TheoreticalAnalysisofDamagedWidth&Instability ...downloads.hindawi.com/journals/ace/2019/6328702.pdf · For mode I cracks, the Westergaard stress equation ‘ ˜ ReZ Ι(z)+ yImZ Ι(z)

Research ArticleTheoretical Analysis of Damaged Width amp InstabilityMechanism of Rib Pillar in Open-Pit Highwall Mining

Rui Wang 12 Shuai Yan 123 Jianbiao Bai14 Zhiguo Chang 24 and Tianhao Zhao12

1State Key Laboratory of Coal Resources amp Safe Mining China University of Mining amp Technology Xuzhou 221116 China2Key Laboratory of Deep Coal Resource Mining School of Mines China University of Mining amp TechnologyXuzhou 221116 China3Key Laboratory for Geomechanics amp Deep Underground Engineering China University of Mining amp TechnologyXuzhou 221116 China4School of Mines and Geology Xinjiang Institute of Engineering Urumqi Xinjiang 830091 China

Correspondence should be addressed to Rui Wang rwangcumteducn and Shuai Yan yanshuai2003cumteducn

Received 13 September 2018 Revised 13 February 2019 Accepted 14 February 2019 Published 11 March 2019

Academic Editor Venu G M Annamdas

Copyright copy 2019 Rui Wang 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

To improve the resourcesrsquo recovery ratio and the economic benefits of open-pit mines the development of highwall mining is used toexploit residual coal e design of rib pillar formed by excavation and mining activities in intact coal seams is crucial to the overallstability and safety of the highwall mining operationsis paper focuses on the damage caused to rib pillars by the large deformationoccurring with the application of highwall mining in an open-pit in China A mechanical model was established to investigate thedamaged width of colinear rib pillars based on HoekndashBrown andMohrndashCoulomb failure criterion e equations for calculating thedamaged width of the rib pillar were obtained respectively by combining theHoekndashBrown failure criterion with theMohrndashCoulombfailure criterione failure mechanism for the width of the rib pillar and the factors affecting the colinear rib pillar were analyzed indetail e results show that the application of theHoekndashBrown criterion has a unique advantage in analyzing the damaged width ofthe colinear rib pillars in open-pit highwall mining e instability mechanism and failure process of the rib pillars are described incombination with the limiting equilibrium method and the ratio of the elastic zonersquos width to the width of the entire rib pillar

1 Introduction

Open-pit mining always plays a considerable role in theworldrsquos coal mining which has many advantages such as largeoutput low cost simple mining technology operating safetyand so on [1ndash3]emain features of Chinese open-pit minesare the thick cover and shovel-truck mining technologysystem which is the most common [4 5] Due to higherstripping ratio under the thick cover a large number of coalswill be left under the highwalls in Chinese open-pit [6]ldquoHighwall miningrdquo is practised when the open-pit coal minesreach their pit limits due to the presence of surface dwellingsor uneconomical stripping ratios It involves driving a seriesof parallel entries separated by rib pillars using a remotelyoperated continuous miner into a coal seam exposed at thehighwall [7] For shovel-truck mining technology systemtransport lines of trucks are usually arranged in the highwallsso the slope angle of highwalls will be designed to be very low

Especially in some regions under poor geological conditionsthe angle of the final highwall will be much lower [8] hencethe residual coal under the highwalls will be more In tra-ditional mining systems the coal under highwalls will beburied by the inner dumping site and coal resources arenormally discarded which are wasted in vain For recoveringthe residual coal around highwalls the system of highwallmining craft which uses the highwall mining coal machine asthe main mining equipment has been improved and de-veloped based on research conducted in the mining industryand research and development conducted by the Fola CoalCompany and Superior Highwall Miners Inc in the UnitedStates Considering the extensive use of this craft by majormining overseas companies it is demonstrated that this craftcan provide a new approach toward the mining of residualcoal in Chinarsquos open-pit mines [9]

Rib pillars in open-pit mines play a key role in providingsupport to the superincumbent strata and maintaining the

HindawiAdvances in Civil EngineeringVolume 2019 Article ID 6328702 15 pageshttpsdoiorg10115520196328702

stability of the highwall mining To prevent the slope fromcollapsing during the process of highwall mining the width ofthe rib pillars is always somewhat large in practice Howeverthe width of the rib pillars is typically kept narrow to achievethe best possible economic benefit frommany open-pit minesin China Once the local damage of rib pillars are destroyeda large area landslide and slope collapse may occur Henceinvestigating a reasonable width for the rib pillar is an im-portant premise and foundation for the successful applicationand popularization of highwall mining technology

Generally methods used to investigate the instability andfailure of rib pillars can be divided into four types the empiricalmethod [10] theoretical method [11 12] fieldmeasuremethod[13] and numerical simulation method [14] However the fieldobservation is time-consuming and costly Numerical simu-lations have great randomness to the field measurements etheoretical equation method has great universality Hence thetheoretical equation method is a powerful tool for studyingrock mechanics in open-pit engineering Many studies haveinvestigated the stability of rib pillars based on the empiricalstrength equation of the MarkndashBienawski coal pillar to de-termine its retaining width In highwall mining the stabilitymodel of the rib pillar can be used in combinationwith the cuspcatastrophe theory and safety factor requirement of the ribpillar to calculate the reasonable width of the pillars [9] eformula for calculating the yield zone width of coal pillar byusing the criterion of MohrndashCoulomb was derived [15] elinear MohrndashCoulomb failure criterion is widely used to an-alyze the stability of rib pillars However although cracksnaturally exist in the rock mass in highwall mining cracks onthe exposed surface are also induced bymining and excavationerefore rib pillars can be defined as a jointed rock massMoreover the linear failure MohrndashCoulomb criterion forjointed rock mass has many shortcomings such as the fol-lowing (i) only the strength condition of the rock mass failurecan be provided but the mechanism of rock mass failure andthe occurrence and development process of rock mass failureare not considered (ii) it is impossible to explain the truestrength characteristics of the rockmass in the tensile stress andlow-stress regions or (iii) fractured rock mass with micro-cracks voids coarse crystals and fractured rock mass withmultiple structural planes and the obtained linear strengthenvelope is quite different from the parabolic strength envelopein actual situations [16] When conducting rock mass stabilityanalysis using the nonlinear HoekndashBrown criterion the dis-turbance factors caused by mining excavation and the dynamicchange of the rock massrsquo mechanical parameters can beconsidered erefore it is important to investigate the dif-ferences in analyzing the damaged width of a rib pillar usingeither the HoekndashBrown criterion or the MohrndashCoulomb cri-terion In this study it was found that using the HoekndashBrowncriterion is advantageouswhen analyzing a rib pillar with joints

erefore this paper focuses on the rib pillars in open-pithighwall mining e damaged width of rib pillars was used tomonitor with borehole camera exploration device (BCED) in anopen-pitmine inChina Amechanicalmodel is used to calculatethe damaged width of rib pillars based on the I-II compositecrack in fracture mechanics using the HoekndashBrown failurecriterion andMohrndashCoulomb failure criterion Accordingly the

formulas for calculating the damagedwidth of the rib pillar wereobtained respectively under different criteria e differences ofthe two criteria were compared and analyzed by considering thedamagemechanism for the width of the rib pillarsemodel isvalidated by fieldmonitoring Finally the instability mechanismof rib pillars in open-pit highwall mining is explained ispaper verifies the advantages of applying the HoekndashBrowncriterion to analyze the damaged width of rib pillars

2 Analysis of a Practical Case

is paper is based on the geological background of an open-pit mine in China e open-pit mine is located inShuozhou Shanxi province China (Figure 1(a)) In this casethe average thickness of the coal seam was 3m and theburial depth of the coal seamwas 122m [9] According to theactual data for the open-pit mine the width of the cavernwas 35m and the width of rib pillar was 40m

As illustrated in Figure 2(a) a horizontal exploratoryborehole 2m deep and 28mm in diameter was drilled in thebelt line of rib pillar A YTJ20 borehole camera explorationdevice (BCED) was used to monitor the damage in rib pillar(Figure 2(b)) Figure 2(c) shows probe images at differentdepths in the borehole ere are many inclined and inter-secting cracks found around the borehole e coal mass nearthe working face has failed and become a fractured zone Inaddition based on the results of the borehole probe it can beinferred that the deformation of the rib pillar adjacent to thegob side is larger or equal to that on the coal mass

Rib pillar deformation is large and approximately sym-metrical during highwall period e collected data on the ribpillar deformation can be summarized as follows (1) the rib-to-rib convergence is larger than the roof-to-floor conver-gence e maximum and average roof-to-floor convergencesare 189mm and 152mm respectively e maximum andaverage rib-to-rib convergences are 873mm and 546mmrespectively (2) e damaged width of rib pillar convergenceis larger e maximum damaged width reaches 916mm andthe average damaged width is 343mme large deformationis mainly caused by abutment stress causing the rib pillar tobecome unstabilityerefore the damaged width of rib pillarshould be studied to determine which rib pillar sizes

e intact parameters obtained by performing com-pression tests and Brazilian tests on small specimens arelisted in Table 1 e mechanical parameters for the coalmaterial and its surrounding rocks are as follows

3 Rib Pillar Bearing-Load Model inHighwall Mining

e damage of rib pillar is a gradual failure process in open-pithighwall mining Because there exist many similarities be-tween the analysis of rib pillars in strip mining and highwallmining the distributive stress characteristics of rib pillars inhighwall mining can be explained by analyzing the stresscharacteristics of rib pillars in strip mining (i) When coal isnot mined the coal body is subjected to the uniformly dis-tributed load of the overburden rock (ii) e supportingpressure zone and the local damaged zone are formed within

2 Advances in Civil Engineering

a certain range of the pillar when the rib pillar is nished onone side and during this period the peak value of the sup-porting pressure does not exceed the limit strength of thepillar (iii) After mining has nished on both sides of the pillarif the pillar can maintain a stable state the vertical stress of thepillar will have a saddle type distribution Damaged zones andelastic core zones exist on both sides of the rib pillar and at thecenter of the rib pillare stress distribution of the core area isapproximately shaped as a parabola (iv)e damaged zone onboth sides of the rib pillar gradually expands under the in-uence of othermining cavernse central stress of the elasticcore area reaches the limit strength of the rib pillar and thestress distribution of the core area is in the plateau stage whichis the critical state for the instability failure of the rib pillar Ifthe central stress of the elastic core area is slightly increasedthe pillar will quickly lose stability [17 18]

A bearing model of the rib pillar was establishedaccording to its stress distribution characteristics duringhighwall mining (Figure 3) e schematic diagram showsthat there is no lateral stress in the rib pillar and that the load ismainly applied from the overburden For a certain rib pillarits interior can be divided into the elastic core zone of themiddle part and the yield zone of the symmetrical distributionon both sides According to the limiting equilibrium methodthe rib pillar supports the weight of the overlying strata whichare equally divided between the upper rib pillars and theadjacent pillars [19] Assuming that the average thickness ofthe overlying strata is H bulk density is c width of the ribpillar isWp and cavern width isWe the (in-plane) load of therib pillar is expressed as follows

P cH WP +We( ) (1)

(a) (b)

(c) (d)

Beijing

Test siteShuozhou City

China

Residual coal

30

14

30

24

12

12

30

15

15

Overlyingstrata

Overlyingstrata

Overlyingstrata

Overlyingstrata

Main roof

Immediateroof

Coal

Immediatefloor

Main floor

RemarksThickness (m)

