The influence of buckling

[Minadin]

When I was back at Clemson, I was on a radio show where a caller made the, "The fires weren't hot enough" argument and I used the thermal expansion and hooke's law to prove that a temperature change of 500 degrees centigrade would result in a 174 psi stress over the entire beam.

The caller challenged me saying that steel failed at 36,000psi and that little stress couldn't hurt anything. I then proceeded to demonstrate that a distributed load of 174psi over 210 feet on an 8 inch beam was the equivalent of 3,000,000 pounds of load at one point on the beam. The caller hung up.

Not to mention, that same 36ksi steel, when heated to about 1000 degrees Fahrenheit (that's close to your 500 Centigrade), becomes roughly 18ksi steel, which doesn't help at all. You have these steel structural members that are loaded at 24ksi failing because of the heat, just due to their design loads, not even considering the stresses and other forces caused by expansion. When they fail, the load they carried has to transfer to other areas where the steel is still intact, but these are now subjected to those new loads in addition to their design loads, and the new loads are being applied in ways and directions that those members weren't designed to handle.

 

[JamesB]

Nice analysis Chippy, it's always great to have a math/science type around to kick a few equations around. When I was back at Clemson, I was on a radio show where a caller made the, "The fires weren't hot enough" argument and I used the thermal expansion and hooke's law to prove that a temperature change of 500 degrees centigrade would result in a 174 psi stress over the entire beam.

The caller challenged me saying that steel failed at 36,000psi and that little stress couldn't hurt anything. I then proceeded to demonstrate that a distributed load of 174psi over 210 feet on an 8 inch beam was the equivalent of 3,000,000 pounds of load at one point on the beam. The caller hung up.

The point is that you can't always use mathematics to prove your point. For people to accept math, they have to demand that their argument be mathematically and scientifically valid. That's something conspiracy theorists simply don't do.

You went to Clemson! Did you study under the great Judy Wood?

 

[The Almond]

You went to Clemson! Did you study under the great Judy Wood?

No, thank God. For people who don't know much about Clemson, the real heavyweight was Robert F. Nowack. I was a civil engineering student, and I had no need to take classes from Wood (and I was strongly advised not to take classes from that loon). Nowack, a professor for nearly 45 years at Clemson, was my professor for the Statics and Dynamics classes I would have had to take from Wood.

Students of Wood, for the most part, hated her. She used to take lecture time to try to convince students that the WTC was part of a controlled demolition. Needless to say, none of the students ever bought into that crap.

 

[JamesB]

Hmm, I can't understand at all why she didn't get picked up for tenure...

 

[RenaissanceBiker]

Just so we all are using the same terminology, beams are horizontal and columns are vertical. The main load in a column is parallel to the column. The main load in a beam is perpendicular to the beam. Euler's formula is for loads parallel to the stress member.

 

[rwguinn]

Just so we all are using the same terminology, beams are horizontal and columns are vertical. The main load in a column is parallel to the column. The main load in a beam is perpendicular to the beam. Euler's formula is for loads parallel to the stress member.

That convention keeps it simple for the non-engineers--I'll buy it...and will comply...or at least try to...

 

[R.Mackey]

Well, that really depends on what you mean by strain.

Most steel structures are over-designed by about 1.5x. Meaning that the actual live+dead load is 2/3 of what the members can take. So, if it's taking 103% of what it should, it would not be a big deal.

At 3% deflection you can visibly see it bend and that is indeed very bad. We normally design buildings first for the structural strength and then doublecheck it for deflection. A deflection of more an 1/4 inch or so for a floor is VERY noticeable and will cause people to feel unsafe even if it isn't structurally true. So oftentimes a structure will be even more oversized for just that reason.

Just to clarify, when I was trained in solid mechanics, strain had a very specific meaning. In the simplest case of homogeneous strain, strain and deflection basically mean the same thing...

Design vs. yield strength is different, of course. The WTC core columns were apparently overdesigned by a factor of 1.67 (from F.R. Greening's letter "To Whom it May Concern", page 11, in the {ahem} Journal of 9/11 Studies, Sept. 2006), slightly higher than the typical 50% safety factor but still pretty average.

Otherwise, we're in good agreement.

 

[Free Thinkr]

I seem to recall seeing video of one guy trying to climb down, presumeably to the storey below..... it doesn't bear thinking about.

