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Refutation of NIST WTC1 Collapse Initiation Scenario

Analysis, observations and theory related to initiation.

Refutation of NIST WTC1 Collapse Initiation Scenario

Postby Major_Tom » Sun Jan 10, 2010 12:15 am

The first 10 posts of this thread are a presentation being written. Comments and criticism are encouraged after reading it carefully.


The argument is only partially complete. I will re-edit the information as I receive input.
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Re: Refutation of NIST WTC1 Collapse Initiation Scenario

Postby Major_Tom » Sun Jan 10, 2010 12:24 am

Collapse initiation scenario consists of:

1) Visible deformations leading into initial buckling sequence, especially inward bowing (IB) of the south face (stage 1).

2) Initial buckling sequence, initial failure of all columns and trajectory over the first 12 ft (stage 2).

3) Initial collision and resulting trajectory and behavior through subsequent collisions(stage 3).



As the NIST interpretation of all three stages has serious flaws, the NIST hypothesis regarding each stage can be refuted independently of the others.

Separate arguments will be given refuting the NIST interpretation of


1) Deformations and IB (NIST interpretation of stage 1)

2) Initial Buckling Sequence (NIST, Bazant interpretation of stage 2)

3) FIrst Collisons and Resultant Trajectory (NIST, Bazant interpretation of stage 3)


This presentation refutes the NIST interpretation of stage 1 and 2. Stage 3 will be treated in a separate study.
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Re: Refutation of NIST WTC1 Collapse Initiation Scenario

Postby Major_Tom » Sun Jan 10, 2010 1:03 am

THe NIST and Bazant interpretations of WTC1 collapse initiation are so interchangable that sometimes it is hard to tell which is which.

Dr Bazant, from Bazant and Verdure:

"Review of Causes of WTC Collapse


Although the structural damage inflicted by aircraft was severe, it
was only local. Without stripping of a significant portion of the
steel insulation during impact, the subsequent fire would likely
not have led to overall collapse (Bažant and Zhou 2002a; NIST
2005). As generally accepted by the community of specialists in
structural mechanics and structural engineering (though not by a
few outsiders claiming a conspiracy with planted explosives), the
failure scenario was as follows:

1. About 60% of the 60 columns of the impacted face of framed
tube (and about 13% of the total of 287 columns) were severed,
and many more were significantly deflected. This
caused stress redistribution, which significantly increased the
load of some columns, attaining or nearing the load capacity
for some of them.

2. Because a significant amount of steel insulation was stripped,
many structural steel members heated up to 600°C, as confirmed
by annealing studies of steel debris (NIST 2005) (the
structural steel used loses about 20% of its yield strength
already at 300°C, and about 85% at 600°C (NIST 2005);
and exhibits significant viscoplasticity, or creep, above
450°C (e.g., Cottrell 1964, p. 299), especially in the columns
overstressed due to load redistribution; the press reports right
after September 11, 2001 indicating temperature in excess of
800°C, turned out to be groundless, but Bažant and Zhou’s
analysis did not depend on that).

3. Differential thermal expansion, combined with heat-induced
viscoplastic deformation, caused the floor trusses to sag. The
catenary action of the sagging trusses pulled many perimeter
columns inward (by about 1 m, NIST 2005). The bowing of
these columns served as a huge imperfection inducing multistory
out-of-plane buckling of framed tube wall. The lateral
deflections of some columns due to aircraft impact, the differential
thermal expansion, and overstress due to load redistribution
also diminished buckling strength.

4. The combination of seven effects—(1) Overstress of some
columns due to initial load redistribution; (2) overheating
due to loss of steel insulation; (3) drastic lowering of yield
limit and creep threshold by heat; (4) lateral deflections of
many columns due to thermal strains and sagging floor
trusses; (5) weakened lateral support due to reduced in-plane
stiffness of sagging floors; (6) multistory bowing of some
columns (for which the critical load is an order of magnitude
less than it is for one-story buckling); and (7) local plastic
buckling of heated column webs—finally led to buckling of
columns (Fig. 1(b)). As a result, the upper part of the tower
fell, with little resistance, through at least one floor height,
impacting the lower part of the tower.
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NIST Record of IB

Postby Major_Tom » Sun Jan 10, 2010 1:56 am

Recording IB:

Image

Image

Image

Four fairly high resolution images of WTC 1 South Side IB.
http://femr2.ucoz.com/photo/6-0-200-3 (1200x1600px/179.7Kb)
http://femr2.ucoz.com/photo/6-0-201-3 (1200x1600px/176.4Kb)
http://femr2.ucoz.com/photo/6-0-202-3 (1200x1600px/187.0Kb)
http://femr2.ucoz.com/photo/6-0-203-3 (1200x1600px/252.8Kb)




NIST Hypothesis of IB Cause:

From OP "Mystery of South wall Thread"

Column damage within the core according to the NIST (right click and open in new window to see the whole image).

Image

Temperature levels according to NIST fire simulations. The IB was witnessed at about 10:07, just a few minutes after large fireballs were seen shooting out of the south face at the same place.


According to their own fire simulations, temperatures were insufficient to cause floors to sag. The floors would have to sag up to 9 ft in some locations in order to create the IB witnessed.

