New artif i cial neural networks for true triaxial stress state analysis and demonstration of intermediate principal stress effects on intact rock strength

Rennie Kaunda

Mining Engineering Department,Colorado School of Mines,Golden,CO,USA

New artif i cial neural networks for true triaxial stress state analysis and demonstration of intermediate principal stress effects on intact rock strength

Rennie Kaunda

Mining Engineering Department,Colorado School of Mines,Golden,CO,USA

A R T I C L EI N F O

Article history:

Received 13 November 2013

Received in revised form

22 April 2014

Accepted 8 May 2014

Available online 24 June 2014

Artif i cial neural networks

Polyaxial loading

Intermediate principal stress

Rock failure criteria

True triaxial test

Simulations are conducted using f i ve new artif i cial neural networks developed herein to demonstrate and investigate the behavior of rock material under polyaxial loading.The effects of the intermediate principal stress on the intact rock strength are investigated and compared with laboratory results from the literature.To normalize differences in laboratory testing conditions,the stress state is used as the objective parameter in the artif i cial neural network model predictions.The variations of major principal stress of rock material with intermediate principal stress,minor principal stress and stress state are investigated.The artif i cial neural network simulations show that for the rock types examined,none were independent of intermediate principal stress effects.In addition,the results of the artif i cial neural network models,in general agreement with observations made by others,show(a)a general trend of strength increasing and reaching a peak at some intermediate stress state factor,followed by a decline in strength for most rock types;(b)a post-peak strength behavior dependent on the minor principal stress, with respect to rock type;(c)sensitivity to the stress state,and to the interaction between the stress state and uniaxial compressive strength of the test data by the artif i cial neural networks models(two-way analysis of variance;95%conf i dence interval).Artif i cial neural network modeling,a self-learning approach to polyaxial stress simulation,can thus complement the commonly observed diff i cult task of conducting true triaxial laboratory tests,and/or other methods that attempt to improve two-dimensional (2D)failure criteria by incorporating intermediate principal stress effects.

©2014 Institute of Rock and Soil Mechanics,Chinese Academy of Sciences.Production and hosting by Elsevier B.V.All rights reserved.

1.Introduction

The case for the signif i cance of the intermediate principal stress, σ2,to rock brittle fracture and rock strength,has been historically well established(Murrell,1963;Mogi,1971;Takahashi and Koide, 1989;Haimson and Chang,2000;Malama,2001;Colmenares and Zoback,2002;Haimson and Rudnicki,2010).A major challenge, however,is that understandably scarce polyaxial laboratory data were obtained from“true triaxial”tests(σ2≠σ3,σ3represents minor principal stress),as opposed to conventional triaxial tests (σ2=σ3)to buttress experimental,analytical or computer models (e.g.Kim and Lade,1984;Christensen et al.,2004;Pan et al.,2012). Conducting true triaxial tests is not trivial,and test machinery capable of independently incorporating all three principal stresses is complicated to be designed.The shortage of data thus makes it a challenge to undertake comprehensive studies to enhance understanding of the true nature of rock failure/strength,as a means of substantiating theoretical models.

