Plate theory used for Shell elements & Free edge boundary conditions

Grasshopper Karamba3D
1

Hi everyone, @karamba3d,

I am currently investigating the internal forces and principal moment trajectories of a simple flat plate (1m X 1m) using Karamba3D’s Shell elements. The plate is only supported at its 4 corners (pinned) and loaded by gravity, meaning all four edges are completely free.

While analyzing the results, I made a few observations regarding the boundary conditions at the free edges that made me wonder about the exact underlying plate theory Karamba3D uses:

  1. Torsional Moment at the free edge: I plotted the torsional moment (Mnt) along the free edge, and it is non-zero (varying linearly, crossing zero in the middle, and reaching its maximum at the supported corners). Consequently, the principal moment trajectories do not intersect the free edge perpendicularly (except in the exact middle), but at a slight angle.

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  1. Bending Moment normal to the free edge: According to classic thin-plate theory, the bending moment normal to a free edge (Mnn) should be exactly 0. However, I am getting a consistent, very small residual value of around -0.02 along the edge.

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My questions to the developers are:

  • Which plate theory is implemented for the Shell elements? Is it the classic Kirchhoff theory (thin plates, no shear deformation), or a shear-deformable theory like Mindlin-Reissner (thick plates)?

  • Does Karamba3D apply the “Kirchhoff equivalent shear force” boundary condition at the free edges, or are shear forces and torsional moments handled independently due to the FEM formulation?

  • Is the small normal bending moment (-0.02) purely a numerical artifact (e.g., due to FEM node extrapolation/interpolation), or is it an effect of Poisson’s ratio and the transverse contraction of the mesh?

I have attached screenshots of the principal moment trajectories and the moment diagrams along the free edge for reference.

Thanks in advance for any responses!

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Hi @Matthias_Hepperle,

thank you very much for your investigation!

The shell element used in Karamba3D is based on the formulation of Argyris’ TRIC element. The main difference is that, in the current version (3.1), Karamba3D neglects the transverse shear deformability terms included in the original TRIC formulation. As a result, the shell element implemented in Karamba3D follows Kirchhoff plate theory.

Within the TRIC formulation, transverse shear forces are derived from the bending moments through equilibrium conditions.

Shell element results are evaluated at the element center points. Nodal results are then obtained by averaging the contributions from all elements connected to the respective node.

-- Clemens