In this article we will be making an analysis of the strength of wood and how it relates to brace dimensions and structural integrity. We will be using some structural engineering formulas to some degree, so it may come a bit confusing, but hang in there.
Since my background is in architectural and structural engineering, I thought I would find out just what factors affect the strength of guitar braces since I would do this on a daily basis with buildings.
There Are Four Structural Factors to Consider:
While there are many formulas that are used in the calculation of wood structural members (and that is exactly what braces are), there are 4 main strength factors that we need to be working with for our bracing.
Section Modulus (S):
The first is called the Section Modulus and it is designated as S
This factor relates directly to the strength that a particular section of wood exhibits and is what we are examining in this article.
Moment of Inertia (I):
The second is call the Moment of Inertia and is designated as I
This factor relates to the amount of deflection a member will exhibit. In a guitar top, that would be both upward crown (or deflection) and cupping in front of the bridge (compression). We will examine that in the article – Wood Brace Deflection.
Extreme Fiber Stress (Fb):
This third factor is a number given to us from the ASTM standards for structural lumber and is designated as Fb. It varies wildly with wood species and wood grading. We will examine that in the article – Wood Brace Species Selection.
Modulus of Elasticity (E):
This fourth factor is a number also given to us from the ASTM standards for structural lumber and is designated as E. It also varies wildly with wood species and wood grading. We will examine that in the article – Wood Brace Species Selection.
Let’s assume we are working with a brace that is 8mm wide x 19mm high , which has a cross-sectional area of 152²mm To calculate the strength of that member:
Our formula for calculation of the Section Modulus of a rectangular section is this:
S = bd²/6
b = the width of the brace
d = the depth of the brace
S = Section Modulus in cubic mm
S = 8 x (19 x 19) / 6 = 481³ mm
Now let’s say you would like to modify the brace to a degree – say make it a bit shorter and wider, of the dimensions 10mm wide x 15mm high, which has a cross-sectional area = 150²mm
Applying this to the formula we get:
S = 10 x (15 x 15) / 6 = 375³ mm
See how much difference minor dimensional changes make in the strength of wood? Buy keeping the basic cross-sectional area of the brace the same and modifying the height and width you can see that the first brace has 28% more strength than the second brace (481 / 375 = 1.28)
Just for the fun of it, lets make it narrower and higher, say 7mm wide x 22 mm high. This has a cross-sectional area of 7 x 22 = 154² mm:
S = 7 x (22 x 22) / 6 = 564³ mm. OK this brace is 17% stronger than the first brace and 50% stronger than the second.
You can use this formula to make general decisions about how much to scallop braces, how to modify them and what effects that has on the brace. Now for a couple of qualifiers.
First: We do not deal with strictly rectangular members. Braces are often sculpted with a curved top and an elliptical cross-section, which drastically modifies our calculations, with a formula that is longer than this article. We won’t go into that. Even if your braces are elliptical in shape, you can use the formula to compare one size vs. the other and make some decisions based on your calcs.
Second: The Section Modulus is the given strength for the cross section of a member only at the given point. If other words, as a member tapers, and bracing usually does, the section modulus will diminish as the brace gets small in cross-section. This is further complicated by the stresses placed on the brace.
The primary stress is placed on the “X” braces of the acoustic guitar, and it comes from the bridge. There are 2 forces at work here. The bridge puts tension on the braces, on the back of the bridge from string force and compression on the braces on the front of the bridge from the same force. So in theory, the greatest amount of Section Modulus required will be at that point, which, of course makes sense. The diagram at the beginning of this article indicates these stresses.
Third: As bracing is scalloped, it reduces the Section Modulus at the point of the minimum brace height. (refer to the diagram at the top of this article). This means that you are taking a lot of strength out of the brace. How much? Let’s take a look, by using the first brace of 8mm x 19mm for our example.
The brace height at the scallop at the lowest part of the scallop is 10mm. That would make the brace dimension 8mm x 10mm
S = bd² / 6
S = 8 x (10 x 10) / 6 = 133³ mm
133 / 481 = .27
This means by lowering the brace by 9mm with the scallop, we have reduced it’s strength by 73% at that point. That is why I say be careful how and how much you scallop.
I hope you aren’t too bored with this discussion. We didn’t get into structural analysis too deeply and if you understand the basic concepts presented here it will give you some insight as to what happens if you change brace height, so you aren’t blindly forging ahead.
It will allow you to make subtle changes. based on successful guitar results and allow you to experiment with making braces slightly higher and thinner to minimize the amount of wood for tone production.