An integrated SRM-multi-gene genetic programming approach for prediction of factor of safety of 3-D soil nailed slopes
Introduction
Soil nailing is one of the economic slope stabilisation technique and is particularly useful for the strengthening of existing slopes. The principle behind soil nailing is to reinforce the soil with passive inclusions which increases the overall shear strength of the soil slope and also restrains its displacements. There are many different design methods for soil nailing, which includes limit equilibrium method (LEM), strength reduction method, several working stress design methods, the Davis method (Shen et al., 1981), the German method (Stocker et al., 1979) and the French method (Schlosser, 1982, Schlosser, 1991). Among them, LEM is commonly used for slope stability analysis by many researchers (Donald and Giam, 1988, Matsui and San, 1992, Griffiths and Lane, 1999, Ugai and Leshchinsky, 1995, Song, 1997, Dawson et al., 1999, Cheng et al., 2007, Cheng et al., 2008). The LEM analysis used in nailed slope analysis is based essentially on Spencer's method, where the effect of a soil nail is considered by applying a concentrated load provided by the nail on the slip surface. FOS for slopes using LEM methods are computed using 2-D or 3-D Finite element method (FEM) or Finite difference method (FDM) is generally used to estimate FOS of nailed slopes (Shen et al., 1981, Juran and Elias,, Plumelle et al., 1990, Thompson and Miller, 1990, Srinivasa Murthy et al., 2002, Sivakumar Babu et al., 2002, Wei and Cheng, 2010).
On the other hand, in recent years use of soft computing methods such as artificial neural network (ANN) is becoming increasingly important in geotechnical engineering (Shahin et al., 2008). Many ANN models have been developed to estimate FOS of slopes with different slope geometry and soil conditions (Li and Liu, 2004, Cho, 2009, Lin et al., 2009, Choobbasti et al., 2009, Li and Wang, 2010, Abdalla et al., 2012). However, all those models did not take into account any effect of nails on FOS of slopes. In addition to this, most of these developed models are based on 2-D boundary conditions without incorporating 3-D effects which are more realistic in nature and is essential for soil nail problems. Since, Wei and Cheng (2010) showed that FOS of nailed slope is greatly affected by cohesion, frictional angle, nail inclination angle, nail length, slope height and slope angle of 3-D nailed slope. These parameters must be taken into account while developing any models using soft computing methods.
Other soft computing methods such as support vector regression (SVR) and genetic programming (GP) can also be used to predict the FOS for the different soil properties. Among these methods, GP possesses the ability to evolve models structure and its coefficients automatically (Cevik and Guzelbey, 2007, Cevik and Sonebi, 2008, Gandomi et al., 2010). Most popular variant of GP used recently is multi-gene genetic programming (MGGP) (Gandomi and Alavi, 2011, Garg and Tai, 2012, Garg and Tai, 2013, Garg et al., 2013a). Despite large applications of MGGP in field of structural and civil engineering, MGGP has limitation for producing models that over-fit on the testing data. The reason can be attributed to the large size models produced during the evolutionary stage in MGGP that captures noise along with relationships from the training data. Over-fitting in MGGP is the popular problem among researchers and have been paid less attention (Chan et al., 2011, Gonçalves et al., 2012, Garg et al., 2013b).
Therefore, in the present work, SRM-based-multi-gene genetic programming approach (SRM-MGGP) is proposed for predicting FOS for 3-D nailed slopes using the results from the LEM analysis. FOS under different six input parameters such as cohesion, frictional angle, nail inclination angle, nail length, slope height and slope angle of 3-D nailed slope were used to train and test newly developed SRM-MGGP model. Unlike standard GP, each model participating in SRM-MGGP approach is a set of combination of genes. Structural risk minimisation (SRM) principle is integrated to improve the generalisation ability of the models during the evolutionary stage of the method. The performance of the proposed SRM-MGGP method is compared to that of the other three potential methods: standardized MGGP, SVR and ANN.
Section snippets
Numerical method for estimating FOS
For model development, FOS data for nailed slopes was selected from a comprehensive study conducted by Wei and Cheng (2010). A parametric study was conducted to investigate FOS using LEM analysis under different combinations of cohesion, frictional angle, nail inclination angle, nail length, slope height and slope angle. In their LEM analysis, only the tensile strength and pull-out capacity of the nail were considered. The tensile forces mobilised in the nails were divided into tangential and
Multi-gene genetic programming
In order to understand the concepts of MGGP method, the basics of GP is first discussed in brief.
GP generates computer programs/models automatically based on the given data using Darwinian principle of “Survival of the fittest” (Koza, 1996). Working principle of GP is same as GA but the only difference between them, is that, GA evolves solutions represented by strings (binary or real number) of fixed length, whereas GP generates solutions represented by tree structures of varying sizes (Garg et
Models evaluation and comparison
The results obtained from the four prediction modelling methods SRM-MGGP, MGGP, SVR and ANN are illustrated in Fig. 9, Fig. 10, Fig. 11, Fig. 12 on training and testing data respectively. The best prediction method that gives highest accuracy is determined by comparing these four modelling methods. The correlation coefficient (R) and relative error (%) between the predicted values and the actual values of the FOS are estimated. The correlation coefficient (R) and relative error (%) are given by:
Conclusion and future work
This study presents an alternative approach using four soft computing methods to investigate FOS of 3-D nailed slope based on certain limitations such as ignoring effects of pore water pressure and overburden pressure. Problem of over-fitting in soft computing method, namely, MGGP has been discussed. To overcome this, a SRM-MGGP methodology is proposed for the prediction of FOS of the 3-D nailed slope. The performance of the proposed SRM-MGGP model is compared to those of MGGP, SVR and ANN
Acknowledgement
This work was partially supported by the Singapore Ministry of Education Academic Research Fund through Research Grant RG30/10, which the authors gratefully acknowledge.
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