Geometric Properties of a Square with a Central Square Hole

Aim: To check the accuracy of the torsional moment of inertia calculation for the cross-section of a rod with a hole.

Name of a file with the initial data: SquareWithHole.cns

Formulation: Check the accuracy of the torsional geometric properties calculation for a rod cross-section in the form of a square with a central square hole.

References: Arutiunian N.H., Abramian B.L., Torsion of Elastic Bodies, M., Fizmatgiz,, 1963 (see § III.1.7).

Initial data:

ν = 0.3 - Poisson’s ratio;
b = 100 cm - geometric dimansions.
d = 40 cm  

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 square, the internal one is a central square. The number of vertices of the contour in a model is 8.

Results Obtained in Consul

Design model, coordinate and principal axes, center of mass, ellipse of inertia, core of the section

Comparison of results:




Deviation, %

Torsional moment of inertia, It cm4




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

\[ I_{t} \approx 8\left[ {\left( {\frac{b}{d}} \right)^{3}-1,2704\left( {\frac{b}{d}} \right)^{2}+0,661\left( {\frac{b}{d}} \right)-0,1043} \right]d^{4}. \]