amethodfordeterminingfeasibilityofminingresidual...

Research ArticleA Method for Determining Feasibility of Mining ResidualCoal aboveOut-FashionedGoafunderVariableLoadACaseStudy

Yuxia Guo12 Honghui Yuan12 Xiaogang Deng3 Yujiang Zhang 1234 Yunlou Du12

Hui Liu3 Guorui Feng12 Jiali Xu4 Renle Zhao2 and Dao Viet Doan5

1College of Mining Engineering Taiyuan University of Technology Taiyuan 030024 China2Shanxi Engineering Research Center for Green Mining Taiyuan 030024 China3Shandong Energy Linyi Mining Group Co Ltd Linyi 276017 China4Postdoctoral Centre Shandong Energy Group Co Ltd Jinan 250014 China5Department of Underground and Mining Construction Hanoi University of Mining and Geology Hanoi 100803 Vietnam

Correspondence should be addressed to Yujiang Zhang ylczyjyeahnet

Received 25 March 2020 Revised 18 July 2020 Accepted 12 August 2020 Published 27 August 2020

Academic Editor Chiara Bedon

Copyright copy 2020 Yuxia Guo 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

Out-fashioned goaf is the protective structure for mining the upper residual coal and its stability is the core problem inmining theupper residual coal According to the upward mining demand for No 5 coal seam above the out-fashioned goaf in Baizi CoalMine a new method is proposed to determine the upward mining safety According to the analysis of the actual situation of themine the coal pillar and suspended roof in the out-fashioned goaf are taken as the objects Furthermore a ldquocoal pillar-suspendedroofrdquo system model based on the variable load induced by abutment pressure of upper coal seam mining is established After themechanical model was solved the parameter acquisition method of the model was established +e basic parameters of Baizi CoalMine were considered to determine the feasibility of mining residual coal above out-fashioned goaf And the effects of variableload on the coal pillar and suspended roof stability were analyzed +e results show that the upper No 5 coal seam in Baizi CoalMine can be mined safely Compared to the traditional method which simplifies all the upper loads to uniform loads the newmethod is safer +e system stability of the suspended roof and coal pillar is influenced by ldquoaLrdquo and ldquoLrdquo Axial stress curves of thecoal pillar and suspended roof appear nearly parabolic with ldquoaLrdquo varying +eir maximum values are obtained when the ldquoaLrdquovalue is around 05sim06 In this situation the combination structure is most easy to to be damaged +e ratio qprimeq has a linearrelationship with all stresses of the system model +e failure sequence of the system model is determined by analyzing therelationship between the tensile strength of the suspended roof and compressive strength of the coal pillar +is study provides areference case for coal resources upward mining under similar conditions

1 Introduction

+e amount of residual coal exceeds 12 Gt [1 2] with a greatmajority resulting from the utilization of the out-fashionedcoal mining method It mainly includes the reamer-pillarcoal mining roadway pillar mining and top-coal cavingmining of the lane pillar +ese goafs could be called as theout-fashioned goafs According to statistical findings it isobserved that there are abundant coal resources left in out-fashioned goafs and the recovery ratio is from 10 to 20[3] +ese out-fashioned mining methods are used for

mining high-quality coal resources in many mines for a longtime In the 1950s to 1980s in order to ensure coal supplyand short-term economic benefits massive lower high-quality thick coal seams have been mined in preference tothe upper coal seams Because of this fact lots of coal re-sources above the out-fashioned goafs are abandoned whichis widespread in China especially in Shanxi and Shaanxiprovinces [4 5] If the upper residual coal remains unmineduntil the mine is closed it will remain forever in the groundresulting in wastage of resources Currently due to shortageof energy resources we need to exploit them to improve the

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 8837657 12 pageshttpsdoiorg10115520208837657

recovery rate of coal resources However the upper residualcoal distinguishes from the common coal in mining con-ditions +e upper residual coal is supported by pillars andsuspended roofs in the out-fashioned goaf When the upperresidual coal working face goes across the coal pillar or thesuspended roof it is very likely to cause the coal pillar in-stability or the suspended roof rupture causing the collapseof personnel or equipment Hence the stability of the coalpillar and suspended roof is the key technique problem inupper residual coal mining above the out-fashioned goafWe must ensure the stability of the lower coal pillar and itssuspended roof during mining upper coal seams

Many researchers analyzed the integrity of the upper coalseam and its surrounding rock stability proposing corre-sponding methods for determining the feasibility of upwardmining +ese methods mainly include the rational ana-lytical method the ldquothree-zonerdquo method mathematicalstatistics method and the method of equilibrium sur-rounding rock +eir determination formulas and applica-tion conditions are shown in Table 1

+ese methods are proposed after statistically analyzingthe field measured data which are derived from longwallmining conditions But the out-fashioned mining differ-entiates from the longwall mining In the out-fashioned goafcompared with the longwall goaf coal pillars with irregularsize and shape were left to support the roof which results ina limited roof caving or even no caving +e upper coal seamand its surrounding rock are integral and stable So if thesemethods are adopted to determine the feasibility of themining upper residual coal seam above the out-fashionedgoaf plentiful coal resources will be wasted due to limita-tions on the applicability of these methods

Analyzing the stability of the system composed of coalpillars and suspended roofs in the out-fashioned goaf ispreferred to determine the feasibility of mining the upperresidual coal seam above the out-fashioned goaf A largenumber of scholars have been studying these relative issues+e residual coal pillar and its roof in the lower out-fash-ioned goaf were simplified to the mechanical model of beamand plate by using elastic theory In the reference thesuspended roof was simplified as a simply supported beamand clamped beam [12] And the limited spans were cal-culated to judge the feasibility of mining the upper coalabove the out-fashioned goaf If the maximum length of thesuspended roof exceeds the limit span the suspended roofwould be judged to be unstable Xie et al [17] established amath analysis model of coal pillars and suspended roofs ingoaf to study the out-fashioned goaf stability throughresearching the bearing character of the coal pillar +esuspended roof above the lane mining goaf was simplified asa consecutive deep seammechanical model under uniformlydistributed load [18] From the mechanical model twofailure modes of antishearing and antistretching exist andthe suspended roof stability determines the feasibility ofupward mining He et al [19] established the interactionsystem model of ldquocoal pillar-roofrdquo In the system the hardroof and coal pillar were respectively regarded as elasticslabs and continuous support spring A strain-softeningmodel considering the damage constituent was established

to describe coal pillar mechanical characteristics Further-more the model was utilized to study the instabilitymechanism of the ldquocoal pillar-roofrdquo interaction system ingoaf Finally mechanical conditions and system stabilitywere confirmed according to the engineering condition ofMajiliang Coal Mine

Furthermore the influence of abutment pressure on theupper coal seam has been considered in upward mining Liet al [20] studied the damage evolution and failure mech-anism of the suspended floor under concentrated load anduniformly distributed load and put forward a floor rockfracture criterion Based on the experiment and field mea-surement in Qiuci Mine Wu et al [21] verified that abut-ment pressure induced by upper coal seam mining hadspecific effect on the stability of the lower goaf suspendedroof in upwardmining Stability conditions of the suspendedroof in lower goaf were calculated through simplifyingabutment pressure as uniformly distributed constant load[22] Based on elastic-plastic theory and practical engi-neering condition in Shirengou Iron Mine China a two-dimensional fixed beam mechanical model was built toanalyze the stability of bi-level goafs and isolated pillars Andthe result shows that a variety of stress transmit caused byincomplete overlapping of bilevel goafs gives rise to dif-ference in different isolated pillar zones causing stress anddisplacement mutation of goafs

In the above studies beam or slab models were built tostudy the stability of the system consisting of suspendedroofs and coal pillars And abutment pressure induced byupper coal seam mining was simplified to static additionalload +e system stability of the ldquosuspended roof and coalpillarrdquo is a criterion determining whether the upper coalseam could be mined

+ese research studies solve the problem of upwardmining to some extent and promote the recovery and uti-lization of upper coal resources above the out-fashioned goaf[23] However coal mining is a dynamic process With theadvance of the upper coal seam working face its abutmentpressure will change periodically +e abutment pressureinduced by upper coal seam mining simplified to staticadditional load is a gross oversimplification +e destructiveeffects on the coal pillar and suspended roof caused byshifting variable abutment pressure were ignored +is maylead to a decline in the accuracy of stability judgment whichmay cause the judgment to be inconsistent with the actualengineering conditions leading to the occurrence of sus-pended roof collapse accident causing damage to personneland equipment in upper coal seam mining

In order to ensure the safety of upper residual coal abovethe out-fashioned goaf a method for determining the fea-sibility of upward mining considering upward mining effectswas proposed first +e method is used to judge the feasi-bility of upward mining by judging the stability of the out-fashioned goaf +en taking the coal pillar and suspendedroof in out-fashioned goaf as an integral object a stabilitymechanical model focusing on the variable load induced byabutment pressure of upper coal seam mining was estab-lished And the parameter acquisition method of the modelwas established +irdly the basic parameters of Baizi Coal

2 Advances in Civil Engineering

Mine were considered to determine the feasibility of upwardmining residual coal above the out-fashioned goaf And theeffects of variable load on the coal pillar and suspended roofstability were analyzed

