[Thomas Utley]

All,

Long post, but bear with me...

I'm in the early stages of designing my second router and focusing much more closely on performance aspects after lessons learned on my first machine.

One key aspect of any moving gantry machine is the gantry beam, what it's made of, and the loads it can resist, especially in torsion. Unless you're cutting strictly parallel to the beam's long axis (or using a non-contact tool like a plasma or laser), torsional loads resulting from the spindle load inducing twisting in the beam via bearing reactions is something we should think very carefully about.

There seems to be fairly healthy debate on the Zone about 8020 vs. steel and the pros and cons of each. I read a couple days ago a post where someone asserted that 8020 t-slot extrusions are terrible in torsion compared to steel. The basis of that person's logic was that 8020 doesn't have a continuous "skin" and the t-slots greatly reduce its ability to resist twisting forces. I bought into his rationale hook, line, and sinker initially and very nearly convinced myself I should sell the 3060 (3" x 6") extrusion I just bought, go find a welder, and start fabricating a "real" beam out of steel.

Of course, what followed naturally was a study of the pros and cons of steel, and like many of you, I quickly came back down to Earth after realizing how challenging it can be for a DIY home-shop builder to deal with welding, and then straightening, steel as the basis for something that has to be fairly straight in three dimensions. I know there will be folks jumping in saying extrusions aren't necessarily straight, either. Fair enough, but I think most of those same people would agree that good quality extrusion in its natural state is likely straighter than steel after significant welding with no access to stress relief ovens or large surface grinders. And yes, I do realize "access" to those kinds of services can be either physical OR financial. Both are tough for most of us DIY types.

Back to the question at hand--8020 vs. steel in torsion. It's been many years since I worked as a mechanical engineer, but I remember enough about the mechanics of materials to be dangerous. If anything I'm about to state is inaccurate, by all means please correct me for the benefit of everyone following the thread. I decided to do the analysis to determine whether it would be worth the pain to deal with steel.

I have the benefit of having a working copy of Pro/ENGINEER 2000i, albeit a student edition from grad school. It is full-featured other than lacking certain import/export functionality that kept me from importing an actual 8020, Inc. section of 3060 extrusion. So I modeled it myself using 8020's radii and web thicknesses, leaving out only the external cosmetic grooves and the 2° drop lock feature on the slot tabs. Otherwise it's a very accurate representation of the real thing, and the CAD system's section properties calculations bear this out when compared to the numbers published on 8020's website.

For non-engineers, there are couple of terms you should be aware of before I go further:

First is the Area Moment of Inertia (aka "First Moment"): this is a measure of how well a beam made of a particular cross section resists bending in one axis. It's common to see rectangular beam specs with two MOIs: "Ix" and "Iy," meaning how well they resist bending in one direction, let's assume the "easy" way vs. a direction 90° to it, let's assume that direction is the "hard" way. Think about a piece of 2x4 lumber. It's much harder to deflect it against the wide dimension than in the thinner dimension. Without getting into the math of it, the more material your beam has farther away from its neutral (center) plane, the stiffer it will be. 8020 provides Ix and Iy values next to each of their profiles online, so finding this info is easy. For steel profiles, there are multiple online calculators that let you enter in the outer dims and wall thickness (or inner dims of the hole through the tube) to get this information.

The second term is Polar Moment of Inertia (aka "Second Moment," or "Ij"). This is a measure of the beam's ability to resist torsional (twisting) loads. While it sounds complicated, in practice the Polar Moment value is nothing more than the sum of Ix and Iy. It's really that simple. If you can get the first two, you can calculate the third one.

Final engineering geek-speak I'll share is this: the UNITS of moments of inertia are in units^4 (in^4, cm^4, etc.). That's a result of the math involved, but for rectangular beams like a 2x4, it's nothing more than a function of the first moments being a product of the base^3 multiplied by the height and divided by a constant. If you're bending in the other direction, it's the base multiplied by the height^3 divided by the same constant. Hence, you wind up with in^4. Don't let that bother you.

