PSI - Issue 84

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

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Table 1 Material properties.

Masonry

Soil

Density

γ m 1800 kg/m 3

Density

2600 kg/m 3

γ

Compressive strength f c

6 MPa

Friction angle

30°

-

φ c

Young modulus

E 4800 MPa

Cohesion

0.3 MPa 1500 MPa

Young modulus

E s K 0

Earth pressure coefficient at rest

1

-

2.1. Simplified approach – Desimon method To evaluate the condition of existing tunnels, this study proposes a simplified approach SA (Sorrentino et al., 2026) based on the graphical design method proposed by Desimon, V. (1939). This methodology is built upon the work of Giovannini, M. (1936) and incorporates the later refinements introduced by Falchi Delitala, G. (1971). The method employs a static graphical approach to determine the line of thrust within the tunnel arch by discretizing the lining into voussoirs and solving the equilibrium through a force polygon. Specifically, in this study, the arch geometry was approximated using thirteen discrete trapezoidal elements (Figure 1a). The model assumes the detachment of a rock mass of height h above the crown, which exerts parabolic vertical pressure h ⋅ γ (Figure 1a). This vertical load triggers a lateral failure mechanism where a soil wedge slides along a failure plane defined by the friction angle  . Consequently, the slip plane generates a horizontal pressure ( V p ) representing the overload of the equivalent rock mass. This pressure increases linearly with depth—proportional to the soil unit weight—reaching a maximum value of V g ⋅ γ at the base of the sidewall. According to the geometric framework revised by Falchi Delitala, G. (1971), horizontal pressure is distributed between the keystone (H 1R ) and the invert arch (H 2R ). Below the invert arch, the soil bears the vertical loads, redistributing the pressures under the sidewall ( σ p ) and the invert (σ inv ) in proportions determined by the rotational equilibrium around the pole O (Figure 1a). However, for soils with a natural slope angle lower than 35°, the resulting line of thrust tends to fall outside the middle third of the cross-section, leading to a theoretical condition of instability. To address this, Desimon introduced a horizontal amplification factor (1+ f) of the active horizontal force. This coefficient is designed to account for the passive resistance provided by the surrounding soil against the outward rotation of the sidewall. By incorporating this additional resisting force, the model mimics the actual confinement provided by the ground, allowing the thrust line to shift back within the stable core of the section. Once the external actions are defined, the static analysis proceeds by calculating the resultant force vector for each voussoir. This vector combines the masonry self-weight, the vertical rock load acting on the element, and the corresponding horizontal soil pressure. The construction of the funicular polygon begins at the keystone, where the line of thrust is constrained to pass below the upper limit of the middle third. From this starting point, the curve is traced progressively towards the invert arch. This procedure determines the point of application of the resultant force at each interface, thus defining the eccentricity relative to the cross-section's centroid. Although f was historically calibrated based on design practice, it remains a significant unknown for existing structures, much like the theoretical load height h. In the analyzed case study, applying standard theoretical values derived from Desimon ( h =24m and f = 0.8) results in very high stress levels compared to those measured in the considered section of the tunnel. Figure 1b highlights the differences between the resulting stresses and those measured. Starting from the current condition, the SA is applied by treating the load height h and the amplification factor f as variables to be calibrated. 3. Calibration of the Simplified approach The calibration of the SA was performed by initializing the analysis with theoretical values for the load height h and the amplification factor f . Subsequently, an optimization procedure based on the Least Squares Method was applied to minimize the error between the stress values calculated by the model and the actual stresses measured in-