From Frames to Tunnels: A Multi-scale Fracture-Mechanics Framework for Progressive Collapse and Robustness of Underground Support Systems


<!-- metadata commented in wiki content


<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<big>'''From Frames to Tunnels: A Multi-scale Fracture-Mechanics Framework for Progressive Collapse and Robustness of Underground Support Systems'''</big></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
'''EUR ING Alessandro Calvi'''<sup>1</sup></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<sup>1</sup>'' Hydropower Engineer, ENEL Green Power S.p.A., Torino, Italy''</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
[mailto:alessandro.calvi84@gmail.com alessandro.calvi84@gmail.com]</div>
-->

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
'''Abstract'''</div>

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.

'''Keywords: '''tunnels; progressive collapse; robustness; fracture mechanics; ductile-to-brittle transition; hierarchical structures; hydropower; lining; discrete-element method

==1. Introduction==

Tunnel collapses combine geological, hydrogeological and structural causes, and they are rarely the consequence of a single event. Case histories such as the roof failure of the Hanekleiv Tunnel in Norway, the landslide-induced failure of the Walibie Tunnel in China, the shallow collapse in an urban setting in Indonesia and the collapse of a hydropower tunnel in a fault-damaged zone share a common feature: a local loss of support or of rock-mass integrity was followed by redistribution of load, by further local failures and, finally, by a failure that was disproportionate to the initiating event (Calvi, 2025b). This is exactly the definition of progressive collapse used for buildings, where robustness is the ability of a structure to avoid damage that is disproportionate to its cause (Eurocode 1, 2004; Knoll and Vogel, 2009).

For buildings, a mature body of work exists on how a sudden loss of columns or beams propagates through a frame, and how the answer depends on redundancy, ductility and structural hierarchy (Starossek, 2009; Izzuddin et al., 2008; Masoero, 2010). In two recent studies the author used a discrete-element method (DEM) to derive closed-form expressions for collapse loads and a robustness ratio of reinforced concrete (RC) frames (Calvi, 2025a), and then showed that the response of hierarchical frames to localised damage can be read as a ductile-to-brittle transition governed by two thresholds, one geometric and one energetic (Calvi, 2026). Tunnels are, however, rarely analysed in this language, although a lining that loses a shotcrete panel, a rock bolt or a precast segment is, conceptually, a structure that has suffered an accidental loss of load-bearing elements.

The objective of this paper is to extend the frame framework to tunnels. The contribution is conceptual and methodological: it defines the correspondence between frame and tunnel quantities, formulates a robustness ratio and two threshold criteria for tunnel support systems, and demonstrates their use on a parametric example inspired by hydropower tunnels. The thresholds themselves are not claimed to be validated for tunnels; they are carried over from the frame study as hypotheses and the conditions for their calibration are stated explicitly.

==2. Background: three previous studies==

===2.1 Tunnel collapse mechanisms===

The review of tunnel collapses in Calvi (2025b) identifies five main mechanisms: roof instability, shear failure of the sidewalls, base heave, wedge failure and progressive collapse. Roof instability and wedge failure are governed by the weight of the detaching block and by the shear resistance on its boundary, with pore pressure acting as a strong destabilising factor (Liu et al., 2021; Lyu et al., 2019). Sidewall shear failure is controlled by the orientation of discontinuities with respect to the principal stress directions (Zhou et al., 2025), while base heave develops when high pore pressure reduces the effective stress below the floor (Lyakhovsky and Mendecki, 2025). Progressive collapse is the mechanism in which fracture propagation and stress relaxation evolve over time and in which the monitoring of deformation and stress is the primary defence.

