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Analytical approach to scattering of P, SV waves by twin tunnels in saturated half-space

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Abstract

During the recent earthquake, a lot of damage was reported from the historical monuments in the adjacent twin tunnels. While scientific research shows, the vulnerability of other monuments is much less than the monuments adjacent twin tunnels. The first type of Hankel function was used to study scattering waves in time domain analysis. The time domain results were transferred to the frequency domain using Fourier transform criteria. Extra stress concentrations were calculated based on the results of frequency domain analysis and developed a new approach to scattering. Arg-Karim Khani citadel and twin subway tunnels in Shiraz were selected as a case study. Three near-field earthquakes to perform the time history analysis were considered. The statistical difference based on Mann–Whitney test illustrates the seismic behavior of monument with and without considering twin tunnels, which shows that there is significant difference between them. The mean of P value amount of this comparison in horizontal and vertical direction is 0.005 and 0.016, respectively (P value < 0.05). The integrated analysis shows the importance of frequencies in creating different scattering around twin tunnels, which depends on the distance tunnels, the diameter of the tunnels, the soil characteristics, the seismic wave types, etc.

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Abbreviations

\(a\left( {t_{k} } \right)\) :

Acceleration of time history analysis

\({A}_{1}\) and \({A}_{2}\) :

The boundary conditions

\(A_{n}\), \(B_{n}\) and \(C_{n}\) :

The boundary coefficients

b:

The viscous coupling factor

C:

Soil adhesion

\(C_{s}\) :

Shear wave velocity

\(C_{L}\) :

Coefficient of fast shear wave

\(C_{T}\) :

Coefficient of shear wave velocity

\({\text{DSCF}}\left( {\sigma_{\theta }^{*} } \right)\) :

Dynamic stress concentration factor

E:

Young's modulus

\(F\left( {\omega_{n} } \right)\) :

Discrete Fourier transform

FEM:

Finite element method

FESCAM:

Finite element scattering method

\(H_{n}\) :

The first kind of cylindrical Hankel function

\(J_{n}\) :

The first kind of Bessel function for cylindrical domain

\(k_{l1}\) :

The ratio of length of velocity wave to width

\(k_{f}\) :

The selected wave numbers at fast elastic modes

\(k_{s}\) :

The selected wave numbers at slow elastic modes

\(k_{t}\) :

The selected wave numbers at shear elastic modes

\(K_{0}\) :

Bulk modulus of the dry skeleton

\(k_{f,s}\) :

The selected wave numbers at fast and slow modes

\({\text{K}}_{F1}\) :

Bulk modulus of saturated fluid

\(K_{s} M\) :

Bulk modulus of elastic materials the number of observations

MDOF:

Multi degree of freedom

MAE:

Mean absolute error

\(N\) :

Degree of discrete Fourier transform

NTHA:

Nonlinear time history analysis

\(O_{i}\) :

The observed values

\(\overline{O}\) :

The average of the observed values

\(P_{i}\) :

The predicted values

\(\overline{P}\) :

The average of the predicted values

PGA:

Peak ground acceleration

\(P_{p}\) :

Fluid pore pressure

R RMSE:

Distance from coupling point to tunnel center

\(R_{1}\) :

Root-mean-square error

\(R^{2}\) :

Tunnel radius

\(t_{k}\) :

Correlation coefficient time history

u:

The displacement of saturated porous media

U:

The displacement of solid media

\({\dot{\text{w}}}_{{\text{r}}}\) :

The rate of filtration

\(\alpha\) :

Angle of incident wave

\(\beta\) :

Angle of reflected SV wave

\({\upgamma }_{{{\text{sat}}}}\) :

Saturation density

\(\varepsilon_{0}\) :

Porosity

\(\theta\) :

Wave angle

\(\theta_{cr}\) :

Critical propagation angle

\(\lambda\) :

Lama’s constant

\(\mu\) :

Shear modulus

ν:

Poisson's ratio

\(\rho\) :

The total mass density of fluid saturated material

\(\rho_{1} { },{ }\) \(\rho_{2} { }\) and \(\rho_{3}\) :

Effective densities with frequency-independent conditions

\(\rho_{e}\) :

Coefficient of average density

\(\sigma_{rr}\) :

Normal stress

\(\sigma_{r\theta }\) :

Tangential stress

\(\sigma_{\theta \theta }\) :

Cyclic stress

\(\emptyset_{{}}\) :

Internal friction angle

\(\emptyset_{0}\) :

The amplitude of P waves

\(\emptyset_{f,s}\) :

The potential function of fast and slow waves

\(\emptyset^{i}\) :

The incident wave of P waves \(\emptyset^{r}\): The reflected wave potential of P waves

\(\psi_{{}}\) :

Dilation angle

\(\psi_{0}\) :

The amplitude of SV waves

\(\psi^{r}\) :

The reflected wave potential of SV waves

\(\omega\) :

Frequency of simulation

\(\omega_{n}\) :

Frequency of Fourier

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Authors and Affiliations

  1. Department of Civil Engineering, Faculty of Civil and Earth Resources Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran

    Amir Rabiefar & Hassan Negahdar

  2. Department of Civil Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran

    Hamidreza Vosoughifar

  3. Department of Civil Engineering, Shahid Rajae Teacher Training University, Lavizan, Tehran, Iran

    Ali Nabizadeh

  4. Department of Civil and Environmental Engineering, University of Hawaii at Monoa, Honolulu, USA

    Hamidreza Vosoughifar

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  1. Amir Rabiefar
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Correspondence to Hamidreza Vosoughifar.

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Rabiefar, A., Vosoughifar, H., Nabizadeh, A. et al. Analytical approach to scattering of P, SV waves by twin tunnels in saturated half-space. Arch Appl Mech 93, 445–466 (2023). https://doi.org/10.1007/s00419-022-02249-4

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