HARDING PRIZE COMPETITION 2023 | BTS
Regarding the interface conditions between the
extrados of the lining and the surrounding ground, full bond conditions were simulated with normal and shear springs acting all around the extrados whereas normal springs only were considered for tangential slip conditions. The magnitudes of the spring stiffnesses were adjusted for the numerical model to reproduce the behaviour of the elastic continuum model. Prior to the application of the soil stresses a bolt preload corresponding to 25% of the yield load was applied. The self-weight of the tunnel ring was neglected, in
accordance with the elastic continuum model. The stress-strain behaviour of GCI was simulated with an appropriate elasto-plastic model. The stiffness and strength parameters of the interface between segments and the bolting system were based on those established in the analyses of the experimental results. The value of the springs simulating the soil reaction was varied parametrically for each combination of tunnel conditions to investigate the influence of the joints under a range of ovalisation levels. Six parametric studies were conducted, each
adopting a different combination of tunnel depth (10m, 20m, and 30m), and soil-tunnel interface condition, adopting either full bond or tangential slip. The same analyses conducted on the segmental ring were conducted with an equivalent continuous ring (i.e. without joints) in order to make comparisons between the two sets of analyses.
Derivation of bending stiffness reduction factors Figure 4 presents the results from the analysis adopting full bond conditions in terms of bending moments (kNm) at the crown and springline (see Figure 2a), with the tunnel squat (% of the internal diameter) for the three tunnel depths considered. The crown and springline are the sections where the maximum positive (tension at intrados) and negative (tension at extrados) bending moments, respectively, occur and therefore are the most critical sections in terms of the assessment of internal forces. Each pair of data points, corresponding to the crown and springline, for each squatting level were obtained from a single analysis. The figures also include the elastic continuum
solution for the full bending stiffness (continuous rigid) and the reduced bending stiffness (continuous flexible) as defined by Muir Wood’s (1975) formula. The latter gives a reduction factor of 0.444 for a tunnel lining with six longitudinal joints. By comparison with the continuous rigid ring result,
it can be observed that the segmental ring exhibits a reduction in bending stiffness that becomes more prominent with the squatting magnitude. This is a result of the increasing opening taking place at the intrados of the crown/invert joints which, being subjected to positive bending, are more prone to opening than the other joints, i.e. the segment’s U cross-section requires a larger bending moment to reach zero stress at the extreme fibre under negative bending.
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1
0.8 7
0.9 8
0.7 0.6 0.5 0.4 0.3 2
0.2 0.1 0
0 025 0.25
Crown
Springline Global
05 0.5 07 0.75 1 125 Tunnel squat (%) 1
0.8 7
0.9 8
0.7 0.6 0.5 0.4 0.3 2
0.2 0.1 0
0 025 0.25 1.25 15 1.5 17 1.75 2
Crown
Springline Global
05 0.5 07 0.75 1 125 Tunnel squat (%) 1
0.8 7
0.9 8
0.7 0.6 0.5 0.4 0.3 2
0.2 0.1 0
0 025 0.25 1.25 15 1.5 17 1.75 2
Crown
Springline Global
05 0.5 07 0.75 1 125 Tunnel squat (%)
Above, figure 5a: Full bond conditions Top: 10m tunnel depth Centre: 20m tunnel depth Bottom: 30m tunnel depth
1.25 15 1.5 17 1.75 2
Reduction factor η
Reduction factor η
Reduction factor η
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