Dynamic Energy Balance and Size Effect in Tunnel Linings under Sudden Support Loss: A Load-Sharing Formulation of Robustness for Hierarchical Support Systems


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<big>'''Dynamic Energy Balance and Size Effect in Tunnel Linings under Sudden Support Loss: A Load-Sharing Formulation of Robustness for Hierarchical Support Systems'''</big></div>

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'''EUR ING Alessandro Calvi'''<sup>1</sup></div>

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<sup>1</sup>'' Hydropower Engineer, ENEL Green Power S.p.A., Torino, Italy''</div>

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[mailto:alessandro.calvi84@gmail.com alessandro.calvi84@gmail.com]</div>
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'''Abstract'''</div>

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.

'''Keywords: '''tunnel lining; sudden support loss; progressive collapse; dynamic energy balance; load sharing; size effect; robustness; hierarchy; hydropower

==1. Introduction==

In a first extension of the author’s frame studies to tunnels, the robustness of a support system was expressed through the ratio between the critical load multipliers of the damaged and intact structure, and the ductile-to-brittle transition through two thresholds, a/D ≈ 0.5 and an energetic ratio of about 0.75, transferred from the results obtained for reinforced concrete (RC) frames (Calvi, 2025a, 2026). That formulation has two limits. It is quasi-static, whereas the loss of a shotcrete panel, of a group of bolts or of a rock wedge is a sudden event whose dynamic effect is the reason why the alternate load path method is used for buildings (GSA, 2003; Izzuddin et al., 2008). And it takes the effect of hierarchy from the frame simulations, without a mechanism that explains why a system with more, smaller elements is less robust.

This paper develops that mechanism. The starting point is the observation that, after a sudden loss of support, the question is whether the elements adjacent to the damaged zone can absorb, in a dynamic way, the load that was carried by the lost ones. The answer depends on three things: how close to their capacity the elements were (utilisation), how much plastic deformation they can undergo before failing (ductility), and over how many elements the redistributed load is spread (load sharing). Each of these has a classical counterpart: the energy balance of an elastic–plastic oscillator (Biggs, 1964), the beam on elastic foundation (Hetényi, 1946), and the size effect law of quasi-brittle materials (Bažant, 1984). The novelty lies in their combination into a closed-form robustness measure for a ring.

The scope is deliberately limited. The model is two-dimensional, analytical, and applies to a single ring in which a contiguous group of elements is lost at the same instant. Parameters used in the application are assumed, not measured, and the results are closed-form consequences of the model and not simulation outcomes.

==2. Starting points from previous work==

Three results of the earlier studies are used as reference. From the review of tunnel collapses (Calvi, 2025b): the initial phase of failure in hydropower tunnels is often a material degradation (hydration–dehydration of laumontite, swelling of clays) followed by stress redistribution and shotcrete detachment (Schlotfeldt et al., 2025), and the support that failed was static. From the study on robustness (Calvi, 2025a): the collapse load decreases when more columns are removed from one storey and frames with fewer, larger primary elements are more resistant after damage. From the multi-scale study (Calvi, 2026): the residual-strength ratio R = Pmax,d/Pmax,i computed from its Tables 1 and 2 is 0.125, 0.090 and 0.051 for the 2 × 2, 5 × 5 and 11 × 11 frames; a power-law fit through the three points gives R ∝ n−0.53, an exponent between zero and one that will be interpreted in Section 5.

==3. Formulation==

===3.1 Dynamic capacity of an element under a sudden load increment===

Consider an element of the ring, idealised as an elastic–perfectly plastic oscillator with yield force Py, yield displacement uy and ductility μ = umax/uy. Before the damage it carries a static force P0 = u0Py, where u0 < 1 is its utilisation. At the instant of the loss of neighbouring elements its load jumps by ΔP. Measured from the initial equilibrium position, the available elastic reserve is Q = Py − P0 and the ductility becomes μ′ = (μ − u0)/(1 − u0). Equating the work of the suddenly applied increment to the strain energy absorbed (Figure 1a), the largest increment that can be sustained is

[[Image:Draft_Calvi_941894204-image1.png|348px]] (1)

