上覆分数阶粘弹性饱和场地土位移地震放大系数

刘林超，闫启方，刘 滕

(信阳师范学院 土木工程学院，河南 信阳 464000)

上覆分数阶粘弹性饱和场地土位移地震放大系数

刘林超，闫启方，刘 滕

(信阳师范学院 土木工程学院，河南 信阳 464000)

考虑土体液相和固相的耦合作用，将基岩上覆场地土视为两相饱和多孔介质。为了考虑饱和场地土的粘弹性特性，其固相土骨架的应力应变关系利用分数阶Kelvin粘弹性模型来描述，建立了上覆分数阶粘弹性饱和场地土在简谐地震波作用下的运动控制方程。运用分数导数的性质并考虑上覆场地土的边界条件和透水性条件求解了上覆分数阶粘弹性场地土在简谐地震波作用下的振动问题，得到了饱和场地土的位移地震放大系数。采用数值算例分析讨论了分数导数的阶数、液固耦合系数、土体模型参数、基岩土体剪切模量比等参数对位移地震放大系数的影响。研究结果表明，分数导数的阶数、液固耦合系数、土体模型参数、基岩土体剪切模量比对饱和场地土的地震响应有较大的影响，通过压实场地土，可以达到增大液固耦合系数减小地震响应的作用，通过增大饱和场地土的粘性和剪切模量也可以减小地震反应。

饱和土；分数导数；地震波；地震放大系数；液固耦合系数

地震往往会造成建筑物和人员的伤亡，在进行建筑抗震设计时要选择有利的建筑场地，这是因为不同的场地条件和场地土的动力学特性对地震的放大效应不同，且十分敏感。因此，研究不同场地条件下场地土的振动特性和地震反应成为地震和岩土工程领域研究的一个重点和难点，且受到了该领域专家的足够重视。从20世纪60年代开始，不少学者就采用各种模型、方法对地震激励作用下场地土的动力学特性进行了研究，Idriss等[1]在一维剪切梁模型的基础上研究了剪切模量为常数和沿深度按幂函数变化的场地土的动力特性和地震反应；Wolf[2]将土层简化为竖杆，研究了剪切模量按指数函数随深度增大的匀质自由场地土层的竖向振动；Zhao[3-4]对剪切模量沿深度按幂函数分布的场地土进行了地面振动分析和横向振动研究；高玉峰[5]在时间域内给出了基岩任意输入地震作用下一维土层地震反应的解析解；栾茂田等[6]基于一维剪切梁模型，运用分离变量方法研究了各层土的剪切模量沿深度按幂函数规律变化的水平成层非均质地基的自振特性和地震动力反应。尚守平等[7]基于一维波动模型，假设土层剪切模量沿深度按指数规律增长，研究了基岩上作用竖直向上传播的稳态剪切地震波激振的场地土的横向自由振动。Chen等[8]在冻融情况下研究了土层的地震反应。Nimtaj等[9]利用时域-频域混合方法研究了土层的非线性地震反应。这些研究对研究场地土的动态特性起到了重要作用。然而，由于实际工程中土体是由固液气组成的三相多孔介质，且具有粘弹性性质，将其视为单相弹性介质势必与工程实际不符。考虑土体的粘弹性特性，李刚等[10]采用工程波动理论，考虑简谐SH波诱发的水平振动和场地土的材料阻尼，假定场地土体系处于反平面应变状态，研究半空间上SH波激励下上覆粘弹性场地土的自由场动力反应；刘林超等[11]借助于一维波动模型和分数导数粘弹性本构关系，分析了在竖直向上传播的剪切地震波激励作用下，基岩上分数导数粘弹性场地土的横向振动问题；段玮玮等[12]基于Biot饱和土理论和分数阶导数理论研究了饱和分数导数型粘弹性土层的竖向振动放大效应。为了考虑液相的影响，笔者基于饱和土的多孔介质理论，将基岩上覆土体视为两相饱和多孔介质，利用分数导数粘弹性模型描述固相土骨架的应力应变关系，研究简谐地震波作用下上覆分数阶粘弹性饱和场地土的地震反应与振动特性。

1 模型与控制方程

(1)