Surface soil

Weatheredsandstone

Sandstone

Mudstone

Siltstone

Sandstone

Coal

Shale

Siltstone

Rib pillar

Lithology

Figure 1 Mine location and highwall of the test site (a) location of the test site (b d) present situation of study site and (c) generalizedstratigraphic column of the test site

Advances in Civil Engineering 3

4 Mechanical Model of Rib PillarrsquosDamaged Width

In actual open-pit mines the mining thickness of the coalseam is much smaller than the inclined length of the facesus the faces can be considered as a series of inclinedcollinear cracks to be analyzed First the Cartesian co-ordinate system shown in Figure 4 was established and itsorigin was placed at the center of the cracks in the modelthat is the origin was placed at the center of the cavernUnder the assumption that the inclined length of theworking faces isWe 2a the vertical stress of the rib pillarimposed by the overlying strata is σ cH and the lateralstress of the rib pillar is λσ e angle between the inclineddirection and the vertical direction of the working face isα and the dip angle of the coal seam is β 90degminus αMoreover the model established in this study assumesthat the edge of the working face is not aected by thesurface force

Rib pillar

2m

Exploring borehole

15m

4m

(a) (b)

038m 048m 049m 055m

(c)

Figure 2 Borehole camera exploration in the rib pillar (a) Monitoring design (b) Test equipment YTJ20-BCED (c) Photographs obtainedfrom borehole at dierent depths

Table 1 Physical and mechanical parameters of coal-rock [9]

Rock strata Burial depth (m) Internal frication angle φ (deg) Poissonrsquos ration (μ) Density c (kNmiddotmminus3) Cohesive force c (MPa)Surface soil 30 12 042 196 0085Weatheredsandstone 44 38 036 230 25

Sandstone 74 39 032 238 30Mudstone 98 38 034 249 20Siltstone 110 36 032 232 35Sandstone 122 40 030 238 40Coal seam 125 36 038 144 162Shale 140 42 033 245 38Siltstone 155 38 032 260 50

H

WpWe Cavern Rib pillarRib pillar

YZ ECZ YZ

YZ yield zoneECZ elastic core zone

Figure 3 Schematic diagram of the bearing model of rib pillarduring highwall mining


At the edge of the rib pillar more cracks will occur whenthe rib pillar is subjected to tunnel excavation and over-burden Under certain tectonic stress eld conditionsfracture cracks will often occur in multiple groups simul-taneously and dierent fracture crack groups will often havedierent fracture mechanisms [20] By establishing a me-chanical model of the intermittent jointed rock mass Liuand Zhu reported that the fracture mechanism of the in-termittent jointed rock mass involves tensioning failure andcompression shear failure [21 22]

By combining eld observations with the results ofpreviously proposed theoretical models and the criteria ofcrack classication from fracture mechanics it can be seenthat the edge crack of the rib pillar investigated in this studyis a composite problem of a mode Ι-ΙΙ crack erefore thestress eld at the crack tip can be obtained by superimposingthe solutions of mode Ι and mode II cracks

41 Calculation of Stress Distribution at Tip of Mode I CrackFor mode I cracks the Westergaard stress equation ϕ ReZΙ(z) + yImZΙ(z) and Biharmonic equation of nabla4φ 0were combined en according to the CauchyndashRiemannconditions the relationship between the stress componentand stress function in elastic theory was applied withconsideration to the boundary conditions for a mode Ιcracks according to fracture mechanics Finally accordingto classical elastic theory the equations of the progressivestress eld at the crack tips can be expressed as follows

σx(r θ) KΙ2πr

radic cosθ2

1minus sinθ2sin

3θ2

( )

σy(r θ) KΙ2πr

radic cosθ2

1 + sinθ2sin

3θ2

( )

τxy(r θ) KΙ2πr

radic sinθ2cos

θ2cos

3θ2

(2)

In this equation KI is dened as the stress intensityfactor In particular the relevant equation KI may beexpressed as KΙ σ

πa

radic[23 24] based on fracture me-

chanics theoryBy substituting We 2a with KΙ σ

πa

radic the stress

intensity factor of the mode Ι crack under normal stressconditions can be expressed as follows

KΙ λ cos2 α + sin2 α( )σπa

radic

λ cos2 α + sin2 α( )σπWe

2

radic (3)

e equations of the progressive stress eld at the cracktips are the combination of equation (3) with equation (2)and are expressed as follows

σx(r θ) cH

2λ cos2 α + sin2 α( )

middotWe

r

radiccos

θ2

1minus sinθ2sin

3θ2

( )

σy(r θ) cH


middotWe

r

radiccos

θ2

1 + sinθ2sin

3θ2

( )

τxy(r θ) cH


We

r

radicsin

θ2cos

θ2cos

3θ2

(4)

42 Calculation of Stress Distribution at Tip of Mode II CrackSimilarly for a mode II crack the Westergaard stressequation ϕ minusyReZ∐(z) is combined with the Biharmonic

2a

ry

x λ σβ

2a

ry

xβ

2a

ry

xβ

Decomposed

Mode I crack

Mode II crack

τ

τ

τ

τ

σ

σ

σ

σ

σσ

λ σ

Figure 4 e analysis of damaged width at the rib pillar


equation nabla4ϕ 0 en according to the CauchyndashRiemannconditions and by considering the boundary conditions of themode II crack from fracture mechanics the relevant equation

of KII may be expressed as KΙΙ τπa

radic Hence the equations

of the progressive stress field at the crack tips can be expressedon the basis of classical elastic theory as follows

σx(r θ) minuscH

2(1minus λ) sin middot α cos middot α

We

r

1113970

sinθ2

2 + cosθ2cos

3θ2

1113888 1113889

σy(r θ) cH

2(1minus λ) sin middot α middot cos middot α

We

r

1113970

sinθ2cos

θ2cos

3θ2

τxy(r θ) cH

2(1minus λ)sin middot α cos middot α

We

r

1113970

cosθ2

1minus sinθ2sin

3θ2

1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

e equations of the progressive stress field at the tip ofthe model Ι-ΙΙ composite crack can be obtained by super-posing the equations of mode Ι crack and the equations of themode II crack according to fracture mechanics theoryHence the progressive stress field equations at the edge of theface are shown in Figure 4 and can be expressed as follows

σx(r θ) K

1r

1113970

[A(1minusB)minusC(2 + D)]

σy(r θ) K

1r

1113970

[A(1 + B) + CD]

τxy(r θ) K

1r

1113970

[E D + F(1minusB)]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

where A (λ cos2 α + sin2 α)cos(θ2) B sin(θ2)sin(3θ2) D cos(θ2)cos(3θ2) K (cH

We

1113968)2 C

(1minus λ)sin α cos α sin(θ2) E (λ cos2 α + sin2 α)sin(θ2)and F (1minus λ)sin α cos α cos(θ2)

43 Calculation of Rib Pillarrsquos Damaged Width In highwallmining the tip region of the rib pillars always exists as adamaged field e strength parameters of the elastic-plasticstress-strain field at the cracksrsquo tip region canbe quantitatively described by applying the theory of elastic-plastic fracture mechanics based on large yield conditionse equations of principal stress can be expressed accordingto elastic mechanics theory as follows

σ12 σx + σy

2plusmn

σx minus σy

21113874 1113875

2+ τ2xy

1113971

σ3 μ σx + σy1113872 1113873

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)

(1) For the I-II composite cracks without consideringthe influence of the adjacent mining faces theequations of the principal stress in the field close tothe crack tips of the damaged face edges are

expressed by the combination of equations (6) and(7) and defined as follows

σ12 K

1r

1113970

(AminusC) plusmn K

1r

1113970

1113864[AB + C(1 + D)]2

+[ED + F(1minusB)]2111386512

σ3 2μK

1r

1113970

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

In this equation A (λ cos2 α + sin2 α)cos(θ2) B

sin(θ2)sin(3θ2) D cos(θ2)cos(3θ2) K (cHWe

1113968)2 C (1minus λ)sin α cos α sin(θ2) E (λ cos2α

+ sin2 α)sin(θ2) and F (1minus λ)sin α cos α cos(θ2)

(a) e damaged width of the rib pillar can be calculatedby applying the HoekndashBrown strength criterionaccording to which the equations of σ1 with σ3 can beexpressed as follows

σ1 σ3 +

mRcσ3 + sR2c

1113969

(9)

where Rc σ1 and σ3 are the uniaxial compressivestrength of the rock and the maximum and mini-mum principal stress of the rock mass failure re-spectively e empirical parameters of m and s areconstants depending on the rock characteristics mdenotes the rockrsquos degree of hardness In the rangebetween 00000001 and 25 the value of 0000000001is considered as a severely disturbed rockmass whilethe value of 25 is considered as a completely hardrock mass s denotes the fracture degree of the rockmass with a range between 0 and 1 wherein thefractured rock mass is considered as 0 and thecomplete rock mass is considered as 1 [25 26]e boundary equation of the damaged field at theedge of the faces is expressed by the combination ofequations (6) and (7) as follows


rH 2mRcμK(AminusC) +

2mRcμK(AminusC)1113858 11138592

+ 4sR2cG

1113969

2G

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(10)

where A (λ cos2 α sin2 α)cos(θ2) B sin(θ2)sin(3θ2) D cos(θ2)cos(3θ2) K (cH

We

1113968)2

C (1minus λ)sin α cos α sin(θ2) E (λ cos2 α+ sin2 α)

sin(θ2) F (1minus λ)sin α cos α cos(θ2) and G 1113864 K

(AminusC) + K1113864[AB + C(1+ D)]2 + [ED + F(1minusB)]2111386512

minus 2μK(AminusC)11138652

Although θ 0 is ensured in equation (10) thedamaged width of the rib pillar along the x-directioncan be expressed by applying the HoekndashBrownstrength criterion rHp1 as follows

rHp1 2mRcμM +

2mRcμM( 11138572

+ 4sR2c[M(1minus 2μ) + N]2

1113969

2K[M(1minus 2μ) + N]2

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(11)

where M λ cos2 α + sin2 α N (1minus λ)sin α cos αand K (cH

We

1113968)2

(b) e damaged width of the rib pillar can be calculatedby applying the MohrndashCoulomb strength criterionaccording to which the equation of σ1 and σ3 can beexpressed as follows

σ1 1minus sinφ1 + sinφ

σ3 +2c cosφ1minus sinφ

(12)

where σ1 and σ3 are the maximum and minimumprincipal stress of the rock mass failure respectivelyc and φ are the cohesion and internal frictional angleof the coal body respectivelye boundary equation of damaged fieldat the edge of the faces can be expressed by

combining equation (8) with equation (12) asfollows

rC 2c cosφ

(1minus sinφ) middot J1113896 1113897

minus2

(13)

where A (λ cos2 α + sin2 α)cos(θ2) B sin(θ2)

sin(3θ2) D cos(θ2)cos(3θ2) K (cHWe

1113968)2

C (1minus λ)sin α cos α sin(θ2) E (λ cos2 α+ sin2α)sin(θ2) F (1minusλ)sin α cos α cos(θ2) and J

K(AminusC)[1minus2μ(1minus sinφ) (1+ sinφ)] + K [AB + C

(1 + D)]2 +[ED + F(1minusB)]212Although θ 0 is ensured in equation (13) thedamaged width of the rib pillar along the x-directioncan be expressed by applying the MohrndashCoulombstrength criterion rCp1 as follows

rCp1 2c cosφ(1 + sinφ)

