高级检索

含裂隙煤体能量耗散特征与冲击倾向性研究

朱志洁, 李瑞琪, 汤国水, 韩军, 王来贵, 吴云龙

朱志洁,李瑞琪,汤国水,等. 含裂隙煤体能量耗散特征与冲击倾向性研究[J]. 煤炭科学技术,2023,51(5):32−44

. DOI: 10.13199/j.cnki.cst.2022-2242
引用本文:

朱志洁,李瑞琪,汤国水,等. 含裂隙煤体能量耗散特征与冲击倾向性研究[J]. 煤炭科学技术,2023,51(5):32−44

. DOI: 10.13199/j.cnki.cst.2022-2242

ZHU Zhijie,LI Ruiqi,TANG Guoshui,et al. Research on energy dissipation characteristics and coal burst tendency of fissured coal mass[J]. Coal Science and Technology,2023,51(5):32−44

. DOI: 10.13199/j.cnki.cst.2022-2242
Citation:

ZHU Zhijie,LI Ruiqi,TANG Guoshui,et al. Research on energy dissipation characteristics and coal burst tendency of fissured coal mass[J]. Coal Science and Technology,2023,51(5):32−44

. DOI: 10.13199/j.cnki.cst.2022-2242

含裂隙煤体能量耗散特征与冲击倾向性研究

基金项目: 

辽宁省自然科学基金计划资助项目(2023-MS-318);煤炭资源高效开采与洁净利用国家重点实验室开放基金课题资助项目(2021-CMCUKF016);安徽省高校学科(专业)拔尖人才学术资助项目(gxbjZD2022134)

详细信息
    作者简介:

    朱志洁: (1986—),男,辽宁调兵山人,副教授,硕士生导师。E-mail: zhuzhijie@lntu.edu.cn

  • 中图分类号: TP315

Research on energy dissipation characteristics and coal burst tendency of fissured coal mass

Funds: 

Natural Science Foundation of Liaoning Province (2023-MS-318); State Key Laboratory of Efficient Mining and Clean Utilization of Coal Resources Open Fund Project (2021-CMCUKF016); Top Talent Academic Program of Anhui Province (gxbjZD2022134)

  • 摘要:

    冲击倾向性是煤岩体能否发生冲击地压的自然属性,裂隙的分布对其有重要影响。为研究煤体原始裂隙对能量耗散特征和冲击倾向性的影响机制,采用PFC2D数值模拟方法,对不同裂隙类型的煤体试件进行了单轴压缩测试。研究表明:①随着裂隙倾角的增大,宏观力学参数抗压强度和弹性模量均表现为先减小后增大的趋势;当裂隙倾角为 30°时,两者都取得最小值。不同裂隙类型宏观力学参数大小关系为:非共面平行双裂隙试件<单裂隙试件<共面断续双裂隙试件。②弹性应变能和总应变能的变化规律与宏观力学参数相似。平行非共面的裂隙试件在裂隙之间形成了能量耗散结构,共面断续双裂隙试件裂隙之间形成了能量集中区,揭示了不同裂隙类型弹性能量大小关系的内在原因。③从煤岩体储存弹性能的能力和破坏后释放弹性能的能力2个角度对冲击倾向性进行分析,提出了弹性能储存率和弹性能释放率2个冲击倾向性指标。④随着裂隙倾角的增大,弹性能储存率和弹性能释放率均表现为先减小后增大的趋势;当裂隙倾角为 30°时,两冲击倾向性指标都取得最小值。不同裂隙类型冲击倾向性大小关系为:非共面平行双裂隙试件<单裂隙试件<共面断续双裂隙试件。裂隙的分布形态对煤体的冲击倾向性具有显著影响,在煤岩体的冲击倾向性评价和冲击地压防治中应考虑裂隙这一因素。

    Abstract:

    Coal burst tendency is the natural property of whether coal rock mass can have coal burst, and the distribution of fissures has an important influence on it. In order to study the influence mechanism of the original coal fissures on the energy dissipation characteristics and coal burst tendency, the PFC2D numerical simulation method was used to conduct uniaxial compression tests on coal specimens with different fracture types. The results show that: ①With the increase of the inclination angle of the fissure, the compressive strength and elastic modulus of the macroscopic mechanical parameters show a trend of decreasing first and then increasing; when the inclination angle of the fissure is 30, both of them reach the minimum value. The relationship between the macro-mechanical parameters of different fracture types is: non-coplanar parallel double-fissure specimen < single-fissure specimen < co-planar discontinuous double-fissure specimen.②The variation law of elastic strain energy and total strain energy is similar to that of macroscopic mechanical parameters. The parallel and non-coplanar fracture specimens form an energy dissipation structure between the fissures, and the coplanar discontinuous double-fissure specimen forms an energy concentration area between the fissures, revealing the intrinsic reason for the relationship between the elastic energy of different fracture types. ③The coal burst tendency is analyzed from the two perspectives of the ability of coal and rock to store elastic energy and the ability to release elastic energy after failure, and two coal burst tendency indicators, elastic energy storage rate and elastic energy release rate, are proposed. ④With the increase of the fissure inclination angle, both the elastic energy storage rate and the elastic energy release rate showed a trend of first decreasing and then increasing; when the fissure inclination angle was 30, the two coal burst propensity indexes both achieved the minimum value. The relationship between the coal burst tendency of different fracture types is: non-coplanar parallel double-fissure specimen < single-fissure specimen < co-planar discontinuous double-fissure specimen. The distribution of fissures has a significant coal burst on the coal burst tendency of coal mass, and the factor of fissures should be considered in the evaluation of the coal burst tendency of coal and rock mass and the prevention and control of rock burst.

  • 煤炭资源是我国能源构成中至关重要的部分,近些年诸多矿区资源开采逐渐由浅部转向深部[1-3],煤炭安全高效开采是保障该资源正常供应的首要问题。采煤过程中会导致原岩应力状态失去平衡[4-6],继而使得底板岩层受到一定程度的扰动,底板产生的裂隙可能贯穿至含水层,直接影响采煤工作面的安全,甚至造成严重的生命财产损失,底板破坏深度的准确判断[7-8]是矿井防治底板突水的有效手段。

    目前国内外学者在煤层底板破坏深度研究方面取得的成果丰硕,理论方面以塑性滑移线场为主。鲁海峰等[9]运用瑞典条分法搜出危险滑面并获取稳定系数和滑面最大深度,为正确使用弹塑性理论求解底板最大破坏深度提供了依据;张金才等[10]根据塑性滑移线场理论构建了底板受力模型,给出了采面底板破坏形态;孙建[11] 以塑性滑移线场为依托建立了倾斜煤层的底板破坏力学模型,分析了沿煤层倾斜方向底板三区破坏形态;李昂等[12-13]针对董家河煤矿带压水上采煤问题,基于塑性滑移线场理论给出了煤层底板破坏深度解析解。上述理论计算方法中利用塑性滑移线场理论进行底板破坏深度分析时均将底板视为单一岩体,然而底板是由不同岩性岩层构成的复合结构[14-16],且塑性滑移线场理论是基于岩层平均内摩擦角和层厚等参数去求解底板最大破坏深度,若按照传统的单一岩层底板塑性滑移线场理论计算会因为忽略各岩层不同内摩擦角和岩性而产生较大误差。

    随着采深的日益增加,煤矿安全开采受强烈开采扰动和高承压水的影响日趋严重化与复杂化[17-20]。虽然李昂等[21-22]对于平煤某矿深部开采建立了3层复合底板塑性滑移线场力学计算模型,并对底板破坏深度解析式进行推导,但该计算过程繁杂、工作量大,部分工况计算结果因泰勒展开式的运用只能取近似值。如果底板隔水层厚度较大、岩性较为复杂,根据柱状把岩性相近或岩层厚度较薄的岩层划分为1层,摩擦角取其平均值;岩层较厚且岩性差异较大的岩层划分为另外层,但最多只能渐划为3层,适用性较差,难以满足深部开采中多岩性结构底板破坏深度的分析计算。

    对于深部煤层开采迫切需要一种计算精度高、方便快捷的多岩性结构底板最大破坏深度计算方式。基于此,在3层复合结构底板的力学模型基础上,对多层结构底板破坏深度解析解进行推导。对各计算工况进行优化,提出了多层结构底板的5种计算工况,形成了计算逻辑流程判别图,利用Matlab语言开发了多层结构底板破坏深度计算系统V1.0软件,并通过应用分析部分验证了该计算系统的严谨性与精确性,对推动深部煤岩开采面底板水害防治技术有较大作用。

    考虑到现场实际底板岩层的复杂性,采用n层模型计算底板最大破坏深度及采面至最大破坏深度的水平距离。而底板最大破坏深度与底板超前塑性破坏长度密切相关,采用理论值与实测值相结合的方法确定底板超前塑性破坏长度。理论值采用经典塑性滑移线力学模型计算:

    $$ x_0=\frac{H_2}{\theta^*}\left\{\left[\left(K\gamma H_1+\frac{C_1}{f_1}\right)\left(\dfrac{f_1}{C_1+f_1\sigma_{\mathrm{c}}}\right)\right]^{\tfrac{\theta^*}{2k_{\mathrm{p}}f_1}}-1\right\} $$ (1)

    式中:x0为底板超前塑性破坏长度;H1为煤层的埋深; H2为煤层的采高;K为支承压力峰值系数(由现场实测得到);$ \gamma $ 为煤层覆岩的平均容重;$ \theta^* $为煤层的变形角;$ C_1 $为煤层与顶底板接触面上的黏聚力;f1为煤层与顶板界面间的摩擦因数;$ \sigma_c $为单轴压缩时煤体的极限抗压强度,$ \sigma_{\mathrm{c}}=\dfrac{2C\cos\ \varphi_{ }}{1-\sin\ \varphi_{ }} $,C为煤体的黏聚力;$\varphi $为内摩擦角;$ k_{\mathrm{p}}=\dfrac{1+\sin\ \varphi_{ }}{1-\sin\ \varphi_{ }} $。

    深部煤岩开采面底板岩性不尽相同,使得计算工况也愈发复杂。以岩体主动极限区深度H0所处岩层位置作为判别依据,对多层结构底板进行5种工况划分。n层结构底板破坏深度计算简图如图1所示。