图1 基岩上覆分数导数粘弹性饱和场地土Fig.1 Fractional derivative viscoelastic saturated site soil on a bedrock

(2)

(3)

(4)

为了更加合理的考虑上覆饱和场地土的粘弹性特性，采用分数阶Kelvin粘弹性本构模型来描述固相土骨架的应力-位移关系，即[14]

(5)

将式(5)代入式(1)，可得简谐地震波作用下上覆分数阶粘弹性饱和场地土的运动控制方程为

(6)

(7)

(8)

2 简谐地震波作用下上覆分数阶粘弹性饱和场地土振动求解

(9)

(10)

(11)

(12)

(13)

(14)

将式(13)代入式(12)整理得

(15)

(16)

求解式(16)有

(17)

由式(13)、(16)、(17)可得

(18)

(19)

式(17)、(18)、(19)中，C1、C2、C3、C4、C5为待定系数，可以从边界条件得到。

3 上覆分数阶粘弹性饱和场地土的位移地震放大系数

将基岩看作弹性体，由基岩的入射地震波位移表达式(1)和基岩剪力与位移的关系可知地震波产生的基岩剪应力幅值为

(20)

(21)

C1+C3=v0

(22)

(23)

(24)

(25)

(26)

求解式(22)～(26)可得

C1=a1v0，C2=a2v0，C3=a3v0，

C4=a4v0，C5=a5v0

(27)

式中：

(28)

由式(17)可得固相土骨架的位移为

(29)

引入分数导数粘弹性饱和场地土的地震位移放大系数[9]

(30)

4 位移地震放大系数分析与讨论

针对由式(30)得到的上覆分数阶粘弹性饱和场地土地震位移放大系数，采用数值算例的形式来分析讨论分数导数的阶数α、液固耦合系数Sv、土体模型参数κ、基岩土体剪切模量比GJ/G0对地震位移放大系数的影响。在未作说明的情况下，有关参数的取值为α=0.5、nS=0.67、nF=0.33、Sv=0.5、κ=0.8、GJ/G0=1 000、ρJ/ρ0=2。图2～6为上覆分数阶粘弹性饱和场地土地震位移放大系数随无量纲频率的变化曲线。很明显，曲线存在着波峰和波谷，也就是说，在简谐地震波作用下，系统将产生共振现象。分数导数的阶数α对地震位移放大系数的影响见图2和图3，随着分数导数的阶数α的增大，上覆分数阶粘弹性饱和场地土地震位移放大系数将减小，且逐渐退化到经典粘弹性饱和土的情形。由图1、图2和图3可以看出，饱和土液固耦合系数Sv对上覆分数阶粘弹性饱和场地土地震位移放大系数的影响相当明显(图4)。当液固耦合系数Sv较小时，地震位移放大系数随频率变化曲线的峰值越大且越尖，随着液固耦合系数Sv的增大，地震位移放大系数随频率变化曲线的峰值将明显减小。可见，在实际工程中，对饱和的软土需要进行辗压，这样将使其液固耦合系数增大，从而减小其地震反应。随着饱和场地土土体模型参数κ(即粘性)的增长，场地土的耗散地震能量的能力将增大，此时，地震位移放大系数将减小(图5)。基岩土体剪切模量比GJ/G0对地震位移放大系数的影响见图6，随着基岩土体剪切模量比GJ/G0的减小，也即土体剪切模量G0的增大，地震位移放大系数减小，这是因为土体剪切模量越大，相应的剪应力将增大，可见在实际工程中提高基岩上覆土的剪切模量对提高地基的抗震性能有利。

图2 分数导数的阶数α对位移地震放大系数的影响(Sv=0.05)Fig.2 The influence of the order of fractional derivative on displacement seismic amplification coefficient

图3 分数导数的阶数α对位移地震放大系数的影响(Sv=0.5)Fig.3 The influence of the order of fractional derivative on displacement seismic amplification coefficient

图4 液固耦合系数Sv对位移地震放大系数的影响Fig.4 The influence of fluid and solid coupling coefficient on displacement seismic amplification coefficient

图5 土体模型参数κ对位移地震放大系数的影响Fig.5 The influence of soil model parameter on displacement seismic amplification coefficient

