In this study, a special fitting technique is proposed to apply the Hoek–Brown failure criterion to availablelaboratory strength data collected on samples of Ankara andesites. The original method of fitting theHoek–Brown failure criterion using the spreadsheet and the fitting method utilized in the RocLab softwareare statistically compared with this new modified nonlinear fitting technique. The Hoek–Brown failure pa-rameters obtained by a simple multiple regression technique, utilizing the solver function within an Excelspreadsheet and the revised least squares procedure significantly improved the fitness of the Hoek–Brownenvelope especially within the tensile and high confining stresses. The Brazilian tensile strength is found tobe a useful parameter to fix a rock failure criterion at low confining stresses, particularly, in the tensile region.If tensile, uniaxial and triaxial test results are available, when compared to two existing fitting methods themodified multiple regression method is found to predict more precise tensile and compressive strengths, andmaterial constant mi for the Hoek–Brown failure criterion.
Probably the determination of whether a rock will fail or not is one ofthe most important issues in engineering geology and rock engineering.Most theoretical criteria do not accurately predict the failure strength ofrocks and often rely on parameters that are difficult to measure experi-mentally (Ramamurthy et al., 1988; Andreev, 1995; Zuo et al., 2008).For these reasons, many empirical criteria have been formulated that tryto capture the important elements ofmeasured rock strengths or attemptto revise theoretical approaches to grasp experimental evidence.
In rock mechanics literature, there exist some empirical failuremodels that estimate the failure envelope of the rock samples in var-ious degrees (Hoek and Brown, 1980; Yudhbir et al., 1983; Sheorey etal., 1989; Yoshida et al., 1990; Ramamurthy, 2001). Of these differentstrength criteria, the Hoek–Brown strength criterion is the most wellknown, most frequently used and has been applied successfully to awide range of intact and fractured rock masses for almost 30 yearsby practitioners in rock engineering. While the Hoek–Brown failurecriterion is currently the most commonly used failure criterion inpractice, further work is needed to refine the failure estimates, espe-cially at low values of confining stress and tension zone as concludedby some researchers (Carter et al., 1991; Douglas, 2002; Cai, 2010).
Shah and Hoek (1992) have found that the simplex reflectiontechnique is better for fitting laboratory strength data to non-linearHoek–Brown failure criterion than ordinary least squares regression.Lade (1993) states that it may be advantageous to include the tensilestrength values in estimation of material parameters as such data
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gives good control on the failure envelope over the low stress ranges.A comprehensive analysis by Douglas (2002) of a large database oftest results highlights the fact that there are inadequacies in determi-nation of Hoek–Brown empirical failure parameters as currently pro-posed for intact rock.
The objective of this study is to propose a modified nonlinear curvefitting approach that is very robust especially in matching a singleHoek–Brown failure model to experimental data that has multiple ten-sile, uniaxial and triaxial compressive strength values. The originalmethod of fitting the Hoek–Brown failure criterion using the spread-sheet (Hoek and Brown, 1997) and the fitting method utilized in theRocLab software (Rocscience, 2007) will be statistically comparedwith a multiple regression procedure originally developed by Douglas(2002) and modified by the author.
2. Background
The Hoek–Brown empirical rock failure criterion (Hoek andBrown, 1980) was developed in the early 1980s for intact rock androck masses, it has been subject to continual refinement for rockmasses. Over the years, the Hoek–Brown rock mass failure criterionhas undergone numerous revisions (Hoek and Brown, 1988; Hoeket al., 1992, 1995; Hoek and Brown, 1997; Hoek et al., 2002). It haseven been tailored to specific rock masses (Hoek et al., 1998). A sum-mary of the changes to the Hoek–Brown failure criterion throughoutits development is given by Hoek and Marinos (2006). For intactrock its form has not changed and is given in Eq. (1).
Similar tomost natural materials, rocks are not ideal materials. Theyare notoriously heterogeneous in strength, with a coefficient of varia-tion of typically 15–20% (Gunsallus and Kulhawy, 1984; Pincus, 2003;Sari and Karpuz, 2006). In view of the variability of rock properties,when adequate samples are available, repeated testing might be war-ranted to determine average values. Unfortunately, eachproperty rarelyhas a unique value but a range and often the large variations in thevalues of the strength of rock samples are attributed to sampling errors,accuracy of sample preparation or testing procedures (Gill, 1963;Yamaguchi, 1970; Gunsallus and Kulhawy, 1984; Grasso et al., 1992;Pincus, 2003; Ruffolo and Shakoor, 2009). Even when samples of iden-tical lithological composition are tested, the existence of microscopicdiscontinuities results in the variation in the value of mechanical prop-erties, especially on the uniaxial tensile and compressive strengths.Ruffolo and Shakoor (2009) explained that for a 95% confidence intervaland a 20% acceptable strength deviation from themean, 9 or 10 sampleswere needed to test for strength.
