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.

TPC & MCTC Calculator

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.

Example and notes

For a 150 m × 22 m vessel at 8 m draught with Cw = 0.82 and Cb = 0.72 in sea water, the waterplane area is about 2,706 m², giving a TPC of roughly 27.7 t/cm — load 27.7 tonnes to sink the mean draught 1 cm. With an inertia coefficient of 0.070 the MCTC works out near 540 t·m/cm. Shifting 100 t aft by 27 m changes the trim about 5 cm by the stern. Remember TPC and MCTC both change with draught: recompute them at the working waterline rather than reusing a single value.