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)稱問題的解析解。解析層元的ij,j

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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);

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