    图  1  n层结构底板破坏深度计算简图
    Figure  1.  Sketch of damage depth calculation of n-story structure floor

    n=1,即将底板看作单一岩层,学者张金才[10]认为当煤层被采出后,采空区四周岩体所承受的垂直应力增大,当垂直应力作用范围的底板 (如图2中I区所示) 下岩体所受采动应力超过其最大强度时,底板岩体结构将发生塑性破坏,由于该岩体在法向方向上受到挤压作用,将会发生水平方向的拉伸,变形后的岩体压迫过渡区域(如图2中Ⅱ区所示)内的底板岩体,并将附加应力传递到该区域。过渡区域岩体受其压迫后,将会继续压迫被动区域内的底板岩体 (如图2中Ⅲ区所示)。其中,主动和被动区各由两条直线构成,过渡区滑移线分别由对数螺旋线和自O点的放射线组成。

    图  2  极限状态下底板破坏深度计算简图
    Figure  2.  Sketch of damage depth calculation for bottom slab in limit condition

    螺线方程为

    $$ r=r_0\mathrm{e}^{\theta\tan\ \varphi_1} $$ (2)

    式中:rO到对数螺旋线BC之间的距离;r0OB之间的距离;$ \theta $为相邻底板岩层间的夹角,此时代表OBOE之间的夹角;$ \varphi_{1} $为底板岩层的平均内摩擦角。

    根据三角函数公式能求出第1段对数螺旋线的起点半径r0

    $$ OB = {r_0} = \frac{{{x_0}}}{{2\cos \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2}} \right)}} $$ (3)

    然后根据得出的起始半径r0rθ值,经过推导得出最终的单层结构底板最大滑移破坏深度的计算公式为

    $$ h_0=\frac{x_0}{2\cos\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}\right)}\mathrm{e}^{\left(\tfrac{\pi}{4}+\frac{\varphi_1}{2}\right)\tan\ \varphi_1}\cos\ \varphi_1 $$ (4)

    图1几何关系知开采面至最大破坏深度水平距离OG为

    $$ OG=\frac{x_0}{2\cos\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}\right)}\mathrm{e}^{\left(\tfrac{\pi}{4}+\frac{\varphi_1}{2}\right)\tan\ \varphi_1}\sin\ \varphi_1 $$ (5)

    此求解方法是将底板简化为单一岩层,但实际工程中底板岩体由多种岩性的岩体组成,岩层之间内摩擦角并不相同,采用单一岩层底板计算最大破坏深度存在较大误差。

    H0H1,即主动极限区深度小于首层岩层厚度时,因结构底板各内摩擦角不同会导致出现多条对数螺旋线半径,采用层数n确定待求对数螺旋线半径的条数,由图1几何关系知底板主动极限区深度H0

    $$ {H_0} = \frac{{{x_0}}}{2} \cdot \tan \left( {\frac{\pi }{4} + \frac{{{\varphi _1}}}{2}} \right) $$ (6)

    O点作为2段对数螺旋线的旋转中心点,且交界处螺旋线半径保持不变,根据三角函数公式能求出第1段对数螺旋线的起点半径r0

    $$ OB = {r_0} = \frac{{{x_0}}}{{2\cos \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2}} \right)}} $$ (7)

    第2段对数螺旋线起始半径r1由方程组(8)联立求解:

    $$ \left\{\begin{gathered}r_1=\frac{H^1}{\sin\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}+\theta_1\right)} \\ r_1=r_0\mathrm{e}^{\theta_1\tan\ \varphi_1} \\ \end{gathered}\right. $$ (8)

    式中:$ \theta_{1}\text { 为对数螺旋线半径 } r_{0} \text { 与 } r_{1} \text { 间夹角。化简为 } $

    $$ H^1=r_0\mathrm{e}^{\theta_1\tan\ \varphi_1}\cdot\sin\left(\frac{\pi}{4}+\frac{\varphi_1}{2}+\theta_1\right) $$ (9)

    对于式(8)中的未知数θ1采用泰勒展开求得近似解,根据泰勒展开式:

    $$ \mathrm{e}^{\theta_1\tan\ \varphi_1}\approx1+\theta_1\tan\ \varphi_1 $$ (10)
    $$ \sin \left( {\frac{\pi }{4} + \frac{{{\varphi _1}}}{2} + {\theta _1}} \right) \approx \left( {\frac{\pi }{4} + \frac{{{\varphi _1}}}{2} + {\theta _1}} \right) - \dfrac{{{{\left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2} + {\theta _1}} \right)}^3}}}{6} $$ (11)

    将式(10)和式(11)代入式(9),可得

    $$ H^1\approx r_0\cdot(1+\theta_1\tan\ \varphi_1)\cdot\left[\begin{array}{*{20}{c}}\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}+\theta_1\right)- \\ \dfrac{\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}+\theta_1\right)^3}{6}\end{array}\right] $$ (12)

    展开得

    $$ \begin{array}{*{20}{c}}\left(-\dfrac{1}{6}\mathrm{tan\ }\varphi_1\right)\theta_1^4+\left[-\dfrac{1}{6}-\dfrac{1}{2}\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}\right)\mathrm{tan}\ \varphi_1\right]\theta_1^3+ \\ \left[\mathrm{\mathrm{tan}}\ \varphi_1-\dfrac{1}{2}\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}\right)-\dfrac{1}{2}\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}\right)^2\mathrm{tan}\ \varphi_1\right]\theta_1^2+ \\ \left[\begin{array}{*{20}{c}}1-\dfrac{1}{2}\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}\right)^2+\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}\right)\mathrm{tan}\ \varphi_1 \\ \\ -\dfrac{1}{6}\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}\right)^3\mathrm{tan\ }\varphi_1\end{array}\right]\theta_1+ \\ \left[\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}-\dfrac{H_1}{r_0}-\dfrac{1}{6}\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}\right)^3\right]=0\end{array} $$ (13)

    在式(13)中,令

    $$ \begin{gathered}a=-\frac{1}{6}\tan\ \varphi_1,b=-\frac{1}{6}-\frac{1}{2}\left(\frac{\pi}{4}+\frac{\varphi_1}{2}\right)\tan\ \varphi_1, \\ c=\tan\ \varphi_1-\frac{1}{2}\left(\frac{\pi}{4}+\frac{\varphi_1}{2}\right)-\frac{1}{2}\left(\frac{\pi}{4}+\frac{\varphi_1}{2}\right)^2\tan\ \varphi_1, \\ d=1-\frac{1}{2}\left(\frac{\pi}{4}+\frac{\varphi_1}{2}\right)^2+\left(\frac{\pi}{4}+\frac{\varphi_1}{2}\right)\tan\ \varphi_1- \\ \frac{1}{6}\left(\frac{\pi}{4}+\frac{\varphi_1}{2}\right)^3\tan\ \varphi_1, \\ e=\frac{\pi}{4}+\frac{\varphi_1}{2}-\frac{H_1}{r_0}-\frac{1}{6}\left(\frac{\pi}{4}+\frac{\varphi_1}{2}\right)^3 \\ \end{gathered} $$

    则式(13)表示为

    $$ a\theta _1^2 + b\theta _1^3 + c\theta _1^4 + d{\theta _1} + e = 0 $$ (14)

    由一元四次方程的费拉里解法可知,式(14)存在4个解,考虑到所求解是对数螺旋线的夹角,因此实际取值需为正数,最后经验算,实际存在的解仅1个,即

    $$ \begin{array}{*{20}{c}} {{\theta _1} = \dfrac{{ - b}}{{4a}} + \dfrac{1}{2}\sqrt {\dfrac{{{b^2}}}{{4{a^2}}} - \dfrac{{2c}}{{3a}} + {\text{\Delta }}} } -\\ { \dfrac{1}{2}\sqrt {\dfrac{{{b^2}}}{{2{a^2}}} - \dfrac{{4c}}{{3a}} - {\text{\Delta }} + \dfrac{{ - \dfrac{{{b^3}}}{{{a^3}}} + \dfrac{{4bc}}{{{a^2}}} - \dfrac{{8 d}}{a}}}{{4\sqrt {\dfrac{{{b^2}}}{{4{a^2}}} - \dfrac{{2c}}{{3a}} + {\text{\Delta }}} }}} } \end{array} $$ (15)

    式中

    $$ \begin{gathered}{\text{\Delta }} = \frac{{\sqrt[3]{2}{{\text{\Delta }}_1}}}{{3\alpha \sqrt[3]{{{{\text{\Delta }}_2} + \sqrt[2]{{ - 4{{\text{\Delta }}_1}^3 + {{\text{\Delta }}_2}^2}}}}}} + \\ \frac{{\sqrt[3]{{{{\text{\Delta }}_2} + \sqrt[2]{{ - 4{{\text{\Delta }}_1}^3 + {{\text{\Delta }}_2}^2}}}}}}{{3\sqrt[3]{2}\alpha }}\\ {{\text{\Delta }}_1} = {c^2} - 3bd + 12ae; \\ {{\text{\Delta }}_2} = 2{c^3} - 9bcd + 27a{d^2} + 27{b^2}e - 72ace \end{gathered} $$

    式中:αGE与对数螺旋线r之间夹角。

    从而计算出第2段螺旋线起始半径r1,将所求θ1r1代入方程组(16):

    $$ \left\{\begin{gathered}r_2=\dfrac{H'+H''}{\sin\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}+\theta_1+\theta_2\right)} \\ r_2=r_1\mathrm{e}^{\theta_2\tan\ \varphi_2} \\ \end{gathered}\right. $$ (16)

    式中:θ2 为对数螺旋线半径r1与r2间夹角。

    依次计算$ \theta_I,\theta_2,\cdots\theta_{\mathrm{\mathit{n}}-2},r_1,r_2,\cdots r_{\mathit{\mathrm{\mathit{n}}}-2} $,将所求$ \theta_I,\theta_2,\cdots\theta_{\mathrm{\mathit{n}}-2},r_{\mathit{\mathrm{\mathit{n}}}-2} $代入方程组(17):

    $$ \left\{\begin{gathered}r_{\mathrm{\mathit{n}}-1}=\dfrac{H^1+H^2+...+H^{n-1}}{\mathrm{sin}\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}+\theta_1+\theta_2+\theta_3+...\theta_{\mathrm{\mathit{n}}-1}\right)} \\ r_{\mathrm{\mathit{n}}-1}=r_{\mathrm{\mathit{n}}-2}\mathrm{e}^{\theta_{\mathit{\mathrm{\mathit{n}}}-1}\mathrm{tan}\ \varphi_{\mathrm{\mathit{n}}-1}} \\ \end{gathered}\right. $$ (17)

