飽和土中部分土塞部分外露管樁的水平動(dòng)力阻抗

劉林超， 陳磊磊， 閆啟方

(信陽(yáng)師范學(xué)院 建筑與土木工程學(xué)院，河南 信陽(yáng) 464000)

飽和土中部分土塞部分外露管樁的水平動(dòng)力阻抗

劉林超， 陳磊磊， 閆啟方

(信陽(yáng)師范學(xué)院 建筑與土木工程學(xué)院，河南 信陽(yáng) 464000)

為了考慮管樁的部分土塞效應(yīng)，將管樁劃分為土塞部分和未考慮土塞部分，同時(shí)考慮外露部分管樁的影響，建立了部分土塞部分外露管樁的水平振動(dòng)模型。在忽略管樁縱向位移的情況下，將樁周飽和土視為由無窮多帶一圓孔的薄土層組成，而樁芯土視為由無窮多圓形薄土層組成，運(yùn)用數(shù)學(xué)物理手段求得了樁周飽和土和樁芯飽和土對(duì)管樁的水平動(dòng)力作用，在此基礎(chǔ)上，借助初參數(shù)法和矩陣傳遞法求解了飽和土中部分土塞部分外露管樁的水平振動(dòng)，得到了管樁的水平動(dòng)力阻抗。通過數(shù)值分析可知，管樁壁厚、管樁各段長(zhǎng)度、管樁長(zhǎng)徑比、管樁與樁周飽和土模量比對(duì)飽和土中部分土塞部分外露管樁的水平振動(dòng)有較大的影響，且管樁壁厚的影響與頻率有關(guān)；研究管樁水平動(dòng)力阻抗時(shí)土塞效應(yīng)不應(yīng)被忽略。

土塞效應(yīng)；飽和土；管樁；水平振動(dòng)；水平動(dòng)力阻抗

由于樁基通常要承受水平、豎向、扭轉(zhuǎn)等動(dòng)態(tài)激勵(lì)的作用，所以有關(guān)樁基振動(dòng)特性的研究得到了一定的重視[1-7]，這些研究要么是將樁周土視為單相介質(zhì)，要么是將樁基視為實(shí)芯桿件，這些模型的簡(jiǎn)化使得模型與實(shí)際工程有一定的差距。隨著管樁技術(shù)的發(fā)展和應(yīng)用，管樁在建筑基礎(chǔ)、橋梁基礎(chǔ)、高鐵基礎(chǔ)等眾多領(lǐng)域得到了應(yīng)用，近年來，針對(duì)管樁動(dòng)態(tài)特性的研究越來越受到眾多學(xué)者的關(guān)注，以飽和土中管樁的振動(dòng)特性為研究對(duì)象，應(yīng)躍龍等[8]在考慮質(zhì)量耦合效應(yīng)的情況下研究了飽和黏彈性地基土中管樁的豎向振動(dòng)；鄭長(zhǎng)杰等[9]對(duì)飽和土地基中現(xiàn)澆大直徑管樁的水平振動(dòng)做了研究；劉林超等[10-11]基于多孔介質(zhì)理論研究了飽和土中單樁的水平和縱向振動(dòng)；靳建明等[12]基于Biot提出的飽和多孔介質(zhì)的波動(dòng)方程，研究了均質(zhì)各向同性飽和土中端承管樁的扭轉(zhuǎn)振動(dòng)問題。由于實(shí)際工程施工中，管樁在成樁的過程中，管樁內(nèi)會(huì)形成一定高度的土塞，且管樁的直徑越大，管樁內(nèi)土塞高度就越大，但土塞高度并不是完全充滿管樁。針對(duì)管樁的土塞效應(yīng)，吳文兵等[13]基于附加質(zhì)量法，研究了考慮土塞效應(yīng)時(shí)成層地基中管樁的縱向振動(dòng)問題。本文將基于多孔介質(zhì)理論，在考慮土塞效應(yīng)的情況下研究飽和土中部分土塞部分外露管樁的水平振動(dòng)問題。

