The 9/11 Forum

I thought of starting a thread for this about two or three years ago. It was a recent post of SanderO's in another thread that convinced me it was time, so as to avoid going off topic. It isn't necessarily destined to go anywhere, but can start.

Fiber Bundle Models (FBM) is what it sounds like, a bundle of 'fibers' used to model phenomena involving some form of cascading failure. They are a relatively new and only lightly explored concept. From what I've seen, not developed to the tool stage yet. Well, all that's about to change, heh heh. More descriptions will follow later.

SanderO posted a "cartoon" (as he called it) here which sketched out a core failure sequence:

SanderO wrote:I suppose the load redistribution can be simulated in a computer model and by removing n # of columns.
...
I sketched out a graphic *simulation* of how the core could likely failed in tower 1. The obvious unknown factors are:
the number of columns destroyed by the plane impacts
the actual Factor of Safety
the amount of heat weakening and where it was applied

The attached cartoon develops a scenario where 4 of row 500 columns are destroyed by the plane impact.
...

and what follows is the response I was going to give there.

It's exactly the sort of thing that's amenable to fiber bundle modeling. Rather complex and groundbreaking from the academic standpoint... maybe not so practically useful, that remains to be seen. You've laid out a very nice scenario and, importantly, done legwork bringing the relevant architectural details together. It's actually quite inspiring and, honestly, it takes a bit to do that these days.

I played with FBM but one impediment to useful results was the lack of meaningful input. I'm not by nature a "prints" kind of person and tend to lose focus/interest if I have to dip into the world of tables, specs, drawings and so forth to assemble a picture. In a word, I'm better suited for variables. Then I see this "cartoon" and it suddenly jumps out - this is a template for building a fiber bundle model of core collapse!

Could be worth pursuing.

To get a simulation on par with Enik's sort of work or better surely seems within reach of anyone having the skills, tools, time and interest.

Crickets.

(I believe Enik lacks the tools at the moment, last I heard. )


What a fiber bundle model can do is quite limited by comparison; there's no attempt to simulate at all in the conventional sense. It's much more a 'what if?' process of an exploratory nature. I'm surely biased, but I think the tradeoff is loss of specificity for gain in honesty.

Finite element simulations at the highest level are indeed impressive, but how accurate? As detailed as some simulations are and powerful the environment may be (e.g., NIST and even Enik), how accurate have the results really been? Did WTC7 wrinkle like a beer can? Do floor-wall connections stretch to the next floor below (an Enik thing in one simulation)? I'm not knocking these works because they are quite ambitious. I'm confident that if someone like Enik were to put the level of detail into just a reasonable approximation of the participating members in your cartoon, it would have the potential to be accurate and be very interesting in any case.

How would you know if it were accurate? Exploring the nearby input parameter space would tell you about the sensitivity to various assumptions and configurations, but here's where time is a factor. Enik's larger simulations took many hours to crank out a few seconds of simulation time. It isn't practical to run many trials without taking a long time or having many machines working.

By contrast, I envision fiber models as being useful for bounding cases. While the calculations of (e.g.) load and response are floating point calculations representing continuous variables, the majority of the process is discretized at a much coarser granularity than FEM, and generally dispenses with time altogether. This makes for very fast computations. So fast you can run millions of trials in a reasonable amount of time. If it is possible to frame the system in terms of rules for state transitions - and get meaningful results - then the problem becomes artfully boxing the possibilities.

In concrete terms, each of your columns is an element, to use the FEA lingo. In FBM, the element is called a fiber, but same thing. In FEA, a column could be thousands or millions of elements to provide a realistic simulation. In FBM, a column is ONE element. It has as complicated a state description as you want, but it needn't be any finer spatial granularity than that. Along with the state schema there are state transition rules which govern the evolution of the element's composite state.

Each element interacts with other elements of the global assembly according to whatever rules you prescribe. The typical FBM I've seen in the literature barely scratches the surface, rarely going deeper than global versus local load sharing. The possibilities, however, are endless. While FEA elements interact almost exclusively with nearest neighbor for mechanics of materials, the FBM elements interact according to rules and associated functions and have virtually unlimited connectivity and behavior.