2 Engineering Situations

21 Geological Conditions As shown in Figure 1 Baizi CoalMine is located in Xunyi County Shaanxi Province +ewidth of the coal mine is 15 kilometers from west to eastand the length is 20 kilometers from south to north [17]

As shown in Figure 2 No 5 coal seam and No 8 coalseam are commercial seams in Baizi Coal Mine +e coalseams and their roof and floor occurrence conditions arefollowed+e coal bed pitches from 1deg to 5deg+e buried depthof No 5 coal seam is 115m to 190m +e thickness of No 5coal seam is from 08m to 25m and the average thickness is16m And the thickness of No 8 coal seam is from 08m to55m and the average thickness is 32m +e distance be-tween No 5 coal seam and No 8 coal seam is from 20m to40m and its average distance is 25m +e detailed roof andfloor occurrence condition distributions are shown inFigure 2

22 Physical and Mechanical Parameters +e physical andmechanical parameters of the rock strata of the roof andfloor are essential for determining the feasibility of as-cending mining +e elasticity volume weight cohesionand tensile strength of lithology are acquired in Table 2

23 Mining Conditions For historical reasons No 8 coalseam has been mined using the reamer-pillar coal miningmethod which is one of the out-fashioned mining methods+e 4m wide coal was mined and 4m wide coal pillars wereleft in No 8 coal seam with this method But the coal pillar isirregular in size and shape (Figure 3) +e mining height ofNo 8 coal seam is 24m and the coal resources of No 8 coalseam have been used up So No 5 coal seam was planned tobe mined +e longwall mining method was planned toutilize to mine No 5 coal seam above No 8 coal seam out-fashioned goafs +e mining engineering plan is shown inFigure 3

3 The Determination Method

Briefly the method is to judge the feasibility of upwardmining by judging the stability of lower out-fashioned goafsconsisting of the coal pillar and their suspended roof +estability of the coal pillar and suspended roof is determinedby using the theoretical analysis method based on structuralmechanics and elastic theory [24] Firstly the pressure above

Table 1 Stability determination methods of the upward mining of longwall goaf

Methods Calculation formula and application condition

Rational analytical method [6 7] Kgt 6(single seam)Kgt 63(multiple seam)

Mathematical statistics method [8] Hgt 114h2 + 414 + h

+e ldquothree-zonerdquo method [6 9ndash13]

Caving zone 1 Hc (h minus Ss)(b minus 1)

Caving zone 2 Hc 1000h(c1h + c2)

Fracture zone 1 Hf 1000h(c3h + c4)

(here (a) strong and hard c1 21 c2 16 c3 12 c4 2 (b) medium strong c1 47 c2 19c3 16 c4 36 (c) soft and weak c1 62 c2 32 c3 31 c4 5)

Fracture zone 2 Hf m + 1113936nminus1i1 hi

(here m is the mining thickness and hi is the stratum thickness i)

+emethod of equilibrium surroundingrock [14ndash16]

HP M(K1 minus 1) + hP

(here K1 M and hp are respectively the bulking coefficient mining thickness and equilibriumsurrounding rock thickness)

Figure 1 Location plan of Baizi Coal Mine

Lithology Rock type

Medium sandstone 19

Mudstone 40

Charcoal mudstone 05

5 coal seam 16


Mudstone 36

Siltite 67


8 coal seam 32


Alumina mudstone 60

Thickness (m)

Figure 2 Bar chart of coal seam geology of the coal seam

Advances in Civil Engineering 3

the suspended roof is simplified as a constant load and theabutment pressure during mining the upper coal seam issimplified as a triangular distributed load which is the maindifference with other methods +en the mechanical modelis established by using the theoretical analysis method themodel is utilized to study the stability of the system con-sisting of the coal pillar and their suspended roof of the lowerout-fashioned goaf +e feasibility of mining the upper re-sidual coal seam above the out-fashioned goaf is determinedbased on unstable conditions under combined action

31 Model Establishment Figure 4 provides a profile alongthe advancing direction of the upper longwall working faceaccording to Figure 3 It indicates the relationship amongthe upper coal seam lower residual coal pillar and sus-pended roofs [25] +e suspended roofs bear the over-burden load and varied distributed load induced by uppercoal seam mining An integral system model of the residualcoal pillar and suspended roof is built to study the stabilityof the system model under combined action of overburdenload and the lead abutment pressure of upper working faceIn the system model the roof is simplified as a girder whichcan bear axial force shearing force and bending momentand coal pillar as elastic fixed support that only bears axialforce overburden load as uniformly distributed constantload and the lead abutment pressure of upper working faceas the triangular distributed load which is varied in lengthand scale It is supposed that the coal pillar only bears axialforce and the simplified mechanical model of the sus-pended roofs and coal pillars is obtained (shown inFigure 4(b))

32 Model Solution +e model was solved by using me-chanics of materials and structural mechanics [24 26] +esolution procedures are as follows first the results bearingonly overburden load are calculated and then the resultsbearing only varied triangular load are calculated finally thefinal results utilizing the superposition principle arecalculated

+e structure and load of the model are symmetricalwhen only overburden load acts Half of the model was usedto simplify the calculation and solve the mechanical model+e model before and after simplification is shown inFigure 5 According to the potential failure form of the coalpillar and suspended roof the axial force and shearing forceof the coal pillar and its roof need to be solved So based onthe static equilibrium moment equilibrium and deforma-tion coordination conditions the displacement equation ofthe simplified model is obtained as follows

X1lprime3

3E2I2+ X2

lprime2

2E2I2+

qL2l2

8E2I2

X1lprime2

2E2I2+ X2

l

E2I2+

lprimeE2I2

1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where X1 is the axial force of the suspended roof MZ is thebending moment of the suspended roof X3 is the maximalshearing force of the suspended roof FN is the axial force ofthe coal pillar I1 is the inertia moment of the suspendedroof I2 is the inertia moment of the coal pillar L is thedistance of two adjacent pillars lʹ is the height of the pillar qis the load intensity of the overburden load of the suspended

Table 2 Physical and mechanical parameters of lithology

Lithology Elasticity modulus E(GPa)

Volume weight ρ(kgmiddotmminus3)

Cohesion C(MPa)

Tensile strength σt(MPa)

Friction angle φ()

No 5 coal seam 51 1400 30 12 30Charcoalmudstone 103 2483 55 41 32

Mudstone 124 2369 35 35 35Siltite 145 2534 122 51 38Charcoalmudstone 171 2463 36 32 32

No 8 coal seam 55 1430 28 13 29

The lower coal pillarsThe roadway of upper coal seam

Figure 3 Mining engineering plan schematic map


roof qprime is the load intensity of the varied triangular load anda is the range of the varied triangular load

+e simultaneous equation (1) is solved and the resultsare as follows

X1 qE2I2L

3

8E1I1lprime2

minus 4E2I2lprime1113872 1113873

X2 qL

3E2I2

24E1I1lprime2

minus 12E2I2lprime1113872 1113873

(2)

According to the force balance in the vertical directionutilizing the above results the axial force of the coal pillarFNC is calculated using the following formula

FNC qL

2 (3)

As shown in Figure 6 when the model only bears variedtriangular load the models before and after simplificationare given In order to obtain the axial stress bending mo-ment and shearing stress of the suspended roof the dis-placement equation of the simplified model is obtained asfollows

X3prime L3

24E1I1+

L2lprime

8E2I21113888 1113889 0

FNC X3prime +qprimea2

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(4)

where X3 is the maximal shearing force of the suspendedroof and Frsquo

NC is the axial force of the coal pillar underuniform load

+e simultaneous equations (4) are solved and the re-sults are shown below

X3prime 2E1I1qprimeaL

2lprime

3E1I1L + 8E2I2lprime( 1113857L2

1113872 1113873

FNCprime 2E1I1qprimeaL

2lprime

3E1I1L + 8E2I2lprime( 1113857L2

1113872 1113873

1113868111386811138681113868111386811138681113868111386811138681113868+ qprimea

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

+en the methods of moment distribution are utilized toobtain the torque equilibrium equation to solve the mo-ments of coal and the suspended roof

M1L

2E1I1+

lprimeE2I2

1113888 1113889 + M2lprime

E2I2

qprimeaL(L minus a)

32E1I1+

qprimelprimea(L minus a)

16E2I2

M1lprime

E2I2+ M2

L

2E1I1+

lprimeE2I2

1113888 1113889 qprimealprime(L minus a)

8E2I2

M1 2MZR 2MZL

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

where MZR is the bending moment of the right suspendedroofMZL is the bending moment of the left suspended roofand MM is the bending moment of the coal pillar

+e simultaneous equations (5) and (6) are solved andthe result is shown as follows

MZL qprimea(L minus a) E

22I

22L + 4E

21I

21lprime1113960 1113961

2 8E22I

22L

2+ 32E1E2I1I2Llprime1113872 1113873

MZR qprimea(L minus a) E

22I

22L

2+ 4E

21I

21lprime1113960 1113961

8E22I

22L

2+ 32E1E2I1I2Llprime1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

As shown in Figure 6 we calculate the stress bendingmoment and shearing force of the suspended roof when itbears inverse varied triangular load +e mechanics equi-librium equations are given as shown below

X3Prime FNPrime

qprimea2

MMPrime

qprimea(L minus a)

