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英文原文The influence of roadway backfill on the coal pillar strength by numerical investigationHongwei Wang a, Brett A. Poulsen b, Baotang Shen b,n, Sheng Xue b, Yaodong Jiang a(a China University of Mining and Technology, Beijing 100083, China ; b CSIRO Division of Earth Science and Resource Engineering, PO Box 883, Kenmore, QLD 4069, Australia)abstract：Stability of coal pillars and immediate rock mass is of importance in the bord and pillar extraction method underlying critical surface infrastructure. This paper describes the influence of roadway backfill on the coal pillar stability. Based on over 120 numerical models of various pillar height and backfill percentage, it is indicated that the pillar strength will increase with increasing roadway backfill, while the increase in pillar strength is greater for tall pillars than squat pillars for both cohesive and non-cohesive backfill. Modelling suggests that cohesive backfill is more effective in increasing both peak and post-peak pillar strength than non-cohesive backfill. It is also observed that the post-peak response will change from softening to hardening at a certain percentage of either cohesive or non-cohesive backfill.Crown Copyright & 2010 Published by Elsevier Ltd. All rights reserved.Keywords:Roadway; backfillPillar;strengthPost-peak;responseSofteningHardening1. IntroductionIn the bord and pillar mining method, pillars are left to controlmine stability and surface subsidence, and hence to prevent damage to surface or near surface features, e.g., buildings, railways, highways, rivers, pipelines, etc. Improvement of pillar stability can be achieved by leaving large pillars or using secondary support methods such as bolts, mesh and roadway backfill. Roadway backfill as a support method may be achieved through boreholes from the surface, eliminating the need for people to work underground and allowing pillar strength improvement post-mine closure.Considerable research in the stability of pillars has been carried out particularly for coal mines. Several aspects of this research are discussed here. The coal pillar design must take into account the mechanical and physical properties of the coal. In the area where extraction of coal is mainly done by the bord and pillar mining method, empirical data on the relationship between the strength of a coal pillar and the size of the pillar is essential for the design of coal pillars. For this purpose, Bieniawski 1 in 1967 established an empirical relationship between in situstrength of coal and the size of the pillar. Furthermore, in a comprehensive study with the objective of establishing a framework for the in situ strength and deformation properties of coal pillars at a range of width-to-height ratios, Medhurst and Brown 2 in 1998 investigated the effects of size on coal strengthby a series of triaxial compression tests to provide engineers witha practical and systematic method for estimating the mechanicalproperties of a coal seam.The coal pillar strength is fundamental to pillar design: Salamon and Munro 3,4 proposed an empirical approach based on pillar width and height for pillar strength calculation in South African coal mines that has found wide international application. In the 1990s a database of Australian coal pillars 5,6 wasanalysed both in isolation and combination with the South African database used by Salamon and Munro and a similar empirical strength formula was proposed. To investigate the complete load deformation behaviour of coal pillars, a set of rectangular and square coal pillars was tested in situ by Wagner 7,8 in 1974 and 1980. The research highlights the distribution of stress in a coal pillar and the importance of the pillar core of yielded coal which may remain effective at peak strength.In a mine panel