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# The Mathematics of a Spiral Stair

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Stan, Joe, et al,

I don't think this is an exact retread (get it, re-"tread"?!?!) of a stair discussion from a few months ago. If it is, slap me and send me on my way.

I just got a call from my steel fabricator asking ME a steel question! How's that for evidence of a parallel universe?! I built a curved front staircase and used 1 5/8" pipe for the handrail. I made some measurements and approximated the radius of curvature for bending the pipe. The steel company bent it to my radius and it actually fit.

Now someone is finishing a stainless steel spiral stair for a client, and they're trying to figure out how to bend the ss pipe for the railing. Figure 5'4" dia and 8" rise. How would you define the curvature for your steel fabricator? Thanks.

Jim

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## (post #163457, reply #1 of 99)

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Dear Cloud - or is it Mr. Hidden,

I think you've already described the thing with 5' 4" dia. and 8" rise.

Any competent machine shop should be able to rigure it out. However they will need a very large lathe - 6' swing and bed - ? - whatever. Requiring the thing to be "pipe", i.e. hollow is problematic. I don't know of any practical method of drilling a curved hole thru a helix.

-Peter

## (post #163457, reply #2 of 99)

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Jim: I will be glad to give you the radius that the pipe needs to be bent, but first I need to know how many degrees does each tread turn. This determines the run of the stairs. If for example the run of the stairs approaches infinity, ie, a zero inclination angle, then the radius of the rail will be 32 inches. As the run of the stairs approaches zero, then the radius of the rail approaches infinity.

## (post #163457, reply #3 of 99)

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> I think you've already described the thing with 5' 4" dia. and 8" rise.

They'd also need to know how many degrees per step, or how many of those 8" rises in a full circle. This isn't a lathe job, they'd probably have to use something like a rail bender, with a set of wheels for the size of pipe involved. Not a standard machine shop item.

-- J.S.

## (post #163457, reply #4 of 99)

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They order their curved pipe out of a place 4 hrs away from here. Apparently they have only ever described length and diameter. But as you all know, a spiral requires one more piece of info. What I don't know, and what this steel company doesn't know, is if there is a typical way of expressing this so that the metal benders will include slope along with radius. I think John is on target with them being rail benders. Now I don't know why the fabricators--the biggest in the area--don't call the benders and say "Hey, how do I describe a spiral handrail to you?". Maybe they've never dealt with this before. Maybe they don't wanna look stupid. Maybe they don't know how to measure the angles. Don't know. But they are really good guys and if there's a standard way of ordering a spiral pipe for a handrail, I figured I'd find it here from you guys.

Cheers,

Jim

## (post #163457, reply #5 of 99)

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Jim: The fabricators will need to know that they are bending this handrail around a cylinder of 32 inch radius and at an inclination angle determined by the pitch of the rail. We know the rise is 8 inches, but how many degrees does each 8 inch rise take? With this info, then the proper inclination angle will be known. The radius of the bend can now be computed. The steeper this angle, the greater the radius that the pipe has to be bent to.

## (post #163457, reply #6 of 99)

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Stan, If I understand, they wouldn't just set the radius for 32" and then hold the pipe at the proper angle as they run it through the roller, would they? That would create too tight a spiral. We'd have to calculate the effective radius of the angled handrail--the greater the angle of inclination, the "more bigger" than 32" it would be--and as they roll that radius they'd also have to hold the pipe at the proper angle. Is that correct? I've never seen one of these rollers--will they roll a piece at an angle, or does that take an entirely different machine?

So if I went to my spiral in the house, measured a chord, determined the radius, it'd be larger than the 36" radius of each tread, yes?

By what formula, though? I've spent a few minutes with my Standard Mathematical Tables book and have yet to reach it.

A "for example". If 5'5" dia. 9" rise. 30 degree treads. That's 12 treads/circle and a 17.25" segment of the circle for each 9" rise. That should determine the inclination, but what next?

Thanks! This is fun. Appreciate all the help.

