# How Buckling Led to "Free Fall" acceleration for part of WTC7's Collapse.

Well I guess in answering how the buckling leads to free fall (back on-topic) we have to look a bit more in detail into the basics of buckling.

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With this much bowing the bar supports just 12 pounds. How would I have to brace it so that it supports 1200 pounds? Theoretically I don't need to use multiple supports, I can just use an 8" section braced at either end and that should be the same as the longer piece braced every 8"?

I've been researching about the theory of buckling and of structural stability the last days, since I have had not much of an idea of it. However, I already learned some things.

When you measure the maximum load you have to account correctly for the boundary conditions i.e. the way the ends of are column are fixed (or not), see (from Wikipedia)

@Mick West measured the fourth case, where the buckle mode has the form of x =x_0 \sin(\pi z / L) (sorry for tex source code here, but the latex feature seems broken). In the building of WTC7 the columns are all fixed, so this corresponds to the first case with fixed ends where the buckle mode takes the form x= x_0/2 (1 + \cos(2\pi z/L ). The sinus and cosinus come from the fact that it is a second order differential equation.

The critical buckling load then should be 4 times as big, as it is proportional to 1/(KL)^2.

Furthermore, when you look not at the total load but on the stress, you get that it is proportional to the ratio of the moment of inertia to the total cross section area, I/A. So the beam @Mick West used had two very different moments of inertia, and it buckled along the axis with the smaller moment of inertia. Such a beam is called a slender beam.

From Wikipedia:
External Quote:
For slender columns, critical stress is usually lower than yield stress, and in the elastic range. A stocky column would, on the contrast, have a critical buckling stress higher than the yield, i.e. it yields in shortening prior the virtual elastic buckling onset.
So Euler buckling (i.e. elastic) is valid only for high slenderness ratios before it yields, here the yellow area

However, close to the border between the yellow and white area, there is apparently another failure mode, which is given by Johnson's parabolic formula, which gives another boundary by a parabola:

I did however not get how they can calculate this critical slenderness ratio, nor what failure mode exactly this corresponds to. I would have guessed that this is a inelastic buckling mode, as the critical stress is the yield for axial compression only. Maybe someone can help here?

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qed
It should be noted that the 7wtc columns were 2 story height lengths. The bracing... beams and girders framed into them where not at their ends. Column to column end conditions were unbraced. So even if you consider multiple column lengths the ends are unbraced.

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Not all columns in the WTC were "equal"... as clearly some carried significantly more loads than others. The columns in the core where the elevators were located had much lower axial loads... the elevator lobbies and perhaps toilets and mechanical closets. Column 79 was carrying a large area of floor loads on the NE corner of the building. It appears that a core column's collapse lielky would not lead to a progressive collapse resulting in building collapse. However the collapse of of col 79 could lead to the failure of the load transfers below it causing the failure to propagate laterally.. Column 79,80 and 81 were critical columns the failure of which would lead to the collapse of entire building. It should be noted that load transfers are no common features of high rise frames... but that are used when called for. In the case of 7wtc building above Con Ed required load transfers.

t should be noted that the 7wtc columns were 2 story height lengths. The bracing... beams and girders framed into them where not at their ends. Column to column end conditions were unbraced. So even if you consider multiple column lengths the ends are unbraced.

I am afraid that I disagree here. The different cases of the Euler buckling I showed above in the table are determined by the boundary conditions: In the first case, the end positions and the derivatives (i.e the angle the column has to the vertical) at the end position are fixed. In the fourth case only the positions are fixed but the angle is free (the column can 'rotate' around its fixation point meaning it can have an angle).

And in WTC7 columns were connected by column splices that fixed the position of the end of the columns as well as their angle to each other (straight). However, I think it might be misleading to discuss how single columns buckle, I think one needs to consider the composite column made out of separate beams connected by splices for the whole length where it is not fixed laterally. Do you agree with that?

++++

Not all columns in the WTC were "equal"... as clearly some carried significantly more loads than others. The columns in the core where the elevators were located had much lower axial loads... the elevator lobbies and perhaps toilets and mechanical closets. Column 79 was carrying a large area of floor loads on the NE corner of the building. It appears that a core column's collapse lielky would not lead to a progressive collapse resulting in building collapse. However the collapse of of col 79 could lead to the failure of the load transfers below it causing the failure to propagate laterally.. Column 79,80 and 81 were critical columns the failure of which would lead to the collapse of entire building. It should be noted that load transfers are no common features of high rise frames... but that are used when called for. In the case of 7wtc building above Con Ed required load transfers.