To compensate for these def i ciencies,some efforts have been redirected to place emphasis on the effect of only the relationship of the major principal stress,σ1,and the minor principal stress,σ3, on rock strength as for example evidenced by a plethora of twodimensional(2D)rock strength criteria in the literature.However, evidence has been accumulating that the role of the intermediate principal stress,σ2,in rock fracture and/or rock strength can neither be trivialized nor ignored(Haimson,2006).For example,Murrell (1963)demonstrated that Carrara marble is stronger under triaxial extension(σ2=σ1)than under triaxial compression (σ2=σ3),i.e.conventional triaxial tests in compression,in which the intermediate principal stress is equal to the minor principalstress,do not lead to a general failure criterion.Handin et al.(1967) showed thatσ2caused the angle between the failure plane at brittle fracture and the direction ofσ1to decrease between triaxial compressionandtriaxialextensioninSolnhofenlimestone. Wiebols and Cook(1968)developed a failure criterion,based on effective strain energy,to show that,for a constant value ofσ3,the strength asσ2is raised from its initial value ofσ2=σ3to where it reaches a peak and then declines toσ2=σ1(Fig.1).Mogi’s experiments on carbonates and silicates(Mogi,1971)demonstrated that the largest effect ofσ2on strength is reached at a level well inside the range betweenσ2=σ3andσ2=σ1.The work on sandstones and shales by Takahashi and Koide(1989)showed that these rock strengths were not only dependent on the absolute value ofσ2,but also dependent on the relative value ofσ2.On the other hand,Cai (2008)demonstrated numerically that little strength increase occurred in rock whenσ2was increased substantially at low values ofσ3.Chang and Haimson(2000)showed that the increase in strength as a function ofσ2for constantσ3is substantial,and in some cases,as much as 50%or more over the commonly used conventional triaxial strength,and that higher intermediate principal stress magnitudes appeared to extend the elastic range of the stress-strain behavior for a givenσ3,thereby retarding the onset of the failure process.Perhaps the mixed effects of the intermediate principal stress were best highlighted by Chang and Haimson (2005)who indicated that,for certain rock types(e.g.hornfels or metapelite),compressive strengthσ1does not vary signif i cantly regardless of the appliedσ2after all.

Interestingly,some more recent studies(Haimson and Rudnicki, 2010;Ma and Rodriguez,2012)proposedsymmetricalfailure envelopes with respect to Mogi’s stressfactor,β(Mogi,1971),different from classical three-dimensional(3D)failure envelopes(such as thosedepictedinFig.1).Thestressfactor,β,def i nedas β=(σ2-σ3)/(σ1-σ3),ranges from 0(σ2=σ3)to 1(σ2=σ1),as shown in Fig.2.In other words,the symmetrical failure envelopes imply that the rock strength at triaxial extension(σ2=σ1)can be equal to the strength at triaxial compression(σ2=σ3).This observation,contradicting prevailing understanding(e.g.Drucker and Prager,1952;Murrell,1963;Wiebols and Cook,1968;Lade and Duncan,1973)that rock strength is always higher at triaxial extension than at triaxial compression,implies that the symmetry imposed by the stress factor could affect the range of applicability of some 3D rock failure criteria commonly used in rock engineering (Ma and Rodriguez,2012).

In terms of mechanism,some studies attribute intermediate principal stress effects to extended evolution of localized deformation that ultimately needs signif i cant additional strain for failure (Haimson and Rudnicki,2010),inhomogeneous distribution of localized shear strains in shear bands with respective localized stresses(Christensen et al.,2004),and 3D interaction of microcracks prior to shear failure(Healy et al.,2006).

The challenge remains,therefore,to enhance current understanding on the effects of the intermediate principal stress on brittle/ductile rock behavior through more laboratory testing.It is therefore imperative to explore/develop new avenues to complement laboratory experiments,such as analytical approaches or computer modeling techniques(Kim and Lade,1984;Christensen et al.,2004;Pan et al.,2012).

In summary,the main objectives of this study are:

(1)To develop new artif i cial neural network(ANN)models which predict stress state factors,β,from laboratory tests for several rock types.The stress state factor,β,def i ned above was selected as the objective parameter,because it not only allows one to normalize the inf l uence ofσ2(Smart et al.,1999; Alexeev et al.,2008;Zhang et al.,2010)on rock strength,but also enables one to equally treat reported laboratory tests subjected to different testing modes or stress states(Ma and Rodriguez,2012).

(2)To show that the output from the new ANN models can act as tools to investigate/substantiate the major effects ofσ2on rock strength discussed above,i.e.:

(a)The characteristic that asσ2is raised fromσ2=σ3toσ2=σ1, the strengthσ1for a constantσ3f i rst increases,reaches a maximum at some intermediate value ofσ2,and then decreases to a value greater than the conventional triaxial equivalent value whenσ2=σ1.

(b)In certain cases there is no clear trend toward an eventual decrease in strength at higherσ2.

