Will Pipe Fit In the Open-Web of a Joist? [Tool]

Have you specified or encountered a specification that asks for the pipe to be "as high as possible" in areas with exposed structure?

If so, does that mean we want to pipe through the open web of a structural joist?

THEORY VERSUS REAL-WORLD
This might be the most classic design versus real-world installation conundrum. Just because something might be possible doesn't necessarily mean it will fit. 

Well, for some years now, I've asked people I respect how they determine whether a pipe can go into the joist.

SHOULD WE ROUTE IN OPEN WEB JOISTS?
We might first want to ask whether we should put the pipe in the open web joist, to begin with.

If the joists are shallow, not going to be aligned, or will they be interrupted by solid beams and the end of each bay? In those cases, then the pipe really shouldn't be up there anyway.

But, assuming we do have some depth to open-web joists, and the joists will be aligned (giving us an open and continuous path to hang the pipe), we still need to know if the pipe will fit.

LENGTH OF PIPE THAT WILL FIT
The answer from an novice consultant might be - well of course it'll fit. Just cut the length of the pipe down so that it'll fit up there.

But where do we draw that line? If we have hundreds of feet of pipe run in an open-structure area, it's going to be a labor and materials nightmare if we have to use 6-ft long sticks of pipe the whole way down. Additional fittings, additional hangers (if we want a hanger on each stick of pipe), additional labor... major cost impact.

If we can use cut lengths of 10'-6" (half of a full-length 21-ft stick of pipe), then maybe that cost impact isn't as bad.

CALCULATED APPROACHES
In asking around, I've found three different calculated methods of determining whether a pipe will fit (mathematically) to slip up and into open web joists.

Those three methods, as I can best identify, is a calculated simple method using exponential relationships of the joist depth and gap-between joists (I called it the Simplified Formula, please inform me of a source if you know it). This is the second calculation.

The third was originally credited to AFSA's Ed Miller from the 1990's, which I've identified third in the list and seems to generally be the most-conservative of the three calculated concepts.

And the main concept is a purely diagrammatical calculated approach based on the visual. The concept is that the slope of the pipe just as it slips past the joist on the right is calculated, the rise of the pipe is calculated and compared against the available open height in the space (can the height of the left-end of the pipe fit underneath an upper-chord?).

SKETCHED APPROACH
Of course, we can always draft or model up an example and see it for ourselves, but my hope in creating this tool is to shed some light on the practicality of putting pipe up into the joists and help see that relationship come together.

​Below is the tool:

TOOLKIT
If you like tools like this - you should check out our Toolkit and MeyerFire University (which includes the Toolkit). Plenty more practical tools for everyday use for the fire protection professional.

YOUR TAKE
Where do you land on this?

Have you used any of these methods before, or do you have your own?

Do you know where these originated, and if so, point me in the right direction so I can credit the right source?

​Comment below - would love to know your thoughts on the topic and where you see something like this helping.