Interaction between the ground mass and a tunnel support is complex. Excavation of the tunnel causes unloading and changes in the ground's original stress state resulting in movements throughout the ground mass. While these changes are still occurring, the support is installed and the interaction of the support with the ground further influences the stresses and movements in the ground mass.

This paper presents a closed-form solution for analyzing the effects of the ground's post-peak stress-strain behavior on tunnel supports. The solution is valuable in identifying the underlying mechanisms and the consequences of yielding ground behavior. Ground characteristic curves can be constructed and used together with support reaction curves to obtain an idea about ground-structure interaction in a strain softening ground. Several studies dealing with the problem of ground softening behavior around openings are available in the literature (Baguelin et al., 1972; Daemen and Fairhurst, 1975; Daemen, 1975; and Prevost, 1975).


2.1 Assumptions

The solution presented herein predicts the displacements, strains and stresses around a circular opening unloaded uniformly in a homogeneous isotropic strain-softening infinite medium where gravity forces are neglected. The solution is valid under the following assumptions:

  • The stress strain relationship of the ground mass can be idealized as shown in Fig.1 FIGURE 1 MODEL USED FOR ANALYSIS (Available in full paper) If the strains resulting from unloading the ground mass are smaller than those required to mobilize the peak strength, Cp, the stress strain relation is elastic (zone III). After the peak strength has been fully mobilized, a further increase in strains will be accompanied by a decrease in the strength (strain softening, zone II) until the strains are large enough to reduce the strength to its residual value, CR. In zone I (perfectly plastic zone), the strength is constant, equal to C R and is independent of the state of strain.

  • The free field principal stresses are equal. The tangential, s¿, and radial, sr, stresses are, therefore, the major and minor principal stresses, respectively, and the problem is one dimensional. Choosing a cylindrical coordinate system in which r denotes the radial distance from the center of the opening, the material is in a plastic state within the region, a r rp, in which a and rp are the radii of the opening and the plastic zone, respectively. Furthermore, as was described above, the plastic region is divided into two zones depending on the state of strain within that zone. For a r rR, the strains due to excavation are large enough to reduce the strength to its residual value. For rR r rp, the ground is in a strain softening state. Beyond the plastic domain (r > rp), the material is in an elastic state.

  • The ground mass obeys the Tresca yield criterion.

  • The ground mass increases in volume at failure. In this solution, it was assumed that upon failure, the material throughout the plastic zone experiences a constant and equal volume increase which is independent of the strains within the plastic zone.

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