Catastrophe analysis on pillar instability considered mining effect

Vol.12 No. 1J. CENT. SOUTH UNIV. TECHNOL.Feb.2005Article ID: 1005 - 9784(2005)01 - 0102 - 05Catastrophe analysis on pillar instability considered mining effectLI Jiang -teng(李江腾), CAO Ping(曹平)(School of Resources and Safety Engineering，Central South University, Changsha 410083，China)Abstract: The instability of the pillar was discussed based on the potential energy principle and the cusp catastro-phe theory, and a simplified mechanical model of the pillar was established considering the mining effect. The neces-sary-sufficient conditions, the jump value of displacement of pillar and the released energy expressions were de-duced. The results show that the instability of the pillar is related to the properties of the rock, the external forceand the relative stiffness of the elastic area to the plastic area. The instability of system is like to occur with the en-larging of the softening area or the decreasing of E/A. The calculation done shows that the estimated results corre-spond to practical experience.Key words: pillar; potential energy; cusp catastrophe; plastic area; relative stiffness; stabilityCLC number: TD324. 1Document code: A1 INTRODUCTIONceed the yield stress due to mining effect, such asthe shock，the impact load and the repeat load,At present， underground mining is the mainwhich make the state of rock changes from elasticapproach for the exploitation of mineral resources.state to plastic state. But, in the middle of the pil-The geologic condition will deteriorate with the ex-lar, the state of rock is still elastic state, and thesetending of the mining depth， which affects therocks in elastic state form elastic area3l. The sof-safety of the mine and the mining layout of oretening area is enlarged abruptly when the elasticbody. Those pillars support the stope, and theirstrain energy stored in elastic area is released, andstability is the most importants. At present， ththe instability of the pillar occurs consequently.analysis of the instability of pillar is developingFor simplifying， a foursquare pillar is only takenfrom the simple strength theory to the instabilityinto account in this work，and the softening areaof the mechanics equilibrium system. However, assize in pillar periphery is constant along height.material properties of rock mass and instabilityThe simplified mechanical model is shown inprocess are quite complicated, it is difficult to sim-Fig. 1.ulate mathematically the occurrence and develop-ment dynamic process by using a uniform model,2.2 Constitutive relationand it is also impossible to arrive at a uniform con-A lot of experiment results show that the me-clusion2. As the ratio of highness to width ischanics properties of rock vary with the deforma-big，and the stability of pillars declines distinctly.tion increasingl4. At the beginning of the load,A tiny disturbance in horizontal direction will re-these initial tiny cracks in the rock close under thesult in the instability of system under certain con-compressive stress， and the rock is in the elasticditions. The catastrophe theory is just a powerfulstage with the increasing of the load, and when thetool for the analysis of the instability phenomena.stress exceeds the elastic limit, these tiny cracks inIn this paper, on the basis of the potential en-rock is propagated quickly until the stress in rockergy principle and the cusp catastrophe theory thereaches the peak stress. Then the mechanics prop-instability of pillars concerning mining effect, a cusperties of rock are changed evidently, and the abilitycatastrophe model of instability of pillars is estab-of resisting deformation is declined with the in-lished. The quantifying of the instability of the systemcreasing of the deformation， that is to say，theis discussed elementarily by an example.strain softening is brought consequently. The2 MECHANICAL MODEL AND CONSTITUTIVEstress- strain curve is shown in Fig. 2.RELATIONAccording to Refs. [5,6]， the constitutive re-latifollows:2.1 Mechanical modelelas中国煤化工There is a softening area(i. e. plastic area) in:TYHC NMH Gee(1)the pillar periphery where the stresses of rock ex-softening stage:①Foundation item: Projeet( 50274074) supported by the National Natural Science Foundation of ChinaReceived date:, 2004 - 02 - 10; Accepted date: 2004 - 10- 09Correspotsponde格.