投审稿入口
Three-dimensional calculation method for sliding stability of rockfall with steep inclined fissures at the rear edge
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摘要:
危岩稳定性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，易导致稳定性系数计算较大误差。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时稳定系数较二维模型降低5.5%；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 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 translational rock masses with a steeply dipping fracture at the rear edge was developed based on the theory of limit equilibrium. Additionally, a calculation method for the water pressure acting on dangerous rocks under the influence of multiple rear-edge fissure water groups in three-dimensional space, along with a calculation method for the buoyant force on the sliding surface of dangerous rocks in three-dimensional spatial configurations, is proposed. The model was applied to the Dazhaikou rock mass 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 can accurately characterize the irregular geometry of the rock mass and the hydro-mechanical coupling effects of multiple fractures. Under heavy rainfall conditions with both fractures filled with water, the stability coefficient calculated using the 3D model is 5.5% lower than that obtained from the 2D model. Numerical simulation validation demonstrates that the discrepancy between the 3D limit equilibrium method and the strength reduction method is less than 0.4%.
ConclusionThe study concludes that the shape of the rock mass significantly influences its stability. Except for regular cubic shapes, a 3D analysis method is generally necessary to ensure the accuracy of the assessment in most cases.
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图 1 滑移式危岩稳定性计算简图
$ {Q}_{h} $为地震力水平分量；$ G $为危岩体自重；$ V $为裂隙水压力；$ {\gamma }_{w} $为水的容重；$ {h}_{w} $为后缘裂隙充水高度；$ U $为滑面水压力；$ \theta $为危岩体与基座接触面倾角；$ L $为滑移长度
Figure 1. 2D Calculation Diagram of sliding rockfall
图 2 后缘陡倾裂隙水压力
x，y，z为直角坐标系的3个方向，x轴正方向为北向，y轴正方向为东，z轴正方向为垂直向上；下同
Figure 2. Water pressure in steeply inclined fissures
图 3 后缘裂隙水压力计算简图[10]
d为后缘裂隙水竖直方向条带划分高度；$ {h}_{wi} $为后缘裂隙水第i个条带至水位面的距离；$ {L}_{i} $为缘裂隙水第i个条带水平方向上的宽度
Figure 3. Water pressure of rockfall in trailing-edge
图 4 后缘裂隙水压力坐标变换图示
x'，y'，z'为滑面所在平面建立的独立坐标系的3个方向，x'轴为沿滑面倾向的水平方向，y'轴为沿滑面走向的水平方向，z'轴正方向为垂直向上方向
Figure 4. Diagram of coordinate transformation of water pressure in steeply inclined fissures
图 11 不同形态的危岩计算模型
a. 后缘裂隙面为三角形；b. 后缘裂隙面为梯形；c. 后缘裂隙面为矩形；d. 顶底面为梯形；e. 二维剖面计算模型
Figure 11. Calculation Models of Unstable Rocks of Various Shapes
表 1 危岩三维稳定性计算
Table 1. Three-dimensional stability calculation table for rockfall
分析工况 危岩容重
γ/(kN·m−3）危岩自重
G/kN滑面内摩擦
角$ \varphi $/(°)滑面黏聚力
C/kPa裂隙L1充水
高度hw1/m裂隙L2充水
高度hw2/m裂隙L1
水压力V1/kN裂隙L2
水压力V2/kN$ {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}_{w} $水压力合力沿滑面向下分量；$ {N}_{w} $为水压力合力垂直于滑面分量；$ U $为滑面水压力；K为稳定系数；下同 
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表 2 计算参数
Table 2. Parameters of rockfall
灰岩 密度/(kg·m–3) 弹性模量/GPa 泊松比 黏聚力/MPa 内摩擦角/(°) 抗拉强度/kPa 26.6 6.8 0.22 2.26 58.9 410.0 结构面 法向刚度/GPa 切向刚度/GPa 黏结力/kPa 摩擦角/(°) 抗拉强度/MPa 19 19 55 20 0
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表 3 不同形态危岩稳定性计算
Table 3. Stability Calculation Table of Unstable Rocks of Various Shapes
危岩形态 体积Vr/m3 滑面面积A/m2 自重G/kN 裂隙水压力V/kN 陡倾裂隙倾角$ {\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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