飽和土體中多排柱腔屏障對(duì)壓縮快波隔離

第一作者徐平男，博士，副教授，1977年2月生

飽和土體中多排柱腔屏障對(duì)壓縮快波隔離

徐平1，鐵瑛2，陳斌3

(1. 鄭州大學(xué)水利與環(huán)境學(xué)院，鄭州450001；2. 鄭州大學(xué)機(jī)械工程學(xué)院，鄭州450001；3.寧波市軌道交通工程建設(shè)指揮部, 浙江寧波315012)

摘要：假定柱腔(圓柱形空腔)長(zhǎng)度遠(yuǎn)大于直徑，將多排柱腔組成的非連續(xù)屏障對(duì)平面P1波(壓縮快波)隔離簡(jiǎn)化成彈性波多重散射的二維平面問題，利用復(fù)變函數(shù)保角映射法及波場(chǎng)勢(shì)函數(shù)展開法，據(jù)柱腔自由(土骨架軸向、切向應(yīng)力為零)邊界條件獲得散射波場(chǎng)勢(shì)函數(shù)展開式的待定復(fù)系數(shù)理論解。通過分析位移比值(屏障后某點(diǎn)由入射、散射彈性波產(chǎn)生的總位移與入射彈性波單獨(dú)產(chǎn)生的位移之比)變化規(guī)律，對(duì)比單、雙及三排柱腔屏障的隔離效果。研究表明，邊界不透水柱腔屏障隔離效果好于透水；多排柱腔屏障對(duì)高頻入射P1波隔離效果好于低頻入射；隨排數(shù)增多柱腔屏障隔離效果明顯提高，有效隔離區(qū)域明顯增大；當(dāng)排數(shù)達(dá)到三排時(shí)柱腔屏障后一定范圍內(nèi)隔離效果達(dá)70%，效果較理想。

關(guān)鍵詞：飽和土體；多排柱腔；非連續(xù)屏障；隔離效果；隔振設(shè)計(jì)

基金項(xiàng)目：國(guó)家自然科學(xué)

收稿日期：2014-02-26修改稿收到日期：2014-08-19

中圖分類號(hào):TU435文獻(xiàn)標(biāo)志碼：A

Isolation of fast longitudinal waves by barriers composed of several rows of cylindrical cavities in saturated soils

XUPing1,TIEYing2,CHENBin3(1. School of Water and Environment, Zhengzhou University, Zhengzhou 450001, China;2.School of Mechanical Engineering, Zhengzhou University, Zhengzhou 450001, China;3.Ningbo Urban Rail Transit Project Construction Headquarters, Ningbo 315012, China)

Abstract:The lengths of cylindrical cavities were considered as much larger than their diameters, so the isolation of incident plane P1 waves (fast compressive wave) by discontinuous barriers composed of several rows of cylindrical cavities was simplified as a two-dimensional plane problem. The conformal mapping method of complex functions and expansion method of wave functions were adopted, the cylindrical cavities were considered to be of free boundaries, where no stresses exist, and then the theoretical solutions about the complex coefficients of scattering wave potential functions were obtained. By studying the normalized displacement amplitudes, which are expressed as the ratios of displacement amplitudes of soils behind the barriers caused by both the incident and scattering waves to those only by the incident P1 waves, the isolation effects of the barriers composed of one, two and three rows of cylindrical cavities were compared. The results reveal that the isolation effects of barriers composed of several waterproof cylindrical cavities are better than those composed of permeable cylindrical cavities; the isolation effects of barriers composed of several rows of cylindrical cavities on incident P1 waves with higher frequencies are much better than on those with lower frequencies; with the increasing of row number of cylindrical cavities, the vibration isolation effects is obviously improved, and the effective isolation area is obviously increased; as the row number of cylindrical cavities reaches three, the isolation effects at some locations behind the barriers are much better and almost increase to 70%.

