM-based failure criterion into a linear failure criterion.

Linearizing Eq. (10):

\begin{array}{}\text{(14a)}& \tan {\phi }_{\text{t}}=\tan \phi ,\end{array}

\begin{array}{}\text{(14b)}& c_{\text{t}}=LSSVM({\sigma }_{\text{n}})-\tan {\phi }_{\text{t}}{\sigma }_{\text{n}},\end{array}

where c_t and φ_t are the linearized cohesion and internal friction angle of the soil.

It should be noticed that pore water pressure u (a function of the wetting front depth s and the unit weight of the water (γ_w), i.e., u=\iint {\gamma }_{\text{w}}s\text{d}x\text{d}y) on the slip surface can also be easily incorporated in Eq. (13). If the pore water pressure is considered, the failure criterion of the soil can be expressed as

\begin{array}{}\text{(15a)}& \tan {\phi }_{\text{t}}=\tan \phi ,\end{array}

\begin{array}{}\text{(15b)}& c_{\text{t}}=LSSVM({\sigma }_n)-\tan {\phi }_{\text{t}}({\sigma }_n-u),\end{array}

Since c_t and {\phi }_{\mathrm{t}} are both functions of the normal stress, it is necessary to use an iterative method to solve the safety factor of the soil slope. Taking the two parameters c and \phi of the linear failure criterion (see Eq. (8)) as the initial iteration values of them, and substituting Eq. (13) into Eq. (12), one can obtain

\begin{array}{rl}& {\lambda }_1\iint \left(-{\mathit{\text{z}}}_x^{\prime }+\frac{1}{F_{\text{s}}}\frac{\omega }{{\omega }_1}\tan \phi \right)xw\text{d}x\text{d}y\\ & +{\lambda }_2\iint \left(-{\mathit{\text{z}}}_x^{\prime }+\frac{1}{F_{\text{s}}}\frac{\omega }{{\omega }_1}\tan \phi \right)w\text{d}x\text{d}y\\ =& \iint K_{\text{c}}w\text{d}x\text{d}y+\iint \left(\frac{1}{F_{\text{s}}}\frac{\omega }{{\omega }_1}(u\tan \phi -c)\right)\text{d}x\text{d}y\end{array},

\begin{array}{rl}& {\lambda }_1\iint \left(1\text{+}\frac{1}{F_{\text{s}}}\frac{\omega }{{\omega }_1}{\mathit{\text{z}}}_x^{\prime }\tan \phi \right)xw\text{d}x\text{d}y\\ & +{\lambda }_2\iint \left(1\text{+}\frac{1}{F_{\text{s}}}\frac{\omega }{{\omega }_1}{\mathit{\text{z}}}_x^{\prime }\tan \phi \right)w\text{d}x\text{d}y\\ =& \iint w\text{d}x\text{d}y+\iint \left(\frac{1}{F_{\text{s}}}{\mathit{\text{z}}}_x^{\prime }\frac{\omega }{{\omega }_1}(u\tan \phi -c)\right)\text{d}x\text{d}y,\end{array}

\begin{array}{rl}& {\lambda }_1\iint \left((x\text{+}{\mathit{\text{z}}}_x^{\prime }\mathit{\text{z}})+\frac{1}{F_{\text{s}}}\frac{\omega }{{\omega }_1}({\mathit{\text{z}}}_x^{\prime }x-\mathit{\text{z}})\tan \phi \right)xw\text{d}x\text{d}y\\ & +{\lambda }_2\iint \left((x\text{+}{\mathit{\text{z}}}_x^{\prime }\mathit{\text{z}})+\frac{1}{F_{\text{s}}}\frac{\omega }{{\omega }_1}({\mathit{\text{z}}}_x^{\prime }x-\mathit{\text{z}})\tan \phi \right)w\text{d}x\text{d}y\\ =& \iint \left(\frac{1}{F_{\text{s}}}\frac{\omega }{{\omega }_1}({\mathit{\text{z}}}_x^{\prime }x-\mathit{\text{z}})(u\tan \phi -c)\right)\text{d}x\text{d}y\\ & +\iint wx\text{d}x\text{d}y-\iint K_{\text{c}}w{z}_{\text{c}}\text{d}x\text{d}y\end{array}

