Maths people, help with calculations?

I built some curved steps as part of the set I'm building. See photos.

I need to make a handrail for it. My first thought was PVC pipe just bent around but any PVC that is large enough (like 1-1/4" or bigger) to look right won't make the bend. What I will likely do is stack two layers of 3/4" ply and cut the radius for this section, then continue on with a the rest as a straight rail.

I know the radii I used for cutting the steps and the rise of the steps. How do I figure out the radius I would need to cut for the angle? How long?

outside radius = 60"
inside radius = 12" (its a 4' wide tread that I drew with 1' inside and 5' outside)
rise = 8" per step, so total of 24" rise for the handrail.
approximate run inside = 6.5", outside = 32" (on the circmference, not straight line)
angle = 30 degrees (times three steps for 90 degrees total)

Need to find radius of the handrail for inside and outside. I may end up only doing outside, but can't hurt to have both.

Unless you all have brilliant ideas on how to make handrail that bends on that radius. I dont' have a tubing bender, nor is there time to farm it out to a fabrication company, and no money in the budget either. Gotta be an in-house thing.

a squared plus b squared is c squared

Its a single radius, but on a new plane. the radius is the "distance" in one plane and the rise is the distance in the other. c^2 is the distance in the new plane, so your new radius.

For the inside: 1 foot radius, 2 ft rise. 1^2+2^2 = 5. sqrt of 5 is your new radius for the handrail.

edit: this works out easy because you are using a 90 deg angle and therefore the radius is the same as the distance from the initial plane.

and now that I think about it I'm not sure the handrail will be a full 90 degree sweep. So I may be wrong about the above.

OK, I think I found something to help: Intro to Differential Geometry

They use something just like this as an example. Using their equations and assuming a 2" wide railing (so an 'inner radius' of 58" for the railing), the inner radius of the flat railing to bend down to form the railing SHOULD be 62.02" with an outer diameter of 63.89" (you could probably just make it 62" & 64" and have it work well enough. You'll need to be twisting the railing as well, so I wouldn't make the wood so thick you can't do that...

Hopefully that is what you need- I had to do some digging for that as it's not something I've touched in better than 2 decades (and even then I don't think we did much in-depth with differential geometry...).

This is where I get lazy and just draw it in CAD.

I'm a little confused on what some of the dimensions you are referencing are. If you want to clarify some of them with me, I can update these numbers.

If your handrail center line is even with the the outer edge of the steps (60" radius when viewed from above), the radius you want to cut when you orient it flat is 65.510" This won't be a perfect match (a line projected onto a cylinder doesn't always result in an arc or circle), but its very close.

This creates a handrail that is 90 degrees when viewed from above, a 60" CL radius when viewed from above, and rises 24" over that distance.

Edit: Its 91.98 degrees in plane and 97.88 inches long

I always do it on a framing square- that probably won't help you.

Others have posted the math right, but the radius is incorrect in all cases. The radius is dependent on the visual look you are trying to achieve.

If it is a traditional style railing with the spindles sitting on the treads, then the radius of the centerline of the rail is about 57" (this would have the centerline of the spindles sitting on the treads, about 3" from the end).

If it is a contemporary style rail, then the centerline radius is 60" + 1/2 the thickness of the spindles. So, if the spindles are 1 1/4" thick, the centerline radius would be 60 5/8". This would allow the spindles to be attached to the outside of the curved stringer (and assumes the treads do not overhang the stringer).

Looks good! Post pics when you get done!!

In reply to curtis73:

BTW, you are gonna need to give your actors a warning...

The tread depth at the inside of the radius is too narrow. If you were building real steps, the building code would require a minimum tread depth with f 6" on a winder stair. Yours look more like 3" (if I'm seeing it correctly)

If they are that narrow, it will invite someone to miss a step on the way down (and try to take 2 at once). This would be a guaranteed recipe for falling down the stairs (especially if someone tries it in a dress).

Tell the actors to stick to the outside radius.

The inside depth of the step is actually about 6". I recalled that from my old contractor days.

Radius of that inside curve is 1' on a 30 degree angle. So if you think about a triangle in that void, its a 30/60/90 with the hypotenuse being 12". Therefore the opposite side of the triangle (tread depth) is 6".

This quick sketch was how I laid it out and measured to make the steps. Mathematically (and actually) the pink line works out to 6"

Were it me. I would cut a diagonal strip of .25 ply(ala a diver flag) and make a plumb cut at either end that matches the rise (kinda like a roof rafter but bendy). Then bend/cut to fit.Add 2 more strips, glue and clamp, to get to .75 inch and at that point one would have a pattern for a cap.

Bustedplug... I think I might take that idea to another level.

I have a bunch of scrap luan. Here's my idea:

screw some 2x4 uprights to the side of the steps to match the curve. Mark them all at the right height. Cut about 10 strips of luan about 2" wide. Then I can stack them with glue and make a glu-lam using clamps and the 2x4s as a jig. Remove, cut, router, sand, reinstall.

Its just bloody labor intensive.

In reply to curtis73:

Yep. Curved railings are labor intensive.

How about if you go back to your PVC idea. 3/4" PVC should take the bend easily. Add pipe insulation for thickness- Voila!

Seems like you get the gist of what I said.if you stack all of the strips and zip-tie one end together tightly and make the plumb cut, the stack will bend together and then you'd have the marks for all the other cuts at once.also disregard the diagonal cutting, luan doesn't seem to care(in my mind i was cutting bendy-ply which only bends one way)

SVreX wrote: In reply to curtis73: Yep. Curved railings are labor intensive. How about if you go back to your PVC idea. 3/4" PVC should take the bend easily. Add pipe insulation for thickness- Voila!

I gave strong consideration to that, but its a small black-box theater. The audience will be close enough that they'll see detail and I'm afraid of what it will look like. Audience will be as close as 12'.

Well, then you are gonna have to invest the labor.

The three variables in any project are time, talent, and money. To reduce one, you have to increase another.

No money, so you're gonna have to put in a bit more time.

Can you find a talented volunteer who might like to take on this project?