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英文原文The dynamic impact of rock burst induced by the fracture ofthe thick and hard key stratumFENG Xiaojun,WANG Enyuan,SHEN RongxiState Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Xuzhou 221008, ChinaAbstract:To reveal the dynamic impact of rock bursting induced by the thick and hard roof of gravel as key stratum fractured,based on the key strata and rock control theories,combined with the mechanical load-based dynamic loading with the Law of Conservation of Energy in a system,this paper analysed the static stability and the processing of energy conversion within the system when the key stratum was fracturing.And the key stratum dynamic model of fracturing was established.The results indicate that large amounts of strain energy would be forced into the rock during the process as the key stratum fractures and becomes unstable.The greater the dynamic load factor is,the more elastic energy is forced into the lower rock strata of the key stratum,and the more obvious the dynamic impact of the lower rock strata would be.The size of the dynamic load factor relates to the aspect ratio of key rock masses,the thickness of the overburden,the mining height,the height of the fragmentized rock at the bottom of key stratum and the compaction coefficient of the fragmentized rock under the static during loading conditions.According to the calculation of actual work,the results are consistent with the field tests approximately.The results can provide references to similar conditions.Keywords:key stratum;rock burst;dynamic loading;energy constant1.IntroductionThe key stratum controls the overlying rock strata,even to the surface,with high strength,high thickness,and overall subsidence of overlying strata happen easily when it fractures1-3.The key stratum that has different lithology,strength and thickness impacts the control of the rock strata and the dynamic impact of rock burst after it fractures4-7.As the key stratum,the thick and hard roof with gravel is controlling the movements of the entire overburden,because of its characteristics of high strength,high thickness,fracture would not happen until the area of mining space reaches a value that is big enough8.The whole overburden even surface sinks the surface instantaneously when it fractures, causing a release of strong dynamic force on the rock strata below the key stratum.With the increasing in the strength and depth of mining exploitation,the dynamic impact of rock burst will be more and more important when the key stratum fractures suddenly,and the support designs and the hazards of rock burst induced by dynamic effects continue to increase9-10.The previous have developed comprehensive and systematic studies on rock control,mining damage and the mechanism of water inrush at the overburden separation area11-14.Reference15discussed the impact on the height of water inrush caused by the location of key stratum,and explained the mechanism of water inrush in some coalmines accidents.Reference16discussed the stress of the compound key stratum with the Elastic Thin Plate Theory and Laminated Plate Theory of the mechanics of composite materials,and obtained the limit load when the key stratum was unstable by using plastic limit analysis methods,regarding the combined motion of rock strata overburden as the combined motion of the key strata based on the hard rock strata.Reference17discussed the mechanical mechanism that it could easily cause mining earthquake and rock burst occurrence when the key stratum broke off in positiveO-Xtype,and by monitoring the rock burst induced by the fracture of the key stratum instantly in three-dimensions using the micro seism monitoring system.As the key stratum fracture is a dynamic loading process to the lower rock strata,causing the loading