指Jiang teng, Associate professor, PhD ceandidate; Tel: + 86-731 8879612; E mail: ljt@ 21cn. com .LI Jiang -teng. et al: Catastrophe analysis on pillar instability considered mining effectn= Os(6)σ。E is elastic modulus, σc and Ec are the peak stress .Fand corresponding the strain respectively, σg andεgare the stress and corresponding the strain at the(a)inflexion point of the curve after the peak of thestress respectively.At the inflexion point G, the absolute value ofWallslope is called a decline modulus, and the value canrockStopePillarbe calculated by formula below:λ= 3AEm2(7)3 CUSP CATASTROPHE ANALYSIS OF INSTA-BILITY OF PILLAR .3.1 Cusp catastrophe theorySoftening areaIn the catastrophe theory[7.8], the normal potential function can be expressed as follows:IT(x) =二x+Px2+ Qx .(8)where x is the state variable， P and Q are thecontrol variables respectively.ElasticareaWhen the derivative of potential functionI(x) is zero，the equilibrium equation of the sys-tem can be obtained by formula below :IT'(x) =x°+Px +Q =0(9)This is a camber of smooth fold as shown inFig. 3 (a), and a point on the camber denotes abequilibrium state of the system. The quadratic de-1-rivative of potential function for x is written as fol-Fig. 1.Mechanical model of pillarlows:(a) - Schematic diagram for mining; (b)- Section I III"(x) = 3x2+ P(10)On the up and down part of the camber inσ|Fig. 3(a)，nI” (x)<0，and the potential energy isσc tthe minimum and the equilibrium is stable. On themiddle part of the camber, I (x)>0，and the po-tential energy is the maximum and the equilibrium\GσG tis instable. On the juncture of the up, down andmiddle part of the camber, i. e. on the smooth foldOA andOB, I”(x)=0，, and this is a critical equi-librium state as shown in Fig. 3(b), and P and Qsatisfy formula below:4P*+27Q2 =0(11)When the system is on the down part of the cam-Fig. 2Stress-strain curve of pillarber, the equilibrium is stable. With further load-ing，the equilibrium point moves to the fold, and= A[(三)8-1]+ B[(∈)2-1]+1,e>e.thelihrium state. Under theEc(2)sma中国煤化工shifts to the middlewherepart:YHCNM H Glle part of the camberA=(3)is unstable and cannot exist in nature， so，the(m-1)(2m2- 2m+1)point must jump into the corresponding stableB=- 3Am(4)equilibrium point on the up part of the camber,which brings the unstability of system consequent-m=ε(5)ly, as shown in Fig. 3(c). .●104●Journal CSUT Vol. 12 No.1 2005Up =号[量15σxε2 d.xdydx =?"Eu2 (14)xJoJ-J_号2?h“jPUs=|w,dv(15)where u is the axial displacement of the pillar, Fis the axial load of the pillar, h is the height of theUppillar, W, is the unit strain energy of the softening10area, and it can be calculated byW。=. o;de;Middle yA4hsu'+ ;Bo。u3+(1-A- B)ou (16)3h°eu° +DownSo, Us can be obtained byU.= Ac.(B3-a2) + Bor.(b2-a2)u+4h°E23h°e2(a(1-A- B)(b2 -a2)σ。u(17)The total potential energy of the system canbe obtained as follows:I= Au'+ B_u°+Cju2 + Du(18)whereA,= Ao.(b3-a?)(19)4h*ε:B,=_Bσ。 (b"-a2 )(20)3h2e2c=a"E(21)2hBD1=(1-A- B)(b2-a')σ。F(22)(b)3.3 Cusp catastrophe analysis of pillarWe assume that u is a state variable, accord-Hing to the cusp catastrophe theory, when ' = 0,the equilibrium surface M can be obtained as fol-lows:II'= 4Aμu + 3Byu2 + 2Cju+ D、(23 )→QWhenII "=0, the cusp can be obtained as follows:(24)DK Middle. u=u= 4ASubstituting B and A1 into Eqn. (24) leads to .u1 = hme。