Ship Form Coefficients

Ship form coefficient is used to describe hull shapes and compare different hulls. During design recommended values for each coefficient can be found in literature to give a starting point when designing the most optimum hull shape for the given application.

Symbols used in the following article

Table 1 - List of symbols
Symbol Description
L Length
B Beam
T Draft or draught
Ax Section area
Az Water plane area
∇ Displacement (displaced volume)
zk z-coordinate for the keel
zw z-coordinate for the waterline
xa The most aft part of the waterline
xf The most forward part of the waterline
⨂ Midship location
Coordinate system
x Ship longitudinal direction
y Transverse direction
z Vertical direction with 0 typically being at the keel

Waterplane Area Az

The waterplane area is determined by integrating y(x,z) from the aft part of the waterline xa to the forward part xf. The integral below is valid for symmetrical hulls with x being the axis of symmetry i.e. normal hull shapes.

Numerical integration can be done using this calculator or by using the trapezoidal rule or the Simpson's rule described in this article.

\begin{align} A_z(z)=2\int _{x_a(z)}^{x_f(z)} y(x,z)dx \end{align}

The geometric center of each waterplane is determined using the integral below. Similarly to above it is only valid for symmetrical hull shapes.

\begin{align} \overline{x}(z)& =\frac{2\int _{x_a(z)}^{x_f(z)} xy(x,z)dx}{A_z(z)}\\ \overline{y}(z)& =0 \end{align}
Waterplane example
Waterplane example

Section Area Ax

The section area Ax is the area of the transverse sections as illustrated by the example below. This can be at the same x-coordinates as the frames but not necessarily.

Exampleof midship section
Example of midship section

The section area at each x is for a symmetrical hull with x being the the axis of symmetry calculated as shown below.

\begin{align} A_x(x,z_w)=2\int _{z_k(x)}^{z_w(x)} y(x,z)dz \end{align}

The geometric center of each section is calculated using the equation below. \( \overline{y}(x,z_w) \) is zero as the hull is assumed symmetrical.

\begin{align} \overline{z}(x,z_w)& =\frac{2\int _{z_k(x)}^{z_w(x)} zy(x,z)dz}{A_x(x,z_w)}\\ \overline{y}(x,z_w)& =0 \end{align}

Below is an example of a cross-sectional area where the section area is calculated at difference stations (x-values) and at different drafts (zw).

Cross-sectional area at different drafts
Cross-sectional area at different drafts

∇ - Displacement

The displacement or volume of the underwater hull can either be determined by integrating the section areas (Ax(x,zw)) or by integrating the the water plane areas (Az(z))

\begin{align} \nabla(z_w)= \int _{x_a(z_w)}^{x_f(z_w)}A_x(x,z_w)dx =\int _{z_k}^{z_w}A_z(z)dz \end{align}

LCB and VCB (KB) - Center of Buoyancy

The longitudinal center of buoyancy (LCB) and the vertical center of buoyancy (VCB or KB) can either be calculated using the sectional areas and the geometric center of each section or by using the water plane area and the geometric center of each water plane. The result will be the same if done correctly. The transverse center of buoyancy (TCB) is equal zero for typical symmetrical hull shapes but can otherwise be calculated using the same approach as the one used to calculate LCB and VCB.

LCB - Longitudinal Center of Buoyancy

\begin{align} LCB(z_w)& =\frac{\int _{x_a(z_w)}^{x_f(z_w)}xA_x(x,z)dx}{\nabla (z_w)}\\ LCB(z_w)& =\frac{\int _{z_k}^{z_w}\overline{x}(z)A_z(z)dz}{\nabla (z_w)} \end{align}

VCB - The Vertical Center of Buoyancy

The vertical center of buoyancy is known as both VCB and KB with KB being named after keel to buoyancy.

\begin{align} VCB(z_w)& = \frac{\int _{x_a(z_w)}^{x_f(z_w)}\overline{z}(x,z_w)A_x(x,z)dx}{\nabla (z_w)}\\ VCB(z_w)& =\frac{\int _{z_k}^{z_w}zA_x(z)dx}{\nabla (z_w)} \end{align}

CB - Block Coefficient

The block coefficient is a dimensionless ratio between the displacement and the displacement of a rectangular with the size of LBT. It is a measure of the ships fullness e.g. a container vessel will have a block coefficient around 0.55 whereas a tanker will have a block coefficient at about 0.85.

\begin{align} C_B = \frac{\nabla }{LBT} \end{align}
Sketch illustrating the block coefficient C<sub>B</sub>
Sketch illustrating the block coefficient CB

CP - Prismatic Coefficient

The prismatic coefficient is the dimensionless ratio between the displacement and the volume of the prism with a transverse section equal to the midship section.

\begin{align} C_p = \frac{\nabla }{A_x(\bigotimes ,z_w)L} \end{align}

The prismatic coefficient can also be calculated from the block coefficient (CB) and the midship section coefficient (CM)

\begin{align} C_p = \frac{C_B}{C_M} \end{align}
Sketch illustrating the prismatic coefficient C<sub>p</sub>
Sketch illustrating the prismatic coefficient Cp

Cw - Water Plane Coefficient

The water plane coefficient is the ratio between the water plane area and the rectangle formed by LxB.

\begin{align} C_w = \frac{A_z(z_w)}{LB} \end{align}
Water plane area coefficient
Water plane area coefficient

CM - Midship Section Coefficient

The midship section coefficient. \( A_x(\bigotimes ,z_w) \) is the section area of the midship section.

\begin{align} C_M=\beta =\frac{A_x(\bigotimes ,z_w)}{BT} \end{align}
Midship section coefficient
Midship section coefficient

SR - Slenderness ratio

The slenderness ratio is a measure of the waterline length to the displacement.

\begin{align} SR = \frac{L}{\nabla ^{1/3}} \end{align}