Curve calculation needed

[BarkityBark]

How to I determine how short I can cut the piece of copper tubing in the attached photo without it slipping off/around the corner of the metal rod please?

 

[mmm4444bot]

To answer the question mathematically, measurements are needed.

Copper tube's inner diameter

Rod diameter (is it constant?)

Rod angles (at each bend)

The maximum amount of force you expect to be applied to the tube

If I had the tools to make precise cuts, I would likely experiment. I would cut the tube shorter and shorter, until I could no longer force it around any corners.

 

[BarkityBark]

Calculation

Was hoping for a calculation to plug my numbers into. But anyway, that being said, my tubing ID is 3/8", the rod is 1/8", wall thickness of the tubing is 1/32", all dimensions remain consistent.

 

[Dr.Peterson]

Was hoping for a calculation to plug my numbers into. But anyway, that being said, my tubing ID is 3/8", the rod is 1/8", wall thickness of the tubing is 1/32", all dimensions remain consistent.

I'm assuming a perfect right angle. The other piece of information needed is the radius of the outside curve of the bend in the rod. Here is a picture of the situation that limits the length of the tube:


I won't go into the specifics of the geometry, but the length that will fit is minimum at this 45 degree orientation, so any length greater than this (4.41 as shown) will not fit around the bend. (My drawing uses units of eighths of an inch, so my diameter is 3/8 and my length is 4.41/8 = .55125"; I supposed the bend radius to be 3/16" for the drawing. The thickness of the tube and the inside radius of the bend don't matter, so they aren't shown.)

Taking L = tube length, D = tube I.D., d = rod O.D., and r = bend radius, it is not hard to determine that

\(\displaystyle D = L/2 + d\sqrt{2} - (\sqrt{2}-1)r\)

and solving for L gives

\(\displaystyle L = 2D - 2d\sqrt{2} + 2(\sqrt{2}-1)r\)

which is the formula you want. (Of course, practical matters may require increasing this a little to make sure it can't squeeze by because edges are a little rounded or something.)

For our example, with D = 3/8, d = 1/8, and r = 3/16, the formula gives

\(\displaystyle L = 2(3/8) - 2(1/8)\sqrt{2} + 2(\sqrt{2}-1)(3/16) = 0.55178\)

agreeing with my picture.

 

[BarkityBark]

Wow, thank you for taking the time with explaining this. Extremely helpful!

I'm assuming a perfect right angle. The other piece of information needed is the radius of the outside curve of the bend in the rod. Here is a picture of the situation that limits the length of the tube:
View attachment 9795

I won't go into the specifics of the geometry, but the length that will fit is minimum at this 45 degree orientation, so any length greater than this (4.41 as shown) will not fit around the bend. (My drawing uses units of eighths of an inch, so my diameter is 3/8 and my length is 4.41/8 = .55125"; I supposed the bend radius to be 3/16" for the drawing. The thickness of the tube and the inside radius of the bend don't matter, so they aren't shown.)

Taking L = tube length, D = tube I.D., d = rod O.D., and r = bend radius, it is not hard to determine that

\(\displaystyle D = L/2 + d\sqrt{2} - (\sqrt{2}-1)r\)

and solving for L gives

\(\displaystyle L = 2D - 2d\sqrt{2} + 2(\sqrt{2}-1)r\)

which is the formula you want. (Of course, practical matters may require increasing this a little to make sure it can't squeeze by because edges are a little rounded or something.)

For our example, with D = 3/8, d = 1/8, and r = 3/16, the formula gives

\(\displaystyle L = 2(3/8) - 2(1/8)\sqrt{2} + 2(\sqrt{2}-1)(3/16) = 0.55178\)

agreeing with my picture.