# Does the exclusion of stiffness from Nordenson's falling girder calculations demonstrate anything?

Discussion in '9/11' started by Tony Szamboti, Jan 9, 2016.

1. ### Tony SzambotiActive Member

Nordenson's failure mechanism for the northeast corner floors below could not have occurred. He errs in not including the much lower stiffness of the falling girder, which will limit the impact load to well below the shear capacity of the girder seat at column 79.

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2. ### econ41Active Member

Don't be coy please Tony. Tell us exactly what part of Nordenson's report you disagree with. And why.

The falling to impact and related stiffness issues are explained in part 5 - page 242 of the pdf as indicated by my document reader - page B30 as per the doc itself.

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3. ### Tony SzambotiActive Member

You are wrong. Nordenson does not include the much lower stiffness of the falling girder. He only uses the stiffness of the girder on the floor below at 10 to 12 inches from the seat, where it is quite high at 7,627 kips/inch. This produces a very high impact force of 4.133 million lbs. (4,133 kips). However, this is fictional as the effective stiffness involved in the impact is actually that of both the stationary girder below and that of the falling girder in series. The falling girder stiffness is about 3.7 kips/inch and it dominates in the equation

1/Keff = 1/K1 + 1/K2

where K1 = 7,627 kips/inch and K2 = 3.7 kips/inch.

The Keff stiffness will limit the impact load to about 100,000 lbs. (100 kips), which is not nearly enough to break the seat connection at column 79 which has a shear capacity of 632,000 lbs. (632 kips).

This problem with the Nordenson report shows a propagating collapse from a falling girder was impossible. Read pages 242 through 245 and do a calculation for the stiffness of the falling girder as a cantilever and then calculate the effective stiffness and impact force.

Last edited: Jan 10, 2016
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4. ### econ41Active Member

An astonishingly silly assertion Tony. All I asked was that you tell us what you object to. How can that be 'wrong'?

5. ### Tony SzambotiActive Member

You are just playing a semantical game here and need to do the math.

When the falling girder stiffness is accounted for the Nordenson report actually shows there can be no cascading floor collapse in the northeast corner of WTC 7 due to a girder falling off its seat.

The NIST WTC 7 report is in shambles.

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6. ### econ41Active Member

Tony I asked you to clarify what part of the Norsdenson report you are criticising.
What part of the paper please.
How does an error in Nordenson's report have the slightest effect on the status of the NIST report?

Interesting logic - I'll leave it for someone to identify the Formal Fallacy. Bill said "A". Fred said "B". Fred is wrong THEREFORE Bill is wrong?

7. ### Tony SzambotiActive Member

I told you pages 242 through 245. You need to determine the effective stiffness which includes that of the falling girder and use the equations Nordenson did to find the actual force of impact with the effective stiffness.

Nordenson tries to calculate the force of impact by the falling girder on the floor below but he does not include the stiffness of the falling girder. When the effective stiffness including both the floor below and falling girder is calculated it shows the impact force to be far too low to shear the seat at column 79. NIST never did this, they just assume the floor below failed.

[...]

Last edited by a moderator: Jan 10, 2016
8. ### econ41Active Member

You did no such thing - why be untruthful. (Apparently I missed an edit) Thanks anyway - you confirm it was the section that I identified.
I don't need to do anything - I have chosen to call on you to support your so far unsupported claim.
Wrong. Try understanding what Nordenson was doing - where "stiffness" fits in his process - then it should be obvious why stiffness other than the failing beam is not relevant. You seem to have a different explanation - BUT you are supposed to be telling us what is wrong with Nordenson's. Don't leave us guessing - either spell yours out OR go along with Nordenson. Then we may have something to discuss.
Of course he doesn't - see previous advice.
That looks like you are playing "Anything to get the numbers you want" Tony. Why not also add in the number of presents Santa gave out at Xmas?
Irrelevant as previously commented.

Last edited: Jan 10, 2016

Tony posted at 12:10 without the page numbers, and edited the post several time, adding the page numbers at 12:17. You would not have seen it unless you reloaded the page. New posts show up in-page while you are replying, but edits don't.

