How can filling the empennage (aluminium tube) with Styrofoam stiffen it to avoid buckling as seen in Whingding ultralight?
A simplified analysis of column buckling shows that the axial load required for buckling is directly proportional to the second moment of area of the column's cross section: $F=\pi^2EI/(KL)^2`$`, where F is the buckling load, E is the elastic modulus of the material, I is the second moment of area, and KL is an effective column length based on the geometric length of the column (L) and a variable factor K that is based on the boundary conditions. The critical term for this question, though, is I.
For example, look at the picture below of a beam that someone is trying to bend. You can do this yourself with a ruler or a paperback book.
In the upper image, you're trying to bend the book/ruler by pressing down on the spine, and it's really hard to do. Rotate the book 90 degrees and press down on the cover...and it's relatively easy (unless you picked a phone book). The material of the book didn't change, only how the area of the cross section that you were bending was distributed in the direction that you were trying to bend the book. In the same way, adding foam to the tube increases the total area that needs to bend in order for the tube to buckle, which, if you'll allow an approximation, increases the strength of the tube by 5% from the baseline case (a hollow, aluminum tube). Double the radius, and that becomes 10%. Double it again, and that goes to around 25%. Don't take those numbers as gospel, but it should give you an idea that adding area -- even with a relatively low stiffness material -- you can increase the strength of a material significantly.
However, buckling is different from strength. Buckling is a bending instability caused by compressive loads, and failures in buckling often occur well below the failure load for the structure. Let's run some numbers for this, using the expression I mentioned back up at the top.
For example, if I run an analysis for a 4.5 ft long, hollow, 1 in diameter tube with a wall thickness of 1/16 of an inch made out of 7075-T6 aluminum (E = 10.3 msi) with both ends effectively pinned, I get a buckling load of 714 lbf (note: the axial compressive load that would make this structure yield would be closer to 8,000 lbf). Naturally, one could fill that in with metal and increase buckling load dramatically to 1,727 lbf, and, while that's not the query, I am adding some bending stiffness to the tube by filling in the center. Buckling of a composite column (insofar as I know) is tricky, but, here is an approximation (which will undershoot the actual buckling strength).
If I add foam and say that the foam will not buckle (as it is fully constrained by the tube and cannot deform in any direction save axially, let us say), and then assume also that the purely axial load on the column will be divided between the foam and the aluminum according to the proportion of their elastic moduli (polystyrene has a Young's modulus of about 2.5 GPa), then the aluminum will only absorb about 88% of the load placed on the column. This increases the buckling load by about 100 lbf (without causing the foam to yield). However, this does not take into account how the foam will add bending stiffness to the structure, which would further increase the benefits of adding the foam.
In short, you're using a low-weight, cohesive filler material to increase the bending stiffness of a structure, dramatically increasing its tolerance for compressive loads.
A simple explanation of buckling failure is a kinking of the tube wall. Force on the tube deforms it out of round. A flat side is weaker than a curved side just as a flat piece of paper bends more easily than a paper tube. With enough force the flat side folds.
Filling the tube with any material that prevents the initial deformation (keeps the tube round) will strengthen the tube. Solid, rigid foam does this and it is lightweight, important for an aircraft.
You could also seal the ends of the tube and inflate it with air, using the pressure of the air to keep the tube round. Commercial airliner fuselages are strengthened in flight this way.
First of all, lets start with a short introduction on buckling. There are two types of buckling, global (or column) buckling, and local buckling.
Global buckling is what happens to long slender (thin) structures, with a compressive load. Say you take a piece of plastic pipe with a diameter of 3/4 inch (2 cm +/-) and a length of 8 feet (2.5 m +/-). You place it vertically on the floor, and you start pushing on it. If it is very straight, you have to push it quite hard, and then it suddenly curves in to a hoop, and you need very little force to push it further. Foam in the wing helps very little against this type of buckling because the increase in stiffness is very small because of the low stiffness of the foam. That also does not matter, because there is no compressive load empennage at all, therefore this type cannot occur there.
Local buckling happens to thin-walled structures, when the thickness is small compared to the diameter. It can happen due to compressive loads, bending or torsion (external pressure can also make it buckle or buckle faster). This is what happens when you crush a cola can. Dents start to form and the structure loses its strength. You can do some interesting experiments with this at home. First take an empty can without any dents, place it on the ground, and start to crush it with your hands. You will notice that it is very hard to crush the can, you may not even succeed (I once put a 35 kg block of steel on an empty can easily). Now take an empty can and make a few big dents in a can and try to crush it again, it will be very easy (with a long stick I made a dent in the can with the block of steel on it, and it fell down so hard the whole building was shaking. I had to explain to my colleagues that i was giving a demonstration on buckling). Now try to crush a full can of cola, I doubt you will get there. The pressure of the cola is making sure that the can cannot dent, therefore it cannot buckle.
The cola in the can is doing the same thing as the foam in the wing; it makes sure that the empennage cannot dent (maybe cannot is an overstatement, but it sure helps!), and therefore it will not buckle.