Having this background, now you can compare beams in both bending and torsion.

Using my 3060 extrusion section in Pro/E, I had the system provide these properties to me. I then compared the result to a steel beam of the same 3" x 6" dimensions with a 1/4" wall thickness. Here's what popped out:

	Ix	Iy	Ij
Steel 3" x 6" Rectangular Tube, 1/4" Wall	6.3385 in^4	19.3385 in^4	25.6771 in^4
8020 3060 3" x 6" T-Slot Aluminum Extrusion (my CAD, not 8020's)	6.5164 in^4	22.0300 in^4	28.5464 in^4
Aluminum as % of Steel	103%	114%	111%

With respect to bending and torsion, the t-slot aluminum beats the steel in this particular 3" x 6" beam envelope.

By the way, for a 72" long beam, the aluminum weighs about 45 lbs vs. 87 lbs for the steel.

So far, it's a home run for the 8020, right? Maybe. On the cost side, 8020 is real loser for raw material cost, BUT once you factor in costs to weld, mill, grind, perhaps stress relieve, etc., 8020 starts looking better again. All depends on what you have access to in the way of tools and services to work with steel.

One last thought I'll throw out there before the "tastes great! less filling!" debate starts: 3" x 6" 8020 is about as big as you can buy (they do offer a metric profile that's just slightly larger in both directions). Steel, however, can be had in many more sizes, and particularly important to a gantry beam is the height dimension because that's what determines how far apart you can spread your horizontal bearing rails. The farther apart they are, the lower the loads on the bearings (assuming you have them spaced appropriately ALONG the beam as well, which eats up travel...but I digress).

Mass, vibration dampening, and rigidity are also important points to consider for a moving gantry, so feel free to offer up any lessons learned there as well. Just don't respond back with "more mass is better" because most of us have learned by now that extra mass is only better if you have motors and power supplies that can accelerate the extra mass well enough to meet your performance needs.

So I'll throw this out there for you to consider. Would love to have somebody check my math, if for no other reason than to help me avoid a mistake on my new machine.

Best regards,

Tom

[Thomas Utley]

All, in my rush to get the original post out before taking the kids to their sports practices yesterday, I neglected one of the most important parts to this entire discussion...strength of the two materials.

Having the MOIs in hand is just half the information you need to assess beam twist. MOIs are a property of the cross-section only, not of the actual beam you'll be using. You also need to know a material property called the Shear Modulus. For rectangular beams like we're talking about here, the angle of twist is proportional to the following formula: ((length of beam * applied torque) / (Polar MOI * Shear Modulus)). This is intuitive...the longer the beam or greater the torque (the two factors in the numerator), the more it will twist. Conversely, the higher the PMOI or Shear Modulus in the denominator, the less it will twist in response to the torque.

For our beams being compared in the original post, three of the factors are equal: beam length, torque being applied, and PMOI. Only one factor is different, the Shear Modulus of aluminum vs. steel.

Since steel has a shear modulus of approximately 3X that of aluminum, and given that the two PMOIs are close enough to 1:1 for the 3060 extrusion vs. the hollow steel, we can conclude the steel will twist only about 1/3 of what the 8020 will for the same twisting load.

Does the extra rigidity of the steel make up for the extra mass? You'll have to answer that one yourself because it comes down to your budget for biggers motors and power supplies to accelerate the steel and get the same speed performance out of your machine.

Thanks, and apologies for not finishing the whole thought before posting the original.

Tom

[ger21]

There was a very good thread a few months ago that went into a lot of detail about gantry beam design. Unfortunately, I couldn't find it when I looked last night.

[dmalicky]

Slow reply -- I haven't had much time to contribute lately. Overall, T-slot extrusion, aluminum tube, steel tube, and built-up wood tube all have their place for a gantry, depending on many factors. I have often said that T-slot extrusion is flexy in torsion (which is true -- see below), but the convenience of it is also true. It is best suited to shorter gantries (~2') and/or wood cutting. If one has the budget, the machined-flat Misumi extrusions are very helpful.

