Skip to main content
SpringerLink
Log in

Anti-plane seismic ground motion above twin horseshoe-shaped lined tunnels

  • Technical paper
  • Published:
Innovative Infrastructure Solutions Aims and scope Submit manuscript

Abstract

The ground surface response was obtained in the presence of twin horseshoe-shaped lined tunnels embedded in a linear elastic half-space, subjected to obliquely propagating incident plane SH-waves. A numerical approach known as direct half-plane time-domain boundary element method was used to prepare the model. First, the heterogeneous problem was decomposed into three parts, two closed linings and a double-hole half-plane. Then, the method was applied to each part to compute the required matrices. Finally, the problem was solved by satisfying the continuity conditions at the interfaces. After solving a verification example, an advanced parametric study was carried out for obtaining the surface response with shallow twin horseshoe-shaped lined tunnels as synthetic seismograms and 3-D amplification patterns. Some intended parameters were also considered to illustrate the ground response including the horizontal/vertical overlapping of the tunnels, depth and distance between them and incident wave angle. To summarize the results, some graphs were presented to obtain the maximum amplification ratio of the surface. Numerical results showed that the existence of twin horseshoe-shaped tunnels could amplify the ground surface response up to three and half times the free-field movement at dimensionless frequencies greater than one. Also, achievements indicated that the greater role of vertically overlapped tunnels in creating safe areas on the surface compared to the horizontally placed tunnels. The results can be practically used in clarifying seismic codes in the field of urban transportation geotechnics.

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

Similar content being viewed by others

Abbreviations

A, B :

The matrices whose elements are corresponding to the unknown and known boundary values, respectively

a max :

The maximum displacement of the Ricker wavelet

b :

The outer radius of the tunnel

b (x, y, t):

Anti-plane body force at the point (x, y) and current time t

c :

Shear wave velocity

c (ξ):

Boundary element coefficients

f p :

Predominant frequency of the Ricker wavelet

G, H :

The matrices whose elements are obtained by integration over u* and q*, respectively

H (.):

Heaviside function

J :

Jacobian transformation

M :

Number of elements used on the boundary

N :

Number of time steps used on the time axis

n :

Normal vector, perpendicular to the boundary

q* :

Traction fundamental solution

q, Q :

Traction fields on the boundary and traction kernels, respectively

R :

The effect of past dynamic history on the current time

t, τ :

Current and preceding time, respectively

t 0 :

Time-shift parameter of the Ricker wavelet

u* :

Displacement fundamental solution

u, U :

Displacement fields on the boundary and displacement kernels, respectively

u ff :

Free-field displacements on the ground surface

X, Y :

The vectors of the unknown and known variables

α inc , α ref :

Phase of incident and reflected waves, respectively

Γ :

Boundary of the problem

Δt :

Time step

η :

Dimensionless frequency

θ :

Incident wave angle

κ :

Local coordinate system component

μ :

Shear modulus of the domain

ρ :

Density of the domain

ω :

Angular velocity

BEM:

Boundary element method

BIE:

Boundary integral equation

DASBEM:

Dynamic analysis of structures by boundary element method

FBEM:

Full-plane boundary element method

FDM:

Finite difference method

FEM:

Finite element method

HBEM:

Half-plane boundary element method

References

  1. Ba Z, Yin X (2016) Wave scattering of complex local site in a layered half-space by using a multi-domain IBEM: incident plane SH waves. Geophys J Int 205:1382–1405

    Google Scholar 

  2. Balendra T, Thambiratnam DP, Koh CG, Lee SL (1984) Dynamic response of twin circular tunnels due to incident SH-waves. Earthq Eng Struct Dyn 12:181–201

    Google Scholar 

  3. Belytschko T, Chang HS (1988) Simplified direct time integration boundary element method. J Eng Mech 114(1):117–134

    Google Scholar 

  4. Benites R, Aki K, Yomogida K (1992) Multiple scattering of SH waves in 2-D media with many cavities. Pure Appl Geophys 138(3):353–390

