Learning objectives
By the end of this chapter you will be able to:
- define the principal dimensions of a ship: LOA, LBP, breadth moulded, depth moulded, draught and freeboard;
- locate the forward and after perpendiculars and amidships;
- define the block coefficient and use ∇ = L × B × d × CB to find volume, displacement or draught;
- define the waterplane area coefficient and use AW = L × B × CW;
- define the midship section and prismatic coefficients and relate them by CP = CB ÷ CM;
- explain how the coefficients of form vary with draught and between ship types;
- calculate MV Ninja's own coefficients from her hydrostatic data.
Chapters 1 and 2 treated the underwater volume ∇ as a number to be looked up or divided out. This chapter looks at its shape. A handful of dimensionless ratios, the coefficients of form, capture how full or how fine a hull is, and they connect the ship's main dimensions to her volume and waterplane in the two formulas this chapter takes from the MCA sheet. They are also the language in which naval architects compare a racing yacht with a crude carrier.
3.1 The principal dimensions
The length overall (LOA) is the extreme length of the hull and is what matters to a pilot judging a lock. Calculations, however, use the length between perpendiculars (LBP). The after perpendicular (AP) is a vertical line through the rudder stock; the forward perpendicular (FP) passes through the point where the summer load waterline crosses the forward side of the stem. Amidships lies halfway between them. For MV Ninja, LOA = 152.00 m and LBP = 148.00 m.
Across the ship, the breadth moulded (B) is the greatest breadth measured to the inside of the shell plating: 24.20 m for MV Ninja. Vertically, the depth moulded (D) runs from the base line to the deck at side, 13.50 m; the draught (d) from the keel to the waterline; and the freeboard (f) from the waterline up to the deck at side. At any waterline the moulded quantities add up: depth = draught + freeboard. The freeboard assigned under the load line rules of Chapter 5 is measured a little differently, from the upper edge of the deck line, which is marked level with the top of the freeboard deck plating, so it exceeds D − d by the thickness of that plating: MV Ninja's summer freeboard is 3920 mm, against a moulded 13.50 − 9.60 = 3.90 m, and the 20 mm difference is the deck plating.
3.2 The block coefficient
Imagine a rectangular box built exactly around the underwater body: L long, B wide and d deep. The hull cannot fill the box, because it must taper to a bow and a stern and round off at the bilge. The fraction of the box the hull actually fills is the block coefficient, CB. Rearranged into the form on the MCA sheet:
A tug or a yacht, built for speed and seakeeping, may have a CB of 0.50: half box, half water. A large tanker or bulk carrier, built to carry, may reach 0.85 or more. Combined with ∆ = ∇ × ρ from Chapter 2, the block coefficient lets us estimate a ship's displacement from nothing more than her main dimensions, which is exactly how quick feasibility sums are done.
Find the block coefficient of MV Ninja at her summer draught (LBP 148.00 m, B 24.20 m, d 9.60 m, summer displacement 30456 t in salt water).
∇ = ∆ ÷ ρ = 30456 ÷ 1.025 = 29713 m³
∇ = L × B × d × CB, so CB = ∇ ÷ (L × B × d) = 29713 ÷ (148 × 24.20 × 9.60) = 29713 ÷ 34383.4 = 0.864
A thoroughly full form, as one expects of a bulk carrier.
A general cargo ship has LBP 120 m, breadth 18 m and block coefficient 0.750 at her load draught of 7.00 m. Find her load displacement in salt water.
∇ = L × B × d × CB = 120 × 18 × 7.00 × 0.750 = 11340 m³
∆ = ∇ × ρ = 11340 × 1.025 = 11623.5 t, quoted, as displacements are, to the nearest tonne: 11624 t
A vessel of LBP 100 m and breadth 16 m displaces 10455 t in salt water. Her block coefficient at this draught is 0.750. Find her draught.
∇ = ∆ ÷ ρ = 10455 ÷ 1.025 = 10200 m³
d = ∇ ÷ (L × B × CB) = 10200 ÷ (100 × 16 × 0.750) = 10200 ÷ 1200 = 8.50 m
3.3 The waterplane area coefficient
Slice the ship horizontally at the waterline and look down: the shape you see is the waterplane, and its area is AW. The waterplane area coefficient CW compares it with the circumscribing rectangle L × B:
The waterplane area is the working end of several tools we meet shortly: the tonnes per centimetre immersion of Chapter 4 comes directly from it, and much later the waterplane's shape governs the ship's resistance to heeling and trimming. Hold on to one physical picture: AW is the area of water the ship must push down through when a weight is loaded.
The hydrostatic table does not list the waterplane area directly, but it lists the TPC, and the two are the same information. A slab one centimetre thick over the whole waterplane has a volume of AW × 0.01 m³ and, in water of density ρ, a mass of AW × 0.01 × ρ tonnes; that mass is what sinks the ship one centimetre, so TPC = AW × ρ ÷ 100 and therefore AW = 100 × TPC ÷ ρ. Chapter 4 uses the first form; here we borrow the second, always with the salt water density, because the TPC column of the table is a salt water column.
At a draught of 6.00 m the hydrostatic table gives MV Ninja a TPC of 33.06 t in salt water. Find (a) her waterplane area and (b) her waterplane area coefficient at this draught.
