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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
Replies
*
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
*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.
*> 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.
*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
*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.
*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
*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.
*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
*CH,
View Image"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
*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 = 77442) multiply the result of step 1 by 3282.8 7744 x 3282.8 = 25422003.23) square the total floor angle 90 x 90 = 81004) multiply the result of step 3 by the floor radius 8100 x 32 = 2592005) divide step 2 result by step 4 result 25422003.2 ÷ 259200 = 98.086) 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
*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.
*Don,
View Image"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
*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
*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 = 3282.8In 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
*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
*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
*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.
*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
*Ken: To be fair, it may have been clever, but you steered me in the right direction.
*Mr Drake,
View Image"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
*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)
*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
* Mr Drake,
View Image"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
* Mr Drake,
View Image"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
*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
*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 not part of a horizontal plane, but part of a "tilted" plane. The term best fits loosely 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 of best fit in the earlier thread and found some consistency in them. The plywood experiment was to validate our theories.
*Ted,
View Image"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
*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
*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
*Mr Drake,
View Image"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
*Joe,I don't see any value in the last 2 AutoCad drawings that you show in this thread. I think it's pretty obvious to anyone who has worked with circular stair handrails, that a helix curve does not lie in a plane, like other curves such as circles and ellipses.Therefore, it is not possible to pass a circle through any 2 points on a helix curve, and have that circle contain all of the points in between the chosen 2 points on the helix. The concepts of radius of curvature and circle of curvature do not require a circle to pass thru more than one point of the helix. The radius of curvature is the radius of a circle that "best fits" the shape of the helix at any point on the curve. What this means is that as you look at the helix curve and the circle near their common point, that they appear to coincide in the near vicinity of the chosen point, since the circle is a very close fit for a small section of the helix. In other words, the shape of the circle, is an indication of the curvature of the helix.And once again, there is a big difference between a mathematical "tool", and a mathematical "proof".The fact that you have the ability to draw something using AutoCad does not constitute a proof. The AutoCad drawing is a visual aid to help people understand what it is that your are talking about, or attempting to prove. If you use your AutoCad abilities to draw what I am actually talking about, I think that you'll find that it CAN draw it. Show a section of a helix curve, and show a circle that has ONE point in common with it, but in the near vicinity of the point, comes very close to fitting the helix.
*Mr Drake,
View Image"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
*Joe,Regarding your statement:"I've never seen ANYTHING related to the "RADIUS of CURVATURE" Just scroll up to the top of the page to the original post by Cloud Hidden. It's addressed to you. Notice the sentence "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."So, I guess you have heard of it.
*Aaaaaack! I don't wanna get in the middle of this. Thanks Ken for the note and the information. I had read the prior discussion, but lost my way in some of the back and forth. Joe, thanks for your input, too. All help is appreciated and the guy at the steel company seems happy with what I told him.I ain't claiming no expertise so the phrase is just colloquial--I only walk up a spiral every day, not build them.Cheers,Jim
*Mr Drake,
View Image"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
*Joe,I own a copy of "A Simplified Guide to Custom Stairbuilding and Tangent Handrailing" by George R. di Cristina. I wouldn't be surprised if you do also.Although I've never built a handrail using these principles, I did read through the material, and am familiar with drawings and terms such as the ones that you posted.I don't see any conflict between those ideas and the ones that I've been talking about in some of these threads. I use the concept of "radius of curvature" to predict the shape and curvature of the rail. It can be very useful for certain aspects of circular stair work.Stan Foster, someone whose stair work is respected by all in here, uses this same concept and formula for certain aspects of his work also if I'm not mistaken. He mentioned to me that it worked perfectly for him in making rail filet for a job a few months back.
*Ken: It was real good timing and luck on my part when you started that post. I knew I had that rail fillet to cut out soon, and I was going to just trace it and do it that way. It would have come out just fine, but there is nothing like also being able to figure it out with a calculator. That post motivated me to figure out a way to compute the radius for the fillet. Had you never started it, I would more than likely still be tracing them out. The rail fillet did fit very nice and is was satisfying to be able to figure it out before hand. Its the same as all those complicated roofs that you layout and cut in your shop. Its very rare to find someone who can layout and precut all the rafters on the ground and have them work. You have proven that you are very capable of doing this day in and day out. Its been most interesting and educational to follow all your posts.
