CorrectionofLine-SamplingBiasofRockDiscontinuity...
10
Research Article CorrectionofLine-SamplingBiasofRockDiscontinuity OrientationsUsingaModifiedTerzaghiMethod HuimingTang, 1 JunrongZhang, 1 LeiHuang , 1 ChangbinYan , 2 ShengmingHu, 3 andMutasimAdamMohamedEzEldin 1 1 Department of Engineering Geology and Geotechnical Engineering, Faculty of Engineering, China University of Geosciences, Wuhan, Hubei 430074, China 2 School of Civil Engineering, Zhengzhou University, Zhengzhou, Henan 450003, China 3 JiangXi Engineering Research Center of Water Engineering Safety and Resources Efficient Utilization, Nanchang Institute of Technology, Nanchang, Jiangxi 330099, China Correspondence should be addressed to Lei Huang; [email protected] and Changbin Yan; [email protected] Received 20 December 2017; Accepted 8 May 2018; Published 29 August 2018 Academic Editor: Rihong Cao Copyright©2018HuimingTangetal.isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e Terzaghi method is widely used to correct the line-sampling bias of rock discontinuity orientations. e method includes four procedures, one of which is meshing the stereographic projection diagram into cells. e method is based on the bias- compensatory factor, 1/sin θ, where θ is the angle between the scanline and the discontinuity defined at each cell center. is paper presents a modified Terzaghi method that eliminates meshing, thereby reducing the method to three steps that (1) count the frequencies, (2) weigh the frequencies by the bias-compensatory factor, and (3) round the weighed frequencies to the nearest integer. Due to the elimination of the mesh, the counting object has changed to the frequency at each pole, and θ in the bias- compensatory factor is redefined as the angle between the scanline and the discontinuity at each pole. e applicability of the redefined bias-compensatory factor is verified through a mathematical logical deduction. e accuracy of the conventional and the modified Terzaghi methods are compared using a case study in Wenchuan, China, revealing improved accuracy for the latter. 1.Introduction Many important engineering structures are built on or in rock, such as tunnels, dam foundations, bridge foundations, tall buildings, etc. Rock is an inhomogeneous natural ma- terial owing to the presence of discontinuities, defined as any significant mechanical break or fracture of negligible tensile strength [1]. Discontinuities include bedding planes, faults, fissures, fractures, joints, etc. e orientation of disconti- nuities is known to greatly affect the kinematical and me- chanical behavior of rock [2–5]. ese orientations can be observed on rock exposures such as outcrops, tunnel faces, pit faces, and drillhole faces. A common observational method is line sampling down a borehole or along a scanline on an exposure. However, such sampling biases the apparent abundance of the orientations, because the probability that a particular discontinuity is intersected by a scanline depends on its orientation [6, 7]. To correct for this bias, Terzaghi [8] proposed a well-known method that weighs the observed frequencies by a bias-compensatory factor, which is the reciprocal of the sine of the intersection angle be- tween the scanline and discontinuity. A detailed de- scriptionofthismethodisavailableinEinsteinandBeacher [9, 10]. e Terzaghi method has been widely used by researchers, including Goodman [11], Park [12], and Fouch´ e and Diebolt [13]. Tangetal.[14]foundthattheTerzaghimethodinvolvesan error, even if optimizing countermeasures are applied. Un- fortunately, the origin of the error is unknown, but the meshing procedure is suspected to be a significant component. is paper presents a modified Terzaghi method that eliminates the meshing procedure. Following a brief in- troduction of the conventional Terzaghi method, we de- scribe this modified method and its internal procedures. Hindawi Advances in Civil Engineering Volume 2018, Article ID 1629039, 9 pages https://doi.org/10.1155/2018/1629039
Transcript of CorrectionofLine-SamplingBiasofRockDiscontinuity...
Research ArticleCorrection of Line-Sampling Bias of Rock DiscontinuityOrientations Using a Modified Terzaghi Method
Huiming Tang1 Junrong Zhang1 Lei Huang 1 Changbin Yan 2 Shengming Hu3
and Mutasim Adam Mohamed Ez Eldin1
1Department of Engineering Geology and Geotechnical Engineering Faculty of Engineering China University of GeosciencesWuhan Hubei 430074 China2School of Civil Engineering Zhengzhou University Zhengzhou Henan 450003 China3JiangXi Engineering Research Center of Water Engineering Safety and Resources Efficient UtilizationNanchang Institute of Technology Nanchang Jiangxi 330099 China
Correspondence should be addressed to Lei Huang huangleicugeducn and Changbin Yan yanchangbinzzueducn
Received 20 December 2017 Accepted 8 May 2018 Published 29 August 2018
Academic Editor Rihong Cao
Copyright copy 2018Huiming Tang et al)is 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
)e Terzaghi method is widely used to correct the line-sampling bias of rock discontinuity orientations)emethod includes fourprocedures one of which is meshing the stereographic projection diagram into cells )e method is based on the bias-compensatory factor 1sin θ where θ is the angle between the scanline and the discontinuity defined at each cell center)is paperpresents a modified Terzaghi method that eliminates meshing thereby reducing the method to three steps that (1) count thefrequencies (2) weigh the frequencies by the bias-compensatory factor and (3) round the weighed frequencies to the nearestinteger Due to the elimination of the mesh the counting object has changed to the frequency at each pole and θ in the bias-compensatory factor is redefined as the angle between the scanline and the discontinuity at each pole )e applicability of theredefined bias-compensatory factor is verified through amathematical logical deduction)e accuracy of the conventional and themodified Terzaghi methods are compared using a case study in Wenchuan China revealing improved accuracy for the latter
1 Introduction
Many important engineering structures are built on or inrock such as tunnels dam foundations bridge foundationstall buildings etc Rock is an inhomogeneous natural ma-terial owing to the presence of discontinuities defined as anysignificant mechanical break or fracture of negligible tensilestrength [1] Discontinuities include bedding planes faultsfissures fractures joints etc )e orientation of disconti-nuities is known to greatly affect the kinematical and me-chanical behavior of rock [2ndash5] )ese orientations can beobserved on rock exposures such as outcrops tunnel facespit faces and drillhole faces A common observationalmethod is line sampling down a borehole or along a scanlineon an exposure However such sampling biases the apparentabundance of the orientations because the probability thata particular discontinuity is intersected by a scanline
depends on its orientation [6 7] To correct for this biasTerzaghi [8] proposed a well-known method that weighs theobserved frequencies by a bias-compensatory factor whichis the reciprocal of the sine of the intersection angle be-tween the scanline and discontinuity A detailed de-scription of this method is available in Einstein and Beacher[9 10] )e Terzaghi method has been widely used byresearchers including Goodman [11] Park [12] andFouche and Diebolt [13]
Tang et al [14] found that the Terzaghimethod involves anerror even if optimizing countermeasures are applied Un-fortunately the origin of the error is unknown but themeshing procedure is suspected to be a significant component
)is paper presents a modified Terzaghi method thateliminates the meshing procedure Following a brief in-troduction of the conventional Terzaghi method we de-scribe this modified method and its internal procedures
HindawiAdvances in Civil EngineeringVolume 2018 Article ID 1629039 9 pageshttpsdoiorg10115520181629039
Owing to mesh elimination some procedural modificationsare required Second the bias-compensatory factor had beenreobtained in the case of the presence of mesh [8] so wedetermine whether this factor is still applicable in the no-mesh case using a mathematical logical deduction Finallywe apply both the conventional and the modified Terzaghimethods to a test case and compare the accuracy of the tworesults
2 Modified Terzaghi Method
21 Procedures )e procedures of the conventional Ter-zaghi method [8] are (Figure 1(a)) as follows
(1) Subdivide the projection net into cells(2) Count the frequencies lying in each cell(3) Weigh the frequencies by the bias-compensatory
factor 1sin θ where θ is the intersection angle be-tween the scanline and the discontinuity defined ateach cell center Since the orientation is composed oftwo elements that is the dip direction and dip anglethe weighing should be executed for dip directionand dip angle respectively
(4) Because the frequencies defined in this manner mustbe integers the weighed frequencies that are roundedto the nearest integer
)e procedures of the modified Terzaghi method are(Figure 1(b)) as follows
(1) Count the frequencies at each pole Owing to meshelimination the counting objects have changed tothe frequencies at each pole
(2) Weigh the frequencies by the bias-compensatoryfactor 1sin θ where θ is redefined as the angle be-tween the scanline and the discontinuity at each pole
(3) Round the weighed frequencies to the nearestinteger
22 Applicability of the Redefined Bias-Compensatory Factor)e kernel of the modified Terzaghi method is the bias-compensatory factor )is factor is known to be 1sin θ forthe conventional Terzaghi method [8] but its applicability tothe modified Terzaghi method with redefinition remainsunknown In this section a mathematical deduction is usedto verify its applicability
For the conventional Terzaghi method the weighingprocedure is obtained by multiplying the bias-compensatoryfactor 1sin θ by the observed frequency defined in the cell[8] )is step can be formulated as follows
PA 1113938α+Δαα 1113938
β+Δββ pAB(α β) dαdβ
sin θ
P
sin θ (1)
where α is the dip direction β is the dip angle ∆α is thecell size on dip direction ∆β is the cell size on dip angle(Figure 2) PA is the corrected frequency in cell pAB(α β) is
the joint probability density that the dip direction and dipangle are intersected by scanline and P is the observedfrequency in cell
For the modified Terzaghi method the weighing pro-cedure is to multiply 1sin θ by the observed frequency ateach orientation pole Such a pole in calculus can beregarded as a special cell that is an infinitesimal cell aroundthe pole (∆α⟶ 0deg ∆β⟶ 0deg) In this case P and PA inEquation (1) become the observed frequency and thecorrected frequency at the orientation pole respectively)erefore despite a slight redefinition of θ the bias-compensatory factor can be also used for the modifiedTerzaghi method
3 Application of theModified TerzaghiMethod
)euse of the modified Terzaghi method is illustrated and itsaccuracy is tested against the conventional Terzaghi methodusing a real example of bedding orientation observations ina roadcut in China )e study area is near Yingxiu town inWenchuan Sichuan Province and located only about1800m east of the epicenter of the 2008 WenchuanEarthquake (Figure 3) )e particular roadcut is 11m long5m wide and 6m high and consists of Upper Triassic lithicarkose of the Xujiahe Formation )e rock has two dis-continuity sets one of which is the bedding plane
A scanline with the trendplunge of 10815deg was fixed toobserve the bedding planes on this outcrop (Figure 4)Table 1 lists 55 observed orientations whose pole diagram isshown in Figure 5
First the sampling bias of the observed orientations wascorrected according to the conventional and the modifiedTerzaghi methods A cell size of 2deg times 2deg was selected for theconventional Terzaghi method because Tang [15] concludedthat this size optimizes the accuracy )e result corrected bythe conventional Terzaghi method is shown in Figure 6(a)and the result corrected by the modified Terzaghi method isshown in Figure 6(b) In addition the volumetric abun-dance diameter and aperture were calculated with theresults listed in Table 2
