The lines plan is a set of drawings that defines the shape of the hull. It consists of three principal views:

• Sheer plan (profile): shows the hull as seen from the side. It contains the waterlines, the deck line, the sheer line and the profile of the stem and stern. The vertical positions of the waterlines and the longitudinal positions of the stations are shown.
• Half-breadth plan (plan view): shows the hull as seen from above, drawn for one half of the ship (port or starboard) because the hull is symmetrical. It contains the waterlines (horizontal sections) and the deck line, showing the half-breadths at each station.
• Body plan (end view): shows the transverse sections (stations) as seen from the bow and stern. The forward half of the stations is drawn on one side and the after half on the other. It contains the buttock lines and the diagonal lines.

The three views are drawn in projection so that every point on the hull appears in all three views, allowing the shape to be fully defined and faired. The lines plan is used to calculate hydrostatic data, displacement, and to construct the ship.

Length L = 120 m, breadth B = 15 m. Half-breadths at stations 0 to 8:

Station: 0 1 2 3 4 5 6 7 8

Half-breadth (m): 1.6 2.8 5.5 6.4 7.3 6.2 4.2 2.0 0

Number of intervals n = 8, so the common interval h = L/n = 120/8 = 15 m.

(i) Waterplane area.

Using Simpson's First Rule with 8 intervals (9 ordinates), multipliers 1,4,2,4,2,4,2,4,1:

Sum of products = 1x1.6 + 4x2.8 + 2x5.5 + 4x6.4 + 2x7.3 + 4x6.2 + 2x4.2 + 4x2.0 + 1x0

= 1.6 + 11.2 + 11.0 + 25.6 + 14.6 + 24.8 + 8.4 + 8.0 + 0 = 105.2.

Area of half waterplane = (h/3) x sum = (15/3) x 105.2 = 5 x 105.2 = 526 m2.

Waterplane area (full) = 2 x 526 = 1052 m2.

(ii) TPC in salt water.

TPC = (waterplane area x density of sea water)/100 = (1052 x 1.025)/100 = 1078.3/100 = 10.78 tonne/cm.

Answer: TPC = 10.78 tonne per cm.

(iii) Cw (waterplane area coefficient).

Cw = waterplane area / (L x B) = 1052 / (120 x 15) = 1052/1800 = 0.584.

Answer: Cw = 0.584.

(iv) LCF from midship.

The centre of flotation is the centroid of the waterplane. Using the first moment about midship (station 4):

Distances from station 4 (in intervals): station 0 = -4, 1 = -3, 2 = -2, 3 = -1, 4 = 0, 5 = +1, 6 = +2, 7 = +3, 8 = +4.

Moment = 1x1.6x(-4) + 4x2.8x(-3) + 2x5.5x(-2) + 4x6.4x(-1) + 2x7.3x0 + 4x6.2x(+1) + 2x4.2x(+2) + 4x2.0x(+3) + 1x0x(+4)

= -6.4 - 33.6 - 22.0 - 25.6 + 0 + 24.8 + 16.8 + 24.0 + 0 = -22.0.

Distance of centroid from midship = (moment/sum) x h = (-22.0/105.2) x 15 = -0.2092 x 15 = -3.14 m.

The negative sign means the LCF is forward of midship.

Answer: LCF is 3.14 m forward of midship.