Earlier work by the author showed that the robustness of ring-type structures (tunnel linings, masonry domes on piers) after the sudden loss of load-carrying elements can be obtained, without imposed thresholds, as the root of an equation that equates the peak load redistributed onto the neighbouring elements to their dynamic capacity. This paper transfers that formulation to arch dams. The dam is represented by a vertical strip whose bending stiffness is that of the cantilever and whose elastic foundation is the hoop stiffness of the horizontal arches; the elements are the arch levels, and the damage is the sudden loss of arch action over ''k'' contiguous levels (abutment wedge, thrust block or foundation failure). The load-sharing length is shown to be purely geometric, ℓ_s = π(''Rt'')1/2/[3(1−ν2)]1/4, and equals 59 m for a reference dam with ''H'' = 100 m, ''R'' = 100 m and ''t'' = 6 m. Numerical verification includes a time-integration check of the dynamic capacity (maximum error 1.2%), a reproduction of two results of the tunnel-ring study, and 87 nonlinear dynamic finite-element cases. The dynamic-efficiency parameter calibrated on tunnel rings (ω = 0.14) is not transferable: it is over-conservative by 16% on average for the dam strip, whereas the isolated-oscillator capacity (ω = 1) is conservative by 10%. For a utilisation of 0.4 the dam tolerates the loss of a height of about 0.3''H'' at mid-height, 0.11–0.15''H'' at the crest and more than 0.5''H'' near the base, nearly independent of the number of elements. A flexural check shows, however, that the unreinforced cantilever strip cannot bridge even a single lost arch level elastically, so that the redistribution bound is attainable only if a vertical flexural capacity exists. All inputs are assumed and the results are hypothesis-level.
Abstract Earlier work by the author showed that the robustness of ring-type structures (tunnel linings, masonry domes on piers) after the sudden loss of load-carrying elements can [...]
Previous work by the author showed that progressive collapse of hierarchical reinforced concrete frames can be read as a ductile-to-brittle transition with a geometric and an energetic threshold, and proposed to carry the same reading over to tunnel linings. That framework is quasi-static and does not explain why robustness depends on the number of structural elements. This paper replaces it with a mechanical model of sudden support loss in a tunnel lining. Three ingredients are combined: (i) a dynamic energy balance for the elements adjacent to the lost ones, which gives a closed-form capacity for a suddenly applied load increment as a function of utilisation and ductility; (ii) a load-sharing length of the ring on elastic ground, derived from the beam-on-elastic-foundation solution, which separates a local-sharing regime (segmental or weakly jointed linings) from a distributed-sharing regime (continuous rings); and (iii) a Bažant-type size effect that links the ductility and the strength of the lining to its thickness through the characteristic length E·Gf/ft². The model gives a critical damaged arc fraction ψcrit that decreases as 1/n for n below a transition value n* = πD/ℓs and becomes independent of n above it. For an illustrative hydropower tunnel (D = 6 m, tl = 0.30 m) the lining tolerates the loss of about 1.8 elements for n ≤ 9 and a damaged arc of about 0.64 diameters in the distributed regime, consistent in order of magnitude with the geometric threshold a/D ≈ 0.5 found for frames. The formulation is analytical and hypothesis-level; a calibration programme with discrete-element simulations is outlined.
Abstract Previous work by the author showed that progressive collapse of hierarchical reinforced concrete frames can be read as a ductile-to-brittle transition with a geometric and [...]
Progressive collapse is usually discussed for buildings, whereas tunnel failures are discussed through case histories, ground-reaction curves and support-design charts. This paper brings the two literatures together. Starting from three previous studies by the author (on tunnel collapse mechanisms, on the robustness of intact and damaged framed structures, and on the multi-scale ductile-to-brittle transition in hierarchical reinforced concrete frames), it proposes a framework in which a tunnel lining and its surrounding ground are treated as a hierarchical load-bearing system subjected to sudden local damage. The framework combines (i) a second-order robustness ratio R = λd/λi between the damaged and intact critical load multipliers, (ii) a geometric criterion based on the ratio between the width of the damaged zone and the tunnel diameter, and (iii) an energetic criterion based on a Griffith-type critical flaw length and on the material characteristic length E·Gf/ft². The transfer of the two thresholds identified for frames (a/H ≈ 0.5 and an energetic ratio of ≈ 0.75) to tunnels is formulated as a working hypothesis. A parametric illustration for a hydropower tunnel crossing a fault-damaged zone shows how mineralogical degradation shrinks the critical flaw length (by more than an order of magnitude for severe degradation), bringing the energetic threshold within the thickness of the lining and shifting the sequence of failure from a purely geometric to a combined energetic–geometric path. Limits of the analogy, in particular ground confinement and three-dimensional arching, and a programme for calibration with discrete-element simulations are discussed.
Abstract Progressive collapse is usually discussed for buildings, whereas tunnel failures are discussed through case histories, ground-reaction curves and support-design charts. This [...]
This research addresses the critical challenge of progressive collapse in Reinforced Concrete (RC) frames by establishing a rigorous multi-scale link between material fracture mechanics and macroscale structural hierarchy. The study investigates the highly non-linear 'Ductile-to-Brittle' transition, identifying two dimensionless ratios as the fundamental governing parameters: the Geometric Ratio (a/H), which marks the onset of brittle behavior at a threshold of approximately 0.5, and the Energetic Ratio (L
Abstract This research addresses the critical challenge of progressive collapse in Reinforced Concrete (RC) frames by establishing a rigorous multi-scale link between material fracture [...]