*Mr Drake,
View Image"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
*Saying I did my formula "using radians" is like saying you did your formula using inches.The formula I presented determined the radius based on the lengths of the inner and outer edges of the handrail. Ken determined the radius of curvature based on the Calculus concept of limits. Both produced very similar values for a very broad range of rise/run/floor-radius combinations.
*
View Image"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
*
View Image"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
*Joe,That's exactly what I've been trying to get through to you for about 5 months now.Now if you take the formula that you show in your last post,and express it so that degrees can be used, making it "useable" for the average guy who pounds nails for a living, you'll get the formula that I've been suggesting all along for radius of curvature, not only in this thread, but in the last one also.Radius of curvature = floor radius + ((total rise)² x 3282.8)/((total floor angle)²x(floor radius)).As I said, and have demonstrated in my posts, the floor angle is in degrees, just like the average carpenter would like it. An interesting feature of this formula is that in place of the "total rise" you could use the riser height of just one tread, as long as you change the "total floor angle" to the floor angle of just one tread.As far as that goes, you could use the rise at ANY point along the curve, as long you used the corresponding floor angle with it, since the ratio of the rise at any point along the curve to the floor angle, remains constant.So here we are, 2 threads, and about 400 posts later, and this is finally making sense to you.CongratulationsKen
*Mr Drake,
View Image "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
*Joe,Regarding your last statement "So, the only thing that has made sense to me is that you still have not posted how you came about the constant 3282.8 and have avoided all requests for such."Sometimes I wonder if you even take time to read my posts. Look back at my post #14 in this thread for an answer to your question.If you have any further questions about 3282.8, let me know, and I'll clarify it further for you.Ken
*Mr Drake,
View Image"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
*Joe,If I take the time to go through it for you, will you respect me for doing so?ken
*Mr Drake,
View Image"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
*Joe,Darn it, I can't get no respect.I'm not quite sure about what the rest of your statement means, about "having enough time to claim it as my own", but let me make a few comments regarding that statement.When I first went to work in San Antonio, on the very first day, my new boss wanted me to help him put a plywood cap on top of a raked circular wall standing alongside a set of circular stairs that were already rough framed. The purpose of the plywood cap was just to be a "nailer" for whatever was to eventually be used to finish it out. (he was not a finish carpenter)He suggested that we just rip a bunch of 12" plywood strips, nail them on one layer after another, overlapping of course, and then cut off all the excess when we were done, which is what we did. I remember him saying to me with a big grin on his face, something like, "unless you know how to rip them to fit"That's how I got involved with this radius of curvature thing. Unaware that mathematical formulas existed to help me, I set out to determine how to rip the strips to fit. Since the helix curve, like a handrail, has a constant inclination as it climbs, I felt fairly confident that the only curve that could be cut from a sheet of plywood would have to be a circular arc. So my next step was to determine the radius of that circle. I knew that it would depend on the floor radius and the pitch of the stairs, or the inclination of the helix.In the process of my derivation, which will follow in the next several posts, I used a theorem from plane geometry, that I will talk about briefly in my next post.Ken
*Here's the theorem from plane geometry that I was referring to in my last post.The theorem states"In a circle, if a diameter is drawn perpendicular to a chord, then the diameter bisects the chord, AND, the product of the segments of the diameter, is equal to the product of the segments of the chord.In the diagram below, GE is a diameter, and BC is a chord of the circle. Since the chord will always be bisected, BM = MCThe segments of the diameter are ME and MG. The equal segments of the chord are BM and MC, soME x MG = BM x MCThis is a very easy theorem to prove, but I won't go through the steps.I'll refer to this theorem, as the DIAMETER/CHORD THEOREM in future posts.
