BTS | HARDING PRIZE COMPETITION 2023
20 40 60
-60 -40 -20 0
0 0.5 1 Tunnel squat (%)
20 40 60
-60 -40 -20 0
0 0.5 1 Tunnel squat (%)
20 40 60
-60 -40 -20 0
0 0.5 1 Tunnel squat (%)
Above, figure 4: Bending moment (kNm) at the crown and invert locations with tunnel squat (%) assuming full bond conditions adapted from Ruiz López (2022) Top: 10m tunnel depth
Centre: 20m tunnel depth Bottom: 30m tunnel depth
Crown Springline Crown Springline Crown Springline
PRELIMINARY ASSESSMENTS ACCOUNTING FOR THE JOINTS Segmental GCI tunnel linings are frequently analysed as continuous (non-jointed) structures using analytical models, such as the elastic continuum model (Duddeck & Erdmann, 1985). In those instances, TfL (2017) recommends starting the structural assessment assuming the full circumferential bending stiffness of the tunnel ring (rigid ring) and changing to a reduced bending stiffness (flexible ring) should the initial assessment fail. Although TfL (2017) does not specify what reduced
1.5 2
bending stiffness should be adopted, it is common practice in industry to employ the reduced second moment of area proposed by Muir Wood (1975). The latter does not consider the nonlinear behaviour of tunnel joints with respect to compression stresses and ovalisation levels and, consequently, its applicability to GCI tunnel linings is somewhat tenuous. Considering the limitations of existing methods, a
new set of bending stiffness reduction factors (η) were derived, based on an extensive numerical investigation replicating the assumptions of the elastic continuum model. The new reduction factors are presented here in a series of design charts, allowing their straightforward application in routine engineering practice.
1.5 2
Methodology The segmental GCI ring considered in the analyses was based on those of the Central Line’s 11ft 81/4
in (3.562m)
running tunnels which can be considered as standard for GCI tunnels in the LU network. Each ring was formed by six segments and a smaller key segment (which was omitted for simplicity). The joint arrangement was the typical adopted in the first few decades of construction of the LU system with the longitudinal joints aligned between rings (i.e. allowing the analysis of a single ring to investigate the whole tunnel response) and one of the longitudinal joints located at the crown (see Figure 2a). The 3D numerical model followed the methodologies
1.5 2
adopted to successfully simulate the experimental tests mentioned above (Ruiz López et al., 2022); Figure 2b presents the FE mesh employed in the analyses. Symmetry conditions were applicable and so it was only necessary to simulate a quarter of the circumference and half the segments’ width, as shown in Figure 2b. The soil stresses were applied via normal and
shear stresses acting on the extrados of the ring. The magnitude of the total vertical stress σv
was based on
the tunnel depth being considered by assuming that the full overburden stress was acting on the tunnel lining. As recommended by Tube Lines (2007), the ratio of the total horizontal stress to vertical stress σh 0.7. Adopting a σh/σv
/σv was taken as <1 implies that the tunnel ring would
deform into squatting (i.e. enlargement of the horizontal diameter along with shortening of the vertical diameter) which is consistent with field observations: Wright (2013) reported that measurements from circularity surveys across the ‘Tube’ indicated that tunnels generally squat between 0.5%-1% of the diameter.
32 | July 2023
Bending moment (kNm)
Bending moment (kNm)
Bending moment (kNm)
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