Image

Image

No fire was seen on the east side of the south wall until the emergence of the fireballs (at 9:59). IB was first witnessed on the east side just 8 minutes later.
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Re: Refutation of NIST WTC1 Collapse Initiation Scenario

Postby Major_Tom » Sun Jan 10, 2010 1:58 am

IB: Arguments against NIST Interpretation

A summery of the NIST argument for pull-in and it's absurdity demonstrated by mmmlink:

http://www.youtube.com/watch?v=WwXLIdMzmG8


1) Insufficient temperatures and the absence of visible fire

2) The NIST's own tests needed ridiculously intense fires over a prolonged time period, and still couldn't reproduce the IB as witnessed.


3) Peterene:

If 50 minutes of relatively mild fire can bow the exterior wall to the point of collapse, why I can't see any sign of deformation in this photo?

http://i711.photobucket.com/albums/ww114/peterene/9_56_8_66.jpg



According to NIST - both of these areas should be stripped of the fireproofing ( east part of floors 78-84 and west part of the floors 98,97.

So, 56 minutes of mild fire acting upon "naked" trusses should cause extensive bowing to the point of failure, but 75 minutes of severe fire acting on another naked trusses did not result in one/two decimeters of bowing?



..............................................................................................................
Reconsidering IB:

Geometrically, there can only be 2 fundamental causes of IB in the locations witnessed:

1) Floor sagging and pull-in
2) 1000 row CC sagging and pull-in

There is a third possibility shown in the video by achimspok at

http://www.youtube.com/watch?v=YoZY7lNRlSI

Here you use the hat truss system to overload the perimeter. Fantastic video, but is depends on the ability of the hat truss outriggers to transfer significant load which affects the IB area. Even if the outriggers could do this, successive spandrel plates will work to transfer that load to the corners just as we see over the airplane hole on the WTC1 north face. I'll include this as a third possibility:

3) Achimspok mechanism.



It seems NIST had no choice but to present a sagging floor theory since the only other logical possibility is 1000 row CC manipulation.

I cannot visualize a 4th choice. Can anyone else? The cause cannot be from anywhere else in the building since IB is a local phenomenon. For example, it cannot come from the 900 CC row since an intact 1000 CC row cannot be bypassed. In the case of IB we don't have many choices. Therefore NIST really had no choice in the theory they present. They chose the only possibility that could explain the IB naturally (though poorly).
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Re: Refutation of NIST WTC1 Collapse Initiation Scenario

Postby Major_Tom » Sun Jan 10, 2010 1:59 am

IB: Better interpretations

Image


It appears that the 95th floor slab where IB begins. The whole perimeter below the 95th floor slab appears straight. IB deviation begins at this point.

There are weld/bolt/rivet column to column connections on the 95th floor (I'd guess about 4 feet up from the floor slab level).

Hat truss cubic reinforcement above fl 105. There is a CC weld/bolt/rivet connection plane on fl 104.


This leaves 3 core column sections between the fl 95 slab and the hat truss with connections at Fls 95, 98, 101 and 104. You can create the IB by attacking connections at fl 95, 98, 104. Max IB would then be at fl 98 as observed. Columns to displace: CC 1002 to 1007. Leave 1001 and 1008 in place and create the IB between them.

Dotted lines are CC connection planes. The breaks are the red circles.

This gives us 2 floating blocks; An interconnected 6 floor hanging block above the max IB line and a 3 floor hanging, floating block below.

It matches the triangular shape of the IB better.

We can see in the available photos there is a visible crease along the 95th floor slab above which the IB pattern takes shape.

This gives us a 6 floor slab hanging block above the IB which is about twice as heavy, twice as high as the one below.


Image

The green slab is the floor in it's original position. The blue line is the floor with the 1000 row column hanging freely, unsupported from both above and below.

The pivot (r= constant= length of 1000-900 lateral brace) is an important contributer to the pull-in (the dominant factor, actually). Here we can see a simple and direct relationship between how far the 1000 row column hangs down from it's original position (d) and perimeter pull-in (h).

Without the 1000-900 lateral bracing and flooring acting as a pull-in pivot, we see the column would have to hang pretty low to attain a 55" (h=4ft +) pull-in.

But with the pivot arm the 1000 row column would only have to hang down about 6 to 8 feet (d=6, to obtain an IB of over 4 ft.

The mechanism is shown below.


Image

Image


..................................


Initial attempts to estimate IB by sagging 100 row CCs.



Tony Szamboti:

In addition to the 42 inch sagging of the main floor trusses allowed to occur due to the removal of the bridging trusses from between the core and perimeter in their model, the NIST added artificial 5 kip lateral loads on the south wall perimeter columns of WTC 1, as the model apparently did not produce the inward bowing with the sagging trusses. I only read 5 kips per column not 5 kips per floor per column, but I'll have to check again. Interestingly, they then say the maximum inward bowing they generated was 31 inches.

