This text addresses how and in what sense Bažant's model, initially stated in [B 2001] and [BZ], and refined in [BV] and [BLGB], with clarifications in [BL] (see below, References), can or can not apply to real world buildings. The author of this text is a layman in the matters in question, yet felt the need for an explanation to exist in view of recent discussions in other threads. Ideally, this analysis should have been done by someone in the structural engineering field who could write such an explanation authoritatively, but apparently noone felt compelled enough to do so, leaving the author the burden to write it despite his lack of qualification. The inaccuracies arising from it may be the result of the lack of author's understanding of engineering principles and practices, and the author welcomes any constructive criticisms aimed at addressing such inaccuracies as long as they are stated with arguments other than incredulity and disbelief, and preferably with sound engineering arguments.
The need for this text arises from the fact that, as stated by another forum member recently, many skeptics of the theory of a government conspiracy misinterpreted Bažant's papers, methods and conclusions, including the author at some point in past, plus there's a new wave of 9/11 deniers who are misinterpreting them for discrediting purposes. So it's important to know how far the scope reaches; where the model can and where it can't be applied to real buildings. Some people, in both sides of the discussion, believe that the models are intended to be applied literally to buildings as a description of what happened. Others, from both sides of the discussion, think that the model is just a theoretical limiting case with no applicability to real buildings. The author has believed both of these things at different points in time, and hopes to prove here that both are incorrect.
A model is an idealized representation of a physical phenomenon in the real world, which can be used to study its behavior and make predictions about it. An example of a model is Newtonian physics. It can describe a wide range of behaviours of our environment and that makes it useful for us.
However, most models, if not all, have limitations. Newtonian physics don't describe the behaviour of objects traveling at a speed close to that of light, for example. That led to the creation of a revised model, special relativity physics, further generalized by general relativity physics. That model still has limitations. And yet, Newtonian physics are still useful as an approximation, as long as we know where the model's limits are and don't ask more of it than what it can give.
When we are confronted with the task of modeling a building collapse, there's one thing that becomes immediately obvious. It's flat impossible to make an accurate model that copes with all possible structural failure causes and modes as they happen in the real world. The reason is simple: the process is too chaotic and even tiny variations in the pressure, inclination, density or other parameters of the structural and non-structural elements can make e.g. a column bend in different angles, with the potential of causing it to impact other columns that would otherwise not be affected by the failure of that column, or to miss them where they would be affected. That's just a very tiny example of how chaos affects a collapse. Even any attempt at a computer simulation of a collapse will be subject to uncertainties in a myriad of tiny parameters such as the exact real-world positioning of a column or girder or the exact distribution of the strength of a welded connection, which will inevitably make the simulation deviate from the real world collapse as it happens, to the point that only a handful of columns, if at all, would show a movement which has any resemblance to the real world. This aspect is clearly demonstrated by the NIST simulations of WTC7 collapse, in which, even if the collapse development exhibits some features of the actual one, the details differ substantially.
To put a bit of order in that chaos, in order to get a workable model which can be rigurously formalized, Bažant made a simplification to his modeling of the WTC collapse: the column impacts would always be axial, regardless of how they would be in a real case.
What is an axial impact:
This brought consequences to his formalization. One of them was that in such a situation, using the parameters for the towers, the collapse progression would happen by first crushing down the bottom part of the building, and then crushing up the top part. This is the infamous crush-down/crush-up sequence which has been in the center of most misunderstandings and criticisms to Bažant's work.
Obviously, these axial impacts between columns would be quite unlikely for many buildings and failure modes. For example, a possible outcome of an impact of a vertical column with an inclined surface is that the column is dislodged rather than bent, depending on surface angle and other factors, resulting in the column acquiring an inclination itself and losing its ability to carry loads, after using probably less energy than a plastic axial deformation would require.
