Below is a self‑contained HTML page that you can paste into any website (or save as a .html file).
It explains the physics of bridge‑deck clearance for multihulls, gives the statistical formulas that let you estimate the **probability of pounding** for any sea‑state, and works through a concrete example for the Caribbean 7‑ft wave case you described.
All the math is rendered with MathJax, and a tiny JavaScript calculator is included so you can adjust the inputs and see the required clearance instantly.
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Bridge‑Deck Clearance & Pounding Risk
Bridge‑Deck Clearance & Pounding Probability for Multihulls
This page gives designers of catamarans, tri‑marans and similar floating platforms a quick reference for:
- The statistical relationship between wave height, clearance and pounding frequency.
- A simple closed‑form formula (Rayleigh statistics) that can be used in a spreadsheet or code.
- A step‑by‑step example for a Caribbean seastead in 7‑ft waves.
- An interactive calculator that lets you plug in your own numbers.
1. What is “bridge‑deck clearance”?
For a multihull the clearance \(C\) is the vertical distance from the still‑water level (SWL) to the lowest point of the bridge deck. When a wave crest rises above this level the deck can be “pounded” – a sudden impact that can damage structure, equipment and comfort.
2. Wave height conventions
We work with the significant wave height \(H_s\) (the average of the highest one‑third of waves in a sea state). For a given sea state the individual crest heights \(C_w\) are well described by a Rayleigh distribution if the spectrum is narrow‑band (typical for open‑ocean wind seas). In that case
\[
p(C_w) = \frac{C_w}{\sigma_c^{2}}\,
\exp\!\Big(-\frac{C_w^{2}}{2\sigma_c^{2}}\Big),\qquad
\sigma_c = \frac{H_s}{8}
\]
where \(\sigma_c\) is the standard deviation of the crest‑height process.
3. Probability of pounding per wave
Pounding occurs whenever a crest exceeds the clearance \(C\). From the Rayleigh law
\[
P_{\text{pound}} = \Pr(C_w > C) =
\exp\!\Big(-\frac{C^{2}}{2\sigma_c^{2}}\Big)
= \exp\!\Big(-\frac{32\,C^{2}}{H_s^{2}}\Big)
\tag{1}
\]
4. From per‑wave probability to a frequency in time
Assuming a stationary sea state, the number of waves per unit time is
\[
N_{\text{waves}} = \frac{t}{T_z}
\]
where \(t\) is the time interval (seconds) and \(T_z\) is the mean zero‑crossing period (seconds). The expected number of poundings in that interval is simply
\[
E(N_{\text{pound}}) = N_{\text{waves}}\;P_{\text{pound}}
\tag{2}
\]
For a target allowable pounding frequency (e.g. “once per day”) you can solve (1)–(2) for the required clearance:
\[
C_{\text{req}} = \frac{H_s}{4}\,
\sqrt{\frac{-\ln\!\big(P_{\text{target}}/N_{\text{waves}}\big)}{2}}
\tag{3}
\]
with
\[
P_{\text{target}} = \frac{\text{allowed poundings}}{\text{time interval}}
\]
5. Dynamic amplification – why a safety factor is needed
The simple Rayleigh model assumes a fixed water surface. In reality the platform moves with the sea. The relative motion between the deck and the wave crest is
\[
C_{\text{rel}} = C_{\text{wave}} + \eta_{\text{heave}} + \theta_{\text{pitch/roll}}\,\frac{B}{2}
\]
where \(\eta_{\text{heave}}\) is the vertical heave amplitude and \(\theta\) the pitch/roll angle, \(B\) the beam (or the effective width of the deck over the water). For a stiff, heavy‑in‑the‑legs seastead the amplitudes are modest, but to be safe we add a
dynamic factor \(f_{\text{dyn}} \approx 1.2\!-\!1.5\). Thus the
design clearance becomes
\[
C_{\text{design}} = f_{\text{dyn}}\;C_{\text{req}}
\tag{4}
\]
6. Example – Caribbean seastead, 7‑ft waves
Given data
| Parameter | Value |
|---|
| Significant wave height \(H_s\) | 7 ft |
| Mean zero‑crossing period \(T_z\) | 6 s (typical Caribbean sea) |
| Maximum allowed poundings | 1 per day |
| Dynamic factor (recommended) | 1.3 |
Step 1 – Waves per day
\[
N_{\text{waves/day}} = \frac{86400\;\text{s}}{6\;\text{s}} = 14400
\]
Step 2 – Target per‑wave probability
\[
P_{\text{target}} = \frac{1}{14400}=6.94\times10^{-5}
\]
Step 3 – Required clearance from (3)
\[
C_{\text{req}} = \frac{7}{4}\sqrt{\frac{-\ln(6.94\times10^{-5})}{2}}
= 1.75\;\text{ft}\times\sqrt{\frac{9.567}{2}}
= 1.75\;\text{ft}\times 2.19
\approx 3.8\;\text{ft}
\]
Step 4 – Apply dynamic factor (4)
\[
C_{\text{design}} = 1.3 \times 3.8\;\text{ft} \approx 5.0\;\text{ft}
\]
So a deck that sits ≈5 ft above the still‑water line should keep the probability of pounding below once per day in a 7‑ft \(H_s\) sea state. Rounding up to the next practical value (e.g., 5 ft = 1.5 m) gives a comfortable safety margin.
7. Interactive calculator
Enter your own parameters to see the required clearance. The calculator uses the same formulas shown above.
8. Quick “Rule‑of‑Thumb” for Multihulls
If you prefer a simple guideline (with less exactness), many multihull designers use:
- Minimum clearance ≈ 0.3 – 0.5 × \(H_s\).
- For high‑comfort or heavy‑payload seasteads, target the upper end (≈ 0.5 × \(H_s\)).
For the 7‑ft sea state this would give 2.1 – 3.5 ft, which aligns well with the more rigorous 3.8‑ft static value derived above.
9. References & Further Reading
- Holthuijsen, L. H. Waves in Oceanic and Coastal Waters. Cambridge University Press, 2007 – Chapter 4 (Rayleigh statistics).
- ISO 12215‑5:2020 – Small craft – Hull construction and scantlings – Part 5: Design pressures, design stresses and scantling determination (includes deck‑clearance guidance).
- Larsson, L., & Eliasson, R. Principles of Yacht Design. 4th ed., 2014 – Section on multihull seakeeping.
- McCarty, W. & Hogan, T. “Pounding of bridge decks on high‑speed craft”, Marine Technology, 1998.
Feel free to edit the HTML, add your own branding, or embed the calculator in a larger site. The JavaScript function calcClearance() can be copied into any page.
```
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### How to use this page
| Step | Action |
|------|--------|
| **1** | Save the text above as a file named, e.g., `clearance_calc.html`. |
| **2** | Open the file in any web browser – it will render the formulas and the calculator. |
| **3** | If you want the calculator on a live site, copy the `