# Helical Stair-Rail Science

Pros puzzle through the complexities of following a spiral stairway's curve using round stair rail. March 5, 2008

Question
Here's something to chew on. In the attached photo you can see we have a rather straightforward curved stair rail to make. The run of the stairs is 10 1/4" and the rise is 7 5/8". Note: don't be concerned about the gaps between the rail and the tread ends... not my plan!

With most any hand rail, we would just simply build a curved wall and do a glue up of the hand rail laminated strips with the proper rise along the curved wall. I have a theory, though, for this one since, as you'll notice, the hand rail is round at 2 1/4" diameter. My thinking is that unlike a more traditional hand rail profile, where the top surface must always be facing up (hence the twist in the rail created by wrapping the laminations on the curved wall), this perhaps is different.

Since this is a round rail, any "side" can be facing up. So, if I built this on a traditional curved lamination glue up wall, and then profiled it, would it then lay flat on a work surface, or would there be a helix twist in the length of the rail, preventing the rail from laying flat in a single plane?

Trying to picture it I think it would have a helix twist and not lay flat. However, part of me also thinks that I could make this without the curved wall. That is, calculate a stretched out radius from the plan view radius of 93 3/4" and the rise of 7 5/8" along the full run of the curve. If this is the case then I could use that extended radius to just make a single plane eyebrow curve that could be tilted into place. Remember, there is no up on the hand rail profile, so the twist to keep the top edge up isn't an issue... Any "edge" can be up.

Forum Responses
(Architectural Woodworking Forum)
From contributor M:
Lay any curved stair rail on the floor and you'll see that it won't lay flat. So unfortunately, you'll still have to do it the old fashioned way. Also, even though the profile is simpler that a traditional hand rail, I think profiling it will be harder, since there is no "bottom" to index off.

From contributor P:
Sounds like a compound curve to me; maybe not helical, though.

From contributor D:
I see what you are thinking. If you logically continue your 7' + radius to 360 degrees, it will not lie anywhere near flat, and if continued, would look like a spring. Progressively shorter segments will lie progressively flatter. Therefore I would rule out your theory that you can just do a flat curve of an as yet unknown radius and tilt it up into place. A compound curve is not it either; compounds are the product of two curved surfaces. The stair is a (so-called) simple helix.

However, in this real world, if you are doing a short enough segment of arc - maybe 3-4 treads or less - you can probably do a flat curve and be done with it. Therefore the helical rail would not be needed.

At what point does the flat curve no longer work? I don't know. How can you predict it? I don't know. How do you calculate the flat curve radius to work on the pitch? I don't know. Am I much help? No. You need a better mathematician than I, plus a good 3D CAD drawing.

From contributor B:
Don't be so sure that you haven't been helpful. Your suggestion that just a few treads as versus a whole stairway might make a difference is a very good point. As to the extended radius, I'm thinking that one might be able to work mathematically with the exiting chord length between the upper and lower points in the plan view distance and then use the overall rise to create a new chord length (actual finished rail chord as installed). The center of the arc would remain at the same location so that would perhaps give you 3 points from which to draw a new arc.

I think I have to agree that this is not the right way to do this. However, this is a temporary railing for getting some wedding pictures done and then this entire curved end of the staircase will get redesigned. Go figure!

From contributor A:
If that is the case... Why not buy some plastic rod (1" or 1 1/2" D) and use the handrail brackets to bend it in place? I will bet you that a piece of LDPE (low density polyethylene) or HDPE will bend that helix/curve. The plastic is cheap.

From contributor B:
Cherry rail, I'm afraid! Plus it is only supported at 3 points... top, middle, and bottom. We'll have to see if that is solid enough or not.

From contributor P:
I guess it's time to put my engineering degree to work. I laid out everything per your sketch and created a helix that will follow the centerline of your rail. I broke the helix at the bottom of the 5th step and projected the helical curve onto a plane. This gives you a straight chord length of 50.83 with a maximum deviation from flat of .319. Basically, this segment will not lie flat within .319 inches.