Dividing by the static force P0 gives the number of ‘element loads’ that an element can additionally take without failing,

[[Image:Draft_Calvi_941894204-image2.png|222px]] (2)

For μ → ∞ the function Λ tends to the static plastic reserve (1 − u0)/u0; for μ = 1 it tends to (1 − u0)/(2u0), i.e. the static reserve halved by the dynamic amplification factor of 2 that applies to the increment but not to the pre-load (Figure 1b). This distinction is important: applying the factor 2 to the total load, as in the simplest alternate-load-path checks, would give no margin at all for u0 = 0.5 and μ = 1, whereas Equation (2) gives a margin of half an element load.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_941894204-image3.png|600px]] </div>

'''Figure 1. '''(a) Energy balance of an elastic–plastic element subjected to a suddenly applied load increment: the work of the increment (rectangle above P0) equals the strain energy absorbed (area under R(u)). (b) Function Λ(u0, μ), Equation (2), for three values of the utilisation.

===3.2 Load-sharing length of the ring===

The load of the lost elements is transferred to the neighbours through the bending stiffness of the ring and the reaction of the ground. A ring of radius R, with bending stiffness EI per unit width, on a ground with radial stiffness kg behaves locally as a beam on elastic foundation, whose characteristic wavelength gives the length over which a local disturbance is transmitted (Hetényi, 1946):

[[Image:Draft_Calvi_941894204-image4.png|276px]] (3)

where Er and νr are the modulus and Poisson ratio of the ground. The length ℓs is the load-sharing length. If the ring is made of n elements of length ℓe = πD/n, the number of elements over which the load is shared on each side of the damaged zone is

[[Image:Draft_Calvi_941894204-image5.png|210px]] (4)

The lower bound m = 1 represents a segmental or weakly jointed lining in which the joints cannot transmit the bending moments needed for the load to spread beyond the adjacent element (Figure 2a); m > 1 represents a lining whose joints are rigid enough, or a continuous ring, so that the distribution follows the foundation solution (Figure 2b). The transition between the two regimes occurs at n* = πD/ℓs, as defined below.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_941894204-image6.png|600px]] </div>

'''Figure 2. '''Load sharing after the loss of elements (hatched). The elements that receive the redistributed load are shown in orange: (a) local sharing, m = 1; (b) distributed sharing over m elements each side.

===3.3 Critical damaged arc===

If k contiguous elements are lost, the 2m neighbours take, on average, k/(2m) additional element loads each, so that their demand-to-capacity ratio is

[[Image:Draft_Calvi_941894204-image7.png|144px]] (5)

The neighbours survive the dynamic event if the increment k/(2m) is not larger than Λ, so that

[[Image:Draft_Calvi_941894204-image8.png|240px]] (6)

The second part of Equation (6) is the critical damaged arc fraction, ψcrit = (lost arc)/(πD). For n ≤ n* (m = 1) the number of elements that can be lost is the constant 2Λ, so that ψcrit decreases as 1/n: the finer the subdivision of the ring, the smaller the arc that can be lost. For n ≥ n* the critical arc tends to a plateau ψ∞, that does not depend on n, because each element is small compared with the transfer length and the load is spread in proportion to n; the plateau and the transition number are

[[Image:Draft_Calvi_941894204-image9.png|174px]] (7)

Since the lost arc has length a = ψπD, the result is expressed in the geometric parameter used for frames as

[[Image:Draft_Calvi_941894204-image10.png|60px]] (8)

which allows the comparison with the threshold a/D ≈ 0.5 of Calvi (2026). The averaging implicit in Equation (5) is non-conservative with respect to a peaked distribution of the load on the neighbours, and the model should be applied only to values of ψcrit small enough for the damage to be regarded as local (here, ψ < 0.5).

