Designer's Notebook:
DISPLACEMENT by Simpson's Rule


Next the designer will estimate where he expects the waterline to fall and calculate the area of the hull below the waterline. If the boat is a simple rectangle the cubic content can be obtained by multiplying the underwater height x width x length (in feet) to obtain the volume in cubic feet. This volume is then multiplied by the weight of a cubic foot of water to obtain the estimated displacement weight of the hull. If this displacement weight does not match the weight that the designer previously calculated, the waterline is is adjusted and the process repeated. Doesn't sound that bad, but remember, a boat is not uniform in shape.

There are many ways to determine the volume of an irregular shape; a computer with a proper program can solve the problem quickly if the proper factors are entered. There are several other methods but let's go back to a way of calculating volume that is basic and old as the hills: Simpson's Rule.

Simpson's Rule is used to determine the square area of the surface of a plane, such as the area of one cross section of a hull. When the areas of a number of sections have been obtained, the cubic contents of the whole can be calculated.

Let's use an example of how this rule is used to determine volume, then displacement. Suppose a boat has a waterline (estimated) length of 20'. This length is divided into an even number of equally spaced parts drawn perpendicular to the waterline. Let's use 10 parts 2' apart and number them from left to right 0 through 10 (the bow).

Next the areas in feet of these sections must be determined. The designers use a tool called a "planimeter", with dials and wheels that determines area when traced around an irregular plane. But the area can also be figured by laying the drawing over graph paper to a scale compatable with the drawing, counting the squares in the plane and estimating partial ones. The area on one half of the boat section is determined and multiplied by two. A typical boat will have no area at station #10.

After the areas of all of the 10 sections have been obtained, list them in columnar form with the station number (#) in the first column and three additional columns, labeled A, B, and C. List the area of each station in column A. In column B enter the following numbers: Station 1# and #10 enter a 1, all odd stations station a 2, and all even stations a 4. You should have a column starting and ending with 1 and alternating 2 and 4. Kinda sounds like voodoo doesn't it?

Next multiply each of the figures in the B column by the area listed in column A. Put the result in column C. When all station rows in column C have been figured, add them together. So far, the point of this exercise has been to get the total of column C. The next step is to multiply the column C total by the station spacing in feet, (2' in our example), and the result divided by 3 to give the underwater volume in cubic feet. Multiply this by the weight of water (62.5 or 64 lbs./cu. ft.) and the result is the displacement in pounds.

That wasn't so bad was it? But, the displacement figured does not equal the calculated weight of the boat. So it's back to the beginning, adjust the waterline up or down, and try again.

There are many ways to estimate the waterline with reasonable accuracy and numerous attempts are seldom required. But our original intent was to present or re-present an old rule that figures displacement. Simpson's Rule is a handy one and can be used to find the area of any irregular plane as well as the volume.