投审稿入口
Three-dimensional calculation method for sliding stability of unstable rocks with steeply inclined fractures at rear edge
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摘要:
危岩稳定系数是崩塌灾害评估的核心指标，传统二维剖面模型因忽略三维形态特征与多裂隙协同作用，易导致稳定系数计算较大误差。基于极限平衡理论，构建了后缘含陡倾裂隙的滑移式危岩三维稳定性计算模型，提出了危岩在三维空间下多组后缘裂隙充水作用下水压力计算方法，以及三维空间形态下危岩滑面浮托力计算方法，并以重庆市涪陵区大闸口危岩为例开展了应用，对比分析三维与二维模型的计算差异。结果表明：三维模型可精确表征危岩不规则形态及多裂隙水力耦合效应，暴雨工况下双裂隙充水（分析工况⑦）时稳定系数较单裂隙充水（分析工况③）降低5.04%；数值模拟验证显示三维极限平衡法与强度折减法结果偏差约为0.4%；危岩形态对稳定性影响显著，除规则立方体外，多数情形需采用三维分析方法以确保评估精度。研究成果可为复杂条件下滑移式危岩精准稳定性评估提供理论与技术支撑。
Abstract:ObjectiveThe stability coefficient of unstable rocks is a critical metric for assessing rockfall hazards. Traditional two-dimensional (2D) cross-sectional models, which fail to account for three-dimensional (3D) geometric characteristics and the synergistic effects of multiple fractures, often result in substantial errors in the calculation of stability coefficients.
MethodsIn this study, a 3D stability calculation model for sliding unstable rocks with steeply inclined fractures at the rear edge was developed based on the theory of limit equilibrium. Additionally, a calculation method for the water pressure acting on unstable rocks under the influence of multiple groups of rear-edge fractures in 3D space, along with a calculation method for the uplift force on the sliding surface of unstable rocks in 3D spatial configurations, was proposed. The model was applied to the Dazhaokou unstable rocks in Fuling District, Chongqing, and the differences between the 3D and 2D model calculations were compared and analyzed.
ResultsThe results indicate that the 3D model could accurately characterize the irregular geometry of the unstable rocks and the hydro-mechanical coupling effects of multiple fractures. Under heavy rainfall conditions, the stability coefficient calculated for both fractures filled with water (case ⑦) was 5.04% lower than that for single fracture filled with water (case ③). Numerical simulation validation demonstrated that the discrepancy between the 3D limit equilibrium method and the strength reduction method was about 0.4%. The shape of unstable rocks significantly influences stability. Except for regular cubic shapes, 3D analysis methods are required in most cases to ensure assessment accuracy.
ConclusionThis research provides theoretical and technical support for accurate stability assessment of sliding unstable rocks under complex conditions.
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图 1 滑移式危岩稳定性计算简图
$ {Q}_{{\mathrm{h}}} $为地震力水平分量；Qv为地震力竖向分量；$ G $为危岩体自重；$ V $为裂隙水压力；$ {\gamma }_{{\mathrm{w}}} $为水的容重；$ {h}_{{\mathrm{w}}} $为后缘裂隙充水高度；$ U $为滑面处水产生的浮托力；$ \alpha $为危岩体与基座接触面倾角；$ L $为滑移长度；下同
Figure 1. Stability calculation of sliding unstable rocks
图 2 后缘陡倾裂隙水压力
x，y，z为直角坐标系的3个方向，x轴正方向为北，y轴正方向为东，z轴正方向为垂直向上；V1，V2，V3分别为后缘裂隙1，2，3的水压力；下同
Figure 2. Water pressure in steeply inclined fractures at rear edge
图 3 后缘裂隙水压力计算简图[11]
d为后缘裂隙水竖直方向条带划分高度；$ {h}_{{\mathrm{w}}i} $为后缘裂隙水第i个条带至水位面的距离；$ {L}_{i} $为后缘裂隙水第i个条带水平方向上的宽度；下同
Figure 3. Water pressure calculation of rear-edge fractures
图 4 后缘裂隙水压力坐标变换图示
x'，y'，z'为滑面所在平面建立的独立坐标系的3个方向，x'轴为沿滑面倾向的水平方向，y'轴为沿滑面走向的水平方向，z'轴正方向为垂直向上方向；θh为直角坐标系x，y，z和滑面所在平面坐标系x'，y'，z'对应坐标轴之间的夹角，即滑面倾向；Vx， Vy分别为水压力合力在直角坐标系x，y方向上的分量；$V'_x $为水压力合力在滑面所在平面坐标系x'方向上的分量；虚线代表各作用力之间的投影关系
Figure 4. Coordinate transformation of water pressure in steeply inclined fractures
图 5 滑面浮托力计算示意图
dSi为后缘裂隙水第i个条带滑面微断长度
Figure 5. Schematic diagram for calculating uplift force on the sliding surface
图 7 倾斜摄影危岩表面点云
Figure 7. Point cloud of unstable rock surface derived from oblique photogrammetry
图 9 危岩体三维模型(a)及相关信息(b)
Figure 9. 3D model (a) and related parameters (b) of unstable rock
图 12 不同形态危岩计算模型
a. 后缘裂隙面为三角形；b. 后缘裂隙面为梯形；c. 后缘裂隙面为矩形；d. 顶底面为梯形；e. 二维剖面计算模型。除V，G，U外，其余字母均为模型端点编号
Figure 12. Calculation models of unstable rocks of various shapes
表 1 危岩三维稳定性计算
Table 1. 3D stability calculation for unstable rocks
分析工况 危岩容重
γ/(kN·m−3）危岩自
重G/kN滑面内摩
擦角$ \phi $/(°)滑面黏聚
力c/kPa左侧边界面L1
充水高度hw1/m右侧边界面L2
充水高度hw2/m左侧边界面L1
水压力V1/kN右侧边界面L2
水压力V2/kN$ {V}_{{\mathrm{w}}} $/kN $ {N}_{{\mathrm{w}}} $/kN U/kN K ①后缘裂隙未充水 26.6 51801.6 20.0 55.0 0 0 0 0 0 0 0 1.454 ②后缘单裂隙L1充水
高度为3.6 m27.0 52580.6 20.0 55.0 3.6 0 795.0 0 619.3 201.2 1770.5 1.354 ③后缘单裂隙L1充水
高度为5.1 m（暴雨）27.8 54138.6 18.0 50.0 5.1 0 1560.6 0 1215.8 395.0 2480.6 1.151 ④后缘单裂隙L2充水
高度为3.6 m27.0 52580.6 20.0 55.0 0 3.6 0 662.5 445.5 144.8 1770.5 1.369 ⑤后缘单裂隙L2充水
高度为5.1 m（暴雨）27.8 54138.6 18.0 50.0 0 5.1 0 1300.5 874.6 284.2 2480.6 1.176 ⑥后缘双裂隙L1和L2充水
高度均为3.6 m27.0 52580.6 20.0 55.0 3.6 3.6 795.0 662.5 1064.9 346.0 1770.5 1.316 ⑦后缘双裂隙L1和L2充水
高度均为5.1 m（暴雨）27.8 54138.6 18.0 50.0 5.1 5.1 1560.6 1300.5 2090.4 679.2 2480.6 1.093 注：$ {V}_{{\mathrm{w}}} $为水压力合力沿滑面向下分量；$ {N}_{{\mathrm{w}}} $为水压力合力垂直于滑面分量；K为稳定系数；下同 
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表 2 危岩数值分析模型物理力学参数
Table 2. Physico-mechanical parameters of numerical analysis model of unstable rock
灰岩 密度/(kg·m–3) 弹性模量/GPa 泊松比 黏聚力/MPa 内摩擦角/(°) 抗拉强度/kPa 26.6 6.8 0.22 2.26 58.9 410.0 结构面 法向刚度/GPa 切向刚度/GPa 黏结力/kPa 摩擦角/(°) 抗拉强度/MPa 19 19 55 20 0
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表 3 不同形态危岩稳定性计算
Table 3. Stability calculation of unstable rocks of various shapes
危岩形态 体积Vr/m3 滑面面积A/m2 自重G/kN 裂隙水压力V/kN 陡倾裂隙倾角$ {\varphi }_{1} $/(°) $ {V}_{{\mathrm{w}}} $/kN $ {N}_{{\mathrm{w}}} $/kN U/kN K 稳定状态 后缘裂隙面为三角形 1744.02 193.19 46390.8 3659.1 90.0 3534.4 947.0 6181.9 1.379 稳定 后缘裂隙面为梯形 2860.04 110.46 76077.1 4232.5 90.0 4088.3 1095.5 3534.8 1.086 欠稳定 后缘裂隙面为矩形 3732.05 207.06 99272.6 4096.0 90.0 3956.4 1060.1 6625.8 1.237 稳定 顶底面为梯形 2776.71 155.29 73860.5 4096.0 90.0 3956.4 1060.1 4969.3 1.182 基本稳定 二维剖面计算模型 186.60 10.35 4963.63 204.8 — — — 331.2896 1.237 稳定
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