===3.4 Size effect, strength and ductility===

The strength and the ductility of the elements are not independent of their size. For a quasi-brittle material the nominal strength of a geometrically similar structure of characteristic size t decreases with size, from the strength criterion for small sizes to the linear elastic fracture mechanics (LEFM) trend for large sizes (Bažant, 1984; Bažant and Planas, 1998):

[[Image:Draft_Calvi_941894204-image11.png|294px]] (9)

where κ is a geometry-dependent constant of order 0.5 that must be calibrated, and ℓch the characteristic length of the material (Carpinteri, 1984). Here the size is the lining thickness tl. β is a brittleness number: β ≪ 1 indicates a behaviour governed by strength, β ≫ 1 a behaviour governed by fracture (Figure 3). To couple the size effect with the ductility used in Equation (2), a simple closure is adopted, which is an assumption and not a derived result: the available ductility decreases from a reference value μ0, for β → 0, to 1, for β → ∞,

[[Image:Draft_Calvi_941894204-image12.png|114px]] (10)

Finally, if the demand is fixed and the lining thickness changes, the utilisation varies with the bending capacity, proportional to σN tl², and therefore

[[Image:Draft_Calvi_941894204-image13.png|210px]] (11)

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_941894204-image14.png|432px]] </div>

'''Figure 3. '''Size effect law, Equation (9): nominal strength normalised by the strength-criterion value as a function of the brittleness number. The points correspond to three lining thicknesses for ℓch = 0.58 m and κ = 0.5.

==4. Parametric application: a hydropower tunnel==

The same illustrative tunnel as in the first extension is used: diameter D = 6 m, shotcrete lining of thickness tl = 0.30 m, with a ground of moderate stiffness. All input values in Table 1 are assumed for illustration and do not refer to a specific project. The reference utilisation is u0 = 0.5, i.e. a global safety factor of 2 on each element, and the reference ductility is μ0 = 5.

'''Table 1. '''Assumed input and derived quantities for the illustrative case.

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Input (assumed)'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Value'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Derived'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''Value'''</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Diameter D</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">6 m</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Ground stiffness kg, Eq. (3)</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.33 GN/m³</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Lining thickness tl</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.30 m</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Bending stiffness EI</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">70.3 MN·m</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Lining modulus Ec; ν</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">30 GPa; 0.2</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Load-sharing length ℓs</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.13 m</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Ground modulus Er; ν</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">5 GPa; 0.25</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Transition n* = πD/ℓs</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">8.9</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Gf; ft (lining)</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">120 N/m; 2.5 MPa</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">ℓch = Ec·Gf/ft²</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.58 m</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">κ; μ0; u0</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.5; 5; 0.5</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">β; μeff; Λ</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.04; 2.96; 0.90</span>
|}


Table 2 and Figure 4 give the critical damaged arc as a function of the number of elements. For n ≤ 8 the ring tolerates the loss of 1.8 elements whatever n is, so that ψcrit falls from 0.45 (n = 4) to 0.23 (n = 8). From n ≈ 9 the tolerance in terms of elements increases with n, and the critical arc stabilises at ψ∞ = 0.20, i.e. a/D = 0.64. The value is of the same order as the geometric threshold of the frames, a/D ≈ 0.5, which corresponds to ψ = 0.16.

'''Table 2. '''Critical damaged arc for different numbers of elements per ring (tl = 0.30 m, u0 = 0.5, μeff = 2.96).

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''n'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''m'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''kcrit (elements)'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''ψcrit'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''a/D = π ψcrit'''</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">4</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.00</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.80</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.449</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.41</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">6</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.00</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.80</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.299</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.94</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">8</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.00</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.80</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.225</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.71</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">12</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.36</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.44</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.203</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.64</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">24</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.71</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">4.87</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.203</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.64</span>
|}


<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_941894204-image15.png|474px]] </div>

'''Figure 4. '''Critical damaged arc fraction as a function of the number of elements per ring for different ductilities (u0 = 0.5, tl = 0.30 m). The right axis gives the equivalent a/D. The dashed line is the geometric threshold of the frame study.

Ductility has a strong effect when it is low and a weak effect once it exceeds about 3 (Table 3): going from μ = 1 to μ = 3 raises the plateau from a/D = 0.35 to 0.64, whereas going from 3 to 5 raises it only to 0.67. A lining with little ductility and a utilisation of 0.5 can lose only about 1.0 element at n ≤ n*, compared with 1.8 for a ductile one. Figure 5a shows that the utilisation is the dominant parameter: the plateau falls rapidly as u0 approaches 1, whatever the ductility.