    式中:θn–1 为对数螺旋线半径rn–2与rn–1间夹角。

    第2段随后的螺旋线起始半径及螺旋线夹角$ r_{\mathrm{2}},\theta_{\mathrm{2}},r_{\mathrm{3}},\theta_{\mathrm{3}},\cdots,r_{\mathrm{\mathit{n}}-1},\theta_{\mathrm{\mathit{n}}-1} $的求解方法与r1相同,最终结果采用数学计算软件Mathematic进行求解。

    在△OEG

    $$ OE=r=r_{\mathrm{\mathit{n}}-1}\mathrm{e}^{\theta\tan\ \varphi_{\mathrm{\mathit{n}}}} $$ (18)
    $$ GH=h=r\cos\ \alpha=r_{\mathrm{\mathit{n}}-1}\mathrm{e}^{\theta t\tan\ \varphi_{\mathrm{\mathit{n}}}}\cos\left(\begin{array}{*{20}{c}}\theta+\theta_1+\theta_2+\cdots+ \\ \theta_{\mathit{\mathrm{\mathit{n}}}-1}+\dfrac{\varphi_1}{2}-\dfrac{\pi}{4}\end{array}\right) $$ (19)

    式中:$ \theta $为相邻底板岩层间的夹角,此时代表对数螺旋线 rrn–1之间夹角。

    当$ \dfrac{\mathrm{d}h}{\mathrm{d}\theta}=0 $ 时,h 达到最大破坏深度h0,解得

    $$ \theta=\varphi_{\mathrm{\mathit{n}}}+\frac{\pi}{4}-\frac{\varphi_1}{2}-\theta_1-\theta_2-\cdots\theta_{\mathrm{\mathit{n}}-1} $$ (20)

    将式(17)、式(18)和式(20)的解代入式(19),得出多层结构底板最大滑移破坏深度的计算公式:

    $$ h_0=r_{\mathrm{\mathit{n}}-1}\mathrm{e}^{\left(\varphi_{\mathrm{\mathit{n}}}+\tfrac{\pi}{4}-\tfrac{\varphi_1}{2}-\theta_1-\theta_2-\theta_3-...-\theta_{\mathrm{\mathit{n}}-1}\right)\mathrm{tan}\ \varphi\mathit{\mathit{\mathit{_{\mathrm{\mathit{n}}}}}}}\mathrm{cos\ }\varphi_{\mathrm{\mathit{n}}} $$ (21)

    开采面至最大破坏深度水平距离OG:

    $$ OG=r_{\mathrm{\mathit{n}}-1}\mathrm{e}^{\left(\varphi\mathit{\mathit{_{\mathrm{\mathit{n}}}}}+\tfrac{\pi}{4}-\tfrac{\varphi_1}{2}-\theta_1-\theta_2-\theta_3-...-\theta_{\mathrm{\mathit{n}}-1}\right)\mathrm{tan}\ \varphi_{\mathrm{\mathit{n}}}}\sin\ \varphi_{\mathrm{\mathit{n}}} $$ (22)

    H0=H1,主动极限区深度等于首层岩层厚度,即主动极限区深度H0处于第1层岩体和第2层岩体交界处。由图1几何关系知底板主动极限区深度H0

    $$ {H_0} = \frac{{{x_0}}}{2} \cdot \tan \left( {\frac{\pi }{4} + \frac{{{\varphi _1}}}{2}} \right) $$ (23)

    O点作为2段对数螺旋线的旋转中心点,且交界处螺旋线半径保持不变,根据三角函数公式能求出第1段对数螺旋线的起点半径r0

    $$ OB = {r_0} = \frac{{{x_0}}}{{2\cos \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2}} \right)}} $$ (24)

    各段螺旋线起始半径及螺旋线夹角$ r_1,\theta_1, r_2,\theta_2,r_3,\theta_3,\cdots,r_{\mathrm{\mathit{n}}-3},\theta_{\mathit{\mathrm{\mathit{n}}}-3} $的求解方法与工况2相同,最终结果采用数学计算软件Mathematic进行求解。

    将所求$ \theta_{\mathrm{1}},\theta_2,\cdots,\theta_{\mathrm{\mathit{n}}-3},r_{\mathit{\mathrm{\mathit{n}}}-3} $代入方程组(25):

    $$ \left\{\begin{gathered}r_{\mathit{\mathrm{\mathit{n}}}-2}=\dfrac{H^1+H^2+...+H^{\mathrm{\mathit{n}}-1}}{\mathrm{sin}\left(\dfrac{\pi}{4}+\dfrac{\varphi_1}{2}+\theta_1+\theta_2+\theta_3+...\theta_{\mathrm{\mathit{n}}-2}\right)} \\ r_{\mathit{\mathit{\mathrm{\mathit{n}}}}-2}=r_{\mathit{\mathit{\mathit{\mathrm{\mathit{n}}}}}-3}\mathrm{e}^{\theta_{\mathrm{\mathit{n}}-2}\mathrm{tan}\ \varphi_{\mathit{\mathrm{\mathit{n}}}-1}} \\ \end{gathered}\right. $$ (25)

    式中:θn-2 为对数螺旋线半径rn-3与rn-2间夹角。

    在△OEG

    $$ OE=r=r_{\mathit{\mathrm{\mathit{n}}}-2}\mathrm{e}^{\theta\tan\ \varphi_{\mathrm{\mathit{n}}}} $$ (26)
    $$ \begin{gathered}\qquad GH=h=r\cos\ \alpha= \\ r_{\mathit{\mathit{\mathrm{\mathit{n}}}}-2}\mathrm{e}^{\theta t\tan\ \varphi_{\mathrm{\mathit{n}}}}\cos\left(\begin{gathered}\theta+\theta_1+\theta_2+\cdots+ \\ \theta_{\mathrm{\mathit{n}}-2}+\frac{\varphi_1}{2}-\frac{\pi}{4} \\ \end{gathered}\right) \\ \end{gathered} $$ (27)

    式中:$ \theta $为相邻底板岩层间的夹角,此时代表对数螺旋线 rrn–2之间夹角。

    当$ \dfrac{\mathrm{d}h}{\mathrm{d}\theta}=0 $ 时,h 达到最大破坏深度h0,解得

    $$ \theta=\varphi\mathit{\mathit{_{\mathrm{\mathit{n}}}}}+\frac{\pi}{4}-\frac{\varphi_1}{2}-\theta_1-\theta_2-\cdots\theta_{\mathrm{\mathit{n}}-2} $$ (28)

    将式(25)、式(26)和式(28)的解代入式(27),得出多层结构底板最大滑移破坏深度的计算公式:

    $$ h_0=r_{\mathrm{\mathit{n}}-2}\mathrm{e}^{\left(\varphi\mathit{\mathit{_{\mathrm{\mathit{n}}}}}+\tfrac{\pi}{4}-\tfrac{\varphi_1}{2}-\theta_1-\theta_2-\theta_3-...-\theta_{\mathrm{\mathit{n}}-2}\right)\mathrm{tan\ }\varphi_{\mathrm{\mathit{n}}}}\mathrm{cos}\ \varphi_{\mathrm{\mathit{n}}} $$ (29)

    开采面至最大破坏深度水平距离OG

    $$ OG=r_{\mathit{\mathrm{\mathit{n}}}-2}\mathrm{e}^{\left(\varphi_{\mathrm{\mathit{n}}}+\tfrac{\pi}{4}-\tfrac{\varphi_1}{2}-\theta_1-\theta_2-\theta_3-...-\theta_{\mathrm{\mathit{n}}-2}\right)\mathrm{tan}\ \varphi_{\mathrm{\mathit{n}}}}\sin\ \varphi_{\mathrm{\mathit{n}}} $$ (30)

    图1几何关系

    $$ \overline{N^{t} A^{\prime}}=x_{0}-\sum_{i=1}^{n-1} 2 H^{{i}} \tan \left(\frac{\pi}{4}-\frac{\varphi_{{i}}}{2}\right) $$ (31)

    在△N’A’B

    $$ \overline{OA'}=\frac{\overline{N'A'}\cdot\tan\left(\dfrac{\pi}{4}+\dfrac{\varphi_{\mathit{\mathit{\mathit{\mathrm{\mathit{i}}}}}+1}}{2}\right)}{2} $$ (32)
    $$ H_0=\overline{O A^{\prime}}+\sum_{i=1}^{n-1} H^{{i}} $$ (33)

    $ \displaystyle\sum_{i=1}^{n-2}H\mathit{^{\mathrm{\mathit{i}}}} < H_0\leqslant\displaystyle\sum_{i=1}^{n-1}H^{\mathrm{\mathit{i}}} $,即主动极限区深度H0处于岩体第2层至第n–1层之间,主动极限区、被动极限区分别为2条直线,但过渡区滑移线由多段对数螺旋线组成,且各段螺旋线起始半径由主动极限区深度H0所处岩层位置确定。该工况底板层数为n层时,主动极限区深度H0所处岩层位置有i’=n–2种情况,需要对主动极限区深度H0每种情况i进行分析计算,后依次计算出相应对数螺旋线起始半径及起始半径间夹角。

    第1段螺旋线起始半径r0的计算公式如下:

    图1几何关系

    $$ ON = \frac{{{H^1}}}{{{\mathrm{sin}}\left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2}} \right)}} $$ (34)

    在△OMN

    $$ MN = {H^1}\left[ {{\mathrm{tan}}\left( {\frac{\pi }{4} + \frac{{{\varphi _1}}}{2}} \right) + {\mathrm{tan}}\left( {\frac{\pi }{4} - \frac{{{\varphi _1}}}{2}} \right)} \right] $$ (35)
    $$ M^{\prime} N^{\prime} = M N+ \sum_{l=2}^{i} \dfrac{H^{{{l}}}}{\tan \left(\dfrac{\pi}{4}+\dfrac{\varphi_{{{l}}}}{2}\right)} + \sum_{l=2}^{i} \dfrac{H^{{{l}}}}{\tan \left(\dfrac{\pi}{4}-\dfrac{\varphi_{{{l}}}}{2}\right)} $$ (36)