This might seem to be an unphysical hodge-podge and, in the wrong hands, I'm sure it is.

For typical FBM: Global Load Sharing (yes, another acronym - GLS) describes the situation where there is one load divided equally amongst all load bearing elements. Local Load Sharing (LLS) is where the load formerly held by a failed element is redistributed to the n nearest neighbors in the connectivity chain, weighted as desired. Between GLS and LLS with only the adjacent elements lies the entire spectrum of interaction between elements in the model.

Morphologically, the FBM is a network but literature I've seen is mostly confined to very simple systems comprised of linear chains of elements with simple load redistribution schemes. To be honest, I classify the connectivity as a network but I don't recall seeing a formal declaration of such nor ever seeing even an example of a '2D' grid of elements, only chains. It's possible I'm taking this a bit further than its exponents envisioned, but is that necessarily a bad thing?

To apply an FBM to SanderO's cartoon, I'd consider the natural connectivity expressed by the building structure. The primary elements would be the columns, core and perimeter. These are assigned attributes (such as load displacement response) according to their specifications and can be logically aggregated into composite elements like corners and walls. The hat truss would be a special element and quite the wildcard.

Each of the columns interact and hat truss interact according to rules specifying how load is to be applied to each element at each step. These rules depend on the state of the prior step and evolution/transition functions. For example, the initial state of the columns is intact in static equilibrium with assigned loads per design. The impact severed columns are then discretely removed for the second step and the load formerly borne by these columns is divvied between the remaining columns in particular way, and the load response is calculated. This response, in turn, drives the calculation of load allocation for the next step. Which determines the response of elements. And so on.

The strength of the method is how fast every conceivable scenario can be examined, where the scenarios can represent one extreme to the other. Don't know how the load gets partitioned between adjacent columns on the same wall versus the hat truss? Try all the possibilities and see what happens. I mean, most interest lies with seeing the paths to global failure, a sort of boolean decision - failure or new static equilibrium. That's pretty simple.


Where FEA tries to apply material mechanics to 'predict' the evolution of the system, given only material properties and initial values for dynamical quantities, FBM tries to chart the course of failure through individual elements given the assumption that the dynamics adheres to certain rules. Thus one does not try to answer the question "what happens to the system over time?" but rather "what happens to the system if <insert clause>?"

Clauses, as referred to in the last sentence of the last post, can be evaluated much more quickly than you can think of them. Where a very complex FEA can take many hours to set up, and many more hours to run, it might take days to figure out how to apply the FBM model to the problem, a few hours to write and test a program, then only minutes to get all the answers sought from that framework.

Interesting OWE. By the way if you want you can place the cartoon as a image into a post. I don't use an online hosting site so it is presented as an attachment for download... an obvious inconvenience.

My cartoon approach has some very simple rules (probably way too simple) which was basically based on the notion that columns will buckle and fail when their yield strength is exceeded. This is only possible from strength reduction due:

to heating
to mechanical destruction of part of the column's cross section (cutting explosion)
to removal of bracing
factor if safety

I assumed that the loads of a failed column were shared equally to only adjacent columns. I also assumed the heat / destruction was the same value / rate and in the cartoon...symmetrical about the north south axis.

A program where you could dial in the various inputs at each iteration... the program could produce snap shots each time there is a failure (iteration???)... Then the iterations could be assembled into a GIFS showing the failure progressing through the core.

I am wondering about how to account for lateral movement of the (top) facades from the core buckling. We did did see the facade of Tower 1 top slip outside the facade of the lower part as it came down at core failure. The whole top seemed to have moved diagonally bit.

This seems like a simple program to write, but not for this architect!