process of rock fracturing is different from the general static loading conditions,it is necessary to conduct researches on the rock burst induced by the fracture of the key stratum.This paper established the dynamic model of the key stratum fractured and derived the dynamic load factor when the dynamic impact loading on the lower strata by combining the theory of the mechanical load-based dynamic loading with the law of conservation of energy of system, we analysed the static stability and the process of energy conversion within the system when the key stratum was fracturing.Confirmatory calculation of actual work was also done,which was consistent with the field tests approximately.The results can provide references to similar conditions.2.Subsidence profile curve of the super mining explorationThe super mining exploration is the critical mining state that the width of goaf is larger than 1.2 to 1.4 times the depth.Compared with the full exploration,the maximum subsidence in the central of the super mining exploration shows:the distribution of the maximum subsidence presents regional characteristic. The fracture interval is closely related to rock mass strength and thickness according to the theory of the model of main roof initial fracture.The greater the intensity and thickness are,the greater the fracture interval is,so the mining conditions are determined by the rock property and thickness of the key stratum. As the thick and hard layer of gravel of the key stratum possesses high strength,high integrity,and it is not easy to fracture,the fracture interval is far greater than the general strength of the key stratum, on which conditions,the movements of rock below the key stratum can be regarded as the super mining exploration.According to the super mining exploration theory,the deformation of the surface subsidence varies from zero to maximum with time,which is similar to the full mining exploration.To facilitate the description of the subsidence profile curve of the mining exploration,the subsidence profile curve of the super mining exploration is divided into three parts“full explorationequal deformationfull exploration”,which is shown in figure 1.The super mining exploration curve equation was described by the sink basin profile function derived by Soviet Unions18.(1)Where Vz is the value of separation area height below the key stratum,Vzm is the maximum value of separation area height below the key stratum,x is the horizontal distance from the centre of the separation area to the expected point of surface subsidence,L is the distance from the centre of the separation area to the boundary of the sink area.Fig.1 The key stratum mining model of the thick and hard layer3.The dynamic impact model of key stratumThe dynamic loading effects are caused to the lower rock strata when the key stratum fractures.The instantaneous stress level is much higher than the static in the stable state,and it forces large strain energy into the rock strata in the form of elastic wave in a short period of time,being superimposed with the original energy.When the key stratum fractures,the more elastic energy is forced into the lower layer of the key stratum,the greater the scope the greater the energy,and the more obvious the dynamic effects to the lower strata.So it can be seen that the greater the dynamic load factor,the higher the stress level,the greater the inputted energy,the more prone to instability for rock.Based on the state of the super mining exploration in the plane model,combined with mechanical mechanism and equilibrium conditions of the main roofbond-beamstructure，it is confirmed that when the lower strata are shocked by the fractureof the key stratum,lots of energy is forced into the lower strata at the earlier stage through the two salient points where they are contact