Three findings of that review are particularly relevant here. First, the stress state around the opening matters: tensile stresses at the crown and compressive stresses at the invert were found in landslide-affected tunnels, and the major principal stress was found to rotate after excavation in a fault-damaged zone. Second, in hydropower tunnels the initial phase of collapse is frequently linked to mineralogical degradation, for example hydration–dehydration cycles of laumontite or swelling of montmorillonite, followed by stress redistribution, crack propagation under transient water pressure, shotcrete detachment and progressive rockfalls (Schlotfeldt et al., 2025). Third, the support systems that failed were generally static, while the failure process was time-dependent.

===2.2 Robustness of intact and damaged frames===

In Calvi (2025a) the critical multiplier of the vertical loads of a frame with movable nodes is obtained from the P–Δ method as the smallest value of λ that makes the matrix κt + λE singular, where κt is the translational stiffness matrix and E the equilibrium matrix of the second-order effects. The same condition written for the damaged structure, with additional contributions Ed, gives the damaged multiplier λd, and the robustness ratio is

[[Image:Draft_Calvi_126091501-a33e2d75ef81e9163e6cb3073839bcaf86400502.png|48px]] (1)

Using a DEM code with beam elements, the alternate load path method and a Hertzian contact law, the study showed that bending collapse is the most severe mode, that the collapse load decreases as more columns are removed from one storey, and that frames with fewer, larger primary elements (lower n) are more resistant to bending collapse after damage; when one-third of the columns of a storey were removed, the residual-strength ratio was about 25%.

===2.3 Multi-scale ductile-to-brittle transition===

The follow-up study (Calvi, 2026) considered 2 × 2, 5 × 5 and 11 × 11 RC frames with the same overall size and a central-bottom damage zone (CB) of width a. By analogy with the strength of a notched beam, the dimensionless ultimate plastic load was written as

[[Image:Draft_Calvi_126091501-2b6e3f03141d08571c43c14c21ab63d615a24b57.png|132px]] (2)

where L, H and t are the length, height and thickness of the frame and σp the tensile strength. Two thresholds were identified. The geometric threshold is reached when the ratio between the damaged width and the storey height approaches a/H ≈ 0.5, beyond which the 2 × 2 frame, although the strongest in the intact state, loses its ability to redistribute load and its fragility converges to that of the less hierarchical frames. The energetic threshold, associated with a critical length derived from fracture mechanics, was found at a ratio of approximately 0.75 and was interpreted as the exhaustion of the energy-dissipation capacity. The two thresholds define a sequential path, from a state of geometric vulnerability (point P) to brittle collapse (point U).

The simulation outputs of Tables 1 and 2 of Calvi (2026) allow the residual ratio R = Pmax,d/Pmax,i to be computed directly: 54 141/433 125 = 0.125 for the 2 × 2 frame, 15 593/173 250 = 0.090 for the 5 × 5 frame and 3 986/78 750 = 0.051 for the 11 × 11 frame (Figure 4a). Robustness therefore decreases monotonically with the number of structural cells, a result that will be used as the reference trend for tunnels.

==3. The frame-to-tunnel analogy==

A frame carries vertical loads through beams and columns, and a tunnel carries ground and water loads through a lining, a bolted rock ring and the surrounding ground arch. The analogy adopted here maps each quantity of the frame study onto a tunnel quantity (Table 1). Three correspondences deserve comment. The sudden removal of elements, which is the alternate load path method of the GSA (2003), corresponds to the sudden loss of a shotcrete panel, of a group of bolts or of a rock wedge. The storey height H, which normalises the width of the damaged zone in the frame study, corresponds to the tunnel diameter or span D. The hierarchical level 1/n corresponds to the organisation of the support: a thick continuous ring with a few primary ribs is a hierarchical system, whereas a lining made of many small segments or of a dense pattern of short bolts is a fragmented one (Figure 2).

'''Table 1. '''Correspondence between the quantities of the frame study and the tunnel formulation.

