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Transmitting boundaries are important for modeling the wave propagation in the finite element analysis of dynamic foundation problems. In this study, viscoelastic boundaries for multiple seismic waves or excitations sources were derived for two-dimensional and three-dimensional conditions in the time domain, which were proved to be solid by finite element models. Then, the method for equivalent forces’ input of seismic waves was also described when the proposed artificial boundaries were applied. Comparisons between numerical calculations and analytical results validate this seismic excitation input method. The seismic response of subway station under different seismic loads input methods indicates that asymmetric input seismic loads would cause different deformations from the symmetric input seismic loads, and whether it would increase or decrease the seismic response depends on the parameters of the specific structure and surrounding soil.

The 1995 Kobe earthquake caused a major collapse of the Daikai subway station in Kobe, which represents the first modern underground structure failure during a seismic event. The seismic design and analysis methods of underground structures have been rapidly developed since then [

To avoid the undesirable reflection of waves at the artificial boundaries of the finite element model, absorbing boundaries or transmitting boundaries should be applied. Problems involving inelastic behavior of the soil in the near field are most readily solved in the time domain. Wolf [

In this study, a viscoelastic boundary for multiple waves or excitation sources is presented and validated. The main work of this study is as follows. First, a two-dimensional viscoelastic boundary for multiple waves or excitation sources is derived based on an approximation of the form of the outward traveling waves. The proposed visc-elastic boundary is validated by a finite element model. Next, the proposed two-dimensional viscoelastic boundary is promoted for three-dimensional condition. Then, input equivalent forces of seismic waves are described when the viscoelastic boundary for multiple waves or excitation sources is used. Finally, the seismic response of a subway station using different seismic loads input methods is discussed.

In this section, a two-dimensional viscoelastic boundary for the multiple input waves or excitation sources would be developed based on an approximation of the form of the outward traveling waves.

According to the elastic wave propagation theory, the two-dimensional shear waves in the elastic media follow

According to Whitham [

Assuming two independent outgoing waves

The velocity of point X is as follows:

The corresponding shear strain and shear stress are expressed as in

Assume that the shear stress at point X is related to the velocity and displacement at point X as

Combining (

An exact solution of (

Equation (_{1} =_{2}. This is unavailable for most conditions. As in (

Assuming_{1} is shorter than_{2}, if shear stiffness parameters are mainly affected by the source at distance of_{1}, the shear stiffness parameter_{1} follows

If shear stiffness parameters are mainly affected by the source at distance of_{2}, the shear stiffness parameter_{2} is given by

If shear stiffness parameters are affected by the both sources, the shear stiffness parameter_{3} is expressed as

To employ the proposed viscoelastic boundary equations in the finite element analysis model, discrete springs and dashpots should be implemented on the corresponding boundary nodes, which would make the shear stresses and displacements approximate to the real ones. Since (

As in Section

As in the viscous boundary, the damping parameter

A homogeneous elastic soli model is constructed to validate the viscoelastic boundary conditions. Shown in Figure ^{3}, and Poisson’s ratio is 0.25. A triangular impulsive excitation is imposed at sites B and C on the top of the model. The input triangular impulsive excitation is displayed in Figure

The 2D model of rectangular soil rectangular: a represents M1 and b represents M2.

Input triangular loads history at points B and C.

Figure

a and b represent the horizontal displacements at site A for different model size and artificial boundary conditions, while c and d represent the vertical displacements.

In Figures

In Figures

From the above calculated results and analysis and considering that the distances between the excitation sources and boundaries could be distinctly different, it is suggested to estimate the shear stiffness parameter

Similar to the proposed 2D viscoelastic boundary equations in Section

The damping parameters

Locations of input seismic loads and observation site on the free surface of the 3D model.

a represents the displacements of O in the x direction for different model size and artificial boundary conditions, b represents the displacements in the y direction, and c represents displacements of O in the z direction.

The calculated results in three-dimensional model are similar to those in two-dimensional model. Using the results of M4FB as a reference, responses of M3EV3 are more accurate than those of M3VB. The proposed viscoelastic boundary could allow the model to recover from elastic deformation and absorb the transmitting waves. Thus, the proposed viscoelastic boundary conditions could be employed for the dynamic calculation when multiple excitation sources are imposed.

The appropriate method for input of seismic waves is important in soil-structure interaction dynamic analysis. For different artificial boundary conditions, different methods for seismic waves input should be adopted. Joyner and Chen [

To achieve accurate input of seismic waves, the input equivalent loads should make the displacements and stress conditions of artificial boundary same as the real ones. Assuming

To employ the proposed viscoelastic boundary conditions, discrete springs and dashpots are implemented.

Internal stress

The input loads on boundary could be estimated from (

For the two-dimensional viscoelastic boundary under multiple excitation sources, (

For the three-dimensional viscoelastic boundary under multiple excitation sources, (

Parameters of the subway station and surrounding soil.

