Research Article A Copula-Based Method for …downloads.hindawi.com/journals/mpe/2014/693062.pdfA Copula-Based Method for Estimating Shear Strength Parameters of Rock Mass DaHuang,

Research ArticleA Copula-Based Method for Estimating Shear StrengthParameters of Rock Mass

Da Huang123 Chao Yang1 Bin Zeng1 and Guoyang Fu4

1 School of Civil Engineering Chongqing University Chongqing 400045 China2 State Key Laboratory of Coal Mine Disaster Dynamics and Control Chongqing University Chongqing 400045 China3 Key Laboratory of New Technology for Construction of Cities in Mountain Area Ministry of Education Chongqing 400045 China4 School of Civil Environmental and Mining Resource Engineering The University of Western Australia CrawleyWA 6009 Australia

Correspondence should be addressed to Da Huang dahuangcqueducn

Received 14 June 2014 Revised 30 September 2014 Accepted 14 October 2014 Published 4 November 2014

Academic Editor Xi Frank Xu

Copyright copy 2014 Da Huang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The shear strength parameters (ie the internal friction coefficient 119891 and cohesion 119888) are very important in rock engineeringespecially for the stability analysis and reinforcement design of slopes and underground caverns In this paper a probabilisticmethod Copula-based method is proposed for estimating the shear strength parameters of rock mass The optimal Copulafunctions between rock mass quality 119876 and 119891 119876 and 119888 for the marbles are established based on the correlation analyses of theresults of 12 sets of in situ tests in the exploration adits of Jinping I-Stage Hydropower Station Although the Copula functions arederived from the in situ tests for the marbles they can be extended to be applied to other types of rock mass with similar geologicaland mechanical properties For another 9 sets of in situ tests as an extensional application by comparison with the results fromHoek-Brown criterion the estimated values of 119891 and 119888 from the Copula-based method achieve better accuracy Therefore theproposed Copula-based method is an effective tool in estimating rock strength parameters

1 Introduction

The shear strength parameters including friction coefficient119891 (the tangent of friction angle 120593) and cohesion 119888 are themost fundamental mechanical parameters for rock massThey are of very high importance for stability evaluation andsupport design of rock mass in rock engineering especiallyfor landslide risk mitigation Different approaches includinglaboratory tests in situ tests back analysis and empiricalequations can be used to estimate these strength parametersIn the laboratory tests scale effects are involved since rockspecimens are of small sizes [1 2] The back analysis methodis also recommended by [3 4] the fact that it might bemore reliable than laboratory or in situ test method Someempirical equations between shear strength parameters andgeological indices are widely used due to simplicity and

practicality [5ndash13] The empirical equations based on famousHoek-Brown criterion are [13]119891

= tansinminus1 [6119886119898119887

(119904 + 119898119887

1205903119899

)119886minus1

2 (1 + 119886) (2 + 119886) + 6119886119898119887

(119904 + 119898119887

1205903119899

)119886minus1

]

119888

=120590119888

[(1 + 2119886) 119904 + (1 minus 119886) 119898119887

1205903119899

] (119904 + 119898119887

1205903119899

)119886minus1

(1 + 119886) (2 + 119886) radic1 + 6119886119898119887

(119904 + 119898119887

1205903119899

)119886minus1

(1 + 119886) (2 + 119886)

(1)

where 119898119887

is a reduction of the rock material constant 119898119894

andobtained by

119898119887

= 119898119894

exp (GSI minus 100

28 minus 14119863) (2)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 693062 11 pageshttpdxdoiorg1011552014693062

2 Mathematical Problems in Engineering

where GSI is the Geological Strength Index and 119863 is adisturbance factor depending on the degree of disturbanceby which the rock mass has been subjected to blast damageand stress relaxationThe 119863 varies from zero for undisturbedin situ rock mass to 1 for very disturbed rock mass 119904 and 119886 in(1) can be calculated by

119886 =1

2+

1

6(119890minusGSI15 minus 119890minus203)

119904 = exp(GSI minus 100

9 minus 3119863)

(3)

In (1)120590119888

is uniaxial compressive strength1205903119899

= 1205903max120590

119888

and 120590

3max is the upper-limit value of confining stress dis-cussed by Hoek et al [13]

It is important to take full advantage of in situ test data inorder to establish the correlativity between geological indicesand mechanical parameters of rock mass With regard tothe correlativity of the above variables an effective way isto establish their joint distribution function The commonstatistical methods for obtaining joint distribution functionof variables are generally based on two assumptions that isthe variables have the samemarginal distributions and followthe normal distribution [14] However rock mass param-eters do not always follow the normal distribution especiallyas the number of samples is small [15] With respect to smallsamples normal information spread estimation method(NISEM) [15] is suitable Meanwhile Copula theory [16ndash21] is a fairly effective method to establish the joint distri-bution function of nonnormal distribution variables Theadvantages of Copula theory are that the marginal distribu-tions and dependence structure of variables can be studiedseparately and the variables can follow any distributions TheCopula theory has been widely used in many fields such asinsurance [16] finance [17] hydrology [18ndash20] and geotech-nicalgeological engineering [21]

In this paper the correlations between the rock massquality indices (ie rock quality designation (RQD) rockmass quality (119876) and Rock Block Index (RBI)) and theshear strength parameters (119891 and 119888) are investigated basedon the results of in situ direct shear tests at Jinping I-StageHydropower Station in China It shows that 119876 presents thestrongest correlation with the shear strength parametersThen the joint distribution functions of 119876 and 119891 119876 and 119888 areproposed based on Copula theory By means of the functionsand the given guarantee rates the shear strength parameterscan be estimated Compared with those estimated by Hoek-Brown criterion the estimated values using the proposedCopula-based method are closer to the in situ tests ones

2 Correlations between Rock Mass QualityIndices and Shear Strength

21 In Situ Test Data The Jinping Hydropower Station islocated on the middle reach of the Yalong River in SichuanChina It has a double-curvature arch dam with a height of305m and a total installed capacity of 3300MW [22 23]

Shear box

Sample

Roller rows

Jack

Steel plate

Mortar

Transfer column

Jack

Steel plate

Roller rows

Marble

J1 Bedding jointsJ2 Cross jointsJ1

J2

Sheared surface

Figure 1 The sketch of the in situ direct shear test

The Hydropower Station is located in the east edge of theTibet Plateau tectonic belt Due to the rapid crust upliftduring the Quaternary the steep slopes exist on both sidesof the valley deeply cut by Yalong River The stratum in thedam zone consists of a series of metamorphic rocks of middleof Triassic period of which the great majority are marbleswith grey orwhite andfine-grained textureThemainmineralcompositions of the marble are calcite and dolomite Theuniaxial compressive strength of the freshmarble is about 60ndash75MPa

In order to investigate the shear strength parameters ofthe representative marbles 12 sets of in situ direct shear testswith five similar specimens in each set were performed inthe exploration adits of Jinping I-Stage Hydropower StationFigure 1 shows the sketch of in situ direct shear test whichwasperformed using flat stacking method [24] The side lengthof each cube specimen is 1m which contains in general twosets of joints (bedding joints and cross joints see Figure 1)RQD and RBI [25 26] of each specimen can be obtainedusing scanline field mapping method The scanline used formeasuring RQD was parallel to shear direction The sheardirection was parallel to the bedding joint along which theshear deformation happened The five similar specimens in aset were tested at different normal stress 120590

119899

from 2 to 10MPaThe shear strength parameters (119891 and 119888) of each set areobtained by linear regression according to Mohr-Coulombcriterion A good fitting straight line between the peak shearstress 120591 and normal stress 120590

119899

can be derived from each set testdata for which the 1198772 of fitting degree is greater than 09 (seeFigure 2) The test results are shown in Table 1 in which therock mass quality indices are the average of each set samplesrespectively

22 Correlation Measure Pearson linear correlation coeffi-cient 120574

119899

[27] Spearmanrsquos rank correlation coefficient 120588119899

[28]and Kendallrsquos rank correlation coefficient 120591

119899

[29] are widely

Mathematical Problems in Engineering 3

0

4

8

12

16

20

0 2 4 6 8 10

120591(M

Pa)

Shea

r stre

ss

120591 = 179120590n + 195

R2 = 09546

Normal stress 120590n (MPa)

Figure 2 Linear regression of normal stress 120590119899

versus shear stress 120591for set number 1 in Table 1

Table 1 Results of in situ direct shear tests and rock mass qualityindices of the specimens

Setnumber

Rock mass quality indices Shear strengthparameters

RQD RBI 119876 119891 119888MPa1 753 176 305 179 1952 893 315 238 138 1983 968 416 387 117 1504 993 318 397 220 2255 964 434 386 139 1806 981 226 265 160 1807 598 80 81 180 2808 560 103 52 117 1209 659 143 89 123 24110 774 132 52 056 30011 679 90 54 107 32512 518 70 46 115 432

used to measure how relevant the variables are They can becalculated by

120574119899

=sum119899

119894=1

(1199091119894

minus 1199091

) (1199092119894

minus 1199092

)

119899radic11990421

sdot 11990422

120588119899

=12

119899 (119899 minus 1) (119899 + 1)

119899

sum119894=1

119877119894

119878119894

minus 3119899 + 1

119899 minus 1

120591119899

= (119899

2)minus1

sum119894lt119895

sign [(1199091119894

minus 1199091119895

) (1199092119894

minus 1199092119895

)]

(4)

where 119899 is the number of observations 1199091119894

(1199091119895

) and 1199092119894

(1199092119895

)are the observed values of variables (119909

1

and 1199092

) 1199091

and 1199092

aresample mean 1199042

1

and 11990422

are sample variance 119877119894

and 119878119894

are therank of 119909

1119894

and 1199092119894

sign = 1 if 1199091119894

lt 1199091119895

and 1199092119894

lt 1199092119895

andsign = 0 if 119909

1119894

= 1199091119895

and 1199092119894

= 1199092119895

otherwise sign = minus1 119894119895 = 1 2 119899

Table 2 shows the results of the 120574119899

120588119899

and 120591119899

of test dataThere are clearly positive correlations (the same variation

Table 2 Correlation coefficients of the rock mass quality indicesand the shear strength

Variables Pearson 120574119899

Spearman 120588119899

Kendall 120591119899

RQD-f 0307 0386 0303RBI-f 0241 0376 0303Q-f 0550 0656 0545RQD-c minus0517 minus0397 minus0333RBI-c minus0573 minus0635 minus0515Q-c minus0563 minus0517 minus0485

trend) between three rock mass quality indices (RQD RBIand 119876) and 119891 The correlation coefficients between 119876 and 119891are much greater than those between RQD and 119891 RBI and 119891There is clearly negative correlation (changes in the oppositetrend) between the three indices and 119888 The correlationbetween RBI and 119888 is slightly better than 119876 and 119888 whileRQD and 119888 have the worst correlation In summary 119876 hasthe best correlation with shear strength parameters (119891 and119888) In fact RQD and RBI evaluate the rock mass quality onlyby the integrality of rock core While many other factorssuch as joint surface roughness and degree of weatheringfor dominantly oriented joint sets joint water reductionstress reduction factor and volumetric joint count are fullyinvolved in 119876-system the 119876-system therefore provides acomprehensive description on rock mass quality [30] Theshear strength parameters of jointed rock mass are generallycontrolled by complex geological conditions and mechanicalbehaviors of rock material and so on The best correlationbetween119876 and the shear strength parameters in this statisticalstudy also indicates that 119876-system associated with multiplefactors of influence on 119888 and119891 of jointed rocksmore preciselydepicts the quality of rock mass