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Frame (Calvi, 2025a, 2026)'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Tunnel (this paper)'''</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Beams and columns</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Lining elements: shotcrete panels, steel ribs, precast segments, rock bolts</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Sudden removal of elements (ALPM)</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Sudden loss of support: shotcrete detachment, bolt failure, local rockfall</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Storey height H</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Tunnel diameter or span D</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Width a of the damaged zone</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Axial or circumferential extent a of the failed support or loosened zone</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Hierarchical level 1/n</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Support organisation 1/n: number n of independent load-bearing elements per ring</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Load multiplier λ</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Multiplier of ground, pore and hydraulic pressure</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Redistribution by frame action</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Ground arching and ground–support interaction</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Crack length a in a beam</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Flaw or spalling depth at in the lining; fracture in the rock mass</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Fragility number</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Tunnel fragility number, same definition</span>
|}


<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_126091501-db37a98ac48075874739baef2091b8975d537b0a.png|600px]] </div>

'''Figure 1. '''Correspondence between (a) an RC frame with a damaged zone of width a and storey height H, (b) a tunnel cross-section with a damaged lining arc of extent a and depth at, and (c) a tunnel longitudinal section with a loosened zone of length a above a crown of diameter D, here intersected by a fault-damaged zone.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_126091501-0235045b051e9c062c1dea4d41555d91a0214da5.png|600px]] </div>

'''Figure 2. '''Hierarchical levels of a tunnel support ring, for the same damaged fraction of the ring: (a) few large primary elements, (b) intermediate organisation, (c) many small elements. Hatched elements are removed.

==4. Tunnel formulation==

===4.1 Stress state around the opening===

The stress redistribution that follows excavation is the starting point of the process. For a circular opening in a homogeneous elastic medium subjected to a vertical stress σv and a horizontal stress σh = Kσv, the Kirsch solution gives the tangential stress on the boundary as a function of the angle θ measured from the horizontal axis (Kirsch, 1898):

[[Image:Draft_Calvi_126091501-07b0962be8edea9c01c805a7c2a5f8a99fb6423b.png|258px]] (3)

which gives σθ = (3K − 1)σv at the crown and σθ = (3 − K)σv at the springline (Figure 3). Tension appears at the crown for K < 1/3, a condition that is relevant for tunnels in steeply dipping or relaxed formations, and it is consistent with the high tangential stress concentrations at the haunch and crown reported for the fault-damaged zone of the hydropower tunnel (Schlotfeldt et al., 2025). When the major principal stress rotates after excavation, as observed in that case, the angular positions of the peaks in Figure 3a shift, and the maxima may coincide with the zones of lowest strength.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_126091501-3434fbd33373464af8713ffc1b8efd9edc4a44ca.png|600px]] </div>

'''Figure 3. '''(a) Normalised tangential stress on the boundary of a circular opening for different stress ratios K (Kirsch solution); the shaded band is the tensile range. (b) Crown and springline stress as a function of K; the crown goes into tension for K < 1/3.

===4.2 Robustness ratio of the support===

The robustness ratio of Equation (1) is transferred to the support ring by keeping its second-order character. A ring subjected to external pressure loses stability when the matrix of the linear stiffness of the lining, Kl, and the geometric matrix Gp that depends on the pressure p become singular. For a thin circular ring of radius R, bending stiffness EI per unit length and Poisson ratio ν, the classical result is pcr = 3EI/[(1 − ν²)R³] (Timoshenko and Gere, 1961). If the multiplier of the ground and hydraulic pressure is λ, the intact and damaged critical multipliers are the smallest roots of

[[Image:Draft_Calvi_126091501-04d47dc8ff88eedf77ea5fcd1567d7192185b612.png|180px]] (4)

where Gd contains the contribution of the lost elements. The tunnel robustness ratio is Rt = λd/λi, as in Equation (1). It should be noted that the ground does not appear explicitly in Equation (4): it enters through the pressure multiplier and through the stiffness of the ground springs added to Kl when a ground–support interaction model is used. This is the first difference with respect to a frame and it is discussed in Section 6.

