Areas of my expertise
May 18, 2026 at 12:25 PM by Dr. Drang
A few years ago, I wrote a post describing how I asked ChatGPT to solve a couple of elementary beam bending problems and how its answers were persistently wrong, even after I told it the mistakes it had made. For the first problem, determining the deflection at the center of a simply supported beam with a uniform load, ChatGPT gave the correct formula—presumably because the correct formula was part of its training corpus—but couldn’t come up with the correct numerical solution. As I said in the post:
Strictly speaking, this wasn’t a good example of a structural analysis homework problem. Students don’t get asked to just look up formulas and plug in numbers. More likely, they’d be asked to derive the equation that ChatGPT started with by either solving the differential equation for beam deflection or using some simplified technique like the moment-area or conjugate beam method. I didn’t think asking ChatGPT to do something like that would be fair.
This got me wondering how many ways I could derive the formula. A handful of ways came to me immediately, and I kept thinking of other methods over the course of the next several weeks.
Here’s a sketch of the problem:
where w is the intensity of the load, in units of force per length, L is the length of the beam, E is the modulus of elasticity of the beam’s material, and I is the moment of inertia of the beam’s cross-section. I’m not going to get into the details of these terms or the assumptions implicit in my derivations. Suffice it to say that I’m using the typical definitions and assumptions described in strength of materials and structural analysis textbooks.
I gave myself some rules for the derivations:
- They had to be truly different methods; slight variations on the same technique didn’t count.
- They had to give an exact formula; no numerical approximations.
- They had to be arrived at by hand; no use of a computer. The upshot of this was that although methods that yielded simultaneous equations were OK, there could be no more than three simultaneous equations. No one in their right mind solves more than three simultaneous equations, and frankly, my skills have deteriorated to the point that more than two equations is iffy.
I scratched out the derivations in my notebook, eventually coming up with twelve ways. They were:
- Second-order differential equation.
- Fourth-order differential equation.
- The moment-area method.
- The conjugate beam method.
- The slope-deflection method.
- The “myosotis” method.
- Energy minimization with polynomials.
- Energy minimization with Fourier series.
- Castigliano’s second method.
- Finite element analysis.
- The dummy unit load method.
- Newmark’s method.
I thought about presenting the derivations here, but I dithered over the best way to organize them. Eventually, other parts of my life intruded, and I gave up on the idea. It wasn’t until I wrote about the definition of “kip” a couple of weeks ago that I decided to just do a brain dump of all the derivations, one post for each. That’s what you’ll see here for the next couple of weeks. I know most of you don’t care about this sort of stuff, but I don’t care that you don’t care. Forewarned is forearmed—you’ll know what each post is about from their titles and can skip as you see fit.
For those few who are interested, this post will serve as a table of contents. The items in the list above will be turned into links as the posts are written.
Let me put a couple of things here that will be common. First, the deflection at the center is
This is the formula each post is aiming towards.
Second, the upward reaction forces at each end of the beam are
which is, as you might expect, half the total applied load.
Third, the shear and moment diagrams for the beam are
The moment diagram is very important to many of the derivations. It’s a parabola with a peak value of
We won’t be using the shear diagram1 directly in any of the derivations, but I tend to draw it whenever I draw a moment diagram. The mathematically inclined might notice that the shear is the derivative of the moment. It passes through zero when the moment is at its peak.
OK, that’s the setup. We’ll start zipping through the derivations next time.
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Shear is usually denoted V because it’s a vertical force in most beams. ↩
Unsound
May 11, 2026 at 2:17 PM by Dr. Drang
I mentioned last week that some wording in a news article struck me as odd. A similar thing happened when I read this Scientific American piece (that’s an Apple News link—I don’t have a SciAm subscription). In this case, I didn’t even have to read the article; the oddity is right there in the headline: “A SpaceX rocket booster is on track to hit the Moon at several times the speed of sound.”
I know what the writer means, of course. He’s telling us that the SpaceX debris will hit the Moon at several thousand feet per second, 1,000 fps being about the speed of sound we’re used to here on Earth. But the Moon isn’t the Earth, so it’s kind of a weird comparison, don’t you think? It’s not as if the booster is going to create a sonic boom. I’d find this less odd if the story were in a normal newspaper or magazine instead of Scientific American.
Again, I realize that the writer is just giving his readers a point of comparison, but is the speed of sound (on Earth) really something most people have an intuitive feel for? I remember learning as a kid to count off the seconds between a lightning flash and the resulting thunder to get the distance to the lightning strike in thousands of feet, but that doesn’t mean I have a strong sense of the speed of sound. Other than “really fast,” which isn’t all that helpful.
It’s mentioned in the body of the article that the speed of the booster at impact, which will be on August 5, is estimated to be about 5,400 mph. That’s about 100 times the posted speed limit on a two-lane highway, which is something most Americans do have a feel for.
By the way, if you have any interest in the problem of space debris coming back and striking the Earth (where the speed of sound might be relevant), you should follow Sam Lawler on Mastodon. She’s a professor of astronomy at the University of Regina and also has a keen interest in the clogging of near-Earth orbital space by the huge number of satellites launched in recent years. Her Mastodon feed is also a great source of farm animal photos, most recently a crop of baby goats.