(c)The observation that in certain cases a steady state is reached when the level of conf i ning stressσ3is nearly equalto the uniaxial compressive strength(UCS)of the intact rock.

Fig.1.The classical Wiebols and Cook curves illustrating the effects of the intermediate principal stressσ2on true triaxial rock strength(after Haimson,2006).c0is the uniaxial compressive strength.

Fig.2.Characteristic symmetry imposed by Mogi’s stress factor,β,on true triaxial rock strength.Note that in this plot the rock strength at triaxial extension can be equal to the strength at triaxial compression due to symmetry(modif i ed after Ma and Rodriguez,2012).

(d)The observation that for some rock types the intermediate principal stress hardly affects strength at certain values of σ3.

(e)The suggestion that the stress factorβimposes symmetry on strength leading to rock strength at triaxial extension being equal to the strength at triaxial compression.

2.How artif i cial neural networks work

The theoretical background for ANNs can be found in several sources(e.g.Wasserman,1989;Bishop,1995;Nielsen,1998; Haykin,1999;Gurney,2009).Unlike other modeling techniques which may produce highly subjective results inf l uenced by prior user assumptions,ANNs learn complex variable/phenomena relationships of a system independent of such prejudice.The fundamental basis for this independence is the ability of ANNs to“learn”the behavior of a system via sets of connection weights modeled after biological neurons of the human nervous system.Their advantage over other approaches is that they can be extremely useful when the exact relationship between system parameters is little understood,making them highly practical to study intermediate principal stress effects.

An ANN consists of a mesh of computing nodes and connections (Fig.3),as the basic processing elements(PEs)which can be trained to map data nonlinearly once they are activated(“turned on”).Each connection is assigned a numericalvalue,known as aweight,which can be changed during neural network training using several learning algorithm options,such as gradient descent method via back propagation(BP)(Rumelhart et al.,1986)summarized as

whereΔwjiis the change in connection weights between hidden (jth)and input(ith)layers,δpjis the derivative of error output with respect to sum between input and hidden layers,xpiis the value received by each input node,Δwkjis the change in connection weights between hidden(kth)and output(jth)layers,ηis the proportion of change inweight to error,δpkis the derivative of error output with respect to the sum between hidden and output layers, andactpjis the activation parameter of the nodes in the output layer.

A search for the global optimum of connection weights that yields the result with minimum error is conducted.For each iteration,the accuracy of an ANN can be assessed by the mean squared difference between actual and predicted or output values (root mean squared error,RMSE):

Fig.3.An example of a feed-forward artif i cial neural network architecture showing processing elements(neurons)in three different layers connected by a mesh of connection weights.

whereTis the total number of presented data,anderroris the difference between the actual and target values.

In supervised ANN learning,input/output pairs must be provided via a training f i le before the model can be tested or utilized. Inputs are factors inf l uencing the target output.When selecting input/out pairs,it is necessary to have some insight into the system dynamics of the problem to set constraints or boundaries for the problem.An input/output suite in the training set constitutes a training pattern.There is a f i xed number of training patterns in the training f i le.The training f i le is developed such that the patterns are statistically similar(i.e.mean,standard deviation and range)to the actual system being modeled.

The ability to learn complex relationships among supplied data sets,and then apply this knowledge to a fresh dataset is a major advantageous feature of ANN over purely analytical/empirical models.As explained above,and because information about the physical parameters of the system is not required,an ANN does not rely upon the physical laws of the system it is modeling. However,being completely dependent on data sets is in a sense a big disadvantage of ANNs,because any errors present in the data are inherited by the ANN model possibly leading ANN models to be in contradiction to physical laws and reality.It is therefore imperative to understand the source and limits of available data sets.

In summary,the steps for supervised ANN learning and testing are:

(1)Select and prepare input and output parameters from raw data and create a training f i le.

(2)Create ANN model architecture.

(3)Input training f i le into the model.

(4)Initiate training by implementing the BP algorithm.