Rock engineers frequently utilize empirical methods to estimateproperties with the help of limited experimental data. For a good esti-mation, the more observations that are provided, the more accuratethe estimate of the parameters will be. To overcome this constraintcomprehensive rock mechanics experiments were conducted on Anka-ra andesites (Sari and Karpuz, 2006, 2008). Rock blocks that gatheredfrom andesite quarries for this purpose were usually unweathered,but sometimes slightly weathered andesites were also observed. Fromthese blocks cylindrical rock samples were prepared in the laboratorywithout considering flow direction of andesites. The samples were ap-proximately 54 mm in diameter but not less than NX core size. Thelength-to-diameter (L/D) ratio of 2–2.5 was applicable to the most ofthe specimens for compression tests and for the indirect tension tests,the thickness of samples were approximately equal to the specimen ra-dius, in accordance with the suggestions of the ISRM (2007).
Results were gathered from a range of tests, including 41 Braziliantensile, 106 uniaxial compressive, and at least 5 replicates for each tri-axial tests at confining pressures of 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20and 25 MPa. Thus, there were 266 data points in all, simulating a verycomplete test program from which it should be possible to determineaccurate estimates of the material properties. A good range of σ3 (i.e.,σ3max−σ3min≥0.3σc) was also fulfilled. The complete data set isgiven in Appendix A1. A very wide scatter was usually observed inthe laboratory strength tests of the rock specimens due to random-ness in the number, orientation and distribution of micro-cracks.The large scatter in the experimental data can also be attributed tothe differences in weathering grades and flow direction anisotropyin the studied rock samples.
It is a prerequisite that the strength data should be normally distrib-uted to accurately carry out least-square procedures. Many authorshave shown that the distribution of strength test results on rock canbe well-represented by a normal probability function (Gill, 1963;Yamaguchi, 1970; Hsu and Nelson, 2002; Gill et al., 2005; Sari andKarpuz, 2006; Ruffolo and Shakoor, 2009), but Grasso et al. (1992)and Pincus (2003) have shown that other types of distributions suchas lognormal, gamma, and Weibull may give a better fit. Fig. 1 showsthe histograms for the unconfined compressive strength (UCS) and Bra-zilian tensile strength (BTS) results for theAnkara andesite samples. It ispossible to quantitatively test how well the distribution of the samplevalues conforms to a theoretical distribution by the goodness-of-fit orchi-square, χ2 procedure. The expected frequency of occurrence withineach interval of a theoretical distribution can be compared with the fre-quency of the actual observations that falls within the same intervals. Ifthe actual number of observations in each interval deviates significantlyfrom that expected it seems unlikely that the sample distribution wasdrawn from an anticipated population (Daniel, 1978). The test statisticsare calculated from the equation χ2=Σ(oi−ei)2/ei where oi is thenumber of actual observations within the ith class, and ei is the numberof observations expected in that class. There are k classes or intervals.
Tables 1 and 2 illustrate the results of the chi-square goodness-of-fittest for the two strength parameters. During this test, 106 UCS and 41BTS values are arbitrarily subdivided into 10 classes, each having awidth of 10 MPa and 1 MPa, respectively as shown in the first columnsof Tables 1 and 2. The second column lists the observed frequencies foreach class, while the third column lists the expected frequencies foreach class as a result of the best fitting normal curves in Fig. 1. Thefourth column provides the χ2 term for each class determined fromχ2=Σ(oi−ei)2/ei and the overall χ2 statistic is calculated as 13.07 forthe UCS data and 23.42 for the BTS data. According to the χ2 test, thelower the χ2 calculated value with respect to the χ2 tabulated value,then the closer the theoretical distribution appears to fit the data.
Given a value ofα and ν=N−1 degrees of freedom (d.f.), a p-valuecan be calculated from this distribution using the Excel commandCHIINV(α, d.f.). For UCS values a reasonably higher p-value (0.160) isobtained with α=0.05 and ν=9, which indicates that they follow anormal distribution. However, a small p-value (0.005) for the BTS
Table 2The results of the chi-square goodness-of-fit test for BTS.