4

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(8)

where X3Prime is the maximal shearing force of the rock stratumof the roof FN

Prime is the axial force of the coal pillar under

The upper longwall goaf

The residualcoal pillar


Suspended roof

(a)

The residual coal pillar

a2L

qprime q Suspended roof

(b)

Figure 4 Diagrammatic figure and simplified mechanical model

The suspended roof

The suspendedroof


2L

Lq

q


Structuresimplified

Figure 5 Model simplification bearing only overburden load


triangular load and MMPrime is the bending moment of the coal

pillarSo far the mechanical equation of the coal pillar and the

suspended roof in the situation of bearing variable triangularload is obtained According to equations (2) (5) (7) and (8)the superposition principle is utilized to calculate the resultsof the system model

X1 qE2I2L3

8E1I1l2primeminus4E2I2Llprime

X3 2qprimeaE1I1L

2lprime

3E1I1L3

minus8E2I2L2lprime

+qL

2+

qa

2

FN 2qprimeaE1I1L

2lprime

3E1I1L3

minus8E2I2L2lprime

+qL

2+ qprimea

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

MZ qL

3E2I2

24E1I1lprime minus12E2I2L+

qprimea(L minus a)L2E22I

22 +4lprime2E2

1I21

8E22I

22L

2+32E1E2I1I2Llprime

MM qprimea(L minus a)

4+


22 +4lprime2E2

1I21

16E22I

22L

2+64E1E2I1I2Llprime

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(10)

where MZ is the moment of the suspended roof MM is themoment of the coal pillars and FN is the combined axialforce of coal pillars

In order to determine the stability of the model themaximal axial stress and shearing stress should be cal-culated [26] Considering the bending stress derivedfrom force and bending moment the maximal axial stressand shearing stress are acquired using the followingequation

σmax Mmax

W

τmax FQSlowastZmax

IZb

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)

where σmax is the maximal axial stress τmax is the maximalshearing stress W is the section modulus in bending mo-ment of the suspended roof W bh36 Mmax is the max-imal moment of beam such as the moment of the suspendedroof MZ FQis the shearing force of the beam the suspendedroof X3 SlowastZmax is the static moment of the suspended roofSlowastZmax bh2 and Iρ is the moment of inertia of the sus-pended roof

+e stress of axial and shearing is correlated with themoment of the beam and the detailed relationship isshown in formula (11) +e moment of the suspended roofis the moment of the beam +e axial stress and shearingstress of the suspended roof and axial stress of the coalpillar (formula (12)) are derived from formulas (9) (10)and (11)

τRmax

2E1I1qprimeaL(L minus a)lprime

3E1I1L3

+ 8E2I2L2lprime1113872 1113873b1h1

+qL

2b1h1+

qa

2b1h1

σRmax

1b1h1

qL3E2I2

4E1I1lprime minus 2E2I2L( 1113857b1+

qE2I2L3

lprime 8E1I1lprime minus 4E2I2L( 1113857b1+3qprime(L minus a)a L

2E22I

22 + 12lprime2E2

1I211113872 1113873

4E2I2L E2I2L + 16E1I1lprime( 1113857b1

⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

σCmax

1b2h2

3qprimea(L minus a)

2b2h2+

qprimea(L minus a) 3E22I

22L

2+ 12E

21I

21lrsquo

21113872 1113873

E2I2L 8E2I2L + 32E1I1lprime( 1113857b2h2+

2qprimeaE1I1L2lprime

3E1I1L3

+ 8E2I2L2lprime1113872 1113873

+ qprimea + qL⎛⎝ ⎞⎠ (13)

The suspended roof

The suspended roof


2L

La

a


Structure simplified

qprime

qprime2

Figure 6 Model simplification bearing only varied triangular load


where τRmax is the maximal shearing stress of the suspended

roof σRmax is the maximal normal stress of the suspended

roof b1 is the width of the transverse section of the sus-pended roof h1 is the height of the transverse section of thesuspended roof σC

max is the bearing normal stress of the coalpillar b2 is the width of the transverse section of the coalpillar and h2 is the height of the transverse section of the coalpillar

+e shearing stress of the coal pillar has little influenceon the strength stability of the out-fashioned goaf So whenwe utilized the abovementioned results of the system sta-bility of the coal pillar and the suspended roof the effect ofshearing stress on the coal pillar can be ignored

33 Model Parameters Determination Method +e basicparameters of the determination model are acquired byengineering geologic reports operational rules and rockmechanical tests In addition the suspended roof load andcompression strength of the coal pillar are determined by thefollowing methods [27]

331 Suspended Roof In the system model the suspendedroof is a key rock stratum +e suspended roof is the mainstructure that bears overburden load and abutment loadAccording to the theory of the key rock stratum [28] the keyrock stratum is the stratum bearing the overburden load and

varied distributed load induced by upper coal seam mining+e stability of the key rock stratum is a key question tostudying ascending mining feasibility So the identificationof the key rock stratum of No 8 coal seam should beimplemented According the theory of key stratum themethod of identifying the key rock stratum is given in thefollowing formula

q1(x)n E1d

31 1113936

ni1 ρidi

1113936ni1 Eid

3i

q1(x)n q1(x)n+1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(14)

where q1(x)n is the load of the nth stratum q1(x)n+1 is theload of the (n+1)th stratum E1 is the elasticity modulus ofthe first stratum Ei is the elasticity modulus of the ithstratum di is the thickness of the first stratum and ρi is thedensity of the ith stratum

332 Load +e loads of overburden and abutment pressureare the key parameters [5] Solving the vital parameters hastwo steps Firstly the pressure of overburden and abutmentshould be obtained +e methods of obtaining the pressureare similar to the stimulated test and field measurements[9 29] And then the load can be calculated by

q Pq times L

qprime qz times L

⎧⎪⎨

⎪⎩(15)

qz 0637 PQ minus Pq1113872 1113873 arctanx0l0

z

4x04

+ l02

+ 4z2

1113969 +l0z

4x04

+ l02

+ 4z2

1113969

minus

l02

+ 4z2

1113969

1113874 1113875

2x0 l02

+ 4z2

1113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)

where q is the load of the overburden qprime is the load of theabutment pressure Pq is the mine ground pressure whenmining upper coal seam PQ is the mine ground pressurewhen periodic weighting of ascending mining occurs x0 andl0 respectively are the width and length of the load ofoverburden and z is the depth of floor rock strata

333 Compression Strength of Coal Pillar +e compressionstrength of the coal pillar correlates with uniaxial com-pressive strength and coal pillar size [30] +e uniaxialstrength of the laboratory coal sample was converted into theuniaxial strength of the critical cube by using the Hustrulidmethod +e Hustrulid method estimates the size effect ofthe coal pillar in the field +e formula of the Hustrulidmethod is shown below

σm σc

D

d

1113970

(17)

where D is the sample diameters d is the sample height σcisthe compression strength of the coal sample and σmis theuniaxial strength of the critical cube

Furthermore the strength of the coal pillar in the field iscalculated using the relationship of uniaxial strength of thecritical cube and coal pillar [31] +e representative strengthconverting formula is as follows

σs1 σm 0778 + 0222W

h1113874 11138751113876 1113877

σs2 σm

W0446

h066

σs min σs1 σs21113864 1113865

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

whereW is the width of the coal pillar h is the height of thecoal pillar and σs is the uniaxial compression strength of thecoal pillar


4 Results and Discussion

41 Parameters +e study data of Baizi Coal Mine [32] areutilized to verify the validity of theoretical formulas in thefollowing According to engineering situation the funda-mental data are followed +e distance of two adjacent coalpillars L is 4m the mining height of No 8 coal seam lprime is24m Physical and mechanical parameters derived fromTable 2 are shown in Table 3

+e key rock stratum of No 8 coal seam the load ofoverburden and abutment pressure and uniaxial com-pression strength of the coal pillar are vital for the feasibilitydetermination of mining upper coal seam So the vitalparameters of the suspended roof load and coal pillar aresolved in the following

Firstly combining the geological data in Table 1 and thekey stratum determination formula (13) the key stratum isdetermined +e cohesion of siltite in No 8 coal seam roof is122MPa the tensile strength is 51MPa+e siltite is the keyrock strata So the siltite is the suspended roof Parametersof the key rock strata were determined +e thickness ofsiltite h1 is 67m So the parameters of the suspended roofare shown in Table 3

Furthermore the vertical pressure data of No 5 coalseam floor measured by stress sensor are collected andanalyzed We find that the pressure of No 5 coal seam flooris equal when the first pressure is approaching the verticalpressure Pq is 267MPa No 8 periodic pressure significantlyoccurs in the procedure of mining upper coal seams +emaximum pressure PQ is 43MPa According to formula(14) the load intensities q and qprime are calculated in Table 3

Finally the compression strength of the coal pillarcorrelates with uniaxial compressive strength and coal pillarsize +e test specimens are round specimen diameter isD 5 centimeters and specimen height is d 10 centimeters+e laboratory experiment was carried out on the coalsample of No 8 coal seam to get the uniaxial compressivestrength σc 157MPa [17] +e uniaxial strength of thecritical cube is converted by using formula (17) and formula(18) +e compression strength of the coal pillar σs is118MPa