The correlation measure indicates that for the in situtests presented in this paper the marbles with higher rockmass quality indices show higher friction angle and lowercohesion Many researchers had also found that for rocksand soils there was a negative correlation between 119891 and 119888[31 32] If this negative correlation is true between 119891 and 119888 itis easily understood that a variable has a positive correlationwith one (119891) of shear strength parameters while a negativecorrelation must appear with the other (119888) based on the puremathematical logic As shown in Figure 1 there are someintact rock segments on the sheared bedding joint in thecase of relatively low 119876 value due to cross joint which mightexplain why higher 119876 has higher 119891 and lower 119888 The shearfailure occurs along the weak bedding joint with a relativelylow 119888 value on condition that the tested specimen has a high119876value (good rock mass quality) When a low 119876 value appearsthese intact rock block segments on the shear surface mustbe thus cut through before reaching peak strength which canincrease the cohesion 119888 of the failure surface

3 Copula-Based Method

31 Copula Theory Considering multivariate random vari-ables 119883

1

119883119894

119883119896

withmarginal distribution functions


Determine the best relevant variable (Q) of target

Target variables (cf) and their relevant variables (Q RQD and RBI)

Reject

Generate a suitable marginal distribution functional form (FX119894(xi)) as (9)

Accept

Determine suitable multivariable joint distributionH (ie Copula functional forms) as (6) and (12)

Confirm the optimal Copula function C

A guarantee rate 120573

Calculate the AIC values as (13) and (14)

Estimate the values of target variables as (17a) and (17b)

Correlation measure as (4)

NISEM as (7) and (8)

K-S test as (10)

Figure 3 The calculation flow chart of Copula-based method

as 119865119883119894

(119909119894

) = 119875119883119894

(119883119894

le 119909119894

) where 119896 is the number ofrandom variables and 119909

119894

is the value of random variable119883119894

(1198831

119883119894

119883119896

) 119865119883119894

(119909119894

) is the marginal distributionfunction and 119875

119883119894(119883119894

le 119909119894

) is the probability when 119883119894

le 119909119894

The joint distribution function 119867 of the multivariate randomvariables 119883

1

1198832

119883119896

can been expressed as [33]1198671198831 119883119894 119883119896

(1199091

119909119894

119909119896

)

= 119875 (1198831

le 1199091

119883119894

le 119909119894

119883119896

le 119909119896

) (5)

The Copula function is to connect multivariate probabil-ity distributions to their marginal probability distributionsThus the multivariate joint distribution 119867 is expressed interms of its marginal distribution function 119865

119883119894(119909119894

) and theassociated dependence structure 119862 as

1198671198831119883119894 119883119896

(1199091

119909119894

119909119896

)

= 119862 (1198651198831

(1199091

) 119865119883119894

(119909119894

) 119865119883119896

(119909119896

)) (6)

where 119862 called Copula function is a uniquely determinedmapping whenever 119865

119883119894(119909119894

) are continuous and capturesthe essential features of the dependence between randomvariables 119862 is essentially the joint distribution of multi-variate random variables of 119883

1

119883119894

119883119896

through theirmarginal distributions 119865

1198831(1199091

) 119865119883119894

(119909119894

) 119865119883119896

(119909119896

) [33]The joint distribution function established using Copula-

based method has twomajor advantages the marginal distri-bution and dependence structure can be studied separately

and the variables can follow any distributions Therefore theproblem of determining 119867 reduces to determining marginaldistribution of variables and Copula function 119862

In order to clearly demonstrate the calculation procedureusing the present Copula-based method the calculation flowchart is shown in Figure 3 The detailed procedures are asfollows

Step 1 The correlation between target variables (119891 119888) andrelevant variables (119876 RQD and RBI) is analyzed accordingto the correlation coefficients obtained by (4) or else Thebest relevant variable (ie 119876) can be selected as shown inSection 22

Step 2 Utilize suitable functional forms and methods (suchas normal information spread estimation method (NISEM))to establish marginal distribution function 119865

119883119894(119909119894

) of vari-ables which could be accepted according to correspondingtest (eg Kolmogorov-Smirnov (K-S) test) It is addressed inSection 32

Step 3 Introduce suitable multivariable joint distributionfunctional forms (ie Copula functions) and confirm theoptimal Copula function 119862 indicated by a minimum AICvalue Such is stated in Section 33

Step 4 Estimate the values of target variables (119891 119888) using theCopula method on condition that a guarantee rate 120573 and


Table 3 Values of 120574 for different values of 119899 [35]

119899 120574 119899 120574

3 0849 321 800 11 1420 835 4434 1273 982 782 12 1420 269 5705 1698 643 675 13 1420 698 7956 1336 252 561 14 1420 669 6717 1445 461 208 15 1420 693 3218 1395 189 816 16 1420 692 2269 1422 962 345 17 1420 693 10110 1416 278 786 gt17 1420 693 101

Table 4 Values of 1(radic2120587119899ℎ) and 12ℎ2

Variables 1(radic2120587119899ℎ) 12ℎ2

RQD 0005 420 722 0013 293 110RBI 0007 073 744 0022 636 592119876 0007 335 735 0024 344 428119891 0157 002 612 11151 315 78119888 0082 527 014 3081 091 687

the best relevant variable 119876 have been given which ispresented in Section 4

32 Verification ofMarginalDistribution Functions For smallsamples the normal information spread estimation method(NISEM) [15] is an effective way to generate probabilitydensity functions The main difference between NISEM andthe available classical methods [15 34] is that the variablesare not assumed to follow a distribution (such as normallog normal and exponential distribution) in advance Ithas been proved that the NISEM used for generating theprobability density functions of geotechnical parameters withsmall samples is more accurate and effective than the otherclassical methods [15] The probability density function ofrandom variables 119883

119894

by using NISEM is expressed as

119891 (119909119894

) =1

radic2120587119899ℎ

119899

sum119895=1

exp[

[

minus(119909119894

minus 119909119894119895

)2

2ℎ2]

]

(7)

where ℎ is the windowwidth of standard normal informationspread function and 119909

119894119895

is the observed value of variable119883119894

By the approaching principle of normal informationdiffusion ℎ can be calculated by

ℎ =120574 (119909119894max minus 119909

119894min)

119899 minus 1 (8)

where 119909119894max and 119909

119894min are the maximal and minimumobserved values of variable 119883

119894

The value of 120574 is related to 119899as shown in Table 3 [35]

For the variables in Table 1 120574 = 1420269570 while 119899 =12 Therefore the estimation of probability density functionscan be obtained from (7) The values of 1[(2120587)05 119899ℎ] and12ℎ2 (parameters of 119891(119909

119894

)) are shown in Table 4

Table 5 Results of Kolmogorov-Smirnov test

Variables 119899 120572 119863119899120572

119863119899

Accept orreject

RQD 12 005 0375 013617 AcceptRBI 12 005 0375 009205 Accept119876 12 005 0375 011670 Accept119891 12 005 0375 012352 Accept119888 12 005 0375 011224 Accept

By means of the probability density functions 119891(119909119894

)(see (7)) the marginal distribution functions 119865

119883119894(119909119894

) can beestablished as

119865119883119894

(119909119894

) = int+infin

minusinfin

119891 (119909119894

) 119889119909119894

(9)

Kolmogorov-Smirnov (K-S) test commonly used to checkwhether a set of data follows a certain distribution [36] isadopted to test the established marginal distribution func-tions The procedure of K-S test involves forming the cumu-lative frequency distribution 119865

119883119894(119909119894

) and computing the teststatistic 119863

119899

119863119899

= max 10038161003816100381610038161003816119865119883119894 (119909119894

) minus 119865119883119894

(119909119894

)10038161003816100381610038161003816 (10)

The cumulative frequency distribution 119865119883119894

(119909119894

) is defined as

119865119883119894

(119909119894

) =1

119899

119899

sum119894=1

119868119909119894119895le119909119894

(11)

where 119868119909119894119895le119909119894

is the indicator function equal to 1 if 119909119894119895

le 119909119894

and equal to 0 otherwiseThe established marginal distribution functions 119865

119883119894(119909119894

)will be accepted if 119863

119899

lt 119863119899120572

otherwise they will be deniedWhere 119863

119899120572

is the critical value of a given significance level 120572(usually120572 = 5[37]) the119863

119899120572

can be obtained using the tableestablished by [38]when 119899 and120572have been determinedHerethe significance level 120572 is 5 and 119899 is 12 so119863

119899120572

= 0375 basedon the relevant table in [38]The K-S test results are shown inTable 5 Obviously all the established marginal distributionfunctions can be accepted

33 Determination of the Optimal Copula Functions In orderto obtain the optimal Copula functions the commonmethodof the goodness-of-fit is adopted Therefore two steps aretaken to determine the optimal Copula function (1) selectingthe suitably qualified Copula functions according to the char-acteristics of marginal distribution functions and dependentstructure of variables and (2) determining the parameters ofCopula functions selected

With respect to two-dimensional Copula functions theparameter 120579 of Copula functions can be obtained by meansof Kendallrsquos rank correlation coefficient 120591

119899

which has beencalculated in (4) The relation between 120591

119899

and 120579 is [39]

120591119899

= 4 int1

0

int1

0

119862 (119906119894

V119894

120579) 119889119862 (119906119894

V119894

120579) minus 1 (12)


00 02 04 06 08 1000

02

04

06

08

10

T2f

T1Q

(a)

00 02 04 06 08 1000

02

04

06

08

10

T3c

T1Q

(b)

Figure 4 The scatter plots of empirical distribution (a) 1198791

119876 versus 1198792

119891 and (b) 1198791

119876 versus 1198793

119888

where 119906119894

and V119894

are the marginal distribution functionalvalues on condition that the values of two variables are equalto 1199091119894

and 1199092119894

(ie 119906119894

= 1198651198831

(1199091119894

) and V119894

= 1198651198832

(1199092119894

)) while119862(119906119894

V119894

120579) is the introduced Copula functional value with theparameter 120579

Akaike information criteria (AIC) [40] are commonlyused for goodness-of-fit test AIC are based on entropyprinciple and are a simple and effectivemethod for goodness-of-fit test [41] And the optimal Copula is the one which hasthe minimum AIC value the value of AIC can be calculatedas

AIC = 119899 ln1

119899 minus 119905

119899

sum119894=1

[119865emp (1199091119894

1199092119894

) minus 119862 (119906119894

V119894

)]2

+ 2119905

(13)

where 119905 is the number of parameters of Copula function(119905 = 1 for two-dimensional Copula function) and 119862(119906

