# Geometric Properties of a Trapezoid with an Acute Angle of 45°

* Aim:* To check the accuracy of the geometric properties calculation for a rod cross-section in the form of a trapezoid with an acute angle of 45°.

* Name of a file with the initial data:* Trapec45.cns

* Formulation:* Check the accuracy of the torsional geometric properties calculation for a rod cross-section in the form of a trapezoid with an acute angle of 45°.

* References: *Arutiunian N.H., Abramian B.L., Torsion of Elastic Bodies, M., Fizmatgiz, 1963 (§ II.4)

**Initial data:**

a = 50 cm | - side length of a trapezoid; |

с = 90 cm | - side length of a trapezoid. |

* Design model: *The design model is formed by triangulation (the number of triangles ≈ 3000) on the basis of a model of the external contour. The external contour is a trapezoid with an acute angle of 45°. The number of vertices of the contour in a model is 4.

## 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 |
3277083,333 |
3258832,976 |
0,556 |

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

\[ I_{t} \approx a^{4}\left( {\frac{1}{3}\frac{a+c}{a}-0.409} \right). \]