分数导数微分算子描述的粘弹性土层中群桩的水平振动研究
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摘要
将桩周土体视为粘弹性介质,利用分数导数粘弹性模型描述土体的力学特性,在Novak平面应变假定的基础上,借助于势函数并考虑土体边界条件求得了分数导数微分算子描述的粘弹性土层水平位移的衰减函数以及分数导数粘弹性土层的刚度和阻尼系数。利用Winkler动力弹簧-阻尼器模型模拟桩-土之间的动力相互作用,并在此基础上利用初始参数法求解了分数导数粘弹性土层中桩-桩水平动力相互作用和群桩的水平振动问题。以数值算例的形式讨论了分数导数微分算子的阶数和土体的模型参数对分数导数微分算子描述的粘弹性土层水平位移的衰减函数和群桩的水平动力阻抗的影响。研究表明:分数导数微分算子的阶数对土层水平位移的衰减函数的影响与桩间距和荷载方向角有关;分数导数粘弹性土中群桩的动力阻抗可以退化到经典粘弹性和弹性情况;分数导数微分算子的阶数和土体模型参数对群桩水平动力阻抗有较大影响。
The soil around piles is regarded as viscoelastic medium and the mechanical properties are described by fractional viscoelastic model.Based on the Novak plane strain hypothesis,the attenuation function of horizontal displacement and the stiffness and damping coefficient of the soil layer described by fractional viscoelastic model were obtained by potential functions and boundary conditions.The pile-soil dynamic interaction was modeled by Winkler dynamic elastic-damping model,the pile to pile dynamic interaction and horizontal vibration of the pile groups in the soil described by fractional derivative viscoelastic model were solved by initial parameter method.The influence of the order of fractional derivative and model parameter of soil on the pile to pile dynamic interaction and horizontal dynamic impedance of pile groups was investigated through numerical examples.The result indicates that the influence of the order of fractional derivative on the attenuation function of horizontal displacement is related to the pile space and load direction angle;in studying the horizontal dynamic impedance of pile groups,the soil model can be simplified to the cases of classic viscoelastic and elastic;the order of fractional derivative and the model parameter of soil have great effect on the lateral dynamic impedance of the pile groups.
The soil around piles is regarded as viscoelastic medium and the mechanical properties are described by fractional viscoelastic model.Based on the Novak plane strain hypothesis,the attenuation function of horizontal displacement and the stiffness and damping coefficient of the soil layer described by fractional viscoelastic model were obtained by potential functions and boundary conditions.The pile-soil dynamic interaction was modeled by Winkler dynamic elastic-damping model,the pile to pile dynamic interaction and horizontal vibration of the pile groups in the soil described by fractional derivative viscoelastic model were solved by initial parameter method.The influence of the order of fractional derivative and model parameter of soil on the pile to pile dynamic interaction and horizontal dynamic impedance of pile groups was investigated through numerical examples.The result indicates that the influence of the order of fractional derivative on the attenuation function of horizontal displacement is related to the pile space and load direction angle;in studying the horizontal dynamic impedance of pile groups,the soil model can be simplified to the cases of classic viscoelastic and elastic;the order of fractional derivative and the model parameter of soil have great effect on the lateral dynamic impedance of the pile groups.
引文
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[10]Zhu Zhengyou,Li Genguo,Chen Changjun.Quasi-staticand dynamical analysis for viscoelastic Timoshenkobeam with fractional derivative constitutive relation[J].Applied Mathematics and Mechanics,2002,23(1):1―7.
