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一种基于弹性理论的大采高区段煤柱合理尺寸计算方法

于远祥, 解智勋, 胡梦玲, 邵红旗, 常石峰, 白永彪

于远祥,解智勋,胡梦玲,等. 一种基于弹性理论的大采高区段煤柱合理尺寸计算方法[J]. 煤炭科学技术,2023,51(3):37−51

. DOI: 10.13199/j.cnki.cst.2021-0590
引用本文:

于远祥,解智勋,胡梦玲,等. 一种基于弹性理论的大采高区段煤柱合理尺寸计算方法[J]. 煤炭科学技术,2023,51(3):37−51

. DOI: 10.13199/j.cnki.cst.2021-0590

YU Yuanxiang,XIE Zhixun,HU Mengling,et al. A calculation method of reasonable size of coal pillar in large mining height section based on elastic theory[J]. Coal Science and Technology,2023,51(3):37−51

. DOI: 10.13199/j.cnki.cst.2021-0590
Citation:

YU Yuanxiang,XIE Zhixun,HU Mengling,et al. A calculation method of reasonable size of coal pillar in large mining height section based on elastic theory[J]. Coal Science and Technology,2023,51(3):37−51

. DOI: 10.13199/j.cnki.cst.2021-0590

一种基于弹性理论的大采高区段煤柱合理尺寸计算方法

基金项目: 

陕西省自然科学基础研究计划资助项目(2014JM2-5052);中国博士后科学基金资助项目(2015M582761XB);陕西省教育厅自然科学研究资助项目(17JK0502)

详细信息
    作者简介:

    于远祥: (1976—),男,重庆梁平人,副教授,博士。E-mail:1374400919@qq.com

  • 中图分类号: TU353

A calculation method of reasonable size of coal pillar in large mining height section based on elastic theory

Funds: 

Natural Science Basic Research Program of Shaanxi Province (2014JM2-5052); Postdoctoral Science Foundation of China (2015M582761XB); Natural Science Research Project of Education Department of Shaanxi Province (17JK0502)

  • 摘要:

    不同工况下区段煤柱两侧支承压力分布及岩体变形存在显著差异,考虑煤柱两侧不同支承压力对煤柱整体稳定性的影响,基于大采高区段煤柱的弹性力学计算模型,分析了支承压力下煤柱任一单元岩体的应力应变分量。通过建立大采高煤柱弹塑性界面上岩体的柱条模型,确定在0.65倍煤柱高度处单元岩体将首先发生水平拉伸破坏,利用虎克定律提出了该单元岩体极限拉应变与煤柱极限平衡区宽度的关系式。依据煤柱破裂区岩体的受力特征,运用摩尔库伦准则推导了煤柱破裂区宽度的计算公式。结果表明:①煤柱极限平衡区宽度与岩体极限拉应变和弹性模量反相关,与煤柱埋深和煤柱高度正相关;②煤柱高度及其与顶底板的界面摩擦角是影响破裂区宽度的关键性因素;③煤柱两侧不同工况下,煤柱岩体极限拉应变与其所受侧压呈正变关系,区段煤柱采空区侧所受侧压较巷道侧偏大,采空区侧岩体的极限拉应变也相应较大,表现为采空侧极限平衡区宽度较巷道侧偏小。最后,将上述理论公式应用于陕北某矿30109工作面大采高区段煤柱极限平衡区和破裂区宽度的分析计算,给出了该工作面两侧区段煤柱的合理宽度及其支护方案。工程应用表明,30109工作面区段巷道围岩变形控制效果良好,满足现场生产需求。

    Abstract:

    There are significant differences in abutment pressure distribution and rock mass deformation on both sides of section coal pillar under different working conditions, the influence of different abutment pressures on the overall stability of coal pillar is considered, based on the elastic mechanics calculation model of coal pillar in large mining height section, the stress-strain components of any unit rock mass of coal pillar under abutment pressure are analyzed. Through the establishment of large mining height pillar elastic-plastic interface on the rock column model, it is determined that unit rock mass will first undergo horizontal tensile failure at 0.65 times the height of coal pillar, based on Hooke's law, the relationship between the ultimate stretching strain of rock mass and the width of limit equilibrium zone of coal pillar is proposed. According to the mechanical characteristics of the rock mass in the coal pillar fracture zone, the calculation formula of the width of the coal pillar fracture zone is derived by using the Mohr-Coulomb criterion. The results show that: ①The width of limit equilibrium zone of coal pillar is inversely related to ultimate tensile strain and elastic modulus of rock mass and positively related to buried depth and height of coal pillar;②The height of coal pillar and the interface friction angle between coal pillar and roof and floor are the key factors affecting the width of fracture zone; ③Under different working conditions on both sides of coal pillar, due to the positive relationship between the ultimate tensile strain of coal pillar rock mass and its lateral pressure, the lateral pressure on the gob side of the section coal pillar is larger than that on the roadway side, and the ultimate tensile strain of the rock mass on the gob side is correspondingly larger, which shows that the width of the limit equilibrium area on the gob side is smaller than that on the side of the gateway. Finally, the above theoretical formula is applied to the analysis and calculation of the width of limit equilibrium zone and fracture zone of coal pillar in large mining height section of 30109 working face in a mine in Northern Shaanxi, and the reasonable width of coal pillar and its supporting scheme under different working conditions on both sides of the working face are given. The engineering application shows that the deformation control effect of surrounding rock along the working face is good, which can meet the demand of field production.

  • 陕北地区作为我国重要的煤炭生产基地,煤层具有埋深浅、倾角近水平、煤层厚等显著特点,普遍采用一次采全高的大采高综采技术。在一次采全高的工艺条件下,合理确定区段煤柱作为采区设计的关键科学问题之一,对于降低吨煤成本、提高煤炭采出率、确保工作面安全高效生产具有重要意义。长期以来,大量专家学者对煤柱宽度留设进行了系统深入研究。侯朝炯等[1]通过建立煤柱巷道侧支承压力作用下的力学模型,运用极限平衡理论研究了对该侧岩体的极限平衡区宽度;高玮[2]通过建立倾斜煤柱倾角在支承压力作用下的力学模型,推导了倾斜煤柱岩体的应力分布及其变形分区情况;于远祥等[3]提出了煤柱巷道侧的支承压力双曲函数力学模型,讨论了影响煤帮塑性变形的主要影响因素;翟锦[4]考虑上覆岩层荷载和煤柱岩体自身体积应力的影响,建立了倾斜区段煤柱塑性区和弹性区的理论宽度公式;王德超等[5]基于煤柱侧向支承压力的现场监测和数值模拟,确定了赵楼矿某工作面区段煤柱的合理尺寸;孙德中等[6]将极限平衡理论和现场监测相结合研究了大采高综放工作面侧向支承压力的分布规律;张念超等[7]基于支护荷载与护巷煤柱尺寸的分析模型,探讨了支护应力与护巷煤柱尺寸的关系;魏臻等[8]分析了煤柱上方支承压力在不同阶段的演变过程,提出了煤柱采空侧和巷道侧的极限平衡区计算式;赵宾等[9]根据Wilson两区约束理论,认为采空侧和巷道侧的塑性区近似相等,确定了蒋家河煤矿某工作面倾斜区段煤柱的合理尺寸;邸帅等[10]假设煤柱采空区和巷道侧的塑性区宽度相同,提出了煤柱塑性区宽度等同理论;宁静[11]考虑煤柱巷道侧和采空侧支承压力集中程度的差异,基于极限平衡理论分析了红庆河某工作面区段煤柱的塑性区宽度;顾士坦等[12]通过对煤柱岩体应力变化规律的数值分析,得到了双侧采空不规则煤柱的最小安全尺寸;王东星[13]、王宝石[14]、薛雄飞[15]基于极限平衡理论和等效圆方法分别计算了煤柱采空侧与巷道侧的塑性区宽度;白进龙[16]、张艳军[17]将区段煤柱两侧的变形简化为塑性对称,计算了煤柱单侧岩体的塑性变形范围。

    上述专家学者从不同角度对煤柱岩体的变形破坏规律进行了系统深入的分析,所得研究成果对确定合理区段煤柱留设尺寸具有积极的指导意义,但在确定煤柱变形范围方面,文献[9-10]忽略了采空侧和巷道侧煤柱变形的差异,文献[1-3]和[13-17]尽管研究单侧支承压力下的煤柱变形规律,但均忽略了两侧支承压力对煤柱整体变形及其稳定性的影响。在工程实际中,煤柱某侧岩体的变形破坏不仅与其具体工况密切相关,而且也受煤柱另一侧支承压力的重要影响。从现有关于煤柱宽度留设的研究中可以看出,考虑两侧工况差异及支承压力对煤柱整体变形影响的研究成果还较少。基于此,在总结前人研究成果的基础上,通过建立不同工况下区段煤柱在支承压力作用下的力学分析模型,运用弹性理论计算近水平大采高区段煤柱岩体在两侧支承压力作用下的应力应变,并提出一种基于煤柱岩体变形特征来确定煤柱留设宽度的反演分析方法,为合理留设大采高工作面区段煤柱提供新的思路。

    工程实际中,区段煤柱包括两侧均为采区、两侧均为巷道、一侧采区和一侧巷道3种工况,煤柱上方支承压力分布状态也因工况差异而不尽相同。在无支护条件下,煤柱两侧岩体在支承压力下将会产生不同程度的变形,最终形成破碎区、塑性区及弹性区3个区域,如图1所示[18]

    图  1  不同工况下煤柱支承压力分布模型
    Figure  1.  Distribution model of coal pillar supporting pressure under different working conditions

