A chief mate's stability sign-off has to survive three separate tests — intact, after credible damage, and sitting on the blocks — and this chapter works the numbers behind each one.
Picture the moment a compartment holes and fills. The volume that floods stops doing the one job it was doing for the ship: displacing water and contributing buoyancy. Nothing else about the ship's weight has changed — no cargo was added, KG has not moved — so the only way the ship can find the buoyancy that compartment used to provide is by sinking bodily and trimming until the remaining intact volume, deeper in the water, makes up the shortfall. That is the whole physical picture behind the lost buoyancy method, and it is worth holding onto, because it is also why the method is the one built into every loading instrument's damage module: it tracks a real volume of water finding its own level, not an abstract weight added at a point.
Permeability (μ) is what stops the calculation being a simple hold-volume sum. A cargo hold, a machinery space and a stores room do not fill completely even when fully open to the sea, because structure, machinery, stowed cargo or stores are already occupying part of the volume and excluding water from it. The figures an officer works with are around 0.95 for an empty hold, 0.85 for a machinery space (engines, floor plates, foundations all displace water that would otherwise flood in), and 0.60 for a stores space packed with dense stowage. A part-loaded hold sits somewhere between the empty and full figures, and a mixed zone — say a hold with an adjoining void — should be split and permeability applied space by space, not averaged across the whole flooded zone as one number.
The denominator is doing real work: the flooded compartment's own waterplane, times its permeability, comes off the intact waterplane too, because that portion is no longer resisting further sinkage the way solid hull is. Drop that term and the sinkage comes out too small — a genuinely easy way to understate how deep the ship actually settles.
Floodable length is the length of ship, centred on a given point, that can flood at the relevant permeability without the resulting waterline touching the margin line — a line set a nominal distance below the bulkhead deck at the side, roughly 76 mm, chosen as the last safe clearance before openings in the deck start taking on water. Plot that permissible length against position along the ship and you get the curve of floodable lengths, and dividing it by a required factor of subdivision gives the maximum spacing the transverse bulkheads are actually allowed. In other words, the bulkhead plan on the midship section drawing is not an arbitrary layout — it is floodable length worked backwards into steel.
That deterministic picture — one compartment, or two adjacent compartments, assumed flooded and checked against the margin line — has largely been superseded by a probabilistic method for the ships it applies to. Instead of testing a small number of prescribed damage cases, the rules consider a wide range of possible damage extents along the ship's length, weight each one by how likely it is (a p-factor) and by how likely the ship is to survive it (an s-factor, built from residual GM, range and area of the GZ curve in that damaged condition), and sum the products into a single attained subdivision index, A. That figure is compared against a required index R, which itself depends on the ship's type and size.
None of that arithmetic is something an officer redoes at sea — A and R are fixed at the design and approval stage, and they live in the ship's damage stability information rather than in a day-to-day calculation. What matters operationally is what falls out of it: the approved list of compartments (or groups of compartments) the ship is certified to survive flooded, and the loading limits that keep an actual voyage condition inside the envelope those calculations assumed.
Free surface correction is not a new idea by the time an officer reaches management level, but its size changes completely once the free surface in question is a damaged, partly flooded compartment rather than a designed tank. A ballast or fuel tank is subdivided precisely to keep its free surface — and so its FSC — small. A holed machinery space or cargo hold has no such subdivision: at a partial level of flooding it behaves as one very large, very shallow tank, and the second moment of area of that free surface, i, can be enormous compared with anything in the tank sounding tables.
Permeability enters here a second time. The second moment used in the FSC should reflect only the area that can actually carry a moving free surface — broadly, the geometric second moment of the compartment reduced for its permeability — rather than the full geometric figure for the whole space. Using the full geometric i overstates the loss slightly; using no permeability adjustment at all, or worse, omitting the free surface term completely because "the compartment is flooded, not carrying liquid cargo," is the error that actually costs marks and, in a real casualty, GM.
Damage takes GM down in two separate ways at once: the extra draught and trim from the flooding itself, and the free surface of the water now sitting inside the ship. A solid GM that looks perfectly adequate on its own can be eaten into badly once that second effect is included — which is exactly why the damage stability criteria ask for residual GM, range of stability and area under the GZ curve all to be positive and adequate, not GM alone.
A ship's hull girder is a beam, and like any beam it has a still water bending moment (SWBM) and a still water shear force (SF) at every point along its length, set by how weight and buoyancy are distributed fore and aft. Class assigns limiting values for both, in hogging and in sagging, and keeping inside them for the vessel's actual loading condition is a condition of class, not a piece of good practice left to the master's judgement. The underlying relationship is the ordinary strength-of-materials one — stress from bending is moment over section modulus — which is why the limits are expressed as bending moment and shear force figures the loading instrument can check directly against the ship's actual section properties.
