Midship Coefficient (CM)

Midship Coefficient (C_M): Formula, Methods and Worked Example

The midship coefficient describes the fullness of a ship's immersed transverse section at midship. It compares the actual underwater sectional area with a rectangle having the same beam and draft at that section.

A section with a flat bottom and nearly vertical sides fills most of the reference rectangle and therefore has a high C_M. A rounded, V-shaped or strongly deadrised section occupies less of the rectangle and gives a lower value.

C_M describes one transverse section. It does not describe how volume is distributed over the full length of the vessel, and it should not be treated as a complete measure of hull fullness.

Midship coefficient formula

The midship coefficient is calculated from the immersed midship section area:

The numerator and denominator are both areas, so C_M has no unit. In the calculator, the beam and draft fields correspond to B_M and T_M.

The area, beam and draft must refer to the same section and loading condition. An area measured at one draft cannot be combined with a draft from another condition.

Worked example: direct calculation from midship area

Calculation methods available on this page

The calculator provides three ways to determine C_M. The suitable method depends on the information available.

1. Direct calculation from A_M, B and T

Use the direct method when the immersed midship section area is available from a body plan, hydrostatic model, sectional offsets or another reliable source. This method applies the defining equation directly.

2. Calculation from C_B and C_P

When the block coefficient and prismatic coefficient use compatible dimensional definitions, the hull coefficients are related by:

For example, if C_B = 0.752 and C_P = 0.800:

This gives the same result as the direct worked example. The comparison is valid because the coefficients have been selected using compatible area, length, beam and draft definitions.

3. Calculation from vertical half-breadth ordinates

When A_M is not already known, it can be calculated from half-breadths measured at equally spaced vertical levels between the keel and waterline.

For equally spaced levels, Simpson's one-third rule estimates the area under the half-section. The result is multiplied by two to obtain the full symmetric transverse area:

In this expression, h is the vertical spacing and y₀ to y_n are half-breadths measured from the centreline to one side of the section.

Simpson's one-third rule requires an even number of intervals, which means an odd number of vertical levels. The ordinates must be entered from keel to waterline in the order requested by the calculator.

• Use equal vertical spacing.
• Enter half-breadths rather than full breadths.
• Include the keel and waterline levels.
• Keep all ordinates in the same unit as the entered beam and draft.
• Check that no half-breadth exceeds the geometric half-beam without a clear reason.

Midship section and maximum transverse section

The midship section is the transverse section at the defined midship location. The maximum transverse section is the section having the greatest immersed area. They are often the same on conventional merchant ships, but this should not be assumed for every hull.

Hulls with unusual volume distribution, significant local shaping or a maximum section located away from midship may require separate treatment. In such cases, the maximum sectional-area coefficient and the midship coefficient are different quantities.

The direct mode on this page calculates the coefficient for the midship area entered by the user. The input area should therefore represent the intended midship section rather than an unrelated maximum section.

Estimating the section area from a simplified shape

The direct calculator includes simplified shape estimates for cases where A_M is not available. These models are useful for preliminary checks, but they do not reproduce the details of a real body plan.

A rectangle with rounded bilges represents a full section with vertical sides and curved lower corners. An elliptic approximation represents a more rounded section. The result can be useful during early dimensioning, but it should be replaced with an area derived from actual offsets or hull geometry when those data become available.

How to interpret C_M

A value close to 1.00 indicates that the immersed section is close to a rectangle over the stated beam and draft. Lower values indicate that more area has been removed by bottom rise, deadrise, bilge curvature, side flare or another departure from a rectangular section.

The ranges below are broad comparison values. They are not acceptance limits and should not replace vessel-specific geometry.

The ranges overlap because vessel type alone does not determine the coefficient. Draft, bilge form, section location, structural arrangement and design priorities also affect the result.

Effect of loading condition and draft

C_M is associated with a particular immersed section. As draft changes, both A_M and the reference rectangle B_M × T_M change. They do not necessarily change in the same proportion.

A coefficient calculated at the design draft should not automatically be applied to a ballast or partial-load condition. The current waterline may intersect a different part of the bilge or side shape, producing a different area ratio.

What the midship coefficient can and cannot show

C_M is useful when describing sectional fullness, checking preliminary dimensions and connecting block and prismatic coefficients. It is also used in several empirical hull-resistance and wetted-surface formulations.

It does not define the complete midship geometry. Two sections can have the same area and the same C_M while having different bilge radii, deadrise, chine arrangements or side flare.

C_M alone cannot determine:

• ship resistance or required propulsion power
• wetted surface area
• initial or large-angle stability
• cargo capacity
• structural strength
• seakeeping behaviour

Common input errors

• Entering full breadths where the ordinate method asks for half-breadths.
• Using unequally spaced vertical levels with an equal-spacing Simpson calculation.
• Entering an even number of levels for a pure Simpson one-third-rule dataset.
• Using moulded depth instead of the immersed draft.
• Combining an area and draft taken from different loading conditions.
• Using the maximum transverse area when the required value is the area at midship.
• Mixing metres, millimetres or feet within the same calculation.
• Using C_B and C_P values based on incompatible reference definitions.

Related NauticalSolver calculators

• Block Coefficient (C_B) Calculator — overall underwater-volume fullness.
• Prismatic Coefficient (C_P) Calculator — longitudinal volume distribution.
• Displacement Calculator — displacement mass and underwater volume.
• Wetted Surface Area Calculator — hull surface used in resistance estimates.

References

The terminology and formula used on this page follow standard naval architecture references:

1. Tupper, E. C., Introduction to Naval Architecture, 5th edition, Butterworth-Heinemann, 2013. Introduction to Naval Architecture
2. Rawson, K. J. and Tupper, E. C., Basic Ship Theory, Volume 1: Hydrostatics and Strength, 5th edition, Butterworth-Heinemann, 2001. Basic Ship Theory, Volume 1

NauticalSolver calculators are intended for preliminary engineering, study and independent checking. Use approved hydrostatic particulars, vessel drawings and vessel-specific calculation software for operational, contractual or statutory work.