FractureMechanicalBehaviorofCrackedCantileverRoofwith … · 2020. 4. 13. · was established. e...

Research ArticleFracture Mechanical Behavior of Cracked Cantilever Roof withLarge Cutting Height Mining

Zenghui Zhao 12 Wei Sun12 Mingzhong Zhang12 Xiaojie Gao12 and Shaojie Chen 12

1State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Provinceand the Ministry of Science and Technology Qingdao 266590 China2College of Energy and Mining Engineering Shandong University of Science and Technology Qingdao 266590 China

Correspondence should be addressed to Zenghui Zhao tgzyzzh163com

Received 15 September 2019 Accepted 3 February 2020 Published 13 April 2020

Academic Editor Roberto Palma

Copyright copy 2020 Zenghui Zhao 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

Accurately predicting the roof collapse span is crucial in ensuring the safe production of thick seam mining with large miningheight which is easy in forming a ldquocantilever beamrdquo structure Considering roof damage caused by roadway excavation and coalseam mining disturbance the fracture mechanics model of large mining height roof cantilever beam with nonpenetrating crackswas established )e roof was divided into two parts the crack-affected area and the crack-unaffected area )e analyticalexpression of the boundary between the two areas was established by fracture mechanics methods Based on the boundaryequation the influences of crack size crack inclination roof lithology and roof thickness on the roof crack-affected area wereanalyzed in detail Finally the accuracy of the theoretical model was verified by numerical experiments using the extended finiteelement method )e results demonstrate that the size of the area affected by the vertical crack increases with the increase of thecrack size and the thickness of the roof )e influence of the crack decreases with the increase of roof lithology )e probability ofearly periodic collapse of a thin roof with the crack is increasedWhen the crack is completely located in the interior of the roof thecrack-affected area shrinks greatly with the decrease of the crack inclinationWhen the crack inclination is small the crack will notcause the early collapse of the roof Overall the conclusions obtained are of great significance for predicting the collapse span of acantilever roof with initial damage in large mining height

1 Introduction

)e total of thick seam the main mining seam in Chinareserves and production accounts for about 45 [1 2]Comprehensive mechanized mining of thick coal seamsmainly contains slice mining top coal caving mining andlarge cutting height mining [3] Large cutting height mininghas been widely used for its high output rate and goodeconomic benefits However there are still many technicalproblems in large cutting height mining For example withthe increase of mining height and the height of overburdencaving zone the low-level key strata that can form articu-lated balance structure can not form stable ldquovoussoir beamrdquostructure because of large rotation but form ldquocantileverrdquostructure to collapse directly [4] So the accurate predictionof the roof periodic collapse span is salient for ensuring thesafe mining of coal seams In fact thanks to complex

geological conditions such as diagenesis high temperatureand high humidity as well as the influence of excavationroadway and mining it is inevitable that there will benonpenetrating cracks in the roof [5ndash9] )e roof is madeinto a discontinuous structure which will collapse earlierthan continuous structure resulting in serious safety acci-dents and economic losses )us it is of profound signifi-cance to clarify the influence of the nonpenetrating cracks onthe roof periodic collapse span with large mining heightwhether for the support design of working face or for the safeand efficient production of coal mine

Nowadays the construction of roof collapse modelmostly follows the assumption of uniformity and continuityOn the basis of these assumptions scholars put forward theclassical hypothesis of mine pressure such as pressure archhypothesis precrack hypothesis and articulated rock blockhypothesis [10 11] )e most influential underground

HindawiShock and VibrationVolume 2020 Article ID 1641382 10 pageshttpsdoiorg10115520201641382

pressure theories in China are the voussoir beam theory [12]the key stratum theory [13] and the transmission rock beamtheory [14] In recent years considerable research effortshave been devoted to underground pressure theories fordifferent mining environments and methods Aiming at theproblem of shallow thin bedrock seammining Huang and Liestablished a cantilever beam model based on Winklerelastic foundation theory )e results show that bedrock willbreak ahead of the coal wall during the period of periodicpressure [15] Some scholars combine Protodyakonovrsquostheory with the voussoir beam theory and then put forwardthe balanced composite structure of ldquoarchrdquo on ldquobeamrdquo )eresearch indicates that the width of ldquoarchrdquo is equal to thebasic roof periodic weighting span [16 17] Relevant studiesdemonstrate that the mining of deeply inclined coal seams isdifferent from that of horizontal coal seams and its periodiccollapse span is affected by the position of working face theway of underhand mining and overhand mining and thecrack inclination of the coal seam )e periodic weightingspan at the top and bottom of the working face is slightlylarger than that in the upper middle middle and lowermiddle of it )e roof periodic collapse span is shorter inoverhand mining than in underhand mining )e periodicweighting span decreases with the increase of the crackinclination of the coal seam [18ndash20] In addition somescholars have considered the damage effect of roof stratumand regarded the roof as a cantilever beam with cracksEquations of the roof periodic collapse span and supportworking resistance are obtained )e effects of crack incli-nation roof length and thickness on the roof periodiccollapse span and support working resistance were analyzed[21 22] Yang et al [23] further established the mechanicalmodel of a cracked plate by using thin plate theory andanalyzed the roof caving motion state of longwall face largecutting height mining Unfortunately these discussions areall aimed at the specific location of the cracks not involvingthe different positions of cracks And there is not any in-depth study of the boundaries of the crack-affected area andthe crack-unaffected area In fact the position of crackdetermines its stress state Under a certain load the differentpositions of cracks have different effects on the roof periodiccollapse span

Much work of the roof periodic collapse span so far hasbeen based on the assumption of uniform continuous mediaAlthough some studies have established the roof model withcracks the range of cracks affecting roof collapse motionremains an unanswered question )e innovations of thisstudy are as follows

(1) Introduce the crack into the analysis of the periodiccollapse of the roof with large cutting height miningand establish a cantilever beam model with thenonpenetrating crack

(2) Release the constraint condition of crack position)e studying content is not limited to the crack atspecific location

(3) Determine the scope of the cracks that cause the roofto collapse periodically in advance

)erefore aiming at the problem of the roof periodiccollapse span with the nonpenetrating crack with largecutting height mining this study proposed to establish acantilever beam model with the nonpenetrating crack de-termined the location of crack which can cause the earlyperiodic collapse of a roof and revealed the influence ofcrack size crack inclination roof lithology and roofthickness on the crack-affected area Necessary supportingtheories for roof support design were provided

