Rock Mechanics and Rock Engineering

, Volume 50, Issue 10, pp 2695–2707 | Cite as

Analysis of the Seismic Performance of a Rock Joint with a Modified Continuously Yielding Model

Original Paper


The problem of seismic wave transmission at a rock joint is a critical issue for aseismic affairs of underground rock engineering. Little attention has been given to the normal cyclic loading behavior of a joint during wave propagation. This paper introduces a modification to Cundall’s continuously yielding (CY) model. The original normal behavior of the CY model is upgraded by adding the capacity to account for sophisticated characteristics during normal cyclic loading. The proposed modified-CY (M-CY) model is verified through comparison with existing experimental data. The effects of various parameters in this M-CY model are discussed, indicating that the M-CY model is capable of fitting realistic test data. Subsequently, P-wave transmission across a M-CY joint is numerically performed and evaluated. The results show that the M-CY model exhibits more transmission and consequently less reflection than the linear model as the velocity magnitude of an incident impulse approaches its peak value. Furthermore, the compatibility of the joint under an incident P-wave can be revealed by the M-CY model. The seismic stability of the tailrace tunnel of the Baihetan Hydropower Plant, which is controlled by a large fault, is studied as an engineering application of the proposed model. Seismic analysis suggests that the tunnel’s main failure mode under an earthquake would be shear slip at the intersection of the tunnel ceiling and fault. A comparison between the proposed model and original CY model illustrates that the original model would give a failure mode that overestimates the fault opening during the earthquake.


Seismic analysis Wave propagation Nonlinear joint Cyclic loading Continuously yielding model 

List of symbols

CY model

Continuously yielding model

\( {\text{en}} \)

Joint normal stiffness exponent of the CY model

\( {\text{es}} \)

Joint shear stiffness exponent of the CY model

\( d_{\text{m}} \)

Joint maximum closure

\( d_{{{\text{r}}\left( j \right)}} \)

Irrecoverable closure of loading path \( j \)

\( E \)

Young’s modulus

\( F \)

Tangent modulus governing parameter of the CY model


Interlayer shear weakness zone

\( k_{\text{n}} \)

Joint normal stiffness

\( k_{{{\text{ni}}\left( j \right)}} \)

Joint initial normal stiffness of loading path \( j \)

\( k_{\text{s}} \)

Joint shear stiffness

\( k_{\text{si}} \)

Joint initial shear stiffness

\( {\text{LP}}_{\left( j \right)} \)

Loading path \( j \)

M-CY model

Modified continuously yielding model

\( u_{\text{n}} \)

Joint normal displacement

\( u_{\text{s}} \)

Joint shear displacement

\( \alpha_{1} ,\alpha_{2} ,\beta_{1} ,\beta_{2} \)

Parameters of the M-CY model

\( \sigma_{\text{n}} \)

Joint normal stress

\( \sigma_{\text{s}} \)

Joint shear stress



This study was financially supported by the National Basic Research Program of China (no. 2015CB057905), the National Key R&D Program of China (no. 2016YFC0401803) and the National Natural Science Foundation of China (Nos. 51409263, 41672319 and 11472292).


