Learn Me: Buckling Force on Square Steel Tube

I'm totally overthinking this, but it now has me really curious... What formulas would I use to calculate the force at which a square steel tube will buckle?

Here's some background to help understand the question: My Miata is a pile of rust. I basically bought it so I could have something meaningful to practice welding on, but at a price where I won't feel bad if I berkeley up & have to drag it to the crusher. The rails that support the floor pan are basically rusted off. I plan to cut them out & weld in some fresh square tube. I could just do this & all would be fine, but curiosity has gotten the best of me. I also recognize that buying a southern non-rusty tub is far more economical than trying to fix this one. That's not the purpose of this exercise though.

The current rails, even when new, will crush if you use them to jack the car up. It would be super convenient to be able to use these as jack points, especially since the actual jack points have rusted away. I am trying to calculate what the minimum size of square tube necessary to support the weight of the car on a jack stand without crushing would be.

For the sake of this question, let's just assume that we're using ASTM A500 Grade A square steel tube.

You need to look up what is called a "web crippling check" for box beams. Its basically a calculation that assumes a point load on the "web", the upright portions, of a structural member. I doubt you would have a problem with 3/16" or greater wall thickness of a box beam but I am too lazy to do the calculation for you. The variables would be the value of the point load, the length you would distribute the load, the wall thickness, the height of the wall, the steel strength. The AISC manual would be the code for structural design.

Most race series roll cages mandate a minimum .090" thick wall thickness on the main hoops. 1/8" thick wall tube would be more than enough. Also a good idea to maybe install a little reinforcement at the point you choose to be the jack point. Your Miata was built using fairly thin stamped steel. Odds are, if you have any clue at all, whatever you do will be an improvement over the factory as far as strength goes.

In similar situations, I have found it a real challenge to MIG weld thicker steel to thinner chassis parts, especially if the chassis is old. Burn through city. I think other folks who have done this on fresher Miata chassis have gone double the stock chassis thickness (14 gauge tube onto 20 gauge chassis steel). That kind of step up would be the limits of my welding ability (but that's not saying much).

I don't know the formulas, but 14 gauge 2" box tube mild steel is something I would think of as "strong" for frame reinforcement work. For cross-braces at the frame-rail ends of unibody cars, I have used 16 gauge 1" tube with good results. But this is as a reinforcement, not a structural use like the chassis framerails.

Since I'm sitting in front of a structural analysis suite, I did some quick (and likely VERY conservative) checking. We don't typically analyze that case directly, but as jharry3 stated it's largely a crippling/buckling calculation and that we have the tools for- at least for flat (or curved, but that's not useful here...) plates- which is where the conservatism comes into play. Assuming a 1" box beam (1" height) and a load of 2000lbs, 14-gauge (.0781" thick) 301 annealed steel, it passes both in strength and crippling/buckling.

In reply to Ashyukun: Umm, for what length of beam? A 6" piece of 1x1 behaves far differently than a 6' piece as it would potentially see 12,000 ft-lb of bending if loaded in the center.

I've been trying to research this & do some theoretical calculations based on the info you guys have given me.

Some of these numbers are assumptions/guesses, but the point is to get it close enough to understand the math. Please point out if/where I make blatantly false assumptions or mistakes. The Criteria for Rectangular HSS suggests that I should calculate the limit of local yielding due to uneven load distribution in a loaded plate.

This is assuming assuming that Mazda initially used a 20 gauge steel:
R = [10Fyt/(B/t)]*Bp
R = Strength
Fy= (39ksi) Yield Stress of HSS material
t = (.034875in) Design Wall Thickness of HSS (.93 * Nominal Wall Thickness of HSS)
B = (2in) Width of HSS
Bp= (2in) Width of Loaded Plate

R=[10 * 39ksi * .034875in / (2in / .034875in)] * 2in
R=474.344 pounds-force

This would explain why the factory rails crush when used as jack points.

Substituting 14 gauge steel in for the wall thickness (t = .07265625in) gives us this:
R=[10 * 39ksi * .07265625in / (2in / .07265625in)] * 2in
R=2058.783 pounds-force

Using 1.58 as the factor of safety means 14 gauge 2" mild steel tube can support 1303 lbs, which should support half of a Miata.