To avoid confusion, I'd recommend substantive addition go in a new post, or are at lease marked as "EDIT: ", or somesuch.

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10. ### econ41Active Member

Thanks Mick. I've put a note in the miscreant post. (Missed by a minute - 7:17 v 7:18 in my +11 time zone)
Agreed - my near universal practice - never edit a post after someone has made a substantial response. Except to correct errors and then with full disclosure.

Last edited: Jan 10, 2016
11. ### Tony SzambotiActive Member

I understand where stiffness fits in and Nordenson does not account for the effective stiffness, which would include the falling girder. He has a concentrated load applied at a point on the girder below as though the falling girder load has infinite mass. With the equation 1/keff = 1/K1 + 1/K2 this would then just give the lower stiffness mass as 1/infinity goes to zero.

This shows that the NIST WTC 7 report claim that the floors below would fail if the one girder came off its seat is bunk. The ARUP analyses have caused the NIST WTC 7 report to completely fall apart.

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12. ### Jeffrey OrlingActive Member

Tony... Are you saying that a falling floor section would not destroy the floor including the girder on the one below "dropping" because the only means to do this is to shear the seat of the girder?

13. ### Tony SzambotiActive Member

I am saying that the falling girder from above alleged to have been pulled off its seat at column 79 could not shear the girder seat on the floor below.

This is the NIST theory and it could not possibly have occurred. It is time for a new investigation.

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14. ### Jeffrey OrlingActive Member

Frankly, I have understood how the single grider off the seat leads to multiple floor collapses leaving a multi story section of col 79 too slender and it buckles.

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15. ### Tony SzambotiActive Member

I know you have often said that you think it was the trusses and that you doubt the single girder fall could have led to column 79 buckling, so it sounds like you meant to say "Frankly, I haven't understood...."

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16. ### econ41Active Member

Your posts show no understanding of what Nordenson's explanation says. Why don't you go and read what he says without trying to overlay it with your preconceived ideas? Start at page #241 - Section B4.3 - and work forward. It is your burden of proof to support your claim that he is wrong Tony. And given your record of ignoring sound advice from me makes me disinclined to spoon feed the explanation of Nordenson's method.

He does not have to - in fact it would be wrong if he did. And that is begging the question of what you mean by "effective stiffness". The stiffness of the falling beam is irrelevant.
Try to understand what he said WITHOUT imposing your own half thought through alternate ideas.

And this:
Is still irrelevant unsupported nonsense. You are still asserting "Bill said 'A'. Fred said 'B'. Fred is wrong THEREFORE Bill is wrong."

...and I'll explain the analogy if anyone cannot interpret it.

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17. ### Jeffrey OrlingActive Member

I left off the negative... sorry... Once floor sections are falling on a heat weakened frame it's reasonable to assume that there would be more damage than falling on a pristine cold structure.

18. ### benthamitemetricActive Member

How did you derive the 3.7 kips/inch figure?

EDIT: To clarify with more background, it seems to me that Nordenson accounts for the energy dissipation of the deformations of the falling girder as described in Section B3.6 and set forth in Table 6.1. How specifically do you see the stiffness of the falling section playing a role in this scenario? We aren't calculating the ability of the falling section to withstand a force and it doesn't appear the stiffness would play a role (other than what is already accounted for in Nordenson's deformation dissipation calculation) in the force it imparts on the lower floor.

Last edited: Jan 11, 2016
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19. ### econ41Active Member

That could be of some interest even though the whole aspect of falling beam stiffness is irrelevant to the Nordenson explanation.

3.7 is very low - I've forgotten which forum it was on (here or ISF) but Tony's earlier attempt to falsify Nordenson missed the critical a2 * b2 factors in the stiffness equation. Stiffness of a simple beam is very sensitive to off centre loading and stiffness rises very quickly as the load moves closer to one end. But 3.7 is very low compared with the 7,627 for the relevant beam.

It appears that - having realised that Nordenson's stiffness calcs are correct - Tony is trying to dig deeper into the explanation to find another error to base his claim on.

So I agree - It would be interesting to know how Tony derived it. It could help identify where his reasoning is going astray.