1 數(shù)學(xué)模型與控制方程

由于實(shí)際工程中樁芯土并不是完全充滿，這里研究圖1所示的飽和土中部分土塞部分外露端承管樁的水平簡(jiǎn)諧振動(dòng)問題，樁底為基巖。部分土塞部分外露管樁在樁頂水平簡(jiǎn)諧荷載P(t)=P0eiωt的作用下作簡(jiǎn)諧振動(dòng)，動(dòng)荷載頻率為ω，i是虛數(shù)單位。土塞深度(即樁芯部分土深度)為L(zhǎng)1，飽和土層地表面到土塞頂面距離為L(zhǎng)2，管樁外露部分長(zhǎng)度為L(zhǎng)3，樁身總長(zhǎng)為L(zhǎng)，且L=L1+L2+L3。

圖1 飽和土中部分土塞部分外露管樁的水平振動(dòng)模型Fig.1 Horizontal dynamic model of part soil plug part exposed pipe pile in saturated soil

將樁周土和樁芯土視為飽和兩相介質(zhì)，為了方便求解，將樁周飽和土視為由無窮多帶一半徑為rI圓孔的薄土層組成，而樁芯土視為由無窮多圓形薄土層組成，各土層之間相互獨(dú)立，忽略樁周飽和土和樁芯飽和土的豎向位移，僅考慮飽和土的徑向位移和環(huán)向位移，同時(shí)徑向位移和環(huán)向位移與z無關(guān)。由多孔介質(zhì)理論[14-15]可知樁周飽和土和樁芯飽和土的控制方程為

(1)

(2)

(3)

(4)

(5)

2 樁周飽和土和樁芯飽和土對(duì)管樁的水平動(dòng)力作用

(6)

(7)

進(jìn)行解耦可得無量綱化后的樁周飽和土和樁芯飽和土的控制方程分別為

(8)

(9)

考慮樁周飽和土徑向位移和環(huán)向位移的奇偶性，以及樁周飽和土無窮遠(yuǎn)處位移為零的邊界條件，求解式(8)可得樁周飽和土的徑向位移、環(huán)向位移和孔隙水壓力分別為

(10)

(11)

(12)

(13)

(14)

(15)

設(shè)樁周飽和土和樁芯飽和土與管樁交界面處不透水，則有如下邊界條件

(16)

由式(12)和式(15)可得

AO3=aO6AO2，AI3=aI6AI2

(17)

將管樁與飽和土的動(dòng)力相互作用利用Winkler彈簧-阻尼器模型來描述，為此需先求出相應(yīng)的剛度系數(shù)和阻尼系數(shù)。當(dāng)管樁產(chǎn)生單位水平位移時(shí)，樁周飽和土和樁芯飽和土對(duì)管樁的水平作用即為管樁-飽和土動(dòng)力相互作用阻抗。當(dāng)管樁產(chǎn)生單位水平位移時(shí)，管樁與樁周飽和土和樁芯飽和土接觸面處的位移邊界條件為

(18)

由樁周飽和土和樁芯飽和土的徑向位移和環(huán)向位移得

(20)

(21)

(22)

由式(19)～式(22)可求得待定系數(shù)AO1、AO2、AI1、AI2分別為

AO1=

(23)

AO2=

(24)

(25)

(26)

由此可以完全確定樁周飽和土和樁芯飽和土的徑向位移和環(huán)向位移，在徑向位移和環(huán)向位移的基礎(chǔ)上根據(jù)飽和土固相土骨架的本構(gòu)關(guān)系可得到樁周和樁芯飽和土的徑向應(yīng)力和環(huán)向應(yīng)力，并可以求出單位厚度樁周飽和土和樁芯飽和土對(duì)管樁的水平作用力分別為

(27)

(28)