Back in the practical world, my welding ability is certainly not better than Jamey_from_Legal. This means that 14 gauge is probably both the thinnest metal I would want & the thickest that I would be capable of doing a good job with.

Also, it would still be prudent to add some additional reinforcement at the locations I actually plan to use as jack points.

I also tried to calculate this using another approach; By looking at the square tube as 2 vertical webs with a flange on top & a flange on bottom. If the weight were distributed uniformly across the flange, that should mean that each web is supporting half of the weight.
R = (5 * k + N) * F * t
R = Strength
k = (.034875in) Flange Thickness
N = (1in) Length of Bearing (The top of my jack stand is 1in wide)
F = (39ksi) Yeild Stress of Web
t = (.034875in) Web Thickness

R = (5 * .034875in + 1in) * 39ksi * .034875in R = 1597 pounds-force

Using 1.50 as the factor of safety here means each web can support 1065 lbs, which means the factory rail could support 2130 pounds if load were distributed uniformly. Since the rail is clearly crushed in numerous spots but this math says it should be fine, I believe I made several false assumptions here. The largest false assumption is probably that load would be distributed evenly when using the rail as a jack point.

erohslc wrote: In reply to Ashyukun: Umm, for what length of beam? A 6" piece of 1x1 behaves far differently than a 6' piece as it would potentially see 12,000 ft-lb of bending if loaded in the center.

I was purely looking at whether the local section would fold at the jack point. Bending over the length of it is going to be a different story of course- and how well it handles that is going to be dependent on how well the rail is welded to the rest of the body/frame and how well it will resist the bending moment. The presumption would likely be that if the new square tube rail could transmit the load across its height and into the rest of the frame/body that it would be fine- so the local loading is more important than the bending over the length of it.

Get some 14 gauge plate bent into the shape of the channel and leave about 4" flanges at the top where it meets the floor. That will do and you might even end up with something that you can weld to.

If you don't have the connections to get it bent up, buy some equivalent rectangular tube and weld a wide flat piece over the top.

I also have to wonder about the sanity of people who buy totally rotted out cars so that they can learn to weld. What are you a glutton for punishment? That's like learning about the birds and the bees by going to an S&M bordello. Everybody knows you cant weld rust. Well, I do now cause I bought a totally rotted out car and taught myself. But just saying...

frecks wrote: I'm totally overthinking this...

Yup.

Since this thread began, I have jacked my Miata up 20 or 30 times. I used the pinch welds under the rocker panels every time without issue.

Miatas are pretty light.

Carry on...

SVreX wrote: Since this thread began, I have jacked my Miata up 20 or 30 times. I used the pinch welds under the rocker panels every time without issue. Miatas are pretty light. Carry on...

The issue with that is his are likely rotted off. I used to jack my capri up by the subframe connectors but im pretty sure 1x3 quarter inch wall may be overkill for what you need.

In reply to dropstep:

I dunno. His build thread photos look like the pinch welds are in much better shape than the floor pan support tubes.

The factory designed these cars to be lifted from the pinch welds. The "tubing" under the pan is not a frame- it is a floor support.

Alot of small unibodys are designed that way. It annoys me often at work. But yeah i assumed since severe rust was in the post that the rockers and pinch welds were gone.

The assumption was my bad.

In reply to frecks: Jaguar XK-E used extremely thin wall tubing for the front half of the chassis.. less than sheet metal thickness! The tubing was about 1 inch by 1 inch.

