轨道附近地面振动模型中的饱和地层动力格林函数

王福彤, 陶夏新, 谢礼立, 郑鑫, 崔高航

1 中国地震局工程力学研究所, 哈尔滨 150001 2 黑龙江大学建筑工程学院, 哈尔滨 150001 3 哈尔滨工业大学土木工程学院, 哈尔滨 150090 4 黑龙江八一农垦大学工程学院, 黑龙江大庆 163319 5 东北林业大学土木工程学院, 哈尔滨 150040



轨道附近地面振动模型中的饱和地层动力格林函数

王福彤1,2, 陶夏新1,3, 谢礼立1,3, 郑鑫1,4, 崔高航5

1 中国地震局工程力学研究所, 哈尔滨 150001 2 黑龙江大学建筑工程学院, 哈尔滨 150001 3 哈尔滨工业大学土木工程学院, 哈尔滨 150090 4 黑龙江八一农垦大学工程学院, 黑龙江大庆 163319 5 东北林业大学土木工程学院, 哈尔滨 150040

列车引起场地振动的建模需要能够表达地层的动力格林函数. 本文兼顾饱和土的流固两相耦合性、场地土的分层性和波动的三维传播性,构建了半解析的场地动力格林函数. 首先,基于Biot方程,在傅里叶变换域求解固体骨架和流体的位移和应力. 然后采用传递矩阵方法建立地表位移和应力间的关系,得到格林函数矩阵. 进而讨论矩阵的一些固有特征,提出改善竖向位移计算效率的措施. 最后利用推导的格林函数计算了几个典型算例. 数值结果与文献中其他方法得到的结果十分接近,与场地振动的现场观测试验基本符合. 软土场地振动的计算结果高于饱和砂土场地,高速列车场地振动强度高于低速列车. 当车速接近场地瑞利波速,模拟结果中显示出马赫锥. 数值结果还显示,即使车速略低于瑞利波速,马赫锥也可能出现. 本文推导的格林函数将有助于深入理解列车等移动激励作用下层状饱和土场地的振动特征.

波动; 多孔弹性介质; 层状半空间; 格林函数； 列车

1 引言

轨道交通运行过程中,列车、轨道结构和场地土层三个子系统之间始终存在着复杂的动态变化的相互作用. 针对三者的耦合振动,一些学者(如Sheng et al.,1999a,1999b,2003,2004; Lombaert et al.,2006,2009)建立了精妙的分析模型.其中场地对简谐荷载的动力响应函数,即频域格林函数,是解耦列车-轨道-场地系统的必要的关键因素.模型中的场地土被抽象为单相的黏弹性连续介质,没有考虑孔隙水与土骨架的流固两相耦合振动问题.天然土的孔隙水可能会分担土中的应力,也可能相对于土骨架运动而产生内摩擦.很多情况下,土中水对场地振动的影响相当大,基于单相介质模型的方法难以给出满意的结果(Cai et al.,2009; Beskou and Theodorakopoulos,2011).

Biot(1956a,1956b,1962)建立了流固两相介质运动方程,被广泛应用于饱和土场地对地表移动荷载的动力响应问题.其中很多研究(Jin et al.,2004; Lu and Jeng,2007; Lefeuve-Mesgouez and Mesgouez,2008; Sun et al.,2010; Cao et al.,2011)关注均匀半空间两相介质的波动,忽略了场地土的分层现象.均匀半空间模型易于获得闭合形式的解析解，有助于理解饱和土场地振动的一些基本特性.但是由于没有分层界面,均匀半空间不能反映土层界面上反射、折射和透射等一些典型的波动现象.

比较而言,两相介质的层状半空间更接近场地的物理实际.针对移动荷载激励问题，一些学者(Xu et al.,2007,2008; Lu et al.,2009; Mesgouez and Lefeuve-Mesgouez,2009; Lefeuve-Mesgouez and Mesgouez,2012)采用透射反射矩阵方法(Lowe,1995; Luco and Apsel，1983; Rokhlin and Wang,2002)计算了场地动力响应.少数的研究采用了2.5维有限元(Gao et al.,2012)和薄层法(高广运等,2013)等数值模型.