119894

V119894

)is the Copula functional value 119865emp(1199091119894 119909

2119894

) is the empiricalprobability functional value it can be obtained as

119865emp (1199091119894

1199092119894

) = 119875 (1198831

le 1199091119894

1198832

le 1199092119894

) =1

119899 + 1

119899

sum119895=1

119899

sum119897=1

119873119895119897

(14)

where 119873119895119897

is the number of joint observations under thecondition of 119883

1

le 1199091119894

as well as 1198832

le 1199092119894

In order to visualize the dependence structure of the

measured data (1199091

1199092

) and to obtain an insight on a suitableCopula the measured data in original space should betransformed into the standard uniform random vector T =(1198791

1198792

) Tang et al [21] defined the random vector T by

adopting the empirical distributions ofmeasured data1198791

and1198792

are expressed as

1199051119894

=rank (119909

1119894

)

119899 + 1

1199052119894

=rank (119909

2119894

)

119899 + 1

(15)

where rank(1199091119894

) (or rank(1199092119894

) (1199093119894

)) denotes the rank of1199091119894

(or 1199092119894

1199093119894

) among 1199091

(or 1199092

1199093119894

) in an ascendingorder (119905

1119894

1199052119894

) are the values of variables (1198791

1198792

) or (1198791

1198793

)corresponding to (119876 119891) or (119876 119888)

The scatter plots of empirical cumulative distribution of(119876 119891) and (119876 119888) are shown in Figure 4 119876 and 119891 119876 and 119888 aresubstantially symmetric about the diagonal lines Althoughthe data points (119876 and 119888) in right Figure 4(b) seem to benot symmetric as well as those in left Figure 4(a) most ofthem are located around diagonal line (ie symmetrical line)in Figure 4(a) Archimedean Copulas have a wide range ofapplication [19ndash21 42] due to several competitive advantages(1) the ease with which they can be constructed (2) the greatvariety of families of copulas which belong to this class and(3) the many nice properties possessed by members of thisclass [39] As results the symmetrical Archimedean CopulasNelsen No 1 Nelsen No 2 Nelsen No 4 Nelsen No 5Nelsen No 12 Nelsen No 14 Nelsen No 15 and NelsenNo 18 [39] are selected to model the positive correlationbetween 119876 and 119891 while Nelsen No 1 Nelsen No 2 NelsenNo 5 Nelsen No 7 Nelsen No 8 and Nelsen No 15 [39] areselected to model the negative correlation between 119876 and 119888By means of AIC calculated by (13) the goodness-of-fit test isused to determine the optimal Copula functions The chosensymmetrical Archimedean Copula functions and the valuesof AIC are shown in Table 6 It shows that Nelsen No 1 has


Table 6 The parameter 120579 of 2D Copula functions and the values of AIC

Copula type 119862(119906 V) 120579 (Q-f ) AIC (Q-f ) 120579 (Q-c) AIC (Q-c)Nelsen No 1 max [(119906minus120579 + Vminus120579 minus 1)

minus1120579

0] 2396 minus67990 minus0653 minus61777

Nelsen No 2 max 1 minus [(1 minus 119906)120579 + (1 minus V)120579]1120579

0 4396 minus19240 1347 minus66184

Nelsen No 4 119890minus[(minus ln 119906)120579+ (minus ln V)120579]

1120579

2198 minus66341 mdash mdash

Nelsen No 5 minus1

120579ln[1 +

(119890minus120579119906 minus 1) (119890minus120579V minus 1)

119890minus120579 minus 1] 4978 minus65175 minus5442 minus59637

Nelsen No 7 max [120579119906V minus (1 minus 120579) (119906 + V minus 1) 0] mdash mdash 0634 minus44658

Nelsen No 8 max[1205792119906V minus (1 minus 119906) (1 minus V)

1205792 minus (120579 minus 1)2 (1 minus 119906) (1 minus V) 0] mdash mdash 1629 minus64809

Nelsen No 12 1 + [(119906minus1 minus 1)120579

+ (Vminus1 minus 1)120579

]1120579

minus1



+ (Vminus1120579 minus 1)120579

]1120579

minus120579


Nelsen No 15 max(1 minus [(1 minus 119906minus1120579)120579

+ (1 minus Vminus1120579)120579

]1120579

120579

0) 2698 minus65360 1173 minus60240

Nelsen No 18 max1 +120579

ln (119890120579119906minus1 + 119890120579Vminus1) 0 2930 minus60246 mdash mdash

theminimumAICvalue for establishing the joint distributionfunction of 119876 and 119891 while Nelsen No 2 shows the minimumfor that of 119876 and 119888 Therefore Nelsen No 1 and Nelsen No 2can be used as the optimal Copula functions to construct thejoint distribution functions 119867 of 119876-119891 and 119876-119888 respectively

4 Application to Estimating ShearStrength Parameters

41 Guarantee Rate Instability would happen if the strengthparameters of rock mass are overvalued while economicwaste could be caused if they are undervalued Assuming thatthe estimated values of shear strength parameters are 119891

1

and1198881

the actual values are 119891 and 119888 the guarantee rates 120573119891

and120573119888

are then defined as the probability of 119891 gt 1198911

and 119888 gt 1198881

respectively which can be expressed as

120573119891

= 119875 (119891 gt 1198911

)

120573119888

= 119875 (119888 gt 1198881

) (16)

The estimated value is overvalued if guarantee rates 120573 arelow while being undervalued if high In consequence theestimated values can be evaluated by means of the guaranteerates 120573 On the other hand the estimated 119891 and 119888 witha certain guarantee rates 120573 can be calculated by solvingthe conditional probability of the optimal Copula functionsBased on the optimal Copula functions 119862 of 119876-119891 and 119876-119888established in Section 33 120573

119891

and 120573119888

can be calculated as

120573119891

= 119875 119891 gt 1198911

| 119876 = 1198760

= 1 minus 119875 119891 le 1198911

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)

= 1 minus limΔVrarr0

119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17a)

120573119888

= 119875 119888 gt 1198881

| 119876 = 1198760

= 1 minus 119875 119888 le 1198881

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)


119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17b)

where 1198760

is the measured value of 119876For direct shear test in general the predominant slope

method and the least square method [43] are used to statisti-cally process the test data Using the least square method thebest linear fitting relations of normal stress 120590

119899

and shear stress120591 for each class of rock mass quality are obtained and theslope of fitting straight line is119891 and the intercept at 120590

119899

axial is119888 based on Mohr-Coulomb criterionThe predominant slopemethod is obviously different from the least square methodalthough they are all based on Mohr-Coulomb criterion andlinear fitting method For the predominant slope methodthe upper- and lower-limit values of cohesion 119888 shouldbe conducted (Figure 5) the lower one is often suggestedConsidering the obvious differences of 119876 values in Table 1 1ndash6 sets are grouped as Class II while 7ndash12 sets being grouped asClass III in rock mass quality Their scatter plots of normalstress 120590

119899

versus shear stress 120591 are shown in Figure 5 Theestimated results using the twomethods are shown inTable 7

As the prescribed standard [44] for the hydroelectricengineering in China the suggested design parameters musthave a guarantee rate 120573 of more than 08 Therefore it couldbe suggested that the 120573 of more than 095 are too high whilethose of less than 020 are too low Relative to the 12 setstest results in Table 1 the guarantee rates 120573

119891

and 120573119888

(by (17a)and (17b)) associated with the estimated 119891 and 119888 (in Table 7)are shown in Table 8 Most of them are not acceptable all120573119888

are too low using least square method which means thatthe estimated 119888 are overvalued the 120573

119888

are larger than 095 for


Upper-limit line

120591 = 135120590n + 640

120591 = 138120590n + 316

120591 = 135120590n + 200

Lower-limit line

Shea

r stre

ss120591

(MPa

)

Normal stress 120590n (MPa)0 5 10

2

5

10

15

20

(a)

Upper-limit line

120591 = 104120590n + 550

120591 = 089120590n + 419

120591 = 107120590n + 150

Lower-limit line

Shea

r stre

ss120591

(MPa

)Normal stress 120590n (MPa)

0 4 8 121

5

10

15

(b)

Figure 5 The scatter plots of 120590119899

versus 120591 (a) Class II and (b) Class III

Table 7 The estimated values of 119891 and 119888 using predominant slopeand least square methods

Class of rockmass quality

Setnumber

Predominantslope method Least square method

119891 119888 119891 119888

II 1ndash6 135 200 138 316III 7ndash12 107 150 089 419

Table 8 The guarantee rates of estimated f and c using thepredominant slope and least square methods

Setnumber 119876

Predominantslope method

Least squaremethod

120573119891

120573119888

120573119891

120573119888

1 305 0762 0397 0721 00372 238 0650 0564 0600 00703 387 0857 0187 0828 00134 397 0868 0206 0839 00155 386 0857 0209 0827 00156 265 0698 0499 0652 00557 81 0796 0963 0963 00548 52 0532 0980 0867 00989 89 0834 0957 0971 004610 52 0532 0980 0867 009811 54 0557 0979 0879 009412 46 0453 0982 0819 0110

7ndash12 sets using predominant slopemethod whichmeans thatthe estimated 119888 are too undervalued The data points mustbe located around the fitted straight line as far as possibleon account that the least square method focuses on the bestlinear fitting of normal stress 120590

119899

and shear stress 120591 (Figure 5)

which can cause the relatively low guarantee rates120573 defined as(17a) and (17b) while the data pairs of 120590

119899

and 120591 are above thelower-limit fitting line (Figure 5) when the lower-limit valuesare adopted using the predominant slope method which canlead to the large guarantee rates 120573 Therefore the guaranteerate of the value 119888 is low by the least square method but highby the predominant slope method

42 Estimated Shear Strength Parameters Using Copula-Based Method The Copula-based fitting formulas estab-lished above are based on the test data of the 12 sets beforetherefore these tested shear strength parameters cannot beused to examine the extensional suitability of the Copulafunction obtainedThe additional in situ direct shear tests arethus performed for the other 9 sets of themarbles with similarproperties to check the application of Copula-based methodto estimating shear strength parameters of rock mass The 119876andGSI of the tested samples are also investigated at the sametime The test results are shown in Table 9

The estimated 119891 and 119888 using Copula-based method havethe guarantee rate 120573 of 08 The Hoek-Brown criterion (see(1) to (3) the disturbance factor 119863 = 0) as a comparativemethod is also adopted to calculate shear strength parame-ters (in Figure 6) The indicator to assess the predicted suit-ability of Copula-basedmethod andHoek-Brown criterion isthe root mean square errors (RMSE) obtained by

RMSE119891

= radic1

119899

119899

sum119894=1

(119891119894

minus 1198911015840119894

)2

RMSE119888

= radic1

119899

119899

sum119894=1

(119888119894

minus 1198881015840119894

)2

(18)


0

02

04

06

08

10

12

14

16

18

20

13 14 15 17 18

f

16 2119 20

Test set number

In situ test Hoek-Brown criterion

RMSEf = 0464 by Hoek-Brown criterionRMSEf = 0220 by Copula-based method with 120573 = 08

Copula-based method with 120573 = 08

(a)

13 14 15 17 1816 2119 20

Test set number

0

1

2

3

4

5

6

7

8

9

10

c(M

Pa)



RMSEc = 4089 by Hoek-Brown criterionRMSEc = 0775 by Copula-based method with 120573 = 08

(b)