Next a three-dimensional model of the rock was con-structed by discrete fracture network modeling as describedby Xu and Dowd [16] Brzovic and Villaescusa [17] andGrenon and Hadjigeorgiou [18] Multiplying the volumetricabundance (10mminus3) by the volume of the simulated zone(400m3) in Table 2 results in a total of 4000 discontinuities)en pseudorandom numbers of these 4000 discontinuitieswere generated for the five elements namely the X-co-ordinate Y-coordinate Z-coordinate diameter and aper-ture )ese pseudorandom numbers are not listed herebecause of space limitations After entering the pseudo-random numbers and the corrected orientation data intomodeling software such as OpenGL or AutoCAD twomodels can be built corresponding to the conventional andthe modified Terzaghi methods (Figure 7) A scanline withthe same orientation as the field scanline was applied to themodel outcrop and the discontinuities that are intersected by
2 Advances in Civil Engineering
1 31211
12
2
14 3414261512
1125
241 31321
13
2
Poles
(1) Meshing (2) Counting
(3) Weighing
(4) Rounding
(a)
Poles 1111
1
11 11
2
2
1 13151214
11
16131211
26
2411
1211
1
21 11
3
2
1
(1) Counting (2) Weighing
(3) Rounding
(b)
Figure 1 Procedures of (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Advances in Civil Engineering 3
this scanline were then ldquomeasuredrdquo )e quantity of thesemeasured discontinuities is set equal to the number ofobserved discontinuities in the field To distinguish betweenthese measured discontinuities and the real discontinuitiesobserved in the field the former are named ldquomodeledrdquodiscontinuities Figure 8 shows the modeled discontinuityorientations
)ird the distribution difference between the observedand the modeled orientations was tested by theKolmogorovndashSmirnov two-sample test )is nonparametric
hypothesis test evaluates the difference between the cumu-lative distribution functions of two sample data vectors )etest can be executed by the software Statistical Product andService Solutions and returns an asymptotic significance tocharacterize the difference )e significance ranges from 0 to1 the higher the significance the lower the difference Moreinformation about this test is given in Ozcomak et al [19] andAl-Labadi and Zarepour [20]
)e test returned two significances corresponding to thedip direction and dip angle )ese two significances werecombined and their average is 0686 for the conventionalTerzaghi method and 0988 for the modified Terzaghimethod It is apparent that the latter value is much higherthan the former value demonstrating that the modifiedTerzaghi method is more accurate than the conventionalmethod even if the optimal cell size of 2deg times 2deg is chosen forthe conventional method
4 Discussion
Tang [14] presented the accuracy of the conventional Terzaghimethod in two cases
Shanghai
Beijing
China
Yingxiu TownWenchuan Country
S
Yingxiu Town
Duwen Expressway
317 National Road
Min River
213 National Road
DaoliupoHejia Mountain
Huangjiacun
Gancang
Madi
Dashuigou
Caijiagangcun
Study area
1km
N
1200
m
1600m
1600
m
1400
m
1400
m 1600
m
1800m
1400
m
1200m1000
m
Epicenter of the WenchuanEarthquake 2008
Figure 3 Map of the Wenchuan area showing the location of the study area )e blue line is the Min River the thick black lines denote the213 National Road and the 317 National Road the medium black lines are rural roads and the thin black lines are topographic contours aslabeled
αβ
Cell
Δα
Δβ
Figure 2 Mesh of projection net
ScanlineTrendplunge
10815deg
Outcrop lithic arkoseDip directionangle
19270deg
Figure 4 Photograph of outcrop and the scanline used forobservation
4 Advances in Civil Engineering
Table 1 Data of observed orientations
Discontinuity Dip directionangle (deg) Discontinuity Dip directionangle (deg)1 12873 29 139662 13470 30 138653 13570 31 135704 13371 32 137725 13075 33 137776 13371 34 129747 13274 35 145788 14572 36 138709 13376 37 1306910 13167 38 1337111 13574 39 1367012 14172 40 1327013 13875 41 1347214 13069 42 1357715 12773 43 1276916 13780 44 1297317 13873 45 1297118 13278 46 1447219 13773 47 1397120 13173 48 1427521 13169 49 1377622 13271 50 1437423 13769 51 1447124 13279 52 1417925 13874 53 1447326 13177 54 1417527 13273 55 1408028 13378 mdash mdash
Fisherconcentration
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
Figure 5 Pole diagram of 55 observed orientations )is diagram was plotted by the software Dips )is is an equal-angle projection of theupper hemisphere showing equal-density rings shaped by Fisher concentrations
Advances in Civil Engineering 5
Case 1 Ifn⟶infin in otherwords if the cell size is infinitesimal
BD
pA(α β) dαdβ 1k
limn⟶infin
1113944
n
i1
Pi
sin θci (2)
Case 2 If n does not approach infin
BD
pA(α β) dαdβ asymp1k
1113944
n
i1
Pi
sin θci (3)
Fisherconcentrations
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
(a)
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 6 Pole diagram of orientations corrected by (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Table 2 Volumetric abundance diameter aperture and size of simulated zone
Volumetric abundancea (mminus3) Diameter (m) Aperture (mm)Simulated zone
Length (m) Width (m) Height (m)4 Exp (05)b Exp (32) 10 10 10aVolumetric abundancenumber of discontinuity centers per rock volume bExp (k) exponential distribution where k is the reciprocal of mean
6 Advances in Civil Engineering
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(a)
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(b)
Figure 7 Rock models corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Fisherconcentrations
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
(a)
Figure 8 Continued
Advances in Civil Engineering 7
where D is a given orientation interval and pA(α β) is thejoint probability density of the dip direction and dip angle inthe rockmass
As mentioned in Section 22 the elimination of mesh inthe modified Terzaghi method is identical to meshing theprojection diagram into infinitesimal cells (Case 1 forEquation (2)) Tang [14] proved that Equation (2) is accu-rate while Equation (3) introduces error Hence themodified Terzaghi method is a more accurate procedure Incontrast for the conventional Terzaghi method the cells canonly approach but can never attain an infinitesimal sizeSuch a mesh can only meet Case 2 and not Case 1 so theconventional Terzaghi method will consequently involveerror )is analysis reinforces the result regarding accuracyobtained in Section 3
As shown in Section 3 the significance of the modifiedTerzaghi method approaches but does not attain unitysuggesting that factors in addition to meshing contributeerror to the Terzaghi method One inferred source of error isrounding the corrected frequencies to the nearest integer)is issue will be studied in the future
5 Conclusions
)e Terzaghi method is widely used to correct the observeddistribution of rock discontinuity orientations A modifiedTerzaghi method that eliminates the meshing procedure wasdeveloped to improve accuracy )is modified method onlyincludes three procedures that (1) count the frequencies ateach pole (2) weigh the observed frequencies by the bias-compensatory factor and (3) round the weighed frequenciesto the nearest integer Mesh elimination requires that thecounting objects are changed to the frequencies at each pole
instead of the frequencies lying in each cell accordingly thebias-compensatory factor has become the reciprocal ofthe sine of the intersection angle between the scanline andthe discontinuity at each pole rather than the reciprocalof the sine of the intersection angle between the scanline andthe discontinuity defined at each cell center A mathematicallogical deduction verifies that this slightly redefined bias-compensatory factor can be used in the modified Terzaghimethod
)e conventional and the modified Terzaghi methodswere both applied to a case study near Yingxiu in SichuanChina )e results indicate that the modified Terzaghimethod performed more accurately than the conventionalmethod
Disclosure
Mutasim Adam Mohamed Ez Eldin is presently at theSection of Engineering Geology Department of AppliedGeology Saudi Geological Survey Jeddah 21514 SaudiArabia
Conflicts of Interest
)e authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
)is research was supported by the Open Foundation ofJiangxi Engineering Research Center of Water EngineeringSafety and Resources Efficient Utilization (Grant noOF201602) Zhejiang Collaborative Innovation Center forPrevention and Control of Mountain Geologic Hazards
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 8 Pole diagram of 55 modeled orientations corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghimethod
8 Advances in Civil Engineering
(Grant no PCMGH-2017-Z03) and the National Key Re-search and Development Program of China (Grant no2017YFC1501300)
References
[1] J Liu X D Zhao and Z Xu ldquoIdentification of rock dis-continuity sets based on a modified affinity propagation al-gorithmrdquo International Journal of Rock Mechanics andMining Sciences vol 94 pp 32ndash42 2017
[2] X Wang L Huang C Yan and B Lian ldquoHKCV rheologicalconstitutive model of mudstone under dry and saturatedconditionsrdquo Advances in Civil Engineering vol 2018 ArticleID 2621658 10 pages 2018
[3] N Bar and N Barton ldquo)e Q-slope method for rock slopeengineeringrdquo Rock Mechanics and Rock Engineering vol 50no 12 pp 3307ndash3322 2017
[4] B H G Brady and E T Brown Rock Mechanics for Un-derground Mining Springer Dordrecht Netherlands 3rdedition 2006
[5] E Hoek Practical Rock Engineering Evert Hoek ConsultingEngineer Inc North Vancouver BC Canada 1st edition2007
[6] N H Tran ldquoFracture orientation characterization mini-mizing statistical modeling errorsrdquo Computational Statisticsand Data Analysis vol 51 no 6 pp 3187ndash3196 2007
[7] L Huang H Tang Q Tan et al ldquoA novel method for correctingscanline-observational bias of discontinuity orientationrdquo Sci-entific Reports vol 2016 no 6 Article ID 22942 2016
[8] R D Terzaghi ldquoSource of error in joint surveysrdquo Geo-technique vol 15 no 3 pp 287ndash304 1965
[9] H H Einstein and G B Baecher ldquoProbabilistic and statisticalmethods in engineering geologyrdquo Rock Mechanics and RockEngineering vol 16 no 1 pp 39ndash72 1983
[10] G B Baechaer ldquoStatistical analysis of rock mass fracturingrdquoMathematical Geology vol 15 pp 329ndash348 1983
[11] R Goodman ldquoToppling a fundamental failure mode indiscontinuous materials description and analysisrdquo in Pro-ceedings of Geo-Congress pp 2338ndash2368 San Diego CAUSA 2013
[12] H J Park T R West and I Woo ldquoProbabilistic analysis ofrock slope stability and random properties of discontinuityparameters Interstate Highway 40 Western North CarolinaUSArdquo Engineering Geology vol 79 no 3-4 pp 230ndash2502005
[13] O Fouche and J Diebolt ldquoDescribing the geometry of 3Dfracture systems by correcting for linear sampling biasrdquoMathematical Geology vol 36 no 1 pp 33ndash63 2004
[14] H M Tang L Huang A Bobet et al ldquoIdentification andmitigation of error in the Terzaghi bias correction for in-homogeneous material discontinuitiesrdquo Strength of Materialsvol 48 no 6 pp 825ndash833 2016
[15] H Tang L Huang C H Juang and J Zhang ldquoOptimizing theTerzaghi estimator of the 3D distribution of rock fractureorientationsrdquo Rock Mechanics and Rock Engineering vol 50no 8 pp 2085ndash2099 2017
[16] C Xu and P Dowd ldquoA new computer code for discretefracture network modelingrdquo Computers and Geosciencesvol 36 no 3 pp 292ndash301 2010
[17] A Brzovic and E Villaescusa ldquoRock mass characterizationand assessment of block-forming geological discontinuitiesduring caving of primary copper ore at the El Teniente mineChilerdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 4 pp 565ndash583 2007
[18] M Grenon and J Hadjigeorgiou ldquoDrift reinforcement designbased on discontinuity network modelingrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 40 no 6pp 833ndash845 2003
[19] M S Ozcomak M Kartal O Senger and A K CelikldquoComparison of the powers of the Kolmogorov-Smirnov two-sample test and the Mann-Whitney test for different Kurtosisand Skewness coefficients using the Monte Carlo simulationmethodrdquo Journal of Statistical and Econometric Methodsvol 2 pp 81ndash98 2013
[20] L Al-Labadi and M Zarepour ldquoTwo-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approachrdquoMathematical Methods of Statistics vol 26 no 3 pp 212ndash2252017
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Owing to mesh elimination some procedural modificationsare required Second the bias-compensatory factor had beenreobtained in the case of the presence of mesh [8] so wedetermine whether this factor is still applicable in the no-mesh case using a mathematical logical deduction Finallywe apply both the conventional and the modified Terzaghimethods to a test case and compare the accuracy of the tworesults