本文可以看做是Sheng et al.(2003,2004)模型的进一步拓展,目的是构建饱和层状半空间的竖向位移格林函数,用以取代该文献中的单相介质格林函数,使其能够模拟列车引起的饱和场地振动.论文首先给出频率-波数域内Biot方程的三维解答,基于传递矩阵方法推导格林函数矩阵,然后探讨改善格林函数数值计算效率的措施,最后通过具体算例验证格林函数的可靠性并展示饱和层状半空间的振动特征.

2 饱和多孔弹性介质运动方程在Fourier变换域的解答

在流体饱和的多孔弹性介质中建立笛卡尔坐标系,用Ui(i=x,y,z)表示固体骨架位移张量,用wi表示流体相对于固体骨架的位移张量,依据Biot (1962) 理论,在忽略孔隙形状影响的条件下,流固两相的运动方程为

(1)

其中：ρ表示流固两相的总密度(对于土介质而言,即土的密度),ρf为流体密度; 参数b表达两相介质的黏性耦合,与渗透系数成反比关系; 似密度参数m=ρf/φ,φ为孔隙率;M和α分别称为Biot第一参数和第二参数,表示流固两相压缩性的对比关系;λc=λ+α2M,λ和μ表示排水条件下骨架的拉梅常数.沿用文献(Lefeuve-MesgouezandMesgouez,2008;Lefeuve-MesgouezandMesgouez,2012;MesgouezandLefeuve-Mesgouez,2009)的方法,这两个拉梅常数以复数形式涵盖介质的阻尼特性.流固两相介质的本构关系可表示为

(2)

其中：σij表示两相介质的应力张量,Pf表示超孔隙水压力,δij为Kroneker delta函数; 单位体积两相介质排出的流体体积ε=wi,i,固体骨架的体应变e=Ui,i.

在频率为ω的谐振条件下,对(1)式两边点乘笛卡尔空间梯度算子,得到体变方程：

(3)

式中，函数上的波浪线“～”表示时间函数的幅值.采用如下定义的空间Fourier变换对：

(4a)

(4b)

将方程(3)的两个水平方向(x,y)变换到波数(β,γ)域,得到体变函数的二阶齐次线性常微分方程组,通解形式：

(6a)

和

(6b)

得到具体数值.求解过程中要注意的是,由于P1波总是快于P2波,必须保证VP1的实部总是大于VP2的实部.

对式 (1) 应用式 (4a) 定义的Fourier变换,将式 (5) 的体应变解代入,得到位移解,

式中A3,B3,A4和B4为积分常数.考虑式本构方程(2),可得应力解

(7b)

将式 (7) 写为矩阵形式,

S=eξp1zAdb,

(8)

其中,S称为状态向量,

积分常数向量b={A1,B1,A2,B2A3,B3,A4,B4}T; 式(6)和式(7)中的d1,d2,t1,t2,g1,g2,b0,b1,b2,tb1,tb2,tc1和tc2等参数以及矩阵A的元素详见附录.

3 层状半空间格林函数矩阵的建构

图1 层状半空间场地模型Fig.1 Layered half-space model for ground

(9)

消去bj得

Sj,1=eξp1,jhjTjSj,0,

(10)

(11)

式中T=TnTn-1…T1,表示从第n层底面到地面的传递矩阵.将这个8×8的矩阵划分为4个4×4的子矩阵:

(12)

则式 (11) 可改写为

(13)

在下卧半空间中,当z趋于无穷远时,位移必趋近于0.据此，可以建立下卧半空间顶面(也就是第n层底面)处的位移-应力关系:

(14)

矩阵Au和Aτ的元素详见附录.

将式 (13) 代入式 (14) 得

(15)

(16)

或展开写为

.

