# Geometric Properties of an Equilateral Triangle

* Aim:* To check the accuracy of the geometric properties calculation for a rod cross-section in the form of an equilateral triangle.

* Name of a file with the initial data:* Triangle.cns

* Formulation:* Check the accuracy of the torsional geometric properties calculation for a rod cross-section in the form of an equilateral triangle.

* References: *Young W.C., Budynas R.G.,

*Roark's Formulas for Stress and Strain*, New York , McGraw-Hill, New York, 2002.

**Initial data:**

ν = 0.3 | - Poisson’s ratio; |

a = 40 cm | - side length of an equilateral triangle. |

* 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 an equilateral triangle. The number of vertices of the contour in a model is 3.

## 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, % |
---|---|---|---|

Torsional moment of inertia, I |
55425,625 |
54477,143 |
1.71 |

Y-coordinate of the shear center, y |
20 |
19,999 |
0,005 |

Z-coordinate of the shear center, z |
11,547 |
11,589 |
0,36 |

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

\[ I_{t} =\frac{\sqrt 3 }{80}a^{4}; \] \[ y_{b} =a/2; \] \[ z_{b} =\frac{a}{2\sqrt 3 }. \]