Dynamic Robustness of Arch Dams under Sudden Loss of Arch Action: A Threshold-Free Load-Sharing Formulation Transferred from Tunnel Linings and Masonry Domes


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<big>'''Dynamic Robustness of Arch Dams under Sudden Loss of Arch Action: A Threshold-Free Load-Sharing Formulation Transferred from Tunnel Linings and Masonry Domes'''</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==

Earlier work by the author showed that the robustness of ring-type structures (tunnel linings, masonry domes on piers) after the sudden loss of load-carrying elements can be obtained, without imposed thresholds, as the root of an equation that equates the peak load redistributed onto the neighbouring elements to their dynamic capacity. This paper transfers that formulation to arch dams. The dam is represented by a vertical strip whose bending stiffness is that of the cantilever and whose elastic foundation is the hoop stiffness of the horizontal arches; the elements are the arch levels, and the damage is the sudden loss of arch action over ''k'' contiguous levels (abutment wedge, thrust block or foundation failure). The load-sharing length is shown to be purely geometric, ℓ_s = π(''Rt'')<sup>1/2</sup>/[3(1−ν<sup>2</sup>)]<sup>1/4</sup>, and equals 59 m for a reference dam with ''H'' = 100 m, ''R'' = 100 m and ''t'' = 6 m. Numerical verification includes a time-integration check of the dynamic capacity (maximum error 1.2%), a reproduction of two results of the tunnel-ring study, and 87 nonlinear dynamic finite-element cases. The dynamic-efficiency parameter calibrated on tunnel rings (ω = 0.14) is not transferable: it is over-conservative by 16% on average for the dam strip, whereas the isolated-oscillator capacity (ω = 1) is conservative by 10%. For a utilisation of 0.4 the dam tolerates the loss of a height of about 0.3''H'' at mid-height, 0.11–0.15''H'' at the crest and more than 0.5''H'' near the base, nearly independent of the number of elements. A flexural check shows, however, that the unreinforced cantilever strip cannot bridge even a single lost arch level elastically, so that the redistribution bound is attainable only if a vertical flexural capacity exists. All inputs are assumed and the results are hypothesis-level.

''Keywords:'' arch dam; progressive collapse; robustness; sudden support loss; load sharing; energy balance; size effect; hydropower

=1. Introduction=

Arch dams carry the hydrostatic load mainly by horizontal arch action and transmit it to the abutments as thrust. Their safety is therefore governed by the abutment rock, and the historical failures of arch dams are failures of support: the Malpasset dam (France, December 1959; about 60 m high, 1.5 m thick at the crest and about 6.8 m at the base) collapsed after the left-abutment rock mass and the thrust block it carried were displaced, with more than 400 casualties (Londe, 1987). The event is the prototype of the scenario considered here: a sudden loss of support over part of the height of a structure that has no alternative load path designed for it.

Robustness is normally assessed with the alternate load path method, in which an element is removed suddenly and the response of the rest is analysed (GSA, 2003; Izzuddin et al., 2008; Starossek, 2009). The method was developed for frames. In earlier studies the author expressed it for reinforced concrete frames (Calvi, 2025a, 2026), for tunnel linings (Calvi, 2025b, 2026b) and, in a threshold-free form, for ring-type structures including masonry domes on piers (Calvi, 2026c). The last formulation has three features that are relevant to dams: the critical damage is the output of an equation and not an imposed threshold; the redistribution is controlled by a load-sharing length; and the material closures (size effect, ductility) are derived from fracture mechanics.

The aim of this paper is to apply that formulation to arch dams, to verify numerically which of its ingredients transfer and which do not, and to identify what is specific to a thin, unreinforced, doubly curved shell. The scope is limited: the model is a two-dimensional analytical idealisation (a vertical strip), the dam is a reference case with assumed parameters, and no project-specific conclusion is drawn.

=2. Results transferred from the previous studies=

Four results are used. (i) From the tunnel-ring study (Calvi, 2026b): an element idealised as an elastic–perfectly plastic oscillator with utilisation ''u''<sub>0</sub> and ductility μ can take, after a sudden load increment, a number of additional element loads Λ(''u''<sub>0</sub>, μ) in which the dynamic amplification applies to the increment and not to the pre-load. (ii) From the same study: the critical damage is reached when the peak additional load on the neighbours equals Λ, and the number of elements ''n'' enters through the ratio between element length and load-sharing length. (iii) From the dome study (Calvi, 2026c): the simple averaging of the redistributed load underestimates the peak demand by a factor of 1.5–5, so that the peak must be computed from the response of the ring (here, of the strip), and the dynamic efficiency ω of the neighbour’s ductility was found to be 0.14 on rings. (iv) From the frame studies (Calvi, 2025a, 2026): robustness decreases when more elements are removed and depends on hierarchy, with a geometric threshold ''a''/''H'' ≈ 0.5 that is an output of one simulation campaign and is not assumed here.