I do not quite understand what the part below the +++ has to do with the buckling / free fall question mentioned in the OP?

The presumption seems to be that there was column buckling.mm meaning that columns which lose capacity can no longer support the design loads and will collapse under those loads. This is a FACT.

There is also the notion that if one column fails for whatever reasons the loads it supported will ALWAYS be transferred to adjacent columns. This is not always the case. A column can fail and the loads ... floor are and column above can collapse down and these loads do not get transferred. However the beams in a multi story structure do provide lateral bracing and this has an impact on load bearing capacity.

However this is not the ONLY only mechanism by which this structure could have collapsed. If axial alignment is destroyed the loads above have no bearing and will "drop"...and will drop at the force of gravity. Axial misalignment is possible from steel beams expanding when heated and pushing the beams they are attached to "apart" and causing their ends to mis align.

The attached sketch show how much the bearing surface is VASTLY reduced by a 1/2" two axis translation.

No one disputes that steel heated will expand... In order for the bearing area to be preserved with expanding beams.... the push would have to be equal on both to upper and lower column.... and this means that the the beams framed into both the upper and lower columns be heated the same way and expand the same. This is possible but not always what occurs in a fire.

If and when bearing area is severely reduced web and or flange crippling will occur in the columns at their end connections. This is a form of buckling of course. But the take away is that when there is buckling there is translation and loss of axial alignment enabling rapid onset of gravitational collapse.

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In reading now NCSTAR1-9A, I found several figures that are of interest to this thread.

Here the show the lateral displacement only, with the x-displacement (east +/blue, west -/red) (east -/red, west +/blue) on the left and the y-displacement (north +/blue, south -/red) (north +/red south -/blue) on the right.This is quite interesting as one can see the buckling modes better in these figures.

So they see here buckling for floors 7 to 14, that is 8 floors, which is consistent with the free-fall time observed; NCSTAR1-9, Ch.12.5.3 p. 602 (668 pdf):
External Quote:

In Stage 2, the north face descended at gravitational acceleration, as exterior column buckling progressed and the columns provided negligible support to the upper portion of the north face. This free fall drop continued for approximately 8 stories (105 ft), the distance traveled between times t = 1.75 s and t = 4.0 s.

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What led to the RELEASE was the failure of those (transfer) trusses. What caused their failure needs to be determined. I suppose it could have initiated in load transfer region. Or it could have initiated on some floor(s) above them... but the "internal movement" would go unseen from outside.
Well, we have this; one transfer truss, I can only guess part of the west truss, due to it's location after being removed from the pile, shows horizontal, not vertical buckling, as does the still attached floor girders. This suggests the failure was at some point above the truss, not the truss itself. I can see no way for that damage to occur other than the mass attached above falling and twisting.
[GALLERY=media, 121]7evidence_Twisted_Steel_Barclay_AnnotatedInset2 by Joe Hill posted Jun 14, 2019 at 12:09 AM[/GALLERY]

For years I pointed out the "soda can test" when discussing the building collapse, but could never find a video of someone showing it.

As Ive gotten older / heavier, i cant do it anymore. But as a 100 pound or so kid, I did this 100s of times, literally every can that my family recycled.

In simplest terms. The can, when not compromised, can hold several times its weight.

Once its compromised, even slightly.....

Thanks again

I think the presentation that the 2 story column 79 buckled on floors 12-14 caused by the loss of bracing from girders framed into it may be true. This is premised that removing those girders rendered the column too slender and could no longer support the imposed loads from above. That was quite a stout section and I would presume it was located in the rubble and shows evidence of buckling. MAYBE. The massive sections... either WF or made from plates or made from rolled sections and plates would take enormous loads to deform. I would expect to see many "intact" column sections in the collapse piles. I believe this to make sense because the weak place in the load path would be the connection of one column to the next one. When a jenga structure falls... the blocks are intact.

I can't find photos of the columns removed from the piles post collapse. Some photos show columns with "clean" ends sticking out of the pile. And of course smaller debris would cover the piles.

