PSI - Issue 84

Daniela Boldini et al. / Procedia Structural Integrity 84 (2026) 175–182

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4. Finite Element results Fig. 3 illustrates the deformed shapes obtained for the full lining thickness and for the 10-cm reduced-thickness configuration, under the Eurocode-based quasi-static shear deformation (EC) and for the two considered values of K 0 . In the reduced-thickness cases, the cavity behind the crown causes an upward distortion of the lining in this zone during the quasi-static phase, owing to its lower stiffness. The imposed shear deformation also induces, independent on the thickness reduction, a main detachment at the left shoulder between the rock mass and the lining, together with a secondary interface opening at the right sidewall. In general, the thickness of the gap decreases as K 0 increases. Overall, the lining undergoes marked ovalisation, which is likely amplified by the absence of the invert and the resulting lower global stiffness. This behaviour is associated with the development of broad regions where the zero tensile strength of the lining is reached, extending from the left shoulder to the crown and along the right sidewall, in good accordance with the location of detachments at the interface (Fig. 4). Compared with the full-thickness lining, the reduced crown thickness causes this plasticised zone to extend further towards the right-hand side of the crown, potentially promoting detachment of the crown slab. In addition, the interface shear strength is also attained in the right portion of the crown. The influence of K 0 in this respect remains relatively limited, although higher confining stresses lead to a slight increase in the extent of the plasticised areas.

(d) K 0 = 1.5 - EC

(a) K 0 = 0.5 - EC

(b) K 0 = 1.5 - EC

(c) K 0 = 0.5 - EC

Fig. 3. Deformed shapes for full section (a,b) and 10 cm residual thickness (c,d) for different value of K 0 (scaled to 1,000). Black arrows indicate the main detachment between the rock-mass and the lining; the red arrow shows the secondary interface opening at the right sidewall.

(c) 10 cm thickness – K 0 = 0.5 (d) 10 cm thickness – K 0 = 1.5

(a) full section – K 0 = 0.5

(b) full section – K 0 = 1.5

Fig. 4. Plastic points distributions: black dots represent tensile cut-off points and red dots represent shear failure (limited to the interface).

The distributions of normal force N and bending moment M during the quasi-static phase are shown in Fig.5 for K 0 = 0.5 and 1.5. The overall range of lining forces is broadly similar for the full-thickness lining and for the configuration with a 10 cm residual crown thickness. However, when the crown thickness is reduced, the normal force becomes more uneven distribution and tends to redistribute more uniformly within the weakened crown region, while bending moments increase at the locations where the largest lining deflections occur. Axial forces and bending moments in the tunnel lining were evaluated against the N – M interaction domains of each section. Fig.6 presents the results for sections with a residual lining thickness of 10 cm and for the considered K 0 values. Hollow symbols correspond to the static phase, while solid symbols refer to the quasi-static phase. The N – M domains were obtained using the fibre method, assuming a parabolic stress–strain curve for concrete based on its characteristic compressive strength, while neglecting tensile strength and without applying reduction factors. Each domain reflects the local lining thickness, indicated with different colours in the figures, with the neutral axis assumed at mid-depth and slight curvature effects ignored. The imposed shear strain also introduces a marked asymmetry between the two halves of the tunnel section. In the crown region (coloured symbols in Fig. 6), the absolute value of the bending moment generally increases during the