*I'd like to mention quickly that the DIAMETER/CHORD THEOREM, which I have just described, is the theorem that is used to determine the radius of a flat arch that we ofter build in rough openings.Using that theorem, you can prove that if H = the height of the arch, and if L = the rough opening, then the Radius of the circle = (L² + 4H²)/8HSee the diagram below.
* Mr Drake,
View Image"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
*Okay, back to finding the radius of curvature.In this next drawing below, I've taken the same circle that I used to describe the DIAMETER/CHORD THEOREM, and used it as the floor radius circle for a circular stair.The curve AFC, is a helix curve associated with this circle. You could think of it as the centerline of a railing, or of a steel pipe, if you like, but I'll just treat it here as a helix curve, not particularly associated with anything, except the circle which lies directly below it.The helix begins at point C on the circle ( on the floor) and winds its way upward around a cylinder to some point A. At this point A, I let the height of point A above the floor, be represented by the letter "h", as shown in the diagram below.Next, I joined the point A to the point C with a straight line, and then drew a plumb line from point M on the floor up to the line. It intersects the line from A to C at its midpoint, which I call the point "N".Likewise, I drew a line from point E, which is the midpoint of the cicular arc BEC, plumb once again, up to the helix. Its also will intersect the helix at the midpoint of the section of the helix between A and C.I called this point "F"Since MN and FE are both equal in length and parallel, the quadrilateral, EMNF, is a rectangle. (We also know that angle NME and angle FEM are right angles)It follows that since EM is perpendicular to BC, that FN is perpendicular to the line AC. Also, the length of NF must be the same as the length of ME.To simplify equations that I am about to use in the next several posts, I let the length of ME be equal to x, and therefore, the length of FN, is also x.Finally, the floor radius is represented by the letter "r" and the floor angle that corresponds to the shown section of the helix, has a measure of "a".At this point, you can think of "a" as the measure of the floor angle in degrees, or as radians, it doesn't matter.The diameter GE would bisect the angle BOC, dividing it into two equal angles, each of whose measure would be a/2 as shown.Okay, those are the preliminaries. I'm going to take a break. When I come back later, I'll look for any questions or comments before I start the derivation for the radius of curvature of the helix.
*Joe,Give it a break.The form of the equation that I showed a few posts back, to determine the radius of a circle, is just one of many ways to write that formula. It often appears in this form, because it's easier for a person that doesn't have a strong math background to work with. All of the parenthesis tend to be confusing to some.It's not a matter of how many steps, it's a matter of "user friendly"Ken
* Mr Drake,
View Image"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
*Okay Joe, I hope you're still with me.Now that my diagram is set up and labeled, I'm going to use it to write some equations. I'll repost the diagram below so you can refer to it as I develop the equations.Since the floor circle is identical to the circle that I used to demonstrate the DIAMETER/CHORD THEOREM, once again, we know,ME x MG = BM x MC (Equation 1)I will temporarily replace the measurement of ME with "x" ( for simplicity). Since the floor radius = r, then MG = 2r - xSo now, we can write equation 1 asx(2r - x) = BM x MC (Equation 2)But BM = MC, and they both are equal to r(sin(a/2))So substituting back into equation 2, we can write it as, x(2r - x) = (rsin(a/2))(rsin(a/2)), and this in turn can be simplified into the quadractic equation,x² - 2rx + r²sin²(a/2) = 0Now, if I apply the quadratic formula, to find the roots of this equation, one of the roots is extraeneous, because it would require the value of x to be greater than the floor radius, which isn't possible. The other root, however, turns out to be,x = 2r - sqrt(4r²-4r²sin²(a/2)), which in turn can be simplified intox = (2r - 2r(sqrt(1 - sin²(a/2)))/2,and finally,x = r(1 - cos(a/2)) (Equation 3)So now I have an equation for the length of that small line segment, ME, or x. That's enough for this particular post. Any comments or questions? If anyone else has questions, I'll be glad to answer them also.