20. ### benthamitemetricActive Member

Yes, was busy writing an edit to that end when you replied (see it now above). I had hoped him explaining how he derived it would help work through his error in thinking it was immediately applicable. After thinking about it some more, however, I decided to skip to the chase.

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21. ### econ41Active Member

I've seen the edit. Agree 100%

Agreed.

Well the status of the OP is straight forward:
OP asks: Does the exclusion of stiffness from Nordenson's falling girder calculations demonstrate anything?
The answer is a definite "No!"

The simple reason is that Tony has not made out his case - so there is strictly "no case to answer."

However the purported claim has not been explicitly debunked. So here is the outline of the debunking reasoning;

Nordenson's Argument in brief:
1) He calculates the energy available for the falling mass derived from the potential energy dissipated;
2) He subtracts the energy lost to deformation of the falling beam, the remaining energy then impacts the lower beam;
3) He then calculates the deformation of that lower beam - the beam which is being impacted and will fail;
4) Having derived the deformation he then "back calculates" the static force which would produce the same deformation
5) That static force is well in excess of what the end connections could support;
6) QED

Given that is Nordenson's explanation the consequences for the OP question are:
The role of "stiffness" is to provide a "pivot" point in the maths for converting energy dissipated >> equivalent force applied.

And the stiffness must be of the impacted beam AND that beam alone.

THEREFORE omission of the falling beam stiffness has no effect on Nordenson's conclusions.

AND any attempt to include falling beam stiffness would be a fatal error to the argument.

[/EndThread]???? - unless our joint position is falsified.

Last edited: Jan 11, 2016
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22. ### Tony SzambotiActive Member

For those here who are wondering about the stiffness of the girder, it is that of a cantilever beam. The equation for stiffness of a cantilever beam is 3EI/L^3. Where E is modulus of elasticity which for steel is 29,000,000 psi, I about X-X is 6,710 in^4, and L = 540 inches. If you do the math you will get 3,707 lbs./inch (3.7 kips/inch).

The girder will contact the next floor down at its column 79 tip and its full length would be involved in how it deflects. The girder's stiffness and deflection is involved in determining the force of impact in the same way a car's stiffness is involved if it strikes something very stiff like a thick brick wall. The stiffness involved is that of both the brick wall and the car. The effective stiffness is found using the equation for springs in series.

1/Keff = 1/K1 + 1/K2

For the impact of the girder and the floor at 10 to 12 inches from the next floor down's column 79 seat it would be

1/Keff = 1/7,627 + 1/3.707

where the stiffness is in kips/inch

Keff = 3.705 kips/inch

Nordenson used a point load to impact the next floor down at 10 to 12 inches from the column 79 seat and that is like using an infinitely stiff item to impact the floor. That is not correct and Nordenson's force of impact is wildly exaggerated because of his error. The actual force of impact would be less than 100 kips and this would not shear the girder seat at column 79 which had a shear capacity of 632 kips. There could not have been a cascade of collapsing floors to the north of column 79.

Last edited: Jan 13, 2016
23. ### econ41Active Member

Members should note that this latest post by Tony Szamboti ignores the debunking of Tony's claims by benthamitemetric and myself and the summary explicit debunk in my previous post.

There are several errors in Tony's repeated bare assertion claim - I will focus on two fatal errors but first a preliminary comment on this admission of cheating with the maths:
Tony has wrongly selected the best case to support his argument - that of a cantilever which gives him the lowest stiffness factor which maximises the error he alleges in Nordenson's work. That trick is cheating the maths. The two fatal errors I will focus on are:

1) He persists with his own false starting assumption which misrepresents the stiffness of the falling beam as somehow negating Nordenson's method. It doesn't.

2) Even within his own false scenario his application of engineering is wrong. I will limit my comments to the first error - the assumption of "cantilever".

"1)" as already been debunked by my previous post.

"2)" Even within Tony's false scenario the error with cantilever is that a cantilever has one free end and relies on bending moment at the fixed end to support the cantilever action . Since the falling beam has tilted it has already failed in bending moment at the fixed end. So - even within the false scenario relevant stiffness is not that of a cantilever - it is closer to simply supported. ( It depends on the residual BM at the already failed "fixed" end - no need to pursue the detail.)

