[TheForge] Spiral staircases

[Aaron Silver]

Just a simple calculation (I'm sure that there are a lot of variables that 
you could throw into this to make it a great deal harder) but for the basic 
handrail, the linear measurement would be the height the railing rises 
divided by the sin of the angle the handrail rises.

This is similar to a cardboard paper towel roll that you unroll. You'll be 
able to make a right triangle with the "handrail" the hypotenuse. Since the 
sin of the rise rate (angle) equals the opposite side (the height the 
handrail rises) divided by the length of the hypotenuse (the length of the 
handrail):

                               Opposite            Height
              sin(angle) =    ----------    =    ---------
                              Hypotenuse         HR Length

You can re-arrange things to get the length of the handrail (HR)

             Hypotenuse * sin(angle) = Opposite

                          or

             HR Length * sin (angle) = Height

Which gets re-re-arranged as:

                           Opposite
            Hypotenuse =  -----------
                           sin(angle)

                          or

                            Height
             HR Length = ------------
                          sin(angle)

The angle will of course change based on how big around the outside of the 
spiral staircase will be. The larger the diameter of the staircase, the 
less the angle could be for a given height, and the longer the handrail length.

There will be some min/maxing going on based on building codes (riser 
rise/run limitations) as well as the need to have the second level of 
stairs being high enough so that someone has enough room to stand on the 
stairs.

The circumference of the circle (outer edge) will be (2 * 3.1415 * radius).

This will be used for figuring out the distance around before the second 
row is right above the first. If you crank that in as the base of the 
triangle, and give a height, you can use the following to give you a 
minimum angle for the rise/run.

                            Opposite               Height
          tan(angle) =     ----------   =    --------------------
                            Adjacent         Base (Circumference)

                      -1( Opposite)           -1 (      Height        )
          angle = tan   ( --------)  =     tan   (--------------------)
                        ( Adjacent)              (Base (Circumference))


Don't forget to include the height of the underlayer of the steps otherwise 
you'll find yourself with that much less clearance.

Please understand that what I've just provided is based on my degree in 
Mathematics, not from real world experience. I've found tape measures like 
to bite my equations right in the butt and therefore you should always 
calculate twice, measure 4 or more times, and it helps if you're not 
cross-eyed. :-)

Aaron

At 08:59 AM 5/15/2003 -0700, you wrote:
>Thought I have yet to do one, I am interested in building spiral 
>staircases.  What is a good method of calculating the length of material 
>needed for, and the manner of bending the hand rail on these?
>
>Also, any info on constructing spiral staircases would be helpful.  Thanks.
>
>                                 Grant Marcoux
>
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