Almost seven years ago in Toronto, a drum technician for Radiohead, Scott Johnson, was killed when a stage structure he was on collapsed. My question is not regarding the incompetence (and possible negligence) that contributed to this tragedy. I am interested in some of the engineering and physics/statics concepts having to do with the failed structure’s loads and the beams (i.e. tubes) that supported them.
HERE is a link to a current CBC news article about the incident and the belated coroner’s inquest which is about to conclude.
Near the beginning of the article there is a photograph of two tubes in cross section. One tube is 2" in outside diameter and the other 3". The caption emphasizes the difference in load-bearing between the two tubes.
Is the load-bearing proportional to the 4th-power of the radius? If so, then the smaller tube would have only about 20 percent of the load-bearing capacity as the larger. More importantly, perhaps, and I may well be wrong about this, ‘load-bearing’ does not include tube buckling, only tube bending. And that (finally) leads to my question - in this type of construction, how much ‘weaker’ is a two-inch tube than a 3-inch one? How (well) can this be quantified?
The area moment of inertia for a tube is proportional to the difference between the the fourth order of the outer and inner diameters. The article doesn’t mention the wall thickness or grade of tube so it is impossible to give a quantitative value for the strenght but generally a 3” diameter tube is going to have about 3.5 to 4 times the strength in bending of 2” diameter tube for typical pipe wall thicknesses.
As the tube was part of a truss structure, you’d have to actually look at the loads of each member. The whole purpose behind truss structures is to minimize bending loads on individual members so as to make more efficient use of material. In an idealized “pin-connected truss” all members are exclusively in tension or compression; in a real world truss with gussetted or welded joints and things sometimes attached in between joints, members may see some combined loads, and in many modern structures members may be preloaded (or “pre-stressed”) to achieve a particular load state to improve stiffness and load transfer.
For long, slender columns the Euler buckling formula can be used to reliably predict the critical buckling load for axial compression. However, for shorter, thin-walled, or complex/offset loaded structures the non-Euler wall or flange buckling is complex and either empirical formulas or finite element analysis is used along with structural verification testing for critical applications.
The article is difficult to follow and gives a very imcomplete description of the testimony but it would appear that specified components (e.g. the 3 inch tube) were replaced or left out, the incorrect number of diagonal braces were used, and various other changes were made to an engineering drawing without review or formal revision. Without performing analysis on the as-built structure it is difficult to say what specifically precipitated the failure but once a truss structure starts to fail it can undergo progressive collapse as members are overloaded or joints are deformed.
Edit: Ninja’d by Stranger On A Train. Criss-cross!
Yes, the stiffness and strength of a tube in bending are both proportional to the fourth power of the radius. “Load-bearing” doesn’t have a precise definition in this context, so it’s impossible to say whether it “includes” buckling.
That said, classical Euler buckling loads are inversely proportional to bending stiffness. Since tube bending stiffness is proportional to the fourth power of the radius, buckling resistance is too (with some minor caveats).
One thing to remember is that the truss’ stiffness isn’t directly related to the component tubes’ bending stiffness. Truss members are essentially loaded only axially—they’re two-force members. In the real world, welded trusses have small bending moments at the welds, but these are trivial compared to the axial loads.
If the truss members aren’t on the verge of buckling, then the overall truss stiffness and strength depends on the cross-sectional area of the members. When the truss is loaded in bending (like a giant beam) then another factor matters a lot: the distance between the extreme members relative to the bending plane. But this matters so much for the same reason the tube radius matters so much in bending. The truss becomes a meta-tube or at least a meta-beam (a beam of beams).
As ever, Stranger, your answer is very informative. Thank you.
I will come back later when I have time, but didn’t want to forget to ask whether the difference between ‘fourth power’ and ‘fourth order’ is that the latter suggests that the ‘fourth’ is the dominant but not the only contributor to the fundamental relationship.
ETA: hadn’t seen EdelweissPirate’s answer when I posted. Thanks.
I don’t get some of the argument in that article. A 3" member didn’t exist, so they didn’t use it? I’m pretty sure 3" pipe exists, and more clamps exist - so the guy’s argument was that “no one” provided all the parts the engineering drawing called for, and it was “no one’s” responsibility to do anything about that? Just gotta go with the parts currently in the pile of stuff initially provided?
That’s a very good explanation of what a truss does; you essentially get a very high area moment of inertia with a minumum amount of material, and it also lets you build a very strong structure with a lot of hollow space, which is good for bridges and buildings. They are, however, very sensitive to the damage of individual members or joints and can fail catastrophically or in rapid cascading fashion as loads are redistributed.
The two terms are synonymous in this context.
It is pretty clear whomever was responsible for assembling the stage structure used the drawing as a suggestion rather than a set of requirements, and further modified the drawing in unauthorized fashion. It seems from the article that because the structure is temporary there was no independent review or verification and the engineer does not have to sign off on the as-built construction, and the company responsible for assembling thr structure had a long history of cutting corners. Think about that next time you hear someone ranting about obtrustive government interference and bureaucratic oversight.
Regarding the article, I agree that it’s confusing. One major reason for the confusion is that Dale Martin, the owner of Optex staging seems to be—pardon the technical jargon—a little nutty. Among other things, the guy decided to represent himself in a major inquest involving a death for which he may well be liable. Something’s not quite right there.
There are more direct indications of Martin’s cognitive dissonance: he claims that the truss with the three-inch members “never existed” and that the drawing itself was a mistake, apparently because he did not already own such a truss. Martin seems to think that the drawing existed to describe the trusses he had rather than the trusses the design requires. That’s an insane understanding of what the drawing’s there to do.
