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Equivalent linear time history analyses are conducted to calculate the seismic response of various types of cut-and-cover single box tunnels. A finite-element numerical model is calibrated against the results of centrifuge tests. The calculated tunnel responses compare favourably with the measurements. A validated model is then used to quantify the seismic response of box tunnels. The flexibility ratio (

Tunnels constitute an integral part of the transportation infrastructure in urban areas. Historically, tunnels have experienced a lower level of damage compared with above-ground structures during strong earthquakes. However, recent earthquakes have demonstrated that even tunnels are vulnerable to structural damage under severe excitation [

The seismic response of rectangular tunnels has been a subject of in-depth research. A wide range of studies using centrifuge tests [

The EQL analysis is widely used in seismic analysis of tunnels [

A series of 2D dynamic analyses were performed using the finite-element (FE) analysis program ABAQUS [

2D FE model computational domain.

A four-node plane strain element with reduced integration (CPE4R) and a two-node beam element (B21) were used to simulate the soil/rock and tunnel lining, respectively. The lateral boundaries of the computational domain were tied with a kinematic constraint to simulate the simple shear condition using the “multipoint constraints” (MPCs) option in ABAQUS. This technique creates periodic boundary conditions and is used to simulate the free-field condition of the soil deposits that extends infinitely in the horizontal direction. The lower boundary was fixed representing a rigid boundary. Slip and normal behaviour of the soil-structure interface was simulated using the surface-to-node interface. An interface slip coefficient was defined as

The nonlinearity of soil under seismic excitation plays an important role in the seismic response of tunnels. The EQL analyses have widely been used for the wave propagation in soil profiles excited by seismic loading. The EQL approach has also widely been adopted in seismic simulation of tunnels (e.g., [

The EQL properties were determined via parallel one-dimensional (1D) nonlinear site response analysis performed using DEEPSOIL version 7.0 [^{st} and 5^{th} modes were used to calculate the Rayleigh damping coefficients [

The process of extracting the EQL soil properties from DEEPSOIL and application to the 2D FEM time history analysis is shown in Figure

EQL history analysis procedure.

The results of the dynamic analysis were compared with the centrifuge test performed by Gillis [_{50} = 0.14 mm, _{u} = 2.07, _{min} = 0.53, _{max} = 0.9, and _{s} = 2.66) was pluviated in the centrifuge container to achieve an initial relative density of ^{3}. The shear wave velocity was measured from the bender element at two depths (8 m and 21.3 m) in the centrifuge. Owing to the limited recording available, the shear wave velocity profile was constructed using the following power law [_{a} is the atmospheric pressure and _{o} is the curve fitting parameter (_{o} = 97348 kPa).

Centrifuge instrumentation layout indicating the accelerometers and strain gauge locations [

The properties of the tunnel structure and the estimated soil shear wave velocity for the centrifuge model are presented in Table

Dimensions and properties of the tunnel in the centrifuge test [

Properties | |
---|---|

Size (m) | 8 × 14 |

Thickness (m) | 0.57 |

Material | 6061 aluminium |

Density (kg/m^{3}) |
2700 |

Young’s modulus (kPa) | 6.89 |

Poisson’s ratio | 0.33 |

Racking stiffness (kN/m/m) | 25,000 (2D frame analysis) |

(a) Measured shear wave velocity profile [

Characteristics of input motion [

Earthquake name | Year | Station name | PGA (g) |
_{a} (m/s) |
_{5–95} (s) |
_{P} (s) |
---|---|---|---|---|---|---|

Loma Prieta | 1989 | Santa Cruz | 0.1 | 0.1 | 11.3 | 0.6 |

The numerical model to simulate the centrifuge test is shown in Figure _{0}) was set to 0.46, and the plasticity index (PI) was assumed as zero (0). The number of cycles of loading (_{max} and damping ratio profiles calculated from the 1D site response analysis are shown in Figure

Calibrated nonlinear soil properties, (a) shear stress versus strain, (b) modulus reduction versus strains, and (c) damping versus strain curves [

The results calculated from 1D nonlinear site response analysis: (a) maximum shear strain profile, (b) _{max} profile, and (c) damping ratio profile.

The computed responses from 2D FE analysis are shown in Figures

Comparison of the measured and calculated free-field responses. (a) PGA profile. (b) Acceleration response spectra at various depths.

Comparison of the measured and calculated tunnel responses. (a) PGA profile. (b) Acceleration response spectra at top slab (4 m), midwall (8 m), and bottom floor (12 m) of the tunnel.

Comparison of measured and calculated dynamic bending moments at maximum thrust. (a) North-side tunnel wall. (b) South-side tunnel wall. (c) Roof slab. (d) Floor slab.

