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<p>Rockfall movement possesses complex and challenging properties that are difficult to determine precisely. Traditional two-dimensional methods have generally overlooked the lateral offset induced by collisions, while three-dimensional approaches often failed to fully capture the intrinsic randomness of rockfall trajectories. To address these issues, we introduced a method for calculating the spatial restitution coefficient in three dimensions. By simplifying the slope into a three-dimensional model, we analyzed the rockfall’s motion as a series of collision and bounce events. A collision time interval threshold of 0.0001 s marked the end of each event. This theoretical framework mathematically coupled the three spatial dimensions with the temporal domain and reconstructed four-dimensional (4D) motion trajectories. We incorporated comparative analyses with the lumped mass method and the DEM. The results indicated that the proposed method exhibits greater stochasticity compared to the lumped mass method. The maximum relative error for maximum movement distance between our proposed model and the DEM is 6.13%, whereas maximum kinetic energy error is 10.46%. The reliability of our methodology was confirmed through validation. A significant benefit of our approach is the capability to extract key rockfall motion characteristics at any moment, including movement offset, travel distance, bouncing height, and impact energy at a range of time scales. Finally, a real-world case study was analyzed using this method. A comprehensive evaluation of rockfall risk was then conducted by synthesizing data related to the random motion energy and spatial distribution of rockfalls. This evaluation provided valuable references for the design of passive protective measures.</p>

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Theoretical Modeling of Stochastic Rockfall Trajectories: A Four-Dimensional Reconstruction Approach

  • 期刊论文

  • 发表日期：

    2025年9月10日

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  • Taijiang Chen,
  • Guangcheng Zhang,
  • Yongning Wu,
  • Xingtao Beng

摘要

Rockfall movement possesses complex and challenging properties that are difficult to determine precisely. Traditional two-dimensional methods have generally overlooked the lateral offset induced by collisions, while three-dimensional approaches often failed to fully capture the intrinsic randomness of rockfall trajectories. To address these issues, we introduced a method for calculating the spatial restitution coefficient in three dimensions. By simplifying the slope into a three-dimensional model, we analyzed the rockfall’s motion as a series of collision and bounce events. A collision time interval threshold of 0.0001 s marked the end of each event. This theoretical framework mathematically coupled the three spatial dimensions with the temporal domain and reconstructed four-dimensional (4D) motion trajectories. We incorporated comparative analyses with the lumped mass method and the DEM. The results indicated that the proposed method exhibits greater stochasticity compared to the lumped mass method. The maximum relative error for maximum movement distance between our proposed model and the DEM is 6.13%, whereas maximum kinetic energy error is 10.46%. The reliability of our methodology was confirmed through validation. A significant benefit of our approach is the capability to extract key rockfall motion characteristics at any moment, including movement offset, travel distance, bouncing height, and impact energy at a range of time scales. Finally, a real-world case study was analyzed using this method. A comprehensive evaluation of rockfall risk was then conducted by synthesizing data related to the random motion energy and spatial distribution of rockfalls. This evaluation provided valuable references for the design of passive protective measures.