Simpson’s Second and first Rules for ship’s waterplane area calculation

Naval architects mostly use Simpson's first rule when calculating the waterplane area, the displacement of ship from curves of sectional areas which had been obtained from bonjean curves plotted against ship's draught and many other curves in ship design. In fact there are two other Simpson's rules which can be used for calculating areas of curve forms. They are Simpson's second and third rule. The latter is also called Simpson's 5, 8 minus 1 Rule. In today's post, I am going to explain how to calculate the area under a curve using Simpson's second rule and then compare the result with the calculation based Simpson's first rule.

The half breadth ordinates of a ship which are 20 meters equidistantly spaced are tabulated as follows:

Simpson's second rule calculation for waterplane area of a ship

We should remember that the sum of the area function calculated above is for half breadth only. To obtain the area of the waterplane for both sides of the ship, we have to multiply it with 2. Obviously the waterplane area of the ship will be:

Area WP = 3/8 x CI x Total of Area Function x 2 (for both sides of the ship)

= 3/8 x 20 x 81.9 x 2 = 1228.5 square meters

If the half breadth same ordinates is calculated using Simpson's first rule, then the tabulation will be as follows:

Simpson's second rule calculation for waterplane area of a ship

Area = 1/3 x CI x Total Area Function x 2 (for both sides of the ship)

= 1/3 x 20 x 92.3 x 2 = 1230.666 square meters

The small difference in the two calculations shows that the area obtained from both of the Simpson's rules are only close approximation to the correct area. We should remember that naval architecture is art and science of designing a ship. It is not an exact science but it is a science. As we dive deeper into the science of ship stability, we will use Simpson's rules more often for determining the area, the displacement and other hydrostatic properties of the ship. by Charles Roring

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