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Here is a tip. One type of carbon rod that is readily available and rather cheap is 10 mm square tubes with an 8 mm hole. They are used for building drones and model airplanes (I think), not that hard to find on eBay. I use them instead of a truss rod to stiffen parlor necks.

Besides being cheap and available, they are surprisingly stiff for its weight. On the net I found this:

"”In a round rod with a 1 unit thick wall and a 3 unit diameter hole: The hollow shape maintains 87% of its ”strength”, with only 63% of its mass (or to put in another way, you only lose 13% of the strength, but lose 37% of the weight).”"

The drawback is that a rod with a hole will split faster than a solid one when bending. Not a problem in bedded in a guitar neck...

Another great thing is that you have two alternatives, leaving the hole empty or filling it up with a 8 mm round rod. I use birch rods as a standard and glue it with thin epoxy. Sometimes when I need more strength I use a solid carbon rod making it a solid 10 mm square rod. I love the idea to have more wood in the mix, but I can't say that I can hear it in the sound of the guitar!

Here is the carbon tube with a birch rod in the hole mounted in a parlor neck. A strip of birch is glued on top (epoxy glue) for the hide glue to bond with

The neck will still bend a little with steel strings at tension. But it's predictable and the fretboard can be sanded to a backbow to compensate.

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