I just calculated the concentrated force required to generate a center span deflection of 55 inches in a 14 inch square steel beam supported at both ends that was five stories in length and with a .289" wall thickness like those at the 98th floor of WTC 1 would have had. I used a story height of 149 inches so the beam is 745 inches long. The equation is

deflection = (force x length^3)/(48 x modulus of elasticity x moment of inertia)

modulus of elasticity for steel = 29 x 10^6 psi
moment of inertia for a 14 inch square beam with .289" wall thickness = 496.8 in^4
length = 745 inches

to find force just manipulate the equation with the deflection being 55 inches

force = (deflection x 48 x modulus of elasticity x moment of inertia)/(length^3)

The answer I get is about 91,989 lbs. or 92 kip. The perimeter columns in the towers would actually require a little more as they would have help from the spandrels.

For 31 inches of inward bowing the lateral force required for five unsupported stories is 51,848 lbs. or 52 kip. Did the NIST make an order of magnitude mistake or did they have 25 kips of catenary force with the trusses and added 5 kips per floor per column to get another 25,000 lbs. per column artificially? I know Dr. Bazant was off by a factor of ten with his calculation of the axial stiffness of the columns in the towers. He shows 71 GN/m and we calculated 7.1 GN/m.

The average weight per floor of a 1000 row column at the 98th floor was about 4,200 lbs.. The floors outside of the core were about 31,000 square feet and weighed about 2,500,000 lbs giving 80 psf. The south wall side between the core and perimeter was 137 foot long x 60 foot wide giving an area of 8,220 square feet and at 80 psf would weigh about 657,600 lbs.

If a five story section of the 1000 row columns was cut and dropped about 1 story, generating an angle of 12 degrees with the floor, then the lateral load on the perimeter columns directly in front of the core would be about 4.7 times that of the vertical load due to catenary action. So given the 4,200 lbs./column per story (33,600 for eight columns) and 657,000 lbs. floor weight between the core and perimeter and five stories of floor you get 690,600 lbs. x 4.7 x 5 = 16,229,100 lbs.

This would then be applied to 40 perimeter columns which are directly across from the core. So you would get about 400,000 lbs./column of lateral load. This could definitely do the job.

So it could be done with cutting the 1000 row columns and leaving them hang and you would only need a couple stories of 1000 row columns completely hanging to get 55 inches. With the spandrels involved the bowing would also be parabolic to some degree with the worst being in the center.

These calculations also show a very large force could be applied to pull in the perimeter columns with a couple of stories of the entire central core and all of the floors going down.

If perimeter column inward bowing did occur on the south wall of WTC 1 for minutes before collapse I believe 1000 row cutting to be the only viable explanation. There is no way that the hat truss could have applied the extraordinarily high vertical load necessary to do it and it again raises the issue of why any inward bowing caused by it wouldn't occur on the 106th floor. The fact that the NIST needed to artificially add lateral loads shows the floor truss sagging could not do it either.


..............................................................

Supporting Evidence


1) 9:59 South Wall FIre Ejections: Visual evidence which demonstrates that "something" big happened to the south wall of WTC1 at 9:59, just as WTC2 was collapsing.

Achimspok
http://www.youtube.com/watch?v=rCWqdMXV6qY

Xenomorph
http://www.youtube.com/watch?v=htVnlp_qg9g&feature=channel_page



NIST mentions the fire ejections briefly:

NCSTAR1-5, Page 15: "Very shortly after the collapse began, fire and smoke were pushed out of the south face of WTC 1, probably due to a pressure pulse transmitted to WTC 1 from the collapsing tower. The most prominent effect was on the 98th floor where flames were pushed out of windows along the west side of the face."

Probably? There is smoke coming from the south face around this region and we can see no movement due to the collapsing tower within the smoke. Any wind pulse transmitted from WTC2 to WTC1 would be visible within movement of the veil of smoke covering the south wall of WTC2.



2) Movement of Antenna

Image

Consider the hat truss design from floors 106 up:

http://www.sharpprintinginc.com/911/ima ... russes.JPG


We have a reinforced cubic structure from floors 106 up.

The tall antenna can be considered as strongly connected to the hat truss structure? It therefore acts as a pretty accurate "angle guage" to measure any hat truss deviation from horizontal.

Important to note the four main diagonal connections between the hat truss and south wall: Across from 1001, 1004, 1005 and 1008. The 1004 and 1005 connections are just above the IB.


3) 10:18 Pressure Pulses from north and west face, 92nd and 95th floors:

http://the911forum.freeforums.org/pressure-pulse-10-18-wtc1-t280.html

NIST briefly mentions them:

NCSTAR 1-5, Page 17: "An event took place withiin the tower at 10:18:48 a.m. that generated a pressure pulse with sufficient magnitude to force a large amount of smoke from the open windows on the 92nd floor, ... While it seems likely that the pressure pulse was generated by some sort of collapse within the tower, e.g., a portion of the core settling or a partial floor collapse, it has not been possible to determine the nature of the event or even its general location based on the visual record."

There are simultaneous ejections from the 92nd and 95th floors as can be seen below.

So NIST states "it has not been possible to determine the nature of the event or even its general location based on the visual record." Can anyone else not see the simultaneous ejections from floor 95 in the image below? How could NIST miss that? SImultaneous pulses from 2 floors with a 3 floor separation between them. Is that not a big clue that this is not just local truss sagging and failure?

Image


............................................................................................