* Mr Drake,
View Image "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
*Joe,I'm glad that you're following along with what I'm doing. Let me explain how I arrived at Equation 3.I'm sure you're familiar with the trig identity,sin²x + cos²x = 1You can rewrite this in an equivalent form as,cos²x = 1 - sin²xTherefore, sqrt(1 - sin²x) = sqrt(cos²x) = cosxDoes that help?Obviously, I factored out 2r from both expressions in the numerator, then canceled out the 2's.
*Joe, I'll be looking to see if my last post clarified your question about equation 3, but in the meantime, I'd like to take this "derivation" one step further.At this point, I'd like you and anyone else reading this, to use your imagination a little. Suppose you built a handrail for a set of circular stairs and laid it down on the floor. It wouldn't lie flat, as you know, because even a small section of a helix curve does not lie in a flat plane.Now, suppose you cut a small piece of the rail out of the middle and placed it on the floor. The small piece still wouldn't completely lie flat, but it would come much closer to doing so, than a large piece of the rail. It would appear to fit closer to the plane of the floor, in other words, not "stick up" as much as a larger piece. If we continued this process of cutting smaller and smaller pieces, eventually it would appear that they are lying flat on the floor.Now look at the diagram below once again. I want to focus on the small portion of the helix, AFC. Generally speaking, it's "sort of" circular, and "sort of" flat. Indeed, if we chose an even smaller section of the helix, it would tend to lie in a single plane even more, and it wouldn't be hard to imagine that "some" circle exists, that would fit very closely to the small section.By picking smaller and smaller sections, the helix will tend to fit better into a single plane, and a circle could be found with some radius, that would fit the small section very well, altho not exactly.So what I intend to do next, is treat the section of the helix shown in the diagram below, AFC, as if it actually were a piece of a circle, and AC as a chord of that circle. Even if it is not perfectly flat, or perfectly round, if I apply the DIAMETER/CHORD THEOREM once again, I should come up with a radius for a circle that fits the small section fairly well.I don't know what the length of this radius is, so I'll just call it "R". Remember, "r" is the floor radius, and "R" is the radius that I'm looking for. I'm looking for a circle whose radius is R, that "best fits" the shape of the helix. Also keep in mind that the small line segment NF, has the same length as ME, or just x, and that this small segment is perpendicular to the chord AC. So if NF = x, is part of some diameter of the unknown circle, then the other segment of the diameter would be =2R - xApplying the DIAMETER/CHORD THEOREM to this situation, I can write the equationx(2R - x) = (AN)(NC)I'll stop here a minute to see if there are any comments or questions from anyone.
* Mr Drake,
View Image "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
* Mr Drake,
View Image "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
*Joe,I ended my last post with an equation that contains the Radius R, of a circle that I would like to know.That equation is,x(2R - x) = (AN)(NC) (Equation 4)In this form, the equation is not very useful. I need to express the lengths of both AN and NC in terms of r, a, and h. Once again, look at the diagram below. In right triangle BOM, sin(a/2) = BM/r, therefore, BM = (r)sin(a/2)and in right triangle COM, CM also = (r)sin(a/2)Therefore, the length of segment BC = 2(r)sin(a/2)Now look at right triangle ABC. We know that the length of one leg is h, and we just found the length of the other leg, BC = 2(r)sin(a/2)We can now use the Pathagorean Theorem to find the length of the hypoteneuse AC AC = sqrt((2(r)sin(a/2))² + h²) = sqrt( 4r²sin²(a/2) + h²)Then half of the length of the hypoteneuse, are the lengths of AN and NC, which would be,=(1/2)sqrt(4r²sin²(a/2) + h²)Once again, using equation 4,x(2R - x) = (AN)(NC), if we substitute the above expression for AN and NC, and in addition, in place of x, we substitute r(1 - cos(a/2)), we end up with the rather ugly equation below.r(1-cos(a/2))(2R - r(1-cos(a/2)))=(1/2sqrt(4r²sin²(a/2) + h²))²In turn, this ugly equation, can be simplified down to this,R = r + h²/(8r(1 - cos(a/2))) Equation 5So now I have an equation for the Radius of the circle that I'm looking for, in terms of quantities, that I now the values of. But were still not done yet.I'll stop here once again, to see if there are any questions or comments.In my next post, I'll simply show how I simplified the "ugly equation" down to Equation 5
