Skip to main content
Erschienen in: Rock Mechanics and Rock Engineering 10/2022

30.06.2022 | Original Paper

Analytical Solution for Seismic Response of Deep Tunnels with Arbitrary Cross-Section Shape in Saturated Orthotropic Rock

verfasst von: Gong Chen, Haitao Yu, Antonio Bobet

Erschienen in: Rock Mechanics and Rock Engineering | Ausgabe 10/2022

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Unified analytical solutions for stresses and displacements of a deep arbitrary shaped tunnel buried in saturated orthotropic rock subjected to seismic loading are presented. A pseudo-static load is used for the analysis, and the tunnel liner is isotropic and elastic and is modeled as a shell. Slip along the ground-liner interface is included. Two extreme conditions, drained and undrained, are considered in the analytical solution. The proposed analytical solution is derived using complex-variable theory and conformal mapping techniques. Validation of the solution is done by comparing its results with those obtained from a FEM, and with results from the literature. Parametric analyses are carried out for a typical section of rock tunnel, i.e. horse-shoe tunnel, to further investigate the influence of the cross-section shape on the tunnel response in orthotropic rock. In addition, the effects of the orientation of the axes of elastic symmetry with respect to the tunnel, as well as the loading conditions, namely drained and undrained, are discussed.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Anhänge
Nur mit Berechtigung zugänglich
Literatur
Zurück zum Zitat Amadei B (1996) Importance of anisotropy when estimating and measuring in situ stresses in rock. Int J Rock Mech Min Sci Geomech 33(3):293–325CrossRef Amadei B (1996) Importance of anisotropy when estimating and measuring in situ stresses in rock. Int J Rock Mech Min Sci Geomech 33(3):293–325CrossRef
Zurück zum Zitat Amorosi A, Boldini D (2009) Numerical modelling of the transverse dynamic behaviour of circular tunnels in clayey soils. Soil Dyn Earthq Eng 29(6):1059–1072CrossRef Amorosi A, Boldini D (2009) Numerical modelling of the transverse dynamic behaviour of circular tunnels in clayey soils. Soil Dyn Earthq Eng 29(6):1059–1072CrossRef
Zurück zum Zitat Asakura T, Yutaka S (1996) Damage to mountain tunnels in hazard area. Soils Found 36(Special):301–311CrossRef Asakura T, Yutaka S (1996) Damage to mountain tunnels in hazard area. Soils Found 36(Special):301–311CrossRef
Zurück zum Zitat Bobet A (2001) Analytical solutions for shallow tunnels in saturated ground. J Eng Mech 127(6):1258–1266 Bobet A (2001) Analytical solutions for shallow tunnels in saturated ground. J Eng Mech 127(6):1258–1266
Zurück zum Zitat Bobet A (2003) Effect of pore water pressure on tunnel support during static and seismic loading. Tunn Undergr Sp Technol 18(4):377–393CrossRef Bobet A (2003) Effect of pore water pressure on tunnel support during static and seismic loading. Tunn Undergr Sp Technol 18(4):377–393CrossRef
Zurück zum Zitat Bobet A (2010) Drained and undrained response of deep tunnels subjected to far-field shear loading. Tunn Undergr Sp Technol 25(1):21–31CrossRef Bobet A (2010) Drained and undrained response of deep tunnels subjected to far-field shear loading. Tunn Undergr Sp Technol 25(1):21–31CrossRef
Zurück zum Zitat Bobet A (2011) Lined circular tunnels in elastic transversely anisotropic rock at depth. Rock Mech Rock Eng 44(2):149–167CrossRef Bobet A (2011) Lined circular tunnels in elastic transversely anisotropic rock at depth. Rock Mech Rock Eng 44(2):149–167CrossRef
Zurück zum Zitat Bobet A (2016) Deep lined circular tunnels in transversely anisotropic rock: complementary solutions. Rock Mech Rock Eng 49(9):3817–3822CrossRef Bobet A (2016) Deep lined circular tunnels in transversely anisotropic rock: complementary solutions. Rock Mech Rock Eng 49(9):3817–3822CrossRef
Zurück zum Zitat Cheng AD (1997) Material coefficients of anisotropic poroelasticity. Int J Rock Mech Min Sci 34(2):199–205CrossRef Cheng AD (1997) Material coefficients of anisotropic poroelasticity. Int J Rock Mech Min Sci 34(2):199–205CrossRef