consisted of many pillars and roadways, the stability is a function of the strength of each coal pillar and the interaction between pillars. In particular, the residual strength will influence the load transfer between pillars if one of them is to fail. Consequently, pre- and post-failure mechanism of a coal pillar is essential to understand the load transfer between pillars in the mine panel. Extensive experimental test and numerical analyses have been carried out to study the post-peak strength of stress-strain response of a coal pillar 7,9-13.The stability of pillars for a period longer than mine life is of great importance to prevent damage of surface features into the failure. Deformation and strength change of a coal pillar during and after mining were analysed to establish long-term stability ofcoal pillars. An integrated design method was proposed by Bieniawski in 1994 14 to provide researchers with means to improve the stability of a coal pillar as well as to promote the gradual reduction rate of surface subsidence. Some important work has been done by Salamon and Ozbay 15 to investigate the extent and rate of scaling of coal pillars to indentify the life of a coal pillar for assessing the long-term stability of workings. Empirical and theoretical methods have been developed for pillar strength and stability, loading condition, post-failure and time dependent effect. However, the determination of influence of roadway backfill on the coal pillar stability is less widely reported in the literature. Therefore, the understanding of roadway backfill in the coal pillar design is a topic of this study. In this paper, the function of roadway backfill and the coal pillar stability is investigated by numerical analysis using the numerical code FLAC3D 17.This paper presents a study on the influence of backfill on the coal pillar stability and the relationship between the coal pillar strength and the percentage of roadway backfill. The arrangement is listed as follows: in Section 2 empirical formulas used to calculate the coal pillar strength will be discussed and in Section 3 the preparation of this numerical study and the comparison between theoretical and numerical results will be conducted. In Section 4 of this paper, numerical results, including pillar strength increase with increasing cohesive and non-cohesive backfill, will be presented. In Section 5 the relationship between non-failed pillar core and pillar strength and the preliminary analysis of post-failure of the pillar will be investigated at various percentages of backfill.2. Empirical study of pillar strengthPillar strength has been the subject of study for many years and is a fundamental step for investigating the influence of roadway backfill on the coal pillar stability. Researchers have proposed empirical formulae to describe the strength of coal pillars. The common feature of many of these formulae is that they define the coal pillar strength in terms of their relationship between width (w) and height (h) of the pillar and the units for strength in these formulae is MPa.In 1966, based on a large number of pillar observations a database comprising stable and unstable coal pillars was devel-oped in South Africa. From this, Salamon and Munro 3,4 reported the results of their comprehensive study and presented the widely used the coal pillar strength formula:Pillar strength 7:20w0:46 =h0:66 1where w and h are the width and height of the pillar, respectively. In 1996, a similar database of Australia pillars was compiled and Galvin et al. 5,6 developed the following coal pillar strength formula:Pillar strength 8:60w0:51 =h0:84 2It was found that the formula derived from the Australian database resemble closely the results obtained by Salamon and Munro. The similarity in the pillar strength formulae derived from the two national