PSI - Issue 84

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

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4. Results Figure 4 shows the comparison between the pressures acting on the lining for the two methods. The approaches lead to different results. While the SRE formulation models vertical pressure as parabolic (decreasing at the sides), the STW method assumes a quasi-uniform distribution that increases on the sidewalls ( Figure 4 a). With respect to horizontal pressures (Figure 4b), the SRE method assumes a triangular reaction at the base, while the STW method employs a trapezoidal distribution.

(a) (b) Figure 4 Pressures acting on tunnel lining, where x=0 indicates the keystone, until arriving to the base of the sidewall. Comparison between the two design approaches. (a) Vertical soil pressure (b) horizontal soil pressure. Static equations, instead of graphical static, were used to combine loads and calculate the line of thrust and stresses on masonry lining intrados ( σ i ) and extrados ( σ e ), as showed in Figure 5 . Despite different load conditions, the two approaches reveal similar stress states ( Figure 5 a); the STW method incorporates the invert contribution—absent in the SRE scheme—allowing for a redistribution of forces. At the keystone, both approaches set the thrust line within the upper middle third. Different horizontal load components lead to slightly higher σ e stresses in the STW method. In both analyses, the thrust line shifts toward the intrados near the springer, reaching comparable values of σ i but at slightly different locations. Along the sidewall, stresses remain low in the SRE approach, as the eccentricity is assumed to be minimal (about 2 cm). Conversely, the STW method, produces slightly higher stress due to the larger equivalent loading body acting on the lining; a marginally smaller eccentricity at the extrados further influences the variation between intrados and extrados stresses. Finally, the axial force transmitted to the ground through the sidewall is lower in the SRE (2091 kN) than in STW (2417.5 kN) one. However, the resulting maximum soil stress is significantly higher in the SRE method (1.59 MPa) compared to the STW one (1.05 MPa) as the latter benefits from the combined area of the sidewall and invert to redistribute the normal force. The soil stresses should be compared against the limit stresses of 1.20 MPa (without invert) and 1.40 MPa (with invert), as defined in section 2.3.

(a) (b) Figure 5 (a) Stresses at intrados and extrados comparison between the two methods (b) In situ and design approaches lining stress comparison (stress computed at 15 cm of depth from the intrados). The x-axis starts from the keystone.