    为便于分析,设区段煤柱岩体为连续、均质且各向同性的弹性体。将图1中煤柱所受的支承压力等效为均布荷载和线性荷载2种工况,并将其分解为若干部分。在煤柱内取任一单元岩体作为研究对象,分别计算该单元岩体在各部分支承压力作用下产生的各应力分量,然后将各应力分量进行叠加即可求得煤柱内任一单元岩体的应力分量[19]。因此,建立煤柱内任一单元岩体在均匀分布荷载和线性分布荷载作用下的力学模型如图2所示。

    图  2  煤柱荷载分布模型
    Figure  2.  Coal pillar load distribution model

    由弹性理论可知[20],均匀分布荷载下煤柱岩体中任一单元岩体的应力分量分别为

    $$ \left\{ \begin{gathered} {\sigma _x} = \frac{p}{\pi }\left[ {\alpha - \sin\; \alpha \cos \left( {\alpha + 2\delta } \right)} \right] \\ {\sigma _y} = \frac{p}{\pi }\left[ {\alpha + \sin\; \alpha \cos \left( {\alpha + 2\delta } \right)} \right] \\ {\tau _{xy}} = \frac{p}{\pi }\sin \;\alpha \sin \left( {\alpha + 2\delta } \right) \\ \end{gathered} \right. $$ (1)

    同理可得线性分布荷载下煤柱岩体中任一单元岩体的应力分量分别为

    $$ \left\{ \begin{gathered} {\sigma _x} = \frac{p}{{2\pi }}\left( {\frac{x}{b}\alpha - \frac{y}{b}\ln \frac{{R_1^2}}{{R_2^2}} + \sin\; 2\delta } \right) \\ {\sigma _y} = \frac{p}{{2\pi }}\left( {\frac{x}{b}\alpha - \sin\; 2\delta } \right) \\ {\tau _{xy}} = \frac{P}{{2\pi }}\left( {1 + \cos \;2\sigma - \frac{y}{b}\delta } \right) \\ \end{gathered} \right. $$ (2)

    式中:$\sigma _x $为煤柱水平方向应力;$\sigma _y $为煤柱垂直方向应力;$\tau _{xy} $为煤柱界面上水平向的剪应

    图1c所示区段煤柱为分析对象。当煤柱一侧为采场,另一侧为巷道时,由于两侧采空区悬顶长度不同,煤柱两侧上方应力集中程度不同[21]。为方便讨论,将煤柱上方支承压力简化为线性荷载,如图3所示。

    图  3  两侧不同工况下煤柱受力变形简化模型
    Figure  3.  Simplified model of mechanical deformation of coal pillar under different conditions on both sides
    $$ 令,\left\{ \begin{gathered} {P_1} = {k_1}\gamma H \\ {P_0} = \gamma H \\ {P_2} = {k_2}\gamma H \\ \end{gathered} \right. $$ (3)

    将式(1)—(2)中所有变量采用直角坐标表示。按荷载分布形式,将图3所示煤柱支承压力简化模型划分为5个部分,各部分应力分量计算如下:

    1)I部分荷载作用下应力分析。煤柱左侧极限平衡区线性荷载作用下的计算模型如图4所示。

    图  4  左侧极限平衡区线性荷载计算模型
    Figure  4.  Linear load calculation model for limit equilibrium zone

    由式(2)可得在左侧极限平衡区线性荷载作用下煤柱内任一点的应力分量为:

    $$ \begin{split} &{\sigma _{x1}} = \frac{{{P_1}}}{\pi }\left[ {\frac{x}{{{x_{01}}}}\left( {\arctan \frac{x}{y} - \arctan \frac{{x - {x_{01}}}}{y}} \right) - \frac{y}{{{x_{01}}}} \times } \right. \\ &\qquad \left. {\ln \frac{{{x^2} + {y^2}}}{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}} + \frac{{\left( {x - {x_{01}}} \right)y}}{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}}} \right] \end{split}$$ (4)
    $$ {\sigma _{y1}} = \frac{{{P_1}}}{\pi }\left[ {\frac{x}{{{x_{01}}}}\left( {\arctan \frac{x}{y} - \arctan \frac{{x - {x_{01}}}}{y}} \right)} \right. \left. { - \frac{{\left( {x - {x_{01}}} \right)y}}{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}}} \right] $$ (5)
    $$ {\tau _{xy1}} = \frac{{{P_1}}}{\pi }\left[ {\frac{{{y^2}}}{{{{\left( {x - {x_{01}}} \right)}^2} - {y^2}}} - } \right. \left. {\frac{y}{{{x_{01}}}}\left( {\arctan \frac{x}{y} - \arctan \frac{{x - {x_{01}}}}{y}} \right)} \right] $$ (6)

    2)II部分荷载作用下应力分析。煤柱左侧弹性区线性荷载计算模型如图5所示。

    图  5  左侧弹性区线性荷载计算模型
    Figure  5.  Calculation model of linear load in elastic zone

    由式(2)可得左侧弹性区线性荷载作用下煤柱内任一点的应力分量为

    $$ \begin{split} &\qquad{\sigma _{x2}} = \frac{{{P_1} - {P_0}}}{\pi }\left[ {\frac{{x - {x_{01}} - {L_{\rm{e}}}}}{{{L_{\rm{e}}}}}\left( {\arctan \frac{{x - {x_{01}}}}{y}} \right.} \right. -\\ &\qquad\left. { \arctan \frac{{x - {x_{01}} - {L_e}}}{y}} \right) - \frac{y}{{{L_{\rm{e}}}}}\ln \frac{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}}{{{{\left( {x - {x_{01}} - {L_e}} \right)}^2} + {y^2}}} + \\ &\qquad\qquad\qquad\qquad \left. { \frac{{\left( {x - {x_{01}}} \right)y}}{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}}} \right] \end{split} $$ (7)
    $$ \begin{split} &{\sigma _{y2}} = \frac{{{P_1} - {P_0}}}{\pi }\left[ {\frac{{x - {x_{01}} - {L_{\rm{e}}}}}{{{L_{{\rm{e}}1}}}}} \right.\left( {\arctan \frac{{x - {x_{01}}}}{y}} -\right. \\ &\qquad\left. {\left. { {\text{arctan}}\frac{{x - {x_{01}} - {L_e}}}{y}} \right) - \frac{{\left( {x - {x_{01}}} \right)y}}{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}}} \right] \end{split} $$ (8)
    $$ \begin{split} &{\tau _{xy2}} = \frac{{{P_1} - {P_0}}}{\pi }\left[ {\frac{{{y^2}}}{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}} - \frac{y}{{{L_{\rm{e}}}}} \times } \right. \\ &\left. {\left( {{\text{arctan}}\frac{{x - {x_{01}}}}{y} - {\text{arctan}}\frac{{x - {x_{01}} - {L_{\rm{e}}}}}{y}} \right)} \right] \end{split} $$ (9)

    3)III部分荷载作用下应力分析。煤柱弹性区均布荷载计算模型如图6所示。

    图  6  弹性区均布荷载计算模型
    Figure  6.  Calculation model of uniformly distributed load in elastic zone

    由式(1)可得弹性区均布荷载作用下煤柱内任一点的应力分量为

    $$\begin{split} & {\sigma _{x3}} = \frac{{{P_0}}}{\pi }\left[ {\arctan \frac{{x - {x_{01}} - {L_{\rm{e}}}}}{y}} \right. - \arctan \frac{{x - {x_{01}}}}{y} - \\ &\qquad\left. { \frac{{\left( {x - {x_{01}}} \right)y}}{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}} + \frac{{\left( {x - {x_{01}} - {L_{\rm{e}}}} \right)y}}{{{{\left( {x - {x_{01}} - {L_{\rm{e}}}} \right)}^2} + {y^2}}}} \right] \end{split}$$ (10)
    $$ \begin{split} &{\sigma _{y3}} = \frac{{{P_0}}}{\pi }\left[ {\arctan \frac{{x - {x_{01}} - {L_{\rm{e}}}}}{y}} \right. - \arctan \frac{{x - {x_{01}}}}{y} + \\ &\qquad\left. { \frac{{\left( {x - {x_{01}}} \right)y}}{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}} - \frac{{\left( {x - {x_{01}} - {L_{\rm{e}}}} \right)y}}{{{{\left( {x - {x_{01}} - {L_{{\rm{e}}}}} \right)}^2} + {y^2}}}} \right] \end{split}$$ (11)
    $$ {\tau _{xy3}} = \frac{{{P_0}}}{\pi }\left[ {\frac{{{y^2}}}{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}} - \frac{{{y^2}}}{{{{\left( {x - {x_{01}} - {L_{\rm{e}}}} \right)}^2} + {y^2}}}} \right] $$ (12)

    4)IV部分荷载作用下应力分析。煤柱右侧弹性区线性荷载计算模型如图7所示。

    图  7  右侧弹性区线性荷载计算模型
    Figure  7.  Calculation model of Linear load in elastic zone

