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

Article
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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.

Keywords

Rockfall Motion characteristics Contact theory Maximum impact force 

Notation

vx

Initial horizontal velocity (m/s)

vy

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)

en

Normal coefficient of restitution

eτ

Tangential coefficient of restitution

vrn

Normal rebound velocity (m/s)

vin

Normal incident velocity (m/s)

v

Tangential rebound velocity (m/s)

v

Tangential incident velocity (m/s)

v0

Initial velocity (m/s)

α

Slope gradient (°)

f1

Rolling friction coefficient

R1

Radius of the rockfall (m)

ß

Energy ratio

Er

Rotational energy (J)

Ev

Translational energy (J)

E

Total energy (J)

m

Quality of rockfall (kg)

vn

Normal velocity (m/s)

vτ

Tangential velocity (m/s)

vn'

Equivalent normal velocity (m/s)

vτ'

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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Notes

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.

References

  1. ASTRA S (2008) Actions due to falling rocks on protective galleries. Guideline, Federal Roads Office, Building Directorate SBB, Swiss Federal Institute of Printing and Materials, Bern. (In Italian)Google Scholar
  2. Azzoni A, Barbera GL, Zaninetti A (1995) Analysis and prediction of rockfalls using a mathematical model. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 32(7): 709–724. https://doi.org/10.1016/0148-9062(95)00018-CCrossRefGoogle Scholar
  3. Bozzolo D, Pamini R (1986) Simulation of rock falls down a valley side. Acta Mech 63(1): 113–130. https://doi.org/10.1007/BF01182543CrossRefGoogle Scholar
  4. Brizmer V, Kligerman Y, Etsion I (2006) The effect of contact conditions and material properties on the elasticity terminus of a spherical contact. International Journal of Solids and Structures 43(18–19): 5736–5749. https://doi.org/10.1016/j.ijsolstr.2005.07.034CrossRefGoogle Scholar
  5. Buzzi O, Spadari M, Giacomini A, et al. (2013) Experimental testing of rockfall barriers designed for the low range of impact energy. Rock Mechanics and Rock Engineering 46(4): 701–712. https://doi.org/10.1007/s00603-012-0295-1CrossRefGoogle Scholar
  6. Calvetti F, Di Prisco C, Vecchiotti M (2005) Experimental and numerical study of rock-fall impacts on granular soils. Rivista Italiana Di Geotecnica 4: 95–109.Google Scholar
  7. Chau KT, Wong RHC, Wu JJ (2002) Coefficient of restitution and rotational motions of rockfall impacts. International Journal of Rock Mechanics and Mining Sciences 39(1): 69–77. https://doi.org/10.1016/S1365-1609(02)00016-3CrossRefGoogle Scholar
  8. Cui YF, Nouri A, Chan D, Rahmati E (2016) A new approach to DEM simulation of sand production. Journal of Petroleum Science and Engineering 147: 56–67. https://doi.org/10.1016/j.petrol.2016.05.007CrossRefGoogle Scholar
  9. Cui YF, Chan D, Nouri A (2017) Discontinuum modeling of solid deformation pore-water diffusion coupling. International Journal of Geomechanics 17: 04017033. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000903CrossRefGoogle Scholar
  10. Delhomme F, Mommessin M, Mougin JP, et al. (2007) Simulation of a block impacting a reinforced concrete slab with a finite element model and a mass-spring system. Engineering Structures 29: 2844–2852. https://doi.org/10.1016/j.engstruct.2007.01.017CrossRefGoogle Scholar
  11. Dorren LKA (2003) A review of rockfall mechanics and modelling approaches. Progress in Physical Geography 27(1): 69–87. https://doi.org/10.1191/0309133303pp359raCrossRefGoogle Scholar
  12. Dorren LKA, Berger F, Putters US (2006) Real-size experiments and 3-D simulation of rockfall on forested and non-forested slopes. Natural Hazards and Earth System Sciences 6(1): 145–153. https://doi.org/10.5194/nhess-6-145-2006CrossRefGoogle Scholar
  13. Genis M, Sakız U, Aydıner BC (2017) A stability assessment of the rockfall problem around the Gökgöl Tunnel (Zonguldak, Turkey). Bulletin of Engineering Geology and the Environment 76(4): 1237–1248. https://doi.org/10.1007/s10064CrossRefGoogle Scholar
  14. Gentilini C, Govoni L, de Miranda S, et al. (2012) Threedimensional numerical modelling of falling rock protection barriers. Computers and Geotechnics 44: 58–72. https://doi.org/10.1016/j.compgeo.2012.03.011CrossRefGoogle Scholar
  15. Giacomini A, Thoeni K, Lambert C, et al. (2012) Experimental study on rockfall drapery systems for open pit highwalls. International Journal of Rock Mechanics and Mining Sciences 56(12): 171–181. https://doi.org/10.1016/j.ijrmms.2012.07.030CrossRefGoogle Scholar