Interval Observed frequency (oi) Expected frequency (ei) (oi−ei)2 /ei
values indicates that they cannot be represented by a normal distribu-tion. This is mostly due to two large values in the 10–11 MPa rangewhere the fitted distribution is well skewed to the right. For triaxialstrength values at different confining pressures, the data are too limitedand it is difficult to discern a single type of probability distribution ascan be seen in Fig. 2.
3. Methodology and data analysis
In this section, in addition to the two existing well-knownmethods, a different fitting method was applied to fit the Hoek–Brown failure criterion to the intact rock data collected from past lab-oratory experiments conducted on the Ankara andesite samples. Theoriginal method of fitting the Hoek–Brown failure criterion using thespreadsheet in Hoek and Brown (1997) and the fitting method uti-lized in the RocLab software (Rocscience, 2007) were statisticallycompared with the new modified non-linear fitting method. Thefirst method is based on the linear regression of the Hoek–Brown fail-ure criterion and the second method uses the non-linear Levenberg–
Fig. 2. Histograms of triaxial compressive strength (TCS
“Curve fitting” is frequently used in many engineering applications toobtain the coefficients of amathematical model that describes the exper-imental data (Billo, 2007). The “best-fit” of a curve to a set of data points isconsidered to be found when the sum of the squares of the deviations ofthe experimental points from the calculated curve is at minimum. Thisprocedure is known as least squares curve fitting or, more generally, asregression analysis. In a non-linear regression analysis the aim is to findthe best-fitting parameters that will minimize the sum of the squared re-sidual values (i.e., the “least squares”). Non-linear regression, like linearregression, assumes that the scatter of data around the ideal curve followsa Gaussian or normal distribution. This assumption leads to the familiargoal of regression: to minimize the sum of the squares of the vertical orY-value distances between the points and the curve.
Curve-fitting through regression is a compromise that does not nec-essarily give the desired result. Often, it is better to force the function tofit the data in a particular range of confining pressure.Most rock failuresoccur at low confining pressure. Slabbing in underground cavities, forexample, proceeds under a stress conditionwhere theminimumprinci-pal stress is practically zero (Martin, 1997). Through ordinary regres-sion analysis, neither the uniaxial tensile nor the uniaxial compressivestrength calculated from the fitted functions coincide with the mea-sured values.
The ordinary regression procedures consider equally the variabilityof all experimental data, including those at high confining pressure. Be-cause the sharpest curvature of the Hoek–Brown failure criterion occursin the tensile to low confining pressure region, the risk of a poor fit is the
) of rock samples at different confining pressures.
30 M. Sari / Engineering Geology 127 (2012) 27–35
greatest in this region. This is particularly true when the usual fittingprocedure of Hoek and Brown (1997) spreadsheet and RocLab software(Rocscience, 2007) is employed for the two-parameter Hoek and Brownsquare root parabola in Eq. (1). Thefitted curves by twoproceduresmaygive an excellent fit at high confining pressure, but will under- or over-estimate the uniaxial tensile and/or the uniaxial compressive strength.The difficulty of fitting a single function to strength data over thewhole range of loading conditions, including tension, could signal thatthe nature of the failure mechanism changes as the minimum principalstress moves from tension to high compression (Cai, 2010).
It is well known that a rock behaves differently in tension and lowstress environments compared to highly confined conditions. Whilecrack initiation and unstable crack propagation completely govern therock failure behavior in tension zone for Griffith type materials, in highconfinement zone, crack closure, initiation and propagation in a stablemannermostly control the failure behavior of the rocks. Therefore, a sin-gle empirical model describing the complete failure envelope both intension and high compression zones may not be sufficient. Accordingly,Cai (2010) has recently suggested bi-segmental Hoek–Brown strengthenvelope for the experimental data by taking into consideration failuremechanisms observed in the different zones. However, defining bi-segmental strength envelopes for a rock may create difficulties in nu-merical modeling.
The regression model of the Hoek–Brown failure criterion is non-linear in the coefficients and so some form of numerical optimizationsuch as the Excel built-in Solver must be applied to minimize thesums of the squares of the error terms. Unlike in linear regression,there are no analytical expressions to obtain the set of regression co-efficients for a fitting function that is non-linear in its coefficients. Toperform non-linear regression, essentially trial and error must used tofind the set of coefficients that minimize the sum of squares of differ-ences between measured σ1 and predicted σ1.