(17)

一般轨道周边地面运动的竖向分量远大于水平向分量,通常建模中仅考虑竖向激励而忽略两个水平向的剪切荷载(Shengetal.,1999;Shengetal.,2003;Shengetal.,2004;Lombaertetal.,2006;LombaertandDegrande,2009).在地面可渗水的条件下,式 (17) 的竖向位移分量

(18)

两边进行式 (4b) 所示的逆Fourier变换,可得

(19)

4 竖向位移格林函数数值计算中可利用的一些属性

(20)

(21)

(22a)

和

(22b)

图2 列车-轨道-场地耦合振动体系Fig.2 The train-track-ground coupling system

5 列车-轨道-场地耦合动力分析模型

Sheng等(2003,2004)建立了列车-轨道-场地耦合动力分析模型,全系统如图2所示.列车子系统用弹簧阻尼器连接的多刚体模拟,质量矩阵为MT,复阻尼刚度矩阵KT.无限长轨道与场地接触面宽度为2bt.钢轨用Euler梁模拟,单位长度质量为mR,截面抗弯刚度EI.轨枕连续化成为沿轨道方向延伸的无抗弯刚度扁梁,单位长度质量ms.轨垫板用竖向弹簧代表,刚度为kp.道碴路堤用线密度mB的阻尼弹性层代表,用一致质量近似方法简化,仅考虑竖向刚度kB.EI、kB和kp均为复数,虚部包含各自的耗散系数.假定轨道与场地接触面上只存在竖向相互作用力,并且沿着垂直轨道的水平方向均匀分布.

6 数值算例

首先计算层状半空间上直接作用矩形分布荷载的简单算例,然后模拟包括列车、轨道和饱和场地全系统的工况.

6.1 矩形分布简谐荷载的算例

Jones等(1998)和 Lefeuve-Mesgouez and Mesgouez(2008)分别计算了矩形分布简谐荷载对单相介质层状半空间和两相介质均匀半空间的激励.前者的荷载作用位置不变,即非移动激励; 后者的荷载在地面匀速直线运动,为移动激励.现采用前面推导的格林函数分别计算这两个简单算例,与文献结果对比.

6.1.1 固定位置荷载

如图3所示,竖向简谐荷载均匀分布于地表矩形区域,坐标原点设在矩形的中心.频率-波数域内荷载可表示为

(23)

其中,a1和a2是矩形两个方向的半边长,a1=a2=0.3 m; 荷载振动频率Ω/2π=64 Hz.

将式(23)代入式(18)得到(β,γ,ω)域中的地表竖向位移表达式：

(24)

Jones等(1998)采用单相介质模型,利用表1中的两套参数区分A、B两种土.A、B土的不同组合形成4个场地模型,如图3所示.场地1为A土构成的均匀半空间; 场地2为7 m厚的A土覆盖层加B土下卧半空间; 场地3为7 m厚A土覆盖层加下卧刚性基岩; 场地4为B土构成的均匀半空间.Biot(1962)指出,当参数ρf,M和b趋近于0时,固相的动力特性将趋近于相同参数值的单相介质.为验证本文推导的格林函数,固相参数值设为表1单相介质的相应值; 其余参数ρf=0.0001 kg·m-3,M=0.0001 Pa,b=0,α=1.0和φ=0.6.

表1 单相介质参数值(Jones et al.(1998))Table 1 Parameters of single phase media in Jones et al.(1998)

图3 四个场地模型及其矩形分布简谐荷载Fig.3 Four ground models with rectangular harmonic load acting on surfaces

图4 当ρf,M和b取值极小时场地1的竖向位移(a) 波数域位移谱; (b) 空间域位移幅值.Fig.4 Vertical displacement of the Ground 1,when values of ρf,M and b are very small(a) Wave number domain; (b) Space domain.

