Tony,
Are you saying that you don't think the graph shows that it takes more energy to further deform the column once it is in the plastic range?
No, I'm not saying anything like that. And neither did you in the paragraph that I quoted.
Here's what you said.
Szamboti_Legge said:
As the steel sags two things will happen: the columns, as they shorten, will become wider, which is obvious; and the inherent strength of the steel will increase, which is not obvious. It is well established however that the yield strength of steel increases as the degree of distortion increases. This tendency increases with rising temperature and is pronounced at the temperatures required for collapse, as can be seen in the graph below. 9 For both of these reasons the initial sag cannot be catastrophic but will be very slow and the rate will depend on the rate of heat input. A rising temperature will be needed to offset both the significant increase in yield strength and the slight increase in cross-section area, if collapse is to progress. It is clear therefore that the upper section should only have moved down slowly and only continued to do so if additional heat was supplied. A slow, protracted, and sagging collapse was not observed however with either tower.
Here's the graph:
These are the most important things that the graph shows:
1. The yield strength (stress level at which the curve first deviates from linear) decreases with increasing temp
2. The ultimate strength (max stress level) decreases with increasing temp.
3. The elastic modulus (slope of the initial linear portion of the curve) decreases with increasing temp.
My gut feel tells me that the third item played the largest role in the collapse of the towers.
Now, I'll leave it to you to explain why you guys published curves for A43 instead of for A36. It's not like the right curves were not available.
A36 Stress Strain Curve
With regard to the comments:
Szamboti_Legge said:
As the steel sags two things will happen: the columns, as they shorten, will become wider, which is obvious and the inherent strength of the steel will increase, which is not obvious.
This is only true if the columns stay almost perfectly straight. As soon as they begin to bow, which in the towers happens very quickly, this effect becomes irrelevant.
Szamboti_Legge said:
It is well established however that the yield strength of steel increases as the degree of distortion increases.
And this is your key claim. It is wrong.
The yield strength, the stress level at which the curve deviates from linear (or for steels with gradual elastic-plastic transition regions, the stress level that produces 0.2% residual strain when unloaded) is unrelated to "distortion levels". Strain is the independent variable. And the yield strength is DEFINED as occurring at one single, specific point on the stress strain curve. (The FIRST such point on a non-monotonic curve.) Ergo, by definition, it does not, and can not change its location on this curve as a function of the independent variable.
Szamboti_Legge said:
This tendency increases with rising temperature and is pronounced at the temperatures required for collapse, as can be seen in the graph below.
Well, the tendency doesn't occur. So it's not at all visible in that graph. If you think that it is, I'd be happy for you to point it out to me & explain your interpretation.
Szamboti_Legge said:
For both of these reasons the initial sag cannot be catastrophic but will be very slow and the rate will depend on the rate of heat input.
"the initial sag cannot be catastrophic...?" Parts made of structural steel cannot fracture? I disagree.
Szamboti_Legge said:
A rising temperature will be needed to offset both the significant increase in yield strength and the slight increase in cross-section area, if collapse is to progress.
Except that there is no "significant increase in yield strength". And when the progressive tilting of the building was occurring, it was happening due to bending of the vast majority of the columns, not pure compression of virtually any of the columns.
And this statement is absolutely, 100% wrong. It is wrong because you have been considering only the short term stress-strain curve. If you were to add a time axis to your stress-strain curve, you'd find that runaway, progressive, catastrophic creep can happen at very low temperatures. As Bazant, et al, showed, you can get progressive, catastrophic creep at temperatures as low as 150°C.
You most certainly do NOT have to put in "rising temperature" to get unlimited creep and failure. The reality is exactly the opposite.
Bazant showed that creep rate is crucially dependent on the stress level. At high stress levels, runaway creep can occur at remarkably, unexpectedly low temperatures.
The consequence of this fact is that, since the stress levels increase as the building's tilt increases, then a constant temperature (at increasing stress levels) will produce a progressively FASTER creep. The building's progressive tile causes it to race towards catastrophe faster & faster, even at a constant temperature.
Szamboti_Legge said:
It is clear therefore that the upper section should only have moved down slowly and only continued to do so if additional heat was supplied. A slow, protracted, and sagging collapse was not observed however with either tower.
Wrong. The tower did not "sag down slowly". It tilted to the side. As it did, it put bending stresses into components that were never designed to withstand them. Pieces fractured continuously, but other parts were able to take up the load for the lost piece. Finally, there was no margin left. Some last part failed, the other components were unable to take up the load of that last, lost part, they failed, more failed and the cascade led to runaway failure.
The last, catastrophic failure was NOT "temperature mediated". It was a physical failure, specifically sudden buckling. Temperature brought the building to this failure point, but the failure itself was a sudden mechanical fracture. And nobody expects "slow, protracted, sagging collapse" out of fractures.
There is no mystery here, Tony.