databases prompted a study based on the combined databases in 1999 5 that produced the formula as below:Pillar strength 6:88w0:50 =h0:70 3By comparing the coal pillar strength calculated using formulae (1)-(3), respectively, for width-to-height ratio from 2 to 4, it is found that the coal pillar strength calculated by these three formulae resembled each other closely and the difference is less than 8.3%. At w/h ratio less than 2 Galvin et al. 5 reported geological structure start to dominate pillar strength and when w/h ratio is over 4, pillar strength increase is greater than that produced by formula (3).The original extensive South African database used by Salamon and Munro is widely accepted under the geological conditions of South Africa and likewise the Australian database is accepted under Australian geological conditions. For the general study undertaken in this paper, formula (3) derived from the combined databases was used to calibrate the numerical model and to analyse the stability of pillar in the subsequent section of this paper.3.Numerical model and determination of parameters3.1.Numerical model and failure criterionFLAC3D (Fast Lagrangian Analysis of Continua in 3-Dimen-sions) is employed for simulating the influence of roadway backfill on the coal pillar stability. The pillar analysed in this simulation is one of an infinite number of pillars extending in both axis directions of plan section. Based on this assumption, only a quarter of pillar needs to be modelled taking advantage of symmetry 16. Fig. 1 is an example of the FLAC3D mesh showing a quarter of pillar as well as roof and floor in this study. The element size of pillar is kept constant as 0.5 m 0.5 m 0.5 m,whilst the thickness of roof and floor is selected as 3 m in the numerical model.The chosen failure criterion is the strain softening model which is based on the FLAC3D Mohr-Coulomb model with non-associated shear and associated tension flow rules 16-19. In this model the cohesion, friction, dilation and tensile strength may soften after the onset of plastic yield by a user defined piecewise linear function 20,21, as shown in Fig. 2. In the standard Mohr-Coulomb model, these properties remain constant.3.2.Boundary conditions of numerical modelBecause the focus of this study is the strength of pillar, in situ stress can be ignored as it has little influence on the peak pillar strength(according to the modelling results, thedifferent is less than 0.63% at 6 m mining height at 100 m depth from the surface). In order to obtain the pillar strength under the condition of uniaxial compression, reasonable bound-ary conditions should be set around the numerical model, as indicated in Fig. 1. Since the area represents a quarter of a pillar with symmetry, the displacement of four vertical symmetry planes of model is restricted in the normal direction, and zero vertical displacement is set at the base of model. A constant velocity is applied to the top of the model in the negative z-direction to generate a vertical loading on this model. In this study, the magnitude of constant velocity has been set as 10-5 m/s.3.3.Determination of relevant material parametersIn this study, formula (3) was used to calibrate the material parameter for width-to-height ratios from 2 to 4. The final calibration results give a coal uniaxial compressive strength (UCS) of 4 MPa and tensile strength is set as 1% of the UCS. Final coal softening rates were 90% cohesion reduction over 5% plastic strain and 61 friction angle reduction over 0.5% plastic strain. The final mechanical parameters and softening rate used (see Fig. 2) in this study are listed in Table 1. The focus of this study is the stability of a coal pillar, hence the roof and floor are assumed to be stiffer than coal and are able to deform but not to yield as recommended by Fama et