PSI - Issue 84

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

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Figure 2 (a) Calibrated simplified approach with the thrust line in red; (b) Beam LE model with concentrated forces.

3.1. Validation through Beam LE model To validate the accuracy of the SA based on graphical statics, a comparative numerical analysis was conducted using a linear elastic Finite Element Method model (LE FEM) shown in Figure 2b. The model analyzed the masonry arch (considering a half-section due to symmetry), subjected it to the external actions derived from the calibrated SA. The properties of the masonry are those described in the paragraph on the case study. The arch was horizontally constrained at the axis to impose symmetry. In the LE FEM model, the tunnel lining section was subdivided into thirteen finite element beam elements, with each node located at the centroid of a voussoir. In this configuration, the external load was applied as discrete nodal forces (vertical and horizontal components) that correspond to the resultant actions calculated for the centroid of each individual voussoir in the simplified discretization. This setup was designed to verify the mathematical precision of the force polygon calculation. The comparison of the results, illustrated in Figure 3, demonstrates a remarkable consistency across the two approaches. The normal stresses at both the intrados (  i ) and extrados (  e ) (Figure 3a), as well as the eccentricity ( e ) (Figure 3b) of the line of thrust related to the SA, follow the same overall trend as the FEM model along the normalized length of the arch. Specifically, in the crown section, the FEM simulation predicts slightly higher compressive stresses at the intrados and lower values at the extrados compared to the simplified analytical solution. Similar stress states and eccentricities are found in the sidewall in both approaches, with FEM model resulting in higher differences at the sidewall top and base. Regarding the invert, a shift in eccentricity results in as well in a shift as well in the invert stress, the stress values calculated by the simplified method are higher at the intrados and lower at the extrados respect to those of the FEM approach.