Free Surface Effect on Ships: Formula, Correction and Worked Examples
How liquid movement in slack tanks produces free surface moment, reduces corrected GM and changes a vessel’s initial stability.
A practical calculation guide to FSM, FSC, virtual KG, tank geometry, subdivision and operational tank management.
A vessel can leave port with an acceptable solid GM and still lose a meaningful part of its initial stability as fuel, ballast and service tanks become slack. One partly filled tank may produce only a modest correction. Several slack tanks acting together can reduce the effective GM enough to change the vessel’s response to wind, turning, lifting operations or cargo movement.
The cause is liquid freedom of movement. As the ship heels, the surface of liquid in a slack tank tends to remain approximately horizontal relative to gravity. It does not remain parallel to the tank top or to the vessel’s structure. Liquid moves toward the lower side, its centre of gravity shifts transversely and a moment develops in the direction of heel. The ship therefore produces less effective righting response than a calculation based only on solid weights would suggest.
Stability calculations normally represent this loss as an equivalent or virtual rise of G. The actual solid-weight KG has not necessarily moved upward, but the calculated stability response is the same as if G had risen by the free surface correction. This distinction between actual KG and virtual KG is central to using free surface data correctly.
For the wider relationship between KG, KM, GM, GZ and righting moment, see Ship Stability Explained: KG, GM and GZ Curves. The present article focuses on free surface effect on ships: how it is calculated, why tank breadth matters, how several tanks accumulate and why operational transfer sequences can create limiting intermediate conditions.
What Is Free Surface Effect?
Free surface effect is the reduction in stability caused by liquid moving within a space that is not completely full. When a ship is upright, the liquid’s centre of gravity is normally close to the tank centreline. As the vessel heels, liquid flows toward the low side. The liquid surface seeks a level plane under gravity, so the liquid’s centre of gravity moves away from the vessel centreline.
That movement creates a transverse heeling influence. Weight still acts downward through the combined centre of gravity, while buoyancy acts upward through the shifted centre of buoyancy. The liquid movement reduces the effective separation that would otherwise form a righting lever. Near upright, the result is treated as a reduction in metacentric height.
Without free surface correction, solid GM is:
After accounting for all relevant slack tanks:
The correction does not describe a new item of weight placed high in the ship. It is a convenient equivalent representation of a stability loss caused by transverse liquid movement. This is why stability books and loading computers may show both an actual KG from weight moments and a higher virtual KG after free surface allowance.
Free surface does not normally lift the ship’s solid weights. It makes the vessel respond as though its centre of gravity were higher.
Empty, Pressed-Up and Slack Tanks
A tank’s free surface condition depends on whether liquid has room to move across the tank. The labels empty, full and slack are operational descriptions, but the actual piping, venting, internal structure and filling condition determine whether a free surface exists.
| Tank condition | Liquid movement | Free surface treatment | Operational qualification |
|---|---|---|---|
| Empty | No tank liquid available to shift | Normally no liquid free surface moment | Residues, bilge contents or retained liquids may still require consideration |
| Pressed up | Little or no effective surface movement | May be treated without a free surface correction when approved criteria are met | Expansion space, trapped air and incomplete filling can prevent a tank from being genuinely pressed up |
| Slack | Liquid can move toward the low side | FSM or FSC must be included | Use approved tank tables, loading-computer treatment and vessel procedures |
A nominally full sounding does not prove that a tank has no effective free surface. An expansion space may be deliberately retained, a vent arrangement may hold air, or the tank top may not be completely covered. Conversely, a small quantity retained in an irregular sump may not produce the same surface geometry assumed for a broad rectangular tank.
The approved stability information determines when a tank may be treated as pressed up. Operators should use the vessel’s defined filling limits rather than assuming that a tank described informally as “full” has zero correction.
Free Surface Moment
Free surface moment, abbreviated FSM, combines the geometry of the liquid surface with liquid density. It is calculated from:
where:
- \(\rho_l\) is the liquid density in \(\mathrm{t/m^3}\)
- \(i\) is the second moment of area of the free surface about the tank’s longitudinal axis, in \(\mathrm{m^4}\)
- \(FSM\) is the free surface moment in \(\mathrm{t\cdot m}\)
FSM is a moment, not a vertical correction. It cannot be subtracted directly from GM. The moment must first be divided by the vessel’s displacement to obtain a correction in metres.