What is a plastic axial deformation:
Despite that and other simplifications, the model was still useful to draw conclusions from it. In particular, already in [B 2001] Bažant stated (p.1):
From there it is immediately obvious that Bažant was perfectly aware that there were important differences between his model and the real behavior since the very beginning. That didn't prevent him from going on drawing a crude estimate from it. Had he modeled crush-up occuring simultaneously with crush-down, that would probably mean a factor of two deviation in some of his results. Yet that wouldn't affect the outcome, as he explained in an addendum of [BZ] (p.370):
The corollary is that it is possible to draw some conclusions even from an inexact model, just not any conclusions, and we have to be careful about which ones can be extracted and which ones won't match the real world to any sufficient degree of accuracy at all.
This approach is used several times in [BLGB] to make comparisons and extract several consequences. For example, the free fall myth is finally put to rest in some fronts, including videos from the start of the collapse and seismic records. Today that myth seems to be losing some strength, fortunately.
In particular, the comparison with videos requires a whole section dedicated to compensating the model for the tilt that it does not provide, because it assumes that the top falls straight (see p.901, "Analysis of Video-Recorded Motion and Correction for Tilt").
In order to fine-tune a model so that it more closely resembles the real world whenever the parameters being considered can't be solved in the theoretical field, measurements are taken if possible and applied to the model. That is what NIST did when several severity cases were elaborated, of which the most severe one was picked in view of the results. This proceeding, which has been widely used to blame NIST for making up data, is just a standard parameter adjusting practice for the model to fit reality as closely as possible.
That proceeding is indeed described in the [BLGB] conclusions (p.905, emphasis added):
Here's an example of such finetuning also proposed in [BV] (from the abstract, p.308, emphasis added):
Finally, there have been some claims that the comments in [BL], in particular points 4 ("Can Crush-Up Proceed Simultaneously with Crush Down?", p.917) and 5 ("Why Can Crush-Up Not Begin Later?", p.919) mean a direct application of the crush direction part of the model to the WTC 1 and 2 as if it was what actually happened. That is wrong. That section is dedicated to discussing the theoretical basis of the crush direction part of the model, about which [G 2008] objected, as this excerpt shows (p. 915):
Since the author's always been not-so-good for writing closing words, please indulge him in his use of [BLGB] to do it:
(Thanks to ozeco41 for raising the point that led to writing this text.)
References
The need for this text arises from the fact that, as stated by another forum member recently, many skeptics of the theory of a government conspiracy misinterpreted Bažant's papers, methods and conclusions, including the author at some point in past, plus there's a new wave of 9/11 deniers who are misinterpreting them for discrediting purposes. So it's important to know how far the scope reaches; where the model can and where it can't be applied to real buildings. Some people, in both sides of the discussion, believe that the models are intended to be applied literally to buildings as a description of what happened. Others, from both sides of the discussion, think that the model is just a theoretical limiting case with no applicability to real buildings. The author has believed both of these things at different points in time, and hopes to prove here that both are incorrect.
A model is an idealized representation of a physical phenomenon in the real world, which can be used to study its behavior and make predictions about it. An example of a model is Newtonian physics. It can describe a wide range of behaviours of our environment and that makes it useful for us.
However, most models, if not all, have limitations. Newtonian physics don't describe the behaviour of objects traveling at a speed close to that of light, for example. That led to the creation of a revised model, special relativity physics, further generalized by general relativity physics. That model still has limitations. And yet, Newtonian physics are still useful as an approximation, as long as we know where the model's limits are and don't ask more of it than what it can give.
When we are confronted with the task of modeling a building collapse, there's one thing that becomes immediately obvious. It's flat impossible to make an accurate model that copes with all possible structural failure causes and modes as they happen in the real world. The reason is simple: the process is too chaotic and even tiny variations in the pressure, inclination, density or other parameters of the structural and non-structural elements can make e.g. a column bend in different angles, with the potential of causing it to impact other columns that would otherwise not be affected by the failure of that column, or to miss them where they would be affected. That's just a very tiny example of how chaos affects a collapse. Even any attempt at a computer simulation of a collapse will be subject to uncertainties in a myriad of tiny parameters such as the exact real-world positioning of a column or girder or the exact distribution of the strength of a welded connection, which will inevitably make the simulation deviate from the real world collapse as it happens, to the point that only a handful of columns, if at all, would show a movement which has any resemblance to the real world. This aspect is clearly demonstrated by the NIST simulations of WTC7 collapse, in which, even if the collapse development exhibits some features of the actual one, the details differ substantially.