42 Judgment +e maximal shearing stress of the sus-pended roof is τR

max 1595MPa and it is under the shearingstrength of suspended roof [τ] 892MPa +e maximaltensile stress of the suspended roof is σR

max 398MPa and itis under the uniaxial tensile strength of the suspended roof[σt] 51MPa +e maximal axial stress of the coal pillar isσCmax 102MPa and it is under the axial strength of the coal

pillar [σ2] 111MPa +ese data mean that the systemstrength of the suspended roof and coal pillar is more thanthe stress on the system Hence the system of the coal pillarand suspended roof is stable which is determined by thismethod And the results corresponded with the practicalengineering projects

43 Effects of Variable Load +e varying values of ldquoaLrdquo areused to measure the influence of load movement on axialforce and shear force in the model +e range of varied

triangular load ldquoardquo is associated with the distance of theadjacent coal pillar According to formulas (11) and (12) andthe Baizi coal minersquos conditions when ldquoaLrdquo is 01 02 03 1 the effect of the varied triangular load for the maximalaxial stress and shearing stress of determined formulas onsystem stability is analyzed With varying ldquoaLrdquo the maximalstress of rock and coal pillar is shown in Figure 7

As shown in Figure 7 the variation trend of axial stress ofthe residual coal pillar and suspended roof is similar As ldquoaLrdquo increases these stresses first go up and then go down Butwith the increase of ldquoaLrdquo the peak value of shear stress inthe coal pillar appears later than that in the suspended roofSynchronous changes in the position and size of variableloads are the reason

As shown in Figure 7 in order to analyze the effects ofthe varied load we used (q + qprime) as the uniform load cal-culation stress of the model when ldquoaLrdquo 0 +e maximalshear and axial stress of the suspended roof as well as themaximal axial stress of the coal pillar are calculated as107MPa 0026MPa and 36MPa Without considering thevaried load the maximal stress of the model is much smallerthan when considering the varied load +is indicates thatwhether the load movement considered has a great influenceon the stress when the load value is the same Obviously theresults obtained by this method are more secure

In general the compressive strength of coal and rockmass is vital in all strength [33] +e shearing strength isgreater than the tensile strength Furthermore the shearingstress of the suspended roof is small and increases slowly Sothe suspended roof is not subject to shear failure Whetherthe suspended roof tensile failure or coal pillar compressionfailure occurs firstly +e calculated stress in the pillar isgreater than that of the rock However for coal and rockmass failure sequence cannot be ensured from the curves Itis determined by the relationship between the tensilestrength of the suspended roof and the compressive strengthof the coal pillar

431 Load Range Effect for Suspended Roof From practicalascending mining engineering project cases [19 21 22] it isnot difficult to find that the system strength stability of theresidual coal pillar and suspended roof can be mainlyinfluenced by the parameters ldquoLrdquo and ldquoaLrdquo Hence theregulations of system stability are studied through thecontrol variables approach +e influence of regulations ofthe stability of the suspended roof is studied in detail firstlyin the following

According to formulas (11) and (12) and Table 3 therelationship of stress of the suspended roof and coal pillarwith ldquoaLrdquo and ldquoLrdquo was analyzed When L 4 8 12 1620meters and ldquoaLrdquo is 01 02 10 the shear stress andaxial stress of the suspended roof and coal pillar are re-spectively calculated and displayed in Figures 8ndash10

As shown in Figure 8 the shearing stress of the sus-pended roof has a positive correlation with ldquoaLrdquo When ldquoLrdquois larger the shearing stress curves of the suspended roofare steeper It indicates that when the length of the sus-pended roof is longer the system strength stability of the


suspended roof and coal pillar is lower whereas other factsare equal

In practical mining engineering projects when thelength of the suspended roof gets wider the system strengthstability will be destroyed more easily And the shearingstress of the suspended roof increases with the increase of thevaried load range on the suspended roof Based on theabovementioned analysis results engineers in coal mine can

calculate the maximal length and the varied load range todetermine whether the suspended roof can be fractured

+e relationship of axial stress of the suspended roofwith ldquoaLrdquo is shown in Figure 9 +e curves show a parabolicshape with downward opening For different ldquoLrdquo the axialstress curves are higher and steeper with increasing ldquoLrdquo Allaxial stress curves of the suspended roof have a positivecorrelation with ldquoLrdquo When ldquoLrdquo increases the curves con-verge more It is indicated that the axial stress increases moreand more quickly than the shear stress For all curves inFigure 9 when ldquoaLrdquo is about 05 the axial stress of thesuspended roof is the maximum For practical engineeringproject cases when ldquoaLrdquo is about 05 the stability of thesystem of the coal pillar and suspended roof will bedestroyed more easily Hence this situation should beavoided in the process of ascending mining

432 Load Range Effect for Coal Pillar +e relationshipcurves of axial stress of the coal pillar with ldquoaLrdquo are shown inFigure 10 +e curves show a parabolic shape with down-ward opening which have a similar trend with the curves ofaxial stress of the suspended roof +e axial stress of the coalpillar has a positive correlation with the value of ldquoLrdquo Hencethe distance of two adjacent coal pillars should be givenpriority consideration

Furthermore when all values of ldquoaLrdquo is about 05 theaxial stress curves reach maximum +ese results can in-struct practical engineering operations For practical miningengineering projects when ldquoaLrdquo equals to about 05 thestrength stability of the system of the suspended roof and

L = 4

Shea

ring

stres

s of s

uspe

nded

roof

(MPa

)

8

6

4

2

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10

Figure 8 Shearing stress of the suspended roof with ldquoaLrdquo

L = 4A

xial

stre

ss o

f sus

pend

ed ro

of (M

Pa)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10

Figure 9 Axial stress of the suspended roof with ldquoaLrdquo

Table 3 Detailed values of parameters in Baizi Coal Mine

qprime (MNm) q (MNm) lprime (m) L (m) b1 (m) b2 (m) h1 (m) h2 (m) I1 (m4) I2 (m4) E1 (GPa) E2 (GPa)372 1068 24 4 4 4 67 4 133 53 14 55

100

75

50

25

00

00 02 04aL

Stre

ss (M

Pa)

06 08 10

Roof rsquos shearing stress under (q + qprime) or varied loadRoof rsquos axial stress under (q + qprime) or varied loadCoal pillarrsquos axial stress under (q + qprime) or varied load

Figure 7 +e relationship between stresses and ldquoaLrdquo in Baizi CoalMine


coal pillar is used to determine the feasibility of upwardmining above the out-fashioned goaf

433 Load Magnitude Effect for the System Model With allother conditions being equal we changed the value ofvariable load qprime ( such as 02q 04q 06q 08q and 10q) Andthe stress curves of the system model are derived with thevarying ratio of qprimeq (Figure 11)

Figure 11 shows that the ratio of qprimeq has a linear re-lationship with all stresses of the system model +e slope ofshearing stress curves is the smallest +e ratio of qprimeq hasleast effect on shearing stress of the suspended roof which isconsistent with the conclusion that shearing force does notaffect roof failure [34] When the ratio of ldquoaLrdquo is the samethe slopes of the curves between axial stress of the suspendedroof and coal pillar are very close indicating that the axialforce of the suspended roof and coal pillar increases nearlysynchronously with the change of ratio of qprimeq +e failuresequence of the system model depends on the relationshipbetween the tensile stress of the suspended roof and thecompressive strength of the coal pillar According to thelinear relationship shown in Figure 11 the failure sequenceof the system model is calculated using the followingformula

σs

σt

fqprimeq

1113888 1113889 kc qprimeq( 1113857 + σsI

kr qprimeq( 1113857 + σtI

(19)

where σsI and σtI are respectively the axial stress of thesuspended roof and coal pillar when qprime equals to zero kC andkr are respectively axial stress curve slopes of the suspendedroof and coal pillar σS is the compression strength of thecoal pillars and σt is the tensile strength of the suspendedroof

For cases of varying ratio of (qprimeq) the relationshipbetween f(qprimeq) and the ratio of σSσt are compared to

determine whether the coal pillar or suspended roof will bedamaged first which is more advantageous for fieldtechnicians

5 Conclusions

A method for determining the feasibility of upward miningresidual coal above the out-fashioned goaf was studied andthe following conclusions were obtained

(1) Variable additional load is mainly considered tosimulate the influence of abutment pressure causedby upper coal seam mining On this basis the sus-pended roof and coal pillar in the lower out-fash-ioned goaf are taken as objects to establish themechanical model +e mechanical model solutionand parameter acquisition method were alsoestab-lished +e practical engineering data of Baizi CoalMine were utilized to analyze the relationship ofstress with ldquoaLrdquo and ldquoLrdquo

(2) Compared with the system without variable load theultimate stresses obtained by the system modelconsidering the variable load are much higher+erefore compared with this method the tradi-tional method which simplifies upper loads touniform loads leaves a greater safety factor to ensurethe safe mining of the upper coal seam above the out-fashioned goaf

(3) +rough the analyses of determining formula of keyparameters the system stability of the suspendedroof and coal pillar is influenced by ldquoaLrdquo and ldquoLrdquoand the effect of ldquoaLrdquo is significant for axial andshearing stress Axial stress curves of the coal pillarand suspended roof appear nearly parabolic with

