层状横观各向同性地基平面应变问题的扩展精细积分解
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摘要
利用扩展精细积分法求解横观各向同性地基的平面应变问题。扩展精细积分法具有高精度和较高计算效率的特点,是求解微分方程的有效方法,相比于解析法可以节省大量理论推导工作量。从直角坐标下弹性力学控制方程出发,推导出Fourier变换域内地基的常微分矩阵方程;之后对地基微层元进行消元合并,进一步得到荷载作用在地基内部时层状地基的扩展精细积解。与已有文献的对比验证了方法的精确性,并分析了横观各向同性参数、层状性质和荷载作用点对计算结果的影响。结果表明:土体竖向位移随着横观各向同性参数n的增大而减小,而随着横观各向同性参数m的增大而增大;荷载作用点z?的变化只对作用点以上的土体有影响,而上层土体的模量对竖向位移计算结果的影响更为显著,土体成层性对沉降的影响要比对竖向应力的影响更为显著。
In this paper, the extended precise integration method is used to solve the plane strain problem of transversely isotropic multilayered soils. Extended precise integration method is of high accuracy and efficiency, which is an effective method to solve differential equations. Compared with analytical methods, it can save a lot of time for theoretical derivation. Starting with the governing equations of elasticity in Cartesian coordinates, the matrix differential equation in the Fourier transform domain is derived. Then, the establishment and combination of adjacent layer elements are introduced, and the extended precise integration solution of multilayered soils subjected to internal loads can be obtained further. The comparison with existing results verifies the accuracy and correctness of this method, and numerical examples are presented to elucidate the influence of transverse isotropy, stratification and load position. It turns out that the vertical displacement decreases with the increase of modulus ratio n, and increases with the increase of modulus ratio m. Besides, soils above the loading point z' is more sensitive to change of load position, while the modulus of upper soils is more important to the results of vertical displacement, and stratification of the soils has more significant influence on vertical displacement compared with vertical stress.
In this paper, the extended precise integration method is used to solve the plane strain problem of transversely isotropic multilayered soils. Extended precise integration method is of high accuracy and efficiency, which is an effective method to solve differential equations. Compared with analytical methods, it can save a lot of time for theoretical derivation. Starting with the governing equations of elasticity in Cartesian coordinates, the matrix differential equation in the Fourier transform domain is derived. Then, the establishment and combination of adjacent layer elements are introduced, and the extended precise integration solution of multilayered soils subjected to internal loads can be obtained further. The comparison with existing results verifies the accuracy and correctness of this method, and numerical examples are presented to elucidate the influence of transverse isotropy, stratification and load position. It turns out that the vertical displacement decreases with the increase of modulus ratio n, and increases with the increase of modulus ratio m. Besides, soils above the loading point z' is more sensitive to change of load position, while the modulus of upper soils is more important to the results of vertical displacement, and stratification of the soils has more significant influence on vertical displacement compared with vertical stress.
引文
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[10]钟阳,耿立涛.多层弹性平面问题解的精确刚度矩阵法[J].岩土力学,2008,29(10):2829-2832.ZHONG Yang,GENG Li-tao.Explicit solution of multiplayer elastic plane by exact stiffness matrix method[J].Rock and Soil Mechanics,2008,29(10):2829-2832.
[2]王林生.用柔度矩阵递推法计算横观各向同性成层地基[J].河海大学学报,1989,17(2):104-110.WANG Lin-sheng.Flexibility matrix recursion method to solve transversely isotropic multilayered soil[J].Journal of Hohai University,1989,17(2):104-110.
[3]陈光敬,赵锡宏,于立.传递矩阵法求解成层横观各向同性弹性体轴对称问题[J].岩土工程学报,1998,20(5):105-108.CHEN Guang-jing,ZHAO Xi-hong,YU Li.Transferring matrix method to solve axisymmetric problems of layered cross-anisotropic elastic body[J].Chinese Journal of Geotechnical Engineering,1998,20(5):105-108.
[4]顿志林,刘干斌,苌向阳.层状横观各向同性地基轴对称问题的位移解法[J].焦作工学院学报(自然科学版),2002,21(6):420-426.DUN Zhi-lin,LIU Gan-bin,CHANG Xiang-yang.Displacement function method of axisymmetrical problem with transversely isotropic layers[J].Journal of Jiaozuo Institute of Technology(Natural Science),2002,21(6):420-426.
[5]艾智勇,李博.横观各向同性层状地基平面应变问题的解析层元解[J].岩土工程学报,2012,34(10):1787-1791.AI Zhi-yong,LI Bo.Analytical layer element solutions to plane strain problem of transversely isotropic multilayered soils[J].Chinese Journal of Geotechnical Engineering,2012,34(10):1787-1791.
[6]ZHONG W X.Duality system in applied mechanics and optimal control[M].Boston:Kluwer Academic Publishers,2004.
[7]ZHONG W X,LIN J H,GAO Q.The precise computation for wave propagation in stratified materials[J].International Journal for Numerical Methods in Engineering,2004,60(1):11-25.
[8]AI Z Y,CHENG Y C.Extended precise integration solution for consolidation of transversely isotropic poroelastic layered media[J].Computers and Mathematics with Applications,2014,68(12):1806-1818.
[9]AI Z Y,WU Q L,WANG L J.Extended precise integration method for axisymmetric thermo-elastic problem in transversely isotropic material[J].International Journal for Numerical and Analytical Methods in Geomechanics,2016,40(2):297-312.
[10]钟阳,耿立涛.多层弹性平面问题解的精确刚度矩阵法[J].岩土力学,2008,29(10):2829-2832.ZHONG Yang,GENG Li-tao.Explicit solution of multiplayer elastic plane by exact stiffness matrix method[J].Rock and Soil Mechanics,2008,29(10):2829-2832.