Engineering help requested: Calculating stiffness of a beam

[TubbaBlubba]

Yes.


See? That doesn't help.

I'm just a theoretical physics student, not a structural engineer, but I think he has a point. In a decreased stiffness scenario, two things would change:
1. It would spread out the area of impact a bit. Steel is sensitive to shattering when subject to shock, but that's not going to happen here, so this is unlikely to be an important factor - the force will distribute throughout the steel anyway.
2. It would "flatten" the impulse a bit. With a perfectly stiff girder the impulse would be described by a delta function, with a girder made out of mercury, you'd have an integral running until every particle has made contact with the impact surface. Again, structural steel is going to be tough enough that I struggle to see this difference mattering, if you apply the impulse over 0.1 or 0.01 seconds. It's going to do a nice bit of plastic deformation during the peak load anyway, the structural integrity will be compromised and things will risk collapsing.

In any realistic scenario, the magnitude of the applied force will be the primary factor, not slight differences in impact time or area of application.

 

[Oystein]

Did I not see you saying that you thought the conspiracy theories had merit.

Not at all.

I merely said that a specific technical claim adcanced by a Conspiracy Theorists against a detail in published technical analysis has merit.
This is no more than admitting a 9/11 Truther is right when he points out that the cloudless daytime sky is blue, after a proponent of the "official story" made a mistake and claimed it is red.

There is no conspiracy theory being debated here, so I made no comment on any, let alone say any had merit.
This is why I opened the thread in the SMMT subforum and not the 9/11 CT subforum.

As to the stiffness issue, it has already been commented by others that the stiffness is essentially irrelevant to the effect of a falling beam on the one below it.

Yes, I have seen the assertions - unfortunately they didn't come with sufficient supporting arguments.

 

[Oystein]

Why is there 0.5F at the left, hinged side? Is that simple mass of the girder load on the hinge? There's no angular momentum at the hinge point as it doesn't move.

I am looking for a video and can't find it right now - it shows how a beam reacts to a force at its end and perpendicular to its long axis when there is no other force acting on it.

You can do this for yourself and get a rough result: Put a straight ruler or something like that on as smooth a surface as you can find, perhaps a large tile. Then snap its end with a finger, push it with a billard queue, whatever - and observe:

You should find that it rotates about a point 2/3 its length away from the end you kicked, such that the other end will move in the opposite direction by half as much as the kicked end does. This means half the acceleration, and implies half the angular impulse / force (from the point of view of some small volume unite at either end).

 

[Oystein]

I'm just a theoretical physics student, not a structural engineer, but I think he has a point. In a decreased stiffness scenario, two things would change:
1. It would spread out the area of impact a bit. Steel is sensitive to shattering when subject to shock, but that's not going to happen here, so this is unlikely to be an important factor - the force will distribute throughout the steel anyway.
2. It would "flatten" the impulse a bit. With a perfectly stiff girder the impulse would be described by a delta function, with a girder made out of mercury, you'd have an integral running until every particle has made contact with the impact surface. Again, structural steel is going to be tough enough that I struggle to see this difference mattering, if you apply the impulse over 0.1 or 0.01 seconds. It's going to do a nice bit of plastic deformation during the peak load anyway, the structural integrity will be compromised and things will risk collapsing.

In any realistic scenario, the magnitude of the applied force will be the primary factor, not slight differences in impact time or area of application.

That's all well thought out, except that your expectation that the "differences in impact time" would be "slight" turns out to be wrong.

Like I said: The original result in the report, where the falling girder implicitly was treated as having infinite stiffness such that that (pretty high) stiffness of the impacted girder equals the stiffness of both girders in series, the peak elastic rebound force turns out to be 7279 kips (sorry, imperial units, that's the gravity force of 7,279,000 US pounds) - in this case way above the vertical shear capacity of the nearest connection - it would have failed in plastic deformation long before the maximum deflection of just under 1 inch has occurred.

If the falling girder had the same stiffness (just by way of example), effective stiffness would be half, and resulting force 1/sqrt(2) of what it was before; deflection would decrease by the same factor.

However, the falling girder is really much less stiff - we are still disagreeing by how much, but suggested factors vary from ca. 100 to 2000! Consequently, force would decrease and deflection increase by a factor of ca. 10 to 50!
I haven't computed the time from begin of impact to maximum deflection, but suggest that it increases by the same factor.

 

[TubbaBlubba]

I am looking for a video and can't find it right now - it shows how a beam reacts to a force at its end and perpendicular to its long axis when there is no other force acting on it.

You can do this for yourself and get a rough result: Put a straight ruler or something like that on as smooth a surface as you can find, perhaps a large tile. Then snap its end with a finger, push it with a billard queue, whatever - and observe:

You should find that it rotates about a point 2/3 its length away from the end you kicked, such that the other end will move in the opposite direction by half as much as the kicked end does. This means half the acceleration, and implies half the angular impulse / force (from the point of view of some small volume unite at either end).

