Wood Brace Deflection:
We discussed the method for calculation of brace strength in the last article Brace Strength Analysis, where we had a discussion of the structural calculation methods we use to make brace sizing decisions and adjustments.
The second factor that we should consider is the amount of deflection or bending the brace will resist. This force is called deflection and the term used to identify it is called the Moment of Inertia and represented by the letter I.
The Deflection Formula:
The formula used in calculation of deflection is as follows:
I = bd³ / 12 where:
b = the width of the brace
d = the depth of the brace
If we look at similar brace sizes as we used in the preceding article on strength analysis, our first example takes a brace of rectangular section that is 8 mm wide x 19 mm high. Plugging these numbers into to formula, we come up with the following:
I = 8 x (19 x 19 x 19) / 12 = 4,572
Again if we decide to modify the brace to 10 mm x 15 mm the calculations would change to:
I = 10 x (15 x 15 x 15) / 12 = 2,812.
This calculation tells us that the second brace has about 61% of the bending resistance as the first brace.
Let’s see what happens when we use a taller brace:
h = 22 mm w = 7 mm
I = 7 x (22 x 22 x 22) / 12 = 6,211,
which is about 35% stronger in bending than the first brace and 221% stronger than the first brace in bending.
Again, these calculations are used only to compare one rectangular brace section to another, without regard to the shape of the brace, the influence the guitar top has on the brace strength and how the remaining braces react with the brace in question.
These calculations are only intended to give you some direction as to what happens to the bending strength of a brace when we change the size and proportions of it.
We can see that by making some minor changes in the width and height of braces, dramatic structural differences take place. This is especially true with deflection. You can see that by keeping the same area of wood and making the brace thinner and taller, it makes the brace much more resistant to the bending forces that are present in the guitar top plate.
There are several popular bracing shapes you can consider when building your next guitar. Let’s look at a few of them. (They are shown in the diagram at the top of this article).
Rectangular with Rounded Top: This is possibly one of the more popular styles of brace shaping, especially with classical guitars. I have even made a special scraper tool from a metal saw blade with a half-round cut-out to shape the top of my Classical Guitar braces.
Trapezoidal: This type of brace shape has a great deal of merit in my estimation. It gives you a wide stance for gluing and tip resistance at the bottom and minimizes the area of wood by it’s shape. It also maximizes the height of the brace, by reducing the amount of wood material as it progresses toward the top. Round the top of this brace as well.
Parabolic: This is my favorite brace shape for Acoustic Guitar construction. It is glued to the top as a rectangular cross-section and you can simply hand shape it with Finger Planes. It too has a wide gluing surface at the bottom and can be slightly taller due to the elliptical shape.