王福彤, 陶夏新, 謝禮立, 鄭鑫, 崔高航

1 中國(guó)地震局工程力學(xué)研究所, 哈爾濱 150001 2 黑龍江大學(xué)建筑工程學(xué)院, 哈爾濱 150001 3 哈爾濱工業(yè)大學(xué)土木工程學(xué)院, 哈爾濱 150090 4 黑龍江八一農(nóng)墾大學(xué)工程學(xué)院, 黑龍江大慶 163319 5 東北林業(yè)大學(xué)土木工程學(xué)院, 哈爾濱 150040

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軌道附近地面振動(dòng)模型中的飽和地層動(dòng)力格林函數(shù)

王福彤1,2, 陶夏新1,3, 謝禮立1,3, 鄭鑫1,4, 崔高航5

1 中國(guó)地震局工程力學(xué)研究所, 哈爾濱 150001 2 黑龍江大學(xué)建筑工程學(xué)院, 哈爾濱 150001 3 哈爾濱工業(yè)大學(xué)土木工程學(xué)院, 哈爾濱 150090 4 黑龍江八一農(nóng)墾大學(xué)工程學(xué)院, 黑龍江大慶 163319 5 東北林業(yè)大學(xué)土木工程學(xué)院, 哈爾濱 150040

列車引起場(chǎng)地振動(dòng)的建模需要能夠表達(dá)地層的動(dòng)力格林函數(shù). 本文兼顧飽和土的流固兩相耦合性、場(chǎng)地土的分層性和波動(dòng)的三維傳播性,構(gòu)建了半解析的場(chǎng)地動(dòng)力格林函數(shù). 首先,基于Biot方程,在傅里葉變換域求解固體骨架和流體的位移和應(yīng)力. 然后采用傳遞矩陣方法建立地表位移和應(yīng)力間的關(guān)系,得到格林函數(shù)矩陣. 進(jìn)而討論矩陣的一些固有特征,提出改善豎向位移計(jì)算效率的措施. 最后利用推導(dǎo)的格林函數(shù)計(jì)算了幾個(gè)典型算例. 數(shù)值結(jié)果與文獻(xiàn)中其他方法得到的結(jié)果十分接近,與場(chǎng)地振動(dòng)的現(xiàn)場(chǎng)觀測(cè)試驗(yàn)基本符合. 軟土場(chǎng)地振動(dòng)的計(jì)算結(jié)果高于飽和砂土場(chǎng)地,高速列車場(chǎng)地振動(dòng)強(qiáng)度高于低速列車. 當(dāng)車速接近場(chǎng)地瑞利波速,模擬結(jié)果中顯示出馬赫錐. 數(shù)值結(jié)果還顯示,即使車速略低于瑞利波速,馬赫錐也可能出現(xiàn). 本文推導(dǎo)的格林函數(shù)將有助于深入理解列車等移動(dòng)激勵(lì)作用下層狀飽和土場(chǎng)地的振動(dòng)特征.

波動(dòng); 多孔彈性介質(zhì); 層狀半空間; 格林函數(shù)； 列車

1 引言

軌道交通運(yùn)行過程中,列車、軌道結(jié)構(gòu)和場(chǎng)地土層三個(gè)子系統(tǒng)之間始終存在著復(fù)雜的動(dòng)態(tài)變化的相互作用. 針對(duì)三者的耦合振動(dòng),一些學(xué)者(如Sheng et al.,1999a,1999b,2003,2004; Lombaert et al.,2006,2009)建立了精妙的分析模型.其中場(chǎng)地對(duì)簡(jiǎn)諧荷載的動(dòng)力響應(yīng)函數(shù),即頻域格林函數(shù),是解耦列車-軌道-場(chǎng)地系統(tǒng)的必要的關(guān)鍵因素.模型中的場(chǎng)地土被抽象為單相的黏彈性連續(xù)介質(zhì),沒有考慮孔隙水與土骨架的流固兩相耦合振動(dòng)問題.天然土的孔隙水可能會(huì)分擔(dān)土中的應(yīng)力,也可能相對(duì)于土骨架運(yùn)動(dòng)而產(chǎn)生內(nèi)摩擦.很多情況下,土中水對(duì)場(chǎng)地振動(dòng)的影響相當(dāng)大,基于單相介質(zhì)模型的方法難以給出滿意的結(jié)果(Cai et al.,2009; Beskou and Theodorakopoulos,2011).