with.With the variation of coefficients,the contact extent increases constantly,and extends to surface gradually.Given that the stress disturbance and the energy release on the fracturing moment make the greatest effects on the surrounding rock masses,we focused on studying the state on the fracturing moment.Based on the above analysis,the key stratum mechanical model on the plane of the thick and hard layer was established.See figure 2.Fig.2 The thick and hard layer of the key stratum model for dynamic impactAccording to the actual engineering conditions,combined with the theory of the key stratum fracture, assuming the key stratum of thick and hard layer as like rigid body,i.e.the effects on the dynamic impact process from the deformation of fracture can be by passed.Considering the variation of the cushion coefficient,the dynamic impacted rock subject to Hookes Law and the elastic modulus is not changed. The key stratum fractures instantly when the mining interval reaches the limit value.The energy releases by fracturing is composed of three parts:within the survey region,external load exerts on the key stratum through the upper boundary of it,which will produce a certain displacement under the action of force.i.e. the energy generated by external load;then the key stratum will produce a displacement downward when it fractures,the change of gravitational potential energy;finally,the kinetic energy of the key stratum before the key stratum fractures.The energy equation was established by the theory of conservation of energy.(2)Where W is the energy generated by external load,T is the kinetic energy before the key stratum fractures,V is the gravitational potential energy,Vd is the energy increase in the rock strata below the key stratum.3.1 Solve the dynamic load factorThe stress and displacement on the moment of the key stratum fracturing is far greater than that in the state of stationary.It is difficult to measure the stress and displacement directly under the actual engineering conditions or by using theoretical arithmetic,while it is easy under the static conditions.The dynamic load factor is defined as ratio of dynamic load stress,displacement and the static stress, displacement.The simultaneous equations that are used to resolve the dynamic load factor are established by combining the static mechanical analysis with the energy constant laws.The concrete steps are as follows.3.1.1 Solve the static mechanical equationThe model shown in figure 2 is analysed with the static mechanical theory,setting up the following equations based on the material mechanical analysis.(3)(4)The static stress is solved based on the above equations,the result is as follow:(5)Where a and b are the length and height of the block mass of the key stratum,?is the maximum angle when the key stratum fractured,?is the depression angle of the block mass of the key stratum,i.e. the arctan a/b shown in figure 2,G is the gravity of the two block masses,Fy is the vertical stress of the boundary of the key block masses below the key stratum,Fst is the static on the static equilibrium conditions, st is the displacement on the static equilibrium conditions.3.1.2 Energy SolutionUnder the action of the static,with the effects of the cushion coefficient,the lower strata occur to some subsidence;and in the process of rock bursting,not only does the static impact the subsidence,but also the displacement subsidence is also impacted by dynamic loading.Taking into account the range of the lower impacted strata is large,and the strata are fragmentized on the whole and fully cracks,it is considered that it is mainly the mutual embedding of the rock fractures in the process of impact,and the real kinetic energy of the impacted strata are small which is seen as zero in the following equations.The formulas