K(1minus sinφ) [(1 + sinφ)minus 2μ(1minus sinφ)]M + N(1 + sinφ)1113864 11138651113896 1113897

minus2

(14)


We

1113968)2

(2) If the interaction between the adjacent mining facesis considered a series of periodic cracks will bedistributed on the infinite plates e boundaryequations of the damaged field at the edge of thefaces have periodicity owing to the periodic dis-tribution of cracks Additionally Z(z) σz(

z2 minus a2

radic) is a complex variable function satisfying

all of the boundary conditions for a single mode Icrack To satisfy the periodic boundary conditions z

must be the periodic function of (Wp + We) in thecomplex variable function presented above It iswell known that the simplest periodic function is

a sinusoidal function therefore in the above complexvariable function sin[πz(2(We + Wp))] is usedinstead of z and sin[πa(2(We + Wp))] is used in-stead of a [27] en the stress intensity factorequations are KΙprime η middot KΙ and KIIprime η middot KΙI for the I-IIcomposite cracks respectively [28] In summary thestress at the tip of several inclined cracks is sepa-rated by a certain distance on the infinite plate andcan be calculated by multiplying the stress of a singleinclined crack with the affected coefficient Addi-tionally the affected coefficient is expressed as followsη [(2(We + Wp)πWe)tan(πWe2(We + Wp))]12Hence the equations of stress distribution at the edgeof the faces can be expressed as follows


σx(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


σy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972

[A(1 + B) + C middot D]

τxy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972

[E middot D + F(1minusB)]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

where A (λ cos2 α sin2 α)cos(θ2) B sin(θ2)sin(3θ2) D cos(θ2)cos(3θ2) C (1minus λ)sin α cos αsin(θ2) E (λ cos2 α+ sin2 α)sin(θ2) and F

(1minus λ)sin α cos α cos(θ2)e equations of the principal stress in the field nearthe crack tips of the facesrsquo damaged edges areexpressed by combining equation (15) with equation(7) as follows

σ12 cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972

middot (AminusC) plusmn 1113864[AB + C(1 + D)]2

+[E D + F(1minusB)]2111386512

σ3 μcH

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)

(a) e damaged width of the rib pillar can be calculatedby applying the HoekndashBrown strength criterion

according to which the boundary equation of damagedfield at the edge of the faces can be expressed bycombining equation (16) with equation (9) as follows

r mRcμcHP +

mRcμcHP( 11138572

+ 4sR2c(GcHP2K)2

1113969

(2GcHPK)2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

minus2

(17)



1113968)2

C (1minus λ)sin α cos α sin(θ2) E (λ cos2 α + sin2 α)

sin(θ2) F (1minus λ)sin α cos α cos(θ2) P (2(We + Wp)π)tan(πWe2(We + Wp))

1113969 and G K(A1113864

minusC) + K [A middot B + C(1 + D)]2 + [ED + F(1minusB)]21113966 111396712minus

2μK(AminusC)2Although θ 0 is ensured in equation (17) the dam-aged width of the rib pillar along the x-direction can beexpressed by applying the HoekndashBrown strength cri-terion rHp2 as follows

rHp2 mRcμcHP +

mRcμcHP( 11138572

+ sR2c KcHP[M(1minus 2μ) + N]1113864 1113865

21113969

2KcHP[M(1minus 2μ) + N]1113864 11138652

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(18)

whereM λ cos2 α+ sin2 αN (1minusλ)sin α cos αK (c

HWe

1113968)2 and P

(2(We + Wp)π)tan(πWe2(We + Wp))

1113969

(b) e damaged width of the rib pillar can be cal-culated by applying the MohrndashCoulomb strengthcriterion according to which the boundary equa-tion of the damaged field at the edge of the faces can

be expressed by combining equation (16) withequation (12) as follows


(1minus sinφ)[cHPQ(1 + sinφ)minus 2μcHP(AminusC)(1minus sinφ)]1113896 1113897

minus2

(19)




1113968)2



1113969 and Q (AminusC) +

[AB + C(1 + D)]2+1113966 [E D + F(1minusB)]212

Although θ 0 is ensured in equation (19) the dam-aged width of rib pillar along the x-direction can beexpressed by applying the MohrndashCoulomb strengthcriterion rCp2 as follows


(1minus sinφ)[cHP(M + N)(1 + sinφ)minus 2μcHPM(1minus sinφ)]1113896 1113897

minus2

(20)

where M λ cos2 α + sin2 α N (1minus λ)sin α cos αand P


1113969

e actual data presented in Table 1 were used to verifythe feasibility of the above formulas with regard to thedamaged width of the rib pillars Using these data it wasfound that in the open-pit mine the cavern width was 35mand the width of the rib pillar was 40m Additionally theinternal friction angle and Poissonrsquos ratio of the coal bodywere φ 36deg and μ 038 respectively e dip angle of thecoal seam was β 11deg the coefficient of lateral pressure wasλ 12 the uniaxial compressive strength of the coal bodywas Rc 278MPa and the average bulk density wasc 228 KNm3 according to the data pertaining to thismine

e abovementioned data were substituted in equation(18) m and s were confirmed as m 12 and s 05 More-over half of the value range of m and s was confirmed toavoid either of them having a prominent influence on thefinal result Hence the damaged width of the rib pillar wascalculated as rHp2 453mm by applying the HoekndashBrownstrength criterion rHp2 Similarly the damaged width of therib pillar was calculated as rCp2 225mm by applying theMohrndashCoulomb strength criterion rCp2 On one side of thefaces the damaged width of the rib pillar was approximately485mm according to the monitoring data obtained byapplying the methods of borehole stress monitoringacoustic wave detection and optical borehole imaging eanalytical solution obtained by applying the HoekndashBrownstrength criterion was closer to the measured value as wasdetermined by comparing the calculated and measuredvalues of the damaged width of the rib pillar under the twostrength criteria e relative error in the application of theHoekndashBrown strength criterion was only 66 However therelative error in the application of the MohrndashCoulombstrength criterion was as high as 536 In addition Chenand Wu [9] derived the calculation formula of pillar yieldwidth based on the bearing model of rib pillar duringhighwall mining and combined with cusp catastrophe theoryand safety factor demand of rib pillar Hence the above-mentioned data were substituted into the calculation for-mula of pillar yield width such as Y 283mm e relativeerror was 416 Although Chenrsquos error is smaller than thatobtained in the application of the MohrndashCoulomb strengthcriterion in this paper it still has a greater accuracy with the

error obtained in the application of the HoekndashBrownstrength criterion in this paper

In conclusion in the analysis and calculation of thejointed rock mass with cracks it was found that the non-linear HoekndashBrown strength criterion is a better approxi-mation of the actual mechanism compared to the linearMohrndashCoulomb strength criterion

5 Analysis of Characteristic Parameters andInstability Mechanism of Rib Pillar

51 Analysis of Characteristic Parameter With regard towhether the rib pillar was affected by the adjacent faces ornot the main independent factors affecting the lateraldamaged width of the rib pillar in highwall mining weredetermined based on the theoretical deduction of applyingtwo different strength values us the factors were de-termined as follows the retaining rib pillar width Wp thecavern width We the burial depth of the coal seam H andthe dip angle of the coal seam β Regarding whether the ribpillar is typically affected by the adjacent faces during theactual production process this study carried out a detailedanalysis with respect to the influencing factors in theboundary equation of the rib pillarrsquos lateral damaged widthMoreover to further clarify the calculation of the rib pillarrsquosdamaged width in open-pit highwall mining the applicationof the HoekndashBrown criterion is more suitable

e boundary equations (18) and (20) which representthe rib pillarrsquos damaged width were fitted with curves usingdifferent parameters to demonstrate the relationship be-tween rp2 andWpWeH and β respectively As described in[29] the value range of the internal friction angle φ istypically 16degndash40deg and the value range of cohesion c is10ndash98MPa Hence when selecting the values of the rele-vant parameters in equation (20) the characteristic valuesrepresenting the different failure states of the coal body canbe confirmed using the abovementioned range According tothe relevant parameters in equation (18) the parametersrepresenting a different hardness degree m different crackdegree s and uniaxial compressive strength Rc were selectedin a descending manner While Poissonrsquos ratio reflected theactual strength the rockrsquos weathering degree and fracturedevelopment degree μ were selected in an ascendingmanner


511 Analysis of Relationship between Width of RetainingRib Pillar and Damaged Width of Rib Pillar Provided thatthe other parameters are fixed the functional relationship ofthe rib pillarrsquos damaged width with the width of the retainingrib pillar becomes inversely logarithmic by applying theHoekndashBrown strength criterion as shown in Figure 5 Inother words the damaged width of the rib pillar graduallydecreases as the width of the retaining rib pillar increasesand tends toward stability Additionally its curvaturegradually decreases and tends toward zero Moreoveraccording to the order of the three curves under the differentparameters shown in Figure 5 it can be seen that the denserthe joints and cracks distributed in the coal body are theeasier it is for the rib pillars to be destroyed and the widthdamaged by the rib pillars becomes greater Overall whenthe width of the retaining rib pillar is 1-2m the damagedwidth of the rib pillar rapidly fluctuates At that moment theelastic core area of the rib pillar is somewhat small and therib pillar is vulnerable to instability under the load of theoverlying strata While the width of the retaining rib pillar is2ndash4m the range of the rib pillarrsquos damaged width graduallydecreases and the range of the rib pillarrsquos elastic core areagradually increases Although the superposition stress andoverburden load of the rib pillar gradually increases the ribpillar is not vulnerable to instability because the elastic corearea of the rib pillar increases When the width of theretaining rib pillar is more than 4m the damaged width ofthe rib pillar changes slightly and remains in a state ofstability and invariance At this moment the range of theelastic core area in the rib pillar is much larger than that ofthe damaged area and it is thus difficult for the rib pillar tobecome unstable owing to the existence of local failure at theedge of the rib pillar

e curve relationships between the damaged width ofthe rib pillar and the width of the retaining rib pillar indifferent states of coal destruction were obtained by applyingto the MohrndashCoulomb strength criterion as shown in Fig-ure 5 e functional relationship of the damaged width ofthe rib pillar and the width of the retaining rib pillar is anapproximately inversely proportional function obtained byobservation and analysis In other words the damaged widthof the rib pillar gradually decreases as the width of theremaining rib pillar increases and its curvature graduallydecreases In the process of coal destruction the internalfriction angle gradually increases but the cohesion graduallydecreases Additionally it can be seen that the smaller theintegrity in the coal body the larger the damaged width ofthe rib pillars However in Figure 5 this observation isslightly inconsistent is suggests that there exist largeerrors in the analysis of the jointed rock mass characteristicswhich were obtained by applying the MohrndashCoulombstrength criterion

Additionally observing the variation of the three curvesshown in Figure 5 under different parameters intuitivelydemonstrates the step-by-step process of applying theHoekndashBrown strength criterion Moreover the selection ofparameters indicates the integrity of the coal body withregard to reflecting the failure mechanism of the actual ribpillar Conversely the three correlation curves produced by

applying the MohrndashCoulomb strength criterion do not fullyreflect the ladder property under different parameter se-lection conditions erefore this indicates that there existsa certain deviation between the failure mechanism of theactual rib pillar and the actual failure of the actual rib pillarFurthermore it can be verified that the application of theHoekndashBrown strength criterion is more reasonable in theanalysis of the jointed rock mass