Key words:saturated soil; several rows of cylindrical cavities; discontinuous barrier; isolation effect; vibration isolation design

非連續(xù)屏障由多個(gè)不連續(xù)個(gè)體(如樁、柱腔)組成，盡管對(duì)振動(dòng)波隔離效果略低于空溝，但施工方便、維護(hù)費(fèi)用低，較空溝應(yīng)用前景更廣。大多采用理論分析及數(shù)值模擬等方法研究非連續(xù)屏障，Avilés等[1-2]首次用波動(dòng)理論獲得均質(zhì)土體中單排樁對(duì)S波隔離的精確解；Kattis等[3-4]通過頻域邊界元技術(shù)建立三維單排圓樁、方樁隔振問題模型研究表明，樁間距為影響隔振效果的決定性因素。李志毅等[5]以瑞利波散射積分方程為基礎(chǔ)，研究多排樁屏障對(duì)瑞利波的三維隔離問題，認(rèn)為多排樁屏障體系隔振效果主要取決于樁的排數(shù)，而排距對(duì)隔振效果影響較小。時(shí)剛等[6]通過推導(dǎo)單排樁屏障對(duì)瑞利波散射的三維邊界元方程，認(rèn)為增加單排樁長(zhǎng)度、提高樁身剪切模量均能有效提高隔振效果；高廣運(yùn)等[7]利用迭代法對(duì)瑞利波散射積分方程精確求解，分析彈性土體中多排樁對(duì)瑞利波隔離。陸建飛等[8]用數(shù)值方法模擬多排樁對(duì)高速列車振動(dòng)隔離表明，小間距、大樁長(zhǎng)多排樁屏障隔離效果最好。徐平[9]用保角映射法及波場(chǎng)位移勢(shì)函數(shù)展開法對(duì)多排空心管樁屏障對(duì)平面SV波隔離問題進(jìn)行理論求解及數(shù)值計(jì)算，并分析管樁壁厚、排數(shù)對(duì)隔離效果影響。李校兵等[10]利用一組可求解多散射問題的圓柱坐標(biāo)系統(tǒng)及界面處位移連續(xù)條件，建立飽和土體中單排柱腔對(duì)壓縮快波的理論解答，認(rèn)為飽和土的滲透性對(duì)隔離效果影響較明顯。天然土體為由固、液、氣三相介質(zhì)組成的集合體，若孔隙被流體(通常意義的水)充滿則為飽和土，故需研究飽和土體中非連續(xù)屏障隔振效果，但關(guān)于飽和土體中多排柱腔組成的非連續(xù)屏障隔振效果研究尚少見。

不同于一般彈性介質(zhì)，飽和土體中可傳播一種剪切波(S波)與兩種壓縮波，即快壓縮(P1)波與慢壓縮(P2) 波[11]。本文用復(fù)變函數(shù)的保角映射法及波函數(shù)展開法，獲得多排柱腔屏障對(duì)平面彈性波隔離問題的理論解，并對(duì)比單、雙、三排柱腔屏障隔離效果。

1飽和土體中多排柱腔波場(chǎng)展開

1.1飽和土體中波場(chǎng)理論

據(jù)Biot波動(dòng)理論[11]，飽和土體基本控制方程為

土體應(yīng)力-應(yīng)變關(guān)系

σij=λui,iδij+2μεij-αpfδij

(1)

滲流連續(xù)性方程

-pf=Mwi,i+αMui,i

(2)

土體運(yùn)動(dòng)方程(不計(jì)體力)

(3)

流體運(yùn)動(dòng)方程

(4)

式中：pf為孔隙流體壓力；λ，μ為固相土骨架Lamé彈性常量；u為固相土骨架位移；w為孔隙流體相對(duì)土骨架位移；M，α分別為表征土顆粒、孔隙流體壓縮性常數(shù)；δij為Kronecker Delta符號(hào)；ρ=fρf+(1-f)ρs為飽和土體總密度，ρs為土顆粒密度，ρf為孔隙流體密度，f為飽和土體孔隙率；m=ρf/f；b=η/kd，η，kd分別為孔隙流體粘滯、滲透系數(shù)。

將式(1)、(2)代入式(3)、式(2)代入式(4)，整理可得飽和土體波動(dòng)問題矢量方程為

(5)