follow：(6)(7)(8)Where Fd is the dynamic loading of rock induced by the key stratum fractured,d is the dynamic displacement of rock induced by the key stratum fractured,h is the mining height,ha is the thickness of roof collapse above the coal seam,K is the dynamic load factor,is the compaction factor under static load,usually between 0.05-0.15.The characteristics of the fracturing process and movements are assayed.Beside the energy from the upper loading,there only exists the transformation of the gravitational energy in the model.The W and V are figured out based on the theory of quality differential.(9)(10)Where y is vertical displacement of the key block mass.3.1.3 Solve the dynamic load factorThe simultaneous equations are solved by using the static stress in formula 5 and the energy generated by external load in formula 9 and the potential energy changes in formula 10.(11)Result of the dynamic load factor（12）The dynamic load factor is resolved according to the above formulas.Adding the relationship between the dynamic load stress,displacement and the static stress,displacement,the dynamic load stress and displacement of the impact from the key stratum to the lower layer could be worked out.Combining with the safety factor,the possible maximum stress and displacement after the key stratum fractures can be developed,which may provide references for support designing and safety mining in the next phase.3.1.4 Analysis of resultsThe analysis of dynamic load factor formula shows that the size of the factor is relate with several parameters including the aspect ratio of the key block masses,the thickness of the overburden,the mining height,the thickness of the bottom of the key stratum and the compaction coefficient on static loading conditions.Because the contribution from each parameter is different,the sensitivity of different parameters from the dynamic load factor is different as well.In order to study the sensitivity of different parameters,the formulas are analysed by using the principle of single-factor changing.On the given actual conditions,this paper focused on the sensitive analysis of the static load compaction factor and the vertical overburden pressure so that it could be applied as much as possible to the similar mining areas.The results are shown as follows:Fig.3 The dynamic load factor changes with different cushion coefficientsFig.4 The dynamic load factor changes with different burial depthFigure 3 shows that the dynamic load factor increased with the decrease of cushion coefficients under the actual mining geology conditions,which suggests that the lower strata collapse are equivalent to the formation of a loose buffer space,making the effect of rock burst is relatively little when the key stratum fractures.On the other hand,it explains that it favour the management principle of the mining pressure when the roof suddenly fractures by increasing the fragmentation of the main roof.Figure 4 shows that the dynamic load factor increases with the burial depth decreasing.The main reasons are:as the depth increased,the incremental value of static stress itself is at a very high magnitude.Although the dynamic load factor trends downward around-0.2 with the depth increasing,it does not mean that the dynamic effect decreased.Although the dynamic load factor is relatively smaller,the static load by each part of the dynamic load factor increased greatly.The increase of static stress makes a major contribution to the dynamic impact of rock burst at this time.That is to say although the relative dynamic load decreases,the absolute static increases greatly,so the management and monitoring should be strengthened to prevent rock burst hazards.4.Checking the actual projectA coal mine located in north-eastern China.In the process of deep mining,the mine ground pressure tended to