Alongside stability and trim, every loading plan has to check longitudinal strength. A ship behaves like a long beam supported by water. When weight and buoyancy are not distributed the same way along the length, the hull carries shear force (SF) and bending moment (BM). This guide explains the concept in planner's terms, how to read the loading computer output and a simple worked example.
From weight distribution to shear force and bending moment
At every slice of the ship there are two forces: weight acting down (lightship, cargo, ballast, bunkers) and buoyancy acting up (set by the immersed hull shape). The difference between them per metre of length is the net load.
- Shear force at a point is the sum of the net load from the end of the ship up to that point. It usually peaks near transverse bulkheads or at the boundary between a heavy and a light hold.
- Bending moment at a point is the sum (integral) of the shear force from the end up to that point. It usually peaks around midships, right where the shear force passes through zero.
Two ships with the same total weight can have very different bending moments depending on where the cargo sits. That is why longitudinal strength cannot be judged from deadweight or mean draught alone.
Hogging vs sagging
| Condition | Typical cause | Hull shape | Deck | Bottom |
|---|---|---|---|---|
| Hogging | Heavy ends (fore/aft holds full, peak tanks filled), light midships | Middle rises | Tension | Compression |
| Sagging | Heavy midships (centre holds full), light ends | Middle drops | Compression | Tension |
A ship with the engine room aft tends to hog when empty or lightly loaded. Waves add to both: a crest amidships increases hogging, a trough amidships increases sagging. The sign convention (positive = hogging, or the other way round) differs between software packages, so always check the legend of your vessel's loading computer.
Seagoing limits, harbour limits and reading points
Strength limits are set by the designer and approved by class, following IACS requirements (for example UR S11 on longitudinal strength and UR S1 on loading manuals and loading instruments) and the rules of the classification society, such as BKI in Indonesia. There are two sets of still-water limits for moment and shear:
- Seagoing: tighter, because room is left for wave loads.
- Harbour: more generous, valid during cargo operations in sheltered water.
The limit is not a single number. The loading manual lists limits at specific reading points, usually at transverse bulkheads and selected frame numbers. Permissible BM is normally greatest amidships and falls off towards the ends. The loading computer shows the actual value at each point and its percentage of the limit. What the planner reads is the highest percentage, not an average.
A simple worked example (illustrative box barge)
Teaching model: a fictional box barge, 100 × 25 m, lightship 1,000 t spread evenly (10 t/m), cargo 2,000 t. Because it is box-shaped and on even keel, buoyancy is uniform: 3,000 t / 100 m = 30 t/m. Illustrative limits: SF 700 t (sea) / 900 t (harbour), BM 14,000 t·m (sea) / 18,000 t·m (harbour).
Case A, cargo in the middle (x = 30–70 m): cargo 2,000 / 40 = 50 t/m. Net load at the ends is 30 − 10 = +20 t/m (excess buoyancy) and in the middle 30 − 60 = −30 t/m. SF rises from 0 to 20 × 30 = 600 t at x = 30 m, then falls to 0 at x = 50 m. BM at x = 30 m = ½ × 30 × 600 = 9,000 t·m; amidships 9,000 + ½ × 20 × 600 = 15,000 t·m, sagging.
Case B, cargo at the ends (1,000 t at x = 0–20 m and 1,000 t at x = 80–100 m): the pattern reverses, with a maximum SF of 600 t at x = 20 m and a BM of 15,000 t·m amidships, but hogging.
Case C, cargo spread evenly (20 t/m): weight of 30 t/m equals buoyancy of 30 t/m, so SF and BM are zero everywhere.
Case D, case A plus 200 t of ballast at each end (x = 0–10 m and 90–100 m): displacement becomes 3,400 t and buoyancy 34 t/m. Maximum SF drops to 520 t and the midship BM to 11,000 t·m sagging.
| Case | Max SF (t) | % sea limit | Max BM (t·m) | Type | % sea limit | % harbour limit |
|---|---|---|---|---|---|---|
| A. Cargo amidships | 600 | 86% | 15,000 | Sagging | 107% | 83% |
| B. Cargo at ends | 600 | 86% | 15,000 | Hogging | 107% | 83% |
| C. Cargo spread evenly | 0 | 0% | 0 | — | 0% | 0% |
| D. A + end ballast | 520 | 74% | 11,000 | Sagging | 79% | 61% |
Case A is acceptable during cargo work (83% of the harbour limit), but the barge must not sail like that because the BM exceeds the seagoing limit. Adding end ballast (case D) brings the BM to 79%, at the cost of about 0.16 m more draught (400 t ÷ (100 × 25 × 1.025)). Check draught and trim effects with the draft & trim calculator.
For case A, the per-point output looks like this loading computer style table (the seagoing limit is kept the same at every point for simplicity; on a real ship it varies by point):
| Position (m from aft end) | SF (t) | % SF | BM (t·m) | % BM |
|---|---|---|---|---|
| 10 | 200 | 29% | 1,000 | 7% |
| 30 | 600 | 86% | 9,000 | 64% |
| 50 | 0 | 0% | 15,000 | 107% |
| 70 | −600 | 86% | 9,000 | 64% |
| 90 | −200 | 29% | 1,000 | 7% |
Real ships are not this tidy: the hull form makes buoyancy uneven, the engine room concentrates weight aft and trim shifts the buoyancy distribution. That is why real calculations are done frame by frame in the loading computer.
How cargo distribution and ballast change the picture
- Spread heavy cargo along the ship. Concentrating heavy containers or bulk in the centre holds drives sagging; concentrating it in the end holds drives hogging.
- Peak tank ballast reduces sagging but adds hogging. Ballast in midship tanks does the opposite.
- Intermediate stages count too. A loading or discharge sequence can pass through conditions worse than the start and end states. This is critical on bulk carriers, see bulk carrier loading sequence.
- Fuel and water consumption during the voyage changes the weight distribution, so check the arrival condition as well as departure.
- Container ships also have torsional moment limits because of their wide hatch openings.
The link between weight, centre of gravity and stability is covered in ship stability basics for planners, and you can work out a combined centre of gravity with the combined CoG calculator.
Where planning software fits
STOWAGE.ID Vessel Stowage estimates SF and BM from the planned cargo distribution and flags reading points that approach their limits, so the planner can move cargo early. It is a shore-side planning and decision-support tool. The official reference remains the vessel's class-approved loading computer and its loading manual or stability booklet. The difference is explained in loading computer vs shore-side planning. For bulk carriers, see also bulk stowage.