Calculation method of semi-ellipsoid loose load on tunnel in weak and fragmented stratum
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摘要: 为有效控制隧道围岩压力，优化隧道支护结构，研究了软弱破碎地层作用在隧道衬砌结构上的松散荷载；根据相关文献中砂性地层隧道周边围岩松动变形试验，采用半椭球体拟合了地层松动范围，提出了软弱破碎地层半椭球体松散荷载模型，给出了松散荷载随隧道收敛变形的计算表达式，分析了松散荷载的分布特征及其随隧道收敛变形的变化特征；为提高松散荷载计算效率，将半椭球体松动边界离散为多个线段，提出了松散荷载离散求和的数值计算流程；为进一步增强计算模型的工程实用性，采用二次多项式拟合松散荷载，得到了松散荷载随隧道收敛变形的简化计算方法。研究结果表明：提出的半椭球体模型可较为准确地描述隧道所受的松散荷载及其随隧道收敛变形的动态变化特征，由于滑动面上摩阻力的作用，松动区范围内的地层竖向应力小于地层初始自重应力，松动区荷载通过滑动面上的摩阻力向周边地层转移，松动区土拱效应显著；随着隧道收敛变形的增加，隧道周边地层松动范围增大，松散椭球体受周边地层的约束作用减弱，地层土拱效应减弱；地层内摩擦角和椭球体偏心率对松散荷载的计算结果影响较大，需根据地质条件测试确定；将半椭圆曲线分为3段时，离散求和的数值计算结果与解析解的相对误差约为3.9%，具有较好的一致性。Abstract: To effectively control the tunnel pressure from the surrounding rock and optimize the support structure of the tunnel, the loose load applied in the tunnel lining structure in the weak and fragmented stratum was studied. Based on an experiment in the literature about the loose deformation of the surrounding rock in sandy stratum, the semi-ellipsoid loose load model for the weak and fragmented stratum was put forward by using the semi-ellipsoid to fit the loose range of the stratum. The calculation equations of the loose load with the tunnel convergence deformation were developed. The distribution characteristics of the loose load and its changing with the tunnel convergence deformation were analyzed. For quickly calculating the loose load, a numerical calculation procedure by scatter and summation of the loose load was developed by dividing the semi-ellipsoid boundary of the loose zone into many short straight lines. To further improve the engineering practicability of the calculation model, the quadratic polynomial was used to fit the loose load, and the simplified calculation method of the loose load with the tunnel convergence deformation was obtained. Research results show that the proposed semi-ellipsoid model can accurately represent the loose load applied to the tunnel and its dynamically developing process with the tunnel convergence deformation. Due to the presence of frictional resistance on the slide surface, the vertical stress of the stratum in the loose range is less than the initial gravity stress, and the load in the loose range is transferred to the surrounding stratum by the frictional resistance on the slide surface, which obviously manifests a soil arching effect in the loose range. The increase in the tunnel convergence deformation causes an extended loose range of the surrounding stratum of the tunnel and weak constraints on the loose ellipsoid from the surrounding stratum, and the soil arching effect of stratum weakens. The internal friction angle of the stratum and the eccentricity of the ellipsoid have great influences on the calculation results of the loose load, so these two parameters should be determined according to the test of geological conditions. When the semi-ellipse boundary is discretized into three segments, the results of the numerical calculation by scatter and summation can have a good consistency with the analytical solution, and the error is about 3.9%.
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图 1 隧道周边地层松动范围随隧道埋深的动态变化
Figure 1. Dynamic changes in loose range of surrounding stratum of tunnel with tunnel buried depth
图 6 松动区外荷载应力竖向分布
Figure 6. Distriubtions of applied loads in loose zone along vertical direction
图 7 松动区合外荷载竖向分布
Figure 7. Applied combined external load distributions in loose zone along vertical direction
图 8 松动区内压应力竖向分布
Figure 8. Distributions of compression stress in loose zone along vertical direction
图 9 隧道收敛变形对松动区内压应力竖向分布的影响
Figure 9. Effects of tunnel convergence deformation on distributions of compression stress along vertical direction in loose zone
图 10 隧道收敛变形对松动区高度和松散荷载的影响
Figure 10. Effects of tunnel convergence on height of loose zone and loose load
图 12 地层内摩擦角对松散荷载的影响
Figure 12. Effect of stratum internal friction angle on loose load
图 16 半椭球体松动区松散荷载计算流程
Figure 16. Calculation process of loose load on semi-ellipsoid loose zone
图 17 松散荷载与隧道收敛变形的关系
Figure 17. Relationship between loose load and tunnel convergence deformation
表 1 离心模型试验结果拟合
Table 1. Fitting of centrifugal model test results
C/D 1 2 3 4 bG/m 3.64 3.89 4.33 4.65 aG/m 5.10 10.15 16.86 23.17 ε 0.70 0.92 0.97 0.98 
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表 2 滑动面上的侧压力系数
Table 2. Lateral pressure coefficients at sliding surface
滑动面位置 侧压力系数 0°≤θ＜φ/2 K≥1(σh≥σv) φ/2≤θ＜90°+φ/2 K≤1(σh≤σv) 90°+φ/2≤θ≤180° K≥1(σh≥σv)
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表 3 不同理论分析结果对比
Table 3. Comparison of analysis results among different theories
C/D 模型试验所得p′c/kPa Terzaghi公式所得p′c/kPa 普氏压力拱所得p′c/kPa 文献[6]计算结果 本文计算结果 m p′c/kPa ε ω/m σa /kPa 1 28.4 37.87 2.281 25.2 0.93 0.21 23.64 1 28.8 38.40 2.281 25.2 0.93 0.21 23.64 2 37.5 51.56 75.35 2.461 34.4 0.94 0.80 41.21 2 37.0 55.52 75.35 2.461 34.4 0.94 0.80 41.21 3 33.7 56.76 75.35 2.513 38.5 0.95 1.64 46.49 3 36.4 58.24 75.35 2.513 38.5 0.95 1.64 46.49
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表 4 隧道周边地层基本参数
Table 4. Basic parameters of stratum around tunnel
参数 取值 参数 取值 P0/MPa 0.5 R/m 6 c/MPa 0.003 ω/m 0.3 φ/(°) 20 β 1.07 γ/(kN·m-3) 18 ε 0.92 γs/(kN·m-3) 17
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表 5 松散荷载离散计算结果
Table 5. Discrete calculation results of loose load
计算方法 数值积分计算结果 不同椭圆曲线沿竖向小区间总数n下的离散求和计算结果 1 2 3 4 5 10 15 20 25 σa(ω=0.70 m) 0.209 2 0.158 9 0.192 7 0.204 0 0.208 8 0.210 9 0.211 7 0.210 7 0.210 0 0.209 7 σa(ω=0.30 m) 0.173 5 0.113 9 0.158 0 0.173 2 0.179 2 0.181 5 0.180 4 0.177 9 0.176 4 0.175 5 σa(ω=0.10 m) 0.144 8 0.077 9 0.130 3 0.148 5 0.155 5 0.157 9 0.155 4 0.151 7 0.149 4 0.148 1 σa(ω=0.05 m) 0.133 4 0.063 4 0.119 2 0.138 6 0.146 0 0.148 5 0.145 4 0.141 2 0.138 6 0.137 1
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