intense,and rock burst occurred many times.The main coal seams are 3#and 9#at present.The hardness of 3#coal is greater(Plats coefficient f=3).The average coal thickness is 3.38meters,and the average depth is-400 meters or so.The average coal seam angle is 30-33.The immediate roof is fine sandstone of 3 to 15meters,which has a 9 meters sandstone thickness.The thickness of the main roof is around 60 meters gravel,which is determined as the key stratum by calculating and analysing.The mining pressure reports show that the main roof?s first weighting interval is around 100 meters under the conditions,so parameteracan be determined to be 50 meters.After the field test on the force of the support when the main roof fractured firstly,it was found that the dynamic pressure coefficient was about 2.2.The following parameters are determined after calculation.The results are shown in table 1.Table.1 The actual project results by using the formula 12ParametersabK2KThe actual dynamic pressure coefficientValue506060.16.722.592.1-2.4The results showed that the theoretical result of the dynamic load factor based on the formula is larger than the measured one.Analysis shows the following reasons:on the one hand,the calculation is completed within the elastic deformation,but the rock plastic deformation would also happen due to the presence of high stress under the actual situations,even though the gravel roofs are hard,thick and strong enough,the joints are still exist in them.The relative motion of joints and the friction would consume energy constantly,which lead to larger calculation result;on the other hand,the energy is inputted to both sides of the key stratum in the form of elastic wave and the energy transfers effectively,which is both ignored in the process of energy calculation,which causes the energy value is larger than the actual result, leading to the K value larger,but some guidance and practice still could be given to engineering references.5.Conclusions1)Under the exploration conditions that the thick and hard roof of gravel is regarded as the key stratum, the separation area matches the subsidence profile curve of the super mining exploration,having the equal deformation at the central region.The model which contacts the lower strata with two salient points when the thick and hard layer of the key stratum fractured is established.Lots of energy is forced into the lower strata at the earlier stage through which they contacted firstly.2)A large amount of strain energy would be forced into the rock during the process of fracture and destabilization of the key stratum.The greater the dynamic load factor was,the more elastic energy that is forced into the lower strata,and the more obvious the dynamic impact on the lower rock strata would be, the more easily it lead to rock burst hazards.3)The size of the dynamic load factor relates to the aspect ratio of key rock masses,the thickness of the overburden,the mining height,the thickness of the rock broken down at the bottom of key stratum and the compaction coefficient on static during loading conditions.The dynamic load factor increases with the cushion coefficients decreasing.The lower strata collapse are equivalent to the formation of a loose buffer space,making the effects of rock shock are relatively little when the key stratum fractured;the dynamic load factor increases with the burial depth decreasing.Increment of the static stress becomes a major contribution to the dynamic impact on rock burst at this time.According to the calculation of actual work, the results are consistent with the field tests.6.References1Qian M G,Miao X X,XU J L,Mao X B.Key strata theory in ground control.Xuzhou:China University of Mining and Technology Press,2003:17-18.