This Chief Mate stability formula guide brings together the calculations used to find KG, GM, free-surface correction, GZ, list and change of trim. Each section gives the formula, explains when to use it and shows a short worked example.
Quick answer: which stability formula do I need?
- To find initial stability: GM = KM − KG.
- After loading or discharging: calculate the final KG from total vertical moments and final displacement.
- After shifting a weight vertically: calculate the rise or fall of G, then update KG and GM.
- For slack tanks: divide total free surface moment by displacement and subtract the correction from GM.
- For a righting lever at a stated angle: GZ = KN − KG × sin θ.
- For a small angle of list: divide the transverse heeling moment by displacement × corrected GM to find tan θ.
- For change of trim: divide the trimming moment by MCTC.
Use the units beside each formula. Displacement and weights are in tonnes; distances are in metres unless stated otherwise. All worked examples use original training figures. They are separate examples, not successive stages of one ship’s loading condition.
1. KG, KM and GM
Use this to find the ship’s initial transverse metacentric height. Read KM for the relevant displacement and trim from the hydrostatic data, and use KG for the same loading condition.
GM = KM − KG
KG = KM − GM
KM = KB + BM
| Term | Meaning |
|---|---|
| K | Keel datum used for the vertical measurements. |
| KG | Height of the ship’s centre of gravity, G, above K. |
| KB | Height of the centre of buoyancy, B, above K. |
| BM | Transverse metacentric radius, from B to the initial metacentre M. |
| KM | Height of the initial transverse metacentre above K. |
| GM | Initial transverse metacentric height, from G to M. |
Short example
KM = 7.20 m and KG = 6.30 m.
GM = 7.20 − 6.30 = 0.90 m
If 6.30 m is the solid KG, this is the GM before free-surface correction.
A positive corrected GM describes initial stability. It does not, by itself, confirm adequate stability at larger angles or compliance with every applicable criterion.
2. Final KG after loading and discharging
Use moments about K. Loading adds weight and its vertical moment; discharging removes both.
Vertical moment = Weight × Height of its centre of gravity above K
Final displacement = Initial displacement + Weights loaded − Weights discharged
Final KG = Final total vertical momentFinal displacement
The starting vertical moment is initial displacement × initial solid KG. Add each loaded weight × its height above K, and subtract each discharged weight × its height above K.
Short example: load one weight and discharge another
A ship initially displaces 10,000 t with KG 6.00 m. She loads 1,000 t at 4.00 m above K and discharges 500 t from 8.00 m above K.
| Item | Weight (t) | Height above K (m) | Moment (t·m) |
|---|---|---|---|
| Initial ship | 10,000 | 6.00 | 60,000 |
| Load cargo | +1,000 | 4.00 | +4,000 |
| Discharge cargo | −500 | 8.00 | −4,000 |
| Final total | 10,500 | — | 60,000 |
Final KG = 60,00010,500 = 5.714 m
At the final displacement, the supplied hydrostatic KM is 7.000 m.
Final GM = 7.000 − 5.714 = 1.286 m
No free-surface correction is included in this example.
Check the direction: loading below the original G and removing weight above it both lower KG in this example.
Use the final displacement in the denominator and obtain the final KM. Loading and discharging may change KM as well as KG. Tank liquid weights belong in the weight-and-moment table; their free-surface effect is an additional correction.
Reference for the moment-table method: MCA example stability booklet, KG and condition table.
3. Rise or fall of G when a weight is shifted vertically
Use this when the weight is already on board. Moving it to another height changes the position of G without changing displacement.
Rise or fall of G = Weight shifted × Vertical distance movedShip’s displacement
Weight moved up: New KG = Old KG + Rise of G
Weight moved down: New KG = Old KG − Fall of G
Short example: moving cargo upwards
A 200 t weight is raised 5.00 m on a ship displacing 10,000 t. Initial KG = 6.00 m; KM remains 7.00 m.
Rise of G = 200 × 5.0010,000 = 0.100 m
New KG = 6.00 + 0.10 = 6.10 m.
New GM = 7.00 − 6.10 = 0.90 m.
