Journal of Mountain Science

, Volume 15, Issue 2, pp 430–443 | Cite as

A theoretical model for the estimation of maximum impact force from a rockfall based on contact theory

  • Shi-lin Zhang
  • Xing-guo Yang
  • Jia-wen Zhou


Rockfall poses a great threat to buildings and personal security. To understand the dynamic characteristics of rockfalls is a prerequisite for disaster prevention and assessment. Models for rockfalls in different forms are established based on the theory of rigid body motion. The equivalent velocity considering the rotational effect is determined by the energy ratio. Besides, considering plastic deformation and nonlinear hardening, the maximum impact force is estimated based on the Hertz contact theory. Then, a case study is carried out to illustrate the applicability of the model and sensitive analyses on some affecting parameters are also made. Calculation results show that the maximum impact force increases with the increasing of incident velocity, angle and slope gradient reflected by the changing of energy ratio. Moreover, the model for the estimation of maximum impact force is validated by two different scales of experiments and compared with other theoretical models. Simulated maximum impact forces agree well with the experiments.


Rockfall Motion characteristics Contact theory Maximum impact force 



Initial horizontal velocity (m/s)


Initial vertical velocity (m/s)


Final velocity (m/s)


Gravitational acceleration (m2/s)


Vertical falling height (m)


Horizontal distance during freefall (m)


Normal coefficient of restitution


Tangential coefficient of restitution


Normal rebound velocity (m/s)


Normal incident velocity (m/s)


Tangential rebound velocity (m/s)


Tangential incident velocity (m/s)


Initial velocity (m/s)


Slope gradient (°)


Rolling friction coefficient


Radius of the rockfall (m)


Energy ratio


Rotational energy (J)


Translational energy (J)


Total energy (J)


Quality of rockfall (kg)


Normal velocity (m/s)


Tangential velocity (m/s)


Equivalent normal velocity (m/s)


Equivalent tangential velocity (m/s)


Incident angle (°)


Contact stress on the contact surface (Pa)


Normal impact force (N)


Maximum indentation depth (m)


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This work was supported by the National Natural Science Foundation of China (41472272) and the Youth Science and Technology Fund of Sichuan Province (2016JQ0011). Critical comments by the anonymous reviewers greatly improved the initial manuscript.


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Copyright information

© Science Press, Institute of Mountain Hazards and Environment, CAS and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.State Key Laboratory of Hydraulics and Mountain River EngineeringSichuan UniversityChengduChina
  2. 2.College of Water Resource and HydropowerSichuan UniversityChengduChina

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