If the departure of the measured σ1 from the predicted σ1 (i.e. theerror) is normally distributedwith a variance that is independent of con-trolled variables (here σ3), then the predictions obtained with leastsquares, either with a simplex or otherwise, will be uniform minimumvariance unbiased estimators, which are highly desirable. However,the consideration of data withmultiple measurements of tensile, uniax-ial and triaxial compressive strengths and unequal variances dependingon the stress zone will indicate that a straight least squares method isnot appropriate for estimating the Hoek–Brown criterion parameters.
Given that a general least squares approach assesses the error inthe tensile region as the observed σ1 (i.e., zero) minus the predictedσ1. By recognizing that in a uniaxial tensile strength test, the con-trolled variable is σ1 and the measured variable is σ3, thus the realerror is the measured σ3 minus the predicted σ3. However, thiserror is scaled in σ3 and needs to be adjusted if it is to have equalorder with measurements in σ1, which are usually 10 times greater.Douglas (2002) has suggested that scaling by mi was a convenientand accurate approach. Given this, Douglas (2002) recommendedthat a least squares procedure be used where the error is defined as:
measuredσ1–predictedσ1ð Þ for σ1 > −3σ3measuredσ3–predictedσ3ð Þ �mi for σ1≤−3σ3:
ð2Þ
Most common non-linear regression schemes, similar to linear re-gression, minimize the sum of the square of the vertical distances ofthe points from the curve. This approach is statistically valid when theexperimental uncertainties do not relate in a systematic way to thevalues of σ3 or σ1. Often this is not true, for example, in many experi-mental situations the experimental uncertainty or variability is, on aver-age, a constant fraction of the value of σ1. In these situations, the usualregression methods are not optimal, such that the points with large σ1
values i.e., high confinement zones, tend to be further from the curvethan the points with small σ1 values i.e., low confinement zones. Inmin-imizing the sum of the squares, the program would therefore, tend to
relatively emphasize those pointswith largeσ1 values, and ignore pointswith small σ1 values. To circumvent this problem, the procedure for thequantization goodness-of-fit can be altered, so that the deviation of eachpoint from the curve is divided by the predicted σ1 or predictedσ3 valueof that point, and then squared. The new error terms will be as follows:
measuredσ1–predictedσ1ð Þ=predictedσ1 for σ1 > −3σ3measuredσ3–predictedσ3ð Þ=predictedσ3 for σ1≤−3σ3:
ð3Þ
It should be noticed that, as carried out by Douglas (2002), scalingwith an unknown parameter is not an easy task. Also, the strengthdata does not show a constant variance along the different zones offailure envelope. Douglas's (2002) original approach given in Eq. (2)did not take into account those differences in the variations stemmedfrom multiple measurements. Therefore, the author's original contri-bution in Eq. (3) is the usage of localized difference scaling as a con-venient practical solution. This approach will take into considerationthe pitfalls resulting from both the localized variance and the usageof unknown parameter for scaling purpose as Douglas (2002) did.
Sometimes experimental mistakes can lead to erroneous valuesknown as outliers. An outlier is an observation that deviates significantlyfrom the bulk of the data, whichmay be due to errors in data collection,recording, or natural causes. The presence of outliers in the data causesdifficulties when fitting a distribution or a curve to the data. Low andhigh outliers are both possible and have different effects on the analysis.Even a single outlier can dominate the sum-of-the-squares calculation,and lead tomisleading results. In the current case, two potential outliersin the tensile strength tests dominated the type of probability distribu-tion fitted in the previous section (Figure 1) and they will also affectthe curve fitting stage. So to overcome this deficiency the proposedapproach also offers a viable solution by taking the error term as a frac-tion of local variability.
The modified least squares (MLS) in Eq. (3), was combined with theextended formulation of the Hoek–Brown criterion in Eq. (1), to esti-mate σci and mi for all the test data given in Appendix A1. For themodified non-linear regression technique, it is possible to proceed inthe following manner: reasonable guesses are used for σc and mi, forσ1>−3σ3 (i.e., confining zone) calculate σ1 at each σ3 data pointfrom Eq. (1), then calculate the sum of squares of residuals, SSR1=Σ((σ1act−σ1est)/σ1est)
2. For σ1≤−3σ3 (i.e., tensile zone) a uniquevalue of σ3 is calculated from Eq. (1) at σ1=0, then the sum of squaresof the residuals is calculated, SSR3=Σ((σ3act−σ3est)/σ3est)
2. Our goal isto minimize these two error square sums (Figure 3). One of the spread-sheet models used in the multiple regression method can be found inAppendix A2 with related cell formulas. In addition, an Excel file con-taining spreadsheet model is provided online to the readers as a supple-ment to electronic version of this article.