=3. Formulation for arch dams=

==3.1 Mapping of the ring formulation onto a dam==

In a thin arch dam of mean radius ''R'' and thickness ''t'', loaded by the water pressure ''p'', the horizontal arches carry the load in hoop compression, with a thrust ''N'' = ''pR'' per unit height, and the circular arch is funicular for a uniform radial pressure. The vertical cantilever carries a small part of the load and behaves as a beam. A unit-width vertical strip is therefore a beam of bending stiffness ''EI'' = ''E''<sub>c</sub>''t''<sup>3</sup>/[12(1−ν<sup>2</sup>)] resting on an elastic foundation whose stiffness is the hoop stiffness of the arches, ''k''<sub>g</sub> = ''E''<sub>c</sub>''t''/''R''<sup>2</sup> (units N/m<sup>3</sup>); this is the classical analogy between the axisymmetric cylindrical shell and the beam on elastic foundation (Hetényi, 1946; Timoshenko and Woinowsky-Krieger, 1959). The correspondence with the earlier cases is given in Table 1. The elements are the ''n'' arch levels of height ℓ<sub>e</sub> = ''H''/''n'', each with stiffness ''K''<sub>s</sub> = ''k''<sub>g</sub>ℓ<sub>e</sub>, yield force ''P''<sub>y</sub> = ''P''<sub>0</sub>/''u''<sub>0</sub> and ''P''<sub>0</sub> = ''p''ℓ<sub>e</sub>; hierarchy is the number of independently failing levels, which is set by the discretisation of the abutment into rock blocks or lifts. The damage is the sudden loss, at ''t'' = 0, of the arch action of ''k'' contiguous levels, the load remaining on the strip.

Table 1. Correspondence between the ring formulation and the arch-dam strip.

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Quantity</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Tunnel lining (ring)</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Dome on piers</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Arch dam (strip)</span>
|-
|  style="border-top: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Continuous element</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Lining ring, ''EI''</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Ring beam, ''EI''</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Cantilever strip, ''EI''</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Elastic support</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Ground, ''k''<sub>g</sub></span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Piers, ''K''<sub>s</sub>/ℓ<sub>e</sub></span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Arch hoop stiffness, ''E''<sub>c</sub>''t''/''R''<sup>2</sup></span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Elements</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Shotcrete/bolt panels</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Piers</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Arch levels (abutment blocks)</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Load path of the element</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Radial bearing</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Compression</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Hoop thrust and abutment</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Closed line / open line</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Closed, length π''D''</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Closed, 2π''R''</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Open, length ''H''; clamped base, free crest</span>
|-
|  style="border-bottom: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Utilisation ''u''<sub>0</sub></span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Section/bolt design</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''P''<sub>0</sub>/(''u''<sub>0</sub>''f''<sub>c</sub>)</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''pR''/(''t f''<sub>c</sub>) or 1/FS<sub>sliding</sub></span>
|}

==3.2 Dynamic capacity of an element==

For an elastic–perfectly plastic element with initial utilisation ''u''<sub>0</sub> and ductility μ, equating the work of a suddenly applied increment Δ''P'' to the strain energy absorbed gives, as in Calvi (2026b), the number of element loads that can be added without failure:

Λ(''u''<sub>0</sub>, μ) = [(1 − ''u''<sub>0</sub>)/''u''<sub>0</sub>] · [1 − (1 − ''u''<sub>0</sub>)/(2(μ − ''u''<sub>0</sub>))] (1)

For μ = 1 it reduces to (1 − ''u''<sub>0</sub>)/(2''u''<sub>0</sub>) and for μ → ∞ to the static reserve (1 − ''u''<sub>0</sub>)/''u''<sub>0</sub>. In a continuous system the ductility of the neighbour is only partly effective, and an effective value μ<sub>eff</sub> = 1 + ω(μ − 1) is used, with ω a dynamic-efficiency parameter to be verified (Section 4). The utilisation of a thin arch has a direct limit-analysis meaning: with a thrust line coinciding with the centreline, the minimum thickness is ''t''<sub>min</sub> = ''pR''/''f''<sub>c</sub> (Heyman, 1966), so that the crushing utilisation is ''u''<sub>0</sub> = ''t''<sub>min</sub>/''t'', the inverse of the geometric factor of safety; in practice the sliding safety factor of the abutment wedge often governs, and ''u''<sub>0</sub> is the larger of the two.