To put a bit of order in that chaos, in order to get a workable model which can be rigurously formalized, Bažant made a simplification to his modeling of the WTC collapse: the column impacts would always be axial, regardless of how they would be in a real case.
What is an axial impact:
This brought consequences to his formalization. One of them was that in such a situation, using the parameters for the towers, the collapse progression would happen by first crushing down the bottom part of the building, and then crushing up the top part. This is the infamous crush-down/crush-up sequence which has been in the center of most misunderstandings and criticisms to Bažant's work.
Obviously, these axial impacts between columns would be quite unlikely for many buildings and failure modes. For example, a possible outcome of an impact of a vertical column with an inclined surface is that the column is dislodged rather than bent, depending on surface angle and other factors, resulting in the column acquiring an inclination itself and losing its ability to carry loads, after using probably less energy than a plastic axial deformation would require.
What is a plastic axial deformation:
Despite that and other simplifications, the model was still useful to draw conclusions from it. In particular, already in [B 2001] Bažant stated (p.1):
The details of the progression to failure after the decisive initial trigger that sets the upper part of the structure in motion are of course more complicated. The upper part of the structure, for example, tilts as it falls; furthermore, because the structure is a framed tube with floor beams of large spans, the impacted floors may collapse ahead of the tube, thus depriving the tube wall of its lateral support against global buckling. Regardless of these and other details, however, we can make the following two simple and crude estimates of the overload ratio of the columns of the floor just below the critical floor that triggered the catastrophic chain of events.
From there it is immediately obvious that Bažant was perfectly aware that there were important differences between his model and the real behavior since the very beginning. That didn't prevent him from going on drawing a crude estimate from it. Had he modeled crush-up occuring simultaneously with crush-down, that would probably mean a factor of two deviation in some of his results. Yet that wouldn't affect the outcome, as he explained in an addendum of [BZ] (p.370):
Once accurate computer simulations are carried out, various details of the failure mechanism will undoubtedly be found to differ from the simplifying hypotheses made. Errors by a factor of 2 would not be terribly surprising. But that would hardly matter since the analysis in the paper reveals order-of-magnitude differences between the dynamic loads and the structural resistance. Crude order-of-magnitude estimates made easily by pencil suffice in this case to rule out various intuitive theories that were advanced to explain the collapse.
The corollary is that it is possible to draw some conclusions even from an inexact model, just not any conclusions, and we have to be careful about which ones can be extracted and which ones won't match the real world to any sufficient degree of accuracy at all.
This approach is used several times in [BLGB] to make comparisons and extract several consequences. For example, the free fall myth is finally put to rest in some fronts, including videos from the start of the collapse and seismic records. Today that myth seems to be losing some strength, fortunately.
In particular, the comparison with videos requires a whole section dedicated to compensating the model for the tilt that it does not provide, because it assumes that the top falls straight (see p.901, "Analysis of Video-Recorded Motion and Correction for Tilt").
In order to fine-tune a model so that it more closely resembles the real world whenever the parameters being considered can't be solved in the theoretical field, measurements are taken if possible and applied to the model. That is what NIST did when several severity cases were elaborated, of which the most severe one was picked in view of the results. This proceeding, which has been widely used to blame NIST for making up data, is just a standard parameter adjusting practice for the model to fit reality as closely as possible.