More resources:

Video clips by Achimspok on the subject of IB

http://redirectingat.com/?id=593X1004&u ... 26fmt%3D18

http://www.youtube.com/watch?v=yTqY_dld08g&fmt=18 (this is the most interesting part)

http://www.youtube.com/watch?v=m3CFlRVMs8g&fmt=18

NCSTAR1-5, pg 14: "Around 9:40 a.m. a flame jet suddenly erupted from the south side of the 98th floor. This new area of fire then spread and grew rapidly, covering most of the west side of the floor within a few minutes. This fire continued to burn to the end of the period[9:59 a.m.]"
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Stage 2: Initial Buckling Sequence, NIST and Bazant Interpr

Postby Major_Tom » Sun Jan 10, 2010 2:00 am

NIST has no visual model of the initial column buckling sequence for WTC1. Why?

The only clue I get is from Bazant and Verdure. Here we see the collapse initiation scenario of the official story summerised. Notice the official reason given for not simulating collapse initiation in in red characters (color added on edit)

From the paper:

"Review of Causes of WTC Collapse


Although the structural damage inflicted by aircraft was severe, it
was only local. Without stripping of a significant portion of the
steel insulation during impact, the subsequent fire would likely
not have led to overall collapse (Bažant and Zhou 2002a; NIST
2005). As generally accepted by the community of specialists in
structural mechanics and structural engineering (though not by a
few outsiders claiming a conspiracy with planted explosives), the
failure scenario was as follows:

1. About 60% of the 60 columns of the impacted face of framed
tube (and about 13% of the total of 287 columns) were severed,
and many more were significantly deflected. This
caused stress redistribution, which significantly increased the
load of some columns, attaining or nearing the load capacity
for some of them.

2. Because a significant amount of steel insulation was stripped,
many structural steel members heated up to 600°C, as confirmed
by annealing studies of steel debris (NIST 2005) (the
structural steel used loses about 20% of its yield strength
already at 300°C, and about 85% at 600°C (NIST 2005);
and exhibits significant viscoplasticity, or creep, above
450°C (e.g., Cottrell 1964, p. 299), especially in the columns
overstressed due to load redistribution; the press reports right
after September 11, 2001 indicating temperature in excess of
800°C, turned out to be groundless, but Bažant and Zhou’s
analysis did not depend on that).

3. Differential thermal expansion, combined with heat-induced
viscoplastic deformation, caused the floor trusses to sag. The
catenary action of the sagging trusses pulled many perimeter
columns inward (by about 1 m, NIST 2005). The bowing of
these columns served as a huge imperfection inducing multistory
out-of-plane buckling of framed tube wall. The lateral
deflections of some columns due to aircraft impact, the differential
thermal expansion, and overstress due to load redistribution
also diminished buckling strength.

4. The combination of seven effects—(1) Overstress of some
columns due to initial load redistribution; (2) overheating
due to loss of steel insulation; (3) drastic lowering of yield
limit and creep threshold by heat; (4) lateral deflections of
many columns due to thermal strains and sagging floor
trusses; (5) weakened lateral support due to reduced in-plane
stiffness of sagging floors; (6) multistory bowing of some
columns (for which the critical load is an order of magnitude
less than it is for one-story buckling); and (7) local plastic
buckling of heated column webs—finally led to buckling of
columns (Fig. 1(b)). As a result, the upper part of the tower
fell, with little resistance, through at least one floor height,
impacting the lower part of the tower. This triggered progressive
collapse because the kinetic energy of the falling upper
part exceeded (by an order of magnitude) the energy that
could be absorbed by limited plastic deformations and fracturing
in the lower part of the tower.

In broad terms, this scenario was proposed by Bažant (2001),
and Bažant and Zhou (2002a,b) on the basis of simplified analysis
relying solely on energy considerations. Up to the moment of
collapse trigger, the foregoing scenario was identified by meticulous,
exhaustive, and very realistic computer simulations of
unprecedented detail, conducted by S. Shyam Sunder’s team at
NIST. The subsequent progressive collapse was not simulated at
NIST because its inevitability, once triggered by impact after column
buckling, had already been proven by Bažant and Zhou’s
(2002a) comparison of kinetic energy to energy absorption capability.

The elastically calculated stresses caused by impact of the
upper part of tower onto the lower part were found to be 31 times
greater than the design stresses (note a misprint in Eq. 2 of Bažant
and Zhou 2002a: A should be the combined cross section area of
all columns, which means that Eq. 1, rather than 2, is decisive)."


Sunder claims to take it to the point of collapse initiation, and Bazant after a one story freefall. How can they not know they skipped over the initial buckling sequence, the part where the entire core fails in less than one second?

Less than 1 second failure of the entire core from south to north and it is unimportant?

All columns fail so quickly that not a single floor to floor collision could take place in the interval. The lowest part of the upper moving portion could fall only 8 feet in that time. This is unimportant?