If a client came to me with Martin’s complaint, I’d put this in writing: the truss’ presence in the drawing indicates that it must be fabricated and used every time the stage is assembled. In other words, if it doesn’t exist, make it. I’m no lawyer, but it’s hard not to think that Martin added massively to his own liability by treating the drawing as a mistake or a suggestion.
The fact that Optex asked to have that part removed from the drawing only to see it added back in annually strongly suggests that the responsible engineer had determined that it was absolutely necessary. So disregarding the drawing is what I imagine would fall under something like “gross negligence.” On top of that, claiming that the drawings were “garbage” implies that you know the design is deeply flawed as drawn. So acknowledging that and then deciding to make the structure weaker than required by the drawings is just…shocking.
It’s also unclear which engineer did the initial design. There were at least two involved: Wikipedia mentions one George Snowden[sup]1[/sup], while the linked article calls Domenic Cugliari “the engineer of record.” Snowden is now deceased, but Cugliari may be personally liable for the collapse if the structure—as drawn—wasn’t strong enough. (Engineers are more tightly regulated in Canada than the US; a Professional Engineer license is optional for most American engineers but mandatory, I believe, for Canadian ones. PEs are usually on the hook personally for any design they sign off on).
Regardless of any design flaws, it sounds like Optex didn’t adhere to the drawings and substituted a weaker truss than the drawings called for. Given that the weaker-than-specified truss is cited as the root cause of the collapse and given and Martin’s strange attitude in court, I’d expect (perhaps naively, as I’m not a lawyer) most or all of the liability to end up in Optex’s lap.
The linked article doesn’t contain enough information to reach a conclusion about who’s at fault, but from an engineering perspective, this seems like a hot mess. The whole thing reminds me of the 1980 Hyatt Regency walkway collapse in Kansas City, MO—except that disaster was the result of non-malicious incompetence. No one involved in that collapse exhibited the flagrant disdain for engineering requirements that Martin seems to have expressed in court. While the walkway engineers weren’t found criminally negligent, they lost their PE licenses and their firm lost its operating license.
[sup]1[/sup] Apparently, Snowden was disciplined at some point when a scaffold he designed also collapsed. But neither engineer’s role is clear to me from the articles I’ve read, so it’s hard to say which designed the stage and the truss.
To get back to the OP’s question, while a 2” tube is much weaker in bending and buckling than a 3” tube of the same proportional wall thickness, it doesn’t follow that the overall truss was weaker in direct proportion to the bending strength of those tubes. But regardless, it was a lot weaker than the truss in the drawing.
Those numbers match mine. I mean, of course, because these are incredibly basic calculations for mechanical and civil engineers. Depending on the loads applied to the truss when it failed, the overall truss’ strength was probably proportional to one of the following: the cross-sectional area of the tubes used (a direct proportion), the bending stiffness of those tubes in particular (about 1/4 that of the truss as drawn) or somewhere in between.
In terms of numbers, I’d speculate that the failed truss was somewhere between 70% and 25% as strong as the truss in the drawing. Keep in mind that (a) I’m speculating and (b) the forensic engineers involved in the investigation were probably able to find the degree of diminished strength for those conditions with reasonable certainty. Unless something very weird happened as the structure collapsed, they’d have access to all the data they’d need to reach a conclusion about whether the truss in the drawing would have prevented this collapse.
Thank you so much (your and Stranger’s answers are why I come here).
When you say that the forensic engineers could determine those things, does that include the catastrophic, essentially discontinuous failures that may ensue from a more proximate failure?
Given a known or suspected condition, a forensic engineer could perform analysis to calculate the reduction in margin or potential for failure under various loading conditions. Once a collapse starts to occur that becomes more difficult because the dynamic loading conditions be perturbative; that is, depending on the way one member starts to fail may change the instantaneous load state, and figuring out exactly what sequence of failures occurred will require looking at the post-failure components and figuring out the sequence of failure (which is the “forensic” part; the rest is just basic structural engineering).
I’ve done work on the periphery of anomaly investigations involving structural failures doing analysis on different load conditions; in one case we were able to pinpoint exactly the mode of failure which had already been determined by physical evidence so we were just verifying that configuration had negative margins. In other failures, there has not been a ‘smoking gun’, although we were able to verify that proposed failure modes were or were not plausible. In this particular case, however, it seems utterly clear that the company responsible for assembling the stage structure displayed gross negligence whether or not the design was adequate. (That the structure was used for years in this non-spec condition argues that it was probably well designed with considerable margins but the only way to verify that would be to look back through the original design analysis or perform independent analysis.)
Stranger’s answer to your question was a fine one—and I was probably a little too glib in my original statement. I didn’t mean to imply that the forensic engineer’s job was trivial, but rather that this is the kind of failure for which the root cause is usually determinable. For one thing, it was part of a concert for a major rock band, and it’s likely that some or all of the collapse was captured on video. That can be helpful in determining which piece of mangled wreckage failed first.
Some failures can only be assigned a root cause with evidence that was destroyed during the failure itself. Other failures may be due to one of two or three possible causes, but it’s difficult to determine which one is the “true” root cause. But this collapse is unlikely to fall into either of those categories (at least from my rather distant vantage point).
One small note that may or may not have been clear: the moment of inertia does go up in proportion to the 4th power, but not necessarily the strength. It depends on the failure mode. Now, in a truss, buckling is likely to control so strength should be proportional to the moment of inertia, but if bending is the failure mode, then strength only goes up in proportion to the 3rd power of the radius.
Of note, it was in use from 1996 so possibly some fatigue from use? I imagine these things are expensive considering their size and one wants to get as much of a ROI as possible.
