# 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, A |
1327,36 |
1332,135 |
0,359 |

Conventional shear area along the principal V-axis, A |
1327,36 |
1332,135 |
0,359 |

Torsional moment of inertia, I |
360000 |
357205,548 |
0,77 |

Y-coordinate of the shear center, y |
20 |
20 |
0 |

Z-coordinate of the shear center, z |
20 |
20 |
0 |

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

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