512 Analysis of Relationship between Width of Cavern andDamaged Width of Rib Pillar Provided that the other pa-rameters are fixed the functional relationship of the ribpillarrsquos damaged width with the width of the retaining ribpillar is the exponential function obtained by applying theHoekndashBrown strength criterion as shown in Figure 6 Inother words the damaged width of the rib pillar graduallyincreases with the increase of the cavern width and thentends toward infinity Additionally its increased curvaturealso gradually increases toward infinity Similarly accordingto the order of the three curves obtained with differentparameters as shown in Figure 6 it can be seen that thedenser the distributions of the joints and cracks is in the coalbody the easier it is to destroy the rib pillars and the greateris the production of damaged width by the rib pillars Whenthe cavern width is 0ndash5m the variation of the rib pillarrsquosdamaged width is very small which indicates that thevariation of the cavernrsquos width has a small effect on thedamage incurred by the rib pillar within this range Whilethe width of cavern is 5ndash85m the damaged width of the ribpillar gradually increases but the change of its growthcurvature is small Additionally when the cavern width ismore than 85m the damaged width of the rib pillar sharplyincreases and tends toward infinitye previous descriptionindicates that the load on the rib pillar is mainly derivedfrom the overburden action According to the effective re-gion theory the rib pillar supports the weight of theoverburden which is equally divided between the upper ribpillar and the adjacent rib pillar erefore the overburdenon the top of rib pillar will gradually increase with the in-crease of the cavern width If the working face is excessivelylarge the rib pillars will become less stable and the overallstability of the open-pit slope will be endangered when therib pillars fail to bear the load of the overburden rock in fullerefore the variation law of the abovementioned curve isvery indicative of the rib pillarrsquos failure and instabilityprocess

e functional relationship between the damaged widthof the rib pillar and the cavern width is also a directlyproportional function obtained by applying the MohrndashCoulomb strength criterion as shown in Figure 6 In otherwords the damaged width of the rib pillar increases ina linear manner with the increase of the cavern width whileits increasing curvature remains constant Similarlyaccording to the order of the three curves obtained withdifferent parameters as shown in Figure 6 it can be seen thatthe denser the joints and crack distribution are in the coalbody the easier it is to destroy the rib pillars and the greateris the production of the damaged width by the rib pillars


In combination with the previous description it can beunderstood that in open-pit highwall mining the rib pillarfails in a gradual manner and is in a period of instability e

vertical stress gradually changes from saddle type to plat-form type rather than maintaining a constant curvaturewith a linear increase erefore it can be inferred that thechange of the rib pillarrsquos damaged width exhibits a linearlyincreasing trend is also indicates that the application ofthe MohrndashCoulomb failure criterion in the analysis of thejointed rock mass is insubrvbarcient

513 Analysis of Relationship between Burial Width andDamaged Width of Rib Pillar Provided that other param-eters do not change the relationship between the rib pillarrsquosdamaged width and the burial width of the coal seam re-sembles an exponential function by applying the HoekndashBrown strength criterion as shown in Figure 7 In otherwords the damaged width of the rib pillar gradually in-creases with the increase of the burial depth and tends to-ward innity When the burial depth of the coal seam isbetween 0m and 800m the damaged width of the rib pillarchanges innitesimally which indicates that the burial depthvariation has little inuence on the rib pillarrsquos damagedwidth However when the burial depth of the coal seam ismore than 800m the damaged width of the rib pillar sharplyincreases and tends toward innity

e damaged width of the rib pillar linearly increaseswith the burial width of the coal seam by applying theMohrndashCoulomb strength criterion shown in Figure 7 Inother words when the burial depth of the coal seam isbetween 0m and 500m the damaged width of the rib pillar

0 1 2 3 4 5 6 7 8 9 10 11070

075

080

085

090

095

100

Wp (m)

000

005

010

015

020

Partial enlarged drawing



0 1 2 3 4 5 6 7 8 9 10

078856

078858

078860

078862

078864

078866

078868

Wpr C

p2 (m

)

0 1 2 3 4 5 6 7 8 9 10

083778

083780

083782

083784

083786

083788

083790

083792

Wp

r Cp2

(m)r H

p2 (m

)

r Cp2

(m)

0 1 2 3 4 5 6 7 8 9 10092974092976092978092980092982092984092986092988092990

Wp

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001

Figure 5 e inuence curve of damaged width of rib pillar by width of retaining rib pillar

0 1 2 3 4 5 6 7 8 9 10 11

00

05

10

15

20

25

30

35

40

We (m)

00

05

10

15

20

25

30

35

40

r Hp2

(m)

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015m = 001 Rc = 05MPa μ = 1 s = 001

Figure 6 e inuence curve of damaged width of rib pillar by thewidth of cavern


increases with the increase of the burial depth but itscurvature is relatively smooth While the burial depth of thecoal seam is more than 500m the damaged width of ribpillar increases in a linear manner with the increase of theburial depth

514 Analysis of Relationship between Dip Angle of CoalSeam and DamagedWidth of Rib Pillar Provided that otherparameters do not change the relationship between thedamaged width of the rib pillar and the dip angle of the coalseam is a periodic uctuating function obtained by applyingthe HoekndashBrown strength criterion as shown in Figure 8 Inother words the damaged width of the rib pillar uctuatesperiodically with the change of the dip angle

As can be seen in the gure the uctuation range ofthe black and blue curves implies that the improved coalintegrity is 1 mm and 18 mm respectively while theuctuation range of the other curve indicates poor coal inthe proximity of 320mm e uctuation period is ap-proximately 3deg according to the periodic variation of thethree curves e crack strain and bulk strain describingthe damage amount uctuate in an approximately peri-odic manner during the damage evolution period of thebrittle rock obtained from marble as described in [30]However the uctuation range is relatively small estress law of the rock instability and failure can be in-vestigated using the crack strain model erefore in thisstudy the curve shape has the same property owing to itsminimal amplitudee change of the rib pillarrsquos damagedwidth with the dip angle of the coal seam can be ap-proximated as an inclined line with smaller curvature ifthe scale of the ordinal coordinate in Figure 8 is expandedseveral times en the tendency of the curves indicates

that the change of the coal seamrsquos dip angle has a smalleect on the damaged width of the rib pillar

e relationship between the damaged width of the ribpillar decreases in a linear manner with the increase of thecoal seamrsquos dip angle by applying the MohrndashCoulombstrength criterion as shown in Figure 8 In other words thedamaged width of the rib pillar gradually decreases with theincrease of the coal seamrsquos dip angle

e vertical decomposition force of the rib pillargradually decreases with the increase of the coal seamrsquos dipangle combined with the mechanical model shown inFigure 2 In other words the main source of the load exertedon the rib pillar decreases and the entire damaged width ofthe rib pillar should gradually decrease However thetransverse decomposition force of the rib pillar graduallyincreases which forces the rib pillar to exhibit stress con-centration in the local area us the change of the ribpillarrsquos damaged width slightly uctuates but does notexhibit a constant downward trend

52 Analysis of Rib Pillarrsquos InstabilityMechanism e stablerib pillar has a saddle type vertical stress distribution asdescribed previously whereas in the transition fromstability to yield the rib pillar has a platform type criticalstress distribution e equation for calculating the widthof the rib pillar in the critical state is expressed as follows[31]

Wp cHWeL + Lminus 2rpz( )rpzσz times 105

σz times 105 Lminus rpz( )minus cHL (21)

In equation (21) c is the average bulk density of theoverburden rock in kgm3 H is the burial depth of the coalseam in m We is the cavern width in m L is the cavernlength inm rpz is the damaged width of the rib pillar inm rpis the damaged width on one side of the rib pillar inm and σzis the ultimate strength of the rib pillar in MPa e ultimatestrength of the rock mass is typically the compressivestrength of the rock mass because the rock mass is typicallycompressive rather than tensile Additionally in equation(21) rpz 2rp

Moreover the safety factor of the rib pillar can be ob-tained by applying the limit strength theory f [9]

f σzWp

cH We +Wp( ) (22)

In highwall mining the safety factor of the rib pillar istypically required to be more than 13 [31 32] Furthermorethe theoretical value and measured value of the rib pillarrsquosdamaged width are compared with the rib pillarrsquos retainingwidth in combination with the abovementioned examplesus it is demonstrated that the rib pillar remains stablewhen the damaged width of the rib pillar is between 30 and35 of the rib pillarrsquos width However instability occurswhen the rib pillar exceeds this range

In conclusion in highwall mining two conditions mustbe simultaneously satised to maintain the rib pillarrsquos sta-bility [33] e rst condition is that the rib pillar safety

0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

r Cp2

(m)

5

6

7

H (m)

0

1

2

3

4

5

r Hp2

(m)

6

7

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1


Figure 7 e inuence curve of damaged width of rib pillar byburial depth


factor must be more than the limited value the secondcondition is that the damaged width of the rib pillar must beless than the limited value

e radial stress of the surrounding rock around thecavern is gradually released the equilibrium stress of theoriginal rock is destroyed and the stress of the overburdenrock is gradually transferred to the rib pillar on both sides ofthe cavern after the excavation of the cavern has completedus the stress of the rib pillar gradually increases Duringthis period the primary ssure of the rib pillar graduallydevelops and extends the secondary crack gradually accu-mulates the internal friction angle of the coal body graduallyincreases and the cohesion force gradually decreases esurroundings of the rib pillar start being destroyed andwhen the face is excavated in the vicinity of the cavern therib pillar is subjected to a mining superimposed stress loadAdditionally the rib pillar between the caverns is notsubjected to lateral stress erefore the rib pillar graduallyproduces a higher stress concentration in this state whichmakes the inner crack of the pillar expand further and passthrough while the surrounding rock around the pillargradually breaks down and falls oe elastic core region ofthe rib pillar under the overburden load gradually decreasesand the area of the yield zone gradually increases Once theelastic core area is below the critical value of the rib pillarrsquosinstability failure the instability failure of the rib pillaroccurs

6 Conclusions

A eld test and theoretical mechanical model were com-bined to investigate the damaged width of rib pillars in open-pit highwall mining and optimize the reasonable width of therib pillar Additionally the instability mechanism of the ribpillars in open-pit highwall mining was analyzed in a simple

manner e following conclusions were drawn from thisstudy

(1) As shown in the images captured by a boreholecamera the monitored rib pillar deformation in-dicates that the 4m rib pillar was damaged com-pletely by the coalescence of internal cracks Usingmechanical model analysis the calculation of the ribpillarrsquos damaged width was compared under dif-ferent failure criteria which further veries theunique advantage of the HoekndashBrown criterion

(2) e formulas obtained by mechanical model analysiswere validated against data obtained by eld mon-itoring Based on the data obtained from the open-pitmine in this paper the relative error value obtainedby applying the HoekndashBrown strength criterion was66 but the relative error value obtained by ap-plying the MohrndashCoulomb strength criterion was ashigh as 536 e result calculated based on theHoekndashBrown criterion is more consistent with theeld observations

(3) Studies have shown that the main factors aectingthe damaged width of the rib pillar rp are as followsWp We H and β However the function re-lationships between rp andWpWe H and β in theboundary equation which was obtained by applyingthe MohrndashCoulomb failure criterion were inverselyproportional directly proportional increasing near-linearly and decreasing near-linearly respectivelye function relationships between rp and Wp WeH and β in the boundary equation obtained byapplying the HoekndashBrown failure criterion were thetanti-logarithm exponent similar exponent andperiodic function of the minimal uctuation am-plitude Generally the application of the Hoekndash

