弹性半无限空间中矩形孔收缩的复变函数解答

Solution of complex function of rectangular hole contraction in elastic semi-infinite space

随着城市建设的发展,矩形隧道的应用越来越多,但针对矩形隧道的理论研究却鲜有见闻。针对矩形隧道,建立了半无限空间矩形隧道的弹性理论计算模型,采用最小二乘迭代方法确定共形映射函数的各项系数,并将计算区域映射为复平面上的一个同心圆环;运用Muskhelishvili复变函数方法,将计算区域内的应力函数展开成为Laurant级数的形式,给定了地表零应力边界和矩形孔口径向位移边界,求得了半无限空间矩形隧道在给定位移条件下的应力场和位移场。分析了不同高宽比、不同泊松比、不同埋深对位移场和应力场的影响,总结了矩形隧道位移场和应力场的一般规律。结果表明:高宽比偏小、泊松比偏大、埋深偏小都会使得沉降槽不再是类高斯曲线的形状,这些参数的变化也会在不同程度上影响应力场和位移场的大小和分布。

With the development of urban construction in China,more and more applications of rectangular tunnel are emerging,but there are few theoretical studies on rectangular tunnels.In this paper,the elastic theoretical calculation model of the semi-infinite space rectangular tunnel was established.The coefficients of conformal mapping function were determined by the least squares iterative method.And the calculated area was mapped to a concentric ring on the complex plane.Afterwards,The Muskhelishvili complex function method was used to expand the stress function in the calculation area into the form of Laurant series,which gives the zero stress boundary on the ground surface and the radial displacement boundary of the rectangular hole.The stress field and displacement field of the rectangular tunnel in the semi-infinite space under the given displacement condition were also obtained by the method.In this paper,the influence of aspect ratios,Poisson's ratios,and buried depths on the displacement field and stress field was analyzed,and the general rules of the displacement field and stress field of rectangular tunnels has been summarized.The results show that a smaller aspect ratio,a larger Poisson's ratio,and a smaller buried depth will make the settlement curve no longer similar to a Gaussian curve.The variation of these parameters will also affect the size and distribution of the stress field and displacement field to varying degrees.