I wont waste effort detail debunking the remainder of this explanation since it is based on fatally flawed foundations.
Nordenson is correct. Tony's assertion of "wildly exaggerated" and "error" are both unproven bare assertions. And Tony's conclusions likewise are unproven:
Still wrong Tony - and debunked on the basis of two fatal errors. Wrong scenario and even within the false scenario the engineering maths is wrong.

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This thread might be enhanced somewhat with a notated diagram.

25. ### econ41Active Member

This is the relevant diagram from the Nordenson paper.

My notations in blue. The "hinge" negates Tony's assumption of "cantilever".

Last edited: Jan 13, 2016
26. ### OysteinActive Member

Tony and econ,

I have followed this thread, and I have read the critical pages 242-245 of the Nordenson report, plus a few pages before and after that. Now I haven't read the entire Appendix B, which starts on page 210 in my PDF reader, and not being an engineer, this is a tough read, so I haven't succeeded yet in wrapping my mind fully around Nordenson's approach and argument. Consequently, I can't decide yet whose arguments I accept as correct here.

First, do you agree that Nordenson's approach, model and values (Sections B1-B4) are nearly enough valid to derive a plausible answer to the question whether or not the falling girder would fail the girder connection below in vertical shear? econ, knowing you, I 'd guess you are in general agreement, or else you wouldn't even start to engage in a discussion of B5. Tony, you have of course raised objections - let me see if I understand them correctly, at least roughly:

B1: Nordenson summarizes his approach thusly: "The basis for the analysis was an energy comparison...
(Potential Energy of Falling Floor Slab) + (Energy Dissipated in Failure of Floor) vs. Energy Required to Fail Girder Connection to Column at Floor Below
" <- Tony, I read you as claiming that Nordenson is missing an additional term, namely (Energy Dissipated by falling Floor through Impact with Floor Below). Correct?
Nordensen lists several energy sinks for the term (Energy Dissipated in Failure of Floor) - they include "Plastic deformation of falling girder tip at impact with floor below", but do not include elastic and plastic bending of the falling girder along its entire length at impact. <- Tony, again, that's what you say is missing and must be included?
And then the deciding step: "This potential energy at impact was then converted to an equivalent static force based on the stiffness of the impact location and the resulting girder deflection. The resulting shear force transferred to the connection at Column 79 was then calculated and compared with the expected shear capacity of the connection to determine whether the failure of one floor would cause the failure of the floor below." <- Tony, you argue that the missing term would be correctly inlcuded into the energy comparison by using the stiffness formula for springs in series, right?

B2: Nordenson makes some assumptions about DL, SDL and LL - reasonable?
He then describes how he modeled the connection failure on fl 13 and shows the geometry of floor deformations that result in SAP2000 - seems all plausible?

B3: Presents several failure modes that the falling girder and supported beams and slab would experience, and long lists of input values for the dimensions and properties of the floor elements result in some numbers for energy dissipation. <- Do you accept, at least tentatively, that Nordenson worked with due diligence and introduces no significant error here?

B4: The weight of the falling girder + beams + floors is determined as 46,000 pounds, and the drop height of its center of gravity at the moment of impact as 83 inches (for floor 13) - roughly half a story height, the product is 3818 kip-inches, from which he subtracts 345 kip-inches, the already dissipated energy derived im B4 (detailed numbers in Table B6-1) - so roughly 90% of the PE is available to wreak havoc going into B5 <- Are you ok with these numbers, at least in a ballpark sort of way?

B5.1: I think we will all agree that the end of the falling girder impacts the girder below impacts a few inches away from the column - 10-12 inches seem plausible?
B5.2: Girder stiffness of the fixed girder below derived correctly?

So those are the premises going into B5.3 - the conversion of kinetic energy to a static force by computing how far the lower girder deflects when it dissipates all that kinetic energy. <- And that's where Tony disagrees: He claims that the falling girder would deflect too and dissipate energy, and even more as it is less stiff - and that reduces the equivalent force. <- And I agree with Tony! Unless, of course, I am missing something that I haven't been explained by econ

Over to you!

