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I can see in my mind's eye that there are 2 different planes of force exerted on the neck join by string tension. The first and largest is the tension that is pulling the neck into the body; like a big C-clamp between the end of the neck and tail of the guitar. The second is 90 degrees to that: The tension pulling the neck vertically, like what happens when you unglue a dovetail neck.
I would assume that if all the tension of example #1 was completely in one plane then there would be none of the tension #2. But since there is a bridge height then there has to be some of tension #2 present.
My question is: Has anyone ever figured out how much tension there is in tension #2? I am thinking it is not a whole lot. I know folks can string up a dovetail joint steel string guitar with no glue and a tight wedge fit. Ruck told me that one can easily hold a classical neck in place with your hands at full tension. In both of these instances, if the vertical tension were significant then neither would work.
I know it varies dependent on height of the strings off the bridge. But as a general measurement, is it so small as to not be a significant factor?

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It's fairly simple to work it out using trig. The angle between the strings and the line going from the nut to the top corner of the heel is about 1.3°. If string tension is about 180lb, the vertical force would be around 4.2lb. Applied at the end of the neck, they produce about 5lbft of torque.

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You've strayed into engineering jargonland, and Todd has answered your question in a free body style .
First, the tension in the strings induces compression at the neck joint, and the fact that the strings are above the neck induces a bending moment, or couple. The bending moment is small.

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A lot depends on the exact geometry, of course. Suppose the strings are 1/2" above the top edge of the box, and the tension is 100#. The compression load is then 100#. The torque as measured at the top edge (which is the tipping point) is 100#*1/2", or 50 inch/pounds. If the nut is a foot or so away from the edge, then the upward force at the nut is around 2#. The force pulling the end of the heel away from the box is greater, but still less than 20# in most cases. You can plug in more realistic numbers if you like, but this gives a feel for it.

In a similar vein, how does the deflection of the nut perpendicular to the top change with a change in the length of the neck? I know my baritone necks show a lot more relief before a truss rod adjustment then do my normal necks... it'd be nice to be able to figure out how much.

As I understand the question, it will depend on the stiffness of the neck and the tension on the strings, and will be different for every guitar.
When tension is applied, the neck bows upward in response. The amount can be calculated, but you need to know the (actual, not statistical)modulus of elasticity, the cross sectional area(which varies a bit along the length of the neck), and the tension on the strings. Long story short, the calculation is useless because the degree of uncertainty of the input parameters is too large for the result to be valid to any useful amount of accuracy.
You can assume some values and get a result, but it will only be an estimate. How about 0.03"? That's a SEG(semi educated guess) that is probably as good as you could calculate with estimated values.

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Put another way... IIRC the deflection of the end of a horizontal cantilevered beam given a weight at the free end is proportional to the cube of the length of the beam. I wonder if a neck will act the same way, so that relief will go up exponentially the longer the neck, or if there are other factors that I'm not thinking of.

I'm thinking the practical implication would be that, if indeed the nut deflection is proportional to the cube of the neck length, then since the stiffness of the neck is proportional to the cube of it's thickness, then to get the same relief with a longer neck you would increase the thickness about the same proportion as the increase in the length. In other words, if a baritone's neck length is 10% longer than a standard neck, then does it also need to be about 10% thicker to keep relief about the same? Materials, tension, etc. all being equal.

That's not exactly correct, but close enough for practical purposes. The canteliver beam analogy is not applicable (wrong loading, it's a bending moment), so the exponential variation may not be correct, but qualitatively that's correct. A longer neck needs to be stiffer to have the same relief, I doubt it's as simple as a direct relationship between length and thickness.

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Indeed.

It's large enough to require the use of truss rods and to cause necks to "sag" forward over time, but some of that sag is due to the "foundations". It's significant enough that it can't just be ignored (evidenced by the use of truss rods and the prevalence of neck re-sets).

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Peter, yours was the figure I was hoping to glean. Trig might be easy for some, but wasn't my strong suit in school.



