Dave Rogers
Bandaged ice that stampedes inexpensively through
Dave, this was obviously a rough back of the envelope calculation to show just why the core was self-supporting. I did not say it could support 37 billion pounds, just that that size column would need that kind of a load to buckle, meaning it would not. It would have failed by compressive rupture first and is out of the realm of buckling.
Here is a calculation for a 137 foot x 87 foot rectangular column x 1440 foot high with just 12 inch walls. The calculation is converted to inches with a 1,644 inch x 1,044 inch rectangular column x 17,280 inch high.
The moment of inertia I = 1/12bh^3. The hollow area MOI about the least radius of gyration is subtracted from that of the exterior dimensions to find the MOI of the hollow section with 12 inch walls and it is 12.628 x 10^9 in.^4.
The critical buckling load equation is F = (Pi^2 x E x I)/(K x L)^2
Since it is fixed at one end and free at the other K = 2.0. The modulus of elasticity is 29 x 10^6 psi for steel. In this case F = 3.02 billion pounds (which is at least 3 times and possibly 5 times the entire weight of a tower). This simply means it is out of the range of buckling, since it would obviously fail by compressive rupture first.
You can cut lots of holes in the side of this 12 inch thick wall column as long as all slenderness ratios are such that local buckling will not occur, which is what the X-Y lattice structure of the core did. It limited buckling of the columns to near their compressive yield stress by keeping their slenderness ratios low. Since the columns would not yield with less than 3 times the load on them the core was self-supporting.
I've edited the above to correct the typo you identified.
The problem I have with this is that the structure you're starting with has virtually no resemblance to the structure you're using it to represent. You've started from the argument that a solid column and a square tube have roughly the same buckling strength, yet when you actually do the calculation you find that they differ by more than a factor of ten. You're now asserting, without calculations to back it up, that a structure still vastly stronger and heavier than the actual core, with all the structural material concentrated at its edges where it can provide the most resistance to buckling, is again similar in strength against buckling ratio. In effect, you're handwaving away a correction factor that, as you've already shown, can amount to more than an order of magnitude.
I'll be the first to hold my hand up and say that I have absolutely no idea how to calculate the stiffness of the WTC cores. However, your bare assertion that they must behave similarly to two different structures whose buckling strength differs by more than an order of magnitude is simply not credible. Given the extreme differences, you can't claim you've proven anything unless you've analysed at least a reasonable approximation to the actual structure.
Dave