'''Table 3. '''Sensitivity of ψcrit to the ductility μ (u0 = 0.5, tl = 0.30 m); a/D refers to the plateau (n = 24).

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''μ'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''ψcrit, n = 4'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''ψcrit, n = 8'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''ψcrit, n = 24'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''a/D (plateau)'''</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.250</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.125</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.113</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.35</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%;">0.417</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.208</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.188</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.59</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.450</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.225</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.203</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.64</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.472</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.236</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.213</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.67</span>
|}


The thickness enters through three channels: it reduces the utilisation (Equation 11), it lengthens the load-sharing length (ℓs ∝ tl3/4 from Equation 3) and, through the size effect, it reduces strength and ductility (Equations 9 and 10). Table 4 and Figure 5b show that the first two prevail in this range, so that thickening the lining increases the tolerated arc even when the size effect is accounted for. The size effect removes only a small part of the gain: at tl = 0.30 m the plateau is 0.203 with the closure and 0.213 without it. For tl ≥ 0.40 m, ψ∞ exceeds 0.5 and the result leaves the range of validity of the model, as the damage can no longer be regarded as local.

'''Table 4. '''Effect of the lining thickness (demand fixed, u0 = 0.5 at tl = 0.30 m).

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''tl (m)'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''β'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''μeff'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''u0'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''ℓs (m)'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''ψ∞'''</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">'''a/D'''</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%;">0.87</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">3.14</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.689</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.86</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.083</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.26</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.30</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.04</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.96</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.500</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.13</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.203</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.64</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.35</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.22</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.81</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.383</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.39</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.357</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.12</span>
|-
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.40</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.39</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.67</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.304</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.64</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.547</span>
|  style="border: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.72</span>
|}


<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_941894204-image16.png|600px]] </div>

'''Figure 5. '''(a) Plateau value ψ∞ as a function of the utilisation for four ductilities (tl = 0.30 m). (b) ψ∞ as a function of the lining thickness with the size-effect closure and with constant ductility and strength. Values are capped at 0.5, the limit of validity of the model.

==5. Comparison with the frame results==

The model reproduces qualitatively the dependence on hierarchy found for the RC frames. In the local-sharing regime the tolerated arc decreases as n−1, in the distributed regime it does not depend on n; the frames of Calvi (2026) give R ∝ n−0.53 over n = 2 to 11 (Figure 6), between the two limits, which is what is expected if the joints transmit part of the moments and the sharing is only partial. The comparison is qualitative: R is a ratio of ultimate loads obtained by DEM simulations and ψcrit is a critical damaged fraction, so the exponents are not directly equivalent, and the frame has no ground restraint.

A second comparison concerns the threshold. In the frame study the geometric threshold a/H ≈ 0.5 was found numerically. Here an equivalent quantity follows from the model, and it varies between a/D ≈ 0.35 for a brittle lining (μ = 1) and 0.67–0.69 for a ductile one (μ = 5 to 10) at u0 = 0.5. The value 0.5 falls in the middle of this range, which supports its use as a first estimate, but the model also shows that it is not a constant: it depends on the utilisation, ductility, thickness and load-sharing length. The energetic criterion of the first extension remains useful, as the characteristic length ℓch enters here through the size effect, but it plays a different role: it fixes the brittleness number rather than a threshold on the damaged length.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_941894204-image17.png|444px]] </div>

'''Figure 6. '''Residual-strength ratio of the frames of Calvi (2026) as a function of the number of cells, with the two limiting trends of the load-sharing model anchored at n = 2.

==6. Discussion==

===6.1 Practical indications===

Four indications follow from the model, to be checked against data. First, the utilisation of the elements before the event is the most effective control: reducing u0 from 0.5 to 0.3 more than doubles the tolerated arc (Figure 5a), so that conservatism in the design of the primary lining pays off more than the choice of the subdivision. Second, the subdivision matters only below the transition n*: for a segmental lining, increasing the number of segments reduces the tolerance in proportion to 1/n unless the joints are made capable of transmitting moments, which brings the system towards the distributed regime. Third, the load-sharing length ℓs can be estimated in the field from the rotational stiffness of the joints and the ground stiffness, and a monitoring programme can compare it with the element length. Fourth, in a lining that is degraded by hydration or swelling, the loss of ductility and the increase in utilisation both act in the unfavourable direction, and they can be followed through the parameters μeff and u0.