0 2 4 6 8 10 12 14 16 18 20

06

08

10

12

14

16

18

00

01

02

03

04

0 2 4 6 8 10 12β (deg)

β (deg)

14 16 18 20

00000

00005

00010

00015

00020

00025

r Hp2

(m)

r Hp2

(m)

r Cp2

(m)


φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001

Figure 8 e inuence curve of damaged width of rib pillar by the dip angle of coal seam


Brown failure criterion in the analysis of jointed rockmass failure is closer to engineering practice than theMohrndashCoulomb failure criterion

Data Availability

(1) e data files of numerical simulation (Figure 3) used tosupport the findings of this study are available from thecorresponding author upon request (2) e data files ofcurve (Figures 4ndash7) used to support the findings of this studyare available from the corresponding author upon request

Conflicts of Interest

e authors declare no conflicts of interest

Authorsrsquo Contributions

RW and S Y conducted a thorough literature search RWdrafted the manuscript Z C J B and T Z reviewed thefinal paper and made important suggestions and recom-mendations for paper revision

Acknowledgments

is work was financially sponsored by the National NaturalScience Foundation of China (nos 51604268 and 51574227)and Independent Research Project of State Key Laboratoryof Coal Resources and Safe Mining CUMT(SKLCRSM18X08) e authors thank the open-pit mine fortheir support during the field test Also this article benefittedfrom valuable comments and suggestions by coeditors andother anonymous reviewers

References

[1] G F Brent ldquoQuantifying eco-efficiency within life cyclemanagement using a process model of strip coal miningrdquoInternational Journal of Mining Reclamation and Environ-ment vol 25 no 3 pp 258ndash273 2011

[2] F Owusu-Mensah and C Musingwini ldquoEvaluation of oretransport options from Kwesi Mensah Shaft to the mill at theObuasi minerdquo International Journal of Mining Reclamationand Environment vol 25 no 2 pp 109ndash125 2011

[3] B Shen and F M Duncan ldquoReview of highwall mining ex-perience in Australia and a case studyrdquo Australian Geo-mechanics vol 36 no 2 pp 25ndash32 2001

[4] Y L Chen Q X Cai T Shang and Z X Che ldquoMining systemfor remaining coal of final highwallrdquo Journal of Mining Sci-ence vol 47 no 6 pp 771ndash777 2011

[5] Q X Cai W Zhou J S Shu et al ldquoAnalysis and applicationon end-slope timeliness of internal dumping under flatdipping ore body in large surface coal minerdquo Journal of ChinaUniversity of Mining amp Technology vol 37 no 6 pp 740ndash7442008

[6] W Cao Y Sheng Y Qin J Li and J Wu ldquoGrey relationprojection model for evaluating permafrost environment inthe Muli coal mining area Chinardquo International Journal ofMining Reclamation and Environment vol 24 no 4pp 363ndash374 2010

[7] J L Porathur S Karekal and P Palroy ldquoWeb pillar designapproach for highwall mining extractionrdquo International

Journal of Rock Mechanics and Mining Sciences vol 64pp 73ndash83 2013

[8] Z M Liu S L Wang and Q X Cai ldquoApplication of top-coalcaving method in Anjialing surface minerdquo Journal of ChinaUniversity of Mining amp Technology vol 30 pp 515ndash517 2001

[9] Y L Chen and H S Wu ldquoCatastrophe instability mechanismof rib pillar in open-pit highwall miningrdquo Journal of ChinaUniversity of Mining ampTechnology vol 45 no 5 pp 859ndash8652016

[10] D Hill ldquoCoal pillar design criteria for surface protectionrdquo inProceedings of the Coal 2005 Conference vol 4 pp 26ndash28Pittsburgh PA USA September 2005

[11] W Gao and M Ge ldquoStability of a coal pillar for strip miningbased on an elastic-plastic analysisrdquo International Journal ofRock Mechanics and Mining Sciences vol 87 pp 23ndash28 2016

[12] B A Poulsen ldquoCoal pillar load calculation by pressure archtheory and near field extraction ratiordquo International Journalof Rock Mechanics and Mining Sciences vol 47 no 7pp 1158ndash1165 2010

[13] B Yu Z Zhang T Kuang and J Liu ldquoStress changes anddeformation monitoring of longwall coal pillars located inweak groundrdquo Rock Mechanics and Rock Engineering vol 49no 8 pp 3293ndash3305 2016

[14] S-L Wang S-P Hao Y Chen J-B Bai X-Y Wang andY Xu ldquoNumerical investigation of coal pillar failure undersimultaneous static and dynamic loadingrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 84pp 59ndash68 2016a

[15] L X Wu and J Z Wang ldquoCalculation of width of yieldingzone of coal pillar and analysis of influencing factorsrdquo Journalof China Coal Society vol 20 no 6 pp 625ndash631 1995

[16] B Z Li ldquoDiscuss and practice of the design theory in con-struction pit engineeringrdquo Masterrsquos thesis Harbin Engi-neering University Harbin China 2004

[17] L X Wu and J Z Wang Mining Ieory and Practice of CoalStripe under Building China University of Mining andTechnology Press Xuzhou China 1994

[18] F Russell and R Guy ldquoCoal pillar design when considereda reinforcement problem rather than a suspension problemrdquoInternational Journal of Mining Science and Technologyvol 28 no 1 pp 11ndash19 2018

[19] L G Wang X X Miao X Z Wang et al ldquoAnalysis ondamaged width of coal pillar in strip extractionrdquo ChineseJournal of Geotechnical Engineering vol 28 no 6 pp 767ndash769 2006

[20] S J Wang ldquoGeological nature of rock and its deduction forrock mechanicsrdquo Chinese Journal of Rock Mechanics andEngineering vol 28 no 3 pp 433ndash450 2009

[21] D Y Liu and K S Zhu ldquoAn experimental study on cementmortar specimens with a central slits under uniaxial com-pressionrdquo Journal of Civil Architectural amp EnvironmentalEngineering vol 16 no 1 pp 56ndash62 1994

[22] D Y Liu Study on compressive shear fracture and strengthcharacteristics of intermittent jointed rock mass PhD thesisUniversity of Chongqing Chongqing Institute of Architec-tural Engineering Chongqing China 1993

[23] C Q Cao and J Hua Engineering Fracture Mechanics XirsquoanJiao Tong University Press Xirsquoan China 2015

[24] Z Q Wang and S H Chen Advanced Fracture MechanicsScience Press Beijing China 2009

[25] J M Zhu X P Peng Y P Yao et al ldquoApplication of SMPfailure criterion to computing limit strength of coal pillarrdquoRock and Soil Mechanics vol 31 no 9 pp 2987ndash2990 2010


[26] B Wang J B Zhu P Yan et al ldquoDamage strength de-termination of marble and its parameters evaluation based ondamage control testrdquo Chinese Journal of Mechanics and En-gineering vol 31 no 2 pp 3967ndash3973 2012

[27] L G Wang and X X Miao ldquoStudy on catastrophe charac-teristics of the destabilization of coal pillarsrdquo Journal of ChinaUniversity of Mining and Technology vol 36 no 1 pp 7ndash112007

[28] R K Zipf and C Mark ldquoGround control for highwall miningin the United Statesrdquo International Journal of Surface MiningReclamation and Environment vol 19 no 3 pp 188ndash2172005

[29] K Matsui H Shimada T Sasaoka et al ldquoHighwall stabilitydue to punch mining at opencut coal minesrdquo InternationalJournal of Surface Mining Reclamation and Environmentvol 18 no 3 pp 185ndash204 2004

[30] M R Shen and J F Chen Mechanics of Rock Masses TongjiUniversity Press Shanghai China 2006

[31] L K Youn and S Pietruszczak ldquoAnalytical representation ofMohr failure envelope approximating the generalizedHoekndashBrown failure criterionrdquo International Journal of RockMechanics and Mining Sciences vol 100 pp 90ndash99 2017

[32] L K Youn S Pietruszczak and C H Byung ldquoFailure criteriafor rocks based on smooth approximations MohrndashCoulomband HoekndashBrown failure functionsrdquo International Journal ofRock Mechanics and Mining Sciences vol 56 pp 146ndash1602012

[33] S Mo I Canbulat C Zhang et al ldquoNumerical investigationinto the effect of backfilling on coal pillar strength in highwallminingrdquo International Journal of Mining Sciences and Tech-nology vol 7 pp 1ndash6 2017


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stability of the highwall mining To prevent the slope fromcollapsing during the process of highwall mining the width ofthe rib pillars is always somewhat large in practice Howeverthe width of the rib pillars is typically kept narrow to achievethe best possible economic benefit frommany open-pit minesin China Once the local damage of rib pillars are destroyeda large area landslide and slope collapse may occur Henceinvestigating a reasonable width for the rib pillar is an im-portant premise and foundation for the successful applicationand popularization of highwall mining technology

Generally methods used to investigate the instability andfailure of rib pillars can be divided into four types the empiricalmethod [10] theoretical method [11 12] fieldmeasuremethod[13] and numerical simulation method [14] However the fieldobservation is time-consuming and costly Numerical simu-lations have great randomness to the field measurements etheoretical equation method has great universality Hence thetheoretical equation method is a powerful tool for studyingrock mechanics in open-pit engineering Many studies haveinvestigated the stability of rib pillars based on the empiricalstrength equation of the MarkndashBienawski coal pillar to de-termine its retaining width In highwall mining the stabilitymodel of the rib pillar can be used in combinationwith the cuspcatastrophe theory and safety factor requirement of the ribpillar to calculate the reasonable width of the pillars [9] eformula for calculating the yield zone width of coal pillar byusing the criterion of MohrndashCoulomb was derived [15] elinear MohrndashCoulomb failure criterion is widely used to an-alyze the stability of rib pillars However although cracksnaturally exist in the rock mass in highwall mining cracks onthe exposed surface are also induced bymining and excavationerefore rib pillars can be defined as a jointed rock massMoreover the linear failure MohrndashCoulomb criterion forjointed rock mass has many shortcomings such as the fol-lowing (i) only the strength condition of the rock mass failurecan be provided but the mechanism of rock mass failure andthe occurrence and development process of rock mass failureare not considered (ii) it is impossible to explain the truestrength characteristics of the rockmass in the tensile stress andlow-stress regions or (iii) fractured rock mass with micro-cracks voids coarse crystals and fractured rock mass withmultiple structural planes and the obtained linear strengthenvelope is quite different from the parabolic strength envelopein actual situations [16] When conducting rock mass stabilityanalysis using the nonlinear HoekndashBrown criterion the dis-turbance factors caused by mining excavation and the dynamicchange of the rock massrsquo mechanical parameters can beconsidered erefore it is important to investigate the dif-ferences in analyzing the damaged width of a rib pillar usingeither the HoekndashBrown criterion or the MohrndashCoulomb cri-terion In this study it was found that using the HoekndashBrowncriterion is advantageouswhen analyzing a rib pillar with joints

erefore this paper focuses on the rib pillars in open-pithighwall mining e damaged width of rib pillars was used tomonitor with borehole camera exploration device (BCED) in anopen-pitmine inChina Amechanicalmodel is used to calculatethe damaged width of rib pillars based on the I-II compositecrack in fracture mechanics using the HoekndashBrown failurecriterion andMohrndashCoulomb failure criterion Accordingly the

formulas for calculating the damagedwidth of the rib pillar wereobtained respectively under different criteria e differences ofthe two criteria were compared and analyzed by considering thedamagemechanism for the width of the rib pillarsemodel isvalidated by fieldmonitoring Finally the instability mechanismof rib pillars in open-pit highwall mining is explained ispaper verifies the advantages of applying the HoekndashBrowncriterion to analyze the damaged width of rib pillars