    由式(2)可得右侧弹性区线性荷载作用下煤柱内任一点的应力分量为:

    $$ \begin{split} &\qquad{\sigma _{x4}} = \frac{{{P_2} - {P_0}}}{\pi }\left[ {\frac{{x - {x_{01}} - {L_{\rm{e}}}}}{{{L_{\rm{e}}}}}\left( {\arctan \frac{{x - {x_{01}}}}{y}} \right.}- \right. \\ &\qquad\left. { \arctan \frac{{x - {x_{01}} - {L_{\rm{e}}}}}{y}} \right) - \frac{y}{{{L_{\rm{e}}}}}\ln \frac{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}}{{{{\left( {x - {x_{01}} - {L_{\rm{e}}}} \right)}^2} + {y^2}}} + \\ &\qquad\qquad\qquad\qquad\left. { \frac{{\left( {x - {x_{01}}} \right)y}}{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}}} \right] \end{split} $$ (13)
    $$\begin{split} & {\sigma _{y4}} = \frac{{{P_2} - {P_0}}}{\pi }\left[ {\frac{{x - {x_{01}} - {L_{\rm{e}}}}}{{{L_{e1}}}}} \right.\left( {\arctan \frac{{x - {x_{01}}}}{y}}- \right. \\ &\qquad\left. {\left. { {\text{arctan}}\frac{{x - {x_{01}} - {L_{\rm{e}}}}}{y}} \right) - \frac{{\left( {x - {x_{01}}} \right)y}}{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}}} \right] \end{split}$$ (14)
    $$\begin{split} &{\tau _{xy4}} = \frac{{{P_2} - {P_0}}}{\pi }\left[ {\frac{{{y^2}}}{{{{\left( {x - {x_{01}}} \right)}^2} + {y^2}}} - \frac{y}{{{L_{\rm{e}}}}} \times } \right. \\ &\left. {\left( {{\text{arctan}}\frac{{x - {x_{01}}}}{y} - {\text{arctan}}\frac{{x - {x_{01}} - {L_{\rm{e}}}}}{y}} \right)} \right] \end{split} $$ (15)

    5)V部分荷载分析。煤柱右侧极限平衡区线性荷载作用下的计算模型如图8所示。

    图  8  右侧极限平衡区线性荷载计算模型
    Figure  8.  Calculation model of Linear load in elastic zone

    由式(2)可得右侧极限平衡区线性荷载作用下该区内任一点的应力分量为

    $$\begin{split} & \qquad {\sigma _{x5}} = \frac{{{P_2}}}{\pi }\left[ {\frac{{x - {x_{01}} - {L_{\rm{e}}} - {x_{02}}}}{{{x_{02}}}}\left( {\arctan \frac{{x - {x_{01}} - {L_{\rm{e}}}}}{y}} \right.} -\right. \\ &\left. { \arctan \frac{{x - {x_{01}} - {L_{\rm{e}}} - {x_{02}}}}{y}} \right) - \frac{y}{{{x_{02}}}}\ln \frac{{{{\left( {x - {x_{01}} - {L_{\rm{e}}}} \right)}^2} + {y^2}}}{{{{\left( {x - {x_{01}} - {L_{\rm{e}}} - {x_{02}}} \right)}^2} + {y^2}}} + \\ &\qquad\qquad\qquad\left. {\frac{{\left( {x - {x_{01}} - {L_{{\rm{e}}1}}} \right)y}}{{{{\left( {x - {x_{01}} - {L_{{\rm{e}}1}}} \right)}^2} + {y^2}}}} \right] \end{split} $$ (16)
    $$ \begin{split} &{\sigma _{y5}} = \frac{{{P_2}}}{\pi }\left[ {\frac{{x - {x_{01}} - {L_{\rm{e}}} - {x_{02}}}}{{{x_{02}}}}} \right.\left( {\arctan \frac{{x - {x_{01}} - {L_{\rm{e}}}}}{y}}- \right. \\ &\left. {\left. { {\text{arctan}}\frac{{x - {x_{01}} - {L_{\rm{e}}} - {x_{02}}}}{y}} \right) - \frac{{\left( {x - {x_{01}} - {L_{\rm{e}}}} \right)y}}{{{{\left( {x - {x_{01}} - {L_{\rm{e}}}} \right)}^2} + {y^2}}}} \right] \end{split} $$ (17)
    $$\begin{split} &\qquad {\tau _{xy5}} = \frac{{{P_2}}}{\pi }\left[ {\frac{{{y^2}}}{{{{\left( {x - {x_{01}} - {L_{\rm{e}}}} \right)}^2} + {y^2}}} - \frac{y}{{{x_{02}}}} \times } \right.\\ &\left. {\left( {{\text{arctan}}\frac{{x - {x_{01}} - {L_{\rm{e}}}}}{y} - {\text{arctan}}\frac{{x - {x_{01}} - {L_{\rm{e}}} - {x_{02}}}}{y}} \right)} \right] \end{split} $$ (18)

    通过确定煤柱岩体内任一点在各部分支承压力作用下的应力分量,由弹性力学理论可得该点的应力分量大小为

    $$ \left\{ \begin{gathered} {\sigma _x} = {\sigma _{x1}} + {\sigma _{x2}} + {\sigma _{x3}} + {\sigma _{x4}} + {\sigma _{x5}} \\ {\sigma _y} = {\sigma _{y1}} + {\sigma _{y2}} + {\sigma _{y3}} + {\sigma _{y4}} + {\sigma _{y5}} \\ {\tau _{xy}} = {\tau _{xy1}} + {\tau _{xy2}} + {\tau _{xy3}} + {\tau _{xy4}} + {\tau _{xy5}} \\ \end{gathered} \right. $$ (19)

    将煤柱岩体简化为均质连续的弹性体,根据虎克定律可得煤柱岩体内任一点的应变分量

    $$ \left\{ \begin{gathered} {\varepsilon _x} = \frac{{1 - {\mu ^2}}}{E}\left( {{\sigma _x} - \frac{\mu }{{1 - \mu }}{\sigma _y}} \right) \\ {\varepsilon _y} = \frac{{1 - {\mu ^2}}}{E}\left( {{\sigma _y} - \frac{\mu }{{1 - \mu }}{\sigma _x}} \right) \\ \end{gathered} \right. $$ (20)

    式中:$ E $$ \mu $分别为煤柱综合弹性模量和泊松比。

    由文献[22]可知,当大采高煤柱两侧采空后,两侧岩体由三向受力状态进入二向受力状态,岩体强度也有所降低。在顶底板集中压力夹持作用下,形成典型的“压缩柱条”结构,如图9所示。

    图  9  区段煤柱岩体压缩柱条分析模型
    Figure  9.  Compression column bar analysis model of section coal pillar rock mass

    当“压缩柱条”岩体所产生的径向拉应变达到其极限拉应变时将发生挠曲断裂。由煤柱表面至该侧弹塑性界面,各“压缩柱条”岩体所发生的水平挠曲变形逐渐减小,多个连续“压缩柱条”岩体的水平挠曲变形叠加后向采空区挤进,最终在煤柱两侧一定范围内形成连续的极限平衡区。

    图9所示区段煤柱左侧破裂区的岩体为研究对象,如图10所示。

    图  10  煤柱极限平衡区柱条分析模型
    Figure  10.  Column bar analysis model for limit equilibrium zone of coal pillars

    将煤柱弹塑性界面上岩体视为下端固定,上端铰支的受压柱条,不考虑岩体自重和及其在垂直方向上的压缩变形,建立该界面岩体的“压缩柱条”计算模型如图11所示。

    图  11  煤柱弹塑性界面岩体柱条模型
    Figure  11.  Fracture-flexural mechanical model of rock mass with elastic-plastic interface of coal pillar

    图11b11c可知,取$ y $截面以上部分为分析对象,以$ y $截面处取矩得:

    $$ M = {F_{\rm{p}}}\omega - \frac{{{M_0}y}}{h} (y\geqslant 0) $$ (21)

    式中:Fp为弹塑性界面柱条岩体所受峰值压力;$\omega $为煤柱弹塑性界面扰度;M为x截面形心的弯矩;Fp为煤柱弹塑性界面所受垂直方向压力;Mo为煤柱弹塑性界面固定端力矩;h为煤柱高度;y为煤柱在弹塑性界面发生挠曲断裂的位置。

    由材料力学理论[23]可得微分方程:

    $$ M = - {{EI}}\omega '' $$ (22)

    式中:${{EI}}$为柱条弯矩刚度。

    联立式(21)和式(22)有:

    $$ {{EI}}\omega '' + {F_{\rm{p}}}\omega = \frac{{{M_0}y}}{h} $$ (23)
    $$ 令\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad{K^2} = \frac{{{F_{\rm{p}}}}}{{{{EI}}}} $$ (24)

    进而可得:

    $$ \omega '' + {K^2}\omega = \frac{{{M_0}y}}{{{{EI}}h}} $$ (25)

    解之得:

    $$ \omega = {c_1}\cos Ky + {c_2}\sin Ky + \frac{{{M_0}y}}{{{P_1}h}} $$ (26)

    式中:c1c2均为微分方程的通解系数;K为常数。

    图11可知,在煤柱弹塑性界面上柱条岩体的边界条件为

    $$ \left\{ \begin{gathered} y = 0,\omega \left( 0 \right) = 0 \\ y = h,\frac{{{\rm{d}}w}}{{{\rm{d}}y}} = 0 \\ \end{gathered} \right. $$ (27)

    联立式(26)和式(27)得:

    $$ \left\{ \begin{gathered} {c_1} = 0 \\ {c_2} = 1.02 \frac{{{M_0}}}{{{F_{\rm{p}}}}} \\ \end{gathered} \right. $$ (28)

    由式(24)、式(26)及式(28)得:

    $$ \omega = \frac{{{M_0}}}{{{F_{\rm{p}}}}}\left[ {\left( {\frac{y}{h} + 1.02\sin \frac{{4.49y}}{h}} \right)} \right] $$ (29)

