# Coefficients of Form

Coefficients of form or Coefficients of fineness are dimensionless numbers, which describe about the fineness of the hull and overall shape characteristics of a vessel.

The coefficients are nothing but the ratios of areas or volumes for the actual hull form of the vessel compared to prisms or rectangles governed by the vessel’s length, breadth, and draft.

The length and breadth on the waterline as well as draft changes with displacement, coefficients of form also vary accordingly with displacement. The most often used dimension is the Length between perpendiculars (LBP) , although some designers prefer length on the waterline. Coefficients of form are used to facilitate the area and volume calculations for stability or strength analysis of the vessel. As hull form of a ship approaches that of a rectangular barge, the coefficients approach their maximum value of 1.0.

Coefficients of form are broadly classified into two categories:-

Area Coefficients:

1. Midship Coefficient (CM)
2. Waterplane coefficient (CWP)

Volume Coefficients:

1. Block Coefficient (CB)
2. Prismatic Coefficient (CP)

• Longitudinal Prismatic Coefficient (CPL)

• Vertical Prismatic Coefficient(CPV)

1. Volumetric Coefficient (CV)

1. Midship Coefficient (CM):-

The ratio of the area of the Midship Section till the Summer Load line (SLL) or Designated water line(DWL) and area of a circumscribing or confined rectangle whose sides are equal to the draught and breadth of the vessel is known as Midship Coefficient.

𝐂𝐌 = area of the Midship Section till SLL or DWL/ area of a circumscribing rectangle with sides equal to B & T

𝐂𝐌 = (𝐀𝐌/𝐁∗𝐓)

Values of CM may range between 0.75 to 0.995 for normal ships. There are some exceptions in which vessels have been built with bulges or blisters below the design waterline. If B is taken at the Summer Load Line (SLL), then CM may be greater than unity on such vessels.

2. Waterplane Coefficient (𝐂𝐖𝐏) or Coefficient of fineness:-

It is the ratio of the area of the waterplane taken at SLL or DWL and area of a circumscribing rectangle whose sides are equal to the Length and the breadth.

𝐂𝐖𝐏 =area of the waterplane taken at SLL or DWL/area of a circumscribing rectangle with sides equal to L & B

𝐂𝐖𝐏 = (𝐀𝐖𝐏/𝐋 ∗𝐁)

The values of CWP at the DWL range from about 0.65 to 0.95, depending upon type of ship, speed, and other factors.

3. Block Coefficient (CB):-

It is the ratio of the volumetric displacement of the ship and the volume of the circumscribing rectangular prism with dimensions Length, Breadth and Draft of the ship.

𝐂𝐁 = volumetric displacement of the ship/volume of the circumscribing rectangular prism with sides L, B & T.

𝐂𝐁 =  (∇ /𝐋 ∗𝐁 ∗𝐓)

The block coefficient indicates whether the form is full or fine and whether the waterlines will have large angles of inclination to the middle line plane at the ends.

Values of CB at design displacement may vary between 0.36 for a fine high-speed vessel to about 0.92 for a slow and full Great Lakes bulk carrier. Relatively high block coefficient is favourable for a slow ship as the vessel’s resistance is predominately due to frictional forces.

A high value is favoured for cargo carrying vessel and is frequently determined by the use of length of parallel middle body, approximately 15–20 per cent of the total length of the ship.

4. Prismatic Coefficient (CP): –

•  Longitudinal Prismatic Coefficient (CP or CPL):-

It is the ratio of the volumetric displacement of the ship and the volume of the prism whose length equals the length of the ship and whose cross section equals the midship section area till SLL remaining constant throughout the length of the ship.

𝐂𝐏𝐋 = Volumetric displacement of the ship/Volume of the prism with ( 𝐀Φ∗𝐋)

𝐂𝐏𝐋 = (∇ /𝐀𝐌∗𝐋)

Normal range of values lies between 0.50 and 0.90. A vessel with a low value of 𝐂𝐏𝐋 (or CB) is said to have a fine hull form, while one with a high value of 𝐂𝐏𝐋 has a full hull form.

• Vertical Prismatic Coefficient (CPV): –

It is the ratio of the volumetric displacement of the ship and the volume of the prism whose depth equals the draft of the ship and whose vertical cross section equals the waterplane area taken at SLL remaining constant throughout the draft of the ship.

𝐂𝐏𝐕 = Volumetric displacement of the ship/Volume of the prism ( 𝐀𝐖𝐏∗𝐓)

𝐂𝐏𝐕 = (∇ /𝐀𝐖𝐏∗𝐓)

5. Volumetric Coefficient (𝐂𝐕) or fatness ratio: –

The volume of displacement divided by the cube of one tenth of the vessel’s length is known as the Volumetric coefficient (or fatness ratio).

𝐂𝐕 = [𝛁/(𝐋/𝟏𝟎)3]

Ships with low volumetric coefficients can be said to be “thin”, while those with a high coefficient are said to be “fat.”

Values of the volumetric coefficient range from about 1.0 for light ships and long ships like destroyer vessels , to about 15 for short heavy ships like trawler vessels.