SanderO wrote:My cartoon approach has some very simple rules (probably way too simple) which was basically based on the notion that columns will buckle and fail when their yield strength is exceeded. This is only possible from strength reduction due:

to heating
to mechanical destruction of part of the column's cross section (cutting explosion)
to removal of bracing
factor if safety

I assumed that the loads of a failed column were shared equally to only adjacent columns. I also assumed the heat / destruction was the same value / rate and in the cartoon...symmetrical about the north south axis.

All perfectly suited to this approach.

A program where you could dial in the various inputs at each iteration... the program could produce snap shots each time there is a failure (iteration???)... Then the iterations could be assembled into a GIFS showing the failure progressing through the core.

That's it, you've got it.

I am wondering about how to account for lateral movement of the (top) facades from the core buckling. We did did see the facade of Tower 1 top slip outside the facade of the lower part as it came down at core failure. The whole top seemed to have moved diagonally bit.

Lateral motion is a bit more difficult to account for, and somewhat arbitrary if it is. We can start with vertical only and see how far that goes.

This seems like a simple program to write, but not for this architect!

You do the pretty pictures and pull down those specs, and I'll write the program.

SanderO's core diagram:

That diagram seems to neglect the inward bowing of the south face, the NIST impact simulation (the core damage was narrower and deeper) and the southward tilt following the initiation.

I tend to look at it more as a ledger sheet where the rectangles are cells for entry. Anything can be plugged in. Want inward bowing as cause? Introduce it at a particular step and ramp it up as desired. Want inward bowing as effect? Twiddle the parameters on other columns until it emerges. Of course, IB here will probably not be distinguished from any other failure mode.

The beauty of this diagram, though, isn't what it depicts but how it depicts it. It's so suggestive of how FBM treats the problem of cascading failure. There is considerable focus on the finite element method in the general engineering community when trying to assess overloading situations, to model and predict actual deformations and so on, and this is good but quite ambitious and fraught with many perils even when one has the resources.

The FBM approach is radically different and in some ways ridiculously simple by comparison, but I think something meaningful can be extracted using this method so long as the interpretions are handled carefully. SanderO's diagram, with the arrows indicating load redistribution (state transitions), is exactly the sort of simple process which is easily described by FBM. Unlike a large FEM simulation which unfolds slowly with great detail, I suspect some FBM runs of this system will be done before the mouse-up on the 'Run' button is complete, if you get my drift. We're talking about arithmetic calculations numbering in the hundreds or thousands, trivial stuff when automated but a huge pain by calculator and hand.


What say you, SanderO? Any particular reason for deviating from NIST, or omitting IB?

It's been a couple of years since I perused the literature on fiber bundle models, and I never looked that deeply in the first place. Turns out a few things need to be mentioned up front.

What I want to do with SanderO's graphic is probably more correctly termed a lattice model, though that's not exactly it, either. Same family of models.

Fiber bundle models were first treated in 1926 by F. T. Peires, Tensile Tests for Cotton Yarns, a most practical and direct application. The idea wasn't picked up again for quite some time but now there has been over 30 years of solid interest and research, to the point where the field is characterized as:

A very important class of models of material failure are the fiber bundle models (FBM), which have been extensively studied during the past years.

http://arxiv.org/pdf/cond-mat/0107284.pdf

I thought of it as a sort of fringe interest but apparently it's considered an important area of study capable of generating constitutive relations. It finds application in a wide range of areas from geoscience to composite materials, and obviously textiles.

This might be the first ever application of this model family to building failure.

So, to start it off, let's talk about the rules. SanderO has offered the following as associated with the diagram:

SanderO wrote:columns will buckle and fail when their yield strength is exceeded. This is only possible from strength reduction due:

to heating
to mechanical destruction of part of the column's cross section (cutting explosion)
to removal of bracing
factor if safety

I assumed that the loads of a failed column were shared equally to only adjacent columns. I also assumed the heat / destruction was the same value / rate and in the cartoon...symmetrical about the north south axis.

This is a good starting point. SanderO describes purely local load sharing between adjacent elements. That can be twiddled in more complex ways is subsequent trials. I can definitely see load distribution being extended in a weighted fashion to many close neighbors, especially along the perimeter, and global sharing entering via the hat truss.