(a) AW = 100 × TPC ÷ ρ = 100 × 33.06 ÷ 1.025 = 3225 m²
(b) L × B = 148 × 24.20 = 3581.6 m², so CW = AW ÷ (L × B) = 3225.4 ÷ 3581.6 = 0.901
Keep the 3225 m² in mind for Chapter 4, which turns it back into the TPC. Notice the effect of rounding when the calculation is reversed: 148 × 24.20 × 0.901 = 3227 m², two square metres more than we started with, because the unrounded coefficient is 0.9005 and 0.901 is slightly too large. Carry a fourth figure whenever a coefficient is an intermediate result rather than the answer.
3.4 The midship section and prismatic coefficients
Two further coefficients complete the family, and although they sit beyond the MCA formula sheet they appear throughout naval architecture. The midship section coefficient CM compares the immersed area of the midship section, AM, with the rectangle B × d around it. Merchant hulls are nearly rectangular amidships, so CM runs from about 0.98 to 0.995 for full ships. The prismatic coefficient CP compares the underwater volume with a prism formed by sliding that midship section along the length: it measures how the volume is distributed along the ship rather than how full the middle is. The three are locked together:
At her summer draught MV Ninja has a block coefficient of 0.864 (Worked example 3.1). The data booklet does not tabulate her midship section coefficient; take it as 0.995, a typical figure for a full bodied bulk carrier. Find (a) her immersed midship section area and (b) her prismatic coefficient.
(a) AM = CM × B × d = 0.995 × 24.20 × 9.60 = 231.2 m²
(b) CP = CB ÷ CM = 0.864 ÷ 0.995 = 0.868
CP is only a little above CB: with the midship section so nearly rectangular, almost all the fining of the hull happens at the ends. As a check, ∇ = CP × AM × L = 0.868 × 231.2 × 148 = 29700 m³, which agrees with the 29713 m³ of Worked example 3.1 within the rounding of the three figure coefficients.
3.5 How the coefficients change with draught
The coefficients are properties of a particular waterline, not of the ship for all time. Deep down at the bilge the hull is rounded and the ends are fine, so at light draughts the box is poorly filled; as the ship sinks deeper the wall sided middle body dominates and every coefficient climbs. The table below is computed directly from the MV Ninja hydrostatic table, and you can and should verify any line of it yourself:
| Draught (m) | ∆ (t) | ∇ (m³) | CB | TPC (t) | AW (m²) | CW |
|---|---|---|---|---|---|---|
| 2.60 | 7280 | 7102 | 0.763 | 30.30 | 2956 | 0.825 |
| 4.00 | 11608 | 11325 | 0.790 | 31.48 | 3071 | 0.857 |
| 6.00 | 18064 | 17623 | 0.820 | 33.06 | 3225 | 0.901 |
| 8.00 | 24850 | 24244 | 0.846 | 34.64 | 3380 | 0.944 |
| 9.60 | 30456 | 29713 | 0.864 | 35.28 | 3442 | 0.961 |
| 10.40 | 33293 | 32481 | 0.872 | 35.54 | 3467 | 0.968 |
This is the deeper reason hydrostatic tables exist at all. If the coefficients were constants, a single formula would replace the whole table; because they change with draught, the naval architect computes the hull properties waterline by waterline and tabulates them, and the officer interpolates. The ∇ column above is the salt water displacement divided by 1.025, and the AW column was recovered from the tabulated TPC by AW = 100 × TPC ÷ 1.025, a small preview of Chapter 4.
Using the MV Ninja hydrostatic table, find her block coefficient at a draught of 8.00 m (∆ = 24850 t in salt water), and compare it with the summer figure of 0.864.
∇ = 24850 ÷ 1.025 = 24244 m³
CB = ∇ ÷ (L × B × d) = 24244 ÷ (148 × 24.20 × 8.00) = 24244 ÷ 28652.8 = 0.846
Nearly two hundredths below the summer figure: the same ship is a measurably finer form at the shallower waterline.
Interactive: the MV Ninja hull form explorer
Slide the draught and watch CB and CW climb as the waterline rises into the fuller body of the ship. Every value is computed live from the hydrostatic table.
Interactive: the displacement from dimensions calculator
The naval architect's quick estimate: enter any ship's main dimensions, block coefficient and water density, and the calculator applies ∇ = L × B × d × CB followed by ∆ = ∇ × ρ.
Chapter summary
- LBP runs from the after perpendicular (rudder stock) to the forward perpendicular (where the summer waterline crosses the stem); LOA is the extreme length.
- Moulded dimensions are measured to the inside of the shell plating. Depth = draught + freeboard for the moulded quantities; the load line freeboard adds the deck plating thickness (MV Ninja: 3920 mm against 3.90 m).
- CB = ∇ ÷ (L × B × d); on the MCA sheet, ∇ = L × B × d × CB.
- AW = L × B × CW: the waterplane area drives sinkage and, next chapter, the TPC, through AW = 100 × TPC ÷ ρ.
- CM = AM ÷ (B × d), CP = ∇ ÷ (AM × L), and CP = CB ÷ CM.
- All the coefficients increase with draught, which is why hull properties are tabulated waterline by waterline.
Self test questions
Work each question with pencil and paper first. MV Ninja questions use LBP 148.00 m and B 24.20 m throughout. Your score appears in the bar below.