    在△O′M′N′

    $$ \begin{array}{c} {O^\prime }{N^\prime } = {M^\prime }{N^\prime } \cdot \cos \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{{i}} + 1}}}}{2}} \right)= \\ \left[ {\begin{array}{*{20}{c}} {{H^\prime }\left[ {\tan \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2}} \right) + \tan \left( {\dfrac{\pi }{4} - \dfrac{{{\varphi _1}}}{2}} \right)} \right]}+ \\ {\left. { \displaystyle\sum\limits_{l = 2}^i {\dfrac{{{H^{{l}}}}}{{\tan \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{l}}}}}{2}} \right)}}} + \sum\limits_{l = 2}^i {\dfrac{{{H^{{l}}}}}{{\tan \left( {\dfrac{\pi }{4} - \dfrac{{{\varphi _{{l}}}}}{2}} \right)}}} } \right]} \end{array}} \right. \times \\ \cos \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{{i}} + 1}}}}{2}} \right) \\ \end{array} $$ (37)
    $$ \begin{array}{c} {O^\prime }P = {O^\prime }{N^\prime } \cdot \sin \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{{i}} + 1}}}}{2}} \right) = \\ \left[ {\begin{array}{*{20}{c}} {{H^\prime }\left[ {\tan \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2}} \right) + \tan \left( {\dfrac{\pi }{4} - \dfrac{{{\varphi _1}}}{2}} \right)} \right]} +\\ {\left. { \displaystyle\sum\limits_{l = 2}^i {\dfrac{{{H^{{l}}}}}{{\tan \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{l}}}}}{2}} \right)}}} + \sum\limits_{l = 2}^i {\dfrac{{{H^{{l}}}}}{{\tan \left( {\dfrac{\pi }{4} - \dfrac{{{\varphi _l}}}{2}} \right)}}} } \right]} \end{array}} \right. \times \\ \dfrac{{\sin \left( {\dfrac{\pi }{2} + {\varphi _{{{i}} + 1}}} \right)}}{2} \\ \end{array} $$ (38)
    $$ N^{\prime} A^{\prime}=x_{0}-\sum_{l=1}^{{{t}}} 2 H^{{{l}}} \tan \left(\frac{\pi}{4}-\frac{\varphi_{{{l}}}}{2}\right) $$ (39)
    $$ N^{\prime} B=\dfrac{N^{\prime} A^{\prime}}{2 \cos \left(\dfrac{\pi}{4}+\dfrac{\varphi_{{{i}}+1}}{2}\right)}=\dfrac{x_{0}-\displaystyle\sum_{i=1}^{t} 2 H^{\prime} \tan \left(\dfrac{\pi}{4}-\dfrac{\varphi_{1}}{2}\right)}{2 \cos \left(\dfrac{\pi}{4}+\dfrac{\varphi_{{{i}}+1}}{2}\right)} $$ (40)
    $$ {r_0} = O'N' + N'B $$ (41)

    将式(37)、式(40)代入式(41),r0

    $$ \begin{gathered} r_{0}=\left[\begin{gathered} H^{\prime}\left[\tan \left(\dfrac{\pi}{4}+\dfrac{\varphi_{1}}{2}\right)+\tan \left(\dfrac{\pi}{4}-\dfrac{\varphi_{1}}{2}\right)\right]+ \\ \displaystyle\sum_{l=2}^{i} \dfrac{H^{{{l}}}}{\tan \left(\dfrac{\pi}{4}+\dfrac{\varphi_{{{l}}}}{2}\right)}+\displaystyle\sum_{l=2}^{i} \dfrac{H^{{{l}}}}{\tan \left(\dfrac{\pi}{4}-\dfrac{\varphi_{{{l}}}}{2}\right)} \end{gathered}\right] \times \\ \cos \left(\dfrac{\pi}{4}+\dfrac{\varphi_{{{i}}+1}}{2}\right)+\dfrac{x_{0}-\displaystyle\sum_{l=1}^{i} 2 H^{{{l}}} \tan \left(\dfrac{\pi}{4}-\dfrac{\varphi_{{{l}}}}{2}\right)}{2 \cos \left(\dfrac{\pi}{4}+\dfrac{\varphi_{{{i}}+1}}{2}\right)} \end{gathered} $$ (42)

    图1几何关系知G′E(h′)段最大值为

    $$ \begin{gathered} h_0^{\prime}=\frac{O^{\prime} P+H^{{{i}}+1}+H^{({{i}}+1)+1}+\cdots+H^{{{n}}-1}}{\sin \left(\dfrac{\pi}{4}+\dfrac{\varphi_{{{i}}+1}}{2}+\theta_1+\theta_2+\cdots+\theta_{{{n}}-{{i}}-1}\right)} \times \\ {\mathrm{e}}^{\left(\varphi_{{n}}+\tfrac{\pi}{4}-\tfrac{\varphi_{{{i}}+1}}{2}-\theta_1-\theta_2-\cdots-\theta_{{{n-i}}-1}\right) \tan\; {{n}}} \cos\; \varphi_{{n}} \end{gathered} $$ (43)

    式中:θn−i−1为对数螺旋线半径rn−i−2rn−i−1间夹角。

    最终得出$ \displaystyle\sum_{i=1}^{n-2} H^{{{i}}}< H_{0} \leqslant \displaystyle\sum_{i=1}^{n-1} H^{\prime} $情况下的多层结构底板最大滑移破坏深度h0

    $$ \begin{gathered} {h_0} = {h^\prime }_0 + \sum\limits_{i = 1}^{n - 2} {{H^{{i}}}} - {O^\prime }P = \\ \frac{{{O^\prime }P + {H^{{{i}} + 1}} + {H^{({{i}} + 1) + 1}} + \cdots + {H^{{{n}} - 1}}}}{{\sin \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{{i }}+ 1}}}}{2} + {\theta _1} + {\theta _2} + \cdots + {\theta _{{{n}} -{{ i}} - 1}}} \right)}} \times \\ {{\mathrm{e}}^{ {\left({\varphi _{{n}}} + \tfrac{\pi }{4} - \tfrac{{{\varphi _{{{i}} + 1}}}}{2} { - {\theta _1} - {\theta _2} - \cdots - {\theta _{{{n - i}} - 1}}} \right)\tan \; {\varphi _{{n}}}} }}\cos\; {\varphi _{{n}}}+ \\[-6pt] \sum\limits_{i = 1}^{n - 2} {{H^{{i}}}} - {O^\prime }P \\ \end{gathered} $$ (44)

    在△O′G′E

    $$ \begin{gathered} {O^\prime }{G^\prime } = r\sin \;\alpha = \\ \frac{{{O^\prime }P + {H^{{{i }}+ 1}} + {H^{({{i}} + 1) + 1}} + \cdots + {H^{{{n}} - 1}}}}{{\sin \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{{i}} + 1}}}}{2} + {\theta _1} + {\theta _2} + \cdots + {\theta _{{{n - i}} - 1}}} \right)}} \times \\ {{\mathrm{e}}^{ {\left({\varphi _{{n}}} + \tfrac{\pi }{4} - \tfrac{{{\varphi _{{{i }}+ 1}}}}{2} { - {\theta _1} - {\theta _2} - \cdots - {\theta _{{{{{n - i}} }}- 1}}} \right)\tan \;{\varphi _{{n}}}} }}\sin\; {\varphi _{{n}}} \\ \end{gathered} $$ (45)

    在△OFN′

    $$ F N^{\prime}=\sum_{1}^{i} H^{\prime} \cdot \tan \left(\frac{\pi}{4}-\frac{\varphi_{{{i}}}}{2}\right) $$ (46)
    $$ PN' = O'P \cdot {\mathrm{tan}}\left( {\frac{\pi }{4} - \frac{{{\varphi _{{{i}} + 1}}}}{2}} \right) $$ (47)
    $$ \begin{gathered} P F=P N^{\prime}-F N^{\prime}= \\[-6pt] O^{\prime} P \cdot \tan \left(\dfrac{\pi}{4}-\dfrac{\varphi_{{{i}}+1}}{2}\right)-\sum_{1}^{i} \tan \left(\dfrac{\pi}{4}-\dfrac{\varphi_{{{i}}}}{2}\right) \end{gathered}$$ (48)

    图1几何关系有

    $$ OG = O'G' + PF $$ (49)

    将式(45)、式(48)代入式(49),知开采面至最大破坏深度水平距离OG

    $$ \begin{gathered} OG = \frac{{{O^\prime }P + {H^{{{i}} + 1}} + {H^{({{i}} + 1) + 1}} + \cdots + {H^{{{n }}- 1}}}}{{\sin \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{{i }}+ 1}}}}{2} + {\theta _1} + {\theta _2} + \cdots + {\theta _{{{n - i}} - 1}}} \right)}} \times \\ {{\mathrm{e}}^{ {\left({\varphi _{{n}}} + \tfrac{\pi }{4} - \tfrac{{{\varphi _{{{i}} + 1}}}}{2} { - {\theta _1} - {\theta _2} - \cdots - {\theta _{{{n - i}} - 1}}} \right)\tan\; {\varphi _{{n}}}} }}\sin \;{\varphi _{{n}}} + \\ {O^\prime }P \cdot \tan \left( {\frac{\pi }{4} - \frac{{{\varphi _{{{i }}+ 1}}}}{2}} \right) - \sum\limits_1^i {\tan } \left( {\frac{\pi }{4} - \frac{{{\varphi _{{i}}}}}{2}} \right) \\ \end{gathered} $$ (50)

    $ \displaystyle\sum_{i=1}^{n-1} H^{{{i}}}< H_{0} $,即主动极限区深度H0处于第n层岩体内,设过渡区滑移线由1段对数螺旋线组成,主动极限区、被动极限区分别为2条直线。

    第1段螺旋线起始半径r0由式(41)可知:

    $$ \begin{gathered} {r_0} = \left[ \begin{gathered} {H^\prime }\left[ {\tan \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{\mathrm{1}}}}}{2}} \right) + \tan \left( {\dfrac{\pi }{4} - \dfrac{{{\varphi _{\mathrm{1}}}}}{2}} \right)} \right] + \\ \sum\limits_2^{n - 1} {\dfrac{{{{\mathrm{H}}^{{i}}}}}{{\tan \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{i}}}}}{2}} \right)}}} + \sum\limits_2^{n - 1} {\dfrac{{{{\mathrm{H}}^{{i}}}}}{{\tan \left( {\dfrac{\pi }{4} - \dfrac{{{\varphi _{{i}}}}}{2}} \right)}}} \\ \end{gathered} \right] \times \\ \cos \left(\dfrac{\pi}{4}+\dfrac{\varphi_{{{i}}+1}}{2}\right)+\dfrac{x_0-\displaystyle\sum_1^{n-1} 2 H^{{i}} \tan \left(\dfrac{\pi}{4}-\dfrac{\varphi_{{l}}}{2}\right)}{2 \cos \left(\dfrac{\pi}{4}+\dfrac{\varphi_{{{i}}+1}}{2}\right)} \end{gathered} $$ (51)