Choosing the starting values is not an easy task in non-linear leastsquares curve fitting. A poor choicemay result in a lengthy computationwithmany iterations. It may also lead to divergence, or to a convergencedue to a local minimum. Therefore, good initial values will result in fastcomputations with few iterations and if multiple minima exist, it willlead to a solution that is a minimum. For this reason, the initial valuesof σc and mi for the multivariate regression analysis could be assumedto be equal to the parameters estimated by spreadsheet or the RocLabprogram.
How precisely a fittingmethod can predict Hoek–Brown failure crite-rion is justified by the coefficient of determination, R2, and the standarderror of the estimates, SE(y). The only problem with the use of Solver toperform least squares regression is that, although the regression coeffi-cients can be readily obtained, their uncertainties and goodness-of-fit sta-tistics are unknown. Amacro is provided by Billo (2007) to perform thesecalculations in a standard spreadsheet model. The SolvStat add-in returnsregression statistics for the regression coefficients obtained from theSolver. The values are the standard deviations of the regression coeffi-cients, plus the R2 and SE(y) statistics. In the Hoek and Brown (1997)
Fig. 3. Flow chart of MLS procedure used in Solver.
0
20
40
60
80
100
120
140
160
180
200
-10 0 10 20 30Minor principal stress σ3 - MPa
Maj
or p
rinci
pal s
tres
s σ 1
- M
Pa
HB (1997)
RocLab
MLS
Fig. 4. Three different fitting methods applied on the Ankara andesite samples excludingtensile strength data.
0
20
40
60
80
100
120
140
160
180
200
-10 0 10 20 30
Minor principal stress σ3 - MPa
Maj
or p
rinci
pal s
tres
s σ 1
- M
Pa
HB (1997)
RocLab
MLS
Fig. 5. Three different fitting methods applied on the Ankara andesite samples whereσ3=σt and σ1=−3σ3 for tensile strength data.
31M. Sari / Engineering Geology 127 (2012) 27–35
spreadsheet, the only provided statistics is the R2, while the RocLab pro-gram calculates just the sum of the squared residuals (SSR). Tomake rea-sonable comparisons between three fitting methods, the same kindmeasure of goodness-of-fit statistics is required. For that, the residualsterm provided by the RocLab program is normalized by calculatingSE(y) as;
SE yð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiSSRn−k
rð4Þ
where n= number of data points, k= number of regression coefficientsto be estimated.
4. Results and discussion
The results of three different fitting methods are compared for threedifferent sets of strength data in Figs. 4 to 6 and Tables 3 to 5. In the re-gression analysis, the first set of data consists of only uniaxial and triaxialcompressive strength test results (Figure 4). The second set of dataincludes tensile, uniaxial and triaxial compressive strength values(Figure 5). Here, Brazilian tensile strength data is represented in the anal-ysis asσ3=σt atminor principal stress axis andσ1=−3σ3 atmajor prin-cipal stress axis by considering the theoretical relationship observedbetween the stress conditions during the failure of a disc specimen.Some researchers (Fairhurst, 1964; Goodman, 1989; Chen et al., 1998;Douglas, 2002; Chou and Chen, 2008) pointed out that at failure, in theBrazilian test, the radial compressive stress (σ1) obtained at the centerof the disc specimen is almost 3 times the maximum tensile stress (σ3)for an isotropic Griffith material. The third set of data considers thesame Brazilian tensile values as equal to the uniaxial tensile strengthwhere σ3=σt atminor principal stress axis and σ1=0 atmajor principalstress axis (Figure 6).
In the tables, the Hoek–Brownparameters estimated by three differ-ent fitting methods are shown for three sets of data. These parametersinclude uniaxial compressive strength (σc), uniaxial tensile strength(σt), material constant mi, and triaxial compressive strength σ1 at
σ3=25MPa. The measure of goodness-of-fit statistics R2 and SE(y) isalso presented in the tables.
