8. Summary of results#

In conclusion of this comparative evaluation of models of elastoplastic shells and pipes, of various thickness-to-radius ratios (from thin shell to thick shell), the following summary can be made:

  • for a complex loading path under internal and external pressures:

    • for displacement: the modeling « COQUE_SOLIDE » thanks to pinching gives values that are almost identical to the 2D axisymmetric calculation; the « DKT » modeling with a behavior model expressed in plane stresses deviates significantly from the 2D axisymmetric calculation in cases of plastic loads. By introducing the internal and external pressures with a corrected value making it possible to correctly represent the true area of the internal and external walls (metric correction according to the thickness), the results are significantly corrected, but this is not enough to get close enough to the reference calculation; the modeling « COQUE_3D » produces results very similar to the modeling « DKT », because these fine membrane elements, in a homogeneous situation, must give the same results that « DKT » with flat meshes; modeling « TUYAU_3M » (modeling « TUYAU_6M » would give the same result) provides results that are very close to modeling « DKT ».

    • for stresses: Radial stresses and residual circumferential stresses are very well evaluated by modeling « COQUE_SOLIDE » thanks to pinching with reference to 2D axisymmetric calculation; modeling « DKT » with a behavior model expressed in plane stresses deviates significantly from 2D axisymmetric calculation even after using internal and external pressures with a corrected value. While in modeling « DKT » and « COQUE_3D » produce constant circumferential stresses in the thickness, modeling « TUYAU_3M » produces constraints that vary in the thickness, but that do not intersect with the values in the reference modeling.

    • for internal variables: the modeling « COQUE_SOLIDE » thanks to pinching evaluates the work-hardening variable very well, while the models in plane constraints (« DKT », « COQUE_3D », « « , « TUYAU_3M ») are very disappointing.

  • for a monotonous loading journey to ruin under internal pressure:

    • we observe the ability of the « COQUE_SOLIDE » modeling to correctly reproduce the evolution of plasticization in thickness and to provide the limit load with great precision.

    • the modeling « DKT » with a behavior model expressed in plane stresses deviates significantly from the 2D axisymmetric calculation, even after correcting the internal pressure with the true area of the inner wall. As expected, it can be seen that this modeling without pinching leaves no margin between initial plastification and the ruin of the structure.

    • The evolution of radial displacement during loading corroborates these observations.

    • However, in terms of plastic ruin limit load, the models « COQUE_SOLIDE », « COQUE_3D », « DKT », « TUYAU_3M », provide conservative values to within 10%.

So, for these types of problems:

  • the hypothesis of plane constraints appearing in the writing of the law of behavior is too restrictive and the enrichment in pinch constraints is decisive;

  • Metric correction according to thickness is also necessary.

In contrast, shell modeling « COQUE_SOLIDE » [r3.07.04] makes it possible to reproduce quite satisfactory results, with reference to the calculation with massive finite elements, cf. Modeling A. The other shell models are too « poor ».