層狀地基與彈性薄板相互作用的邊界元解*

艾智勇，蔡建邦

(1.同濟大學 地下建筑與工程系，上海 200092；2.同濟大學 巖土及地下工程教育部重點實驗室，上海 200092)

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層狀地基與彈性薄板相互作用的邊界元解*

艾智勇1, 2?，蔡建邦1, 2

(1.同濟大學 地下建筑與工程系，上海 200092；2.同濟大學 巖土及地下工程教育部重點實驗室，上海 200092)

將無限大薄板的基本解作為薄板邊界積分方程的核函數(shù)，對薄板的內部和邊界進行離散，并假定薄板內部和邊界上的節(jié)點與地基反力的分布情況，得到薄板的邊界元方程組；同時基于層狀地基的解析層元解，通過Guass-Legendre積分得到地基柔度矩陣；結合地基與薄板接觸面上的位移協(xié)調條件，得到層狀地基與薄板共同作用問題總的邊界元法方程組；求解該方程組，得到層狀地基與薄板共同作用問題的解答.基于本文理論，編制了相應的FORTRAN程序，通過與已有文獻結果對比驗證本文理論及程序的正確性，數(shù)值分析結果表明：方形基礎薄板情況下，離板中心越近，垂直于坐標軸y(x)方向、距離相等的2條線段的豎向位移差越小，且該位移差隨著板-土剛度比減小而減?。浑S著板長寬比的增大，板中心點與長邊中點位移差變化不明顯，而短邊中心與邊界角點的位移差也有相類似的規(guī)律.

邊界元；層狀地基；薄板；解析層元

筏板基礎具有剛度大、整體性好、能較好地抵抗不均勻沉降的優(yōu)點，因此在高層建筑中得到了廣泛的應用.目前，地基與板相互作用分析的方法主要有：有限差分法[1]、有限單元法[2-4]、邊界單元法[5]、邊界單元-有限單元耦合法[6]、廣義微分求積法[7]、半解析數(shù)值方法[8-10]，以及有限網(wǎng)格法[11]等.相比于有限元、有限層等方法，邊界單元法能將求解過程的維數(shù)降低一維，并具有計算時間短、精度高等優(yōu)點.因此，很多學者運用邊界元法來研究筏板與地基的相互作用問題.佘穎禾和朱萬寧[12]將地基效應歸并到地基板的彎曲微分方程內，得到了含有第三類復變量的Bessel函數(shù)的基本解，再根據(jù)該問題的邊界積分方程，建立了Winkler和雙參數(shù)地基上薄板的無奇異邊界單元法.王建國和黃茂光[13]提出了雙參數(shù)地基上薄板問題的邊界單元解法.鄧安福等[14]采用邊界單元法研究了雙參數(shù)地基上的厚板問題.Rashed等[15]通過邊界元法研究了Winkler地基上的厚板問題.閆富有等[16]基于Reissner 板的邊界積分方程，建立了有限壓縮層地基上厚筏基礎與地基相互作用分析的邊界元法.

由以上研究可知，目前板-土相互作用的邊界單元法研究所采用的地基模型大多是Winkler和雙參數(shù)地基模型.Winkler地基模型將地基對板的作用看做是一系列相互獨立的彈簧，忽略了彈簧之間的剪切作用，因此只適用于很軟弱的地基土.雙參數(shù)地基模型雖然在獨立彈簧之間引入力學的相互作用以消除其不連續(xù)性，但其參數(shù)較難獲取，因而限制了它的工程應用.天然地基由于沉積而常常呈層狀分布，因此采用層狀地基模型更加符合工程實際.而層狀地基上板的邊界元研究還很少見諸報道.為此本文對層狀地基與彈性薄板的共同作用問題進行邊界元分析，以便精確、高效地求解基礎板問題.

1 彈性薄板的邊界積分方程

考慮橫向分布荷載p時，彈性薄板的控制方程為：

D22s=p.

(1)

式中：s為板中面的橫向位移；2為Laplace算子；)]，為薄板的抗彎剛度，其中，hp,Ep和νp分別為板的厚度、彈性模量和泊松比.

由式(1)可得無限大平面薄板(如圖1所示)的基本解為[17]：

(2)

(3)

2(1+νp)(lnr+1)},

(4)

圖1 薄板邊界元法示意圖Fig.1 The diagram of the BEM for a thin plate

采用由式(2)—(5)得到的基本解作為邊界積分方程的核函數(shù)，于是可以得到具有光滑邊界的薄板撓度邊界積分方程：

(6)

式中：當源點在板內部時，即ξ1∈Ω時，c(ξ1)=1，當源點在板邊界時，即ξ1∈Γ時，c(ξ1)=0.5；Mn,Vn和θ分別表示薄板邊界的彎矩、橫向力、轉角；Γ和Ω分別表示薄板的邊界域和內域；α為nξ與x的夾角，θ為幅角，β=α-θ，其中，tξ和nξ分別為ξ2點處的切線方向和外法線方向.

2 層狀地基與薄板的共同作用

假設板四邊自由，有：

Vn(ξ2)=0,

(7)

Mn(ξ2)=0.

(8)

結合地基反力分布假設和式(7)與式(8)，當源點為內點及邊界點時，式(6)可分別表示為式(9)和式(10)，即：

(9)

(10)

式中：Γi(i=1,2,…,Ne)對應于各段邊界的單元；Ωi對應于反力qi(i=1,2,…,Nc)的作用面積；sc和se分別表示薄板內點、邊界點的豎向位移.