對(duì)于L1段樁身，樁周飽和土和樁芯飽和土對(duì)管樁都有作用，此時(shí)樁周飽和土和樁芯飽和土對(duì)管樁的水平作用力為

(29)

對(duì)于L2段樁身，僅有樁周飽和土對(duì)管樁有作用，此時(shí)樁周飽和土對(duì)管樁的水平作用力為

(30)

式中：k1h、c1h、k2h、c2h分別為L(zhǎng)1段和L2段管樁與飽和土動(dòng)力相互作用的剛度系數(shù)和阻尼系數(shù)。而L3段樁身由于外露所以沒有飽和土的作用，即k3h=c3h=0。

3 飽和土中部分土塞部分外露管樁水平振動(dòng)求解

對(duì)于圖1所示的飽和土中部分土塞部分外露管樁，考慮式(29)和式(30)，可以建立無量綱化的飽和土中部分土塞部分外露管樁的水平振動(dòng)控制方程為

(31)

(32)

式中，Uβp(0)、φβp(0)、Qβp(0)、Mβp(0)為第β段管樁樁底端的位移和內(nèi)力，

(33)

其中，lβ=Lβ/L、Uβp(lβ)、φβp(lβ)、Qβp(lβ)、Mβp(lβ)為第β段管樁管樁樁底端的位移和內(nèi)力

[T]β=

(34)

運(yùn)用矩陣傳遞法，可以得到整個(gè)部分土塞部分外露管樁樁頂和樁底位移、轉(zhuǎn)角、剪力和彎矩的關(guān)系為

(35)

其中，

(36)

由于圖1所示的部分土塞部分外露管樁為端承管樁，且樁底與基巖完全固定，所以管樁樁底位移和轉(zhuǎn)角為零。根據(jù)管樁水平動(dòng)力阻抗的定義可知，對(duì)于上部為剛性承臺(tái)的管樁，當(dāng)樁頂約束轉(zhuǎn)角(轉(zhuǎn)角為零)時(shí)，產(chǎn)生單位水平位移時(shí)所需的剪力即為所求的管樁水平動(dòng)力阻抗，由式(35)可得

(37)

由此可得部分土塞部分外露管樁樁頂?shù)乃絼?dòng)力阻抗

(38)

4 數(shù)值算例與討論

圖2 管樁壁厚對(duì)水平動(dòng)力阻抗的影響Fig.2 Influence of wall thickness on horizontal dynamic impedance of part soil plug part exposed pipe pile

圖3 管樁長(zhǎng)徑比對(duì)水平動(dòng)力阻抗的影響Fig.3 Influence of length to diameter ratio on horizontal dynamic impedance of part soil plug part exposed pipe pile

圖4 管樁各段長(zhǎng)度對(duì)水平動(dòng)力阻抗的影響Fig.4 Influence of length of each segment on horizontal dynamic impedance of part soil plug part exposed pipe pile

圖5 管樁樁身與樁周飽和土模量比對(duì)水平動(dòng)力阻抗的影響Fig.5 Influence of modulus ratio between pipe pile and saturated soil around pile on horizontal dynamic impedance of part soil plug part exposed pipe pile

圖6 飽和土液固耦合系數(shù)對(duì)水平動(dòng)力阻抗的影響Fig.6 Influence of liquid and solid coupling coefficient on horizontal dynamic impedance of part soil plug part exposed pipe pile

5 結(jié) 論

本文在多孔介質(zhì)理論的基礎(chǔ)上，考慮部分土塞效應(yīng)，運(yùn)用初始參數(shù)法和矩陣傳遞法對(duì)飽和土中部分土塞部分外露管樁的水平振動(dòng)進(jìn)行了求解，并通過數(shù)值算例對(duì)水平動(dòng)力阻抗進(jìn)行了研究。主要得到以下主要結(jié)論：