    由式(29)可以看出,当$ \sin \dfrac{{4.49y}}{h} = 1 $时弹塑性界面上柱条挠度取得最大值,此时有:

    $$ \frac{{4.49y}}{h} = 2k\pi + \frac{\pi }{2} $$ (30)

    式中,k为非负整数。

    进而有:

    $$ y = \left( {1.4k + 0.35} \right)h $$ (31)

    $ k = 0 $时,由式(30)有:

    $$ y=0.35h $$ (32)

    图10及式(32)可以看出,煤柱弹塑性界面上产生挠曲断裂的单元岩体为$ {M_l}\left( {{x_{01}},0.35h} \right) $,这与现场实际监测的实际情况[24-25]相一致。

    结合图3图9建立两侧不同工况下大采高区段煤柱极限平衡区宽度计算模型如图12所示。由文献[26]可知,峰值支承压力作用下煤柱左右两侧弹塑性界面上岩体的极限拉应变与该界面所受侧压成正变关系。为便于讨论,设区段煤柱弹性区宽度$ {L_{\rm{e}}}{\text{ = 2}}h $

    图  12  煤柱极限平衡区宽度计算模型
    Figure  12.  Calculation model of the width of coal pillar limit equilibrium zone

    将单元岩体$ {M_l}\left( {{x_{01}},0.35h} \right) $的坐标代入式(4)、式(5)、式(7)、式(8)、式(10)、式(11)、式(13)、式(14)、式(16)及式(17)得该岩体在各部分支承压力作用下产生的应力分量分下产生的应力分量分别为:

    $$ \left\{ \begin{gathered} {\sigma _{x1}} = \frac{{{P_1}}}{\pi }\left( {\arctan \frac{{2.86{x_{01}}}}{h} - \frac{{0.35h}}{{{x_{01}}}}\ln \frac{{x_{01}^2 + 0.12{h^2}}}{{0.12{h^2}}}} \right) \\ {\sigma _{y1}} = \frac{{{P_1}}}{\pi }\arctan \frac{{2.86{x_{01}}}}{h} \\ \end{gathered} \right. $$ (33)
    $$ \left\{ \begin{gathered} {\sigma _{x2}} = \frac{{ - 0.79 \left( {{P_1} - {P_0}} \right)}}{\pi } \\ {\sigma _{y2}} = \frac{{ - 1.4 \left( {{P_1} - {P_0}} \right)}}{\pi } \\ \end{gathered} \right. $$ (34)
    $$ \left\{ \begin{gathered} {\sigma _{x3}} = - 1.57\frac{{{P_0}}}{\pi } \\ {\sigma _{y3}} = - 1.23\frac{{{P_0}}}{\pi } \\ \end{gathered} \right. $$ (35)
    $$ \left\{ \begin{gathered} {\sigma _{x4}} = \frac{{ - 0.79 \times \left( {{P_2} - {P_0}} \right)}}{\pi } \\ {\sigma _{y4}} = \frac{{ - 1.4 \times \left( {{P_2} - {P_0}} \right)}}{\pi } \\ \end{gathered} \right. $$ (36)
    $$ \left\{ \begin{gathered} \begin{gathered} {\sigma _{x5}} = \frac{{{P_2}}}{\pi }\left[ \frac{{{x_{02}} + 2h}}{{{x_{02}}}}\left( {1.4 - \arctan \frac{{{x_{02}} + 2h}}{{0.35h}}} \right) - \right. \\ \quad \left. \frac{{0.35h}}{{{x_{02}}}} \times \ln \frac{{4.12{h^2}}}{{{{\left( {{x_{02}} + 2h} \right)}^2} + 0.12{h^2}}} - 0.17 \right] \\ \end{gathered} \\ {\sigma _{y5}} = \frac{{{P_2}}}{\pi }\left[ {\frac{{{x_{02}} + 2h}}{{{x_{02}}}}\left( {1.4 - \arctan \frac{{{x_{02}} + 2h}}{{0.35h}}} \right) + 0.17} \right] \\ \end{gathered} \right. $$ (37)

    将式(33)—式(37)代入式(18)得煤柱弹塑性界面发生最大水平位移的单元岩体应力分量为

    $$ \begin{split} &{\sigma _x} = \frac{{{P_1}}}{\pi }\left( {\arctan \frac{{2.86{x_{01}}}}{h} - 0.35\frac{h}{{{x_{01}}}} \ln \frac{{x_{01}^2 + 0.12{h^2}}}{{0.12{h^2}}} - 0.79} \right) + \\ &\qquad\frac{{{P_2}}}{\pi }\left[ {\frac{{{x_{02}} + 2h}}{{{x_{02}}}}\left( {1.4 - \arctan \frac{{{x_{02}} + 2h}}{{0.35h}}} \right) - \frac{{0.35h}}{{{x_{02}}}} \times } \right. \\ &\qquad\left. {\ln \frac{{4.12{h^2}}}{{{{\left( {{x_{02}} + 2h} \right)}^2} + 0.12{h^2}}} - 0.96} \right] + 0.01\frac{{{P_0}}}{\pi } \end{split} $$ (38)
    $$\begin{split} & {\sigma _y} = \frac{{{P_1}}}{\pi }\left( {\arctan \frac{{2.86{x_{01}}}}{h} - 1.4} \right) + \frac{{{P_2}}}{\pi }\left[ {\frac{{{x_{02}} + 2h}}{{{x_{02}}}}}\times \right. \\ &\qquad\left. {\left( {1.4 - \arctan \frac{{{x_{02}} + 2h}}{{0.35h}}} \right) - 1.23} \right] + 1.57\frac{{{P_0}}}{\pi } \end{split} $$ (39)

    对于峰值支承压力下的煤柱左侧弹塑性界面而言,发生最大水平应变的单元体的水平拉应变将达到其极限拉应变,由式(20)、式(38)及式(39)得:

    $$ \begin{split} &{\varepsilon _{{\text{tmax}}{\text{.BC}}}} = \frac{{1 - {\mu ^2}}}{E}\left\{ {\left( {1 - \frac{\mu }{{1 - \mu }}} \right) \frac{{{P_1}}}{\pi }\arctan \frac{{2.86{x_{01}}}}{h} - } \right.\frac{{{P_1}}}{\pi } \times \\ &\qquad\left[ {\frac{{0.35h}}{{{x_{01}}}} \ln \frac{{x_{01}^2 + 0.12{h^2}}}{{0.12{h^2}}} + 1.4\frac{\mu }{{1 - \mu }} - 0.79} \right] + \\ &\quad\left( {1 - \frac{\mu }{{1 - \mu }}} \right) \frac{{{P_2}}}{\pi } \frac{{{x_{02}} + 2h}}{{{x_{02}}}}\left( {1.4 - \arctan \frac{{{x_{02}} + 2h}}{{0.35h}}} \right) - \\ &\frac{{{P_2}}}{\pi }\left[ {\frac{{0.35h}}{{{x_{02}}}} \ln \frac{{4.12{h^2}}}{{{{\left( {{x_{02}} + 2h} \right)}^2} + 0.12{h^2}}} + 1.23\frac{\mu }{{1 - \mu }} - 0.96} +\right] \times\\ &\qquad\qquad\left. { \left( {0.01 - 1.57\frac{\mu }{{1 - \mu }}} \right)\frac{{{P_0}}}{\pi }} \right\} \end{split} $$ (40)

    同理,将单元岩体$ {M_{\rm{r}}}\left( {{x_{01}} + {L_{\rm{e}}},0.35h} \right) $的坐标代入式(4)、式(5)、式(7)、式(8)、式(10)、式(11)、式(13)、式(14)、式(16)及式(17)可得:

    $$ \left\{ \begin{gathered} \begin{gathered}{\sigma _{x1}} = \frac{{{P_1}}}{\pi }\left[ \frac{{{x_{01}} + 2h}}{{{x_{01}}}}\left( {\arctan \frac{{{x_{01}} + 2h}}{{0.35h}} - \arctan \frac{{2h}}{{0.35h}}} \right)- \right. \\ \left. \qquad \frac{{0.35h}}{{{x_{01}}}} \ln \frac{{{{\left( {{x_{01}} + 2h} \right)}^2} + 0.12{h^2}}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}} +\right. \\ \qquad\quad\left. \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}\right] \\ \end{gathered} \\ \begin{gathered}{\sigma _{y1}} = \frac{{{P_1}}}{\pi }\left[ \frac{{{x_{01}} + 2h}}{{{x_{01}}}}\left( {\arctan \frac{{{x_{01}} + 2h}}{{0.35h}} - \arctan \frac{{2h}}{{0.35h}}} \right)- \right. \\ \qquad\quad\left. \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}} \right] \\ \end{gathered} \\ \end{gathered} \right. $$ (41)
    $$ \left\{ \begin{gathered} {\sigma _{x2}} = \frac{{{P_1} - {P_0}}}{\pi }\left[ { - 0.18 \ln \frac{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}{{0.12{h^2}}} + \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] \\ {\sigma _{y2}} = \frac{{{P_1} - {P_0}}}{\pi }\left[ { - \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] \\ \end{gathered} \right. $$ (42)
    $$ \left\{ \begin{gathered} {\sigma _{x3}} = \frac{{{P_0}}}{\pi }\left[ { - \arctan \frac{{2h}}{{0.35h}} - \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] \\ {\sigma _{y3}} = \frac{{{P_0}}}{\pi }\left[ { - \arctan \frac{{2h}}{{0.35h}} + \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] \\ \end{gathered} \right. $$ (43)
    $$ \left\{ \begin{gathered} {\sigma _{x4}} = \frac{{{P_2} - {P_0}}}{\pi }\left[ { - 0.18 \ln \frac{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}{{0.12{h^2}}} + \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] \\ {\sigma _{y4}} = \frac{{{P_2} - {P_0}}}{\pi }\left[ { - \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] \\ \end{gathered} \right. $$ (44)
    $$ \left\{ \begin{gathered} {\sigma _{x5}} = \frac{{{P_2}}}{\pi }\left( { - \arctan \frac{{{x_{02}}}}{{0.35h}} - \frac{{0.35h}}{{{x_{02}}}} \ln \frac{{0.12{h^2}}}{{x_{02}^2 + 0.12{h^2}}}} \right) \\ {\sigma _{y5}} = \frac{{{P_2}}}{\pi } - \arctan \frac{{{x_{02}}}}{{0.35h}} \\ \end{gathered} \right. $$ (45)