圖2 層狀地基與薄板相互作用圖Fig.2 The draft of the interaction betweenlayered soils and a thin plate

圖3 彈性薄板單元及節(jié)點圖Fig.3 The elements and nodes of an elastic thin plate

當板上作用均布荷載p時，薄板的邊界積分方程可表示為：

(11)

其中，

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

式中:se,θ,sc,q分別為薄板邊界節(jié)點位移、邊界節(jié)點轉角、內節(jié)點位移、地基反力矩陣；Ic為元素是D的Nc階對角矩陣.

根據(jù)層狀各向同性彈性地基的解析層元解[18]，通過兩維Guass-Legendre積分可得單位均布荷載作用在反力qj區(qū)域時，板節(jié)點i處的沉降為：

δij=?δ(i,ψ,R)dΓj.

(25)

式中:δ(i,ψ,R)為地基的基本解，它表示層狀地基表面任意點ψ作用單位點荷載在點i引起的沉降，R為點ψ到點i的距離.

地基反力引起的板內點及邊界點的豎向位移可表示為：

(26)

(27)

假設加載前后板與地基不脫離，則有：

we=se,

(28)

wc=sc．

(29)

將式(26)-(29)代入式(11)，可得：

(30)

求解式(30)，得到地基與彈性薄板相互作用的板邊界節(jié)點轉角和地基反力，再通過式(26)和(27)可求得薄板內部點和邊界點的豎向位移.

3 數(shù)值計算與分析

3.1 理論與程序驗證

為了驗證本文理論及程序的正確性，將本文的計算結果與Wang&Cheung[4]的結果進行對比(如圖4所示)，其中，bp=4.0 m，νs=0.4，hp=0.2 m，Es=0.343×103MPa，νp=0.167，Ep=0.343×105MPa，均布荷載p=0.98 MPa(見圖5).本文采用厚度為1 000m的單層土來模擬彈性半空間地基.由圖5可知，本文的結果與Wang&Cheung[4]的結果吻合較好，這表明本文理論與程序的正確性.

圖4 方形基礎薄板邊界的撓度對比圖Fig.4 The comparison of vertical displacementsof the boundary of a square foundation thin plate

3.2 板-土剛度比的影響

圖5 彈性半空間上的方形薄板Fig.5 A square thin plate on an elastic half-space

圖6 方形薄板平面圖Fig.6 The plane graph of a square thin plate

圖7 當K=0.05(實線)和0.01(虛線)時 各線段的豎向位移曲線Fig.7 The settlement curves of the lines on the plate when K=0.05(solid line) and 0.01(imaginary line)

3.3 矩形板長寬比η的影響

圖8 三層地基上矩形薄板Fig.8 A rectangular plate on three-layered soils

η圖9 板邊界角點、短邊中點、長邊中點 和板中心點的撓度隨η的變化曲線Fig.9 The settlement-η curves of the boundaryangular dot, the midpoits of the long sideand the wide side, and the center of the plate

4 結 論

本文基于更加符合工程實際的層狀地基模型，采用邊界單元法來求解地基與彈性薄板的共同作用問題，并通過與已有文獻結果對比，驗證了本文理論與計算程序的正確性.數(shù)值分析結果表明：

1)方形基礎薄板情況下，離板中心越近，垂直于y(x)方向、距離相等的2條線段的豎向位移差越小，且其隨板-土剛度比減小而減小.

2)隨著長寬比的增大，短邊中點與長邊中點的撓度差也增大；但板中心點與長邊中點位移差變化不明顯，且短邊中心與邊界角點位移差也有相類似的規(guī)律.

相比于有限元、有限層等方法，邊界單元法能將求解過程的維數(shù)降低一維，并具有計算時間短、精度高等優(yōu)點.另外，基于層狀地基的動力解析層元解[19]及達朗貝爾原理[9]，還可進一步將本文工作拓展，用以分析層狀地基上彈性薄板的動力響應.

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Foundations and an Elastic Thin Plate

AI Zhiyong1, 2?, CAI Jianbang1, 2

(1. Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China; 2. Key Laboratoryof Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China)

The kernel functions of the boundary integral equations for thin plate were determined by the fundamental solutions for an infinite thin plate. By the discretization of the plate interior and boundary as well as the assumption of the distribution states of plate nodes and foundation reaction forces, the BEM equations of the plate can be established. Meanwhile, based on the analytical layer element solutions for layered foundations, the flexibility matrix of the foundation was obtained by a two-dimensioned Guass-Legendre quadrature. Taking into account the compatible conditions of the displacements at the soils-plate interface, the global BEM equations for the interaction problem between the layered foundation and the thin plate were then established. The solutions for the problem were further obtained by solving the global BEM equations. The accuracy of the present method was verified by comparing existing solutions with the numerical results obtained from the corresponding FORTRAN program in this study. It is observed from numerical examples that when a square thin plate is placed on a foundation, the settlement difference between the two lines perpendicular to y or x coordinate decreases as they approach the center of the plate, and the difference decreases with the decrease of the plate-soil stiffness ratio. Furthermore, the settlement discrepancy between the plate center and the midpoint of the long side is unapparent with the increasing length-width ratio, and the similar variation trend can be found between the midpoint of the wide side and angular point.

boundary element; layered soils; thin plates; analytical layer element

2015-11-26

國家自然科學基金資助項目(50578121), National Natural Science Foundation of China(50578121)

艾智勇(1966-)，男，江西余江人，同濟大學教授，博士 ?通訊聯(lián)系人，E-mail:zhiyongai@#edu.cn

1674-2974(2017)03-0120-06

10．16339/j．cnki．hdxbzkb．2017．03．015

TU443

AA BEM for Interaction between Layered