Tony,
When I asked you about the assumptions built into your "solutions", you facilely & unimpressively answered "the same assumptions that NIST used."
I can assure you that you are wrong about even this vapid non-response.
tfk said:
1. Please draw the stress-strain curves (one curve for each temperature) implied by your values for E(Temp) from a strain of zero up to the max strain that you think is appropriate for your assumptions. (The assumptions that you haven't provided yet.)
Here they are.
Note well that the equation that you used assumes that the correct stress strain curve is the dashed lines on the left of the graph.
The fact that, when materials
creep yield due to temperature, they are no longer operating in the linear mode. You'd need to calculate the stress conditions at each point in the beam to figure out if the material could actually support that stress level at that particular temperature.
To the extent that any particular element was shown to not be able to support the stress assumed by the linear analysis, your results will be wrong.
tfk said:
2. Do you see any problem with using end constraints of "simple supports" for very low temps?
Yes, at low temps, the bolts haven't broken. They are therefore holding the ends of the beam such that the slopes of the beams (∂y/∂x) are zero. This results in a completely different curve shape than the simply supported beams, which allows there to be a non-zero slope at the ends. This changes the solution of the "boundary condition differential equation", because the boundary conditions are not met.
The proper solution is to use a "fixed support / fixed support" at 20°C (before the bolts break), a "fixed support / simple support" for when one bolt breaks, and a "simple support / simple support" after both bolts break.
tfk said:
3. Why did you not build your model considering construction loads?
This is highly significant, because with the construction loads considered, you'll START your thermal deflections from an outer fiber stress state in your beams that is (for a factor of safety of 2) somewhere between 50% & 100% of yield strength in the beams, depending on the engineers' design criteria.
As you can see from the stress-strain curves above, even a conservative design loading will cause the outer fiber stress to drop considerably when heated, and cause the outer fibers to elongate dramatically.
Even in a rectangular cross-section beam (like the concrete), the loss of stress carrying capability in the outer fibers cause the fibers towards the neutral axis to assume stresses that are much, much higher than they would have to if you were still in the linear stress/strain range, because the outer fibers (with their high "moment arm" from the neutral axis) are the most efficient at counteracting the externally applied moments.
As the heat soaks into the beam, the inner fibers can also no longer support the necessary stresses, and the whole situation cascades in a positive feedback loop.
Note that the tensile strength of the concrete is much lower than the compressive strength, and this causes the neutral axis to shift towards the upper surface.
But the situation in an I-beam is far, far worse than described, because of the cross-section. The load carrying capability of the web is insignificant compared to the load carrying capability of the upper & lower flange, simply due to the webs thin width. Once the outer fiber start yielding, then there is way too little material in the web to take up the load shed by the outer fibers.
tfk said:
4. What are the underlying assumptions associated with the deflection curve for distributed load equation that you used? Paying primary attention to where the equation breaks down & gives wrong answers.
The important assumptions of this solution are the following:
a. You are staying within the linear stress-strain region.
This is wildly violated for your analysis.
b. Plane sections stay planar.
This is violated for your solution.
c. Boundary conditions are as stated above.
y
0 = y
L = ∂x/∂y
0 = ∂x/∂y
L = 0
One or both of your slope boundary conditions are violated, depending on how many bolts are fractured.
d. The beams are carrying all of the load.
This is significantly violated.
As soon as the beams
START to yield from heat (i.e., without any physical deflections), they are going to shed their load back into the concrete, because 1. the concrete is also a solid beam, but a much, much wider one and 2. the concrete heats up much much slower than the steel.
This changes your load condition on the beams HUGELY. Their load fairly quickly will drop to self-weight, if the concrete can support itself over those spans.
tfk said:
5. What is the fundamental theoretical justification used in the generation of the distributed load equation that you used?
Cripes, Tony. This one was a bone that I threw you, and you still couldn't get it.
It's "Newton's 3rd law".
You guys are so fond of saying this in vapid meaningless contexts, yet the one time that it was appropriate, you drop the ball...?
One might come to the conclusion that you don't understand the flexure of beams at any deep level.
Not very good, Tony.
Why don't you gather back a couple of shards of engineering dignity & explain to everyone exactly how this equation is based on Newton's 3rd? Or would you rather that I did
that for you, too?
___
The ultimate conclusion from all of the above is that, for the highly complex, high strain, & highly non-linear conditions that occur in beam sagging due to heating, the simple beam equations that are applicable to low stress & low strain are WILDLY inappropriate.
And that
a competent FEA is the only way to get accurate results.
And this, right here, is my main message when looking at your spreadsheet, based on equations used, & assumptions made, that are wholly inappropriate for the analysis intended.
Is that "engineer-y" enough for ya, Tony?
Or does it still sound like "sales" to you?
Tom
