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I've built the deflection testing jig that Brian Burns wrote about in a Tonewood Testing PDf I have, that was published in 2010.

Eveything is making great sense, except my Young's Modulus / Modulus of Elasticty calculations don't seem accurate based on the averages he shares in the document. I've tested five sets of Spruce tops, and clearly something is wrong with my equation.

In the pdf he has the equation written:

E= (.25 x Load x Span) / (Deflection x Width x Depth )

For example

Top #4
Thickness: .170"
Width: 8.267"
Length: 21.732"
Weight: 6.75 oz

Long Grain:

Span: 18"
Load: 2.436 lbs
Deflection: 0.058"

Short Grain:

Span: 6 in
Load: 4.25 lbs
Deflection: 0.037"

I calculate a Density of 23.87 lbs/cubic ft, and a Specific Gravity of 0.38

For the Youngs Modulus of long grain, I get a number that doesn't seem reasonable based on the fact that Burns has seen a range of 2.45 - .97 over 200 tops.

E = ( .25 x 2.436 x 18 ) / ( 0.058 x 8.267 x .170 )
E= 134.48

Am I misunderstanding the equation? Doing my math incorrectly?

Any thoughts greatly appreciated

What units does his equation use?

Most equations are in SI - meters, Newtons, kg

_________________
I'm not a luthier.
I'm just a guy who builds guitars in his basement.
It's better than playing golf.

Check your equation....I think it's...

E=(Load x Length)/(Area x delta Length)........

you show 0.25 constant? good luck..my $.02

Brian Burns was kind enough to follow up to a few of my questions, and I've sorted it out.

He noted that .25 is a constant for center loaded beams.
I had the equation wrong, its (((0.25)×((Load×Span^3))÷(Deflection×Width×Depth^3))÷10^6)

Which helps, as now I'm getting E for long grain, for the 5 tops, as:
2.19
1.51
1.76
1.88
1.98

And cross grain, for the 5 tops, as:
0.06
0.06
0.18
0.08
0.12


E Long /Specific Gravity:
4.66
3.97
4.00
4.27
3.54

Cross Grain E / Sp. Gr.:
0.13
0.16
0.41
0.18
0.21

Not yet sure yet what I'll do with these numbers, but they are at least in an expected range.

Here's a copy I found online of exactly the same table I have in my strength of materials textbook... The equations you've posted don't look right:



One of the easier to explain uses for the modulus is designing/optimizing tops and braces (depending on how much of a background in engineering/physics you have or want to learn.) Modulus can be thought of as a measure of stiffness (the degree to which a material will resist flexure when stressed.) Among other things, you have a true empirical strength to weight ratio for each piece now. These data can be used to tune the thickness of the top to get a obtain a consistent response while safely doing so. As you can see from the beam bending formulae, all else being equal the resistance to flexure is a cubic function of thickness... Without first knowing the modulus if you were to make every top the same thickness, some could fail while others would have terrible response and tone.

I'm not personally a big fan of calculating every last element in a design even though I have the tools and education to do so. At the very most I personally suggest calculating the top thickness plus 15% to 20% as a design safety factor as well as a few of a the most critical braces. For example, the two members of an X-bracing system, or the cross member of a classical guitar that uses a fan bracing pattern. it gives you improved consistency and a good starting point for any build.

One last thing: The "long grain" measurements are much more useful than "cross grain", simply because that's where most of the load bearing takes place. If you're plugging everything into Finite Element Analysis software it will make a slight difference.

Hugh, What value are you using for allowable stress?
Jim

_________________
Jim Watts
http://jameswattsguitars.com

Substituting I = (1/12) * (width * depth^3) into the 4th case (simply supported, center load), gives Brian's equation: (((0.25)×((Load×Span^3))÷(Deflection×Width×Depth^3))÷10^6)



Yes, modulus is a material's stiffness. But remember that stiffness and strength are different concepts:
- Stiffness (or modulus) refers to how much load it takes to elastically deform an object (or material) by a given amount. "Elastic" means that after the load is removed, it completely recovers to its original shape and length.
- Strength refers to how much load it takes to permanently deforms an object (or material). "Permanent" means that after the load is removed, it takes a set, yields, fractures, cracks, or breaks.

For example:
A bungee cord has low stiffness, but relatively decent strength.
A glass rod has fairly high stiffness, but relatively low strength.

There is another important (and confusingly similar) measure for guitar wood: the creep or viscoelastic behavior over time. After detensioning strings, a newer guitar should not show any permanent deformation. But that same guitar with 50 years of string tension may have a permanently sunken top. That long-term creep behavior is different than strength, although the net result is similar.

For guitars, we design the top and braces to meet various targets related to stiffness (deflections, bridge rotation, resonant frequencies). If we do those in the normal range, the guitar should have plenty of strength, although maybe not resistance to creep. So we wouldn't normally need to know the "allowable stress" (a safe stress for strength), although that would be a useful measure in a sophisticated FEA model to predict creep. (The FEA models I've run show that the main 'hot spots' of high stress are in the scalloped X-brace and top just in front of the bridge.)


Yes, long grain stiffness is the main measure for resisting string tension. But many luthiers think that high cross-grain stiffness is more important for acoustics.

_________________
David Malicky

David wrote: "So we wouldn't normally need to know the "allowable stress" (a safe stress for strength), although that would be a useful measure in a sophisticated FEA model to predict creep"

This is exactly why I was wondering what he was using as an allowable stress. Hugh stated a factor of safety of 15-20% so I figured he must have some value in mind.
Jim

_________________
Jim Watts
http://jameswattsguitars.com