# Geometric Properties of Regular Polygons

 pentagon hexagon octagon

Aim: To check the accuracy of the torsional moment of inertia calculation for a rod cross-section in the form of a regular polygon.

 Names of files with the initial data: Pentagon.cns Hexagon.cns Octagon.cns

Formulation: Check the accuracy of the torsional geometric properties calculation for a rod cross-section in the form of a regular pentagon, hexagon and octagon.

References: Hassenpflug W.C., Torsion of uniform bars with polygon cross-section, Computers & Mathematics with Applications, 2003, 46, No. 2-3, 313–392.

Kovář A., Moment tuhosti v kroucení pravidelného pětiúhelníka, Aplikace matematiky, 1957, 2, No. 1, 58-65.

Initial data:

 ν = 0.3 - Poisson’s ratio; r = 10 cm - radius of a circumscribed circle.

Design model: The design model is created by triangulation (the number of triangles ≈ 3000) on the basis of a model of the external contour. The external contour is a regular polygon. The number of vertices of the contour in a model is 5 (6, 8).

## Results Obtained in Consul

Design model, coordinate and principal axes, center of mass, ellipse of inertia, core of the section

Comparison of results:

Parameter

Theory

CONSUL

Deviation, %

pentagon

Torsional moment of inertia, It cm4

8478,1

8312,915

1,98

hexagon

10384

10215,966

1,61

octagon

12556,6

12453,297

0,822

Notes: Geometric properties can be determined analytically by the following formulas:

$\text {пятиугольник }\quad I_{t} \approx \mbox{0,84781}r^{4};$

$\text {шестиугольник}\quad I_{t} \approx \mbox{1,03877}r^{4};$

$\text {воcьмиугольник}\quad I_{t} \approx \mbox{1,25566}r^{4};$