Tank capacity alone does not determine FSM. Two tanks can hold the same volume but have very different effects. A deep, narrow tank restricts transverse movement, while a broad, shallow tank allows the liquid centre of gravity to move farther across the vessel. The relevant geometry is the shape of the free surface at the liquid level, not simply the total tank volume.
Density also matters. For identical free-surface geometry, seawater produces a greater FSM than a lighter fuel oil because more liquid mass is involved in the transverse shift. Approved tank tables may already incorporate an assumed density, so the convention used by the source data must be checked before any additional density correction is applied.
Rectangular Tank Free Surface Formula
For a rectangular free surface with length \(l\) and breadth \(b\), the transverse second moment of area is:
Combining this with liquid density gives:
Breadth is measured transversely across the vessel and appears to the third power. Length is measured fore and aft and appears only to the first power. If tank length doubles while breadth and density remain unchanged, free-surface inertia doubles. If breadth doubles while length and density remain unchanged, inertia increases by \(2^3 = 8\).
| Geometry change | Inertia relationship | Resulting factor |
|---|---|---|
| Double length | \(i_{\text{new}} = (2l)b^3/12\) | 2 times the original inertia |
| Double breadth | \(i_{\text{new}} = l(2b)^3/12\) | 8 times the original inertia |
This cubic breadth relationship explains why longitudinal subdivision is so effective and why broad slack tanks deserve particular attention. It also explains why replacing approved data with tank volume or a length-times-breadth estimate is technically wrong.
Free Surface Correction and Corrected GM
Each relevant tank produces an FSM. The tank moments are added, then divided by total vessel displacement:
where \(\Delta\) is the ship’s total displacement in tonnes. It is not deadweight. Displacement includes lightship, cargo, fuel, ballast, stores, people and every other weight forming the ship’s actual mass condition.
The resulting FSC is measured in metres and is deducted from solid GM:
For the same total FSM, a heavier displacement produces a smaller correction because the liquid-shift moment is being compared with a larger total ship mass. This does not mean free surface can be ignored on a large ship; broad cargo or ballast tanks can also produce very large moments.
| Quantity | Formula | Typical units |
|---|---|---|
| Free-surface inertia | \(i = lb^3/12\) | m⁴ |
| Free surface moment | \(FSM = \rho_l i\) | t·m |
| Total correction | \(FSC = \sum FSM/\Delta\) | m |
| Corrected GM | \(GM_{\text{corrected}} = GM_{\text{solid}}-FSC\) | m |
| Virtual KG | \(KG_{\text{virtual}} = KG_{\text{actual}}+FSC\) | m |
Actual KG Versus Virtual KG
Actual KG is obtained from solid-weight moments. Loading, discharging or shifting a real weight changes those moments and can physically move the combined centre of gravity. Free surface correction is different: it represents the stability consequence of liquid being able to move as heel develops.
The virtual KG method adds FSC to actual KG:
Corrected GM can then be found from:
This is mathematically equivalent to subtracting FSC from solid GM:
Virtual KG should therefore be described as a calculation convention, not as a new physical mass location without qualification. It allows hydrostatic and righting-lever calculations to account for the reduced stability response using a familiar KG-based method.
Density and Unit Conventions
Free surface data is not presented in one universal format. A tank table may provide geometric inertia in m⁴, FSM in t·m at a stated density, a moment based on seawater density, several corrections at different sounding levels or FSC directly in metres for a specified displacement.
Before using a value, identify exactly what the table contains:
- Is the entry free-surface inertia, FSM or FSC?
- What are the stated units?
- Has liquid density already been applied?
- Is the reference density seawater, a standard specific gravity or the tank’s actual liquid?
- Does the value correspond to the actual sounding, ullage or volume?
- Is interpolation required between tabulated levels?
- Does the loading computer automatically adjust the value for entered density?
In a mass-based calculation, use compatible units: density in t/m³ multiplied by inertia in m⁴ gives FSM in t·m. Dividing by displacement in tonnes gives FSC in metres. If righting moment is then calculated as \(\Delta \times GZ\), the result is in t·m.
For a force-based moment, convert tonnes of mass consistently:
Mixing a force in kilonewtons with a mass-based tonne-metre value without applying gravitational acceleration produces an inconsistent result.
Is a Half-Full Tank Always the Worst Case?
