Following a discussion at Eng-Tips and resulting comments I have updated my Section Properties spreadsheet to include calculation of the area and centroid of a “Lune” (the shape of a crescent moon). I have also added a “Re-draw Shape” button because in at least one case the automatic re-draw was not working. The download spreadsheet includes full open source code.
The screen-shot below shows the input and results for a Lune:
The results at the moment only include the angles of the two arc segments, the areas of the smaller radius segment and the lune, and the centroid position of the lune, measured from the centre of the larger arc. The second moments of area can be calculated using the “Group” function, as shown below.
1. Enter the section details (radius and arc angle from the x-axis, Theta) for the smaller radius circular segment, using Shape Type 4:
2. Scroll down to the Group Properties table and enter the shape X offset as the offset between the two segments. The “Elastic Modulus” should be entered as 1. After entering these details, click the “Create New Group” button:
3. Enter the section details for the larger radius segment:
4. Go back down to the Group Properties table, enter X offset as 0 and Elastic Modulus as -1 (so the shape properties are subtracted from the total) then click “Add shapes to group”. The Group Properties table will update, including Ix and Iy values:
A similar procedure allows any combination of the standard shapes to be generated, including multiple copies of any shape with different offsets and/or rotations from the origin. Shapes may also be rotated about their own centroid.