I think I speak for just about everyone when I say that those videos are some of the most horrendously disturbing/depressing videos of all time. Even when I see the stills the bile resurfaces, and I instantly become enraged. The worst part is that a sizable portion of our own population is willing to exonerate those responsible so that they can pursue their own deranged theories for entertainment.

 

[PerryLogan]

One does get repulsed. But keep in mind the conspiracy folks are mentally ill. They can't be held to normal standards of moral discernment. Facts bounce off of them; logic escapes them. This thread alone has debunked their claims a hundred times--but of course they will never know.

The Truthers live in a bad comic book in which the government is always out to get you and everyone is an agent or a shill. They are completely alienated from their society and have no real connection with other human beings. The horrors of 9/11 are just a movie to them.

 

[William Rea]

How many of you were aware of the fact that buckling was actually a major factor in the collapse of the World Trade Center? From a layman's point of view, you'll likely say "well duh, of course it buckled, that's how it fell on itself, silly!".....And please, stop arguing that the fire wasn't hot. If it wasn't hot, people wouldn't have chosen to jump down to the ground 100 stories, which is one of the most terrifying ways you can die.

And what you fail to point out is that Euler applies to slender columns where the length is a minimum of 30 times the width and no bending moments are applied at the joints. Where moments are applied at the pin joints (the columns are restrained at each end) the critical load rises by a factor of 4. For shorter columns P crit becomes so insignificant that they are more likely to fail in compression.

The factors in your favour are a) the eccentric loading from the non-uniform damage across the floors (which should have resulted in the floors above the damage toppling off) and b) the Young's Modulus E element of the equation which will be affected by the heat during the elastic strain portion of the deformation, once plastic strain is achieved then most equations break down. Most people are happy to talk about the drop in strength from the heat but nobody talks about the ductility and toughness imparted into the structure by the heat.

I'll leave the calculations up to you!

 

[William Rea]

Just to clarify, when I was trained in solid mechanics, strain had a very specific meaning. In the simplest case of homogeneous strain, strain and deflection basically mean the same thing...

Just to clarify further in this context, Strain in a single axis can also be a % reduction in length under compression and not just an increase as seen under tension. In volumetric terms it becomes more complex than is required in the context of this thread.

Deflection necessitates both tensile and compressive strain in the member under load and the distance of the material from the primary or neutral axis of bending is why the Moment of Inertia (I) is important. It can be roughly approximated to be the thickness of the beam.

The material in the cross section that is furthest away from the neutral axis will need to strain the most to allow deflection.

 

[gumboot]

The factors in your favour are a) the eccentric loading from the non-uniform damage across the floors (which should have resulted in the floors above the damage toppling off)

Please tell me you're joking...

-Gumboot

 

[William Rea]

Please tell me you're joking...

-Gumboot

I wouldn't joke about something as serious as this.

 

[Gravy]

Most people are happy to talk about the drop in strength from the heat but nobody talks about the ductility and toughness imparted into the structure by the heat.

I'll leave the calculations up to you!

William, are you suggesting that the WTC buildings may have been strengthened by the fires in them? If so, that's certainly a novel argument and I'd appreciate it if you could expand on it for a layman like me. And if you could explain how the top floors of the towers could have "toppled," that would be great.

 

[rwguinn]

And what you fail to point out is that Euler applies to slender columns where the length is a minimum of 30 times the width and no bending moments are applied at the joints. Where moments are applied at the pin joints (the columns are restrained at each end) the critical load rises by a factor of 4. For shorter columns P crit becomes so insignificant that they are more likely to fail in compression.

The factors in your favour are a) the eccentric loading from the non-uniform damage across the floors (which should have resulted in the floors above the damage toppling off) and b) the Young's Modulus E element of the equation which will be affected by the heat during the elastic strain portion of the deformation, once plastic strain is achieved then most equations break down. Most people are happy to talk about the drop in strength from the heat but nobody talks about the ductility and toughness imparted into the structure by the heat.

I'll leave the calculations up to you!

Pardon me, but your ignorance is showing...
Any engineers out there (besides me) wonder how you impose a moment at a pinned joint?

the joints were not "pinnned", they were fully restrained (more than 1 fastener), which allows a moment input, which decreases the necessary vertical load required to initiate buckling.
It was a "self-eating watermelon" situation--the worse it got, the worse it became.
Also, Young's Modulus decreases with temperature, which reduces stiffness, which increases deflection under load, which increases relative strain, which enhances the onset of buckling.