The detail that catches people out is that the ship's final, departure condition is not the only condition that has to pass. During a loading or discharging sequence, particularly on a bulk carrier working a homogeneous cargo through alternate holds, or a tanker discharging parcels in a particular order, an intermediate stage can produce a higher bending moment or shear force than either the fully loaded or fully empty condition either side of it. The loading instrument is there precisely to be run stage by stage through the intended sequence, not just typed in once for the finished condition, and a sequence that fails at an intermediate stage is a genuine non-compliance even if the departure figures look comfortable.
Shear force limits are usually tightest at or near bulkhead positions, where local strength and the change in loading either side of the bulkhead combine, so a sequence check needs to look at the whole envelope the instrument produces, not just the single worst number it reports. Treating the loading instrument as a report generated after the fact, rather than a limit checked before and during loading, is the underlying mistake behind most strength-related non-compliances.
Entering dry dock removes stability in a way that has nothing to do with weight distribution and everything to do with where the ship is supported. As the water level falls, or the ship is lowered onto the blocks, the keel blocks begin to take part of the ship's weight, developing an upward reaction, P, at the keel. That upthrust behaves, for stability purposes, as if a weight P had been effectively removed from support at the ship's lowest point — and removing support at a point that low has an out-of-proportion effect on the ship's righting arm, which is why the loss is measured against KM, the height of the metacentre, rather than against the ship's own KG.
P itself is zero the instant the keel first touches the blocks and grows as the tide falls, reaching its maximum — the critical instant — at the point where the blocks have taken up the trimming moment that was previously being carried by the water, and the ship is, in effect, about to come level along the keel. That is why trim drives P: the practical formula used to estimate it ties P directly to the trim still to be removed and the distance between the centre of flotation and the point where the ship first takes the blocks.
The operational precautions all fall directly out of this. Minimum trim on entering dock keeps P, and therefore the virtual loss of GM, as small as possible. Entering upright matters because the ship's remaining GM at the critical instant is at its lowest point of the whole docking evolution, and any list combined with a reduced GM is a far more dangerous combination than either on its own. Slack tanks are avoided for the same reason free surface matters everywhere else — they would stack an avoidable FSC loss on top of a loss that is already unavoidable.
At management level, "the stability is fine" has to mean something more specific than a single GM figure on a printout. Approving a loading condition means confirming, for that specific condition, that intact stability criteria are met (GM, the area and range under the GZ curve, the angle of maximum GZ), that any applicable damage stability requirement is still satisfied for the compartments the ship is certified against, that longitudinal strength stays within the class SWBM and SF envelope at every stage of getting to that condition, and — if docking is imminent — that the docking condition itself has been separately checked.
In practice that is a sequence, not a single lookup. Run the intended loading or discharging plan through the instrument stage by stage and confirm the strength envelope holds throughout, not only at the end. Confirm the final GM against the required criteria, remembering to include free surface from every slack tank actually expected to be slack, not the tank plan's nominal arrangement. If the voyage or the cargo operation puts the ship within a scenario the damage stability information covers, check that the condition sits inside it. If a docking follows, check trim and tank status against the docking calculation on its own terms, since it is a different limit with a different governing formula.
Every one of the common failure points already covered in this topic — checking only the final condition, applying one permeability figure across a whole flooded zone, forgetting free surface in a damaged space, using KG instead of KM in a docking calculation — is really the same failure wearing different clothes: treating one convenient number as the whole picture. The habit that actually protects marks, and protects the ship, is checking each limit on its own terms, at every stage a condition passes through.
The three examples below carry the chapter's formulas through complete, numerical problems — a lost buoyancy sinkage against the margin line, a docking GM loss, and a residual GM check that combines lost buoyancy with free surface in the damaged space.
A bulk carrier grounds and holes an empty cargo hold. At the waterline at the time of flooding, the intact ship's waterplane area is 1995 m², the hold's own waterplane area is 100 m², and the volume of the hold below the original waterline is 1200 m³. The hold is empty (μ = 0.95). Before the incident, the freeboard from the waterline up to the margin line was 1.05 m. Find the bodily sinkage and state whether the margin line remains clear.
A_W (intact waterplane area) = 1995 m² a (waterplane area of the hold) = 100 m² v (volume of hold below original WL) = 1200 m³ μ (permeability, empty hold) = 0.95 Freeboard, WL to margin line (before flooding) = 1.05 m
Find the bodily sinkage and state whether the margin line remains clear
Find the effective loss of waterplane area.