27. ### OysteinActive Member

Now, Tony computes the stiffness of the falling girder as that of a cantilevered beam, and finfds that, using Nordenson's energy->equivalent force conversion via system stiffness results in less that 100 kip equivalent force.
Econ argues that "cantilever" is the wrong model, and I agree - the bending of that girder is that of a beam supported on both ends. What's the formula here, what's the result?

Nordenson didn't need to do this, as already his equivalent static load exceeded capacity.

28. ### Tony SzambotiActive Member

Econ41 shows a diagram of a cantilever and says it isn't a cantilever. He cautiously says it is closer to simply supported, but doesn't go any further. It will behave like a cantilever when it impacts. If he wants to call it simply supported then fine we can do that. It is a simply supported cantilever while it is falling and takes a load on one end. Its stiffness will not be that of a simply supported beam that is supported on both ends and it will deflect tremendously.

The load is dynamic and Nordenson computed an equivalent static load based on energy and so did I. The problem with Nordenson's calculation is that he did not account for the overall stiffness and deflection of the impact. His amplification is about 50 times too high due to his use of an extremely exaggerated stiffness.

When the stiffness of both the lower floor girder at 10 to 12 inches away from its seat and the falling 45 foot long simply supported cantilevered girder are taken into account it is clear that the load produced cannot cause the lower floor to collapse. The energy is absorbed over a longer duration due to the deflection of the falling girder and the deceleration is much lower than what would occur with Nordenson's extremely high but erroneous stiffness.

Last edited: Jan 13, 2016
29. ### econ41Active Member

Then may I suggest that should be Step 1 - comprehend the two very different arguments. Nordenson's explanation which is explicitly spelled out in the referenced paper and which I summarised in my recent post. Versus Tony's implied alternate case which he has not explained.

And Tony is playing "mix and match" - taking bits out of his own explanation or misunderstanding and forcing them into the middle of Nordenson's explanation. And that is not valid. Recall "mix and match" is SOP for Tony eg "Missing Jolt". So we need IMNSHO to be explicitly clear which explanation or model we are discussing AND hold Tony to the same standard of rigour. I will stay with my SOP - I will call him on errors, explain them if necessary but I will not pursue derailing red herrings off focus.

So - at this stage - I'll take the rain check on your discussion of details till we are sure which model we are discussing.

He is right within the context of his own so far undefined model. He is wrong in the context of Nordenson's explanation. So make up your mind which horse you are riding - and don't change horses in mid stream.

I thought I had explained the Nordenson model explicitly, simply and clearly.

Which bit do you disagree with?
"Nordenson's Argument in brief:
1) He calculates the energy available for the falling mass derived from the potential energy dissipated;
2) He subtracts the energy lost to deformation of the falling beam, the remaining energy then impacts the lower beam;
3) He then calculates the deformation of that lower beam - the beam which is being impacted and will fail;
4) Having derived the deformation he then "back calculates" the static force which would produce the same deformation
5) That static force is well in excess of what the end connections could support;
6) QED"

Nordenson extracts the falling beam and other energies at stage "2)" - Tony effectively wants to add it in again. More accurately he ignores Nordenson's explanation and puts up points of disagreement which come from a different scenario. He pre-emptively denies the central theme of Nordenson THEN claims Nordenson is wrong. So in the normal simple terms used on forums he is relying on a strawman.

Last edited: Jan 13, 2016
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30. ### econ41Active Member

Neither formula nor result matter here. They are moot WRT the Nordenson explanation and Tony has not shown where they fit in his implied explanation. We cannot progress discussion of Tony's claim until he clarifies it. And we should not be falling for the "mix and match" trap of applying the bits Tony cannot explain in his own scenario as if they fit into the middle of Nordensons. They don't fit.

Could be right for Tony's model if he ever defines it. Definitely wrong for Nordenson who carefully worked past those issues. Remember Nordenson only needed to translate "NET applied energy == equivalent static force"
Correct - Nordenson did not need to do it. His method - explicitly described - is different to Tony's implied and so far not described method.