Equations (16a) and (16b) can be rewritten as

(17)\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right] \left[\begin{array}{c}{\lambda }_1\\ {\lambda }_2\end{array}\right]=\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right],

where the expressions of the coefficients A₁₁, A₁₂, A₂₁, A₂₂, B₁ and B₂ are

A_{11}=\iint \left(-{\mathit{\text{z}}}_x^{\prime }\text{+}\frac{1}{F_{\text{s}}}\frac{\omega }{{\omega }_1}\tan \phi \right)xw\text{d}x\text{d}y

A_{12}=\iint \left(-{\mathit{\text{z}}}_x^{\prime }\text{+}\frac{1}{F_{\text{s}}}\frac{\omega }{{\omega }_1}\tan \phi \right)w\text{d}x\text{d}y

A_{21}=\iint \left(1\text{+}\frac{1}{F_{\text{s}}}\frac{\omega }{{\omega }_1}{\mathit{\text{z}}}_x^{\prime }\tan \phi \right)xw\text{d}x\text{d}y

A_{22}=\iint \left(1\text{+}\frac{1}{F_{\text{s}}}\frac{\omega }{{\omega }_1}{\mathit{\text{z}}}_x^{\prime }\tan \phi \right)w\text{d}x\text{d}y

B_1=\iint K_{\text{c}}w\mathrm{d}x\mathrm{d}y+\iint \left(\frac{1}{F_{\text{s}}}\frac{\omega }{{\omega }_1}(u\tan \phi -c)\right)\text{d}x\text{d}y

B_2=\iint w\text{d}x\text{d}y+\iint \left(\frac{1}{F_{\text{s}}}{\mathit{\text{z}}}_x^{\prime }\frac{\omega }{{\omega }_1}(u\tan \phi -c)\right)\text{d}x\text{d}y

So, the undetermined parameters λ₁ and λ₂ can be obtained by

(18)\left[\begin{array}{c}{\lambda }_1\\ {\lambda }_2\end{array}\right]={\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]}^{-1}\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right].

where λ₁ and λ₂ are expressions related to F_s. Thus, substituting them into Eq. (16c), F_s can be obtained.

Specifically, λ₁ and λ₂ obtained by Eq. (18) can be substituted into Eq. (13), and linearized c_t and {\phi }_{\mathrm{t}} can be calculated using Eq. (15). The two calculated values, instead of their initial values c and \phi, can then be subsitiuted into Eq. (18). The iterative process is repeated until the relative calculation errors of λ₁ and λ₂ between two adjacent iterations are both less than a certain threshold which can be set as 10⁻⁵ to ensure the calculation accuracy. Then, the F_s based on the failure criterion refined by LSSVM and the overall limit equilibrium method can be precisely determined. The calculation flow chart of the proposed method is given in Fig.2. The specific form of linearization has negligible effect on the calculated safety factor.

In summary, only the following parameters are required in proposed method: 1) measured normal stress and shear strength, which are used to determine the parameters of the LSSVM-based failure criterion (hyperparameters δ and C) and 2) unit weight of soil. The above parameters are deterministic parameters and no uncertain external parameter exists, so the proposed method is completely data-driven. Note that the K-fold cross-validation method can be used to determine the hyperparameters of LSSVM-based failure criteria. As shown in Fig.3, the data sets are randomly divided into K parts, of which one part is selected as the test set and the others are selected as the training sets. In other words, in the K-fold cross-validation method, each sample can serve as a test set. Using the K-fold cross-validation method to determine the hyperparameters can prevent LSSVM from falling into over-learning. Following some existing studies [56], K is set to 5 in this work.