For Ankara andesite, the MLS method gives the best fit overallbased on the lowest SE(y) (13.51) and the highest R2 (0.92) valueswhen available tensile data are used in the analysis (Table 5). Besides,if Brazilian tensile strength data are included as σ3=σt and σ1=−3σ3 in the analysis the RocLab software underestimates the uniaxialcompressive strength and substantially overestimates the uniaxialtensile strength. An underestimation in the high confinement zoneby RocLab is another reason for developing a fitting procedure thatwill fit closely the Hoek–Brownmodel for the whole range of strengthdata. However, it is not possible to do a direct comparison betweenHoek and Brown (1997) spreadsheet and RocLab software results,since two methods use different kind of goodness-of-fit measure.
When tensile strength data are completely excluded in the anal-ysis all three methods estimate closer strength parameters with sim-ilar goodness-of-fit statistics as shown in Fig. 4 and Table 3. Anotherimportant finding is that if Brazilian tensile values are included in theanalysis whether σ3=σt and σ1=−3σ3 in Fig. 5 or σ3=σt andσ1=0 in Fig. 6, the R2 values always are higher than for the case inFig. 4 where tensile values are completely excluded in the analysis.This clearly indicates that inclusion of tensile strength data in theanalysis has a pronounced effect on the accurate parameterestimation.
Table 3Results of the regression analysis for the data set where tensile values are excluded.
Approximating mi to the ratio of the uniaxial compressive strengthto the uniaxial tensile strength (usually estimated from the Braziliantest) of the rock is an important concept in rock mechanics. Some re-searchers have used this approximation for many years when triaxialtest data are not available in their analysis and recent studies byStewart (2007), Brown (2008), Zuo et al. (2008), Cai (2010) and Sari(2010) also have shown the closer relationship between mi constantand the ratio ofσc/σt. The proposed fittingmethod satisfies this approx-imation sufficiently. As can be followed in Table 5, the ratio between es-timated uniaxial compressive strength and tensile strength, which is10.55, is almost equal to the ratio betweenmeasured uniaxial compres-sive strength and tensile strength, which is 10.54.
In summary, the regression method originally developed by Douglas(2002) and the modified least squares fitting procedure applied in thisstudy result in a vastly improved parameter estimation in the applicationof the Hoek–Brown criterion to the Ankara andesite data. In particular,good fits are obtained both in the tensile stress region and high compres-sion zone. It was seen that the parameters obtained from new procedureis an improvement on those provided by only using triaxial data. The fit-tingmethod of the criterion to different zones of the test data has amajoreffect on the estimates obtained from the material properties. The modi-fied fitting method utilized in this study solves the problem of definingbi-segmental Hoek–Brown failure envelopes for tensile and compressiveregions separately. By assigning equalweight to the values of tensile, uni-axial and triaxial strength data on the fitting process, it is possible to ob-tain a single but sufficiently good failure envelope.
4.1. Why rock strength properties vary
Often large variations in values of strength of rock samples are at-tributed to: 1) natural variability, 2) sampling errors, 3) accuracy ofsample preparation or testing procedures.
1) Natural variability: Rocks are not ideal materials. They are notoriouslyheterogeneous, with a coefficient of variation of typically 15–20%. Inbulk specimens of intact rock, the mechanical properties depend notonly on the properties of the individual minerals, but also upon theway in which the minerals are assembled. Andesite is a fine-grainedigneous, volcanic (extrusive) rock, of intermediate composition, withaphanitic to porphyritic texture. It is intermediate because it containssomeminerals that are common to rhyolite and some common to ba-salt. Therefore, the variability in the characteristics of andesite is con-siderably larger than in the similar igneous rock groups. Even whensamples of identical lithological composition are tested, the existenceof microscopic discontinuities results in variation in the value of me-chanical properties. The wide scatter in laboratory values of strength
Table 5Results of the regression analysis where σ3=σt and σ1=0 for the tensile data.
values is essentially due to randomness in the number, orientationand distribution of micro-cracks.
2) Sampling errors: Color is a distinguishing characteristic of the rockunits in the area where rock blocks are gathered for laboratory exper-iments and the difference in colors is attributed to the degree ofweathering by some researchers. For the Golbasi district, Karpuz andPasamehmetoglu (1997) have distinguished three types of andesite,which are (a) brownish gray-to-grayish purple, fine-grained glassyandesite; (b) pale red, fine-grained andesite; and (c) gray, fine-grained andesite. Rock blocks gathered from the quarry are includedusually fresh ones, but sometimes slightly weathered andesites arealso observed. The difference between weathering grades may resultin variation in the test results.