 

[RenaissanceBiker]

I'm a civil engineer specializing in water resources. I took a lot of classes in structures in college (B.S. and M.Eng. from U. of Louisville). I still have my books in my office in Charlotte. I'm working in Atlanta right now so I can't look anything up. However, I believe rwguinn is correct. I don't know of any common construction material that gets stronger under intense heat.

 

[chippy]

There is actually an equation I can use to determine whether the Euler buckling formula or another buckling formula is to be used. I'll see if I can dig it up.

 

[The Almond]

Most people are happy to talk about the drop in strength from the heat but nobody talks about the ductility and toughness imparted into the structure by the heat.

If we're going to take the modulus of toughness as the area under the entirety of the stress-strain curve for steel, then toughness will not increase as the modulus of elasticity decreases.

And what you fail to point out is that Euler applies to slender columns where the length is a minimum of 30 times the width and no bending moments are applied at the joints.

I pounded out the math on this one. No only is this "30 times the width" not in any of my literature, it just doesn't make sense. My Mechanics of Materials book does mention that Euler's formula fails for short, thick beams, but also adds, "if a short or intermediate-length stocky column is considered, then the applied load, as it is increased, may eventually cause the material to yield, and the column will begin to behave in an inelastic manner." It seems to me that the objection to using Euler's formula for thick columns is that they behave inelastically, not elastically. My analysis shows that the number is far closer to 20*W for thick columns and 10*W for thin columns. An I beam with a y-y moment of inertia of 259 in^4 exceeds Euler's buckling formula at 119' tall, but the ridiculous 30*W rule would apply at a mere 60'. It seems to arbitrarily define a column as being unusually thick at 60' in length.

 

[R.Mackey]

And what you fail to point out is that Euler applies to slender columns where the length is a minimum of 30 times the width and no bending moments are applied at the joints. Where moments are applied at the pin joints (the columns are restrained at each end) the critical load rises by a factor of 4. For shorter columns P crit becomes so insignificant that they are more likely to fail in compression.

Well... you're not entirely wrong. In treating beams, the slenderness ratio is significant in that above about 30 you can simplify the expression (length >> width being one of the assumptions, and 30 being a good rule-of-thumb threshold). And it is also true that a column pinned at both ends has four times the yield strength of a column that is not constrained at the ends -- the factor of four being the energy difference between first-order bending and triple-curvature.

However, none of these observations apply to the WTC cases. As the NIST report demonstrates, sagging floors -- those members that would have kept the columns pinned -- led to deflection as though the column ends were free. And the slenderness ratio is greater than 30 in that case.

The factors in your favour are a) the eccentric loading from the non-uniform damage across the floors (which should have resulted in the floors above the damage toppling off)

No. Just... no. We can get into this if you want, but please put some thought into it before making this assertion again!

and b) the Young's Modulus E element of the equation which will be affected by the heat during the elastic strain portion of the deformation, once plastic strain is achieved then most equations break down. Most people are happy to talk about the drop in strength from the heat but nobody talks about the ductility and toughness imparted into the structure by the heat.

If you're talking about strain hardening and annealing, it doesn't help you in this case. You wind up with materials that are more rigid, but more prone to fracture, and with decreased ultimate strength.

Also missing is that the heating itself creates significant displacement, and non-uniform as well. As the floors heat and cool, it stresses the connections between floors and columns immensely, further induces deflection in the columns, and contributes to the failure. Again, read the NIST report.

 

[William Rea]

Pardon me, but your ignorance is showing...
Any engineers out there (besides me) wonder how you impose a moment at a pinned joint?

the joints were not "pinnned", they were fully restrained (more than 1 fastener), which allows a moment input, which decreases the necessary vertical load required to initiate buckling.
It was a "self-eating watermelon" situation--the worse it got, the worse it became.
Also, Young's Modulus decreases with temperature, which reduces stiffness, which increases deflection under load, which increases relative strain, which enhances the onset of buckling.

All I can suggest is that you read what I wrote over again.

I cannot take responsibility for your errant response but I will accept your apology for calling me ignorant when you have fully read my post.