Can anyone direct me to links of recovered columns from the collapse?

Was column 79 recovered?... from floors 12-14?

In reading now NCSTAR1-9A, I found several figures that are of interest to this thread.

Here the show the lateral displacement only, with the x-displacement (east +/blue, west -/red) (east -/red, west +/blue) on the left and the y-displacement (north +/blue, south -/red) (north +/red south -/blue) on the right.This is quite interesting as one can see the buckling modes better in these figures.

View attachment 31852
View attachment 31851 View attachment 31850 View attachment 31849
So they see here buckling for floors 7 to 14, that is 8 floors, which is consistent with the free-fall time observed; NCSTAR1-9, Ch.12.5.3 p. 602 (668 pdf):
External Quote:

In Stage 2, the north face descended at gravitational acceleration, as exterior column buckling progressed and the columns provided negligible support to the upper portion of the north face. This free fall drop continued for approximately 8 stories (105 ft), the distance traveled between times t = 1.75 s and t = 4.0 s.
You beat me to it.

The NIST simulations suggest that one reason for multistorey freefall was was the entire structure of stories 7-14 shifting bodily northwards due to

- loss of restraint from the east wall from east floor collapse

- southward sway of the upper stories

- loss of restraint as west wall buckled

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My conception was that the interior literally collapsed inside the exterior shell. The EPH drop was the evidence of this. The interior collapsing also left the exterior with no lateral bracing and as it dropped parts of the moment frame were likely experiencing lateral forces as the interior pulled away from it. The massive debris from the interior collapse most likely undermined the columns supporting the moment frame.

The moment frame was "supported" by a 3 story (IIRC) truss structure which was at floors 5-7 encircling the building. Composite with that 3 story belt truss were the 26 of the 57 columns at grade.

The 2+ second collapse at close to FF is explained by the collapse of the truss and columns from grade to floor 8 which was caused by the massive debris of the buildings interior pile up. The entire "collapse sequence" was initiated by structural failures in the NE quadrant around col. 79, 80 (blue area in plan below).

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@Jeffrey Orling failure of the transfer structure is one possibility. Unfortunately, it has yet to be seriously investigated.

And the NIST simulation does not suggest substantial floor collapse beyond the east. Rather it suggests a sideways sliding of largely intact floor structures which caused the storeys to fold along the north-south axis like a square truss. So columns bent in sync with those to the south and north. Hence caption "Northward shifting of structure spreads to buckling of exterior columns"

My conception was that the interior literally collapsed inside the exterior shell. The EPH drop was the evidence of this. The interior collapsing also left the exterior with no lateral bracing and as it dropped parts of the moment frame were likely experiencing lateral forces as the interior pulled away from it. The massive debris from the interior collapse most likely undermined the columns supporting the moment frame.
What's interesting is that the collapse of the twin towers seemed to progress in the opposite way, especially for the south tower:

the exterior columns and floors collapsed downward around the inner core at slower than free fall in ROOSD, leaving standing portions of the inner core column structure left with no lateral bracing from the missing floors, and the debris from the initial collapse that accumulated at the base of the core (from the exterior columns and crushed floor trusses) appears to have caused further undermining of the base of the core columns followed by the interior, hollowed-out remains of the core columns falling down more or less vertically at near free fall.

The core column to column connections were un braced and between floors in the twin towers. I don't recall, but I think the same for 7WTC.

The twins collapse were first collapses / destruction of the towers above the plane strike zones. That created and freed the destructive masses which then drove down through the towers inside the cage of the perimeter panels... which lost lateral support and fell away.

The "sequence" with respect to 7wtc seems to be as follows:

Apparently fires on a low floor (12-14) in the NE quadrant caused expansion/distortion of beams and girders framed into cols 79, 80. This framing was pushed off the beam seats causing local floor collapses and leaving these columns with inadequate lateral bracing. The columns were below the EPH and did buckle leading to the EPH collapsing through the building.

This likely led to the massive transfer trusses at the mechanical floors 5-7 to loose their support....or were displaced. The failures of the these transfer trusses likely pulled the girders on the north side of the core all the way to TT3 on the west side of the core. These structures also failed.... causing the floors on the north of the core to collapse down through the tower... massive pile up of debris at the base which then undermined the structure at the base supporting the moment frame. It had 27 columns composite with a belt truss. This collapsed and the moment frame dropped down from the 8th floor... 100'+.... accounting for the apparent almost FF collapse of the moment frame and facade.

