1. Jedo

    Jedo Member

    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.

    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)
    [​IMG]
    @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:
    So Euler buckling (i.e. elastic) is valid only for high slenderness ratios before it yields, here the yellow area
    [​IMG]

    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:
    [​IMG]
    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?
     
    Last edited: Feb 12, 2018
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  2. Jeffrey Orling

    Jeffrey Orling Active Member

    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.

    ++++

    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.
     
  3. Jedo

    Jedo Member

    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?
     
  4. Jedo

    Jedo Member

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

    Jeffrey Orling Active Member

    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.
     

    Attached Files:

  6. Jedo

    Jedo Member

    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.

    global17.
    global19. global22. global24.
    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):
     
    Last edited: Feb 14, 2018
  7. Mick West

    Mick West Administrator Staff Member

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  8. Joe Hill

    Joe Hill Member

    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.
     
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