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Dr. G wrote:The simple momentum transfer solution yields the result that the raindrop should fall at 1/3g, which is not observed for WTC 1.
Major_Tom wrote:If the lower block were taller (200 floors) rather than 100, do you think the yellow velocity graph will level off to some near constant "terminal acceleration" like the blue and red examples?
It is nice the way you compare the zone C structure as if it is in a falling elevator. (Mathematically the same as being in a reduced gravitational field).
Major_Tom wrote:You introduce an interesting balance of forces that the analytical approach has ignored. How does the analytical approach treat zone C connections? It doesn't.
BL wrote:The acceleration (v_B dot) rapidly decreases because of mass accretion of zone B and becomes much smaller than g, converging to g/3 near the end of crush down (Bažant et al. 2007).
my boldAbstract wrote:A standard undergraduate mechanics problem involves a raindrop which grows in size as it falls through a mist of suspended water droplets. Ignoring air drag, the asymptotic drop acceleration is g/7, independent of the mist density and the drop radius. Here we show that air drag overwhelms mist drag, producing drop accelerations of order 10-3g. Analytical solutions are facilitated by a new empirical form of the air drag coefficient C = 12R-1/2, which agrees with experimental data on liquid drops in the Reynolds-number range 10<R<1000 relevant to precipitating spherical drops. Solutions including air drag are within reach of students of intermediate mechanics and nonlinear dynamics.
Major_Tom wrote:It is interesting to see "upper block" momentum being applied to the lower portion only through connections. It is so much more believable and workable than some oversimplified differential equations which have been treated like little gods.
There are many factors to consider, like connections for example, that cannot be treated through analytical equations with a very finite number of variables.
Major_Tom wrote:You follow my point?
- constant mass
- varying strength
- true 1D in 3D
- dynamics calculated using spheres
- rendered as slabs with height = diameter of spheres
- camera tracks with topmost member
- sequence ends at arrest
The assumption of the prevailing theory, which is a continuum model, is that the upper block remains relatively undamaged until the end. I've read the part of B&L that addresses this (many times) and, while I understand the majority, I'm not in a position to check the validity of the work. I'll assume it to be correct and conclude that this is a significant difference with a discrete floor model - the upper block crushes up because it would if dropped onto a moving platform accelerating downward at a significant fraction of g, whereas in the case of continuous media this is not the case
If this style of simulation offers anything of value by way of comparison to physical progressive collapse, at least in these two examples, it is:
1) mixed crush direction should be distinguishable from exclusive crush down via roofline measurements of sufficient duration, by virtue of a faster descent
2) having mixed crush direction does little to alter the energetics of the overall problem, so ignoring it is OK (sometimes)
The simulator is not junk. It's a little difficult to reconcile a figure of 0.6g with 0.3g, however. Maybe WTC1 starts at 0.6g and goes to 0g, whereas this starts at g and goes to 0.3g, and it all comes out in the wash. ???
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