(1)管樁壁厚對(duì)飽和土中部分土塞部分外露管樁水平動(dòng)力阻抗的影響與頻率有一定的關(guān)系，低頻和高頻時(shí)的影響規(guī)律不太一樣。

(2)隨著管樁長(zhǎng)徑比的增大，飽和土中部分土塞部分外露管樁水平動(dòng)力阻抗的實(shí)部和虛部越小。

(3)各段長(zhǎng)度對(duì)飽和土中部分土塞部分外露管樁水平動(dòng)力阻抗實(shí)部的影響較大，土塞部分深度最大時(shí)水平動(dòng)力阻抗要比其他情況大，土塞效應(yīng)較為明顯。

(4)提高管樁樁身混凝土強(qiáng)度對(duì)提飽和土中部分土塞部分外露管樁的水平動(dòng)力阻抗作用非常明顯。

[1] NOVAK M,NOGAMI T. Soil-pile interaction in horizontal vibration[J]. Earthquake Engineering and Structural Dynamics, 1977,5(3): 263-281.

[2] GAZETAS G,MAKRIS N. Dynamic pile-soil-pile interaction: analysis of axial vibration [J]. Earthquake Engineering and Structural Dynamics,1991,20(3): 115-132.

[3] AI Zhiyong , LIU Chunlin.Vertical vibration of a pile in transversely isotropic multilayered soils[J].Journal of Sound and Vibration, 2015,357: 145-155.

[4] Lü Shuhui,WANG Kuihua,WU Wenbing. Longitudinal vibration of pile in layered soil based on Rayleigh-Love rod theory and fictitious soil-pile model[J].Journal of Central South University,2015,22(5): 1909-1918.

[5] SHI Li, XU Changjie, CAI Yuanqiang,et al. Dynamic impedances and free-field vibration analysis of pile groups in saturated ground[J].Journal of Sound and Vibration, 2014,333 (16): 3709-3731.

[6] 聞敏杰, 徐金明.非完全黏結(jié)下飽和土中樁扭轉(zhuǎn)振動(dòng)的解[J]. 工程力學(xué),2014,31(7): 156-161.

WEN Minjie,XU Jinming.Analytical solution for torsional vibration of a pile in saturated soil considering imperfect contact[J].Engineering Mecahanics,2014,31(7): 156-161.

[7] 吳文兵,蔣國(guó)盛,鄧國(guó)棟,等.黏彈性地基中基于虛土樁模型的樁頂縱向振動(dòng)阻抗研究[J].振動(dòng)與沖擊,2015,34(7): 192-198.

WU Wenbing, JIANG Guosheng,DENG Guodong,et al. Vertical dynamic impedance of pile embedded in viscoelastic soil based on fictitious soil pile model[J]. Journal of Vibration and and Shock,2015,34(7): 192-198.

[8] 應(yīng)躍龍,羅海亮,聞敏杰.飽和黏彈性地基土中管樁縱向振動(dòng)研究[J].巖土力學(xué),2013,34(增刊1): 103-108.

YING Yuelong, LUO Hailiang,WEN Minjie.Study of vertical vibration of pipe pile in saturated viscoelastic soil[J].Rock and Soil Mechanics, 2013,34(Sup1): 103-108.

[9] 鄭長(zhǎng)杰,劉漢龍,丁選明，等.飽和黏性土地基中現(xiàn)澆大直徑管樁水平振動(dòng)響應(yīng)解析解[J].巖土工程學(xué)報(bào),2014,36(8): 1447-1454.

ZHENG Changjie,LIU Hanlong,DING Xuanming,et al. Analytical solution of horizontal vibration of cast-in-place large-diameter pipe piles in saturated soils[J]. Chinese Journal of Geotechnical Engineering, 2014,36(8): 1447-1454.

[10] 劉林超,閆啟方. 飽和土中管樁的水平動(dòng)阻抗研究[J].巖土力學(xué),2014,35(5): 1348-1356.