(In Chinese)2Qian M G,Miao X X,Xu J L.Study of key stratum theory for control.Journal of China Coal society,1996,21(3):225-229(In Chinese).3Xu J L,Zhu W B,Wang X Z,Yi M S.Classification of key strata structure of overlying strata in shallow coal seam.Journal of China Coal Society,2009,34(7):865-869(In Chinese).4Wei M Y,Wang E Y,Liu X F,Song D Z,Zhang Y.Numerical Simulation of Roof Fracture Under Dynamic Disturbance.Journal of Mining&Safety Engineering,2010,27(4):532-536(In Chinese).5Jia J Q,Wang H T,Tang J X,Li X H,Li K X,Hu G Z.Determination of key strata and interval of roofing breaking of hard and soft composite roofs.Chinese Journal of Rock Mechanics and Engineering,2006,25(5):974-978(In Chinese).6Phillipson S E.Texture,mineralogy,and rock strength in horizontal stress-related coal mine roof falls.International Journal of Coal Geology,2008,75(3):175184(In English).7Shen B,King A,Guo H.Displacement stress and seismicity in roadway roofs during mining-induced failure.International Journal of Rock Mechanics&Mining Sciences,2008,45(5):672688(In English).8Wang L,Guo G L,ZHANG X N,LIU Y X.Evaluation on stability of old goaf foundation by long wall caving based on the key stratum theory.Journal of Mining&Safety Engineering,2010,27(3):57-61(In Chinese).9Tian J S,Gao S.Deformation and failure study of surrounding rocks of dynamic pressure roadways in deep mines.Mining Science and Technology,2010，20(6):850-854(In English).10Xu X F,Dou L M,Lu C P,Zhang Y L.Frequency spectrum analysis on micro-seismic signal of rock bursts induced by dynamic disturbance.Mining Science and Technology,2010，20(5):682-685(In English).11Li S G,Zhang T J.Catastrophic mechanism of coal and gas outbursts and their prevention and control.Mining Science and Technology,2010，20(2):209-214(In English).12Song D Z,Wang E Y,Wang C,Xu F L.Electromagnetic radiation early warning criterion of rock burst based on statistical theory.Mining Science and Technology,2010，20(5):686-690(In English).13Tang J H,Bai H B,Yao B H,Wu Y.Theoretical analysis on water-inrush mechanism of concealed collapse pillars in floor. Mining Science and Technology,2004，23(4):1301-1306(In English).14Zuo Y J,Li S C,Qin S F,Li L P.A catastrophe model for floor water-resisting key stratum instability induced by dynamicdisturbance.Rock and Soil Mechanics,2010,31(8):2361-2366(In Chinese).15Zhu W B,Wang X Z,Kong X,Liu W T.Study of mechanism of stope water inrush caused by water accumulation in overburden separation areas.Chinese Journal of Rock Mechanics and Engineering,2009,8(2):36-311(In Chinese).16Liu K Y,Qiao C S,Zhou H,Teng W Y.Research on combined motion characteristics of overlying rock stratum and position of key stratum.Chinese Journal of Rock Mechanics and Engineering,2004，23(4):1301-1306(In Chinese).17He H,Dou L M,GONG S Y,ZHOU P,XUE Z J.Rock burst rules induced by cracking of overlying key stratum.Chinese Journal of Geotechnical Engineering,2010,32(8):1260-1265(In Chinese).18Kratzsch H.The Mining damages and protection.Beijing:Coal Industry Press,1984:154-155(In Chinese).中文译文由于厚硬关键层断裂所引起的冲击矿压的动态影响冯晓军,王恩源 ,沈荣喜（煤炭资源与安全开采国家重点实验室，中国矿业大学，徐州221008，中国）摘要: 为了揭示以厚硬顶板为关键层的断裂所引起的冲击矿压的动态影响，以关键层和岩层控制理论为依据，结合机械负荷为基础的动态加载系统能量守恒定律，本文分析了关键层断裂系统内的静态稳定性和能量转换的处理。还有关键层断裂的动态模型的建立。结果表明：在关键层断裂而变得不稳定的过程中，会对周围的岩层产生较大的应变能。动态负载因子越大，关键层对下部岩层产生的弹性能越大，下部岩层的动态影响越显著。动态负载因子的大小涉及到关键层岩体的宽高比，覆盖层的厚度，采高，关键层下部破碎岩石的高度，加载条件下的静态破碎岩石的压实系数。根据实际工作中的计算，结果与现场试验的数据相一致，可以作为类似条件的参考依据。关键词:关键层;冲击矿压;动载荷;能量常数1.引言关键层控制着部分或直至地表的上覆岩层，当高强度，高厚度的关键层断裂时，上覆岩层极易发生整体下沉。1-3具有不同岩性，强度，厚度的关键层影响着岩层控制，以及关键层断裂时冲击矿压的动态影响。4-7作为关键层，坚硬厚顶板控制着整个上覆岩层移动,由于它高强度，高厚度的特点，断裂不会发生直至采空区达到足够大的范围。8当关键层断裂时整个上覆岩层直至地表发生塌陷，关键层以下的岩层瞬时释放出强大的动力。.随着矿山开采的强度和深度的增加，当关键层瞬时断裂时冲击矿压的动态影响将变得越来越严重，冲击的动态影响而产生的冲击灾害持续增加。9-10先前已经对岩层控制，采动损害，上覆隔离区的突水机制进行了全面、系统的研究。11-14参考文献15讨论了关键层的位置所引发的突水高度的影响因数，解释了一些煤矿的突水机制。参考文献16讨论了复合关键层的应力与复合材料力学弹性薄板理论和层合板理论，并得到了当关键层不稳定时采用塑性极限分析方法时的极限载荷，将上覆岩层的整体移动看作基于坚硬岩层的关键层的联合移动。