The 9/11 Forum

War Wheel wrote:Didn't you show (on the other thread) that even a 50/50 split demonstrated this effect?

In these last tests, top slab massed differed but the split point and fall distance were the same, so the velocities were different. One example between 1x and 8x is collapse times of 13.166 and 11.558. These translate directly into the velocity differences since the distance is the same, 100 stories in those times. The normalized position coordinate is the same as the normalized time coordinate.

OneWhiteEye wrote:Fascinating! This I want to explore.

The source is:
The Domino Effect: Successive Destabilization by Cooperative Neighbours Proc. R. Soc. Lond. A July 8, 1988 418:155-163 (see http://www.jstor.org/pss/2398320 )

I can access the journal until I'm at the uni in the morning, but the relevent part describes the tendency of domino chains with a fixed number of units leaning on the leading domino at any given time, to "approach and then fluctuate around a natural speed of collapse". If you don't have access to JSTOR, I can post the full text here.

Edit: The general effect we observed is very similar to the effect associated with the tautochrone curve.

War Wheel wrote:Didn't you show (on the other thread) that even a 50/50 split demonstrated this effect?

Almost. The reason 50/50 doesn't work out so well is there's not enough distance to fall with a split before a certain point. If you think about it, splitting at the first floor (bottom), even though there's 109 stories above as a driving mass, and it freefalls through one story, there's only one story to drop.

Did a series of runs on a 100 story structure with splits every tenth floor:

Is that seconds on the X?

Yes. Meters per second on the vertical axis.

I've done some interesting stuff with terminal velocity, acceleration, drag force, etc. No time to put it up, unfortunately. ASAP.

Been working on a step-wise crush calculator, here and there. The two processes can help validate each other. In working out some of the kinks with both the 1D physics engine simulation and the complementary stepwise algebraic version, a few interesting things popped up.

Most of the time there's a really good match between the two methods:


Overlays of positions
- physics simulation and stepwise computation
- varying top impactor velocity




Sometimes there is a significant divergence:


Comparison of velocities
- physics simulation and stepwise computation
- 1000 m/s impactor


The jagged blue line headed south is the physics engine model arresting a 1000 m/s single slab impactor six seconds into crush. The arresting model had only an equivalent average resistive force of less than 0.25m(t)g from supports; the same in the stepwise computation begins accelerating again during this time. The difference is that the debris zone in the phyics simulation is free to wiggle about up and down, though the slabs are inelastic (zero restitution) they still slap about a little and approximate 'loose(r) debris' if only in a 1D sense. This is in contrast to the fusing which occurs by definition in the stepwise model - the debris zone as well as the upper block is rigid. This turns out to matter more in some cases than the rigidity of the upper block.

A number of things came out of this calibration exercise, aside from a few minor bug fixes and a maxwell line of 0.0237*mg*fos determined for the fixed joints in the simulator. The two images are the bookends of the exercise.

What is the point of investigating the results of 1000 m/s drop speeds? Curiosity and calibration. Based on more everyday (? by that I mean one day out of all of history versus never) conditions of the speed of a single story drop, I'd ascertained the values of a couple of parameters that are key in step-wise calculations, the average equivalent resistive force over a story and the effective compaction ratio. By showing these values to be consistent even into extreme conditions, the stability of the simulator is confirmed. Having reliable values allows direct comparison of trials using both methods with the same input, as was done for the graphs above.

The stepwise calculator is in work, but usable. There were small disagreements between it and the physics simulation, primarily stemming from ill-defined story counts in the zones. I was forced to think about this problem and resolve it in order to calibrate against the analytical model. For example, a crush down involving a rigid upper block of 1 story leaves the top story intact at the end. Since one story is as small as the upper block can be, it means a collapse can never be complete under this scheme if exclusively crushdown. The top story can crush up (but as Bazant notes, even this cannot fully complete the crushing of the story!) but an exclusive crushdown leaves a minimum of one story behind, an intellectually untidy affair to me but one that was never a concern until trying to relate results to the slab model in the physics engine. Since the physics sim starts with slabs already 'compressed' and just lets them collide, a single story upper block is already compacted and the final pile height from the y_top coordinate is different.