2 Analysis of a Practical Case

is paper is based on the geological background of an open-pit mine in China e open-pit mine is located inShuozhou Shanxi province China (Figure 1(a)) In this casethe average thickness of the coal seam was 3m and theburial depth of the coal seamwas 122m [9] According to theactual data for the open-pit mine the width of the cavernwas 35m and the width of rib pillar was 40m

As illustrated in Figure 2(a) a horizontal exploratoryborehole 2m deep and 28mm in diameter was drilled in thebelt line of rib pillar A YTJ20 borehole camera explorationdevice (BCED) was used to monitor the damage in rib pillar(Figure 2(b)) Figure 2(c) shows probe images at differentdepths in the borehole ere are many inclined and inter-secting cracks found around the borehole e coal mass nearthe working face has failed and become a fractured zone Inaddition based on the results of the borehole probe it can beinferred that the deformation of the rib pillar adjacent to thegob side is larger or equal to that on the coal mass

Rib pillar deformation is large and approximately sym-metrical during highwall period e collected data on the ribpillar deformation can be summarized as follows (1) the rib-to-rib convergence is larger than the roof-to-floor conver-gence e maximum and average roof-to-floor convergencesare 189mm and 152mm respectively e maximum andaverage rib-to-rib convergences are 873mm and 546mmrespectively (2) e damaged width of rib pillar convergenceis larger e maximum damaged width reaches 916mm andthe average damaged width is 343mme large deformationis mainly caused by abutment stress causing the rib pillar tobecome unstabilityerefore the damaged width of rib pillarshould be studied to determine which rib pillar sizes

e intact parameters obtained by performing com-pression tests and Brazilian tests on small specimens arelisted in Table 1 e mechanical parameters for the coalmaterial and its surrounding rocks are as follows

3 Rib Pillar Bearing-Load Model inHighwall Mining

e damage of rib pillar is a gradual failure process in open-pithighwall mining Because there exist many similarities be-tween the analysis of rib pillars in strip mining and highwallmining the distributive stress characteristics of rib pillars inhighwall mining can be explained by analyzing the stresscharacteristics of rib pillars in strip mining (i) When coal isnot mined the coal body is subjected to the uniformly dis-tributed load of the overburden rock (ii) e supportingpressure zone and the local damaged zone are formed within




P cH WP +We( ) (1)

(a) (b)

(c) (d)

Beijing


China

Residual coal

30

14

30

24

12

12

30

15

15

Overlyingstrata

Overlyingstrata

Overlyingstrata

Overlyingstrata

Main roof

Immediateroof

Coal

Immediatefloor

Main floor


Surface soil

Weatheredsandstone

Sandstone

Mudstone

Siltstone

Sandstone

Coal

Shale

Siltstone

Rib pillar

Lithology





Rib pillar

2m

Exploring borehole

15m

4m

(a) (b)

038m 048m 049m 055m

(c)





H


YZ ECZ YZ







σx(r θ) KΙ2πr

radic cosθ2

1minus sinθ2sin

3θ2

( )

σy(r θ) KΙ2πr

radic cosθ2

1 + sinθ2sin

3θ2

( )

τxy(r θ) KΙ2πr

radic sinθ2cos

θ2cos

3θ2

(2)


πa



πa

radic the stress



radic


2

radic (3)


σx(r θ) cH


middotWe

r

radiccos

θ2

1minus sinθ2sin

3θ2

( )

σy(r θ) cH


middotWe

r

radiccos

θ2

1 + sinθ2sin

3θ2

( )

τxy(r θ) cH


We

r

radicsin

θ2cos

θ2cos

3θ2

(4)


2a

ry

x λ σβ

2a

ry

xβ

2a

ry

xβ

Decomposed

Mode I crack

Mode II crack

τ

τ

τ

τ

σ

σ

σ

σ

σσ

λ σ







σx(r θ) minuscH


We

r

1113970

sinθ2

2 + cosθ2cos

3θ2

1113888 1113889

σy(r θ) cH


We

r

1113970

sinθ2cos

θ2cos

3θ2

τxy(r θ) cH


We

r

1113970

cosθ2

1minus sinθ2sin

3θ2

1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)


σx(r θ) K

1r

1113970


σy(r θ) K

1r

1113970

[A(1 + B) + CD]

τxy(r θ) K

1r

1113970

[E D + F(1minusB)]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)


We

1113968)2 C



σ12 σx + σy

2plusmn

σx minus σy

21113874 1113875

2+ τ2xy

1113971

σ3 μ σx + σy1113872 1113873

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)



σ12 K

1r

1113970

(AminusC) plusmn K

1r

1113970

1113864[AB + C(1 + D)]2

+[ED + F(1minusB)]2111386512

σ3 2μK

1r

1113970

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)






σ1 σ3 +

mRcσ3 + sR2c

1113969

(9)




2mRcμK(AminusC)1113858 11138592

+ 4sR2cG

1113969

2G

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(10)


We

1113968)2






rHp1 2mRcμM +

2mRcμM( 11138572


1113969


⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(11)


We

1113968)2




(12)



rC 2c cosφ


minus2

(13)



1113968)2






minus2

(14)


We

1113968)2


z2 minus a2






σx(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


σy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


τxy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)



σ12 cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


+[E D + F(1minusB)]2111386512

σ3 μcH

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)



r mRcμcHP +

mRcμcHP( 11138572

+ 4sR2c(GcHP2K)2

1113969

(2GcHPK)2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

minus2

(17)



1113968)2



1113969 and G K(A1113864



rHp2 mRcμcHP +

mRcμcHP( 11138572


21113969

2KcHP[M(1minus 2μ) + N]1113864 11138652

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(18)


HWe

1113968)2 and P


1113969





minus2

(19)




1113968)2




[AB + C(1 + D)]2+1113966 [E D + F(1minusB)]212




minus2

(20)



1113969




















0 1 2 3 4 5 6 7 8 9 10 11070

075

080

085

090

095

100

Wp (m)

000

005

010

015

020




0 1 2 3 4 5 6 7 8 9 10

078856

078858

078860

078862

078864

078866

078868

Wpr C

p2 (m

)

0 1 2 3 4 5 6 7 8 9 10

083778

083780

083782

083784

083786

083788

083790

083792

Wp

r Cp2

(m)r H

p2 (m

)

r Cp2

(m)

0 1 2 3 4 5 6 7 8 9 10092974092976092978092980092982092984092986092988092990

Wp

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001


0 1 2 3 4 5 6 7 8 9 10 11

00

05

10

15

20

25

30

35

40

We (m)

00

05

10

15

20

25

30

35

40

r Hp2

(m)

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1















f σzWp

cH We +Wp( ) (22)



0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

r Cp2

(m)

5

6

7

H (m)

0

1

2

3

4

5

r Hp2

(m)

6

7

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1






6 Conclusions






0 2 4 6 8 10 12 14 16 18 20

06

08

10

12

14

16

18

00

01

02

03

04

0 2 4 6 8 10 12β (deg)

β (deg)

14 16 18 20

00000

00005

00010

00015

00020

00025

r Hp2

(m)

r Hp2

(m)

r Cp2

(m)


φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001




Data Availability






Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




P cH WP +We( ) (1)

(a) (b)

(c) (d)

Beijing


China

Residual coal

30

14

30

24

12

12

30

15

15

Overlyingstrata

Overlyingstrata

Overlyingstrata

Overlyingstrata

Main roof

Immediateroof

Coal

Immediatefloor

Main floor


Surface soil

Weatheredsandstone

Sandstone

Mudstone

Siltstone

Sandstone

Coal

Shale

Siltstone

Rib pillar

Lithology





Rib pillar

2m

Exploring borehole

15m

4m

(a) (b)

038m 048m 049m 055m

(c)





H


YZ ECZ YZ







σx(r θ) KΙ2πr

radic cosθ2

1minus sinθ2sin

3θ2

( )

σy(r θ) KΙ2πr

radic cosθ2

1 + sinθ2sin

3θ2

( )

τxy(r θ) KΙ2πr

radic sinθ2cos

θ2cos

3θ2

(2)


πa



πa

radic the stress



radic


2

radic (3)


σx(r θ) cH


middotWe

r

radiccos

θ2

1minus sinθ2sin

3θ2

( )

σy(r θ) cH


middotWe

r

radiccos

θ2

1 + sinθ2sin

3θ2

( )

τxy(r θ) cH


We

r

radicsin

θ2cos

θ2cos

3θ2

(4)


2a

ry

x λ σβ

2a

ry

xβ

2a

ry

xβ

Decomposed

Mode I crack

Mode II crack

τ

τ

τ

τ

σ

σ

σ

σ

σσ

λ σ







σx(r θ) minuscH


We

r

1113970

sinθ2

2 + cosθ2cos

3θ2

1113888 1113889

σy(r θ) cH


We

r

1113970

sinθ2cos

θ2cos

3θ2

τxy(r θ) cH


We

r

1113970

cosθ2

1minus sinθ2sin

3θ2

1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)


σx(r θ) K

1r

1113970


σy(r θ) K

1r

1113970

[A(1 + B) + CD]

τxy(r θ) K

1r

1113970

[E D + F(1minusB)]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)


We

1113968)2 C



σ12 σx + σy

2plusmn

σx minus σy

21113874 1113875

2+ τ2xy

1113971

σ3 μ σx + σy1113872 1113873

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)



σ12 K

1r

1113970

(AminusC) plusmn K

1r

1113970

1113864[AB + C(1 + D)]2

+[ED + F(1minusB)]2111386512

σ3 2μK

1r

1113970

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)






σ1 σ3 +

mRcσ3 + sR2c

1113969

(9)




2mRcμK(AminusC)1113858 11138592

+ 4sR2cG

1113969

2G

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(10)


We

1113968)2






rHp1 2mRcμM +

2mRcμM( 11138572


1113969


⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(11)


We

1113968)2




(12)



rC 2c cosφ


minus2

(13)



1113968)2






minus2

(14)


We

1113968)2


z2 minus a2






σx(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


σy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


τxy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)



σ12 cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


+[E D + F(1minusB)]2111386512

σ3 μcH

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)



r mRcμcHP +

mRcμcHP( 11138572

+ 4sR2c(GcHP2K)2

1113969

(2GcHPK)2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

minus2

(17)



1113968)2



1113969 and G K(A1113864



rHp2 mRcμcHP +

mRcμcHP( 11138572


21113969

2KcHP[M(1minus 2μ) + N]1113864 11138652

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(18)


HWe

1113968)2 and P


1113969





minus2

(19)




1113968)2




[AB + C(1 + D)]2+1113966 [E D + F(1minusB)]212




minus2

(20)



1113969




















0 1 2 3 4 5 6 7 8 9 10 11070

075

080

085

090

095

100

Wp (m)

000

005

010

015

020




0 1 2 3 4 5 6 7 8 9 10

078856

078858

078860

078862

078864

078866

078868

Wpr C

p2 (m

)

0 1 2 3 4 5 6 7 8 9 10

083778

083780

083782

083784

083786

083788

083790

083792

Wp

r Cp2

(m)r H

p2 (m

)

r Cp2

(m)