Ignoring for the moment both the hat truss and floor assemblies (they will probably be treated as external input, though clearly constrained by the couplings to columns), there are two types of elements, core and perimeter columns. A formal characterization of these elements is necessary.

A column has a capacity which determines the load under which it will fail. We shall not at this point discriminate between the different yield points, elastic or plastic response, or geometric nature of failure mode. Rather, we'll simply say that, when a column is loaded above capacity it fails and is removed from the model. An intact column starts with a given design capacity - which for now is its only explicit attribute - and from that is derived an effective capacity which is some fraction of that value.

The manner in which the effective capacity is determined may as well be arbitrary. At some point, it will be nice to plug in expressions for capacity reduction due to thermal response, eccentricity, increase of unsupported length, etc. But for now it probably suffices to just simply assign reductions with various distributions and see the response. Broad swaths of uninteresting results can be identified and swept aside.

For the first pass, then, it's a pretty simple system. The only explicit state variable is capacity. Under the hood, we can talk about how heating is applied in some gradient or whatever, but the factors merely result in calculated capacity based on simple fractional reductions.

Perimeter columns will have uniform design capacity, core columns have differing specific relative capacities of the different core columns. I'll ask you to gin up some numbers, SanderO, when you get the chance.

I propose working with synthetic dimensionless units, most likely normalizing the total load imposed by the upper section to 1. There is no ideal set of units in that everything comes out to a pretty rounded number, unfortunately, so the global load may as well be a unit load. Then, in a uniform capacity scheme with global load sharing, each column of a model with N columns will bear 1/Nth of the load. With 236 perimeter columns (I noticed you omitted corner columns, which I have seen mentioned but know nothing about) and 47 core columns, that makes 283 total columns so each would support 0.35% of the load and, with an FOS of 3, have capacities roughly of 1% of the load.

As mentioned before, uniform capacity is very dull and can make for trivial results. In a global load sharing scheme with uniform capacities, all members will survive if total capacity exceeds loads, and all members will fail in one step if load exceeds capacity. Dull!

Local load sharing can be interesting, even with uniform capacity, so I may start with that, but it will be necessary to move quickly on to reasonably accurate core column capacities.

Since there isn't (or at least needn't be) a time dimension involved, it's natural to ask what meaning a step has. In FEA, a step is a time step, and generally a small one. The FEA system evolves by calculation of physical quantities such displacements/velocities/strain for each element over sequential differential time slices.

Here, a step is a simply logical contrivance. It doesn't represent a length of time but, even if it did, successive steps would not represent equal time intervals. About the only thing it has in common with time is that successive steps do represent the action of the system going forward in time. A step here is a logical iteration of the transformation of the system according to applied rules.

For a typical, simple FBM it goes like this:

Step 1: Load is applied in a specified fashion to a collection of elements with a distribution of capacities. If any applied load exceeds the capacity of a given member, it is failed and removed from the collection, with the load it formerly bore allocated to other members according to the load sharing rules.

Step 2: Repeat step 1 with the new collection of remaining members and load distribution.

And so on until no change occurs from one step to another (equilibrium). Really straightforward, it is. The failures may stop, which in this context means no initiation, or they may proceed to completion by failing all members.

In global load sharing, the load formerly borne by failed members is allocated equally to the remaining members. Local sharing amongst only adjacent columns splits the load of a failed column with the two (or more, depending on geometry) nearest columns. If there is an FOS of two, and the middle column of three is removed, then the load of the two adjacent columns increases by 50%, and the actual margin of safety for those columns is halved from the design point. In a row of four perimeter columns, the removal of two in the middle leaves the outer two with no margin of safety.

It's obvious that global sharing is the most conservative for survival, and that even local sharing can be dull in the sense that the threshold of catastrophic failure is more easily reached. Neither is purely local sharing any more realistic than global, and the true interest is in the shades of gray between the two.