Figure 6 The shear strength parameters derived from Copula-based method and Heok-Brown criterion (a) friction coefficient 119891 and(b) cohesion 119888

Table 9 The results of other 9 sets of in situ direct shear tests

Setnumber

Rock mass quality indices Shear strength parameters119876 GSI 119891 119888MPa

13 47 578 104 1614 53 596 098 2415 67 615 085 23116 254 758 143 2117 60 603 105 3918 73 621 107 21419 146 702 125 26620 234 763 132 2221 331 803 193 168

where 119891119894

and 119888119894

are obtained from the in situ shear test and1198911015840119894

and 1198881015840119894

are derived from Copula-based method or Hoek-Brown criterion Clearly the smaller the RMSE is the moresuitable the estimated result is

Figure 6 indicates that the estimated shear strengthparameters by Copula-based method are closer to in situtest values than those by Hoek-Brown criterion for the rockmass specimens in Table 9 which means that the proposedCopula-based method would be a more effective tool inestimating shear strength parameters of the marbles involvedin this paper The Hoek-Brown criterion with a wide rangeof applications is empirically established on the basis of thetest results of a large number of different types of rock massthe calculated accuracies of 119891 and 119888 are however seriouslyaffected by the uncertain test conditions of 120590

3119899

(see (1)) andundetermined disturbance factor 119863 and so forth (see (1)ndash(3))[45 46]However the optimal Copula-based fitting functionsestablished in the paper are on the basis of probabilistic

analysis on the in situ direct shear test results of the 12 sets ofsimilar marbles under the same test conditions which shouldshow better estimations of shear strength parameters of therock mass with similar geological andmechanical properties

5 Conclusion

Using the proposedCopula-basedmethod the shear strengthparameters with a given 120573 can be estimated provided that 119876has been obtained The calculated guarantee rate 120573 can bealso used to analyze the suitability of the estimated valuesfrom the statistical methods for direct shear test data (such aspredominant slopemethod and the least squaremethod) Forthe marbles tested in this paper compared with Hoek-Browncriterion the Copula-basedmethod is amore effective tool inestimating shear strength parameters

The paper only considers the relationships between twoof the parameters Actually the Copula functions could be ofmultidimensions and can be used to evaluate the parametersrelated with multiple variables Furthermore the Copulafunctions can be used for reliability analysis of estimated rockmass parameters

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study is supported by the National Natural ScienceFoundation of China (41172243 and 41472245) and theFundamental Research Funds for the Central Universities(CDJZR12205501 and 106112014CDJZR200009)


References

[1] H Sonmez C Gokceoglu H A Nefeslioglu and A KayabasildquoEstimation of rock modulus For intact rocks with an artificialneural network and for rock masses with a new empirical equa-tionrdquo International Journal of Rock Mechanics and Mining Sci-ences vol 43 no 2 pp 224ndash235 2006

[2] J ShenM Karakus and C Xu ldquoA comparative study for empir-ical equations in estimating deformation modulus of rockmassesrdquo Tunnelling and Underground Space Technology vol 32pp 245ndash250 2012

[3] K Zhang P Cao and R Bao ldquoRigorous back analysis of shearstrength parameters of landslide sliprdquo Transactions of Nonfer-rous Metals Society of China (English Edition) vol 23 no 5 pp1459ndash1464 2013

[4] J-H Wu and P-H Tsai ldquoNew dynamic procedure for back-calculating the shear strength parameters of large landslidesrdquoEngineering Geology vol 123 no 1-2 pp 29ndash147 2011

[5] E Hoek P K Kaiser andW F Bawden Support of UndergroundExcavations in Hard Rock Balkema Rotterdam The Nether-lands 1995

[6] M Cai P K Kaiser H Uno Y Tasaka and M MinamildquoEstimation of rock mass deformation modulus and strength ofjointed hard rock masses using the GSI systemrdquo InternationalJournal of RockMechanics andMining Sciences vol 41 no 1 pp3ndash19 2004

[7] X-L Yang and J-H Yin ldquoSlope equivalent Mohr-Coulombstrength parameters for rockmasses satisfying theHoek-Browncriterionrdquo Rock Mechanics and Rock Engineering vol 43 no 4pp 505ndash511 2010

[8] B Singh M N Viladkar N K Samadhiya and V K MehrotraldquoRock mass strength parameters mobilised in tunnelsrdquo Tun-nelling and Underground Space Technology vol 12 no 1 pp 47ndash54 1997

[9] M Hashemi S Moghaddas and R Ajalloeian ldquoApplicationof rock mass characterization for determining the mechanicalproperties of rock mass a comparative studyrdquo Rock Mechanicsand Rock Engineering vol 43 no 3 pp 305ndash320 2010

[10] J L Justo E Justo J M Azanon P Durand and A MoralesldquoThe use of rock mass classification systems to estimate themodulus and strength of jointed rockrdquoRockMechanics andRockEngineering vol 43 no 3 pp 287ndash304 2010

[11] R J Watters D R Zimbelman S D Bowman and J KCrowley ldquoRock mass strength assessment and significance toedifice stability Mount Rainier and Mount Hood CascadeRange volcanoesrdquo Pure and Applied Geophysics vol 157 no 6-8pp 957ndash976 2000

[12] N Barton ldquoSome newQ-value correlations to assist in site char-acterisation and tunnel designrdquo International Journal of RockMechanics andMining Sciences vol 39 no 2 pp 185ndash216 2002

[13] E Hoek C Carranza-Torres and B Corkum ldquoHoek-Brownfailure criterion-2002 editionrdquo in Proceedings of the 5th NorthAmerican Rock Mechanics Symposium vol 1 pp 267ndash273Toronto Canada 2002

[14] L Zhang Multivariate hydrological frequency analysis and riskmapping [PhD thesis] The Louisiana State University andAgricultural and Mechanical College Baton Rouge La USA2005

[15] F Gong X Li and J Deng ldquoProbability distribution of smallsamples of geotechnical parameters using normal informationspread methodrdquo Chinese Journal of Rock Mechanics and Engi-neering vol 25 no 12 pp 2559ndash2564 2006 (Chinese)

[16] A J Patton ldquoModelling asymmetric exchange rate depen-dencerdquo International Economic Review vol 47 no 2 pp 527ndash556 2006

[17] J C Rodriguez ldquoMeasuring financial contagion a copulaapproachrdquo Journal of Empirical Finance vol 14 no 3 pp 401ndash423 2007

[18] J T Shiau ldquoFitting drought duration and severity with two-dimensional copulasrdquoWater ResourcesManagement vol 20 no5 pp 795ndash815 2006

[19] F Serinaldi ldquoCopula-based mixed models for bivariate rainfalldata an empirical study in regression perspectiverdquoHandbook ofFinancial Time Series vol 23 no 5 pp 767ndash785 2009

[20] S Song and V P Singh ldquoFrequency analysis of droughts usingthe Plackett copula and parameter estimation by genetic algo-rithmrdquo Stochastic Environmental Research and Risk Assessmentvol 24 no 5 pp 783ndash805 2010

[21] X-S Tang D-Q Li G Rong K-K Phoon and C-B ZhouldquoImpact of copula selection on geotechnical reliability underincomplete probability informationrdquo Computers and Geotech-nics vol 49 pp 264ndash278 2013

[22] SW Qi F QWu F Z Yan andH X Lan ldquoMechanism of deepcracks in the left bank slope of Jinping first stage hydropowerstationrdquo Engineering Geology vol 73 no 1-2 pp 129ndash144 2004

[23] N W Xu C A Tang L C Li et al ldquoMicroseismic monitoringand stability analysis of the left bank slope in Jinping firststage hydropower station in southwestern Chinardquo InternationalJournal of Rock Mechanics and Mining Sciences vol 48 no 6pp 950ndash963 2011

[24] P K Robertson ldquoIn situ testing and its application to foundationengineeringrdquo Canadian Geotechnical Journal vol 23 no 4 pp573ndash594 1986

[25] A Palmstroslashm ldquoCharacterizing rock masses by the RMi for Usein Practical Rock Engineering Part 1 the development of theRock Mass index (RMi)rdquo Tunnelling and Underground SpaceTechnology vol 11 no 2 pp 175ndash188 1996

[26] A Palmstroslashm ldquoCharacterizing rock masses by the RMi for usein practical rock engineering Part 2 somepractical applicationsof the rock mass index (RMi)rdquo Tunnelling and UndergroundSpace Technology vol 11 no 3 pp 287ndash303 1996

[27] L Legendre and P Legendre ldquoNumerical ecologyrdquo in Devel-opments in Environmental Modelling Elsevier Scientific BVAmsterdam The Netherlands 2nd edition 1998

[28] E L Lehmann ldquoSome concepts of dependencerdquo The Annals ofMathematical Statistics vol 37 pp 1137ndash1153 1966

[29] E L Lehmann Nonparametrics Statistical Methods Based onBanks Test Holden-Day San Francisco Calif USA 1975

[30] B Singh andRGoel Engineering RockMass Classification Tun-nelling Foundations and Landslides EIsevier 2011

[31] G Mollon D Dias and A-H Soubra ldquoProbabilistic analysisof circular tunnels in homogeneous soil using response surfacemethodologyrdquo Journal of Geotechnical and GeoenvironmentalEngineering vol 135 no 9 pp 1314ndash1325 2009

[32] H F Schweiger R Thurner and R Pottler ldquoReliability analysisin geotechnics with deterministic finite elements theoreticalconcepts and practical applicationrdquo International Journal ofGeomechanics vol 1 no 4 pp 389ndash413 2001

[33] S C Kao and R S Govindaraju ldquoA copula-based joint deficitindex for droughtsrdquo Journal of Hydrology vol 380 no 1-2 pp121ndash134 2010

[34] J Deng X B Li and G S Gu ldquoA distribution-free methodusing maximum entropy and moments for estimating proba-bility curves of rock variablesrdquo International Journal of Rock


Mechanics and Mining Sciences vol 41 supplement 1 pp 1ndash62004

[35] XWang Y You and Y Tang ldquoAn approach to calculate optimalwindow-width serving for the information diffusion techniquerdquoInternational Journal of General Systems vol 33 no 2-3 pp223ndash231 2004

[36] F A Andrade I I Esat and M N M Badi ldquoGear conditionmonitoring by a new application of the Kolmogorov-Smirnovtestrdquo Proceedings of the Institution of Mechanical Engineers PartC Journal of Mechanical Engineering Science vol 215 no 6 pp653ndash661 2001

[37] H X Chen P S K Chua and G H Lim ldquoFault degradationassessment ofwater hydraulicmotor by impulse vibration signalwith Wavelet Packet Analysis and Kolmogorov-Smirnov TestrdquoMechanical Systems and Signal Processing vol 22 no 7 pp1670ndash1684 2008

[38] F J Massey ldquoTheKolmogorov-Smirnov test for goodness of fitrdquoJournal of the American Statistical Association vol 46 no 253pp 68ndash78 1951

[39] R B NelsenAn instruction to Copulas Springer NewYork NYUSA 2006

[40] H Akaike ldquoA new look at the statistical model identificationrdquoIEEE Transactions on Automatic Control vol 19 no 6 pp 716ndash723 1974