图5 x=0处的竖向位移绝对值Fig.5 Amplitudes of vertical displacements along x=0

6.1.2 移动激励

前述矩形分布简谐荷载沿x轴以速度c移动,式 (24) 变为

图6 在随矩形分布荷载中心点一起移动的坐标系中，y=0位置处的地面竖向位移(荷载速度为122.5 m·s-1)Fig.6 Vertical displacement of ground surface at y=0 in the moving frame of reference bound with the center of a harmonic square load traveling at the speed of 122.5 m·s-1

图7 移动坐标系中，地面y=0处的竖向位移绝对值Fig.7 Modulus of the vertical displacement of ground surface at y=0 in the moving frame of reference

图8 地表观测点平面布置图Fig.8 Plane layout of the observation points on the ground surface

6.2 列车引起的场地振动

6.2.1 模拟计算结果与观测数据的对比

笔者所在课题组曾对北京城轨13号线地面运行区间进行过现场实测(王福彤等，2011b),地表测点的平面布局如图8所示.列车4节编组,行驶速度为60 km·h-1(或17 m·s-1).利用强震仪以200 Hz的采样频率记录地表测点的竖向加速度; 根据加速度时程进行功率谱估计,并去除了本底振动成分(王福彤等,2011a; 郑鑫等,2013); 按照国家标准《城市区域环境振动量测方法》(GB10071-88)将功率谱转换为振动加速度级VAL谱,如图9所示.

采用第5节模型模拟计算地表振动.表2、表3分别为车辆和轨道结构的模型参数表,数值基于设计资料综合确定.表4数据为场地土介质参数,源于钻探取样和原位波速测结果.现场地下水位埋藏较深,因此Biot模型参数ρf,M和b近于0值.土骨架泊松比取为0.3.图10为计算所需的轮轨不平顺功率谱密度,根据现场实测数据反演得来(王福彤等，2012).为便于比较,模型计算结果也展示在图9中.从中可见,在远近不同的位置处,在环境振动主要频率范围内,计算结果与试验结果都很接近,说明本文方法能够很好模拟预测场地振动水平.

表2 机车车辆参数值Table 2 Parameter values for the vehicles

表3 轨道系统基本参数Table 3 Parameter values for the track structure

图9 地表测点上振动加速度级的模型计算值与观测值的对比Fig.9 Comparison between the simulated values and the observed values of the vibration acceleration levels (VALs) at the observation points on the ground surface

表4 地层模型基本参数Table 4 Parameter values for the ground

6.2.2 饱和软土场地与饱和砂土场地计算结果

前述观测试验的局限是没有得到饱和土场地的振动数据,现采用本文模型模拟计算饱和土场地的环境振动.众所周知,饱和软土与饱和密砂的动力学特性存在很大差异.陈龙珠等(1998)提供了饱和软土和饱和砂土的典型参数值,如表5所示.本算例采用这两套数值表示两种覆盖层,分别与一个模量较高、孔隙率极小的近似不透水坚硬下卧层搭配,形成两个典型的饱和层状场地.列车和轨道结构参数仍旧保持表2和表3中数值不变.

表5 饱和软土场地和饱和砂土场地的模型参数Table 5 Parameter values in the models for the grounds of saturated soft clay and saturated sand

图11和图12展示了距离轨道中线10、20 m和30 m处各地表点的振动级模拟计算值.图11的车速为17 m·s-1,模拟慢速列车; 图12的车速为101 m·s-1(约等于表5软土的120 Hz以内的瑞利波速),模拟高速列车.对比饱和砂土场地和软土场地可见,无论距轨道远近,在几乎所有频段上,软土场地的振动水平均高于饱和砂土场地.

多篇文献(Adolfsson et al.,1999; Madshus and Kania 2000; Takemiya,2003; Sheng et al.,2003)报道,瑞典X2000高速列车行经西部海岸线Ledsgard路段的软土场地时,车速接近瑞利波速,场地振动急剧增加.通过对比图11和图12的各条曲线可以发现,本文模拟结果反映了这个现象.对于软土场地来说,图12的车速已达瑞利波速,几乎所有频段上的振动级都明显高于图11中相应数值.

