周期分布桩体对平面SH波隔振效应的解析求解

将排桩对平面SH波的隔振简化为弹性波散射的二维平面问题，基于全空间中无限周期结构的周期特性，给出了一种求解无限周期分布桩体对平面SH波隔振效应的解析方法。该方法采用波函数展开法并结合Graf加法定理，利用全空间中相邻周期单元的散射波场在频域内相差一个相位的特性，仅选取一个周期单元，将入射波场和所有散射波场的贡献叠加后，根据边界条件求解待定系数，从而求得整个散射波场。该解析解能够精确求解无限周期分布桩体的散射问题，分析周期分布桩体数量较多时的隔振规律，弥补了以往理论分析中桩体个数较多时难以求解的不足。重点讨论了桩体个数、桩体刚度、桩体间距和桩体类型等因素对隔振效果的影响，结果表明：（1）该方法显著降低了求解大量桩体问题时的存储量和计算量，有限周期模型计算结果随桩体个数增多收敛于无限周期模型，反映了该方法的正确性；（2）整体上桩体刚度增大有利于提高隔振效果，但桩体刚度对隔振效果的提升有限，桩体剪切波速为土体5倍时已具有足够的隔振效果；（3）桩间距对隔振效果有着直接影响，间距越小则低频禁带宽度越大；（4）桩体类型对隔振效果有着显著影响，实心桩有着良好的隔振效果，而具有柔性内填充的管桩在低频段有着更佳的隔振效果。

Analytical solution for isolation effect of plane SH waves by periodically distributed piles

The vibration isolation of piles on plane SH waves is simplified to a two-dimensional (2-D) plane problem of elastic wave scattering. Based on the periodic characteristics of the infinite structure in 2-D full-space, an analytical solution for isolation effect of plane SH waves by periodically distributed piles is presented. In the solution, the wave function expansion method together with Graf addition theorem are utilized. Only one periodic element is selected by using the characteristic that the scattering wave field of adjacent periodic elements is one phase different in the frequency domain. Assembling the contribution of the incident wave field and all scattering wave fields, the whole wave fields are obtained by introducing the boundary conditions to determine the unknown coefficients. The analytical solution can accurately solve the scattering problem of infinite periodic distributed piles, and analyze the vibration isolation law when the number of periodic distributed piles is large. It makes up for the shortage that is difficult to solve when there are many piles in the previous theoretical analysis. In this paper, the influences of pile number, pile stiffness, pile spacing and pile type on the vibration isolation effect are discussed. The results show that: 1) The method significantly reduce the amount of storage and calculation in the solving process. The finite period model converges to the infinite period model with the increase of the number of piles, which reflects the correctness of the method. 2) In general, the increase of pile stiffness is beneficial to improve the isolation effect, but it can’t improve vibration isolation effect infinitely. When the shear wave velocity of pile is five times of soil, the isolation effect is enough. 3) The pile spacing has a direct impact on the isolation effect, and reducing the spacing can increase the low frequency band gap width. 4) The type of pile has a significant impact on the isolation effect, the solid pile as a whole has a good isolation effect, and the pipe pile with flexible inner filling has a better isolation effect in the low-frequency segment.