2 Modified Terzaghi Method
21 Procedures )e procedures of the conventional Ter-zaghi method [8] are (Figure 1(a)) as follows
(1) Subdivide the projection net into cells(2) Count the frequencies lying in each cell(3) Weigh the frequencies by the bias-compensatory
factor 1sin θ where θ is the intersection angle be-tween the scanline and the discontinuity defined ateach cell center Since the orientation is composed oftwo elements that is the dip direction and dip anglethe weighing should be executed for dip directionand dip angle respectively
(4) Because the frequencies defined in this manner mustbe integers the weighed frequencies that are roundedto the nearest integer
)e procedures of the modified Terzaghi method are(Figure 1(b)) as follows
(1) Count the frequencies at each pole Owing to meshelimination the counting objects have changed tothe frequencies at each pole
(2) Weigh the frequencies by the bias-compensatoryfactor 1sin θ where θ is redefined as the angle be-tween the scanline and the discontinuity at each pole
(3) Round the weighed frequencies to the nearestinteger
22 Applicability of the Redefined Bias-Compensatory Factor)e kernel of the modified Terzaghi method is the bias-compensatory factor )is factor is known to be 1sin θ forthe conventional Terzaghi method [8] but its applicability tothe modified Terzaghi method with redefinition remainsunknown In this section a mathematical deduction is usedto verify its applicability
For the conventional Terzaghi method the weighingprocedure is obtained by multiplying the bias-compensatoryfactor 1sin θ by the observed frequency defined in the cell[8] )is step can be formulated as follows
PA 1113938α+Δαα 1113938
β+Δββ pAB(α β) dαdβ
sin θ
P
sin θ (1)
where α is the dip direction β is the dip angle ∆α is thecell size on dip direction ∆β is the cell size on dip angle(Figure 2) PA is the corrected frequency in cell pAB(α β) is
the joint probability density that the dip direction and dipangle are intersected by scanline and P is the observedfrequency in cell
For the modified Terzaghi method the weighing pro-cedure is to multiply 1sin θ by the observed frequency ateach orientation pole Such a pole in calculus can beregarded as a special cell that is an infinitesimal cell aroundthe pole (∆α⟶ 0deg ∆β⟶ 0deg) In this case P and PA inEquation (1) become the observed frequency and thecorrected frequency at the orientation pole respectively)erefore despite a slight redefinition of θ the bias-compensatory factor can be also used for the modifiedTerzaghi method
3 Application of theModified TerzaghiMethod
)euse of the modified Terzaghi method is illustrated and itsaccuracy is tested against the conventional Terzaghi methodusing a real example of bedding orientation observations ina roadcut in China )e study area is near Yingxiu town inWenchuan Sichuan Province and located only about1800m east of the epicenter of the 2008 WenchuanEarthquake (Figure 3) )e particular roadcut is 11m long5m wide and 6m high and consists of Upper Triassic lithicarkose of the Xujiahe Formation )e rock has two dis-continuity sets one of which is the bedding plane
A scanline with the trendplunge of 10815deg was fixed toobserve the bedding planes on this outcrop (Figure 4)Table 1 lists 55 observed orientations whose pole diagram isshown in Figure 5
First the sampling bias of the observed orientations wascorrected according to the conventional and the modifiedTerzaghi methods A cell size of 2deg times 2deg was selected for theconventional Terzaghi method because Tang [15] concludedthat this size optimizes the accuracy )e result corrected bythe conventional Terzaghi method is shown in Figure 6(a)and the result corrected by the modified Terzaghi method isshown in Figure 6(b) In addition the volumetric abun-dance diameter and aperture were calculated with theresults listed in Table 2
Next a three-dimensional model of the rock was con-structed by discrete fracture network modeling as describedby Xu and Dowd [16] Brzovic and Villaescusa [17] andGrenon and Hadjigeorgiou [18] Multiplying the volumetricabundance (10mminus3) by the volume of the simulated zone(400m3) in Table 2 results in a total of 4000 discontinuities)en pseudorandom numbers of these 4000 discontinuitieswere generated for the five elements namely the X-co-ordinate Y-coordinate Z-coordinate diameter and aper-ture )ese pseudorandom numbers are not listed herebecause of space limitations After entering the pseudo-random numbers and the corrected orientation data intomodeling software such as OpenGL or AutoCAD twomodels can be built corresponding to the conventional andthe modified Terzaghi methods (Figure 7) A scanline withthe same orientation as the field scanline was applied to themodel outcrop and the discontinuities that are intersected by
2 Advances in Civil Engineering
1 31211
12
2
14 3414261512
1125
241 31321
13
2
Poles
(1) Meshing (2) Counting
(3) Weighing
(4) Rounding
(a)
Poles 1111
1
11 11
2
2
1 13151214
11
16131211
26
2411
1211
1
21 11
3
2
1
(1) Counting (2) Weighing
(3) Rounding
(b)
Figure 1 Procedures of (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Advances in Civil Engineering 3
this scanline were then ldquomeasuredrdquo )e quantity of thesemeasured discontinuities is set equal to the number ofobserved discontinuities in the field To distinguish betweenthese measured discontinuities and the real discontinuitiesobserved in the field the former are named ldquomodeledrdquodiscontinuities Figure 8 shows the modeled discontinuityorientations
)ird the distribution difference between the observedand the modeled orientations was tested by theKolmogorovndashSmirnov two-sample test )is nonparametric
hypothesis test evaluates the difference between the cumu-lative distribution functions of two sample data vectors )etest can be executed by the software Statistical Product andService Solutions and returns an asymptotic significance tocharacterize the difference )e significance ranges from 0 to1 the higher the significance the lower the difference Moreinformation about this test is given in Ozcomak et al [19] andAl-Labadi and Zarepour [20]
)e test returned two significances corresponding to thedip direction and dip angle )ese two significances werecombined and their average is 0686 for the conventionalTerzaghi method and 0988 for the modified Terzaghimethod It is apparent that the latter value is much higherthan the former value demonstrating that the modifiedTerzaghi method is more accurate than the conventionalmethod even if the optimal cell size of 2deg times 2deg is chosen forthe conventional method
4 Discussion
Tang [14] presented the accuracy of the conventional Terzaghimethod in two cases
Shanghai
Beijing
China
Yingxiu TownWenchuan Country
S
Yingxiu Town
Duwen Expressway
317 National Road
Min River
213 National Road
DaoliupoHejia Mountain
Huangjiacun
Gancang
Madi
Dashuigou
Caijiagangcun
Study area
1km
N
1200
m
1600m
1600
m
1400
m
1400
m 1600
m
1800m
1400
m
1200m1000
m
Epicenter of the WenchuanEarthquake 2008
Figure 3 Map of the Wenchuan area showing the location of the study area )e blue line is the Min River the thick black lines denote the213 National Road and the 317 National Road the medium black lines are rural roads and the thin black lines are topographic contours aslabeled
αβ
Cell
Δα
Δβ
Figure 2 Mesh of projection net
ScanlineTrendplunge
10815deg
Outcrop lithic arkoseDip directionangle
19270deg
Figure 4 Photograph of outcrop and the scanline used forobservation
4 Advances in Civil Engineering
Table 1 Data of observed orientations
Discontinuity Dip directionangle (deg) Discontinuity Dip directionangle (deg)1 12873 29 139662 13470 30 138653 13570 31 135704 13371 32 137725 13075 33 137776 13371 34 129747 13274 35 145788 14572 36 138709 13376 37 1306910 13167 38 1337111 13574 39 1367012 14172 40 1327013 13875 41 1347214 13069 42 1357715 12773 43 1276916 13780 44 1297317 13873 45 1297118 13278 46 1447219 13773 47 1397120 13173 48 1427521 13169 49 1377622 13271 50 1437423 13769 51 1447124 13279 52 1417925 13874 53 1447326 13177 54 1417527 13273 55 1408028 13378 mdash mdash
Fisherconcentration
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
Figure 5 Pole diagram of 55 observed orientations )is diagram was plotted by the software Dips )is is an equal-angle projection of theupper hemisphere showing equal-density rings shaped by Fisher concentrations
Advances in Civil Engineering 5
Case 1 Ifn⟶infin in otherwords if the cell size is infinitesimal
BD
pA(α β) dαdβ 1k
limn⟶infin
1113944
n
i1
Pi
sin θci (2)
Case 2 If n does not approach infin
BD
pA(α β) dαdβ asymp1k
1113944
n
i1
Pi
sin θci (3)
Fisherconcentrations
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
(a)
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 6 Pole diagram of orientations corrected by (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Table 2 Volumetric abundance diameter aperture and size of simulated zone
Volumetric abundancea (mminus3) Diameter (m) Aperture (mm)Simulated zone
Length (m) Width (m) Height (m)4 Exp (05)b Exp (32) 10 10 10aVolumetric abundancenumber of discontinuity centers per rock volume bExp (k) exponential distribution where k is the reciprocal of mean
6 Advances in Civil Engineering
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(a)
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(b)
Figure 7 Rock models corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Fisherconcentrations
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
(a)
Figure 8 Continued
Advances in Civil Engineering 7
where D is a given orientation interval and pA(α β) is thejoint probability density of the dip direction and dip angle inthe rockmass
As mentioned in Section 22 the elimination of mesh inthe modified Terzaghi method is identical to meshing theprojection diagram into infinitesimal cells (Case 1 forEquation (2)) Tang [14] proved that Equation (2) is accu-rate while Equation (3) introduces error Hence themodified Terzaghi method is a more accurate procedure Incontrast for the conventional Terzaghi method the cells canonly approach but can never attain an infinitesimal sizeSuch a mesh can only meet Case 2 and not Case 1 so theconventional Terzaghi method will consequently involveerror )is analysis reinforces the result regarding accuracyobtained in Section 3
As shown in Section 3 the significance of the modifiedTerzaghi method approaches but does not attain unitysuggesting that factors in addition to meshing contributeerror to the Terzaghi method One inferred source of error isrounding the corrected frequencies to the nearest integer)is issue will be studied in the future
5 Conclusions
)e Terzaghi method is widely used to correct the observeddistribution of rock discontinuity orientations A modifiedTerzaghi method that eliminates the meshing procedure wasdeveloped to improve accuracy )is modified method onlyincludes three procedures that (1) count the frequencies ateach pole (2) weigh the observed frequencies by the bias-compensatory factor and (3) round the weighed frequenciesto the nearest integer Mesh elimination requires that thecounting objects are changed to the frequencies at each pole
instead of the frequencies lying in each cell accordingly thebias-compensatory factor has become the reciprocal ofthe sine of the intersection angle between the scanline andthe discontinuity at each pole rather than the reciprocalof the sine of the intersection angle between the scanline andthe discontinuity defined at each cell center A mathematicallogical deduction verifies that this slightly redefined bias-compensatory factor can be used in the modified Terzaghimethod
)e conventional and the modified Terzaghi methodswere both applied to a case study near Yingxiu in SichuanChina )e results indicate that the modified Terzaghimethod performed more accurately than the conventionalmethod
Disclosure
Mutasim Adam Mohamed Ez Eldin is presently at theSection of Engineering Geology Department of AppliedGeology Saudi Geological Survey Jeddah 21514 SaudiArabia
Conflicts of Interest
)e authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
)is research was supported by the Open Foundation ofJiangxi Engineering Research Center of Water EngineeringSafety and Resources Efficient Utilization (Grant noOF201602) Zhejiang Collaborative Innovation Center forPrevention and Control of Mountain Geologic Hazards