第5节模型可以计算单一波长的谐波不平顺激励下的地表位移场.设不平顺幅值为1 mm,波长2 m,则对于速度c=17 m·s-1的慢速列车,轮轨动态相互作用力的频率为17/2=8.5 Hz.图13a为慢速列车运行引起的砂土场地表面的竖向位移幅值,列车位置在移动坐标x-ct轴的0到-76 m处.整体

图11 慢速轨道交通地表点的振动加速度级Fig.11 VALs at some points on the ground surface caused by a low speed train

图12 高速铁路地表点的振动加速度级Fig.12 VALs at some points on the ground surface caused by a high speed train

图13 c=17 m·s-1的列车在轨道上运行引起的地表竖向位移幅值(a) 饱和砂土场地; (a) 饱和软土场地.Fig.13 Vertical displacement of ground surface when a train moving on a track with speed of 17 m·s-1(a) Sandy ground; (b) Soft clay ground.

看饱和砂土场地振动幅值极小,各组车轮正下方的位移略为明显.图13b为慢速列车下的饱和软土场地,振动强度比砂土场地剧烈得多; 各个轮对激发环状波动,相互干涉,传播很远; 由于车速远小于瑞利波速,列车前方地表可见向前传播的振动.

图14为高速列车计算结果,此时轮轨力频率为101/2=50.5 Hz.饱和砂土场地与软土场地的振动幅值仍然迥异.50 Hz的移动激扰,在砂土地面激发了向远处传播的环状的波动,如图14a所示.尽管车速高达101 m·s,仍然远小于饱和砂土的瑞利波速(0～120 Hz频段为278～300 m·s-1),所以前方可见领先于列车的表面波.

无论从位移幅值,还是从影响范围看,图14b的软土场地位移反应都比图14a的砂土场地剧烈.最大峰值出现在列车前部第一转向架下面,振动向列车侧后方向传播,形成典型的马赫锥.振动局限于马赫锥之内,前方场地几乎没有地表波动.

众所周知,如果移动荷载直接作用于半空间表面,与瑞利波同行达到临界速度,马赫角应该等于90°.图14b的马赫角明显小于90°,显示车速已经超越了临界速度.这说明,由于轨道的存在,临界速度并不等于场地瑞利波速,而是有所降低.保持其他条件不变,仅将车速降至90 m·s-1,地表位移波场中仍然出现了马赫锥,如图15所示.这个结果提醒软土场地工程实践注意,即使设计车速低于场地的瑞利波速,剧烈振动的马赫锥效应也可能出现.

图14 c=101 m·s-1的列车在轨道上运行引起的地表竖向位移幅值(a) 饱和砂土场地; (a) 饱和软土场地.Fig.14 Vertical displacement of ground surface when a train moving on a track with speed of 101 m·s-1(a) Sandy ground; (b) Soft clay ground.

图15 c=90 m·s-1时的地面竖向位移幅值Fig.15 Vertical displacement of ground surface when a train speed is 90 m·s-1

7 结论

动力格林函数是表达场地的数学模型,在列车-轨道-场地系统的耦合振动分析中起到关键作用.本文构建的格林函数,能够考虑饱和岩土体的流固两相动力相互作用,场地分层沉积的几何物理特性以及弹性波在场地中的三维传播.数值算例的结果与文献结果十分接近,与现场观测试验基本符合.模型反映了高速列车的场地振动强度大于低速列车、软土场地振动强度大于饱和砂土场地的规律.当车速接近场地瑞利波速时,基于格林函数计算的地表振动幅值能够反映出马赫锥效应; 即使车速略低于瑞利波速,剧烈振动的马赫锥亦可能出现,这为软土场地高铁工程实践提供了有益的参考.

应该说明的是,还有很多影响因素值得进行数值分析,例如土层厚度，土体的渗透性、孔隙比、密度等对场地振动的影响.但本文意在提出格林函数的计算方法并数值验证,限于篇幅,影响因素将另行撰文讨论.