Yeah, you get torque throughout the beam due to the applied force at one end, proportional to the distance to that end. But you also have differing moments of inertia about various axes, and the moment of inertia (proportional to the amount of torque needed to induce some rotation about an axis) is proportional to the square of the distance from a given point to every point containing mass - it thus has its minimum about the midpoint. Essentially, the torque prefers rotation about the far end, but the moment of inertia more strongly prefers rotation around the midpoint of the beam, and this results in the 2/3rds compromise.

 

[TubbaBlubba]

That's all well thought out, except that your expectation that the "differences in impact time" would be "slight" turns out to be wrong.

Like I said: The original result in the report, where the falling girder implicitly was treated as having infinite stiffness such that that (pretty high) stiffness of the impacted girder equals the stiffness of both girders in series, the peak elastic rebound force turns out to be 7279 kips (sorry, imperial units, that's the gravity force of 7,279,000 US pounds) - in this case way above the vertical shear capacity of the nearest connection - it would have failed in plastic deformation long before the maximum deflection of just under 1 inch has occurred.

If the falling girder had the same stiffness (just by way of example), effective stiffness would be half, and resulting force 1/sqrt(2) of what it was before; deflection would decrease by the same factor.

However, the falling girder is really much less stiff - we are still disagreeing by how much, but suggested factors vary from ca. 100 to 2000! Consequently, force would decrease and deflection increase by a factor of ca. 10 to 50!
I haven't computed the time from begin of impact to maximum deflection, but suggest that it increases by the same factor.

Yeah, it would be interesting to see what the involved times are. I would be surprised if, in real life, the stiffness of the impacting girder plays as large a role as those equations (whatever conditions they be valid under) seem to imply.

What I struggle is to imagine the rebound of the lower girder reacting quickly enough to prevent catastrophic failure, and not failing due to internal forces during the exchange. Of course, I've been wrong before, and with the dimensions and quantities involved intuition often fails.

 

[rwguinn]

Yeah, it would be interesting to see what the involved times are. I would be surprised if, in real life, the stiffness of the impacting girder plays as large a role as those equations (whatever conditions they be valid under) seem to imply.

What I struggle is to imagine the rebound of the lower girder reacting quickly enough to prevent catastrophic failure, and not failing due to internal forces during the exchange. Of course, I've been wrong before, and with the dimensions and quantities involved intuition often fails.

The stiffness of the impactor is pretty irrelevant. The stiffness and mass of the impacted are extremely relevant, as the transmissibility of the impact through Dynamic Ringing (or amplification ) is a function of its natural frequency.
The term of interest is a function of z (structural damping), and is of the form 1/(wn^2-wF^2+2*i*z*wn*wF) where wn is the natural frequency, wF is the forcing frequency (where the stiffness of the impactor does enter in, a bit), and z is damping (for riveted structure, around 8%. For a chunk of steel, around 2%.
As can be seen, the only thing keeping that impactor from dumping an infinite load into the impacted beam is that little 2*i*z term
For a full treatment of the basics, see http://people.duke.edu/~hpgavin/cee541/sdof-dyn.pdf. Dynamic magnification is round about page 20, or section 3.2...

 

[Elind]

Yes, I have seen the assertions - unfortunately they didn't come with sufficient supporting arguments.

One doesn't need cribbed equations from textbooks to understand BS when one sees it, and I mean yours.

 

[Elind]

Not at all.

I merely said that a specific technical claim adcanced by a Conspiracy Theorists against a detail in published technical analysis has merit.

Except for the fact that it doesn't, as countless smarter people than you, or me, have long since pointed out. This is classic "creationist" or AGW denier attempts at logic, where they cherry pick sentences that they don't understand and think they can prove something ridiculous, like you are doing.

 

[TubbaBlubba]

The stiffness of the impactor is pretty irrelevant. The stiffness and mass of the impacted are extremely relevant, as the transmissibility of the impact through Dynamic Ringing (or amplification ) is a function of its natural frequency.
The term of interest is a function of z (structural damping), and is of the form 1/(wn^2-wF^2+2*i*z*wn*wF) where wn is the natural frequency, wF is the forcing frequency (where the stiffness of the impactor does enter in, a bit), and z is damping (for riveted structure, around 8%. For a chunk of steel, around 2%.
As can be seen, the only thing keeping that impactor from dumping an infinite load into the impacted beam is that little 2*i*z term
For a full treatment of the basics, see http://people.duke.edu/~hpgavin/cee541/sdof-dyn.pdf. Dynamic magnification is round about page 20, or section 3.2...

This seems in line with my intuition - the important thing is typically how the impacted girder tends to react to an impulse, not exactly how the impulse is delivered.