Biot(1956a,1956b,1962)建立了流固兩相介質(zhì)運(yùn)動(dòng)方程,被廣泛應(yīng)用于飽和土場(chǎng)地對(duì)地表移動(dòng)荷載的動(dòng)力響應(yīng)問題.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)關(guān)注均勻半空間兩相介質(zhì)的波動(dòng),忽略了場(chǎng)地土的分層現(xiàn)象.均勻半空間模型易于獲得閉合形式的解析解，有助于理解飽和土場(chǎng)地振動(dòng)的一些基本特性.但是由于沒有分層界面,均勻半空間不能反映土層界面上反射、折射和透射等一些典型的波動(dòng)現(xiàn)象.

比較而言,兩相介質(zhì)的層狀半空間更接近場(chǎng)地的物理實(shí)際.針對(duì)移動(dòng)荷載激勵(lì)問題，一些學(xué)者(Xu et al.,2007,2008; Lu et al.,2009; Mesgouez and Lefeuve-Mesgouez,2009; Lefeuve-Mesgouez and Mesgouez,2012)采用透射反射矩陣方法(Lowe,1995; Luco and Apsel，1983; Rokhlin and Wang,2002)計(jì)算了場(chǎng)地動(dòng)力響應(yīng).少數(shù)的研究采用了2.5維有限元(Gao et al.,2012)和薄層法(高廣運(yùn)等,2013)等數(shù)值模型.

本文可以看做是Sheng et al.(2003,2004)模型的進(jìn)一步拓展,目的是構(gòu)建飽和層狀半空間的豎向位移格林函數(shù),用以取代該文獻(xiàn)中的單相介質(zhì)格林函數(shù),使其能夠模擬列車引起的飽和場(chǎng)地振動(dòng).論文首先給出頻率-波數(shù)域內(nèi)Biot方程的三維解答,基于傳遞矩陣方法推導(dǎo)格林函數(shù)矩陣,然后探討改善格林函數(shù)數(shù)值計(jì)算效率的措施,最后通過具體算例驗(yàn)證格林函數(shù)的可靠性并展示飽和層狀半空間的振動(dòng)特征.

2 飽和多孔彈性介質(zhì)運(yùn)動(dòng)方程在Fourier變換域的解答

在流體飽和的多孔彈性介質(zhì)中建立笛卡爾坐標(biāo)系,用Ui(i=x,y,z)表示固體骨架位移張量,用wi表示流體相對(duì)于固體骨架的位移張量,依據(jù)Biot (1962) 理論,在忽略孔隙形狀影響的條件下,流固兩相的運(yùn)動(dòng)方程為

(1)

其中：ρ表示流固兩相的總密度(對(duì)于土介質(zhì)而言,即土的密度),ρf為流體密度; 參數(shù)b表達(dá)兩相介質(zhì)的黏性耦合,與滲透系數(shù)成反比關(guān)系; 似密度參數(shù)m=ρf/φ,φ為孔隙率;M和α分別稱為Biot第一參數(shù)和第二參數(shù),表示流固兩相壓縮性的對(duì)比關(guān)系;λc=λ+α2M,λ和μ表示排水條件下骨架的拉梅常數(shù).沿用文獻(xiàn)(Lefeuve-MesgouezandMesgouez,2008;Lefeuve-MesgouezandMesgouez,2012;MesgouezandLefeuve-Mesgouez,2009)的方法,這兩個(gè)拉梅常數(shù)以復(fù)數(shù)形式涵蓋介質(zhì)的阻尼特性.流固兩相介質(zhì)的本構(gòu)關(guān)系可表示為

(2)

其中：σij表示兩相介質(zhì)的應(yīng)力張量,Pf表示超孔隙水壓力,δij為Kroneker delta函數(shù); 單位體積兩相介質(zhì)排出的流體體積ε=wi,i,固體骨架的體應(yīng)變e=Ui,i.