The statement “a half-full tank is always worst” is not universally correct. In an ideal rectangular tank with vertical sides, the free-surface length and breadth can remain nearly constant across a broad range of filling levels. While a continuous surface extends across the same plan dimensions, its geometric inertia can remain similar whether the tank is one-quarter, one-half or three-quarters full.
Real tanks behave differently. In a wing tank, hopper tank or double-bottom tank, sloped plating can cause the surface breadth to change sharply with level. Curved boundaries, internal girders and suction wells can divide or reshape the surface. At low levels, liquid may pocket into separate areas; at high levels, the liquid may contact the tank top and reduce the effective surface.
Deep tanks and irregular cargo tanks can therefore have peak FSM at a level that is not 50% full. Approved free-surface tables map the actual geometry through the filling range and should be used to identify the critical condition.
Cumulative Effect of Several Slack Tanks
Free surface moments are cumulative:
A fuel settling tank, ballast tank and freshwater tank may each appear harmless when considered alone, yet their moments all reduce the same corrected GM. Lubricating-oil tanks, drain tanks and service tanks can add further corrections. Small tanks should not be omitted merely because each individual effect is modest.
Bilge water, floodwater and water retained on deck can also reduce stability, but they should not automatically be treated as ordinary enclosed rectangular tanks. Their boundaries, communication paths, accumulation patterns and changing quantities may require vessel-specific methods. Water on deck can also add weight high, affect drainage and move across a much broader surface than an internal service tank.
Longitudinal Subdivision and Centreline Bulkheads
A longitudinal subdivision runs fore and aft and reduces the transverse breadth over which liquid can move. For an undivided rectangular tank:
If a complete centreline bulkhead creates two equal longitudinal compartments, each has breadth \(b/2\). The total inertia is:
The total liquid capacity has not necessarily changed. The reduction occurs because each liquid surface has only half the original transverse breadth, and breadth is cubed. The liquid centre of gravity cannot travel as far across the vessel in either compartment.
The theoretical quarter-value assumes a complete effective division, rectangular geometry, correct evaluation of both compartments and no open cross-connection or large opening that allows free communication. Real arrangements must be assessed using approved data. A fore-and-aft centreline bulkhead is a longitudinal subdivision; calling it transverse subdivision reverses the terminology.
Swash Bulkheads and Perforated Divisions
Swash bulkheads restrict liquid flow and reduce rapid surging, but they do not automatically eliminate free surface effect. If liquid can communicate through openings, the separate regions may still behave partly or fully as one connected surface.
Effectiveness depends on opening size and distribution, liquid level, tank geometry, flow resistance and the time scale of the vessel’s motion. A perforated division that damps short-period sloshing may not provide the same static free-surface reduction as a complete watertight centreline bulkhead.
No arbitrary percentage reduction should be assigned. Use the vessel’s approved tank data or a validated engineering method that represents the actual internal arrangement.
The Double Penalty From Consuming Low Fuel or Ballast
Consumption from low double-bottom tanks can reduce stability in two ways at the same time:
- Removing weight from below the existing centre of gravity can raise the actual KG.
- The partly consumed tank becomes slack and adds free surface correction.
A vessel can therefore be lighter on arrival but have a higher virtual KG and lower corrected GM than at departure. Several nearly empty tanks may still retain broad liquid surfaces, so a small remaining quantity does not automatically mean a small FSM.
Consumption sequencing matters. Operators may seek to keep fewer tanks slack, but any plan must also respect trim, longitudinal strength, draft, propeller immersion, pump arrangements and approved operating procedures. Free surface cannot be managed in isolation from the rest of the loading condition.
Free Surface Effect During Tank Transfers
Tank transfers create temporary conditions that may be more limiting than either the start or finish. During ballasting, deballasting, bunkering, fuel transfer, freshwater transfer, cargo-tank operations, tank cleaning or stripping, several tanks may be partly filled at once.
The initial condition may have one tank pressed up and another empty. The final condition may reverse those states. Midway through the operation, both tanks may be slack and both FSM values may apply. Opening a third receiving or settling tank can add another correction.
Sequential filling or emptying can reduce the number of simultaneous slack tanks: one tank is completed before the next is opened. That principle is useful, but the transfer plan must also consider:
- Longitudinal strength and local structural limits
- Trim, list and operating drafts
- Tank pressure and venting limits
- Pump capacity and stripping limitations
- Stability limits throughout the sequence
- Valve alignment and cross-connections
- Approved loading guidance and company procedures
A loading computer check should represent relevant intermediate stages rather than only departure and arrival snapshots.
