Formulation of beam elements

[TidalGuy]

Of the main 3D beam elements available there appears to be the relatively simple B31, B32 and B33 elements. Abaqus claims that they respectively use linear, quadratic and cubic interpolation. Can anyone please tell me what it is that they are interpolating to this order though? I am assuming displacement but cannot be sure.
I have tried making a simple single element cantilever model with either a point load at the end or a line load along it's length. The section is rectangular. For the output I am looking at the end displacement (U), the section moments (SF) and the stresses (S). If I am not mistaken then the moments and stresses should require linear and quadratic interpolation for the point and line load respectively. The displacement should require cubic and quartic interpolation for the point and line load respectively.
For the point load the B31 provides incorrect answers for all three outputs when compared to the analytical solution. Presumably this is because it is outputting the moment and stress from the single central integration point and the displacement is too high order for a linear element. The B32 and B33 elements output all three correctly.
When the beam is loaded using a line load the B32 and B33 still get the displacements correct but the section moments and stresses come out incorrect. Are they simply giving me the section moments and stresses at the integration points and if so why is it correct for the point load and not for the line load? It appears that for these problems, the B32 and B33 are the same, except the fact that B32 does shear deflections and has 3 nodes. Can anyone enlighten me please?

 

[xerf]

I believe the cantiliver beam problem is in the Benchmark or Verification Manuals tested for multiple element types.

See for example, in Verification Manual:
-Love-Kirchhoff beams and shells
-Shear flexible beams and shells

Best,

 

[TidalGuy]

Thanks for the post xerf but I still don't think that solves the problem. The cantilever beam problem has been benchmarked in the verification manual, but only for a point load not a uniformly distributed load. Colleagues of mine have mentioned that FE programs tend to lump distributed loads at the ends of the elements so in theory quadratic elements should be able to model it with a single element but this is not the case. Not only does the quadratic element calculate the incorrect moment (and therefore stress) but the two node cubic element also calculates the incorrect moment, despite the fact that a cubic shape function should be able to get the exact solution for the quadratic distribution of bending moment.