Zurück zum Zitat Cheng AH (1998) On generalized plane strain poroelasticity. Int J Rock Mech Min Sci 35(2):183–193CrossRef Cheng AH (1998) On generalized plane strain poroelasticity. Int J Rock Mech Min Sci 35(2):183–193CrossRef
Zurück zum Zitat Cui C, Meng K, Xu C, Liang Z, Li H, Pei H (2021) Analytical solution for longitudinal vibration of a floating pile in saturated porous media based on a fictitious saturated soil pile model. Comput Geotech 131:103942CrossRef Cui C, Meng K, Xu C, Liang Z, Li H, Pei H (2021) Analytical solution for longitudinal vibration of a floating pile in saturated porous media based on a fictitious saturated soil pile model. Comput Geotech 131:103942CrossRef
Zurück zum Zitat Deeks AJ, Randolph MF (1994) Axisymmetric time-domain transmitting boundaries. J Eng Mech 120(1):25–42 Deeks AJ, Randolph MF (1994) Axisymmetric time-domain transmitting boundaries. J Eng Mech 120(1):25–42
Zurück zum Zitat Einstein H, Schwartz C (1979) Simplified analysis for tunnel supports. J Geotech Eng Div 105(GT4):499–518CrossRef Einstein H, Schwartz C (1979) Simplified analysis for tunnel supports. J Geotech Eng Div 105(GT4):499–518CrossRef
Zurück zum Zitat Gao Z, Zhao M, Du X, Zhong Z (2021) A generalized response spectrum method for seismic response analysis of underground structure combined with viscous-spring artificial boundary. Soil Dyn Earthq Eng 140:106451CrossRef Gao Z, Zhao M, Du X, Zhong Z (2021) A generalized response spectrum method for seismic response analysis of underground structure combined with viscous-spring artificial boundary. Soil Dyn Earthq Eng 140:106451CrossRef
Zurück zum Zitat Givoli D (2004) High-order local non-reflecting boundary conditions: a review. Wave Motion 39(4):319–326CrossRef Givoli D (2004) High-order local non-reflecting boundary conditions: a review. Wave Motion 39(4):319–326CrossRef
Zurück zum Zitat Gordo Monsó C, González Galindo J, Olalla Marañón C (2019) A closed-form solution for the seismic racking and rocking behavior of rectangular tunnels. Tunn Undergr Sp Technol 88(September 2018):87–97CrossRef Gordo Monsó C, González Galindo J, Olalla Marañón C (2019) A closed-form solution for the seismic racking and rocking behavior of rectangular tunnels. Tunn Undergr Sp Technol 88(September 2018):87–97CrossRef
Zurück zum Zitat Grechka V, Tsvankin I (1999) 3-D moveout velocity analysis and parameter estimation for orthorhombic media. Geophysics 64(3):820–837CrossRef Grechka V, Tsvankin I (1999) 3-D moveout velocity analysis and parameter estimation for orthorhombic media. Geophysics 64(3):820–837CrossRef
Zurück zum Zitat Hashash YM, Hook JJ, Schmidt B, Yao JI-C (2001) Seismic design and analysis of underground structures. Tunn Undergr Sp Technol 16(4):247–293CrossRef Hashash YM, Hook JJ, Schmidt B, Yao JI-C (2001) Seismic design and analysis of underground structures. Tunn Undergr Sp Technol 16(4):247–293CrossRef
Zurück zum Zitat Hendron AJ, Fernández G (1983) Dynamic and static design considerations for underground chambers. In: Seismic design of embankments and caverns, ASCE symposium, Philadelphia, Pennsylvania, p 157–197 Hendron AJ, Fernández G (1983) Dynamic and static design considerations for underground chambers. In: Seismic design of embankments and caverns, ASCE symposium, Philadelphia, Pennsylvania, p 157–197
Zurück zum Zitat Huo H, Bobet A, Fernández G, Ramírez J (2006) Analytical solution for deep rectangular structures subjected to far-field shear stresses. Tunn Undergr Sp Technol 21(6):613–625CrossRef Huo H, Bobet A, Fernández G, Ramírez J (2006) Analytical solution for deep rectangular structures subjected to far-field shear stresses. Tunn Undergr Sp Technol 21(6):613–625CrossRef
Zurück zum Zitat Lekhnitskii S (1963) Theory of elasticity of an anisotropic elastic body. Holden-Day Inc, San Francisco Lekhnitskii S (1963) Theory of elasticity of an anisotropic elastic body. Holden-Day Inc, San Francisco
Zurück zum Zitat Monsees JE, Merritt JL (1988) Seismic modeling and design of underground structures. In: Numer methods geomech. Balkema, Rotterdam Holland, p 1833–1842 Monsees JE, Merritt JL (1988) Seismic modeling and design of underground structures. In: Numer methods geomech. Balkema, Rotterdam Holland, p 1833–1842