===4.3 Section capacity of a damaged lining===

The local counterpart of Equation (2) for a lining section of thickness tl, loaded in bending and with a flaw, spalled layer or delamination of depth at, is obtained from the net-section argument used for the notched beam. With δ = at/tl, the ratio between the bending capacities of the damaged and the intact section is

[[Image:Draft_Calvi_126091501-2e7c46ac1dc7a90e1c9a45e58d45ed8e1705578c.png|126px]] (5)

which is the through-thickness analogue of the factor (1 − a/H)² of the frame. A loss of one third of the thickness reduces the section capacity to 45%, and a loss of one half to 25% (Figure 4b). This reduction acts at the section level and it explains why shotcrete spalling, which is often regarded as a surface defect, is an early indicator of capacity loss.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_126091501-44d088a3541047cbfd806558006edb0efd74900f.png|600px]] </div>

'''Figure 4. '''(a) Residual-strength ratio R = Pmax,d/Pmax,i of the RC frames of Calvi (2026), computed from the values of Tables 1 and 2 of that study. (b) Bending-capacity ratio of a lining section as a function of the relative damage depth δ = at/tl, Equation (5).

===4.4 Geometric criterion===

For frames, the geometric criterion is the ratio between the width of the damaged zone and the storey height. For tunnels the analogous parameter is a/D, where a is the axial length of the failed support, or the angular extent expressed as arc length, and D is the diameter. The working hypothesis is that, as in the frames, the capacity of the system to redistribute load by ground arching and by the remaining lining is lost when a/D approaches 0.5, the point P of the sequential path. The value is carried over from the frame study; physically, it indicates that the damaged zone has become as wide as the radius, so that the unsupported span can no longer be bridged by the adjacent rings. Its validity for tunnels must be tested and is likely to depend on the ground class, on the overburden and on the stiffness of the support.

===4.5 Energetic criterion===

The energetic criterion is based on the Griffith balance between the elastic energy released and the energy needed to create new fracture surfaces. For a flaw of length ac in a medium of Young modulus E, fracture energy Gf and acting tensile stress σ, the critical flaw length and the instability index are

[[Image:Draft_Calvi_126091501-394b1137ef937daadfc4d0d02f316bd98fd71514.png|174px]] (6)

so that Φ = 1 marks unstable propagation. In Calvi (2026) an energetic ratio of about 0.75 marked the transition to brittle collapse of the frames; here this value is adopted as the energetic threshold, Φ ≈ 0.75, corresponding to point U. The expression of Equation (6) is written in terms of the acting stress, whereas the characteristic length of the material, which is independent of the loading, is

[[Image:Draft_Calvi_126091501-ed8467b043228e5d081fd2038320f776feafe08f.png|78px]] (7)

Equation (7) is the length used in the frame study in its normalised form. The two lengths are related by acrit = ℓch (ft/σ)²/π, so that a flaw that is critical at a tensile stress equal to the strength is smaller than ℓch/π, whereas at a stress equal to half the strength it is four times larger. Representative order-of-magnitude values of the material parameters are given in Table 2; they are assumed for illustration and must be replaced by project data.

'''Table 2. '''Assumed representative material parameters and resulting characteristic length (illustrative values, not measured data).

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Material'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''E (GPa)'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''ft (MPa)'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Gf (N/m)'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''ℓch = E·Gf/ft² (m)'''</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Concrete or shotcrete</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">30</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.5</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">120</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.58</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Intact hard rock</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">40</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">6.0</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">100</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.11</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Altered rock in a fault-damaged zone</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.5</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">20</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.16</span>
|}


<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_126091501-11a7514dad40b6e55418678a5def1cdc94447a32.png|600px]] </div>

'''Figure 5. '''(a) Instability index Φ as a function of the flaw length in the lining for different degradation levels d, with E and Gf both reduced by the factor (1 − d); E0 = 30 GPa, Gf0 = 120 N/m and σ = 1.5 MPa. (b) Critical flaw length as a function of the driving tensile stress (logarithmic scale).