Next, I did a best fit radius to the helix that was projected onto the plane. I get a radius of 147.31 with a max deviation of .003. Pretty darn good. So, if you could live with making the rail in a bunch of sections that are .319 from flat, you could just round profile them with a 147 in rad on your stock. Pretty standard process for you. I am not sure if that is what you wanted for info. I can provide you with other numbers if you wish.

From the original questioner:
Good stuff. Thanks. You said you ran the helix to the bottom of the 5th step. I'm not clear on whether that is the entire arc length of the rail as shown in the image or not. If it is almost the full arc length, why not just extend a little further to get the entire curve in one piece since the lift out of the plane is only .319" anyway? If we made it as a single plane eyebrow, it would be made full length in one piece for delivery.

We do this by breaking the 2 1/4" thickness into 2 layers 1 1/8" each. Segments in each layer are butt jointed together and then the 2 layers are glued together while offsetting all butt joints by half the segment lengths. Makes for a rock solid girder like rail.

Is it easy for you to determine the out of plane rise for the entire radius section of the rail? I have very little experience with 3D drawing. Don't even know if it's something I can do in AutoCAD LT.

From contributor P:
Extending it a bit longer greatly increases the out of plane deviation. I assumed 6 total rises of 7.625 (is this right?) This gives a chord length of 75.77 inch with a max dev of 1.005 off plane. The best fit rad is now 149.05 in.

When you glue your two halves together, maybe you could deflect the glue-up by the 1.00 dim. Even with spring back it may buy you what you are looking for. Since it is a round profile it will be very forgiving to the eye.

From the original questioner:
Thank you for doing this for me. I've spent the past 30 minutes in Enroute trying to generate a 3d version of your information. I've crashed the program about 6 times and have only gotten close to doing what you've done for me. I truly owe you a favor.

I think they can live with the 1" deviation since it is going to be temporary. Doing it in a single plane like this will cut the price more than in half, so I suspect that would be acceptable. I will contact the customer to discuss your results.

We really couldn't do anything about forcing an out of plane deviation during lamination of the two 1 1/8" thick layers. The butt joints, designed to hold until the layers are brought together, would never take that kind of stress. I'm moving forward here, thanks to your time and help! It is very much appreciated.

From contributor B:
Actually, 5 risers are involved... not 6. Close enough, though. My customer is going to take a 149" radius x 76" chord MDF 2" wide rail template to the project site tomorrow and see how it looks.

From contributor P:
No problem. It actually only took about 5 minutes to calculate once I figured out how to lay out everything. Keep me posted. This could be a simple way to do short helical handrails.

From contributor I:
I have actually produced round rail on the rise by projecting that inclined radius and cutting the parts on the CNC. This was a 3" full round maple that was broken into short sections and end capped between metal stanchions. It was laid up in two halves and where the glue joint had started level, at the downward end had rolled approximately 3/8" of an inch in 5' from what I remember. It was not detectable to the untrained eye and I feel more favorable looking than strip laminated. Below is a drawing of how I used Autocad in 3D to lay out the rise and run and stretch an arc through the points. Going to properties gives me the arc even though it is on an incline.

From contributor J:
The helical curve of a full round handrail can easily be approximated and made from flat radius segments. Metal pipe-rail is often made this way. The bends are made flat per the required developed radius and then cut into segments. The segments can then be laid out on the stair and welded back together in the best position. The secret, of course, is that each piece is rejoined to the next, but the joint is twisted. The flat plane-curve of each segment then becomes part of adjoining oblique planes, that closely approximates the curve of the true helix.

Even the most complex helical curves can be reduced to single-plane bends and then combined to yield sweeping rails and graceful geometric transitions. This forms the basis of the study of "Tangent Handrailing from Single Plane Developments." You can read about it in some very old reprinted books and at least one newer one.

Moulded wood handrails start with the same single plane layout, but the twist and profile must be cut into the curve. These tangent made rails usually follow true helical paths but their graceful curves are the result of the single plane.

I probably have confused a few of you here, but I am not making this stuff up. The math for determining simple helical curvature was first published in the 1750's and is found on this website. It works and it's all you'll need to lay out full round railing pieces that can approximate the true helix. Moulded handrailing and complex helical forms, however, will require a knowledge of tangent handrailing.

From the original questioner:
Contributor J, nice stuff! Thanks.

From contributor B:
It's an ellipse.