3 Examples and verifications

In order to verify the proposed method, two examples are used to compare the difference between the proposed results and those by the pre-existing methods.

3.1 Example 1

Fig.4(a) shows a three-dimensional homogeneous slope composed of Upper Lias clay [57] with a slope angle of 38° and sufficiently long slope surface. Formula of the slope surface and a potential slip surface are both given in Fig.4(a). The strength test data (there are 19 sets of data) of the Upper Lias clay is shown in Fig.5. The parameters of the classical linear and power-law failure criteria obtained through the cftool toolbox [58] in MATLAB are shown in Tab.1.

Fig.5 shows the results of the linear failure criterion, power-law failure criterion, and the proposed LSSVM-based failure criterion (\left[C,\delta \right]=\left[\begin{array}{cc}50.85& 0.29\end{array}\right] in LSSVM). the performance of the LSSVM-based failure criterion is better than those of the classical two criteria. Three evaluation indexes including coefficient of determination (R²), mean absolute percentage error (MAPE) and mean square error (MSE), expressed individually as Eqs. (19)− (21), are used to quantify the error for the three criteria.

(19)R^2=1- {\displaystyle \frac{{\displaystyle \sum {\left({\tau }_{\text{s}}^{\text{obs}}-{\tau }_s^{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}\right)}^2}}{{\displaystyle \sum {\left({\tau }_{\text{s}}^{\text{obs}}-{\overline{\tau }}_{\text{s}}^{\text{obs}}\right)}^2}}},

(20)MAPE= \frac{1}{n}{\sum }_{k=1}^n \left|\frac{{\tau }_{\text{s}}^{\text{pred}}(k)-{\tau }_{\text{s}}^{\text{obs}}(k)}{{\tau }_{\text{s}}^{\text{obs}}(k)}\right|\times 100 \mathrm{\%},

(21)MSE=\frac{1}{n}{\sum }_{k=1}^n{\left({\tau }_{\text{s}}^{\text{pred}}(k)-{\tau }_{\text{s}}^{\text{obs}}(k)\right)}^2,

where τ_s^obs is the test value, and τ_s^pred is the calculated value using various criteria. The larger R², the better the accuracy of the related criterion. The smaller MAPE and MSE, the higher the accuracy of the criterion. The precision statistics of the three criteria are given in Tab.2, where the three evaluation indexes show that the results of the proposed LSSVM-based failure criterion are better than the others.

The calculated safety factors of the soil slope in light of linear failure criterion, power-law failure criterion and the LSSVM-based failure criterion are 1.2572, 1.2243, and 1.1735, respectively (see Tab.3). For emphasis, existing methods for slope stability analysis (including limit equilibrium method, limit analysis method, and numerical simulation method) are all based on failure criteria for simplifications of reality, making it impossible to obtain a meaningful slope safety factor. Considering the LSSVM-based failure criterion can better fit the strength test data, the results of the proposed method may be regarded as the standard solution, the relative errors of the classical linear and nonlinear failure criteria-based methods are 7.12% and 4.32%, respectively. The proposed method achieves convergence in only 5 iterations. Although the computational efficiency of the proposed method is lower than that of the linear failure criterion-based method, it is close to the efficiency of the power-law failure criterion-based method. The calculating time of the linear criterion-based method is about one-fifth of that of the other two methods.

3.2 Example 2

A three-dimensional homogeneous slope composed of Oxford clay [59] with slope angle 35° and adequately long slope surface is taken herein as another example (see Fig.4(b)). Formulas of a slope surface and a potential slip surface are both given in Fig.4(b). There are 6 sets of strength test data. The parameters corresponding to the classical linear and power-law failure criteria are also listed in Tab.1. The results of the three evaluation indexes can also be found in Tab.2. It can be seen from Tab.2 and Fig.6 that the LSSVM-based failure criterion still produces the best results.

