An analytic solution describing the visco-elastic deformation of coal pillars in room and pillar min

Mining Science and Technology( China)21(2011)885-890Contents lists available at SciVerse ScienceDirectMining Science and Technology( China)ELSEVIERjournalhomepagewww.elsevier.com/locate/mstcAn analytic solution describing the visco-elastic deformation of coal pillars inroom and pillar mineLi Qianga b xuHui. Bu Wankui". c," Zhao GuozhenSchool of Mechanics 6 Civil Engineering China University of Mining 6 Technology, Xuzhou 221116, ChinaState Key Laboratory of Geomechanics 8 Deep Underground Engineering. China University of mining 6 Technology, Xuzhou 221008, China"Department of Mechanical and Electrical Engineering, Heze University, Heze 274015, ChinaARTICLE INFOABSTRACTCoal pillar deformation is typically nonlinear and time-dependent. The accurate prediction of this defor-March 2011mation has a vital importance for the successful implementation of mining techniques. These methodsvised form 12 April 2011Accepted 7 May 2011have proven very important as a way to excavate coal resources from under buildings, railways, or waterAvailable online 24 December 2011bodies. Elastic and visco-elastic theory are employed with a maxwell model to formulate an analyticsolution for displacement of coal pillars in room and pillar mine. These results show that the visco-elasticsolution adequately predicts the coal pillar deformation over time. we conclude that the visco-elasticsolution can predict the coal pillar and roadway displacement from the measured geological parametersof the conditions in situ. furthermore, this method would be useful for mine design, coal pillar supportVisco-elastic mediumoptimization, ground subsidence prediction, and coal pillar stability analysis.Coal pillao 2011 Published by Elsevier B V on behalf of China University of Mining Technology1 Introductiondeformation are balanced and these values are assumed to be timehe energy shortage makes coal resources under buildings, rail-Obviously the reality is different. Gray has pointed out that itways, and water-bodies more and more important now. It is con- may take many years for the coal mining induced affects to reachservatively estimated that in China the total reserves under these equilibrium [2]. For instance, in Scotland the ground surfacehree locations are more than 13. 79 billion tons. Hence, exploiting started to subside after a mine was abandoned for 118 years. thethese resources efficiently has a great significance to both China wieliczka salt mine in Poland subsided suddenly 140 years afterand other countries. The room and pillar mine techniques have being caved resulting in all buildings being totally destroyed.proven to be a good option for solving this problem. This technique Abundant other evidence also proves that coal pillar deformationdivides the mining field into several rectangular shapes. The rect- is a transient process [3, 4]. This process can be understood byangular areas are caved alternately and the remaining rectangles considering that the displacement and stress increase dramaticallysupport the overburden load from the overlying strata In this just after mining. the increase rate reduces over time and forcesway ground surface subsidence can be uniform and surface defor- become in balance after some period of time However, the localmation can be controlled within the permitted range, thereby failure continues and finally large scale structures fail a considerThe key issue with this technique is to ensure that the reserved able time after mining.achieving the Green Mining Thought [1]Since coal pillar deformation is a nonlinear process and stronglycoal pillars have the strength to support the overburden after exctime dependent it is extremely important to consider rheologicalvation. the common approach to investigate the stability of a coal behavior when the performance of the coal pillars is estimated.pillar assumes coal deformation is quasi static. In other words, the The estimates then can give useful predictions for pillar behaviordeformation is not time dependent during caving the initial stress under in situ conditions.in the overlying