==3.3 Load-sharing length==

The characteristic wavelength of the strip on its elastic foundation, λ = [''k''<sub>g</sub>/(4''EI'')]<sup>1/4</sup>, gives the load-sharing length ℓ<sub>s</sub> = π/λ. Substituting the expressions of Section 3.1, the modulus cancels:

ℓ<sub>s</sub> = π (''Rt'')<sup>1/2</sup> / [3(1 − ν<sup>2</sup>)]<sup>1/4</sup> (2)

The sharing length of an arch dam is therefore purely geometric and increases with the square root of the product of radius and thickness. For ''R'' = 100 m, ''t'' = 6 m and ν = 0.2 it is 59 m, which equals 0.59''H''; the transition number ''n = ''H*/ℓ<sub>s</sub> = 1.7 is below any practical subdivision, so that a dam is always in the distributed-sharing regime of the earlier studies: a lost arch level is shared over many neighbours, and the number of elements has little influence on the result expressed as a lost height.

==3.4 Critical condition==

Let Ψ(''n'', ''k'') be the peak additional reaction, in units of ''P''<sub>0</sub>, on the neighbouring arch levels after the loss of ''k'' contiguous levels. The neighbours survive if the peak does not exceed their dynamic capacity, and the critical damage is the root of

Ψ(''n'', ''k''<sub>crit</sub>) = Λ(''u''<sub>0</sub>, μ<sub>eff</sub>),     ''a''<sub>crit</sub> = ''k''<sub>crit</sub> ℓ<sub>e</sub> (3)

where ''k''<sub>crit</sub> is continuous (linear interpolation between integer losses) and ''a''<sub>crit</sub> is the critical lost height. No value of ''a''/''H'' is prescribed. The averaged expression of the first tunnel study is not used. The closed form of Ψ calibrated in Calvi (2026c) was obtained for 0.18 ≤ ℓ<sub>e</sub>/ℓ<sub>s</sub> ≤ 2.2, whereas for a dam ℓ<sub>e</sub>/ℓ<sub>s</sub> is 0.07–0.28, so Ψ is evaluated here directly by the static finite-element solution of the strip (Section 4). Because the strip is open, Ψ depends on the position of the damage; three positions are considered: crest zone, mid-height and base zone.

==3.5 Material closures for mass concrete==

The characteristic length of the concrete, ℓ<sub>ch</sub> = ''E''<sub>c</sub>''G''<sub>f</sub>/''f''<sub>t</sub><sup>2</sup>, is 1.1 m for ''E''<sub>c</sub> = 30 GPa, ''G''<sub>f</sub> = 150 N/m and ''f''<sub>t</sub> = 2.0 MPa, so that the brittleness number of a section 6 m thick is about 10: the size effect (Bažant, 1984; Bažant and Planas, 1998) places the strip in the fracture-governed range. Two ductilities are considered for the element. If the failure is governed by compression or sliding, μ is the strain ratio μ<sub>c</sub> = ε<sub>mu</sub>''E''<sub>c</sub>/''f''<sub>c</sub> = 3.0, with ε<sub>mu</sub> = 0.0035 and ''f''<sub>c</sub> = 35 MPa. If it is governed by localised cracking, the crack-band closure of Calvi (2026c), μ<sub>f</sub> = 1 + 2ℓ<sub>ch</sub>/ℓ<sub>e</sub>, gives 1.14–1.54 for ℓ<sub>e</sub> = 17–4 m, i.e. an almost brittle element whose ductility increases with the number of elements.

==3.6 Flexural check of the strip==

The continuous element must also survive the bending produced by the redistribution. The incremental moment Δ''M''(''k'') of the strip is amplified by 2 in the dynamic event, and an unreinforced strip must satisfy

2 Δ''M''(''k'') ≤ ''M''<sub>Rd</sub> = (''f''<sub>t,j</sub> + σ<sub>v</sub>) ''t''<sup>2</sup>/6 (4)

where ''f''<sub>t,j</sub> is the tensile strength of the lift joints and σ<sub>v</sub> the vertical compression. For a gap that is short compared with ℓ<sub>s</sub> the moment tends to that of a point load on an infinite beam, Δ''M'' ≈ ''P''<sub>0</sub>/(4λ).