(5)Continue ANN training until errors are minimized.

(6)Test model on new data for validation.

(7)Ref i ne model if necessary.

(8)Model ready for use.

Despite their usefulness in many situations,ANNs have been found to be limited in the amount of extractable knowledge they provide concerning their mechanics,limited in their ability to generalize/extrapolate beyond the range of the data used for model training,and limited in dealing with data uncertainty(Shahin et al., 2008).

3.ANNs in rock mechanics

The application of ANNs to rock mechanics is not new.For example,Sirat and Talbot(2001)used ANNs to recognize,classify and predict patterns of different fracture sets in the top 450 m in crystalline rocks at the Äspö Hard Rock Laboratory(HRL),Southeastern Sweden.Using two hidden layers with tan-sigmoid and linear transfer functions,a series of trials were carried out using BP neural networks for supervised classif i cation,and the BP networks recognized different fracture sets accurately.Sonmez et al.(2006) constructed ANNs to prepare a chart for a generalized prediction of the elastic modulus of intact rock using a large database including UCS,unit weight and modulus of elastic of intact rock(Ei). Mohammadi and Rahmannejad(2010)used ANNs to obtain amodel for estimating rock mass deformation modulus based on the radial basis function(RBF).The model displayed high accuracy levels when compared to in-situ tests from the elastic modulus of Karun IV dam.Majdi and Beiki(2010)used a genetic algorithm to optimize the architecture and heuristics of a BP ANN for predicting the deformation modulus of rock masses.Using a database obtained from four dam sites and powerhouses,the superiority of the ANN technique in comparison to typical regression methods was demonstrated.Beiki et al.(2010)employed an ANN as a tool for conducting a parametric study to determine the sensitivity of the rock mass deformation modulus to the modulus of elasticity of intact rock,UCS,rock mass quality designation,joint frequency, porosity,dry density,and geological strength index(GSI).Raf i ai and Jafari(2011)trained ANNs to predict the value of major principal stress at failure from uniaxial compressive stress and minor principal stress.They found that on average,for different rock types, using ANN models led to about 30%decrease in prediction error relative to state-of-the-art empirical models.

Table 1Range and distributions of laboratory test data used to develop and test the ANN models.

4.ANN heuristic guidelines

The ANN modeling reported herein utilized guidelines from the literature to select appropriate parameters used in the models. Prior to model construction,real fi eld or laboratory data are required to train and validate the models.Hammerstrom(1993) recommended using two-thirds of the overall dataset for model training and one-third for validation.The statistical properties(e.g. mean and standard deviation)of the training and validation data need to be similar to ensure that each subset represents the same statistical population(Masters,1993).In addition,ANN datasets should be preprocessed to ensure that all variables receive equal attention during the training process(Maier and Dandy,2000). Although several preprocessing techniques are reported in the literature,application of a scale factorappears tobe popular.For the fi nal output data,scaling is essential,as the data have to be commensurate with the limits of the transfer functions used in the output layer(e.g.-1.0 to 1.0 for the hyperbolic transfer function and 0.0-1.0 for the sigmoid transfer function)(Shahin et al.,2008). Although scaling the input data is not critical,it is almost always recommended(Masters,1993).

An additional important heuristic in ANN architecture is the number of hidden layers,plus theirassociated number of nodes(i.e. processing elements).Although there is substantial debate in the literature,it has been shown that one hidden layer is suf fi cient to approximate any continuous function provided that suf fi cient connectionweights/processingelementsaregiven,andthe appropriate activation function is used(Cybenko,1989;Hornik et al.,1989;Bishop,1995).A popular choice for a widely applicable activation function is the sigmoid(logistic)function:y=1/ (1+e-x)(Nielsen,1998).The problem with ANNs containing too many free parameters(i.e.connection weights)is that these ANNS are more subjected to over fi tting of the data and poor generalization(Maren et al.,1990;Masters,1993;Rojas,1996).Shahin et al. (2008)reported that keeping the number of hidden nodes to a minimum:(a)reduces the computational time needed for training; (b)helps the network achieve better generalization performance; (c)helps to avoid the problem of over f i tting;and(d)allows one to analyze the trained network more easily.