al. 16. Fig. 3 shows the comparison between calibrated numerical result and analytical solutions calculated by formulae (1)-(3). To achieve a reasonable match with formula (3), it is found necessary to have yielding interface at pillar boundary with the roof and floor. The cohesion of interface is 0.5 MPa and the friction angle is 201. The normal and shear stiffness of this interface are 2.0 GPa.Fig. 2. Variation of cohesion and friction angle with plastic strain. Fig. 3. Comparison of pillar strength between numerical and analytical results3.4. Property of backfillA literature review of backfill 22-24 in a coal mining context identified several examples from China where backfill is used to reduce surface subsidence and from America where backfill is used to dispose fly-ash and reduce acid mine drainage. There are two different types of backfill considered in this study, a cohesive backfill of 1 MPa UCS and a non-cohesive backfill with friction angle of 421. The property of backfill used in this study is shown in Table 2. It is believed that a typical backfill strength will be within this range.4. Results of the numerical analysisIn order to quantify the influence of backfill on pillar strength in detail, 120 models (60 models for cohesive backfill and 60 models for non-cohesive backfill) are analysed for coal pillars of square cross section with pillar width of 20 m, mining height of 5-10 m, equivalent to w/h ratio of 2-4 and percentage of backfill from 0% to 90%. Table 3 shows the results of pillar strength for all the cases in this study.4.1. Relationship between pillar strength and cohesive backfillRelationship between pillar strength and cohesive backfill in roadway is presented in Fig. 4.The raw data from Table 3 have been calculated to uniform 10% backfill as shown in Fig. 5.The modelling results suggest that the pillar strength will increase with increasing roadway backfill, while the percentage increase of pillar strength is greater for the taller pillars than squat pillars. For example, modelling predicts that a pillar with w/h ratio of 2 is 97.9% stronger with 80% roadway backfill, while apillar of w/h ratio of 4 is 45.6% stronger with the same relativeamount of backfill. From Fig. 5 it is observed that when the backfill is less than 40-50% the pillar strength increase is less sensitive to pillar height or w/h ratio, while over 40-50% backfill, pillar strength rapidly increase with pillar height. Modelling results suggest this is more pronounced for tall pillars than squat pillars. This observation will be further discussed in section 5.1.4.2. Relationship between pillar strength and non-cohesive backfillRelationship between pillar strength and non-cohesive backfill in roadway is presented in Fig. 6. The raw data from Table 3 have been calculated to uniform 10% backfill as shown in Fig. 7.Modelling results predict a similar response from non-cohesive backfill although the effectiveness is reduced in comparison with cohesive backfill. For example, at w/h ratio of 2 with 80% non-cohesive backfill, pillar strength is increased by 27.8% compared to 97.9% with cohesive backfill. From Fig. 7, it can be seen that when the backfill is less than 40-50% the pillar strength increase is largely independent of pillar height or w/h ratio, while over 40-50% backfill pillar strength rapidly increase at lower w/h ratio.4.3. Comparison of effects of cohesive and non-cohesive backfillAs seen in Fig. 8, it is observed that the pillar strength increase is greater for cohesive backfill than that for non-cohesive backfill either with percentage of backfill or pillar height. The difference between cohesive and non-cohesive backfill increases gradually with increasing backfill as well as coal pillar height. As an example, if the factor of safety (FOS) of a pillar need to be increased from 1.4 to 