===6.2 Limits of the model===

The model is a first-order representation. It considers a single ring in two dimensions, and ignores the longitudinal load transfer to the adjacent rings, which is a major source of robustness in real tunnels. The ground is represented by a linear radial spring, whereas it can unload and yield, and its inertia and the dynamic response of the rock mass are neglected. Elements are elastic–perfectly plastic, with no softening, and the interaction between axial force and bending is not included. The averaging of the load over the 2m neighbours, the closure of Equation (10), the constant κ and the assumption that m switches abruptly from 1 to ℓs/ℓe are assumptions, and the plateau obtained for the illustrative case depends on them. For these reasons the numerical values must not be used for design.

===6.3 Calibration and validation===

Each part of the model can be tested separately. Equations (1) and (2) can be checked with dynamic simulations of a ring with beam elements, using the DEM code of the previous studies with the sudden removal of contiguous elements and a variable utilisation. The load-sharing length of Equation (3) can be checked with the same simulations by recording the distribution of the forces in the neighbouring elements, and with elastic finite-element models. The size-effect closure needs fracture tests on shotcrete specimens of different depths, from which κ and ℓch can be obtained. Finally, back-analysis of documented collapses, such as the hydropower tunnel analysed by Schlotfeldt et al. (2025), would allow the plateau to be compared with the observed extent of damage.

===6.4 Algebraic, dimensional and numerical validation===

The proposed equations were independently re-derived and evaluated with direct numerical calculations. This section distinguishes verification of the mathematics from validation of the physical idealisation. Verification asks whether the closed forms reproduce their governing assumptions; physical validation requires experiments, calibrated finite-element or discrete-element analyses, and field evidence.

===6.4.1 Exact energy balance for the suddenly applied increment===

Let r = P0/Py and δ = ΔP/P0. At the maximum displacement umax = μuy, the velocity is zero. Conservation of energy relative to the pre-damage equilibrium gives (P0 + ΔP)(umax − u0) = U(umax) − U(u0). For an elastic–perfectly plastic resistance, U(umax)/(Pyuy) = μ − 1/2 and U(u0)/(Pyuy) = r²/2. Solving for δ gives δ = [(1 − r)/r][1 − (1 − r)/(2(μ − r))], exactly Equation (2). Thus the expression is not an empirical dynamic amplification factor; it is the exact response of the adopted single-degree-of-freedom model to an ideal load step.

The limiting cases are mechanically consistent. For μ = 1, Λ = (1 − r)/(2r); for μ → ∞, Λ = (1 − r)/r; and for r → 1, Λ → 0. Differentiation also gives ∂Λ/∂μ > 0 and ∂Λ/∂r < 0 over 0 < r < 1 and μ ≥ 1, confirming the trends plotted in Figures 1 and 5.

'''Table 5'''. Independent numerical check of Equation (2).

{| style="width: 100%;margin: 1em auto 0.1em auto;" 
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''u0'''</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''μ'''</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Λ, Eq. (2)'''</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Λ, energy root'''</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''absolute error'''</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.3</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1.166667</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1.166667</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2.2e-16</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.3</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1.852941</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1.852941</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">4.4e-16</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.3</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2.030864</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2.030864</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">4.4e-16</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.3</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2.159574</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2.159574</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.0e+00</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.3</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">10</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2.249141</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2.249141</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.0e+00</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.5</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.500000</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.500000</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.0e+00</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.5</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.833333</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.833333</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.0e+00</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.5</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.900000</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.900000</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.0e+00</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.5</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.944444</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.944444</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.0e+00</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.5</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">10</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.973684</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.973684</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.0e+00</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.214286</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.214286</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.0e+00</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.379121</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.379121</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.0e+00</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.400621</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.400621</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.0e+00</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.413621</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.413621</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1.1e-16</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.7</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">10</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.421659</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.421659</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.0e+00</span>
|}


Across the 15 checks, the largest absolute discrepancy is below 10⁻¹⁵, i.e. floating-point round-off. This verifies the algebra but does not by itself validate the elastic–perfectly plastic constitutive assumption.

