分数阶三维积分型黏弹性土体中单桩的水平动力阻抗研究
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摘要
基于土动力学、黏弹性理论和分数阶导数理论建立了分数阶三维积分型黏弹性土体水平振动控制方程。利用势函数对分数阶三维积分型黏弹性土体的水平振动方程进行解耦,借助分离变量法在频率内求解了分数阶三维积分型黏弹性土体的水平振动,得到了土体对单桩的水平作用,进而建立了分数阶三维积分型黏弹性土体中单桩的水平振动方程。考虑单桩的边界条件和桩顶水平动力阻抗的定义,求解了单桩的水平振动,并对桩顶水平动力阻抗进行了数值分析和讨论。研究表明:高频时,分数导数的阶数对水平动力阻抗有影响,土体黏性系数较小时水平动力阻抗实部和虚部随频率的变化曲线存在波动现象;长径比越大,水平动力阻抗越小,长径比达到一定程度时其对水平动力阻抗几乎没有影响。
The lateral dynamic equations of viscoelastic soil described by fractional derivative three-dimensional integral constitutive relationship has been established based on soil dynamics,viscoelastic theory and theory of fractional derivative. The horizontal dynamic equations of the viscoelastic soil are decoupled by using potential functions,the lateral vibration of the viscoelastic soil is solved and obtained the horizontal force of the soil acting on the pile with the method of separation of variables,and the horizontal dynamic equation of the single pile in the viscoelastic soil is described by fractional derivative three-dimensional integral constitutive relationship. The lateral vibration of the single is solved by considering the boundary conditions and the definition of dynamic impedance of the pile at pile head,and the numerical analysis and discussion of the dynamic impedance of the pile is also investigated. The research indicates that the order of fractional derivative has effect on the dynamic impedance of the pile at pile head for higher frequency, and the curves of the real part and image part of lateral dynamic impedance varying with frequency have fluctuations when the soil viscosity coefficient is smaller;the larger the ratio of length to diameter,the smaller lateral dynamic impedance will be,and the ratio of length to diameter has almost no effect on the lateral dynamic impedance when it increases to a certain extent.
The lateral dynamic equations of viscoelastic soil described by fractional derivative three-dimensional integral constitutive relationship has been established based on soil dynamics,viscoelastic theory and theory of fractional derivative. The horizontal dynamic equations of the viscoelastic soil are decoupled by using potential functions,the lateral vibration of the viscoelastic soil is solved and obtained the horizontal force of the soil acting on the pile with the method of separation of variables,and the horizontal dynamic equation of the single pile in the viscoelastic soil is described by fractional derivative three-dimensional integral constitutive relationship. The lateral vibration of the single is solved by considering the boundary conditions and the definition of dynamic impedance of the pile at pile head,and the numerical analysis and discussion of the dynamic impedance of the pile is also investigated. The research indicates that the order of fractional derivative has effect on the dynamic impedance of the pile at pile head for higher frequency, and the curves of the real part and image part of lateral dynamic impedance varying with frequency have fluctuations when the soil viscosity coefficient is smaller;the larger the ratio of length to diameter,the smaller lateral dynamic impedance will be,and the ratio of length to diameter has almost no effect on the lateral dynamic impedance when it increases to a certain extent.
引文
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[2]Yao S,Nogami T.Lateral cyclic response of a pile in viscoelastic Winkler subgrade[J].Journal of Engineering Mechanics,1994,120(4):758-775.
[3]胡昌斌,黄晓明.成层粘弹性土中桩土耦合纵向振动时域响应研究[J].地震工程与工程振动,2006,26(4):205-211.
[4]周绪红,蒋建国,皱银生.粘弹性介质中考虑轴力作用时桩的动力分析[J].土木工程学报,2005,38(2):87-91.
[5]杨骁,潘元.饱和粘弹性土层中端承桩纵向振动的轴对称解析解[J].应用数学和力学,2010,31(2):193-204.
[6]李军强,刘宏昭,王忠民.线性粘弹性本构方程及其动力学应用研究综述[J].振动与冲击,2005,24(2):116-121.
[7]杨挺青,罗文波,徐平,等.黏弹性理论与应用[M].北京:科学出版社,2004.
[8]刘林超,姚庆钊,闫启方,等.基于积分型分数导数本构方程的黏弹性土层中单桩的竖向复刚度与导纳研究[J].水利学报,2012,43(7):796-802.
[9]刘林超,杨骁.竖向集中力作用下分数导数型半无限体粘弹性地基变形分析[J].工程力学,2009,26(1):13-17.
[10]Miller K S,Ross B.An introduction to the fractional calculus and fractional differential equations[M].New York,Wiley,1993.
[11]Igor Podlubny.Fractional Differential Equations:An Introduction to Fractional Derivative,Fractional Differential Equations,to Methods of their Solution and some of their Applications[M].Academic Press,San Diego,1999.
[12]黄茂松,吴志明,任青.层状地基中群桩的水平振动特性[J].岩土工程学报,2007,29(1):32-38.
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