The World Trade Center Twin Towers (ROOSD) Propagation model is a descriptive model of WTC1 and 2 collapse progression process. The reader does not need to read technical literature to understand the observations upon which the model is based.

Mathematical models of collapse front progression rates have been published in technical journals. The collapse progression mappings presented can also be used to review the concepts within these published papers for accuracy and application to the actual collapses of WTC1 and 2.

This section will probably be too difficult for those without much experience in calculus to understand. If so, it is not necessary to understand this subsection to grasp the OOS propagation model described earlier or other sections of the book.

But if one wishes to understand and skeptically analyze the technical literature or the actual written history of discussions on WTC1 and 2 collapse progression mechanics, it is necessary to be able to read and understand the technical literature that exists. My apologies in advance if the reader finds this section too difficult to read.




Mathematical Basis of ROOSD Propagation in WTC1 and 2




Mathematical approach to the study of ROOSD propagation as a simple 5 step process:


Step 1: GAIN AN OVERALL CONCEPTUAL AND VISUAL UNDERSTANDING OF WHAT ONE IS LOOKING AT

Step 2: RESEARCH AVAILABLE LITERATURE ON THE SUBJECT OF FLOORS IMPACTING FLOORS

Step 3: MEASURE THE COLLAPSE PROPAGATION RATE AS ACCURATELY AND COMPLETELY AS POSSIBLE

Step 4: EXAMINE A VARIETY OF PHYSICS-BASED MATHEMATICAL APPROACHES TO THE COLLAPSE OF WTC1 AND 2 AND OF STACKED SYSTEMS IN GENERAL

Step 5: COMPARE MODELS IN STEP 4 TO INFORMATION IN STEPS 1 - 3 to see which models could match propagation behavior or teach something about it.







Each step in detail:



Step 1: GAIN AN OVERALL CONCEPTUAL AND VISUAL UNDERSTANDING OF WHAT ONE IS LOOKING AT

I'd recommend using parts 2.1 and 2.2 in my book.




There are two distinct ways to view the physics of a ROOSD-type process, as a progressor or as an initiator.




Obviously, this structure is composed of 3 main components:




The perimeter
The core
Flooring


The initiation of collapse, in general, can be classified as

1) Perimeter failure initiated
2) Core failure initiated
3) Floor slab failure initiated



The ROOSD process is vital to understanding the collapse progressions of WTC1 and 2, but it is also vital to understanding the initiation sequence. Progressive floor collapse initiated by failed floor slabs has always been one of the 3 basic logical possibilities for the initial failures of WTC1 and 2.




Some defining characteristics of the unique ROOSD progressions in WTC1 and 2:




The OOS progressions were confined by guide rails: Perimeter caging, combined with the outermost core columns, provide a perfectly vertical and very strong confinement of the OOS crush front.


The strength of the guide rails created a concentrated energy funnel: The crushing energy was concentrated on and funnelled toward the 208x208 ft floor space below. The unique perimeter architecture assured that crushing energy was funnelled inward to maximize the destructive power of the crush front.


The unique architecture of WTC1 and 2 created a physical condition I will call a ROOSD basket. A ROOSD basket is essentially a containment vessel consisting of the flooring not yet destroyed and the 4 perimeter walls that act like caging. It is the dynamic and unique shape and nature of the 'ROOSD basket' containment vessel which steered and trapped the progressing crush fronts that explains so much of what is witnessed in the visual record.


Steady state progression rates were characterized by a near constant 8 floors per second crush rate with a steady state acceleration of zero.



The qualities of strong confinement, terminal velocity, and zero steady state acceleration means that these OOS steady state progressions are highly regulated, very controllable, and very predictable processes.




Step one done correctly will lead to a recognition of a ROOSD process. If it is not done correctly every subsequent step will general and vague (see Seffen and BV, BL, BLGB for examples) or skipped completely. A researcher will then go to step 2. It is possible to see how well the actual WTC collapse propagation mechanism was recognized within the academic and engineering community simply by doing step 2.







Step 2: RESEARCH AVAILABLE LITERATURE ON THE SUBJECT OF FLOORS IMPACTING FLOORS


General Conditions of ROOSD mechanics will take the form:

for rubblized driving mass > M(1) moving at a downward velocity > V(1) , the runaway process is assured. (threshold limits to initiate ROOSD)

If driving mass M is large enough and the downward velocity V is large enough the conditions will probably escalate. (conditions in which ROOSD is sustained)


What is the minimum threshold to set off a ROOSD progression in the most vulnerable OOS floor spaces in the buildings?