[41] N Sugiura ldquoFurther analysts of the data by akaikersquos informationcriterion and the finite corrections further analysts of the databy akaikersquosrdquoCommunications in StatisticsmdashTheory andMethodsvol 7 no 1 pp 13ndash26 1978

[42] O Okhrin Y Okhrin and W Schmid ldquoOn the structure andestimation of hierarchical Archimedean copulasrdquo Journal ofEconometrics vol 173 no 2 pp 189ndash204 2013

[43] N Barton ldquoReview of a new shear-strength criterion for rockjointsrdquo Engineering Geology vol 7 no 4 pp 287ndash332 1973

[44] Design Specification for Roller Compacted Concrete Dams SL314-2004 China Water amp Power Press Beijing China 2004(Chinese)

[45] E Hoek C Carranza-torres and B Corkum ldquoHoek-Brownfailure criterionrdquo in Proceedings of the NARMS-TACConferencepp 267ndash273 Toronto Canada 2002

[46] Y H Song and G H Ju ldquoDetermination of rock mass shearstrength based on in-situ tests and codes and comparison withestimation by Hoek-Brown criterionrdquo Chinese Journal of RockMechanics and Engineering vol 31 no 5 pp 1000ndash1006 2012(Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


where GSI is the Geological Strength Index and 119863 is adisturbance factor depending on the degree of disturbanceby which the rock mass has been subjected to blast damageand stress relaxationThe 119863 varies from zero for undisturbedin situ rock mass to 1 for very disturbed rock mass 119904 and 119886 in(1) can be calculated by

119886 =1

2+

1

6(119890minusGSI15 minus 119890minus203)

119904 = exp(GSI minus 100

9 minus 3119863)

(3)

In (1)120590119888

is uniaxial compressive strength1205903119899

= 1205903max120590

119888

and 120590

3max is the upper-limit value of confining stress dis-cussed by Hoek et al [13]

It is important to take full advantage of in situ test data inorder to establish the correlativity between geological indicesand mechanical parameters of rock mass With regard tothe correlativity of the above variables an effective way isto establish their joint distribution function The commonstatistical methods for obtaining joint distribution functionof variables are generally based on two assumptions that isthe variables have the samemarginal distributions and followthe normal distribution [14] However rock mass param-eters do not always follow the normal distribution especiallyas the number of samples is small [15] With respect to smallsamples normal information spread estimation method(NISEM) [15] is suitable Meanwhile Copula theory [16ndash21] is a fairly effective method to establish the joint distri-bution function of nonnormal distribution variables Theadvantages of Copula theory are that the marginal distribu-tions and dependence structure of variables can be studiedseparately and the variables can follow any distributions TheCopula theory has been widely used in many fields such asinsurance [16] finance [17] hydrology [18ndash20] and geotech-nicalgeological engineering [21]

In this paper the correlations between the rock massquality indices (ie rock quality designation (RQD) rockmass quality (119876) and Rock Block Index (RBI)) and theshear strength parameters (119891 and 119888) are investigated basedon the results of in situ direct shear tests at Jinping I-StageHydropower Station in China It shows that 119876 presents thestrongest correlation with the shear strength parametersThen the joint distribution functions of 119876 and 119891 119876 and 119888 areproposed based on Copula theory By means of the functionsand the given guarantee rates the shear strength parameterscan be estimated Compared with those estimated by Hoek-Brown criterion the estimated values using the proposedCopula-based method are closer to the in situ tests ones

2 Correlations between Rock Mass QualityIndices and Shear Strength

21 In Situ Test Data The Jinping Hydropower Station islocated on the middle reach of the Yalong River in SichuanChina It has a double-curvature arch dam with a height of305m and a total installed capacity of 3300MW [22 23]

Shear box

Sample

Roller rows

Jack

Steel plate

Mortar

Transfer column

Jack

Steel plate

Roller rows

Marble

J1 Bedding jointsJ2 Cross jointsJ1

J2

Sheared surface

Figure 1 The sketch of the in situ direct shear test

The Hydropower Station is located in the east edge of theTibet Plateau tectonic belt Due to the rapid crust upliftduring the Quaternary the steep slopes exist on both sidesof the valley deeply cut by Yalong River The stratum in thedam zone consists of a series of metamorphic rocks of middleof Triassic period of which the great majority are marbleswith grey orwhite andfine-grained textureThemainmineralcompositions of the marble are calcite and dolomite Theuniaxial compressive strength of the freshmarble is about 60ndash75MPa

In order to investigate the shear strength parameters ofthe representative marbles 12 sets of in situ direct shear testswith five similar specimens in each set were performed inthe exploration adits of Jinping I-Stage Hydropower StationFigure 1 shows the sketch of in situ direct shear test whichwasperformed using flat stacking method [24] The side lengthof each cube specimen is 1m which contains in general twosets of joints (bedding joints and cross joints see Figure 1)RQD and RBI [25 26] of each specimen can be obtainedusing scanline field mapping method The scanline used formeasuring RQD was parallel to shear direction The sheardirection was parallel to the bedding joint along which theshear deformation happened The five similar specimens in aset were tested at different normal stress 120590

119899

from 2 to 10MPaThe shear strength parameters (119891 and 119888) of each set areobtained by linear regression according to Mohr-Coulombcriterion A good fitting straight line between the peak shearstress 120591 and normal stress 120590

119899

can be derived from each set testdata for which the 1198772 of fitting degree is greater than 09 (seeFigure 2) The test results are shown in Table 1 in which therock mass quality indices are the average of each set samplesrespectively

22 Correlation Measure Pearson linear correlation coeffi-cient 120574

119899

[27] Spearmanrsquos rank correlation coefficient 120588119899

[28]and Kendallrsquos rank correlation coefficient 120591

119899

[29] are widely


0

4

8

12

16

20

0 2 4 6 8 10

120591(M

Pa)

Shea

r stre

ss

120591 = 179120590n + 195

R2 = 09546





Setnumber


RQD RBI 119876 119891 119888MPa1 753 176 305 179 1952 893 315 238 138 1983 968 416 387 117 1504 993 318 397 220 2255 964 434 386 139 1806 981 226 265 160 1807 598 80 81 180 2808 560 103 52 117 1209 659 143 89 123 24110 774 132 52 056 30011 679 90 54 107 32512 518 70 46 115 432


120574119899

=sum119899

119894=1

(1199091119894

minus 1199091

) (1199092119894

minus 1199092

)

119899radic11990421

sdot 11990422

120588119899

=12

119899 (119899 minus 1) (119899 + 1)

119899

sum119894=1

119877119894

119878119894

minus 3119899 + 1

119899 minus 1

120591119899

= (119899

2)minus1

sum119894lt119895

sign [(1199091119894

minus 1199091119895

) (1199092119894

minus 1199092119895

)]

(4)


(1199091119895

) and 1199092119894

(1199092119895


1

and 1199092

) 1199091

and 1199092


1

and 11990422


and 119878119894


1119894

and 1199092119894

sign = 1 if 1199091119894

lt 1199091119895

and 1199092119894

lt 1199092119895


1119894

= 1199091119895

and 1199092119894

= 1199092119895



120588119899

and 120591119899




Spearman 120588119899

Kendall 120591119899






1

119883119894

119883119896





Reject


Accept








K-S test as (10)


as 119865119883119894

(119909119894

) = 119875119883119894

(119883119894

le 119909119894


119894


(1198831

119883119894

119883119896

) 119865119883119894

(119909119894


119883119894(119883119894

le 119909119894


le 119909119894


1

1198832

119883119896


(1199091

119909119894

119909119896

)

= 119875 (1198831

le 1199091

119883119894

le 119909119894

119883119896

le 119909119896

) (5)


119883119894(119909119894


1198671198831119883119894 119883119896

(1199091

119909119894

119909119896

)

= 119862 (1198651198831

(1199091

) 119865119883119894

(119909119894

) 119865119883119896

(119909119896

)) (6)


119883119894(119909119894


1

119883119894

119883119896


1198831(1199091

) 119865119883119894

(119909119894

) 119865119883119896

(119909119896







119883119894(119909119894






119899 120574 119899 120574

3 0849 321 800 11 1420 835 4434 1273 982 782 12 1420 269 5705 1698 643 675 13 1420 698 7956 1336 252 561 14 1420 669 6717 1445 461 208 15 1420 693 3218 1395 189 816 16 1420 692 2269 1422 962 345 17 1420 693 10110 1416 278 786 gt17 1420 693 101



RQD 0005 420 722 0013 293 110RBI 0007 073 744 0022 636 592119876 0007 335 735 0024 344 428119891 0157 002 612 11151 315 78119888 0082 527 014 3081 091 687



119894


119891 (119909119894

) =1

radic2120587119899ℎ

119899

sum119895=1

exp[

[

minus(119909119894

minus 119909119894119895

)2

2ℎ2]

]

(7)


119894119895



ℎ =120574 (119909119894max minus 119909

119894min)

119899 minus 1 (8)

where 119909119894max and 119909


119894



119894



Variables 119899 120572 119863119899120572

119863119899

Accept orreject




119883119894(119909119894


119865119883119894

(119909119894

) = int+infin

minusinfin

119891 (119909119894

) 119889119909119894

(9)


119883119894(119909119894


119899

119863119899

= max 10038161003816100381610038161003816119865119883119894 (119909119894

) minus 119865119883119894

(119909119894

)10038161003816100381610038161003816 (10)


(119909119894

) is defined as

119865119883119894

(119909119894

) =1

119899

119899

sum119894=1

119868119909119894119895le119909119894

(11)

where 119868119909119894119895le119909119894


le 119909119894


119883119894(119909119894


119899

lt 119863119899120572


119899120572


119899120572


119899120572




119899


119899

and 120579 is [39]

120591119899

= 4 int1

0

int1

0

119862 (119906119894

V119894

120579) 119889119862 (119906119894

V119894

120579) minus 1 (12)


00 02 04 06 08 1000

02

04

06

08

10

T2f

T1Q

(a)

00 02 04 06 08 1000

02

04

06

08

10

T3c

T1Q

(b)


119876 versus 1198792

119891 and (b) 1198791

119876 versus 1198793

119888

where 119906119894

and V119894


and 1199092119894

(ie 119906119894

= 1198651198831

(1199091119894

) and V119894

= 1198651198832

(1199092119894

)) while119862(119906119894

V119894



AIC = 119899 ln1

119899 minus 119905

119899

sum119894=1

[119865emp (1199091119894

1199092119894

) minus 119862 (119906119894

V119894

)]2

+ 2119905

(13)


119894

V119894


2119894


119865emp (1199091119894

1199092119894

) = 119875 (1198831

le 1199091119894

1198832

le 1199092119894

) =1

119899 + 1

119899

sum119895=1

119899

sum119897=1

119873119895119897

(14)

where 119873119895119897


1

le 1199091119894

as well as 1198832

le 1199092119894



1199092


1198792



and1198792

are expressed as

1199051119894

=rank (119909

1119894

)

119899 + 1

1199052119894

=rank (119909

2119894

)