在頻率為ω的諧振條件下,對(duì)(1)式兩邊點(diǎn)乘笛卡爾空間梯度算子,得到體變方程：

(3)

式中，函數(shù)上的波浪線“～”表示時(shí)間函數(shù)的幅值.采用如下定義的空間Fourier變換對(duì)：

(4a)

(4b)

將方程(3)的兩個(gè)水平方向(x,y)變換到波數(shù)(β,γ)域,得到體變函數(shù)的二階齊次線性常微分方程組,通解形式：

(6a)

和

(6b)

得到具體數(shù)值.求解過程中要注意的是,由于P1波總是快于P2波,必須保證VP1的實(shí)部總是大于VP2的實(shí)部.

對(duì)式 (1) 應(yīng)用式 (4a) 定義的Fourier變換,將式 (5) 的體應(yīng)變解代入,得到位移解,

式中A3,B3,A4和B4為積分常數(shù).考慮式本構(gòu)方程(2),可得應(yīng)力解

(7b)

將式 (7) 寫為矩陣形式,

S=eξp1zAdb,

(8)

其中,S稱為狀態(tài)向量,

積分常數(shù)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等參數(shù)以及矩陣A的元素詳見附錄.

3 層狀半空間格林函數(shù)矩陣的建構(gòu)

圖1 層狀半空間場(chǎng)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,表示從第n層底面到地面的傳遞矩陣.將這個(gè)8×8的矩陣劃分為4個(gè)4×4的子矩陣:

(12)

則式 (11) 可改寫為

(13)

在下臥半空間中,當(dāng)z趨于無窮遠(yuǎn)時(shí),位移必趨近于0.據(jù)此，可以建立下臥半空間頂面(也就是第n層底面)處的位移-應(yīng)力關(guān)系:

(14)

矩陣Au和Aτ的元素詳見附錄.

將式 (13) 代入式 (14) 得

(15)

(16)

或展開寫為

.

(17)

一般軌道周邊地面運(yùn)動(dòng)的豎向分量遠(yuǎn)大于水平向分量,通常建模中僅考慮豎向激勵(lì)而忽略兩個(gè)水平向的剪切荷載(Shengetal.,1999;Shengetal.,2003;Shengetal.,2004;Lombaertetal.,2006;LombaertandDegrande,2009).在地面可滲水的條件下,式 (17) 的豎向位移分量

(18)

兩邊進(jìn)行式 (4b) 所示的逆Fourier變換,可得

(19)

4 豎向位移格林函數(shù)數(shù)值計(jì)算中可利用的一些屬性

(20)

(21)

(22a)

和

(22b)

圖2 列車-軌道-場(chǎng)地耦合振動(dòng)體系Fig.2 The train-track-ground coupling system

5 列車-軌道-場(chǎng)地耦合動(dòng)力分析模型

Sheng等(2003,2004)建立了列車-軌道-場(chǎng)地耦合動(dòng)力分析模型,全系統(tǒng)如圖2所示.列車子系統(tǒng)用彈簧阻尼器連接的多剛體模擬,質(zhì)量矩陣為MT,復(fù)阻尼剛度矩陣KT.無限長(zhǎng)軌道與場(chǎng)地接觸面寬度為2bt.鋼軌用Euler梁模擬,單位長(zhǎng)度質(zhì)量為mR,截面抗彎剛度EI.軌枕連續(xù)化成為沿軌道方向延伸的無抗彎剛度扁梁,單位長(zhǎng)度質(zhì)量ms.軌墊板用豎向彈簧代表,剛度為kp.道碴路堤用線密度mB的阻尼彈性層代表,用一致質(zhì)量近似方法簡(jiǎn)化,僅考慮豎向剛度kB.EI、kB和kp均為復(fù)數(shù),虛部包含各自的耗散系數(shù).假定軌道與場(chǎng)地接觸面上只存在豎向相互作用力,并且沿著垂直軌道的水平方向均勻分布.

