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originally posted by: LaBTop
Originally posted by: wmd_2008
a reply to: LaBTop
Do you understand at all the quintessence of it?
NO, we say that a PROTECTED steel frame can't collapse only due to fire in 101 minutes and many on here with that belief post the Windsor Tower 18 hours / 1080 minutes fire pictures as evidence of that.
So lets break it down into simple terms for you and the others.
''"with steel perimeter columns""
Windsor Tower Floor Construction
"" Originally, the perimeter columns and internal steel beams were left unprotected in accordance with the Spanish building code at the time of construction""
Did that UNPROTECTED steelwork COLLAPSE only due to fire ALTHOUGH it was not IMPACTED or DAMAGED ? YES
The unprotected steel-glass facade ABOVE the 17th floor was completely destroyed, exposing the concrete perimeter columns. The UNPROTECTED steel PERIMETER columns above the 17th floor suffered PARTIAL (NOT complete) collapse, partially coming to rest on the upper 16th technical floor.
So can a fire ONLY, cause UNDAMAGED unprotected steelwork to collapse? YES.
FINE. All Windsor tower perimeter steel above the 17th floor collapsed PARTIALLY, BECAUSE it was UNPROTECTED with fire insulation foam. That was the reason they started the renovation project work, to protect all that steel above floor 16.
The failing floor at the WTC 1N tower had ALL its steel solidly protected by good insulation.
SO, WHY SHOULD IT FAIL MASSIVELY IN 101 MINUTES / 1 hr 41 minutes?
We discuss two non-obvious inconsistencies between the mathematical models of progressive collapse based on the NIST scenario, and the practical realizations of collapse in WTC 1 and 2 :
(i), the average avalanche pressure is 3 orders of magnitude smaller than the pressure the vertical columns are able to withstand, and
(ii), the intact vertical columns can easily absorb through plastic deformation the energy of the falling top section of the WTCs. (LT : a NATURALLY falling WTC top section. )
We propose a collapse scenario that resolves these inconsistencies, and is in agreement with the observations and with the mathematical models.
To obtain the values for our models we proceed as follows. For each structural element, the PCs and the CCs, we find the scaled ultimate yield force f = f(z), at the top (z = 0) and at the bottom (z = 1). The parameters r and s are then obtained
In Eq. (1) the term R* = R*(Z) comes from the average resistive force produced by the building. It measures how the building does oppose its own destruction by the avalanche, and
is intimately related to the so called, yield strength of the structural members that provide the vertical support to the building.
Page 7 / 15 : Now that we have μ and ν for each building, we recall that the collapse initiation occurred at the top of the primary zone, at position z0 = 1 - F0/FT ,
where for WTC 1, 1z0 = 0.1, and for WTC 2, 2z0 = 0.23.
This allows us to construct the collapse initiation lines. The collapse starts at point z0 because the yield force of the compromised building, ˜R(z0), is not sufficient to resist the weight of the building above,
-˜R (z0) < m(z0) *g. (18)
Please observe, the collapse initiation lines are derived from static properties of the buildings, thus they do not depend on the mathematical model used to describe the dynamics of collapse.
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.
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 [20], [21], [42]) aimed at understanding the behavior of structures during Progressive Collapse (figure 6.1).
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.
originally posted by: LaBTop.
Look-up ultimate yield strength of A514 structural steel, in Wikipedia, or at the Acelor site..
Ultimately, for that 110 - 130 KSI ultimate yield strength range, an at least 5200 times (130/0.025, N.T.) and 2600 times (130/0.050, S.T.) stronger calculated collapse initiating yield force outcome would have been needed in the by Beck proposed NIST worst case scenarios.
Realize that what Beck offered as an additional extra to the existing NIST OFFICIAL scenario, are the only extra losses that in fact, he WILDLY overestimated.
originally posted by: LaBTop
Your answers are so ridiculously simplistic
Lexyghot : But what you're saying is that they were so inept and so wrong in what they were doing that they designed a building with a FOS of somewhere between 2600 and 5200..... .
originally posted by: lexyghot
arxiv.org...
""Now, as the avalanche enters the storey, it first encounters the resistive force from the vertical columns, call it R1. Over the fractional length λ1, the vertical columns maintain ultimate yield force under compression"".
(1) And this is why this paper can be ignored regarding collapse times.
(2) He assumes infinitely strong floors and connections, and that all resistive force is from the columns.
