Analysis of Stress Distribution Law of Surrounding Rock of Rectangular Roadway with Different Specifications Based on Complex Variable Function
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摘要:
鉴于矩形巷道应用的广泛性，为深入剖析矩形巷道尺寸、侧压力系数等因素变化对巷道围岩应力的影响程度及规律，将矩形巷道围岩应力求解视为弹性力学孔口问题，采用复变函数理论与Schwartz−Chistoffel映射函数，结合保角变换进行分析得到ζ复平面单位圆ζ的解析函数φ0（ζ）、ψ0（ζ），再进行Laurent级数展开后，求得复势函数φ（ζ）、ψ（ζ）。经过进一步推导最终确定矩形巷道围岩环向应力解析解，在此基础上通过环向应力的极坐标表达，深入剖析了矩形巷道尺寸与侧压力系数变化下矩形巷道周边围岩应力变化规律以及影响程度。结果表明：取映射函数前三项计算，映射巷道断面已逼近于理论断面，可保证精度要求；随着巷道宽高比增大，以宽高比等于1为分界点，巷道围岩周边应力峰值先增大后减小；巷道帮部围岩应力随宽高比的增大逐渐减小，巷道顶底板围岩应力随宽高比的增大逐渐增大；侧压力系数越大，侧帮围岩应力峰值越大，顶底板应力峰值越小；侧帮围岩应力随侧压力的增大而增大，两者呈正相关，顶底板应力随侧压力增大而减小，两者呈负相关。
Abstract:Given the widespread use of rectangular tunnels, in order to deeply analyze the degree and pattern of the influence of factors such as the size and lateral pressure coefficient of rectangular tunnels on the stress of the surrounding rock of the tunnels. This article regards the solution of the surrounding rock of a rectangular roadway as an elastic mechanics orifice problem, and uses the theory of complex variables and Schwartz Chistoffel mapping function, combined with conformal transformation for analysis ζ Complex plane unit circle ζ Analysis function of φ0(ζ)、ψ0(ζ), After expanding the Laurent series, obtain the complex potential function φ(ζ)、ψ(ζ). After further deduction, the analytical solution for the circumferential stress of the surrounding rock of the rectangular tunnel was finally determined. Based on this, the polar coordinate expression of the circumferential stress was used to deeply analyze the stress variation law and degree of influence of the surrounding rock of the rectangular tunnel under the changes in the size and lateral pressure coefficient of the rectangular tunnel. The results showed that taking the first three calculations of the mapping function, the mapping roadway section had approached the theoretical section, which could ensure the accuracy requirements; with the increase of the width−height ratio of the roadway, the peak stress around the surrounding rock of the roadway increased first and then decreased with the width−height ratio of 1 as the dividing point. The stress of surrounding rock of roadway side decreased with the increase of width−height ratio, and the stress of surrounding rock of roadway roof and floor increased with the increase of width−height ratio. The larger the lateral pressure coefficient was, the larger the peak stress of the surrounding rock was, and the smaller the peak stress of the roof and floor was. The stress of side wall surrounding rock increased with the increase of lateral pressure, and the two were positively correlated. The stress of roof and floor decreased with the increase of lateral pressure, and the two were negatively correlated.
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图 5 不同宽高比下的围岩应力分布(a—侧压力系数为0.6; b—侧压力系数为1.0; c—侧压力系数为1.4; d—侧压力系数为1.8; e—侧压力系数为2.2)
Figure 5.
表 1 不同宽高比情况下矩形巷道保角变换参数
Table 1. Parameter of conformal transformation for rectangular chambers under different aspect ratios
方案 1 2 3 4 5 高度/2b 4.5 m 4.5 m 4.5 m 4.5 m 4.5 m 宽度/2a 2.7 m 4.5 m 6.3 m 8.1 m 9.9 m 宽高比/w 0.6 1.0 1.4 1.8 2.2 k 0.291 0.252 0.234 0.214 0.201 R 2.147 2.700 3.233 3.730 4.226 m1 −0.255 −0.013 0.100 0.224 0.321 m3 −0.156 −0.167 −0.165 −0.158 −0.151 
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表 2 环向应力集中系数Kc最大值对应位置
Table 2. Corresponding positions for the maximum value of circumferential stress concentration coefficient Kc
宽高比 侧压力系数 0.2 0.6 0.8 1.0 1.4 0.6 46°/3.728 48°/4.540 49°/5.082 50°/5.687 51°/7.014 1.0 41°/4.150 44°/4.909 44°/5.432 45°/6.016 46°/7.292 1.4 38°/4.118 40°/4.856 41°/5.322 42°/5.841 43°/7.009 1.8 35°/4.114 37°/4.726 38°/5.137 39°/5.612 40°/6.666 2.2 33°/4.066 35°/4.590 36°/4.953 37°/5.371 38°/6.326
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表 3 岩体力学参数
Table 3. Rock mechanics parameters
岩体密度
/(kg·m−3)内聚力
/MPa地应力
/MPa内摩擦
角/(°)弹性
模量/GPa泊松比 3.34 1.35 11 32 5 0.25
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