Effect of pore water pressure on tunnel support during static and seismic loading
Introduction
The support system of underground facilities in seismic zones must be designed to withstand static overburden loads as well as to accommodate the additional deformations imposed by the earthquake induced motions. Seismic-induced deformations of tunnel liners can be produced by direct shearing displacements of active faults intersecting the tunnels, by ground failure, or by ground shaking around the opening.
There are two basic approaches in present seismic design. One approach is to carry out dynamic, non-linear soil–structure interaction analyses using numerical methods. Input motions are applied to the boundaries of a ‘soil island’ to represent vertically propagating waves. This analysis is complex, requires a significant effort, and thus can only be justified for very special cases. The second approach provides relationships to evaluate the magnitude of seismic-induced solicitations in tunnel liners (Merritt et al., 1985, Penzien, 2000, Hashash et al., 2001). These relationships are based on the premise that tunnel liners under seismic loading will tend to deform with the surrounding ground, and thus the liner must accommodate the ground deformations without loss of its structural integrity. Seismic-induced stresses in the liner need to be added to the stresses imposed by the ground loads and construction process in order to evaluate the structural behavior of the liner.
Complicating all this is the fact that deep tunnels are often found below the ground water table and thus the surrounding ground is fully saturated. Both the static and dynamic loads on the liner depend on the drainage conditions at the contact between the support and the ground. If there is no-drainage at the contact, the liner must support the pressures generated by the water as well as the pressures generated by the ground. With full drainage the pore water pressures become zero at the contact and the liner only needs to support the pressures from the ground. Although the concept is clear, quantification of the loads on the support under any of the two drainage conditions described is not trivial. If there is no-drainage, the pore pressures at the contact will induce deformations of the liner; as a consequence, the soil surrounding the tunnel will move with the liner and thus carry additional load. Hence there is a transfer of the loading from the support to the ground and the support may not carry the full load from the water. If there is full drainage the pore pressures are zero at the contact, but additional seepage forces are generated in the ground due to the movement of the water towards the tunnel (Lambe and Whitman, 1969). The seepage forces produce deformations in the ground which in the end are transmitted to the support because of the compatibility of deformations between the ground and the support. As a result the support will carry additional load due to the water draining towards the tunnel. Thus, the assumption that the liner will carry all the load from the water if there is no-drainage is conservative, and the assumption that the liner will carry only the load from the ground if there is full drainage is unsafe.
During an earthquake the loading is very rapid compared with the permeability of the ground; as the soil tries to deform, excess pore pressures are generated because the ground cannot change volume (i.e. there is no time for the water to move in or out of the pores); because of the rapid loading, no dissipation of pore pressures occur. Thus an ‘undrained’ analysis should be performed. Once the seismic loading ends all the excess pore pressures generated dissipate with time and the initial state is recovered (at least for an elastic ground, which is the focus of this work).
This paper concentrates primarily on the evaluation of the effects of pore water pressure on the loads of the support for a deep circular tunnel located below the water table. A new analytical tool, based on the relative stiffness method (Einstein and Schwartz, 1979, Penzien, 2000), is presented to determine static and seismic loading on the support. The solution can account for: (1) drainage conditions at the ground–liner interface and (2) effect of groundwater pressure on ground and support response. As with the relative stiffness method, it is assumed that the ground and the liner are elastic, and that plane strain conditions apply at any cross-section of the tunnel. These assumptions may limit the practical applicability of the results obtained; however, the objective is not to find a general solution, which is unrealistic given the complexity of tunnels, but rather to develop a simple formulation for preliminary estimates that can be used for further analysis, or even to acquire additional insight into a problem with a small computational effort. At the same time the solution includes the most relevant variables of the problem.
Section snippets
The relative stiffness method
There are a significant number of empirical, analytical and numerical procedures to obtain stresses in the ground and in the liner for a tunnel, given the liner and the ground properties, pore water pressures and drainage conditions at the tunnel perimeter (see for example Einstein and Schwartz, 1979, Aristorenas, 1992, Einstein et al., 1995, Einstein and Bobet, 1997, Suwansawat, 1998, Bobet et al., 1999, ITA, 2000, Bobet, 2001). Of the analytical procedures, the most widely used is the
Static analysis
Two extreme conditions at the ground–liner interface are possible: full drainage, or no-drainage (Fig. 2). For the full drainage case the pore pressures, u, at the interface are zero and there is water flow towards the opening; for the no-drainage case the pore pressures are equal to the far-field pore pressures, uf, and there is no flow. Full drainage or no-drainage at the interface depend on the relative permeabilities of the ground and the liner. In some cases full drainage or no-drainage
Seismic analysis
Shear and pressure waves propagating in the plane of the cross-section of the tunnel generate ground distortions which tend to cause oval deformations of the lining. The resulting change of the shape of the tunnel section generates circumferential strains in the lining which can cause cracking and/or crushing of the concrete and reduce the carrying capacity of the lining. For concrete linings, this deformation is usually the most critical and generally controls the seismic design. Thus, special
Boundary conditions
For the analyses, two assumptions have been made relative to the boundary conditions: (1) the external effective stresses and pore pressures are applied far from the tunnel (i.e. R≫ro) and (2) the ground is weightless and the external stresses are applied uniformly to the boundary. For the static analysis, the magnitude of the far-field stresses is obtained from the unit weight (buoyant unit weight of the ground for effective stresses or the water unit weight for pore pressures) times the
Discussion and conclusions
The effects of drainage conditions at the ground–liner interface are investigated in this paper. It is assumed that the water table is far from the tunnel opening and that the excavation of the tunnel does not lower the water table. This may not be the case for shallow tunnels close to the surface (Ohtsu et al., 1999). The most important conclusion is that for tunnels in which ground stresses and pore pressures are applied far from the tunnel center (i.e. R≫ro), the drainage conditions at the
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