0 1 2 3 4 5 6 7 8 9 10092974092976092978092980092982092984092986092988092990

Wp

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001


0 1 2 3 4 5 6 7 8 9 10 11

00

05

10

15

20

25

30

35

40

We (m)

00

05

10

15

20

25

30

35

40

r Hp2

(m)

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1















f σzWp

cH We +Wp( ) (22)



0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

r Cp2

(m)

5

6

7

H (m)

0

1

2

3

4

5

r Hp2

(m)

6

7

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1






6 Conclusions






0 2 4 6 8 10 12 14 16 18 20

06

08

10

12

14

16

18

00

01

02

03

04

0 2 4 6 8 10 12β (deg)

β (deg)

14 16 18 20

00000

00005

00010

00015

00020

00025

r Hp2

(m)

r Hp2

(m)

r Cp2

(m)


φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001




Data Availability






Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Rib pillar

2m

Exploring borehole

15m

4m

(a) (b)

038m 048m 049m 055m

(c)





H


YZ ECZ YZ







σx(r θ) KΙ2πr

radic cosθ2

1minus sinθ2sin

3θ2

( )

σy(r θ) KΙ2πr

radic cosθ2

1 + sinθ2sin

3θ2

( )

τxy(r θ) KΙ2πr

radic sinθ2cos

θ2cos

3θ2

(2)


πa



πa

radic the stress



radic


2

radic (3)


σx(r θ) cH


middotWe

r

radiccos

θ2

1minus sinθ2sin

3θ2

( )

σy(r θ) cH


middotWe

r

radiccos

θ2

1 + sinθ2sin

3θ2

( )

τxy(r θ) cH


We

r

radicsin

θ2cos

θ2cos

3θ2

(4)


2a

ry

x λ σβ

2a

ry

xβ

2a

ry

xβ

Decomposed

Mode I crack

Mode II crack

τ

τ

τ

τ

σ

σ

σ

σ

σσ

λ σ







σx(r θ) minuscH


We

r

1113970

sinθ2

2 + cosθ2cos

3θ2

1113888 1113889

σy(r θ) cH


We

r

1113970

sinθ2cos

θ2cos

3θ2

τxy(r θ) cH


We

r

1113970

cosθ2

1minus sinθ2sin

3θ2

1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)


σx(r θ) K

1r

1113970


σy(r θ) K

1r

1113970

[A(1 + B) + CD]

τxy(r θ) K

1r

1113970

[E D + F(1minusB)]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)


We

1113968)2 C



σ12 σx + σy

2plusmn

σx minus σy

21113874 1113875

2+ τ2xy

1113971

σ3 μ σx + σy1113872 1113873

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)



σ12 K

1r

1113970

(AminusC) plusmn K

1r

1113970

1113864[AB + C(1 + D)]2

+[ED + F(1minusB)]2111386512

σ3 2μK

1r

1113970

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)






σ1 σ3 +

mRcσ3 + sR2c

1113969

(9)




2mRcμK(AminusC)1113858 11138592

+ 4sR2cG

1113969

2G

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(10)


We

1113968)2






rHp1 2mRcμM +

2mRcμM( 11138572


1113969


⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(11)


We

1113968)2




(12)



rC 2c cosφ


minus2

(13)



1113968)2






minus2

(14)


We

1113968)2


z2 minus a2






σx(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


σy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


τxy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)



σ12 cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


+[E D + F(1minusB)]2111386512

σ3 μcH

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)



r mRcμcHP +

mRcμcHP( 11138572

+ 4sR2c(GcHP2K)2

1113969

(2GcHPK)2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

minus2

(17)



1113968)2



1113969 and G K(A1113864



rHp2 mRcμcHP +

mRcμcHP( 11138572


21113969

2KcHP[M(1minus 2μ) + N]1113864 11138652

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(18)


HWe

1113968)2 and P


1113969





minus2

(19)




1113968)2




[AB + C(1 + D)]2+1113966 [E D + F(1minusB)]212




minus2

(20)



1113969




















0 1 2 3 4 5 6 7 8 9 10 11070

075

080

085

090

095

100

Wp (m)

000

005

010

015

020




0 1 2 3 4 5 6 7 8 9 10

078856

078858

078860

078862

078864

078866

078868

Wpr C

p2 (m

)

0 1 2 3 4 5 6 7 8 9 10

083778

083780

083782

083784

083786

083788

083790

083792

Wp

r Cp2

(m)r H

p2 (m

)

r Cp2

(m)

0 1 2 3 4 5 6 7 8 9 10092974092976092978092980092982092984092986092988092990

Wp

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001


0 1 2 3 4 5 6 7 8 9 10 11

00

05

10

15

20

25

30

35

40

We (m)

00

05

10

15

20

25

30

35

40

r Hp2

(m)

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1















f σzWp

cH We +Wp( ) (22)



0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

r Cp2

(m)

5

6

7

H (m)

0

1

2

3

4

5

r Hp2

(m)

6

7

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1






6 Conclusions






0 2 4 6 8 10 12 14 16 18 20

06

08

10

12

14

16

18

00

01

02

03

04

0 2 4 6 8 10 12β (deg)

β (deg)

14 16 18 20

00000

00005

00010

00015

00020

00025

r Hp2

(m)

r Hp2

(m)

r Cp2

(m)


φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001




Data Availability






Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





σx(r θ) KΙ2πr

radic cosθ2

1minus sinθ2sin

3θ2

( )

σy(r θ) KΙ2πr

radic cosθ2

1 + sinθ2sin

3θ2

( )

τxy(r θ) KΙ2πr

radic sinθ2cos

θ2cos

3θ2

(2)


πa



πa

radic the stress



radic


2

radic (3)


σx(r θ) cH


middotWe

r

radiccos

θ2

1minus sinθ2sin

3θ2

( )

σy(r θ) cH


middotWe

r

radiccos

θ2

1 + sinθ2sin

3θ2

( )

τxy(r θ) cH


We

r

radicsin

θ2cos

θ2cos

3θ2

(4)


2a

ry

x λ σβ

2a

ry

xβ

2a

ry

xβ

Decomposed

Mode I crack

Mode II crack

τ

τ

τ

τ

σ

σ

σ

σ

σσ

λ σ







σx(r θ) minuscH


We

r

1113970

sinθ2

2 + cosθ2cos

3θ2

1113888 1113889

σy(r θ) cH


We

r

1113970

sinθ2cos

θ2cos

3θ2

τxy(r θ) cH


We

r

1113970

cosθ2

1minus sinθ2sin

3θ2

1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)


σx(r θ) K

1r

1113970


σy(r θ) K

1r

1113970

[A(1 + B) + CD]

τxy(r θ) K

1r

1113970

[E D + F(1minusB)]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)


We

1113968)2 C



σ12 σx + σy

2plusmn

σx minus σy

21113874 1113875

2+ τ2xy

1113971

σ3 μ σx + σy1113872 1113873

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)



σ12 K

1r

1113970

(AminusC) plusmn K

1r

1113970

1113864[AB + C(1 + D)]2

+[ED + F(1minusB)]2111386512

σ3 2μK

1r

1113970

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)






σ1 σ3 +

mRcσ3 + sR2c

1113969

(9)




2mRcμK(AminusC)1113858 11138592

+ 4sR2cG

1113969

2G

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(10)


We

1113968)2






rHp1 2mRcμM +

2mRcμM( 11138572


1113969


⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(11)


We

1113968)2




(12)



rC 2c cosφ


minus2

(13)



1113968)2






minus2

(14)


We

1113968)2


z2 minus a2






σx(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


σy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


τxy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)



σ12 cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


+[E D + F(1minusB)]2111386512

σ3 μcH

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)



r mRcμcHP +

mRcμcHP( 11138572

+ 4sR2c(GcHP2K)2

1113969

(2GcHPK)2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

minus2

(17)



1113968)2



1113969 and G K(A1113864



rHp2 mRcμcHP +

mRcμcHP( 11138572


21113969

2KcHP[M(1minus 2μ) + N]1113864 11138652

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(18)


HWe

1113968)2 and P


1113969





minus2

(19)




1113968)2




[AB + C(1 + D)]2+1113966 [E D + F(1minusB)]212




minus2

(20)



1113969




















0 1 2 3 4 5 6 7 8 9 10 11070

075

080

085

090

095

100

Wp (m)

000

005

010

015

020




0 1 2 3 4 5 6 7 8 9 10

078856

078858

078860

078862

078864

078866

078868

Wpr C

p2 (m

)

0 1 2 3 4 5 6 7 8 9 10

083778

083780

083782

083784

083786

083788

083790

083792

Wp

r Cp2

(m)r H

p2 (m

)

r Cp2

(m)

0 1 2 3 4 5 6 7 8 9 10092974092976092978092980092982092984092986092988092990

Wp

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001


0 1 2 3 4 5 6 7 8 9 10 11

00

05

10

15

20

25

30

35

40

We (m)

00

05

10

15

20

25

30

35

40

r Hp2

(m)

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1















f σzWp

cH We +Wp( ) (22)



0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

r Cp2

(m)

5

6

7

H (m)

0

1

2

3

4

5

r Hp2

(m)

6

7

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1






6 Conclusions






0 2 4 6 8 10 12 14 16 18 20

06

08

10

12

14

16

18

00

01

02

03

04

0 2 4 6 8 10 12β (deg)

β (deg)

14 16 18 20

00000

00005

00010

00015

00020

00025

r Hp2

(m)

r Hp2

(m)

r Cp2

(m)


φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001




Data Availability






Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






σx(r θ) minuscH


We

r

1113970

sinθ2

2 + cosθ2cos

3θ2

1113888 1113889

σy(r θ) cH


We

r

1113970

sinθ2cos

θ2cos

3θ2

τxy(r θ) cH


We

r

1113970

cosθ2

1minus sinθ2sin

3θ2

1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)


σx(r θ) K

1r

1113970


σy(r θ) K

1r

1113970

[A(1 + B) + CD]

τxy(r θ) K

1r

1113970

[E D + F(1minusB)]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)


We

1113968)2 C



σ12 σx + σy

2plusmn

σx minus σy

21113874 1113875

2+ τ2xy

1113971

σ3 μ σx + σy1113872 1113873

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)



σ12 K

1r

1113970

(AminusC) plusmn K

1r

1113970

1113864[AB + C(1 + D)]2

+[ED + F(1minusB)]2111386512

σ3 2μK

1r

1113970

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)






σ1 σ3 +

mRcσ3 + sR2c

1113969

(9)




2mRcμK(AminusC)1113858 11138592

+ 4sR2cG

1113969

2G

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(10)


We

1113968)2






rHp1 2mRcμM +

2mRcμM( 11138572


1113969


⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(11)


We

1113968)2




(12)



rC 2c cosφ


minus2

(13)



1113968)2






minus2

(14)


We

1113968)2


z2 minus a2






σx(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


σy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


τxy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)



σ12 cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


+[E D + F(1minusB)]2111386512

σ3 μcH

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)



r mRcμcHP +

mRcμcHP( 11138572

+ 4sR2c(GcHP2K)2

1113969

(2GcHPK)2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

minus2

(17)



1113968)2



1113969 and G K(A1113864



rHp2 mRcμcHP +

mRcμcHP( 11138572


21113969

2KcHP[M(1minus 2μ) + N]1113864 11138652

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(18)


HWe

1113968)2 and P


1113969





minus2

(19)




