# Geometric Properties of an Isosceles Right Triangle

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

* Name of a file with the initial data:* Triangle90.cns

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

* References: *Galerkin B.G.,

*Torsion of a Triangular Prism*, Izv. Akad. Nauk. VI series, 13:1 (1919), 111–118

Polya G., Szego G., *Isoperimetric Inequalities in Mathematical Physics*, — M., Fizmatgiz, 1962.

**Initial data:**

a = 40 cm | - leg length of the triangle. |

* 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 an isosceles right 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 |
66816 |
66251,348 |
0,845 |

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

\[ I_{t} =a^{4}\left[ {\frac{1}{12}-\frac{16}{\pi^{5}}\sum\limits_{n=1}^\infty {\frac{1}{(2n-1)^{5}}\coth \left( {\frac{\pi (2n-1)}{2}} \right)} } \right]\approx 0.0261\cdot a^{4}. \]