1 · Bottom line up front
Recommended clearance
Build to 12 ft (3.66 m) from the design waterline to the lowest point of the platform underside — 13 ft if the cost is trivial.
Condition: enlarge the waterplane of the three legs (waterline collars/bulges) to roughly Awl ≈ 240 ft² total, so the heave natural period lands at Tn ≈ 4.6 s, safely below the Caribbean sea-period band.
The trap in the current sketch
With slim 10-ft-chord × 4-ft-thick columns (Awl ≈ 100 ft²) and ~150 kip displacement, the heave natural period computes to ≈ 7.2 s — almost exactly the period of Caribbean trade-wind seas. Near heave resonance the platform moves out of phase with the waves, relative motion triples, and no practical clearance saves you: you would need ≈ 15 ft just to reach “once per day,” and the ride would be violent. Tuning beats height.
Your three questions, answered
- What rules/formulas exist for bridge-deck clearance? Heuristics (§3) exist, but the engineering answer is the spectral/Rayleigh exceedance method of §4 — clearance is chosen so the relative vertical motion of deck vs. water surface exceeds it less often than your target rate.
- Can you get probability per hour/day vs. sea state and geometry? Yes — Eq. (10)–(12) in §4, tabulated in §6, and live in the §7 calculator. Height (clearance) enters exponentially; width (waterplane area, leg spacing) enters through the platform’s tuned response.
- What should your clearance be? 12 ft, with the waterplane fix. At 9.5 ft (leg-top datum) the statistics are already acceptable in the central model, but 12 ft buys robustness against model uncertainty, short steep seas, and breaking crests.
2 · Why multihull decks pound
Pounding (slamming) of a bridgedeck happens when three things coincide:
- The relative vertical motion R(t) = wave elevation − deck point elevation exceeds the clearance c; and
- The relative vertical velocity Ṙ at that instant is high enough that the impact is energetic (Ochi’s threshold criterion); and
- There is somewhere for the water to go — flat, parallel surfaces trap air and double peak pressures.
Impact pressure scales with relative velocity squared (P ≈ ½ρCsVr²), so a slow, heavy, well-tuned platform that rarely exceeds a small relative velocity produces gentle touches, not structural slams. Everything below is built on the statistics of R(t).
3 · Existing rules of thumb
| Rule | Typical form | Value for this project (Hs=7 ft) | Comment |
|---|---|---|---|
| Cruising-catamaran heuristic | c ≥ 1.5 × Hs (design sea) | ≥ 10.5 ft | Widely used by sailing-cat designers; assumes a pitching, sailing platform. Reasonable sanity check. |
| Offshore “air-gap” practice (semi-subs, floatels) | c ≥ most-probable max crest + margin (≈ 1.5 m) | (Hs/4)√(2 ln 13 500) + 4.9 ft ≈ 7.6 + 4.9 = 12.5 ft | Deterministic-flavor rule used by class societies for floating structures; converges remarkably with our probabilistic answer. |
| Percentage-of-length heuristics | c ≈ 4–6% LOA; or 7–10% BOA | 3.2–4.8 ft; 5.6–8 ft | Coastal-cat numbers. Too low for open-ocean service — ignore for this design. |
| SWATH / semi-sub practice | Strut-deck clearance from relative-motion statistics (Rayleigh exceedance) | Method of §4 | The rigorous approach; what we apply below. |
All heuristics share a weakness: they ignore the platform’s tuned dynamic response. A stiff barge and a pitchy racing cat with identical clearance have wildly different pounding rates. Hence §4.
4 · The probabilistic method (the formulas you asked for)
Step 1 — Describe the sea
Tz ≈ Tp/1.41 (Pierson–Moskowitz / Bretschneider) ; waves per day N = 86 400 / Tz
Step 2 — Tune the platform (this is where “width” enters)
Handy form: Tn[s] ≈ 0.139 √( W[lb]·(1+Ca) / Awl[ft²] )
Roll/pitch: Tn,r ≈ 5.69 √( W / (Awl·R²) ), R = corner radius
One-number tuning rule (Caribbean, Tp ≈ 6–11 s): keep W·(1+Ca)/Awl ≤ ~1 500 (stiff follower, Tn ≤ 5.2 s) or ≥ ~6 500 (deeply decoupled, Tn ≥ 11.4 s — needs enormous weight). Between those values lies the resonance trap.
Step 3 — Relative-motion spectrum (three contributors)
(a) Heave tracking error: Kh = |1 − RAO·eiφ| = √(1 + RAO² − 2·RAO·cosφ); σtrack = Kh·Hs/4
(b) Rotation at corners: σrot = Kr·σslope·R, σslope ≈ πHs/(√2·Lp), Lp = gTp²/(2π)
(c) Rigid-body span mismatch (wave curvature between legs): σcurv ≈ 0.06·(2π/Lp)²·(Hs/2)·S², S = leg spacing
σR = √( σtrack² + σrot² + σcurv² )
Step 4 — Exceedance statistics (Rayleigh peaks)
(11) Event rate: λ = N · exp( −c² / (2σR²) ) [per day; ÷24 for per hour]
(12) Clearance for a target rate λ*: c* = σR · √( 2·ln( N / λ* ) )
(13) Ochi hard-slam refinement: λhard = N · exp( −c²/2σR² − vt²/2σv² ), σv ≈ 2πσR/Tz, vt ≈ 2–6 ft/s
Eq. (12) is the direct answer to “what chance per hour/day in different sea states for different height/width numbers”: feed in Hs, Tp (sea state), W, Awl, S (width numbers) and solve for c (height number) at your target λ*. Assumptions: stationary platform, linear theory, narrow-banded Rayleigh peaks, long-crested seas. Good to roughly a factor of 3 in rate — hence the robustness margin in §1.
5 · Applied to your design
Geometry & mass assumptions
- Platform
- Equilateral triangle, 80 ft per side; corner radius R = 46.2 ft; leg spacing S = 80 ft
- Legs (as sketched)
- 3 × vertical columns, NACA section 10 ft chord × 4 ft thick, 19 ft tall, ≈ 9.5 ft draft → ≈ 9.5 ft exposed
- Speed
- 4 mph (5.9 ft/s) → Froude ≈ 0.10: wavemaking negligible, so waterplane area costs essentially nothing in drag — choose it for seakeeping, not speed
- Mass
- Nominal W = 150 000 lb (68 t) incl. batteries, canned stores, water/waste tanks low in the legs; added-mass coeff Ca = 0.8
- Batteries/stores in legs
- Lowers CG (good), raises rotational inertia (good), but adds mass → lengthens Tn — recheck tuning after every weight change. Also: each ton added sinks the platform ≈ 0.13 ft (at Awl=240), eating clearance.
- Environment
- Hs ≤ 8 ft, Tp ≈ 5–12 s (§10)
Three tuning configurations compared (Hs = 7 ft, Tp = 9 s)
| Configuration | Awl | W | Tn heave | Tn roll | σR | c* for <1/day | Verdict |
|---|---|---|---|---|---|---|---|
| A — Stiff follower ( |