==3.7 Assessment procedure==

The formulation is used in six steps, each of which can be repeated for several levels and damage positions. (1) Select the levels at risk and the support elements, from the abutment geology (blocks bounded by joint sets, thrust blocks, foundation wedges), and fix ''n'' and ℓ<sub>e</sub>. (2) Compute the pre-event utilisation ''u''<sub>0</sub> of each element as the larger of the crushing and sliding utilisations, with the uplift pressures actually acting. (3) Compute ℓ<sub>s</sub> from Equation (2) at the level considered and the ratio ℓ<sub>e</sub>/ℓ<sub>s</sub>. (4) Choose μ from the governing failure mode (compression or sliding, or fracture) and ω between 0.14 and 1; ω = 1 is the verified conservative value for the strip. (5) Compute Ψ(''n'', ''k'') by the elastic strip model for the position of the damage and solve Equation (3) for ''k''<sub>crit</sub> and ''a''<sub>crit</sub>. (6) Check the strip against Equation (4); if it is not satisfied, ''a''<sub>crit</sub> is an upper bound and the result must be complemented by a model that includes the cracked section and the plan-wise load transfer.

=4. Numerical verification=

Each ingredient was verified against an independent calculation, with the same logic as in Calvi (2026c). The strip was modelled with Euler–Bernoulli beam elements (six per arch level) on elastic–perfectly plastic springs, consistent mass 14,400 kg/m (2400 kg/m<sup>3</sup> × 6 m, hydrodynamic added mass neglected), Newmark average-acceleration integration with Newton iterations (time step 1 ms, duration 0.3 s, about 1.7 periods of the breathing mode, ''T'' = 0.18 s), and the critical utilisation was found by bisection (resolution about 0.002) as the value at which the peak ductility demand of the remaining springs equals μ. Two configurations were used: a long free–free strip, which represents the interior of a ring and allows comparison with the earlier studies, and the dam strip of height ''H'' with clamped base and free crest.

''Dynamic capacity.'' Equation (1) was compared with the time integration of a single oscillator for 20 combinations of ''u''<sub>0</sub> (0.2–0.8) and μ (1–5); the maximum difference is 1.2%. ''Beam on foundation.'' The discrete-spring strip with 48 elements reproduces the Hetényi deflection under a point load within 4×10<sup>−4</sup> of the maximum. ''Cross-check with the tunnel rings.'' With the data of the ring of Calvi (2026b, 2026c) (''D'' = 6 m, ''EI'' = 70.3 MN·m<sup>2</sup>, ''k''<sub>g</sub> = 1.33 GN/m<sup>3</sup>, 720 kg/m) the code gives a reaction of 2.64''P''<sub>0</sub> on the neighbours of two lost supports and a critical utilisation of 0.248 for ''n'' = 12, ''k'' = 2, μ = 2, equal to the published values (2.64 and 0.248), so the finite-element implementation is consistent with the one used in the earlier work.

''Dynamic efficiency.'' The critical utilisation predicted by Equations (1)–(3) was compared with the nonlinear dynamic solution for 51 cases of the interior strip (''n'' = 6–24, ''k'' = 1–4, μ = 1, 2, 4) and 36 cases of the dam strip (''n'' = 8, 12, 16; crest and mid-height damage; ''k'' = 1–3; μ = 1, 3). Table 2 and Figure 3a give the result. For μ = 1 the formula is essentially exact (mean error 2–4%). For μ > 1 the formula with ω = 1 is conservative, by 11–20% on average, and the value ω = 0.14 calibrated on tunnel rings is over-conservative by a further 6–12 points (maximum error 58%). A fit of ω on the interior strip does not converge to a finite optimum below the isolated-oscillator limit: the mean error decreases only from 15% (ω = 1) to 12% (ω → ∞). The interpretation is that in the rings the sharing length was of the order of the element length (ℓ<sub>s</sub>/ℓ<sub>e</sub> ≈ 1.4), so that the neighbours’ plastic deformation moved load onto supports that were themselves close to overload, whereas in the dam ℓ<sub>s</sub>/ℓ<sub>e</sub> is 3.5–14 and the plastic redistribution is spread over many levels. The parameter ω is therefore not a constant of the ring-on-supports mechanism, as was tentatively proposed, but depends on ℓ<sub>s</sub>/ℓ<sub>e</sub>, and ω = 1 is adopted for the dam as a conservative choice, with ω = 0.14 as a lower bound.

Table 2. Error of the formula for the critical utilisation against nonlinear dynamic finite elements (bias: negative = conservative).

