Surface settlements due to deformation of a tunnel in an elastic half plane

Abstract

Deformations of a tunnel may result in settlementsof the ground surface. In engineering practice thesesurface settlements are often described by empiri-cal formulae, based upon field observations, forinstance a normal (Gaussian) distribution curve(Peck, 1969; Attewell & Woodman, 1982). In thisnote an approximate analytical solution is presented,considering the soil as a linear elastic material.Although it is realized that this is a very poorschematization of the real behaviour of soils, anelastic solution may well serve to investigate someof the characteristics of the resulting fields of stressand strain, and it may also serve as a reference formore refined (numerical) computations.Two basic deformation mechanisms of thetunnel are considered: a uniform radial displace-ment (representing, in first approximation, theground loss that may occur during constructionof the tunnel), and an ovalization of the tunnel,see Fig. 1. Even though in practice the tunnellingprocess may be executed very carefully andappropriate engineering techniques may be appliedto minimize the deformations, for instance theinjection of grout into the soil surrounding thetunnel, it remains of interest to study the defor-mations and stresses caused by both ground lossand ovalization.The method used is an extension of a methodsuggested by Sagaseta (1987) for the case ofground loss in an incompressible soil. The startingpoint is the solution for a singularity at a point ofan elastic half plane (at the axis of the tunnel). Byadding the image solution for a singularity at apoint located symmetrically above the soil surface,the shear stresses at the surface are made equal tozero. In order to balance the normal stresses at thesurface, a third solution is added, by solving aBoussinesq-type problem. The approximation inestablishing the solution is that, in balancing thenormal and tangential stresses on the surface, thepresence of the tunnel, with its prescribed dis-placements, is disregarded. Although an exactsolution might well be obtainable by the methodsoutlined by Mindlin (1939), using bipolar coordi-nates, such a solution is very complicated. Thepresent approximate solution will be seen to bevery simple and easy to apply.The solution given in this technical note is ageneralization of Sagaseta’s solution in that it givesthe solution for the case of ground loss not onlyfor the incompressible case (with Poisson’s ratioequal to 0·5), but for arbitrary values of Poisson’sratio, and that it includes the effect of ovalization.The process of ovalization has been studied beforefor a tunnel in an infinite elastic medium (MuirWood, 1975). Here the surface displacements for atunnel in a semi-infinite medium are considered, aswell as the displacements and stresses throughoutthe half space.