Roof rsquos shearing stressaL = 01

Roof rsquos shearing stressaL = 05Roof rsquos axial stressaL = 01Roof rsquos axial stressaL = 05

Coal pillarrsquos axial stressaL = 01Coal pillarrsquos axial stressaL = 05

16

12

Stre

ss (M

Pa)

8

σs = kc (qprimeq) + σsIσt = kr (qprimeq) + σtI

4

0

qprimeq02 04 06 08 10

Figure 11 +e relationship of the stress ratio qprimeq

L = 4

Axi

al st

ress

of s

uspe

nded

roof

(MPa

)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10

Figure 10 Axial stress of the residual coal pillar with ldquoaLrdquo


varying ldquoaLrdquo +eir maximum values are obtainedwhen the values of ldquoaLrdquo are 05sim06 In this situa-tion the combination system consisted of the coalpillar and suspended roof is most easy to be dam-aged With the increase of the distance L between thetwo adjacent coal pillars the stress of the modelincreases and the axial stress of the coal pillar in-creases the fastly +e ratio of qprimeq has a linear re-lationship with all stresses of the system model +efailure sequence of the system model is determinedby analyzing the relationship between the tensilestrength of the suspended roof and compressivestrength of the coal pillar

(4) Under the action of variable load caused by No 5coal seam mining in Baizi Coal Mine the suspendedroof and coal pillar of the lower out-fashioned goafremain stable which is consistent with the results ofengineering practice +is study provides a referencefor the mining of coal resources under similarconditions

Data Availability

+e method was validated by the data provided in Ref [32]

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is work was funded by the Joint Research Fund undercooperative agreement between the National Natural Sci-ence Foundation of China (NSFC) and Funds for Coal-Based Low-Carbon Technology of Shanxi (No U1710258)Shanxi Province Applied Basic Research Program (No201801D221342) the National Natural Science Foundationof China (Nos 51422404 and 51574172) Science andTechnology Innovation Project of Shanxi Colleges andUniversities (2019L0181) Program for Key Research Projectof Shanxi Province in the Field of Social Development (No201803D31044) Yanrsquoan Industry Key Core TechnologyInnovation Project (2019ZCGY-025) Program for ldquo1331rdquoKey Team of Scientific and Technology Innovation in ShanxiProvince Program for the Excellent Innovation Team ofHigher Learning Institutions of Shanxi Province Programfor the Leading Talents of Emerging Industry of ShanxiProvince and Shanxi Province Technology InnovationProgram for the Key Team

References

[1] Y Zhang G Feng M Zhang et al ldquoResidual coal exploitationand its impact on sustainable development of the coal in-dustry in Chinardquo Energy Policy vol 96 pp 534ndash541 2016

[2] G R Feng Y J Zhang T Y Qi and L X Kang ldquoStatus andresearch progress for residual coal mining in Chinardquo Journalof China Coal Society vol 45 pp 151ndash159 2020

[3] Y J Zhang G R Feng and T Y Qi ldquoExperimental researchon internal behaviors of caved rocks under the uniaxial

confined compressionrdquo Advances In Materials Science AndEngineering vol 2017 Article ID 6949264 8 pages 2017

[4] L Tang and S Liang ldquoStudy on the feasibility of the upwardmining above the goaf of the irregular roadway mining underthe influence of hard and thick stratardquo Geotechnical andGeological Engineering vol 37 no 6 pp 5035ndash5043 2019

[5] D F Zhu X M Song H C Li Z H Liu C Wang andY M Huo ldquoCooperative load-bearing characteristics of apillar group and a gob pile in partially caved areas at shallowdepthrdquo Energy Science amp Engineering vol 8 no 1 2020

[6] W B Sun ldquo+e ascending mining technology of closedmultiple-seams with a large mining depthrdquo Electronic Journalof Geotechnical Engineering vol 21 pp 6337ndash6346 2016

[7] L Wang ldquo+e technology and application of ascendingmining in coal seamsrdquo in Proceedings of the 21st CenturyEfficient Intensive Mine Symposium pp 76ndash79 ShaowuChina June 2012

[8] G R Feng Basic Deory and Practice of Upward Mining inCoal Mine Residual Area Taiyuan University Of TechnologyBeijing China 2010

[9] S S Peng Longwall Mining p 818 Springer Berlin Ger-many 2014

[10] M Bai F Kendorski and D Roosendaal ldquoChinese and NorthAmerican high-extraction underground coal mining stratabehaviour and water protection experience and guidelinesrdquo inProceedings of the 14th International Conference on GroundControl in Mining pp 209ndash217 Morgantown WV USAAugust 1995

[11] Y Xu J Li S Liu and L Zhou ldquoCalculation formula of ldquotwo-zonerdquo height of overlying strata and its adaptability analysisrdquoCoal Mining Technology vol 16 pp 4ndash11 2011

[12] W Guo G Zhao G Lou and S Wang ldquoA new method ofpredicting the height of the fractured water-conducting zonedue to high-intensity longwall coal mining in Chinardquo RockMechanics and Rock Engineering vol 52 pp 2789ndash2802 2018

[13] D Zhang W Sui and J Liu ldquoOverburden failure associatedwith mining coal seams in close proximity in ascending anddescending sequences under a large water bodyrdquoMine WaterAnd the Environment vol 37 no 2 pp 322ndash335 2018

[14] F Xue ldquoFeasibility study on ascending mining of near coalseam grouprdquo Coal Mine Modernization vol 1 pp 19ndash212013

[15] S F Wang X B Li J R Yao et al ldquoExperimental investi-gation of rock breakage by a conical pick and its application tonon-explosive mechanized mining in deep hard rockrdquo In-ternational Journal Of Rock Mechanics And Mining Sciencesvol 122 Article ID 104063 2019

[16] K Du C Z Yang R Su M Tao and S F Wang ldquoFailureproperties of cubic granite marble and sandstone specimensunder true triaxial stressrdquo International Journal Of Rock Me-chanics And Mining Sciences vol 130 Article ID 104309 2020

[17] X Z Xie ldquoStudy on stability of roof-coal pillar in room andpillar mining goaf in shallow depth seamrdquo Coal Science andTechnology vol 42 pp 1ndash9 2014

[18] Y D Jiang Y M Yang Z Q Ma and Y W Li ldquoBreakagemechanism of roof strata above widespread mined-out areawith roadway mining method and feasibility analysis of up-ward miningrdquo Meitan XuebaoJournal of the China CoalSociety vol 41 pp 801ndash807 2016

[19] G L He D C Li Z W Zhai and G Y Tang ldquoAnalysis ofinstability of coal pillar and stiff roof systemrdquo Journal of theChina Coal Society vol 32 pp 897ndash901 2007

[20] H L Li Study on Coal Seam Floor Failure Rule under MiningDynamic Loading Effect and Control Technique of Water-


Inrush China University of mining and technology XuzhouChina 2016

[21] B Y Wu Z G Deng Y F Feng and F Li ldquoAnalysis of theinfluence of interlayer rock on ascending mining underspecial conditionsrdquo Journal of the China Coal Society vol 42pp 842ndash848 2017

[22] J X Fu W D Song and Y Y Tan ldquoStudy on influence ofoverlapping rate of double layer gobs on stability of isolationroof and its mechanical modelrdquo Journal of Mining and SafetyEngineering vol 35 pp 58ndash63 2018

[23] N Jiang C X Wang H Y Pan D W Yin and J B MaldquoModeling study on the influence of the strip filling miningsequence on mining-induced failurerdquo Energy Science amp En-gineering vol 8 no 6 pp 1ndash17 2020

[24] Y Q Long and S H Bao Structural Mechanics HigherEducation Press Beijing China 2000

[25] B-Y Jiang S-T Gu L-G Wang G-C Zhang and W-S LildquoStrainburst process of marble in tunnel-excavation-inducedstress path considering intermediate principal stressrdquo Journalof Central South University vol 26 no 4 pp 984ndash999 2019

[26] H W Liu and J X Lin Mechanics of Materials HigherEducation Press Bejing China 1998

[27] B H G Brady and E T Brown Rock Mechanics for Un-derground Mining Allen amp Unwin Crows Nest Australia1985

[28] M G Qian X X Miu and J L Xu ldquo+eoretical study of keystratum in ground controlrdquo Journal of China Coal Societyvol 3 pp 2ndash7 1996

[29] H Moomivand ldquoEffect of size on the compressive strength ofcoalrdquo in Proceedings of the 99th International Symposium onMining Science and Technology pp 399ndash404 Beijing China1999

[30] C X Wang B T Shen J T Chen et al ldquoCompressioncharacteristics of filling gangue and simulation of mining withgangue backfilling an experimental investigationrdquo GeomechEngineering vol 20 no 6 pp 485ndash495 2020

[31] Z T Bieniawali and H H Sun Strata Control in MineralEngineering China MiningUniversity Press Bejing China1990

[32] X P Shao E S Liu X L Cai J T Wu and J ZhangldquoExperimental study on stability of pillar in upward mining inpillar mined out areardquo Coal Engineering vol 50 pp 5ndash92018

[33] D Zhu Y Wu Z Liu X Dong and Y Jin ldquoFailuremechanism and safety control strategy for laminated roofof wide-span roadwayrdquo Engineering Failure Analysisvol 111 pp 1ndash14 2020