===4.6 Sequential path in tunnels===

In the frame study the geometric threshold P was reached first and the energetic threshold U second. In tunnels the order is not universal. In an intact lining the critical flaw length is comparable to or larger than the lining thickness, so that the energetic instability cannot develop inside the section and the failure is controlled by the geometry of the loosened zone. When the material degrades, as in hydration and swelling processes, E and Gf decrease and the flaw length required to reach U becomes smaller than the thickness, so that local brittle failure can initiate well before the damaged zone becomes wide. The energetic threshold then governs the initiation of damage and the geometric threshold governs its propagation. The resulting multi-scale path is summarised in Figure 6 and mirrors the sequence described for hydropower tunnels by Schlotfeldt et al. (2025): mineralogical degradation, stress redistribution, crack propagation under transient pressure, shotcrete detachment and progressive rockfall.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_126091501-2c5f7ae7bee4df2e74b9e96998fe52990cca41ee.png|600px]] </div>

'''Figure 6. '''Multi-scale failure path of a tunnel support system. The energetic threshold U governs the initiation of local brittle failure at the material scale, and the geometric threshold P governs the propagation of the damaged zone at the tunnel scale.

==5. Parametric illustration: a hydropower tunnel==

The framework is illustrated for a circular pressure tunnel with diameter D = 6 m and a shotcrete lining with thickness tl = 0.30 m, in a fault-damaged zone with a tensile driving stress at the crown σ = 1.5 MPa resulting from stress redistribution and transient pressure. The lining is characterised by E0 = 30 GPa and Gf0 = 120 N/m. The degradation induced by hydration and swelling is represented by an index d, with E = E0(1 − d) and Gf = Gf0(1 − d). All the figures are the outcome of the closed-form expressions of Section 4 and are not simulation results; they do not represent a specific project.

'''Table 3. '''Critical flaw length and energetic threshold for the illustrative case (D = 6 m, tl = 0.30 m, σ = 1.5 MPa).

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''d'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''E (GPa)'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Gf (N/m)'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''acrit (m)'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''aU at Φ = 0.75 (m)'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''aU / tl'''</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">30</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">120</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.51</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.38</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.27</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.25</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">22.5</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">90</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.29</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.21</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.72</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.50</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">15</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">60</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.13</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.10</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.32</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.75</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">7.5</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">30</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.03</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.02</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.08</span>
|}


Three observations follow from Table 3. First, the critical flaw length decreases with (1 − d)², so that a moderate loss of stiffness and fracture energy produces a much larger loss of tolerance to flaws: a degradation index of 0.5 reduces acrit from 0.51 m to 0.13 m. Second, in the intact case the energetic threshold is reached at a flaw length of 0.38 m, larger than the thickness of the lining (aU/tl = 1.27), so the threshold cannot be reached inside the section; at d = 0.5 it is reached at about one third of the thickness, where Equation (5) gives a residual bending capacity of 0.46. Third, the geometric threshold a/D ≈ 0.5 corresponds to an unsupported length of 3 m, almost one order of magnitude larger than the flaw size at which the energetic instability starts in a degraded lining. In such a lining the process is therefore expected to be initiated at the energetic threshold and propagated up to the geometric one, which is the path shown in Figure 6.

A sensitivity check on the driving stress (Figure 5b) shows that the same conclusion is obtained by raising the stress: since acrit varies with σ⁻², a doubling of the tensile stress caused by transient pressure or stress rotation has the same effect on the flaw tolerance as a loss of about half of the combined stiffness and fracture energy.

