Prediction of tunnel lining loads using correction factors
Introduction
Prediction of lining loads is one of the major issues to be addressed in the design of a tunnel. However, the problem is not easily solved due to uncertainties of the ground conditions, the redistribution of the in situ stresses related to the ground deformation before and after lining installation, and the differences in construction procedures. Therefore, most tunnels are often built too conservatively.
The lining loads can be calculated using many existing lining design methods which may be divided into four groups: empirical and semi-empirical methods, ring and plate models, ring and spring models, and numerical models. Most of these methods have been reviewed in detail by a number of authors (Craig and Muir Wood, 1978, O'Rourke, 1984, Duddeck and Erdmann, 1985, Negro, 1988, Whittaker and Frith, 1990, Kim and Eisenstein, 1998) and will not be repeated here again.
There are several basic requirements of a good design method. First, the design method should be simple to use. Duddeck and Erdmann (1985) reviewed the progress of the development of design models. They concluded that the available design methods are simple enough for practical applications. In other words, if a design method is very complex or time consuming to apply, the method will not be widely used by practical engineers. Second, the design method should consider the stress release occurring before the installation of a liner in some way. Muir Wood (1975) presented a closed form solution, recommending a 50% reduction of the full overburden pressures to account for support delay. The 50% stress reduction is an arbitrary value, and various suggestions have been given by others, e.g. about a 33% stress reduction as suggested by Panet (1973). Einstein and Schwartz (1980) also suggested that the stress reduction factor could be between 15% and 100% according to simple analytical and numerical techniques and case study data. Third, the method should take into account the plastic behaviour of the ground as well as that of elastic ground. However, most of the existing methods were not satisfactory for the estimation of lining loads because they could not consider all the factors mentioned above.
Schwartz and Einstein, 1980a, Schwartz and Einstein, 1980b suggested using correction factors to take into account the decrease of lining loads due to the stress release before lining installation and the increase of lining loads due to development of ground yielding. The method is reviewed in detail in this paper with the emphasis on the delay factor λd. The main purpose of this study is to verify the applicability of the method for actual tunnel design.
Section snippets
Effect of delay in liner placement
Einstein and Schwartz (1979) presented closed form solutions for the estimation of loads on the liner, solutions which depend mainly on the relative support stiffness and in situ stress ratio. Einstein and Schwartz's method has been widely used by practical engineers due to its simplicity. However, the original Einstein and Schwartz method generally overestimates the lining loads because the method is calculated based on the full overburden pressure without consideration of the stress reduction
Comparison with case study data
Schwartz and Einstein (1980a) applied the procedure to five tunnel projects in order to verify the accuracy of the proposed method. These case histories are summarized in Table 1. The normalized lining loads, (T / PR)predic, calculated using the method were compared with field measurements of the case histories and gave errors in the predicted support loads that ranged between the extremes of − 68% (underestimated) and 62% (conservative), with an average error of 32%, as shown in Table 2.
Discussion of results
Schwartz and Einstein, 1980a, Schwartz and Einstein, 1980b compared the actual and simulated tunnelling sequences as shown in Fig. 2. Excavation and support occur at one step in a finite element analysis even though the excavation and support construction do not take place simultaneously in the actual tunnelling sequence. Therefore, the authors considered a delay length Ld for actual tunnelling sequences as the delay length Ld in the finite element analyses. This is true if the liner is
Recalculation of the lining loads
Lining loads of case histories calculated by Schwartz and Einstein were recalculated using Eq. (8) and delay length Ld′ as shown in Table 3. The yield factor λy was obtained following the procedure suggested by Schwartz and Einstein because the derivation of λy was very reasonable. There is an another advantage to using Ld′ over Ld. The method suggested by Schwartz and Einstein could not be used for the Kielder Experimental tunnel because the delay length Ld was 4 m with the normalized delay
Application of the method to tunnels in Edmonton
Schwartz and Einstein's method was applied to tunnels in Edmonton using Eq. (8) and delay length Ld′ to verify the accuracy and applicability of the proposed method. Tunnels in Edmonton used for this study are summarized in Table 4 and described briefly in this section. The City of Edmonton has been developing a Light Rail Transit (LRT) system since the early seventies. The development of the system has been in stages, the first of which is termed the Northeast Line, connecting the city center
Conclusions
Schwartz and Einstein included correction factors considering non-linear ground behaviour and stress reduction occurring prior to lining installation in their original closed form solutions. The proposed method of Schwartz and Einstein is reviewed in detail. From the review of the method, it was concluded that a different λd could be obtained for the same delay length of Ld depending on the round length if Ld is used as suggested by Schwartz and Einstein. Therefore, it was suggested using the
Acknowledgements
The authors wish to acknowledge the support provided by the City of Edmonton.
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