[34] QWang B Jiang LWang et al ldquoControl mechanism of rooffracture in no-pillar roadways automatically formed by roofcutting and pressure releasingrdquo Arabian Journal of Geo-sciences vol 13 p 10 2020


recovery rate of coal resources However the upper residualcoal distinguishes from the common coal in mining con-ditions +e upper residual coal is supported by pillars andsuspended roofs in the out-fashioned goaf When the upperresidual coal working face goes across the coal pillar or thesuspended roof it is very likely to cause the coal pillar in-stability or the suspended roof rupture causing the collapseof personnel or equipment Hence the stability of the coalpillar and suspended roof is the key technique problem inupper residual coal mining above the out-fashioned goafWe must ensure the stability of the lower coal pillar and itssuspended roof during mining upper coal seams

Many researchers analyzed the integrity of the upper coalseam and its surrounding rock stability proposing corre-sponding methods for determining the feasibility of upwardmining +ese methods mainly include the rational ana-lytical method the ldquothree-zonerdquo method mathematicalstatistics method and the method of equilibrium sur-rounding rock +eir determination formulas and applica-tion conditions are shown in Table 1

+ese methods are proposed after statistically analyzingthe field measured data which are derived from longwallmining conditions But the out-fashioned mining differ-entiates from the longwall mining In the out-fashioned goafcompared with the longwall goaf coal pillars with irregularsize and shape were left to support the roof which results ina limited roof caving or even no caving +e upper coal seamand its surrounding rock are integral and stable So if thesemethods are adopted to determine the feasibility of themining upper residual coal seam above the out-fashionedgoaf plentiful coal resources will be wasted due to limita-tions on the applicability of these methods

Analyzing the stability of the system composed of coalpillars and suspended roofs in the out-fashioned goaf ispreferred to determine the feasibility of mining the upperresidual coal seam above the out-fashioned goaf A largenumber of scholars have been studying these relative issues+e residual coal pillar and its roof in the lower out-fash-ioned goaf were simplified to the mechanical model of beamand plate by using elastic theory In the reference thesuspended roof was simplified as a simply supported beamand clamped beam [12] And the limited spans were cal-culated to judge the feasibility of mining the upper coalabove the out-fashioned goaf If the maximum length of thesuspended roof exceeds the limit span the suspended roofwould be judged to be unstable Xie et al [17] established amath analysis model of coal pillars and suspended roofs ingoaf to study the out-fashioned goaf stability throughresearching the bearing character of the coal pillar +esuspended roof above the lane mining goaf was simplified asa consecutive deep seammechanical model under uniformlydistributed load [18] From the mechanical model twofailure modes of antishearing and antistretching exist andthe suspended roof stability determines the feasibility ofupward mining He et al [19] established the interactionsystem model of ldquocoal pillar-roofrdquo In the system the hardroof and coal pillar were respectively regarded as elasticslabs and continuous support spring A strain-softeningmodel considering the damage constituent was established

to describe coal pillar mechanical characteristics Further-more the model was utilized to study the instabilitymechanism of the ldquocoal pillar-roofrdquo interaction system ingoaf Finally mechanical conditions and system stabilitywere confirmed according to the engineering condition ofMajiliang Coal Mine

Furthermore the influence of abutment pressure on theupper coal seam has been considered in upward mining Liet al [20] studied the damage evolution and failure mech-anism of the suspended floor under concentrated load anduniformly distributed load and put forward a floor rockfracture criterion Based on the experiment and field mea-surement in Qiuci Mine Wu et al [21] verified that abut-ment pressure induced by upper coal seam mining hadspecific effect on the stability of the lower goaf suspendedroof in upwardmining Stability conditions of the suspendedroof in lower goaf were calculated through simplifyingabutment pressure as uniformly distributed constant load[22] Based on elastic-plastic theory and practical engi-neering condition in Shirengou Iron Mine China a two-dimensional fixed beam mechanical model was built toanalyze the stability of bi-level goafs and isolated pillars Andthe result shows that a variety of stress transmit caused byincomplete overlapping of bilevel goafs gives rise to dif-ference in different isolated pillar zones causing stress anddisplacement mutation of goafs

In the above studies beam or slab models were built tostudy the stability of the system consisting of suspendedroofs and coal pillars And abutment pressure induced byupper coal seam mining was simplified to static additionalload +e system stability of the ldquosuspended roof and coalpillarrdquo is a criterion determining whether the upper coalseam could be mined

+ese research studies solve the problem of upwardmining to some extent and promote the recovery and uti-lization of upper coal resources above the out-fashioned goaf[23] However coal mining is a dynamic process With theadvance of the upper coal seam working face its abutmentpressure will change periodically +e abutment pressureinduced by upper coal seam mining simplified to staticadditional load is a gross oversimplification +e destructiveeffects on the coal pillar and suspended roof caused byshifting variable abutment pressure were ignored +is maylead to a decline in the accuracy of stability judgment whichmay cause the judgment to be inconsistent with the actualengineering conditions leading to the occurrence of sus-pended roof collapse accident causing damage to personneland equipment in upper coal seam mining

In order to ensure the safety of upper residual coal abovethe out-fashioned goaf a method for determining the fea-sibility of upward mining considering upward mining effectswas proposed first +e method is used to judge the feasi-bility of upward mining by judging the stability of the out-fashioned goaf +en taking the coal pillar and suspendedroof in out-fashioned goaf as an integral object a stabilitymechanical model focusing on the variable load induced byabutment pressure of upper coal seam mining was estab-lished And the parameter acquisition method of the modelwas established +irdly the basic parameters of Baizi Coal

























Lithology Rock type

Medium sandstone 19

Mudstone 40


5 coal seam 16


Mudstone 36

Siltite 67


8 coal seam 32


Alumina mudstone 60

Thickness (m)







X1lprime3

3E2I2+ X2

lprime2

2E2I2+

qL2l2

8E2I2

X1lprime2

2E2I2+ X2

l

E2I2+

lprimeE2I2

1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(1)





Cohesion C(MPa)


Friction angle φ()



No 8 coal seam 55 1430 28 13 29






X1 qE2I2L

3

8E1I1lprime2


X2 qL

3E2I2

24E1I1lprime2

minus 12E2I2lprime1113872 1113873

(2)


FNC qL

2 (3)


X3prime L3

24E1I1+

L2lprime

8E2I21113888 1113889 0


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(4)





2lprime

3E1I1L + 8E2I2lprime( 1113857L2

1113872 1113873


2lprime

3E1I1L + 8E2I2lprime( 1113857L2

1113872 1113873

1113868111386811138681113868111386811138681113868111386811138681113868+ qprimea

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)


M1L

2E1I1+

lprimeE2I2

1113888 1113889 + M2lprime

E2I2

qprimeaL(L minus a)

32E1I1+


16E2I2

M1lprime

E2I2+ M2

L

2E1I1+

lprimeE2I2


8E2I2

M1 2MZR 2MZL

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)




22I

22L + 4E

21I

21lprime1113960 1113961

2 8E22I

22L

2+ 32E1E2I1I2Llprime1113872 1113873


22I

22L

2+ 4E

21I

21lprime1113960 1113961

8E22I

22L

2+ 32E1E2I1I2Llprime1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)


X3Prime FNPrime

qprimea2

MMPrime

qprimea(L minus a)

4

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(8)






Suspended roof

(a)


a2L


(b)


The suspended roof

The suspendedroof


2L

Lq

q


Structuresimplified






X1 qE2I2L3


X3 2qprimeaE1I1L

2lprime

3E1I1L3

minus8E2I2L2lprime

+qL

2+

qa

2

FN 2qprimeaE1I1L

2lprime

3E1I1L3

minus8E2I2L2lprime

+qL

2+ qprimea

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

MZ qL

3E2I2



22 +4lprime2E2

1I21

8E22I

22L

2+32E1E2I1I2Llprime


4+


22 +4lprime2E2

1I21

16E22I

22L

2+64E1E2I1I2Llprime

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(10)



σmax Mmax

W

τmax FQSlowastZmax

IZb

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)



τRmax


3E1I1L3

+ 8E2I2L2lprime1113872 1113873b1h1

+qL

2b1h1+

qa

2b1h1

σRmax

1b1h1

qL3E2I2


qE2I2L3


2E22I

22 + 12lprime2E2

1I211113872 1113873


⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

σCmax

1b2h2

3qprimea(L minus a)

2b2h2+


22L

2+ 12E

21I

21lrsquo

21113872 1113873



3E1I1L3

+ 8E2I2L2lprime1113872 1113873

+ qprimea + qL⎛⎝ ⎞⎠ (13)

The suspended roof

The suspended roof


2L

La

a



qprime

qprime2











q1(x)n E1d

31 1113936

ni1 ρidi

1113936ni1 Eid

3i

q1(x)n q1(x)n+1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(14)



q Pq times L

qprime qz times L

⎧⎪⎨

⎪⎩(15)


z

4x04

+ l02

+ 4z2

1113969 +l0z

4x04

+ l02

+ 4z2

1113969

minus

l02

+ 4z2

1113969

1113874 1113875

2x0 l02

+ 4z2

1113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)



σm σc

D

d

1113970

(17)



σs1 σm 0778 + 0222W

h1113874 11138751113876 1113877

σs2 σm

W0446

h066

σs min σs1 σs21113864 1113865

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)




