6 數(shù)值算例

首先計(jì)算層狀半空間上直接作用矩形分布荷載的簡(jiǎn)單算例,然后模擬包括列車、軌道和飽和場(chǎng)地全系統(tǒng)的工況.

6.1 矩形分布簡(jiǎn)諧荷載的算例

Jones等(1998)和 Lefeuve-Mesgouez and Mesgouez(2008)分別計(jì)算了矩形分布簡(jiǎn)諧荷載對(duì)單相介質(zhì)層狀半空間和兩相介質(zhì)均勻半空間的激勵(lì).前者的荷載作用位置不變,即非移動(dòng)激勵(lì); 后者的荷載在地面勻速直線運(yùn)動(dòng),為移動(dòng)激勵(lì).現(xiàn)采用前面推導(dǎo)的格林函數(shù)分別計(jì)算這兩個(gè)簡(jiǎn)單算例,與文獻(xiàn)結(jié)果對(duì)比.

6.1.1 固定位置荷載

如圖3所示,豎向簡(jiǎn)諧荷載均勻分布于地表矩形區(qū)域,坐標(biāo)原點(diǎn)設(shè)在矩形的中心.頻率-波數(shù)域內(nèi)荷載可表示為

(23)

其中,a1和a2是矩形兩個(gè)方向的半邊長(zhǎng),a1=a2=0.3 m; 荷載振動(dòng)頻率Ω/2π=64 Hz.

將式(23)代入式(18)得到(β,γ,ω)域中的地表豎向位移表達(dá)式：

(24)

Jones等(1998)采用單相介質(zhì)模型,利用表1中的兩套參數(shù)區(qū)分A、B兩種土.A、B土的不同組合形成4個(gè)場(chǎng)地模型,如圖3所示.場(chǎng)地1為A土構(gòu)成的均勻半空間; 場(chǎng)地2為7 m厚的A土覆蓋層加B土下臥半空間; 場(chǎng)地3為7 m厚A土覆蓋層加下臥剛性基巖; 場(chǎng)地4為B土構(gòu)成的均勻半空間.Biot(1962)指出,當(dāng)參數(shù)ρf,M和b趨近于0時(shí),固相的動(dòng)力特性將趨近于相同參數(shù)值的單相介質(zhì).為驗(yàn)證本文推導(dǎo)的格林函數(shù),固相參數(shù)值設(shè)為表1單相介質(zhì)的相應(yīng)值; 其余參數(shù)ρf=0.0001 kg·m-3,M=0.0001 Pa,b=0,α=1.0和φ=0.6.

表1 單相介質(zhì)參數(shù)值(Jones et al.(1998))Table 1 Parameters of single phase media in Jones et al.(1998)

圖3 四個(gè)場(chǎng)地模型及其矩形分布簡(jiǎn)諧荷載Fig.3 Four ground models with rectangular harmonic load acting on surfaces

圖4 當(dāng)ρf,M和b取值極小時(shí)場(chǎng)地1的豎向位移(a) 波數(shù)域位移譜; (b) 空間域位移幅值.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.

