TPC, tonnes per centimetre immersion, is the mass that must be loaded or discharged to change the mean draught by one centimetre. It equals the waterplane area times water density divided by 100, so a larger or fuller waterplane gives a higher TPC.

MCTC, the moment to change trim one centimetre, is the trimming moment in tonne-metres needed to alter the trim by one centimetre. It is found from the displacement and the longitudinal metacentric height, MCTC equals W times GM_L divided by 100 times LBP.

Why does MCTC use the longitudinal metacentric height?

Trim is a rotation about a transverse axis, so it is governed by the longitudinal metacentric height GM_L, which is dominated by the longitudinal BM_L. BM_L is the longitudinal second moment of the waterplane area divided by the displaced volume, and it is very large, which makes ships stiff in trim.

How accurate is this estimate?

TPC is exact given the true waterplane area. MCTC here uses an inertia coefficient to approximate the longitudinal second moment of area, so it is an estimate within a few percent for normal hull forms. Always cross-check against the vessel's hydrostatic tables for operational work.

What density should I use?

Use 1.025 tonnes per cubic metre for standard sea water and 1.000 for fresh water. Brackish or dock water sits in between, so use the measured dock water density when working out fresh water allowance and dock water allowance.

What is the TPC & MCTC Calculator?

Derive TPC and MCTC from waterplane geometry: LBP, beam, draught, waterplane and block coefficients, and water density. Naval architects and ship officers use this to verify hydrostatic tables and solve ballasting and trim problems. It runs free in your browser on Gera Tools, with nothing uploaded.

TPC & MCTC Calculator

Name: TPC & MCTC Calculator
Creator: Gera Tools
License: https://creativecommons.org/licenses/by/4.0/

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TPC and MCTC are two of the core hydrostatic quantities every ship officer and naval architect reaches for when loading, ballasting, or trimming a vessel. This calculator derives both from the waterplane geometry so you can verify a hydrostatic table, sanity-check a loading computer, or work a textbook problem.

How it works

The waterplane area sets TPC directly, while the longitudinal second moment of that area sets the trim stiffness:

A_w  = Cw × LBP × B                 (waterplane area, m²)
TPC  = A_w × ρ / 100                (tonnes per cm)
∇    = Cb × LBP × B × d             (displaced volume, m³)
W    = ∇ × ρ                        (displacement, t)
I_L  = k × LBP³ × B                 (longitudinal inertia of waterplane, m⁴)
BM_L = I_L / ∇
MCTC = W × GM_L / (100 × LBP) ≈ ρ × I_L / (100 × LBP)

Because longitudinal BM_L is so large, KB and KG are negligible against it, so GM_L ≈ BM_L is an accepted approximation for trim moments. The inertia coefficient k (typically 0.060–0.075) captures how the waterplane shape concentrates area toward the ends.

Worked example

For a vessel 150 m LBP, 22 m beam, at 8 m mean draught, with Cw = 0.82, Cb = 0.72, operating in sea water (ρ = 1.025 t/m³):

Waterplane area: 0.82 × 150 × 22 = 2,706 m²
TPC: 2,706 × 1.025 / 100 ≈ 27.7 t/cm — every 27.7 tonnes loaded or discharged changes mean draught by 1 cm
Displaced volume: 0.72 × 150 × 22 × 8 = 19,008 m³
Displacement: 19,008 × 1.025 ≈ 19,483 t
Longitudinal inertia (k = 0.070): 0.070 × 150³ × 22 = 52,987,500 m⁴
BM_L: 52,987,500 / 19,008 ≈ 2,788 m
MCTC: 1.025 × 52,987,500 / (100 × 150) ≈ 3,620 t·m/cm… but applying the full W × BM_L / (100 × LBP): 19,483 × 2,788 / 15,000 ≈ 3,622 t·m/cm

These are estimates within a few percent of the actual hydrostatic table values for a conventional vessel of this form. Always cross-check against the vessel’s official stability booklet for operational work.

How to use TPC and MCTC on passage

Adjusting mean draught with TPC: If you load 500 tonnes of bunkers, the change in mean draught is 500 / TPC = 500 / 27.7 ≈ 18 cm by the stern if loaded aft of the centre of flotation, or a bodily sinkage of 18 cm if loaded exactly at the centre of flotation.

Adjusting trim with MCTC: Trimming moment is mass × shift distance. If you shift 100 t of ballast 30 m aft, the trimming moment is 3,000 t·m. Trim change = 3,000 / MCTC. Using the example MCTC above, that is approximately 0.8 cm by the stern per metre of LBP — trivially small for this large vessel, confirming that large ships are very stiff in trim.

Why TPC and MCTC change with draught

Both quantities depend on the waterplane, which changes shape and area as a ship rises or sinks. A fuller hull form at a deeper draught will have a larger waterplane area (higher TPC) and a larger longitudinal second moment (higher MCTC). The ship’s hydrostatic tables provide these values at half-centimetre or centimetre draught intervals precisely because using a single value across a wide draught range introduces cumulative error in ballasting calculations.

For voyage planning calculations, always interpolate from the hydrostatic table at the departure and arrival draught rather than using a single figure from this estimator.