strata will be disturbed and the excavationCoal pillar deformation and stability have been widely studiedinduced extra load over the goaf transfers onto the remaining coal theoretical, numerically, and experimentally Oravecz proposedillars simultaneously. the surrounding stresses, and coal pillar method based on elastic theory to determine the stress and displacement in bord and pillar workings of coal mines. SimilarlyUnlu used a linear relationship to investigate pillar stability 5.6Donovan and Karfakis, discussed coal pillar stress and displaceCorresponding author Tel +86 18252115782ment performance under conditions where back fill was used. Theymailaddress:tbhbwk239@gmaiL.com(w.Bu)used earth pressureYH中国煤化工7 n contrast1674-5264/5-see front matter o 2011 Published by Elsevier B V on behalf of China University of Mining TechrCNMHGdoi:10.1016/msc201105028Q u et aL/Mining Science and Technology( China)21(2011)885-890elastic-plastic displacement and stress distributions were investi-gated using a Weibull distribution in the analysis of nonlinear coalpillar performance. Different yield and failure criteria were used topredict coal pillar stability in these investigations [8-10]. SimilarlyLi et al. chose total strain theory as a way to analyze mechanicalCoal pillarob areadeformation and stress distribution of a narrow pillar [11]. A con-Coal pillarsiderable number of experiments and simulations have been con-ducted in this area. all these investigations focused on the effectscoal strength or other parameters have on coal pillar stability12-17]. For example, the effects of size and width-to-height ratioof the pillar have been discussed. Heretofore, no investigation ofthe overall coal pillar stability and deformation has been reportedrigorThese numerical, theoretical, or experimental observations haveillustrated that coal pillar deformation is typically time dependent.Fig. 1 Schematic diagram of room and pillar mineHence it is necessary to find an analytic solution for the visco-elastic deformation of a coal pillar. Most of the prior studies focus onthe elastic or elastic-plastic behavior of the coal pillar. Few haveworked out the analytic solution for visco-elastic deformation ofcoal pillar because of the difficulties involved in the theoreticalderivations. This is true even thought the importance of this solu-tion has been widely realized In this paper, we first consider coalas a visco-elastic medium and then use the maxwell model withb doa dor dfelastic and visco-elastic theories to derive an analytic solution forcoal pillar displacement over time. Then the validity of theproposed solution is tested by predicting in situ coal pillar dis-placement from the combined measured geological parametersh2. A mechanical model of coal pillarsThe following assumptions are made for this analysis(a) The coal pillar is homogeneous, isotropic, and a continuum.(b)The coal seam is connected with the roof and floor rock sohat the pressure is continuous among them(c) The system is isothermal.Because the width of the coal pillars is far smaller than thelength, and because the shape and load distribution on a pillar is2. A simplifiedsymmetric, the analytic model can be simplified into a plain strainproblem. A diagram of room and pillar mine is shown in Fig. 1The coal pillar size is evenly distributed and 2h high and 2b wide. lxh=-qThe burden on the pillar is from the weight of the roof bearingn the floor rock. Themechanical and geometric boundaryu-o=0conditions are symmetric so only one quarter of the overall modelneed be analyzed, as shown in Fig. 2.The boundary condition at the left side is assumed to be hori-zontal con finement and that at the bottom a vertical confinement.0The upper side is restricted in the horizontal direction due to theinteraction between the coal pillar and the roof rock. A free defor- where u and v are the displacement vectors in the horizontal andmation boundary is used at the right side. Assuming plastic effects vertical directions respectivelyalong the top of the pillar may be ignored the overburden stresswill be equally distributed along the top surface of the pillar stress are very subtle and may be neglected They are assumed tooverburdenbe x=y=obining the boundary conditions for displacement allowsthe potential displacement expression to be written aswhere y is the average volume weight of the overlying strata, andH u(x, y)=8(1-5)(A1+A2 02)the mining depth.