Zurück zum Zitat Muskhelishvili N (1963) Some basic problems of the mathematical theory of elasticity. Noordhoff, Groningen 17404(6.2): 1 Muskhelishvili N (1963) Some basic problems of the mathematical theory of elasticity. Noordhoff, Groningen 17404(6.2): 1
Zurück zum Zitat Penzien J (2000) Seismically induced racking of tunnel linings. Earthq Eng Struct Dyn 29(5):683–691CrossRef Penzien J (2000) Seismically induced racking of tunnel linings. Earthq Eng Struct Dyn 29(5):683–691CrossRef
Zurück zum Zitat Penzien J, Wu CL (1998) Stresses in linings of bored tunnels. Earthq Eng Struct Dyn 27(3):283–300CrossRef Penzien J, Wu CL (1998) Stresses in linings of bored tunnels. Earthq Eng Struct Dyn 27(3):283–300CrossRef
Zurück zum Zitat Pitilakis K, Tsinidis G (2014) Performance and seismic design of underground structures. Springer, Cham, pp 279–340 Pitilakis K, Tsinidis G (2014) Performance and seismic design of underground structures. Springer, Cham, pp 279–340
Zurück zum Zitat Ruger A (1998) Variation of P-wave reflectivity with offset and azimuth in anisotropic media. Geophysics 63(3):935–947CrossRef Ruger A (1998) Variation of P-wave reflectivity with offset and azimuth in anisotropic media. Geophysics 63(3):935–947CrossRef
Zurück zum Zitat Sandoval E, Bobet A (2017) Effect of frequency and flexibility ratio on the seismic response of deep tunnels. Undergr Sp 2(2):125–133CrossRef Sandoval E, Bobet A (2017) Effect of frequency and flexibility ratio on the seismic response of deep tunnels. Undergr Sp 2(2):125–133CrossRef
Zurück zum Zitat Schoenberg M, Helbig K (1997) Orthorhombic media: modeling elastic wave behavior in a vertically fractured earth. Geophysics 62(6):1954–1974CrossRef Schoenberg M, Helbig K (1997) Orthorhombic media: modeling elastic wave behavior in a vertically fractured earth. Geophysics 62(6):1954–1974CrossRef
Zurück zum Zitat St. John CM, Zahrah TF (1987) Aseismic design of underground structures. Tunn Undergr Sp Technol Inc Trenchless 2(2):165–197CrossRef St. John CM, Zahrah TF (1987) Aseismic design of underground structures. Tunn Undergr Sp Technol Inc Trenchless 2(2):165–197CrossRef
Zurück zum Zitat Stamos AA, Beskos DE (1995) Dynamic analysis of large 3-D underground structures by the bem. Earthq Eng Struct Dyn 24(6):917–934CrossRef Stamos AA, Beskos DE (1995) Dynamic analysis of large 3-D underground structures by the bem. Earthq Eng Struct Dyn 24(6):917–934CrossRef
Zurück zum Zitat Wang JN (1993) Seismic design of tunnels. A simple state-of-the-art approach. Monograph, monograph 7. Parsons, Brinckerhoff, Quade and Douglas Inc, New York Wang JN (1993) Seismic design of tunnels. A simple state-of-the-art approach. Monograph, monograph 7. Parsons, Brinckerhoff, Quade and Douglas Inc, New York
Zurück zum Zitat Wang WL, Wang TT, Su JJ, Lin CH, Seng CR, Huang TH (2001) Assessment of damage in mountain tunnels due to the Taiwan Chi-Chi earthquake. Tunn Undergr Sp Technol 16(3):133–150CrossRef Wang WL, Wang TT, Su JJ, Lin CH, Seng CR, Huang TH (2001) Assessment of damage in mountain tunnels due to the Taiwan Chi-Chi earthquake. Tunn Undergr Sp Technol 16(3):133–150CrossRef
Zurück zum Zitat Wegmann R (2005) Chapter 9. Methods for numerical conformal mapping. In: R Kühnau, Ed. Handbook of complex analysis, vol 2, Elsevier, Amsterdam Wegmann R (2005) Chapter 9. Methods for numerical conformal mapping. In: R Kühnau, Ed. Handbook of complex analysis, vol 2, Elsevier, Amsterdam
Zurück zum Zitat Xue Y, Kong F, Li S, Zhang Q, Qiu D, Su M, Li Z (2021) China starts the world’s hardest “sky-high road” project: challenges and countermeasures for sichuan-tibet railway. Innovation 2(2):100105 Xue Y, Kong F, Li S, Zhang Q, Qiu D, Su M, Li Z (2021) China starts the world’s hardest “sky-high road” project: challenges and countermeasures for sichuan-tibet railway. Innovation 2(2):100105