*Joe,In my last post, I was able to write an equation that contained R, the radius of the circle that would be "close" to fitting a section of the helix. Here's how I simplified that equation down to Equation 5
*Joe,Now I can simplify Equation 5 even further by using another trig identity,sin²x = (1 - cos2x)/2Using this identity, Equation 5 can be wrtten as,R = r + h²/(16(r)sin²a/4)which in turn, can also be written as,R = r + h²/((4sina/4)²(r)) Equation 6Now, this is where the Calculus part begins, but it's not difficult to follow.At this point, I have an expression for the radius R of a circle that should fit a small section of the helix "somewhat close", but it will fit better and better as I choose smaller and smaller pieces of the helix to work with, for reasons that I have already explained.I can bring these very small pieces of the helix into consideration, by simply making the floor angle smaller and smaller and smaller. As I continue this process, the Radius will get closer and closer to one definite value. Another way to say this is, As the floor angle APPROACHES zero,the value of R approaches some limiting value. So how do I find what that limit is?See my next post
*Ken: Thanks for all the effort you are putting into this post. It is most interesting watching the proof that you are laying out. You have my respect.
*One of the limits that you encounter early on in the study of Calculus, is the following,If A is an angle in radians, thenThe Limit, as A Approaches zero of sinA/A = 1.In other words, if A is expressed in radian measure, as you approach zero radians, the value of sinA approaches the value of ARadians are just another way to describe the measure of an angle, for those of you who may be following along. Look at the diagram below. The radius of the circle shown is 8. There is a certain angle that when drawn from the center of the circle, "cuts off" a piece of the circumference of the circle exactly equal to the radius. The measure of that angle is said to be one radian.Since the circumference of a circle is 2(pi)r, this means that there are 2(pi) radii in the circumference a circle. It follows, that there are 2(pi) radians in a circle also.Since there are 360º in a circle, then2(pi)radians = 360º,so, one radian = 360º/2(pi) = about 57.3º (pi is equal to about 3.1416)You can change radians to degrees, by multiplying by180/pi, and you can change degrees to radians by multiplying by pi/180. Now look back at Equation 6 in the last post. I want to know what the limiting value of R is, as the floor angle, a, approaches zero. In the equation is the expression,4sin(a/4) Given what I've shown at the top of this post, we know that,The Limit, as a approaches zero, of (sin(a/4))/(a/4),=1, where a is in radians. This can also be written asThe Limit, as a approaches zero, of (4sin(a/4))/a =1And it implies, that as we let a approach zero, that the value of 4sin(a/4) appoaches "a", as a Limit.So, in Equation 6, if I replace 4sin(a/4) by a, the value of R, as a approaches zero, isR = r + h²/((a²)r)
*In my last post, I finished with this formula:R = r + h²/((a²)r)R is what is known as the "Radius of Curvature" of the helix. It is the radius of a circle, that "best fits" the curvature of the helix at any given point along its curve.r = the floor radiush = the height of the helix, between any 2 points along its patha = the floor angle corresponding to the same 2 points, measured in radians.You could use the height of one riser along with the floor angle of one tread, or you could use the total height along with the total floor angle, or for that matter, you could use the height difference between any 2 points, as long as you use the floor angle corresponding to those 2 points.To make the formula more useful to the average guy working in construction, I wanted to eliminate radians completely from the formula. As I explained in my last post, you can convert degrees to radians, by multiplying the number of degrees by pi, and dividing by 180.Now suppose a stair framer wants to find the radius of curvature, and he finds the floor angle he wants to work with in degrees. If he uses it in the formula in degrees, it won't work, because it needs to be in radians.However, if we multiply the number of degrees by pi, and then divide by 180, and then square that number and invert it so it can appear in the numerator of the fraction, we would get(180/pi)² = (180/3.141592654)² = 3282.8So if the floor angle is found in degrees, the formula can be written,Radius of curvature == floor radius + (rise)²(3282.8)/((floor angle in degrees)²(floor radius) For example, if the floor radius = 32" at the center of a handrail, and we know that the floor angle is 90² as the rail rises 88", then the Radius of Curvature at the center of the rail would be,R = 32 + (88² x 3282.8)/((90)² x 32) = 10' 10 1/16"