The distance is the change in the weight’s height, not its final height above K. This example assumes unchanged free-surface effects.
4. Free-surface correction and corrected GM
Use this when slack tanks contribute a free surface moment. Add the applicable tank moments before dividing by displacement.
Free surface correction = Total free surface momentShip’s displacement
Corrected GM = Uncorrected GM − Free surface correction
Fluid KG = Solid KG + Free surface correction
FSM means free surface moment, in t·m. FSC means free surface correction, in metres. Use the tank moments and density adjustment specified in the question or tank tables.
Short example: two slack tanks
Displacement = 12,000 t; tank moments = 800 t·m and 400 t·m; uncorrected GM = 0.90 m.
FSC = 800 + 40012,000 = 0.100 m
Corrected GM = 0.90 − 0.10 = 0.80 m
Apply FSC once. If GM was calculated as KM − fluid KG, it is already corrected. Do not subtract FSC again. If a given FSM already includes liquid density, do not multiply it by density again.
Read the full Free-Surface Effect explanation and tank calculation →
5. GZ, KN and righting moment
Use KN to calculate GZ at a specified angle of heel. Obtain KN from the correct displacement and trim data, interpolating when necessary. For a ship with G on the centreline:
GZ = KN − KG × sin θ
Righting moment = Displacement × GZ
GZ is the righting lever in metres; θ is the angle of heel. Use the KG and free-surface method specified in the question. In the conventional fluid-KG method, use fluid KG = solid KG + FSC.
Short example: GZ at 30°
At the required displacement, KN at 30° = 3.50 m. The applicable KG, already corrected as required, is 6.00 m. Displacement = 12,000 t.
GZ = 3.50 − 6.00 × sin 30°
= 3.50 − 3.00 = 0.50 m
Righting moment = 12,000 × 0.50 = 6,000 t·m
The moment is written in the usual exam convention of tonne-metres. If a question asks for kN·m, convert the displacement weight to kilonewtons before multiplying by GZ.
What about GZ = GM × sin θ?
At small angles: GZ ≈ GM × sin θ
This is an initial-stability approximation, with G on the centreline and the metacentre treated as fixed. For example, corrected GM 0.80 m at 5° gives GZ ≈ 0.80 × sin 5° = 0.070 m. Use the KN data or GZ curve for larger angles.
Use degree mode when θ is in degrees. Do not subtract FSC from GZ as a plain distance, and do not apply a further free-surface correction when the supplied KG or GZ is already corrected.
Read the full GZ and KN calculation, interpolation and curve example →
Reference for the KN relationship and fluid KG: MCA example stability booklet, KN tables.
6. Angle of list after a transverse weight shift
Use this for a small equilibrium list caused by moving a weight across an initially upright ship. The weight remains on board and moves horizontally, so displacement and KG are unchanged.
Heeling moment = Weight shifted × Transverse distance moved
tan θ = Weight shifted × Transverse distance movedDisplacement × Corrected GM
Find θ using the inverse tangent, tan⁻¹.
Short example: shifting cargo to starboard
Displacement = 10,000 t; corrected GM = 0.80 m. A 100 t weight is moved 4.00 m to starboard at the same height.
tan θ = 100 × 4.0010,000 × 0.80 = 0.050
θ = tan⁻¹(0.050) = 2.86° to starboard
0.050 is the tangent, not the angle. If the weight also moves vertically, calculate the new GM first. If a weight is loaded or discharged off-centre, calculate the final displacement, final corrected GM and net transverse heeling moment for that condition.
This formula assumes positive corrected GM and a small angle. An angle of loll caused by negative initial GM is a different problem; do not use this list calculation to diagnose or correct it.
The same moment–tangent relationship underlies the inclining experiment: NAVSEA, Weights and Stability, section 096-2.1.2.1.
7. Change of trim when a weight is shifted longitudinally
Use MCTC for a fore-and-aft weight movement. MCTC is the moment to change trim by one centimetre, in t·m/cm.
Trimming moment = Weight shifted × Longitudinal distance moved
Change of trim (cm) = Trimming momentMCTC
Shifting a weight aft produces a change of trim by stern; shifting it forward produces a change by head. Use the distance moved for an onboard shift.
