# Geometric Properties of a Square

Aim: To check the accuracy of the geometric properties calculation for a square cross-section of a rod.

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

Formulation: Check the accuracy of the shear and torsional geometric properties calculation for a square cross-section of a rod.

References:     Timoshenko S.P., Goodier J., Theory of Elasticity, M., Nauka, 1975.
Gruttmann F., Wagner W., Shear correction factors in Timoshenko’s beam theory for arbitrary shaped cross-sections // Comput. Mech. — 2001. — 27; No. 3 — 199–207.

Initial data:

 ν = 0.25 - Poisson’s ratio; a = 40 cm - side length of a square.

Design model: The design model is created by triangulation (the number of triangles ≈ 3000) on the basis of a model of the external contour. The external contour is a square. 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, %

Conventional shear area along the principal U-axis, Av,y cm2

1327,36

1332,135

0,359

Conventional shear area along the principal V-axis, Av,z cm2

1327,36

1332,135

0,359

Torsional moment of inertia, It cm4

360000

357205,548

0,77

Y-coordinate of the shear center, yb cm

20

20

0

Z-coordinate of the shear center, zb cm

20

20

0

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

$I_{t} =\frac{a^{4}}{3}\left[ {1-\frac{192}{\pi ^{5}}\sum\limits_{n=1}^{\infty \infty } {\frac{1}{(2n-1)^{5}}\tanh \left( {\frac{\pi (2n-1)}{2}} \right)} } \right]\approx 2,25\left( {\frac{a}{2}} \right)^{4};$ $y_{b} =a/2;$ $z_{b} =a/2;$