31. ### benthamitemetricActive Member

Do you have an example of this approach being used to calculate the force imparted by a falling object elsewhere? It seems to me that if you discount force in this way, falling objects can never impart much force.

33. ### jaydeehessSenior Member

Is a rotating beam actually considered as a "cantilever" beam? Doesn't "cantilever" imply a situation that acts against that rotation?

34. ### econ41Active Member

No and Yes - respectively. Tony was just looking for the "worst case" he could find for Nordenson's explanation - best case for Tony's false claim. With zero regard for whether it was valid engineering. It wasn't. Benthamitemetric and I had already discussed the range of "stiffnesses" - but remember the stiffness of the falling beam is not relevant to Nordenson the way Tony tries to argue it. Nordenson had already isolated that aspect from consideration. Correctly IMO.

Tony is wrong at the two levels I have identified:
1) At the "big picture" level he is ignoring what Nordenson actually said and imposing his (Tony's) undefined alternate model. So shades of "Missing Jolt" repeated; AND

2) Even within the false setting of his so far not made explicit claim he is wrong on the engineering.

There are several errors at that level of engineering detail - I simply falsified the "cantilever" issue because it was the false foundation of Tony's claim within his own false scenario. It has no reference - no meaning - within the context of the Nordenson explanation which is the purported topic of discussion.

With the OP topic the situation is clear - "Does the exclusion of stiffness from Nordenson's falling girder calculations demonstrate anything?" - No it doesn't is my position - supported by outlined reasoning - and Tony so far has not supported his counter claim either in detail OR by putting it into a valid context.

35. ### Tony SzambotiActive Member

There is some talking out of the hat going on here. You can't extract the falling beam's stiffness and deflection characteristics from the impulse. Nordenson just ignores it and uses an essentially infinite stiffness point load applied to the stiffness of the lower beam only. This causes his impulse to be applied at the lower beam stiffness only and that is about 2,000 times greater than than the effective stiffness would be. This causes his deflection to be about 40 times less than it would have been and his force to be about 50 times greater than it would have been in reality.

The impact force Nordenson calculates is ridiculously wrong and the actual force generated by that falling girder on impact with the lower floor would not have been nearly enough to shear the girder seat at column 79. This can be proven in no uncertain terms and is another serious nail in the coffin of the ignominious NIST WTC 7 report. Others have been alerted to it.

Last edited: Jan 13, 2016
36. ### Tony SzambotiActive Member

More talking out of a hat. It isn't surprising that you provide no calculations to support your claims. I can guarantee that the situation would be quite cantilever like in an FEA. The beam connections on the north and east sides are not fully broken when the girder falls and impacts the lower floor.

The reality and bottom line is that the falling girder will deflect significantly at impact, because it is not an infinitely stiff "point load" like Nordenson uses, and the force generated will be significantly less than that needed to shear the girder seat at column 79 and the collapse will not continue.

Last edited: Jan 13, 2016
37. ### Tony SzambotiActive Member

The force is a function of the stiffness and deflection. If there is high stiffness the force will be high due to a rapid deceleration. The deceleration in an impact due to a velocity drop is

G or deceleration = 2 x height / deflection

Deflection is found by using the potential energy and the spring energy generated by it

mgh = 1/2 x stiffness x deflection^2

then F = ma or F = stiffness x deflection

This is general science and Nordenson himself uses it. He just didn't use the right stiffness.

38. ### OysteinActive Member

*big sigh*

Neither of you has answered my questions in #26. My purpose was two-fold:
1. To check with you guys if my understanding of Nordenson's approach, and his steps of work, matches your understanding
2. To find out if you agree that you both agree that Nordenson's approach, his model assumptions and values are okay with you within reasonable bounds.

In the process, I identified one element where Tony explicitly deviates from Nordenson's model - I identified it several times, but it really is the same element each time: Tony wants to add one term - the stiffness of the falling girder.

So my question to you is:
econ: Do you think that the Nordenson model is a correct model to answer the question of whether or not the girder connection below would fail?
Tonyo you think that the Nordenson model plus adding the stiffness of the falling girder is a correct model to answer the question of whether or not the girder connection below would fail?

