Skip to main content
SpringerLink
Log in

Simple Equations for Estimating the Large Convergence and the Longitudinal Displacement Profile of a Tunnel

  • Original Paper
  • Published:
Rock Mechanics and Rock Engineering Aims and scope Submit manuscript

Abstract

This paper presents a modified relationship between small and large strain convergence by applying a small (residual) dilation angle, which behaves accurately for rock mass with the strain-softening behavior and complex dilatancy model. The modified relationship extends its applicability to a more general two- and three-dimensional excavation problems than previous study, irrespective of the constitutive behavior and rock dilatancy. On this basis, the longitudinal displacement profile (LDP) considering finite-strain is proposed, which can be corrected simply from the small strain analysis or obtained according to the actual parameters related to large deformation. The capability of the finite-strain LDP for estimating large convergences during tunnel advancement is verified by several comparisons with the existing numerical simulation and theoretical analysis results. The developed finite-strain LDP is promising to improve the convergence − confinement method when it applied to the rock-support interaction analysis in problems involving large deformation. The applicability of the displacement-release coefficient obtained from the small strain analysis for quantifying the stress relief to squeezing ground conditions is also investigated.

Highlights

  • A relationship between small and large strain convergence is modified to accurately calculate large convergences from small strain analysis.

  • Two equivalent formulas for the longitudinal displacement profile in squeezing ground considering finite strain are proposed.

  • The applicability of displacement-release coefficient by small strain analysis for quantifying the stress relief to squeezing ground is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Data Availability and Materials

Data will be made available on request.

References

  • Alejano LR, Alonso E (2005) Considerations of the dilatancy angle in rocks and rock masses. Int J Rock Mech Min Sci 42:481–507

    Article  Google Scholar 

  • Alejano LR, Rodriguez-Dono A, Alonso E, Fdez-Manín G (2009) Ground reaction curves for tunnels excavated in different quality rock masses showing several types of post-failure behaviour. Tunn Undergr Space Technol Inc Trenchless Technol Res 24:689–705

    Article  Google Scholar 

  • Alejano LR, Alonso E, Rodríguez-Dono A, Fernández-Manín G (2010) Application of the convergence−confinement method to tunnels in rock masses exhibiting Hoek-Brown strain-softening behaviour. Int J Rock Mech Min Sci 47:150–160

    Article  Google Scholar 

  • Alejano LR, Rodríguez-Dono A, Veiga M (2012) Plastic radii and longitudinal deformation profiles of tunnels excavated in strain-softening rock masses. Tunn Undergr Space Technol Inc Trenchless Technol Res 30:169–182

    Article  Google Scholar 

  • Anagnostou G (2007) Practical consequences of the time-dependency of ground behavior for tunneling. Rapid Excavation and Tunneling Conference: 2007 proceedings, SME

  • Brown ET, Bray JW, Ladanyi B, Hoek E (1983) Ground response curves for rock tunnels. J Geotech Eng 109:15–39

    Article  Google Scholar 

  • Cai W, Zhu H, Liang W, Wang X, Su C, Wei X (2022) A post-peak dilatancy model for soft rock and its application in deep tunnel excavation. J Rock Mech Geotech Eng. https://doi.org/10.1016/j.jrmge.2022.05.014

    Article  Google Scholar 

  • Carranza-Torres C, Diederichs M (2009) Mechanical analysis of circular liners with particular reference to composite supports. For example, liners consisting of shotcrete and steel sets. Tunn Undergr Space Technol 24:506–532

    Article  Google Scholar 

  • Carranza-Torres C, Fairhurst C (2000) Application of the convergence−confinement method of tunnel design to rock masses that satisfy the Hoek-Brown failure criterion. Tunn Undergr Space Technol Inc Trenchless Technol Res 15:187–213

    Article  Google Scholar 

  • Chu Z, Wu Z, Liu B, Liu Q (2019) Coupled analytical solutions for deep-buried circular lined tunnels considering tunnel face advancement and soft rock rheology effects. Tunn Undergr Space Technol 94:103111

    Article  Google Scholar 

  • Costamagna E, Oggeri C, Vinai R (2021) Damage and contour quality in rock excavations for quarrying and tunnelling assessment for properties and solutions for stability. IOP Conf Ser: Earth Environ Sci IOP Publ. https://doi.org/10.1088/1755-1315/833/1/012137

    Article  Google Scholar 

  • Cui L, Zheng JJ, Dong YK, Zhang B, Wang A (2017) Prediction of critical strains and critical support pressures for circular tunnel excavated in strain-softening rock mass. Eng Geol. https://doi.org/10.1016/j.enggeo.2017.04.022