2 Analysis Model of the Roof with theNonpenetrating Crack in LargeCutting Height

)e schematic diagram of the roof with the nonpenetratingcrack in large cutting height is given as the principle dis-played in Figure 1 Under the condition of fully mechanizedmining with large cutting height or extra-large cuttingheight the working face advances from right to left Due tothe obvious increase in mining height and caving zoneheight the fractured rock produced by the immediate roofcollapse can not fill the goaf Because of the large amount ofrotary subsidence the main roof and low-level key strata cannot contact the caved rock in the goaf after breaking to forma stable masonry beam structure Low-level key strata mayenter the caving zone and break in the form of a cantileverbeam which forms periodic weighting [4])e l in the figurerepresents the roof periodic collapse span Under the actionof long-term geological structure and mining disturbancerandom cracks at different lengths and directions will occurin rocks In order to facilitate the analysis these cracks areregarded as a main crack It is assumed that the effect of theprincipal crack on roof fracture represents the effect of allcracks on it When the crack satisfies the fracture conditionunder the external force the crack begins to crack whichcauses the roof to collapse ahead of time Under the sameexternal load this effect is closely related to the location ofthe crack )e zone where the crack that causes the roof tocollapse ahead of time is located is called the crack-affectedarea and the other zones are called the crack-unaffectedarea

3 Analysis of the Boundary of the Crack-Affected Area in Large Cutting Height

31 Analysis of the Crack-Free Roof Periodic Collapse SpanFigure 2 is a simplified diagram of the crack-free Only thebending deformation of the roof under dead weight isconsidered )e thickness of the roof is h and the length is LUnder the action of distributed load q q hc and c is thespecific weight of the roof

)e bending moment equation of the roof is

M(x) q

2(L minus x)

2

hc

2(L minus x)

2 (1)

)e expression of normal stress at the point A is

2 Shock and Vibration

σA My

Iz

3cL2

h (2)

)e working face is propelled from right to left Whenthe maximum bending tensile stress (point A) reached thetensile strength that is σA [σt] the root of cantileverbeam is cracked )e roof periodic collapse span iscalculated

L

σt1113858 1113859h

3c

1113971

(3)

As can be seen from equation (3) L is affected by thephysical properties and thickness of the rock

32 Analysis of the Location of the Internal Crack0at Causedthe Roof Periodic Collapse in Advance Cracks on the com-pression side of beams do not expand and have little effect onthe bending deformation of beams )e tensile strength ofrock is far lower than the compressive strength [24])erefore we focus on the boundary of the crack-affectedarea when the crack is located on the tension side of thebeam

Without loss of generality a roof model with randomcracks is established as shown in Figure 3)e crack length is2a and the coordinate of the crack center is (x0 y0) Asshown in Figure 4 unit body with the internal crack is taken

It can be seen that the kind of this crack is a combinationof open mode (mode I) and slide mode (mode II) ie I-IIcomposite mode crack In Figure 4 β is included angle of theinclined direction of crack and horizontal direction θ is aninitiation angle and σ is the tensile stress at the midpoint ofthe crack which is approximately used to represent the

tensile stress on the unit body )e stress σ is decomposedinto normal stress and shear stress along the crack surface)e corresponding stress intensity factors are respectively[25 26]

KΙ σπa

radicsin2 β

KΙΙ σπa

radicsin β cos β

(4)

Stress field near the crack tip is

σr cos(θ2)

22πr

radic KΙ(3 minus cos θ) +sin(θ2)

22πr

radic KΙΙ(3 cos θ minus 1)

σθ cos(θ2)

22πr

radic KΙ(1 + cos θ) minus 3KΙΙ sin θ1113858 1113859

τθ

cos(θ2)

22πr

radic KΙ sin θ minus KΙΙ(3 cos θ minus 1)1113858 1113859

(5)

According to the maximum circumferential stress the-ory when the circumferential stress along the direction ofinitiation angle θ reaches the critical value σθC the crack willexpand in an unstable way

σθ KΙ KΙΙ θ( 1113857 σθC (6)

)e critical instability condition of the crack is obtainedby equations (8) and (9)

cosθ2

KΙcos2θ2

minus32KΙΙ sin θ1113888 1113889 KΙC (7)

Initiation angle θ is

l

Coal seam

Immediate roof

Gangue

Figure 1 Roof periodic collapse in large mining height workface

A

x

y

L

h

q

Figure 2 Stress analysis diagram of crack-free roof

A B

x

y

L

h

q

2ay0

x0

Figure 3 Stress analysis diagram of roof with internal crack

2aσ σ

θ

θ

β

Figure 4 Stress analysis diagram of internal crack

Shock and Vibration 3

cos θ cos2 β +

sin2 β + 8 cos2 β1113969

1 + 8 cos2 β (8)

Combining equations (7) and (8) the critical stress σBc

can be obtained

σBc

πa

radiccos

θ2sin β sin β cos2

θ2

minus32cos β sin θ1113888 1113889 KΙC

(9)

)e normal stress at the point B is obtained by equation (1)

σB My

Iz

6cy0 L minus x0( 1113857

2

h2 (10)

)e boundary equation of the crack-affected area byequations (9) and (10) is as follows

y0

σt1113858 1113859h

3c

1113971

minus x0⎛⎝ ⎞⎠

2

KΙC

6ch2( 1113857πa

radiccos(θ2)sin β sin β cos2(θ2) minus (32)cos β sin θ( 1113857

(11)

33 Analysis of the Location of the Edge or Neutral LayerCrack 0at Caused the Roof Periodic Collapse in AdvanceWhen a crack appears at the upper boundary or theneutral layer it becomes an edge crack problem which isdifferent from an internal crack problem Since there is noexact analytical expression for the stress intensity factorof the crack not perpendicular to the boundary only theproblem of the crack perpendicular to the boundary isstudied here

Figures 5(a) and 5(b) are roof models with cracks at theedge and at the neutral layer respectively Taking the edgecrack as an example unit body with the edge crack is takenas shown in Figure 6

)is crack belongs to a pure mode I crack and its stressintensity factor is [27]

KΙ 112σπ2a

radicasymp 158σ

πa

radic (12)

)e normal stress at the point B is obtained fromequation (1)

σB My

Iz

6cy0 L minus x0( 1113857

2

h2 (13)

where y0 takes a or (h(2 minus a)) and σ is equal to the normalstress at the midpoint of the crack


y0

σt1113858 1113859h

3c

1113971


2

KΙC

948ch2( 1113857πa

radic (14)