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Set</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Cases</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">ω</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Bias / mean |error|</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Maximum |error|</span>
|-
|  style="border-top: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Interior strip, all μ</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">51</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">−10.3% / 10.9%</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">26.1%</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Interior strip, all μ</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">51</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.14</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">−16.3% / 16.9%</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">37.7%</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Dam strip, all μ</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">36</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">−9.7% / 12.0%</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">44.4%</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Dam strip, all μ</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">36</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.14</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">−15.6% / 17.9%</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">57.8%</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Dam strip, μ = 1 (elastic)</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">18</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1 or 0.14</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">+0.9% / 3.7%</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">25.2%</span>
|-
|  style="border-bottom: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Dam strip, μ = 3</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">18</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">−20.3% / 20.3%</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">44.4%</span>
|}


The largest errors occur for crest damage with ''k'' = 3 at ''n'' = 8, where the lost height exceeds 35% of ''H'' and the critical utilisation is below 0.15; this is outside the range of interest. In all other cases the formula with ω = 1 is on the safe side or within 6% for μ = 1.

=5. Application to a reference arch dam=

==5.1 Reference case==

The reference dam is a constant-thickness idealisation of the central part of a thin double-curvature arch dam: ''H'' = 100 m, mean radius ''R'' = 100 m, thickness ''t'' = 6 m, ''E''<sub>c</sub> = 30 GPa, ν = 0.2 (Table 3). The water pressure is taken as uniform over the strip at its mid-height value (0.5 MPa) and the thickness as constant, which is a simplification; the utilisation is taken as uniform and equal to 0.40, i.e. a safety factor of 2.5 on the abutment, higher than the crushing utilisation of the arches (0.24 at mid-height). All values are assumed and do not refer to a specific project. The derived quantities are ''EI'' = 5.63×10<sup>11</sup> N·m per metre of width, ''k''<sub>g</sub> = 18 MN/m<sup>3</sup>, λ = 0.0532 m<sup>−1</sup>, ℓ<sub>s</sub> = 59.1 m and Λ = 1.33 for μ<sub>eff</sub> = 3 (0.75 for μ = 1).

Table 3. Assumed input of the reference arch dam.

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Input</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Value</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Input</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Value</span>
|-
|  style="border-top: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Height ''H''; radius ''R''; thickness ''t''</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">100; 100; 6 m</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''E''<sub>c</sub>; ν; density</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">30 GPa; 0.2; 2400 kg/m<sup>3</sup></span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''G''<sub>f</sub>; ''f''<sub>t</sub>; ''f''<sub>c</sub></span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">150 N/m; 2.0; 35 MPa</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">ε<sub>mu</sub>; μ<sub>c</sub>; ℓ<sub>ch</sub></span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.0035; 3.0; 1.13 m</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''u''<sub>0</sub>; ω; μ</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.40; 1; 3</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''n'' (arch levels)</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">6–24</span>
|-
|  style="border-bottom: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">ℓ<sub>s</sub>; ''n''*</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">59.1 m; 1.7</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Breathing period ''T''</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.18 s</span>
|}

==5.2 Critical lost height==

Figure 2a shows that the peak redistributed load, plotted against the lost height normalised by ℓ<sub>s</sub>, collapses on a single curve for ''n'' ≥ 8: the redistribution is a property of the lost height, not of the number of elements. Together with the nearly constant Λ this gives a plateau of the critical height, Figure 2b and Table 4. At mid-height the dam tolerates the loss of 2.4 levels at ''n'' = 6 and 7.2 at ''n'' = 24, which is a height of 39 m and 30 m, that is 0.39''H'' and 0.30''H''. The critical height depends strongly on the position of the damage: the clamped base absorbs the load, so that damage in the lowest 0.55''H'' is tolerated, whereas in the crest zone, where the strip has only one neighbour, the tolerated height is 0.11–0.15''H''. A threshold ''a''/''H'' = 0.5, taken from the frame studies, would therefore be unconservative by a factor of 4 for the crest zone and only 10–20% conservative near the base, which confirms the objection to imposed thresholds.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_382246594-d3f3055e486b1f50f842f552086807f9a16d6384.png|600px]] </div>

Figure 1. Mapping of the arch dam onto the strip model: (a) horizontal arches and the zone of lost arch action; (b) vertical strip on discrete arch springs, clamped at the base and free at the crest.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_382246594-ef29f33bb440b3bcc60789f635a58bbb9984f16f.png|600px]] </div>

Figure 2. (a) Peak additional load on the neighbouring arch levels against lost height/ℓ<sub>s</sub> (interior strip), with the capacity Λ for ''u''<sub>0</sub> = 0.4; (b) critical lost height ''a''/''H'' against the number of arch levels for four damage positions (μ = 3, ω = 1) and, dashed, for the fracture-governed ductility μ<sub>f</sub> = 1 + 2ℓ<sub>ch</sub>/ℓ<sub>e</sub>.