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 8 Pole diagram of 55 modeled orientations corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghimethod
8 Advances in Civil Engineering
(Grant no PCMGH-2017-Z03) and the National Key Re-search and Development Program of China (Grant no2017YFC1501300)
References
[1] J Liu X D Zhao and Z Xu ldquoIdentification of rock dis-continuity sets based on a modified affinity propagation al-gorithmrdquo International Journal of Rock Mechanics andMining Sciences vol 94 pp 32ndash42 2017
[2] X Wang L Huang C Yan and B Lian ldquoHKCV rheologicalconstitutive model of mudstone under dry and saturatedconditionsrdquo Advances in Civil Engineering vol 2018 ArticleID 2621658 10 pages 2018
[3] N Bar and N Barton ldquo)e Q-slope method for rock slopeengineeringrdquo Rock Mechanics and Rock Engineering vol 50no 12 pp 3307ndash3322 2017
[4] B H G Brady and E T Brown Rock Mechanics for Un-derground Mining Springer Dordrecht Netherlands 3rdedition 2006
[5] E Hoek Practical Rock Engineering Evert Hoek ConsultingEngineer Inc North Vancouver BC Canada 1st edition2007
[6] N H Tran ldquoFracture orientation characterization mini-mizing statistical modeling errorsrdquo Computational Statisticsand Data Analysis vol 51 no 6 pp 3187ndash3196 2007
[7] L Huang H Tang Q Tan et al ldquoA novel method for correctingscanline-observational bias of discontinuity orientationrdquo Sci-entific Reports vol 2016 no 6 Article ID 22942 2016
[8] R D Terzaghi ldquoSource of error in joint surveysrdquo Geo-technique vol 15 no 3 pp 287ndash304 1965
[9] H H Einstein and G B Baecher ldquoProbabilistic and statisticalmethods in engineering geologyrdquo Rock Mechanics and RockEngineering vol 16 no 1 pp 39ndash72 1983
[10] G B Baechaer ldquoStatistical analysis of rock mass fracturingrdquoMathematical Geology vol 15 pp 329ndash348 1983
[11] R Goodman ldquoToppling a fundamental failure mode indiscontinuous materials description and analysisrdquo in Pro-ceedings of Geo-Congress pp 2338ndash2368 San Diego CAUSA 2013
[12] H J Park T R West and I Woo ldquoProbabilistic analysis ofrock slope stability and random properties of discontinuityparameters Interstate Highway 40 Western North CarolinaUSArdquo Engineering Geology vol 79 no 3-4 pp 230ndash2502005
[13] O Fouche and J Diebolt ldquoDescribing the geometry of 3Dfracture systems by correcting for linear sampling biasrdquoMathematical Geology vol 36 no 1 pp 33ndash63 2004
[14] H M Tang L Huang A Bobet et al ldquoIdentification andmitigation of error in the Terzaghi bias correction for in-homogeneous material discontinuitiesrdquo Strength of Materialsvol 48 no 6 pp 825ndash833 2016
[15] H Tang L Huang C H Juang and J Zhang ldquoOptimizing theTerzaghi estimator of the 3D distribution of rock fractureorientationsrdquo Rock Mechanics and Rock Engineering vol 50no 8 pp 2085ndash2099 2017
[16] C Xu and P Dowd ldquoA new computer code for discretefracture network modelingrdquo Computers and Geosciencesvol 36 no 3 pp 292ndash301 2010
[17] A Brzovic and E Villaescusa ldquoRock mass characterizationand assessment of block-forming geological discontinuitiesduring caving of primary copper ore at the El Teniente mineChilerdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 4 pp 565ndash583 2007
[18] M Grenon and J Hadjigeorgiou ldquoDrift reinforcement designbased on discontinuity network modelingrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 40 no 6pp 833ndash845 2003
[19] M S Ozcomak M Kartal O Senger and A K CelikldquoComparison of the powers of the Kolmogorov-Smirnov two-sample test and the Mann-Whitney test for different Kurtosisand Skewness coefficients using the Monte Carlo simulationmethodrdquo Journal of Statistical and Econometric Methodsvol 2 pp 81ndash98 2013
[20] L Al-Labadi and M Zarepour ldquoTwo-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approachrdquoMathematical Methods of Statistics vol 26 no 3 pp 212ndash2252017
Advances in Civil Engineering 9
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
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Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
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Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Navigation and Observation
International Journal of
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wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
1 31211
12
2
14 3414261512
1125
241 31321
13
2
Poles
(1) Meshing (2) Counting
(3) Weighing
(4) Rounding
(a)
Poles 1111
1
11 11
2
2
1 13151214
11
16131211
26
2411
1211
1
21 11
3
2
1
(1) Counting (2) Weighing
(3) Rounding
(b)
Figure 1 Procedures of (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Advances in Civil Engineering 3
this scanline were then ldquomeasuredrdquo )e quantity of thesemeasured discontinuities is set equal to the number ofobserved discontinuities in the field To distinguish betweenthese measured discontinuities and the real discontinuitiesobserved in the field the former are named ldquomodeledrdquodiscontinuities Figure 8 shows the modeled discontinuityorientations
)ird the distribution difference between the observedand the modeled orientations was tested by theKolmogorovndashSmirnov two-sample test )is nonparametric
hypothesis test evaluates the difference between the cumu-lative distribution functions of two sample data vectors )etest can be executed by the software Statistical Product andService Solutions and returns an asymptotic significance tocharacterize the difference )e significance ranges from 0 to1 the higher the significance the lower the difference Moreinformation about this test is given in Ozcomak et al [19] andAl-Labadi and Zarepour [20]
)e test returned two significances corresponding to thedip direction and dip angle )ese two significances werecombined and their average is 0686 for the conventionalTerzaghi method and 0988 for the modified Terzaghimethod It is apparent that the latter value is much higherthan the former value demonstrating that the modifiedTerzaghi method is more accurate than the conventionalmethod even if the optimal cell size of 2deg times 2deg is chosen forthe conventional method
4 Discussion
Tang [14] presented the accuracy of the conventional Terzaghimethod in two cases
Shanghai
Beijing
China
Yingxiu TownWenchuan Country
S
Yingxiu Town
Duwen Expressway
317 National Road
Min River
213 National Road
DaoliupoHejia Mountain
Huangjiacun
Gancang
Madi
Dashuigou
Caijiagangcun
Study area
1km
N
1200
m
1600m
1600
m
1400
m
1400
m 1600
m
1800m
1400
m
1200m1000
m
Epicenter of the WenchuanEarthquake 2008
Figure 3 Map of the Wenchuan area showing the location of the study area )e blue line is the Min River the thick black lines denote the213 National Road and the 317 National Road the medium black lines are rural roads and the thin black lines are topographic contours aslabeled
αβ
Cell
Δα
Δβ
Figure 2 Mesh of projection net
ScanlineTrendplunge
10815deg
Outcrop lithic arkoseDip directionangle
19270deg
Figure 4 Photograph of outcrop and the scanline used forobservation
4 Advances in Civil Engineering
Table 1 Data of observed orientations
Discontinuity Dip directionangle (deg) Discontinuity Dip directionangle (deg)1 12873 29 139662 13470 30 138653 13570 31 135704 13371 32 137725 13075 33 137776 13371 34 129747 13274 35 145788 14572 36 138709 13376 37 1306910 13167 38 1337111 13574 39 1367012 14172 40 1327013 13875 41 1347214 13069 42 1357715 12773 43 1276916 13780 44 1297317 13873 45 1297118 13278 46 1447219 13773 47 1397120 13173 48 1427521 13169 49 1377622 13271 50 1437423 13769 51 1447124 13279 52 1417925 13874 53 1447326 13177 54 1417527 13273 55 1408028 13378 mdash mdash
Fisherconcentration
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
Figure 5 Pole diagram of 55 observed orientations )is diagram was plotted by the software Dips )is is an equal-angle projection of theupper hemisphere showing equal-density rings shaped by Fisher concentrations
Advances in Civil Engineering 5
Case 1 Ifn⟶infin in otherwords if the cell size is infinitesimal
BD
pA(α β) dαdβ 1k
limn⟶infin
1113944
n
i1
Pi
sin θci (2)
Case 2 If n does not approach infin
BD
pA(α β) dαdβ asymp1k
1113944
n
i1
Pi
sin θci (3)
Fisherconcentrations
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
(a)
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 6 Pole diagram of orientations corrected by (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Table 2 Volumetric abundance diameter aperture and size of simulated zone
Volumetric abundancea (mminus3) Diameter (m) Aperture (mm)Simulated zone
Length (m) Width (m) Height (m)4 Exp (05)b Exp (32) 10 10 10aVolumetric abundancenumber of discontinuity centers per rock volume bExp (k) exponential distribution where k is the reciprocal of mean
6 Advances in Civil Engineering
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(a)
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(b)
Figure 7 Rock models corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Fisherconcentrations
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
(a)
Figure 8 Continued
Advances in Civil Engineering 7
where D is a given orientation interval and pA(α β) is thejoint probability density of the dip direction and dip angle inthe rockmass
As mentioned in Section 22 the elimination of mesh inthe modified Terzaghi method is identical to meshing theprojection diagram into infinitesimal cells (Case 1 forEquation (2)) Tang [14] proved that Equation (2) is accu-rate while Equation (3) introduces error Hence themodified Terzaghi method is a more accurate procedure Incontrast for the conventional Terzaghi method the cells canonly approach but can never attain an infinitesimal sizeSuch a mesh can only meet Case 2 and not Case 1 so theconventional Terzaghi method will consequently involveerror )is analysis reinforces the result regarding accuracyobtained in Section 3
As shown in Section 3 the significance of the modifiedTerzaghi method approaches but does not attain unitysuggesting that factors in addition to meshing contributeerror to the Terzaghi method One inferred source of error isrounding the corrected frequencies to the nearest integer)is issue will be studied in the future
5 Conclusions
)e Terzaghi method is widely used to correct the observeddistribution of rock discontinuity orientations A modifiedTerzaghi method that eliminates the meshing procedure wasdeveloped to improve accuracy )is modified method onlyincludes three procedures that (1) count the frequencies ateach pole (2) weigh the observed frequencies by the bias-compensatory factor and (3) round the weighed frequenciesto the nearest integer Mesh elimination requires that thecounting objects are changed to the frequencies at each pole
instead of the frequencies lying in each cell accordingly thebias-compensatory factor has become the reciprocal ofthe sine of the intersection angle between the scanline andthe discontinuity at each pole rather than the reciprocalof the sine of the intersection angle between the scanline andthe discontinuity defined at each cell center A mathematicallogical deduction verifies that this slightly redefined bias-compensatory factor can be used in the modified Terzaghimethod
)e conventional and the modified Terzaghi methodswere both applied to a case study near Yingxiu in SichuanChina )e results indicate that the modified Terzaghimethod performed more accurately than the conventionalmethod
Disclosure
Mutasim Adam Mohamed Ez Eldin is presently at theSection of Engineering Geology Department of AppliedGeology Saudi Geological Survey Jeddah 21514 SaudiArabia
Conflicts of Interest
)e authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
)is research was supported by the Open Foundation ofJiangxi Engineering Research Center of Water EngineeringSafety and Resources Efficient Utilization (Grant noOF201602) Zhejiang Collaborative Innovation Center forPrevention and Control of Mountain Geologic Hazards
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 8 Pole diagram of 55 modeled orientations corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghimethod
8 Advances in Civil Engineering
(Grant no PCMGH-2017-Z03) and the National Key Re-search and Development Program of China (Grant no2017YFC1501300)
References
[1] J Liu X D Zhao and Z Xu ldquoIdentification of rock dis-continuity sets based on a modified affinity propagation al-gorithmrdquo International Journal of Rock Mechanics andMining Sciences vol 94 pp 32ndash42 2017
