## Sunday, February 10, 2008

### Torsion box

Core:
I = 1/12 x (W x T^3); W = rib width, T = rib height

Skin:
I = A x D^2; A = breadth x thickness of skin, D = centerline of skin to neutral axis of box.
Itotal = I top skin + I bottom skin + I each core element

Deflection:
Y = (.021 x W x L^3) / (MOE x I), where;
Y = deflection (inches)
L = span
MOE = modulus of elasticity (1.5 E6 is avg for wood)
I = Itotal from above.

Torsion box pictured: 61" span, 3" x 3/8" web, 3/8" skins. Deflection: .018" (~1/64") @ 1000lb.

Anonymous the Younger said...

We're making a PVC Multi-Wall Panel torsion box for a roof and I've searched all over the intertubes for this math. Thank you so much! Where did you find it? If you got it out of a book, what book? Thanks again for the great info!

-aty

Anonymous the Younger said...

Forgot to click the box.

John Bass said...

I am math challenged and would like some advice on torsion boxes. I am making a table 4'x7' only supported at ends. I would like the table to be 2" thick. When finished I will be putting a thin (3/8" including thin lath) concrete veneer over it.

My initial plans are to use strips 1 1/4h x 1"w on a 4"x4" grid. Then enclose it with 3/8" sheets.

This is of course guess work, a math solution would be greatly appreciated.

Thanks, my brain thanks you in advance.

Steve, thanks for posting the math for a Torsion box. You're the only place I've found it thus far, but I don't understand your algorithm, specifically the skin calculation.

I=AxD^2
D=Centerline of skin to neutral axis of box

I don't understand what A and D are. I tried to back into it using your example, but no luck.

I think most examples of torsion boxes I see are over engineered, but I need the math to prove it.

Thanks.

steve said...

I need to do a better job attending to this post - sorry to all.

1. I got this math from this book: http://www.amazon.com/More-Woodworkers-Essential-Short-Cuts-Essentials/dp/1892836211.

2. I believe the math posted for THE BOX I MADE is wrong. I vaguely recall a miscalculation/ recalculation at some point, and I don't recall updating the blog post to reflect it. Backing stuff out of it probably won't work. AFAIK, the equations are correct, tho.

3. The math suggested by Mr. Horner jives with my aging memories of engineering classes:

The equation for deflection of a beam is as stated.

"I" is the moment of intertia.

Since the "beam" is composed (in a section view looking through the side of the box anyway) of a rib with a skin on each side, you must calculate I for both the rib and the skins and add them together to get the TOTAL moment for the assembly which is then chugged into the deflection equation.

IIRC, the "thickness" of the "rib" is the total for all ribs. IOW, if you have 5 ribs at 3/4" thick (in the span direction), that's like having a 1.25" thick rib.

On calculating I for the skins:
The equation is different from the rib moment calculation because 1) the skins are separated from the centerline; 2) the skins have breadth, whereas the ribs only have thickness. I specifically use "breadth" so as not to be confused with "width".

If you study the rib moment equation (more confusion because what I call "width" is really "height"), note that the width and thickness are accounted for... NOT LENGTH. So, for the skins, breadth is analogous to "width" of the rib; you're capturing the amount of skin going "into the page" in your section view.
Note that each I calculation takes a dimension to the 4th power - and NONE of those dimension are LENGTH (that comes into play in the actual deflection calculation). For the beam (rib) I, it's a height X thickness cubed (4th power); for the skins, it's an area (already a square) times a distance squared (4th power).

Now, "D": The "neutral axis" of the box is the centerline, IF the skins on both sides are the same thickness. So, a 4" rib with 1/2" skins on both sides would have "D" = 2" (half the rib) + 1/4" (half the skin). If you draw the section view, visualizing "D" becomes trivial.

Also note: there's usually TWO skins, note that in the Itotal that gets plugged into the deflection equation.

Lastly, note that a T box is usually a GRIDWORK of ribs. The ribs that SPAN should be continuous (unbroken, not half-lapped into the perpendicular ribs). THOSE are the ribs that carry load, and for which we're calculating all this jazz. Ribs running perpendicular to the span DO NOT contribute to stiffness (preventing deflection), they ONLY contribute to keeping the span ribs from twisting/ folding under load.
If you're building a box to be FLAT, this doesn't really matter. If you're building a box to carry a LOAD, I'd pay attention.

4. DISCLAIMER: It's entirely possible I don't know WTF I'm talking about. I went to school for engineering a thousand years ago and have never actually done it for a living. PLEASE don't rely on me as an "expert" on this.
Like those who seem to have landed on this page, I was a dude frustrated by trying to find math online to calculate T boxes. When I saw a reference to a book that contained the math, I bought it, and posted it on my blog so it would be easy for me to find should I need it again (I still own the book, but if my life depended on it, I doubt I could find it).

Hope all this helps someone.

steve said...

Oh, also, FWIW:

The T box(s) related to this post I made for an "elevated platform" bed which I delivered to my sister shortly after this original post (in 2008).

The T box sits on two cleats on the inside of the long rails of the bed... that is, the T box SPANS the distance between the rails. So, the important (unbroken) ribs run in the direction across the bed, not from headboard to footboard.

As I write this, it's 2012. I'd rather not speculate on how much, uh... let's say "dynamic loading" my sister and her husband put on this bed, but it's still in one piece and keeping them off the ground these few years later.