The hierarchy effect enters in two ways. Through the redistribution it is weak, since the dam is in the distributed regime (''n = 1.7): ''a''_crit_/''H'' at mid-height decreases from 0.39 to 0.30 when ''n'' goes from 6 to 24. Through the material it can be opposite: if the element is fracture-governed, μ_f_ rises from 1.14 to 1.54 with ''n'' and partly offsets the decrease, so that ''a''_crit_/''H'' stays between 0.26 and 0.27 (dashed line in Figure 2b). A dam with brittle elements tolerates about 18% less height than one with ductile elements, at any ''n*.

Table 4. Critical lost height for ''u''<sub>0</sub> = 0.40, μ = 3, ω = 1 (last column: μ<sub>f</sub> = 1 + 2ℓ<sub>ch</sub>/ℓ<sub>e</sub> at mid-height).

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''n''</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">ℓ<sub>e</sub> (m)</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">μ<sub>f</sub></span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''k''<sub>crit</sub> (mid)</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''a''/''H'' mid</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''a''/''H'' crest</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''a''/''H'' base</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''a''/''H'' mid, μ<sub>f</sub></span>
|-
|  style="border-top: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">6</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">16.7</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.14</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.35</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.39</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.15</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.60</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.27</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">8</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">12.5</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.18</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.89</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.36</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.13</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.57</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.26</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">12</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">8.3</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.27</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">3.82</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.32</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.12</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.55</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.26</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">16</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">6.3</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.36</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">5.04</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.31</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.12</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.54</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.26</span>
|-
|  style="border-bottom: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">24</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">4.2</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">1.54</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">7.23</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.30</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.11</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.53</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.26</span>
|}

==5.3 Sensitivity==

Table 5 gives one-at-a-time variations for ''n'' = 12 at mid-height. The utilisation dominates: a change of ±20% of ''u''<sub>0</sub> changes the critical height by +32% and −19%. The ductility closure is second: a brittle element (μ = 1) reduces it by 35%, and the ring value ω = 0.14 by 18%. A ±20% change of thickness or radius changes ℓ<sub>s</sub> by ±10% and the critical height by ±9–11%; because only the product ''Rt'' matters, a thicker dam and a larger radius act identically. A strip with the bending stiffness reduced to a quarter (cracked section) has ℓ<sub>s</sub> = 42 m and tolerates 21% less. Figure 3b shows the dependence on ''u''<sub>0</sub> for the two critical positions: reducing ''u''<sub>0</sub> from 0.4 to 0.3 raises the tolerated mid-height from 0.32''H'' to about 0.45''H'', but at the crest the tolerated height is only 0.17''H'' at ''u''<sub>0</sub> = 0.3 and remains below 0.21''H'' even at ''u''<sub>0</sub> = 0.25.

Table 5. One-at-a-time sensitivity of the critical lost height (''n'' = 12, mid-height).

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Case</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">ℓ<sub>s</sub> (m)</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''a''<sub>crit</sub> (m)</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''a''<sub>crit</sub>/''H''</span>
|-
|  style="border-top: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Reference (''u''<sub>0</sub> = 0.40, μ = 3, ω = 1)</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">59.1</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">31.9</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.32</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''u''<sub>0</sub> = 0.32 / 0.48</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">59.1</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">42.2 / 25.8</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.42 / 0.26</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">μ = 2 / μ = 1 (brittle)</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">59.1</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">30.0 / 20.8</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.30 / 0.21</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">ω = 0.14 (value from rings)</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">59.1</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">26.1</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.26</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''Rt'' × 0.8 / × 1.2</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">52.8 / 64.7</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">29.2 / 35.5</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.29 / 0.35</span>
|-
|  style="border-bottom: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Cracked strip, ''EI'' × 0.25</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">41.8</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">25.2</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.25</span>
|}


<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Calvi_382246594-14eec4c3472d6e769b8f62fdfd40d31777396cc1.png|600px]] </div>

Figure 3. (a) Critical utilisation, formula (ω = 1) against nonlinear dynamic finite elements, 87 cases; the dotted lines are ±20%. (b) Critical lost height against utilisation for the dam strip (''n'' = 12), mid-height and crest zone.