    图1几何关系知G′E(h′)段最大值为

    $$ \begin{gathered} {h_0}^\prime = r\cos \;\alpha = {r_0}{{\mathrm{e}}^{\theta \tan \;{\varphi _{{n}}}}}\cos \;\left( {\theta + \dfrac{{{\varphi _{{n}}}}}{2} - \dfrac{\pi }{4}} \right) = \\ \left\{ \begin{gathered} \left[ \begin{gathered} {H^1}\left[ {\tan \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2}} \right) + \tan \left( {\dfrac{\pi }{4} - \dfrac{{{\varphi _1}}}{2}} \right)} \right] + \\ \displaystyle\sum_{i = 2}^{n - 1} {\dfrac{{{H^{{i}}}}}{{\tan \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{i}}}}}{2}} \right)}}} + \sum\limits_{i = 2}^{n - 1} {\dfrac{{{H^{{i}}}}}{{\tan \left( {\dfrac{\pi }{4} - \dfrac{{{\varphi _{{i}}}}}{2}} \right)}}} \\ \end{gathered} \right] \\ \cos \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{{i }}+ 1}}}}{2}} \right) + \dfrac{{{x_0} - \displaystyle\sum\limits_{i = 1}^{n - 1} 2 {H^{{i}}}\tan \left( {\dfrac{\pi }{4} - \dfrac{{{\varphi _{{i}}}}}{2}} \right)}}{{2\cos \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{{i }}+ 1}}}}{2}} \right)}} \\ \end{gathered} \right\}\times \\ {{\mathrm{e}}^{\left( {\tfrac{\pi }{4} + \tfrac{{{\varphi _{{n}}}}}{2}} \right)\tan\; {\varphi _{{n}}}}}\cos\; {\varphi _{{n}}}_{} \\ \end{gathered} $$ (52)

    最终得出$ \displaystyle\sum_{i=1}^{n-1} H^{{{i}}}< H_{0} $情况下的多层结构底板最大破坏深度h0

    $$ \begin{gathered} {h_0} = {h_0}^\prime + \sum\limits_{i = 1}^{n - 1} {{H^{{i}}}} - {O^\prime }P = \\ \left\{ \begin{gathered} \left[ \begin{gathered} {H^\prime }\left[ {\tan \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2}} \right) + \tan \left( {\dfrac{\pi }{4} - \dfrac{{{\varphi _1}}}{2}} \right)} \right] + \\ \sum\limits_{i = 2}^{n - 1} {\dfrac{{{H^{{i}}}}}{{\tan \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{i}}}}}{2}} \right)}}} + \sum\limits_{i = 2}^{n - 1} {\dfrac{{{H^{{i}}}}}{{\tan \left( {\dfrac{\pi }{4} - \dfrac{{{\varphi _{{i}}}}}{2}} \right)}}} \\ \end{gathered} \right] \times \\ \cos \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{{i }}+ 1}}}}{2}} \right) + \dfrac{{{x_0} - \displaystyle\sum\limits_{i = 1}^{n - 1} 2 {H^{{i}}}\tan \left( {\dfrac{\pi }{4} - \dfrac{{{\varphi _{{i}}}}}{2}} \right)}}{{2\cos \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _{{{i }}+ 1}}}}{2}} \right)}} \\ \end{gathered} \right\}\times \\ {{\mathrm{e}}^{\left( {\tfrac{\pi }{4} + \tfrac{{{\varphi _{{n}}}}}{2}} \right)\tan \;{\varphi _{{n}}}}}\cos\; {\varphi _{{n}}} + \sum\limits_{i = 1}^{n - 1} {{H^{{i}}}} - {O^\prime }P \\ \end{gathered} $$ (53)

    开采面至最大破坏深度水平距离OG

    $$ \begin{gathered}OG=\left\{\begin{array}{l}\left[\begin{array}{l}{H}^{1}\left[\mathrm{tan}\left(\dfrac{\pi }{4}+\dfrac{{\varphi }_{1}}{2}\right)+\mathrm{tan}\left(\dfrac{\pi }{4}-\dfrac{{\varphi }_{1}}{2}\right)\right]+\\ {\displaystyle \sum _{i=2}^{n-1}\dfrac{{H}^{{{i}}}}{\mathrm{tan}\left(\dfrac{\pi }{4}+\dfrac{{\varphi }_{{{i}}}}{2}\right)}}+{\displaystyle \sum _{i=2}^{n-1}\dfrac{{H}^{{{i}}}}{\mathrm{tan}\left(\dfrac{\pi }{4}-\dfrac{{\varphi }_{{{i}}}}{2}\right)}}\end{array}\right]\\ \mathrm{cos}\left(\dfrac{\pi }{4}+\dfrac{{\varphi }_{{{i}}+1}}{2}\right)+\dfrac{{x}_{0}-{\displaystyle \sum _{i=1}^{n-1}2}{H}^{{{i}}}\mathrm{tan}\left(\dfrac{\pi }{4}-\dfrac{{\varphi }_{{{i}}}}{2}\right)}{2\mathrm{cos}\left(\dfrac{\pi }{4}+\dfrac{{\varphi }_{{{i}}+1}}{2}\right)}\end{array}\right\}\text{×}\\ {{\mathrm{e}}}^{\left(\tfrac{\pi }{4}+\tfrac{{\varphi }_{{{n}}}}{2}\right)\mathrm{tan}\;{\varphi }_{{{n}}}}\mathrm{sin}\;{\varphi }_{{{n}}}+\\ {O}^{\prime }P\cdot \mathrm{tan}\left(\dfrac{\pi }{4}-\dfrac{{\varphi }_{{{i}}+1}}{2}\right)-{\displaystyle \sum _{1}^{i}\mathrm{tan}}\left(\dfrac{\pi }{4}-\dfrac{{\varphi }_{{{i}}}}{2}\right)\end{gathered} $$ (54)

    将各工况推导公式整理后利用主动极限区深度进行逻辑流程判别,后进行底板破坏深度及开采面至底板破坏深度水平距离的计算,如图3所示。

    图  3  多层 结构底板 破坏深度 逻辑流程 判别
    Figure  3.  Multi-storey structural substrate damage depth technical flow chart

    因煤岩开采面底板根据柱状进行分层时各岩层内摩擦角不同,采用塑性滑移线理论分析时会产生多条对数螺旋曲线半径。对对数螺旋曲线半径的求解采用工况2加以说明,其他工况对数螺旋曲线半径采用类似的求解方式,不再赘述。

    n=2,即假设底板层数为2层时,如图4所示,当滑移线主动极限区深度H0小于首层底板岩层厚度H′时,对数螺旋线将穿过上层底板岩层,此时O点将作为2段对数螺旋线的旋转中心点,r1同时作为第1段对数螺旋线的终线和第2段对数螺旋线的起点线。

    图  4  双层结构底板计算简图
    Figure  4.  Calculation sketch of double-storey structure base plate

    对数螺旋曲线起始半径r0图4几何关系知:

    $$ {r_0} = \dfrac{{{x_0}}}{{2\cos \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2}} \right)}} $$ (55)

    第2段对数螺旋线起始半径$ r_{1} $可由方程组(56)联立求解:

    $$ \left\{ \begin{gathered} {r_1} = \dfrac{{{H^1}}}{{\sin \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2} + {\theta _1}} \right)}} \\ {r_1} = {r_0}{{\mathrm{e}}^{{\theta _1}\tan\;{\varphi _1}}} \\ \end{gathered} \right. $$ (56)

    化简为

    $$ {H^1} = {r_0}{{\mathrm{e}}^{{\theta _1}{\mathrm{tan}}\;{\varphi _1}}} \cdot \sin \left( {\frac{\pi }{4} + \frac{{{\varphi _1}}}{2} + {\theta _1}} \right) $$ (57)

    对于式(57)中的未知数$ \theta_{1} $和工况2中式(9)的求解方法相同,采用泰勒展开求得近似解$ \theta_{1} $,最终可求得对数螺旋线起始半径$ r_{1} $。

    n=3,即假设底板层数为3层时,如图5所示,此时O点将作为3段对数螺旋线的旋转中心点。

    图  5  3层结构底板计算简图
    Figure  5.  Calculation sketch of three-story structure floor

    对数螺旋半径$ r_{1}, r_{2} $求解如下

    $$ \left\{ \begin{gathered} {r_1} = \dfrac{{{H^1}}}{{\sin \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2} + {\theta _1}} \right)}} \\ {r_1} = {r_0}{{\mathrm{e}}^{{\theta _1}{\mathrm{tan}}\;{\varphi _1}}} \\ \end{gathered} \right. $$ (58)
    $$ \left\{ \begin{gathered} {r_2} = \dfrac{{{H^1} + {H^2}}}{{\sin \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2} + {\theta _1} + {\theta _2}} \right)}} \\ {r_2} = {r_1}{{\mathrm{e}}^{{\theta _2}{\mathrm{tan}}\;{\varphi _2}}} \\ \end{gathered} \right. $$ (59)

    对$ r_{1}, r_{2} $求解方式与n=2时$ r_{1} $的求解方法相同,即通过泰勒展开式后进行一元四次方程求解,且$ r_{2} $求解需经过2次泰勒式转化分别求出$ \theta_{1} $、$ \theta_{2} $。

    n=4,即假设底板层数为4层时,如图6所示,此时O点将作为4段对数螺旋线的旋转中心点。

    图  6  4层结构底板计算简图
    Figure  6.  Sketch of four-story structural floor calculation