One would think the NIST would have looked into this question. A review of available literature on this subject brings up this paper, published in 2007:

"Progressive Collapse of Multi-story Buildings Due to Failed Floor Impact"

by A.G. Vlassis, B.A. Izzuddin, A.Y. Elghazouli and D.A. Nethercot, available at this link.



ABSTRACT

This paper proposes a new design-oriented methodology for progressive collapse assessment of floor systems within multi- storey buildings subject to impact from an above failed floor. The conceptual basis of the proposed framework is that the ability of the lower floor for arresting the falling floor depends on the amount of kinetic energy transmitted from the up per floor during impact. Three principal independent stages are employed in the proposed framework, including : (a) determination of the nonlinear static response of the impacted floor system, (b) dynamic assessment using a simplified energy balance approach, and (c) ductility assessment at the maximum level of dynamic deformation attained upon impact. In order to calibrate the proposed method, the part of the kinetic energy of the impacting floor that is transferred to the impacted floor is first theoretically determined for the two extreme impact possibilities, namely fully rigid and fully plastic impact. Moreover, a series of numerical studies is carried out to further refine the accuracy of this new approach with respect to different impact scenarios, whilst the effects of detailed joint modelling and redundancy are also investigated. The application of the proposed methodology is demonstrated by means of a case study, which considers the impact response of a floor plate with in a typical multi-storey steel-framed composite building. Several possibilities regarding the location of the impacted floor plate, the nature of the impact event and the intensity of the gravity loads carried by the falling floor are examined. The application study illustrates the extremely onerous conditions imposed on the impacted floor resulting in an increased vulnerability to progressive collapse for structures of this type. Importantly, the likelihood of shear failure modes in addition to inadequate ductility supply under combined bending/axial actions is identified, thus establishing mthe need for further research work on the dynamic shear capacity of various connection types subject to extreme events.





The best mathematical approach to the study of ROOSD as initiator I have found within all available literature is contained within this paper.




From the paper a deforming slab responding to impact:



From the introduction:

"It is concluded that such structures are susceptible to progressive collapse initiated by impact of a failed floor, mainly due to insufficient ductility supply under combined bending and axial deformation modes. Moreover, the development of shear failure modes is identified, thus further increasing the observed vulnerability of the studied floor system. Since these shear modes of failure are expected to be even more pronounced when the actual dynamic rather than the static response of the impacted floor is considered, the need for further research work focussing on the shear capacity of a variety of connection types subject to extreme events is established. Finally, practical design recommendations that can improve the impact response of floor systems exposed to impact from the floor above are made."

page 24: "Hence, it can be easily concluded that in the event of failure and subsequent impact of a single floor plate onto the floor plate below, the lower impacted system, modeled using a grillage-type approximation, is highly unlikely to possess sufficient dynamic load carrying capacity to resist the imposed dynamic loads and prevent progressive collapse."

page 26: "Thus, although assessment is based on a simplified grillage-type approximation rather than a detailed slab model, the explicitness of the results leads to the conclusion that a floor system within a steel-framed composite building with a typical structural configuration has limited chances to arrest impact of an upper floor. This is particularly true when the falling floor completely disintegrates and falls as debris without retaining any residual strength or spanning capability."

page 27: "To conclude, although there is room for further improvements with respect to its accuracy and applicability, the proposed assessment methodology provides an effective platform to rationally tackle the scenario of floor impact, which is one of the most prevalent progressive collapse initiation mechanisms."



Also consider the Ph.D thesis by the same author: Vlassis, A.G. (2007). Progressive Collapse Assessment of Tall Buildings, PhD Thesis, Department of Civil and Environmental Engineering, Imperial College London.

















Step 3: MEASURE THE COLLAPSE PROPAGATION RATE AS ACCURATELY AND COMPLETELY AS POSSIBLE: A collapse front down the west face of WTC1 remained visible and measurable down to the lower floors. The movement shows a relatively constant velocity after quickly leveling off after collapse initiation. The velocity is approximately 25m/s, or about 8 floors per second being destroyed. To the author's knowledge this is the first time that the propagation rate of a progressive floor collapse was measured.



Mathematical approach to ROOSD as progressor



Fundamental conditions:

for rubblized driving mass > M(1) moving at a downward velocity > V(1) , the runaway process is assured. The threshold conditions require a minimum M(1) traveling downward at velocity V(1).

The process requires progressive and runaway breakability of successive floor-to-column connections.