This is, at its computational heart, less of a philosophical issue than an off-by-one index thing, but I decided to address it in a philosophical way by making it so the top story could crush itself and an exclusive crushdown could then go to completion as an exclusive crushup can, a thing of wonderful symmetry. Physically, one can think of it as having all the story mass concentrated in the compacted slab which is now top-justified within the story height, instead of bottom-justified as I'd done earlier, and which is perhaps more intuitive but not necessarily more realistic: the slab and office contents for the bottom story is at ground level and gets crushed but does not drop; the topmost slab is really the roof. Of course, this is what happens when trying to pigeonhole non-uniform structure into a uniform model. Both models allow each story unit to be configured in any desired way, but most basic experimentation is done with as much regularity as possible to keep it simple and explore the basic parameter space completely before moving on to greater complexity in structure.

The floor indexing strategy is just a bookkeeping thing to get the position-time values to match between the models and allow the roofline to reach ground level in the case of infinite compaction. None of the mechanics has changed. The strategy is not necessarily optimum or even finalized, but it will do for the time being.


It is quite satisfying to see two radically different means for computing solutions agree so precisely. The processes are not the same, though, so they should not be identical under all conditions, and they are not as the second graph above indicates. Exploration of the differences and reasoning about causes should be a fruitful avenue. Along the way to obtaining calibrated sim parameters, the difference between the two processes became evident under certain conditions where loose debris slabs and elastic structure differed significantly from rigid bodies (in ways quite intuitive). I'll post some of the highlights next.

In summary, in a 1D full entrainment model, the dynamics of collapse do not differ much between strictly rigid zones versus slab-spring/loose slab, but the dynamics of arrest can be radically different.

Caution is always advised in generalizing from the very limited conclusion drawn, however this is is solidly on track with the observation that a sack of flour will do more damage if dropped on your head as opposed to poured out. It starts to put numbers to the notion, though with the crude granularity of only a few mono-sized particles. The most severe limitation is the 1D aspect, with the particles constrained to interact along a line with only their nearest neighbors. I'm afraid the rules will change drastically with the addition of spatial dimensions, and the possibility of flipping (many times) on this general principle based on conditions will be high.

If collapse goes to completion in a momentum-only (no supports) inelastic configuration, then the entire event can be considered a single inelastic collision in the sense that the final velocity of the debris pile just before hitting ground is that given by a collision between the upper block and the lower block as point masses - ignoring gravitational effects, of course. So, in a horizontal arrangement of inelastic slabs, a single impactor hitting a line of 99 masses has the collective debris zone speed reduced to 1/100th that at the final collision. Makes perfect sense.

It also provides another means of validation of calibration obtained under other conditions. Here, the initial downward speed can jacked up to ridiculous amounts, so long as the time step is made sufficiently small. I chose a single top slab v0 of 1000 m/s to slam into 99 floors below under a variety of circumstances. In the stepwise model, it's only necessary to set support energy to zero to run with momentum-only, but the physics sim is like real life in that a stucture with no supports will fall uniformly at g. It would be possible to make the supports disappear at the right moment but it's more effort than I care to expend on it. The solution to get momentum-only is to turn gravity off. This is one of the cases presented next.

The graph below is a plot of top slab velocity vs time for 7 different impact scenarios all at 1000 m/s (plus delta for drop height of 0.8*3.7m in gravity cases). The green lines are with support in 1 g field, the red line is no support in zero g, and the blue are with support in zero g. Three support capacities corresponding to FOS of 1,5 and 10 were used in zero and 1g. The stretch and maxwell numbers are 0.196 and 0.0237 (@ FOS = 1).




Of the seven cases, the momentum only red line cannot arrest because it's the only one which has no support. This exemplifies the kind of collision transaction alluded to above. By completion, it's equivalent to a single inelastic collision between two bodies of masses m and 99m. When support fail energy is added, the results change in a regular way about the special trajectory of the red line except for arrest.