参考文献17讨论了当关键层断裂成“O-X”型时，容易导致矿山地震和冲击矿压发生的力学机理，及其通过微地震监测系统以三维的方式监测因关键层突然断裂所引发的冲击矿压的内容。由为关键层断裂是对较低位岩层动态加载的过程，造成岩层压裂加载过程与一般静载条件不同，所以有必要对关键层断裂引发的冲击矿压进行研究分析。本文建立了关键层断裂的动态模型，以及在对下部岩层动态加载的过程中依据加载系统中能量守恒定律推导出了动态冲击载荷的动载系数，同时我们分析了关键层断裂时系统中的静态稳定性和能量转换过程。.另外，我们做了实际工作的验证计算，结果与现场试验相吻合，这个结果可以为类似的条件提供参考。2. 超充分采动的下沉剖面曲线超充分采动是指采空区的长度和宽度均大于平均采深的1.21.4倍的开采状态。与充分采动相比，超充分采动中心的最大下沉量表明：最大沉降分布呈现区域特征。根据老顶初次垮落的理论模型，岩层破断距与岩体的厚度与强度有关。岩体的强度和厚度越大，岩层破断距越大，因此开采条件是由岩石性质和关键层的厚度决定的。因为厚硬关键层拥有高强度，高整体性，岩层不容易断裂，岩层破断距比一般强度的关键层大，在这种情况下关键层下方的岩层移动可看作超充分采动。根据超充分采动理论，随着时间的推移，地面沉降变形的变化从零到最大，这一点和充分采动相似。为了便于采动沉降剖面曲线的描述，将超充分采动的下沉剖面曲线划分为三个阶段：“充分采动-等变形-充分采动”见图1。超充分采动曲线方程的描述是根据苏联的下沉盆地曲线描述的。18(1)其中Vz是关键层下方离层区域的高度值，Vzm是关键层下方离层区域高度的最大值，x是从离层区域的中心到地面沉降的预期点之间的水平距离，L是从离层区的中心到地面下沉边界之间的距离。图1 厚硬关键层采动模型3.关键层的动态影响模型当关键层断裂时，动态负载效应对底部岩层的影响。瞬时应力水平比稳定状态下的静态值高的多，在极短的时间内对岩层产生大量弹性波形式的应变能，并与原始的能量叠加。当关键层断裂时，关键层下部岩层层位越低，所受的弹性能越大，范围愈大，能量越大，岩层层位越低，动态载荷的影响越明显。由此看来动态负载因子越大，应力水平越高，输入的能量越大，岩石越容易产生不稳定性。基于超充分开采的平面模型，结合“砌体梁”结构的力学机制和平衡条件，证实表明当关键层断裂对下部岩层产生震动影响时，在早期通过接触的两个关键点对下部岩层产生大量的能量。随着系数的变化，接触程度不断提高，逐渐延伸到地表。由于应力干扰和断裂时刻的能量释放对围岩产生的影响最大，我们专注研究断裂时刻的状态。基于上述分析，建立了厚硬岩层关键层力学平面模型。见图2。图2 厚硬关键层动态影响模型根据工程的实际条件，结合关键层断裂理论，假设厚硬关键层为刚体结构，则由变形断裂所引起的动态冲击载荷的影响是可以互相传递的。考虑缓冲系数的变化，动态冲击岩石服从胡克定律和弹性模量是没有改变的。当开采区间达到临界值时，关键层瞬时断裂。岩层断裂释放的能量分为三部分：在调查区域内，通过上部边界对关键层施加的外部载荷，将产生一定力的作用下的位移，即外部负载所产生的能量；关键层断裂时会产生向下方向的位移即重力势能发生改变；关键层断裂前关键层的动能。由能量守恒定律建立的能量方程。(2)其中W是由外部负载产生的能量，T是关键层断裂前的动能，V为重力势能，是关键层以下的岩层中的能量增加3.1 求解动态负载因子关键层断裂时应力和位移比静止状态下大的多。应力和位移在实际工程条件下很难直接测量或通过理论计算，但静态条件下很容易计算。动态负载因子定义为动态应力、位移与静态应力、位移的比值。结合静态力学分析与能量常数法建立的联立方程是用来求解动态负载因子的。具体步骤如下：3.1.1 求解静态力学方程图2所示的模型分析了静态力学理论，并依据材料力学分析建立了如下方程：(3)(4)根据上述方程求解静态力学方程，如下：(5)其中a和b是关键层块体的长度和高度，是关键层断裂时的最大角度，是关键层的块体的俯角,如图2所示的a / b表示反正切，G是两个岩块的重力，Fy是关键层以下的关键块群边界的垂直应力，Fst是静态平衡条件下的静力，st是在静力平衡条件下的位移。3.1.2 求解能量静态条件下，受缓冲系数的影响，底部岩层发生沉降，在岩石破裂的过程中，不仅静态影响沉降，动态加载也影响位移的沉降。考虑到下部受影响地层的范围很大以及地层整体上破碎化，充满裂缝，认为主要是由于在冲击的过程中岩石裂隙的相互嵌入引起的。而且真正受影响的地层动能很小，在下面的公式中看作0.公式如下： (6)(7)(8)其中Fd 是由于关键层断裂所引起的动态载荷，d 是由于关键层断裂所引起的动态位移，h是采高，ha是煤层上方顶板塌陷的厚度，K是动态负载因子，是静载荷下的压实系数，介于0.050.15之间。检测压裂过程和运动的特点。除了上部加载的能量，只存在模型中重力势能的转换，根据质量差理论计算出W和V。(9)(10)其中y是关键块体的垂直位移。3.1.3 求解动态负载因子根据公式5中的静态应力，公式9中外部负载产生的能量，公式10中的位能变化联立方程求解。(11)求解动态负载因子（12）根据上述公式求解动态负载因子。添加动态荷载应力，位移和静态应力，位移之间的关系，则从关键层到下部地层冲击影响产生的动态荷载应力和位移就可以计算出来。结合安全系数，关键层断裂后产生的最大可能应力与位移就可以计算出来。这些数据可能会给下阶段的支护设计与安全开采提供参考。3.1.4 分析结果动态负载因子公式分析表明，因子的大小与几个参数相关包括：关键层岩层块体的宽高比，上覆岩层的厚度，采高，关键层底部的厚度，以及静载条件下的压实系数。因为每个参数的影响指标不同，所以动态负载因子不同参数间敏感性也不同。为了研究不同参数的敏感性，应用单因素变量原则进行公式分析。在给定的实际条件下，本文侧重于静载荷压实系数与垂直方向的载重应力的敏感性分析，这样它可以适用于尽可能多的类似的采动区域。结果如下：图3表明动态负载因子随着不同的开采地质条件下的缓冲系数的减小而增大，这说明了下部岩层的垮落相当于形成了一个松散的缓冲空间，当关键层断裂时冲击矿压的影响相对较小。另一方面，它解释了当老顶破碎使顶板突然垮落时这有利于矿山压力的管理。图4表明动态负载因子随着埋藏深度的减小而增加。其主要原因是：随着深度的增加，静态应力本身的增量值有很大的幅度。虽然动态负载因子随着深度的增加，有下降大约0.2的趋势，但这不表明动态效果下降了。虽然动态负载因子相对较小，每一部分的动态负载因子的静态载荷大大增加。这时静载应力的增加促进了冲击矿压的冲击影响。也就是说虽然相对的动载负荷减少，绝对的静载应力大大增加，因此需要加强管理和控制，防止冲击矿压危害的发生。图3动态负载因子随着不同的缓冲系数的变化图4动态负载因子随着不同的埋深的变化4.实际工程检验位于中国东北的一个煤矿。在深部开采的过程中矿山地压不断增加，多次发生冲击矿压。目前主采煤层为3#和9#煤。3#煤的硬度较大（f=3）。平均煤层厚度的3.38米，平均深度为400米左右。煤层的平均倾角为30-33。直接顶为细砂岩315m，其中砂岩厚9m。 老顶的厚度为60m左右的砾石，通过计算分析将此作为关键层。矿山压力报告显示，老顶的初次来压步距约为100m，这样参数“a”可以定为50m. d当老顶初次断裂进行支撑压力现场检测，我们发现动态压力系数约为2.2. 以下参数经过计算后确定。结果如表1。表1使用公式12的实际工程项目成果参数abK2K实际的动态压力系数数值506060.16.722.592.1-2.4结果表明基于公式的动态负载因子的理论结果大于实测。分析表明有以下原因：一方面，计算是在弹性变形过程中进行的，由于实际中存在高应力，岩石会发生塑性变形。即使是坚硬厚层的砂岩顶板裂隙也是存在的。裂隙的相对运动和摩擦会消耗能量从而导致较大的计算结果。另一方面计算过程中忽略了一些能量转移，从而导致能量值大于实际结果，K值加大，但仍然可以为一些工程提供参考。5.总结1)在开采条件下，厚硬岩层顶板被定为关键层，离层区域和超充分开采的下沉剖面曲线相吻合，在中间区域具有平等变形。当关键层破裂时通过两个关键点联系下部岩层的模型的建立。早期通过初次接触点时对下部岩层产生大量能量。2)当关键层断裂和不稳定时期对下部岩层产生大量的应变能。动态负载因子越大，对下部岩层产生的弹性势能越大，对下部岩层产生更明显的动态影响，更有可能发生冲击矿压的危险。3) 动态负载因子的大小涉及到的关键岩体的宽高比，覆盖层的厚度，开采高度，关键层下方破坏岩层的厚度和动载条件下的压实系数。这时静载应力增量对于冲击矿压的动态影响是个很关键的因数。根据实际工作计算与实测结果相一致。6.参考文献1Qian M G,Miao X X,XU J L,Mao X B.Key strata theory in ground control.Xuzhou:China University of Mining and Technology Press,2003:17-18.(In Chinese)2Qian M G,Miao X X,Xu J L.Study of key stratum theory for contr