Stress analyses that take into account the spatial effects in the vicinity of the tunnel face produce under certain conditions results for ground pressure and deformation that are considerably higher than the ones that are produced when we assume plane strain conditions. The differences are due to path dependency in the mechanical behaviour of the ground and to the inability of the plane strain model to map radial stress history, which involves a complete unloading (and, later, a re-loading) of the tunnel boundary over the unsupported span. This inherent weakness of any plane strain analysis is relevant from the design standpoint particularly for heavily squeezing conditions that require a yielding support. For the majority of tunneling conditions and methods, however, involving as they do support completion within a few meters of the face, the errors introduced by the plane strain assumption are not important from a practical point of view.
In addition to unstable rock blocks ("loosening") or long-term processes such as creep, consolidation or swelling, the spatial redistribution of stress in the ground around the working face also leads to the development of a pressure upon the lining, as the latter partially hinders the convergence of the tunnel walls. The interaction between the ground and the tunnel lining is well understood in principle today (cf. Lombardi 1971). The magnitude of the loading depends on the magnitude of the deformations constrained by the lining (i.e. on magnitude of the deformations that would occur in the absence of a lining) and thus on the distance between the working face and the location of the lining installation (e in Fig. 1a). The smaller this distance, the higher will be the load that develops due to the continuing excavation. Furthermore, as in any statically undetermined system, the magnitude of the ground pressure depends on the load-deformation characteristics both of the lining and of the ground. The ground-lining interaction can be studied through threedimensional numerical models that take into account the sequence of lining installation and excavation work. For practical reasons, however, tunnel design calculations are based in most cases upon plane strain models that consider only a lining cross section and the surrounding ground. The principle of such calculations can be illustrated best by considering the axisymmetric case of a deep cylindrical tunnel. Fig. 1b shows the characteristic lines of the ground and of the lining. The intersection point of the two lines (the "ground response point") fulfils the conditions of equilibrium and compatibility, and shows the radial pressure p ∞ acting upon the lining far behind the face and the respective convergence u ∞ of the ground. For the determination of the intersection point, an a priori assumption must be made concerning the ground displacement u 0 that occurs before the lining is installed ("pre-deformation"). Note that small variations in the assumed pre-deformation u 0 lead to large variations in rock pressure, particularly in the case of a highly non-linear ground response.