附录A

(A1)

(A3)

(A4)

Aτ=

(A5)

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(本文编辑 汪海英)

Dynamic Green′s function of stratified ground with saturated soil layers for modeling ground vibration near railway track

WANG Fu-Tong1,2, TAO Xia-Xin1,3, XIE Li-Li1,3, ZHENG Xin1,4, CUI Gao-Hang5

1InstituteofEngineeringMechanics,ChinaEarthquakeAdministration,Harbin150001,China2SchoolofCivilEngineeringandArchitecture,HeilongjiangUniversity,Harbin150001,China3SchoolofCivilEngineering,HarbinInstituteofTechnology,Harbin150090,China4SchoolofEngineering,HeilongjiangBayiAgriculturalUniversity,Daqing,Heilongjiang163319,China5SchoolofCivilEngineering,NortheastForestryUniversity,Harbin150040,China

The modeling of ground vibration from trains requires a Green′s function to represent the dynamic characteristics of the ground. Because of the existence of groundwater, dynamic coupling vibration of soil skeleton and pore water may propagate in a naturally stratified ground. This paper proposes a semi-analytical Green′s function that is able to model the fluid-solid coupling in saturated soil, stratification of ground configuration and three-dimensional propagation of waves.The Biot′s equation was Fourier transformed with respect to time and two horizontal cartesian components. General solutions of displacements and stresses of solid skeleton and pore fluid were worked out in the Fourier transformed domain. The relationship between displacement and stress on the ground surface was formulated by the transfer matrix technique, so that the matrix of the Green′s function was derived. Improvements of computational efficiency for vertical displacement were achieved by taking advantage of some relevant matrices′ properties. The proposed Green′s function was added into a sophisticated train-track-ground interaction model to include the ground water effect. Validation of the Green′s function was shown by computing several typical examples in the literature and simulating a field observation near the Beijing urban railway. Vibrations of two typical layered grounds with saturated clay and sandy soil, excited by low speed and high speed trains, respectively, were analyzed based on computation results.The calculated amplitudes of ground vibrations were very close to those of some references, in both cases of a harmonic load with a fixed position and a moving load. For the field test in Beijing, the simulated ground vibration levels agreed largely with the observational data. The comparison of numerical results for the two kinds of saturated soil shows that the vibrations of soft clay were higher than those of saturated sand, and the vibration intensities of ground caused by high speed train are larger than those by the low-speed train. The Mach cone appeared in the simulated wave field of the ground surface in the case that train speed approached the phase velocity of Rayleigh waves. The numerical results also show that the Mach cone can still be generated even the train speed is slightly lower than the Rayleigh wave velocity.The proposed Green′s function is able to represent such mechanisms as fluid-solid interaction between two phases of saturated soil, geometric and physical stratification of soil deposit and three-dimensional propagation of viscoelastic waves in a ground. Some train-induced vibration features of the water-saturated layered ground can be simulated by the train-track-ground vibration model based on the Green′s function, which will be helpful to understand the propagation and attenuation of the ground-borne vibrations caused by moving trains.

Wave Propagation; Poroelastic Medium; Layered Half-space; Green′s Function; Train

国家自然科学基金项目(50538030，51108163)，黑龙江省自然科学基金项目(E201221，E201330)，中国博士后科学基金项目(2013M531084)资助.

王福彤，男，1972年生，博士，副教授，主要从事岩土地震工程的研究.E-mail:wang-futong@126.com

10.6038/cjg20150827.

10.6038/cjg20150827

P315

2014-03-13，2015-07-12收修定稿

王福彤, 陶夏新, 谢礼立等.2015.轨道附近地面振动模型中的饱和地层动力格林函数.地球物理学报，58(8):2948-2961,

Wang F T, Tao X X, Xie L L, et al. 2015. Dynamic Green′s function of stratified ground with saturated soil layers for modeling ground vibration near railway track.ChineseJ.Geophys. (in Chinese),58(8):2948-2961,doi:10.6038/cjg20150827.