Zurück zum Zitat Yang Y, Yu H, Yuan Y, Zhao M (2020) Analytical solution for longitudinal seismic response of long tunnels subjected to Rayleigh waves. Int J Numer Anal Methods Geomech 44(10):1371–1385CrossRef Yang Y, Yu H, Yuan Y, Zhao M (2020) Analytical solution for longitudinal seismic response of long tunnels subjected to Rayleigh waves. Int J Numer Anal Methods Geomech 44(10):1371–1385CrossRef
Zurück zum Zitat Yu H, Chen G (2021) Pseudo-static simplified analytical solution for seismic response of deep tunnels with arbitrary cross-section shapes. Comput Geotech 137:104306CrossRef Yu H, Chen G (2021) Pseudo-static simplified analytical solution for seismic response of deep tunnels with arbitrary cross-section shapes. Comput Geotech 137:104306CrossRef
Zurück zum Zitat Yu H, Yuan Y (2014) Analytical solution for an infinite euler-bernoulli beam on a viscoelastic foundation subjected to arbitrary dynamic loads. J Eng Mech 140(3):542–551 Yu H, Yuan Y (2014) Analytical solution for an infinite euler-bernoulli beam on a viscoelastic foundation subjected to arbitrary dynamic loads. J Eng Mech 140(3):542–551
Zurück zum Zitat Yu H, Cai C, Guan X, Yuan Y (2016a) Analytical solution for long lined tunnels subjected to travelling loads. Tunn Undergr Sp Technol 58:209–215CrossRef Yu H, Cai C, Guan X, Yuan Y (2016a) Analytical solution for long lined tunnels subjected to travelling loads. Tunn Undergr Sp Technol 58:209–215CrossRef
Zurück zum Zitat Yu H, Chen J, Bobet A, Yuan Y (2016b) Damage observation and assessment of the Longxi tunnel during the Wenchuan earthquake. Tunn Undergr Sp Technol 54:102–116CrossRef Yu H, Chen J, Bobet A, Yuan Y (2016b) Damage observation and assessment of the Longxi tunnel during the Wenchuan earthquake. Tunn Undergr Sp Technol 54:102–116CrossRef
Zurück zum Zitat Yu H, Cai C, Yuan Y, Jia M (2017) Analytical solutions for Euler-Bernoulli Beam on Pasternak foundation subjected to arbitrary dynamic loads. Int J Numer Anal Methods Geomech 41(8):1125–1137CrossRef Yu H, Cai C, Yuan Y, Jia M (2017) Analytical solutions for Euler-Bernoulli Beam on Pasternak foundation subjected to arbitrary dynamic loads. Int J Numer Anal Methods Geomech 41(8):1125–1137CrossRef
Zurück zum Zitat Yu H, Yang Y, Yuan Y (2018) Analytical solution for a finite Euler-Bernoulli beam with single discontinuity in section under arbitrary dynamic loads. Appl Math Model 60:571–580CrossRef Yu H, Yang Y, Yuan Y (2018) Analytical solution for a finite Euler-Bernoulli beam with single discontinuity in section under arbitrary dynamic loads. Appl Math Model 60:571–580CrossRef
Zurück zum Zitat Yu H, Sun Y, Li P, Zhao M (2020) Analytical solution for dynamic response of underground rectangular fluid tank subjected to arbitrary dynamic loads. J Eng Mech 146(8):04020077 Yu H, Sun Y, Li P, Zhao M (2020) Analytical solution for dynamic response of underground rectangular fluid tank subjected to arbitrary dynamic loads. J Eng Mech 146(8):04020077
Zurück zum Zitat Zhang J, Yuan Y, Yu H (2019) Shaking table tests on discrepant responses of shaft-tunnel junction in soft soil under transverse excitations. Soil Dyn Earthq Eng 120(January):345–359CrossRef Zhang J, Yuan Y, Yu H (2019) Shaking table tests on discrepant responses of shaft-tunnel junction in soft soil under transverse excitations. Soil Dyn Earthq Eng 120(January):345–359CrossRef
Zurück zum Zitat Zhang C, Li W, Zhu W, Tan Z (2020) Face stability analysis of a shallow horseshoe-shaped shield tunnel in clay with a linearly increasing shear strength with depth. Tunn Undergr Sp Technol 97(January):103291CrossRef Zhang C, Li W, Zhu W, Tan Z (2020) Face stability analysis of a shallow horseshoe-shaped shield tunnel in clay with a linearly increasing shear strength with depth. Tunn Undergr Sp Technol 97(January):103291CrossRef
Metadaten
Titel
Analytical Solution for Seismic Response of Deep Tunnels with Arbitrary Cross-Section Shape in Saturated Orthotropic Rock
verfasst von
Gong Chen
Haitao Yu
Antonio Bobet
Publikationsdatum
30.06.2022
Verlag
Springer Vienna
Erschienen in
Rock Mechanics and Rock Engineering / Ausgabe 10/2022
Print ISSN: 0723-2632
Elektronische ISSN: 1434-453X
DOI
https://doi.org/10.1007/s00603-022-02935-3

Weitere Artikel der Ausgabe 10/2022

Rock Mechanics and Rock Engineering 10/2022 Zur Ausgabe