1.6, then at 100 m depth from the surface, the required strength increase of a 6 m high coal pillar is approximate 14.0% and modelling suggests it will need 33.3% cohesive roadway backfill compared with 66.7% non-cohesive backfill. However, although cohesive backfill is more effective to improve the stability of a coal pillar than that of non-cohesive backfill, the latter could still satisfy the needs of preventing coal pillar failure and surface subsidence depending on geology or other conditions.Regression of modelling results has generated general for-mulas for pillar strength from cohesive and non-cohesive road-way backfill. These formulas are applicable to the pillar dimensions and backfill strengths considered in this study and care should be exercised in using the results outside these parameters.Given the general pillar strength equationPillar strength Kwa =hb4If cohesive roadway backfill of UCS 1.0 MPa is placed to height hbf in roadways surrounding a pillar of width w 20 m and height h such that:then pillar strength can be expressed asPillar strength Kwa =hb 0:842e2:361hbf =h 5If non-cohesive backfill is placed in roadways surrounding pillars to greater than 50 percent roadway height then pillarstrength can be expressed asPillar strength Kwa =hb 0:334e2:356hbf =h 65. Post-failure behaviour of a coal pillar5.1. Relationship between pillar core and pillar strengthWagner 7,8 in 1974 highlighted the progressive failure of a coal pillar from the pillar boundary towards the centre. His work indentified that a coal pillar may have an intact core when the peak pillar strength has been exceeded. In his research, Wagner indicated that theFig. 9. Variation of pillar core under compressive loading for 9 m coal pillar.central portion of a coal pillar plays a significant role on loading capacity of pillars. In this study, the intact core of pillar is defined as pillar core.Fig. 9 shows the variation of the pillar core during the process of loading. A gradual decrease of volume of pillar core can be observed in this figure. The load-bearing capacity of a coal pillar would also be reduced with decreasing volume of pillar core. Fig. 10 presents two linear segments which illustrate the variation of the coal pillar strength with 0-90% of cohesive roadway backfill at 9 m mining height versus the volume of pillar core calculated at approximate 2% of strain. The turning point is at 40% of backfill. Modelling predicts that the volume of pillar core rapidly increases above 40-50% of backfill as shown in Fig. 10. In this example the pillar strength and core volume are linearly related above and below this value. Similar results are also observed at other mining height.From Fig. 10, it can be observed that the volume of the pillar core would increase with the increase in the coal pillar strength. As observed previously, modelling suggests the increase in pillar strength is approximately constant when the percentage of cohesive backfill is less than 40-50%, while more than 40-50% backfill pillar strength increases rapidly. From Fig. 11 it is observed that the volume of pillar core is largely independent of mining height and approximately constant less than 40-50% of backfill. It is concluded that the roadway backfill is more effectivewhen the percentage of backfill is more than 40-50%. According to the relationship between the volume of the pillar core and the pillar strength of 5-10 m coal pillar with 0-90% backfill, it can be seen that the proportion of pillar core volume would increase with increasing roadway backfill. It is also found that volume of the pillar core is largely independent of pillar height when the percentage of backfill is less than 40-50% of backfill. Since a large amount of backfill lead to a significant improvement of pillar stability, the volume of pillar core would increase with