Free Surface Effect and the GZ Curve
At small heel angles, corrected GM determines the initial slope of the GZ curve. A reduced GM gives a shallower initial rise:
For the same displacement and heel angle, lower corrected GM produces a smaller GZ and therefore a smaller righting moment. This is the principal reason free surface correction is essential in initial-stability calculations.
Large-angle liquid behaviour is more complex. As heel grows, the liquid may contact the tank top or bottom, the available breadth can change, liquid may pocket behind structure, and connected spaces may exchange liquid. A constant small-angle FSC may not represent every finite-angle effect.
Large-angle stability should therefore come from approved KN data, approved cross-curves, approved loading software, vessel-specific tank data or validated numerical analysis. The expression \(GZ = GM\sin\phi\) must not be extended across the complete heel range as though the metacentre and free-surface geometry remained unchanged.
Free Surface Effect Versus Free Communication Effect
Free surface effect describes liquid shifting inside an enclosed, partially filled space. The liquid quantity is normally treated as fixed while its transverse centre of gravity moves.
Free communication effect involves liquid communicating with the sea, another tank, another compartment, an open deck area or a damaged space through an opening. The quantity and distribution of water may change as the vessel heels, trims or changes draft.
| Feature | Free surface effect | Free communication effect |
|---|---|---|
| Typical space | Enclosed slack tank | Flooded or connected space with an open flow path |
| Liquid quantity | Usually fixed for the calculation | May increase, decrease or redistribute |
| Primary changes | Liquid CG and effective righting response | Weight, buoyancy, displacement, trim, list and free surface |
| Simple FSC formula | Often applicable at small angles with suitable data | May be insufficient on its own |
Examples include a flooded compartment open to the sea, open cross-connections between tanks, water entering an open hold, water retained on a vehicle deck and water accumulated on an exposed deck. These cases can change watertight boundaries and buoyancy as well as create a free surface. They require the damage-control or vessel-specific method applicable to the situation.
Operational Methods to Reduce Free Surface Effect
Practical control aims to limit liquid freedom of movement while maintaining an acceptable overall loading condition. Depending on the vessel and operation:
- Keep tanks pressed up or empty where permitted and practical.
- Minimise the number of tanks that are slack simultaneously.
- Follow planned sequential filling and emptying procedures.
- Use longitudinally subdivided tanks as designed.
- Maintain effective tank boundaries and internal divisions.
- Close cross-connections when required by approved procedures.
- Confirm valve alignment before and during transfers.
- Include every relevant tank in the loading computer.
- Use correct liquid densities and accurate tank quantities.
- Account for fuel, water and ballast consumption throughout the voyage.
- Check arrival and important intermediate conditions.
- Maintain freeing ports and deck drainage.
- Avoid unnecessary water accumulation on exposed or vehicle decks.
These are operating principles, not a substitute for vessel-specific instructions. The approved stability booklet, loading computer, transfer plan, structural limits and flag or class requirements remain controlling.
Worked Example 1 — Single Rectangular Slack Tank
This illustrative calculation estimates the correction from one rectangular fuel tank. Approved tank data must be used for an actual ship.
| Input | Value |
|---|---|
| Tank length, \(l\) | 12.0 m |
| Tank breadth, \(b\) | 8.0 m |
| Liquid density, \(\rho_l\) | 0.85 t/m³ |
| Vessel displacement, \(\Delta\) | 10,000 t |
| Solid GM | 0.75 m |
Step 1: Calculate Free-Surface Inertia
Step 2: Calculate Free Surface Moment
Step 3: Calculate Free Surface Correction
Step 4: Calculate Corrected GM
Step 5: Interpret the Result
The GM reduction is:
One slack tank reduces the illustrative solid GM by about 4.35 cm, leaving a corrected GM of approximately 0.7065 m. The percentage change is meaningful for calculation, but its operational acceptability can only be judged against the vessel’s approved limits, complete GZ curve and loading condition.