  1. Aydan Ö, Ohta Y, Genis M, Tokashiki N, Ohkubo K (2010) Response and stability of underground structures in rock mass during earthquakes. Rock Mech Rock Eng 43(6):857–875CrossRefGoogle Scholar
  2. Bandis SC, Lumsden AC, Barton NR (1983) Fundamentals of rock joint deformation. Int J Rock Mech Min Sci Geomech Abstr 20(6):249–268CrossRefGoogle Scholar
  3. Barbero M, Barla G, Zaninetti A (1996) Dynamic shear strength of rock joints subjected to impulse loading. Int J Rock Mech Min Sci Geomech Abstr 33(2):141–151CrossRefGoogle Scholar
  4. Boulon M, Armand G, Hoteit N, Divoux P (2002) Experimental investigations and modelling of shearing of calcite healed discontinuities of granodiorite under typical stresses. Eng Geol 64:117–133CrossRefGoogle Scholar
  5. Cui Z, Sheng Q, Leng XL (2016) Control effect of a large geological discontinuity on the seismic response and stability of underground rock caverns: a case study of the Baihetan#1 surge chamber. Rock Mech Rock Eng 49(6):2099–2114CrossRefGoogle Scholar
  6. Cui Z, Sheng Q, Leng XL (2017a) Analysis of S wave propagation through a nonlinear joint with the continuously yielding model. Rock Mech Rock Eng 50(1):113–123CrossRefGoogle Scholar
  7. Cui Z, Sheng Q, Leng XL (2017b) Effects of a controlling geological discontinuity on the seismic stability of an underground cavern subjected to near-fault ground motions. Bull Eng Geol Environ. doi: 10.1007/s10064-016-0936-9
  8. Cundall PA, Hart RD (1984) Analysis of block test no.1 inelastic rock mass behavior: phase 2-a characterization of joint behavior (final report). Itasca Consulting Group Report, Rockwell Hanford Operations, Subcontract SA-957Google Scholar
  9. Cundall PA, Lemos JV (1990) Numerical simulation of fault instability with the continuously yielding joint model. In: Fairhurst S (ed) Rockbursts and seismicity in mines. Balkema, Rotterdam, pp 147–152Google Scholar
  10. Dang W, Konietzky H, Frühwirt T (2016) Direct shear behavior of a plane joint under dynamic normal load (DNL) conditions. Eng Geol 213:133–141CrossRefGoogle Scholar
  11. Divoux P, Boulon M, Bourdarot E (1997) A mechanical constitutive model for rock and concrete joints under cyclic loading. In: Proceedings of the first int. conf. on damage and failure of interfaces (DFI-1), Wien, Austria. Balkema Pub., Rotterdam, pp 443–450Google Scholar
  12. Hahasha YM, Hook JJ, Schmidt B, Yao JIC (2001) Seismic design and analysis of underground structures. Tunn Undergr Space Technol 16(4):247–293CrossRefGoogle Scholar
  13. Itasca Consulting Group (2013) 3DEC user’s manual (version 5.0). MinneapolisGoogle Scholar
  14. Jing L, Stephansson O, Nordlund E (1993) Study of rock joints under cyclic loading conditions. Rock Mech Rock Eng 26(3):215–232CrossRefGoogle Scholar
  15. Jing L, Nordlund E, Stephansson O (1994) A 3-D constitutive model for rock joints with anisotropic friction and stress dependency in shear stiffness. Int J Rock Mech Min Sci Geomech Abstr 31(2):173–178CrossRefGoogle Scholar
  16. Lee YK, Park JW, Song JJ (2014) Model for the shear behavior of rock joints under CNL and CNS conditions. Int J Rock Mech Min Sci 70(9):252–263Google Scholar
  17. Li JC (2013) Wave propagation across non-linear rock joints based on time-domain recursive method. Geophys J Int 193(2):970–985CrossRefGoogle Scholar
  18. Li JC, Ma GW, Zhao J (2011) Analysis of stochastic seismic wave interaction with a slippery rock fault. Rock Mech Rock Eng 44(1):85–92CrossRefGoogle Scholar
  19. Li HB, Liu TT, Liu YQ, Li JC, Xia X (2016) Numerical modeling of wave transmission across rock masses with nonlinear joints. Rock Mech Rock Eng 49(3):1115–1121CrossRefGoogle Scholar
  20. Ma GW, Li JC, Zhao J (2011) Three-phase medium model for filled rock joint and interaction with stress waves. Int J Numer Anal Methods Geomech 35(1):97–110CrossRefGoogle Scholar
  21. Makurat A, Ahola M, Khair K, Noorishad J, Rosengren L, Rutqvist J (1995) The decovalex test-case one. Int J Rock Mech Min Sci Geomech Abstr 32(5):399–408CrossRefGoogle Scholar
  22. Malama B, Kulatilake PHSW (2003) Models for normal fracture deformation under compressive loading. Int J Rock Mech Min Sci 40(6):893–901CrossRefGoogle Scholar
  23. Mirzaghorbanali A, Nemcik J, Aziz N (2014) Effects of cyclic loading on the shear behaviour of infilled rock joints under constant normal stiffness conditions. Rock Mech Rock Eng 47(4):1373–1391CrossRefGoogle Scholar
  24. Saeb S, Amadei B (1992) Modelling rock joints under shear and normal loading. Int J Rock Mech Min Sci Geomech Abstr 29(3):267–278CrossRefGoogle Scholar
  25. Souley M, Homand F, Amadei B (1995) An extension to the Saeb and Amadei constitutive model for rock joints to include cyclic loading paths. Int J Rock Mech Min Sci Geomech Abstr 32(2):101–109CrossRefGoogle Scholar
  26. Thirukumaran S, Indraratna B (2016) A review of shear strength models for rock joints subjected to constant normal stiffness. J Rock Mech Geotech Eng 8(3):405–414CrossRefGoogle Scholar
  27. Usefzadeh A, Yousefzadeh H, Salari-Rad H, Sharifzadeh M (2013) Empirical and mathematical formulation of the shear behavior of rock joints. Eng Geol 164(18):243–252CrossRefGoogle Scholar
  28. Xia CC, Yue ZQ, Tham LG, Lee CF, Sun ZQ (2003) Quantifying topography and closure deformation of rock joints. Int J Rock Mech Min Sci 40(2):197–220CrossRefGoogle Scholar
  29. Yin XJ, Wang GL (2005) Study of constitutive model for rock interfaces and joints under normal cyclic loading. Chin J Rock Mech Eng 24(7):1158–1163 (in Chinese) Google Scholar
  30. Zhao J, Cai JG (2001) Transmission of Elastic P-waves across Single Fractures with a Nonlinear Normal Deformational Behavior. Rock Mech Rock Eng 34(1):3–22CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  • Zhen Cui
    • 1
    • 2
  • Qian Sheng
    • 1
  • Xianlun Leng
    • 1
  • Yalina Ma
    • 1
  1. 1.State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil MechanicsChinese Academy of SciencesWuhanChina
  2. 2.PowerChina Huadong Engineering Corporation LimitedHangzhouChina

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