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[13]刘林超,杨骁.基于严格平面应变模型的土中群桩水平振动研究[J].工程力学,2010,27(7):168―174.Liu Linchao,Yang Xiao.Lateral vibration of pile groupsin soil based on plane strain model[J].EngineeringMechanics,2010,27(7):168―174.(in Chinese)
[14]李耀庄,李博.单桩水平及水平-摇摆耦合振动分析[J].世界地震工程,2006,22(4):160―165.Li Yaozhuang,Li Bo.Analysis on horizontal vibrationsand horizontal-rocking vibrations of pile[J].WorldEarthquake Engineering,2006,22(4):160―165.(inChinese)
[15]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.Huang Maosong,Wu Zhiming,Ren Qing.Lateralvibration of pile groups in layered soil[J].ChineseJournal of Geotechnical Engineering,2007,29(1):32―38.(in Chinese)
[2]Nogami T,Novak M.Soil-pile interaction in verticalvibration[J].Earthquake Engineering&StructuralDynamics,1976,4(3):277―293.
[3]Novak M,Nogami T.Soil-pile interaction in horizontalvibration[J].Earthquake Engineering&StructuralDynamics,1977,5(3):263―281.
[4]Veletsos A S,Dotson K W.Vertical and torsionalvibration of foundations in inhomogeneous media[J].Journal of Geotechnical Engineering,1988,114(9):1002―1021.
[5]Veletsos A S,Dotson K W.Impedances of soil layerwith disturbed boundary zone[J].Journal ofGeotechnical Engineering,1986,112(3):363―368.
[6]Chau K T,Yang X.Nonlinear interaction of soil-pile inhorizontal vibration[J].Journal of EngineeringMechanics,2005,131(8):847―858.
[7]程昌钧,胡育佳.受轴力作用的桩基在水平振动时的抗力[J].上海大学学报(自然科学版),2005,11(3):275―281.Chen Changjun,Hu Yujia.Resistance of pile againsthorizontal vibration under axial force[J].Journal ofShanghai University(Natural Science),2005,11(3):275―281.(in Chinese)
[8]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,2006,26(4):205―211.Hu Changbin,Huang Xiaoming.A quasi-analyticalsoluteon to soil-pile interaction in longitudinal vibrationin layered soils considering vertical wave effect on soils[J].Earthquake Engineering and Engineering Vibration,2006,26(4):205―211.(in Chinese)
[9]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.Zhou Xuhong,Jiang Jianguo,Zhou Yinsheng.Dynamicanalysis of piles under axial loading and lateral dynamicforce in visco-elastic medium[J].China CivilEngineering Journal,2005,38(2):87―91.(in Chinese)
[10]Zhu Zhengyou,Li Genguo,Chen Changjun.Quasi-staticand dynamical analysis for viscoelastic Timoshenkobeam with fractional derivative constitutive relation[J].Applied Mathematics and Mechanics,2002,23(1):1―7.
[11]Miller K S,Ross B.An introduction to the fractionalcalculus and fractional differential equations[M].NewYork:Wiley,1993.
[12]孙海忠,张卫.一种分析软土粘弹性的分数导数开尔文模型[J].岩土力学,2007,28(9):1983―1986.Sun Haizhong,Zhang Wei.Analysis of soft soil withviscoelastic fractional derivative Kelvin model[J].Rockand Soil Mechanics,2007,28(9):1983―1986.(inChinese)
[13]刘林超,杨骁.基于严格平面应变模型的土中群桩水平振动研究[J].工程力学,2010,27(7):168―174.Liu Linchao,Yang Xiao.Lateral vibration of pile groupsin soil based on plane strain model[J].EngineeringMechanics,2010,27(7):168―174.(in Chinese)
[14]李耀庄,李博.单桩水平及水平-摇摆耦合振动分析[J].世界地震工程,2006,22(4):160―165.Li Yaozhuang,Li Bo.Analysis on horizontal vibrationsand horizontal-rocking vibrations of pile[J].WorldEarthquake Engineering,2006,22(4):160―165.(inChinese)
[15]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.Huang Maosong,Wu Zhiming,Ren Qing.Lateralvibration of pile groups in layered soil[J].ChineseJournal of Geotechnical Engineering,2007,29(1):32―38.(in Chinese)
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