圖5 x=0處的豎向位移絕對(duì)值Fig.5 Amplitudes of vertical displacements along x=0

6.1.2 移動(dòng)激勵(lì)

前述矩形分布簡(jiǎn)諧荷載沿x軸以速度c移動(dòng),式 (24) 變?yōu)?/p>

圖6 在隨矩形分布荷載中心點(diǎn)一起移動(dòng)的坐標(biāo)系中，y=0位置處的地面豎向位移(荷載速度為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

圖7 移動(dòng)坐標(biāo)系中，地面y=0處的豎向位移絕對(duì)值Fig.7 Modulus of the vertical displacement of ground surface at y=0 in the moving frame of reference

圖8 地表觀測(cè)點(diǎn)平面布置圖Fig.8 Plane layout of the observation points on the ground surface

6.2 列車引起的場(chǎng)地振動(dòng)

6.2.1 模擬計(jì)算結(jié)果與觀測(cè)數(shù)據(jù)的對(duì)比

筆者所在課題組曾對(duì)北京城軌13號(hào)線地面運(yùn)行區(qū)間進(jìn)行過現(xiàn)場(chǎng)實(shí)測(cè)(王福彤等，2011b),地表測(cè)點(diǎn)的平面布局如圖8所示.列車4節(jié)編組,行駛速度為60 km·h-1(或17 m·s-1).利用強(qiáng)震儀以200 Hz的采樣頻率記錄地表測(cè)點(diǎn)的豎向加速度; 根據(jù)加速度時(shí)程進(jìn)行功率譜估計(jì),并去除了本底振動(dòng)成分(王福彤等,2011a; 鄭鑫等,2013); 按照國(guó)家標(biāo)準(zhǔn)《城市區(qū)域環(huán)境振動(dòng)量測(cè)方法》(GB10071-88)將功率譜轉(zhuǎn)換為振動(dòng)加速度級(jí)VAL譜,如圖9所示.

采用第5節(jié)模型模擬計(jì)算地表振動(dòng).表2、表3分別為車輛和軌道結(jié)構(gòu)的模型參數(shù)表,數(shù)值基于設(shè)計(jì)資料綜合確定.表4數(shù)據(jù)為場(chǎng)地土介質(zhì)參數(shù),源于鉆探取樣和原位波速測(cè)結(jié)果.現(xiàn)場(chǎng)地下水位埋藏較深,因此Biot模型參數(shù)ρf,M和b近于0值.土骨架泊松比取為0.3.圖10為計(jì)算所需的輪軌不平順功率譜密度,根據(jù)現(xiàn)場(chǎng)實(shí)測(cè)數(shù)據(jù)反演得來(王福彤等，2012).為便于比較,模型計(jì)算結(jié)果也展示在圖9中.從中可見,在遠(yuǎn)近不同的位置處,在環(huán)境振動(dòng)主要頻率范圍內(nèi),計(jì)算結(jié)果與試驗(yàn)結(jié)果都很接近,說明本文方法能夠很好模擬預(yù)測(cè)場(chǎng)地振動(dòng)水平.

表2 機(jī)車車輛參數(shù)值Table 2 Parameter values for the vehicles

表3 軌道系統(tǒng)基本參數(shù)Table 3 Parameter values for the track structure

圖9 地表測(cè)點(diǎn)上振動(dòng)加速度級(jí)的模型計(jì)算值與觀測(cè)值的對(duì)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 地層模型基本參數(shù)Table 4 Parameter values for the ground

6.2.2 飽和軟土場(chǎng)地與飽和砂土場(chǎng)地計(jì)算結(jié)果

前述觀測(cè)試驗(yàn)的局限是沒有得到飽和土場(chǎng)地的振動(dòng)數(shù)據(jù),現(xiàn)采用本文模型模擬計(jì)算飽和土場(chǎng)地的環(huán)境振動(dòng).眾所周知,飽和軟土與飽和密砂的動(dòng)力學(xué)特性存在很大差異.陳龍珠等(1998)提供了飽和軟土和飽和砂土的典型參數(shù)值,如表5所示.本算例采用這兩套數(shù)值表示兩種覆蓋層,分別與一個(gè)模量較高、孔隙率極小的近似不透水堅(jiān)硬下臥層搭配,形成兩個(gè)典型的飽和層狀場(chǎng)地.列車和軌道結(jié)構(gòu)參數(shù)仍舊保持表2和表3中數(shù)值不變.