    将式(41)—式(45)代入式(18)得煤柱右侧弹塑性界面上具有最大水平挠曲位移的单元岩体应力分量为:

    $$ \begin{split} &{\sigma _x} = \frac{{{P_1}}}{\pi }\left[ {\frac{{{x_{01}} + 2h}}{{{x_{01}}}}\left( {\arctan \frac{{{x_{01}} + 2h}}{{0.35h}} - \arctan \frac{{2h}}{{0.35h}}} \right)} \right. - \frac{{0.35h}}{{{x_{01}}}} \times\\ &\qquad\left. {\ln \frac{{{{\left( {{x_{01}} + 2h} \right)}^2} + 0.12{h^2}}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}} + \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] + \frac{{{P_1} - {P_0}}}{\pi } \times \\ &\qquad\left[ { - 0.18 \ln \frac{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}{{0.12{h^2}}} + \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] + \frac{{{P_0}}}{\pi } \times \\ &\qquad\left[ { - \arctan \frac{{2h}}{{0.35h}} - \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] + \frac{{{P_2} - {P_0}}}{\pi } \times \\ &\qquad\left[ { - 0.18 \ln \frac{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}{{x_{02}^2 + 0.12{h^2}}} + \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] + \frac{{{P_2}}}{\pi } \times \\ &\qquad\left( { - \arctan \frac{{{x_{02}}}}{{0.35h}} - \frac{{0.35h}}{{{x_{02}}}} \ln \frac{{0.12{h^2}}}{{x_{02}^2 + 0.12{h^2}}}} \right) \end{split}$$ (46)
    $$\begin{split} & {\sigma _y} = \frac{{{P_1}}}{\pi }\left[ {\frac{{{x_{01}} + 2h}}{{{x_{01}}}}\left( {\arctan \frac{{{x_{01}} + 2h}}{{0.35h}} - \arctan \frac{{2h}}{{0.35h}}} \right)}- \right. \\ &\left. { \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] - \frac{{{P_1} - {P_0}}}{\pi }\left[ {\frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] + \frac{{{P_0}}}{\pi } \times \\ &\qquad\left[ { - \arctan \frac{{2h}}{{0.35h}} + \frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] - \frac{{{P_2} - {P_0}}}{\pi } \\ &\qquad\quad\left[ {\frac{{2h \times 0.35h}}{{{{\left( {2h} \right)}^2} + 0.12{h^2}}}} \right] -\frac{{{P_2}}}{\pi } \arctan \frac{{{x_{02}}}}{{0.35h}} \end{split} $$ (47)

    在峰值支承压力下,煤柱右侧弹塑性界发生最大水平挠曲位移的煤柱单元岩体的水平拉应变将达到其极限拉应变,由式(46)、式(47)及式(20)得:

    $$ \begin{split} &\quad{\varepsilon _{{\text{tmax}}{\text{,EH}}}} = \frac{{1 - {\mu ^2}}}{E}\left\{ {\left( {1 - \frac{\mu }{{1 - \mu }}} \right) \frac{{{P_1}}}{\pi } \frac{{{x_{01}} + 2h}}{{{x_{01}}}} \times } \right. \\ &\quad\left( {\arctan \frac{{{x_{01}} + 2h}}{{0.35h}} - 1.4} \right) + \left( {1 + \frac{\mu }{{1 - \mu }}} \right) \frac{{{P_1}}}{\pi } 0.17 - \\ &\frac{{0.35h}}{{{x_{01}}}} \frac{{{P_1}}}{\pi } \ln \frac{{{{\left( {{x_{01}} + 2h} \right)}^2} + 0.12{h^2}}}{{4.12{h^2}}} + \left( {0.17\frac{\mu }{{1 - \mu }} - 0.47} \right) \times \\ &\frac{{{P_1} + {P_2} - {P_0}}}{\pi } + \left( { - 1.57 - 1.23\frac{\mu }{{1 - \mu }}} \right) \frac{{{P_0}}}{\pi } + \left( {\frac{\mu }{{1 - \mu }} - 1} \right) \frac{{{P_2}}}{\pi } \times \\ &\left. {\arctan \frac{{{x_{02}}}}{{0.35h}} - \frac{{{P_2}}}{\pi } \frac{{0.35h}}{{{x_{02}}}} \ln \frac{{0.12{h^2}}}{{x_{02}^2 + 0.12{h^2}}}} \right\} \end{split}$$ (48)

    联立式(40)和式(48)即可求得两侧不同工况下煤柱岩体的极限平衡区宽度。

    当煤柱两侧同工况时,有:

    $$ \left\{ \begin{gathered} {k_1} = {k_2} \\ {P_1} = {P_2} \\ {x_{01}} = {x_{02}} \\ \end{gathered} \right. $$ (49)

    将式(49)代入式(40)得两侧同工况条件下煤柱弹塑性界面上岩体极限水平拉应变与煤柱极限平衡区宽度的关系为:

    $$\begin{split} & {\sigma _{{{t,{\rm{max}}}}{\text{,BC}}}} = \frac{{1 - {\mu ^2}}}{E}\left\{ {\left( {1 - \frac{\mu }{{1 - \mu }}} \right) \frac{{{P_1}}}{\pi }\left[ {\arctan \frac{{2.86}}{h}} \right.{x_0} + } \right. \\ &\quad \frac{{{x_0} + 2h}}{{{x_0}}} \left. {\left( {1.4 - \arctan \frac{{{x_0} + 2h}}{{0.35h}}} \right)} \right] - \frac{{{P_1}}}{\pi }\left( {0.35\frac{h}{{{x_0}}} \times } \right. \\ &\quad\left. {\ln \frac{{34.33x_0^2 + 4.12{h^2}}}{{{{\left( {{x_0} + 2h} \right)}^2} + 0.12{h^2}}} + 1.75 - 2.63 \frac{\mu }{{1 - \mu }}} \right) + \\ &\qquad\qquad\left. {\left( {0.01 - 1.57\frac{\mu }{{1 - \mu }}} \right)\frac{{{P_0}}}{\pi }} \right\} \end{split} $$ (50)

    由式(40)、式(48)及式(50)可以看出,当其它参数一定时,区段煤柱极限平衡区宽度是关于煤柱弹塑性界面上岩体极限拉应变的隐函数,其大小不仅取决于煤柱岩体的极限拉应变、弹性模量及泊松比,而且与煤柱埋深、煤柱高度及支承压力峰值等因素密切相关。

    煤柱一侧采空后,其上方出现应力集中。当煤柱表面岩体所受集中应力超过其极限抗压强度时,煤柱开始出现破裂区。取图10所示煤柱极限平衡区分析模型中左侧破裂区岩体为研究对象,如图12所示。

    为简化计算,忽略破裂区煤柱岩体自重,由图13可知,煤柱破裂区任一界面上支承压力大小[3]可表示为:

    图  13  煤柱破裂区简化计算模型
    Figure  13.  Simplified calculation model for coal pillar fracture zone
    $$ {P_i} = \left[ {\frac{{N\sqrt \beta \times \cosh \left( {\sqrt \beta x} \right)}}{{2\sinh \left( {\sqrt \beta {x_0}} \right)}} - c} \right]\cot\; \varphi $$ (51)

    由破裂岩体在水平方向上的静力平衡关系得:

    $$ {\sigma _{\rm{h}}} h = \int_0^{{L_{\rm{s}}}} {\left( {\tan\; {\varphi _{\rm{u}}}{\text{ + }}\tan\; {\varphi _{\rm{d}}}} \right)} {P_i}{\rm{d}}x $$ (52)

    式中:$ {\varphi _{\rm{u}}} $$ {\varphi _{\rm{d}}} $分别为煤柱与顶底板岩体接触面上的摩擦角。

    由莫尔−库伦强度准则可知,在煤柱破裂区与塑性区分界面上,岩体所受垂直支承压力和水平侧压力满足:

    $$ {\sigma _{\rm{h}}} = \frac{{1 - \sin\; \varphi }}{{1 + \sin \;\varphi }}\gamma H - \frac{{2c\cos \;\varphi }}{{1 + \sin \;\varphi }} $$ (53)

    式中:cφ为煤柱岩体的黏聚力和内摩擦角。

    联立式(51)—式(53)可得:

    $$ {L_{\rm{s}}} = \ln \left( {A + \sqrt {{A^2} + 1} } \right) $$ (54)

    式中:系数A由下式确定:

    $$\begin{split} & A = \left[ {\frac{{h\left( {1 - \sin\; \varphi } \right)\gamma H - 2hc\cos \;\varphi }}{{\left( {\tan\; {\varphi _{\rm{u}}} + \tan\; {\varphi _{\rm{d}}}} \right)\left( {1 + \sin\; \varphi } \right)\cot \;\varphi }} + c} \right] \frac{{2\sinh \left( {\sqrt \beta {x_0}} \right)}}{{\beta N}} \times\\ &\qquad\qquad\qquad\sin \;hx = \frac {{\rm{e}}^x - {\rm{e}}^{ - x} }{2} \end{split}$$ (55)

