I think I mentioned it on physorg many moons ago, before Gregory Urich appeared on the scene, but at the end of Love's treatment of problem 281, 'Rod fixed at one end and struck longitudinally at the other', he states,
It may be shown also that the greatest compression of the rod occurs at the fixed end, and that, if m < 5, its value is 2(1 + exp(-2/m)V/a, but, if m > 5, its value is approximately equal to (1 + sqrt(m))V/a
V is the impact velocity, taken to be 8.5 m/s in the BZ scenario.
I used Greg's spreadsheet at http://www.cool-places.0catch.com/911/calcMassAndPeWtc1_v2.5.xls
to calculate m, which in the formulas above is the mass ratio between the striking mass (considered rigid) and the rod (considered elastic, with bar velocity a)
total mass 98 - 111
from Greg's posts here:
"Mass of columns B6-97 = 55,700 metric tons."
which is 55,700,000 kg
so taking m to be the ratio of all the stories above floor 97 to the mass of the columns in floor 97 and below, we get
m = 32,815,800 / 55,700,000 = .589
being less than 5, we use the first formula mentioned above, viz.,
2(1 + exp(-2/m))V/a
substituting, we get
2(1 + exp (-2/ .589 )) V/a
=2(1 + .0335) V / a
= 2.067 (8.5 / 5200)
While this is > .002, considering that cross area of the steel columns in the basement level is at least 4x the cross area at floor 97, and spring constants add for springs in parallel, if we ignore the fact that the problem is for an "unloaded" (gravity free) rod, then surpassing the elastic limit at the base is impossible. (Not sure about the rest of the rod.)
For a more realistic "loaded" solution, this has to solved numerically.
Now, as to the peculiar sub-title I have chosen, I am confused as to the difference in stated results between A. E. H. Love's A Treatise on the Mathematical Theory of Elasticity, problem 281, and the results stated by Johnson in Contact Mechanics, Section 11.1
As far as I can see, they are describing the same problem - rigid mass hitting an elastic rod, the rod is fixed at the opposite end. Gravity is ignored.
However, as I mentioned above, Love gives maximum compression at the fixed end, for all mass ratios, while Johnson states that, for low mass ratio, maximum stress is at the end of the rod being struck.
Love doesn't actually show the derivation of his maximum compression claim, though he does state his boundary condition explicitly in his initial statement of the problem in a manner that looks "well posed", and he also actually solves the wave equation, mathematically. The boundary condition at the free, impacted end is : lim t-> +0 ( d_ w / d_ t) = - V
( "d_" is for partial derivatives) . Does this at least imply that the lim t-> +0 stress -> - rho * c0 * v, for small m? (see below).
Johnson, OTOH, doesn't even do a full derivation of the wave equation in 11.1 (though he gives some references). He uses a momentum argument to show that stress = - rho * c0 * v, where v is the (local) velocity of the rod itself, c0 is the velocity of the wave front, and rho is density. In a scenario such as this, where you consider the impactor to be rigid, it does seem like a good first approximation to take v = V, where V is the velocity of the impactor, at the instance of impact.
However, absent a full treatment, I am wondering if Johnson is simply wrong in his claims.
I also wonder if I am simply wrong in assuming that maximum stress is completely determined by maximum compression!