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, It cm4 |
3277083,333 |
3258832,976 |
0,556 |
Notes: Geometric properties can be determined analytically by the following formulas:
\[ I_{t} \approx a^{4}\left( {\frac{1}{3}\frac{a+c}{a}-0.409} \right). \]