34 Analysis of the Location of the Vertical Crack0at Causedthe Roof Periodic Collapse in Advance According to theanalysis of Sections 32 and 33 when included angle of theinclined direction of crack and horizontal direction is 90degthe boundary equation of the crack-affected area is

y0

σt1113858 1113859h

3c

1113971


2

KΙC6ch2( 1113857

πa

radic alty0 lth

2minus a

KΙC948ch2( 1113857

πa

radic y0 a y0 h

2minus a

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(15)

In this chapter the relationship between stress and strainis not involved in the process of analysis and derivation)erefore the derivation results are applicable to plane stressand plane strain problems

4 Analysis and Discussion

41 Parameter It can be seen from equation (9) that theboundary equation is affected by crack size crack inclina-tion roof thickness and roof lithology A pertinent study ofthese factors is to reveal the laws of their impact )e pa-rameters of the roof are shown in Table 1

42 Influence of Vertical Crack Size and Roof Lithology onthe Crack-Affected Area )e crack length is 02m 03m04m 05m and 06m respectively )e roof thickness is6m)e roofs are taken from the four types of strata listed inTable 1 Figure 7 shows the variation rules of different rooflithologies and crack sizes on the roof crack-affected areaFigures 7(a)ndash7(d) correspond to the roofs from the hard roofto the soft roof On the premise of only considering the deadweight of the roof the crack-free roof periodic collapse spandecreases with the softening of the roof from 2635m to2052m )e upper-left zone of tension side of the roof isdefined as the crack-affected area while the lower right zoneis defined as the crack-unaffected area For the same lith-ologic roof the boundary lines of the crack-affected area areroughly of the same shape under different crack sizes theupper and lower parts of the line are straight while themiddle part is the curve convex outward to the lower rightWith the increase in crack size the crack-affected area in-creases continuously but the growth rate slows down As


shown in Figure 7(d) for the softer roof with the large crackthe crack-affected area will expand to the neutral layer andits proportion of the total crack-affected area will graduallyincrease

Figure 8 shows the change rules of the boundary of thecrack-affected area with the same roof thickness and cracksize in various lithologies )e crack length is 02m and theroof thickness is 6m For the same crack size and roofthickness the crack-affected area gradually expands with thesoftening of the roof)e upper and lower parts of the crack-affected areas of gabbro roof granite roof and marble roofare roughly of the same shape )e area of their middle partincreases slightly in turn while the size of the crack-affectedarea by the oil shale roof increases significantly )e softcrack-free roof has a short periodic collapse span and thecrack-affected area of the soft roof with a crack is large)erefore the soft roof is more likely to early periodiccollapse due to the crack which entails a more considerablerisk

43 Influence of Roof 0ickness on the Roof Crack-AffectedArea with Vertical Crack Take the marble roof as an ex-ample )e crack length is 02m and the roof thickness is2ndash6m Figure 9 shows the influence law of roof thickness on

the boundary of the crack-affected area of the marble roofwith a vertical crack As the roof thickness increases theperiodic collapse span of the roof without a crack increasesChange trends of the boundaries of the crack-affected arearemain the same

Figure 10 shows charts of the crack-affected area and itsproportion to total roof area As the roof thickness increasesthe size of the crack-affected area gradually increases but itsproportion to the total area gradually decreases which isbetween 3 and 55 It demonstrated that for the sameroof with the same crack the smaller the thickness of theroof is the higher the probability that the roof collapses inadvance due to the crack

44 Influence of the Internal Crack Inclination on the RoofCrack-Affected Area Figure 11 shows the effect of variousinternal crack inclinations on the crack-affected areas )ecrack length is 02m the roof thickness is 6m and β 75deg80deg 85deg 90deg respectively It can be viewed from the figurethat the size of crack-affected area shrinks significantly withthe decrease of the crack inclination When the crack in-clination is lower than a certain value the size of the crack-affected area is 0 )is indicates that when the crack appearsat any position inside the roof with this inclination the roofwill not collapse in advance due to the crack As shown inFigure 12 the initiation angle decreases with the increase ofthe crack inclination

5 Numerical Simulation

In order to further verify the conclusions of theoreticalanalysis the fracture mechanics model of the roof with acrack is established by XFEM method

51 Basic Principles of Extended Finite Element MethodFor the numerical solution of cracks the traditional finiteelement method has strict requirements (1) geometricsegmentation of the model is required at the crack location(2) high density mesh is required near the crack tip (3) it isnecessary to have a rough prediction of the crack propa-gation path and to simulate crack propagation by

A B

x

y

L

h

q

2ax0

(a)

A B

x

y

L

h

q

2ax0

(b)

Figure 5 Stress analysis diagrams of roof with crack (a) Roof with crack at edge (b) Roof with crack at neutral layer

2a

σ σ

Figure 6 Stress analysis diagram of edge crack

Table 1 Parameters of roofs

c (Nm3) σc (times106Pa) KΙC (times106Nmiddotmminus 32) h (m)

Gabbro 28224 98 362

2 3 4 5 6Granite 25872 7 233Marble 26460 65 199Oil shale 15680 33 037


continuous mesh reconstruction )e extended finite ele-ment method is a very effective method in solving dis-continuous problems like cracks Cracks can occur inside themesh )e cracks are independent of the mesh and theinitiation and development of the cracks are entirely fromthe results of the model calculations )e essence of theextended finite element method is to describe the discon-tinuous displacement field by introducing discontinuousdisplacement mode using finite element shape function aspartitions of unity [28ndash30]

)e extended finite element method is similar to theconventional finite element method )e displacementfunction is substituted for the virtual work equation to

derive the basic governing equation of the finite elementmethod )e biggest difference and the most critical point isthe construction of displacement function )e approximatedisplacement interpolation function of the extended finiteelement method is [31ndash33]

U 1113944iisinN1

Ni(x)ui + 1113944jisinN2

Nj(x)H(x)aj

+ 1113944kisinN3

Nk(x) 1113944

4

α1Fα(x)b

αk

(16)

2635

a = 01a = 015a = 02

a = 025a = 03

0

1

2

3Th

ickn

ess o

f hal

f-roo

f (m

)

5 10 15 20 25 300Length of roof (m)

(a)

a = 01a = 015a = 02

a = 025a = 03

2326

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)

(b)

a = 01a = 015a = 02

a = 025a = 03

2217

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)

(c)

a = 01a = 015a = 02

a = 025a = 03

2052

5 10 15 20 25 300Length of roof (m)

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

(d)

Figure 7 Effect of different crack sizes on the boundary of crack-affected area (a) Gabbro (b) Granite (c) Marble (d) Oil shale

26352326

2217

2052

GabbroGranite

MarbleOil shale

5 10 15 20 25 300Length of roof (m)

