有限元三維層狀橫觀各向同性地基解析層元解

胡亞 ：從彈性力學(xué)的基本方程出發(fā)，采用解耦變換并利重Fourier變換以及Cayley-Hamilton定理，推：橫觀各向同性Fourier變換傳遞矩陣解析層元yticallayer-elementsolutionforthree-dimensionalproblemoftransverselyisotropicmultilayeredfoundation(DepartmentofGeotechnicalEngineering,KeyLaboratoryofGeotechnicalandUndergroundEngineeringofMinistryofEducation,TongjiUniversrty,Shanghai200092,China):Startingfromthebasicequationsofelasticity,usingadecoupledtransformtechniqueandadoubleFouriertransform,aswellasCayley-Hamiltontheorem,thetransfermatrixsolutionofthree-dimensionaltransverselyisotropicsingle-layerfoundationisdeducedinCartesiancoordinatesystem.Then,throughrestructuringtheelementsofthematrix,theyticallayer-elementofsingle-layerfoundationisobtained.Followingtheprincipleofthefiniayermethod，andintroducingthecorrespondingboundaryconditions,thetotalstiffnessmatrixisassembledandsolved.Inordertoverifytheproposedmethod,numericalresultsareobtainedbythecompiledprocedure,andcomparedwiththosebyABAQUS,whichshowsagoodagreement.Otherresultsdemonstratedthatthetransverselyisotropicandmultilayeredpropertiesofthefoundationhaveagreatimpactonitsverticaldisplacements.Keywords:transverselyisotropy,doubleFouriertransform,transfermatrix,yticallayerelement,multilayered引（水平面內(nèi)通常呈現(xiàn)各向同性，將土體簡(jiǎn)化成橫觀定的實(shí)用價(jià)值。國(guó)內(nèi)外已有許多學(xué)者對(duì)橫觀各向同性介質(zhì)進(jìn)行了研究。Lekhntski1]首先求[2]ekhnitskii[3]過(guò)引入應(yīng)（位移函數(shù)得到了此問(wèn)題的解答；和[4]通過(guò)Hankle變換方法得到了橫觀各向同性彈性半空間非軸對(duì)稱問(wèn)題的解析解。解析層元的ij,j

x

ij(ui,juj,i)/

0000cc000000x

y

yz

z

yz yz

(c11c12)

xy 表示應(yīng)力分量對(duì)坐標(biāo)的偏導(dǎo),

n(n2)ij,

c

(1)，c(12)，

G，

/((1)(12n2))，n

/E

EhEvGvh為水平vh為豎直向應(yīng)力引起的水平向應(yīng)變的泊松比。Fourierf() g(x,y)ei(xxyy)

42g(x,y) f()ei(xxyy)d

g(xy)f(x,yFourierFourierU1(uv

X1(xzyz

V1(uv

Y1(xzyz W

Z z22z yUdX1

Xd2U1d

Vd

Yd

WdUd

Z 2 c2

34 34

c

11

13，

11 13，

，

，

12，2

Laplace對(duì)式（7）FourierdG(,z)A()G(,

dx dG(,z)A()G(,

dx 式中：，式中：，

d4d0

0

d4A()

A2()d 0

d20d 0 （8（9

式中：exp[zA1(和exp[zA2(z=0z（10aIaA()aA2()aA3 1 2 3

1(,z)exp[zA()]bI'bA 1II4×42×2A1(A24(2ddd)22(d2dd)4

2 2 42dd2 4

解方程（15）2（14

d4d5d2d 22((dd2d2d 22 d2d 22((dd2d2d 21則22221

2 4222（1）當(dāng)2=01（2）當(dāng)2>0時(shí)，令，則或 1（3）當(dāng)2=0時(shí)，令w22，則(wi)1（10V(,z) g12V(, g

U(,z)

W(,z)W(, Z(,z) Z(,0)

（16Y(,0) T X U(,0) Z(,0) K12W X(,

U(,z) 22 Z(,z)

1

K

12，

，

23

22

23

33 23

1 K 12 32 33 23

22X(,

k14U(,0) Z(, , 24W

X(,z)

k12U(,z) Z(,z)

k22W(, （18三維成層地基的解析層元nX(,0)U(,0) Z(,0)W(,0) ..