    对数螺旋半径$ r_{1}, r_{2}, r_{3} $用方程组(60)求解

    $$ \left\{ \begin{gathered} {r_1} = \dfrac{{{H^1}}}{{\sin \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2} + {\theta _1}} \right)}} \\ {r_1} = {r_0}{{\mathrm{e}}^{{\theta _1}{\mathrm{tan}}\;{\varphi _1}}} \\ \end{gathered} \right. $$ (60)
    $$ \left\{ \begin{gathered} {r_2} = \dfrac{{{H^1} + {H^2}}}{{\sin \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2} + {\theta _1} + {\theta _2}} \right)}} \\ {r_2} = {r_1}{{\mathrm{e}}^{{\theta _2}{\mathrm{tan}}\;{\varphi _2}}} \\ \end{gathered} \right. $$ (61)
    $$ \left\{ \begin{gathered} {r_3} = \frac{{{H^1} + {H^2} + {H^3}}}{{\sin \left( {\dfrac{\pi }{4} + \dfrac{{{\varphi _1}}}{2} + {\theta _1} + {\theta _2} + {\theta _3}} \right)}} \\ {r_3} = {r_2}{{\mathrm{e}}^{{\theta _3}{\mathrm{tan}}\;{\varphi _3}}} \\ \end{gathered} \right. $$ (62)

    则对数螺旋半径$ r_{3} $求解需经过3次泰勒式转化求出近似解$ \theta_{1} $、$ \theta_{2} $和$ \theta_{3} $,且3个近似解误差在逐次累积。可以预见若底板层数持续增加,则对数螺旋曲线半径的求解需经过更多次泰勒式转化,其产生的累积误差会随之加大,导致其计算的破坏深度误差偏大,且计算量急剧加重,不易进行人工手算。因此考虑采用Matlab内置函数编程语言进行对数螺旋线半径求解,避免泰勒展开式产生的累积误差,并且计算结果精度不受多层结构底板层数的影响。

    多层结构底板破坏深度计算系统V1.0 (简称计算系统)主要采用Matlab语言中的math operations模块开发而成。Matlab是用于算法开发、数据可视化、数据分析及数值计算的高级技术计算语言和交互式环境的商业数学软件,运用guide指令搭接的GUI界面主要控件见表1

    表  1  界面主要控件
    Table  1.  Interface main controls
    英文名称中文名称控件作用
    Button按钮控件在程序中显示按钮
    Entry输入控件用于显示简单的文本内容
    Label标签控件可以显示文本
    Text文本控件用于显示多行文本
    Toplevel容器控件用来提供一个单独对话框
    MessageBox对话框控件用于显示应用程序的消息框
    下载: 导出CSV 
    | 显示表格

    使用Matlab内置函数计算多层结构底板破坏深度过程中,利用EXCEL计算部分已知参数的底板破坏深度及开采面至底板最大破坏深度水平距离的理论值,佐证Matlab编码的准确性及人为因素影响所计算出的软件值。

    if strcmp(f{i}, 'MaxDegree')

    v = sym(v);

    elseif strcmp(f{i}, 'ReturnConditions')

    f{i} = 'OutputType';

    if v == true

    v = evalin(symengine, '"FullMode"');

    else

    v = evalin(symengine, '"CompatibleMode"');

    end

    elseif v == true

    v = evalin(symengine, 'TRUE');

    else

    % v is false

    v = evalin(symengine, 'FALSE');

    end

    entries(i) = feval(symengine, '_equal', sym(f{i}), v);

    end

    entries(end) = evalin(symengine, 'VectorFormat = TRUE');

    entries = feval(symengine, 'op', entries);

    T = feval(symengine, 'table', entries);

    local function warnIfParams

    warn if the solution depends on parameters and conditions

    function warnIfParams(parameters, conditions)

    If isempty(parameters)

    paramstring = char(parameters(1));

    for i = 2:numel(parameters)

    paramstring = [paramstring ', ' char(parameters(i))]; %#ok

    end

    以上为多层结构底板破坏深度计算系统软件计算过程中的部分核心代码。

    表2中可以看到对于工况1由于不涉及泰勒展开式及方程组的求解其最终计算结果与Matlab软件计算结果相同。当底板层数分别为3层和7层时,对于工况4中的理论值 (利用EXCEL对多层结构底板破坏深度推导公式进行求解)和软件值结果误差也随之加大,其原因在于求解对数螺旋线半径时多次使用了泰勒展开式,且误差随结构底板岩层层数增加误差也随之累积,符合预期设想。

    表  2  多层结构底板破坏深度及开采面至破坏深度水平距离误差对比
    Table  2.  Comparison of damage depth and horizontal distance error from mining surface to damage depth of multi-layered structure floor
    输入参数 输出结果 工况
    h0 /m OG/m
    底板层数n 底板超前塑性破坏长度X0 /m 底板层厚Hi/m 内摩擦角φi /(°) 理论值 软件值 理论值 软件值
    1 7.9 25 37 16.318 16.2975 12.295 12.281 工况1
    2 7.9 5.5 39 16.59 16.5868 12.528 12.5088 工况5
    19.5 36
    3 9.2 6.8 30 16.74 16.0903 10.494 10.8452 工况4
    2.3 35
    10.4 36
    3 9.54 10 39 19.404 19.3943 14.103 14.0908 工况2
    3.6 35
    7.3 36
    7 22.7 4.3 39 44.351 42.8537 30.562 29.5707 工况4
    3 34
    5.1 38
    3.5 35
    6 30
    2.4 32
    21 33
    3 4.4 5 37 7.197 8.04518 5.026 5.0271 工况3
    0.6 33
    8.7 32
    下载: 导出CSV 
    | 显示表格

    图7所示,将相应参数输入到输入框,点击“计算”按钮进行最终计算。如输入底板层数“3”,页面中会显示相应层数,后在列表中依次输入相应的底板层厚和内摩擦角;然后输入底板超前塑性破坏长度x0 (实测值为空,计算取理论值;实测值有实值情况,系统默认实测值)。如输入实测值“5.6”后,点击“计算”按钮,界面会显示该3层底板最大破坏深度和开采面至最大破坏深度的水平距离分别为10.557、7.012 m。

    图  7  多层结构底板破坏深度计算页面
    Figure  7.  Multi-storey structure bottom slab damage depth calculation page

    若现场因客观原因无法测得底板超前塑性破坏长度时,需计算底板超前塑形破坏长度的理论值,则可以分别输入H1(煤层的埋深,m);H2(煤层的采高,m);K(支撑压力峰值系数);θ(煤层的变形角,(°));φc(煤体的内摩擦角,(°));φc*(煤体的残余内摩擦角,(°));C1(煤层与顶底板接触面上的黏聚力,MPa);C2(煤体的黏聚力,MPa);C*(煤体的残余黏聚力,MPa);f1(煤层与顶板界面间的摩擦力,N);r(煤层覆盖的平均容重,N/m3)。如输入图8页面中的地质参数,可计算底板超前塑形破坏长度的理论值为18.697 m。

    图  8  底板超前塑性破坏长度理论值计算页面
    Figure  8.  Calculation page for theoretical values of overrunning plastic damage length of subgrade

    根据平煤某矿工作面底板岩体地质情况,工作面超前塑性区长度现场实测值为7.9 m。虽然下保护层层位与寒武系灰岩含水层之间有平均64 m厚的隔水层,在正常情况下隔水层可以阻隔承压水的侵入,但受开采扰动和寒武系高水头压力的影响,底板将形成扰动破坏带和承压水导升带,使得有效隔水层厚度减小,若遇大的断裂构造,承压水可通过隔水层较薄地带或构造破碎带进入工作面。参考周边矿井煤岩层工作面底板扰动破坏深度,取下保护层开采工作面底板岩层厚度H=25 m内的岩体底板进行最大破坏深度理论计算。

    在《深部煤岩层复合结构底板破坏机制及应用研究》[21]中李昂等针对此工程把相近岩性或岩层厚度较薄的岩层划分为1层,摩擦角取其平均值,岩层较厚且岩性差异较大时划分为另外层,分别进行单层、双层及三层底板最大破坏深度计算,并通过现场监测数据判断底板最大破坏深度为17.8 m。

    本章节将通过多层结构底板破坏深度计算系统V1.0软件对该理论值、拟合值与现场实测值进行对比分析,表3中可以看到1~3层结构底板软件计算值与理论值结果误差极小。

    表  3  数值计算结果对比
    Table  3.  Comparison of numerical calculation results
    底板层数n 底板超前塑性破坏长度X0/m 底板层厚Hi/m 内摩擦角φi/(°) 底板最大破坏深度h0/m
    理论值 软件值 拟合值 实测值
    1 7.9 25 37 16.3 16.298 18.12 17.9
    2 7.9 5.5 39 16.59 16.587
    19.5 36
    3 7.9 5.5 39 17.08 17.077
    7.2 34
    12.3 38
    5 7.9 4.6 36 17.33 17.761
    7.17 39
    6.58 35
    6.19 34
    2.29 39
    下载: 导出CSV 
    | 显示表格

    因以往研究中以1~3层作为结构底板破坏深度分析计算,不足以应对复杂的底板岩层。因此将该开采工作面底板岩层厚度H=25 m内的岩体进行5层分析计算,依据柱状图(图9)把L8灰岩及厚度较小的煤线和砂质泥岩作为首层,厚度和为4.6 m,内摩擦角取该层位岩性平均内摩擦角36°;第2层以L7灰岩为主,层厚为7.17 m,内摩擦角39°;第3层由厚度较小的砂质泥岩、煤和泥岩构成,总厚度为6.58 m,平均内摩擦角为35°;第4层以厚度为6.19的黑灰色泥岩为主,内摩擦角34°;第5层是厚度为2.29 m的石灰岩,内摩擦角39°。将以上各参数输入到该计算系统,得到底板最大破坏深度软件值为17.761 m。从表3中可看到其软件值更接近现场实测值和拟合值,计算精度更高。虽然拟合值[23]和实测值相近,但拟合值往往取自不同矿井数据所得到的经验公式,当具体到实际工程时,可能得到较大的偏差。

    图  9  钻孔综合柱状图
    Figure  9.  Drill hole composite histogram

    李昂在《带压开采下底板渗流与应力耦合破坏突水机理及其工程应用》[12]中将董家河煤矿5号煤层22507工作面作为工程研究背景,相关力学岩性指标取值为:x0=5.7 m、H=350 m、φd=35°。计算出单层底板破坏深度为10.9 m,开采面至底板破坏深度水平距离为7.6 m,现场实测底板破坏深度结果为10.8 m。