    其中:

    $$ \left\{ \begin{gathered} \beta = \frac{{{K_{\rm{s}}}}}{{hE}} \\ N{\text{ = }}\lambda k\gamma H h \\ \end{gathered} \right. $$ (56)

    式中:$ c $$ \varphi $$ {K_{\rm{s}}} $分别为煤柱岩体与顶底板界面的黏聚力、内摩擦角及剪切刚度系数;N为煤帮弹塑性界面上所受的水平推力;$ \lambda $为煤帮弹塑性界面上的侧压系数。

    综上所述,大采高区段煤柱的理论宽度为

    $$ B = \left( {{x_{01}} + {x_{02}}} \right) + 2h $$ (57)

    考虑开采扰动影响,则大采高区段煤柱的理论宽度为

    $$ B = \left( {{x_{01}} + {x_{02}}} \right) d + 2h $$ (58)

    式中:d为开采扰动因子,1.5≤ d ≤3.0。

    1)极限平衡区宽度与煤体弹性模量的关系。设某区段煤柱埋深H=250 m,采高h=4.0 m,其上覆岩层容重$ \gamma {\text{ = }}25 $ kN/m3,煤柱岩体的泊松比和极限拉应变分别为$ \mu {\text{ = 0}}{\text{.3}} $${\varepsilon _{{\rm{t}},\max }} = 0.6 \times {10^{ - 3}}$,煤帮弹塑性界面上应力集中系数k=1.5,将上述参数代入式(50)可得:

    $$\begin{split} &\qquad\qquad E = - 2.58 \left[ {\arctan \left( {0.715{x_0}} \right)} \right. + \frac{{{x_0} + 8}}{{{x_0}}} \times \\ &\left. {\left( {1.4 - \arctan \frac{{{x_0} + 8}}{{1.4}}} \right)} \right] + 4.52 \left[ {\frac{{1.4}}{{{x_0}}}\ln \frac{{34.33x_0^2 + 65.92}}{{{{\left( {{x_0} + 8} \right)}^2} + 1.92}}}+ \right. \\ &\qquad\qquad\qquad\qquad\left. { 0.62} \right] + 2 \end{split} $$ (59)

    由式(59)可得煤柱极限平衡区宽度与煤体弹性模量的关系如图14所示。

    图  14  煤柱极限平衡区与煤体弹性模量的关系
    Figure  14.  Relationship between coal pillar limit equilibrium zone and coal elastic modulus

    2)极限平衡区与岩体极限拉应变的关系。同理,当区段煤柱的相关参数分别为:H=250 m,h=4.0 m,$ \gamma {\text{ = }}25 $ kN/m3$ \mu {\text{ = 0}}{\text{.3}} $k=1.5,$ E = 3.6 $ GPa,由式(50)可得:

    $$ \begin{split} &\qquad\varepsilon = - 0.425 \left[ {\arctan \left( {0.715{x_0}} \right)} \right. + \frac{{{x_0} + 8}}{{{x_0}}} \times \\ &\left. {\left( {1.4 - \arctan \frac{{{x_0} + 8}}{{1.4}}} \right)} \right] + 0.745 \left[ {\frac{{1.4}}{{{x_0}}}\ln \frac{{34.33x_0^2 + 65.92}}{{{{\left( {{x_0} + 8} \right)}^2} + 1.92}}}+ \right. \\ &\qquad\qquad\qquad\qquad\left. { 0.62} \right] + 0.33 \end{split} $$ (60)

    由式(60)可得煤柱极限平衡区宽度与岩体极限拉应变的关系如图15所示。

    图  15  煤柱极限平衡区宽度与煤体极限拉应变的关系
    Figure  15.  Relationship between the ultimate equilibrium zone of coal pillar and ultimate tensile strain of coal

    3)极限平衡区与煤柱埋深的关系。当区段煤柱相关参数分别为:h=4.0 m,$ \gamma {\text{ = }}25 $ kN/m3${\varepsilon _{t,\max }}{\text{ = }} {\text{0.6}} \times {\text{1}}{{\text{0}}^{ - 3}}$$ \mu {\text{ = 0}}{\text{.3}} $k=1.5,$ E = 3.6 $GPa,由式(50)可得:

    $$\begin{split} &\qquad \frac{{2.4 \times {{10}^3}}}{H} = 6.8 \left[ {\arctan \left( {0.715{x_0}} \right)} \right. + \frac{{{x_0} + 8}}{{{x_0}}} \times \\ &\left. {\left( {1.4 - \arctan \frac{{{x_0} + 8}}{{1.4}}} \right)} \right] - 11.94 \left[ {\frac{{1.4}}{{{x_0}}}\ln \frac{{34.33x_0^2 + 65.92}}{{{{\left( {{x_0} + 8} \right)}^2} + 1.92}}} +\right. \\ &\qquad\qquad\qquad\qquad\left. { 0.62} \right] - 5.27 \end{split} $$ (61)

    由式(61)可得极限平衡区宽度与煤柱埋深的关系如图16所示。

    图  16  煤柱极限平衡区宽度与煤柱埋深的关系
    Figure  16.  Relationship between coal pillar limit equilibrium zone and buried depth of roadway

    4)极限平衡区宽度与煤柱高度的关系。当浅埋煤柱的相关参数分别为:H=250 m,$ {\varepsilon _{t\max }}{\text{ = 0}}{\text{.6}} \times {\text{1}}{{\text{0}}^{ - 3}} $$ \gamma {\text{ = }}25 $ kN/m3$ \mu {\text{ = 0}}{\text{.3}} $k=1.5,$ E = 3.6 $ GPa,由式(50)可得:

    $$ \begin{split} &\qquad- 0.81 = - 1.7 \left[ {\arctan \left( {\frac{{2.86{x_0}}}{h}} \right)} \right. + \frac{{{x_0} + 2h}}{{{x_0}}} \times \\ &\left. {\left( {1.4 - \arctan \frac{{{x_0} + 2h}}{{0.35h}}} \right)} \right] + \frac{{1.04h}}{{{x_0}}} \ln \frac{{34.33x_0^2 + 4.12{h^2}}}{{{{\left( {{x_0} + 2h} \right)}^2} + 0.12{h^2}}} \end{split}$$ (62)

    由式(62)可得煤柱极限平衡区宽度与煤柱高度的关系如图17所示。

    图  17  煤柱极限平衡区宽度与煤柱高度的关系
    Figure  17.  Relationship between coal pillar limit equilibrium zone and coal pillar height

    图14图17可以看出,极限平衡区宽度随煤柱岩体弹性模量和极限拉应变的增加而减少,随煤柱埋深和煤柱高度的增加而增大。

    陕北某矿主采3号煤层,该矿30108工作面倾向长度299.1 m,走向长度3 993 m,目前在处于回采阶段。煤层开采深度H=250 m,煤层厚度7.8~8.6 m,平均厚度8.3 m,平均倾角$ 5^\circ $,属于近水平煤层。煤层结构复杂,一般含多层夹矸。直接顶为泥质粉砂岩和粉砂岩,平均厚度4.1 m;基本顶为中粗砂岩、细砂岩及粉砂岩,平均厚度32.2 m;底板岩性以铝质泥岩和粉砂岩为主。现拟在该工作面西侧布置30109和30110两个分层大采高综放工作面,开采高度与巷道高度一致,采用双巷布置方式,运输巷和辅运巷道高h=4.0 m,宽L=5.4 m,工作面布置如图18所示。

    图  18  工作面布置
    Figure  18.  Working face distribution

    由于30109工作面左侧区段煤柱两侧均为相同工况的巷道,仅需对煤柱一侧变形进行分析。经现场测试,30109工作面左侧区段煤柱岩体物理力学参数见表1

    表  1  煤帮各计算参数取值
    Table  1.  Value of calculating parameters of coal wall
    $ \gamma $/(kN·m−3K/GPa$ \lambda $$ \varphi $/(º)$ c $/MPa$ {\varphi _{\rm{u}}} $/(º)$ {\varphi _{\rm{d}}} $/(º)$ E $/GPa$ \mu $${\varepsilon _{ {\rm{t} ,}\max } }$/10−3D
    250.150.8301.825253.60.30.681.6
    注:D为直接顶与煤层弹量比。
    下载: 导出CSV 
    | 显示表格

    1)应力集中系数的确定。由文献[27]可知,左侧煤柱上方峰值应力集中系数为

    $$ k = \left( { - 841 + 445h + 13L + 84D + 3.275H - 20\alpha } \right) \times {10^{{-}3}} $$ (63)

    由式(63)可得30109工作面左侧煤柱上方应力集中系数为

    $$\begin{split} &{k_l}{\text{ = }}\left( { - 841 + 445 \times 4 + 13 \times 5.4 + 84 \times 1.6} +\right. \\ &\left. {3.275 \times 250 - 20 \times 0.017 \times 5} \right) \times {10^{ - 3}}{\text{ = }}1.96 \end{split} $$ (64)

    2)煤柱极限平衡区宽度计算。将上述相关参数代入式(50)得:

    $$\begin{split} & \qquad 1.37 = - 2.22\left[ {\arctan \left( {0.715{x_0}} \right) + \frac{{{x_0} + 8}}{{{x_0}}} \times } \right.\left( {1.4 - } \right. \\ &\left. { {\arctan \frac{{{x_0} + 8}}{{1.4}}} )} \right] + 3.9\left[ {\frac{{1.4}}{{{x_0}}}\ln \frac{{34.33x_0^2 + 65.92}}{{{{\left( {{x_0} + 8} \right)}^2} + 1.92}}} { + 0.62} \right] \end{split} $$ (65)

