Armour Damage at an Exposed Rubble-Mound Breakwater

Hudson check. W = ρr·H³ / (KD·Δ³·cotθ), with ρr = 2,650 kg/m³, ρw = 1,025 kg/m³, Δ = 1.585, Δ³ = 3.98, cotθ = 2.

• For H = 5.5 m and KD = 2 (rock, no damage): W = 2,650 × 166.4 / (2 × 3.98 × 2) = 440,900 / 15.9 ≈ 27.7 t.
• Depth-limited check: at a toe depth of 6.0 m the maximum wave at the structure is roughly 0.78 × 6.0 ≈ 4.7 m, giving W ≈ 2,650 × 103.8 / 15.9 ≈ 17.3 t.
• Reverse the equation for the 4.0 t stone: H = (4,000 × 15.9 / 2,650)^(1/3) ≈ 2.9 m.

On any reading the 4 t armour is between 3 and 6 times too light for a no-damage design at this wave height, and corresponds to a stable sea state of about 2.9 m.

Why has it survived? Several possibilities, and the honest answer is that we cannot yet distinguish them:

• The design H may be an offshore or deep-water value rather than the height at the toe; if the real toe height is 3.5–4.0 m, 4 t rock is marginal but not absurd.
• KD = 2 is conservative. Non-breaking, non-overtopping rock can carry KD = 4; with H = 4.7 m that would still call for about 8.6 t.
• The 20-year storm may not have been the design event; the structure may be living on borrowed time and the recent damage is the first sign.

Critical clarification before any repair design: confirm whether 5.5 m is Hsig or H1/10, and at what location. Using Hsig where the KD table assumes H1/10 under-predicts by about 2× in weight. The damage pattern - waterline concentrated - also points at breaking-wave attack and toe scour rather than uniform overloading, which is a separate design issue.

Capacity of each candidate (H for no damage, cotθ = 2 unless noted, from H = (W·KD·Δ³·cotθ/ρr)^(1/3)):

Option	KD	Stable H
Existing 4 t rock	2	2.9 m
6 t rock (local maximum)	2	3.3 m
6 t rock, slope flattened to 1:3	2	3.8 m
12 t rock (400 km haul)	2	4.2 m
20 t rock (400 km haul)	2	4.9 m
4 t concrete cube	8	4.6 m
6 t concrete cube	7 at 1:2	5.2 m
6 t concrete cube, slope 1:2.5	8	5.6 m

Assessment.

• Same 4 t rock: not a repair, a repeat. The exposed underlayer will strip in the next storm and the damage will propagate.
• 6 t local rock: still only stable to about 3.3 m; even with a flatter slope it does not cover the design event. It is a temporary measure at best.
• Large rock: to reach 5.5 m needs roughly 28 t, which is beyond normal quarry production, requires 400 km haulage at 2.6× unit cost, and needs plant that may not be available. Rejected.
• Concrete cubes: 6–7 t units on a 1:2.5 slope cover the design condition at KD 8, are locally produced, and can be placed by the same plant. The weight is manageable and the units can be designed with a specific gravity and interlock suited to the site.

Recommendation.

1. Rebuild the damaged 120 m with 6–7 t concrete cubes on a 1:2.5 seaward slope, with a correctly sized two-layer underlayer (W/10 and W/15) and a properly founded toe berm.
2. Keep rock on the less exposed sections, but armour the exposed length consistently - mixed armour along a trunk creates weak points.
3. Verify with a physical model test covering overtopping, toe stability and wave breaking on the foreshore, because Hudson says nothing about any of those.
4. Before the next storm season, place temporary 1–3 t rock over the exposed underlayer and fill the scour hole - an emergency measure, not a design.

Yes, probably the proximate cause. The evidence supports it:

• 1.2 m of scour at the toe means the armour has lost its support and can rotate and settle seaward; displaced stones at the waterline are often the visible end of a toe failure, not its beginning.
• A 1:2 slope with the toe only 6 m below still-water level sits in the breaking zone; plunging waves and reflection scour the seabed directly in front of the structure.
• Once scour lowers the bed, the effective water depth at the toe increases and the waves reaching the armour get larger, so damage accelerates.

Toe design principles.

• Toe berm of 1–3 t rock, at least two layers thick and with a crest width of at least three armour diameters, founded on a geotextile or granular filter.
• Extend the berm to below the credible scour depth - at least 1.5 × the observed 1.2 m, so a base at about −8 m CD unless the toe is raised instead.
• Check toe stability with the depth-limited wave at the scoured depth, not the design depth, and check that the berm does not simply sink into the mobile bed.
• Consider raising the toe and flattening the lower slope; a 1:3 lower slope is cheaper than adding armour weight.

Repair sequence.

1. Immediate (before next storm season): place 1–3 t rock to fill the scour hole and cover the exposed underlayer along the 120 m; survey and record as-built.
2. Design: physical model of the toe and armour; finalise cube size, underlayer, berm geometry and crest level, including overtopping discharge against the acceptable limit for the harbour behind.
3. Construction: work in the calm season, from the seaward side with a crane and split barges, building the toe berm first so the armour always has support; do not place armour from the crest downward onto an unsupported toe.
4. Post-repair: annual bathymetric survey of the toe, plus a survey after every significant storm, with a documented trigger for topping up the berm.