Long story short... heat cause beam and girder expansion... which left key columns with inadequate bracing and they collapsed.

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@Jeffrey Orling The furst problem I have with that explanation that very few exterior columns depended on transfer structures. Only 8 on the north and 1 in the West.

The second problem is that the west penthouse was still visible 2 seconds after the roofline moved. Meaning the interior probably didn't collapse all that much

@Jeffrey Orling The furst problem I have with that explanation that very few exterior columns depended on transfer structures. Only 8 on the north and 1 in the West.

The second problem is that the west penthouse was still visible 2 seconds after the roofline moved. Meaning the interior probably didn't collapse all that much
The WPS was above the core... the core was shafts... what collapsed was the outside the core floors... and then the core columns I suspect.

@Jeffrey Orling one important thing I forgot to mention. Falling from the 8tg floor you will land on the 2 storey Con Edison substation

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yes

@Jeffrey Orling

Why did the collapse of truss 1 and 2 cause the collapse of the transfer girders?

And why did the floors in the West and south collapse?

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Col 73 the west side of TT1 and the girder along the north side of the core (magenta).. that girder was displaced laterally causing the failure of the 6 MG 27s which carried the north side of the moment frame.
Cols 74 and 75 moved laterally and undermined the south side of the core and the floors to the south of the core.

@Jeffrey Orling the perimeter girders were separated at each column. I don't understand how they would move as one

There was bucking in the process.
Bucking was not the initiating failure. As noted in another thread the collapse involved a series of failures which apparently began with beams and girders expanding which LEAD to floor collapse and local column buckling.
Unless there is a progression... buckling of a column will only be a "local"... not global failure.

As I said, the disconnection to the east wall left the rest of the floor vulnerable to horizontal rotation and consequent buckling of columns

As I said, the disconnection to the east wall left the rest of the floor vulnerable to horizontal rotation
rotation requires a means for a lateral impulse to "develop"

@Jeffrey Orling correct, but no one rises that objection about buckling. Just accepting that the column is unrestrained is sufficient.

It is important to understand how one thing (failure) led to another and that there is little in steel frame designs to prevent this.. or maybe there is. Few steel frames are built with structures to span over something like an power substation. This required transferring axial loads with trusses etc.. and these transfers enabled a global rather than a local (NE quadrant) collapse. Note it took many floors of mass to dislodge the transfers.

@Jeffrey Orling are there any analyses to back up the idea of widespread transfer failure as the main cause of global collapse?

@Jeffrey Orling are there any analyses to back up the idea of widespread transfer failure as the main cause of global collapse?
I don't know if @Jeffrey Orling has drifted into claiming that transfer truss failure was THE cause of global collapse. I have many times agreed with Jeffrey that TT Failure is a plausible cause of the free-fall descent of the perimeter shell through some 7 or 8 storeys. That is a long way short of claiming it was the cause of GLOBAL collapse.

@Jeffrey Orling are there any analyses to back up the idea of widespread transfer failure as the main cause of global collapse?
There is no claim by the engineering community that the transfer structures were THE cause of the collapse. But the building's engineer and others did suggest that they played a large role in the global nature of the collapse... ie the structures whose failure cause involvement of the entire footprint. Just the nature (and name) implies that they involve directing forces laterally and this is what was needed to spread destruction throughout the plan.
At one point I attributed the failure to transfer truss failure and tagged the explanation TTF... transfer truss failure. However it is more likely that they were not the cause but the means to make the collapse go "global". FEMA seemed to find the design down there concerning.

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@econ41 he thinks that transfer failure was the main mechanism of horizontal collapse progression.

@econ41 he thinks that transfer failure was the main mechanism of horizontal collapse progression.
He seems correct. How else would "damage" propagate laterally?
You could have column failure from load redistribution.
Yet it looks like the WTC collapse phase mostly involved floor collapse... floors coming free from the columns... or failing of beams, girders and floor trusses.

@Jeffrey Orling check out this snapshot from the NIST simulation and see if it changes your mind

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