PSI - Issue 84

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

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An iterative procedure based on the shear-stiffness reduction curve proposed by Bardet et al. (2000) for rocks was applied to determine the shear stiffness, G qs , and the corresponding code-based shear strain,  EC , compatible with the seismic event. The procedure yielded to a maximum expected shear strain,  EC , of 0.00237%, a shear stiffness G qs of 7.07 GPa and a corresponding Young’s modulus, E qs , of 19.09 GPa. As an additional term of comparison, local seismic response analyses were also carried out using the MARTA code (v. 1.2.3). Two acceleration time histories, namely the Kalamata (1986) and L’Aquila (2009) records, were selected to be compatible with the reference response spectrum at the fundamental period of the deposit T 1 = 0.235 s. The equivalent-linear visco-elastic analyses provided the depth-dependent profiles of maximum shear strain, operational dynamic shear modulus and damping ratio. On this basis, a second quasi-static shear strain demand  SR was defined as the average shear strain at tunnel depth obtained from the local seismic response analyses. The resulting value was 0.0042% with a substantially unvaried shear stiffness (Spaggiari et al., 2026). 3. Finite Element model The tunnel lining was analysed using 2D plane strain finite element models developed in Plaxis 2D (Brinkgreve et al. 2024). The role of the in-situ stress ratio K 0 was preliminary assessed by applying realistic bounding values of 0.5 and 1.5 in a first set of analyses. At this stage, the lining was assumed to be intact and the seismic action was introduced through the Eurocode-based shear strain demand  EC . The lining was represented by volume elements, while reduced thickness crown configurations were simulated by deactivating the corresponding lining clusters (Figs. 2b and c). The ground-lining contact was described by no-tension elasto-plastic interface elements, adopting R inter = 1.0 at peak strength and R inter,res = 0.1 in the residual condition. In this second stage additional analyses were performed to identify the onset of failure. The lining model was refined by introducing a degraded inner layer representative of pre-existing diffuse cracking, and the seismic demand was increased to the site-specific shear strain  SR obtained from local seismic response analyses. The analysis proceeded in sequential stages (Boldini et al. 2026). First, the in-situ stress state was established. Tunnel excavation was then simulated by deactivating elements in the excavation zone and applying a stress release factor of  = 90%. The lining was subsequently activated, and  was increased to 100%. Finally, a quasi-static phase applied a shear strain,  max , to the domain to represent seismic loading, assuming the surrounding rock-mass to be incompressible (e.g., Kontoe et al. 2008, Amorosi et al. 2011). Rock-mass stiffness was set to E s during the static phases and E qs for the quasi-static phase. Boundary conditions fixed the model base and restricted horizontal displacements along the sides. During the quasi static phase, a horizontal displacement of u max =  max  Z was applied at the top, while lateral displacements increased linearly from zero at the base to u max at the top (Fig. 2a).

Fig. 2. (a) computational mesh; (b–c) lining detail with cracked inner lining (in pink) and cluster boundaries for a residual thickness of 10 cm.