# 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, It cm4

66816

66251,348

0,845

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

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