BTS | HARDING PRIZE COMPETITION 2023
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
Although the stiffness reduction affects the global
response of the ring, the impact of joint opening is mostly local and more exacerbated around the crown joint where the bending moments start deviating from the continuous rigid solution at tunnel squats of about 0.15%, 0.25% and 0.35% for tunnel depths of 10m, 20m, and 30m, respectively. At the springline, which is the section farthest from
Crown
Springline Global
05 0.5 0.75 07 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
the joint undergoing opening, the bending moments are similar to those of the continuous rigid solution up to squatting levels of around 0.4%, 0.7% and 1% for tunnel depths of 10m, 20m, and 30m. It is evident that the segmental ring shows a stiffer response the deeper the tunnel, which is related to smaller joint openings with increasing compression. The tunnel response is stiffer than that predicted
Crown
Springline Global
05 0.5 0.75 07 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
adopting Muir Wood’s (1975) reduction factor for all tunnel depths, even at squatting levels greatly exceeding the range typically observed in the field (0.5%-1%). This suggests that the application of Muir Wood’s reduction factor can result in non-conservative estimates when assessing the internal forces of a segmental GCI lining. Similar conclusions can be obtained for the cases with tangential slip conditions. As an alternative to the standard methodology of
Crown
Springline Global
05 0.5 0.75 07 1 125 Tunnel squat (%)
Above, figure 5b: Numerically-derived bending stiffness reduction factors – adapted from Ruiz López (2022) Top: 10m tunnel depth
Centre: 20m tunnel depth Bottom: 30m tunnel depth
1.25 15 1.5 17 1.75 2
adopting either a rigid ring or a flexible ring following Muir Wood’s formula, a new set of bending stiffness reduction factors were derived from the numerical results. As explained above, the extent of the stiffness degradation varies with the tunnel squatting level and the location around the ring and consequently, three reduction factors varying with squatting magnitude were proposed for each tunnel depth and soil-tunnel interface condition: two of them representative of the stiffness reduction taking place at the crown/invert and the springline, which provide the upper and lower bounds of a ring’s bending stiffness and a global reduction factor which represents an average measure of the bending stiffness all around the ring. Figure 5 presents the charts displaying the reduction
factors for different tunnel depths with full bond and tangential slip conditions. The reduction factor at the crown ηc
was determined as the ratio between the
bending moment given by the segmental ring at the crown and that given by the continuous rigid ring solution for the same tunnel squat:
1 ηc
= Ms c
Mr c Similarly, the reduction factor at the springline ηs
established as: 2 ηs
= Ms s
Mr s where and denote the bending moments at the
springline from the segmental ring and the continuous rigid ring, respectively. The global reduction factor ηg was defined as the ratio between the bending stiffness required for a continuous ring to match, for equal soil stiffness, the tunnel squat of the segmental ring and the bending stiffness of the segment cross-section. The abovementioned bending stiffness of the continuous
34 | July 2023 was
Reduction factor η
Reduction factor η
Reduction factor η
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