层状地基三维问题的解析层元解
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摘要
从弹性力学三维问题的基本控制方程出发,通过双重Fourier变换及解耦变换技术,推导出相应问题在积分变换域的解析解;根据该解析解,进而推导出三维弹性地基问题的精确刚度矩阵,即解析层元;然后根据有限层法原理,组装得到总体刚度矩阵;通过求解总体刚度矩阵形成的代数方程,得到三维层状地基问题在变换域内的解答;最后应用Fourier逆变换技术,得到其物理域内的解。与经典Bounsinesq解及有限元软件ABAQUS的结果进行比较,验证了本文理论及数值计算方法的正确性;计算分析结果还表明:土体的分层特性对地基沉降具有较显著的影响。
Starting from the basic equations of elasticity for three-dimensional problem,the analytical solution in the integral transform domain is obtained by use of the decoupled transform technique and the Fourier transform method.Based on the analytical solution,the exact stiffness matrix which is called analytical layer-element for the three-dimensional problem of multilayered elastic foundation is derived.The global stiffness matrix is assembled by means of the principle of finite layer method,and the solutions for the three-dimensional problem of multilayered foundation in the transform domain are obtained by solving the algebraic equations of the global stiffness matrix.Finaly,the solutions in the physical domain are acquired by inverting the Fourier transform.The validity of the proposed theory and numerical method is examined by comparing the results in this study with those of Boussinesq solution and the finite element soft ABAQUS,and numerical calculation shows that the influence of stratification on the displacement of foundation is remarkable.
Starting from the basic equations of elasticity for three-dimensional problem,the analytical solution in the integral transform domain is obtained by use of the decoupled transform technique and the Fourier transform method.Based on the analytical solution,the exact stiffness matrix which is called analytical layer-element for the three-dimensional problem of multilayered elastic foundation is derived.The global stiffness matrix is assembled by means of the principle of finite layer method,and the solutions for the three-dimensional problem of multilayered foundation in the transform domain are obtained by solving the algebraic equations of the global stiffness matrix.Finaly,the solutions in the physical domain are acquired by inverting the Fourier transform.The validity of the proposed theory and numerical method is examined by comparing the results in this study with those of Boussinesq solution and the finite element soft ABAQUS,and numerical calculation shows that the influence of stratification on the displacement of foundation is remarkable.
引文
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[6]张问清,赵锡宏,宰金珉.任意力系作用下的层状弹性半空间的有限层分析方法[J].岩土工程学报,1982,3(2):27-42.(ZHANG Wen-qing,ZHAO Xi-hong,ZAI Jin-min.Finite layer analysis for layered and elastic half-space underarbitrary force system[J].Chinese Journal of GeotechnicalEngineering,1982,3(2):27-42.(in Chinese)).
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[8]ZIENKIEWICZ O C,TAYLOR K L.The finite elementmethod[M].4th ed.New York:McGraw-Hill Inc,1987.
[2]王林生.求解成层地基空间轴对称问题的初参数法[J].力学学报,1986,18(6):528-537.(WANG Lin-sheng.Theinitial parameter method for solving three-dimensionalaxisymmetrical problems of layered foundations[J].ActaMechanica Sinica,1986,18(6):528-537.(in Chinese)).
[3]钟阳,王哲人,郭大智.求解多层弹性半空间轴对称问题的传递矩阵法[J].土木工程学报,1992,25(6):38-43.(ZHONG Yang,WANG Zhe-ren,GUO Da-zhi.The transfermatrix method for solving axisymetrical problems inmuti-layered elastic half space[J].China Civil EngineeringJournal,1992,25(6):38-43.(in Chinese)).
[4]AI Z Y,YUE Z Q,THAM L G,et al.Extended Sneddon andMuki solutions for multilayered elastic materials[J].International Journal of Engineering Science,2002,40(13):1453-1483.
[5]艾智勇,吴超.三维直角坐标系下分层地基的传递矩阵解[J].重庆建筑大学学报,2008,30(2):43-46.(AIZhi-yong,WU Chao.Transfer matrix solutions formulti-layered soils in rectangular coordinate system[J].Journal of Chongqing Jianzhu University,2008,30(2):43-46.(in Chinese)).
[6]张问清,赵锡宏,宰金珉.任意力系作用下的层状弹性半空间的有限层分析方法[J].岩土工程学报,1982,3(2):27-42.(ZHANG Wen-qing,ZHAO Xi-hong,ZAI Jin-min.Finite layer analysis for layered and elastic half-space underarbitrary force system[J].Chinese Journal of GeotechnicalEngineering,1982,3(2):27-42.(in Chinese)).
[7]王复明,林皋.粘弹性层状地基动力柔度系数的半解析解[J].地震工程与工程振动,1988,8(2):27-36.(WANGFu-ming,LIN Gao.A semi-analytical solution of the dynamicflexibility coefficients of viscoelastic layered foundation[J].Earthquake Engineering and Engineering Vibration,1988,8(2):27-36.(in Chinese))
[8]ZIENKIEWICZ O C,TAYLOR K L.The finite elementmethod[M].4th ed.New York:McGraw-Hill Inc,1987.
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