0

1

2

3

ic

knes

s of h

alf-r

oof (

m)

Figure 8 Effect of different roof lithology on the boundary ofcrack-affected area

2217

12801567

18092023

h = 2h = 3h = 4

h = 5h = 6

0

1

2

3

ic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)

Figure 9 Effect of different roof thicknesses on the boundary ofcrack-affected area


As shown in Figure 13 the function divides all the nodesnear the crack into three categories)e function classifies allnodes near the crack into three categories )e first itemdescribes the node displacement of the standard elementswithout the crack (set N1) the second item describes thenode displacement of the elements penetrated by the crack(set N2) the third term describes the node displacement ofthe elements including the crack tip (set N3) aj and bαk areextra degrees of freedom at nodes And H(x) is the en-richment function of the elements penetrated by the crack

and it is the jump function Fα(x) is the enrichment functionof the crack tip elements

Fα(r θ) α 1 sim 41113858 1113859

r

radicsin

θ2

r

radiccos

θ2

r

radicsin

θ2sin θ

r

radiccos

θ2sin θ1113896 1113897

(17)

where (r θ) is the polar coordinate system of the crack tip

52Numerical Simulation of theRoofwith theNonpenetratingCrack )e results of theoretical analysis were numericallyverified by ABAQUS Taking themarble roof as an example thetwo-dimensional plane stress model was used for analysiswithout considering the width of the roof)e roof was set to anelastoplastic constitutive model Elastic modulus is 558GPaPoissonrsquos ratio is 025 tensile strength is 65MPa density is2700kgm3 length is 2217m and thickness is 6m )e cracklength is 02m )e left side of the roof is completely fixed theupper part is affected by the distributed load and the loadcollection degree (q) is 158760kNm )e element type isCPS4R and the number of elements is 380

Figure 14 shows the simulation results of cracks atdifferent locations Figures 14(a)ndash14(d) correspond to crack-free the crack located at the edge of crack-affected area thecrack located inside crack-affected area and the crack lo-cated at the edge of the crack-unaffected area respectively)e dotted line is the boundary line between the crack-af-fected area and the crack-free area and the circle indicates apicture magnified 100 times in the vicinity of the crack )eroof in Figure 14(a) has no cracks and the maximum tensilestress exists in the upper-left corner of the roof)e cracks inFigures 14(b) and 14(c) both expand which can cause theroof to collapse in advance )e edge crack in Figure 14(b)

AreaProportion

2 3 4 5 6 71Thickness (m)

0

1

2

3

4

5

6A

rea o

f dan

gero

us ar

ea (m

2 )

30

35

40

45

50

55

60

Prop

ortio

n

Figure 10 Size of crack-affected area vs proportion of crack-af-fected area

β = 75degβ = 80deg


0

1

2

3

ic

knes

s of h

alf-r

oof (

m)

1 2 3 4 5 60Length of roof (m)

Figure 11 Effect of different crack inclinations on the boundary ofcrack-affected area

05

10152025303540

Crac

king

angl

e (deg)

80 85 9075Crack inclination (deg)

Figure 12 Relation between crack inclination and initiation angle

Crack tip elements

Interior elementsN2

N3

N1

Figure 13 Node and element near crack


S S11 (Avg 75)

ndash6408e + 06ndash4564e + 06ndash2719e + 06ndash8748e + 05+9696e + 05+2814e + 06+4658e + 06+6503e + 06

(a)

S S11 (Avg 75)


(b)

S S11 (Avg 75)


(c)

S S11 (Avg 75)


(d)

Figure 14 Normal stress of roof under different crack positions (a) Crack-free (b) At the edge of crack-affected area (c) In the interior ofcrack-affected area (d) In the crack-unaffected area


extends to the inside )ere are large tensile stresses near thelower crack tip and the upper-left corner of the roof )einternal crack in Figure 14(c) expands inward and outwardRestricted by XFEM itself when the crack reaches the edgeof the roof the propagation stops Large tensile stresses existat the tips on both sides of the crack In Figure 14(d) thecrack is in the crack-unaffected area the crack does notexpand there is no stress concentration at the crack tip andthere is a maximum tensile stress at the upper-left corner ofthe roof )eoretical analysis suggests that cracks in thecrack-affected area will expand causing the roof to collapsein advance )e cracks located in the crack-unaffected areawill not expand and the stress distribution of the roof isbasically similar to the roof with no crack In summary thenumerical simulation results are consistent with the theo-retical analysis results

6 Conclusion

In view of the periodic collapse of the roof with the non-penetrating crack with large cutting height a fracture me-chanics model of the cantilever beam with thenonpenetrating crack was established )e quantitativefunctional relationship between crack size crack inclinationroof thickness roof lithology and the boundary of the crack-affected area was deduced )e influence of the crack lo-cation on roof failure was revealed and further numericalsimulations were carried out to verify the analysis resultsRelevant conclusions are as follows

(1) For vertical cracks the influence range of cracks onthe roof periodic collapse expands with the increaseof the crack size but its growth rate slows down)iseffect decreases with the increase of the roof li-thology In the soft roof with a large size crack thepossibility of the crack-affected area near the neutrallayer is greater

(2) For vertical cracks the crack-affected area increaseswith the increase of the roof thickness but theproportion of the crack-affected area to the total roofarea decreases For thin roof the probability of earlyroof periodic collapse caused by the crack is higher

(3) For the internal crack the crack-affected area de-creases greatly with the decrease of the crack incli-nation )e initiation angle decreases with theincrease of the crack inclination )e initiation angleof the vertical crack is 0 and the crack is cracked inthe vertical direction When the crack inclination isless than a certain angle the roof is all the crack-affected area And the roof always cracks at its endwhich is consistent with the crack-free roof

(4) )e numerical simulation shows that the crack in thecrack-affected area can expand and there is a largetensile stress at the crack tip which will lead to theroof collapse in advance )e crack in the crack-unaffected area can not expand )e stress distri-bution of the roof is similar to that of the crack-freeroof and the roof will not collapse in advance

Data Availability

)e calculation data used to support the findings of thisstudy are included within the article

Conflicts of Interest

)e authors have no conflicts of interest

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (51774196 and 51774194) NationalNatural Science Foundation of China and ShandongProvince joint Program (U1806209) SDUST Research Fund(GrantAward Number 2019TDJH101) and China Post-doctoral Science Foundation (No 2016M592221)

References

[1] J C Wang and S H Zhong ldquo)e present status and the keyissues to be resolved of thick seam mining technique inChinardquo Sciencepaper Online vol 3 no 11 pp 829ndash834 2008

