# Thread: Beam and column theory equations

1. ## Beam and column theory equations

I am trying to model the curvature of a musical instrument neck under string tension, for example, an electric guitar neck.

What I specifically need is the deflection of the neck as a function of distance from the fixed end, an equation for the curve.
I've been able to find the equation for a horizontal beam, one end fixed, and one and free. This resembles the instrument
neck in it's rest position. But in all the examples I have found, the load is always vertical, and perpendicular to the beam
The string pull on an instrument is almost parallel to the neck, though. In this case the load would be nearly horizontal.

The configurations for columns and the load directions resemble the instrument neck much more closely. But I can't find
anything related to deflection (or at least deflection as a function of distance). All of it deals with critical load for buckling.
Also the load in the case of an instrument neck would be different than what is normally considered in column theory as
it relates to buckling; the load from the strings would only cause a portion of the deflection that leads to buckling. The load
would never actually reach that point.

For the case of a horizontal beam, one end fixed, and one end free, would the same equation apply to a load vector that
was not vertical, aside from the deflection as a function of load being different? If not, do equations exist for non vertical

Or for the case of a column with the bottom end fixed and the top end free, and an eccentric load, are there equations for
deflection?

2. The strings are offset from the axis of bending, therefore the string tension and offset distance create moment at the end of the beam (neck). So use an equation for moment at the end of the beam:

Deflections and Slopes of Beams

Use equation 6. Note that as the neck deflects upward, the string tension and moment will change, but since the deflection is very small, you can neglect this and just apply string tension to offset distance.

3. Epy -

Thanks!
I ought to be able to derive an equation from that.

Is the simple form with a vertical load at the end of the beam just a special case of zero moment?

4. Assuming you're referring to a beam with a vertical point load at the end, the point load causes internal moment throughout the beam with the maximum at the fixed end. As long as the load is acting directly through the beam (not offset), there shouldn't be any *external* moment applied.

So, I'll throw a curveball at you: the neck of a guitar becomes smaller with distance, that is to say that cross-sectional area is changing and the area moment of inertia I isn't constant, so you can't use these equations. You have to use this bastard of a thing called Castigliano's method which gives me the HIV (I didn't enjoy the class that had this).

Luckily, it's not that hard in this case. Check out the wiki page: Castigliano's method - Wikipedia, the free encyclopedia. You basically have to take the equation shown in the examples section and replace I with I(l), area moment of inertia as a function of position l. M(l), moment with respect to position l, is constant in your case, (offset distance times total string tension).

Page 13 of this pdf shows a bending example with a changing cross-sectional area with a point load.

5. Yes, I did mean zero applied torque at the end.

I was wondering how things differed, depending on whether the end moment was the result of the deflection
or the cause of it. (if that makes sense)

Thanks for the additional information. I do want to figure the neck taper into the model at some point.

6. No problem.

Moment causes the bending. Like I said, that point load at the end causes moment throughout the beam internally, there's just no moment applied externally. The result would be the stress and strain in the beam/member.

7. I didn't ask the question too well, but that makes sense.

Ok, I believe I have something that will work, at least for a uniform, non-tapered neck. Will improve on later, as I would like to get the
graphic display part of the program working with a simpler model first.

First I should explain what I am doing. The end deflection, or some intermediate deflection, will be a given. So I am not actually trying to
calculate the actual deflection based on string tension. I am calculating the various deflections (for each fret on the neck) based on one
given deflection.

Using the equation for end moment, v = -Mo x² / 2EI, I should able to combine Mo, 2, E, I into a single constant k, and would now have
v = -kx². I would determine k by the given end deflection, k = -v / x². Then the other varios deflections are simply v = -kx².

So would it in fact be this simple?

8. Yes, you can do just that. It becomes much more complicated once you take the taper into consideration.