===6.4.2 Dimensional verification and reproduction of the illustrative case===

With EI expressed per unit tunnel width, kg has units N/m³ and kg/(4EI) has units m⁻⁴; therefore λ has units m⁻¹ and ℓs = π/λ is a length. The characteristic length ℓch = EGf/ft² also has units of length because Gf is energy per crack area (N/m). Hence β, m, Λ and ψ are dimensionless. This check is important because alternative foundation conventions use a spring stiffness per unit longitudinal length; mixing those conventions would change λ and the plateau.

'''Table 6'''. Reproduction of the baseline calculations from the stated inputs.

{| style="width: 100%;margin: 1em auto 0.1em auto;" 
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''quantity'''</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''recomputed value'''</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">'''rounded value in article'''</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">EI</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">70.312 MN·m² per m</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">70.3</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">kg</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1.3333 GN/m³</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1.33</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">ℓs</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2.1291 m</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2.13</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">ℓch</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.5760 m</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.58</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">β</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1.0417</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">1.04</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">μeff</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2.9592</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">2.96</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">Λ</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.8983</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.90</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">n*</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">8.8535</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">8.9</span>
|-
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">ψ∞</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.2029</span>
|  style="text-align: center;"|<span style="text-align: center; font-size: 75%;">0.203</span>
|}


All baseline quantities reproduce the published rounded values. The numerical chain is therefore internally consistent under the stated convention EI = Ectl³/[12(1 − ν²)] and kg = Er/[(1 + νr)R].

===6.4.3 Integer damage states and a conservative interpretation===

Equation (6) returns a continuous kcrit, whereas the number of lost segments is discrete. For decision use, the admissible integer loss should be kallow = floor(kcrit) when failure at equality is excluded. In the baseline local-sharing regime kcrit = 1.797, so the strict guaranteed result is loss of one complete element, not 1.8 elements. The continuous value remains useful as a normalized resistance index and for comparing layouts. For a continuous lining, k may instead be interpreted through the damaged arc length, provided that localization and joint opening are represented consistently.

The quantity in Equation (5) is the post-redistribution total force normalized by Py. It is therefore a static utilization measure, not the dynamic demand-to-capacity ratio used in Equation (6). To avoid ambiguity, it is preferable to denote it ηpost = u0(1 + k/2m), while retaining the energy condition k/(2m) ≤ Λ for sudden loss.

===6.5 Uncertainty propagation and sensitivity===

A reproducible screening calculation was performed with 20,000 independent samples. The assumed ranges were uref = 0.35–0.65, μ0 = 3–7, Er = 2–10 GPa sampled log-uniformly, Gf = 80–180 N/m, ft = 2.0–3.0 MPa, and tl = 0.27–0.33 m; the remaining baseline inputs were unchanged. These ranges are illustrative and are not statistical descriptions of a specific tunnel.

The resulting uncapped plateau had a median ψ∞ = 0.209, with a 5–95% interval of 0.093–0.402. The spread is wide enough to confirm that a single deterministic plateau should not be used as a design resistance. The one-at-a-time calculation in Figure 7 shows that utilization and thickness dominate this assumed range, while ground modulus primarily changes the transfer length and fracture parameters act through both strength and the ductility closure.

'''Figure 7'''. Numerical verification and sensitivity of the analytical model.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_941894204-image18.png|600px|alt='6.6 Broader literature context and model positioning'|6.6 Broader literature context and model positioning]] </div>

6.6 Broader literature context and model positioning

The load-sharing concept belongs to the broader family of ground–support interaction models. Classical convergence–confinement methods describe the evolution of radial pressure and displacement as the face advances, whereas the present formulation addresses an instantaneous local loss after an initial equilibrium has already formed. The two descriptions are complementary: convergence–confinement can define the pre-event utilization u0, while the transient energy balance estimates the additional reserve following loss.