That proceeding is indeed described in the [BLGB] conclusions (p.905, emphasis added):
Several of the parameters of the present mathematical model have a large range of uncertainty. However, the solution exhibits small sensitivity to some of them, and the values of others can be fixed on the basis of observations or physical analysis. One and the same mathematical model, with one and the same set of parameters, is shown to be capable of matching all of the observations, including: (1) the video records of the first few seconds of motion of both towers; (2) the seismic records for both towers; (3) the mass and size distributions of the comminuted particles of concrete; (4) the energy requirement for the comminution that occurred; (5) the wide spread of the fine dust around the tower; (6) the loud booms heard during collapse; (7) the fast expansion of dust clouds during collapse; and (8) the dust content of the cloud implied by its size. At the same time, the alternative allegations of some kinds of controlled demolition are shown to be totally out of range of the present mathematical model, even if the full range of parameter uncertainties is considered.
Here's an example of such finetuning also proposed in [BV] (from the abstract, p.308, emphasis added):
The parameters are the compaction ratio of a crushed story, the fracture of mass ejected outside the tower perimeter, and the energy dissipation per unit height. The last is the most important, yet the hardest to predict theoretically. It is argued that, using inverse analysis, one could identify these parameters from a precise record of the motion of floors of a collapsing building. Due to a shroud of dust and smoke, the videos of the World Trade Center are only of limited use. It is proposed to obtain such records by monitoring with millisecond accuracy the precise time history of displacements in different modes of building demolitions.
Finally, there have been some claims that the comments in [BL], in particular points 4 ("Can Crush-Up Proceed Simultaneously with Crush Down?", p.917) and 5 ("Why Can Crush-Up Not Begin Later?", p.919) mean a direct application of the crush direction part of the model to the WTC 1 and 2 as if it was what actually happened. That is wrong. That section is dedicated to discussing the theoretical basis of the crush direction part of the model, about which [G 2008] objected, as this excerpt shows (p. 915):
Applying Newton's third law to the collapse of the Twin Towers, it is clear that the downward force imposed on Part B by the upper Part C generates an equal but opposite upward force. It logically follows that if the downward force generated when Part C impacts Part B is destructive, then the equal and opposite upward force generated in accordance with Newton's third law will be destructive. Instead of embracing this basic law of physics, the paper treats Part C as a rigid body during the crush-down phase, then allows Part C to start deforming only at the start of the crush-up phase
Since the author's always been not-so-good for writing closing words, please indulge him in his use of [BLGB] to do it:
These conclusions show the allegations of controlled demolition to be absurd and leave no doubt that the towers failed due to gravity-driven progressive collapse triggered by the effects of fire.
(Thanks to ozeco41 for raising the point that led to writing this text.)
References
- [B 2001]: Bažant, Z.P. (2001), Why Did the World Trade Center Collapse?. Siam News, Vol. 34, No. 8, pp. 1 and 3.
- [BZ]: Bažant, Z.P.; Zhou, Y. (2002), Why Did the World Trade Center Collapse?—Simple analysis. Journal of Engineering Mechanics ASCE 128 (No. 1) pp. 2-6, with addendum March (No. 3), pp. 369-370. It's an expansion of the former with corrections, appendices and an addendum published later.
- [BV]: Bažant, Z.P.; Verdure, M. (2007), Mechanics of Progressive Collapse: Learning from World Trade Center and Building Demolitions. Journal of Engineering Mechanics ASCE 133 (No. 3), pp. 308-319.
- [BLGB]: Bažant, Z.P.; Le, J.-L.; Greening, F.R.; Benson, D.B. (2008), What did and did not cause collapse of World Trade Center twin towers in New York. Journal of Engineering Mechanics ASCE 134 (No. 10), pp. 892-906.
- [BL]: Bažant, Z.P.; Le, J.-L. (2008), Closure to "Mechanics of Progressive Collapse: Learning from World Trade Center and Building Demolitions" by Zdeněk P. Bažant and Mathieu Verdure. Journal of Engineering Mechanics ASCE 134 (No. 10), pp. 917-923.
- [G 2008]: Gourley, J.R. (2008), Discussion of "Mechanics of Progressive Collapse: Learning from World Trade Center and Building Demolitions" by Zdeněk P. Bažant and Mathieu Verdure. Journal of Engineering Mechanics ASCE 134 (No. 10), pp. 915-916. It's the discussion which is followed up by the above closure. The PDF file in which both appear is the same.