The simulations of the initial core failure were probably never performed because the results would show the earliest physical trajectory of the WTC1 cap (total core failure in less than 0.8 seconds) is impossible to achieve through a natural mechanism. As the collapse simulations of WTC7 should be considered an embarrassment to the NIST WTC7 reports, so a visual model of the initial lateral collapse propagation of WTC1 would show the official initiation scenario to be extremely flawed..
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Collapse Initiation as Pure Angular Motion: Stick Visualiza

Postby Major_Tom » Sun Jan 10, 2010 2:02 am

I propose a humble 2-D stick model approach to analysis of lateral collapse propagation and to visualize the initial separation of the building into two distinct parts: Falling and stationary.


In the 2-D model the east and west walls of WTC1 are missing. They don't exist. This allows us to visualize the simpler mechanism of pure south to north core column failure all by itself.

This can later be expanded to a simple 3-D model by adding depth and the east-west perimeter walls.

For WTC1 the short-lived hinge about the north wall axis (fl 98) gave after the moving portion of the building rotated less than 2 degrees (or 0.8 seconds) about it's axis. All core and perimeter columns had failed by that time.

Image

This image shows what a 2 degree tilt around the north wall 98th floor axis looks like. In a perfectly rigid model we see that each column must shorten simultaneously for this to occur. The yellow area shows the overlap at 2 degrees.

Every column in the tower had failed within 0.8 seconds of the earliest detectable movement. The stick graphic shows the south wall falls less than 8 feet during a 2 degree tilt. There could have been no significant collisions during this time interval since the floor slabs along the southernmost part of the building are aren't close to making their first contact when the pivot hinge broke.

....................................................................................

The original buckling would happen during purely rotational motion around the north wall. Therefore our variable is "theta" as theta moves from 0 to 2 degrees.

We consider the case of the south wall "disappearing". How does the core react?


For a typical column with pivot arm "a", the shortening as the angle moves through theta is

a*sin(theta), or if we express theta in radians and use a small angle approximation, the shortening is a*theta.

Within the elastic limit this shortening will meet with a stiff upwards force. It is basically a spring force with a very stiff spring (because of the linear relation between shortening (u) and F in this region).


The 2-D physics problem is this: If you remove the south wall quickly, would the resulting motion cause enough downward displacement on the 1000 row CCs to shorten the columns outside of their elastic limit?

The original problem is set up in 2-D which is very generous towards collapse continuation. The core receives no support from the east and west perimeter walls and the loss of the south wall strength cannot be transferred to the east and west perimeter walls as it would in reality. The hat truss system transfers all south wall and OOS south weight to the core instantaneously which is also absurdly generous towards core failure and collapse continuation.
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Basic Models of Initial Buckling for pure rotational motion

Postby Major_Tom » Sun Jan 10, 2010 2:03 am

By what mechanism is the initial damage front propagating northward? How can we visualize how the whole core failed in less than one second?


In this 2-D stick model, or any model, the initial lateral collapse can propagate damage south to north in only three ways:

1) Successive overload: A column fails transferring it's weight to nearby columns., which are overloaded and fail, and so on.

2) Push and pull: Failing columns tug or push other columns down through lateral braces or flooring

3) Some type of shock through material by which damage can propagate.

Using combinations of these 3 types of movement we will look for a natural way to destroy the whole core in less than one second to match the observed duration of the hinge.

This is a first attempt at a horizontal propagation model which let's us visualize the propagation component by component. For WTC1 we want to be able to visualize how the south wall can fail, which leads to the 1000 CC failing, which leads to the 900, then to the 800 row, to the 700 row, the 600 row to the 500 row CC failure...........all in less than one second.

Is there another way by which lateral destruction can propagate that I didn't include?


Consider this a game or a challenge. Can you find a way to propagate column failures from the south wall to the 500 row columns in less than one second using combinations of the three known ways that lateral destruction can progress listed above?

..................................................

Extreme cases: Rigid and Non-rigid Models

1) RIgid model: The entire south wall fails and transfers it load to the core. All CCs shorten simultaneously in order to allow the upper portion to rotate and fail as a rigid block through the first 2 degrees.

2) Non-rigid model: The entire south wall fails and transfers it's load only to the 1000 CCs. No other CCs help carry the extra load.

...............................................................

Problems for the rigid model:

Image

1) In the yellow region notice all those floor slabs just hanging in mid air supported only by the hat truss system after the south wall gives.

Can it transfer all that weight to the 1000 CCs?

There has to be a limit to the number of dead weight floor slabs the outriggers can hold suspended in mid air if the 1000 CCs temporarily hold.

Image

The dead weight of the slabs in the yellow area is hanging from the outriggers. Will the yellow area sag and start a partial collapse? Will the whole load get transferred to the core? Could the 1000 CCs receive the extra load and remain in their elastic range (handle the load)?

If we know the peak load F0 value in the B&V graph for the 1000 CCs, we can answer the last question.

Towards a possible answer Tony wrote:

The load would predominantly transfer to the east and west perimeter walls through the spandrels and the NIST analysis itself shows the additional load on the southern columns of the east and west walls is only 69% on columns which can take 500% of their design load. Additionally, a rigorous analysis would also show that the central core columns wouldn't be affected much by a south wall failure.




In our 2-D model the east and west walls do not exist so the south OOS would slouch or the total load would be transferred to the core, Would this seem to overload the 1000 CCs?


If the rotating portion is treated as a rigid object and the hat truss system is infinitely strong, all CCs would shorten simultaneously, thus each would contribute to resist rotation caused by the loss of the south wall proportional to their shortening in the elastic range.