The accurate characterization of in-situ stress states within rock masses is a critical prerequisite for the safe design and construction of road and railway tunnels. This paper synthesizes and extends the reservoir geomechanics framework developed by Zoback (2010, Stanford University) — originally conceived for petroleum wellbore applications — and rigorously adapts it to the problem of underground rock excavation in civil engineering contexts. The paper presents a new analytical methodology, the Integrated Stress Analysis Framework for Rock Tunnels (ISAF-RT), which combines in-situ stress characterization techniques (extended leak-off tests, hydraulic fracturing in pilot boreholes, wellbore breakout analysis) with classical rock mechanics tools (Kirsch analytical solution, Mohr-Coulomb and Hoek-Brown failure criteria, ConvergenceConfinement Method) within a unified probabilistic design workflow. The ISAF-RT framework introduces a novel six-phase sequential procedure that explicitly accounts for tectonic stress regime (normal, strike-slip, thrust faulting), anisotropic stress ratios (K₀), rock mass quality (GSI index), and time-dependent deformation, enabling the engineer to select appropriate support systems with quantified reliability. Application of the framework is demonstrated through numerical examples representative of Alpine railway tunnels and highway tunnels in folded mountain belts, where thrust-faulting regimes impose horizontal stresses up to 1.8 times the vertical overburden. The results highlight the importance of pre-excavation stress measurement campaigns and show that neglecting stress anisotropy can lead to underdesign of support by factors of 2-3. The ISAF-RT framework provides a rigorous, systematic, and practically applicable tool for geotechnical engineers engaged in subsurface infrastructure in challenging rock conditions.
Abstract The accurate characterization of in-situ stress states within rock masses is a critical prerequisite for the safe design and construction of road and railway tunnels. This [...]
The excavation of road and railway tunnels through inclined fractured zones represents one of the most challenging scenarios in underground engineering. The stress state within the rock mass undergoes complex perturbations when the tunnel axis intersects fault zones, shear bands, or preexisting fracture sets at non-perpendicular angles. Drawing on the theoretical framework developed by Zoback (2010) for failure analysis of deviated wells — specifically the generalised borehole wall stress equations, drilling-induced tensile fracture (DITF) diagnostics, and shear velocity anisotropy measurements — this article extends these principles to the three-dimensional geomechanical problem of inclined tunnel excavation. A critical review of stress transformation theory, breakout analysis for inclined openings, and the role of fracture dip in governing tunnel stability is presented. On this basis, the article proposes a new geomechanical methodology: the Integrated Stress Analysis Framework for Fractured Zones (ISAF-FZ). The ISAF-FZ is a sixphase, adaptive workflow that integrates in-situ stress characterisation, fracture network analysis, inclined opening stability assessment, support design, and real-time monitoring. Application to two case studies — the Gotthard Base Tunnel and the A1 Apennine Motorway Tunnel — demonstrates the framework's predictive capacity for support pressure requirements and excavation sequencing in crossed fractured zones. The results show that fracture dip angle is a primary control on required support pressure, and that the ISAF-FZ framework reduces prediction uncertainty by up to 35% compared to conventional isotropic approaches.
Abstract The excavation of road and railway tunnels through inclined fractured zones represents one of the most challenging scenarios in underground engineering. The stress state within [...]
Fault zones represent the most challenging geological discontinuities encountered during road and railway tunnel construction. When tunnel excavation intersects active or reactivated faults, the in-situ stress state within the rock mass undergoes profound perturbations: the fault core and damage zone display sharply contrasting mechanical and hydraulic properties compared with intact host rock, preferential slip planes exist at critically stressed orientations, and elevated pore pressures may transiently destabilise the excavation face. Conventional tunnel design methods, which treat the rock mass as a homogeneous elastic medium, systematically underestimate the support requirements and the risk of sudden failure events — including face collapse, large-displacement squeezing, and rockburst — in these settings. This article presents a rigorous geomechanical analysis of fault zone behaviour during tunnel excavation, grounded in the critically-stressed fault theory of Zoback (2010) and in the extensive body of evidence from deep borehole studies, seismic reflection surveys, and fault zone hydraulics. The Mohr-circle framework for evaluating fault criticality, the Stress Criticality Index (SCI) quantifying the proximity of a fault to frictional reactivation, the fracture density powerlaw model governing damage zone architecture, and the permeability-stress coupling equations are all translated into tools applicable to the tunnel design workflow. On this basis, the article proposes a new predictive framework — the Fault Zone Stress-Excavation Interaction (FZSEI) framework — that integrates SCI computation, damage zone geometry characterisation, pore-pressure transient modelling, and adaptive support classification into a unified design methodology
Abstract Fault zones represent the most challenging geological discontinuities encountered during road and railway tunnel construction. When tunnel excavation intersects active or [...]
The Gorges du Verdon (Haute-Provence, SW French Alps) constitute a spectacular deep fluvial canyon incised into Upper Jurassic to Lower Cretaceous Tithonian limestones of the Castellane fold-and-thrust belt, a region characterised by active thin-skinned compression (~0.1 mm/yr uplift), a complex network of N–S and E–W fault systems, and topographic relief of 300–700 m. Recent cosmogenic ³⁶Cl exposure dating by Cardinal et al. (2024) has quantified river incision rates of 0.06–0.20 mm/yr for the Late Pleistocene (60–15 ka), with a marked acceleration to 0.90
Abstract The Gorges du Verdon (Haute-Provence, SW French Alps) constitute a spectacular deep fluvial canyon incised into Upper Jurassic to Lower Cretaceous Tithonian limestones of [...]
Dam Engineering & Hydroelectric Energy 
Dynamic Robustness of Arch Dams under Sudden Loss of Arch Action: A Threshold-Free Load-Sharing Formulation Transferred from Tunnel Linings and Masonry Domes 
