PSI - Issue 84

Gerardo Sorrentino et al. / Procedia Structural Integrity 84 (2026) 1071–1078

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the case study, a tunnel built in 1935. In fact, by analyzing the method proposed just before the tunnel’s construction (Kommerell 1912) and the one codified by Desimon (1939) after the end of the construction of the tunnel under analysis, this study provides a comprehensive view of the historical design assumptions that defined the lining’s final geometry. 2. Design approaches comparison 2.1. Sidewall Rotation Equilibrium Method The SRE method assumes symmetric loads and constraints on the lining, negligible shear forces and a soil with compressive strength and stiffness high enough to provide a horizontal reaction along the upper portion of the sidewall. The method, developed by Kommerell, does not take into account the presence of an invert arch. The SRE theory applies to cohesive soils. It requires three hinges in the arch so that the system becomes statically determined. Their exact positions are not known in advance but are typically taken at the keystone and at the springer. The soil above the lining is supposed to load on the tunnel lining in coherence with the detachment of an equivalent loading body with parabolic shape of height h above the central axis of the tunnel ( Figure 2 a). Given the symmetry, only half of the arch is analyzed. The vault—from keystone to springer—is subdivided into voussoirs to determine the individual contributions of self-weight and external loads ( h ⋅ γ ) ( Figure 2 b). Once the vertical forces on the voussoirs are defined, a force polygon is constructed to establish the magnitude, direction, and line of action of the total resultant force acting on the half-arch. As a first trial, the thrust line is assumed to pass through the upper limit of the middle third at the crown and near the section centroid, at the base. These points are iteratively adjusted to ensure the curve remains entirely within the middle third region along the arch, satisfying the condition of a purely compressed arch. Finally, the horizontal thrust at the springing is fully balanced by the horizontal soil reaction ( E ). During the rotation mechanism, the soil reaction generates a frictional force R along the sidewall as: = ∙ (1) Where is the friction coefficient (ranging from 0 to 0.3). Consequently, the vertical load transmitted to the foundation is reduced by this friction component R . The point of application of this reduced load (P f ) is assumed to lie between the foundation centroid and the outer limit of the middle third. Once the point of application is defined, the stresses at the intrados ( σ i ) and the extrados ( σ e ) and be determined. Moreover, the height of the resulting triangular load for horizontal soil reaction can be found as: = 13 √2 ∙ ∙ − (2) Where, b is the sidewall base length. After combining this load with the vertical forces of the voussoirs, the thrust line construction is repeated, incorporating the horizontal reaction from the soil. 2.2. Soil–Thrust Wedge Method The STW method analyzes the tunnel lining under the assumption of symmetric loading and constraints, neglecting shear forces at the interface. The STW theory works for cohesive soils. The approach assumes the detachment of a loading body of height h above the crown, which produces a parabolic pressure (h ⋅ γ) ( Figure 3 b). This pressure triggers the sliding of a lateral soil wedge along a failure plane defined by the friction angle ( φ ). The resulting slip plane produces an overload associated with the equivalent loading body above it. This is modeled as a horizontal pressure