3) Accuracy of sample preparation or testing procedures: In the laborato-ry works, carefully prepared rock samples should be used in the ex-periments. However, the laboratory technicians carry out thepreparation of the cylindrical rock samples from the blocks and theydo not consider the flow direction of andesites. The degree of isotropyor anisotropy imposed by flow direction is an important factor andthis may lead another variation in the rock strength values. Also, thestudy is based on the extensive data collected from the results oftests performed on the past laboratory sessions for a period of foursuccessive years. Core samples are prepared in advance in these ses-sions and waiting of the samples in laboratory ambient conditionsfor a long time might alter the results.
Appendix A1. Laboratory strength test results for the Ankara andesite
Most of the rock mechanics experiments conducted in a laboratoryare generally used to determine the properties of intact rock. Comparedto a jointed rock mass, to estimate the strength parameters of intactrock is relatively simple, but in practice, for large-scale engineering pro-jects, jointed rockmasses are the norm and presentmajor challenges toengineering geologists. A rock mass is basically an assemblage of intactrock blocks separated by different types of geological discontinuities.Besides micro-cracks in intact rocks, the presence of numerous discon-tinuities (fractures, joints, faults, etc.) in macro scale virtually puts fur-ther scatter on the strength data of a rock mass. How we can treat thescatter in the values of rock masses is beyond the scope of this study,but the interested readers are directed to two papers of the author relat-ed with this subject. A practical solution on account of the uncertaintiesassociated with predicting the rock mass strength parameters detailedby Sari (2009) for an ignimbritic rock mass and Sari et al. (2010) foran andesitic rock mass. In these studies, the probability distributionsof rock mass strength parameters were calculated from the knownprobability distributions of the input parameters using the MonteCarlo method. How to incorporate the variability of such a scattereddata into numerical modeling has been recently shown by Cai (2011).He proposed a probabilistic design approach which utilizes the GSI sys-tem to characterize the rock mass and obtain input parameters for de-sign using numerical methods. Applying the point estimate method,Cai (2011) reflected rock mass property uncertainties in the numericalanalysis of tunnel/cavern stability.
The proposed fitting method is advantageous compared to otherexisting methods. Inclusion of the uniaxial tensile and compressivestrength values in the Hoek–Brown parameters calculation give amuch better fit than using the triaxial data alone. The multiple re-gression method devised by Douglas (2002) and the revised non-linear curve fitting approach applied in this study is found to be themost suitable procedure for estimating the Hoek–Brown parametersof the Ankara andesite. The following conclusions can be derivedfrom the study:
• At least one set of triaxial, uniaxial and tensile tests are required be-fore a reliable envelope can be justified.
• Tensile data is an essential part of fitting process to provide improvedestimates of the tensile strength from the Hoek–Brown failure criteri-on. If there are no tensile data then the Hoek–Brown envelope asfitted by three methods will give similar failure parameters.
• The modified least squares result in an accurate estimation of theparameters and does so in almost all circumstances where thereare special requirements for the treatment of local variability ofthe measured strengths.
• The limitations of the Hoek–Brown failure criterion should be fullyrecognized and a sensitivity analysis should be conducted to assess
Appendix A2. Spreadsheet used in Solver for MLS analysis where σ3=
A B C D E F1 72.11 MPa2 mi 10.55 B4 = SUM(E7:E272)3 − 6.83 MPa4 8.005 B3 = −B1 / B2 Fro
the likely effect of local variations in the σt, σc and mi estimates indifferent types of rock.
In further studies, the author suggests that it may be a plausible al-ternative intuitively to add a confidence interval (90% C.I.) to the fail-ure curve of an intact rock instead of drawing a single mean line. Thevalue of this study in terms of improving the state of the art in deter-mining rock mass strength parameters is that in order to estimaterock mass strength parameters, at first it is required to estimate thestrength parameters of intact rock correctly. A Turkish saying cansimply explain this: if shirt's first button is correctly placed, the sub-sequent buttons will also be correct or vice versa.
Acknowledgment
The Author thanks the Editor and two anonymous reviewers for thetime they have taken in reviewing the paper and the comments and con-structive criticisms, which certainly improve the quality of the paper.
Appendix B. Supplementary data
Supplementary data to this article can be found online at doi:10.1016/j.enggeo.2011.12.011.
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