(x, y)= By3. Elastic displacement and stress distributionwhere Al. A2, and B are coefficients that depend upon the boundaryand initial condition中国煤化工Ritz method wasFig. 2 shows the boundary conditions, which can be expressed solution [21].the analyticalFor aby Eqs.(2)-(5)CN MHgmation energya coal pillar can beQ uf et aL/Mining Science and Technology( China)21(2011)885-890Table 1Basic equations: elasticity and visco-elasticitya=-1008h4-15a4+1683h2-26002h2a2+88013h2a2ElasticVisco-elastic-21842h+75a4k-12042+21602h2+60043+x1=0+3024h14-5400h2Geometry equation Ey=y(u4+uu)Physical equation剩m2.m=3a=20,a=3Kap=-356h4-54012h2a2-72013h2a2+72Boundary condition l= Ti, u=pIaul=Ti Q=p1008h4-352h2+25a4-30d412-20a42+1008h4-180a2h2+720“h2a2+672h1Substituting Eqs. 9 and 11 into 6 gives the expressions fordisplacemen(+2y)(7)4 Analytic visco-elastic solutionIving Eq.(7)requires that the following conditions be4.1. A comparison of elastic and visco-elastic principlessatisfiedA comparison of elastic and visco-elastic models is presented inTable 1Xu, dsIt is easy to prove thatUTurdsPs=20YvesIf we use the Laplace transform to change the elastic governingequations and substitute Q/P for G the solutions for these twocases become identical. Thus, three steps are required to solve(1-)3(1-;),u=ythe visco-elastic problem. The first step is to obtain the solutionombining Eqs. (6)-(8)allows the analytical displacement to be for the elastic case and perform a Laplace transform on thatexpressed astion. Then g is replaced by Q/P and K is kept fixed. Finally, theinverse Laplace transform is applied to obtain the visco-elasticA,_agh(-1+ +2p )(-1260 -22a2+225a+12600) solution. This process is known as the correspondence principleA2=-2￥udqn(4p2+1-31The relationships between G, K, E, and u are:9KGB==3B3K(12)6K+2Gwhere E is the elastic modulus of a coal pillar and u poissons ratio.According to the correspondence principle the elastic solutionThe expressions for a and B shown belotmay be transformed into Eq (13)using a Laplace transform.Commonly used visco-elastic models.NameSchematicField of applicationdiagraMaxwell bodyEK&+ neRarely used individuallyGeneralized Kelvin body ExModerate firmness of rockEHJoseph Thomsond+0na+EHEModerate firmness of rock中国煤化工CNMHGQ ui et aL/Mining Science and Technology(China)21 (2011)885-890=x(1-Azms=504hk'nG+672h KG'n+270a2hKG'n(25)a(13)m=252hK2G2(26)Substituting Eqs. ( 19)into(13)and applying the inverse Laplacetransform gives the displacement of a coal pillar as4.2. Visco-elastic analytic solutionu=5(1-h)(g1+g2()2)a reasonable analytical solution requires the proper visco-elas-D=gaytic model. the most widely used visco-elastic models are listed inTable 2wherein Table 2 EH is the modulus for a Hook model, Ex the modulusfor a Kelvin model, EM the modulus for a Maxwell model, n the vis-Roo(m23+m2+m2+m)4卯acosity, o the stress, and a the strain.The Maxwell model which has been proven valid for coal defonation, will be used herein [22]. It is formulated asm(3m1r2+ 2m2+m exp(r)+2835gh'aK2 G+z=2e(14)2=2,om2+m2m么15750n2aGmm+G(3_rKn-2.rGn + 3KG)The other expressions required for this analytical solution are(12m2+8m2r+4mep(m)(P)181F-Root of(m -2+m2-2+m,-Z+m4)G at r9gm3m2+2m2J+m少exp(m)-男m,Gn2Q}=8tm1=168G3Km4h2n+75602k2m4h2n-450Km4a2n-252C3K2mh2(17)+756rGmhKn2-90. G'm4a2nm2K+30-rG'm4a2n2Q(18)24rG'm h'nmiSubstituting G=Q/P into Eq(9)gives+336_,h'n?K-252_rGm2hK-224rmahG'n252rmh22n3A=-4a(+O(-3K7+25G-3K+30-2m4a2Gn-45-m4a2GmK+168_m h"GrK(11hGm+84sh2Kn-150G7+84h2KG)mIs3+ m252 +m3S+m4252]2m, K'G1575(-3sKn+25G7-3KG)n2=manny+m4Gm6-m3 Gm7 +rm4nm6 +rm,Gmsqh"asGn(sn+G+s3+m2s2+3s+marm2Gm,+r m4nms-'m1GmrB=l-9q( n+G)(ms s2+mes+m7)mIS+ m252+m3S+m4where k is the volume modulus, G the shear modulus, n the viscos. 5. An applicationity coefficient. The expressions for the other parameters are shownThe trial district for the room and pillar mine technique was loas eqs.