12. Summary of results

Beam models:

The results of the 5 beam models are very close to the reference solution, which is an average of the results of 8 codes. There is a small effect due to warping since the results of the C and E models (POU_D_TG/POU_D_TGM) are slightly different from the others, while remaining within \(\text{2\%}\) of the reference.

Although the last two models (D and E) implement the nonlinear buckling criterion of CRIT_STAB in the nonlinear statics operator, they do not use the same approach.

In the D modeling, it is the Euler criterion that is used because we are in small movements (DEFORMATION =” PETIT “) and the tangent matrix contains only the material stiffness matrix.

In modeling E, we have placed ourselves in large displacements (DEFORMATION =” REAC_GEOM “, although this is not necessary to deal with this problem), the tangent matrix includes both material and geometric rigidities, which makes it possible to illustrate the generalized nonlinear criterion in which we test the singularity of the tangent matrix. We therefore no longer predict the critical load but we detect it (by observing the change in sign of the smallest eigenvalue of the tangent matrix, confer [R7.05.01]).

Hull models:

In these models, it is the Euler criterion that is used because we are in small movements (DEFORMATION =” PETIT “) and the tangent matrix contains only the material stiffness matrix.

The results of the DKT (QUAD4, TRIA3) and COQUE_3D (QUAD9, TRIA7) models are far from the reference solution, with a maximum difference of 15.5% for the first buckling mode. On the other hand, the results are satisfactory for the second mode, with a difference of 1.2%.