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Exact Section Properties

Basic Polygon

Consider a triangle with the following vertices

closedPolygon converts these points to a series of parametric curves as pure functions. The curves are given the head "path". The resulting "pure" functions are a bit ugly to look at, but for polygons, there is no need to understand anything about the curve definitions.

Using this list of curves, all of the section property functions can be invoked.

Note that ixx, ix, iyy, ect are relative to the origin used to define the vertices. For quantities about the cg, use ixxbar, iyybar, ixybar.

All the common properties can be obtained with tabulateProperties

Ixx | |

Iyy | |

Ixy | |

Xcg | |

Ycg | |

Ixx(cg) | |

Iyy(cg) | |

Ixy(cg) | |

Radius of Gyration-x | |

Radius of Gyration-y | |

Area | |

Max. Inertia | |

Min. Inertia |

Finally, show the polygon, with the cg marked as a black dot

Symbolic Usage

Consider the equilateral triangle with leg length w

All the section properties functions work with symbols

Curved Paths

Any curve that can be described as a series of pure functions can be used. The parameter always runs from 0 to 1 for each line segment. Consider the ellipse generated from a single line segment.

To plot, we have to substitute in some real numbers for a and b.

tabulate works with symbolic quantities

Ixx | |

Iyy | |

Ixy | 0 |

Xcg | 0 |

Ycg | 0 |

Ixx(cg) | |

Iyy(cg) | |

Ixy(cg) | 0 |

Radius of Gyration-x | |

Radius of Gyration-y | |

Area | a b π |

Max. Inertia | |

Min. Inertia |

Section with holes

Outer boundary (counter clockwise)

Inner boundaries (each run clockwise)

showPath has the option Direction (->True, False). If True, arrows are drawn to show the path directions. This is useful for verifying that all the curve segments have been entered properly, The default is False

Ixx | 8.9977 |

Iyy | 8.9977 |

Ixy | 0 |

Xcg | 0 |

Ycg | 1. |

Ixx(cg) | 2.56849 |

Iyy(cg) | 8.9977 |

Ixy(cg) | 0 |

Radius of Gyration-x | 1.18301 |

Radius of Gyration-y | 1.18301 |

Area | 6.4292 |

Max. Inertia | 8.9977 |

Min. Inertia | 2.56849 |

Example with straight and curved segments ( I- Beam)

Ixx | 17.1212 |

Iyy | 3.18513 |

Ixy | 0 |

Xcg | 0 |

Ycg | 1.69656 |

Ixx(cg) | 8.22832 |

Iyy(cg) | 3.18513 |

Ixy(cg) | 0 |

Radius of Gyration-x | 2.35405 |

Radius of Gyration-y | 1.01534 |

Area | 3.0896 |

Max. Inertia | 8.22832 |

Min. Inertia | 3.18513 |

Regular Polygons

Create a function that returns the vertices of a regular polygon with n sides, and the length of the side is "side".

The section property functions can then be used to get symbolic results for the properties of the polygon sections.

Create a table of formulae for the area of the polygons going from 3 sides to 8 sides.

n | Area |

3 | |

4 | |

5 | |

6 | |

7 | |

8 |

The result for 7 uses the Mathematica Root functions. These are implicit notations for the root of a polynomial that cannot be expressed in terms of combinations of radicals. The result for n=7 can be easily expressed as an approximate (Real) value

Created by Mathematica (March 7, 2004)