Lesson 2: Water-Plane Area Calcuations using Simpson's First Rule

Finding the area of a water plane is one of the important topics that you will learn in the ship stability course. You will be using Simpson's Rules for finding areas under a curve for this purpose. There are three different variations of this rule. We will review them one by one.

Simpson's Rules use ordinates to calculate the waterplane area. The rules also require that one side of the area we are trying to calculate must be a straight line.

First of all let us recall that an ordinate is the y-coordinate of a point which defines the vertical distance from a horizontal axis.
When calculating water plane areas, an ordinate is the vertical distance between two points on the water plane, as shown in the diagram below.
The midline will divide the ordinates in two equal parts. Hence the ordinates on either side of the midline are refered to as half ordinates or semi-ordinates.

The shape of the water plane is symmetrical about the midline, therefore it will be sufficient to calculate the area of one half of the shape and then double the answer to find the total area. The midline satisfies the straight line boundary requirement of Simpson's Rules.

The following steps are common for all three versions of Simpson's Rules:

Divide the water plane into two halves using the midline.
Choose one half to work with.
Sub-divide the chosen half into area strips or sections of equal width, 'h'.
Based on the number of area sections and hence the number of half ordinates, choose which version of Simpson's rules to apply.
In some cases you may have to combine two rules to calculate an area.

Simpson's First Rule

This rule is also called the 3-ordinate rule because you need at least three ordinates to be able to apply this rule.
When you have an even number of strips and hence an odd number of half ordinates, you can use Simpson's First Rule.

If a given curve can be divided into an even number of strips of equal width, then the area under the curve can be found using the formula for Simpson's First Rule:

\begin{array}{l} {For 3 ordinates y}_{1} {, y}_{2} {and y}_{3} equally spaced at h units: A = \frac{h}{3} (y_{1} + 4 y_{2} + y_{3}) \\ {For 5 ordinates y}_{1} {, y}_{2} {, y}_{3} {, y}_{4} {and y}_{5} : A = \frac{h}{3} (y_{1} + 4 y_{2} + 2 y_{3} + 4 y_{4} + y_{5}) \\ For 7 ordinates: A = \frac{h}{3} (y_{1} + 4 y_{2} + 2 y_{3} + 4 y_{4} + 2 y_{5} + 4 y_{6} + y_{7}) and so on ... \end{array}

Note that 'h' is also called the common interval or CI, and represents the width of each of the area sections or strips.

Let's try an example to better understand how to use this formula:

Example 1: The length of a ship's water-plane area is 70 m. The lengths of the equidistantly spaced half ordinates commencing from forward are as follows: 0, 5.2, 6.4, 7.0, 6.0, 4.9, 0.3
Find the area of the water-plane.

Solution:

First of all, we need to check if we can apply Simpson's First Rule to calculate the area:
Number of ordinates = 7 and since 7 is an odd number, we can use Simpson's First Rule to find the area of this water-plane.
Next, let us find the value of the common interval 'h' which can be calculated by dividing the length of the water-plane (70m) by the number of area sections (6): $h = \frac{70 m}{6} = 11.7 m$
The following figure will help us visually represent the information: example1

Half-ordinates (1)	Simpson's Multiplier (2)	Area Function (3)=(1)x(2)
0	1	0
5.2	4	20.8
6.4	2	12.8
7.0	4	28.0
6.0	2	12.0
4.9	4	19.6
0.3	1	0.3
$(T o t a l) Σ_{1}$		93.23

Table 5.1

As noted above, the Simpson's First Rule formula requires that we multiply the half-ordinates by a series of constants called Simpson's Multipliers.
For 3 ordinates, the Simpson's Multipliers are 1, 4, 1.
For 5 ordinates, the Simpson's Multipliers are 1, 4, 2, 4, 1.

For 7 ordinates, the Simpson's Multipliers are 1, 4, 2, 4, 2, 4, 1.

For 9 ordinates, the Simpson's Multipliers are 1, 4, 2, 4, 2, 4, 2, 4, 1.

For 11 or more ordinates, the same pattern of multiplier values is applied.

Table 5.1 shows the half-ordinate values and their corresponding multipliers for this example.

For each half-ordinate, an Area Function is calculated by multiplying the half-ordinate by its corresponding Simpson's Multiplier. Then, a Total Area Function is calcuated by summing the individual area functions. The Total Area Function for this example is 93.23, as shown on Table 5.1

Finally, we apply the First Rule formula to calculate the area of half of the water-plane, as bounded by the curve and the midline. Note that in the calculation below we then multiply the formula by 2 in order to obtain the entire water-plane area, since the area caluculated using Simpson's First Rule is for one half of the ship's water-plane area.