==6. Discussion==

===6.1 Implications for design and monitoring===

The framework suggests four practical indications. The first is to monitor the extent of the damaged zone, expressed as a/D, and not only the convergence of the opening, because the geometric threshold identifies the point at which load redistribution is no longer effective. The second is to track the degradation of the lining and of the rock in fault-damaged zones, since the energetic threshold depends quadratically on it; this agrees with the recommendation of detailed mineralogical surveys and adaptive support for hydropower tunnels (Calvi, 2025b). The third concerns the hierarchy of the support. If the trend of the frames, R decreasing from 0.125 to 0.051 as n increases, is confirmed, support systems with fewer, larger primary elements should be preferred where local loss of elements is credible, but the frame study also warns that hierarchy is a double-edged sword, because hierarchical systems cross the critical threshold more abruptly. The fourth is that drainage and pore-pressure control, which reduce the driving stress σ, act on the energetic threshold with the same weight as the improvement of material properties.

===6.2 Limits of the analogy===

Several differences between frames and tunnels limit the direct transfer of the thresholds. The ground confines the lining, so that the pressure on the support decreases as the opening converges, in the sense of the convergence–confinement method; in a frame, the load does not decrease with the deformation. Ground arching is a three-dimensional mechanism, similar to the membrane action of slabs that, as stated in Calvi (2026), would delay the transition in three-dimensional frames. The frame thresholds were obtained in two dimensions for a strongly ductile RC frame and for a central-bottom damage scheme; for tunnels the damage may be located at the crown, at the haunch or at the invert, with different consequences (Figure 3). Finally, the load in tunnels is not purely gravitational: pore pressure, swelling pressure and transient hydraulic loads add time-dependent components to the multiplier λ, and the failure process is influenced by creep and relaxation (Calvi, 2025b). For these reasons the thresholds a/D ≈ 0.5 and Φ ≈ 0.75 should be regarded as initial values, and probably as a conservative lower bound, in line with the way the frame thresholds were presented.

===6.3 Calibration programme===

The proposed calibration has three steps. In the first, the DEM code used for the frames would be extended to a lining ring with beam elements and ground springs, using the alternate load path method with the removal of a variable number of elements at the crown, haunch and invert, for different levels of hierarchy n and of ground stiffness. In the second, the geometric threshold would be searched for as the value of a/D at which the fragility number departs from the trend of the intact structure, as done for frames. In the third, the energetic threshold would be compared with fracture parameters measured on cores of shotcrete and of altered rock, and with post-collapse back-analysis of documented cases, such as the 12 000 m³ collapse in a hydropower tunnel analysed by Schlotfeldt et al. (2025). Fully non-linear hydromechanical models and the machine-learning surrogate approaches mentioned by Thai (2022) would then allow the framework to be used for monitoring.

==7. Conclusions==

A framework has been proposed that extends to tunnels the robustness and ductile-to-brittle transition concepts developed by the author for reinforced concrete frames. The main points are the following.

* A tunnel lining and its ground arch can be treated as a hierarchical system subjected to sudden local loss of elements, and the frame quantities map onto tunnel quantities (Table 1), with the diameter D replacing the storey height H and the organisation of the support replacing the hierarchical level 1/n.

* The robustness ratio Rt = λd/λi, the section-capacity factor (1 − δ)² and the instability index Φ = ac/acrit provide, respectively, global, local and material-scale measures of the damage tolerance of the support.

* The thresholds identified for frames, a/H ≈ 0.5 and an energetic ratio of ≈ 0.75, are transferred to tunnels as working hypotheses (a/D ≈ 0.5 and Φ ≈ 0.75) that require calibration.

* In the parametric illustration, a degradation index of 0.5 reduces the critical flaw length by a factor of four and brings the energetic threshold within the lining thickness, so that local brittle failure can initiate before the damaged zone becomes wide; the failure path of a degraded hydropower tunnel is then energetic at initiation and geometric at propagation.

* Ground confinement, three-dimensional arching, time-dependent loads and the position of the damage are the main factors that limit the direct transfer of the frame results, and they define the programme of DEM simulations and back-analyses proposed in Section 6.3.