[2] Z Zhao W Sun S Chen WWang and QWang ldquoCouplingmodel of jointed rock mass and rock bolt in offshore LPGunderground storagerdquo Energy Science amp Engineering pp 1ndash16 2019

[3] J H Wang ldquoKey technology for fully-mechanized top coalcaving with large mining height in extra-thick coal seamrdquoJournal of China Coal Society vol 38 no 12 pp 2089ndash20982013

[4] J L Xu and J F Ju ldquoStructural morphology of key stratumand its influence on strata behaviors in fully-mechanized facewith super-large mining heightrdquo Chinese Journal of RockMechanics and Engineering vol 30 no 8 pp 1547ndash15562011 in Chinese

[5] M S Diederichs P K Kaiser and E Eberhardt ldquoDamageinitiation and propagation in hard rock during tunnelling andthe influence of near-face stress rotationrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 41 no 5pp 785ndash812 2004

[6] T Xu T-h Yang C-f Chen H-l Liu and Q-l Yu ldquoMininginduced strata movement and roof behavior in undergroundcoal minerdquo Geomechanics and Geophysics for Geo-Energy andGeo-Resources vol 1 no 3-4 pp 79ndash89 2015

[7] X Z Lyu Z H Zhao X J Wang andWMWang ldquoStudy onthe permeability of weakly cemented sandstonesrdquo Geofluidsvol 2019 Article ID 8310128 14 pages 2019

[8] Z Zhao Q Ma Y Tan and X Gao ldquoLoad transfer mech-anism and reinforcement effect of segmentally yieldable an-chorage in weakly consolidated soft rockrdquo Simulation vol 95no 1 pp 83ndash96 2019

[9] Z H Zhao Y L Tan S J Chen Q Ma and X J Gaoldquo)eoretical analyses of stress field in surrounding rocks ofweakly consolidated tunnel in a high-humidity deep envi-ronmentrdquo International Journal of Rock Mechanics andMining Sciences vol 122 Article ID 104064 2019

[10] H Shi and F X Jiang ldquoSturctural theories of overlying stratain longwall faces and their new developmentrdquo Journal ofShandong University of Science and Technology (NaturalScience) vol 24 no 1 pp 21ndash25 2005 in Chinese


[11] S M Ma W Y Lyu and B Y Sun ldquoCurrent situation andtrend of research on ground pressure theory in deep stoperdquoCoal Engineering vol 10 pp 87ndash89 2010

[12] M G Qian and T C LiuMining Pressure and Strata ControlChina Coal Industry Publishing House Beijing China 1984

[13] M G Qian X X Miao J L Xu and X B Mao Key Strata0eory in Ground Control China University of Mining andTechnology Press Beijing China 2003

[14] Z Q Song Practical Mine Pressure Control China Universityof Mining and Technology Press Beijing China 1988

[15] Q X Huang and S G Li ldquoBreakage of coal seam roof inshallow and thin basement rock and its controlrdquo GroundPressure and Strata Control no Z1 pp 22ndash25 1995 inChinese

[16] H F Huang Tectonic motion and control of overburden in thinbedrock fully-mechanized sublevel caving face PhD thesisChina University of Mining and Technology Beijing China2012

[17] F Du H B Bai H F Huang and G H Jiang ldquoMechanicalanalysis of periodic weighting of main roof in longwall topcoal caving face with thin-bedrock roofrdquo Journal of ChinaUniversity of Mining amp Technology vol 42 no 3 pp 362ndash3692013

[18] Y F Zhao X Y Zhang and M Tu ldquoRoof caving charac-teristic and srata behavior in exploiting steep coal seamsrdquoJournal of Mining amp Safety Engineering vol 24 no 2pp 231ndash234 2007

[19] Y D Zhang J Y Cheng X X Wang Z J Feng and M Jildquo)in plate model analysis on roof break of up-dip or down-dip mining stoperdquo Journal of Mining amp Safety Engineeringvol 27 no 4 pp 487ndash493 2010 in Chinese

[20] Z Y Ti H Y Qin and Y M Chen ldquoInfluence of coal seamdip angle on periodic weighting feature in fully-mechanizedsublevel caving miningrdquo Journal of China University ofMining amp Technology vol 44 no 5 pp 817ndash822 2015 inChinese

[21] Z H Chen J J Feng C C Xiao and R H Li ldquoFracturemechanical model of key roof for fully-mechanized top-coalcaving in shallow thick coal seamrdquo Journal of China CoalSociety vol 32 no 5 pp 449ndash452 2007

[22] D F Yang L F Zhang M Chai B Li and Y F Bai ldquoStudy ofroof breaking law of fully mechanized top coal caving miningin ultra-thick coal seam based on fracture mechanicsrdquo Rockand Soil Mechanics vol 37 no 7 pp 2033ndash2039 2016

[23] D F Yang Z H Chen J W Sun L N Wang and Q GaoldquoCracked plate mechanical model for large mining height andlangwall face of roof cavingrdquo Journal of Southeast University(Natural Science Edition) vol 46 no S1 pp 210ndash216 2016

[24] Z Zhao M Zhang Q Ma and B Chen ldquoDeviation effect ofcoaxiality on the rock Brazilian splitrdquo Advances in Mathe-matical Physics vol 2020 Article ID 5782457 8 pages 2020

[25] J Cheng and S S Zhao Fracture Mechanics Science PressBeijing China 2006

[26] J F Knott Fundamentals of Fracture Mechanics GruppoItaliano Frattura Cassino Italy 1973

[27] S Al Laham and S I Branch Stress Intensity Factor and LimitLoad Handbook British Energy Generation Limited LondonUK 1998

[28] X J Fang and F Jin ldquoExtended finite element method basedon abaqusrdquo Engineering Mechanics vol 24 no 7 pp 6ndash102007 in Chinese

[29] Z L Ru C R Zhu Y L Zhang and H B Zhao ldquoStudy offracture problem with extended finite element methodrdquo Rockand Soil Mechanics vol 32 no 7 pp 2171ndash2176 2011

[30] L L Guo Z F Chen J Y Luo and G Chen ldquoA review of theextended finite element method and its applicationsrdquo ChineseQuarterly of Mechanics vol 32 no 4 pp 612ndash625 2011

[31] T Belytschko and T Black ldquoElastic crack growth in finiteelements with minimal remeshingrdquo International Journal forNumerical Methods in Engineering vol 45 no 5 pp 601ndash6201999

[32] N Moes J Dolbow and T Belytschko ldquoA finite elementmethod for crack growth without remeshingrdquo InternationalJournal for Numerical Methods in Engineering vol 46 no 1pp 131ndash150 1999