For single hidden layer networks,the optimal number of nodes may be obtained using several guidelines(e.g.Hecht-Nielsen,1987; Caudill,1989;Berke and Hajela,1991;Salchenberger et al.,1992). Perhaps the most eff i cient approach,espoused by Nawari et al. (1999),is to commence with a small number of nodes and iteratively increase that number until no signif i cant improvement in model performance is achieved.

The main criteria used toevaluate the predictionperformance of ANN models are the coeff i cient of correlation,R2,RMSE,and the mean absolute error,MAE,between the predicted and observed data(Shahin et al.,2008).Smith(1986)suggested the following guide for values of|R|between 0.0 and 1.0:

(1)|R|≥0.8:strong correlation exists between two sets of variables;

(2)0.2＜|R|＜0.8:correlation exists between the two sets of variables;and

(3)|R|≤0.2:weak correlation exists between the two sets of variables.

TheRMSEis the most popular measure of error and has the advantage that large errors receive much greater attention than small errors(Hecht-Nielsen,1990).In contrast withRMSE,MAEeliminates the emphasis given to large errors.However,bothRMSEandMAEare desirable when the evaluated output data are smooth or continuous(Twomey and Smith,1997).

5.Training and test data

As stated in Section 1,true triaxial laboratory data are extremely scarce due to the somewhat cumbersome nature of the test.In this study,to help validate the ANN models,published rock strength data spanning uniaxial,conventional triaxial and true triaxial stress states from awide variety of sources were utilized(Colmenares and Zoback,2002;Al-Ajmi and Zimmerman,2005;Zhang,2008).The rock strength data,selected to be as widely representative aspossible of various failure envelopes,included KTB amphibolite (KTBA)(Chang and Haimson,2000),Westerly granite(WG) (Haimson and Chang,2000),Dunham dolomite(DD)(Mogi,1971), Mizuho trachyte(MT)(Mogi,1971),and Solnhofen limestone(SL) (Colmenares and Zoback,2002).A summary of the statistics for the rock strength data used in the neural network training is provided in Table 1 to show their distribution and range.

Table 2Summary of ANN model parameters.

Table 3Summary of processing element coeff i cients for the f i ve ANN models after training.

Fig.4.Summary of the ANN model test results for f i ve different rock types:(a)DD;(b)KTBA;(c)MT;(d)SL;and(e)WG.Note that only 20%of original data were shown for each case because approximately 80%of the data were used to develop and validate each neural network model.

Fig.5.Comparison of ANN model predictions,modif i ed Mohr-Coulomb predictions(Singh et al.,2011),and experimental data for f i ve different rock types:(a)DD;(b)KTBA;(c) MT;(d)SL;and(e)WG.Note that only 20%of original data were shown for each case because approximately 80%of the data were used to develop and validate each neural network model.

6.Implementation and application

In this study,the data sets described in Section 5 were divided into f i ve classes by rock type,i.e.KTBA,WG,DD,MT and SL. Henceforth these data sets are referred to by their initials(e.g.DD ANN model/DD data means the Dunham dolomite training data set was used to create the neural network model and was tested onDunham dolomite test data set).For each rock type,approximately 80%of the data were used to develop the ANN model via training, while approximately 20%were used to test the model as discussed in Sections 2 and 5.The data used to develop each model were further subdivided into training data and validation data sets at an approximate ratio of 4:1 respectively,using a random function to prevent bias.Upon each passing through the training sets,models were evaluated with the validation sets.