式中：λc=λ+α2M；u，w為位移矢量。

引入標(biāo)量勢(shì)φ，φ及矢量勢(shì)ψ，χ表示土骨架與孔隙流體中波函數(shù)，則波場(chǎng)可分解為

(6)

將式(6)代入式(5)，得飽和土體的波動(dòng)方程為

(7)

式中：ω為入射頻率；Q=mω2+iωb。

經(jīng)求解可得飽和土體中P1波、P2波及S波波數(shù)k1，k2，ks計(jì)算公式為

(8)

用勢(shì)函數(shù)φ1，φ2，ψ分別表示土骨架中P1、P2、S波波場(chǎng)，φ1，φ2，χ分別表示孔隙流體中P1、P2、S波波場(chǎng)，據(jù)式(7)、(8)可得勢(shì)函數(shù)φ1，φ2，χ與φ1，φ2，ψ之間轉(zhuǎn)換關(guān)系式為

φ1=γ1φ1,φ2=γ2φ2,χ=γsψ

(9)

式中：γ1，γ2，γs表達(dá)式為

(10)

1.2入射波場(chǎng)勢(shì)函數(shù)展開

飽和土體中P2波衰減較快，且占總彈性波能量比例較小，因此對(duì)爆炸、交通、重工業(yè)廠房等常見振動(dòng)問題，通常不考慮入射P2波。入射P1及S波會(huì)產(chǎn)生耦合散射，即飽和土體散射波中同時(shí)含P1、P2、S波成分，因此P1波與S波求解過程完全相似，限于篇幅，本文僅對(duì)入射P1波解答。

圖1　柱腔分布及坐標(biāo)系設(shè)置 Fig.1 Distribution of cylindrical cavities and coordinate systems

取入射平面P1波幅值為φ0，在參考直角坐標(biāo)系(x,y)下，飽和土體土骨架中入射平面P1波勢(shì)函數(shù)為

φinc=φ0exp[ik1(xcosβ+ysinβ)]

(11)

式中：上標(biāo)inc表示入射；β為入射P1波與水平方向(即x軸)夾角。

(12)

(13)

1.3散射波場(chǎng)勢(shì)函數(shù)展開

(14)

(15)

通過疊加，觀測(cè)點(diǎn)M處飽和土體土骨架中總散射P1波的勢(shì)函數(shù)展開式及M處總散射S波勢(shì)函數(shù)為

2待定復(fù)系數(shù)求解

引入保角變換zj=rjexp(iθj)，由彈性介質(zhì)中應(yīng)力、位移極坐標(biāo)表達(dá)式[12]，得飽和土體中應(yīng)力、位移極坐標(biāo)表達(dá)式為

設(shè)飽和土體在柱腔邊界處(rj=aj)自由，邊界分不透水(柱腔建成后立即插入PVC管材，阻斷水滲透)、透水(柱腔建成后保持土體原有滲透性)兩種，即

(18)

(19)

邊界不透水為

邊界透水為

(20)

(21)

由于式(21)為Fourier-Bessel函數(shù)的無窮級(jí)數(shù)解，n的取值為-∞～∞，實(shí)際計(jì)算時(shí)無法實(shí)現(xiàn)，更無必要，只需將n截取到N，保證N已對(duì)應(yīng)力、位移不再產(chǎn)生明顯影響即可，由文獻(xiàn)[14]，規(guī)定本文計(jì)算的容許誤差為0.005%。

3隔離效果分析

為簡(jiǎn)化算例計(jì)算，取所有柱腔半徑a相同，且均勻布置，設(shè)入射波垂直屏障入射，即β=π/2。為分析屏障隔離效果引入位移比值(屏障后某點(diǎn)由入射P1波與散射P1波、P2波、S波產(chǎn)生的位移之和vy與未設(shè)屏障時(shí)由入射P1波單獨(dú)產(chǎn)生的位移v0比值)|vy/v0|，該比值越小，說明屏障隔離效果越好。由文獻(xiàn)[13]取飽和土體物理力學(xué)參數(shù)見表1。

