NATARAJAThe ship's form — Simpson's rules & the coefficients
Learning outcomes & permitted supports
- Work Simpson's first rule on a set of half-ordinates — area, centroid (the LCF) and the second moments — and say why it is EXACT for a cubic
- Know when the second rule (3h/8) is the one to use, and why it exists at all
- Turn a waterplane into TPC, I_T for BM, and I_L for MCTC — the first lines of every hydrostatic table
- Define Cb, Cw, Cm and Cp, and use Cp·Cm = Cb as a CHECK on a set of figures
- Apply Morrish's KB and Denny's wetted surface, and state where each approximation is exact and where it is not
Supports in this lab: Calculator. Simpson's rules are proved exact on cubics in the golden suite rather than asserted; Morrish's KB is pinned exact on a box and Denny's wetted surface carries its known 0.3·L·d shortfall. Golden 961/961.
Glossary: Simpson's rules · block coefficient · prismatic coefficient · waterplane coefficient · TPC · metacentre
Simpson's first rule — and why it is not an approximation
Measure the half-breadth at equally spaced stations, multiply by 1, 4, 2, 4, … 4, 1, add, and take a third of the spacing. That is the whole rule, and it works because it lays a PARABOLA through every three ordinates. A parabola is a cubic with no cube term — and the rule turns out to integrate any cubic exactly. Ship's lines are fair curves close to cubic over three stations, which is why the whole industry still uses it.
| station | half-ord y (m) | SM | product |
|---|---|---|---|
| 0 | 0.000 | 1 | 0.000 |
| 1 | 5.494 | 4 | 21.974 |
| 2 | 7.734 | 2 | 15.469 |
| 3 | 9.113 | 4 | 36.453 |
| 4 | 9.845 | 2 | 19.689 |
| 5 | 10.000 | 4 | 40.000 |
| 6 | 9.611 | 2 | 19.222 |
| 7 | 8.686 | 4 | 34.745 |
| 8 | 7.197 | 2 | 14.394 |
| 9 | 4.990 | 4 | 19.961 |
| 10 | 0.000 | 1 | 0.000 |
| Σ products | 221.907 | ||
| Half area = (h/3)·Σ | 1035.6 m² → whole waterplane 2071.1 m² | ||
| Centroid from aft | 69.04 m — the LCF, drawn on the plan | ||
Fine the ends and the area shrinks, the LCF walks, and TPC falls with it: every hydrostatic quantity on this ship is downstream of that column of products. Simpson's SECOND rule (3h/8 · 1,3,3,2,3,3,1) does the same job when the station count is 3k+1 instead of odd — the engine carries both, and the golden suite proves each exact on a cubic.
The four coefficients — three facts and one consequence
Cb = ∇/(L·B·d): how much of her box she fills — 0.55 for a fine liner, 0.85 for a bulk carrier. Cw = Aw/(L·B) for the waterplane, Cm = Am/(B·d) for the midship section, and Cp = ∇/(Am·L) for how the volume is spread along her length. They are not four independent numbers: Cp · Cm = Cb, always, and the lab prints the residual so you can watch it stay at zero.
| Displacement Δ = L·B·d·Cb·ρ | 16531 t (volume 16128 m³) |
|---|---|
| Cb · Cw · Cm · Cp | 0.720 · 0.740 · 0.960 · 0.750 |
| Identity check Cp·Cm − Cb | 0.0e+0 — zero, by construction |
| KB by Morrish | 4.071 m — d − ⅓(d/2 + ∇/Aw) |
| BM_T = I/∇ | 3.118 m → KM 7.189 m |
| Wetted surface (Denny) | 3920 m² — 1.7·L·d + ∇/d |
| FWA = Δ/(4·TPC) | 195 mm |
Honesty about the approximations: Morrish's KB is a SHAPE guess that happens to be exact on a box (it returns d/2) and lands within a centimetre or two on real forms — the golden suite pins the box case at machine precision. Denny's wetted surface under-counts a box's sides by exactly 0.3·L·d; on a real hull the curvature makes up most of that, which is why the formula survives.
Why the shipyard cares, and why you do
Cb is a commercial decision drawn as a hull: full ends carry cargo, fine ends carry speed cheaply. Cw decides TPC and therefore how fast she sinks as you load. Cm and Cp decide where the volume sits and how she behaves in a seaway. And the same column of Simpson products that gave you the area gives the LCF that ST-05's trim corrections need, the I that ST-01's BM needs, and the hydrostatic curve that every other stability lab reads from. This is the lab underneath the other eight.
| Sinking her 1 cm costs | 21.23 t of cargo (that IS the TPC) |
|---|---|
| Ship type by this Cb | full — product tanker, handysize bulker |
| Waterplane vs the Cb rule of thumb | Cw ≈ (2Cb + 1)/3 = 0.813 against this plan's 0.740 |
| Waterplane area (Simpson) | 2071.1 m² |
|---|---|
| LCF from aft | 69.04 m (abaft midships) |
| TPC | 21.23 t/cm |
| I about the centreline (BM_T numerator) | 50288 m⁴ |
| I about the LCF (BM_L numerator) | 2309345 m⁴ |