图6 基岩土体剪切模量比GJ/G0对位移地震放大系数的影响Fig.6 The influence of shear modulus ratio of rock and soil on displacement seismic amplification coefficient

5 结 论

为了使对基岩上覆场地土动力特性的研究更加符合工程实际，必须要考虑液相影响和土体的粘弹性特性。笔者在饱和土理论、粘弹性理论、分数导数理论等理论的基础上，研究了上覆分数阶粘弹性饱和场地土的地震位移响应。与将上覆土视为单相弹性介质一样，上覆分数阶粘弹性饱和场地土在简谐地震激励的作用下同样存在有共振现象。通过分析发现，饱和土的液固耦合系数和土体模型参数对场地土的地震位移放大系数有较大的影响，可见，忽略液相和固相的影响以及土体的粘性来研究场地土的地震响应问题将与工程实际存在差异。为了降低地震造成的地震响应放大效应的影响，需要对场地土进行压实，达到增大液固耦合系数和土体剪切模量进而减小地震响应的目的。

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(编辑 胡英奎)

Displacement seismic amplification coefficient of overlying fractional derivative viscoelastic saturated site soil

LiuLinchao，YanQifang，LiuTeng

(School of Civil Engineering,Xinyang Normal University,Xinyang 464000,Henan,P. R. China)

Considering the coupling effect between fluid phase and solid phase,the site soil on bedrock is considered as two-phase saturated porous media. In order to consider the viscoelastic characteristics of the saturated site soil,the stress-strain relationship of soil skeleton is described by fractional derivative Kelvin viscoelastic model,and the motion control equations of overlying fractional derivative viscoelastic saturated site soil under the action of harmonic earthquake wave are established. Using the properties of fractional derivative and considering the boundary conditions and permeable conditions,the vibration problem of overlying fractional derivative viscoelastic saturated site soil under the action of harmonic earthquake wave is solved,and the seismic displacement amplification coefficient of the saturated site soil is also obtained. The influences of the order of fractional derivative,liquid and solid coupling coefficient,model parameters of soil,and bedrock and soil shear modulus ratio on the seismic displacement amplification coefficient are analyzed and discussed by numerical examples. The results show that the order of fractional derivative,liquid and coupling coefficient,model parameters of soil,bedrock and soil shear modulus ratio have great effects on the seismic response of the site soil. The fluid and solid coupling coefficient could the increa of be increased and the seismic response can be decreased by compacting the saturated soil. Meanwhile,the viscosity and shear modulus of the saturated site soil can also reduce the seismic response.

saturated soil; fractional derivative; seismic wave; seismic amplification coefficient; fluid and soil coupling coefficient

10.11835/j.issn.1674-4764.2015.05.007

2015-07-16 基金项目：国家自然科学基金(U1504505)；河南省科技发展计划项目(142102210063)；河南省高等学校重点科研项目(15A560036)；河南省高等学校青年骨干教师资助计划项目(2013GGJS-121)；信阳师范学院重大课题预研项目(2013ZDYY19)；信阳师范学院青年科研基金项目(2014-QN-063)；信阳师范学院青年骨干教师资助计划(2012007)。

刘林超(1979-)，副教授，博士，主要从事多孔介质理论、粘弹性理论、桩基动力学研究，(E-mail)llc109@126.com。

Foundation item：National Natural Science Foundation of China(No.U1504505); Development of Science and Technology Plan Project of Henan Province(No.142102210063); Key Scientific Research Project of Henan Province(15A560036); Youth Backbone Teacher Plan Project of Colleges and Universities in Henan Province(2013GGJS-121); Major Beforehand Research Project of Xinyang Normal University(2013ZDYY19); Scientific Research Fund for Young of Xinyang Normal University(2014-QN-063); Youth Backbone Teacher Plan Project of Xinyang Normal University(2012007).

TU573.12

A

1674-4764(2015)05-0048-06

Received:2015-07-16

Author brief：Liu Linchao(1979-),associate professor,PHD,main research interests: theory of porous medium,viscoelastic theory,pile dynamics,(E-mail) llc109@126.com.