[2] X Wang L Huang C Yan and B Lian ldquoHKCV rheologicalconstitutive model of mudstone under dry and saturatedconditionsrdquo Advances in Civil Engineering vol 2018 ArticleID 2621658 10 pages 2018
[3] N Bar and N Barton ldquo)e Q-slope method for rock slopeengineeringrdquo Rock Mechanics and Rock Engineering vol 50no 12 pp 3307ndash3322 2017
[4] B H G Brady and E T Brown Rock Mechanics for Un-derground Mining Springer Dordrecht Netherlands 3rdedition 2006
[5] E Hoek Practical Rock Engineering Evert Hoek ConsultingEngineer Inc North Vancouver BC Canada 1st edition2007
[6] N H Tran ldquoFracture orientation characterization mini-mizing statistical modeling errorsrdquo Computational Statisticsand Data Analysis vol 51 no 6 pp 3187ndash3196 2007
[7] L Huang H Tang Q Tan et al ldquoA novel method for correctingscanline-observational bias of discontinuity orientationrdquo Sci-entific Reports vol 2016 no 6 Article ID 22942 2016
[8] R D Terzaghi ldquoSource of error in joint surveysrdquo Geo-technique vol 15 no 3 pp 287ndash304 1965
[9] H H Einstein and G B Baecher ldquoProbabilistic and statisticalmethods in engineering geologyrdquo Rock Mechanics and RockEngineering vol 16 no 1 pp 39ndash72 1983
[10] G B Baechaer ldquoStatistical analysis of rock mass fracturingrdquoMathematical Geology vol 15 pp 329ndash348 1983
[11] R Goodman ldquoToppling a fundamental failure mode indiscontinuous materials description and analysisrdquo in Pro-ceedings of Geo-Congress pp 2338ndash2368 San Diego CAUSA 2013
[12] H J Park T R West and I Woo ldquoProbabilistic analysis ofrock slope stability and random properties of discontinuityparameters Interstate Highway 40 Western North CarolinaUSArdquo Engineering Geology vol 79 no 3-4 pp 230ndash2502005
[13] O Fouche and J Diebolt ldquoDescribing the geometry of 3Dfracture systems by correcting for linear sampling biasrdquoMathematical Geology vol 36 no 1 pp 33ndash63 2004
[14] H M Tang L Huang A Bobet et al ldquoIdentification andmitigation of error in the Terzaghi bias correction for in-homogeneous material discontinuitiesrdquo Strength of Materialsvol 48 no 6 pp 825ndash833 2016
[15] H Tang L Huang C H Juang and J Zhang ldquoOptimizing theTerzaghi estimator of the 3D distribution of rock fractureorientationsrdquo Rock Mechanics and Rock Engineering vol 50no 8 pp 2085ndash2099 2017
[16] C Xu and P Dowd ldquoA new computer code for discretefracture network modelingrdquo Computers and Geosciencesvol 36 no 3 pp 292ndash301 2010
[17] A Brzovic and E Villaescusa ldquoRock mass characterizationand assessment of block-forming geological discontinuitiesduring caving of primary copper ore at the El Teniente mineChilerdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 4 pp 565ndash583 2007
[18] M Grenon and J Hadjigeorgiou ldquoDrift reinforcement designbased on discontinuity network modelingrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 40 no 6pp 833ndash845 2003
[19] M S Ozcomak M Kartal O Senger and A K CelikldquoComparison of the powers of the Kolmogorov-Smirnov two-sample test and the Mann-Whitney test for different Kurtosisand Skewness coefficients using the Monte Carlo simulationmethodrdquo Journal of Statistical and Econometric Methodsvol 2 pp 81ndash98 2013
[20] L Al-Labadi and M Zarepour ldquoTwo-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approachrdquoMathematical Methods of Statistics vol 26 no 3 pp 212ndash2252017
Advances in Civil Engineering 9
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
this scanline were then ldquomeasuredrdquo )e quantity of thesemeasured discontinuities is set equal to the number ofobserved discontinuities in the field To distinguish betweenthese measured discontinuities and the real discontinuitiesobserved in the field the former are named ldquomodeledrdquodiscontinuities Figure 8 shows the modeled discontinuityorientations
)ird the distribution difference between the observedand the modeled orientations was tested by theKolmogorovndashSmirnov two-sample test )is nonparametric
hypothesis test evaluates the difference between the cumu-lative distribution functions of two sample data vectors )etest can be executed by the software Statistical Product andService Solutions and returns an asymptotic significance tocharacterize the difference )e significance ranges from 0 to1 the higher the significance the lower the difference Moreinformation about this test is given in Ozcomak et al [19] andAl-Labadi and Zarepour [20]
)e test returned two significances corresponding to thedip direction and dip angle )ese two significances werecombined and their average is 0686 for the conventionalTerzaghi method and 0988 for the modified Terzaghimethod It is apparent that the latter value is much higherthan the former value demonstrating that the modifiedTerzaghi method is more accurate than the conventionalmethod even if the optimal cell size of 2deg times 2deg is chosen forthe conventional method
4 Discussion
Tang [14] presented the accuracy of the conventional Terzaghimethod in two cases
Shanghai
Beijing
China
Yingxiu TownWenchuan Country
S
Yingxiu Town
Duwen Expressway
317 National Road
Min River
213 National Road
DaoliupoHejia Mountain
Huangjiacun
Gancang
Madi
Dashuigou
Caijiagangcun
Study area
1km
N
1200
m
1600m
1600
m
1400
m
1400
m 1600
m
1800m
1400
m
1200m1000
m
Epicenter of the WenchuanEarthquake 2008
Figure 3 Map of the Wenchuan area showing the location of the study area )e blue line is the Min River the thick black lines denote the213 National Road and the 317 National Road the medium black lines are rural roads and the thin black lines are topographic contours aslabeled
αβ
Cell
Δα
Δβ
Figure 2 Mesh of projection net
ScanlineTrendplunge
10815deg
Outcrop lithic arkoseDip directionangle
19270deg
Figure 4 Photograph of outcrop and the scanline used forobservation
4 Advances in Civil Engineering
Table 1 Data of observed orientations
Discontinuity Dip directionangle (deg) Discontinuity Dip directionangle (deg)1 12873 29 139662 13470 30 138653 13570 31 135704 13371 32 137725 13075 33 137776 13371 34 129747 13274 35 145788 14572 36 138709 13376 37 1306910 13167 38 1337111 13574 39 1367012 14172 40 1327013 13875 41 1347214 13069 42 1357715 12773 43 1276916 13780 44 1297317 13873 45 1297118 13278 46 1447219 13773 47 1397120 13173 48 1427521 13169 49 1377622 13271 50 1437423 13769 51 1447124 13279 52 1417925 13874 53 1447326 13177 54 1417527 13273 55 1408028 13378 mdash mdash
Fisherconcentration
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
Figure 5 Pole diagram of 55 observed orientations )is diagram was plotted by the software Dips )is is an equal-angle projection of theupper hemisphere showing equal-density rings shaped by Fisher concentrations
Advances in Civil Engineering 5
Case 1 Ifn⟶infin in otherwords if the cell size is infinitesimal
BD
pA(α β) dαdβ 1k
limn⟶infin
1113944
n
i1
Pi
sin θci (2)
Case 2 If n does not approach infin
BD
pA(α β) dαdβ asymp1k
1113944
n
i1
Pi
sin θci (3)
Fisherconcentrations
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
(a)
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 6 Pole diagram of orientations corrected by (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Table 2 Volumetric abundance diameter aperture and size of simulated zone
Volumetric abundancea (mminus3) Diameter (m) Aperture (mm)Simulated zone
Length (m) Width (m) Height (m)4 Exp (05)b Exp (32) 10 10 10aVolumetric abundancenumber of discontinuity centers per rock volume bExp (k) exponential distribution where k is the reciprocal of mean
6 Advances in Civil Engineering
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(a)
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(b)
Figure 7 Rock models corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Fisherconcentrations
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
(a)
Figure 8 Continued
Advances in Civil Engineering 7
where D is a given orientation interval and pA(α β) is thejoint probability density of the dip direction and dip angle inthe rockmass
As mentioned in Section 22 the elimination of mesh inthe modified Terzaghi method is identical to meshing theprojection diagram into infinitesimal cells (Case 1 forEquation (2)) Tang [14] proved that Equation (2) is accu-rate while Equation (3) introduces error Hence themodified Terzaghi method is a more accurate procedure Incontrast for the conventional Terzaghi method the cells canonly approach but can never attain an infinitesimal sizeSuch a mesh can only meet Case 2 and not Case 1 so theconventional Terzaghi method will consequently involveerror )is analysis reinforces the result regarding accuracyobtained in Section 3
As shown in Section 3 the significance of the modifiedTerzaghi method approaches but does not attain unitysuggesting that factors in addition to meshing contributeerror to the Terzaghi method One inferred source of error isrounding the corrected frequencies to the nearest integer)is issue will be studied in the future
5 Conclusions
)e Terzaghi method is widely used to correct the observeddistribution of rock discontinuity orientations A modifiedTerzaghi method that eliminates the meshing procedure wasdeveloped to improve accuracy )is modified method onlyincludes three procedures that (1) count the frequencies ateach pole (2) weigh the observed frequencies by the bias-compensatory factor and (3) round the weighed frequenciesto the nearest integer Mesh elimination requires that thecounting objects are changed to the frequencies at each pole
instead of the frequencies lying in each cell accordingly thebias-compensatory factor has become the reciprocal ofthe sine of the intersection angle between the scanline andthe discontinuity at each pole rather than the reciprocalof the sine of the intersection angle between the scanline andthe discontinuity defined at each cell center A mathematicallogical deduction verifies that this slightly redefined bias-compensatory factor can be used in the modified Terzaghimethod
)e conventional and the modified Terzaghi methodswere both applied to a case study near Yingxiu in SichuanChina )e results indicate that the modified Terzaghimethod performed more accurately than the conventionalmethod
Disclosure
Mutasim Adam Mohamed Ez Eldin is presently at theSection of Engineering Geology Department of AppliedGeology Saudi Geological Survey Jeddah 21514 SaudiArabia
Conflicts of Interest
)e authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
)is research was supported by the Open Foundation ofJiangxi Engineering Research Center of Water EngineeringSafety and Resources Efficient Utilization (Grant noOF201602) Zhejiang Collaborative Innovation Center forPrevention and Control of Mountain Geologic Hazards
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 8 Pole diagram of 55 modeled orientations corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghimethod
8 Advances in Civil Engineering
(Grant no PCMGH-2017-Z03) and the National Key Re-search and Development Program of China (Grant no2017YFC1501300)
References
[1] J Liu X D Zhao and Z Xu ldquoIdentification of rock dis-continuity sets based on a modified affinity propagation al-gorithmrdquo International Journal of Rock Mechanics andMining Sciences vol 94 pp 32ndash42 2017
[2] X Wang L Huang C Yan and B Lian ldquoHKCV rheologicalconstitutive model of mudstone under dry and saturatedconditionsrdquo Advances in Civil Engineering vol 2018 ArticleID 2621658 10 pages 2018
[3] N Bar and N Barton ldquo)e Q-slope method for rock slopeengineeringrdquo Rock Mechanics and Rock Engineering vol 50no 12 pp 3307ndash3322 2017
[4] B H G Brady and E T Brown Rock Mechanics for Un-derground Mining Springer Dordrecht Netherlands 3rdedition 2006
[5] E Hoek Practical Rock Engineering Evert Hoek ConsultingEngineer Inc North Vancouver BC Canada 1st edition2007
[6] N H Tran ldquoFracture orientation characterization mini-mizing statistical modeling errorsrdquo Computational Statisticsand Data Analysis vol 51 no 6 pp 3187ndash3196 2007
[7] L Huang H Tang Q Tan et al ldquoA novel method for correctingscanline-observational bias of discontinuity orientationrdquo Sci-entific Reports vol 2016 no 6 Article ID 22942 2016
[8] R D Terzaghi ldquoSource of error in joint surveysrdquo Geo-technique vol 15 no 3 pp 287ndash304 1965
[9] H H Einstein and G B Baecher ldquoProbabilistic and statisticalmethods in engineering geologyrdquo Rock Mechanics and RockEngineering vol 16 no 1 pp 39ndash72 1983