increasing pillar height when the percentage of backfill is more than 40-50%. This means that the high percentage of backfill has a great efficiency for improving the stability of a coal pillar.Fig. 10. Variation of coal pillar strength versus the volume of pillar core of 2% strain at 9 m mining height with increasing cohesive backfill.5.2. Analysis of post-peak strength of a coal pillarThe post-failure response of pillar can be classified as strain softening or strain hardening 25 depending on whether the strength of a coal pillar reduces or increases after the peak of the stress-strain curve. Pillar softening or hardening is of great interest since it may influence the stability of the mining panel in that softening pillars will shed most of their load to adjacent pillars while hardening pillars will continue to carry load even after the pillar has failed 25. The load transfer to adjacent pillars will be reduced for hardening pillars. In Fig. 12 a series of average axial stress-strain curve of 6 m coal pillar with cohesive backfill and non-cohesive backfill is shown. The amount of backfill to achieve the transition between strain softening and hardening is presented in Table 4 for all mining height. The values in this table are estimated from the stress-strain curve analysed to 3.5% of plastic strain and are depended on the backfill increment which is in turn depended on the ratio of mining height to the calibrated element size 0.5 m.Fig. 11. Variation of the volume of pillar core with different percentage of backfill.The modelling results suggest that the average axial stress of a coal pillar increases with compression until peak stress is reached. In the post-failure region, pillar stress is reduced as strain increases for all models without backfill. All the model results predict that there would be a change to strain hardening behaviour as percentage of backfill increases. For example, this transition point is 33.3% for cohesive and 50% for non-cohesive backfill with 6 m mining height as shown in Fig. 12 and Table 4. It can also be seen from Table 4 that tall pillars need more backfill than squat pillars to achieve the change from strain softening to hardening for either cohesive backfill or non-cohesive backfill.Fig. 12. Stress-strain curve of 6 m coal pillar with cohesive backfill (a) and non-cohesive backfill (b) for 6 m mining height.6. ConclusionThe observations developed in this paper are intended to provide engineers with initial estimate of the stability of a coal pillar with and without roadway backfill. In satisfying this objective, it is necessary to choose an empirical formula for a general study of the strength of a coal pillar. The well known pillar strength formula of Salamon and Munro, modified by Galvin et al., is the base of the calibration of the numerical model used in this study.The focus of this study is the roadway backfill. Two different types of backfill have been analysed in this study, a cohesive backfill of 1 MPa UCS and a non-cohesive backfill with 421 of friction angle. Results of modelling investigation are that the pillar strength will increase with increasing roadway backfill, while the percentage increase of pillar strength is greater for the tall pillars than squat pillars for both cohesive and non-cohesive backfill. Model results suggest that the increase in pillar strength is approximately constant when percentage of cohesive backfill is less than 40-50%, and when cohesive backfill is more than 40-50% pillar strength increases rapidly. As far as the post-failure behaviour of a coal pillar in this study is concerned, modelling suggests that there exists a certain value of backfill to make a change from strain softening to hardening of a coal pillar. This value is greater for tall pillars than squat pillars and cohesive backfill is more