A Key followup
May 8, 2026 at 8:26 PM by Dr. Drang
Earlier this week, I saw this article in Apple News. It’s from the San Francisco Chronicle and discusses a recent report on the safety of the Golden Gate Bridge. The report is one of several reports spurred by the Francis Scott Key Bridge collapse a couple of years ago.
Image from Wikipedia.
Spoiler: the report finds the Golden Gate Bridge safe—quite unlikely to suffer damage from the impact of a ship. This has to do with the Golden Gate’s two towers:
- The south tower (in the foreground) is surrounded by a reinforced concrete protective structure that a ship would have to strike and destroy before coming in contact with the tower and affecting the bridge’s integrity. We discussed the lack of protective structures around the piers of the Key Bridge.
- The north tower is right up against the Marin County shoreline, where the water is shallow. A large vessel would run aground before striking the tower, scrubbing away most, if not all, of its kinetic energy. A ship that wouldn’t run aground before hitting the tower would be too light to bring it down.
A striking omission from the Chronicle story is a link to the report itself. But that appears to be the fault of Apple News. I can’t read the story on the Chronicle’s website because I’m not a subscriber, but this reprint on Yahoo! News includes the report embedded as a PDF and available to download. So it looks like Apple removed the report itself, which is pretty poor form.
The nice thing about finding the actual report, written by HDR, Inc., is that it confirmed some suspicions I had regarding this paragraph in the Chronicle story:
The resilience of the Golden Gate Bridge partly comes from sheer strength. The south tower, on the San Francisco side, is described in the report as a “robust structural feature like no other” and is surrounded by a reinforced concrete protective shell up to 28 feet thick. It can withstand about 50,000 kips of force, or roughly 25,000 tons. In many cases, engineers found, a ship would crumple and absorb its own impact energy before it could seriously damage the structure.
First, it’s not “sheer strength,” it’s “shear strength.” HDR calculated the strength of the protective structure around the south tower to be at least 50,000 kips when that force is attempting to shear through the reinforced concrete wall. They specifically mention shearing capacity, shearing interfaces, and shearing area when discussing this calculation on pp. 59–60 of the report. While it’s true that engineers tend to be crummy writers, we definitely know the difference between “shear” and “sheer.”
Second, I suspect the writer, Brooke Park, doesn’t know what a kip is. When writing for a general audience1 most people wouldn’t use the word “kip” without saying it’s short for “kilopound.” Which is to say,
There’s no “roughly” about it.2
Still, this oddball paragraph doesn’t affect the overall story and provides the engineers who read it some amusement. Too bad about Apple’s redaction of a link to the report itself, though. That’s sheer incompetence.
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Which HDR isn’t, so I don’t blame them for using structural engineering terms without further explanation. ↩
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You could argue that Park’s “roughly” was meant to parallel the “about” in the previous clause. I think that’s an overly generous interpretation. Also, don’t write to me about long tons or metric tonnes—there’s no way Park was talking about those. ↩
Blocking sender addresses from list view
May 4, 2026 at 11:54 AM by Dr. Drang
After yesterday’s post, I got an email from reader Ron Sprague outlining two ways to block sender addresses from the list view in iOS Mail. Both of these methods are efficient, and neither starts with choosing a command with a stupid name like View Contact Card. I knew about one of these methods but not the other; this post will describe both.
(My first thought was that this would be an update to yesterday’s post, but I soon realized it was going to be too long for that.)
The first method starts with a swipe to the left on the message whose sender you want to block. This brings up three options:
The options are
- Trash, which deletes the message.
- Move…, which lets you file the message in a folder.
- More, which brings up the following set of commands:
You may need to swipe up to see the whole list.
Update 4 May 2026 2:48 PM
I am reminded by Leon Cowle that the buttons you see when swiping left may not be what my screenshot shows. The possibilities for the middle button are given in Settings→Apps→Mail→Swipe Options as None, Mark as Read, Flag, and Move Message. I have it set to the latter, even though I have no memory of doing so.
Two of these commands repeat what was in the previous screenshot, but a little redundancy never hurt anyone. What we’re interested in is the Block Contact command at the bottom, which does exactly what we want.1
This is a pretty efficient way to block the sender if you’re in list view and can tell that that’s what you want to do from just the first couple of lines of the message. The next method, which I didn’t know about before Ron’s email, gives you a chance to see more of the message before deciding to block.
Starting again in list mode, long press on the message whose sender you think you want to block. This will bring up two windows, a larger one with a more complete view of the message and a small one with a list of commands.
If you were unsure whether to block the sender from the list view’s abbreviation, you’ll probably decide after seeing this. Tapping on the small list of commands makes it bigger and reveals the Block Contact command.
Why Block Contact is red in this command list and not in the others is one of the wonderful mysteries of Apple interface design.
These blocking methods (thanks again, Ron!) prove that Apple can make reasonably efficient interfaces when it wants to. Too bad it didn’t put the same effort into the blocking method you have to use from message view.
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Yes, there’s a confirmation dialog after you choose Block Contact. There’s a similar dialog after the Block Contact command described in yesterday’s post. ↩