[10] G B Baechaer ldquoStatistical analysis of rock mass fracturingrdquoMathematical Geology vol 15 pp 329ndash348 1983
[11] R Goodman ldquoToppling a fundamental failure mode indiscontinuous materials description and analysisrdquo in Pro-ceedings of Geo-Congress pp 2338ndash2368 San Diego CAUSA 2013
[12] H J Park T R West and I Woo ldquoProbabilistic analysis ofrock slope stability and random properties of discontinuityparameters Interstate Highway 40 Western North CarolinaUSArdquo Engineering Geology vol 79 no 3-4 pp 230ndash2502005
[13] O Fouche and J Diebolt ldquoDescribing the geometry of 3Dfracture systems by correcting for linear sampling biasrdquoMathematical Geology vol 36 no 1 pp 33ndash63 2004
[14] H M Tang L Huang A Bobet et al ldquoIdentification andmitigation of error in the Terzaghi bias correction for in-homogeneous material discontinuitiesrdquo Strength of Materialsvol 48 no 6 pp 825ndash833 2016
[15] H Tang L Huang C H Juang and J Zhang ldquoOptimizing theTerzaghi estimator of the 3D distribution of rock fractureorientationsrdquo Rock Mechanics and Rock Engineering vol 50no 8 pp 2085ndash2099 2017
[16] C Xu and P Dowd ldquoA new computer code for discretefracture network modelingrdquo Computers and Geosciencesvol 36 no 3 pp 292ndash301 2010
[17] A Brzovic and E Villaescusa ldquoRock mass characterizationand assessment of block-forming geological discontinuitiesduring caving of primary copper ore at the El Teniente mineChilerdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 4 pp 565ndash583 2007
[18] M Grenon and J Hadjigeorgiou ldquoDrift reinforcement designbased on discontinuity network modelingrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 40 no 6pp 833ndash845 2003
[19] M S Ozcomak M Kartal O Senger and A K CelikldquoComparison of the powers of the Kolmogorov-Smirnov two-sample test and the Mann-Whitney test for different Kurtosisand Skewness coefficients using the Monte Carlo simulationmethodrdquo Journal of Statistical and Econometric Methodsvol 2 pp 81ndash98 2013
[20] L Al-Labadi and M Zarepour ldquoTwo-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approachrdquoMathematical Methods of Statistics vol 26 no 3 pp 212ndash2252017
Advances in Civil Engineering 9
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Table 1 Data of observed orientations
Discontinuity Dip directionangle (deg) Discontinuity Dip directionangle (deg)1 12873 29 139662 13470 30 138653 13570 31 135704 13371 32 137725 13075 33 137776 13371 34 129747 13274 35 145788 14572 36 138709 13376 37 1306910 13167 38 1337111 13574 39 1367012 14172 40 1327013 13875 41 1347214 13069 42 1357715 12773 43 1276916 13780 44 1297317 13873 45 1297118 13278 46 1447219 13773 47 1397120 13173 48 1427521 13169 49 1377622 13271 50 1437423 13769 51 1447124 13279 52 1417925 13874 53 1447326 13177 54 1417527 13273 55 1408028 13378 mdash mdash
Fisherconcentration
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
Figure 5 Pole diagram of 55 observed orientations )is diagram was plotted by the software Dips )is is an equal-angle projection of theupper hemisphere showing equal-density rings shaped by Fisher concentrations
Advances in Civil Engineering 5
Case 1 Ifn⟶infin in otherwords if the cell size is infinitesimal
BD
pA(α β) dαdβ 1k
limn⟶infin
1113944
n
i1
Pi
sin θci (2)
Case 2 If n does not approach infin
BD
pA(α β) dαdβ asymp1k
1113944
n
i1
Pi
sin θci (3)
Fisherconcentrations
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
(a)
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 6 Pole diagram of orientations corrected by (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Table 2 Volumetric abundance diameter aperture and size of simulated zone
Volumetric abundancea (mminus3) Diameter (m) Aperture (mm)Simulated zone
Length (m) Width (m) Height (m)4 Exp (05)b Exp (32) 10 10 10aVolumetric abundancenumber of discontinuity centers per rock volume bExp (k) exponential distribution where k is the reciprocal of mean
6 Advances in Civil Engineering
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(a)
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(b)
Figure 7 Rock models corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Fisherconcentrations
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
(a)
Figure 8 Continued
Advances in Civil Engineering 7
where D is a given orientation interval and pA(α β) is thejoint probability density of the dip direction and dip angle inthe rockmass
As mentioned in Section 22 the elimination of mesh inthe modified Terzaghi method is identical to meshing theprojection diagram into infinitesimal cells (Case 1 forEquation (2)) Tang [14] proved that Equation (2) is accu-rate while Equation (3) introduces error Hence themodified Terzaghi method is a more accurate procedure Incontrast for the conventional Terzaghi method the cells canonly approach but can never attain an infinitesimal sizeSuch a mesh can only meet Case 2 and not Case 1 so theconventional Terzaghi method will consequently involveerror )is analysis reinforces the result regarding accuracyobtained in Section 3
As shown in Section 3 the significance of the modifiedTerzaghi method approaches but does not attain unitysuggesting that factors in addition to meshing contributeerror to the Terzaghi method One inferred source of error isrounding the corrected frequencies to the nearest integer)is issue will be studied in the future
5 Conclusions
)e Terzaghi method is widely used to correct the observeddistribution of rock discontinuity orientations A modifiedTerzaghi method that eliminates the meshing procedure wasdeveloped to improve accuracy )is modified method onlyincludes three procedures that (1) count the frequencies ateach pole (2) weigh the observed frequencies by the bias-compensatory factor and (3) round the weighed frequenciesto the nearest integer Mesh elimination requires that thecounting objects are changed to the frequencies at each pole
instead of the frequencies lying in each cell accordingly thebias-compensatory factor has become the reciprocal ofthe sine of the intersection angle between the scanline andthe discontinuity at each pole rather than the reciprocalof the sine of the intersection angle between the scanline andthe discontinuity defined at each cell center A mathematicallogical deduction verifies that this slightly redefined bias-compensatory factor can be used in the modified Terzaghimethod
)e conventional and the modified Terzaghi methodswere both applied to a case study near Yingxiu in SichuanChina )e results indicate that the modified Terzaghimethod performed more accurately than the conventionalmethod
Disclosure
Mutasim Adam Mohamed Ez Eldin is presently at theSection of Engineering Geology Department of AppliedGeology Saudi Geological Survey Jeddah 21514 SaudiArabia
Conflicts of Interest
)e authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
)is research was supported by the Open Foundation ofJiangxi Engineering Research Center of Water EngineeringSafety and Resources Efficient Utilization (Grant noOF201602) Zhejiang Collaborative Innovation Center forPrevention and Control of Mountain Geologic Hazards
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 8 Pole diagram of 55 modeled orientations corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghimethod
8 Advances in Civil Engineering
(Grant no PCMGH-2017-Z03) and the National Key Re-search and Development Program of China (Grant no2017YFC1501300)
References
[1] J Liu X D Zhao and Z Xu ldquoIdentification of rock dis-continuity sets based on a modified affinity propagation al-gorithmrdquo International Journal of Rock Mechanics andMining Sciences vol 94 pp 32ndash42 2017
[2] X Wang L Huang C Yan and B Lian ldquoHKCV rheologicalconstitutive model of mudstone under dry and saturatedconditionsrdquo Advances in Civil Engineering vol 2018 ArticleID 2621658 10 pages 2018
[3] N Bar and N Barton ldquo)e Q-slope method for rock slopeengineeringrdquo Rock Mechanics and Rock Engineering vol 50no 12 pp 3307ndash3322 2017
[4] B H G Brady and E T Brown Rock Mechanics for Un-derground Mining Springer Dordrecht Netherlands 3rdedition 2006
[5] E Hoek Practical Rock Engineering Evert Hoek ConsultingEngineer Inc North Vancouver BC Canada 1st edition2007
[6] N H Tran ldquoFracture orientation characterization mini-mizing statistical modeling errorsrdquo Computational Statisticsand Data Analysis vol 51 no 6 pp 3187ndash3196 2007
[7] L Huang H Tang Q Tan et al ldquoA novel method for correctingscanline-observational bias of discontinuity orientationrdquo Sci-entific Reports vol 2016 no 6 Article ID 22942 2016
[8] R D Terzaghi ldquoSource of error in joint surveysrdquo Geo-technique vol 15 no 3 pp 287ndash304 1965
[9] H H Einstein and G B Baecher ldquoProbabilistic and statisticalmethods in engineering geologyrdquo Rock Mechanics and RockEngineering vol 16 no 1 pp 39ndash72 1983
[10] G B Baechaer ldquoStatistical analysis of rock mass fracturingrdquoMathematical Geology vol 15 pp 329ndash348 1983
[11] R Goodman ldquoToppling a fundamental failure mode indiscontinuous materials description and analysisrdquo in Pro-ceedings of Geo-Congress pp 2338ndash2368 San Diego CAUSA 2013
[12] H J Park T R West and I Woo ldquoProbabilistic analysis ofrock slope stability and random properties of discontinuityparameters Interstate Highway 40 Western North CarolinaUSArdquo Engineering Geology vol 79 no 3-4 pp 230ndash2502005
[13] O Fouche and J Diebolt ldquoDescribing the geometry of 3Dfracture systems by correcting for linear sampling biasrdquoMathematical Geology vol 36 no 1 pp 33ndash63 2004
[14] H M Tang L Huang A Bobet et al ldquoIdentification andmitigation of error in the Terzaghi bias correction for in-homogeneous material discontinuitiesrdquo Strength of Materialsvol 48 no 6 pp 825ndash833 2016
[15] H Tang L Huang C H Juang and J Zhang ldquoOptimizing theTerzaghi estimator of the 3D distribution of rock fractureorientationsrdquo Rock Mechanics and Rock Engineering vol 50no 8 pp 2085ndash2099 2017
[16] C Xu and P Dowd ldquoA new computer code for discretefracture network modelingrdquo Computers and Geosciencesvol 36 no 3 pp 292ndash301 2010
[17] A Brzovic and E Villaescusa ldquoRock mass characterizationand assessment of block-forming geological discontinuitiesduring caving of primary copper ore at the El Teniente mineChilerdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 4 pp 565ndash583 2007
[18] M Grenon and J Hadjigeorgiou ldquoDrift reinforcement designbased on discontinuity network modelingrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 40 no 6pp 833ndash845 2003
[19] M S Ozcomak M Kartal O Senger and A K CelikldquoComparison of the powers of the Kolmogorov-Smirnov two-sample test and the Mann-Whitney test for different Kurtosisand Skewness coefficients using the Monte Carlo simulationmethodrdquo Journal of Statistical and Econometric Methodsvol 2 pp 81ndash98 2013
[20] L Al-Labadi and M Zarepour ldquoTwo-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approachrdquoMathematical Methods of Statistics vol 26 no 3 pp 212ndash2252017
Advances in Civil Engineering 9
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Case 1 Ifn⟶infin in otherwords if the cell size is infinitesimal
BD
pA(α β) dαdβ 1k
limn⟶infin
1113944
n
i1
Pi
sin θci (2)
Case 2 If n does not approach infin
BD
pA(α β) dαdβ asymp1k
1113944
n
i1
Pi
sin θci (3)
Fisherconcentrations
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
(a)
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 6 Pole diagram of orientations corrected by (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Table 2 Volumetric abundance diameter aperture and size of simulated zone
Volumetric abundancea (mminus3) Diameter (m) Aperture (mm)Simulated zone
Length (m) Width (m) Height (m)4 Exp (05)b Exp (32) 10 10 10aVolumetric abundancenumber of discontinuity centers per rock volume bExp (k) exponential distribution where k is the reciprocal of mean
6 Advances in Civil Engineering
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(a)
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(b)
Figure 7 Rock models corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Fisherconcentrations
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
(a)
Figure 8 Continued
Advances in Civil Engineering 7
where D is a given orientation interval and pA(α β) is thejoint probability density of the dip direction and dip angle inthe rockmass
