Learning objectives
By the end of this chapter you will be able to:
- define light displacement, load displacement and deadweight, and state the relationship DWT = ∆ − ∆Light;
- use the symbols of the MCA examination formula sheet: ∆ for displacement and ∇ for volume of displacement;
- read a deadweight scale and explain how it is built from the hydrostatic table;
- distinguish total deadweight from cargo deadweight, allowing for fuel, fresh water, stores and the constant;
- calculate cargo loaded or discharged from the change of displacement between two draughts;
- define stowage factor and broken stowage, and calculate the mass of cargo a compartment can take;
- decide whether a cargo is limited by deadweight or limited by space.
Chapter 1 established that a floating ship displaces her own weight of water, so if we know the draught we know the weight. This chapter turns that physics into the daily commercial arithmetic of a working ship: how much can she lift, how much has she loaded, and will the cargo run out of weight or run out of space first?
2.1 Light displacement, load displacement and deadweight
The light displacement, written ∆Light, is the mass of the ship as the builders hand her over: hull, machinery, spare parts and permanent equipment, with water in the boilers, condensers and pipework at working level, but with no cargo, fuel or lubricating oil in tanks, ballast, fresh water, stores or crew on board. For MV Ninja, ∆Light = 4950 t. It is established at the inclining experiment, which we meet in Chapter 10, and it changes very little through a ship's life.
The load displacement is her displacement when floating at the summer load line in salt water: for MV Ninja, 30456 t. It is a mass, not a draught: the same 30456 t floats her at 9.600 m in salt water and deeper in fresh water. Between the two lies everything she can carry. The deadweight (DWT) is the difference between the displacement at any draught and the light displacement:
The summer deadweight, the figure quoted whenever a ship is described as, say, a 25000 tonner, is the deadweight at the summer load line: 30456 − 4950 = 25506 t for MV Ninja.
2.2 The displacement equation in MCA symbols
From this chapter onward the series uses the symbols of the MCA examination formula sheet, so that what you practise here is exactly what you will write in the examination room. Displacement is written ∆ (capital delta) and the volume of displacement is written ∇ (called nabla, or del). In Chapter 1, while the ideas were new, we wrote these as W and V; the physics is unchanged, only the dress is more formal. The fundamental equation of flotation becomes:
| Symbol | Meaning | Units |
|---|---|---|
| ρ | density (= mass ÷ volume) | t/m³ |
| ∆ | displacement | t |
| ∇ | volume of displacement | m³ |
| ∆Light | light displacement | t |
| DWT | deadweight | t |
| w | a weight loaded, discharged or shifted | t |
| RD | relative density (= ρSubstance ÷ ρFW) | none |
Key point
The examiners accept any symbols that are used clearly and consistently, but adopting the formula sheet symbols from the start means the sheet in front of you in the examination reads like your own notes. Every formula in this series that has a counterpart on the MCA sheet is quoted in the sheet's own symbols.MV Ninja floats on an even keel at a draught of 7.00 m in salt water. Find (a) her displacement, (b) her volume of displacement and (c) her deadweight at this draught.
(a) From the hydrostatic table at 7.00 m: ∆ = 21415 t
(b) ∆ = ∇ × ρ, so ∇ = ∆ ÷ ρ = 21415 ÷ 1.025 = 20893 m³
(c) DWT = ∆ − ∆Light = 21415 − 4950 = 16465 t
2.3 The hydrostatic table and the deadweight scale
The hydrostatic table answers the question one row at a time; the deadweight scale answers it at a glance. It is nothing more than the hydrostatic table redrawn as parallel vertical scales, draught alongside displacement alongside deadweight, with the light displacement already subtracted from the third column. Lay a ruler horizontally across the scale at the ship's draught and every figure you need stands on the line. Printed deadweight scales usually also carry the TPC and the load line marks; we add the TPC to our toolkit in Chapter 4. The scale is drawn for salt water and must be read that way: a draught fixes the underwater volume, not the mass, so at 7.00 m the same 20893 m³ supports 21415 t in salt water but only 20893 t in fresh water, and the deadweight on board is then 15943 t, not 16465 t.
Interactive: the deadweight scale explorer
Slide the draught and read the whole line of the scale at once, exactly as a ruler across the printed scale would.
2.4 What the deadweight is made of
Deadweight is not all cargo. Out of the total must come the fuel and diesel oil for the voyage, the fresh water, the stores and provisions, the crew and their effects, and the constant, that quietly accumulating tonnage of spare gear, mud in the ballast tanks, paint and sundries that every ship carries and every draught survey rediscovers. What remains is the cargo deadweight, the figure the charterer actually pays for.