The safety factors determined based on linear failure criterion, power-law failure criterion and the LSSVM-based failure criterion are 1.1352, 1.0808, and 1.0126, respectively (see Tab.3). In this case, both the proposed method and the method based on nonlinear failure criterion obtain results in 5 iterations. The proposed safety factor herein is smaller than that by the two classical criteria. The relative errors of the linear and nonlinear failure criteria-based method are 12.11% and 6.74%, respectively.

4 Parametric study and discussion

4.1 Parametric study

As mentioned above, the proposed LSSVM-based failure criterion stems from strength test data (i.e., a data-driven approach [60]). In addition, the LSSVM-based failure criterion does not have a simple explicit expression. The two aspects make it impossible for us to analyze the influence of strength parameters on the safety factor. Therefore, herein we only analyze the influence of the slope angle (α) and the ratio of the width B over the height H of the slide mass on the safety factor (see Fig.7 for the definition of the ratio B/H).

Fig.7 displays the results of the safety factor of the Upper Lias clay slope with the range of slope angles from 10° to 60° and the ratios B/H from 1 to 8. The slip surface is represented by an ellipsoid [61]. It can be seen from Fig.7 that the safety factor is gradually decreasing with the increase of the slope angle, which is consistent with the results by Xiao et al. [62]. As the ratio B/H increases, the safety factor gradually decreases and tends to be close to the safety factor under two-dimensional conditions, which agrees with the results obtained by Wang et al. [5].

Fig.8 shows the results of the safety factor of the Oxford clay slope with the range of slope angles from 10° to 60° and the ratios B/H from 1 to 8. The characteristics shown in Fig.8 are fairly similar to those in Fig.7.

4.2 Discussion

LSSVM, based on statistical learning theory [63], has been proven to be one of the most effective machine learning methods for solving small sample problems [43]. The cost of a series of soil strength tests is generally high, so it is promising for practitioners to obtain reasonable failure criterion from a small number of samples. Use of machine learning methods including neural networks is usually difficult for solving small sample problems. However, the parameters of the failure criterion based on LSSVM can be accurately estimated by using 5 sets of test data. Compared with the use of linear or non-linear failure criteria, the cost of using the failure criterion based on LSSVM does not increase significantly. The difficulty in implementing the proposed method is to construct the failure criterion based on LSSVM. Fortunately, this can be easily overcome with some open source software. LSSVM and the K-fold cross-validation method are implemented using the LS-SVMlab Toolbox [64]. After constructing the failure criterion based on LSSVM, the calculation of the corresponding safety factor can be transformed into the problem considering the linear failure criterion via Eqs. (14) and (15).

Although many studies claim that a strict analysis method has been established, these studies actually have only established a method for determining the safety factor of slope stability considering a linear or nonlinear failure characteristics of the soil, which is not certainly true safety factor of the slope in many cases. There are also some studies that have directly matched the relationship between input variables (e.g., slope angle, slope height, soil unite weight, cohesion, and internal friction angle) and safety factor through machine learning techniques. However, these procedures are concerned with mathematical calculation instead of physical significance of the problem in soil mechanics. Compared with the existing estimation methods of the safety factor of slopes based on machine learning, the proposed approach has strict physical implication for soil mechanics. Certainly, the proposed method depends on a few shear strength test results of the soils, and it cannot provide a briefly generalized formula for the soil failure criterion. However, as mentioned above, the implementation of the proposed method is also simple, and more accurate results will be obtained based on shear strength test results.

5 Conclusions

The safety factor of slopes depends not only on their limit equilibrium conditions, but also on the failure criterion of the soils. The coefficient of determination, mean absolute percentage error, and mean square error of two slope examples with strength test data of the soil show that the fitting precision of the proposed machine learning based failure criterion is better than that of the classical linear and power-law failure criteria. It indicates that the proposed method is acceptable for extensive application.

