Question
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.
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.
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.
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".
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.
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.
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.
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.
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.
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).
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.
Common Lumber Name | A | B | C |
Hardwoods | |||
Alder, Red | 9.9 | 19.2 | 2506 |
Apple | 10.9 | 31.7 | 4132 |
Ash, Black | 9.3 | 23.4 | 4132 |
Ash, Green | 14.3 | 27.6 | 3590 |
Aspen, Bigtooth | 10.3 | 18.7 | 2439 |
Aspen, Quaking | 10.3 | 18.2 | 2373 |
Basswood | 6.2 | 16.6 | 2174 |
Beech, American | 8.9 | 29.1 | 3793 |
Birch, Paper | 8.8 | 25.0 | 3260 |
Birch, Sweet | 11.9 | 31.2 | 4065 |
Birch, Yellow | 9.2 | 28.6 | 3723 |
Buckeye | 8.9 | 17.2 | 2235 |
Butternut | 11.3 | 18.7 | 2440 |
Cherry | 13.8 | 24.4 | 3184 |
Chesnut, American | 11.6 | 20.8 | 2708 |
Cottonwood | 8.5 | 16.1 | 2102 |
Dogwood | 6.8 | 33.3 | 4331 |
Elm, American | 10.2 | 23.9 | 3116 |
Elm, Rock | 12.2 | 29.6 | 3860 |
Elm, slippery | 11.5 | 25.0 | 3251 |
Hackberry | 11.8 | 25.5 | 3319 |
Hickory, Bitternut (Pecan) | 14.7 | 31.2 | 4062 |
Hickory (True) | |||
Hickory, Mockernut | 9.1 | 33.3 | 4332 |
Hickory, Pignut | 9.3 | 34.3 | 4332 |
Hickory, Shagbark | 10.9 | 33.3 | 4333 |
Hickory, Shellbark | 6.6 | 32.2 | 4195 |
Holly, American | 8.3 | 26.0 | 3387 |
Hophornbeam, Eastern | 7.9 | 32.8 | 4266 |
Laurel, California | 15.1 | 26.5 | 3456 |
Locust, Black | 21.2 | 34.3 | 4470 |
Madrone, Pacific | 7.8 | 30.2 | 3925 |
Maple (Soft) | |||
Maple, Bigleaf | 12.8 | 22.9 | 2980 |
Maple, Red | 13.1 | 25.5 | 3318 |
Maple, Silver | 12.4 | 22.9 | 2981 |
Maple (Hard) | |||
Maple, Black | 12.3 | 27.0 | 3523 |
Maple, Sugar | 12.3 | 29.1 | 3793 |
Oak (Red) | |||
Oak, Black | 11.7 | 29.1 | 3792 |
Oak, California black | 16.4 | 26.5 | 3455 |
Oak, Laurel | 6.3 | 29.1 | 3791 |
Oak, Northern red | 13.6 | 29.1 | 3793 |
Oak, Pin | 13.0 | 30.2 | 3928 |
Oak, Scarlet | 13.2 | 31.2 | 4065 |
Oak, Southern red | 9.6 | 27.0 | 3520 |
Oak, Water | 10.4 | 29.1 | 3793 |
Oak, Willow | 6.4 | 29.1 | 3790 |
Oak (White) | |||
Oak, Bur | 15.4 | 30.2 | 3928 |
Oak, Chestnut | 10.1 | 29.6 | 3858 |
Oak, Live | 17.5 | 41.6 | 5417 |
Oak, Overcup | 10.7 | 29.6 | 3860 |
Oak, Post | 11.0 | 31.2 | 4063 |
Oak, Swamp chestnut | 10.7 | 31.2 | 4063 |
Oak, White | 10.8 | 31.2 | 4062 |
Persimmon | 7.0 | 33.3 | 4332 |
Sweetgum | 8.9 | 23.9 | 3115 |
Sycamore | 10.7 | 23.9 | 3115 |
Tanoak | 9.0 | 30.2 | 3926 |
Tupelo, Black | 10.4 | 23.9 | 3116 |
Tupelo, Water | 12.4 | 23.9 | 3115 |
Walnut | 13.4 | 26.5 | 3454 |
Willow, Black | 8.6 | 18.7 | 2438 |
Yellow-poplar | 10.6 | 20.8 | 2708 |
Common Lumber Name | A | B | C |
Softwoods | |||
Baldcypress | 13.2 | 21.9 | 2844 |
Cedar, Alaska | 14.4 | 21.9 | 2844 |
Cedar, Atlantic white | 10.9 | 16.1 | 2100 |
Cedar, eastern red | 16.4 | 22.9 | 2981 |
Cedar, Incense | 13.1 | 18.2 | 2371 |
Cedar, Northern white | 11.1 | 15.1 | 1964 |
Cedar, Port-Orford | 12.6 | 20.2 | 2641 |
Cedar, Western red | 12.2 | 16.1 | 2100 |
Douglas-fir, Coast type | 12.3 | 23.4 | 3049 |
Douglas-fir, Interior west | 13.2 | 23.9 | 3116 |
Douglas-fir, Interior north | 14.0 | 23.4 | 3048 |
Fir, Balsam | 9.9 | 17.2 | 2236 |
Fir, California red | 10.6 | 18.7 | 2437 |
Fir, Grand | 10.7 | 18.2 | 2371 |
Fir, Noble | 10.1 | 19.2 | 2507 |
Fir, Pacific silver | 10.4 | 20.8 | 2711 |
Fir, Subalpine | 10.5 | 16.1 | 2101 |
Fir, White | 12.2 | 19.2 | 2506 |
Hemlock, Eastern | 12.6 | 19.8 | 2573 |
Hemlock, Western | 11.5 | 21.8 | 2847 |
Larch, Western | 11.3 | 25.0 | 3251 |
Pine, Eastern white | 12.3 | 17.7 | 2303 |
Pine, Lodgepole | 11.5 | 19.8 | 2576 |
Pine, Ponderosa | 12.6 | 19.8 | 2573 |
Pine, Red | 12.2 | 21.3 | 2777 |
Southern yellow group | |||
Pine, Loblolly | 12.9 | 24.4 | 3183 |
Pine, Longleaf | 15.0 | 28.1 | 3658 |
Pine, Shortleaf | 12.9 | 24.4 | 3183 |
Pine, Sugar | 12.6 | 17.7 | 2302 |
Pine, Western white | 10.0 | 18.2 | 2370 |
Redwood, Old growth | 14.9 | 19.8 | 2573 |
Redwood, Second growth | 13.2 | 17.7 | 2302 |
Spruce, Black | 11.3 | 19.8 | 2575 |
Spruce, Engelmann | 10.0 | 17.2 | 2234 |
Spruce, Red | 10.6 | 19.2 | 2506 |
Spruce, Sitka | 10.8 | 19.2 | 2506 |
Tamarack | 12.0 | 25.5 | 3318 |