Three of the six remaining cases arrest, two under zero g and one in gravity. The one in gravity is the arrest case portrayed in the second graph of the earlier post. The similar case in the stepwise model does not arrest. What are the differences?

1) In the physics simulation, the slabs are connected by joints that are short travel non-linear springs. Several things remain in characterizing their dynamic behavior but this exercise has shown their energy consumption under failure to be consistent under a variety of settings. In high FOS trials, the bottom slabs are observed to recoil significantly with each impact above, indicating the joints absorb energy. They seem to be highly damped, like a shock absorber. This is an energy sink not accounted for in the stepwise crush calculator, but it could be in a generic way.

2) The slabs collide mostly inelastically in the physics simulation, but not perfectly so. There is a repulsive skin depth because singularities don't solve well in any system, and there is some elasticity. Because of this, the slabs in the debris zone will be loosely packed instead of artificially and unrealistic welded on collision. This results in a vertical newton's cradle sort of affair when impacts occur, and these repeated inelastic collisions dissipate energy.

Item 1 is not really characterized in any way other than the appearance of damping. The propagation of shock waves through this medium is undoubtedly peculiar to this bizarre structure of 1D semi rigid spheres and highly dissimilar to any real macroscopic system of interest.

Item 2 is quite a bit more realistic for physical slabs like tiles than the analytical model of accretion, however the analytical model may be more like real buildings than rigid slab approximations. To the extent that debris was loose, it applies but doesn't go far enough, it just hints at what the characteristics are in a discrete 1D model with relatively small numbers of particles.

The area under the velocity curves above are the total distances traveled, obviously. Those trials going to completion all have the same area.

The green lines, with gravitational attraction added as a driving force, are dramatically different in nature from the zero g (red and blue) in which initial kinetic energy of the impactor is the only energy input to the system. At only 0.5 seconds in, the velocity has decreased tremendously in all cases, but by 1 second the greens are already splitting off noticeably. By 2.5 - 3.0 seconds, the green lines have hit a minimum of velocity and are beginning to accelerate again!

This is the difference gravitational potential energy makes.

An average force at each story of 0.1185m(t)g (FOS=5) can arrest a unit mass impactor in just over 7 seconds but, with gravity, the same structure goes to completion in slightly less time. The oddball, though, is the FOS 10 case with gravity. It arrests in 6 seconds.

How do the two FOS 10 cases above, 1g and 0g, both of which arrest, compare to a single story drop of one slab on 99? The two cases above are shown below in the same color along with a third line which is a single slab drop initiation in a gravity-driven collapse. It goes to completion!

The previous three cases will be examined in greater detail shortly. In the meantime, here is a similar pair of graphs for the same configurations done in the stepwise calculator. The big difference is that these are in the spatial domain; that is, plots of velocity versus position instead of versus time, because the step increment is spatial in the stepwise calculation. Kind of an interesting twist. The two zero g cases of FOS 5 and 10 arrest as in the physics sim, but none of the 1g runs arrest.

Revisiting some earlier questions and comments from Major_Tom.

Major_Tom wrote:Conservation of momentum in a defined system with applied outside force during impact cannot be assumed.

What do you think about that?

I think I gave a far too wordy and roundabout answer to this before, in an attempt to dance around saying momentum is not conserved. You're absolutely right, and it's all in the definition of the system. Momentum is conserved, but not necessarily in the system being examined.

If the structure is floating in space, an impact between upper and lower conserves momentum for the entire structure whether support fail energy is included or not. Couple the structure to the ground and momentum is not conserved in the structure (ignoring continuous momentum increase from falling in gravity) because the supports represent an external force on the slabs.

Major_Tom wrote:I have not seen any cases of terminal velocity in OWE's examples.

No, most converge quickly on a constant acceleration, which means the force is scaling along with the mass accretion, and this is expected here. There are some avenues to explore in these simulations but it's never going to be as fertile ground for terminal velocity as direct computation.