L = 4

Shea

ring

stres

s of s

uspe

nded

roof

(MPa

)

8

6

4

2

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10


L = 4A

xial

stre

ss o

f sus

pend

ed ro

of (M

Pa)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10




100

75

50

25

00

00 02 04aL

Stre

ss (M

Pa)

06 08 10







σs

σt

fqprimeq

1113888 1113889 kc qprimeq( 1113857 + σsI


(19)




5 Conclusions








16

12

Stre

ss (M

Pa)

8


4

0

qprimeq02 04 06 08 10


L = 4

Axi

al st

ress

of s

uspe

nded

roof

(MPa

)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10





Data Availability




Acknowledgments


References






























































Lithology Rock type

Medium sandstone 19

Mudstone 40


5 coal seam 16


Mudstone 36

Siltite 67


8 coal seam 32


Alumina mudstone 60

Thickness (m)







X1lprime3

3E2I2+ X2

lprime2

2E2I2+

qL2l2

8E2I2

X1lprime2

2E2I2+ X2

l

E2I2+

lprimeE2I2

1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(1)





Cohesion C(MPa)


Friction angle φ()



No 8 coal seam 55 1430 28 13 29






X1 qE2I2L

3

8E1I1lprime2


X2 qL

3E2I2

24E1I1lprime2

minus 12E2I2lprime1113872 1113873

(2)


FNC qL

2 (3)


X3prime L3

24E1I1+

L2lprime

8E2I21113888 1113889 0


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(4)





2lprime

3E1I1L + 8E2I2lprime( 1113857L2

1113872 1113873


2lprime

3E1I1L + 8E2I2lprime( 1113857L2

1113872 1113873

1113868111386811138681113868111386811138681113868111386811138681113868+ qprimea

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)


M1L

2E1I1+

lprimeE2I2

1113888 1113889 + M2lprime

E2I2

qprimeaL(L minus a)

32E1I1+


16E2I2

M1lprime

E2I2+ M2

L

2E1I1+

lprimeE2I2


8E2I2

M1 2MZR 2MZL

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)




22I

22L + 4E

21I

21lprime1113960 1113961

2 8E22I

22L

2+ 32E1E2I1I2Llprime1113872 1113873


22I

22L

2+ 4E

21I

21lprime1113960 1113961

8E22I

22L

2+ 32E1E2I1I2Llprime1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)


X3Prime FNPrime

qprimea2

MMPrime

qprimea(L minus a)

4

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(8)






Suspended roof

(a)


a2L


(b)


The suspended roof

The suspendedroof


2L

Lq

q


Structuresimplified






X1 qE2I2L3


X3 2qprimeaE1I1L

2lprime

3E1I1L3

minus8E2I2L2lprime

+qL

2+

qa

2

FN 2qprimeaE1I1L

2lprime

3E1I1L3

minus8E2I2L2lprime

+qL

2+ qprimea

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

MZ qL

3E2I2



22 +4lprime2E2

1I21

8E22I

22L

2+32E1E2I1I2Llprime


4+


22 +4lprime2E2

1I21

16E22I

22L

2+64E1E2I1I2Llprime

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(10)



σmax Mmax

W

τmax FQSlowastZmax

IZb

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)



τRmax


3E1I1L3

+ 8E2I2L2lprime1113872 1113873b1h1

+qL

2b1h1+

qa

2b1h1

σRmax

1b1h1

qL3E2I2


qE2I2L3


2E22I

22 + 12lprime2E2

1I211113872 1113873


⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

σCmax

1b2h2

3qprimea(L minus a)

2b2h2+


22L

2+ 12E

21I

21lrsquo

21113872 1113873



3E1I1L3

+ 8E2I2L2lprime1113872 1113873

+ qprimea + qL⎛⎝ ⎞⎠ (13)

The suspended roof

The suspended roof


2L

La

a



qprime

qprime2











q1(x)n E1d

31 1113936

ni1 ρidi

1113936ni1 Eid

3i

q1(x)n q1(x)n+1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(14)



q Pq times L

qprime qz times L

⎧⎪⎨

⎪⎩(15)


z

4x04

+ l02

+ 4z2

1113969 +l0z

4x04

+ l02

+ 4z2

1113969

minus

l02

+ 4z2

1113969

1113874 1113875

2x0 l02

+ 4z2

1113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)



σm σc

D

d

1113970

(17)



σs1 σm 0778 + 0222W

h1113874 11138751113876 1113877

σs2 σm

W0446

h066

σs min σs1 σs21113864 1113865

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)




























L = 4

Shea

ring

stres

s of s

uspe

nded

roof

(MPa

)

8

6

4

2

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10


L = 4A

xial

stre

ss o

f sus

pend

ed ro

of (M

Pa)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10




100

75

50

25

00

00 02 04aL

Stre

ss (M

Pa)

06 08 10







σs

σt

fqprimeq

1113888 1113889 kc qprimeq( 1113857 + σsI


(19)




5 Conclusions








16

12

Stre

ss (M

Pa)

8


4

0

qprimeq02 04 06 08 10


L = 4

Axi

al st

ress

of s

uspe

nded

roof

(MPa

)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10





Data Availability




Acknowledgments


References











































X1lprime3

3E2I2+ X2

lprime2

2E2I2+

qL2l2

8E2I2

X1lprime2

2E2I2+ X2

l

E2I2+

lprimeE2I2

1113888 1113889

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(1)





Cohesion C(MPa)


Friction angle φ()



No 8 coal seam 55 1430 28 13 29






X1 qE2I2L

3

8E1I1lprime2


X2 qL

3E2I2

24E1I1lprime2

minus 12E2I2lprime1113872 1113873

(2)


FNC qL

2 (3)


X3prime L3

24E1I1+

L2lprime

8E2I21113888 1113889 0


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(4)





2lprime

3E1I1L + 8E2I2lprime( 1113857L2

1113872 1113873


2lprime

3E1I1L + 8E2I2lprime( 1113857L2

1113872 1113873

1113868111386811138681113868111386811138681113868111386811138681113868+ qprimea

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)


M1L

2E1I1+

lprimeE2I2

1113888 1113889 + M2lprime

E2I2

qprimeaL(L minus a)

32E1I1+


16E2I2

M1lprime

E2I2+ M2

L

2E1I1+

lprimeE2I2


8E2I2

M1 2MZR 2MZL

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)




22I

22L + 4E

21I

21lprime1113960 1113961

2 8E22I

22L

2+ 32E1E2I1I2Llprime1113872 1113873


22I

22L

2+ 4E

21I

21lprime1113960 1113961

8E22I

22L

2+ 32E1E2I1I2Llprime1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)


X3Prime FNPrime

qprimea2

MMPrime

qprimea(L minus a)

4

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(8)






Suspended roof

(a)


a2L


(b)


The suspended roof

The suspendedroof


2L

Lq

q


Structuresimplified






X1 qE2I2L3


X3 2qprimeaE1I1L

2lprime

3E1I1L3

minus8E2I2L2lprime

+qL

2+

qa

2

FN 2qprimeaE1I1L

2lprime

3E1I1L3

minus8E2I2L2lprime

+qL

2+ qprimea

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

MZ qL

3E2I2



22 +4lprime2E2

1I21

8E22I

22L

2+32E1E2I1I2Llprime


4+


22 +4lprime2E2

1I21

16E22I

22L

2+64E1E2I1I2Llprime

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(10)



σmax Mmax

W

τmax FQSlowastZmax

IZb

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)



τRmax


3E1I1L3

+ 8E2I2L2lprime1113872 1113873b1h1

+qL

2b1h1+

qa

2b1h1

σRmax

1b1h1

qL3E2I2


qE2I2L3


2E22I

22 + 12lprime2E2

1I211113872 1113873


⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

σCmax

1b2h2

3qprimea(L minus a)

2b2h2+


22L

2+ 12E

21I

21lrsquo

21113872 1113873



3E1I1L3

+ 8E2I2L2lprime1113872 1113873

+ qprimea + qL⎛⎝ ⎞⎠ (13)

The suspended roof

The suspended roof


2L

La

a



qprime

qprime2











q1(x)n E1d

31 1113936

ni1 ρidi

1113936ni1 Eid

3i

q1(x)n q1(x)n+1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(14)



q Pq times L

qprime qz times L

⎧⎪⎨

⎪⎩(15)


z

4x04

+ l02

+ 4z2

1113969 +l0z

4x04

+ l02

+ 4z2

1113969

minus

l02

+ 4z2

1113969

1113874 1113875

2x0 l02

+ 4z2

1113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)



σm σc

D

d

1113970

(17)



σs1 σm 0778 + 0222W

h1113874 11138751113876 1113877

σs2 σm

W0446

h066

σs min σs1 σs21113864 1113865

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)




























L = 4

Shea

ring

stres

s of s

uspe

nded

roof

(MPa

)

8

6

4

2

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10


L = 4A

xial

stre

ss o

f sus

pend

ed ro

of (M

Pa)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10




100

75

50

25

00

00 02 04aL

Stre

ss (M

Pa)

06 08 10







σs

σt

fqprimeq

1113888 1113889 kc qprimeq( 1113857 + σsI


(19)




5 Conclusions








16

12

Stre

ss (M

Pa)