(20)}(27cated between Industrial Square and the protection coal pillar of am1=19802h2k2Gn3+4120m2h2Gn3+4256h'Cn3Chinese coal. This area is located 220 m north of the mine shaftbottom It is 480 m long in the running direction and 328 m long+9072hK2n3G+70802h2kC2n3in the inclination direction the run of the coal seam is Es 30 to+378h1K3n314616h1KC2n3+1350KGn3NW 60 and the inclination direction is Ne. This coal seam isknown as number Ji 16-17 and the absolute mining depth is from+180dcn3(20) 747 to 847 m, approximately. The average height of the coal seamis about 6 m. The geological conditions are very good and this coalm2=1134hK3n72G+18144K2n2G2+135d4KG3n2seam is very stable. the coal excavation is 6 by 6 m, which means+70802hKG2n72+39602h2K2G2n2that the entire height was mined at one time. The reserved coal pil-lar is also 6 m wide the values of the other parameters, measured14616hKG3n2(21) in situ, are listed belowThe volume weight of the overlying strata is 19.8 kN/m,them=1134h'K3mc2+19802h2k2Cn+9072h1k2nc3(2shear modulus is 0.3 GPa, the volume modulus is 0.54 GPa, andthe viscosity coefficient is 1015 Pas. Substituting these parametersm4=378hKG3(23) nto Eq(28)givesg1=1.3208+0008ms=15dC22+252hk2m2+672h'KGm23602h?G22+0021261exp中国煤化工+2702h2KGn2+448hG2n2xexp(-1.8659CNMHGQ ui et al/ Mining Science and Technology(China)21(2011)885-890Elastic solution目Fitting curve for56f(a)(a)Horizontal displacementb)vertical displacementFlg. 3. Maximum horizontal and vertical displacements versus tig2=0005502eXp(-23144×107)-002165significant insight into mine design, optimizing coal pillar supports,x exp(-16053 x 100)+0.021089 exp(-1.8659x 10-6)(30 potential ground subsidence, and coal pillar stabllity. This will helg3=-03522200045699eXp(-23144×107t)6. Conclusions83475×10-exp(-1.6053×10t)0.30988exp(-1.8659×10°t)The visco-elastic behavior of a coal pillar was investigated byusing visco-elastic theory with a Maxwell model. The followingCombining Eqs. (29)-(31)with 27 gives the displacement conclusions may be drawn.expressions in the vertical and horizontal directions. Eq. (27)shows that the maximum horizontal displacement occurs at the(1)Stress re-distribution happens synchronously with theoutermost point of the horizontal mid-axis. The maximum verticalmining work. Displacement increases over time until equidisplacement is along the upper boundary. The maximum horizon-librium is re-established However the reason for the con-tal and vertical displacements(umax and Umax)versus time aretinued increase in strain is not because of a stress increaseshown in Fig 3 respectivelybut because the coal pillar is visco-elasticThese analytic solutions show that coal pillar displacement in-(2)Analytical results show that there is an exponential relation-volves three different stageship between displacement and time. they also show that80% of the displacement occurs within the first 3 years. Thi(1)At time zero, uo-004 m, to-0.11 m. These are the initialis called the main stage of displacement. It is more reason-values induced by the mining activity. Essentially, this isble, and reliable, to predict coal pillar deformation withthe elastic solution to the problem. Because the initial bal-visco-elastic theory. The solutions provide an important ref-ance is destroyed by coal excavation the stress distributionerence for choosing the support style, predicting surfacehanges until a new stress balance is reached. This resultsubsidence and arranging the working procedures. All thesein coal pillar deformation.have a significant influence on the application of the room(2) After t has increased to 3a umax and vmax have increased toand pillar mine technique.1.09 and 0.89 m respectively. this is because coaldeformation is affected by the visco-elastic behavior ofmaterial: It increases with time. The values in this case Acknowledgmentsincreased by 1.05 and 0. 78 m. It can be easily shown thathe annual increase is predicted to be 0.35 and 0. 26 m/aFinancial support for this work was provided by the National(3)When t is 8 years umax and vmax have increased to 1.31 and Basic Research Program of China(No 2005CB221502), the Major1.05 m, which is an increase of 0. 