1113968)2




[AB + C(1 + D)]2+1113966 [E D + F(1minusB)]212




minus2

(20)



1113969




















0 1 2 3 4 5 6 7 8 9 10 11070

075

080

085

090

095

100

Wp (m)

000

005

010

015

020




0 1 2 3 4 5 6 7 8 9 10

078856

078858

078860

078862

078864

078866

078868

Wpr C

p2 (m

)

0 1 2 3 4 5 6 7 8 9 10

083778

083780

083782

083784

083786

083788

083790

083792

Wp

r Cp2

(m)r H

p2 (m

)

r Cp2

(m)

0 1 2 3 4 5 6 7 8 9 10092974092976092978092980092982092984092986092988092990

Wp

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001


0 1 2 3 4 5 6 7 8 9 10 11

00

05

10

15

20

25

30

35

40

We (m)

00

05

10

15

20

25

30

35

40

r Hp2

(m)

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1















f σzWp

cH We +Wp( ) (22)



0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

r Cp2

(m)

5

6

7

H (m)

0

1

2

3

4

5

r Hp2

(m)

6

7

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1






6 Conclusions






0 2 4 6 8 10 12 14 16 18 20

06

08

10

12

14

16

18

00

01

02

03

04

0 2 4 6 8 10 12β (deg)

β (deg)

14 16 18 20

00000

00005

00010

00015

00020

00025

r Hp2

(m)

r Hp2

(m)

r Cp2

(m)


φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001




Data Availability






Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



2mRcμK(AminusC)1113858 11138592

+ 4sR2cG

1113969

2G

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(10)


We

1113968)2






rHp1 2mRcμM +

2mRcμM( 11138572


1113969


⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(11)


We

1113968)2




(12)



rC 2c cosφ


minus2

(13)



1113968)2






minus2

(14)


We

1113968)2


z2 minus a2






σx(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


σy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


τxy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)



σ12 cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


+[E D + F(1minusB)]2111386512

σ3 μcH

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)



r mRcμcHP +

mRcμcHP( 11138572

+ 4sR2c(GcHP2K)2

1113969

(2GcHPK)2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

minus2

(17)



1113968)2



1113969 and G K(A1113864



rHp2 mRcμcHP +

mRcμcHP( 11138572


21113969

2KcHP[M(1minus 2μ) + N]1113864 11138652

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(18)


HWe

1113968)2 and P


1113969





minus2

(19)




1113968)2




[AB + C(1 + D)]2+1113966 [E D + F(1minusB)]212




minus2

(20)



1113969




















0 1 2 3 4 5 6 7 8 9 10 11070

075

080

085

090

095

100

Wp (m)

000

005

010

015

020




0 1 2 3 4 5 6 7 8 9 10

078856

078858

078860

078862

078864

078866

078868

Wpr C

p2 (m

)

0 1 2 3 4 5 6 7 8 9 10

083778

083780

083782

083784

083786

083788

083790

083792

Wp

r Cp2

(m)r H

p2 (m

)

r Cp2

(m)

0 1 2 3 4 5 6 7 8 9 10092974092976092978092980092982092984092986092988092990

Wp

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001


0 1 2 3 4 5 6 7 8 9 10 11

00

05

10

15

20

25

30

35

40

We (m)

00

05

10

15

20

25

30

35

40

r Hp2

(m)

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1















f σzWp

cH We +Wp( ) (22)



0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

r Cp2

(m)

5

6

7

H (m)

0

1

2

3

4

5

r Hp2

(m)

6

7

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1






6 Conclusions






0 2 4 6 8 10 12 14 16 18 20

06

08

10

12

14

16

18

00

01

02

03

04

0 2 4 6 8 10 12β (deg)

β (deg)

14 16 18 20

00000

00005

00010

00015

00020

00025

r Hp2

(m)

r Hp2

(m)

r Cp2

(m)


φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001




Data Availability






Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


σx(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


σy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


τxy(r θ) cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)



σ12 cH

2

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972


+[E D + F(1minusB)]2111386512

σ3 μcH

2 We + Wp1113872 1113873

πrtan

πWe

2 We + Wp1113872 1113873

11139741113972

(AminusC)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(16)



r mRcμcHP +

mRcμcHP( 11138572

+ 4sR2c(GcHP2K)2

1113969

(2GcHPK)2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

minus2

(17)



1113968)2



1113969 and G K(A1113864



rHp2 mRcμcHP +

mRcμcHP( 11138572


21113969

2KcHP[M(1minus 2μ) + N]1113864 11138652

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭

minus2

(18)


HWe

1113968)2 and P


1113969





minus2

(19)




1113968)2




[AB + C(1 + D)]2+1113966 [E D + F(1minusB)]212




minus2

(20)



1113969




















0 1 2 3 4 5 6 7 8 9 10 11070

075

080

085

090

095

100

Wp (m)

000

005

010

015

020




0 1 2 3 4 5 6 7 8 9 10

078856

078858

078860

078862

078864

078866

078868

Wpr C

p2 (m

)

0 1 2 3 4 5 6 7 8 9 10

083778

083780

083782

083784

083786

083788

083790

083792

Wp

r Cp2

(m)r H

p2 (m

)

r Cp2

(m)

0 1 2 3 4 5 6 7 8 9 10092974092976092978092980092982092984092986092988092990

Wp

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001


0 1 2 3 4 5 6 7 8 9 10 11

00

05

10

15

20

25

30

35

40

We (m)

00

05

10

15

20

25

30

35

40

r Hp2

(m)

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1















f σzWp

cH We +Wp( ) (22)



0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

r Cp2

(m)

5

6

7

H (m)

0

1

2

3

4

5

r Hp2

(m)

6

7

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1






6 Conclusions






0 2 4 6 8 10 12 14 16 18 20

06

08

10

12

14

16

18

00

01

02

03

04

0 2 4 6 8 10 12β (deg)

β (deg)

14 16 18 20

00000

00005

00010

00015

00020

00025

r Hp2

(m)

r Hp2

(m)

r Cp2

(m)


φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001




Data Availability






Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




1113968)2




[AB + C(1 + D)]2+1113966 [E D + F(1minusB)]212




minus2

(20)



1113969




















0 1 2 3 4 5 6 7 8 9 10 11070

075

080

085

090

095

100

Wp (m)

000

005

010

015

020




0 1 2 3 4 5 6 7 8 9 10

078856

078858

078860

078862

078864

078866

078868

Wpr C

p2 (m

)

0 1 2 3 4 5 6 7 8 9 10

083778

083780

083782

083784

083786

083788

083790

083792

Wp

r Cp2

(m)r H

p2 (m

)

r Cp2

(m)

0 1 2 3 4 5 6 7 8 9 10092974092976092978092980092982092984092986092988092990

Wp

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001


0 1 2 3 4 5 6 7 8 9 10 11

00

05

10

15

20

25

30

35

40

We (m)

00

05

10

15

20

25

30

35

40

r Hp2

(m)

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1















f σzWp

cH We +Wp( ) (22)



0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

r Cp2

(m)

5

6

7

H (m)

0

1

2

3

4

5

r Hp2

(m)

6

7

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1






6 Conclusions






0 2 4 6 8 10 12 14 16 18 20

06

08

10

12

14

16

18

00

01

02

03

04

0 2 4 6 8 10 12β (deg)

β (deg)

14 16 18 20

00000

00005

00010

00015

00020

00025

r Hp2

(m)

r Hp2

(m)

r Cp2

(m)


φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001




Data Availability






Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia













0 1 2 3 4 5 6 7 8 9 10 11070

075

080

085

090

095

100

Wp (m)

000

005

010

015

020




0 1 2 3 4 5 6 7 8 9 10

078856

078858

078860

078862

078864

078866

078868

Wpr C

p2 (m

)

0 1 2 3 4 5 6 7 8 9 10

083778

083780

083782

083784

083786

083788

083790

083792

Wp

r Cp2

(m)r H

p2 (m

)

r Cp2

(m)

0 1 2 3 4 5 6 7 8 9 10092974092976092978092980092982092984092986092988092990

Wp

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001


0 1 2 3 4 5 6 7 8 9 10 11

00

05

10

15

20

25

30

35

40

We (m)

00

05

10

15

20

25

30

35

40

r Hp2

(m)

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1















f σzWp

cH We +Wp( ) (22)



0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

r Cp2

(m)

5

6

7

H (m)

0

1

2

3

4

5

r Hp2

(m)

6

7

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1






6 Conclusions






0 2 4 6 8 10 12 14 16 18 20

06

08

10

12

14

16

18

00

01

02

03

04

0 2 4 6 8 10 12β (deg)

β (deg)

14 16 18 20

00000

00005

00010

00015

00020

00025

r Hp2

(m)

r Hp2

(m)

r Cp2

(m)


φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001




Data Availability






Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






0 1 2 3 4 5 6 7 8 9 10 11070

075

080

085

090

095

100

Wp (m)

000

005

010

015

020




0 1 2 3 4 5 6 7 8 9 10

078856

078858

078860

078862

078864

078866

078868

Wpr C

p2 (m

)

0 1 2 3 4 5 6 7 8 9 10

083778

083780

083782

083784

083786

083788

083790

083792

Wp

r Cp2

(m)r H

p2 (m

)

r Cp2

(m)

0 1 2 3 4 5 6 7 8 9 10092974092976092978092980092982092984092986092988092990

Wp

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001


0 1 2 3 4 5 6 7 8 9 10 11

00

05

10

15

20

25

30

35

40

We (m)

00

05

10

15

20

25

30

35

40

r Hp2

(m)

r Cp2

(m)

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1















f σzWp

cH We +Wp( ) (22)



0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

r Cp2

(m)

5

6

7

H (m)

0

1

2

3

4

5

r Hp2

(m)

6

7

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1






6 Conclusions






0 2 4 6 8 10 12 14 16 18 20

06

08

10

12

14

16

18

00

01

02

03

04

0 2 4 6 8 10 12β (deg)

β (deg)

14 16 18 20

00000

00005

00010

00015

00020

00025

r Hp2

(m)

r Hp2

(m)

r Cp2

(m)


φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001




Data Availability






Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia













f σzWp

cH We +Wp( ) (22)



0 100 200 300 400 500 600 700 800 900 1000

0

1

2

3

4

r Cp2

(m)

5

6

7

H (m)

0

1

2

3

4

5

r Hp2

(m)

6

7

φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1






6 Conclusions






0 2 4 6 8 10 12 14 16 18 20

06

08

10

12

14

16

18

00

01

02

03

04

0 2 4 6 8 10 12β (deg)

β (deg)

14 16 18 20

00000

00005

00010

00015

00020

00025

r Hp2

(m)

r Hp2

(m)

r Cp2

(m)


φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001




Data Availability






Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




6 Conclusions






0 2 4 6 8 10 12 14 16 18 20

06

08

10

12

14

16

18

00

01

02

03

04

0 2 4 6 8 10 12β (deg)

β (deg)

14 16 18 20

00000

00005

00010

00015

00020

00025

r Hp2

(m)

r Hp2

(m)

r Cp2

(m)


φ = 20 c = 98MPa μ = 035

φ = 28 c = 62MPa μ = 065

φ = 36 c = 16MPa μ = 1

m = 20 Rc = 15MPa μ = 035 s = 1

m = 2 Rc = 3MPa μ = 065 s = 015

m = 001 Rc = 05MPa μ = 1 s = 001




Data Availability






Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Data Availability






Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia













RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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