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

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Abstract

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.

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Abbreviations

v x :

Initial horizontal velocity (m/s)

v y :

Initial vertical velocity (m/s)

v :

Final velocity (m/s)

g :

Gravitational acceleration (m2/s)

H :

Vertical falling height (m)

L :

Horizontal distance during freefall (m)

e n :

Normal coefficient of restitution

e τ :

Tangential coefficient of restitution

v rn :

Normal rebound velocity (m/s)

v in :

Normal incident velocity (m/s)

v rτ :

Tangential rebound velocity (m/s)

v iτ :

Tangential incident velocity (m/s)

v 0 :

Initial velocity (m/s)

α:

Slope gradient (°)

f 1 :

Rolling friction coefficient

R 1 :

Radius of the rockfall (m)

ß :

Energy ratio

E r :

Rotational energy (J)

E v :

Translational energy (J)

E :

Total energy (J)

m :

Quality of rockfall (kg)

v n :

Normal velocity (m/s)

v τ :

Tangential velocity (m/s)

v n ':

Equivalent normal velocity (m/s)

v τ ':

Equivalent tangential velocity (m/s)

θ :

Incident angle (°)

σ (r) :

Contact stress on the contact surface (Pa)

P :

Normal impact force (N)

δ :

Maximum indentation depth (m)

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Acknowledgement

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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Authors and Affiliations

  1. State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, 610065, China

    Shi-lin Zhang & Jia-wen Zhou

  2. College of Water Resource and Hydropower, Sichuan University, Chengdu, 610065, China

    Xing-guo Yang

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  1. Shi-lin Zhang
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Correspondence to Jia-wen Zhou.

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Zhang, Sl., Yang, Xg. & Zhou, Jw. A theoretical model for the estimation of maximum impact force from a rockfall based on contact theory. J. Mt. Sci. 15, 430–443 (2018). https://doi.org/10.1007/s11629-017-4606-6

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