X(,z)U(,z) n nZ(,z)W(,zn n式中：(i)i

Fourier數(shù)值計(jì)算與本文根據(jù)10點(diǎn)復(fù)合二維積分的方法，在被積函數(shù)兩個(gè)零點(diǎn)之間進(jìn)行分段積分，實(shí)Fourier數(shù)值逆變換，具體的處理細(xì)節(jié)和精度評(píng)價(jià)可參考文獻(xiàn)[6]。Fortran程序計(jì)算獲得的豎向荷載作用下任意深度的沉降量，與ABAQUS模擬的結(jié)果進(jìn)行比較。由圖1可知，本文的計(jì)算結(jié)果與ABAQUS圖1豎向位移對(duì)比結(jié) 圖2兩層地基中的豎向位Fig.1Comparisonofvertical Fig.2Verticaldisplacementintwo-layered1Table1Thelistofsoil土層 土層EEEEE123 下面以兩層土為例土的橫觀各向同性特性與成層特性對(duì)地基沉降的影響土層參數(shù)11123況1和工況2各參數(shù)的平均值，是一各向同性的均勻土層。計(jì)算結(jié)果見(jiàn)圖2，由圖2可對(duì)于工況1還是工況2，采用傳統(tǒng)各向同性來(lái)簡(jiǎn)化計(jì)算是不安全的，這一點(diǎn)在實(shí)際工程中應(yīng)4.結(jié)從彈性力學(xué)的基本方程出發(fā)，采用解耦變換并利重Fourier變換以及Cayley-Hamilton定理，推導(dǎo)出三維直角坐標(biāo)系下橫觀各向同性單層地基的傳遞矩陣，通過(guò)Fourier逆變換技術(shù)獲得物理域內(nèi)的真實(shí)解答。將所ABAQUS參考文LekhniskiiSG.TheoryofElasticityofanAnisotropicbody[M].MirPublishers,胡海昌.橫觀各向同性體的彈性力學(xué)的空間問(wèn)題[J].物理學(xué)報(bào),1953,9(2):130-144.(HUHai-chang.Theproblemoftransverselyisotropicelasticmechanicsofthespace[J].JournalofPhysical,1953,9(2):130-144.(inChinese)).橫觀各向同性彈性力學(xué)空間問(wèn)題[M].杭州:浙江大學(xué),1997.(DINGHao-jiang.Mechanicsoftansverselyisotropicelasticity[M].Hangzhou:UniversityPress,1997.(inChinese)),.橫觀各向同性彈性半空間非軸對(duì)稱問(wèn)題解析解[J].巖土工程學(xué)報(bào),2004,26(3)：331-334.(LIPei-hao,ZHUXiang-rong.yticalsolutionofnon-axisymmetricproblemsintransverselyistropicelastichalfspace[J].ChineseJournalofGeotechnicalEngineering,2004,26(3)：331-334.(inChinese))艾智勇,沖.三維橫觀各向同性成層地基的傳遞矩陣解[J].巖土力學(xué).2010,31(2):25-30.(AIZhiyong,CHENGYi-chong.Transfermatrixsolutionsofthree-dementionaltransverselyisotropicmultilayeredsoils[J].RockandSoilMechanics,2010,31(2):25－30.(inChinese))艾智勇,.三維直角坐標(biāo)系下分層地基的傳遞矩陣解[J].重慶建筑大學(xué)學(xué)報(bào),2008,30(2)：43-46.Zhi-yong,WUChao.Transfermatrixsolutionsformultilayeredsoilsinretangularcoordinatesystem[J].JournalofChongqingJianzhuUniversity,2008,30(2)：43-46.(inChinese))附錄一：三種情況下的解析層元元素（該矩陣為對(duì)稱矩陣k2(2AA2(8d3d k8z2Adez(2ddd)/ 2 2k16ez(8Ad 2A(1e4z);

(1e2z);AA2(8d3dd)2;

z22d2d2e2z(2d其中：

2

2

31A 31k2ezAA/;k2ezAAA 5 19 k AAAAA k2ezAA/ 1 2 6 AAezez;Aezez;Aez其中： Aezez

(

);

(

A2d;A2d;A22;

ddd

AAA2A2

1

5 1 13k2 13 (A(dAd 3k4ezAS/A 137 (A2(d3A3d4)S1k4ezA(dAd)S 34(1e2z

；

5；T22422d4

(1e2z)z(1e2z)d。44Ssin[z];Scos[z];A(1e2z);A(1e2z);A