Trajectory of rockfall on the uniform slope
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摘要: 现有的落石运动轨迹研究均将斜坡视为半无限均匀的坡面，而实际的工况下，坡面由不均一材料组成。根据落石不同的运动模式，将坡面简化为多层岩土体材料的结构形式，在滚动阶段，基于Hertz接触理论得到落石在不同坡面条件下运动特征，并给出坡面分层条件下落石切向摩擦系数的计算公式，在碰撞阶段，基于能量守恒定理，采用准静态接触力学理论得到法向恢复系数公式，理论推导得到任意层厚岩土体材料坡面条件下落石法向和切向恢复系数的解析解，然后求解得到碰撞后落石运动参数，并根据碰撞后落石的速度及回弹量关系，给出碰撞后落石运动模式转换的判别条件，最后结合运动学定理，得到3种运动模式任意分层坡面条件落石的运动轨迹，将理论公式应用于某山区落石运动轨迹预测，验证了理论公式的适用性和有效性。Abstract: Existing studies on the trajectory of the rockfall all regard the slope as a semi-infinite and uniform slope, but in actual working conditions, the slope is composed of heterogeneous materials.According to the different motion modes of rockfall, the slope surface is simplified to the structure of multi-layer rocks and soil materials.In the rolling stage, based on Hertz contact theory, the movement characteristics of falling rocks under different slope conditions are obtained, and the calculation formula for the tangential friction coefficient of the rockfall under the condition of the slope layering is given.In the collision phase, based on the law of conservation of energy, the normal recovery coefficient formula obtained by the quasi-static contact mechanics theory is used.The analytical solutions of the normal and tangential recovery coefficients of the rockfall under the condition of the arbitrary rock and soil material structure slope are derived, and then the motion parameters of the rockfall after the collision are obtained, According to the relationship between the speed of the rockfall and the amount of rebound after the collision, the judgment conditions for the transition of the motion mode of the rockfall after the collision are given.Finally, combined with the kinematics theorem, the motion trajectory of the falling rock under any layered slope condition in three motion modes is obtained. The theoretical formula is applied to the prediction of the motion trajectory of a rolling rock in a mountainous area, which verifies the applicability and validity of the theoretical formula.
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Key words:
- rockfall /
- layered slope /
- contact theory /
- recovery coefficient /
- motion trajectory
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图 5 分层坡面下落石碰撞过程图
Figure 5. Collision process diagram of rockfall under the layered slope
图 7 危岩体WYT2形态图
Qcol.崩积物；C1b.下石炭统大塘组砂岩
Figure 7. WYT2 morphology of dangerous rock mass
表 1 滚动摩擦系数[24]
Table 1. Rolling friction coefficient
坡面特征 滚动摩擦系数 光滑岩面、混凝土表面 0.40~0.60 块石堆积坡面 0.55~0.70 密实碎石堆积坡面、硬土坡面 0.55~0.85 松散碎石坡面、软土坡面 0.50~0.85 
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表 2 落石运动计算参数
Table 2. Calculation parameters of rockfall movement
材料 ρ/(g·cm-3) E/MPa υ μ 落石 2.05 22 000 0.22 - 堆积物 - 46 0.30 0.6 灰岩 - 280 0.28 0.5
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表 3 落石运动碰撞点速度
Table 3. Rockfall velocity of the point
碰撞点 碰撞前速度/(m·s-1) 碰撞后速度/(m·s-1) q1 36.3 13.5 P1 36.3 13.6 P2 17.1 12.8 P3 13.9 10.9 P4 11.3 10.0
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