表5 飽和軟土場(chǎng)地和飽和砂土場(chǎng)地的模型參數(shù)Table 5 Parameter values in the models for the grounds of saturated soft clay and saturated sand

圖11和圖12展示了距離軌道中線10、20 m和30 m處各地表點(diǎn)的振動(dòng)級(jí)模擬計(jì)算值.圖11的車速為17 m·s-1,模擬慢速列車; 圖12的車速為101 m·s-1(約等于表5軟土的120 Hz以內(nèi)的瑞利波速),模擬高速列車.對(duì)比飽和砂土場(chǎng)地和軟土場(chǎng)地可見,無論距軌道遠(yuǎn)近,在幾乎所有頻段上,軟土場(chǎng)地的振動(dòng)水平均高于飽和砂土場(chǎng)地.

多篇文獻(xiàn)(Adolfsson et al.,1999; Madshus and Kania 2000; Takemiya,2003; Sheng et al.,2003)報(bào)道,瑞典X2000高速列車行經(jīng)西部海岸線Ledsgard路段的軟土場(chǎng)地時(shí),車速接近瑞利波速,場(chǎng)地振動(dòng)急劇增加.通過對(duì)比圖11和圖12的各條曲線可以發(fā)現(xiàn),本文模擬結(jié)果反映了這個(gè)現(xiàn)象.對(duì)于軟土場(chǎng)地來說,圖12的車速已達(dá)瑞利波速,幾乎所有頻段上的振動(dòng)級(jí)都明顯高于圖11中相應(yīng)數(shù)值.

第5節(jié)模型可以計(jì)算單一波長(zhǎng)的諧波不平順激勵(lì)下的地表位移場(chǎng).設(shè)不平順幅值為1 mm,波長(zhǎng)2 m,則對(duì)于速度c=17 m·s-1的慢速列車,輪軌動(dòng)態(tài)相互作用力的頻率為17/2=8.5 Hz.圖13a為慢速列車運(yùn)行引起的砂土場(chǎng)地表面的豎向位移幅值,列車位置在移動(dòng)坐標(biāo)x-ct軸的0到-76 m處.整體

圖11 慢速軌道交通地表點(diǎn)的振動(dòng)加速度級(jí)Fig.11 VALs at some points on the ground surface caused by a low speed train

圖12 高速鐵路地表點(diǎn)的振動(dòng)加速度級(jí)Fig.12 VALs at some points on the ground surface caused by a high speed train

圖13 c=17 m·s-1的列車在軌道上運(yùn)行引起的地表豎向位移幅值(a) 飽和砂土場(chǎng)地; (a) 飽和軟土場(chǎng)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.

看飽和砂土場(chǎng)地振動(dòng)幅值極小,各組車輪正下方的位移略為明顯.圖13b為慢速列車下的飽和軟土場(chǎng)地,振動(dòng)強(qiáng)度比砂土場(chǎng)地劇烈得多; 各個(gè)輪對(duì)激發(fā)環(huán)狀波動(dòng),相互干涉,傳播很遠(yuǎn); 由于車速遠(yuǎn)小于瑞利波速,列車前方地表可見向前傳播的振動(dòng).

圖14為高速列車計(jì)算結(jié)果,此時(shí)輪軌力頻率為101/2=50.5 Hz.飽和砂土場(chǎng)地與軟土場(chǎng)地的振動(dòng)幅值仍然迥異.50 Hz的移動(dòng)激擾,在砂土地面激發(fā)了向遠(yuǎn)處傳播的環(huán)狀的波動(dòng),如圖14a所示.盡管車速高達(dá)101 m·s,仍然遠(yuǎn)小于飽和砂土的瑞利波速(0～120 Hz頻段為278～300 m·s-1),所以前方可見領(lǐng)先于列車的表面波.