If all CCs shortened together, hence resist rotation together,the upward torque provided by the spring restorative forces in all columns would act simultaneously and be massive, increasing proportionally to theta (within the elastic range of the 1000 CCs).


(Actually, in a 3-D model containing the east and west walls, I think the building just may hold considering how well it held on the WTC1 north face after having more than half it's perimeter columns cut. But it holds as a result of the spandrel plates, not because the outriggers can hold all the weight.)

In our 2-D model the outriggers probably cannot hold all the weight. If they cannot transfer all that weight after the south wall looses strength, then they limit the excessive load that can be transferred to the 1000 CCs.

This supports the idea of "slouching" (in our 2-D case). There in no guarantee that loss of the south wall can overload the 1000 CCs if the outriggers have limited capacity to transfer the load.

Instead, the south wall can sag or start to fall on it's own. (In the 2-D model where we have no east or west walls.)

....................................................

Problems for the non-rigid model:

How to do a rough solution? For the building not to rotate the total torque lost when the south wall failed must now be supplied by the core. What if the total torque the south wall used to supply was transferred to only the 1000 CCs?

let's call the upwards force the south wall supplies under normal structural conditions F(s). Around our north wall axis the torque would be 207 ft multiplied by F(s).

For the 1000 CCs to supply the same torque the 1000 CCs would have to supply (207/147)F(s), or 1.4F(s). 1.4 is just the ratio of the torque arms. Because the 1000 CCs have a shorter torque arm than the south wall, they have to supply a larger force to supply the same torque. About 1.4x the force is needed

This means that if the south wall load was transferred as a static load applied to the 1000 CCs and no other CCs s helped carry the new load, the 1000 CCs would experience an additional 1.4F(s) of force. If the 1000 CCs can handle the additional load, the building wouldn't start to rotate.


We find the torque arm ratio between the south perimeter and 1000 row CCs to be about 1.4. And, since we consider the 1000 CCs and south perimter to share the OOS south region load equally, loss of the south wall doubles the load on the 1000 row CCs. Combining these two factors increases the 1000 CC load to about 2.8.

In this case the east and west walls do not help carry the load at all and the rest of the core doesn't shorten or help at all.

Even in this extreme case most favorable to core failure we still can't seem buckle the 1000 CCs.

........................................................
Some Questions


Q1: What is the maximum shortening the 1000 row CCs can experience before leaving the "Hooke's Law" phase? What tilt angle does this correspond to?

Q2: Upon loosing strength in the south wall, what is the new equilibrium angle theta? (Damped case. No rocking, no overshoot, just settling).

..................................................................

COLUMN RESPONSE TO SHORTENING

To find answers we need to know how columns react to being overloaded. A force a typical column can transmit is shown as a function of column shortening below.

Image

The elastic region is in yellow. The plastic region in pink. Columns have strong spring-back abilities when compressed. In order to buckle one from overload it is necessary to compress it past it's natural ability to spring back. The high spike in the graph between the yellow and pink regions shows the minimum downward force necessary to buckle the column (Fo). It is quite high relative to the normal load the column handles (Fc), meaning we need substantial additional force (2x to 5x Fc) applied downwards to buckle a column.

Dr Bazant explains his graph (from the paper):

"Let u denote the vertical displacement of the top floor relative to the floor below
(Figs. 3 and 4), and F(u) the corresponding vertical load that all
the columns of the floor transmit. To analyze progressive collapse,
the complete load-displacement diagram F(u) must be
known (Figs. 3 and 4 top left). It begins by elastic shortening and,
after the peak load F0, curve F(u) steeply declines with u due to
plastic buckling, combined with fracturing (for columns heated
above approximately 450°C, the buckling is viscoplastic). For
single column buckling, the inelastic deformation localizes into
three plastic (or softening) hinges (Sec. 8.6 in Bažant and Cedolin
2003; see Figs. 2b,c and 5b in Bažant and Zhou 2002a). For
multistory buckling, the load-deflection diagram has a similar
shape but the ordinates can be reduced by an order of magnitude;
in that case, the framed tube wall is likely to buckle as a plate,
which requires four hinges to form on some columns lines and
three on others (see Fig. 2c of Bažant and Zhou)."

The description of F(u) seems good enough and it may be perfect for our stick model.

Notice that the elastic range, from u=0 to just before the point F0 (yellow), is a linear relation as expected.

...........................................................................................

The difference between Bazant's use of column overloading and this approach:


Bazant examines the compressive forces of the columns only after an assumed 12 ft freefall and axial collision. We want to examine the compressive forces needed to create the initial buckling instead.


Stated simply, Bazant deals with the secondary group of column buckles (rebuckling). He claims the inevitability of rebuckling of the lower columns only after the columns have already buckled once.

We are interested in the initial buckling.

These are the buckles, given the actual trajectory of WTC1 during the "hinging", that seem physically impossible to make.

Bazant's focus is on successive rebuckling after the columns have already given way and the top of the building has already started to move.

..........................................................................


So either way you look at it, we have a problem getting the core to fail, especially in less than one second.

