The 9/11 Forum

Ozzie makes a very good point! The FOS is really a range or value of lower limit. Each structural member has fixed performance characteristics aside from conditions of heat. Engineers size member based on presumed load stresses and will select a section that provides a margin of safety. If 6 bolts will support a connection, they may spec 8 or high strength bolts if the location does not permit more bolts. Or they may add welds. Not only can you not design a structure to a specific FOS because you don't have the range of sizes of structural elements, but the loads are in a state of flux somewhat... live loads vary based on occupancy and winds. So the loading is dynamic and changes. Designers will likely pick a worse case scenario as they had to in the CitiCorp building with the trusses which were to resist wind shear. They failed to consider that at first and had to retro fit the trusses for the 100 yr storm.

It's probable and predictable that the individual FOS for various member will not be the same because the dynamic nature of the loads and the fixed nature of the chosen steel's properties.

So we use *averages*. One member with a lower FOS will unlikely fail the entire structure. If it does fail its loads are redistributed and this is usually well within the FOS of the members which pick up the redistributed loads/stresses.

OWLie's GIFs show that as supporting members are removed the stresses in remaining ones are increased and FOS is drops. If the process continues the structure reaches a point of no return where the FOS has dropped below 1 and a global collapse occurs with the appearance of *rapid onset*.

For the twins this is basically a matter of determining which columns were *removed* on impact and the additional factors in play which lower the performance (yield strength) of the steel. We know that among them are heat (from fires) and loss of bracing. The CD scenario would include the addition of means of destroying column strength such as placed or delivered by the plane incendiaries or placed explosives.

However the CD proponents take their theories BEYOND initiation and propose that the towers were / had to have been destroyed sequentially from top to bottom by placed explosives.

Their claim is based in a few beliefs and "observations".

1. That the speed of collapse was too rapid approaching free fall acceleration
2. That the collapse could not crush and pulverize the concrete to *dust* with no slab sections present post collapse
3. That the collapse could not crush all the contents, fluted decking, piping, furniture, fixtures and occupants to such tiny fragments or dust.
4. The presence of a high percentage iron micro spheres in the dust explained only by incendiaries or explosions which produce high temps and the atomization and dispersal of liquid iron.
5. The extreme heat under the pile post collapse where apparently molten iron was seen.
6. Heavy material ejected considerable distances from the *collapsing* structures.
7. And in the case of Bldg 7 no significant structural damage from a plane impact.
8. The notion that the asymmetrical plane impacts could not lead to sub symmetry of collapse
9. The appearance that building 7 collapses much like a typical building demolition (in on itself)
10. The reports of witnesses of explosive sounds after the plane strikes up till collapse.
11. That the FOS was at least 3 to as much as 20 and the buildings were too strong to fall as no columns were crushed or could be as the weight was static (no pile driver)
12. The appearance of *squibs* in the twin towers
13. The appearance of molten iron pouring from the NE corner of tower 2 at the 8th floor.

All of the above represent the basis for the belief that the towers were taken down by some manner of engineered intervention and not by the plane impacts and ensuing fires.

SanderO wrote:Ozzie makes a very good point! The FOS is really a range or value of lower limit. Each structural member has fixed performance characteristics aside from conditions of heat.

It's an excellent point. There are (sometimes large) design variations in FOS between member types and individual members of a given type in different locations, and statistical variation of manufacture, assembly and in-service loading. The FOS of the entire structure is an almost meaningless idea. Some places it may have been under 2, other places higher than 6. All bets are off after initial damage is imposed.

For modeling purposes the choice(s) come down to estimation, desired accuracy and expediency. Talking about FBM specifically now in what follows. If ones assumes a nominal average FOS of some value, which is different from the FOS of the building as a whole, it makes sense to apply the nominal value to each member on initialization. The capacity of a member then dictates actual load applied in the beginning of the trial, being derived from the estimated capacity and the assumed FOS.

Two observations:

1) the real loading probably varied significantly (therefore non-constant FOS)
2) the actual FOS/load values are probably dependently clustered about any well-estimated capacity

So, while any particular model configuration has about a nil chance of representing the actual conditions present in the intact structure, all reasonable well chosen parameter ranges should be fairly close to whatever the actual was. In a model of this sort, with discrete steps and cascades, it's possible - even likely - to have (many) very sensitive bifurcation points, where the results become drastically different because of some small change in structure or applied damage. Therefore, while it can be said that good engineering estimates should narrow the input domain, the output domain may be all over the map, anyway.

