1. General
In FEM-Design we can calculate isotropic plates (same parameters in all directions) or orthotropic plates (different parameters in x and y directions, while the x and y are perpendicular to each other). Since the regular plane plates have constant section (or have linearly changing sections if we use the variable thickness plates) we can just set the ratio between two elastic moduli (E1 in x direction and E2 in y direction).
Figure 1. Orthotropic plane plate stiffness modifiers
With profiled plates, however, the thickness is not constant (we use sections instead of just the thickness) and thus cannot be easily set by setting just one ratio.
2. Profiled plate stiff direction
For profiled plates, we instead calculate the stiffnesses in both directions separately. The stiffness along the section (x direction or the "stiff" direction) and the transverse stiffness (in y direction).
Figure 2. Profiled plate stiffness directions
The regular stiffness (the "stiff" or the x-direction) is easy to calculate. First, we separate one of the "beams" in the plate, then we calculate the section properties for it and use them. Since the section is the same along the x-direction, then the stiffness is the same along the plate in that direction. Here is a picture to illustrate this:
Figure 3. Stiff direction meaning
3. Transverse stiffness
The transverse stiffness (or the stiffness in y direction) is more complicated. Let’s look at the following picture:
Figure 4. Transverse direction meaning
Here we also separate a "beam" from the plate in y-direction. This "beam" does not have a constant section anymore. The "beam" consists of segments and each segment is shaped like the section user has selected for the plate. Here in this example, each segment of the "beam" is hollow core shaped. To get the stiffness for the whole plate, we would need to know the stiffness of this segment.
The problem here is that the segment has changing section, and thus we cannot select one section and calculate parameters to find the stiffness for it.
Figure 5. Transverse direction sections along the "beam"
4. Transverse stiffness calculation method
If we look at the stiffness matrix in this form (picture below), we can see that we would need a constant section to calculate any stiffness:
Figure 6. Fictitious shell stiffness matrix components
We have opted for similar logic in the transverse stiffness calculation. We take one of the segments (1m wide for easier calculation) and clamp it from one end (rigid end connection). Then we apply a force on it and bend it. We calculate the deformation using finite element method (in depth description is in the next chapter).
Then we do a backward calculation and find analogues plate with constant thickness that has the exact same deformations. This constant thickness is easy to use now in the stiffness calculation.
Figure 7. Finding analogue plate
So, in short: first we find deformations by using external forces and stiffness; later we find stiffness by using external forces and deformations.
In the end we use this constant section stiffness as the transverse stiffness for the whole profiled plate.
5. In-depth and ignore some parts of the section
There is a setting in the profiled plate called "Ignored distance in transverse stiffness calculation". To explain this, we need to first look how the finite element method is used in the transverse stiffness calculation.
Figure 8. Profiled plate settings with the "Ignored distance in the transverse stiffness calculation" option
In previous chapter we saw that the segment is clamped by one end. In order to calculate that segment, we now divide it into finite elements. We can then calculate stiffness for each finite element and use this data to find the deformations we need.
Figure 9. Behind the scenes in finding analogue plate
Note: The same thing happens with regular beam calculation where we divide the beam into finite elements and then use the stiffness of each element to find deformations. In regular beams, however, the stiffness is the same in each finite element, but here the stiffnesses are different in each finite element.
Please note how small the finite element in the beginning of the segment is (the rectangle that connects the segment to the clamp). Because of this small element, the whole segment will have big deformation in this clamped calculation which results in thin analogue plate.
In this finite element calculation step, we can choose to ignore some parts of the section shape to make the section stiffer. Below is a picture where this is shown.
Figure 10. Ignore some parts of the section for transverse stiffness calculation
We can see from the picture that if we would ignore these parts then the clamped section would have a stiffer connection to the clamping wall and thus have less deformation and thus give us thicker analogue plate.
This is somewhat true to reality, where the joints between hollow cores are grouted and thus increase some of the stiffnesses.