    将该地质相关参数输入到计算系统软件中,如图10所示,可以看到底板破坏深度与开采面至底板破坏深度水平距离计算结果与该文中理论值相同,该软件的计算精度可达到98%。

    图  10  系统计算结果
    Figure  10.  System calculation results

    不同钻孔揭露的岩性存在差异性,岩层厚度、层数往往存在较大差异,导致底板采动破坏不同。开展现场测试过程中,往往钻取岩芯或者借助周边较近钻孔柱状。软件分析结果与现场实测数据进行对比分析,通过软件分析所得到的结果与实测值一致。不可否认,如果换作其他矿区进行现场实测,在岩性差别较大时计算结果也会有所不同。在许多矿井没有实测值或很难开展实测的情况下,该软件依据柱状和基本力学参数便可以计算底板破坏深度的取值,且可以代替繁琐的手工计算过程,优势是明显的。

    1)在3层复合结构底板塑性滑移线场力学模型基础上,推导出n层结构底板破坏深度解析解。利用手算进行底板最大破坏深度计算时部分工况计算过程需多次利用泰勒展开式转化为一元四次方程式。随着层数的增加,会逐次累积误差,且计算值只能取近似解。以主动极限区深度H0所处底板位置为判别依据,对多层结构底板计算工况进行了优化,提出5种计算工况,并形成了逻辑判别流程图。

    2)首次采用了Matlab内置函数运用编程语言进行多层结构底板破坏深度计算,其自带内置函数可避免手算过程中因泰勒式展开式多次利用而产生的累积误差。借助逻辑判别图开发了多层结构底板破坏深度计算系统V1.0软件,该软件通过输入相应参数可快速地计算出底板破坏深度及开采面至底板破坏深度的水平距离。

    3)通过将平煤某矿及董家河煤矿的岩性力学指标参数值代入该计算系统,得到该软件计算结果与理论值相一致。并将平煤某矿底板岩层进行五层计算分析,得到底板最大破坏深度理论值与实测结果偏差0.57 m,软件值与实测值相差0.14 m,即根据岩性层数划分越细致,计算精度更高。可以应对更加复杂的底板岩性。

    4)该软件具有普适性,所采用的理论是以不同实际柱状和力学参数为依据进行的分层计算,在矿井没有实测值或很难开展实测的情况下,仅仅根据柱状和基本力学参数便可以快捷准确地计算底板破坏深度的取值,且可以代替繁琐的手工计算过程,相比常规单一岩层塑性滑移线理论准确性更高。对研究我国华北型煤田煤岩层底板破坏规律具有重要的工程价值。

  • 图  1   含裂隙煤体裂隙演化过程的应力−应变曲线[36]

    Figure  1.   Stress-strain curve of fracture evolution process of fissured coal body[36]

    图  2   PFC中模拟节理的2种方法示意[37]

    Figure  2.   Schematic of two methods for simulating joints in PFC[37]

    图  3   完整与含裂隙试件数值模型的应力应变关系验证

    Figure  3.   Verification of stress-strain relationship of numerical model of intact and cracked specimens

    图  4   数值模型与裂隙参数

    Figure  4.   Numerical model and fissure parameters

    图  5   单轴压缩下不同裂隙类型试件的应力−应变曲线对比

    Figure  5.   Comparison of stress-strain curves of specimens with different fissure type under uniaxial compression

    图  6   单轴压缩下不同裂隙类型试件宏观力学参数对比

    Figure  6.   Comparison of macro-mechanical parameters of coal specimens with different fissure type

    图  7   不同裂隙倾角煤体试件接触力分布

    Figure  7.   Contact force distribution of coal specimens with different fissure type

    图  8   无裂隙煤体试件表面弹性应变能分布

    Figure  8.   Distribution of elastic strain energy on surface of coal specimens with no fissure

    图  9   不同裂隙类型煤体试件表面弹性应变能分布

    Figure  9.   Distribution of elastic strain energy on surface of coal specimens with different fissure types

    图  10   不同裂隙类型试件的能量分布特征 (β=45°)

    Figure  10.   Comparison of energy distribution characteristics for different type fissured specimens (β=45° )

    图  11   基于应力−应变曲线的能量关系示意

    Figure  11.   Schematic of energy relationship based on stress-strain curve

    图  12   相同储存弹性能的条件下不同峰值应变的应力−应变曲线对比

    Figure  12.   Comparison of stress-strain curves of different peak strains under same stored elastic energy

    图  13   不同裂隙类型煤体试件的冲击倾向性对比

    Figure  13.   Comparison of coal burst tendency of coal specimens with different fracture types

    表  1   宏观力学参数对比

    Table  1   Comparison of macro mechanical parameters

    测试类型PFC模拟实验室测试
    单轴抗压强度/MPa完整试件18.918.7
    30°单裂隙试件14.514.2
    30°平行双裂隙试件11.211.3
    30°断续双裂隙试件14.714.3
    弹性模量/GPa完整试件4.24.6
    30°单裂隙试件3.843.80
    30°平行双裂隙试件2.382.41
    30°断续双裂隙试件3.853.88
    下载: 导出CSV

    表  2   不同裂隙类型试件峰值点处能量耗散特征

    Table  2   Characteristics of energy dissipation at peak point of specimens with different fissure types

    裂隙类型裂隙倾
    角/(º)
    峰值点总能量/
    (104J·m-3)
    峰值点弹性应变
    能/(104J·m-3)
    峰值点耗散
    能/(104J·m-3)
    单裂隙试件04.434.190.24
    152.902.690.20
    302.171.950.23
    452.532.390.14
    603.212.910.31
    753.763.450.31
    904.334.180.15
    共面断续双
    裂隙试件
    04.534.280.25
    153.273.050.22
    302.532.350.18
    453.273.040.23
    602.932.760.17
    753.523.390.14
    904.354.200.16
    平行非共面
    双裂隙试件
    04.283.980.30
    152.061.890.18
    301.541.370.17
    451.551.450.10
    601.961.840.12
    752.902.760.13
    904.234.080.15
    下载: 导出CSV
  • [1] 宫凤强,赵英杰,王云亮,等. 煤的冲击倾向性研究进展及冲击地压“人-煤-环”三要素机理[J]. 煤炭学报,2022,47(5):1974−2010. doi: 10.13225/j.cnki.jccs.2022.0165

    GONG Fengqiang,ZHAO Yingjie,WANG Yunliang,et al. Research progress of coal bursting liability indices and coal burst “Human-Coal-Environment” three elements mechanism[J]. Journal of China Coal Society,2022,47(5):1974−2010. doi: 10.13225/j.cnki.jccs.2022.0165

    [2] 潘俊锋,宁 宇,毛德兵,等. 煤矿开采冲击地压启动理论[J]. 岩石力学与工程学报,2012,31(3):586−596. doi: 10.3969/j.issn.1000-6915.2012.03.017

    PAN Junfeng,NING Yu,MAO Debing,et al. Theory of rockburst start-up during coal mining[J]. Chinese Journal of Rock Mechanics and Engineering,2012,31(3):586−596. doi: 10.3969/j.issn.1000-6915.2012.03.017

    [3] 赵毅鑫,姜耀东,张 雨. 冲击倾向性与煤体细观结构特征的相关规律[J]. 煤炭学报,2007,32(1):64−68. doi: 10.3321/j.issn:0253-9993.2007.01.014

    ZHAO Yixing,JIANG Yaodong,ZHANG Yu. The relationship between bump prone property and microstructure characteristics of coal[J]. Journal of China Coal Society,2007,32(1):64−68. doi: 10.3321/j.issn:0253-9993.2007.01.014

    [4] 赵同彬,尹延春,谭云亮,等. 基于颗粒流理论的煤岩冲击倾向性细观模拟试验研究[J]. 煤炭学报,2014,39(2):280−285.

    ZHAO Tongbin,YIN Yanchun,TAN Yunliang,et al. Bursting liability of coal research of heterogeneous coal based on particle flow microscopic test[J]. Journal of China Coal Society,2014,39(2):280−285.

    [5] 冯增朝, 赵阳升. 岩石非均质性与冲击倾向的相关规律研究[J]. 岩石力学与工程学报, 2003, 22(11): 1863–1865.

    FENG Zengchao, ZHAO Yangsheng. Correlativity of rock inhomogeneity and rock burst trend[J]. Chinese Journal of Rock Mechanics and Engineering, 2003, 22(11): 1863–1865.

    [6] 孟召平,潘结南,刘亮亮,等. 含水量对沉积岩力学性质及其冲击倾向性的影响[J]. 岩石力学与工程学报,2009,28(S1):2637−2643.

    MENG Zhaoping,PAN Jienan,LIU Liangliang,et al. Influence of moisture contents on mechanical properties of sedimentary rock and its bursting potential[J]. Chinese Journal of Rock Mechanics and Engineering,2009,28(S1):2637−2643.

    [7] 张志镇, 高 峰, 刘治军. 温度影响下花岗岩冲击倾向及其微细观机制研究[J]. 岩石力学与工程学报, 2010, 29(8): 1591–1602.

    ZHANG Zhizhen, GAO Feng, LIU Zhijun. Research on rock bursts proneness and its microcosmic mechanism of granite considering temperature effect[J]. Chinese Journal of Rock Mechanics and Engineering, 2010, 29(8): 1591–1602.

    [8] 张广辉,欧阳振华,齐庆新,等. 瓦斯对煤冲击倾向性影响的试验研究[J]. 煤炭学报,2017,42(12):3159−3165.

    ZHANG Guanghui,OUYANG Zhenhua,QI Qingxin,et al. Experimental research on the influence of gas on coal burst tendency[J]. Journal of China Coal Society,2017,42(12):3159−3165.

    [9]

    LIU X,WANG X,WANG E,et al. Effects of gas pressure on bursting liability of coal under uniaxial conditions[J]. Journal of Natural Gas Science and Engineering,2017,39:90−100. doi: 10.1016/j.jngse.2017.01.033

    [10]

    YI X,FENG G,TENG T,et al. Effect of gas pressure on rock burst proneness indexes and energy dissipation of coal samples[J]. Geotechnical and Geological Engineering,2016,34(6):1−12.

    [11] 左建平,陈 岩,崔 凡. 不同煤岩组合体力学特性差异及冲击倾向性分析[J]. 中国矿业大学学报,2018,47(1):81−87.

    ZUO Jianping,CHEN Yan,CUI Fan. Investigation on mechanical properties and rock burst tendency of different coal-rock combined bodies[J]. Journal of China University of Mining & Technology,2018,47(1):81−87.

    [12] 邓志刚. 冲击倾向性煤体强度尺寸效应的影响因素研究[J]. 煤炭科学技术,2019,47(8):59−63.

