PSI - Issue 84

Marialorenza Vescovi et al. / Procedia Structural Integrity 84 (2026) 1063–1070

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a) b) Figure 3 (a) Comparison of the stresses of the simplified approach at the intrados (Sa int ) and at the extrados (Sa est ) with those of the finite element model, both at the intrados (LE FEM int) and at the extrados (LE FEM est). (b) Comparison of the eccentricity with simplified approach SAand the FEM LE FEM.

3.2. Shell non-linear model

While the previous linear elastic comparison with beam elements validated the static consistency of the simplified method’s internal algorithms, a more rigorous assessment is required to verify the physical plausibility of the calibrated parameters ( h and f ). For this purpose, a comprehensive 2D non-linear finite element model was developed to simulate the full soil-structure interaction. The numerical domain was defined to ensure that external boundary conditions do not influence the stress distribution around the excavation. The soil domain extends horizontally for a width equal to five times the tunnel diameter and vertically reaches a height of 60 meters, providing sufficient soil volume for the correct redistribution of geostatic and excavation-induced stresses. To ensure accurate numerical results while optimizing computational efficiency, the domain was discretized using plane strain elements with a variable mesh density. The masonry arch features a refined mesh with a uniform element size of approximately 0.3m, selected to accurately resolve the stress gradients and eccentricity variations across the lining thickness. Moving outward from the tunnel-soil interface, the mesh density gradually decreases; the element size increases linearly from the refined zone adjacent to the excavation up to a coarser dimension of 3.0m at the external boundaries. Regarding the constitutive behavior, non-linear laws were adopted for the two materials. The rock mass was modeled as an elastic-perfectly plastic material governed by the Mohr-Coulomb yield criterion. This law enables the simulation of plastic zones in the ground, utilizing strength parameters (cohesion c' and friction angle  ) derived from the geotechnical characterization illustrated in Table 1. The non-linear behaviour of the masonry lining was simulated using the Concrete Damage Plasticity (CDP) constitutive model. The parameters defining the flow potential and yield surface were selected consistent with established literature values for historical brick masonry according to D’Altri, A.M. et al. (2019): a dilation angle  of 10°, eccentricity  of 0.1, ratio of biaxial to uniaxial compressive strength f b0 /f c0 about 1.16, the shape factor K c =0.667 and viscosity  = 10 -4 were adopted. The interaction between the masonry lining and the surrounding ground was managed through a frictional contact formulation. A Coulomb friction coefficient of  =0.3 was assigned to the interface to simulate the mechanical interlocking and roughness between the masonry extrados and the excavated rock surface. The numerical analysis was executed through a sequential staged construction process, defined to replicate the stress history of the tunnel and the conservative loading scenario adopted for the validation. First, the initial lithostatic stress field was established by applying gravity loads to the full domain, generating geostatic stresses based on K 0 . Next, the excavation was simulated by removing the soil elements. Equivalent nodal forces were applied to the boundary nodes to limit deconfinement, replicating a scenario with minimal pre convergence. Finally, the masonry lining was activated in a stress-free state, and the confining forces were released. This transferred the rock load to the lining, allowing the system to settle into its final equilibrium. Forces exchanged between lining and soil in this condition were used to load the lining with regular increments, up to finding the best fitting with the in-situ measured stress.