[33] J Dolbow N Moes and T Belytschko ldquoDiscontinuous en-richment in finite elements with a partition of unity methodrdquoFinite Elements in Analysis and Design vol 36 no 3-4pp 235ndash260 2000


pressure theories in China are the voussoir beam theory [12]the key stratum theory [13] and the transmission rock beamtheory [14] In recent years considerable research effortshave been devoted to underground pressure theories fordifferent mining environments and methods Aiming at theproblem of shallow thin bedrock seammining Huang and Liestablished a cantilever beam model based on Winklerelastic foundation theory )e results show that bedrock willbreak ahead of the coal wall during the period of periodicpressure [15] Some scholars combine Protodyakonovrsquostheory with the voussoir beam theory and then put forwardthe balanced composite structure of ldquoarchrdquo on ldquobeamrdquo )eresearch indicates that the width of ldquoarchrdquo is equal to thebasic roof periodic weighting span [16 17] Relevant studiesdemonstrate that the mining of deeply inclined coal seams isdifferent from that of horizontal coal seams and its periodiccollapse span is affected by the position of working face theway of underhand mining and overhand mining and thecrack inclination of the coal seam )e periodic weightingspan at the top and bottom of the working face is slightlylarger than that in the upper middle middle and lowermiddle of it )e roof periodic collapse span is shorter inoverhand mining than in underhand mining )e periodicweighting span decreases with the increase of the crackinclination of the coal seam [18ndash20] In addition somescholars have considered the damage effect of roof stratumand regarded the roof as a cantilever beam with cracksEquations of the roof periodic collapse span and supportworking resistance are obtained )e effects of crack incli-nation roof length and thickness on the roof periodiccollapse span and support working resistance were analyzed[21 22] Yang et al [23] further established the mechanicalmodel of a cracked plate by using thin plate theory andanalyzed the roof caving motion state of longwall face largecutting height mining Unfortunately these discussions areall aimed at the specific location of the cracks not involvingthe different positions of cracks And there is not any in-depth study of the boundaries of the crack-affected area andthe crack-unaffected area In fact the position of crackdetermines its stress state Under a certain load the differentpositions of cracks have different effects on the roof periodiccollapse span

Much work of the roof periodic collapse span so far hasbeen based on the assumption of uniform continuous mediaAlthough some studies have established the roof model withcracks the range of cracks affecting roof collapse motionremains an unanswered question )e innovations of thisstudy are as follows

(1) Introduce the crack into the analysis of the periodiccollapse of the roof with large cutting height miningand establish a cantilever beam model with thenonpenetrating crack

(2) Release the constraint condition of crack position)e studying content is not limited to the crack atspecific location

(3) Determine the scope of the cracks that cause the roofto collapse periodically in advance

)erefore aiming at the problem of the roof periodiccollapse span with the nonpenetrating crack with largecutting height mining this study proposed to establish acantilever beam model with the nonpenetrating crack de-termined the location of crack which can cause the earlyperiodic collapse of a roof and revealed the influence ofcrack size crack inclination roof lithology and roofthickness on the crack-affected area Necessary supportingtheories for roof support design were provided

2 Analysis Model of the Roof with theNonpenetrating Crack in LargeCutting Height

)e schematic diagram of the roof with the nonpenetratingcrack in large cutting height is given as the principle dis-played in Figure 1 Under the condition of fully mechanizedmining with large cutting height or extra-large cuttingheight the working face advances from right to left Due tothe obvious increase in mining height and caving zoneheight the fractured rock produced by the immediate roofcollapse can not fill the goaf Because of the large amount ofrotary subsidence the main roof and low-level key strata cannot contact the caved rock in the goaf after breaking to forma stable masonry beam structure Low-level key strata mayenter the caving zone and break in the form of a cantileverbeam which forms periodic weighting [4])e l in the figurerepresents the roof periodic collapse span Under the actionof long-term geological structure and mining disturbancerandom cracks at different lengths and directions will occurin rocks In order to facilitate the analysis these cracks areregarded as a main crack It is assumed that the effect of theprincipal crack on roof fracture represents the effect of allcracks on it When the crack satisfies the fracture conditionunder the external force the crack begins to crack whichcauses the roof to collapse ahead of time Under the sameexternal load this effect is closely related to the location ofthe crack )e zone where the crack that causes the roof tocollapse ahead of time is located is called the crack-affectedarea and the other zones are called the crack-unaffectedarea

3 Analysis of the Boundary of the Crack-Affected Area in Large Cutting Height

31 Analysis of the Crack-Free Roof Periodic Collapse SpanFigure 2 is a simplified diagram of the crack-free Only thebending deformation of the roof under dead weight isconsidered )e thickness of the roof is h and the length is LUnder the action of distributed load q q hc and c is thespecific weight of the roof

)e bending moment equation of the roof is

M(x) q

2(L minus x)

2

hc

2(L minus x)

2 (1)

)e expression of normal stress at the point A is


σA My

Iz

3cL2

h (2)


L

σt1113858 1113859h

3c

1113971

(3)






KΙ σπa

radicsin2 β

KΙΙ σπa

radicsin β cos β

(4)


σr cos(θ2)

22πr


22πr


σθ cos(θ2)

22πr


τθ

cos(θ2)

22πr


(5)


σθ KΙ KΙΙ θ( 1113857 σθC (6)


cosθ2

KΙcos2θ2



l

Coal seam

Immediate roof

Gangue


A

x

y

L

h

q


A B

x

y

L

h

q

2ay0

x0


2aσ σ

θ

θ

β



cos θ cos2 β +

sin2 β + 8 cos2 β1113969

1 + 8 cos2 β (8)


can be obtained

σBc

πa

radiccos


θ2


(9)


σB My

Iz

6cy0 L minus x0( 1113857

2

h2 (10)


y0

σt1113858 1113859h

3c

1113971


2

KΙC

6ch2( 1113857πa


(11)




KΙ 112σπ2a

radicasymp 158σ

πa

radic (12)


σB My

Iz

6cy0 L minus x0( 1113857

2

h2 (13)



y0

σt1113858 1113859h

3c

1113971


2

KΙC

948ch2( 1113857πa

radic (14)


y0

σt1113858 1113859h

3c

1113971


2

KΙC6ch2( 1113857

πa

radic alty0 lth

2minus a

KΙC948ch2( 1113857

πa

radic y0 a y0 h

2minus a

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(15)















A B

x

y

L

h

q

2ax0

(a)