The analyses were conducted using the BP algorithm as discussed in Section 2.Five separate ANN models were created by rock type,i.e.KTBA-ANN,WG-ANN,DD-ANN,MT-ANN,and SL-ANN using 80%of the data from each group.The rest of the data (about 20%for each group)were reserved for testing each ANN model.The ANNs were trained with two network inputs(σ2/UCSandσ3/UCS)scaled in the range[0,1],and one output parameter: stress state factor,β,de fi ned in Section 1 and also scaled[0,1]via a sigmoidal function.To avoid over-learning during ANN training,the early stopping technique(i.e.gauging stopping point with no further improvement on model performance using the validation sets)was applied to improving the generalization of the ANNs.The architecture was varied with different numbers of hidden neurons to obtain the optimum solution,and the initial weights and biases were randomly generated.For each ANN,the learning rate,momentum,stopping error criteria,and maximum learning cycle were set at 0.7,0.8,0.01 and 4000 respectively via trial and error.The rest of the training parameters used are summarized in Table 2.The fi nal coef fi cients of each processing node in the ANN models at the end of training(determined via the BP algorithm)are shown in Table 3.After the models were trained and validated,they were tested within their respective rock groups using 20%of the original data to gauge their performance and ability to generalize.

The ANN test results are summarized in Fig.4.When compared with the performance of recent3D failure criteriawhich account for polyaxial strength such as the modi fi ed Mohr-Coulomb(MC) model(Singh et al.,2011),the ANN models measure up favorably as indicated in Fig.5.The results are discussed in detail in Section 7.

Table 4Summary of power functions used to f i t the ANN results from the f i ve rock groups displayed in Fig.5.

7.Results and discussion

Based on the results shown in Fig.4,the WG-ANN model on WG test data(RMSE=0.02,R2=0.96),and SL-ANN model on SL test data performed best(RMSE=0.02,R2=0.99),while the KTBA ANN model on KTBA test data performed least favorably(RMSE=0.09,R2=0.66).In general,however,all the ANN models tested performed very well(i.e.all RMSE less than 0.1)for the same rock type.

When compared with published studies,the ANNs simulate the intermediateprincipalstress effectson rockstrengthquite reasonablyas indicated in Fig.5.The ANN results from the f i ve ANN models displayed in Fig.5 may be f i tted with power functions as summarized in Table 4.The output from plots of the f i tted power functions for each of the f i ve rock groups can be used for further analysis,as shown in Fig.6 with the initial boundary conditions for the stress state factorβdisplayed in dashed lines.For example, when the functions are plotted inσ1-βspace for each rock type,the classical convex contour curves are observed.Fig.6 shows that there is a general trend of increasing strengthσ1for a constantσ3, reaching a peak at some intermediate stress state factorβ(and henceσ2),followed by a decline in strength.The exception is the curves for the DD rock group which show a trend of continued increasingstrength evenuptotriaxial extension(i.e.β=1).The rest of the rock groups predicted declines past the peak strength for all ranges of minor principal stresses,with more pronounced declines at relatively higher conf i nements.As discussed in Section 1,there is some debate in the literature as to what occurs for post-peak in terms of strength behavior inσ1-σ2space.Reports range from nothing,a sharp decrease,a drop to a value that is still greater than the conventional triaxial compression equivalent value(when σ2=σ3orβ=0)at the beginning,to a drop to a value that is at least equal to the conventional triaxial compression equivalent value. The results in this study suggest that post-peak behavior with respect to intermediate stress effects is dependent on rock type, and the relative value of the minor principal stress,σ3.For instance, Fig.6 shows signif i cant drops in strength for KTBA and WG at all conf i ning stresses investigated and signif i cant drops for MT and SL at conf i nements greater thanσ3=40 MPa.In none of the cases evaluated do strength values remain steady after reaching the peak point.The location of the peak strengths in Fig.6 appears to be dependent on rock type and stress state.

For instance,for KTBA and WG,the peak strength occurs at a stress state factor(i.e.β)of approximately 0.1,and at 0.3 for MTand SL.In general,none of the rock types investigated in this study displayed strength behavior independent ofσ2effects.