For shotcrete and fibre-reinforced shotcrete, post-cracking residual strength, bond to the rock, thickness variability, and bolt anchorage govern whether a nominally continuous shell can mobilize the assumed transfer length. Experimental and numerical work on hard-rock tunnel linings shows that load from a loose block is transferred through a relatively narrow bond-stress band and that thickness and bond variability materially affect capacity. This supports treating ℓs and joint continuity as calibration parameters rather than purely elastic constants.

The size-effect component should also be interpreted carefully. Bažant’s law is an asymptotic energetic scaling for geometrically similar quasi-brittle structures. Equation (10), by contrast, is a proposed closure linking brittleness to deformation capacity. A stronger calibration route is to derive μeff from a measured tensile stress–crack opening law, integrate the post-cracking work to obtain Gf, and map the resulting rotation or displacement capacity into the oscillator. This would avoid fitting strength and ductility independently and would permit explicit treatment of fibre dosage and ageing.

Dynamic material enhancement is intentionally omitted from the current baseline. Concrete and shotcrete strengths can increase with strain rate, but a sudden support loss is not defined by a single strain rate: it depends on the removed arc, modal participation, damping, ground inertia and contact opening. Introducing a generic dynamic increase factor without a computed strain-rate history could therefore give false precision. A validated extension should obtain strain rates from the transient model and then apply a rate-sensitive constitutive law.

===6.7 Proposed validation matrix and acceptance metrics===

A staged programme can turn the hypothesis-level model into a calibrated engineering tool:

Level 1, component tests: determine E, ft, Gf, residual tensile law, bond strength and joint moment–rotation curves at relevant age and moisture state.

Level 2, ring benchmark: apply a known radial preload, remove one support sector rapidly, and measure displacement, acceleration and force redistribution around the circumference.

Level 3, numerical blind prediction: calibrate material parameters only on component tests, then predict ring tests without adjusting ℓs or μeff.

Level 4, field back-analysis: reconstruct the pre-event utilization, damaged arc and support continuity for documented incidents, with uncertainty bounds rather than point estimates.

Recommended quantitative acceptance metrics are: error in peak adjacent-element force below 15%; error in the circumferential e-folding or half-power transfer length below 15%; error in maximum displacement below 20%; correct prediction of the first failed element or sector; and coverage of at least 90% of observed outcomes by the stated uncertainty interval. These are proposed research targets, not code requirements.

==7. Conclusions==

A mechanical model of sudden support loss in a tunnel lining has been formulated, based on the dynamic energy balance of the elements adjacent to the damage, on a load-sharing length derived from the beam on elastic foundation, and on a size-effect law for the strength and ductility of the lining. The main results are:

* A neighbouring element can take, under a suddenly applied increment, Λ = [(1 − u0)/u0][1 − (1 − u0)/(2(μ − u0))] additional element loads; the dynamic amplification applies to the increment and not to the pre-load.

* The critical damaged arc fraction is ψcrit = 2mΛ/n, with m = max(1, ℓs n/πD). It decreases as 1/n up to n* = πD/ℓs and is constant above it, which provides a mechanism for the dependence of robustness on hierarchy found in the frame simulations.

* In the illustrative case (D = 6 m, tl = 0.30 m) the ring tolerates the loss of 1.8 elements for n ≤ 8, and a plateau a/D ≈ 0.64 for n ≥ 9; the equivalent a/D varies between 0.35 and 0.69 for ductilities between 1 and 10, so that the frame threshold a/D ≈ 0.5 is a reasonable first estimate but not a constant.

* Utilisation is the dominant parameter, ductility matters mainly below μ ≈ 3, and the size effect reduces only slightly the benefit of a thicker lining in the range examined.

* The results are consequences of an analytical model with assumed parameters; their validation requires dynamic DEM simulations of rings, fracture tests on shotcrete and back-analysis of documented collapses.

The numerical verification added in this revision reproduces the baseline values and confirms Equation (2) against the exact work–energy equation to machine precision. It also identifies two interpretation rules: distinguish the static post-redistribution utilization from the dynamic energy criterion, and round continuous kcrit downward when it is used for discrete segment loss.

The uncertainty screening demonstrates that the plateau is a distribution rather than a fixed threshold. For the illustrative ranges adopted here, ψ∞ has a 5–95% interval of 0.093–0.402, reinforcing the need for project-specific calibration.

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