For an FEA simulation, many of the same types of sensitivity to initial input exists. I've had more than one (simple!) FEM be chugging merrily along and then blow up spectacularly in one time step. Tweak one parameter 10%, rerun, no problem. More often than not it's a singularity in the calculations, which is a different thing, but the Rube Goldberg effect can plague FEA, too.

FBM is a much lighter weight calculation. When my program is optimized, it will be possible to run thousands or even millions of trials with different configurations in a reasonable period of time. Why do that? Because it allows a fairly comprehensive statistical profile of the results obtained by random variations about the nominal. It allows exploration of the uncertainty in a systematic and unbiased fashion. There are likely to be clusters of relative insensitivity and vice versa, which will characterize the true nature of global capacity with the tower column geometry.

Tendencies and traits, not simulation of actual events.

Despite how fast the computations can be done, the solution space is far too large to explore by brute force in excruciating detail. Fortunately, that's probably not necessary. With sensible restrictions on input, a first test series can check the major parameters over a coarse granularity and the interesting input ranges selected for closer inspection. The process will be heuristic and the path determined largely by intermediate results.

A model has a set of static input parameters. By static, I mean these are supplied in the setup of a trial and remain constant throughout a trial. These include all the initial capacities assigned to members, the multiplicative factors like FOS, local redistribution coefficients and the like. Even in a small model like this, the number of such parameters runs into a few hunded, mostly because of per-column attributes (capacity). A combinatorial explosion happens as parameters with uncertainty are added.

Taking nominal global FOS as an example. FOS is applied to everything. To try 10 separate FOS values means 10 trials with all other values held constant, to see what the sensitivity to FOS is with all other things equal. Then you can tweak one other parameter of the hundreds and rerun all FOS again but to check all viable parameter ranges in this way is not possible. That's why it makes sense to run a few thousand random variations in large groups of parameters (like capacities) and systematically check only the primary values like FOS.

Even with the perimeters grouped in threes, and with the initial damage, that's still over a hundred member elements. Add in the rest of the static parameters plus dynamic ones like fire damage mappping, and then there are potentially hundreds more parameters, each subject to uncertainty from small to large. It pays to reduce uncertainty as much as possible, of course, as a starting point. Those factors with irreducibly large uncertainty (capacity loss on column X at time T from heating ) need to be explored first in a coarse subdivision to eliminate uninteresting results in broad swaths. Subequent trials will be restrict to subdivisions of interesting regions only.

Those familiar with the Newon-Raphson method of finding zeros of a function will understand that it is possible to make a pretty stupid program execute an effective search of a given space without doing N equal subdivisions. The technique can be augmented by Monte Carlo methods. What I'll do will probably be somewhere between the vanilla half-interval recursion and this, since half of that I don't understand. The model engine can select interesting regions based on narrowing down the transition points to collapse / no collapse on the globals like nominal FOS.

Getting IronRuby on .NET up and running. It's a bit faster that the 1.8.6 MRI engine I've been using, but that's not the reason for using it, it's the access to .NET libraries and interoperability with C# and C++ (there's the speed, if needed). Ruby libraries cover a decent amount, and are generally preferable, but .NET is pretty big and offers a lot more facilities.

With IronRuby, supposedly, all of both worlds is available. I say 'supposedly' because it hasn't been a snap to get things running, and I'm not sure the WPF support is ready for prime time.

I can justify the investment of time quite simply as I need to get a good scripting framework together for a separate data analysis project, and there's a great deal of overlap. Reduction of data tables, preparation of graphs, dump to PDF, etc. The pace here will be slow, but augmented by spinoffs from the work that nudges this aside on a near daily basis.

OWE, have you seen this?

http://www.luxinzheng.net/publication3/Progressive_Collapse_ICCCBE2008.pdf

No, I hadn't. How excellent. Guess I'm not the first to think of this (damn). Very interesting find Major_Tom, thank you!

The link opens up a door to all manners of related study. Seems I'm well behind the curve. Some study to do. Thanks again for the gold.