Image

Even if all the floor slabs in yellow can hang suspended from the outriggers, and even if all the load is transferred only to the 1000 CCs and no other CCs help at all, I'm still not sure if we have overloaded the 1000 CCs.

F(1000, new)= F(1000, old) +1.4F(s). Are we buckling yet?
Major_Tom
 
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Joined: Wed Jul 09, 2008 5:04 pm

Pure Rotational fixed hinge basic models (Stage 2)

Postby Major_Tom » Sun Jan 10, 2010 2:07 am

We consider the additional torque CCs must provide to make up for the loss of the south wall and provide simple analytical solutions:

For movement d(theta), theta is in radians, small angle approximation:

500 row CCs shorten a*d(theta) where a=60 ft.
600 row CCs shorten b*d(theta) where b=75 ft.
700 row CCs shorten c*d(theta) where c=95 ft
800 row CCs shorten d*d(theta) where d=112 ft
900 row CCs shorten e*d(theta) where e=132 ft
1000 row CCs shorten f*d(theta) where f=147 ft (some numbers wrong, will be edited soon)

We have 6 stiff elastic springs pushing upwards with different torque arms and gravity pushing down with force Mg and a torque arm through the center of the building.

Initial conditions: position theta=0, angular velocity=0, and then we quickly remove the south wall.


The building will oscillate in the variable theta (angular oscillations). The question is (in 2-D) will the maximum downward displacement push the 1000 CC row outside of it's elastic limit?

To do the problem we need to know the moment of inertia of the upper part around the north wall pivot point. We ignore the mass of the east and west walls. We also need the restorative spring force k of each group of columns (only three values since the building is symmetrical).

If the core doesn't fail in this case, how can it fail with the east and west walls included?

..................................................................

Defining the three spring constants.............................................

k(500)=k(1000)=k1
k(600)=k(900)=k2
k(700)=k(800)=k3

Total Torque........................................................

Torque down=Mg*104 ft

Torque upwards=k1*theta*a+k2*theta*b+k3*theta*c+k3*theta*d+k2*theta*e+k1*theta*f

=(k1*(a+f)+k2*(b+e)+k3*(c+d))*theta


but note (a+f)/2=(b+e)/2=(c+d)/2= half the width of the building, which we will call m(=104 ft)


So

Torque upwards =2*m*(k1+k2+k3)*theta

and total torque is (Mg-2*(k1+k2+k3)*theta)*m (small angle approximation)



Our equation of motion is

(Mg-2*(k1+k2+k3)*theta)*m=I*angular acceleration, where angular acceleration is just the second derivative of theta wrt time.



Just as we would expect, we have a constant torque downwards but an upwards torque which increases linearly with theta. This is a harmonic oscillator with no damping which is absurdly favorable to core failure.


When the oscillations stop, what is our new equilibrium position? When Mg-2*(k1+k2+k3)*theta=0, or

theta=Mg/( 2*(k1+k2+k3))


And what is the maximum theta displacement downwards? Double the new equilibrium value, or when

theta(max)=Mg/(k1+k2+k3)

.................................................

Does the 1000 row CC compression exceed it's elastic limit? (Can you compress the 1000 CCs by f*theta(max) and have them bounce back?


In reality the south wall didn't disappear instantaneously, so theta will not overshoot it's new equilibrium value by a factor of 2 and the top portion will not wiggle back and forth before coming to rest (like in an old Pop-eye cartoon). In the case of a damped harmonic oscillator where the building settles to rest in it's new angle of equilibrium (near critical damping) the new theta(max)

=Mg/2*(k1+k2+k3)

meaning a 1000 CC shortening of f*theta(max)

The extreme (cartoon) case of no damping will double this value.

I like the way we loose the moment of inertia in this term because I didn't want to calculate it anyway. All we need are the spring constant values.

.........................................................................

How do we calculate our k values?

We have 47 k values, one for each column. K values for parallel springs are additive.

For example, k1 for the 1000 row columns............

k1=k(1001)+k(1002)+k(1003)+.....+k(1008)


We reduce the problem to finding k values for individual columns.
Major_Tom
 
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Outline of a solution for rigid and non-rigid cases

Postby Major_Tom » Sun Jan 10, 2010 10:29 pm

The first attempt at a solution for rigid and non-rigid cases is given by achimspok. From his posts:

If 2*(k1+k2+k3) is the k-sum for all 6 rows CC then

2*(k1+k2+k3) = 3194921166 N/m

for the cross section area "below" floor 51. I used the "51" data because according to the data of 501 the average cross section B1...98 is 0.293m² and this corresponds best to the 48...51 part of CC501.

