# Shell vs frame element model stiffness differences

I have a model of a tall, slender structure that I am investigating using both shell and $3D$ frame elements.

The shell elements are type MITC4, $4$-node membrane elements. The frame elements are the basic line ($1D$) elements found in any introductory structural analysis / stiffness method book - they include axial, bending, and torsion, but no shear deformation.

Both analysis are linear-elastic (no geometric or material non-linearties), small-deformations using the direct-stiffness formulation.

As a test case, I loaded the structure with a unit load at the top node (or nodes, for the shell element model), once in a direction perpendicular to the long axis of the structure ($X$-axis), and once down the axis of the structure for pure axial loading ($Z$-axis). I neglected all self-weight for this test case.

[I also compared $X$ and $Y$ direction bending, which matched exactly as expected, since the structure is symmetric.]

I compared these results with those found by simple hand computations for a linear-elastic beam:

Axial deformation ($Z$-axis) = $PL/(AE)$ Bending deformation ($X$-axis) = $PL^3 / (3EI)$

For the frame element model, these results matched exactly.

For the shell element model, the axial value was nearly identical (which I take as a good sign that the area is correct).

The bending deformation for the shell element model were somewhat smaller (~8%).

What I would like to know is, is this expected behavior? Should I expect the shell element model to be "stiffer" than the frame element in general?

I understand that the choice of Poisson's ratio in the shell element has an effect here as well. I'm using $0.30$ for steel.

• What are the dimensions of your cantilever beam test case? Did you try Poisson's ratio equal zero to make the plate more beam-like? – Bill Greene Apr 24 '15 at 12:41
• The structure is a pole about 30 meters tall and about 0.6m in diameter. I did not set Poisson's ratio to 0, but I will try that now. – Madeleine P. Vincent Apr 24 '15 at 13:09
• How did you calculate an equivalent plate thickness for the circular cross section? – Bill Greene Apr 24 '15 at 13:28
• Regarding changing Poisson's ratio, setting it to 0 had very little effect on the bending displacements, maybe 0.1%. – Madeleine P. Vincent Apr 24 '15 at 13:43
• Regarding the equivalent thickness, in fact, it's not a round tube, but a multi-sided pole. One shell element per side, going around the pole. The standard method for calculating moment of inertia (Ixx, Iyy, J) for the frame elements were used (bh^3/12 + ad^2 -- or there's a formula for arbitrary polygons). I'm fairly confident in these values. – Madeleine P. Vincent Apr 24 '15 at 13:47

There really aren't enough details about either your structure or your shell FE model to be definitive but, from what you have provided, I would say it is not all surprising that the results from your beam element and shell element models differ significantly.

You didn't say, but if your objective is to be able to accurately analyze this structure using a beam model, I think that is definitely possible. However, it might require a somewhat different approach than your current one.

Here are some things to consider:

1. How sure are you that your shell element model is giving accurate (converged) results? In particular, are you sure your mesh is sufficiently refined? I suggest building three shell FE models with 10, 20, and 30 shell elements down the length of the tube. These results should give you a good idea of the correct displacement from the shell model.

2. I believe that the formula you are using to calculate the equivalent moment of inertia for your polygonal-cross section tube assumes that the wall thickness of each segment in the cross section is thin. That is, the bending stiffness of the individual segments is ignored. Is this really a valid assumption?

3. If the wall thickness of the cross section segments is not small, this also has a consequence in your shell model. At each node around the cross section where shell elements connect, you have a small bit of overlapping material. This seems like it should be possible to ignore this but it can become important if you are trying to get close agreement between shell and beam models.

4. For hollow "tubes", like this one, the transverse shear stiffness is low so you might need to include shear deformation in your beam equation to compute an accurate deflection.