119899 + 1

(15)

where rank(1199091119894

) (or rank(1199092119894

) (1199093119894


(or 1199092119894

1199093119894

) among 1199091

(or 1199092

1199093119894


1119894

1199052119894


1198792

) or (1198791

1198793






minus1120579



0 4396 minus19240 1347 minus66184


1120579


Nelsen No 5 minus1

120579ln[1 +








]1120579

minus1



+ (Vminus1120579 minus 1)120579

]1120579

minus120579



+ (1 minus Vminus1120579)120579

]1120579

120579

0) 2698 minus65360 1173 minus60240






1

and1198881


and120573119888


and 119888 gt 1198881


120573119891

= 119875 (119891 gt 1198911

)

120573119888

= 119875 (119888 gt 1198881

) (16)


119891

and 120573119888


120573119891

= 119875 119891 gt 1198911

| 119876 = 1198760

= 1 minus 119875 119891 le 1198911

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)


119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17a)

120573119888

= 119875 119888 gt 1198881

| 119876 = 1198760

= 1 minus 119875 119888 le 1198881

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)


119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17b)

where 1198760



119899


119899


119899



119891

and 120573119888



119888



Upper-limit line

120591 = 135120590n + 640

120591 = 138120590n + 316

120591 = 135120590n + 200

Lower-limit line

Shea

r stre

ss120591

(MPa

)


2

5

10

15

20

(a)

Upper-limit line

120591 = 104120590n + 550

120591 = 089120590n + 419

120591 = 107120590n + 150

Lower-limit line

Shea

r stre

ss120591

(MPa


0 4 8 121

5

10

15

(b)





Setnumber


119891 119888 119891 119888



Setnumber 119876


Least squaremethod

120573119891

120573119888

120573119891

120573119888

1 305 0762 0397 0721 00372 238 0650 0564 0600 00703 387 0857 0187 0828 00134 397 0868 0206 0839 00155 386 0857 0209 0827 00156 265 0698 0499 0652 00557 81 0796 0963 0963 00548 52 0532 0980 0867 00989 89 0834 0957 0971 004610 52 0532 0980 0867 009811 54 0557 0979 0879 009412 46 0453 0982 0819 0110


119899



119899




RMSE119891

= radic1

119899

119899

sum119894=1

(119891119894

minus 1198911015840119894

)2

RMSE119888

= radic1

119899

119899

sum119894=1

(119888119894

minus 1198881015840119894

)2

(18)


0

02

04

06

08

10

12

14

16

18

20

13 14 15 17 18

f

16 2119 20

Test set number




(a)

13 14 15 17 1816 2119 20

Test set number

0

1

2

3

4

5

6

7

8

9

10

c(M

Pa)




(b)



Setnumber


13 47 578 104 1614 53 596 098 2415 67 615 085 23116 254 758 143 2117 60 603 105 3918 73 621 107 21419 146 702 125 26620 234 763 132 2221 331 803 193 168

where 119891119894

and 119888119894


and 1198881015840119894



3119899



5 Conclusion





Acknowledgments



References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

4

8

12

16

20

0 2 4 6 8 10

120591(M

Pa)

Shea

r stre

ss

120591 = 179120590n + 195

R2 = 09546





Setnumber


RQD RBI 119876 119891 119888MPa1 753 176 305 179 1952 893 315 238 138 1983 968 416 387 117 1504 993 318 397 220 2255 964 434 386 139 1806 981 226 265 160 1807 598 80 81 180 2808 560 103 52 117 1209 659 143 89 123 24110 774 132 52 056 30011 679 90 54 107 32512 518 70 46 115 432


120574119899

=sum119899

119894=1

(1199091119894

minus 1199091

) (1199092119894

minus 1199092

)

119899radic11990421

sdot 11990422

120588119899

=12

119899 (119899 minus 1) (119899 + 1)

119899

sum119894=1

119877119894

119878119894

minus 3119899 + 1

119899 minus 1

120591119899

= (119899

2)minus1

sum119894lt119895

sign [(1199091119894

minus 1199091119895

) (1199092119894

minus 1199092119895

)]

(4)


(1199091119895

) and 1199092119894

(1199092119895


1

and 1199092

) 1199091

and 1199092


1

and 11990422


and 119878119894


1119894

and 1199092119894

sign = 1 if 1199091119894

lt 1199091119895

and 1199092119894

lt 1199092119895


1119894

= 1199091119895

and 1199092119894

= 1199092119895



120588119899

and 120591119899




Spearman 120588119899

Kendall 120591119899






1

119883119894

119883119896





Reject


Accept








K-S test as (10)


as 119865119883119894

(119909119894

) = 119875119883119894

(119883119894

le 119909119894


119894


(1198831

119883119894

119883119896

) 119865119883119894

(119909119894


119883119894(119883119894

le 119909119894


le 119909119894


1

1198832

119883119896


(1199091

119909119894

119909119896

)

= 119875 (1198831

le 1199091

119883119894

le 119909119894

119883119896

le 119909119896

) (5)


119883119894(119909119894


1198671198831119883119894 119883119896

(1199091

119909119894

119909119896

)

= 119862 (1198651198831

(1199091

) 119865119883119894

(119909119894

) 119865119883119896

(119909119896

)) (6)


119883119894(119909119894


1

119883119894

119883119896


1198831(1199091

) 119865119883119894

(119909119894

) 119865119883119896

(119909119896







119883119894(119909119894






119899 120574 119899 120574

3 0849 321 800 11 1420 835 4434 1273 982 782 12 1420 269 5705 1698 643 675 13 1420 698 7956 1336 252 561 14 1420 669 6717 1445 461 208 15 1420 693 3218 1395 189 816 16 1420 692 2269 1422 962 345 17 1420 693 10110 1416 278 786 gt17 1420 693 101



RQD 0005 420 722 0013 293 110RBI 0007 073 744 0022 636 592119876 0007 335 735 0024 344 428119891 0157 002 612 11151 315 78119888 0082 527 014 3081 091 687



119894


119891 (119909119894

) =1

radic2120587119899ℎ

119899

sum119895=1

exp[

[

minus(119909119894

minus 119909119894119895

)2

2ℎ2]

]

(7)


119894119895



ℎ =120574 (119909119894max minus 119909

119894min)

119899 minus 1 (8)

where 119909119894max and 119909


119894



119894



Variables 119899 120572 119863119899120572

119863119899

Accept orreject




119883119894(119909119894


119865119883119894

(119909119894

) = int+infin

minusinfin

119891 (119909119894

) 119889119909119894

(9)


119883119894(119909119894


119899

119863119899

= max 10038161003816100381610038161003816119865119883119894 (119909119894

) minus 119865119883119894

(119909119894

)10038161003816100381610038161003816 (10)


(119909119894

) is defined as

119865119883119894

(119909119894

) =1

119899

119899

sum119894=1

119868119909119894119895le119909119894

(11)

where 119868119909119894119895le119909119894


le 119909119894


119883119894(119909119894


119899

lt 119863119899120572


119899120572


119899120572


119899120572




119899


119899

and 120579 is [39]

120591119899

= 4 int1

0

int1

0

119862 (119906119894

V119894

120579) 119889119862 (119906119894

V119894

120579) minus 1 (12)


00 02 04 06 08 1000

02

04

06

08

10

T2f

T1Q

(a)

00 02 04 06 08 1000

02

04

06

08

10

T3c

T1Q

(b)


119876 versus 1198792

119891 and (b) 1198791

119876 versus 1198793

119888

where 119906119894

and V119894


and 1199092119894

(ie 119906119894

= 1198651198831

(1199091119894

) and V119894

= 1198651198832

(1199092119894

)) while119862(119906119894

V119894



AIC = 119899 ln1

119899 minus 119905

119899

sum119894=1

[119865emp (1199091119894

1199092119894

) minus 119862 (119906119894

V119894

)]2

+ 2119905

(13)


119894

V119894


2119894


119865emp (1199091119894

1199092119894

) = 119875 (1198831

le 1199091119894

1198832

le 1199092119894

) =1

119899 + 1

119899

sum119895=1

119899

sum119897=1

119873119895119897

(14)

where 119873119895119897


1

le 1199091119894

as well as 1198832

le 1199092119894



1199092


1198792



and1198792

are expressed as

1199051119894

=rank (119909

1119894

)

119899 + 1

1199052119894

=rank (119909

2119894

)

119899 + 1

(15)

where rank(1199091119894

) (or rank(1199092119894

) (1199093119894


(or 1199092119894

1199093119894

) among 1199091

(or 1199092

1199093119894


1119894

1199052119894


1198792

) or (1198791

1198793






minus1120579



0 4396 minus19240 1347 minus66184


1120579


Nelsen No 5 minus1

120579ln[1 +








]1120579

minus1



+ (Vminus1120579 minus 1)120579

]1120579

minus120579



+ (1 minus Vminus1120579)120579

]1120579

120579

0) 2698 minus65360 1173 minus60240






1

and1198881


and120573119888


and 119888 gt 1198881


120573119891

= 119875 (119891 gt 1198911

)

120573119888

= 119875 (119888 gt 1198881

) (16)


119891

and 120573119888


120573119891

= 119875 119891 gt 1198911

| 119876 = 1198760

= 1 minus 119875 119891 le 1198911

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)


119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17a)

120573119888

= 119875 119888 gt 1198881

| 119876 = 1198760

= 1 minus 119875 119888 le 1198881

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)


119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17b)

where 1198760



119899


119899


119899



119891

and 120573119888



119888



Upper-limit line

120591 = 135120590n + 640

120591 = 138120590n + 316

120591 = 135120590n + 200

Lower-limit line

Shea

r stre

ss120591

(MPa

)


2

5

10

15

20

(a)

Upper-limit line

120591 = 104120590n + 550

120591 = 089120590n + 419

120591 = 107120590n + 150

Lower-limit line

Shea

r stre

ss120591

(MPa


0 4 8 121

5

10

15

(b)





Setnumber


119891 119888 119891 119888



Setnumber 119876


Least squaremethod

120573119891

120573119888

120573119891

120573119888

1 305 0762 0397 0721 00372 238 0650 0564 0600 00703 387 0857 0187 0828 00134 397 0868 0206 0839 00155 386 0857 0209 0827 00156 265 0698 0499 0652 00557 81 0796 0963 0963 00548 52 0532 0980 0867 00989 89 0834 0957 0971 004610 52 0532 0980 0867 009811 54 0557 0979 0879 009412 46 0453 0982 0819 0110


119899



119899




RMSE119891

= radic1

119899

119899

sum119894=1

(119891119894

minus 1198911015840119894

)2

RMSE119888

= radic1

119899

119899

sum119894=1

(119888119894

minus 1198881015840119894

)2

(18)


0

02

04

06

08

10

12

14

16

18

20

13 14 15 17 18

f

16 2119 20

Test set number




(a)

13 14 15 17 1816 2119 20

Test set number

0

1

2

3

4

5

6

7

8

9

10

c(M

Pa)




(b)



Setnumber


13 47 578 104 1614 53 596 098 2415 67 615 085 23116 254 758 143 2117 60 603 105 3918 73 621 107 21419 146 702 125 26620 234 763 132 2221 331 803 193 168

where 119891119894

and 119888119894


and 1198881015840119894



3119899



5 Conclusion





Acknowledgments



References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Reject


Accept








K-S test as (10)


as 119865119883119894

(119909119894

) = 119875119883119894

(119883119894

le 119909119894


119894


(1198831

119883119894

119883119896

) 119865119883119894

(119909119894


119883119894(119883119894

le 119909119894


le 119909119894


1

1198832

119883119896


(1199091

119909119894

119909119896

)

