PSI - Issue 84

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

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1. Introduction The structural assessment of existing masonry infrastructures is a primary challenge, especially because these assets have reached the end of their nominal design life. Italy has widespread heritage of masonry infrastructure, characterized by tunnels and bridges that have been in service for one or more centuries. Preserving the safety of these infrastructures is therefore essential. To accomplish this, it is necessary to understand how these structures were originally conceived and designed. The aim is to review historical design methods to establish a basis for comparing them with current verification approaches. Examining these methods is a key step in understanding how the structures were meant to carry loads, since their assumptions shaped their geometry. Comparing the original design with the current state of the structure provides key insights into its structural performance and enables the development of strategies for updating or assessing its safety. Historically, the design of masonry tunnel linings was governed by the principles of graphical statics and the theory of the arch-thrust line. However, determining the loads acting on a tunnel lining is a complex geotechnical problem influenced by several interacting factors that are often difficult to capture within theoretical models. The magnitude and direction of the forces depend strongly on soil and rock quality, its properties, and its sensitivity to weathering or water. Subsurface conditions can also vary significantly along the tunnel alignment. Faults, folds, discontinuities, and sliding planes with different orientations may induce highly heterogeneous stress fields that change markedly along the structure. To deal with this complexity, in the past, numerous calculation methods have been developed over the decades, each aiming to provide an a priori estimate of the loads acting on tunnel linings. In the following paragraph a brief overview of the evolution of the theoretical models to estimate the loads and the consequent design of the tunnel lining is reported. 1.1. Historical approaches to tunnel design Culmann (1866) was one of the first to address tunnel lining stability, using graphical statics to study shallow excavations in cohesionless soils with horizontal ground surfaces, deriving load exclusively from the self-weight of the soil and internal friction. Some years later, working on the Giovi railway tunnel, Curioni (1887) shifted the focus to deep tunnels in scaly clays. He described the load on the lining as the combination of the lining’s weight and the soil’s self-weight, treating the soil as a pseudo-fluid whose density reflected its plastic state. Following typical assumption for masonry constructions, Curioni assumed the thrust line remained within the middle third of the cross section of the lining to calculate stresses, in order to account for a completely compressed section. Later, following the Sempione’s Tunnel construction, Heim (1878) suggested that loads should be split into two distinct components: mountain pressure (depth-dependent hydrostatic loads from the rock mass) and rock pressure (local stress concentrations caused by structural irregularities). He further assumed the strength of the rock mass to be 1/3 to 1/10 of the intact rock strength. Ritter (1879) subsequently formulated a theory for squared or circular section tunnels in coherent soils. He modeled a separation curve between the overlying ground mass, or loading body, (term used to indicate the height of the solid of rock, or soil, weighing on the linings) and the soil as a parabolic cylinder extending to the sidewalls base. He considered the tunnel as shallow, accounting for an additional load on the lateral soil portions equal to the weight of the soil below the separation curve and converting the lining self-weight into an equivalent soil volume. According to the Ritter’s theory, Gröeger (1881) established experimentally a ratio of the pressures f:f/2:f/3 applied to the keystone, sidewalls, and bottom, respectively. Engesser (1882) later showed through experiments that in loose, cohesionless soils only part of the overlying material transfers load to the lining. Moreover, he introduced a theory to identify the separation curve, describing it as a circular arc tangent to the line where the sliding planes meet the horizontal plane through the tunnel crown. Afterwards, experimental evidence on cohesionless soils was provided by Janssen (1895), who performed extensive tests on granular soils, proving that once the soil height exceeds a critical limit, the resulting separation surface becomes independent of the overburden height. A widely applied theoretical framework for deep tunnels was proposed by Kirsch (1898), who examined the stress field around a circular opening in a homogeneous, perfectly elastic rock mass. He suggested that, moving toward the tunnel boundary, the material shifts from an elastic response to a state close to failure. Willmann (1911) investigated a rectangular opening in a rock mass subjected to a uniform vertical load, assuming that once the counter-pressure is