  16. Guzzetti F, Reichenbach P, Ghigi S (2004) Rockfall hazard and risk assessment along a transportation corridor in the Nera Valley, central Italy. Environmental Management, 34(2):191–208. https://doi.org/10.1007/s00267-003-0021-6CrossRefGoogle Scholar
  17. He S, Wu Y, Li X (2009) Research on restitution coefficient of rock fall. Rock and Soil Mechanics 30(3): 623–627 (In Chinese)Google Scholar
  18. Hertz H (1882) Über die Berührung fester elastischer Körper. Journal Für Die Reine Und Angewandte Mathematik 92: 156–171Google Scholar
  19. Hou TX, Xu Q, Zhou JW (2015) Size distribution, morphology and fractal characteristics of brittle rock fragmentations by the impact loading effect. Acta Mechanica 226: 3623–3637. https://doi.org/10.1007/s00707CrossRefGoogle Scholar
  20. Hou TX, Xu Q, Xie HQ, et al. (2017). An estimation model for the fragmentation properties of brittle rock block due to the impacts against an obstruction. Journal of Mountain Science 14: 1161–1173. https://doi.org/10.1007/s11629-017-4398-8CrossRefGoogle Scholar
  21. Hou TX, Yang XG, Huang C, et al. (2015) A calculation method based on impulse theorem to determine impact force of rockfall on structure. Journal of rock mechanics and Engineering 34: 3116–3122 (In Chinese)Google Scholar
  22. Huang RQ, Liu W (2009) In-situ test study of characteristics of rockfall rock blocks based on orthogonal design. Journal of rock mechanics and Engineering 28: 882–891. (In Chinese)Google Scholar
  23. Japan Road Association (JRA) (1983) Rockfall Handbook. Tokyo:Maruzen Publisher (Japan): 1–359.Google Scholar
  24. Kawahara S, Muro T (2006) Effects of dry density and thickness of sandy soil on impact response due to rockfall. Journal of Terramechanics 43 (3): 329–340. https://doi.org/10.1016/j.jterra.2005.05.009CrossRefGoogle Scholar
  25. Labiouse V, Descoeudres F, Montani S (1996) Experimental study of rock sheds impacted by rock blocks. Structural Engineering International 6 (3): 171–176CrossRefGoogle Scholar
  26. Mangwandi C, Cheong YS, Adams MJ, et al. (2007) The coefficient of restitution of different representative types of granules. Chemical Engineering Science 62(1–2): 437–450. https://doi.org/10.1016/j.ces.2006.08.063CrossRefGoogle Scholar
  27. Matsukura Y (2001) Rockfall at Toyohama Tunnel, Japan, in 1996: effect of notch growth on instability of a coastal cliff. Bulletin of Engineering Geology and the Environment 60(4): 285–289. https://doi.org/10.1007/s100640100123CrossRefGoogle Scholar
  28. Ministry of Transport of the People’s Republic of China (1995) Specifications for Design of Highway Subgrades (JTJ013-95). (In Chinese)Google Scholar
  29. Parise M (2002) Landslide hazard zonation of slopes susceptible to rock falls and topples. Natural Hazards and Earth System Sciences 2(1/2): 37–49. https://doi.org/10.5194/nhess-2-37-2002CrossRefGoogle Scholar
  30. Pichler B, Hellmich C, Mang HA (2005) Impact of rocks onto gravel Design and evaluation of experiments. International Journal of Impact Engineering 31(5): 559–578. https://doi.org/10.1016/j.ijimpeng.2004.01.007CrossRefGoogle Scholar
  31. Preh A, Mitchell A, Hungr O, et al. (2015) Stochastic analysis of rock fall dynamics on quarry slopes. International Journal of Rock Mechanics and Mining Sciences 80(8): 57–66. https://doi.org/10.1016/j.ijrmms.2015.09.010CrossRefGoogle Scholar
  32. Ritchie AM (1963) Evaluation of rockfall and its control. Journal of Experimental Psychology Applied 2(4): 291–304.Google Scholar
  33. Spadari M, Giacomini A, Buzzi O, et al. (2012) In situ rockfall testing in New South Wales, Australia. International Journal of Rock Mechanics and Mining Sciences 49: 84–93. https://doi.org/10.1016/j.ijrmms.2011.11.013CrossRefGoogle Scholar
  34. Thornton C (1997) Coefficient of restitution for collinear collisions of elastic-perfectly plastic spheres. Journal of Applied Mechanics 64(2): 383. https://doi.org/10.1115/1.2787319CrossRefGoogle Scholar
  35. Yuan JK, Li YR, Huang RQ, et al. (2015) Impact of rockfalls on protection measures: an experimental approach. Natural Hazards and Earth System Sciences 15: 885–893. https://doi.org/10.5194/nhessd-3-337-2015CrossRefGoogle Scholar
  36. Zhang G, Tang H, Xiang X, et al. (2015) Theoretical study of rockfall impacts based on logistic curves. International Journal of Rock Mechanics and Mining Science 78: 133–143. https://doi.org/10.1016/j.ijrmms.2015.06.001CrossRefGoogle Scholar
  37. Zhou JW, Jiao MY, Xing HG, et al. (2016) A reliability analysis method for rock slope controlled by weak structural surface. Geosciences Journal 21: 453–467. https://doi.org/10.1007/s12303-016-0058-1CrossRefGoogle Scholar

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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