= 119875 (1198831

le 1199091

119883119894

le 119909119894

119883119896

le 119909119896

) (5)


119883119894(119909119894


1198671198831119883119894 119883119896

(1199091

119909119894

119909119896

)

= 119862 (1198651198831

(1199091

) 119865119883119894

(119909119894

) 119865119883119896

(119909119896

)) (6)


119883119894(119909119894


1

119883119894

119883119896


1198831(1199091

) 119865119883119894

(119909119894

) 119865119883119896

(119909119896







119883119894(119909119894






119899 120574 119899 120574

3 0849 321 800 11 1420 835 4434 1273 982 782 12 1420 269 5705 1698 643 675 13 1420 698 7956 1336 252 561 14 1420 669 6717 1445 461 208 15 1420 693 3218 1395 189 816 16 1420 692 2269 1422 962 345 17 1420 693 10110 1416 278 786 gt17 1420 693 101



RQD 0005 420 722 0013 293 110RBI 0007 073 744 0022 636 592119876 0007 335 735 0024 344 428119891 0157 002 612 11151 315 78119888 0082 527 014 3081 091 687



119894


119891 (119909119894

) =1

radic2120587119899ℎ

119899

sum119895=1

exp[

[

minus(119909119894

minus 119909119894119895

)2

2ℎ2]

]

(7)


119894119895



ℎ =120574 (119909119894max minus 119909

119894min)

119899 minus 1 (8)

where 119909119894max and 119909


119894



119894



Variables 119899 120572 119863119899120572

119863119899

Accept orreject




119883119894(119909119894


119865119883119894

(119909119894

) = int+infin

minusinfin

119891 (119909119894

) 119889119909119894

(9)


119883119894(119909119894


119899

119863119899

= max 10038161003816100381610038161003816119865119883119894 (119909119894

) minus 119865119883119894

(119909119894

)10038161003816100381610038161003816 (10)


(119909119894

) is defined as

119865119883119894

(119909119894

) =1

119899

119899

sum119894=1

119868119909119894119895le119909119894

(11)

where 119868119909119894119895le119909119894


le 119909119894


119883119894(119909119894


119899

lt 119863119899120572


119899120572


119899120572


119899120572




119899


119899

and 120579 is [39]

120591119899

= 4 int1

0

int1

0

119862 (119906119894

V119894

120579) 119889119862 (119906119894

V119894

120579) minus 1 (12)


00 02 04 06 08 1000

02

04

06

08

10

T2f

T1Q

(a)

00 02 04 06 08 1000

02

04

06

08

10

T3c

T1Q

(b)


119876 versus 1198792

119891 and (b) 1198791

119876 versus 1198793

119888

where 119906119894

and V119894


and 1199092119894

(ie 119906119894

= 1198651198831

(1199091119894

) and V119894

= 1198651198832

(1199092119894

)) while119862(119906119894

V119894



AIC = 119899 ln1

119899 minus 119905

119899

sum119894=1

[119865emp (1199091119894

1199092119894

) minus 119862 (119906119894

V119894

)]2

+ 2119905

(13)


119894

V119894


2119894


119865emp (1199091119894

1199092119894

) = 119875 (1198831

le 1199091119894

1198832

le 1199092119894

) =1

119899 + 1

119899

sum119895=1

119899

sum119897=1

119873119895119897

(14)

where 119873119895119897


1

le 1199091119894

as well as 1198832

le 1199092119894



1199092


1198792



and1198792

are expressed as

1199051119894

=rank (119909

1119894

)

119899 + 1

1199052119894

=rank (119909

2119894

)

119899 + 1

(15)

where rank(1199091119894

) (or rank(1199092119894

) (1199093119894


(or 1199092119894

1199093119894

) among 1199091

(or 1199092

1199093119894


1119894

1199052119894


1198792

) or (1198791

1198793






minus1120579



0 4396 minus19240 1347 minus66184


1120579


Nelsen No 5 minus1

120579ln[1 +








]1120579

minus1



+ (Vminus1120579 minus 1)120579

]1120579

minus120579



+ (1 minus Vminus1120579)120579

]1120579

120579

0) 2698 minus65360 1173 minus60240






1

and1198881


and120573119888


and 119888 gt 1198881


120573119891

= 119875 (119891 gt 1198911

)

120573119888

= 119875 (119888 gt 1198881

) (16)


119891

and 120573119888


120573119891

= 119875 119891 gt 1198911

| 119876 = 1198760

= 1 minus 119875 119891 le 1198911

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)


119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17a)

120573119888

= 119875 119888 gt 1198881

| 119876 = 1198760

= 1 minus 119875 119888 le 1198881

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)


119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17b)

where 1198760



119899


119899


119899



119891

and 120573119888



119888



Upper-limit line

120591 = 135120590n + 640

120591 = 138120590n + 316

120591 = 135120590n + 200

Lower-limit line

Shea

r stre

ss120591

(MPa

)


2

5

10

15

20

(a)

Upper-limit line

120591 = 104120590n + 550

120591 = 089120590n + 419

120591 = 107120590n + 150

Lower-limit line

Shea

r stre

ss120591

(MPa


0 4 8 121

5

10

15

(b)





Setnumber


119891 119888 119891 119888



Setnumber 119876


Least squaremethod

120573119891

120573119888

120573119891

120573119888

1 305 0762 0397 0721 00372 238 0650 0564 0600 00703 387 0857 0187 0828 00134 397 0868 0206 0839 00155 386 0857 0209 0827 00156 265 0698 0499 0652 00557 81 0796 0963 0963 00548 52 0532 0980 0867 00989 89 0834 0957 0971 004610 52 0532 0980 0867 009811 54 0557 0979 0879 009412 46 0453 0982 0819 0110


119899



119899




RMSE119891

= radic1

119899

119899

sum119894=1

(119891119894

minus 1198911015840119894

)2

RMSE119888

= radic1

119899

119899

sum119894=1

(119888119894

minus 1198881015840119894

)2

(18)


0

02

04

06

08

10

12

14

16

18

20

13 14 15 17 18

f

16 2119 20

Test set number




(a)

13 14 15 17 1816 2119 20

Test set number

0

1

2

3

4

5

6

7

8

9

10

c(M

Pa)




(b)



Setnumber


13 47 578 104 1614 53 596 098 2415 67 615 085 23116 254 758 143 2117 60 603 105 3918 73 621 107 21419 146 702 125 26620 234 763 132 2221 331 803 193 168

where 119891119894

and 119888119894


and 1198881015840119894



3119899



5 Conclusion





Acknowledgments



References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899 120574 119899 120574

3 0849 321 800 11 1420 835 4434 1273 982 782 12 1420 269 5705 1698 643 675 13 1420 698 7956 1336 252 561 14 1420 669 6717 1445 461 208 15 1420 693 3218 1395 189 816 16 1420 692 2269 1422 962 345 17 1420 693 10110 1416 278 786 gt17 1420 693 101



RQD 0005 420 722 0013 293 110RBI 0007 073 744 0022 636 592119876 0007 335 735 0024 344 428119891 0157 002 612 11151 315 78119888 0082 527 014 3081 091 687



119894


119891 (119909119894

) =1

radic2120587119899ℎ

119899

sum119895=1

exp[

[

minus(119909119894

minus 119909119894119895

)2

2ℎ2]

]

(7)


119894119895



ℎ =120574 (119909119894max minus 119909

119894min)

119899 minus 1 (8)

where 119909119894max and 119909


119894



119894



Variables 119899 120572 119863119899120572

119863119899

Accept orreject




119883119894(119909119894


119865119883119894

(119909119894

) = int+infin

minusinfin

119891 (119909119894

) 119889119909119894

(9)


119883119894(119909119894


119899

119863119899

= max 10038161003816100381610038161003816119865119883119894 (119909119894

) minus 119865119883119894

(119909119894

)10038161003816100381610038161003816 (10)


(119909119894

) is defined as

119865119883119894

(119909119894

) =1

119899

119899

sum119894=1

119868119909119894119895le119909119894

(11)

where 119868119909119894119895le119909119894


le 119909119894


119883119894(119909119894


119899

lt 119863119899120572


119899120572


119899120572


119899120572




119899


119899

and 120579 is [39]

120591119899

= 4 int1

0

int1

0

119862 (119906119894

V119894

120579) 119889119862 (119906119894

V119894

120579) minus 1 (12)


00 02 04 06 08 1000

02

04

06

08

10

T2f

T1Q

(a)

00 02 04 06 08 1000

02

04

06

08

10

T3c

T1Q

(b)


119876 versus 1198792

119891 and (b) 1198791

119876 versus 1198793

119888

where 119906119894

and V119894


and 1199092119894

(ie 119906119894

= 1198651198831

(1199091119894

) and V119894

= 1198651198832

(1199092119894

)) while119862(119906119894

V119894



AIC = 119899 ln1

119899 minus 119905

119899

sum119894=1

[119865emp (1199091119894

1199092119894

) minus 119862 (119906119894

V119894

)]2

+ 2119905

(13)


119894

V119894


2119894


119865emp (1199091119894

1199092119894

) = 119875 (1198831

le 1199091119894

1198832

le 1199092119894

) =1

119899 + 1

119899

sum119895=1

119899

sum119897=1

119873119895119897

(14)

where 119873119895119897


1

le 1199091119894

as well as 1198832

le 1199092119894



1199092


1198792



and1198792

are expressed as

1199051119894

=rank (119909

1119894

)

119899 + 1

1199052119894

=rank (119909

2119894

)

119899 + 1

(15)

where rank(1199091119894

) (or rank(1199092119894

) (1199093119894


(or 1199092119894

1199093119894

) among 1199091

(or 1199092

1199093119894


1119894

1199052119894


1198792

) or (1198791

1198793






minus1120579



0 4396 minus19240 1347 minus66184


1120579


Nelsen No 5 minus1

120579ln[1 +








]1120579

minus1



+ (Vminus1120579 minus 1)120579

]1120579

minus120579



+ (1 minus Vminus1120579)120579

]1120579

120579

0) 2698 minus65360 1173 minus60240






1

and1198881


and120573119888


and 119888 gt 1198881


120573119891

= 119875 (119891 gt 1198911

)

120573119888

= 119875 (119888 gt 1198881

) (16)


119891

and 120573119888


120573119891

= 119875 119891 gt 1198911

| 119876 = 1198760

= 1 minus 119875 119891 le 1198911

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)


119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17a)

120573119888

= 119875 119888 gt 1198881

| 119876 = 1198760

= 1 minus 119875 119888 le 1198881

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)


119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17b)

where 1198760



119899


119899


119899



119891

and 120573119888



119888



Upper-limit line

120591 = 135120590n + 640

120591 = 138120590n + 316

120591 = 135120590n + 200

Lower-limit line

Shea

r stre

ss120591

(MPa

)


2

5

10

15

20

(a)