The torsional moment of inertia J is indeed equal to Ix + Iy for solids and simple tubes (that can be decomposed into a 'positive' and 'negative' (the hole) solid, each with their centroid in the center). The J = Ix + Iy formula does *not* work for "open" or most complex cross sections. (In the derivation, that formula requires that "plane sections remain plane". T-slot sections experience shear strain along the slots – this distorts the plane of the cross-section.) Some examples of J not equal to Ix + Iy:
- Take a sheet of paper, coil it into a tube, and tape the seam. Except for buckling, it will be stiff in both bending (I) and torsion (J). Now do the same but do not tape the seam. ~Same bending stiffness, but very low torsional stiffness--actually about the same torsional stiffness as the flat sheet!
- Angle iron, especially with thin wall. E.g., 2 x 2 x 16 gauge. Good bending stiffness, but low torsional stiffness.
- Most all T-slot extrusion makers publish Ix and Iy, but almost none publish their torsional J (probably because it is so low). Here is one from Item; J for torsion is ~25% of (Ix + Iy), fairly good for T-slot:
Attachment 251220

Why? Torsion creates a loop of shear stress (and strain) in the cross-section:
- In a continuous closed tube, these stresses (and strains) are small because the stress has a lot of leverage around the tube center. See the left fig below. Low strain = high stiffness.
- In any open cross-section (untaped paper tube, tube with a longitudinal slit, angle iron), the shear stress loop is *confined to the wall thickness*. Right fig. Cross-section shape matters little: the loop always has poor leverage. The only way to get better leverage is to increase the thickness. Stresses and strains are much higher to resist the same torque.
Attachment 251222

- A T-slot extrusion usually has both open and closed cross-sections in it. The *largest closed section* is mostly what matters for the torsional shear stress loop (fig below). Appendages will definitely help I, internals will help prevent collapse-of-the-cross-section, but neither will do much for J. For example, below are the approx outer dimensions relevant to J, for three 8020 cross-sections. Recall J is proportional to size^4, so even a small change in size makes a big difference.
Attachment 251224

More theory in this link, slides 12 and 13: http://ocw.mit.edu/courses/aeronauti...tes/unit12.pdf

For a gantry CNC machine, the gantry's contribution to deflection-at-the-tool is due to both torsion and bending. Torsion dominates for short gantries. Bending is increasingly important for longer gantries (beam deflection goes by L^3). The rail loads can also distort/collapse the cross-section, especially if the wall is “thin” – FEA is needed to account for that (the mathematical I and J will overestimate stiffness). Internal reinforcement is often needed to maintain the full potential of J -- see link below, or the easiest solution is to use thick walls. T-slot cross-sections usually have some internal reinforcement, so probably have little or modest collapse.

This was likely the thread Gerry referred to (See my posts on p 1 and 2 for FEA results): http://www.cnczone.com/forums/diy-cn...questions.html
Especially see the attached image at the end of post #48 (here's a direct link): http://www.cnczone.com/forums/attach...2&d=1389432375
3x6 inch and 100x200mm T-slot extrusions are mostly closed, and so are relatively decent in torsion, but can’t beat a regular 3x6 or 4x8 rectangular tube of the same weight/foot. Compare trial 2 (Misumi GFS8_100200) to trial 12 (4x8x0.5 alum tube): The tube is a bit less weight, but nearly twice as stiff and less than half the cost --> stiffness/cost is 5x better.

A post on J, steel, and alum: http://www.cnczone.com/forums/diy-cn...ml#post1419700
For steel, I’d not suggest welding up a cross-section—yes, too much distortion. A500 steel tubing is quite straight and flat, but the density and wall thickness leads to a very heavy gantry unless it’s fairly short (2’ or less). Heavy may be fine if speed is not a high priority, and heavy is good if the gantry is fixed. Large alum tube is better suited to a long moving gantry, but it may not be flat enough, so leveling of some kind is probably needed, especially if using profile rail. Misumi GFS8_100200 is the stiffest T-slot available, and if pre-machined flat, by far the easiest for profile rail.