Evidence of the influence of

The measured and calculated peak accelerations at roof slab, midwall, and floor slab are compared in Figure

The free-field and tunnel peak accelerations are compared in Figure _{m} is average strain-dependent shear modulus of the free-field ground along the height of the tunnel,

Because

The calibrated numerical model was used to investigate the effect of

Study matrix representing the properties of tunnel structure used in this study.

Structure | Thickness (m) | Young’s modulus (kPa) | Racking stiffness (kN/m/m) | Remarks |
---|---|---|---|---|

Stiff | 1.5 | 3.93 |
233,395 | |

Baseline | 0.8 | 2.5 |
25,842 | Centrifuge |

Flexible | 0.4 | 2.35 |
3,263 |

Characteristics of earthquake ground motion used in the study.

Earthquake name | Year | Station name | PGA (g) |
_{a} (m/s) |
_{5–95} (s) |
_{P} (s) |
---|---|---|---|---|---|---|

Nahanni | 1985 | Site 3 | 0.17 | 0.2 | 6 | 0.06 |

Chi-Chi | 1999 | CHY-047 | 0.18 | 1.1 | 34.9 | 0.54 |

Additional earthquake ground motions. (a) Acceleration-time histories. (b) Acceleration response spectra.

The raking deformation of the tunnel (

Comparison of racking and flexibility ratio (

Figure

Surface RRS (a) comparison for various

The dynamic thrust, which is calculated by integrating the seismically induced normal pressure along the tunnel wall at each time step, is an important parameter in the seismic design of underground structures. In this the study, the maximum dynamic thrust is extracted and plotted against the free-field surface PGA in Figure

Comparison of computed dynamic increment of thrust with available analytical solution.

The calculated thrusts were also compared with the simplified analytical solutions for yielding walls, which were obtained via the Mononobe–Okabe method (Okabe [

Figure

Comparison of free-field acceleration at tunnel middepth, shear strain, and dynamic increment of thrust in stiff soil-tunnel configuration for (a) Loma Prieta, (b) Nahanni, and (c) Chi-Chi motions.

Ratio of extracted maximum responses of tunnel at time step of maximum thrust and maximum shear strain.

Figure

Surface settlement envelope for (a) Loma Prieta, (b) Nahanni, and (c) Chi-Chi motions.

(a) Undeformed shape. Deformed shapes (scale factor = 50 magnification) plotted at time step of maximum thrust for (b) Loma Prieta, (c) Nahanni, and (d) Chi-Chi motions.

The peak shear stress around the tunnel lining is shown in Figure

Shear stress distribution (a) at soil elements adjacent to tunnel at time of maximum thrust for (b) Loma Prieta, (c) Nahanni, and (d) Chi-Chi motions.

A series of EQL dynamic analyses were performed to evaluate the seismic response of cut-and-cover rectangular tunnels. The numerical model was validated against the measurements from the centrifuge experiment. The comparisons showed that both the measured free-field and tunnel acceleration response spectra fit favourably with numerical simulation. A series of numerical analyses were performed to investigate the influence of

_{m}) computed from 1D site response analysis is used to calculate

None of the simplified earth pressure solutions for yielding walls or embedded box structure was shown to provide reliable estimate of the dynamic increment of thrust regardless of peak acceleration level and

Increase in

Flexibility ratio

Racking ratio

Equivalent linear

National Cooperative Highway Research Program

Two dimensional

Four-node plane strain element with reduced integration

Two-node beam element

Multipoint constraints

Friction angle of soil

One dimensional

_{0}:

Horizontal at-rest earth pressure factor

Plasticity index

Excitation frequency

Hertz

Modulus reduction and damping curve fitting procedure

_{a}:

Arias intensity

_{5–95}:

Significant duration

_{P}:

Predominant period

_{max}:

Normalized shear modulus

_{50}:

Particle diameter at 50% in the cumulative distribution

_{u}:

Uniformity coefficient

_{min}:

Minimum void ratio

_{max}:

Maximum void ratio

_{s}:

Specific gravity of soil

_{r}:

Relative density of soil

Reference shear modulus of soil

_{max}:

Small strain shear modulus of soil

Atmospheric pressure

Mean confining stress

Soil shear wave velocity

Numerical mesh size

Wavelength of input ground motion

Tunnel racking displacement

Free-field racking displacement

Peak ground acceleration

Ratio of response spectra

_{m}:

Strain-dependent shear modulus

Racking stiffness of the tunnel

Height of the tunnel

Width of the tunnel

Maximum free-field shear strain.

The measured data of Gillis [

The authors declare no conflicts of interest.

This research was supported by a grant (18SCIP-B146946-01) from the Construction Technology Research Program funded by Ministry of Land, Infrastructure and Transport of Korean Government.