As mentioned in Section 22 the elimination of mesh inthe modified Terzaghi method is identical to meshing theprojection diagram into infinitesimal cells (Case 1 forEquation (2)) Tang [14] proved that Equation (2) is accu-rate while Equation (3) introduces error Hence themodified Terzaghi method is a more accurate procedure Incontrast for the conventional Terzaghi method the cells canonly approach but can never attain an infinitesimal sizeSuch a mesh can only meet Case 2 and not Case 1 so theconventional Terzaghi method will consequently involveerror )is analysis reinforces the result regarding accuracyobtained in Section 3
As shown in Section 3 the significance of the modifiedTerzaghi method approaches but does not attain unitysuggesting that factors in addition to meshing contributeerror to the Terzaghi method One inferred source of error isrounding the corrected frequencies to the nearest integer)is issue will be studied in the future
5 Conclusions
)e Terzaghi method is widely used to correct the observeddistribution of rock discontinuity orientations A modifiedTerzaghi method that eliminates the meshing procedure wasdeveloped to improve accuracy )is modified method onlyincludes three procedures that (1) count the frequencies ateach pole (2) weigh the observed frequencies by the bias-compensatory factor and (3) round the weighed frequenciesto the nearest integer Mesh elimination requires that thecounting objects are changed to the frequencies at each pole
instead of the frequencies lying in each cell accordingly thebias-compensatory factor has become the reciprocal ofthe sine of the intersection angle between the scanline andthe discontinuity at each pole rather than the reciprocalof the sine of the intersection angle between the scanline andthe discontinuity defined at each cell center A mathematicallogical deduction verifies that this slightly redefined bias-compensatory factor can be used in the modified Terzaghimethod
)e conventional and the modified Terzaghi methodswere both applied to a case study near Yingxiu in SichuanChina )e results indicate that the modified Terzaghimethod performed more accurately than the conventionalmethod
Disclosure
Mutasim Adam Mohamed Ez Eldin is presently at theSection of Engineering Geology Department of AppliedGeology Saudi Geological Survey Jeddah 21514 SaudiArabia
Conflicts of Interest
)e authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
)is research was supported by the Open Foundation ofJiangxi Engineering Research Center of Water EngineeringSafety and Resources Efficient Utilization (Grant noOF201602) Zhejiang Collaborative Innovation Center forPrevention and Control of Mountain Geologic Hazards
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 8 Pole diagram of 55 modeled orientations corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghimethod
8 Advances in Civil Engineering
(Grant no PCMGH-2017-Z03) and the National Key Re-search and Development Program of China (Grant no2017YFC1501300)
References
[1] J Liu X D Zhao and Z Xu ldquoIdentification of rock dis-continuity sets based on a modified affinity propagation al-gorithmrdquo International Journal of Rock Mechanics andMining Sciences vol 94 pp 32ndash42 2017
[2] X Wang L Huang C Yan and B Lian ldquoHKCV rheologicalconstitutive model of mudstone under dry and saturatedconditionsrdquo Advances in Civil Engineering vol 2018 ArticleID 2621658 10 pages 2018
[3] N Bar and N Barton ldquo)e Q-slope method for rock slopeengineeringrdquo Rock Mechanics and Rock Engineering vol 50no 12 pp 3307ndash3322 2017
[4] B H G Brady and E T Brown Rock Mechanics for Un-derground Mining Springer Dordrecht Netherlands 3rdedition 2006
[5] E Hoek Practical Rock Engineering Evert Hoek ConsultingEngineer Inc North Vancouver BC Canada 1st edition2007
[6] N H Tran ldquoFracture orientation characterization mini-mizing statistical modeling errorsrdquo Computational Statisticsand Data Analysis vol 51 no 6 pp 3187ndash3196 2007
[7] L Huang H Tang Q Tan et al ldquoA novel method for correctingscanline-observational bias of discontinuity orientationrdquo Sci-entific Reports vol 2016 no 6 Article ID 22942 2016
[8] R D Terzaghi ldquoSource of error in joint surveysrdquo Geo-technique vol 15 no 3 pp 287ndash304 1965
[9] H H Einstein and G B Baecher ldquoProbabilistic and statisticalmethods in engineering geologyrdquo Rock Mechanics and RockEngineering vol 16 no 1 pp 39ndash72 1983
[10] G B Baechaer ldquoStatistical analysis of rock mass fracturingrdquoMathematical Geology vol 15 pp 329ndash348 1983
[11] R Goodman ldquoToppling a fundamental failure mode indiscontinuous materials description and analysisrdquo in Pro-ceedings of Geo-Congress pp 2338ndash2368 San Diego CAUSA 2013
[12] H J Park T R West and I Woo ldquoProbabilistic analysis ofrock slope stability and random properties of discontinuityparameters Interstate Highway 40 Western North CarolinaUSArdquo Engineering Geology vol 79 no 3-4 pp 230ndash2502005
[13] O Fouche and J Diebolt ldquoDescribing the geometry of 3Dfracture systems by correcting for linear sampling biasrdquoMathematical Geology vol 36 no 1 pp 33ndash63 2004
[14] H M Tang L Huang A Bobet et al ldquoIdentification andmitigation of error in the Terzaghi bias correction for in-homogeneous material discontinuitiesrdquo Strength of Materialsvol 48 no 6 pp 825ndash833 2016
[15] H Tang L Huang C H Juang and J Zhang ldquoOptimizing theTerzaghi estimator of the 3D distribution of rock fractureorientationsrdquo Rock Mechanics and Rock Engineering vol 50no 8 pp 2085ndash2099 2017
[16] C Xu and P Dowd ldquoA new computer code for discretefracture network modelingrdquo Computers and Geosciencesvol 36 no 3 pp 292ndash301 2010
[17] A Brzovic and E Villaescusa ldquoRock mass characterizationand assessment of block-forming geological discontinuitiesduring caving of primary copper ore at the El Teniente mineChilerdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 4 pp 565ndash583 2007
[18] M Grenon and J Hadjigeorgiou ldquoDrift reinforcement designbased on discontinuity network modelingrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 40 no 6pp 833ndash845 2003
[19] M S Ozcomak M Kartal O Senger and A K CelikldquoComparison of the powers of the Kolmogorov-Smirnov two-sample test and the Mann-Whitney test for different Kurtosisand Skewness coefficients using the Monte Carlo simulationmethodrdquo Journal of Statistical and Econometric Methodsvol 2 pp 81ndash98 2013
[20] L Al-Labadi and M Zarepour ldquoTwo-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approachrdquoMathematical Methods of Statistics vol 26 no 3 pp 212ndash2252017
Advances in Civil Engineering 9
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(a)
ScanlineTrendplunge
10815deg
OutcropDip directionangle
19270deg
Rockmass
X (north)Y (west)Z (upward)
2m
2m
55 observed discontinuities
ick lines
(b)
Figure 7 Rock models corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghi method
Fisherconcentrations
of total per 10 area
000~800
800~1600
1600~2400
2400~3200
3200~4000
4000~4800
4800~5600
5600~6400
6400~7200
7200~8000
EW
S
N
(a)
Figure 8 Continued
Advances in Civil Engineering 7
where D is a given orientation interval and pA(α β) is thejoint probability density of the dip direction and dip angle inthe rockmass
As mentioned in Section 22 the elimination of mesh inthe modified Terzaghi method is identical to meshing theprojection diagram into infinitesimal cells (Case 1 forEquation (2)) Tang [14] proved that Equation (2) is accu-rate while Equation (3) introduces error Hence themodified Terzaghi method is a more accurate procedure Incontrast for the conventional Terzaghi method the cells canonly approach but can never attain an infinitesimal sizeSuch a mesh can only meet Case 2 and not Case 1 so theconventional Terzaghi method will consequently involveerror )is analysis reinforces the result regarding accuracyobtained in Section 3
As shown in Section 3 the significance of the modifiedTerzaghi method approaches but does not attain unitysuggesting that factors in addition to meshing contributeerror to the Terzaghi method One inferred source of error isrounding the corrected frequencies to the nearest integer)is issue will be studied in the future
5 Conclusions
)e Terzaghi method is widely used to correct the observeddistribution of rock discontinuity orientations A modifiedTerzaghi method that eliminates the meshing procedure wasdeveloped to improve accuracy )is modified method onlyincludes three procedures that (1) count the frequencies ateach pole (2) weigh the observed frequencies by the bias-compensatory factor and (3) round the weighed frequenciesto the nearest integer Mesh elimination requires that thecounting objects are changed to the frequencies at each pole
instead of the frequencies lying in each cell accordingly thebias-compensatory factor has become the reciprocal ofthe sine of the intersection angle between the scanline andthe discontinuity at each pole rather than the reciprocalof the sine of the intersection angle between the scanline andthe discontinuity defined at each cell center A mathematicallogical deduction verifies that this slightly redefined bias-compensatory factor can be used in the modified Terzaghimethod
)e conventional and the modified Terzaghi methodswere both applied to a case study near Yingxiu in SichuanChina )e results indicate that the modified Terzaghimethod performed more accurately than the conventionalmethod
Disclosure
Mutasim Adam Mohamed Ez Eldin is presently at theSection of Engineering Geology Department of AppliedGeology Saudi Geological Survey Jeddah 21514 SaudiArabia
Conflicts of Interest
)e authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
)is research was supported by the Open Foundation ofJiangxi Engineering Research Center of Water EngineeringSafety and Resources Efficient Utilization (Grant noOF201602) Zhejiang Collaborative Innovation Center forPrevention and Control of Mountain Geologic Hazards
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 8 Pole diagram of 55 modeled orientations corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghimethod
8 Advances in Civil Engineering
(Grant no PCMGH-2017-Z03) and the National Key Re-search and Development Program of China (Grant no2017YFC1501300)
References
[1] J Liu X D Zhao and Z Xu ldquoIdentification of rock dis-continuity sets based on a modified affinity propagation al-gorithmrdquo International Journal of Rock Mechanics andMining Sciences vol 94 pp 32ndash42 2017
[2] X Wang L Huang C Yan and B Lian ldquoHKCV rheologicalconstitutive model of mudstone under dry and saturatedconditionsrdquo Advances in Civil Engineering vol 2018 ArticleID 2621658 10 pages 2018
[3] N Bar and N Barton ldquo)e Q-slope method for rock slopeengineeringrdquo Rock Mechanics and Rock Engineering vol 50no 12 pp 3307ndash3322 2017
[4] B H G Brady and E T Brown Rock Mechanics for Un-derground Mining Springer Dordrecht Netherlands 3rdedition 2006
[5] E Hoek Practical Rock Engineering Evert Hoek ConsultingEngineer Inc North Vancouver BC Canada 1st edition2007
[6] N H Tran ldquoFracture orientation characterization mini-mizing statistical modeling errorsrdquo Computational Statisticsand Data Analysis vol 51 no 6 pp 3187ndash3196 2007
[7] L Huang H Tang Q Tan et al ldquoA novel method for correctingscanline-observational bias of discontinuity orientationrdquo Sci-entific Reports vol 2016 no 6 Article ID 22942 2016
[8] R D Terzaghi ldquoSource of error in joint surveysrdquo Geo-technique vol 15 no 3 pp 287ndash304 1965
[9] H H Einstein and G B Baecher ldquoProbabilistic and statisticalmethods in engineering geologyrdquo Rock Mechanics and RockEngineering vol 16 no 1 pp 39ndash72 1983
[10] G B Baechaer ldquoStatistical analysis of rock mass fracturingrdquoMathematical Geology vol 15 pp 329ndash348 1983
[11] R Goodman ldquoToppling a fundamental failure mode indiscontinuous materials description and analysisrdquo in Pro-ceedings of Geo-Congress pp 2338ndash2368 San Diego CAUSA 2013
[12] H J Park T R West and I Woo ldquoProbabilistic analysis ofrock slope stability and random properties of discontinuityparameters Interstate Highway 40 Western North CarolinaUSArdquo Engineering Geology vol 79 no 3-4 pp 230ndash2502005
[13] O Fouche and J Diebolt ldquoDescribing the geometry of 3Dfracture systems by correcting for linear sampling biasrdquoMathematical Geology vol 36 no 1 pp 33ndash63 2004
