2. Benchmark solution#

2.1. Calculation method used for the reference solution#

Two reference solutions can be used to calculate the deformation:

  • the theory of LOVE - KIRCHHOFF, commonly used for so-called « thin » plates, which will be retained for the A, B, C, D, E and I models,

  • the theory of MINDLIN - REISSNER, including the effects of shear for so-called « thick » plates, which will be used for the F, G, H and J models.

At any point that is \(r\) away from the center of the plate (\(r\le R\)), the arrow is expressed:

_images/Object_3.svg

For the calculation of moments, the two theories lead to the same expressions:

_images/Object_4.svg

In the center of the plate:

_images/Object_5.svg

Note:

Code_Aster calculates the moments at the nodes of each finite element in the reference frame defined by the outer normal and the reference axes defined on the shell (see AFFE_CARA_ELEM) .

The value of the moment

_images/Object_6.svg

(or

_images/Object_7.svg

) in a node belonging to several finite elements can be considered to be the average of the values calculated on the elements that have this node in common. This average can be obtained by the procedure POST_RELEVE.

For each node, we have:

_images/Object_8.svg _images/Object_9.svg

2.2. Benchmark results#

Arrow and moments at points \(O,A,B,C,D,E,F\). Extraction of the mean values of the components

_images/Object_10.svg

from the “EFGE_ELNO” field.

2.3. Uncertainty about the solution#

Analytical solution.

2.4. Bibliographical references#

  1. TIMOSHENKO and WOINOWSKY - KRIEGER. Plates and shells. Béranger edition, (1961).

  2. BATOZ and DHATT. Modeling of structures by finite elements. Cases. Univ.Laval Press, 1992.