無論從位移幅值,還是從影響范圍看,圖14b的軟土場(chǎng)地位移反應(yīng)都比圖14a的砂土場(chǎng)地劇烈.最大峰值出現(xiàn)在列車前部第一轉(zhuǎn)向架下面,振動(dòng)向列車側(cè)后方向傳播,形成典型的馬赫錐.振動(dòng)局限于馬赫錐之內(nèi),前方場(chǎng)地幾乎沒有地表波動(dòng).

眾所周知,如果移動(dòng)荷載直接作用于半空間表面,與瑞利波同行達(dá)到臨界速度,馬赫角應(yīng)該等于90°.圖14b的馬赫角明顯小于90°,顯示車速已經(jīng)超越了臨界速度.這說明,由于軌道的存在,臨界速度并不等于場(chǎng)地瑞利波速,而是有所降低.保持其他條件不變,僅將車速降至90 m·s-1,地表位移波場(chǎng)中仍然出現(xiàn)了馬赫錐,如圖15所示.這個(gè)結(jié)果提醒軟土場(chǎng)地工程實(shí)踐注意,即使設(shè)計(jì)車速低于場(chǎng)地的瑞利波速,劇烈振動(dòng)的馬赫錐效應(yīng)也可能出現(xiàn).

圖14 c=101 m·s-1的列車在軌道上運(yùn)行引起的地表豎向位移幅值(a) 飽和砂土場(chǎng)地; (a) 飽和軟土場(chǎng)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.

圖15 c=90 m·s-1時(shí)的地面豎向位移幅值Fig.15 Vertical displacement of ground surface when a train speed is 90 m·s-1

7 結(jié)論

動(dòng)力格林函數(shù)是表達(dá)場(chǎng)地的數(shù)學(xué)模型,在列車-軌道-場(chǎng)地系統(tǒng)的耦合振動(dòng)分析中起到關(guān)鍵作用.本文構(gòu)建的格林函數(shù),能夠考慮飽和巖土體的流固兩相動(dòng)力相互作用,場(chǎng)地分層沉積的幾何物理特性以及彈性波在場(chǎng)地中的三維傳播.數(shù)值算例的結(jié)果與文獻(xiàn)結(jié)果十分接近,與現(xiàn)場(chǎng)觀測(cè)試驗(yàn)基本符合.模型反映了高速列車的場(chǎng)地振動(dòng)強(qiáng)度大于低速列車、軟土場(chǎng)地振動(dòng)強(qiáng)度大于飽和砂土場(chǎng)地的規(guī)律.當(dāng)車速接近場(chǎng)地瑞利波速時(shí),基于格林函數(shù)計(jì)算的地表振動(dòng)幅值能夠反映出馬赫錐效應(yīng); 即使車速略低于瑞利波速,劇烈振動(dòng)的馬赫錐亦可能出現(xiàn),這為軟土場(chǎng)地高鐵工程實(shí)踐提供了有益的參考.

應(yīng)該說明的是,還有很多影響因素值得進(jìn)行數(shù)值分析,例如土層厚度，土體的滲透性、孔隙比、密度等對(duì)場(chǎng)地振動(dòng)的影響.但本文意在提出格林函數(shù)的計(jì)算方法并數(shù)值驗(yàn)證,限于篇幅,影響因素將另行撰文討論.

附錄A

(A1)

(A3)

(A4)

Aτ=

(A5)

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(本文編輯 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

國(guó)家自然科學(xué)基金項(xiàng)目(50538030，51108163)，黑龍江省自然科學(xué)基金項(xiàng)目(E201221，E201330)，中國(guó)博士后科學(xué)基金項(xiàng)目(2013M531084)資助.

王福彤，男，1972年生，博士，副教授，主要從事巖土地震工程的研究.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, 謝禮立等.2015.軌道附近地面振動(dòng)模型中的飽和地層動(dòng)力格林函數(shù).地球物理學(xué)報(bào)，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.