Math for Helical Stair Parts

      Pros find and discuss a formula for figuring helixes (it's a real head-scratcher). April 11, 2005

I am looking for a formula to calculate a radius for parts of a staircase. What we want to be able to do is this: we have a 1.5cm thick by 5cm wide fillet piece for a handrail that will follow up a winding stair that is a true radius in plan. This part is thin enough, and the radius of the stair in plan is large enough, that we should be able bend it as it winds up the cylinder shape, if we just knew what radius to make it. So the question is how to calculate the radius of the part that would be cut while sitting flat, so that it can wrap around a cylinder and gain the proper given rise. Imagine a slinky stretched out. As you stretch a spring, the radius decreases. This is the opposite. We need to cut the part with a larger radius than the cylinder. I have looked in “A Treatise on Stairbuilding and Handrailing” but cannot find an explanation that I can understand. Worst case, I can wing it with hit or miss, but I have been wondering about a formula for this one for quite some time. Any math whizzes out there?

Forum Responses
(Architectural Woodworking Forum)
From contributor A:
It seems to me that you would treat it like any bent stair rail.

From the original questioner:
This webpage has a good explanation of what I want to do. See related web page: Differential Geometry

The sections “Example: Strakes and spiral" staircases” and “Curvature of curves” apply directly to the problem. Figure 1 illustrates it well. At the very end of the “Curvature of curves” section is a formula that should give me what I want, but I cannot get it to give the proper results.

From contributor B:
It doesn't give me the proper results either.

From contributor C:
Reverse the equation and it works. That is, divide the top into the bottom instead of the bottom into the top.

From contributor D:
An equation seems like a pretty good way to go. I use a different method. I will model it in 3D and then explode the 3D solid and list one of the helix lines that define the outer length and look for that number in the window. I also have to unroll conical shapes occasionally.

From contributor E:
What does the p represent in the formula? Is it the pitch in degrees or rads?

From contributor C:
The "p" is "Pi"...3.1416.

I emailed Allen Hatcher in the math department at Cornell University and this is a portion of the response I received from him.

"I think all that's wrong is that you forgot that the radius and the curvature are reciprocals, so curvature = 1/radius and radius = 1/curvature. The formula 4pi^2R/[H^2+(2piR)^2] computes the curvature of the helical curve, not the radius. For R=1, H=10 the formula gives 39.4685/139.47 which equals .283. This is the curvature of the helix. Now since radius = 1/curvature we get r = 1/.283 = 3.533. So the article is right after all."

How kind of him to suggest I merely "forgot" the "radius and the curvature are reciprocals".

He also stated that that great article is by another Cornell professor, David Henderson.

From contributor E:
It is still an unclear formula here for me on this. I need someone to write a step by step calculation here with all the numbers so readers can follow along. I have a mockup here, a 4.5" diameter pipe. That would be 2.25 R. And a length of wire that is 28.28" long, that would be 9" diameter (4.5" R). So when I wrap this wire around the pipe in helical fashion, the top end is 24.38" above the bottom end. So using 24.38 as the height
and 2.25 as the radius, what is the formula to get the answer of 4.5 - which is what is needed to do the job here.

From contributor C:
To contributor E: Is your wire wrapping once around the pipe in a height of 24.38"? If it is going around twice then you would need to use 1/2" of the 24-3/8" height.

How do you arrive at 4.5" radius for the wire?

When I use a cylinder radius of 2.25" and a height of 24.38" I come up with a wire radius of ~8.9".

From contributor E:
A wire length of 28.28 is the circumference of a circle with a 4.5" radius, (9" diameter).
This wire goes around a 4.5" diameter pipe once. I drew a line down the pipe, the wire ends are both on the line, 24.38" apart.

That is the answer, we just need a formula that takes height (24.38) and radius (2.25) and you end up with the needed radius to be cut (4.5) to make the handrail. Then the formula can be reversed to calculate backwards if needed.

From contributor C:
To contributor E: The formula on the Cornell website does not use the "circumference" in its calculation. You can ignore the 28.28" circumference of your cylinder with regard to the formula. I'm not sure what you are looking for here now.

The Cornell formula calculates the radius of the spiraling wire up the 2.25" radius cylinder. You again state that the wire radius is 4.5" (9" diameter). How do you arrive at that?

It's easy to measure the radius of your cylinder (2.25"), and the height of your cylinder (28.28"). But if you are using the circumference of that 2.25" radius cylinder to determine wire length for the upward spiraling wire then that could be your error. I'm afraid I'm a bit lost in what path you are taking in this calculation.

So, if you can give further clarification on where the 9" diameter wire length comes from that would help.

