Ship Stability and Trim Exercises

These exercises practise ship stability and trim as naval architecture evidence. They cover displacement, draft change, tons per centimeter immersion, trim, center-of-gravity shifts, metacentric height, free-surface correction, righting arm, righting moment, roll period, water-density effects, and loading documentation.

The goal is not only to calculate a hydrostatic number. The goal is to decide whether a loading condition remains safe, understandable, and traceable to vessel data. A stability result is weak if the displacement, tank state, draft readings, water density, openings, and center of gravity are not controlled.

Assume simplified screening models unless an exercise states otherwise. Final vessel decisions require approved hydrostatic data, stability criteria, class rules, loading manuals, inclining-experiment evidence, and vessel-specific operating procedures.

How to Use These Exercises

For each exercise, define:

1. the loading condition and water density;
2. the vessel displacement, draft, trim, and tank state;
3. the vertical, longitudinal, and transverse center-of-gravity assumptions;
4. the hydrostatic approximation being used;
5. the operational decision supported by the result.

The common mistake is treating a single stability number as universal. The same vessel can be safe in one loading condition and unsafe after cargo movement, fuel burn, ballast transfer, slack tanks, icing, trapped water, or damage.

Exercise 1: Displacement from Displaced Volume

A vessel displaces:

of seawater with density:

Estimate displacement mass:

Solution

Substitute:

Convert to metric tonnes:

Engineering Comment

The displacement result is tied to water density and immersed volume. A loading record should state whether the vessel is in seawater, brackish water, or freshwater, and whether the volume comes from hydrostatic curves or draft readings.

Exercise 2: Draft Change from Added Load

A small load is added near the current waterline. The added mass is:

The vessel waterplane area is:

Use seawater density:

Estimate draft increase:

Solution

Convert added mass:

Compute:

Therefore:

Engineering Comment

This is a local waterplane approximation. Large loading changes, nonuniform loading, trim changes, and hull-form nonlinearity require updated hydrostatic data rather than one constant waterplane area.

Exercise 3: Tons per Centimeter Immersion

Using:

and:

estimate tons per centimeter immersion:

Solution

Substitute:

Engineering Comment

TPC is useful for quick draft estimates, but it changes with draft and trim. It should be read from hydrostatic data for formal loading decisions.

Exercise 4: Mean Draft and Trim

A vessel has forward draft:

and aft draft:

Compute mean draft and trim by stern.

Solution

Mean draft:

Trim by stern:

Engineering Comment

Mean draft is not enough for vessel operation. Trim can affect resistance, propeller immersion, sonar or sensor alignment, bridge visibility, steering, loading limits, and clearance under the keel.

Exercise 5: Center of Gravity After Added Weight

A vessel has displacement:

and vertical center of gravity:

A deck load of:

is added at:

Compute the new KG:

Solution

Substitute:

Engineering Comment

The deck load raises KG by about 0.10\ \text{m}. That may look small, but stability margins can be controlled by small changes in vertical center of gravity, especially when free-surface effects or high deck loads are present.

Exercise 6: Metacentric Height

For a loading condition:

Compute initial metacentric height:

Solution

Substitute:

Engineering Comment

A positive GM indicates positive initial stability, but it does not prove adequate large-angle stability, downflooding margin, damage stability, or acceptable roll motion.

Exercise 7: Free-Surface Correction

The uncorrected metacentric height is:

Two slack tanks create free-surface corrections:

Compute corrected metacentric height:

Solution

Total correction:

Corrected value:

Engineering Comment

Free-surface effects reduce stability without necessarily changing total liquid mass. Loading procedures should control slack tanks, bilges, fish holds, ballast transfers, and damaged spaces.

Exercise 8: Small-Angle Righting Arm

Using corrected:

estimate righting arm at heel angle:

using:

Solution

Compute:

Engineering Comment

This small-angle approximation is useful near upright. At larger heel angles, use the full righting-arm curve and check downflooding angle, deck immersion, reserve stability, and regulatory criteria.

Exercise 9: Righting Moment

A vessel has displacement mass:

and righting arm:

Estimate righting moment:

Solution

Convert displacement:

Compute:

Therefore:

Engineering Comment

Righting moment gives a physical sense of restoring capacity. It must be compared with heeling moments from wind, cargo shift, turning, towing, lifting, flooding, or wave action.

Exercise 10: Roll Period Estimate

Estimate roll period using:

with roll radius of gyration:

and:

Solution

Substitute:

Engineering Comment

The estimate links stability to motion. A higher GM usually shortens roll period and can make the vessel feel stiff. Comfort and operability require seakeeping review, not only intact stability.

Exercise 11: Freshwater Draft Increase

A vessel displaces:

At the current condition, waterplane area is:

Estimate how much deeper the vessel floats when moving from seawater:

to freshwater:

Use displaced volume:

and:

Solution

Convert displacement:

Seawater volume:

Freshwater volume:

Volume increase:

Draft increase:

Therefore:

Engineering Comment

Freshwater allowance matters for port entry, river operation, loading limits, and under-keel clearance. Water density should be recorded with draft readings when margins are tight.

Exercise 12: Loading Decision Evidence

A loading change produces:

estimated roll period:

and stern trim:

List the evidence needed before approving operation.

Solution

Minimum evidence should include:

1. loading condition and weight report;
2. draft readings forward, aft, and midship if available;
3. water density or port condition;
4. tank soundings and free-surface status;
5. cargo, deck load, and ballast centers of gravity;
6. corrected stability calculation from approved data;
7. downflooding and freeboard check;
8. trim and propeller-immersion check;
9. operational limitations for weather, speed, and deck work;
10. crew-facing record in the stability booklet or loading computer.

Engineering Comment

The arithmetic supports a review, but it is not the review. Stability management succeeds when calculations, loading records, onboard procedures, and crew decisions describe the same vessel state.