Cross section area [m²]

here are the numbers for copy/paste:
# \ row 500 600 700 800 900 1000
1 0,292862318 0,10451592 0,10951591 0,112812274 0,080607497 0,28306395
2 0,195179852 0,102217538 0,114500416 0,114500416 0,094576585 0,18435447
3 0,185957289 0,09483852 0,072973644 0,072459533 0,09483852 0,226360434
4 0,13741908 0,072973644 0,029782082 0,049886881 0,058215609 0,12967716
5 0,139959398 0,07661275 0,028242214 0,064697754 0,064757935 0,122439271
6 0,20177379 0,105876804 0,080607497 0,104064824 0,085978925 0,213890701
7 0,191249618 0,098538109 0,104064824 0,106118739 0,104064824 0,18435447
8 0,292862318 0,08400588 0,10951591 0,000000000 0,080607497 0,28306395
sum row 1,637263662 0,739579166 0,649202498 0,624540422 0,663647392 1,627204407
sum total 5,941437547 all in [m²]

according to Tony:


k = Ma/l

l = 384,6576 m (length between B6 and 98)
a = cross section as table above
M = 30E+6 psi = 2,06844E+11 N/m²

stiffness (k) [N/m]


Here are the numbers for copy/paste:
stiffness [N/m] 500 600 700 800 900 1000
1 157482429,3 56201907,77 58890579,28 60663151,02 43345502,8 152213500,2
2 104955111,3 54965986,18 61570924,58 61570924,58 50857175,72 99133920,64
3 99995813,32 50998027,42 39240510,25 38964054,11 50998027,42 121722013,8
4 73895100,95 39240510,25 16014879,33 26825940,75 31304592,72 69731996,67
5 75261119,54 41197386,09 15186837,89 34790271,47 34822632,67 65839927,83
6 108500905,3 56933703,44 43345502,8 55959337,56 46233909,65 115016597,1
7 102841685,4 52987427,51 55959337,56 57063800,45 55959337,56 99133920,64
8 157482429,3 45172933,85 58890579,28 0 43345502,8 152213500,2
sum row 880414594,4 397697882,5 349099151 335837479,9 356866681,3 875005377,1
sum total 3194921166

I checked the angles for a spring length of "one floor height" and the "98 cross section" and the spring length of "98 floors" - "50 cross section" for the "whole core - mass of upper block" (A) as well as for "1000 row - south wall load"(B).

B)
Image
B)
Image

The graphs are not the damped case but...
Case A)

Image

Torque upwards=tilt*(k1*a+k2*b+k3*c+k4*d+k5*e+k6*f)
Torque downwards=r*m*g*sin(ThetaA)
m=upper block

Case B)

Image

Torque upwards=tilt*k6*f
Torque downwards=r*m*g*sin(ThetaB)
m=south wall load

You see I used your formulas but without the simplifications. I calculated both cases for the short (1floor height) and the long (B6...98) spring. In the end it doesn't matter. The angles are much to small for all cases.
Major_Tom
 
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Joined: Wed Jul 09, 2008 5:04 pm

Re: Refutation of NIST WTC1 Collapse Initiation Scenario

Postby Major_Tom » Thu Jan 14, 2010 3:11 pm

space reserved
Major_Tom
 
Posts: 3277
Joined: Wed Jul 09, 2008 5:04 pm

Re: Refutation of NIST WTC1 Collapse Initiation Scenario

Postby Major_Tom » Thu Jan 14, 2010 3:12 pm

space reserved
Major_Tom
 
Posts: 3277
Joined: Wed Jul 09, 2008 5:04 pm

Re: Refutation of NIST WTC1 Collapse Initiation Scenario

Postby Major_Tom » Thu Jan 14, 2010 3:20 pm

Thanks for waiting. All on topic comments and criticism encouraged.
Major_Tom
 
Posts: 3277
Joined: Wed Jul 09, 2008 5:04 pm

Re: Basic Models of Initial Buckling for pure rotational motion

Postby Heiwa » Thu Jan 14, 2010 5:21 pm

Major_Tom wrote:
COLUMN RESPONSE TO SHORTENING

To find answers we need to know how columns react to being overloaded. A force a typical column can transmit is shown as a function of column shortening below.

Image

The elastic region is in yellow. The plastic region in pink. Columns have strong spring-back abilities when compressed. In order to buckle one from overload it is necessary to compress it past it's natural ability to spring back. The high spike in the graph between the yellow and pink regions shows the minimum downward force necessary to buckle the column (Fo). It is quite high relative to the normal load the column handles (Fc), meaning we need substantial additional force (2x to 5x Fc) applied downwards to buckle a column.

Dr Bazant explains his graph (from the paper):

"Let u denote the vertical displacement of the top floor relative to the floor below
(Figs. 3 and 4), and F(u) the corresponding vertical load that all
the columns of the floor transmit. To analyze progressive collapse,
the complete load-displacement diagram F(u) must be
known (Figs. 3 and 4 top left). It begins by elastic shortening and,
after the peak load F0, curve F(u) steeply declines with u due to
plastic buckling, combined with fracturing (for columns heated
above approximately 450°C, the buckling is viscoplastic). For
single column buckling, the inelastic deformation localizes into
three plastic (or softening) hinges (Sec. 8.6 in Bažant and Cedolin
2003; see Figs. 2b,c and 5b in Bažant and Zhou 2002a). For
multistory buckling, the load-deflection diagram has a similar
shape but the ordinates can be reduced by an order of magnitude;
in that case, the framed tube wall is likely to buckle as a plate,
which requires four hinges to form on some columns lines and
three on others (see Fig. 2c of Bažant and Zhou)."



I think you have misunderstood. It is evidently the upper part columns that buckle first and then 'initiation' is arrested.
Heiwa
 
Posts: 427
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