    由式(65)即可解得30109工作面左侧区段煤柱的极限平衡区宽度为$ {x_0} = 6.78 $ m。

    3)煤帮破裂区宽度分析。将表1中相关参数代入式(55)和式(56)可得:

    $$ \sqrt \beta = \sqrt {\frac{{1.5 \times {{10}^8}}}{{4 \times 3.6 \times {{10}^9}}}} = 0.102 $$ (66)
    $$ N{\text{ = 0}}{\text{.8}} \times {\text{1}}{\text{.96}} \times {\text{25}} \times {\text{1}}{{\text{0}}^3} \times 250 \times 4 = 3.92 \times {10^7}\;{\rm{ N}} $$ (67)
    $$ \begin{split} & A = \left[ {\frac{{\left( {1 - \sin \;30^\circ } \right) \times 4 \times 25 \times {{10}^3} \times 250 - 8 \times 1.8\cos \;30^\circ }}{{\left( {\tan\; 25^\circ + \tan \;25^\circ } \right)\left( {1 + \sin\; 30^\circ } \right)\cot 30^\circ }} + 1.8} \right] \times\\ &\qquad \qquad\frac{{2\sin h \left( {0.102 \times 6.78} \right)}}{{0.010\;4 \times 3.92 \times {{10}^7}}} = 19.59 \end{split}$$ (68)

    将式(68)代入式(54)得30109工作面左侧区段煤柱破裂区宽度为:

    $$L_s=\ln \left(19.59+\sqrt{19.59^2+1}\right)=3.66 \mathrm{~m}$$ (69)

    综上分析,由式(57)得30109工作面左侧区段煤柱理论宽度为:

    $$ B = 2 \times 6.78 + 2 \times 4.0{\text{ = }}21.56\; {\rm{m}} $$ (70)

    考虑工作面开采扰动,取扰动因子d=1.5,同时为提高煤炭采出率,煤柱弹性核宽度取工作面端头采高4 m,此时30109工作面左侧区段煤柱的理论宽度为:

    $$ B = 2 \times 1.5 \times 6.78 + 4.0{\text{ = }}24.34\;{\rm{ m}} $$ (71)

    由于30109工作面右侧区段煤柱一侧为巷道,一侧为采空区,需对煤柱两侧变形分别进行分析。经现场测试,30109工作面左侧区段煤柱岩体物理力学参数见表2

    表  2  煤帮各计算参数取值
    Table  2.  Value of calculating parameters of coal wall
    $ \gamma $/(kN·m−3K/GPa$ \lambda $$ \varphi $/(º)$ c $/MPa$ {\varphi _{\rm{u}}} $/(º)$ {\varphi _{\rm{d}}} $/(º)$ E $/GPa$ \mu $$ {\varepsilon _{{\rm{t}}\;\max }}_{,{\rm{BC}}} $/×10−3${\varepsilon _{ {\rm{t} },\max } }_{,{\rm{EH} } }$/10−3D
    250.180.8322.027273.40.30.630.681.6
    下载: 导出CSV 
    | 显示表格

    1)应力集中系数的确定。由于右侧煤柱两侧工况存在差异,取采空区宽度为40 m,巷道宽度为5.4 m。同理,由式(60)可得30109工作面右侧煤柱巷道侧和采空区侧上方应力集中系数分别为

    $$ {k_{r1}} = 1.96 $$ (72)
    $$ {k_{r2}} =2.41 $$ (73)

    2)煤柱极限平衡区宽度分析。将上述相关参数代入式(40)可得:

    $$ \begin{aligned} &\qquad 1.03=-2.22 \times \arctan \left(0.715 x_{01}\right)+3.9 \times\left[\frac{1.4}{x_{01}} \times\right. \\ & \left.\ln \frac{x_{01}^2 + 1.92}{1.92} - 0.19\right] - 2.73 \times \frac{x_{02}+8}{x_{02}} \times \left(1.4 - \arctan \frac{x_{02}+8}{1.4}\right)+ \\ & \qquad4.8 \times\left[\frac{1.4}{x_{02}} \times \ln \frac{65.92}{\left(x_{02}+8\right)^2+1.92}-0.43\right] \end{aligned} $$ (74)

    将上述相关参数代入式(48)可得:

    $$\begin{split} & - 2.84{\text{ = }} - 2.13 \times \frac{{{x_{01}} + 8}}{{{x_{01}}}} \times \left( {\arctan \frac{{{x_{01}} + 8}}{{1.4}} - 1.4} \right) +\\ &\quad \frac{{5.46}}{{{x_{01}}}} \ln \frac{{{{\left( {{x_{01}} + 8} \right)}^2} + 1.92}}{{65.92}} + 2.74 \arctan \frac{{{x_{02}}}}{{1.4}} +\\ &\qquad\qquad\quad\frac{{6.72}}{{{x_{02}}}}\ln \frac{{1.92}}{{x_{02}^2 + 1.92}} \end{split} $$ (75)

    联立式(74)和(76)可得30109工作面右侧区段煤柱极限平衡区$ {x_{01}} = 8.4 $ m,$ {x_{02}} = 7.2 $ m。

    可以看出,由于煤柱两侧岩体的极限拉应变与其所受侧压成正变关系,采空侧极限平衡区宽度较巷道侧的偏小,该结果与表3中现场实测相吻合。

    表  3  神东矿区综采面护巷煤柱塑性区宽度统计表[10]
    Table  3.  Plastic zone width of coal pillars in mining face in Shendong mining area[10]
    工作面塑性区宽度/m
    采空区侧巷道侧平均值
    寸草塔431151.54.73.1
    大柳塔224042.410.06.2
    活鸡兔123172.55.13.8
    补连塔313043.912.78.3
    大柳塔523062.78.15.4
    下载: 导出CSV 
    | 显示表格

    2)巷道侧破裂区宽度计算。将表2中相关参数代入式(55)和式(56)可得:

    $$ \sqrt \beta = \sqrt {\frac{{1.8 \times {{10}^8}}}{{4 \times 3.4 \times {{10}^9}}}} = 0.115 $$ (76)
    $$ N{\text{ = 0}}{\text{.8}} \times {\text{1}}{\text{.96}} \times {\text{25}} \times {\text{1}}{{\text{0}}^3} \times 250 \times 4 = 3.92 \times {10^7} \;{\rm{N}} $$ (77)
    $$ \begin{split} & A = \left[ {\frac{{4\left( {1 - \sin \;32^\circ } \right) \times 25 \times {{10}^3} \times 250 - 8 \times 2\cos\; 32^\circ }}{{\left( {\tan\; 27^\circ + \tan\; 27^\circ } \right)\left( {1 + \sin\; 32^\circ } \right)\cot \;32^\circ }} + 2} \right] \times \\ & \qquad\qquad\frac{{2\sinh \left( {0.115 \times 8.4} \right)}}{{0.013 \times 3.92 \times {{10}^7}}}{\text{ = }}20.67 \;{\rm{m}} \end{split} $$ (78)

    将式(66)代入式(54)得巷道侧煤柱破裂区宽度为:

    $$ {L_s} = \ln \left( {20.67 + \sqrt {{{20.67}^2} + 1} } \right){\text{ = }}3.72 \;{\rm{m }}$$ (79)

    3)采空区侧破裂区宽度计算。将表2中相关参数代入式(55)和(56)可得:

    $$ \sqrt \beta = \sqrt {\frac{{1.8 \times {{10}^8}}}{{4 \times 3.4 \times {{10}^9}}}} = 0.115 $$ (80)
    $$ N{\text{ = 0}}{\text{.8}} \times {\text{2}}{\text{.41}} \times {\text{25}} \times {\text{1}}{{\text{0}}^3} \times 250 \times 4 = 4.82 \times {10^7} \;{\rm{N}} $$ (81)
    $$ \begin{split} & A = \left[ {\frac{{4\left( {1 - \sin\; 32^\circ } \right) \times 25 \times {{10}^3} \times 250 - 8 \times 2\cos\; 32^\circ }}{{\left( {\tan\; 27^\circ + \tan \;27^\circ } \right)\left( {1 + \sin \;32^\circ } \right)\cot \;32^\circ }} + 2} \right] \times\\ & \qquad\qquad\frac{{2\sinh \left( {0.115 \times 7.2} \right)}}{{0.013 \times 4.82 \times {{10}^7}}}{\text{ = }}13.9\; {\rm{m}} \end{split} $$ (82)

    将式(66)代入式(54)得采空侧煤柱破裂区宽度为

    $$ {L_s} = \ln \left( {13.9 + \sqrt {{{13.9}^2} + 1} } \right){\text{ = }}3.32 \;{\rm{m}} $$ (83)

    综上分析,由式(57)得30109工作面右侧区段煤柱的理论宽度为

    $$ B = 8.4 + 2 \times 4.0 + 7.2{\text{ = }}23.6 \;{\rm{m}} $$ (84)

    考虑工作面开采扰动,同理可得30109工作面右侧区段煤柱理论宽度为

    $$ B = \left( {8.4 + 7.2} \right) \times 1.5 + 4.0{\text{ = }}27.4 \;{\rm{m}} $$ (85)