From contributor E:
The length of the wire is just to make things rather simple in my mockup. I used twice the radius of the pipe, which is 2.25, so a circle with a radius of 4.5, has a circumference of 28.28, the length of my wire. I am not using this number in the formula.
You could use a wire that is longer or shorter. That would affect the height of the helix.
Maybe I should go through the theoretical procedure for making a helix handrail - that goes along with the formula we are trying to discover here. You want to build a stair that goes up a round tower – let’s say the tower is 30' in diameter. It has 3' stairs, has 2 handrails on each side. Let’s build the inner one. In plan view, it’s about 24' in diameter (12' radius). To make a clamping form for the handrail, start with a 12' radius column in your shop tall enough to make a one revolution helix. Draw a plumb line up the column. At the bottom of the line measure over the rise and run of the stairs (slope). From the bottom of the line, start wrapping wire that matches the slope that you drew on the column. Wrap at that slope until you get to the plumb line again. Cut the wire at that point, and trace the wire on the column with a marker. This is the line which you build you clamping forms to. Take down the wire, and measure the length. In theory, the wire put end to end in a circle will have the radius you need to cut your wood that will wrap around the column in a helix. So the length of the wire divided by pi, then divided by 2, will give you the radius.

The answer we are looking for when using a formula where you know the height of the helix that is drawn on the clamping column that was built, and the radius of that column.

In my mockup, the numbers are: column radius - 2.25, helix height - 24.38. The answer is - 4.5 (the radius of the wood needed to be cut to make a hand rail that wraps around my pipe mockup.

I hope all the readers now clearly understand my explanation here.

From contributor C:
To contributor E: I'm no expert on this, and have spent a lot of time contemplating how to come up with the radius of a spiraling line. Hence I was really glad to find the Cornell formula.

I think (and I emphasize "think") you are making an error in assuming:

"In theory, the wire put end to end in a circle will have the radius you need to cut your wood that will wrap around the column in a helix."

I think the whole point here is that the spiraling wire up the column does NOT have the same radius as when it is laid down end to end in a circle on the floor. The fact that it is rising up the column is what is altering the radius. I could easily be wrong here, but I think the spiraling upward radius is not the same as when brought down to the floor.

I can see how one would think that as a fixed length of wire completing a circle gives only one radius. However, I think that things change when you are spiraling upward, and the two wire ends are separated by the column height.

If you disregard this assumption that the length of wire gives the final circumference (and subsequent radius) then does the Cornell formula work for you? It seems to me that this formula is what you are looking for.

From contributor F:
I’ve worked as a railing /stair builder/installer for years, and having completed up to first year university calculus I probably could do the math but why? I know lots of guys who can do the work with a lot less math skills than me. Assemble the stair as on site, place clamps on treads, glue up the rip rail as you would a regular curved stair. The rip rail will have to be cut thin and you’ll need lots of clamps. Use clamping culls "strips" when you glue up. This work is easy if you let it be!

From contributor C:
To contributor F: I'd say that 90% to 95% of the time I agree with you. Occasionally though there are times when it's helpful to be able to do the math.

A case in point is a few years ago when someone wanted a 1-1/2" round dowel hand rail on a spiral stair. There was no need to do a laminated glue up since on a round rail it doesn't matter which edge is up.

Because of this we were able to make a simple eyebrow curved dowel. The trick was in finding out what the radius needed to be without doing a full scale wall mockup. Since I don't have experience in drawing full 3D in AutoCAD I had to estimate the radius and hope there would be enough flex in the rail to accommodate the installation, which turned out fine.

However, I wish I'd stumbled on the Cornell formula back then. It would have saved time and effort.

From contributor E:
In my explanation of the 30' diameter tower - the plan view of the helix handrail - the radius is 12'. If you squished the handrail flat to the floor, the radius would be larger.
Exactly what, I don't know without making a mockup with column and wire. I have not been able to get the formula to work for me. I believe this is what the original questioner was asking for in the original post.

Finding out the curvature of a helix seems to have no relevant meaning to me. It must be something like the slope of the helix (run and rise of the stair).

From contributor F:
I did one inside handrail on a spiral that we double cut the glue up and used hose clamps and a drill with a nut driver to clamp it.

From the original questioner:
To contributor C: Thank for emailing Cornell University and getting the answer. This will be a well used formula. I see now that the article does explain about the curvature. Of course to me, it could have been in Swahili and I wouldn’t know the difference. Thanks everyone.

From contributor G:
I, like contributor C, have been in need of such a formula on a few occasions. How about making a flat cap that lies on top of a curb stair as it descends? I see this as providing me with the radius I need to make my laminations to then stand up across the stair form to make such a cap.

From contributor E:
Since my example of wrapping wire around a cylinder is causing confusion, I decided to use a paper example to clear things up. I grabbed a smaller pipe, measured with my calipers as being 2.3" diameter (1.19" radius), (7.48" circumference). I drew a line up and down the pipe, 5". I drew a right triangle on paper, the vertical leg at 5"; the horizontal leg at 7.48" - circumference of the pipe. The hypotenuse was 9". I cut out the triangle with scissors, and wrapped it around the pipe. The hypotenuse is a perfect helix on the pipe.

Put simply, a helix is a right triangle wrapped around a cylinder, where the horizontal leg is equal to the circumference of your cylinder.

The hypotenuse will give you the length of a helix handrail and if you make a circle with the circumference that equals the hypotenuse. The radius of that circle is what you need to cut from flat stock to make your helical handrail.

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