Examination of available literature: None



Measurements and observations of a first known documented case: WTC1 and 2 (roofline and collapse front)


A collapse front down the west face of WTC1 remained visible and measurable down to the lower floors. The movement shows a relatively constant velocity after quickly leveling off after collapse initiation. The velocity is approximately 25m/s, or about 8 floors per second being destroyed.



Initial data for linear ejecta traversal from West face of WTC 1 (Femr2, 2009):


http://femr2.ucoz.com/photo/linear_2/6-0-217
http://femr2.ucoz.com/photo/6-0-217-3 (1280x720px/74.6Kb)

Source video in H264 format (1280x720x25fps):
http://femr2.ucoz.com/ffdemhd_264.avi

Crop of West Face Ejecta:



http://www.youtube.com/watch?v=BMeTGfCZWMI
http://www.youtube.com/watch?v=2F5Tw2ITMF8

Position⁴:

http://femr2.ucoz.com/photo/6-0-219-3 (1234x731px/67.0Kb)

Velocity:

http://femr2.ucoz.com/photo/6-0-220-3 (1224x730px/87.3Kb)







What are the characteristic features of the progressor (steady state) portions of the curves? (Measured along the OOSsw crush front)

• Tendency toward terminal velocity
• A steady state acceleration of zero

• Crush front seems to start on floor 95
• Quick take off velocity
• Short transitory phase (reaches steady state velocity and acceleration rapidly)

• A discrete model, even highly idealized, is a dynamic problem with more than one degree of freedom if upper is not rigid
• The simplest representation, which conforms to typical discrete inelastic models, has TWO aggregate, generalized coordinates
• A 1D solid slab and spring model with many members, and which fails, has a propensity for crush-up whether or not crush down occurs

Abstract:

The collapse behavior of the World Trade Centre (WTC) towers is considered formally as a propagating instability phenomenon. The application of associated concepts enables the residual capacities of both towers after collapse to be formally estimated. This information is combined into a simplified variable-mass collapse model of the overall dynamic behavior. The resulting, non-linear governing equation of motion can be solved in closed form, to yield compact information about the overall collapse conditions.

Abstract:

Progressive collapse is a failure mode of great concern for tall buildings, and is also typical of building demolitions. The most infamous paradigm is the collapse of the World Trade Center towers. After reviewing the mechanics of their collapse, the motion during the crushing of one floor or group of floors and its energetics are analyzed, and a dynamic one-dimensional continuum model of progressive collapse is developed. Rather than using classical homogenization, it is found more effective to characterize the continuum by an energetically equivalent snap-through. The collapse, in which two phases—crush-down followed by crush-up—must be distinguished, is described in each phase by a nonlinear second-order differential equation for the propagation of the crushing front of a compacted block of accreting mass. Expressions for consistent energy potentials are formulated and an exact analytical solution of a special case is given. It is shown that progressive collapse will be triggered if the total internal energy loss during the crushing of one story equal to the energy dissipated by the complete crushing and compaction of one story, minus the loss of gravity potential during the crushing of that story exceeds the kinetic energy impacted to that story. Regardless of the load capacity of the columns, there is no way to deny the inevitability of progressive collapse driven by gravity alone if this criterion is satisfied for the World Trade Center it is satisfied with an order-of-magnitude margin. 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. The monitoring could be accomplished by real-time telemetry from sacrificial accelerometers, or by high-speed optical camera. The resulting information on energy absorption capability would be valuable for the rating of various structural systems and for inferring their collapse mode under extreme fire, internal explosion, external blast, impact or other kinds of terrorist attack, as well as earthquake and foundation movements.

Eqs. (12) and (17) show that Fc(z) can be evaluated from precise monitoring of motion history z(t) and y(t), provided that m(z) and lamda(z) are known. A millisecond accuracy for
z(t) or y(t) would be required. Such information can, in theory, be extracted from a high-speed camera record of the collapse. Approximate information could be extracted from a regular video of collapse, but only for the first few seconds of collapse because later all of the moving part of the WTC towers became shrouded in a cloud of dust and smoke (the visible
lower edge of the cloud of dust and debris expelled from the tower was surely not the collapse front but was moving ahead of it, by some unknown distance)."

Implications and Conclusions

1. If the total internal energy loss during the crushing of one story representing the energy dissipated by the complete crushing and compaction of one story, minus the loss of
gravity potential during the crushing of that story exceeds the kinetic energy impacted to that story, collapse will continue to the next story. This is the criterion of progressive collapse trigger Eq. 5. If it is satisfied, there is no way to deny the inevitability of progressive collapse driven by gravity alone regardless of by how much the combined strength of columns of one floor may exceed the weight of the part of the tower above that floor . What matters is energy, not the strength, nor stiffness.