    DENG Zhigang. Study on influencing factors of strength size effect based on bump-prone coal[J]. Coal Science and Technology,2019,47(8):59−63.

    [13]

    SONG H,JIANG Y,ELSWORTH D,et al. Scale effects and strength anisotropy in coal[J]. International Journal of Coal Geology,2018,195:37−46. doi: 10.1016/j.coal.2018.05.006

    [14]

    GAO F,STEAD D,KANG H. Numerical investigation of the scale effect and anisotropy in the strength and deformability of coal[J]. International Journal of Coal Geology,2014,136:25−37. doi: 10.1016/j.coal.2014.10.003

    [15] 卢志国,鞠文君,王 浩,等. 硬煤冲击倾向各向异性特征及破坏模式试验研究[J]. 岩石力学与工程学报,2019,38(4):757−768.

    LU Zhiguo,JU Wenjun,WANG Hao,et al. Experimental study on anisotropic characteristics of impact tendency and failure model of hard coal[J]. Chinese Journal of Rock Mechanics and Engineering,2019,38(4):757−768.

    [16] 郝宪杰,袁 亮,王少华,等. 硬煤冲击倾向性的层理效应研究[J]. 煤炭科学技术,2018,46(5):1−7.

    HAO Xianjie,YUAN Liang,WANG Shaohua,et al. Study on bedding effect of bump tendency for hard coal[J]. Coal Science and Technology,2018,46(5):1−7.

    [17] 李 磊,李宏艳,李凤明,等. 层理角度对硬煤冲击倾向性影响的实验研究[J]. 采矿与安全工程学报,2019,36(3):987−994. doi: 10.13545/j.cnki.jmse.2019.05.016

    LI Lei,LI Hongyan,LI Fengming,et al. Experimental study on anisotropic characteristics of impact tendency and failure model of hard coal[J]. Chinese Journal of Rock Mechanics and Engineering,2019,36(3):987−994. doi: 10.13545/j.cnki.jmse.2019.05.016

    [18]

    SZECOWKA Z, DOMZAL J, OZANA P. Energy index of natural bursting ability of coal [J]. Transactions of the Central Mining Institute, 1973, 594.

    [19]

    GIL H, DRZEZLA B. Methods for determining bursting liability of coal [J]. Przegl Górn, 1973: 12.

    [20]

    SINGH S P. Burst energy release index[J]. Rock Mechanics & Rock Engineering,1988,21(2):149−155.

    [21] 唐礼忠,潘长良,王文星. 用于分析岩爆倾向性的剩余能量指数[J]. 中南工业大学学报(自然科学版),2002,33(2):129−132.

    TANG Lizhong,PAN Changliang,WANG Wenxing. Surplus energy index for analysing rockburst proneness[J]. Journal of Central South University of Technology,2002,33(2):129−132.

    [22] 张绪言,冯国瑞,康立勋,等. 用剩余能量释放速度判定煤岩冲击倾向性[J]. 煤炭学报,2009,34(9):1165−1168. doi: 10.3321/j.issn:0253-9993.2009.09.003

    ZHANG Xuyan,FENG Guorui,KANG Lixun,et al. Method to determine burst tendency of coal rock by residual energy emission speed[J]. Journal of China Coal Society,2009,34(9):1165−1168. doi: 10.3321/j.issn:0253-9993.2009.09.003

    [23] 陈卫忠,吕森鹏,郭小红,等. 基于能量原理的卸围压试验与岩爆判据研究[J]. 岩石力学与工程学报,2009,28(8):1530−1540. doi: 10.3321/j.issn:1000-6915.2009.08.003

    CHEN Weizhong,LYU Senpeng,GUO Xiaohong,et al. Research on unloading confining pressure tests and rockburst criterion based on enrgy theory[J]. Chinese Journal of Rock Mechanics and Engineering,2009,28(8):1530−1540. doi: 10.3321/j.issn:1000-6915.2009.08.003

    [24] 卢志国, 鞠文君, 高富强, 等. 基于非线性储能与释放特征的煤冲击倾向性指标[J]. 岩石力学与工程学, 2021, 40(8): 1559−1569 .

    LU Zhiguo, JU Wenjun, GAO Fuqiang, et al. Bursting liability index of coal based on nonlinear storage and release characteristics of elastic energy, 2021, 40(8): 1559−1569.

    [25] 宫凤强, 闫景一, 李夕兵. 基于线性储能规律和剩余弹性能指数的岩爆倾向性判据[J]. 岩石力学与工程学报, 2018, 37(9): 1993–2014.

    GONG Fengqiang, YAN Jingyi, LI Xibing. A new criterion of rock burst proneness based on the linear energy storage law and the residual elastic energy index[J]. Chinese Journal of Rock Mechanics and Engineering, 2018, 37(9): 1993–2014.

    [26] 王 超. 基于 有效冲击能量速率的煤层冲击倾向性指数研究[J]. 煤矿开采,2017,22(5):9−12.

    WANG Chao. Study of coal seam rock burst tendency index under effectively impact energy rate[J]. Coal Mining Technology,2017,22(5):9−12.

    [27] 唐礼忠,王文星. 一种新的岩爆倾向性指标[J]. 岩石力学与工程学报,2002,21(6):874−878. doi: 10.3321/j.issn:1000-6915.2002.06.022

    TANG Lizhong,WANG Wenxing. New rock burst proneness index[J]. Chinese Journal of Rock Mechanics and Engineering,2002,21(6):874−878. doi: 10.3321/j.issn:1000-6915.2002.06.022

    [28]

    GONG F,YAN J,LI X,et al. A peak-strength strain energy storage index for rock burst proneness of rock materials[J]. International Journal of Rock Mechanics and Mining Sciences,2019,117:76−89. doi: 10.1016/j.ijrmms.2019.03.020

    [29]

    GONG Fengqiang,WANG Yunliang,WANG Zhiguo,et al. A new criterion of coal burst proneness based on the residual elastic energy index[J]. International Journal of Mining Science and Technology,2021,31(4):553−563. doi: 10.1016/j.ijmst.2021.04.001

    [30] 姚精明,闫永业,李生舟,等. 煤层冲击倾向性评价损伤指标[J]. 煤炭学报,2011,36(S2):353−357.

    YAO Jingming,YAN Yongye,LI Shengzhou,et al. Damage index of coal seam rock burst proneness[J]. Journal of China Coal Society,2011,36(S2):353−357.

    [31]

    HOMAND F,PIGUET J P,REVALOR R. Dynamic phenomena in mines and characteristics of rocks[J]. International Journal of Rock Mechanics & Mining Sciences & Geomechanics Abstracts,1992,29(1):A69.

    [32] 张万斌,王淑坤,吴耀焜,等. 以动态破坏时间鉴定煤的冲击倾向[J]. 煤炭科学技术,1986,14(3):31−34.

    ZHANG Wanbin,WANG Shukun,WU Yaokun,et al. To determine proneness of coal burst by dynamic failure time[J]. Coal Science and Technology,1986,14(3):31−34.

    [33] 潘一山,耿 琳,李忠华. 煤层冲击倾向性与危险性评价指标研究[J]. 煤炭学报,2010,35(22):1975−1978.

    PAN Yishan,GENG Lin,LI Zhonghua. Research on evaluation indices for impact tendency and danger of coal seam[J]. Journal of China Coal Society,2010,35(22):1975−1978.

    [34] 代树红,王晓晨,潘一山,等. 模量指数评价煤的冲击倾向性的实验研究[J]. 煤炭学报,2019,44(6):1726−1731.

    DAI Shuhong,WANG Xiaochen,PAN Yishan,et al. Experimental study on the evaluation of coal burst tendency utilizing modulus index[J]. Journal of China Coal Society,2019,44(6):1726−1731.

    [35] 齐庆新,彭永伟,李宏艳,等. 煤岩冲击倾向性研究[J]. 岩石力学与工程学报,2011,30(S1):2736−2742.

    QI Qingxin,PENG Yongwei,LI Hongyan,et al. Study of bursting liability of coal and rock[J]. Chinese Journal of Rock Mechanics & Engineering,2011,30(S1):2736−2742.

    [36] 彭 俊,蔡 明,荣 冠,等. 裂纹闭合应力及其岩石微裂纹损伤评价[J]. 岩石力学与工程学报,2015,34(6):1091−1100.

    PENG Jun,CAI Ming,RONG Guan,et al. Stresses for crack closure and its application to assessing stress-induced microcrack damage[J]. Chinese Journal of Rock Mechanics and Engineering,2015,34(6):1091−1100.

    [37]

    KULATILAKE P H S W, MALAMA Bwalya, WANG Jialai. Physical and particle flow modeling of jointed rock block behavior under uniaxial loading[J]. International Journal of Rock Mechanics and Mining Sciences, 2001, 38(5): 641−657.

    [38]

    DEISMAN N, IVARS D M, PIERCE M. PFC2D Smooth joint contact model numerical experiments[C]// Edmonton, Canda: GeoEdmonton’08 Drganizing Committee, 2008.

    [39]

    BAHAADDINI M,HAGAN P C,MITRA R,et al. Parametric Study of Smooth Joint Parameters on the Shear Behaviour of Rock Joints[J]. Rock Mechanics and Rock Engineering,2015,48(3):923−940.

    [40] 李文洲,司林坡,卢志国,等. 煤单轴压缩起裂强度确定及其关键因素影响分析[J]. 煤炭学报,2021,46(S2):670−680.

    LI Wenzhou,SI Linpo,LU Zhiguo,et al. Determination of coal cracking initiation strength under uniaxial compression and analysis of its key factors[J]. Journal of China Coal Society,2021,46(S2):670−680.

    [41] 刘新荣,邓志云,刘永权,等. 岩石节理峰前循环直剪试验颗粒流宏细观分析[J]. 煤炭学报,2019,44(7):2103−2115.

    LIU Xinrong,DENG Zhiyun,LIU Yongquan,et al. Macroscopic and microscopic analysis of particle flow in pre-peak cyclic direct shear test of rock joint[J]. Journal of China Coal Society,2019,44(7):2103−2115.

图(13)  /  表(2)
计量
  • 文章访问数:  115
  • HTML全文浏览量:  18
  • PDF下载量:  50
  • 被引次数: 0
出版历程
  • 收稿日期:  2022-11-11
  • 网络出版日期:  2023-05-17
  • 刊出日期:  2023-05-30

目录

/

返回文章
返回