Jim

## (post #163457, reply #7 of 99)

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My take on this problem is that the 3rd piece of info (rise) has not been factored in to the suggested ways of fabricating the handrail. The finished handrail is a 3 dimensional piece and it is easy to visualize that it will not lie flat on the ground. So it cannot be formed by figuring a radius and setting up a 3 wheeled jig to accomplish the bend. This would only yield a 2 dimensional object.

What I visualize(I have to visualize this because I have never done anything like this - so take this for what it's worth) is another set of 3 wheels immediately following the 3 that create the "outside" radius. This second set of wheels would be fixed at 90 degrees to the first set and would impart the bend required to "climb" the stairway. Wheels 3 & 4 would have to be set very close together. Wheels 5 & 6 would present a challenge in positioning as they would encounter the handrail after it had been curved by the first set of wheels.

I would be very interested in how this is finally accomplished. Good luck.

## (post #163457, reply #8 of 99)

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A fabricator a coupla hours away does this, but says it's hard to get accurate and may need some tugging and pulling to fit. He needs stair diameter and pitch. The guys in TX who built my sprial will roll a railing, and also say it may need on-site tugging and pulling. They use custom equipment that can deal with 1 1/2" or 2" pipe.

So, I guess I didn't learn what I hoped to, but nonetheless found an answer that suffices.

Cheers,

Jim

## (post #163457, reply #9 of 99)

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CH,

"Pitch" requires two variables to calculate, the rise and "run" of the tread. No matter how you slice or dice this you need three componets to calculate the rail "twist", the

radiusto the center of the rail, and the"pitch"(riseandrun) at the center of the rail."The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it."Aristotle

## (post #163457, reply #10 of 99)

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Mr. Cloud,

Many moons ago there was a very lively and interesting discussion about this topic.

In that thread, I offered a formula to calculate the radius of curvature of a rail such as yours.

You say that the "floor diameter" at the centerline of the rail is 5' 4", or, 64". That would make the "floor radius" at the centerline = 32"

Suppose that the "total floor angle" for all of the treads is 1/4 of a circle, or 90º, and that the "total rise" for the rail is 88".

If you make the following calculations, it will give you the radius of curvature, at the centerline of the rail.

1) square the total rise

88 x 88 = 7744

2) multiply the result of step 1 by 3282.8

7744 x 3282.8 = 25422003.2

3) square the total floor angle

90 x 90 = 8100

4) multiply the result of step 3 by the floor radius

8100 x 32 = 259200

5) divide step 2 result by step 4 result

25422003.2 ÷ 259200 = 98.08

6) add the floor radius to the result of step 5. The result is the radius of curvature at the centline of the rail.

98.08 + 32 = 130.08" or 10' 10 1/16"

Be careful not to include the first riser of the stairs in your total rise. The rail begins on top of the first tread, so that's where the total rise starts.

The exact measurements for your set of stairs are undoubtedly different from the ones I used in the above example. Just substitute your values in place of the ones that I used.

Ken

## (post #163457, reply #11 of 99)

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Ken , In step 2 you multiply by 3282.8 , where does that no. come from ,or why use that number other than it makes your calculations come out ? mathamaticaly challenged Don.

## (post #163457, reply #12 of 99)

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Don,

That's the magic number and we've been waiting at least 5 months for the work on how he came up with it. You might have to wait a bit longer. . .

Thinking about this a bit more, I don't believe that formula will give the correct radius for bending this pipe. It seems to me that how they bend steel to make springs is the process we're looking for here.

"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it."Aristotle

## (post #163457, reply #13 of 99)

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Ken,

Seems like that kinda looks right for your example. 90* total floor angle with 88" rise is a very steep staircase though.

Are you going to tell us where the magic number came from or do you want to remain a mathemagician.

Mike

## (post #163457, reply #14 of 99)

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Don,

In the process of deriving the formula, I found it necessary to change an angle expressed in degree measure, into radian measure..

To achieve this, I needed to multiply the degree measure by "pi" (3.141592654) and then divide by 180.

Also, this number occured in the denominator of the fraction, so I inverted it to get it into the numerator. It also needed to be squared.

The result is (180/3.141592654)² = 3282.8

In regards to Joe Fusco's comments, I will say this.

I'm not sure how they go about bending a piece of pipe to fit the rail, or what machinery they may use to do so. I've never seen it done, nor have I ever needed to order such a rail.