*That's about it Joe,Did I answer your question from your post#45?Ken
*Ken,Thanks for taking the time and effort. Same to all the other posters. What I like is seeing all the variety of approaches to solving a problem.Jim
*Jim,And thank you for taking the time to post that.I enjoy that topic, and was happy to find the time to show that information.Ken
*Mr. Drake,
View Image"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
*Joe,The formula that you show in your last post is equivalent to mine. The form that I have it written in is just "slightly" more accurate, because I rounded 3282.8 off to the nearest tenth after I squared it. In the form you show it, you're rounding first, then squaring after, which will lead a to slightly less accurate end result.In practice, it really wouldn't matter however, because we don't normally work to that level of accuracy in the field of construction.Whether it is more, or less, user friendly, I guess would depend on who the user was. But I have no problem with you presenting it in that form. As always, just as with the formula for finding the radius for a flat arch in a rough opening, (which we discussed earlier in this thread), there are always a variety of ways to write any formula. Ken
*Mr. Drake,
View Image "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
*Mr. Drake,
View Image "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
*Joe,Let's go over your numbers.The formula that you show in post #73 which uses the rounded number of 57.3 results in an answer of 130.09336. Mine, using 3282.8, results in 130.07872The "most accurate", if we used an extremely large number of decimal places to approximate pi, would be130.07891All of these results are rounded to 5 decimal places.The difference between yours and the "most accurate" is .01445The difference between mine and the "most accurate" is .00019As I pointed out, because I rounded AFTER squaring, my result would be slighty more accurate than yours, which it is.Notice that, in my formula, because I rounded 3282.806 slightly DOWN, and that you rounded 57.296 slightly UP to get 57.3 in the formula that you show, that your formula will always slightly overestimate the Radius of Curvature, whereas mine, will always slightly underestimate it. But as I mentioned earlier, in practical field work, that degree of accuracy is not needed.As far as your statement about "your original", I have no idea what you are talking about. What exactly is "your original"? If it's some different formula, please show it. Ken
*Mr. Drake,
View Image "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
*What an amazing thread! I have been building stairs for over 35 years and can say at this point in time nothing in the trade baffels me. However the mind numbing formulas and graphics presented here have me confused, if you guys can actually build something using this information my hats off to you. As for myself I'm going to finish my beer, go down to the shop and bend some wood.My wife read the posts (she works with me) and said "Gee's a woman can make a baby in less timethan it takes these guys to figure out what to do next!"
*Armin,
View Image "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
*Armin: I can guarantee Ken and Joe can build very complex projects. I find it very interesting following the intense topics that are posted here from time to time.
*Stan,
View Image"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
*Stan,
View Image"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
*Joe,If we're going to have a competition for who can state Ken's formula in the most accurate way, then I'll just present it using an expression with the symbol for PI instead of typing some numeral. That way I'll have zero round-off error. :)Lighten up. Stating Ken's formula six different ways means nothing to a mathematician, and probably means nothing to a builder. We're interested in the merits of different formulas, and like to see how the formulas were discovered.
*
View Image"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
*
View Image"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
*Joe: Kens picture in post #64 best describes the way I arrived at the answer. I used the chord height formulae to find the chord height of the floor radius, then figured the new inclined radius by plugging in the chord height just computed for the floor radius and using the quadratic formulae to compute the new radius. I had all the numbers posted and I cant get into the archives to retrieve them. Right now my son is having some severe health problems and I just am not in the frame of mind to refigure, please understand. Also, I know nothing about how to produce a spreadsheet, so when I figure something, it takes me awhile.