Worked Example 2 — Several Slack Tanks
This example uses the same 10,000 t displacement and 0.75 m solid GM, but includes three slack tanks. Rectangular geometry is assumed for each free surface.
| Tank | Length | Breadth | Density | Inertia | FSM |
|---|---|---|---|---|---|
| Fuel settling tank | 12 m | 8 m | 0.85 t/m³ | 512.000 m⁴ | 435.200 t·m |
| Ballast tank | 10 m | 6 m | 1.025 t/m³ | 180.000 m⁴ | 184.500 t·m |
| Freshwater tank | 8 m | 5 m | 1.000 t/m³ | 83.333 m⁴ | 83.333 t·m |
Step 1: Verify Each Tank
Step 2: Sum the Moments
Step 3: Calculate Total FSC and Corrected GM
Rounded sensibly, the total correction is 0.0703 m and corrected GM is 0.6797 m. The three corrections reduce solid GM by approximately 9.37%. The calculation demonstrates accumulation: the two smaller tank moments add 267.833 t·m beyond the fuel-tank moment and reduce GM by another 2.68 cm.
Worked Example 3 — Effect of Longitudinal Subdivision
Consider a rectangular tank 12 m long and 8 m wide. Compare the undivided tank with two equal compartments formed by a complete effective centreline bulkhead.
Undivided Tank
Two Equal Longitudinal Compartments
Each compartment is 12 m long and 4 m wide:
The divided arrangement has one-quarter of the undivided inertia because the liquid can move across only half the breadth in each compartment. The result assumes rectangular geometry, a complete effective division, no free communication and correct treatment of both slack compartments. Not every real centreline structure produces exactly this reduction.
Common Free Surface Calculation Errors
| Error | Why it is wrong | What to check instead |
|---|---|---|
| Using deadweight as \(\Delta\) | Deadweight excludes lightship and is not total ship mass. | Use displacement for the actual loading condition. |
| Using tank volume instead of \(i\) | Capacity does not describe transverse surface geometry. | Use approved inertia or FSM data at the liquid level. |
| Forgetting liquid density | Geometric inertia alone is not a mass-based moment. | Apply density once when the source provides m⁴. |
| Applying density twice | An FSM table may already include its reference density. | Read table headings, notes and loading-computer conventions. |
| Treating FSM as FSC | FSM is in t·m; FSC is in metres. | Divide total FSM by displacement. |
| Treating FSC as a physical weight shift | FSC is an equivalent stability correction. | Distinguish actual KG from virtual KG. |
| Omitting small slack tanks | Several small moments can accumulate materially. | Include every tank required by approved data. |
| Assuming half-full is always worst | Critical surface geometry depends on tank shape and level. | Use the approved level-dependent table. |
| Ignoring open cross-connections | Connected liquid may have a much larger effective breadth. | Verify valve state and communication paths. |
| Applying a rectangular formula to an irregular tank | Slopes, frames and wells change the free surface. | Use approved tank-specific values. |
| Using uncorrected GM | Solid GM overstates initial restoring response when tanks are slack. | Use corrected GM or virtual KG. |
| Extending \(GZ=GM\sin\phi\) to large angles | The small-angle model does not capture changing hull and liquid geometry. | Use approved KN data, cross-curves or loading software. |
| Ignoring transfer stages | Intermediate conditions may contain more slack tanks. | Check the full transfer sequence. |
| Using data from another displacement | FSC changes with displacement and hydrostatics change with loading. | Use one consistent condition and interpolate as required. |
| Assuming a tank is pressed up | Expansion space or trapped air may leave a surface. | Apply the approved pressed-up definition. |
| Ignoring deck water or flooding | These can add weight, free surface and communication effects. | Use the applicable vessel-specific or damage-stability method. |
How the NauticalSolver Stability Calculators Fit Together
The Free Surface Correction Calculator is the principal related tool. It can sum tank FSM values, estimate rectangular-tank FSM, divide total moment by displacement and apply the resulting correction to solid GM. Its geometry route is suitable for learning and preliminary estimates; approved tank tables remain controlling for vessel operations.
The GM Calculator can compare solid and corrected GM and show how KM, KG and FSC combine. The Transverse Stability Calculator connects corrected GM with approximate small-angle GZ and righting moment.
The GZ Curve Generator helps illustrate how a higher virtual KG reduces the initial slope of a curve. It should not replace approved KN data or a class-approved loading computer. Readers who need the wider stability framework can return to the main guide, Ship Stability Explained: KG, GM and GZ Curves.
Frequently Asked Questions
What is free surface effect on a ship?