Sorry, LaBTop, but you've hitched your cart to a lame pony.
Beck, Page 1 : We examine static features ( collapse initiation lines, derived from the ultimate yield strength of the structural steel) and dynamic features ( duration of collapse, lines computed using mathematical models); two features of events that comprised the collapse in WTC 1 and 2.
We show that :
(a), the dynamic and static aspects of the collapse are mutually consistent and weakly dependent on the class or type of mathematical model used, and
(b), that the NIST scenario, in which the buildings collapse after a sequence of two damaging events (airplane impact and subsequent ambient fires), is inconsistent with respect to the structural strength of the buildings.
Our analysis shows that the force that resisted the collapse in WTC 1 and 2 came from a single structural element, the weaker perimeter columns, while the second structural element, the stronger core columns, did not contribute.
We discuss two non-obvious inconsistencies between the mathematical models of progressive collapse based on the NIST scenario, and the practical realizations of collapse in WTC 1+2 :
(i), the average avalanche pressure is 3 orders of magnitude smaller than the pressure the vertical columns are able to withstand, and
(ii), the intact vertical columns can easily absorb through plastic deformation the energy of the falling top section of the WTCs.
C. Resistive Force: discrete vs. continuous.
The resistive force R is the force the building opposes its own destruction, by the avalanche. Bazant et al.(ref.10) call it the “crushing force,” and for the most part of their calculation they assume that its magnitude is a constant with respect to height.
For our investigation we relax that assumption by allowing R to vary with height in a simple linear fashion,
f(z) = - R(z) / M*g = r + s*z, -----> Eq.(7)
where M is the total mass of the building, and z = Z/H is the scaled height. As we will see later, when we discuss the load-bearing capacity of the structure of WTC 1 and 2, this will turn out to be a more accurate description.
Let us now discuss the resistive force within the discrete (floor) model.
We posit that the building is rigid, i.e., that the floor that the avalanche is currently penetrating sits on an infinitely strong structure.
( LT : because he is first describing the DISCRETE NIST/Bazant MODEL, not the CONTINUOUS MODEL. In which discrete model he observes one floor only, starting to collapse, assuming it is SITTING on an infinite strong structure, like for example the Earth. )
With that assumption the interaction between the building and the avalanche is localized to the floor of contact between the two and not beyond.
[the above cut-out lexyghot quote: ]
Now, as the avalanche enters the storey, it first encounters the resistive force from the vertical columns, call it R1. Over the fractional length λ1, the vertical columns maintain ultimate yield force under compression.
[ /lexyghot quote]
In other words, for the first phase of floor destruction the fractional distance λ1 corresponds to the yield strain λY , while R1 corresponds to the ultimate yield force.
Assuming that avalanche has passed the fractional distance λ1, the columns fail. The failure mode of the columns is debatable: if the columns are compressed by the avalanche than they fail by buckling. However, if the avalanche front consists of crushed material, so that the ends of the vertical columns facing the avalanche are free, then the failure mode may be bending, as well. Either way, we assume that during this phase the column does not offer any resistance. In other words, for the fractional distance of λ2 the avalanche falls freely, R2 ≡ 0.(LT : ≡ means identical to)
Rather than guessing, we proposed a procedure for estimating R [ref. 4], where one first finds the ultimate yield strength Y of the vertical columns using their specifications, following which the resistive force R is estimated from a simple linear model, R = ε · Y , with ε = 0.25 being the ultimate yield strain of structural steel under compression. Applying this to WTC 1 and 2 led to an initial estimate of R(x2)/(M.g) = r + s · (x2/H), with r ≃ 0.2 and s ≃ 0.7.
Consider the average pressure created by the avalanche at the bottom of the primary zone, z1 = 1 − F1/FT, which is given by
p = [M g / a² ] · z1 (Eq.22)
Here, a = 206′ = 2472′′ is an approximate length of the side of the building. We get (1)p = 0.025 KSI for WTC 1, and (2)p = 0.050 KSI for WTC 2, which is three orders of magnitude smaller than the nominal 36-100 KSI (ultimate 58-110 KSI) the vertical columns were able to maintain while yielding in plastic deformation.
Bazant et al.10 argued that an avalanche propagating through the primary zone would get sufficiently compacted so that it could provide necessary pressure. We see two insurmountable problems with this suggestion :
First, the avalanche front can only “grow” thicker - it cannot expand laterally in such a fashion that would allow its edges to be strong enough to crush the vertical columns.