All that I am saying is that the formula that I presented, will give you what is known as the radius of curvature, for a helix curve such as the centerline of the rail, with the measurements that I mentioned. The radius of curvature, is the radius of a circle that best fits the helix curve at any point along its constant path.

If Cloud Hidden were to trace out this radius on a small piece of poster board, or cardboard, and then lay it on top of the rail once it were in place, I'm confident that he will see that it fits the curvature of the rail nicely, provided that he uses the measurements that actually exist for his situation.

As far as Mr. Fusco's other comment, namely, "That's the magic number and we've been waiting at least 5 months for the work on how he came up with it. You might have to wait a bit longer. . ."

The math involved in the derivation is a bit complicated, with some Calculus involved, so although I wish to present it, I've been hesitant to do so.

Ken

## (post #163457, reply #15 of 99)

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Michael...He is a mathemjician....Used to do my site math in Lake George till his skin got to thin...Now he's a Texas mathemajician.

near the stream,

aj

## (post #163457, reply #16 of 99)

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Michael,

I more or less picked those numbers for total rise and floor angle just as an example, to demonstrate the formula. They may not be practical measurements in real life (perhaps too steep), but once again, I was just demonstating the formula with them.

Ken

## (post #163457, reply #17 of 99)

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Ken: Welcome back! This was a very interesting topic and I remember scratching my head real hard on this. We all had formulaes that were very close to each other, with each one arriving from a different direction.

## (post #163457, reply #18 of 99)

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Hi Stan,

Yes, that's true. But actually, the clever method that you used to arrive at your result was very similar to the direction that I took. I simply went one step further, by examining what happens to the curvature as the arc length under consideration approached zero.

Good to hear from you.

You to AJ.

Ken

## (post #163457, reply #19 of 99)

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Ken: To be fair, it may have been clever, but you steered me in the right direction.

## (post #163457, reply #20 of 99)

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Mr Drake,

"

The radius of curvature?""The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it."Aristotle

## (post #163457, reply #21 of 99)

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Stan,

If I find the time tomorrow, I may post a deeper explanation of the formula. Most of the math is just high school algebra, geometry, and trigonometry, but as I mentioned, I used a tiny bit of calculus in the last steps, but I think I can explain it.

Maybe our old friend Ted LaRue will stop by and take a look at it if he gets time. (For those of you who are new here, Ted is our "senior mathematician" in Breaktime)

## (post #163457, reply #22 of 99)

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Joe,

Yes, the radius of curvature. As I already mentioned, it's the radius of the circle that best fits the curve at any point along the curve.

In the case of a helix, (the curve that's involved in circular stair work), the change along the curve is constant, so there is only one radius of curvature for any given helix. In other words, there is a circle that "best fits the curve", and it fits equally well at any point along the curve.

Does that answer your question?

Ken

## (post #163457, reply #23 of 99)

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Mr Drake,

No it does not. The distance from

anypointon any plane of a helix to its plane center is equal to the radius of its plan circle.What you did was find an approximation for an applied material, mainly plywood. The only validation of this "formula" was trying to bend pieces of plywood around a form. It would stand to reason that the thickness and malleability of the plywood would play a large roll in how well it would bend.

Lastly, since no two points on a single helical path exsist on the plane, you can't describe or define that path with a plane figure like a circle.

Aristotle

## (post #163457, reply #24 of 99)

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Mr Drake,

Two graphics to help illustrate the point.

Aristotle

## (post #163457, reply #25 of 99)

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Originally I thought Mr. Drake's reply in #2 was excessive. He said the radius of the handrail should be 32" which would give it a diameter of 64" or 5' 4". They would be pretty hard to grab onto. A handrail should be about 1 1/2" in diameter or, if you want to put it that way, 3/4" radius. It's like saying that the size of your main sewer line is 2" radius or refering to 3/4" EMT at 3/8" radius EMT. In any event, a handrail 5' in diameter wouldn't leave any room to walk on and would be very heavy.