8


4

0

qprimeq02 04 06 08 10


L = 4

Axi

al st

ress

of s

uspe

nded

roof

(MPa

)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10





Data Availability




Acknowledgments


References









































X1 qE2I2L

3

8E1I1lprime2


X2 qL

3E2I2

24E1I1lprime2

minus 12E2I2lprime1113872 1113873

(2)


FNC qL

2 (3)


X3prime L3

24E1I1+

L2lprime

8E2I21113888 1113889 0


⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(4)





2lprime

3E1I1L + 8E2I2lprime( 1113857L2

1113872 1113873


2lprime

3E1I1L + 8E2I2lprime( 1113857L2

1113872 1113873

1113868111386811138681113868111386811138681113868111386811138681113868+ qprimea

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)


M1L

2E1I1+

lprimeE2I2

1113888 1113889 + M2lprime

E2I2

qprimeaL(L minus a)

32E1I1+


16E2I2

M1lprime

E2I2+ M2

L

2E1I1+

lprimeE2I2


8E2I2

M1 2MZR 2MZL

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)




22I

22L + 4E

21I

21lprime1113960 1113961

2 8E22I

22L

2+ 32E1E2I1I2Llprime1113872 1113873


22I

22L

2+ 4E

21I

21lprime1113960 1113961

8E22I

22L

2+ 32E1E2I1I2Llprime1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)


X3Prime FNPrime

qprimea2

MMPrime

qprimea(L minus a)

4

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(8)






Suspended roof

(a)


a2L


(b)


The suspended roof

The suspendedroof


2L

Lq

q


Structuresimplified






X1 qE2I2L3


X3 2qprimeaE1I1L

2lprime

3E1I1L3

minus8E2I2L2lprime

+qL

2+

qa

2

FN 2qprimeaE1I1L

2lprime

3E1I1L3

minus8E2I2L2lprime

+qL

2+ qprimea

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

MZ qL

3E2I2



22 +4lprime2E2

1I21

8E22I

22L

2+32E1E2I1I2Llprime


4+


22 +4lprime2E2

1I21

16E22I

22L

2+64E1E2I1I2Llprime

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(10)



σmax Mmax

W

τmax FQSlowastZmax

IZb

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)



τRmax


3E1I1L3

+ 8E2I2L2lprime1113872 1113873b1h1

+qL

2b1h1+

qa

2b1h1

σRmax

1b1h1

qL3E2I2


qE2I2L3


2E22I

22 + 12lprime2E2

1I211113872 1113873


⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

σCmax

1b2h2

3qprimea(L minus a)

2b2h2+


22L

2+ 12E

21I

21lrsquo

21113872 1113873



3E1I1L3

+ 8E2I2L2lprime1113872 1113873

+ qprimea + qL⎛⎝ ⎞⎠ (13)

The suspended roof

The suspended roof


2L

La

a



qprime

qprime2











q1(x)n E1d

31 1113936

ni1 ρidi

1113936ni1 Eid

3i

q1(x)n q1(x)n+1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(14)



q Pq times L

qprime qz times L

⎧⎪⎨

⎪⎩(15)


z

4x04

+ l02

+ 4z2

1113969 +l0z

4x04

+ l02

+ 4z2

1113969

minus

l02

+ 4z2

1113969

1113874 1113875

2x0 l02

+ 4z2

1113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)



σm σc

D

d

1113970

(17)



σs1 σm 0778 + 0222W

h1113874 11138751113876 1113877

σs2 σm

W0446

h066

σs min σs1 σs21113864 1113865

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)




























L = 4

Shea

ring

stres

s of s

uspe

nded

roof

(MPa

)

8

6

4

2

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10


L = 4A

xial

stre

ss o

f sus

pend

ed ro

of (M

Pa)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10




100

75

50

25

00

00 02 04aL

Stre

ss (M

Pa)

06 08 10







σs

σt

fqprimeq

1113888 1113889 kc qprimeq( 1113857 + σsI


(19)




5 Conclusions








16

12

Stre

ss (M

Pa)

8


4

0

qprimeq02 04 06 08 10


L = 4

Axi

al st

ress

of s

uspe

nded

roof

(MPa

)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10





Data Availability




Acknowledgments


References










































X1 qE2I2L3


X3 2qprimeaE1I1L

2lprime

3E1I1L3

minus8E2I2L2lprime

+qL

2+

qa

2

FN 2qprimeaE1I1L

2lprime

3E1I1L3

minus8E2I2L2lprime

+qL

2+ qprimea

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

MZ qL

3E2I2



22 +4lprime2E2

1I21

8E22I

22L

2+32E1E2I1I2Llprime


4+


22 +4lprime2E2

1I21

16E22I

22L

2+64E1E2I1I2Llprime

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(10)



σmax Mmax

W

τmax FQSlowastZmax

IZb

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)



τRmax


3E1I1L3

+ 8E2I2L2lprime1113872 1113873b1h1

+qL

2b1h1+

qa

2b1h1

σRmax

1b1h1

qL3E2I2


qE2I2L3


2E22I

22 + 12lprime2E2

1I211113872 1113873


⎛⎝ ⎞⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

σCmax

1b2h2

3qprimea(L minus a)

2b2h2+


22L

2+ 12E

21I

21lrsquo

21113872 1113873



3E1I1L3

+ 8E2I2L2lprime1113872 1113873

+ qprimea + qL⎛⎝ ⎞⎠ (13)

The suspended roof

The suspended roof


2L

La

a



qprime

qprime2











q1(x)n E1d

31 1113936

ni1 ρidi

1113936ni1 Eid

3i

q1(x)n q1(x)n+1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(14)



q Pq times L

qprime qz times L

⎧⎪⎨

⎪⎩(15)


z

4x04

+ l02

+ 4z2

1113969 +l0z

4x04

+ l02

+ 4z2

1113969

minus

l02

+ 4z2

1113969

1113874 1113875

2x0 l02

+ 4z2

1113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)



σm σc

D

d

1113970

(17)



σs1 σm 0778 + 0222W

h1113874 11138751113876 1113877

σs2 σm

W0446

h066

σs min σs1 σs21113864 1113865

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)




























L = 4

Shea

ring

stres

s of s

uspe

nded

roof

(MPa

)

8

6

4

2

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10


L = 4A

xial

stre

ss o

f sus

pend

ed ro

of (M

Pa)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10




100

75

50

25

00

00 02 04aL

Stre

ss (M

Pa)

06 08 10







σs

σt

fqprimeq

1113888 1113889 kc qprimeq( 1113857 + σsI


(19)




5 Conclusions








16

12

Stre

ss (M

Pa)

8


4

0

qprimeq02 04 06 08 10


L = 4

Axi

al st

ress

of s

uspe

nded

roof

(MPa

)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10





Data Availability




Acknowledgments


References















































q1(x)n E1d

31 1113936

ni1 ρidi

1113936ni1 Eid

3i

q1(x)n q1(x)n+1

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(14)



q Pq times L

qprime qz times L

⎧⎪⎨

⎪⎩(15)


z

4x04

+ l02

+ 4z2

1113969 +l0z

4x04

+ l02

+ 4z2

1113969

minus

l02

+ 4z2

1113969

1113874 1113875

2x0 l02

+ 4z2

1113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)



σm σc

D

d

1113970

(17)



σs1 σm 0778 + 0222W

h1113874 11138751113876 1113877

σs2 σm

W0446

h066

σs min σs1 σs21113864 1113865

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)




























L = 4

Shea

ring

stres

s of s

uspe

nded

roof

(MPa

)

8

6

4

2

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10


L = 4A

xial

stre

ss o

f sus

pend

ed ro

of (M

Pa)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10




100

75

50

25

00

00 02 04aL

Stre

ss (M

Pa)

06 08 10







σs

σt

fqprimeq

1113888 1113889 kc qprimeq( 1113857 + σsI


(19)




5 Conclusions








16

12

Stre

ss (M

Pa)

8


4

0

qprimeq02 04 06 08 10


L = 4

Axi

al st

ress

of s

uspe

nded

roof

(MPa

)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10





Data Availability




Acknowledgments


References
































































L = 4

Shea

ring

stres

s of s

uspe

nded

roof

(MPa

)

8

6

4

2

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10


L = 4A

xial

stre

ss o

f sus

pend

ed ro

of (M

Pa)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10




100

75

50

25

00

00 02 04aL

Stre

ss (M

Pa)

06 08 10







σs

σt

fqprimeq

1113888 1113889 kc qprimeq( 1113857 + σsI


(19)




5 Conclusions








16

12

Stre

ss (M

Pa)

8


4

0

qprimeq02 04 06 08 10


L = 4

Axi

al st

ress

of s

uspe

nded

roof

(MPa

)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10





Data Availability




Acknowledgments


References










































σs

σt

fqprimeq

1113888 1113889 kc qprimeq( 1113857 + σsI


(19)




5 Conclusions








16

12

Stre

ss (M

Pa)

8


4

0

qprimeq02 04 06 08 10


L = 4

Axi

al st

ress

of s

uspe

nded

roof

(MPa

)

20

15

10

5

0

L = 8L = 12

L = 16L = 20

aL00 02 04 06 08 10





Data Availability




Acknowledgments


References









































Data Availability




Acknowledgments


References







































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