An efficient iterative procedure based on the proposed machine learning-based failure criterion and global limit equilibrium method is put forward to calculate the safety factor of three-dimensional soil slopes. The analysis results of the two examples show that the soil failure criterion has a great impact on the slope safety factor. The maximum relative error caused by the classical linear and nonlinear failure criterion can reach about 12% and 7%, respectively.

References

[1] Collins I F, Gunn C I M, Pender M J, Yan W. Slope stability analyses for materials with a non-linear failure envelope. International Journal for Numerical and Analytical Methods in Geomechanics, 1988, 12(5): 533–550

[2] Kang F, Li J J. Artificial bee colony algorithm optimized support vector regression for system reliability analysis of slopes. Journal of Computing in Civil Engineering, 2016, 30(3): 04015040

[3] Chen W F, Scawthorn C R. Limit analysis and limit equilibrium solutions in soil mechanics. Soil and Foundation, 1970, 10(3): 13–49

[4] Sun D A, Wang L, Li L. Stability of unsaturated soil slopes with cracks under steady-infiltration conditions. International Journal of Geomechanics, 2019, 19(6): 04019044

[5] Wang L, Sun D A, Yao Y, Tan Y. Seismic stability of 3D piled unsaturated earth slopes using kinematic limit analysis. Soil Dynamics and Earthquake Engineering, 2019, 126: 105821

[6] Lian J, Wu J. Surficial stability analysis of soil slope under seepage based on a novel failure mode. Frontiers of Structural and Civil Engineering, 2021, 15(3): 712–726

[7] Sheng K, Hong B N, Liu X, Shan H. Modified Bishop method for stability analysis of weakly sloped subgrade under centrifuge model test. Frontiers of Structural and Civil Engineering, 2021, 15(3): 727–741

[8] Matsui T, San K C. Finite element slope stability analysis by shear strength reduction technique. Soil and Foundation, 1992, 32(1): 59–70

[9] Yuan W, Li J X, Li Z H, Wang W, Sun X Y. A strength reduction method based on the Generalized Hoek-Brown (GHB) criterion for rock slope stability analysis. Computers and Geotechnics, 2020, 117: 103240

[10] FelleniusW. Calculation of the stability of earth dams. In: Transactions of The 2nd Congress on Large Dams, International Commission on Large Damsof The World Power Conference. 1936, 4: 445–462

[11] Bishop A W. The use of the slip circle in the stability analysis of earth slopes. Geotechnique, 1955, 5(1): 7–17

[12] JanbuN. Slope Stability Computations. Hoboken: John Wiley, 1973

[13] Morgenstern N R, Price V E. The analysis of the stability of general slip surfaces. Geotechnique, 1965, 15(1): 79–93

[14] Spencer E. A method of analysis for stability of embankments using parallel inter-slice forces. Geotechnique, 1967, 17(1): 11–26

[15] BellJ M. General slope stability analysis. Journal of the Soil Mechanics and Foundations Division, 1968, 94(6): 1253−1270

[16] Zhu D Y, Lee C F. Explicit limit equilibrium solution for slope stability. International Journal for Numerical and Analytical Methods in Geomechanics, 2002, 26(15): 1573–1590

[17] Zheng H, Tham L G. Improved Bell’s method for the stability analysis of slopes. International Journal for Numerical and Analytical Methods in Geomechanics, 2009, 33(14): 1673–1689

[18] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[19] RabczukTBelytschko T. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29−30): 2777−2799

[20] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37−40): 2437–2455

[21] Ren H, Zhuang X, Cai Y, Rabczuk T. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476

[22] Ren H, Zhuang X, Rabczuk T. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782

[23] Areias P, Msekh M A, Rabczuk T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143

[24] Areias P, Reinoso J, Camanho P P, César de Sá J, Rabczuk T. Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation. Engineering Fracture Mechanics, 2018, 189: 339–360

