SanderO wrote:I raise this issue because of how heat would impact a steel column. All stress must *pass through* the cross sectional area of the column. If the column is 3 stories tall but heated at a location say 1-5' above the bottom of the column as it was set about 3' off the floor on top of the column below... the very bottom of the heat column is being weakened by the heat. The length above may remain cold enough to have most of its strength. But that wouldn't matter at all. If the lower 5 feet has lost 50% of its strength that governs how much axial load that column can support....
Let me initially limit my answer to the key point of this paragraph. It contains a misunderstanding about column strength which underpins much of the remainder of your post.
And my comments are again "top of the head" on subjects which I have not studied for 50years plus.
The strength of a column for our purposes here is the measure of the maximum axial load it can carry. The limit is reached when the column is on the verge of buckling. Once it buckles strength is lost very rapidly - instantaneous for our purposes here. So it is a "trigger" or "threshold" situation.
The theoretical work to determine that strength was done way back 1700's IIRC by Euler.
The main point is that for any given column cross section and presuming constant cross section the strength is inversely proportional to the square of the length of the column. So doubling the length of column reduces its strength to one quarter. Hence (whether we accept the full NIST WTC7 hypothesis or not) the removal of horizontal bracing of column 79 of WTC7 over (say) three floors would result in a ninefold weakening of the column. Based on the raw length factor alone.
Another factor is "E" the elastic modulus. It is linear in the strength equation - double the "E" and you double the strength. And the effect of heat is to modify "E".
There is another relevant factor which is the "end conditions". Columns may be free to rotate at their ends - referred to as "pin-jointed" or they may be fixed so they cannot rotate. By "rotate" I mean free to lean from side to side versus not free to do so. A pin ended column is effectively twice as long as a fixed ended column. And the length is squared. So freeing up the ends of a column to rotate weakens it fourfold.
Go back to our example the strength over one floor is "X" and the ends of the column are(nearly) fixed at each floor level (depends on the detail of the joints)
If we triple the length the strength is reduced by a factor of 3 squared = 9 (Or X/9) we have freed up two levels of "column ends" but that factor does not come into play.
Let us leave the rough outline of theory and go to your situation. If only part of the column is heated we change the elastic modulus which is one factor which has a linear effect on strength (It is on the top side of the equation and not squared or any other power. So a partly heated column becomes two columns joined end for end - one heated and one not heated. (Lets leave the "transition" temperature grading out at this stage for simplicity.) And if the heated bit is at floor level the weakening due to heating could have the effect of turning the "fixed end" of the column into "pin jointed".
The net result is not that the whole column is reduced in strength to that of the weakest part. That is where I think your reasoning goes off track.
Rather the part heated column has a strength part way between the full strength unheated column and a fully heated weakened column. Exactly where it falls between the two is more complicated than we need address at present. ditto the "end effects".
Now with that bit of principle addressed let's see if we can comment on the rest of your post.
SanderO wrote:...Again, here is where Factor of Safety or reserve strength becomes a critical factor in the structure to withstand all sorts of weakening. If we assume a FOS of 2... then the entire core heated to 600°C would remove ALL the reserve strength. The FOS is 1... nada. But let's consider that not all the columns were the same cross section. Perhaps the thinner ones would fail sooner from the same applied heat as they larger ones might be able to conduct the heat away from the heat sort. Maybe. If that is the case... could the smaller section buckle before the larger ones?...
There are too many interacting factors for a simple answer. Let me pick the dominating factor. That is that all these columns are part of a framed structure. And what fails first will be the result of the overall mechanics of the structure where some members have had their "E" (elastic modulus) changed due to temperature. So the thinner ones won't fail faster because they lack the heat sinking of the thicker ones. Rather they may fail first because in the overall scheme of things they are closest to overload and something trips them into overload. That something being a combination of all the factors adding up in the cascade failure mechanism where every structural elements strength has been modified from original design by the effects of raised temperature.
Apologies for the denseness of that. Let it stand for now and see if you have further queries.
SanderO wrote:...To me it sounds like it wouldn't take too much to weaken row 500 to the point of collapse. First, only every third floor would need to be *attacked* with AP or whatever heat source was used. Second.. a dead center strike would take out 3 or likely 4 of the columns of row 500 causing the north side of the core to lose about 30-40% of its strength on impact....
We are getting lost between hypothetical considerations and the reality of what actually happened. So no comment at this stage.
SanderO wrote:...How about the notion that some of the columns were destroyed by the plane impact... and several were assumed to be damaged. Does a damage columns lose some of it's strength? Half?...
If cut it loses 100%, if bent out of line it loses "most" because the bending initiates buckling orders of magnitude easier (At least one decimal order - I would need to refresh my knowledge)
SanderO wrote:...If the columns on the row 500 or 1000 side were destroyed their loss would be proportionately greater than the first or last column in row, 600, 700, 800 and 900. And those corner columns were carry much larger loads than any other core column. Note it was 501 (a corner column) which stood the tallest after the floor collapse. If 3 or 4 row 500 columns were destroyed by the plane impact... it could mean a loss of 20% of the core's axial load capacity. In the 4 center columns of row 500 were destroyed by the plane row 500 would have lost over 40% of its capacity to support the OOS flooring to the north. And with the facade opposite those columns damaged, it likely that partial floor collapse occurred on impact.....
We need another bit of theory here. Let me try for very simple.
Removing a percentage of columns does not reduce the strength of the assembly by that percentage. Stated differently removing any percentage does not redistribute the load evenly over what remains.
Try this as a simple exmple - looking at a building with three rows of columns.
So we have a number of columns in row A ditto row B and also row C The total load on each row would be something like the 1--2---1 I showed.
Pretend there are no end walls and end wall columns.
If we cut all the "C" row the load carried by "C" is not shared uniformly. The most likely result is:
now that is an extreme example but the principle is of major importance in understanding the cascade failure of the impact zone which actually occurred on 9/11 (Whether or not we consider there to have been some human assistance.) And every time you see a post saying something like "It cut 30% of the columns so there is a 30% load increase....." No way José!
I have probably raised some issues which need to be taken into account before any comments I may have on your final paragraphs so I will pause here.
SanderO wrote:...Now with heat beginning to work the remaining columns which are now at a much lower safety factor having taken on the axial loads of the destroyed columns from above... are growing weaker as their temps rise. Right away the FOS was well below 2 (perhaps 1.5 or so) with the plane strike affecting 4 of the 8 columns on the north side of the core (Tower 1). Now if these remaining columns lose 1/3 of their strength... they will buckle. it looks like 500° C might be enough temperature to buckle the north side of the core if columns 503, 504, 505, and 506 were destroyed on impact.
The next two columns 502 and 507 lose their strength from the increasing heat that leaves the two strong corner columns 501 and 508 holding up the entire north side of the OOS floors and all those above it. YIKES!
Is it possible to raise the temps of the remaining 4 columns to cause them to buckle? This looks like not as complex a take down as one would think.