    Article  Google Scholar 

  • Fahimifar A, Tehrani FM, Hedayat A, Vakilzadeh A (2010) Analytical solution for the excavation of circular tunnels in a visco-elastic Burger’s material under hydrostatic stress field. Tunn Undergr Space Technol 25:297–304

    Article  Google Scholar 

  • Guan K, Zhu W, Wei J, Liu X, Niu L, Wang X (2018) A finite strain numerical procedure for a circular tunnel in strain-softening rock mass with large deformation. Int J Rock Mech Min Sci 112:266–280

    Article  Google Scholar 

  • Guan K, Zhu W, Liu X, Wei J (2020) Finite strain analysis of squeezing response in an elastic-brittle-plastic weak rocks considering the influence of axial stress. Tunn Undergr Space Technol 97:103254

    Article  Google Scholar 

  • Guan K, Zhang Q, Liu H, Zhu W (2022a) A new numerical procedure for the excavation response in mohr-coulomb rock mass exhibiting strain-softening behavior. Front Earth Sci. https://doi.org/10.3389/feart.2022.872792

    Article  Google Scholar 

  • Guan K, Zhu W, Yu Q, Cui L, Song F (2022b) A plastic-damage approach to the excavation response of a circular opening in weak rock. Tunn Undergr Space Technol 126:104538

    Article  Google Scholar 

  • Lee Y-K, Pietruszczak S (2008) A new numerical procedure for elasto-plastic analysis of a circular opening excavated in a strain-softening rock mass. Tunn Undergr Space Technol 23:588–599

    Article  Google Scholar 

  • Li E, Han Y, Tan Y, Wang C, Duan J, Pu S, Wang J (2017) Field measuring test on internal displacement of surrounding rock during whole excavation process of Beishan exploration tunnel. Chin J Rock Mech Eng 36:2741–2754

    Google Scholar 

  • Luo Y, Chen J, Chen Y, Diao P, Qiao X (2018) Longitudinal deformation profile of a tunnel in weak rock mass by using the back analysis method. Tunn Undergr Space Technol 71:478–493

    Article  Google Scholar 

  • Man K, Liu X, Song Z (2021) Rock blasting vibration velocity and excavation damaged zone for the high-level radioactive waste geological disposal. Geomat Nat Haz Risk 12:2590–2606

    Article  Google Scholar 

  • Oggeri C, Oreste P (2012) Tunnel static behavior assessed by a probabilistic approach to the back-analysis. Am J Appl Sci 9:1137

    Article  Google Scholar 

  • Oke J, Vlachopoulos N, Diederichs M (2018) Improvement to the convergence−confinement method: inclusion of support installation proximity and stiffness. Rock Mech Rock Eng 51:1–25

    Article  Google Scholar 

  • Panet M (1982) Analysis of convergence behind the face of a tunnel. Proceedings of the International Symposium Tunnelling. London, IMM: 197–204

  • Paraskevopoulou C, Diederichs M (2018) Analysis of time-dependent deformation in tunnels using the convergence-confinement method. Tunn Undergr Space Technol 71:62–80

    Article  Google Scholar 

  • Park KH, Tontavanich B, Lee JG (2008) A simple procedure for ground response curve of circular tunnel in elastic-strain softening rock masses. Tunn Undergr Space Technol 23:151–159

    Article  Google Scholar 

  • Schuerch R, Anagnostou G (2012) The applicability of the ground response curve to tunnelling problems that violate rotational symmetry. Rock Mech Rock Eng 45:1–10

    Article  Google Scholar 

  • Silva CCCd, Real MdV, Maghous S (2021) A simplified approach to reliability evaluation of deep rock tunnel deformation using First-Order Reliability Method and Monte Carlo simulations. Rev IBRACON De Estrute Mater 15:e15104

    Article  Google Scholar 

  • Song F, Rodriguez-Dono A, Olivella S, Zhong Z (2020) Analysis and modelling of longitudinal deformation profiles of tunnels excavated in strain-softening time-dependent rock masses. Comput Geotech 125:103643

    Article  Google Scholar 

  • Song F, Rodriguez-Dono A, Olivella S (2021a) Hydro-mechanical modelling and analysis of multi-stage tunnel excavations using a smoothed excavation method. Comput Geotech 135:104150

    Article  Google Scholar 

  • Song F, Rodriguez-Dono A, Olivella S, Gens A (2021b) Coupled solid-fluid response of deep tunnels excavated in saturated rock masses with a time-dependent plastic behaviour. Appl Math Model 100:508–535