Upper-limit line

120591 = 104120590n + 550

120591 = 089120590n + 419

120591 = 107120590n + 150

Lower-limit line

Shea

r stre

ss120591

(MPa


0 4 8 121

5

10

15

(b)





Setnumber


119891 119888 119891 119888



Setnumber 119876


Least squaremethod

120573119891

120573119888

120573119891

120573119888

1 305 0762 0397 0721 00372 238 0650 0564 0600 00703 387 0857 0187 0828 00134 397 0868 0206 0839 00155 386 0857 0209 0827 00156 265 0698 0499 0652 00557 81 0796 0963 0963 00548 52 0532 0980 0867 00989 89 0834 0957 0971 004610 52 0532 0980 0867 009811 54 0557 0979 0879 009412 46 0453 0982 0819 0110


119899



119899




RMSE119891

= radic1

119899

119899

sum119894=1

(119891119894

minus 1198911015840119894

)2

RMSE119888

= radic1

119899

119899

sum119894=1

(119888119894

minus 1198881015840119894

)2

(18)


0

02

04

06

08

10

12

14

16

18

20

13 14 15 17 18

f

16 2119 20

Test set number




(a)

13 14 15 17 1816 2119 20

Test set number

0

1

2

3

4

5

6

7

8

9

10

c(M

Pa)




(b)



Setnumber


13 47 578 104 1614 53 596 098 2415 67 615 085 23116 254 758 143 2117 60 603 105 3918 73 621 107 21419 146 702 125 26620 234 763 132 2221 331 803 193 168

where 119891119894

and 119888119894


and 1198881015840119894



3119899



5 Conclusion





Acknowledgments



References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00 02 04 06 08 1000

02

04

06

08

10

T2f

T1Q

(a)

00 02 04 06 08 1000

02

04

06

08

10

T3c

T1Q

(b)


119876 versus 1198792

119891 and (b) 1198791

119876 versus 1198793

119888

where 119906119894

and V119894


and 1199092119894

(ie 119906119894

= 1198651198831

(1199091119894

) and V119894

= 1198651198832

(1199092119894

)) while119862(119906119894

V119894



AIC = 119899 ln1

119899 minus 119905

119899

sum119894=1

[119865emp (1199091119894

1199092119894

) minus 119862 (119906119894

V119894

)]2

+ 2119905

(13)


119894

V119894


2119894


119865emp (1199091119894

1199092119894

) = 119875 (1198831

le 1199091119894

1198832

le 1199092119894

) =1

119899 + 1

119899

sum119895=1

119899

sum119897=1

119873119895119897

(14)

where 119873119895119897


1

le 1199091119894

as well as 1198832

le 1199092119894



1199092


1198792



and1198792

are expressed as

1199051119894

=rank (119909

1119894

)

119899 + 1

1199052119894

=rank (119909

2119894

)

119899 + 1

(15)

where rank(1199091119894

) (or rank(1199092119894

) (1199093119894


(or 1199092119894

1199093119894

) among 1199091

(or 1199092

1199093119894


1119894

1199052119894


1198792

) or (1198791

1198793






minus1120579



0 4396 minus19240 1347 minus66184


1120579


Nelsen No 5 minus1

120579ln[1 +








]1120579

minus1



+ (Vminus1120579 minus 1)120579

]1120579

minus120579



+ (1 minus Vminus1120579)120579

]1120579

120579

0) 2698 minus65360 1173 minus60240






1

and1198881


and120573119888


and 119888 gt 1198881


120573119891

= 119875 (119891 gt 1198911

)

120573119888

= 119875 (119888 gt 1198881

) (16)


119891

and 120573119888


120573119891

= 119875 119891 gt 1198911

| 119876 = 1198760

= 1 minus 119875 119891 le 1198911

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)


119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17a)

120573119888

= 119875 119888 gt 1198881

| 119876 = 1198760

= 1 minus 119875 119888 le 1198881

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)


119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17b)

where 1198760



119899


119899


119899



119891

and 120573119888



119888



Upper-limit line

120591 = 135120590n + 640

120591 = 138120590n + 316

120591 = 135120590n + 200

Lower-limit line

Shea

r stre

ss120591

(MPa

)


2

5

10

15

20

(a)

Upper-limit line

120591 = 104120590n + 550

120591 = 089120590n + 419

120591 = 107120590n + 150

Lower-limit line

Shea

r stre

ss120591

(MPa


0 4 8 121

5

10

15

(b)





Setnumber


119891 119888 119891 119888



Setnumber 119876


Least squaremethod

120573119891

120573119888

120573119891

120573119888

1 305 0762 0397 0721 00372 238 0650 0564 0600 00703 387 0857 0187 0828 00134 397 0868 0206 0839 00155 386 0857 0209 0827 00156 265 0698 0499 0652 00557 81 0796 0963 0963 00548 52 0532 0980 0867 00989 89 0834 0957 0971 004610 52 0532 0980 0867 009811 54 0557 0979 0879 009412 46 0453 0982 0819 0110


119899



119899




RMSE119891

= radic1

119899

119899

sum119894=1

(119891119894

minus 1198911015840119894

)2

RMSE119888

= radic1

119899

119899

sum119894=1

(119888119894

minus 1198881015840119894

)2

(18)


0

02

04

06

08

10

12

14

16

18

20

13 14 15 17 18

f

16 2119 20

Test set number




(a)

13 14 15 17 1816 2119 20

Test set number

0

1

2

3

4

5

6

7

8

9

10

c(M

Pa)




(b)



Setnumber


13 47 578 104 1614 53 596 098 2415 67 615 085 23116 254 758 143 2117 60 603 105 3918 73 621 107 21419 146 702 125 26620 234 763 132 2221 331 803 193 168

where 119891119894

and 119888119894


and 1198881015840119894



3119899



5 Conclusion





Acknowledgments



References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












minus1120579



0 4396 minus19240 1347 minus66184


1120579


Nelsen No 5 minus1

120579ln[1 +








]1120579

minus1



+ (Vminus1120579 minus 1)120579

]1120579

minus120579



+ (1 minus Vminus1120579)120579

]1120579

120579

0) 2698 minus65360 1173 minus60240






1

and1198881


and120573119888


and 119888 gt 1198881


120573119891

= 119875 (119891 gt 1198911

)

120573119888

= 119875 (119888 gt 1198881

) (16)


119891

and 120573119888


120573119891

= 119875 119891 gt 1198911

| 119876 = 1198760

= 1 minus 119875 119891 le 1198911

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)


119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17a)

120573119888

= 119875 119888 gt 1198881

| 119876 = 1198760

= 1 minus 119875 119888 le 1198881

| 119876 = 1198760

= 1 minus 119862 (119880 le 119906 | 119881 = V)


119862 (119906 V + ΔV) minus 119862 (119906 V)

ΔV

= 1 minus120597

120597V119862 (119906 V)

10038161003816100381610038161003816100381610038161003816V=V

(17b)

where 1198760



119899


119899


119899



119891

and 120573119888



119888



Upper-limit line

120591 = 135120590n + 640

120591 = 138120590n + 316

120591 = 135120590n + 200

Lower-limit line

Shea

r stre

ss120591

(MPa

)


2

5

10

15

20

(a)

Upper-limit line

120591 = 104120590n + 550

120591 = 089120590n + 419

120591 = 107120590n + 150

Lower-limit line

Shea

r stre

ss120591

(MPa


0 4 8 121

5

10

15

(b)





Setnumber


119891 119888 119891 119888



Setnumber 119876


Least squaremethod

120573119891

120573119888

120573119891

120573119888

1 305 0762 0397 0721 00372 238 0650 0564 0600 00703 387 0857 0187 0828 00134 397 0868 0206 0839 00155 386 0857 0209 0827 00156 265 0698 0499 0652 00557 81 0796 0963 0963 00548 52 0532 0980 0867 00989 89 0834 0957 0971 004610 52 0532 0980 0867 009811 54 0557 0979 0879 009412 46 0453 0982 0819 0110


119899



119899




RMSE119891

= radic1

119899

119899

sum119894=1

(119891119894

minus 1198911015840119894

)2

RMSE119888

= radic1

119899

119899

sum119894=1

(119888119894

minus 1198881015840119894

)2

(18)


0

02

04

06

08

10

12

14

16

18

20

13 14 15 17 18

f

16 2119 20

Test set number




(a)

13 14 15 17 1816 2119 20

Test set number

0

1

2

3

4

5

6

7

8

9

10

c(M

Pa)




(b)



Setnumber


13 47 578 104 1614 53 596 098 2415 67 615 085 23116 254 758 143 2117 60 603 105 3918 73 621 107 21419 146 702 125 26620 234 763 132 2221 331 803 193 168

where 119891119894

and 119888119894


and 1198881015840119894



3119899



5 Conclusion





Acknowledgments



References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Upper-limit line

120591 = 135120590n + 640

120591 = 138120590n + 316

120591 = 135120590n + 200

Lower-limit line

Shea

r stre

ss120591

(MPa

)


2

5

10

15

20

(a)

Upper-limit line

120591 = 104120590n + 550

120591 = 089120590n + 419

120591 = 107120590n + 150

Lower-limit line

Shea

r stre

ss120591

(MPa


0 4 8 121

5

10

15

(b)





Setnumber


119891 119888 119891 119888



Setnumber 119876


Least squaremethod

120573119891

120573119888

120573119891

120573119888

1 305 0762 0397 0721 00372 238 0650 0564 0600 00703 387 0857 0187 0828 00134 397 0868 0206 0839 00155 386 0857 0209 0827 00156 265 0698 0499 0652 00557 81 0796 0963 0963 00548 52 0532 0980 0867 00989 89 0834 0957 0971 004610 52 0532 0980 0867 009811 54 0557 0979 0879 009412 46 0453 0982 0819 0110


119899



119899




RMSE119891

= radic1

119899

119899

sum119894=1

(119891119894

minus 1198911015840119894

)2

RMSE119888

= radic1

119899

119899

sum119894=1

(119888119894

minus 1198881015840119894

)2

(18)


0

02

04

06

08

10

12

14

16

18

20

13 14 15 17 18

f

16 2119 20

Test set number




(a)

13 14 15 17 1816 2119 20

Test set number

0

1

2

3

4

5

6

7

8

9

10

c(M

Pa)




(b)



Setnumber


13 47 578 104 1614 53 596 098 2415 67 615 085 23116 254 758 143 2117 60 603 105 3918 73 621 107 21419 146 702 125 26620 234 763 132 2221 331 803 193 168

where 119891119894

and 119888119894


and 1198881015840119894



3119899



5 Conclusion





Acknowledgments



References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

02

04

06

08

10

12

14

16

18

20

13 14 15 17 18

f

16 2119 20

Test set number




(a)

13 14 15 17 1816 2119 20

Test set number

0

1

2

3

4

5

6

7

8

9

10

c(M

Pa)




(b)



Setnumber


13 47 578 104 1614 53 596 098 2415 67 615 085 23116 254 758 143 2117 60 603 105 3918 73 621 107 21419 146 702 125 26620 234 763 132 2221 331 803 193 168

where 119891119894

and 119888119894


and 1198881015840119894



3119899



5 Conclusion





Acknowledgments



References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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