The discussion veered off into neck deflection, which is a separate issue. And a neck re-set has nothing to do with the force #2 I was describing, unless the neck comes loose, which is rare indeed.
The reason I asked about this is I am revising my adjustable neck angle joint. I was looking for a loose idea how much force is pulling the neck vertically when up to pitch. It is probably moot since the compressive tension of the strings pushing the neck into the body is significant and effectively cancels out the small amount of tension working 90 degrees to that.

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Well, something causes the whole neck structure to rotate.... . Ruck pointed that out.
Ah, the "compressive tension"... and orthogonal forces don't cancel, I'm afraid.

Joking aside, if you wish to put up a sketch of your ideas I'll be happy (as I'm sure others will be, too) to help you work around any issues it may have, if you haven't already figured them out. The reaction torque you need to supply is not huge (as Ruck and Peter pointed out), but it still needs to be supplied by a suitable bit of structure.

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Neck resets are necessary when the string tension bellies the top and relaxes the curve in the back. There could be a bit of a hinge factor in the sides folding from the upper bout to the neck join. Flat-shouldered dreads are more likely to exhibit this as compared to slope-shouldered dreads. There is nothing in the neck joint itself that causes the need for a neck reset unless it fails.
I wasn't implying that the two forces cancel each other out. An analogy would be: clamp two pieces of wood together with a c-clamp to the equivalent of string tension, say 125 lbs, and then tie a string around the c-clamp screw and hang a weight off the string perpendicular to the clamp screw. It really would not make any difference to the function of the c-clamp holding the pieces together whether the weight you hung from the screw was 1 ounce, 1 pound, or 10 pounds.
Ruck's comments related to the fact that the string tension did the lion's work of holding the neck in place. Hand pressure was sufficient to keep the unglued neck from rotating forward (in the stress Al mentioned).

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As Roger points out, it's a bending moment (moment of inertia) which is torsion. The neck wants to rotate around the point where the neck meets the body at the top. It also wants to pull away from the body down toward the back. There are a lot of forces on a neck!

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TRein, I think you are saying that there is significant friction to prevent the neck sliding upwards since the tension in the strings is compressing the neck against the body (at least the upper portion of the neck near the the fretboard)........and the resistance of the friction exceeds the lifting force. Is that what you are saying?

Note the lower portion of the neck is actually being lifted, or rotated away from the body.

As for neck resets, couldn't they be caused by any of these factors: wood compression and/or glue creep. Any other factors contributing to a neck reset? String tension is the force that causes neck resets (and the force varies with string gauge, scale length, etc.) but if the wood didn't compress and if the glue didn't creep over time......no neck reset would ever be needed would it? Isn't body distortion a result of wood compression and/or glue creep?

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Darryl,
You got it. You may I call it friction and I call it tension, but there is much more of it pushing the neck into the body at the soundboard edge than the amount of force trying to lift the neck vertically, which is defined by the triangle formed by the bridge height and the plane of the neck surface. The force that is trying to rotate the neck forward at the end of the heel is the same force that is pulling the neck into the body. It is just moved out along the heel and the heel becomes a lever of sorts. My original post was to see if any engineering folks could quantify the amount of pull upward at the body join based on the aforementioned triangle. Peter answered it elegantly.
Yes, you could avoid a neck reset by making the guitar so rigid that it the plates did not distort. We all know that this is a great way to make a terrible sounding guitar. Wood compression/stretching is another way of saying top bellying and the curve being pulled out of the back. Cold creep in my opinion is a bit more dicey of a conjecture since most old Martins need neck resets at some point and they were put together with HHG, the best adhesive for resisting cold creep. A guitar put together with Elmer's would certainly have cold creep and Elmer's could be considered a factor in the plates distorting.

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Physics, unlike sounds, have words with very specific meanings to allow clear communication of ideas. You have to understand the jargon to understand what's being expressed in a discussion. The joke was Todd's answer refered to forces in a free body diagram...

edit:by the way, there is no vertical force on the neck joint. The bending moment is what causes the neck to rotate upward, and is equal to the distance from the strings to the pivot point(top of body-neck joint) times the tension on the strings.
Divide that number by the distance below the pivot to get the force required to resist the bending moment.

Typical numbers: 1/2" x 180lb = 90 in-lb ==> 90 in/lb / 3" = 30 lb

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