The engineers crashed it into a wall during the early days of crash testing and it survived remarkably undamaged.. The last Trans-Am car's used thin wall square tubing for the whole chassis. 650-700+ horsepower engines, big fat slicks, ground pounding torque etc.. While the roll cage hoops were mandated to be round most of the chassis proper looked like the Jaguar that was made in 1954 called a "D" type

frenchyd wrote: In reply to frecks: Jaguar XK-E used extremely thin wall tubing for the front half of the chassis.. less than sheet metal thickness! The tubing was about 1 inch by 1 inch. The engineers crashed it into a wall during the early days of crash testing and it survived remarkably undamaged.. The last Trans-Am car's used thin wall square tubing for the whole chassis. 650-700+ horsepower engines, big fat slicks, ground pounding torque etc.. While the roll cage hoops were mandated to be round most of the chassis proper looked like the Jaguar that was made in 1954 called a "D" type

The Jag front frame is chrome moly. Very thin wall. Lot of the frames were compromised by people doing something as simple as drilling a hole to mount an air-horn trumpet or compressor or some other random accessory.

dropstep wrote:

SVreX wrote: Since this thread began, I have jacked my Miata up 20 or 30 times. I used the pinch welds under the rocker panels every time without issue. Miatas are pretty light. Carry on...

The issue with that is his are likely rotted off. I used to jack my capri up by the subframe connectors but im pretty sure 1x3 quarter inch wall may be overkill for what you need.

Yeah... Unfortunately, dropstep is right on. I would use the pinch welds but I don't have any in the back & I'm pretty sure that the front pinch welds would just fold up if I tried to use them.

NOHOME wrote: Get some 14 gauge plate bent into the shape of the channel and leave about 4" flanges at the top where it meets the floor. That will do and you might even end up with something that you can weld to. If you don't have the connections to get it bent up, buy some equivalent rectangular tube and weld a wide flat piece over the top. I also have to wonder about the sanity of people who buy totally rotted out cars so that they can learn to weld. What are you a glutton for punishment? That's like learning about the birds and the bees by going to an S&M bordello. Everybody knows you cant weld rust. Well, I do now cause I bought a totally rotted out car and taught myself. But just saying...

Thankfully I never claimed to have a shred of sanity. And your analogy is amusing. It's also kinda accurate with regard to my interest in cars, which was mostly piqued by some friends who convinced me to go buy this crappy 1989 Accord & race 24 Hours of Lemons with them.

So now that all of my motorsports experience has been with 24 Hours of Lemons, I can't help but look at things through that lens. The Lemons attitude now influences far too many of my decisions, which means I now somehow view things like "everyone knows you can't ..." as "this will be a fun challenge & I'll learn a ton ..." So yes, I'm a glutton for punishment, but I (mostly) enjoy it. Life is too short to be boring & predictable.

Also, I really just wanted a rear wheel drive manual transmission car & a welder. I had little justification for either other than "I want it & it will be fun". By buying them at the same time & telling myself that I would learn to weld on the car, I can justify it all to myself. This should make absolutely no sense to people with sanity, but that doesn't matter because I'm not one of those people. The only part of this decision that I figure actually makes some sense is that $600 for a car that runs and drives means I pretty much bottomed out the deflation curve for something I can actually use. I'm reducing the wear on my nice GTI by driving this & this cost less than the taxes on my GTI did. I've already put nearly 1k miles on the Miata since I bought it a couple weeks ago.

Did you really have to go 4 inches out to the sides to reach metal that you could weld to? I was thinking I would be through to clean metal if I just cut off the rails flush with the floorpan. The floorpan itself doesn't look too bad at the moment. I'm probably being super naive. At least now I have a solid idea of what to do if I realize that my plan won't work.

Probably dont need to go so far in search of solid metal. When I said 4" I was going by memory about how rusty the underside was.

You know flying miata makes exactly what you are proposing and it bolts in through the floor?

NOHOME wrote: Probably dont need to go so far in search of solid metal. When I said 4" I was going by memory about how rusty the underside was. You know flying miata makes exactly what you are proposing and it bolts in through the floor?

Yeah, those look awesome. The Flying Miata stuff is definitely the way I'd go if I had a nice car. But I want to learn to fabricate things myself & I'd like to keep the car within 24 Hours of Lemons budget so I can potentially run in the future, so that rules out most of Flying Miata's stuff for this car.

Guys, the beam-bending and buckling calculations are all right, but my main concern would be the point load from the edge of the jack causing Hertzian contact stresses. Those bulk calculations work so long as your jack has a piece of U-channel that fits tightly (<.003" difference) against the outside of the square tube.

You can tell that the racing season has just ended, these responses look like half of Detroit is bored right now.