It is the reduction in stability caused when liquid in a partially filled space moves toward the lower side as the vessel heels. The liquid surface tends to remain level relative to gravity, shifting the liquid centre of gravity transversely and reducing effective righting response.
What is free surface moment?
FSM is the product of liquid density and the second moment of area of the free surface: \(FSM=\rho_l i\). With density in t/m³ and inertia in m⁴, FSM is expressed in t·m.
What is free surface correction?
FSC is total free surface moment divided by vessel displacement: \(FSC=\sum FSM/\Delta\). It is expressed in metres and is deducted from solid GM or added to actual KG as a virtual correction.
How does free surface reduce GM?
Liquid movement creates a heeling influence that weakens the initial restoring response. The effect is represented by \(GM_{\text{corrected}}=GM_{\text{solid}}-FSC\). A lower corrected GM gives a shallower initial GZ-curve slope.
Is free surface correction an actual rise of KG?
Not normally. The solid-weight centre of gravity has not necessarily moved upward. Adding FSC to actual KG produces a virtual KG that represents the equivalent stability reduction caused by liquid movement.
Is a half-full tank always the worst condition?
No. In a rectangular tank, surface dimensions may remain similar across several filling levels. In an irregular tank, slopes and internal structure can make another level critical. Approved free-surface tables identify the actual variation.
Why does tank breadth matter so much?
Rectangular free-surface inertia contains breadth cubed: \(i=lb^3/12\). Doubling length doubles inertia, while doubling breadth increases it eightfold. Broad slack tanks therefore tend to have a strong effect on transverse stability.
Do empty tanks have free surface effect?
A genuinely empty tank has no liquid surface and normally no liquid FSM. Residues, retained bilge liquid or an inaccurate empty-tank assumption may still require consideration under approved procedures.
Do pressed-up tanks have free surface effect?
A genuinely pressed-up tank has little or no effective liquid movement and may be treated without FSC when the approved criteria are satisfied. Nominal fullness, trapped air or expansion space can prevent this treatment.
Why can several slack tanks be dangerous?
Their FSM values are added before calculating FSC. Several modest corrections can therefore produce a material reduction in corrected GM, particularly in a light displacement condition.
How do centreline bulkheads reduce free surface effect?
A complete longitudinal centreline division halves the available breadth in two equal rectangular compartments. Because breadth is cubed, their combined ideal inertia is one-quarter of the undivided value, subject to no free communication and the stated geometric assumptions.
What is the difference between free surface and free communication?
Free surface concerns liquid shifting inside an enclosed slack space. Free communication involves a flow path to the sea or another space, so liquid quantity, buoyancy, displacement, trim and watertight boundaries may also change.
Can free surface affect the GZ curve?
Yes. FSC reduces corrected GM and therefore lowers the initial slope of the GZ curve. At larger heel angles, tank geometry and liquid behaviour become more complex and require approved vessel-specific analysis.
Which displacement belongs in the FSC formula?
Use total vessel displacement for the loading condition being assessed, not deadweight, cargo weight or tank contents alone. All source values must refer to a consistent condition and unit system.
Final Remarks
Free surface effect is created by liquid movement, not by the tank merely containing liquid. A slack surface allows the liquid centre of gravity to shift toward the low side as the ship heels, reducing the vessel’s effective righting response.
FSM combines liquid density with free-surface geometry. In a rectangular tank, the cubic breadth term makes transverse width especially influential. Total FSC is obtained by adding all relevant tank moments and dividing by displacement. That correction must be applied to solid GM or represented through virtual KG.
Several slack tanks can create a significant cumulative correction, and transfer operations can produce temporary stages that are more limiting than the start or finish. Consumption of low fuel or ballast can add a second penalty by raising actual KG while also creating a free surface.
Rectangular formulas are useful for education and preliminary estimates, but they do not reproduce every real tank. Approved free-surface tables, vessel-specific loading software, stability booklets and operating procedures remain controlling for the actual ship.
References and Further Reading
- International Maritime Organization, Ship Design and Stability and the International Code on Intact Stability, 2008.
- UK Maritime and Coastguard Agency, Instructions to Surveyors – Intact Stability.
- Transport Canada, Stability Study Guide for Applicants to the Fourth-Class Engineer Certificate.
- Transport Canada, Adequate Stability and Safety Guidelines for Fishing Vessels.