LIU Linchao, YAN Qifang. Study of lateral dynamic impedance of pipe pile in saturated soil [J].Rock and Soil Mechanics, 2014,35(5): 1348-1356.

[11] 劉林超, 閆啟方. 飽和土中管樁的縱向振動(dòng)特性[J]. 水利學(xué)報(bào), 2011, 42(3): 366-372.

LIU Linchao, YAN Qifang. Vertical vibration characteristics of pipe pile in saturated soil[J]. Journal of Hydraulic Engineering,2011, 42(3): 366-372.

[12] 靳建明,張智卿,吳章土,等.飽和土中管樁的扭轉(zhuǎn)振動(dòng)特性研究[J].振動(dòng)與沖擊,2015,34(11): 52-57.

JIN Jianming,ZHANG Zhiqing,WU Zhangtu,et al. Torsional vibration of a pipe pile embedded in a porous- saturated soil[J].Journal of Vibration and Shock, 2015,34(11): 52-57.

[13] 吳文兵,蔣國(guó)盛,王奎華,等.土塞效應(yīng)對(duì)管樁縱向動(dòng)力特性的影響研究[J].巖土工程學(xué)報(bào),2014,36(6): 1129-1141.

WU Wenbing, JIANG Guosheng,WANG Kuihua,et al. Influence of soil plug effect on vertical dynamic response of pipe piles [J]. Chinese Journal of Geotechnical Engineering, 2014,36(6): 1129-1141.

[14] DE BOER R, LIU Z. Propagation of acceleration waves in incompressible saturated porous solids[J].Transport in Porous Media,1995,21(2): 163-173.

[15] DE BOER R, LIU Z. Plane waves in a semi-infinite fluid saturated porous medium[J]. Transport in Porous Media, 1994,16(2): 147-173.

Horizontaldynamicimpedanceofpartsoilpartexposedpipepileinsaturatedsoil

LIULinchao,CHENLeilei,YANQifang

(SchoolofAchitectureandCivilEngineering,XinyangNormalUniversity,Xinyang464000,China)

In order to consider soil plugging effect of pipe piles, a pipe pile is divided into soil-plug part and non-soil-plug part, the horizontal vibration model for the soil-plug part exposed pipe pile was established considering the effect of the exposed part of the pipe pile. In the case of ignoring the longitudinal displacement of the pipe pile, saturated soil around the pile was regarded as composed by infinite thin soil layers with a round hole, and the pile inner saturated soil was regarded as composed by infinite circular thin soil layers, and the horizontal dynamic forces of saturated soil around the pile and pile inner saturated soil on the pipe pile were obtained by using mathematical and physics methods. Based on the initial parameter method and matrix transfer method, the horizontal vibration of soil plug part exposed pipe pile was solved, and the horizontal dynamic impedance of the pipe pile was also obtained. The numerical analysis shows that the wall thickness, length of each segment, length to diameter ratio, modulus ratio between the pipe pile and saturated soil around the pile have great effect on the lateral vibration of part soil plug part exposed pipe pile in saturated soil, and the influence of wall thickness of pipe pile is related to frequency; the soil plugging effect should not be neglected in studying the lateral dynamic impedance of pipe piles.

soil plugging effect; saturated soil; pipe pile; horizontal vibration; horizontal dynamic impedance

國(guó)家自然科學(xué)基金(U1504505)；河南省科技發(fā)展計(jì)劃項(xiàng)目(1423 00410200)；河南省高等學(xué)校青年骨干教師資助計(jì)劃項(xiàng)目(2013GGJS-121)；河南省高等學(xué)校重點(diǎn)科研項(xiàng)目(15A560036)；信陽(yáng)師范學(xué)院南湖學(xué)者獎(jiǎng)勵(lì)計(jì)劃青年項(xiàng)目(201606)

2016-06-27 修改稿收到日期： 2016-08-26

劉林超 男，博士，副教授，1979年生

TU435

A

10.13465/j.cnki.jvs.2017.20.025