基坑开挖引起下卧隧道竖向变形的Hermite谱分析方法

Hermite spectral solution for vertical deformation of underlying tunnel induced by excavation of foundation pit

针对基坑开挖引起下卧隧道的竖向变形问题,通过引入Hermite函数,建立基坑开挖所引起土体附加应力及下卧隧道竖向变形的Hermite谱分析方法。基于Mindlin解答与Pasternak地基上的Euler-Bernoulli梁理论,建立了基坑开挖所引起土体附加应力的计算表达式与下卧隧道竖向变形的控制方程,并采用有限项Hermite级数展开,将下卧隧道竖向变形控制方程转化为线性代数系统进行求解,获得其竖向变形解答;在此基础上,结合工程案例,验证所建立解答的收敛性与可靠性,并探讨隧道埋深、隧道与基坑水平距离及基坑平面尺寸对下卧隧道竖向变形特性的影响。结果表明:随着隧道埋深及隧道与基坑水平距离的增加,基坑开挖对下卧隧道竖向变形的影响逐渐降低,下卧隧道的竖向位移逐渐变小;基坑开挖水平尺寸的增大会使下卧隧道竖向变形增大,且平行隧道方向的尺寸变化,影响更为明显。研究成果为分析基坑开挖所引起的下卧隧道竖向变形特性提供了一种新的计算方法。

Aiming at the vertical deformation issue of the underlying tunnel induced by an excavation of the foundation pit,this paper adopts the Hermite functions to develop a novel Hermite-spectral analysis method for estimating the soil additional stress and the vertical deformation of the underlying tunnel.Based on Mindlin's solution and the theory of Euler-Bernoulli beam that lies on the Pasternak foundation,the calculation expression of soil additional stress induced by an excavation of the foundation pit and the governing equation of the underlying tunnel for its vertical deformation are successively developed.Taking advantage of the Hermite functions,both the soil additional stress and the vertical deformation of the underlying tunnel are approximated as two truncated series of Hermite functions and further the developed governing equation above is simplified into a linear algebraic system.By solving this linear algebraic system,a novel solution is developed for calculating the vertical displacement of the underlying tunnel.Then,some calculations on two typical cases are performed to verify the convergence and validity of the developed solution.Finally,the deformation behavior of the underlying tunnel due to the excavation of the foundation pit is investigated,and the influences from the tunnel depth,the horizontal distance between the tunnel and foundation pit as well as the plane sizes of the foundation pit on the induced vertical deformation behavior of underlying tunnel are discussed.The results indicate that as the increase in tunnel depth and that in the horizontal distance between the tunnel and the foundation pit,the effect from the excavation of the foundation pit decreases,and the vertical deformation of the underlying tunnel becomes smaller.The increase in plane size of the foundation pit generally enlarges the vertical deformation of the underlying tunnel,and the size changing parallel to the tunnel shows a more obvious effect.The developed Hermite-spectral solution proposes a novel approach for studying the vertical deformation behavior of the underlying tunnel due to the excavation of foundation pits.