    通过上述研究,为安全起见,30109工作面左侧和右侧区段煤柱宽度分别为25 m和28 m。

    基于上述分析可以看出,由于30109工作面两侧区段煤柱岩体在物理力学参数和工况方面均存在较大差异,要实现煤柱在掘采过程中的安全稳定,需要对30109工作面左右两侧煤柱的变形范围进行差异性研究,并据此对煤柱及其所在巷道围岩进行针对性的支护设计。

    1)锚杆间排距及长度的确定。根据式(69)、(79)及(83)的计算结果可知,各煤柱表面破裂范围均超过1.5 m,应采用大围岩松动圈理论进行支护设计。锚杆采用A22螺纹钢,间排距800 mm×800 mm,设锚固体组合拱厚度1.2 m,锚杆在破裂岩体中压应力作用角45°,则按照挤密加固原理可得锚杆有效长度[28]为:

    $$ L = \frac{{1.2 \tan\; 45^\circ + 0.8}}{{\tan \;45^\circ }} = 2.0 \;{\rm{m}} $$ (86)

    2)锚索间排距及长度的确定。在巷道顶板失稳形成极限平衡拱,其矢高[29]为:

    $$ b = {a_1}\left[ {\sqrt {{{\left( {\frac{f}{K}} \right)}^2} + 1} + \frac{f}{K}} \right] $$ (87)

    式中:$ {a_1} $为巷道半宽;$ f $为各岩层厚度和坚固性系数;$ K $为安全系数。

    此时巷道半宽为2.7 m,取顶板岩层的坚固性系数均值为1.6,安全系数为1.8,由式(87)可得:

    $$ b = 2.7 \times \left[ {\sqrt {{{\left( {\frac{{1.6}}{{1.8}}} \right)}^2} + 1} + \frac{{1.6}}{{1.8}}} \right] = 6.01 \;{\rm{m}} $$ (88)

    取锚索外露长度0.3 m,锚入顶板稳定岩层的长度1.5 m,由式(88)可知顶板锚索长度设计为8 m。考虑回采时的动压影响,顶板单根锚索承受岩体的质量为

    $$ G = {k_1} \gamma B S $$ (89)

    式中:$ {k_1} $为动压影响系数;$ \gamma $为顶板岩体容重;$ S $为冒落拱包络线内岩体截面积;$ B $为锚索排距。

    $ {k_1} = 1.5 $$ \gamma = 25\;{{{\rm{kN}}} /{{{\rm{m}}^3}}} $$ B = 1.2 $ m,则顶板单根锚索承受岩体的重量为

    $$ G = 1.5 \times 25 \times 6.01 \times 5.4 \times 1.2{\text{ = 1\;460}}{\text{.43}} \;{\rm{kN}} $$ (90)

    设锚索由直径ø21.6 mm钢铰线构成,最大破断力为504 kN,则30109工作面两侧巷道顶板每排锚索根数为:

    $$ n = \frac{{1\;460.43}}{{504}} = 2.89 $$ (91)

    为安全考虑,每排宜设3根锚索,顶板锚索材料选用ø21.6 mm×8 000 mm钢绞线,间排距为1 600 mm×1 200 mm。各巷道锚杆(索)支护方案及参数如图19所示:

    图  19  巷道围岩支护方案
    Figure  19.  Support plan for surrounding rock along the channel

    为验证上述区段煤柱宽度留设及其支护参数的合理性,在掘采期间,对各巷道围岩变形进行了长时间的现场监测,如图20所示。

    图  20  工作面巷道围岩变形监测
    Figure  20.  Deformation monitoring results of surrounding rock along the channel of working face

    可以看出:掘进期间,30109工作面运输巷煤柱帮、直接顶与实体煤帮分别在35、25和30 d后趋于稳定,最大变形量依次为55、4和36 mm;该工作面辅运巷道煤柱帮、直接顶与实体煤帮也分别在30、25和35 d后趋于稳定,各自最大变形量依次为68、62和46 mm;受回采扰动的显著影响,30108工作面运输巷围岩稳定后的最大变形量分别为94、76和43 mm,较30109工作面巷道围岩最大变形量显著增加,但仍能满足现场生产需求。

    1)区段煤柱两侧采空后支承压力的分布及其范围与煤柱两侧工况密切相关,当两侧工况相同时,支承压力对称分布;当两侧不同工况时,两侧支承压力呈双峰非对称分布。

    2)大采高区段煤柱岩体在支承压力作用下的劈裂破坏具有典型的柱条特征,弹塑性界面上岩体发生劈裂的临界位置集中在0.35倍的采高处,煤柱极限平衡区宽度随岩体弹性模量和极限拉应的增加而减少,随煤柱埋深和煤柱高度的增大而增加。

    3)煤柱岩体的极限拉应变随侧压增加而增大,由于区段煤柱采空区侧应力集中程度较巷道侧偏大,所以采空区侧的极限拉应变较大,工程实际中表现为巷道的极限平衡区宽度较采空区侧偏大。

    4)运用上述理论确定陕北某矿30109工作面左右侧大采高区段煤柱的合理尺寸及其变形范围,在此基础上确定了巷道与煤柱围岩合理的支护方案,现场监测表明,围岩变形控制效果良好,满足现场生产要求。

  • 图  1   不同工况下煤柱支承压力分布模型

    Figure  1.   Distribution model of coal pillar supporting pressure under different working conditions

    图  2   煤柱荷载分布模型

    Figure  2.   Coal pillar load distribution model

    图  3   两侧不同工况下煤柱受力变形简化模型

    Figure  3.   Simplified model of mechanical deformation of coal pillar under different conditions on both sides

    图  4   左侧极限平衡区线性荷载计算模型

    Figure  4.   Linear load calculation model for limit equilibrium zone

    图  5   左侧弹性区线性荷载计算模型

    Figure  5.   Calculation model of linear load in elastic zone

    图  6   弹性区均布荷载计算模型

    Figure  6.   Calculation model of uniformly distributed load in elastic zone

    图  7   右侧弹性区线性荷载计算模型

    Figure  7.   Calculation model of Linear load in elastic zone

    图  8   右侧极限平衡区线性荷载计算模型

    Figure  8.   Calculation model of Linear load in elastic zone

    图  9   区段煤柱岩体压缩柱条分析模型

    Figure  9.   Compression column bar analysis model of section coal pillar rock mass

    图  10   煤柱极限平衡区柱条分析模型

    Figure  10.   Column bar analysis model for limit equilibrium zone of coal pillars

    图  11   煤柱弹塑性界面岩体柱条模型

    Figure  11.   Fracture-flexural mechanical model of rock mass with elastic-plastic interface of coal pillar

    图  12   煤柱极限平衡区宽度计算模型

    Figure  12.   Calculation model of the width of coal pillar limit equilibrium zone

    图  13   煤柱破裂区简化计算模型

    Figure  13.   Simplified calculation model for coal pillar fracture zone

    图  14   煤柱极限平衡区与煤体弹性模量的关系

    Figure  14.   Relationship between coal pillar limit equilibrium zone and coal elastic modulus

    图  15   煤柱极限平衡区宽度与煤体极限拉应变的关系

    Figure  15.   Relationship between the ultimate equilibrium zone of coal pillar and ultimate tensile strain of coal

    图  16   煤柱极限平衡区宽度与煤柱埋深的关系

    Figure  16.   Relationship between coal pillar limit equilibrium zone and buried depth of roadway

    图  17   煤柱极限平衡区宽度与煤柱高度的关系

    Figure  17.   Relationship between coal pillar limit equilibrium zone and coal pillar height

    图  18   工作面布置

    Figure  18.   Working face distribution

    图  19   巷道围岩支护方案

    Figure  19.   Support plan for surrounding rock along the channel

    图  20   工作面巷道围岩变形监测

    Figure  20.   Deformation monitoring results of surrounding rock along the channel of working face

    表  1   煤帮各计算参数取值

    Table  1   Value of calculating parameters of coal wall

    $ \gamma $/(kN·m−3K/GPa$ \lambda $$ \varphi $/(º)$ c $/MPa$ {\varphi _{\rm{u}}} $/(º)$ {\varphi _{\rm{d}}} $/(º)$ E $/GPa$ \mu $${\varepsilon _{ {\rm{t} ,}\max } }$/10−3D
    250.150.8301.825253.60.30.681.6
    注:D为直接顶与煤层弹量比。
    下载: 导出CSV

    表  2   煤帮各计算参数取值

    Table  2   Value of calculating parameters of coal wall

    $ \gamma $/(kN·m−3K/GPa$ \lambda $$ \varphi $/(º)$ c $/MPa$ {\varphi _{\rm{u}}} $/(º)$ {\varphi _{\rm{d}}} $/(º)$ E $/GPa$ \mu $$ {\varepsilon _{{\rm{t}}\;\max }}_{,{\rm{BC}}} $/×10−3${\varepsilon _{ {\rm{t} },\max } }_{,{\rm{EH} } }$/10−3D
    250.180.8322.027273.40.30.630.681.6
    下载: 导出CSV

    表  3   神东矿区综采面护巷煤柱塑性区宽度统计表[10]

    Table  3   Plastic zone width of coal pillars in mining face in Shendong mining area[10]

    工作面塑性区宽度/m
    采空区侧巷道侧平均值
    寸草塔431151.54.73.1
    大柳塔224042.410.06.2
    活鸡兔123172.55.13.8
    补连塔313043.912.78.3
    大柳塔523062.78.15.4
    下载: 导出CSV
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  • 被引次数: 12
出版历程
  • 收稿日期:  2022-06-01
  • 网络出版日期:  2023-04-26
  • 刊出日期:  2023-03-14

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