1. Role of Compaction Ratio in the Mathematical Model of Progressive Collapse
Charles M. Beck, 14 April, 2008.
Submitted to ASCE J. of Engineering Mechanics.
2. Descent curve and the phases of collapse of WTC 7
Charles M. Beck, February 18, 2013.
Submitted on 29 Jun 2008 (v1), last revised 20 Jan 2009 (this version, v4), to Cornell University Library (arXiv.org/physics).
Page 9/24 Beck remarks :
An apparent weight the top section exerts during the collapse on the Earth’s crust,
W′/(M*g), is given by Eq.(16) --snip--
We believe that W′, and in particular its time derivative, can be used in interpretation of
the seismic signal of the building’s collapse. As an attempt to connect the two brings forth
numerous additional complications which need to be properly addressed, we leave this topic
to future publications.
Time to show Charles M. Beck my seismic work. He can see them in my signature LINKS under each post of mine in this website. Especially my workup of
the WTC7 seismogram, coupled with the timestamps in it, and the photo with NIST timestamp of the onset of collapse. I thus proved that the first
biggest peak in the WTC7 seismogram happened in Manhattan a few seconds before anyone saw any movement in, on and around WTC7. About two seconds
later, the eastern penthouse on top of WTC7 started to dent, then sunk away, then the west penthouse sunk away too, then the whole parapet roof line
started to fall in free fall for 2.3 seconds over ~26 meters (~8 floors). Then that huge chunk of the building met the underlaying construction, and
the second much lower seismic peak arrives on the seismogram, since deceleration hit in. That peak and the rest of the much lower seismic peaks depict
the rest of the following natural collapse crushing forces.
Note : Review the first few of his References...he reads the911forums too. Open that einsteen link and read that whole page, then thread.
3. Fill in a Google search : FrancescoBarni_PhD_Thesis.pdf
pages, 2011, Francesco Barni PhD Thesis about Progressive Collapse initiators and how to prevent them.
Try this link too.
Barni's first interesting steps towards a total valid hypothesis and its implementation in future to understand and then prevent progressive
collapses. 152 PDF-pages.
Very interesting prototypes of some theoretical, mathematically correct steps to understand how progressive collapses do start, and how to prevent
them in the future.
There's room for improvements in the methodology, and these aspects are further discussed.
Page 110/152 it's getting interesting. See also Fig 6.1
6.1 Description of the used structural models
The presented examples are implemented using Finite Element models derived from one developed by the National Institute of Standards and Technology
(NIST) for an ongoing research program (Refs , , ) aimed at understanding the behavior of structures during Progressive Collapse (figure
The models can be used to perform both dynamic and static analyses, with geometrical and material nonlinearities; furthermore, they are able to spot
the moment in which a section becomes detached, which corresponds to Collapse condition in the first proposed methodology.
In most studies carried on so far, element behavior derived from seismic engineering studies has been used to model Progressive Collapses. The NIST
tests highlight that, in some cases, the actual behavior can be very different. Thus, for certain aspects the used models are advanced. On the other
side, they still need improvement under several aspects, which will be highlighted in the text.
From page 136 on, Chapter 7.1 and 7.2 you find the Summary and Conclusions.
Chapter 8 lists several aspects that need to be further considered to improve the proposed
methodologies and elaborates on some of them. Other ideas that might be developed are also
4. An Overview of Progressive Collapse in Structural Systems by Phillip J. Georgakopoulos.
(2005, Abstract from his 55 pages MIT Thesis)
9.025Mb, Preview, non-printable (open to all)
9.022Mb, Full printable version (MIT only,
His first assessment on the long road to a precise description of P.C., with 51 drawings.
He considers the Oklahoma City Murrah building blast, and the WTC buildings, as progressive collapses. And how to possibly prevent the likes in the
future. Again, not conclusive at all at the moment of publication. We are still far from setting iron-cast mechanical and mathematical rules for
P.C.'s, and how to prevent them.
He seriously discusses possible blast loads from page 13 to 25.
And design issues with steel structures from page 44 on.
See his reference list on page 54 too, he mentions these links a lot :
The ASCE Library : www.ascelibrary.org...
Houghton & Karnes : www.mhpse.com...
Note : retrofitting methods are for building owners far too costly and time-consuming.
The new (page 48) SidePlate connection plate solutions are sporadically implemented only in a few new builds (page 50), and those are all three,
military/government objects. Thus, all the old & new public-accessible high rise buildings in private hands (greed : please, no extra reinforcements
costs) still have the same progressive collapse blast risks as before 9/11.
I wonder if the new WTC-7 have these SidePlate column-beam/girder connects implemented?
5. The Lagrangian Formalism, the Principle of least action, a.k.a. Principle of stationary action, a.k.a. Hamilton's principle :
Lagrange (and first Euler) proposed a theorem where the classic 3 Newtonian vectors were changed to a full set of vectorial probabilities.
This appears in a perfectly elastic scenario, as proposed by Beck under Eq.1 on page 2/15, in his main ref.
Note the minus signs in front of the values U and L, for potential energy, and energy associated with the structure of the building.
To fulfill the Lagrangian formulation in the formula : Lagrangian l = K − U − L , where K is the kinetic energy of the building.