Boundary | Left | Right | Front | Back | Bottom | |
---|---|---|---|---|---|---|

Input in the x direction | Stress term | z+ | z- | x+ | x+ | x+ |

Velocity and displacement term | x+ | x+ | - | - | x+ | |

Input in the y direction | Stress term | z+ | z- | - | - | y+ |

Velocity and displacement term | y+ | y+ | - | - | y+ | |

Input in the z direction | Stress term | x- | x+ | y- | y+ | z- |

Velocity and displacement term | z- | z- | z- | z- | z- |

The letter “x+” in the table means that the corresponding term is in the positive x direction and other letters follow the same rule. The minus signs “-” means that there are no corresponding terms for these boundaries. Stress, velocity, and displacement terms are the third, second, and first term in (

To validate the proposed method for input equivalent forces of seismic waves, a homogeneous elastic model of 20 m × 20 m × 20 m is constructed. Except for the free surface, the other five artificial faces are modeled as proposed viscoelastic boundaries, shown in Figure ^{3} kg/m^{3}, Young’s modulus is 3 MPa, shear wave velocity is 25 m/s, and Poisson’s ratio is 0.25.

Three-dimensional model of the soil and viscoelastic artificial boundary condition.

Figure

Input sinusoidal displacement waveform.

Figure

The displacement waveforms of observation point O under different conditions.

Comparisons between numerical and analytical results validate the proposed viscoelastic boundary under multiple excitation sources and corresponding equivalent seismic loads input method. If equivalent forces are only imposed on the bottom of the model, it would underestimate the numerical results.

For dynamic analysis of underground structures, appropriate equivalent seismic loads input methods should be used for different artificial boundary conditions. In Sections 2–4, the accuracy of proposed viscoelastic boundary and corresponding equivalent seismic loads input method have been validated. In real seismic events, the ground motions would vary at different sites when the structure has a large size. The difference of ground motions could induce the difference of stress condition and motion on different parts of the structure, especially for the structure with a large size. In the following section, the seismic response of a subway station subjected to different seismic waves will be analyzed. A group of different seismic waves are used as incident loads.

A 2-storey and 4-span subway station is selected as the target structure, with a cross section displayed in Figure

The cross section of the subway station.

Both subway station and surrounding soil are assumed to be elastic in the following analysis. Velocities of shear wave and primary wave speed in the surrounding soil are 200 m/s and 350 m/s, respectively. The detailed parameters of subway station and surrounding soil are summarized in Table

Parameters of the subway station and surrounding soil.

Density ^{3}) | Young’s modulus (MPa) | Poisson’s ratio | |
---|---|---|---|

Station | 2.5 × 10^{3} | 2.8 × 10^{4} | 0.20 |

Soil | 2.0 × 10^{3} | 3.0 × 10^{2} | 0.25 |

The EW components of MXN wave in Wenchuan earthquake are selected as the input seismic wave in lateral direction, the peak acceleration of which is amplified to 5 m/s^{2}. Seismic response analyses of the subway station under three different seismic loads input conditions are conducted. Table ^{2}, as displayed in Figure

The imposed waves on the boundaries for the three cases.

Case | Bottom | Left | Right | Front | Back |
---|---|---|---|---|---|

1 | MXN | - | - | - | - |

2 | MXN | MXN | MXN | MXN | MXN |

3 | MXN | MXN | SFB | LDD | JYC |

The selected acceleration waveforms of Wenchuan earthquake.

JYC wave

LDD wave

MXN wave

SFB wave

Story drift displacement histories of the floor 1 are compared among different cases, as shown in Figure

Story drift displacements of left floor 1 columns in the subway station under different conditions.

case 1 and case 2

case 2 and case 3

Figure

Maximum story drift angels of columns in the subway station for different cases.

Variable equivalent seismic forces on vertical boundaries would not always induce a larger story drift displacement than that induced by uniform equivalent seismic forces on vertical boundaries. The seismic analysis in case 3 could only indicate that asymmetric seismic forces input would cause asymmetric deformation. Compared to the symmetric seismic forces input, whether it would increase or decrease the seismic response depends on the parameters of the specific structure and surrounding soil.

In this study, the viscoelastic boundaries for multiple seismic waves or excitations sources have been derived and validated for two-dimensional and three-dimensional conditions in the time domain. The corresponding method for equivalent seismic loads input has also been presented.

First, a two-dimensional viscoelastic boundary for multiple waves or excitation sources was derived based on an approximation of the form of the outward traveling waves. The proposed two-dimensional viscoelastic boundary was promoted to the three-dimensional condition. The proposed viscoelastic boundary was validated using a finite element model by the numerical calculation.

Then, the method for input equivalent forces of seismic waves was described when the viscoelastic boundary for multiple waves or excitation sources was used.

Finally, the seismic responses of a subway station using different seismic loads input methods were discussed. Compared with the symmetric seismic forces input, whether the asymmetric seismic forces input would increase or decrease the seismic response depends on the parameters of the specific structure and surrounding soil.

All the data used to support the findings of this study are included within the article. Requests for data are available from the corresponding author after this article is published.

The authors declare that they have no conflicts of interest.

This study was supported by the National Natural Science Foundation of China (Grant no. 51708018) and Beijing Postdoctoral Science Foundation (Grant no. 2017ZZ063).