*Stan,
*Joe: Thanks, its been rough for all around here. I just am getting on here to try to get my mind back. I can hardly quote the formulae for the circumference of a circle right now.
*Sorry, Joe..I wasn't trying to upset you....I was just suggesting that we concentrate on providing content that the general public can use rather than competing for some undefined "accuracy" goal.You think the formula I presented was "a dog"? I hope you were just "lashing out". I'd be wounded if I thought you really meant that. What did you think was wrong with my formula?
*Ken...Very nice sequence of posts on the development of your formula.The graphics were very clear and helped with your explanations. How do you find the time? In one post you expressed doubts about the rigor of you math. It looked rigorous to me, and my students tell me I'm pretty "picky". Nice blend of rigor and "layman's" explanation. I'm making a word processing file containing all of your posts relating to the developement of the formula including the graphics. I'll mail it to you when I'm finished. Sorry I didn't post sooner, but the Web hasn't quite yet consumed all of my life... :)
*Boy,
View Image"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
*I officially declare this thread Fuscoinized, and am invoking Article 5.
*the motion is seconded,near the scream,ajMath fun....bullshit...not so fun.
*Mr. Drake,
View Image"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
*Joe....Your graphics abilities are super....Fo get about the personal griping.If all of you worked together, ya could build spaceshuttles.near the stream but off to work soon,aj
*One,
View Image"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
*Joe,I really don't feel it necessary to respond to your remark in post #95, but I would like to say the following regarding this topic in general.About 10 months ago, when I first posted the topic of this thread, I already had derived the formula that we are talking about, some 6 years earlier. I also had opportunities to apply the formula in rough framing a plywood cap for several sets of circular stairs, with excellent results. If you look back through my derivation, you can see that the end conclusion of it is a formula, with the floor angle in radians, for the Radius of Curvature of a helix. The formula, in that form, is exact, but as I mentioned, is not user friendly. How many carpenters know what a radian is Joe?To help others who might want to try to use the formula in certain types of circular rail work, I alterered the formula so that it could be used with degrees instead. Once again, that's the reason for the 3282.8 constant in the formula.That constant, results in an extremely small error in the result, as I rounded the constant slightly. You'd need a finely tuned micrometer to measure the error.That's really all that needs to be said Joe.I showed the formula, and, as YOU REQUESTED, I showed, in some detail, how I arrived at the formula. I also explained the 3282.8 constant in the formula, as you also requested.So stop spinning your wheels. You're just going around in circles. The information, and the formulas are clear, and I can assure you, accurate.Ken
*
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
Spiral stair handrail bending
You need three pieces of information to convey how you need your handrail bent
1.) Diameter eg. 6'
2.) Riser height eg. 8"
3.) Treads per circle eg. 12
Once you have these measurments, your spiral stair handrail can be accuratly bent. For a curved stair railing, there will be two railings, inner and outer, but the same principle holds true.
Here is a video of us bending a 6' spiral handrail on our adjustable bender
http://www.youtube.com/watch?v=DCGU1CgE2rQ
If you have any questions about bending spiral or curved handrails, feel free to e-mail me at [email protected]
curved stair rails
Hello all; Have read a lot of the posts here,and i would like to contribute this. I am a metal fabricator and welder,i build a lot of my rolling and bending equipment myself includeing design,machineing,and welding.
In my buisness i design and build gates deck railings and superstructers for all types of buildings.Stair railings can become dificult,especialy the curved type.
Here is what i call Okie fabrication for curved stair railings. First measure the total length of railing,next total drop from top floor level to bottom landing,next straight line from begining edge at top to bottom last step,then mesure ofset from actual straight to curved finished point,if wall or other obstruction is in line of site then use laser and do in two or more stages and add reseults.
Now you can use angle finder to get degree of cut on ends of tubeing and also ends of posts next do a layout on fabrication table or concrete slab then start rolling tubeing until you match layout from above measurments.Check for fit.
OK, figure out the math of this one: http://www.bbc.com/travel/slideshow/20120314-the-worlds-coolest-staircases