$Area of the water-plane = 2 \times \frac{h}{3} \times Σ_{1} = 2 \times \frac{11.7 m}{3} \times 93.23 m = 727.2 m^{2}$

Loading the player...

Example 2: Find the area of a water-plane which is 72 metres long, using Simpson's First Rule with the following half-ordinates commencing from forward: 0.2, 2.2, 5.0, 5.8, 6.0, 5.9, 4.9, 2.0, 0.2

Solution:

We are given 9 half-ordinates and therefore we can use Simpson's First Rule to calculate this water-plane area.
Find the value of "h" :
$h = \frac{72}{8} = 9 m$ Draw a diagram:

Complete the table:

Half-ordinates (1)	Simpson's Multiplier (2)	Area Function (3)=(1)x(2)
0.2	1	0.2
2.2	4	8.8
5.0	2	10.0
5.8	4	23.2
6.0	2	12.0
5.9	4	23.6
4.9	2	9.8
2.0	4	8.0
0.2	1	0.2
$(T o t a l) Σ_{1}$		95.8

Table 5.2

Calculate the area: $Area of the water-plane = 2 \times \frac{h}{3} \times Σ_{1} = 2 \times \frac{9 m}{3} \times 95.8 m = 574.8 m^{2}$

Note that in the above calculation we must multiply by 2 in order to obtain the entire water-plane area, since the area caluculated using Simpson's First Rule is for one half of the ship's water-plane area.

Loading the player...

Example 3: The lengths of the half-ordinates of an 80 m long water-plane commencing from forward as follows: 0, 3.6, 5.0, 5.3, 4.8, 0.5
There is a 2.8 m long half-ordinate right midway between the last two half-ordinates. Find the area of the water-plane.

Solution:

In this case there are two different values of 'h' to consider. The green area (Area 1) has 4 strips whose 'h' value is twice that of the 2 strips in the blue area (Area 2). To obtain the total water-plane area, we can divide the water-plane into 2 parts, apply Simpson's First Rule separately to each partial area, and then add the results to get the total area of water-plane. example3

Area 1:
Find the value of h : $h = \frac{80 m}{5} = 16 m$
Complete the table :

Half-ordinates (1)	Simpson's Multiplier (2)	Area Function (3)=(1)*(2)
0	1	0
3.6	4	14.4
5.0	2	10.0
5.3	4	21.2
4.8	1	4.8
$(T o t a l) Σ_{1}$		50.4

Calculate the partial green area (Area 1):

Area  = 2 \times \frac{h}{3} \times Σ_{1} = 2 \times \frac{16 m}{3} \times 50.4 m = 537.6 m^{2}

Area 2:
Find the value of h : $h = \frac{16 m}{2} = 8 m$ Complete the table:

Half-ordinates (1)	Simpson's Multiplier (2)	Area Function (3)=(1)*(2)
4.8	1	4.8
2.8	4	11.2
0.5	1	0.5
$(T o t a l) Σ_{2}$		16.5

Calculate the partial blue area (Area 2):

Area  = 2 \times \frac{h}{3} \times Σ_{2} = 2 \times \frac{8 m}{3} \times 16.5 m = 88 m^{2}

Hence,

T o t a l A r e a = A r e a 1 + A r e a 2 = 537.6 m^{2} + 88 m^{2} = 625.6 m^{2}

Loading the player...

Lesson 2: Water-Plane Area Calcuations using Simpson's First Rule

Simpson's First Rule

Example 1: The length of a ship's water-plane area is 70 m. The lengths of the equidistantly spaced half ordinates commencing from forward are as follows: 0, 5.2, 6.4, 7.0, 6.0, 4.9, 0.3 Find the area of the water-plane.

Example 2: Find the area of a water-plane which is 72 metres long, using Simpson's First Rule with the following half-ordinates commencing from forward: 0.2, 2.2, 5.0, 5.8, 6.0, 5.9, 4.9, 2.0, 0.2

Example 3: The lengths of the half-ordinates of an 80 m long water-plane commencing from forward as follows: 0, 3.6, 5.0, 5.3, 4.8, 0.5 There is a 2.8 m long half-ordinate right midway between the last two half-ordinates. Find the area of the water-plane.

Example 1: The length of a ship's water-plane area is 70 m. The lengths of the equidistantly spaced half ordinates commencing from forward are as follows: 0, 5.2, 6.4, 7.0, 6.0, 4.9, 0.3
Find the area of the water-plane.

Example 3: The lengths of the half-ordinates of an 80 m long water-plane commencing from forward as follows: 0, 3.6, 5.0, 5.3, 4.8, 0.5
There is a 2.8 m long half-ordinate right midway between the last two half-ordinates. Find the area of the water-plane.