A B

x

y

L

h

q

2ax0

(b)


2a

σ σ




Gabbro 28224 98 362






U 1113944iisinN1


Nj(x)H(x)aj

+ 1113944kisinN3

Nk(x) 1113944

4

α1Fα(x)b

αk

(16)

2635

a = 01a = 015a = 02

a = 025a = 03

0

1

2

3Th

ickn

ess o

f hal

f-roo

f (m

)

5 10 15 20 25 300Length of roof (m)

(a)

a = 01a = 015a = 02

a = 025a = 03

2326

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)

(b)

a = 01a = 015a = 02

a = 025a = 03

2217

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)

(c)

a = 01a = 015a = 02

a = 025a = 03

2052

5 10 15 20 25 300Length of roof (m)

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

(d)


26352326

2217

2052

GabbroGranite

MarbleOil shale

5 10 15 20 25 300Length of roof (m)

0

1

2

3

ic

knes

s of h

alf-r

oof (

m)


2217

12801567

18092023

h = 2h = 3h = 4

h = 5h = 6

0

1

2

3

ic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)





Fα(r θ) α 1 sim 41113858 1113859

r

radicsin

θ2

r

radiccos

θ2

r

radicsin

θ2sin θ

r

radiccos

θ2sin θ1113896 1113897

(17)




AreaProportion


0

1

2

3

4

5

6A

rea o

f dan

gero

us ar

ea (m

2 )

30

35

40

45

50

55

60

Prop

ortio

n




0

1

2

3

ic

knes

s of h

alf-r

oof (

m)



05

10152025303540

Crac

king

angl

e (deg)



Crack tip elements

Interior elementsN2

N3

N1



S S11 (Avg 75)


(a)

S S11 (Avg 75)


(b)

S S11 (Avg 75)


(c)

S S11 (Avg 75)


(d)




6 Conclusion






Data Availability




Acknowledgments


References




































σA My

Iz

3cL2

h (2)


L

σt1113858 1113859h

3c

1113971

(3)






KΙ σπa

radicsin2 β

KΙΙ σπa

radicsin β cos β

(4)


σr cos(θ2)

22πr


22πr


σθ cos(θ2)

22πr


τθ

cos(θ2)

22πr


(5)


σθ KΙ KΙΙ θ( 1113857 σθC (6)


cosθ2

KΙcos2θ2



l

Coal seam

Immediate roof

Gangue


A

x

y

L

h

q


A B

x

y

L

h

q

2ay0

x0


2aσ σ

θ

θ

β



cos θ cos2 β +

sin2 β + 8 cos2 β1113969

1 + 8 cos2 β (8)


can be obtained

σBc

πa

radiccos


θ2


(9)


σB My

Iz

6cy0 L minus x0( 1113857

2

h2 (10)


y0

σt1113858 1113859h

3c

1113971


2

KΙC

6ch2( 1113857πa


(11)




KΙ 112σπ2a

radicasymp 158σ

πa

radic (12)


σB My

Iz

6cy0 L minus x0( 1113857

2

h2 (13)



y0

σt1113858 1113859h

3c

1113971


2

KΙC

948ch2( 1113857πa

radic (14)


y0

σt1113858 1113859h

3c

1113971


2

KΙC6ch2( 1113857

πa

radic alty0 lth

2minus a

KΙC948ch2( 1113857

πa

radic y0 a y0 h

2minus a

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(15)















A B

x

y

L

h

q

2ax0

(a)

A B

x

y

L

h

q

2ax0

(b)


2a

σ σ




Gabbro 28224 98 362






U 1113944iisinN1


Nj(x)H(x)aj

+ 1113944kisinN3

Nk(x) 1113944

4

α1Fα(x)b

αk

(16)

2635

a = 01a = 015a = 02

a = 025a = 03

0

1

2

3Th

ickn

ess o

f hal

f-roo

f (m

)

5 10 15 20 25 300Length of roof (m)

(a)

a = 01a = 015a = 02

a = 025a = 03

2326

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)

(b)

a = 01a = 015a = 02

a = 025a = 03

2217

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)

(c)

a = 01a = 015a = 02

a = 025a = 03

2052

5 10 15 20 25 300Length of roof (m)

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

(d)


26352326

2217

2052

GabbroGranite

MarbleOil shale

5 10 15 20 25 300Length of roof (m)

0

1

2

3

ic

knes

s of h

alf-r

oof (

m)


2217

12801567

18092023

h = 2h = 3h = 4

h = 5h = 6

0

1

2

3

ic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)





Fα(r θ) α 1 sim 41113858 1113859

r

radicsin

θ2

r

radiccos

θ2

r

radicsin

θ2sin θ

r

radiccos

θ2sin θ1113896 1113897

(17)




AreaProportion


0

1

2

3

4

5

6A

rea o

f dan

gero

us ar

ea (m

2 )

30

35

40

45

50

55

60

Prop

ortio

n




0

1

2

3

ic

knes

s of h

alf-r

oof (

m)



05

10152025303540

Crac

king

angl

e (deg)



Crack tip elements

Interior elementsN2

N3

N1



S S11 (Avg 75)


(a)

S S11 (Avg 75)


(b)

S S11 (Avg 75)


(c)

S S11 (Avg 75)


(d)




6 Conclusion






Data Availability




Acknowledgments


References




































cos θ cos2 β +

sin2 β + 8 cos2 β1113969

1 + 8 cos2 β (8)


can be obtained

σBc

πa

radiccos


θ2


(9)


σB My

Iz

6cy0 L minus x0( 1113857

2

h2 (10)


y0

σt1113858 1113859h

3c

1113971


2

KΙC

6ch2( 1113857πa


(11)




KΙ 112σπ2a

radicasymp 158σ

πa

radic (12)


σB My

Iz

6cy0 L minus x0( 1113857

2

h2 (13)



y0

σt1113858 1113859h

3c

1113971


2

KΙC

948ch2( 1113857πa

radic (14)


y0

σt1113858 1113859h

3c

1113971


2

KΙC6ch2( 1113857

πa

radic alty0 lth

2minus a

KΙC948ch2( 1113857

πa

radic y0 a y0 h

2minus a

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(15)















A B

x

y

L

h

q

2ax0

(a)

A B

x

y

L

h

q

2ax0

(b)


2a

σ σ




Gabbro 28224 98 362






U 1113944iisinN1


Nj(x)H(x)aj

+ 1113944kisinN3

Nk(x) 1113944

4

α1Fα(x)b

αk

(16)

2635

a = 01a = 015a = 02

a = 025a = 03

0

1

2

3Th

ickn

ess o

f hal

f-roo

f (m

)