    Article  Google Scholar 

  • Song F, Rodriguez-Dono A, Sanchez Farfan P (2022) Modelling underground excavations in rock masses with anisotropic time-dependent behaviour. Geomech Geophys Geo-Energy Geo-Resour 8:146

    Article  Google Scholar 

  • Varas F, Alonso E, Alejano LR, Fdez G (2005) Study of bifurcation in the problem of unloading a circular excavation in a strain-softening material. Tunn Undergr Space Technol Inc Trenchless Technol Res 20:311–322

    Article  Google Scholar 

  • Vlachopoulos N, Su Y (2019) Longitudinal displacement profiles for convergence−confinement analysis of excavations: applicability to tunnelling, limitations and current advancement. John’s, At: St John’s, Newfoundland, Canada.

  • Vlachopoulos N, Diederichs MS (2009) Improved longitudinal displacement profiles for convergence confinement analysis of deep tunnels. Rock Mech Rock Eng 42:131–146

    Article  Google Scholar 

  • Vrakas A, Anagnostou G (2014) A finite strain closed-form solution for the elastoplastic ground response curve in tunnelling. Int J Numer Anal Meth Geomech 38:1131–1148

    Article  Google Scholar 

  • Vrakas A, Anagnostou G (2015) A simple equation for obtaining finite strain solutions from small strain analyses of tunnels with very large convergences. Géotechnique 65:936–944

    Article  Google Scholar 

  • Vrakas A (2016) Analysis of ground response and ground-support interaction in tunnelling considering large deformations, ETH Zurich.

  • Xu C, Xia C (2021) A new large strain approach for predicting tunnel deformation in strain-softening rock mass based on the generalized Zhang-Zhu strength criterion. Int J Rock Mech Min Sci 143:104786

    Article  Google Scholar 

  • Zhang C, Zeng K (2015) Comparisons and applications of displacement release coefficients for a circular rock tunnel subjected to isotropic geostresses. Chin J Rock Mech Eng 34:498–509

    CAS  Google Scholar 

  • Zhang Q, Wang X-F, Jiang B-S, Liu R-C, Li G-M (2021) A finite strain solution for strain-softening rock mass around circular roadways. Tunn Undergr Space Technol 111:103873

    Article  Google Scholar 

  • Zhao X, Cai M (2012) A rock dilation angle model and its application to underground excavation. ISRM International Symposium-EUROCK 2012, OnePetro.

  • Zhu W, Wei J, Zhao J, Niu L (2014) 2D numerical simulation on excavation damaged zone induced by dynamic stress redistribution. Tunn Undergr Space Technol 43:315–326

    Article  Google Scholar 

  • Zou JF, Li C, Wang F (2017) A new procedure for ground response curve (GRC) in strain-softening surrounding rock. Comput Geotech 89:81–91

    Article  Google Scholar 

Download references

Acknowledgements

This work is funded by the National Natural Science Foundation of China (Grant Nos. 52004053 and U1906208), and Natural Science Foundation of Liaoning Province (Grant No. 2021-BS-052). These supports are gratefully acknowledged.

Funding

This article is funded by National Natural Science Foundation of China, 52004053, Kai Guan, National Natural Science Foundation of China, 52004053, Wancheng Zhu, and Natural Science Foundation of Liaoning Province, 2021-BS-052, Kai Guan.

Author information

Authors and Affiliations

  1. Center for Rock Instability and Seismicity Research, School of Resource and Civil Engineering, Northeastern University, Shenyang, 110819, China

    Kai Guan, Wancheng Zhu, Hongping Li, Quanyun Zhang, Qinglei Yu & Xige Liu

Authors
  1. Kai Guan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wancheng Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hongping Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Quanyun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Qinglei Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Xige Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Kai Guan: Supervision, Conceptualization, Methodology, Formal analysis, Data curation, Writing—Original draft preparation, Writing—Review & Editing. Wancheng Zhu: Methodology, Writing—Review & Editing. Hongping Li: Formal analysis, Writing—Review & Editing. Quanyun Zhang: Formal analysis, Data curation, Writing—Review & Editing. Qinglei Yu: Validation, Writing—Review & Editing. Xige Liu: Conceptualization, Methodology.

Corresponding author

Correspondence to Kai Guan.

Ethics declarations

Conflict of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guan, K., Zhu, W., Li, H. et al. Simple Equations for Estimating the Large Convergence and the Longitudinal Displacement Profile of a Tunnel. Rock Mech Rock Eng (2024). https://doi.org/10.1007/s00603-024-03895-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00603-024-03895-6

Keywords

Search

Navigation