[25] Hedayat A A, Ahmadi Afzadi E, Kalantaripour H, Morshedi E, Iranpour A. A new predictive model for the minimum strength requirement of steel moment frames using artificial neural network. Soil Dynamics and Earthquake Engineering, 2019, 116: 69–81

[26] Rivera J I, Bonilla C A. Predicting soil aggregate stability using readily available soil properties and machine learning techniques. Catena, 2020, 187: 104408

[27] Oommen T, Baise L G, Vogel R. Validation and application of empirical liquefaction models. Journal of Geotechnical and Geoenvironmental Engineering, 2010, 136(12): 1618–1633

[28] Bikmukhametov T, Jäschke J. Combining machine learning and process engineering physics towards enhanced accuracy and explainability of data-driven models. Computers & Chemical Engineering, 2020, 138: 106834

[29] Merghadi A, Yunus A P, Dou J, Whiteley J, ThaiPham B, Bui D T, Avtar R, Abderrahmane B. Machine learning methods for landslide susceptibility studies: A comparative overview of algorithm performance. Earth-Science Reviews, 2020, 207: 103225

[30] Smiti A. When machine learning meets medical world: Current status and future challenges. Computer Science Review, 2020, 37: 100280

[31] Sheil B B, Suryasentana S K, Cheng W C. Assessment of anomaly detection methods applied to microtunneling. Journal of Geotechnical and Geoenvironmental Engineering, 2020, 146(9): 04020094

[32] Zhang N, Shen S L, Zhou A, Jin Y F. Application of LSTM approach for modelling stress–strain behaviour of soil. Applied Soft Computing, 2021, 100: 106959

[33] Samui P. Slope stability analysis: A support vector machine approach. Environmental Geology (Berlin), 2008, 56(2): 255–267

[34] Garg A, Garg A, Tai K, Sreedeep S. An integrated SRM-multi-gene genetic programming approach for prediction of factor of safety of 3-D soil nailed slopes. Engineering Applications of Artificial Intelligence, 2014, 30: 30–40

[35] Koopialipoor M, Jahed Armaghani D, Hedayat A, Marto A, Gordan B. Applying various hybrid intelligent systems to evaluate and predict slope stability under static and dynamic conditions. Soft Computing, 2019, 23(14): 5913–5929

[36] Ferentinou M, Fakir M. Integrating Rock Engineering Systems device and artificial neural networks to predict stability conditions in an open pit. Engineering Geology, 2018, 246: 293–309

[37] Zhou J, Li E, Yang S, Wang M, Shi X, Yao S, Mitri H S. Slope stability prediction for circular mode failure using gradient boosting machine approach based on an updated database of case histories. Safety Science, 2019, 118: 505–518

[38] Huang F M, Cao Z, Guo J, Jiang S H, Li S, Guo Z Z. Comparisons of heuristic, general statistical and machine learning models for landslide susceptibility prediction and mapping. Catena, 2020, 191: 104580

[39] Wang Y K, Huang J S, Tang H M, Zeng C. Bayesian back analysis of landslides considering slip surface uncertainty. Landslides, 2020, 17(9): 2125–2136

[40] Wei X, Zhang L L, Yang H Q, Zhang L M, Yao Y P. Machine learning for pore-water pressure time-series prediction: Application of recurrent neural networks. Geoscience Frontiers, 2021, 12(1): 453–467

[41] Safa M, Sari P A, Shariati M, Suhatril M, Trung N T, Wakil K, Khorami M. Development of neuro-fuzzy and neuro-bee predictive models for prediction of the safety factor of eco-protection slopes. Physica A: Statistical Mechanics and its Applications, 2020, 550: 124046

[42] Suykens J A K, Vandewalle J. Least squares support vector machine classifiers. Neural Processing Letters, 1999, 9(3): 293–300

[43] SuykensJ A KGestelT VBrabanter J DMoorB DVandewalleJ. Least Squares Support Vector Machines. Singapore: World Scientific, 2002