==References==

Calvi A (2010) The Twin Towers Collapse: Event Analyses and Structural Lessons. Master’s thesis, Politecnico di Torino, Italy.

Calvi A (2025a) Robustness interpretation of intact and damaged framed structures. Proceedings of the Institution of Civil Engineers – Forensic Engineering 179(1): 1–7. [https://doi.org/10.1680/jfoen.25.00008. https://doi.org/10.1680/jfoen.25.00008.]

Calvi A (2025b) Tunnels collapses: case studies and key failure mechanisms. Preprint. [https://doi.org/10.13140/RG.2.2.10172.76165. https://doi.org/10.13140/RG.2.2.10172.76165.]

Calvi A (2026) Multi-scale progressive collapse in hierarchical RC frames via fracture mechanics. Proceedings of the Institution of Civil Engineers – Forensic Engineering. [https://doi.org/10.1680/jfoen.25.00033. https://doi.org/10.1680/jfoen.25.00033.]

Carpinteri A (1984) Meccanica dei Materiali e della Frattura. Pitagora Editrice, Bologna, Italy.

Danava D, Prassetyo SH and Wattimena RK (2022) Determination of the height and width of the collapse zone for shallow tunnels. International Symposium on Earth Science and Technology.

Eurocode 1 (2004) EN 1991-1-7 Actions on structures – Part 1-7: General actions – Accidental actions.

General Services Administration (GSA) (2003) Progressive Collapse Analysis and Design Guidelines for New Federal Office Buildings and Major Modernization Projects. GSA, Washington, DC, USA.

Gordon JE (1978) Structures: Or Why Things Don’t Fall Down. Zanichelli, Bologna, Italy.

Izzuddin BA, Vlassis AG, Elghazouli AY and Nethercot DA (2008) Progressive collapse of multi-storey buildings due to sudden column loss – Part I: Simplified assessment framework. Engineering Structures 30(5): 1308–1318.

Kirsch G (1898) Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre. Zeitschrift des Vereines Deutscher Ingenieure 42: 797–807.

Knoll F and Vogel T (2009) Design for Robustness. IABSE Structural Engineering Documents 11, Zurich, Switzerland.

Liu J, Chen T, Xie C, Tian J and Wei Y (2021) Limit analysis of collapse mechanisms for tunnel roofs subjected to pore water pressure: a numerical approach. Advances in Civil Engineering.

Lyakhovsky V and Mendecki AJ (2025) Dynamic modelling of tunnel failure. Pure and Applied Geophysics.

Lyu C, Zeng Z and Dong Y (2019) Limit analysis of progressive asymmetrical collapse failure of tunnels in inclined rock stratum. Symmetry.

Masoero E (2010) Progressive Collapse and Robustness of Framed Structures. PhD thesis, Politecnico di Torino, Italy.

Sarioletlagh Fard S and Atiabi M (2020) A study of factors influencing tunnel collapses and how they affected the Norwegian Hanekleiv Tunnel. Conference paper.

Schlotfeldt P, Carvalho J and Panton B (2025) Root cause analysis of a collapse in a hydropower tunnel. Applied Sciences.

Starossek U (2009) Progressive Collapse of Structures. Thomas Telford, London, UK.

Thai HT (2022) Machine learning for structural engineering: a state-of-the-art review. Structures 38: 448–491.

Timoshenko SP and Gere JM (1961) Theory of Elastic Stability, 2nd edn. McGraw-Hill, New York, USA.

Vlassis AG (2007) Progressive Collapse Assessment of Tall Buildings. PhD thesis, Imperial College London, UK.

Zhou W, Fan J, Qiu S and Zhang Y (2025) Failure modes and interaction mechanisms of tunnels under active landslide conditions. Scientific Reports.

<div id="1"></div>
[[#cite-1|1]]

<div id="2"></div>
[[#cite-2|2]]