Calculating the forces on a guitar neck (caution: mathematics ahead)

In idle moments I had worried that the string tension on my project guitar would be too much for the glue in the neck joint. To put my mind at rest I sat down with paper and pencil and put my Applied Maths qualification to use. I just knew it would come in handy eventually.

There are two main forces in play here: the tension of the strings trying to pull the neck out of the pocket and the adhesion of the glue keeping the neck in the pocket. Since the joint is in equilibrium (i.e. not breaking) they must be matching each other.

Here’s a simplified diagram to illustrate the forces.

We can calculate the tension in the strings using the calculator at McDonald Strings. With a set of 0.013s on (the heaviest gauge I’m ever likely to use), and a wound G string, the total tension is 73.9kg. At the most these will be 12mm above the pivot point which equates to a turning force of approx 8.7Nm.

To stop that turning force, the adhesive must be capable of exactly matching that. If we assume it applies the force at the heel of the neck, this is 90mm from the pivot, so it has got to be able to withstand a 9.85kg pull. If, more accurately (but still a simplification) we say that this force is applied across the neck pocket, we can centre it at 45mm from the pivot, it needs to fight against a pull of 19.7kg.

Head over to the spec page for Titebond original and we can see that, in ideal conditions it can withstand 3,600 psi, or converting to SI, 2.53 kg/mm^2. The neck pocket is 90mm by 54mm so therefore, if the wood could keep up, the glue could theoretically hold close to 12.3 metric tonnes! And that is ignoring the additional glue areas at the side of the neck pocket. Even if the glue joint was formed in the least opportune conditions it can hold 1,200 psi, therefore across the neck pocket, about 4 tonnes. Well within operating tolerances!

I’d often heard people say that the glue is stronger than the wood to which it attaches, and I sort of believed it, but I never realised to what extent.

BTW: The last time I tried to use my applied maths was in 1981, so there is a more than even chance that I’ve misplaced a decimal point or two in my calculations, but I am heartened that even if I’m out by a factor of 600 this guitar is still going to stay in one piece.

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