(25)According to the physical meaning of the stiff-(C)ness in one dimension condition, the stiffness is de-fined as:Fig.3 Cusp catastrophe modelp= dU(a)- Equilibrium camber;(26)du2(b)一Projection of equilibrium camber on plane PQ;(c)- Jump of equilibrium stateThe stiffness of the elastic area is defined as:dUp_ a2 E(27)3.2 Potential function of systemThe potential function of the system can beftening area is written中国煤化工expressed as follows:as fII =- W. +Up+Us(12)YHCNMHGwhere W. is the work done by the external force;1641u2+6B: u(28)U: denotes the strain energy of the elastic area ;When u=u，the stiffness of the softening ar-Us is the strain energy of the softening area.ea can be obtained byλ(b°-a2)一」。Fdz= 一Fiu(13)(29)LI Jiang -teng. et al: Catastrophe analysis on pillar instability considered mining effectIn order to write Eqn. (23) as normal formula: Aui+Bui+cui+Du1where(42)of Eqn. (9)，Taylor series expansion on Eqn. (23)Aulare taken at the cusp and intercepted to thriceterm，and the following formula can be gained.4 ANALYSIS AND DISCUSSION4Amui+ 3Bui十2Ciu十D.+(12Aul +6Bu +2C)(u- u1)+1) According to Eqn. (36)，when k,- 1≤0,24Au1 +6B1._.24A1(u-u])°=0the system is likely to span the bifurcation set,2-(u-u)°+6consequently，the catastrophe occurs probably.(30)So，k,0(the right of the bifurcation set)， x安don't jump; but when Q<0(the left of the bifurca-tion set)，x jumps, in this case, the state is insta-bility. So， the necessary sufficient condition of in-stability of system is described as follows:j4(k,-1)*+(3k,-←- 1)*=0(36)l(3k,-ξ- -1)<01.82.0 2.2b/a3.4 Released energy in instability processThe released energy can be estimated by calcu-Fig. 4 Relationship between k, and b/alating the potential energy difference of point F andpoint H (seen in Fig. 3(c)).5 EXAMPLEWhen P<0 and Eqn. (11) are satisfied, wecan obtain following formula by Eqn. (9).Some parameters obtained by experiment arelisted in Table 1. According to above mention, we(37)can obtain some estimated results listed in Table 2.)t=2(1-k,)士(38)Table 1 Material parametersAs spanning the bifurcation set，the state var-E/GPaσc/ MPaiable appears jump, and the jump value is36. 8190.32X10-31.40.7Qx=xs-x1=3(1-k;)言(39)b/ma/mh/mMiningdepth/mThe jump value of displacement of the pillar中国煤化工100550can be calculated byOu=ulOx=3u(1-k;)言(40)HCNM H G。resultsSo，the released energy can be estimated bykOu/moII/Jusing formula below:0.8531.673 -0. 38340. 77680.31 2.18X 106△II= I(x;)- I(x|) = Au{(x'+Px2+4Qx+c)(41)In underground openings, the critical load p is●106●Journal CSUT Vol. 12 No. 12005decided according to the softening area size， anddroelectric Power, 1998.19(1): 45- 47. (in Chinese)thus the mining order and the stope are arranged[2]Adhikary D P, SHEN B, Duncan Fama M E. A studyof highwall mining panel stability[J]. Internationalproperly.Journal of Rock Mechanics & Mining Sciences, 2002,39: 643 - 659.6 CONCLUSIONS[3] LI Tong-in， TAN Xue-shu, LIU Chuan- wei. MineRock Mechanics[ M]. Chongqing: Chongqing Univer-1) The instability of system is due to the in-sity Press, 1991. (in Chinese)teraction of multi- factors, such as the properties ofZHANG Meng- tao. The mechanism and numericalrock，the size of structure and external force.simulation of rockburst[J]. Chinese Journal of Rock2) Based on the catastrophe theory, the insta-Mechanics and Engineering， 1987， 6(3): 197 - 204.(in Chinese)bility of system is studied when the softening area[5]PAN Yi-shan， ZHANG Meng-tao， LI Guo- zheng.is formed in the pillar by the mining effect, and theThe study of chamber rockburst by the cusp model ofnecessary condition of instability of system and thecatastrophe theory[J ]. Applied Mathematics and Me-jump value of displacement of the pillar and the re-chanics, 1994，15(10): 893 - 900. 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