Support capacity can be assigned in any way in step-wise computations, down to zero, and the calculations can proceed oblivious to whether or not the structure would stand statically. In the physics engine, more realistic constraints must be observed or the structure will fail without any help. I've tended to use a simple assignment rule that each joint/connection/support have the capacity to hold the load above times some factor of safety. The energy expended on support failure then results in a constant resistive force to upper mass ratio all the way down, i.e. a constant acceleration when considered in isolation. We've seen inelastic collisions of slabs also approach a constant acceleration, so naturally these models converge on constant acceleration.

The capacity at any given point can only be reduced to a level that still supports the static load above, and can only be increased to the point of arrest, of course. Adjusting joint strength immediately prior to collision is an option, but I'd rather move on to more complex configurations (i.e., get out of 1D) to achieve capacity reduction.

Always accelerations and one phenomenon called "terminal acceleration". How can we explain terminal velocities witnessed moving down the southwest corner?

Pretty much the way you indicate:

OWE may argue that if we consider resistive forces that are proportional to v or v^2 we can be expected to reach a terminal velocity.

The complement of that is mass accretion/shedding balance, subject to realistic constraints. I'm currently playing with this analytically to try to characterize the SW corner ejection front. One can only adjust mass when at the limit of no support energy (momentum-only floor slabs)!

True, but after only 10 floors? This means that the resistive term with v^2 dominates?

I'm getting pretty sharp transitions from only a v dependency, haven't even explored higher powers. I think, then, the issue is not so much turning the corner so abruptly but rather turning so soon with such a high velocity.

Variable stretch has not even entered my world yet, let alone variable entrainment cross section - which is not entirely independent of mass.

For OODRD we guess that a constant resistive force will dominate with smaller contributions from terms with v or v^2 (why not). How can the v^2 resistive term dominate so early?

I agree with this for the momentum component. With support energy, the uniformity of floor connections rapidly becomes insignificant as the collapse progresses, slightly tending the opposite direction.

It's in how quickly it comes on... sharpness of the transition. Parametrically, it can be made to transition very quickly. The key is in getting to the target velocity by the time it kicks in. The (alleged) velocity /time is barely possible to accomplish with the stepwise model, under some questionable assumptions, it would be a waste of time to try it in the physics engine environment with this particular configuration.

However, I had also largely abandoned getting any 'flexibility' or 'give' in the engine models, and now I find that there are conditions where even the simple 1D lineup gives compressible fluid behavior - the arrest mentioned above on which I'm going to comment. This relates to:

Major_Tom wrote:There is reason to suspect that if zone C ceases to exist as an independent, intact structure, the flow crushing down on zone A changes it's nature.

...

I do not beleive that zones B and C masses are of the same nature and would "pack the same punch" to the structure below. They may not be so easily interchangable as your models treat them.

Now, even this model exhibits this effect of smeared impulse. It's why it could arrest a 1000 m/s top slab impact, and I'll explain.

Demos do fail and "peter out" from time to time and there is no way to explain by merging zones B and C together as if both have the same interchangable effect.

The distribution and lack of rigidity of the debris zone is an important factor. It is not inevitable that a (sufficient) upper portion in motion should lead to total collapse, not at all. I think it might be for the towers because of a variety of factors, but not for buildings in general. Particularly not so for the low-story count concrete cell structures typical in verinage. These structures differ from the towers in that the mode of vertical crushing is fracture and refracture to crushing, the cross sectional area of impact is relatively much higher, and the horizontal coupling between adjacent building elements is proportionally much higher. Also less initial driving mass.

To recap, there were a series of physics engine simulation where one slab was 'fired' downward at 1000 m/s, instead of simply dropped, onto 99 slabs below. As the support capacity was increased incrementally, the progression was arrested as shown in the graph above which compares against a pure analytical result of the same configuration which does not arrest:



The 1000 m/s impactor cases were compared to a simple top slab drop of one story with the same support capacity as the case which arrested.



The passive drop collapsed to completion! This is curious and demands an explanation. Is the engine screwing up? No. It is an artifact of the configuration but the engine is performing correctly. It turns out to be the two factors I mentioned above - lack of rigidity in the debris zone and the lower portion, interacting in subtle but important ways to reach a condition where the impulse train from above can be resisted by the lower portion and collapse arrested. Detail to follow.