Abstract: A model for analyzing the magnification effect of the action of seismic wave at the head of a single pile in soil foundation was established. The soil foundation on bedrock was regarded as a viscoelastic medium with the constitutive relation of fractional derivative viscoelasticity (FDV). The horizontal displacements of the FDV soil foundation under the action of SH harmonic seismic wave were obtained by employing the orthogonality of trigonometric functions and the boundary conditions of the soil foundation. Considering a single pile of circular sections with its end supported on the bedrock, the interaction between the pile and the soil foundations was represented by a series of equivalent dynamic Winkler spring-damper elements. On the basis of Novak plane assumptions, the radial and circumferential displacements of the FDV soil were determined, and then the coefficients of stiffness and damping of the Winkler spring-damper elements were obtained. Furthermore, the lateral vibration of the single pile in FDV soil under SH harmonic seismic wave was solved, and the magnification factor at the head of the single pile in the FDV soil was obtained. The results indicate that the order of fractional derivative and model parameters of the soil foundation has a great effect on the resonance of single pile in FDV soil under the action of the SH harmonic seismic wave, while the density ratio and modulus ratio of pile to soil have very small influence on the magnification factor at lower frequencies, and have some effect at higher frequencies.

Key words: vibration and wave, seismic wave, amplification factor, viscoelasticy, interaction, spring-damper model

摘要： 将基岩上的土体视为黏弹性介质，且土体的本构关系利用分数导数黏弹性(FDV)模型来描述。借助三角函数的正交性和土体的边界条件求解SH简谐地震波作用下FDV场地土的水平振动位移。以土体中的圆形截面端承桩为研究对象，将桩土之间的相互作用等效为动力Winkler弹簧－阻尼器，在Novak平面假定的基础上求得由于桩土相互作用引起的FDV土层的径向和环向位移，进而得到了等效动力Winkler弹簧－阻尼器的刚度系数和阻尼系数，在此基础上求解SH简谐地震波作用下FDV土层中单桩的水平振动，得到了FDV土层中单桩桩顶的放大因子。研究表明：分数导数的阶数和土体模型参数对SH简谐地震波作用下FDV土中单桩的共振有较大的影响；在低频时桩土的密度比和模量比对FDV土中单桩桩顶放大因子影响很小，在高频时有一定的影响。

关键词: 振动与波, 地震波, 放大因子, 黏弹性, 相互作用, 弹簧-阻尼器模型