    Google Scholar 

  5. Besharat V, Davoodi M, Jafari MK (2012) Effect of underground structures on free-field ground motion during earthquakes. In: 15th World conference on earthquake engineering, Lisbon, Portugal

  6. Beskos DE (1987) Boundary element methods in dynamic analysis. Appl Mech Rev 40(1):1–23

    Google Scholar 

  7. Brebbia CA, Dominguez J (1989) Boundary elements an introductory course. Computational Mechanics Publications, Southampton

    Google Scholar 

  8. Crouch SL, Starfield AM (1983) Boundary elements methods in solid mechanics. Department of Civil and Mineral Engineering, University of Minnesota

  9. Datta SK, Shah AH (1982) Scattering of SH waves by embedded cavities. Wave Motion 4:265–283

    Google Scholar 

  10. Dominguez J (1993) Boundary elements in dynamics. Computational Mechanics Publications, Southampton

    Google Scholar 

  11. Dominguez J, Meise T (1991) On the use of the BEM for wave propagation in infinite domains. Eng Anal Bound Elem 8(3):132–138

    Google Scholar 

  12. Faccioli E, Maggio F, Paolucci R, Quarteroni A (1997) 2D and 3D elastic wave propagation by a pseudo-spectral domain decomposition method. J Seismolog 1:237–251

    Google Scholar 

  13. Gao Y, Dai D, Zhang N, Wu Y, Mahfouz AH (2016) Scattering of plane and cylindrical SH waves by a horseshoe shaped cavity. J Earthq Tsunami 10(2):1–23

    Google Scholar 

  14. Gelagoti F, Kourkoulis R, Anastapoulos I, Gazetas G (2012) Nonlinear dimensional analysis of Trapezoidal valleys subjected to vertically propagating SV waves. Bull Seismol Soc Am 102(3):999–1017

    Google Scholar 

  15. Hasheminejad SM, Kazemirad S (2008) Dynamic response of an eccentrically lined circular tunnel in poroelastic soil under seismic excitation. Soil Dyn Earthq Eng 28:277–292

    Google Scholar 

  16. Hirai H (1988) Analysis of transient response of SH wave scattering in a half-space by the boundary element method. Eng Anal 5(4):189–194

    Google Scholar 

  17. Jiang LF, Zhou XL, Wang JH (2009) Scattering of a plane wave by a lined cylindrical cavity in a poroelastic half-space. Comput Geotech 36:773–786

    Google Scholar 

  18. Kamalian M, Gatmiri B, Sohrabi-Bidar A, Khalaj A (2007) Amplification pattern of 2D semi-sine shaped valleys subjected to vertically propagating incident waves. Int J Numer Methods Biomed Eng 23(10):871–887

    Google Scholar 

  19. Kamalian M, Jafari MK, Sohrabi-Bidar A, Razmkhah A (2008) Seismic response of 2D semi-sine shaped hills to vertically propagating incident waves: amplification patterns and engineering applications. Earthq Spectra 24(2):405–430

    Google Scholar 

  20. Kamalian M, Jafari MK, Sohrabi-Bidar A, Razmkhah A, Gatmiri B (2006) Time-domain two-dimensional site response analysis of non-homogeneous topographic structures by a hybrid FE/BE method. Soil Dyn Earthq Eng 26(8):753–765

    Google Scholar 

  21. Kattis SE, Beskos DE, Cheng HD (2003) 2D dynamic response of unlined and lined tunnels in poroelastic soil to harmonic body waves. Earthq Eng Struct Dyn 32:97–110

    Google Scholar 

  22. Kolymbas D (2005) Tunneling and tunnel mechanics. Springer, Berlin

    Google Scholar 

  23. Lee VW (1977) On the deformations near circular underground cavity subjected to incident plane SH-waves. In: Proceedings of symposium of applications of computer methods in engineering conference University of Southern California, Los Angeles, pp 951–962