[14] H M Tang L Huang A Bobet et al ldquoIdentification andmitigation of error in the Terzaghi bias correction for in-homogeneous material discontinuitiesrdquo Strength of Materialsvol 48 no 6 pp 825ndash833 2016
[15] H Tang L Huang C H Juang and J Zhang ldquoOptimizing theTerzaghi estimator of the 3D distribution of rock fractureorientationsrdquo Rock Mechanics and Rock Engineering vol 50no 8 pp 2085ndash2099 2017
[16] C Xu and P Dowd ldquoA new computer code for discretefracture network modelingrdquo Computers and Geosciencesvol 36 no 3 pp 292ndash301 2010
[17] A Brzovic and E Villaescusa ldquoRock mass characterizationand assessment of block-forming geological discontinuitiesduring caving of primary copper ore at the El Teniente mineChilerdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 4 pp 565ndash583 2007
[18] M Grenon and J Hadjigeorgiou ldquoDrift reinforcement designbased on discontinuity network modelingrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 40 no 6pp 833ndash845 2003
[19] M S Ozcomak M Kartal O Senger and A K CelikldquoComparison of the powers of the Kolmogorov-Smirnov two-sample test and the Mann-Whitney test for different Kurtosisand Skewness coefficients using the Monte Carlo simulationmethodrdquo Journal of Statistical and Econometric Methodsvol 2 pp 81ndash98 2013
[20] L Al-Labadi and M Zarepour ldquoTwo-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approachrdquoMathematical Methods of Statistics vol 26 no 3 pp 212ndash2252017
Advances in Civil Engineering 9
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
where D is a given orientation interval and pA(α β) is thejoint probability density of the dip direction and dip angle inthe rockmass
As mentioned in Section 22 the elimination of mesh inthe modified Terzaghi method is identical to meshing theprojection diagram into infinitesimal cells (Case 1 forEquation (2)) Tang [14] proved that Equation (2) is accu-rate while Equation (3) introduces error Hence themodified Terzaghi method is a more accurate procedure Incontrast for the conventional Terzaghi method the cells canonly approach but can never attain an infinitesimal sizeSuch a mesh can only meet Case 2 and not Case 1 so theconventional Terzaghi method will consequently involveerror )is analysis reinforces the result regarding accuracyobtained in Section 3
As shown in Section 3 the significance of the modifiedTerzaghi method approaches but does not attain unitysuggesting that factors in addition to meshing contributeerror to the Terzaghi method One inferred source of error isrounding the corrected frequencies to the nearest integer)is issue will be studied in the future
5 Conclusions
)e Terzaghi method is widely used to correct the observeddistribution of rock discontinuity orientations A modifiedTerzaghi method that eliminates the meshing procedure wasdeveloped to improve accuracy )is modified method onlyincludes three procedures that (1) count the frequencies ateach pole (2) weigh the observed frequencies by the bias-compensatory factor and (3) round the weighed frequenciesto the nearest integer Mesh elimination requires that thecounting objects are changed to the frequencies at each pole
instead of the frequencies lying in each cell accordingly thebias-compensatory factor has become the reciprocal ofthe sine of the intersection angle between the scanline andthe discontinuity at each pole rather than the reciprocalof the sine of the intersection angle between the scanline andthe discontinuity defined at each cell center A mathematicallogical deduction verifies that this slightly redefined bias-compensatory factor can be used in the modified Terzaghimethod
)e conventional and the modified Terzaghi methodswere both applied to a case study near Yingxiu in SichuanChina )e results indicate that the modified Terzaghimethod performed more accurately than the conventionalmethod
Disclosure
Mutasim Adam Mohamed Ez Eldin is presently at theSection of Engineering Geology Department of AppliedGeology Saudi Geological Survey Jeddah 21514 SaudiArabia
Conflicts of Interest
)e authors declare that there are no conflicts of interestregarding the publication of this article
Acknowledgments
)is research was supported by the Open Foundation ofJiangxi Engineering Research Center of Water EngineeringSafety and Resources Efficient Utilization (Grant noOF201602) Zhejiang Collaborative Innovation Center forPrevention and Control of Mountain Geologic Hazards
Fisherconcentrations
of total per 10 area
000~750
750~1500
1500~2250
2250~3000
3000~3750
3750~4500
4500~5250
5250~6000
6000~6750
6750~7500
EW
S
N
(b)
Figure 8 Pole diagram of 55 modeled orientations corresponding to (a) the conventional Terzaghi method and (b) the modified Terzaghimethod
8 Advances in Civil Engineering
(Grant no PCMGH-2017-Z03) and the National Key Re-search and Development Program of China (Grant no2017YFC1501300)
References
[1] J Liu X D Zhao and Z Xu ldquoIdentification of rock dis-continuity sets based on a modified affinity propagation al-gorithmrdquo International Journal of Rock Mechanics andMining Sciences vol 94 pp 32ndash42 2017
[2] X Wang L Huang C Yan and B Lian ldquoHKCV rheologicalconstitutive model of mudstone under dry and saturatedconditionsrdquo Advances in Civil Engineering vol 2018 ArticleID 2621658 10 pages 2018
[3] N Bar and N Barton ldquo)e Q-slope method for rock slopeengineeringrdquo Rock Mechanics and Rock Engineering vol 50no 12 pp 3307ndash3322 2017
[4] B H G Brady and E T Brown Rock Mechanics for Un-derground Mining Springer Dordrecht Netherlands 3rdedition 2006
[5] E Hoek Practical Rock Engineering Evert Hoek ConsultingEngineer Inc North Vancouver BC Canada 1st edition2007
[6] N H Tran ldquoFracture orientation characterization mini-mizing statistical modeling errorsrdquo Computational Statisticsand Data Analysis vol 51 no 6 pp 3187ndash3196 2007
[7] L Huang H Tang Q Tan et al ldquoA novel method for correctingscanline-observational bias of discontinuity orientationrdquo Sci-entific Reports vol 2016 no 6 Article ID 22942 2016
[8] R D Terzaghi ldquoSource of error in joint surveysrdquo Geo-technique vol 15 no 3 pp 287ndash304 1965
[9] H H Einstein and G B Baecher ldquoProbabilistic and statisticalmethods in engineering geologyrdquo Rock Mechanics and RockEngineering vol 16 no 1 pp 39ndash72 1983
[10] G B Baechaer ldquoStatistical analysis of rock mass fracturingrdquoMathematical Geology vol 15 pp 329ndash348 1983
[11] R Goodman ldquoToppling a fundamental failure mode indiscontinuous materials description and analysisrdquo in Pro-ceedings of Geo-Congress pp 2338ndash2368 San Diego CAUSA 2013
[12] H J Park T R West and I Woo ldquoProbabilistic analysis ofrock slope stability and random properties of discontinuityparameters Interstate Highway 40 Western North CarolinaUSArdquo Engineering Geology vol 79 no 3-4 pp 230ndash2502005
[13] O Fouche and J Diebolt ldquoDescribing the geometry of 3Dfracture systems by correcting for linear sampling biasrdquoMathematical Geology vol 36 no 1 pp 33ndash63 2004
[14] H M Tang L Huang A Bobet et al ldquoIdentification andmitigation of error in the Terzaghi bias correction for in-homogeneous material discontinuitiesrdquo Strength of Materialsvol 48 no 6 pp 825ndash833 2016
[15] H Tang L Huang C H Juang and J Zhang ldquoOptimizing theTerzaghi estimator of the 3D distribution of rock fractureorientationsrdquo Rock Mechanics and Rock Engineering vol 50no 8 pp 2085ndash2099 2017
[16] C Xu and P Dowd ldquoA new computer code for discretefracture network modelingrdquo Computers and Geosciencesvol 36 no 3 pp 292ndash301 2010
[17] A Brzovic and E Villaescusa ldquoRock mass characterizationand assessment of block-forming geological discontinuitiesduring caving of primary copper ore at the El Teniente mineChilerdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 4 pp 565ndash583 2007
[18] M Grenon and J Hadjigeorgiou ldquoDrift reinforcement designbased on discontinuity network modelingrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 40 no 6pp 833ndash845 2003
[19] M S Ozcomak M Kartal O Senger and A K CelikldquoComparison of the powers of the Kolmogorov-Smirnov two-sample test and the Mann-Whitney test for different Kurtosisand Skewness coefficients using the Monte Carlo simulationmethodrdquo Journal of Statistical and Econometric Methodsvol 2 pp 81ndash98 2013
[20] L Al-Labadi and M Zarepour ldquoTwo-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approachrdquoMathematical Methods of Statistics vol 26 no 3 pp 212ndash2252017
Advances in Civil Engineering 9
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
(Grant no PCMGH-2017-Z03) and the National Key Re-search and Development Program of China (Grant no2017YFC1501300)
References
[1] J Liu X D Zhao and Z Xu ldquoIdentification of rock dis-continuity sets based on a modified affinity propagation al-gorithmrdquo International Journal of Rock Mechanics andMining Sciences vol 94 pp 32ndash42 2017
[2] X Wang L Huang C Yan and B Lian ldquoHKCV rheologicalconstitutive model of mudstone under dry and saturatedconditionsrdquo Advances in Civil Engineering vol 2018 ArticleID 2621658 10 pages 2018
[3] N Bar and N Barton ldquo)e Q-slope method for rock slopeengineeringrdquo Rock Mechanics and Rock Engineering vol 50no 12 pp 3307ndash3322 2017
[4] B H G Brady and E T Brown Rock Mechanics for Un-derground Mining Springer Dordrecht Netherlands 3rdedition 2006
[5] E Hoek Practical Rock Engineering Evert Hoek ConsultingEngineer Inc North Vancouver BC Canada 1st edition2007
[6] N H Tran ldquoFracture orientation characterization mini-mizing statistical modeling errorsrdquo Computational Statisticsand Data Analysis vol 51 no 6 pp 3187ndash3196 2007
[7] L Huang H Tang Q Tan et al ldquoA novel method for correctingscanline-observational bias of discontinuity orientationrdquo Sci-entific Reports vol 2016 no 6 Article ID 22942 2016
[8] R D Terzaghi ldquoSource of error in joint surveysrdquo Geo-technique vol 15 no 3 pp 287ndash304 1965
[9] H H Einstein and G B Baecher ldquoProbabilistic and statisticalmethods in engineering geologyrdquo Rock Mechanics and RockEngineering vol 16 no 1 pp 39ndash72 1983
[10] G B Baechaer ldquoStatistical analysis of rock mass fracturingrdquoMathematical Geology vol 15 pp 329ndash348 1983
[11] R Goodman ldquoToppling a fundamental failure mode indiscontinuous materials description and analysisrdquo in Pro-ceedings of Geo-Congress pp 2338ndash2368 San Diego CAUSA 2013
[12] H J Park T R West and I Woo ldquoProbabilistic analysis ofrock slope stability and random properties of discontinuityparameters Interstate Highway 40 Western North CarolinaUSArdquo Engineering Geology vol 79 no 3-4 pp 230ndash2502005
[13] O Fouche and J Diebolt ldquoDescribing the geometry of 3Dfracture systems by correcting for linear sampling biasrdquoMathematical Geology vol 36 no 1 pp 33ndash63 2004
[14] H M Tang L Huang A Bobet et al ldquoIdentification andmitigation of error in the Terzaghi bias correction for in-homogeneous material discontinuitiesrdquo Strength of Materialsvol 48 no 6 pp 825ndash833 2016
[15] H Tang L Huang C H Juang and J Zhang ldquoOptimizing theTerzaghi estimator of the 3D distribution of rock fractureorientationsrdquo Rock Mechanics and Rock Engineering vol 50no 8 pp 2085ndash2099 2017
[16] C Xu and P Dowd ldquoA new computer code for discretefracture network modelingrdquo Computers and Geosciencesvol 36 no 3 pp 292ndash301 2010
[17] A Brzovic and E Villaescusa ldquoRock mass characterizationand assessment of block-forming geological discontinuitiesduring caving of primary copper ore at the El Teniente mineChilerdquo International Journal of Rock Mechanics and MiningSciences vol 44 no 4 pp 565ndash583 2007
[18] M Grenon and J Hadjigeorgiou ldquoDrift reinforcement designbased on discontinuity network modelingrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 40 no 6pp 833ndash845 2003
[19] M S Ozcomak M Kartal O Senger and A K CelikldquoComparison of the powers of the Kolmogorov-Smirnov two-sample test and the Mann-Whitney test for different Kurtosisand Skewness coefficients using the Monte Carlo simulationmethodrdquo Journal of Statistical and Econometric Methodsvol 2 pp 81ndash98 2013
[20] L Al-Labadi and M Zarepour ldquoTwo-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approachrdquoMathematical Methods of Statistics vol 26 no 3 pp 212ndash2252017
Advances in Civil Engineering 9
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
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