==5.4 Flexural capacity of the strip==

The load redistributed onto the neighbours has to pass through the bending of the cantilever strip, and this is the weak point of an unreinforced arch dam. For the reference case the moment increment of the elastic strip is Δ''M'' = 23, 43, 69 and 98 MN·m per metre of width for the loss of 1, 2, 3 and 4 arch levels (8.3–33 m), close to ''P''<sub>0</sub>/(4λ) = 20 MN·m for one level. With the dynamic factor of Equation (4) the demand for a single lost level is 47 MN·m/m, equivalent to a stress of 7.8 MPa on the 6 m section, against a resistance of 6 MN·m/m for lift joints of 1.0 MPa and 19 MN·m/m if ''f''<sub>t,j</sub> + σ<sub>v</sub> = 3.2 MPa. The elastic strip would therefore crack at the loss of one arch level, and the critical heights of Table 4 are an upper bound that is attainable only if the section can transfer the moment (vertical prestress or reinforcement, a thicker section, or plan-wise arch action that relieves the strip). The comparison between the two criteria is the main practical indication of the study: in a thin unreinforced shell the limit is not the capacity of the neighbouring arches but the ability of the cantilever to bring the load to them.

=6. Discussion=

==6.1 Practical indications==

Four indications follow, to be checked against data. First, the utilisation of the abutment and arches before the event controls the result, as in tunnels and domes; since the sliding safety factor of the abutment wedge is its inverse, a safety factor of 3 instead of 2.5 raises the tolerated height at mid-height from 0.32''H'' to about 0.40''H'' and, at the crest, from 0.12''H'' to 0.15''H''. Second, the position of the damage matters more than the number of elements: the crest zone, which carries little load, is the most vulnerable in the model, and the base is the most robust. Third, the load-sharing length is geometric and can be computed at design from the radius and thickness at the level considered; a long ℓ<sub>s</sub> is favourable for the neighbours but increases the bending demand on the strip. Fourth, material degradation (alkali–silica reaction, thermal cracking, loss of lift-joint strength) acts through ''E'', ''f''<sub>t</sub>, ''G''<sub>f</sub> and the cracked stiffness: it reduces μ and ''EI'' and increases ''u''<sub>0</sub> at the same time, and each of the three reduces the tolerated damage in Table 5.

==6.2 Comparison with tunnel linings and domes==

Table 6 puts the three structures side by side. The dam differs from the others in the length scale and in the regime. The sharing length of a tunnel lining or of a dome on piers is of the order of the element length, so that ''n is 8–9 and the hierarchy of the ring decides whether the sharing is local or distributed; in a dam ℓ_s_ is of the order of the height, ''n** is below 2 and the result is a plateau of the lost height. Consistently, the dynamic efficiency is low where the sharing is local (neighbours that yield move the load onto supports already near overload) and high where it is spread over many elements. The load path also differs: the dome of Calvi (2026c) has a second path through the shell above the gap, and the tunnel loads are carried by bearing on the ground, whereas in the dam the main path is the hoop thrust and the strip is a secondary path whose strength, as shown in Section 5.4, is small. The common result is the role of the utilisation, which is the dominant variable in the three cases.

Table 6. Comparison of the three ring-type structures (illustrative cases, ''u''<sub>0</sub> as indicated).

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Quantity</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Tunnel lining</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Dome on 12 piers</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Arch dam</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Note</span>
|-
|  style="border-top: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">ℓ<sub>s</sub> (m); ''n''*</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.13; 8.9</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">7.83; 8.0</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">59.1; 1.7</span>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Geometry-controlled</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''u''<sub>0</sub> (reference)</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.5</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.25</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.40</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Assumed</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Critical damage</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''a''/''D'' = 0.14 (plateau)</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">2.15 piers, 65°</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.32''H'' (mid)</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Dam: ''n'' = 12</span>
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Dynamic efficiency ω</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.14 (fitted)</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">0.14 (transferred)</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">≥ 1 (verified)</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Depends on ℓ<sub>s</sub>/ℓ<sub>e</sub></span>
|-
|  style="border-bottom: 1pt solid black;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Validation cases</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">63 dynamic</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">22 dynamic</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">87 dynamic</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">Same logic</span>
|}

==6.3 Limits of the model==

The model is a first-order idealisation. It is a vertical strip and assumes loss of the arch action over the whole width at the levels considered, which is a conservative simplification of a wedge failure localised at one abutment, where the horizontal arches and the shell would also redistribute the load in plan; a three-dimensional shell or discrete-element model is the necessary next step. The dam is idealised with constant thickness, uniform pressure and a rigid clamped base, which overestimates the base moment of the intact strip (about 88 MN·m/m) and does not represent the thickening of real dams. The water is represented by its static pressure, without hydrodynamic added mass, which would lower the frequency and change the dynamic factor; the abutment rock is not modelled and failure is represented by an elastic–plastic spring; the dynamic efficiency was verified on one family of strips; damping is neglected and is favourable (Calvi, 2026c); Ψ is computed numerically and not in closed form; and the flexural check uses an elastic uncracked moment. The numerical values must not be used for the design or the assessment of a real dam.