Two-way analysis of variance(ANOVA),used to statistically test parameter independence and interaction,was also conducted by the ANNs with respect to the UCS and the stress state factorβ.For a two-way ANOVA test,the null hypothesis is rejected if thep-value is less thanα,(typicallyα=0.05 implying a 95%conf i dence level). Thep-value indicates the statistical likelihood of obtaining a particular result given that the null hypothesis is true.The analysis results,summarized in Table 5,indicate that the response and performance of the ANN models are sensitive to the stress state, and to the interaction between the stress state and UCS of the test data given that their respectivep-values are less than 0.05.The performance of the ANN models is however independent of the UCS alone.

8.Conclusions

Several previous studies have demonstrated that the inf l uence of the intermediate principal stress in rock mechanics and rock engineering is without question.However the nature and character of this inf l uence have not been comprehensively studied primarily due to a dearth of empirical data,even though laboratory results can be very reliable.In addition,the design and conduction of polyaxial or true triaxial laboratory experiments are not trivial.It is therefore imperative to explore/develop new avenues to complimentlaboratoryexperiments.Inthisstudy,aself-learning approach via ANN modeling is employed to investigate the inf l uence of the intermediate principal stress on the rock strength.To normalize differences such as different laboratory testing conditions and natural variations within rock types,the intermediate principal stressσ2is mathematically converted to a stress state factorβ.The ANN experiments compared favorably with recent 3D rock strength criteria in the prediction of the major principal stress based on the UCS,minor principal stress and intermediate principal stress.The ANN models performed well within their rock type. Two-way ANOVA tests for parameter independence and interactionat 95%conf i dence interval indicate that the ANN models are sensitive to the stress state,and to the interaction between the stress state and the UCS of the test data.

Fig.6.Plots of major principal stress versus stress state factorβobtained from power functions for each of the f i ve rock groups predicted by the neural network models:(a)DD;(b) KTBA;(c)MT;(d)SL;and(e)WG.The dashed lines indicate the limits of the stress state factors used in the development of the ANN models from the empirical data for each rock group.

In addition,the results discussed in this study have important implications for 2D failure criteria in rock engineering.Despite their success and popularity,conventional 2D strength criteria have been known to fall short when applied to 3D problems(such as in situations of multiple failure modes)because 2D strength criteria typically ignore the effects of the intermediate principal stressσ2. Following observations that the intermediate principal stress does have substantial effect on the rock strength,there has been a growing consensus on the need for polyaxial strength criteria.One popular solution has been attempted to upgrade existing 2D strength criteria to 3D by quantitative/mathematical modif i cation. A major disadvantage of solutions that attempt to“improve”conventional 2D strength criteria(e.g.2D Hoek-Brown to 3D Hoek-Brown)is that they inadvertently inherit the fundamental assumptions/shortcomings of the parent criterion.It is therefore important for solutions to explore other avenues to help overcome these short comings.Self-learning techniques such as ANNs provide a fresh approach that is fast,eff i cient,easy to apply,and more importantly also accounts forσ2effects.The caveat,however,is to be aware of the shortcomings and limitations of ANNs as outlined in Sections 2 and 4.

Table 5Summary of two-way analysis of variance(ANOVA)results.

Conf l ict of interest

The author wishes to conf i rm that there are no known conf l icts of interest associated with this publication and there has been no signif i cant f i nancial support for this work that could have inf l uenced its outcome.

Acknowledgments

Comments from the reviewers that helped improve the quality of this manuscript are hereby acknowledged.

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Rennie Kaundagraduated as a Mining Engineer from the University of Arizona in Tucson,Arizona(1999).He got his M.Sc.in Geological Engineering(Rock Mechanics)at the same University,and his Ph.D.in Slope Stability and Geotechnics at Western Michigan University in Kalamazoo, Michigan.He has worked with consulting f i rms offering geotechnical engineering services for over 7 years.He has been involved in more than 50 global projects in Africa, Asia,South America and North America.He has performed or coordinated geotechnical engineering services related to open pit and underground mines,dams,foundations and landslides.He is a licensed Professional Engineer in the State of Colorado,USA.

E-mail address:renniek@hotmail.com.

Peer review under responsibility of Institute of Rock and Soil Mechanics,Chinese Academy of Sciences.