# Simpson's rules (ship stability)

Simpson's rules are a set of rules used in ship stability and naval architecture, to calculate the areas and volumes of irregular figures.[1]

Simpson's First Rule

## Simpson's 1st rule

Also known as the 1–4–1 rule (after the multipliers used ).[2]

${\displaystyle {\text{Area}}={\frac {h}{3}}(a+4b+c).}$

## Simpson's 2nd rule

Also known as the 1–3–3–1 rule, Simpson's second rule is a simplified version of Simpson's 3/8 rule.[3]

${\displaystyle {\text{Area}}={\frac {3h}{8}}(a+3b+3c+d).}$

Simpson's Second Rule

## Simpson's 3rd rule

Also known as the 5–8–1 rule[4], SImpson's third rule is used to find the area between two consecutive ordinates when three consecutive ordinates are known.[5]

${\displaystyle {\text{Area}}={\frac {h}{12}}(5a+8b-c).}$

## Use of Simpsons rules

Simpson's rules are used to calculate the volume of lifeboats,[6] and by surveyors to calculate the volume of sludge in a ship's oil tanks. For instance, in the latter, Simpson's 3rd rule is used to find the volume between two co-ordinates. To calculate the entire area / volume, Simpson's first rule is used.[7]

Simpson's rules are used by a ship's officers to check that the area under the ship's GZ curve complies with IMO stability criteria.

## References

1. ^ Bryan Barass; D.R.Derett. Ship stability for Masters and Mates (PDF). United Kingdom: Elsevier Butterworth. p. 69.
2. ^ Rhodes, Martin (2003). Ship Stability for Mates/Masters. United Kingdom: Seamanship International. p. 70.
3. ^ Subramaniam, Capt. Harry. Nutshell series – Ship Stability III. Mumbai, India: Vijaya Publications.
4. ^ Donkum, Enkhuizen, Van (2010). ShipStability. Dokmar publications. ISBN 978-90-71500-15-2.
5. ^ Bhange, Archana Ashish (February 2017). "Simpson's Rules and It's Application in Ship Stability" (PDF). International Journal of Computer & Mathematical Sciences. 6: 7.
6. ^ Kitching, Capt. R.C.E. Introduction to Ship Stability. Canada: Starpath Publications.
7. ^ V.L. Belenky; N.B Sevastianov (2007). Stability and Safety of Ships. USA: The Society of Naval Architects and Marine Engineers. ISBN 0-939773-61-9.