Second, for compaction to happen the floor material has to be compressed between two solid surfaces, and we see that there are no such surfaces on either end of the avalanche front.
In fact, the strength of the vertical columns will redirect the avalanche (which now consists only of destroyed floor material) to the region in-between the columns.
The formation of such avalanche is promoted by the relative weakness of the floors, the resistive force ƒ of which is ƒ ∼ 0.02 (Ref.16 NIST), per each floor, as compared to the resistive force of the intact vertical columns, ƒCC + ƒPC ≃ 0.8 + 2.7 · z
originally posted by: LaBTop.
Look-up ultimate yield strength of A514 structural steel, in Wikipedia, or at the Acelor site..
Ultimately, for that 110 - 130 KSI ultimate yield strength range, an at least 5200 times (130/0.025, N.T.) and 2600 times (130/0.050, S.T.) stronger calculated collapse initiating yield force outcome would have been needed in the by Beck proposed NIST worst case scenarios.
LT : Realize that what Beck offered as an additional extra to the existing NIST OFFICIAL scenario, are the only extra losses that in fact, he WILDLY overestimated.
This image from the documentary Up From Zero shows the base of a core column, whose dimensions, minus the four flanges, are apparently 52 by 22 inches, with walls at least 5 inches thick.
page 5 : PCs, the external dimensions of which were 14”-by-14”, were made of structural steel the yield strength of which varied from nominal 36 KSI (ultimate 58 KSI) and thickness 1/4” at the top of the building to nominal 100 KSI (ultimate 110 KSI) at the bottom.[ref.2] The thickness of the steel plates at the bottom, to the best of author’s knowledge, is not yet publicly available. CCs were made of structural steel which varied from nominal 36 KSI (ultimate 58 KSI) at the top to nominal 42 KSI (ultimate 60 KSI) at the bottom. Neither the dimensions or the thickness of the steel plates used for CCs, to the best of the author’s knowledge, are yet publicly available. The NIST report [ref.2] claims that there were two types of CCs: “standard” and “massive,” where the four “massive” columns, one at each core’s corner, “together provided 20% of load-bearing capacity of the core.”
This was recently proven to be misleading [ref.13] : information released to the public shows that of 51 CCs at least 16 (along two longer sides) were of external dimensions 22”-by-55” and of thicknesses up to 5”. To obtain the values for our models we proceed as follows. For each structural element, the PCs and the CCs, we find the scaled ultimate yield force f = f(z), at the top (z = 0) and at the bottom (z = 1).
The parameters r and s are then obtained using --snip-- Eq.10a and 10b.
For the PCs we assume that the thickness of the plates at the bottom was 1/2”, yielding --snip-- Eq.11
where M = 4.5 · 10^8 kg is the estimated mass of the building, while g is gravity.
(LT: plates at the bottom of the tower! And you know why he uses that? Because of his statement about the ultimate yield strain has to be applied/calculated over the whole length of a column, ~ the height of the buildings.! )
As for the CCs, we do base our expectations on the following: all 51 column were of dimensions 22”-by-55”, while their thickness and strength varied from 1 1/4” and nominal 36 KSI (ultimate 58 KSI) at the top, to 5” and nominal 42 KSI (ultimate 60 KSI) at the bottom. This yields --snip-- Eq.12
Please note, the assumption of all 51 columns being the same appears to be corroborated by the floor plans : [ref.2] all CCs appear to have the same footprint. Also, while this estimate is purely hypothetical, it turns out that, it is also irrelevant for the analysis of collapse that follows.
We observe that from Eqs. (11) and (12) the ultimate safety factor of the WTCs is ƒCC(1) + ƒPC(1) ∼ 3.5. This is a reasonable estimate considering the dimensions of the buildings and the other safety requirements that entered the structural calculation (ability to withstand hurricane winds and an airplane impact).
From the properties of structural steel [ref.14] it is known that the yield strain under tension and compression are fairly similar, and is ∼ 21 − 25%. In our model this is represented by λ1, which we take to be λ1 = 0.2. The value of compaction limit we take from Bazant,[ref.10] λ∞ = 0.2, which leaves λ2 = 1 − λ1 − λ∞ = 0.6. From there, (r∗, s∗) in the continuous model are related to (r, s) in the discrete values as
r∗ = 0.25 · r,
s∗ = 0.25 · s. (13)