Only the permeable fraction of the hold's waterplane stops contributing buoyancy as the ship settles, so the intact ship is left working with A_W − a·μ, not the full A_W − a.
Apply the lost buoyancy formula.
The flooded volume v, times its permeability, is the buoyancy the intact hull must now find elsewhere; dividing by the reduced waterplane gives the bodily sinkage.
Check the margin line.
Treating this as pure sinkage (the hold's centroid close enough to the centre of flotation to ignore trim for this check), the sinkage comes straight off the freeboard that was in hand.
AnswerSinkage ≈ 0.60 m; 0.45 m of freeboard remains to the margin line, so the margin line stays clear.
The trap: dividing v by the full A_W instead of the reduced waterplane (A_W − a·μ) — that understates the sinkage and can turn a fail into a pass on paper.
A vessel of 12,000 t displacement is about to dock with 30 cm of trim by the stern still to be taken up. MCT1cm is 150 t·m/cm and the centre of flotation is 60 m from the after perpendicular. KM at the docking draft is 8.00 m and the solid GM (already corrected for free surface) is 0.65 m. Find the upthrust P at the critical instant, the virtual loss of GM, and the GM the ship actually has at that instant.
Δ (displacement) = 12000 t Trim still to be taken up = 30 cm MCT1cm = 150 t·m/cm Distance, F to AP = 60 m KM (docking draft) = 8.00 m Solid GM (FSC already applied) = 0.65 m
Find the upthrust P at the critical instant, the virtual loss of GM, and the GM the ship actually has at that instant
Find the upthrust P at the critical instant.
As the tide falls the blocks take an increasing share of the ship's weight; P grows until the moment it removes equals the moment that was holding that last 30 cm of trim.
Convert P into a virtual loss of GM.
The upthrust acts at the keel, the lowest point in the ship, so its effect on stability is measured against KM, not KG — using KG here is the standard mark-losing slip.
Apply the loss to the GM in hand.
AnswerP ≈ 75 t; virtual loss of GM ≈ 0.05 m; GM at the critical instant ≈ 0.60 m — positive, so the docking may proceed as planned.
The trap: using KG instead of KM in P·KM/Δ — the loss is measured to the metacentre, not to the ship's own centre of gravity.
Following damage, a machinery space (μ = 0.85) is partly flooded. The ship's displacement is unchanged at 10,250 t (lost buoyancy, not added weight). At the resulting waterline, KM is 9.20 m and KG is 8.30 m. The free surface of the floodwater in the machinery space, its second moment already reduced for permeability, is 3000 m⁴. Find the residual (fluid) GM.
Δ (displacement, unchanged) = 10250 t KM (at damage waterline) = 9.20 m KG (solid, unaffected by flooding) = 8.30 m i (permeability-adjusted second moment of flooded free surface) = 3000 m⁴ ρ (seawater) = 1.025 t/m³
Find the residual (fluid) GM
Start from the solid GM.
Lost buoyancy adds no weight, so KG does not move; the solid GM is simply KM less KG at the new, deeper waterline.
Add the free surface loss.
A partly flooded machinery space behaves as one very large, very shallow tank — the correction uses the permeability-adjusted second moment i, not the full geometric one, but it is still large next to an intact ballast tank's.
Take the FSC off the solid GM.
The result is the fluid GM the ship actually has in the damaged condition.
AnswerGM(fluid) ≈ 0.60 m — positive, so the immediate check is satisfied, though the residual range and area under the GZ curve still need confirming against the ship's damage stability information before the condition is accepted.
The trap: quoting the solid GM (0.90 m) as the damaged-condition figure — the free surface in a slack, partly flooded space is exactly the loss the rules are testing for.
Sinkage = v·μ/(A_W − a·μ)Lost buoyancy method — bodily sinkage in a flooded compartmentμ: 0.95 hold (empty), 0.85 machinery, 0.60 storesPermeability by use of spaceMargin line ≈ 76 mm below the bulkhead deck at sideFloodable length reference lineAttained index A ≥ Required index RProbabilistic damage stability, summed over damage casesVirtual loss of GM = P·KM/ΔDocking — measured to KM, not KGP = MCT1cm × trim / (dist. F to point of contact)Upthrust at the critical instant in dockFSC = i·ρ/ΔFree surface; use the permeability-adjusted i in a flooded spaceσ = M/ZBasis of the class SWBM/SF limits checked by the loading instrumentGM(fluid) = KM − KG − ΣFSCCheck at every loading stage, not only the final conditionResidual GM, range and GZ area must all be positive/adequate after damageDamage stability criteria — describe qualitatively unless an exact figure is certain