On the other hand, if you're talking about the radius from the centerline of the spiral staircase to the centerline of the handrail, I see your point. What I would do is wrap a sheet of legal-size typewriter paper around a carton of Quaker Oats cereal. Cut exactly to fit. Unroll and scale the bottom edge to your actual circumference [floorplan view]. Use the same scale to plot your total rise, whatever it is [88" ? You never said but it's important.] Then you can use Mr. Phythagorus's theorum to find the total length of your handrail.

Or scale it. I don't care.

As for bending it, you a dealing with a three dimensional curve. You need a warped pipe bender. Perhaps you can find a cheap, used 1 1/2" EMT bender and drive a Mac truck over it.

Or you could bend it into a complete circle - which would have a larger diameter - and then pull it into the spring shape with a forklift. Since the length of the rail remains constant but it is now stretched into 3 dimensions, the circumference should shrink to exactly fit the calculations. If it doesn't either the calculations, the measurements or my theory may be wrong.

Hope this helps in some way.

Near the steam.

-Peter

## (post #163457, reply #26 of 99)

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Joe,

"Lastly, since no two points on a single helical path exsist on the plane, you can't describe or define that path with a plane figure like a circle"Actually, using a simple curve to "describe" or approximate a complex curve is quite common, and the two curves don't need to be in the same plane and don't even have to intersect.

Ken is referring to the circle which "best fits" the helix. Such a circle is

notpart of a horizontal plane, but part of a "tilted" plane. The termbest fitsloosely means that if one uses an increasingly powerful microscope, the shape of the circle approaches the shape of the helix.In the real world, "lines" have thickness, and solids don't bend freely. Thus there are various ways of defining

best fit. Ken chose a definition for best fit and then developed a formula for that definition using sound mathematics. We examined Ken's formula and other formulas developed for slightly different definitions ofbest fitin the earlier thread and found some consistency in them. The plywood experiment was to validate our theories.## (post #163457, reply #27 of 99)

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Ted,

"Actually, using a simple curve to "describe" orThis is only done to illustrate a complex curve through some view/perspective or angle. No plane curve defines any complex curve, if it did the curve wouldn't be complex.approximatea complex curve is quite common, and the two curves don't need to be in the same plane and don't even have to intersect."To further illustrate this draw a helix in AutoCad and then calculate the "Radius of Curvature" and see if you can get them to line up. . . Good luck.

Aristotle

## (post #163457, reply #28 of 99)

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pm,

In the last paragraph of your post your write,

"Or you could bend it into a complete circle - which would have a larger diameter -

and then pull it into the spring shape with a forklift. Since the length of the rail remains

constant but it is now stretched into 3 dimensions, the circumference should shrink to

exactly fit the calculations"

I agree.

What I am saying, is that the radius of that "larger circle" that you refer to, according to my calculations, would be 10' 10 1/16".

If you used a forklift, to pull it into its "spring shape", as you mention, the radius of the circle on the floor would shrink to 32" directly below the centerline of the pipe, when the top of the pipe, cut to the proper length, reaches an elevation of 88". ( a steel slinky )

Whether or not this method is actually practical or not, is another matter. I'm just commenting on the mathematics involved.

Ken

## (post #163457, reply #29 of 99)

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Joe,

AutoCad sketches are very impressive visual aids, but that's all they are.. They don't prove or disprove anything. You do that with mathematics.

Think about it this way Joe. Doesn't it seem logical to you that you could take a piece of pipe, bend it into a circle, and then stretch it upward with a forklift, as pm suggested, to form a handrail???

It would be very much like a slinky toy being stretched. As you stretch the slinky, the diameter of a cylinder that could fit inside of it would decrease.

Also, as the slinky is strectched out, it becomes straighter, and not as "curved". Still, it's possible to find a circle that "best fits its shape". This circle that "best fits" the helix at any given point on it, is known as the circle of curvature, and the radius of curvature, is the radius of that circle.

What is it about that thinking that you find so hard to accept?

Ken

## (post #163457, reply #30 of 99)

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Mr Drake,

Reading that only confirms you really know very little about what makes things work. AutoCad is an absolutely mathematical tool. If it's not possible mathematically, AutoCad won't draw it. Everything AutoCad does is mathematical unlike some of the things you do. . .

Aristotle