## Friday, October 16, 2009

### Waterplane Area Calculation with Simpson’s Rule

Naval architects will always have to deal with ship form calculations. These include the calculations of areas of waterplane, transverse sections to obtain the curve of sectional areas, the calculation of hydrostatic properties of the ship such as the displacement, the center of buoyancy, the center of flotation, the transverse metacenter above the keel and etc. It is a tedious calculation if carried out manually. With spreadsheet software such as Microsoft Excel, we can do the job easily. There are also ship design software such as Delftship, Maxsurf that can help naval architects to do those calculations.

• Basic theories about waterplane area calculation can be found in a number of textbooks such as:
• Basic Ship theory by K.J. Rawson and E.C. Tupper
• Statics and Dynamics of the Ships by Semyonov (Mir Publisher)
• Introduction to Naval Architecture by Eric Tupper

In this post, I am going to explain how in Naval Architecture studies we use Simpson's rules to calculate the waterplane area and the position of its centroid of area and finally, the second moments of area for a boat.
For example, we get the ordinates (in meters) of the waterplane of a boat which is defined by a series of half breadth ordinates at 2 meters separation, as follows:

For illustration, I have drawn or sketched the waterplane using pencil on a piece of paper.
To carry out the ship or boat form calculation, I use Microsoft Excel. I entered the half bread ordinates of the boat in a table and perform the calculation as depicted on the table below:
As explained by E. C. Tupper on page 55 of Introduction to Naval Architecture, The column F(A) represents SM x y; F(M) = SM x lever x y; F(I)long = SM x lever x lever x y and F(I) trans = SM x y 3. From the summations in the table:
The area of the waterplane (for both sides of the boat) = 2/3 x 2 x 67.2 = 89.599 square meters.
The centroid of the area is 2 x 21/67.2 = 0.625 m forward of amidships.
The longitudinal second moment of area about amidships = 2/3 x 2 x 2 x 2 x 284.4 = 1516.799 m4
The minimum longitudinal second moment will be about the centroid of area and given by:
IL = 1516.8 - 89.599 (0.625)2 = 1481.647 m4
The ship form calculations can be carried out not only by Simpson's rules but also by Trapezoidal, Tchebycheff's rules. With the use of computer spreadsheet program like Microsoft Excel and Lotus 123, the calculation can be done and modified in real-time easily. by Charles Roring
Watch the video of ship displacement calculation on the following posts: