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Numerical study on the boundary effect of rigid model boxes in shaking table tests in underground engineering

The boundary effect of rigid model boxes in shaking table tests in underground engineering directly affects the reliability of test data. The lining of flexible materials in the side walls of model boxes has become an important method to reduce the boundary effect. However, the arrangement of flexible materials is mainly based on experience, and its rationality is unclear. Therefore, the reasonable design of flexible layers is important for reducing the boundary effect of rigid model boxes. In this work, the influencing factors of the lateral boundary effect of rigid model boxes under the action of S waves were studied. A flexible layer with different parameters was set on the boundary of a model box, and the reduction degree of the flexible layer was established relative to the boundary effect. The results can provide a reference for the selection and design of flexible boundary materials for rigid model box tests.


Introduction
Given the frequent occurrence of earthquakes and the resulting damage in underground engineering structures, an increasing number of studies have focused on the failure mechanisms and seismic performance of such structures. The shaking table test is one of the most important methods in the study of the structural response of underground engineering structures. According to their structural forms, main model boxes can be divided into three categories, namely, rigid model boxes, flexible model boxes, and laminar shear model boxes [1][2][3][4][5]. Flexible and laminar shear model boxes can effectively reduce the reflection of seismic waves on model boundaries, but their application is restricted because of the limited load-bearing capacity of flexible boxes and the complex fabrication and high cost of layered shear boxes [6]. Rigid model boxes are widely used in shaking table model tests because of their simple design, strong adaptability, and low cost. However, rigid model boxes  2 have high rigidity and small deformation; hence, boundary reflection waves occur when the input ground vibration reaches the side walls of these model boxes. In addition, the horizontal deformation of rocks or soil mass is limited.
To improve the boundary conditions of rigid model boxes, scholars usually set a certain thickness for the flexible material on the side wall of the model box perpendicular to the vibration direction of the shaking table. For example, Shen et al. [7], Li et al. [8], Ding et al. [9], and others arranged expanded polystyrene (EPS) foam layers with thicknesses ranging from 100 to 200 mm at the boundaries of model boxes. Li et al. [10] used a flexible sponge with unknown thickness, Sun et al. [11] utilized organic rubber with a thickness of 4 mm), and Zhou et al. [12] combined 100 mm-thick EPS foam and 50 mm-thick rubber to improve the boundary conditions. However, the arrangement of these flexible materials is mainly based on experience, and its rationality is rarely discussed. Bathurst et al. [13] tested three different geofoam buffer materials retaining sand soil under idealized dynamic loading conditions and found that a reduction in dynamic load increases with decreasing seismic buffer density. Through numerical simulation, Lou et al. [14] concluded that adding a rubber layer to the boundaries can effectively reduce the boundary effect of model boxes and that the reasonable elastic modulus ratio of rubber to soil layer is 2.5 times. Huang et al. [15] concluded that foam thickness exerts a great influence on the boundary effect and that a foam cushion offers the best effect at a thickness of 2%-4% of the model box width. On the basis of the principle of vibration mode superposition, Zhang et al. [16]

Evaluation method of rigid model box boundary effect
At present, four methods are commonly used to evaluate rigid model boxes [15,17]: the first one is to compare the acceleration time histories of the boundary and center points of the rock or soil model directly, i.e., the closer they are, the smaller the boundary effect of the model box will be; the second is to compare the ratios of the peak acceleration of the boundary and center points of the model, i.e., the closer they are to 1, the smaller the boundary effect will be; the third is to subject the acceleration time histories of the boundary and center points to Fourier transform and compare the frequencies, i.e., the closer they are, the smaller the boundary effect of the model box will be; the fourth one is to compare the 2-norm deviations of the response signals between the boundary point and the center point, i.e., the smaller the value is, the smaller the boundary effect will be. Among these methods, the first one is the most intuitive and easiest to understand, but it lacks a quantitative analysis. The second method allows quantitative characterization, but it is not comprehensive enough to evaluate the boundary effect on the basis of the information of the peak acceleration. The third method is a quantitative analysis method in the frequency domain, but it only reflects the differences in signals' Fourier amplitude spectra, and it does not directly include the characteristics of amplitude and duration in the time domain. The fourth method is a comprehensive and quantitative method for studying the boundary effect of model boxes because it can reflect the response characteristics of particles in multiple parts of model boxes [16]. Therefore, the fourth method was used in the current work to study the influencing factors of the boundary effect of rigid model boxes and to evaluate the influence of flexible material parameters on the boundary effect.
The evaluation formula of the 2-norm deviation method is where xiM is the reference signal of the vibration response of the rock or soil mass at the center point of the model box and xkM is the contrast signal of the vibration response of the rock or soil mass at the boundary point of the model box. xiM and xkM can be a time history of acceleration; a spectral signal; or peak acceleration, velocity, displacement, strain, etc. In this work, xiM and xkM are the peak accelerations of the center and boundary points of the rock or soil mass in the model box.
To make the results increasingly accurate, this study also analyzes the vibration response characteristics of the rock or soil mass in the model box and the vibration response in the free field by using the two-norm deviation method. The evaluation formula [16] is where the xiF is the reference signal of the vibration response of the rock or soil mass at the center point in the free field. In this work, xiF is the peak acceleration of the center point of the rock or soil mass in the free field.  however, when KE is less than 5e-3, the lateral boundary effect of the rigid model box is large and cannot be ignored. When the value of KE is less than 1e-3, the value of u1 is more than 8%, the value of u2 is more than 29%, and the boundary effect is very obvious.

Discussion
In this work, the 2-norm method was used to study the influencing factors of the boundary effect of Given the limitation in the number of pages of the manuscript, the research only considered one type of seismic wave frequency (20 Hz) and one type of model box scale. Therefore, the characteristic conditions of seismic waves and the scale conditions of model boxes should be further discussed. In addition, this study mainly involved a numerical simulation, and thus, some deviations with real shaking table test results are inevitable. Readers can refer to this method to prepare experiments, but the results of this work should not be directly used in actual experiments.

Conclusions
The