Structures For Beginners and CTers
Jazzz said:
Game, set and match to me, I believe, Mr. Architect. Don't you feel a bit foolish that I am explaining redundacy calculations to you? After all, I just started learning about them.
Ahh, such confidence. It almost seems a shame to have to bring us all back to planet Earth and point out the many, many ways which Jazz gets this all wong.
Firstly it may be helpful (not least for Jazz) if we briefly cover the mechanical properties of structural steel. In particular we need to understand the difference between yield point or strength (there are minor differences which ar enot particularly relevant to this discussion) and tensile strength.
Yield Pointis the load at which a material begins to plastically deform. Prior to the yield point the material will deform elastically, returning to its original shape when the load is removed. However once the yield point is passed the deformation will be permanent. The yield point is vital when designing structural steelwrok since it generally represents the upper load limit.
A yield failure will not necessarily result in rapid structural failure, however resistance to buckling will typically decrease. As loads continue to rise beyond yield point, there is an increasing risk of wider failure.
Tensile Strength is quite different. As the name suggests, it is the maximum load which the material can sustain in tension before it fractures and fails. As steel approaches tensile failure it will deform, concentrating the tensile loads across a smaller area and increasing the risk of failure.
Amongst a long list other properties which we need to take account of are shear, where forces are acting parallel to the component (tensile and compressive forces being in the same plane). So, for example, bolt or splice failures might typically be due to shear.
The steel
specified at WTC typically had a yield value of 36ksi however like many materials it has a natural variability; depending upon quality of materials, manufacture, and the like there can be significant variations even within one component. 36ksi should therefore be considered as a minimum value.
NIST tested the steel recovered from WTC (which in itself is of interest, as CTers usually claim it was all whisked away to China with unseemly haste). NIST NCS STAR 1-3D (
http://www.fire.nist.gov/bfrlpubs/fire05/PDF/f05158.pdf) confirms a range of actual values:
- Core webs ranged from as low as 31.1 to 41.9 ksi, ie. 86 to 116% of specificed strength.
- Core flanges ranged from 32.4 to a high 53.4 ksi, ie. 90 to 146% of specified strength.
Setting to one side the 31.1 and 32.4 ksi results, inasmuch as a small proprtion of columns below failure point are unlikely to lead to any wider problem, let's take the lower maximum of 116% specified value.
Now, the NIST Demand to Capacity Ratios (DCR) are based upon specified strengths and NIST themselves note that there is effectively spare capacity up to actual (but varying) yield point/strength.
Core columns in WTC typically had a Demand to Capacity Ratio (DCR) of 0.83, ie a safety factor of 1/0.83=1.20. Now let's assume assume that the steel has an additional 16% beyond minimum yield value. This would reduce the DCR to 1.16/.83=1.4.
In other words we could increase the loads in these areas by up to
40% before yield point was reached and plastic (permanent) deformation begins. Of course this figure has lots of variables - most of the steel webs did not have such a high yield factor, some areas had DCRs well in excess of 0.83, and so on.
What we don't do is then add an additional allowance for tensile strength because (a) yield failure is already occuring and (b) gravity loads will be compressive, not tensile.
One thing we also have to appreciate is that the structure of WTC is complex; in addition to dead and live loads, it will be dealing with (for example) transverse and shear loadings from the wind. There will be a degree of torsion due to differential loading. And so on. We would therefore have to look at the exact steelwork design in considerable detail before we could determine a safety factor for each. That's why engineers earn a lot of cash, and why complex modelling software was developed.
Nevertheless it is clear that the actual capacity of the core is not going to be anything like 200% before irreversible damage and failure begin to occur.
So where does Jazz actually go wrong?
And let's also bring in an estimate for the redistribution of load via the hat truss - on a cold day, the shell would tend to contract, and the purpose of the hat truss was to hold it up. In doing so this would produce an extra demand on the core, and I surmise that this was accounted for when they calculated demand figures. So let's say the maximum demand for the core was calculated with an extra 15% from the expected demand with a fully-loaded WTC. (I do not know exactly what extra tolerance was introduced but this may be quite reasonable). This gives us a redundacy figure of... wait for it bees...
(jazzz's maximum demand ratio) * (1/DCR) * (steel yield point) * (maximum steel tensile strength/steel yield strength)
= 1.15 * (1/0.83) * 1.9 * 80/36
= 5.85
= 585%
Well firstly there's no evidence (as far as I can see) that the designers would have added on an "extra" 15% to core loadings - it's just Jazz's guess, and little weight can be attached to the figure.
1/DCR is correct, which is more than can be said for yield point. Jazz has carefully ignored the test figures from the NIST report, which provide hard data.
Thereafter we lapse into a rather strange world where tensile strength is applied to a compressive load failure (ie. additional weight being transferred to the core columns).
And that's before we notice Jazz's new (and unsubstantiated) claim that the purpose of the hat trusses was not to restribute wind loadings betwixt external envelope and core, but also to "hold up" the former when it contracted during cold weather.
Where does this take us?
- There is no substantiated for Jazz's claim of 600% core redundancy
- Jazz' revised calculations giving figures of 386% to 585% are wrong
But in any event the above calculations all assume an intact core, and we know from the various NIST studies and eyewitness evidence that the cores suffered damage - around a third. This will obviously have reduced loadbearing capacity still further, and a simple pro-rata reduction of (say) 30% is likely to be wrong because the damage is concentrated in localised areas and hence these areas will be susceptible to accelerated failure under loads.
I shall, as ever, await Jazz's next attempt to display his intuitive grasp of mechanics and structures with the greatest of interest.
Edited: Clarity/wording