Rock slope stability analyses using extreme learning neural network and terminal steepest descent algorithm

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Highlights

  • Extreme learning algorithm is used to train rock slope stability evaluation system.

  • Terminal steepest descent algorithm (TSDA) based on finite-time theory is adopted.

  • The developed package can either estimate slope stability or perform back analyses.

  • The errors converge to zero which makes analyses more accurate and time-effective.

  • Multiple parameters can be optimized simultaneously during the back analysis.

Abstract

The analysis of rock slope stability is a classical problem for geotechnical engineers. However, for practicing engineers, proper software is not usually user friendly, and additional resources capable of providing information useful for decision-making are required. This study developed a convenient tool that can provide a prompt assessment of rock slope stability. A nonlinear input–output mapping of the rock slope system was constructed using a neural network trained by an extreme learning algorithm. The training data was obtained by using finite element upper and lower bound limit analysis methods. The newly developed techniques in this study can either estimate the factor of safety for a rock slope or obtain the implicit parameters through back analyses. Back analysis parameter identification was performed using a terminal steepest descent algorithm based on the finite-time stability theory. This algorithm not only guarantees finite-time error convergence but also achieves exact zero convergence, unlike the conventional steepest descent algorithm in which the training error never reaches zero.

Introduction

It is often difficult to measure the strength of rock masses accurately because it is variable. Features of rock masses generally include joints, faults, and naturally occurring discontinuities and anisotropies. These features result in difficult analyses using simple theoretical solutions, such as the limit equilibrium method (LEM). Although many researchers have investigated the assessment of the stability of rock slopes [1], [2], [3], [4], an accurate assessment continues to pose a major challenge to geotechnical engineers.

Although it has been known that the failure envelope of rock masses is nonlinear [5], [6], [7], the conventional linear Mohr–Coulomb criterion has been widely used. It would be due to the fact that most computer programs allow only the conventional Mohr–Coulomb strength parameters, cohesion and friction angle, to be used as inputs in slope stability analyses. In fact, the nonlinearity is more pronounced at low confining stresses, which exist in slope stability problems [8]. The studies of Li et al. [9], [10] have shown that a linear failure criterion is not suitable for assessing rock slope stability. Therefore, the application of a nonlinear failure criterion, such as that proposed by Hoek [5], is necessary.

As discussed by Merifield et al. [11], the Hoek–Brown failure criterion is one of the few nonlinear criteria used by practicing engineers for estimating rock mass strength. In the current study, this failure criterion was adopted for determining the failure envelope of rock masses on slopes. Hoek et al. [12] expressed the latest version of the Hoek–Brown failure criterion as:σ1=σ3+σcimbσ3σci+sα,wheremb=miexpGSI1002814D,s=expGSI10093D,α=12+16eGSI15e203.

The magnitudes of mb, s, and α depend on the geological strength index (GSI), which indicates the rock mass quality and ranges between 5 and 100. GSI was introduced to estimate the rock mass strength for different geological conditions because Bieniawski's rock mass rating system [13] and the Q-system [14] had been found to be unsuitable for poor rock masses. As indicated by Hoek and Brown [15], GSI can be adjusted for incorporating the effects of surface weathering. The variables σci and mi represent the intact uniaxial compressive strength and material constant, respectively. The disturbance factor D, which ranges between 0 and 1, may result from the slope cutting process, for which there are several methods. Additional details on estimating the Hoek–Brown strength parameters have been presented by Wyllie and Mah [16] and Marinos et al. [17].

Li et al. [18] showed that the evaluation of the factor of safety for rock slopes can differ considerably if the rock mass disturbance is considered, particularly for cases with a low GSI. In addition, Hoek et al. [12] indicated that the disturbance factor should be determined with caution. The importance of estimating D is therefore evident. For evaluations of D, some recommended magnitudes can be found in the study of Hoek et al. [12]; these magnitudes may serve as starting points for the initial assessment. These authors also stated that a more accurate estimate of the disturbance factor can be obtained by using field observations or measurements.

As highlighted by Burland [19], some of the geotechnical parameters used in the analysis may not be accurately measured in laboratory tests because of the effects of sample disturbance and errors of tests. Therefore, back analysis or the observational method, as suggested by Peck [20], is often applied to determine representative and/or dominant soil parameters based on actual field observations. To obtain more accurate assessments of the rock mass strength parameters, an artificial neural network (ANN) [21], [22], [23] is applied to back calculate failed rock slopes on the basis of the Hoek–Brown failure criterion.

Section snippets

Limit equilibrium analysis

In general, LEMs such as those proposed by Bishop [24] and Janbu [25] are the most widely used methods for estimating slope stability. Because of their simplicity, these methods are also used to obtain the uncertain parameters during slope failure [26], [27]. In addition, the stability charts proposed by Hoek and Bray [1] can be used by practicing engineers to back analyze rock slopes. These chart solutions contain information on the water table and are suitable for the analysis of uniform rock

Finite element upper and lower bound limit analysis methods

To train the back analysis tool, the finite element upper and lower bound limit analysis (LA) methods developed by Lyamin and Sloan [48], [49] and Krabbenhoft et al. [50] were employed. These techniques can be used to bracket the true stability solutions for geotechnical problems [51], [52], [53], [54] and are suitable for both linear and nonlinear failure criteria. Based on the Hoek–Brown failure criterion [12], these techniques have been used to assess rock slope stability and their

ANN training and validation

In this study, as mentioned in Section 3.1, β, GSI, mi, and D were chosen as the inputs and Nr was the output of the ANN. Thus, the ANN had four inputs and one output. For the training, 2094 training data were randomly selected. Each data was an average stability number generated by the numerical upper and lower bound LA methods. The ranges considered were 15°  β  75°, 10  GSI  90, 5  mi  35, and 0  D  1. The solutions were randomly arranged as inputs before the ANN training. Only few hours were

Single uncertain parameter

A series of case studies of rock slopes in mines was presented by Douglas [4], and they were considered in the current study to verify the developed techniques. Table 1 lists the strength parameters and geometries for 12 cases. Information regarding the unit weight (γ) and disturbance factor (D) was not available in the study of Douglas [4]. Therefore, the unit weight of the rock mass was assumed to be fairly close to 27 kN/m3 on the basis of the study of Rzhevsky and Novik [62]. Because F and γ

Conclusion

This study used an ANN and the TSDA to develop a fast evaluation package for rock slope stability analyses based on the Hoek–Brown failure criterion [12]. The package was trained using solutions obtained from the finite element upper and lower bound LA methods. In this study, various functions of the developed package were investigated and verified by using case studies from different countries.

The results obtained demonstrate that this package can provide a direct estimate of the slope factor

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