[44] Duncan J M. State of the art: Limit equilibrium and finite-element analysis of slopes. Journal of Geotechnical Engineering, 1996, 122(7): 577–596

[45] Li S G. A reflection on the Mohr failure criterion. Mechanics of Materials, 2020, 148: 103442

[46] JohnstonI W. Strength of intact geomechanical materials. Journal of Geotechnical Engineering, 1985, 111(6): 730–749

[47] Zhang X J, Chen W F. Stability analysis of slopes with general nonlinear failure criterion. International Journal for Numerical and Analytical Methods in Geomechanics, 1987, 11(1): 33–50

[48] Drescher A, Christopoulos C. Limit analysis slope stability with nonlinear yield condition. International Journal for Numerical and Analytical Methods in Geomechanics, 1988, 12(3): 341–345

[49] Jiang J C, Baker R, Yamagami T. The effect of strength envelope nonlinearity on slope stability computations. Canadian Geotechnical Journal, 2003, 40(2): 308–325

[50] Yang X L, Yin J H. Slope stability analysis with nonlinear failure criterion. Journal of Engineering Mechanics, 2004, 130(3): 267–273

[51] Rafiai H, Jafari A, Mahmoudi A. Application of ANN-based failure criteria to rocks under polyaxial stress conditions. International Journal of Rock Mechanics and Mining Sciences, 2013, 59: 42–49

[52] Wu S, Zhang S, Guo C, Xiong L. A generalized nonlinear failure criterion for frictional materials. Acta Geotechnica, 2017, 12(6): 1353–1371

[53] Xue X H, Xiao M. Deformation evaluation on surrounding rocks of underground caverns based on PSO-LSSVM. Tunnelling and Underground Space Technology, 2017, 69: 171–181

[54] Barzegar R, Ghasri M, Qi Z, Quilty J, Adamowski J. Using bootstrap ELM and LSSVM models to estimate river ice thickness in the mackenzie river basin in the northwest territories, Canada. Journal of Hydrology (Amsterdam), 2019, 577: 123903

[55] Adnan R M, Liang Z, Heddam S, Zounemat-Kermani M, Kisi O, Li B. Least square support vector machine and multivariate adaptive regression splines for streamflow prediction in mountainous basin using hydro-meteorological data as inputs. Journal of Hydrology (Amsterdam), 2020, 586: 124371

[56] Wei J, Chen H. Determining the number of factors in approximate factor models by twice K-fold cross validation. Economics Letters, 2020, 191: 109149

[57] Chandler R J. Lias clay: The long-term stability of cutting slopes. Geotechnique, 1974, 24(1): 21–38

[58] MATLAB. Version 9.4.0. Natick, MA: The Mathworks, Inc., 2018

[59] Perry J. A technique for defining non-linear shear strength envelopes, and their incorporation in a slope stability method of analysis. Quarterly Journal of Engineering Geology and Hydrogeology, 1994, 27(3): 231–241

[60] Tordesillas A, Zhou Z Z, Batterham R. A data-driven complex systems approach to early prediction of landslides. Mechanics Research Communications, 2018, 92: 137–141

[61] Zhou X P, Huang X C, Zhao X F. Optimization of the critical slip surface of three-dimensional slope by using an improved genetic algorithm. International Journal of Geomechanics, 2020, 20(8): 04020120

[62] Xiao S G, Guo W D, Zeng J X. Factor of safety of slope stability from deformation energy. Canadian Geotechnical Journal, 2018, 55(2): 296–302

[63] VapnikV N. Statistical Learning Theory. New York: Wiley-Interscience, 1998

[64] BrabanterK DKarsmakersPOjedaF AlzateCBrabanter J DPelckmansKMoorB DVandewalle JSuykensJ A K. LS-SVMlab Toolbox User’s Guide. Heverlee: Katholieke Universiteit Leuven, 2010

RIGHTS & PERMISSIONS

Higher Education Press 2022