==6.4 Further validation==

The elements that remain open are the following. A three-dimensional discrete-element or finite-element model of an arch dam with sudden loss of a wedge at the abutment would test the strip idealisation and the plan-wise redistribution. Dynamic tests on scaled models with the sudden release of supports would give ω for concrete shells. Fracture tests on mass-concrete specimens of different depth and on lift joints would calibrate ℓ<sub>ch</sub> and ''f''<sub>t,j</sub>. Back-analysis of documented abutment failures such as Malpasset, with the pre-event utilisation and the extent of the lost support estimated with uncertainty bounds, would allow the critical height to be compared with the observed damage.

=7. Conclusions=

The threshold-free formulation of dynamic robustness developed for tunnel linings and masonry domes was applied to arch dams, represented by a vertical strip on the arch hoop stiffness. The main results are the following.

* The critical damage is the root of Ψ(''n'', ''k'') = Λ(''u''<sub>0</sub>, μ<sub>eff</sub>), with the dynamic capacity Λ of Equation (1) verified against time integration (error 1.2%) and with no imposed value of ''a''/''H''.

* The load-sharing length of an arch dam is geometric, ℓ<sub>s</sub> = π(''Rt'')<sup>1/2</sup>/[3(1−ν<sup>2</sup>)]<sup>1/4</sup> = 59 m for the reference dam, which places every practical subdivision in the distributed-sharing regime: the tolerated damage is a lost height almost independent of the number of arch levels.

* The dynamic efficiency ω = 0.14 calibrated on tunnel rings does not transfer: on 87 nonlinear dynamic cases it is conservative by 16% on average, whereas ω = 1 is conservative by 10% and exact for the elastic case. ω depends on ℓ<sub>s</sub>/ℓ<sub>e</sub> and cannot be taken as a universal constant.

* For ''u''<sub>0</sub> = 0.4 the reference dam tolerates the loss of about 0.3''H'' at mid-height, 0.11–0.15''H'' in the crest zone and more than 0.5''H'' near the base; a single geometric threshold would be unconservative by a factor of 4 at the crest.

* Utilisation is the dominant parameter (−19%/+32% for ±20%), followed by the ductility closure (−35% for a brittle element, −18% for ω = 0.14) and the load-sharing length (±10%).

* The elastic cantilever strip cannot bridge even one lost arch level without cracking (demand 47 MN·m/m against 6–19 MN·m/m), so that the redistribution bound is attainable only with a vertical flexural capacity or with plan-wise load transfer, which are not in the model.

The results are consequences of an analytical and numerical model with assumed parameters. Their validation requires three-dimensional models, scaled dynamic tests, fracture tests on mass concrete and back-analysis of documented failures.

==References==

Bažant ZP (1984) Size effect in blunt fracture: concrete, rock, metal. ''Journal of Engineering Mechanics'' 110(4): 518–535.

Bažant ZP and Planas J (1998) ''Fracture and Size Effect in Concrete and Other Quasibrittle Materials''. CRC Press, Boca Raton, FL, USA.

Calvi A (2025a) Robustness interpretation of intact and damaged framed structures. ''Proceedings of the ICE – 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 ICE – Forensic Engineering''. [https://doi.org/10.1680/jfoen.25.00033. https://doi.org/10.1680/jfoen.25.00033.]

Calvi A (2026b) Dynamic energy balance and size effect in tunnel linings under sudden support loss: a load-sharing formulation of robustness for hierarchical support systems. Manuscript.

Calvi A (2026c) A threshold-free theory of dynamic robustness for ring-type structures: tunnel linings and masonry spherical domes under sudden support loss, with numerical validation. Manuscript.

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

Hetényi M (1946) ''Beams on Elastic Foundation''. University of Michigan Press, Ann Arbor, MI, USA.

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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.

Londe P (1987) The Malpasset dam failure. ''Engineering Geology'' 24(1–4): 295–329.

Newmark NM (1959) A method of computation for structural dynamics. ''Journal of the Engineering Mechanics Division, ASCE'' 85(3): 67–94.

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

Timoshenko S and Woinowsky-Krieger S (1959) ''Theory of Plates and Shells'', 2nd edn. McGraw-Hill, New York, USA.

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