Page 33 - Introduction to Naval Architecture

SHIP FORM CALCULATIONS 21

middle line planes, and areas such as waterplanes can be treated as two
halves,
Referring to Figure 3.2, the curve ABC has been replaced by two
straight lines, AB and BC with ordinates y 0, y 1 and y 2 distance h apart.
The area is the sum of the two trapezia so formed:

The accuracy with which the area under the actual curve is calculated
will depend upon how closely the straight lines mimic the curve. The
accuracy of representation can be increased by using a smaller interval
h. Generalizing for n+1 ordinates the area will be given by:

In many cases of ships' waterplanes it is sufficiently accurate to use ten
divisions with eleven ordinates but it is worth checking by eye whether
the straight lines follow the actual curves reasonably accurately.
Because warship hulls tend to have greater curvature they are usually
represented by twenty divisions with twenty-one ordinates. To calculate
the volume of a three dimensional shape the areas of its cross sectional
areas at equally spaced intervals can be calculated as above. These areas
can then be used as the new ordinates in a curve of areas to obtain the
volume.

SIMPSON'S RULES

The trapezoidal rule, using straight lines to replace the actual ship
curves, has limitations as to the accuracy achieved. Many naval
architectural calculations are carried out using what are known as
Simpson's rules. In Simpson's rules the actual curve is represented by
a mathematical equation of the form:

The curve, shown in Figure 3.3, is represented by three equally spaced
ordinates y 0, y 1 and y 2. It is convenient to choose the origin to be at the
base of y 1 to simplify the algebra but the results would be the same