  24. Lee VW, Chen S, Hsu IR (1999) Anti-plane diffraction from canyon above subsurface unlined tunnel. J Eng Mech 125(6):668–675

    Google Scholar 

  25. Lee VW, Trifunac MD (1979) Response of tunnels to incident SH waves. J Eng Mech Div 105(4):643–659

    Google Scholar 

  26. Liu Z, Liu L (2015) An IBEM solution to the scattering of plane SH-waves by a lined tunnel in elastic wedge space. Earthq Sci 28(1):71–86

    Google Scholar 

  27. Liu Z, Wang Y, Liang J (2016) Dynamic interaction of twin vertically overlapping lined tunnels in an elastic half space subjected to incident plane waves. Earthq Sci 29(3):185–201

    Google Scholar 

  28. Luco JE, De-barros FCP (1994) Dynamic displacements and stresses in the vicinity of a cylindrical cavity embedded in a half-space. Earthq Eng Struct Dyn 23(3):321–340

    Google Scholar 

  29. Manolis GD, Beskos DE (1983) Dynamic response of lined tunnels by an isoparametric boundary element method. Comput Methods Appl Mech Eng 36:291–307

    Google Scholar 

  30. Manoogian ME (2000) Scattering and diffraction of SH waves above an arbitrarily shaped tunnel. ISET J Earthq Technol 37(1–3):11–26

    Google Scholar 

  31. Narayan JP, Kumar D, Sahar D (2015) Effects of complex interaction of rayleigh waves with tunnel on the free surface ground motion and the strain across the tunnel-lining. Nat Hazards 79(1):479–495

    Google Scholar 

  32. Okumura T, Takewaki N, Shimizu K, Fukutake K (1992) Dynamic response of twin circular tunnels during earthquakes. Earthq Eng Struct Dyn 840:181–191

    Google Scholar 

  33. Panji M, Ansari B (2017) Modeling pressure pipe embedded in two-layer soil by a half-plane BEM. Comput Geotech 81:360–367

    Google Scholar 

  34. Panji M, Ansari B (2017) Transient SH-wave scattering by a lined tunnel embedded in an elastic half-plane. Eng Anal Bound Elem 84:220–230

    Google Scholar 

  35. Panji M, Asgari Marnani J, Tavousi Tafreshi Sh (2011) Evaluation of effective parameters on the underground tunnel stability using BEM. J Struct Eng Geotech 1(2):29–37

    Google Scholar 

  36. Panji M, Hasanlouyi SM (2018) Time-history responses on surface by regularly distributed enormous embedded cavities: incident SH-waves. Earthq Sci 31:137–153

    Google Scholar 

  37. Panji M, Kamalian M, Asgari Marnani J, Jafari MK (2013) Transient analysis of wave propagation problems by half-plane BEM. Geophys J Int 194(3):1849–1865

    Google Scholar 

  38. Panji M, Kamalian M, Asgari Marnani J, Jafari MK (2013) Amplification pattern of semi-sine shaped valleys subjected to vertically propagating incident SH waves. J Comput Methods Eng 32(2):87–111

    Google Scholar 

  39. Panji M, Kamalian M, Asgari Marnani J, Jafari MK (2014) Analyzing seismic convex topographies by a half-plane time-domain BEM. Geophys J Int 197(1):591–607

    Google Scholar 

  40. Panji M, Kamalian M, Asgari Marnani J, Jafari MK (2014) Anti-plane seismic response from semi-sine shaped valley above embedded truncated circular cavity: a time-domain half-plane BEM. Int J Civ Eng 12(2):193–206

    Google Scholar 

  41. Panji M, Koohsari H, Adampira M, Alielahi H, Asgari Marnani J (2016) Stability analysis of shallow tunnels subjected to eccentric loads by a boundary element method. J Rock Mech Geotech Eng 8:480–488

    Google Scholar 

  42. Parvanova SL, Dineva PS, Manolis GD, Wuttke F (2014) Seismic response of lined tunnels in the half-plane with surface topography. Bull Earthq Eng 12(2):981–1005

    Google Scholar 

  43. Rabeti MM, Baziar MH (2016) Seismic ground motion amplification pattern induced by a subway tunnel: shaking table testing and numerical simulation. Soil Dyn Earthq Eng 83:81–97

    Google Scholar 

  44. Ricker N (1953) The form and laws of propagation of seismic wavelets. Geophysics 18(1):10–40

    Google Scholar 

  45. Shen Y, Gao B, Yang X, Tao SJ (2014) Seismic damage mechanism and dynamic deformation characteristic analysis of mountain tunnel after Wenchuan earthquake. Eng Geol 180:85–98

    Google Scholar 

  46. Shi S, Han F, Wang Z, Liu D (1996) The interaction of plane SH-waves and non-circular cavity surfaced with lining in anisotropic media. Appl Math Mech 17(9):855–867

    Google Scholar 

  47. Smerzini C, Aviles J, Paolucci R, Sanchez-Sesma FJ (2009) Effect of underground cavities on surface earthquake ground motion under SH wave propagation. Earthq Eng Struct Dyn 38(12):1441–1460

    Google Scholar 

  48. Stamos AA, Beskos DE (1995) Dynamic analysis of large 3-D underground structures by the BEM. Earthq Eng Struct Dyn 24:917–934

    Google Scholar 

  49. Takemiya H, Fujiwara A (1994) SH-wave scattering and propagation analyses at irregular sites by time domain BEM. Bull Seismol Soc Am 84(5):1443–1455

    Google Scholar 

  50. Telles JCF, Brebbia CA (1980) Boundary element solution for half-plane problems. Int J Solids Struct 12:1149–1158

    Google Scholar 

  51. Wang WL, Wang TT, Su JJ, Lin CH, Seng CR, Huang TK (2001) Assessment of damage in mountain tunnels due to the Taiwan Chi–Chi earthquake. Tunn Undergr Space Technol 16(3):133–150

    Google Scholar 

  52. Wu G, Finn WDL (1997) Dynamic nonlinear analysis of pile foundations using finite element method in the time domain. Can Geotech J 34(1):44–52

    Google Scholar 

  53. Wu R, Xu JH, Li C, Wang ZL, Qin S (2015) Stress distribution on mine roof with the boundary element method. Eng Anal Bound Elem 50:39–46

    Google Scholar 

  54. Xiao B, Carter JP (1993) Boundary element analysis of anisotropic rock masses. Eng Anal Bound Elem 11:293–303

    Google Scholar 

  55. Yang L, Sterling RL (1989) Back analysis of rock tunnel using boundary element method. J Geotech Eng 115(8):1163–1169

    Google Scholar 

  56. Ye GW, Sawada T (1989) Some numerical properties of boundary element analysis using half-plane fundamental solutions in 2-d elastostatics. J Comput Mech 4:161–164

    Google Scholar 

  57. Yi C, Zhang P, Johansson D, Nyberg U (2014) Dynamic analysis for a circular lined tunnel with an imperfectly bonded interface impacted by plane SH-waves. In: Proceeding of the Word Tunnel Congress, Brazil, pp 1–7

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Zanjan Branch, Islamic Azad University, Zanjan, Iran

    Mehdi Panji

  2. Department of Civil Engineering, University of Zanjan, Zanjan, Iran

    Bahman Ansari

Authors
  1. Mehdi Panji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bahman Ansari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mehdi Panji.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Panji, M., Ansari, B. Anti-plane seismic ground motion above twin horseshoe-shaped lined tunnels. Innov. Infrastruct. Solut. 5, 7 (2020). https://doi.org/10.1007/s41062-019-0257-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s41062-019-0257-5

Keywords

Search

Navigation