If you both answer in the affirmative, then I'd suggest that
- contrary to econ's assertion that Tony's model is undefined, it is very well defined
- contrary to econ's assertion that Tony is playing mix and match, Tony has merely identified what he thinks is a flaw in Nordenson's model - a missing term - and added what he thinks is a proper amend.
- We need to agree on which model is the correct one (or that both are flawed), and work from there.

Now Nordenson has explained all the things he has included in his model, but he has not explained all the things he has not included, so looking at Nordenson's report to find an explanation why the falling girder's stiffness is not included yields nothing, for trivial reasons.

Tony needs to explain why he wants to include it.
Actually, I think I can provide an easy explanation, but I want to give Tony one chance to provide an explanation within the framework of Nordenson's logic.

Then econ would have to explain why Tony's explanation doesn't cut it.

But Step 1 is your answers to whether or not I have framed the models you agree with correctly.

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39. ### Jeffrey OrlingActive Member

This whole stiffness issue is confusing... Obviously a flexible material can absorb impact and perhaps not fracture... This is intuitive. So a beam made of rubber which could support x pounds and a one made of glass which supports x pounds would respond differently to a static and dynamic load. The rubber being flexible would deflect. on impact... the glass would shatter... obviously depending on the load.

The confusing thing here for me is how stiffness comes into play. How much stiffness is presumed to have been lost or remains... and what does this mean for the collision of the two members? Are they assumed the same stiffness? Is the stiffness assumed to be a constant across the entire length of both beams? Does it matter?

If the collisions/collapse collapses the beam below... what is "failing"... the beam, the connection, the seat... the welds... the bolts? How much mass is "connected" to the beam... floor slab... or is this model isolated the steel from the concrete?

40. ### OysteinActive Member

Jeffrey,

"stiffness" is the ability of a member to resist being deflected by a force. This depends on intrinsic material properties (elastic modulus) and on the geometry of the assembly. In case of a beam, the important parameters are
- length
- effective thickness
- where and how is it supported
- where is the force applied

As you say, a rubber beam is less stiff than a glass beam of equal geometry because it has a lower elastic modulus.
A slender beam is less stiff than a sturdy one.
And stiffness is larger when force is applied closer to the supports: When you walk out on a diving board on your local public swimming pool, it bends down more the further you move away from the pinned end. A book shelf will bow down less if you place heavy books closer to the end than smack in the middle.

Look at ozeco's drawing from post #25:

The falling girder and the girder below have the same length and elastic modulus [1], but they are loaded differently as the impact occurs:
• The girder below, which is initially at rest, is simply supported on both ends, and experiences a vertical point force F merely 10 to 12 inches from the support at column 79. The latter is what gives this girder a very high stiffness in this situation.
• The girder above is rotating about its support on column 44, and experiences an upward point force F at its far end. In my opinion, this can be approximated by a situation where the girder is simply supported on both ends and experiences a force F/2 along its entire length - as if gravity acceleration had a value of F/2m (m=mass of girder). The engineers here are still debating how to correctly compute the stiffness of that girder in this situation, but I trust that it will turn out to be significantly lower, as half the effective force is applied at a larger average distance from the supports.
(I am ignoring the continued force of actual gravity on both girders for the time being; not significant at this point, but ought not be forgotten when we do the sums in the end).

The idea now in Nordenson's model is to compute how far the girder below will deflect under a point load 10 to 12 inches from the column such that its elastic strain energy is equal to the kinetic energy of the falling girder. That deflection is proportional to the applied force.
He then converts that force on a point 10 to 12 inches from the column to the resulting vertical force on the connection itself, using the girder's moment of inertia, and compares with the connection's capacity.

Nordenson does not consider the bending/deflection of the falling girder which inhales strain energy as it bends down in the middle. This strain energy is not available to strain the girder below - Tony argues that if you take into account the lower stiffness of the falling girder, much less energy is dissipated by the girder below, resulting in a lower force.
And I have said already that I think Tony is correct.

[1] Well, not exactly, as they are conneted to tributory beams and floor slabs, but the geometry of those has changed for the falling girder - but let's disregard that here

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