MV Ninja is to load to her summer marks. On sailing she will have on board 850 t of fuel oil, 180 t of fresh water and 95 t of stores, and her constant is reckoned at 120 t. Find the cargo deadweight available.
| Summer deadweight | 25506 t |
| less fuel oil | 850 t |
| less fresh water | 180 t |
| less stores | 95 t |
| less constant | 120 t |
| Cargo deadweight | 24261 t |
2.5 Cargo calculations by difference of displacement
Because one draught fixes one displacement, the weight taken on board or put ashore between two moments is simply the difference between the two displacements, adjusted for anything else that came or went in the meantime. This is the principle of the draught survey, which Volume Two develops in full; here we practise it at tabulated draughts.
MV Ninja arrives in port at an even keel draught of 5.00 m in salt water. She sails at 8.60 m, also in salt water. While in port she received 320 t of bunkers and 40 t of fresh water, and consumed 45 t of fuel and water at the berth. Find the cargo loaded.
| Displacement on sailing (8.60 m) | 26941 t |
| Displacement on arrival (5.00 m) | 14798 t |
| Total weight received | 12143 t |
| less bunkers received | 320 t |
| less fresh water received | 40 t |
| add fuel and water consumed in port | 45 t |
| Cargo loaded | 11828 t |
The consumption is added back because it left the ship during the stay: had nothing been burned, the sailing draught would have been deeper still for the same cargo.
MV Ninja lies at 8.00 m in salt water and is to complete loading to her summer displacement. Before sailing she must also lift 400 t of bunkers. How much more cargo can she load?
| Summer displacement | 30456 t |
| Present displacement (8.00 m) | 24850 t |
| Total deadweight remaining | 5606 t |
| less bunkers still to lift | 400 t |
| Cargo still to load | 5206 t |
Interactive: the cargo loaded calculator
Enter the arrival and departure draughts (salt water, even keel, 2.60 m to 10.40 m) and the port figures. The calculator interpolates the MV Ninja table and casts the same ledger as Worked example 2.3.
2.6 Stowage factor and broken stowage
Weight is only half of the loading problem; the other half is space. The stowage factor (SF) of a cargo is the volume, in cubic metres, occupied by one tonne of it as stowed, including the unavoidable spaces within the stow itself:
Iron ore stows at around 0.40 m³/t, grain at about 1.30, and baled wool at 3.00 or more. The reciprocal relationship matters: a hold of given volume holds volume ÷ SF tonnes. Dense cargoes exhaust the deadweight long before the holds are full; light, bulky cargoes fill the holds while deadweight goes begging.
For packaged cargo there is a further loss. Broken stowage is the space that cannot be filled: the voids between packages, around frames and brackets, and at the turn of the bilge, expressed as a percentage of the capacity of the compartment. Bagged and baled cargoes commonly lose 8 to 12 per cent this way; a bulk cargo that flows, such as grain, loses essentially nothing.
No.3 hold of MV Ninja has a grain capacity of 6549 m³. How many tonnes of wheat of SF 1.30 m³/t will it hold?
mass = volume ÷ SF = 6549 ÷ 1.30 = 5037.7 t
Grain flows to fill the space, so no allowance for broken stowage is made; that is precisely why grain capacities are tabulated separately from bale capacities in the data booklet.
No.1 hold of MV Ninja has a bale capacity of 5553 m³. It is to be filled with baled goods of SF 1.90 m³/t, allowing 8 per cent broken stowage. Find the mass of cargo the hold will take.
usable volume = 5553 × (1 − 0.08) = 5108.8 m³
mass = usable volume ÷ SF = 5108.8 ÷ 1.90 = 2688.8 t
A light cargo, the smaller bale capacity and 8 per cent broken stowage together mean this hold takes 2689 t against 5854 ÷ 1.30 = 4503 t of wheat on its grain capacity, about 60 per cent of that tonnage.
Interactive: the stowage factor calculator
Choose a compartment of MV Ninja, a stowage factor and a broken stowage allowance, and see what the space will take.
Chapter summary
- ∆Light is the ship empty as built; DWT = ∆ − ∆Light; the summer deadweight is the DWT at the summer load line in salt water.
- The series now uses the MCA formula sheet symbols: ∆ = ∇ × ρ.
- A deadweight scale is the hydrostatic table drawn as parallel scales with ∆Light subtracted.
- Cargo deadweight = total deadweight less fuel, water, stores and the constant.
- Cargo loaded or discharged = change of displacement, adjusted for other weights received or consumed.
- SF = volume ÷ mass. Cargo a space will take = usable volume ÷ SF, where usable volume allows for broken stowage.
Self test questions
Work every question with pencil, paper and the MV Ninja data booklet before answering. All draughts are even keel in salt water and fall on tabulated values. Your score appears in the bar below.