表1　飽和土體的物理力學(xué)參數(shù)

由文獻(xiàn)[1-2]知，d/a=3.0時(shí)單排樁屏障的隔離效果最佳，故本文取柱腔間距d/a=3.0，柱腔按正三角形布置，見圖2。計(jì)算時(shí)單排柱腔N=8，雙排柱腔N=17，三排柱腔N=25。

圖2　多排柱腔布置 Fig.2 Arrangement of several rows of cylindrical cavities

3.1計(jì)算結(jié)果正確性驗(yàn)證

與文獻(xiàn)[10]對(duì)比分析，取單排柱腔屏障(N=8、d/a=3.0)，繪制屏障中心線(x/a=10.5)|vy/v0|沿y/a的變化曲線，見圖3。由圖3看出，本文與文獻(xiàn)[10]結(jié)果相同，從而驗(yàn)證本文理論解答與計(jì)算程序(Matlab軟件編制)的正確性。

圖3　單排柱腔屏障中心線|v y/v 0|沿y/a變化曲線 Fig.3 Curves of|v y/v 0|changing with y/a at the center of one row of cylindrical cavities

3.2柱腔排數(shù)及邊界透水對(duì)隔離效果影響

取入射P1波無量綱頻率Re(k1a)=0.2(低頻)，繪制單、雙、三排柱腔屏障后一定區(qū)域內(nèi)(0≤x/a≤21, 0≤y/a≤300)的|vy/v0|等值線，見圖4～圖6。比較圖4～圖6看出，①對(duì)相同柱腔屏障，不透水邊界的柱腔屏障隔離效果好于透水邊界，因不透水邊界，孔隙流體相對(duì)土骨架位移wr在柱腔邊界上為零，孔隙流體位移不能繼續(xù)傳遞，屏障后位移有所減小，即不透水邊界更能有效耗散振動(dòng)波能量。②隨柱腔排數(shù)增多，屏障隔離效果明顯提高，有效隔離區(qū)域明顯擴(kuò)大，達(dá)到三排時(shí)屏障后一定區(qū)域內(nèi)(不透水邊界y/a≤110、透水邊界y/a≤90)|vy/v0|≤0.3，即隔離效果超過70%，較理想。

圖4 單排8個(gè)柱腔后位移比值vy/v0等值線(低頻)Fig.4Contoursofvy/v0behindarowof8cylindricalcavitiesforincidentP1waveswithlowfrequencies圖5 雙排17個(gè)柱腔后vy/v0的等值線(低頻)Fig.5Contoursofvy/v0behindtworowsof17cylindricalcavitiesforincidentP1waveswithlowfrequencies圖6 三排25個(gè)柱腔后vy/v0的等值線(低頻)Fig.6Contoursofvy/v0behindthreerowsof25cylindricalcavitiesforincidentP1waveswithlowfrequencies

圖7　三排25個(gè)柱腔后|v y/v 0|等值線(高頻) Fig.7 Contours of |v y/v 0|behind three rows of 25 cylindrical cavities for incident P 1 waves with high frequencies

3.3入射頻率對(duì)隔離效果影響

取入射P1波無量綱頻率Re(k1a)=1.0(高頻)，繪制三排柱腔屏障后一定區(qū)域內(nèi)(0≤x/a≤21, 0≤y/a≤300)的|vy/v0|等值線，見圖7。

4結(jié)論

用復(fù)變函數(shù)保角映射法與波場(chǎng)勢(shì)函數(shù)展開法獲得飽和土體中多排柱腔屏障對(duì)各種平面P1波隔離問題的理論解。通過數(shù)值計(jì)算，結(jié)論如下：

(1)邊界不透水時(shí)，柱腔屏障隔離效果好于透水；柱腔屏障對(duì)高頻入射P1波隔離效果好于低頻；屏障后一定區(qū)域內(nèi)隔離效果超過70%，較理想。

(2)柱腔排數(shù)達(dá)到三排時(shí)屏障后y/a≤100長(zhǎng)度范圍內(nèi)隔離效果達(dá)到或超過70%，隔振效果較理想。

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