Short example: shifting a weight aft
A 200 t weight is shifted 20.00 m aft. MCTC = 200 t·m/cm.
Trimming moment = 200 × 20.00 = 4,000 t·m
Change of trim = 4,000200 = 20.00 cm by stern
That is a 0.200 m change of trim. It is not automatically the final trim.
How is the change divided between forward and aft drafts?
For the small-change calculation, the ship trims about LCF. On this page, LCF is measured forward from AP.
Aft draft change due to trim = Change of trim × LCF from APLBP
Forward draft change due to trim = Change of trim × (LBP − LCF from AP)LBP
These give the magnitudes. For a change by stern, add at AP and subtract at FP. For a change by head, reverse those signs.
Continue the example: calculate final drafts
LBP = 150.00 m; LCF = 70.00 m forward of AP. Initial drafts at the perpendiculars are F 6.000 m and A 6.200 m. Assume MCTC and LCF remain constant over this small change.
- Aft increase = 20.00 × 70.00 ÷ 150.00 = 9.333 cm = 0.09333 m.
- Forward decrease = 20.00 × 80.00 ÷ 150.00 = 10.667 cm = 0.10667 m.
- Final F = 6.000 − 0.10667 = 5.893 m.
- Final A = 6.200 + 0.09333 = 6.293 m.
Check: initial trim 0.200 m + change by stern 0.200 m = 0.400 m by stern.
Do not divide the trim change equally unless LCF is amidships. Keep both the change of trim and the resulting draft changes in the same unit, then convert centimetres to metres before updating drafts in metres.
If the weight is loaded or discharged instead
There is also a change in displacement. For a small weight change, calculate the parallel sinkage or rise using TPC, as well as the trim effect about LCF:
Parallel sinkage or rise (cm) = Weight loaded or dischargedTPC
TPC is tonnes per centimetre immersion. Use the weight’s distance from LCF to calculate its trimming moment when loading or discharging. For larger changes, follow the question’s hydrostatic procedure instead of assuming TPC, MCTC and LCF stay constant.
A useful order for a combined stability question
- Find final displacement after all loading and discharging.
- Find final solid KG from the weight-and-moment table, including any vertical shifts.
- Obtain final KM from the hydrostatic data for that condition.
- Calculate FSC from the applicable slack-tank moments.
- Find corrected GM or fluid KG, applying FSC once.
- Calculate the quantity asked for: GZ, righting moment, list, or trim and final drafts.
Common Chief Mate exam mistakes
- Using initial displacement after loading or discharging. Recalculate the final total.
- Using a weight’s height instead of its distance moved. Height above K belongs in a vertical-moment table; distance moved belongs in a shifting calculation.
- Keeping KM unchanged without checking. Obtain the hydrostatic value for the new condition.
- Correcting twice for free surface. Distinguish solid KG, fluid KG and corrected GM.
- Using GM × sin θ at a large angle. Use the KN values or GZ curve supplied.
- Writing tan θ as the answer in degrees. Take the inverse tangent.
- Confusing list with loll. The small-angle list formula assumes positive corrected GM.
- Treating change of trim as final trim. Combine it with the initial trim, keeping the direction clear.
- Mixing centimetres and metres. MCTC and TPC normally give changes in centimetres.
- Rounding at every step. Keep calculator precision and round the final answer to the question’s requirement.
Quick oral-exam answers
Does loading a weight always reduce GM?
No. Its effect depends on the position of the added weight and the change in KM. Calculate the new KG and use the new hydrostatic KM.
Does shifting a weight change displacement?
No, provided it remains on board and no other weight is added or removed. Its movement changes the position of G.
Are GM and GZ the same?
No. GM describes initial stability near the upright condition. GZ is the righting lever at a particular angle of heel.
Does a positive GM mean the loading condition is acceptable?
It is one check. The relevant GZ curve, stability limits, downflooding and other required criteria must also be considered using the ship’s approved information.
For the role of approved loading instructions and stability limits, see the US Coast Guard guidance on trim and stability booklets. This page is a revision guide to core calculations, not a complete stability syllabus.
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