2. One-dimensional continuum idealization of progressive collapse is amenable to a simple analytical solution which brings to light the salient properties of the collapse process.
The key idea is not to use classical homogenization, leading to a softening stress-strain relation necessitating nonlocal fi-
nite element analysis, but to formulate a continuum energetically equivalent to the snapthrough of columns.

3. Distinction must be made between crush-down and crush-up phases, for which the crushing front of a moving block with accreting mass propagates into the stationary stories below, or into the moving stories above, respectively. This leads to a second-order nonlinear differential equation for propagation of the crushing front, which is different for the crush-down phase and the subsequent crush-up phase.

4. The mode and duration of collapse of WTC towers are consistent with the present model, but not much could be learned because, after the first few seconds, the motion became obstructed from view by a shroud of dust and smoke.
`
5. The present idealized model allows simple inverse analysis which can yield the crushing energy per story and other properties of the structure from a precisely recorded history
of motion during collapse. From the crushing energy, one can infer the collapse mode, e.g., single-story or multistory buckling of columns.

6. It is proposed to monitor the precise time history of displacements in building demolitions—for example, by radio telemetry from sacrificial accelerometers, or high-speed optical
camera—and to engineer different modes of collapse to be monitored. This should provide invaluable information on the energy absorption capability of various structural systems, needed for assessing the effects of explosions, impacts,
earthquake, and terrorist acts.

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. The monitoring could be accomplished by real-time telemetry from sacrificial accelerometers, or by high-speed optical camera. The resulting information on energy absorption capability would be valuable for the rating of various structural systems and for inferring their collapse mode under extreme fire, internal explosion, external blast, impact or other kinds of terrorist attack, as well as earthquake and foundation movements.

4. The mode and duration of collapse of WTC towers are consistent with the present model, but not much could be learned because, after the first few seconds, the motion became obstructed from view by a shroud of dust and smoke.

5. The present idealized model allows simple inverse analysis which can yield the crushing energy per story and other properties of the structure from a precisely recorded history
of motion during collapse. From the crushing energy, one can infer the collapse mode, e.g., single-story or multistory buck ling of columns.

6. It is proposed to monitor the precise time history of displacements in building demolitions—for example, by radio telemetry from sacrificial accelerometers, or high-speed optical
camera—and to engineer different modes of collapse to be monitored. This should provide invaluable information on the energy absorption capability of various structural systems, needed for assessing the effects of explosions, impacts,
earthquake, and terrorist acts.

• Purpose of paper
• Name of model
• Key equations, choice of variable and parameters
• Representation of key equations
• Derivation of key equations
• Application of key equations to predict collapse features

• Inability to verify predictions
• Limited understanding of WTC collapse features
• Inability to identify a specific WTC collapse mode
• Misrepresentation of the actual crushing process along the collapse front contact interface

Abstract:

The collapse behavior of the World Trade Centre (WTC) towers is considered formally as a propagating instability phenomenon. The application of associated concepts enables the residual capacities of both towers after collapse to be formally estimated. This information is combined into a simplified variable-mass collapse model of the overall dynamic behavior. The resulting, non-linear governing equation of motion can be solved in closed form, to yield compact information about the overall collapse conditions.

Progressive collapse is a failure mode of great concern for tall buildings, and is also typical of building demolitions. The most infamous paradigm is the collapse of the World Trade Center towers. After reviewing the mechanics of their collapse, the motion during the crushing of one floor or group of floors and its energetics are analyzed, and a dynamic one-dimensional continuum model of progressive collapse is developed.

The mode and duration of collapse of WTC towers are consistent with the present model, but not much could be learned because, after the first few seconds, the motion became obstructed from view by a shroud of dust and smoke.

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.

• Inability to verify predictions
• Limited understanding of WTC collapse features
• Inability to identify a specific WTC collapse mode
• Misrepresentation of the actual crushing process along the collapse front contact interface

• Purpose of paper
• Name of model
• Key equations, choice of variable and parameters
• Representation of key equations
• Derivation of key equations
• Application of key equations to predict collapse features

• Inability to verify predictions
• Limited understanding of WTC collapse features
• Inability to identify a specific WTC collapse mode
• Misrepresentation of the actual crushing process along the collapse front contact interface

1) For technical accuracy

2) For perception: As an indication of how each author understood the WTC1 and 2 collapse modes at the time they wrote their respective papers

3) As a study of propagating memes within a meme-plex: as memes and memetics