5 10 15 20 25 300Length of roof (m)

(a)

a = 01a = 015a = 02

a = 025a = 03

2326

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)

(b)

a = 01a = 015a = 02

a = 025a = 03

2217

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)

(c)

a = 01a = 015a = 02

a = 025a = 03

2052

5 10 15 20 25 300Length of roof (m)

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

(d)


26352326

2217

2052

GabbroGranite

MarbleOil shale

5 10 15 20 25 300Length of roof (m)

0

1

2

3

ic

knes

s of h

alf-r

oof (

m)


2217

12801567

18092023

h = 2h = 3h = 4

h = 5h = 6

0

1

2

3

ic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)





Fα(r θ) α 1 sim 41113858 1113859

r

radicsin

θ2

r

radiccos

θ2

r

radicsin

θ2sin θ

r

radiccos

θ2sin θ1113896 1113897

(17)




AreaProportion


0

1

2

3

4

5

6A

rea o

f dan

gero

us ar

ea (m

2 )

30

35

40

45

50

55

60

Prop

ortio

n




0

1

2

3

ic

knes

s of h

alf-r

oof (

m)



05

10152025303540

Crac

king

angl

e (deg)



Crack tip elements

Interior elementsN2

N3

N1



S S11 (Avg 75)


(a)

S S11 (Avg 75)


(b)

S S11 (Avg 75)


(c)

S S11 (Avg 75)


(d)




6 Conclusion






Data Availability




Acknowledgments


References













































A B

x

y

L

h

q

2ax0

(a)

A B

x

y

L

h

q

2ax0

(b)


2a

σ σ




Gabbro 28224 98 362






U 1113944iisinN1


Nj(x)H(x)aj

+ 1113944kisinN3

Nk(x) 1113944

4

α1Fα(x)b

αk

(16)

2635

a = 01a = 015a = 02

a = 025a = 03

0

1

2

3Th

ickn

ess o

f hal

f-roo

f (m

)

5 10 15 20 25 300Length of roof (m)

(a)

a = 01a = 015a = 02

a = 025a = 03

2326

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)

(b)

a = 01a = 015a = 02

a = 025a = 03

2217

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)

(c)

a = 01a = 015a = 02

a = 025a = 03

2052

5 10 15 20 25 300Length of roof (m)

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

(d)


26352326

2217

2052

GabbroGranite

MarbleOil shale

5 10 15 20 25 300Length of roof (m)

0

1

2

3

ic

knes

s of h

alf-r

oof (

m)


2217

12801567

18092023

h = 2h = 3h = 4

h = 5h = 6

0

1

2

3

ic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)





Fα(r θ) α 1 sim 41113858 1113859

r

radicsin

θ2

r

radiccos

θ2

r

radicsin

θ2sin θ

r

radiccos

θ2sin θ1113896 1113897

(17)




AreaProportion


0

1

2

3

4

5

6A

rea o

f dan

gero

us ar

ea (m

2 )

30

35

40

45

50

55

60

Prop

ortio

n




0

1

2

3

ic

knes

s of h

alf-r

oof (

m)



05

10152025303540

Crac

king

angl

e (deg)



Crack tip elements

Interior elementsN2

N3

N1



S S11 (Avg 75)


(a)

S S11 (Avg 75)


(b)

S S11 (Avg 75)


(c)

S S11 (Avg 75)


(d)




6 Conclusion






Data Availability




Acknowledgments


References







































U 1113944iisinN1


Nj(x)H(x)aj

+ 1113944kisinN3

Nk(x) 1113944

4

α1Fα(x)b

αk

(16)

2635

a = 01a = 015a = 02

a = 025a = 03

0

1

2

3Th

ickn

ess o

f hal

f-roo

f (m

)

5 10 15 20 25 300Length of roof (m)

(a)

a = 01a = 015a = 02

a = 025a = 03

2326

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)

(b)

a = 01a = 015a = 02

a = 025a = 03

2217

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)

(c)

a = 01a = 015a = 02

a = 025a = 03

2052

5 10 15 20 25 300Length of roof (m)

0

1

2

3

Thic

knes

s of h

alf-r

oof (

m)

(d)


26352326

2217

2052

GabbroGranite

MarbleOil shale

5 10 15 20 25 300Length of roof (m)

0

1

2

3

ic

knes

s of h

alf-r

oof (

m)


2217

12801567

18092023

h = 2h = 3h = 4

h = 5h = 6

0

1

2

3

ic

knes

s of h

alf-r

oof (

m)

5 10 15 20 25 300Length of roof (m)





Fα(r θ) α 1 sim 41113858 1113859

r

radicsin

θ2

r

radiccos

θ2

r

radicsin

θ2sin θ

r

radiccos

θ2sin θ1113896 1113897

(17)




AreaProportion


0

1

2

3

4

5

6A

rea o

f dan

gero

us ar

ea (m

2 )

30

35

40

45

50

55

60

Prop

ortio

n




0

1

2

3

ic

knes

s of h

alf-r

oof (

m)



05

10152025303540

Crac

king

angl

e (deg)



Crack tip elements

Interior elementsN2

N3

N1



S S11 (Avg 75)


(a)

S S11 (Avg 75)


(b)

S S11 (Avg 75)


(c)

S S11 (Avg 75)


(d)




6 Conclusion






Data Availability




Acknowledgments


References






































Fα(r θ) α 1 sim 41113858 1113859

r

radicsin

θ2

r

radiccos

θ2

r

radicsin

θ2sin θ

r

radiccos

θ2sin θ1113896 1113897

(17)




AreaProportion


0

1

2

3

4

5

6A

rea o

f dan

gero

us ar

ea (m

2 )

30

35

40

45

50

55

60

Prop

ortio

n




0

1

2

3

ic

knes

s of h

alf-r

oof (

m)



05

10152025303540

Crac

king

angl

e (deg)



Crack tip elements

Interior elementsN2

N3

N1



S S11 (Avg 75)


(a)

S S11 (Avg 75)


(b)

S S11 (Avg 75)


(c)

S S11 (Avg 75)


(d)




6 Conclusion






Data Availability




Acknowledgments


References




































S S11 (Avg 75)


(a)

S S11 (Avg 75)


(b)

S S11 (Avg 75)


(c)

S S11 (Avg 75)


(d)




6 Conclusion






Data Availability




Acknowledgments


References





































6 Conclusion






Data Availability




Acknowledgments


References




































FractureMechanicalBehaviorofCrackedCantileverRoofwith … · 2020. 4. 13. · was established. e...

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