22 and 0.16 m. The increase Program of National Natural Science Foundation of China(Noin this case is 0.04 and 0.03 m/a. This shows that the rate of 50490273), the Postdoctoral Subject Foundation of the State Keydeformation is decreasing and the pillar is becoming stable. Laboratory of Geomechanics Deep Underground Engineeringalthough the absolute value of the deformation is still(No. PD1005), the Research Foundation of Heze University(NoXY10BS04), the Trans-Century Training Program Foundation forhe Talents by the State Education Commission(No. NCET-08-4. This shows that coal deformation is mainly due to its viscous 0837)and the National Natural Science Foundation of China(Nosavior. The elastic deformation is really very small compared 50834005). This support is gratefully acknowledgedto the viscous deformation For instance the original 6 by 6 m coalpillar deforms to a 8.62 by 3.90 m one(width by height). The seam Referencesroadway changes from 6 by 6 m to 3.38 by 3.90 mA comparison of measured values to the visco-elastic solution (11 Xu JL Qian MC Concept of green mining and its technical framework. Scithe deformation tendency of the pillar over time But joints and 12] Gray RE. Mining sule-past, present, future. Int J Min Geol Engcracks decrease the deformation modulus of the coal pillar so the (3 Guo GL Deng KZ. Tan ZX. L FC.Studymeasured deformation is greater than the predicted visco-elasticsolution中国煤化工ion measures forTherefore it is necessary and critical to investigate the time[41 Qiao ZC, Xia Jw. Gdependent strain of a coal pillar. the predictions may provideC NMHGChina Univ MinQ u et aL/Mining Science and Technology(China)21(2011)885-890[5] Oravecz Kl. Analogue modeling of stresses and displacements in bord andinternational workshop on coal pillar mechanics and design. Vail: Nationalof coal mines. Int J Rock Mech Min Sci Geomech Abstr977;141):7-23.Institute for Occupational Safery and Health: 1999. p 5-13.[14] Pan Y. Li Aw. Q YS. Fold catastrophe meynamic pillar failure in[6] Unlu T. Critical dimension concept in pillar stability. In: Proceedings of theasymmetric mining. Min Sci Technol 2009: 19(1): 49-57,Ofset Malbaacilik San Etic l sress and exhibition of Turkey. Ankara:: Kozan [15] Zhang HL Wang LG. Gao F. Yang HB. Numerical simulation study on thestress field on the stability of roadways. Min Sci[7] Donovan JG. Karfakis MG. Design of back filled thin-seam coal pillars usingechnol2010:205):707-11ssure theory. Geotech Geol Eng 2004: 22(4): 627-42[8] Qin SQ, Jiao 1. Tang CA Li ZG. Instability leading to coal bumps and nonlinear[161 Wang JX. Lin M, Tian DX, Zhao CL Deformation characteristics of surroundingrock of broken and soft rock roadway. Min Sci2009:192):205-9evolutionary mechanisms for a coal-pillar-and-roof system Int Solids Struc[17] Tian JS, GaoS. Deformation and failure study of2006:43(25-26):7407-23pressure roadways in deep mines. Min Sci Tech1020(6)850-4[9 Duncan Fama METwo and three-dimensional elast[18I Xu JH, Liu KG, Lu AH. Visco-elastic analysis and application of key strata ofplastic analysis for coal pillar design and its application to highwall mining. nRock Mech Min Sci Geomech Abstr 1995: 32(3): 215-25igers rocK to short wall mining Chin J Rock Mech Eng 2006: 25(6)[10] Murali Mohan G, Sheorey PR. Kushwaha A. Numerical estimation of pillar [191 Kobayashi H, Takahashi H. Hiki Y. Aines. Int J Rock Mech Min Sci 2001: 38(8): 1185-92ratus for mescosity of solids. Int J Thermophys 1995: 16(2): 577-84.1111 u SC. Bai JB, Dong Zz Mechanical deformation and stress analyses of the [20] Zou YF. Chai HB. Researchrch status of strip coal pillar stability and its mainnarrow pillar of road driven along next goaf for fully-mechanized top-coalems in China. J Min Saf Eng 2006: 23(2): 141-145, 150ving face Ground Pressure Strata Control 2004: 21(3): 17-9I Landau LD, Lifshitz EM. Theory of elasticity New York: Pergamon Press: 1986112] Prasad RK, Kejriwal BK. Effect of specimen size on the long-term strength ofent for finite elementcoal pillars. IntJ Min Eng 1984: 2(4): 355-8[131 Biswas K, Mark C. Peng SS. A unique approach to determining thebased on a Maxwell model. Mining R&D 2008: 28(1): 19-20, 51dependent in situ strength of coal pillars. In: Proceedings of the中国煤化工CNMHG