effective than non-cohesive backfill in achieving this transition.AcknowledgementsThe authors wish to express their appreciation to Division of Earth Science and Resource Engineering of CSIRO for providing authors with advanced research conditions. The authors would like to thank Dr. Yucang Wang from CSIRO Division of Earth Science and Resource Engineering for his valuable contribution during this study. The authors are also indebted to the support of CSIRO-MOE Visiting Scholar Fellowship Program, National Basic Research Program (973) of China (no. 2010CB226801), National Natural Science Foundation of China (no. 50704034) and State KeyLaboratory of Coal Resources and Save Mining.中文译文关于巷道回填对煤柱影响强度的数据研究与分析王宏伟a，沈宝堂b（a 中国矿业大学北京校区，邮编100083b 澳大利亚地球科学与资源工程科，昆士兰市，邮箱883，4069号）摘要：煤柱和岩体的稳定性在柱式采煤方法中占据至关重要的位置。本文介绍了巷道回填对煤柱稳定性的影响，数据来源于对120个回填和煤柱模型的系统分析，研究显示煤柱强度会随着回填的加大而加强，然而无论回填与否，高大煤柱的增长比较矮的煤柱增长都要大，模拟显示，凝聚力回填比非凝聚力回填能更有效的增加支柱力量。同时，研究发现，充填后的强度会有一个从软化到硬化的反应关键词：巷道回填，支柱力量，充填后反应，软化，硬化。1引言在房柱式采煤法中，煤柱是用来保持煤体稳定和防止地表沉陷，防止破坏地面建筑物，比如楼房、铁路、公路、河流、管道等。支护强度的提高可通过留设更大的煤柱或者回填技术，比如巷道回填。巷道回填作为一种支护方法可以通过地面钻孔来实现，这种方法省去了人们在地下工作并且等支护完成以后可以及时关闭矿井。稳定支柱的研究已经开展了很多，特别是在煤矿方面，下面讨论下这些研究的几个方面。煤柱的设计尺寸必须考虑煤的物理、化学性质。在这一领域，煤炭开采主要取决于煤柱和煤房尺寸，而煤柱和煤房尺寸设计主要来源于经验。因为这个原因，Bieniawski在1967年建立了一个关于煤的强度和煤柱尺寸的模型。此外，在建立一个煤柱宽度和高度尺寸的研究中，Medhurst和Brown在1998年通过一系列三轴压缩试验阐明了煤柱尺寸对煤炭强度的影响，以此为房柱式采煤提供一个系统、使用的设计思路。煤柱强度对于支柱设计来说是最基本的：Salamon和Munro提出一种基于经验来设计煤柱宽度和高度的方法，此方法已经在南部非洲的煤矿开采中获得广泛应用。在20世纪90年代，Salamon和Munro通过比对澳大利亚和南非煤炭开采的煤柱数据提出了一个煤柱强度计算公式。通过调查煤柱在负载状态下的变形现象，Wagner在1974年1978年观测了一系列长方形煤柱，这项研究重点研究了应力在煤柱中的分布，以及强度峰值在煤柱应力中的重要性。在一个包含支柱和巷道的专题研究中显示，稳定性是每个煤柱和煤柱间相互作用的基础，特别是如果一个煤柱被破坏以后，残余强度会影响煤柱间的负荷转移，因此，煤柱破损前和破损后的特征是了解顶板负载转移所不可或缺的，因此，对煤柱破坏的所需的最大应力及破坏后的应力-应变状况已进行了大量实验和数值分析。比矿山开采年限更长的保护煤柱是确保地表不塌陷所必须的，因此科学家研究了矿山开采期间和开采完毕之后的煤柱强度变化以建立长期、稳定的煤柱。Bieniawski在1994年提出了一个完整的煤柱设计方法来改善煤柱的稳定性同时减少地表沉降速率。Salamon和Ozbay做了很多重要的工作来调查煤柱的缩放率，以求对煤柱长期稳定工作的可能性进行评估。关于支柱的强度和稳定性、负载条件煤柱破坏后的影响已进行了很多研究，然而巷道回填对煤柱稳定性的研究结果却鲜有报道。因此，巷道回填对煤柱设计的影响是本研究的主题。在本文中，巷道回填和煤柱稳定性是用FLAC3D软件分析数据来进行研究的。本文描述了航道回填对煤柱稳定性影响的研究和煤柱强度和巷道回填百分比之间的联系。文章具体编排如下：在第二节将讨论由经验得来的计算煤柱强度的公式，第三节将对数值研究、理论和数值模拟结果进行比较，第四节将会阐述数值计算结果，包括随着粘性和非粘性而引起的支柱力量增加，第五节将阐述煤柱损坏和煤柱强度之间的关系。2煤柱支撑强度的研究煤柱支护强度已经是一个经过多年研究的课题了，并且是巷道回填对煤柱稳定性影响的基本数据。研究者经过多年的研究已经提出具体的公式来计算煤柱强度。这些公式的共同点是它们通过煤柱宽度和高度来定义煤柱强度，煤柱强度的单位是MPa，在1966年，通过对南非境内煤矿中稳定和不稳定煤柱的观测记录，研究者们建立了一个数据库，基于此，Salamon和Munro发表了此项研究成果，并提出了目前广泛使用的煤柱强度的计算公式：煤柱强度=7.20w0.46/h0.66 （1）w煤柱宽度，h煤柱高度1966年，类似的一个基于澳大利亚煤炭支柱的数据库同时建立了起来，Galvin et al提出了下面这个计算煤柱强度的计算公式：煤柱强度=8.60w0.51/h0.70 （2）这个基于澳大利亚方面数据库得出来的公式和Salamon还有Munro提出的公式极其类似，1999年，通过类比基于两个国家数据库得出来的公式，研究者提出了下面计算煤柱强度的公式：煤柱强度=6.88W0.50/h0.70 （3）将宽度和高度从2变换到四，再分别用公式（1）（3）计算煤柱强度，通过对比发现，三个公式的计算结果差别不大，差异率小于8.3%。在W/h小于2时，Galvin发现地质构造决定煤柱强度，当W/h大于4的时候，煤柱强度的增加大于有公式（3）的计算。由Salamon和Munro基于南非数据库总结出来的公式广泛适用于南非的地质条件，同样，基于澳大利亚数据库总结出来的公式也广泛适用于澳大利亚的地址条件。在本文所叙述的研究中，公式（3）被用来进行数值模型的计算，其计算结果被用来分析煤柱的稳定性。3.数学模型及其参数的确定3.1数学模型和煤柱破坏条件研究者采用FLAC3D巷道回填对煤柱稳定性的影响。这个被模拟分析的煤柱是向多方向延伸的模型的其中一个。基于这个假设，仅有四分之一的支柱需要需要建模展示的对称性条件，图一是利用FLAC3D模拟出来的这四分之一巷道支柱展示。模型中煤柱的尺寸为0.5m0.5m0.5m，顶板、底板为3m，研究者选择的破坏条件是基于FLAC3D莫尔 - 库仑模型和非关联的剪切原则得出来的。在这个模型中，凝聚力、摩擦力、张力和抗拉强度在分段线性函数 20,21可能出现塑性变形，如图二所示。而在标准的Mohr-Coulomb模型中，这些属性保持不变。图1 在顶底板作用下，四分之一煤柱出现应力反应的模拟图3.2 数值模型的边界条件由于本研究的重点是煤柱的支护强度，由于它的峰值对顶底板影响很小，所以可以忽略不计（根据模拟结果，所不同的是在地下100m深度，采高6m的情况下，变化幅度小于0.63%）。为了获得在单周压缩条件下的煤柱支撑力量，应根据数值模型设置合理的边界条件，如图一所示。由于该地区的煤柱四分之一具有对称性，四种模式的纵向位移应设置在对称平面的正交方向且垂直位移为零。这个模型产生一个负Z方向的垂直载荷，载荷增加的幅度恒定。在这项研究中，载荷等幅增速被设定为105 m/s。图. 2. 内聚的变异和摩擦渔以塑料张力 图. 3. 柱子力量比较在数字和分析结果之间3.3 有关材料参数的测定在这项研究中，公式（3）被用来校准宽度/高度有2到4的材料的参数。最终结果显示煤柱的单轴抗压强度为4MPa，单轴拉伸强度为单轴压缩强度的1%，如果塑性变形为5%，则凝聚力降低90%；塑性变形为0.5%时，摩擦角减少6度，最终的参数和软化率如表一所示。本项研究的重点是煤柱的稳定性，所以，顶底板变形更能引起煤柱变形，但是并不是像Fama等人模拟的那样。图三显示了标准数值和由公式（1）（3）计算出的结果对照。由此可见，公式（3）的应用条件是煤柱必须和顶底板在煤柱边界有接触。煤柱边界和顶底板的作用力为0.5MPa、摩擦角为20度；这个接触面的正常和剪切强度为2.0 GPa。3.4回填一个在煤炭回填方面的文章列举了几个地区回填的用途，如中国用回填来减少地面沉降，美国则用回填来处理矿渣和减少矿山酸性水的排放，在本项研究中有两种不同类型的回填，一个凝聚力为1MPa的巷道回填和一个摩擦角为42度的非粘性