Old Postcard

Estimating the Stiffness
Ukulele and Guitar Necks

The neck under consideration is a composite assembly of three components: 1) the neck itself, usually made of wood, and comprising largest proportional volume of the composite; 2) the fretboard, usually made of a wood different from the neck wood for reasons of durability or looks; 3) the insert, generally made of material many times stiffer than either the neck or fretboard, and smallest in proportional volume. as shown in the first figure:

Neck Cross-Section

The analytical model assumes an elliptical cross-section for the neck itself. The symbols are defined below both in the figure and a separate listing:

Symbols for calculations defined

The first equation to bring together the varying geometries and properties of the components is the calculation of the neutral axis of the cross-section. If a beam is bent by placing a weight in the center, the neutral axis is that point in the cross-section, above which the material is in compression and below which the material is in tension.

The location of the neutral axis (ybar) is given by:

Neutral Axis Equation

The next equation uses the value of ybar and earlier data for the calculation of EI for a given composite.

EI for Composite Neck

We can now calculate the deflection of an end-loaded beam (our instrument neck) using the equation:

90 degree defl = (P * L3) / (3 * E * I)

In this equation the tension of the strings P is assumed to be at right angles to the neck. Needless to say this would be near the ultimate case of the string action being too high, but it does give an initial deflection number. To approximate the pull of the strings at their more normal angle, the spreadsheet then calculates the sin of the small angle formed by the surface of the fretboard and the strings in their normal position, using the string length (nut to body) divided into the action (fret to string distance at the neck-body join).

sin (string-fretboard angle) = L / (String Action)

This small number is then multiplied times the 90 degree deflection to obtain a more realistic actual deflection. The resulting number is the change in action at the neck to body join. Smaller deflection numbers relative to the neck-only example mean that the neck is stiffer.

Deflection at neck/body join = sin (string-fretboard angle) * 90 degree deflection

The following Microsoft Excel spreadsheet which performs the above calculations is offered as a downloadable file.

Since most folks won't have Young's moduli at their fingertips, the following table is offered for convenience. The woods listed are those mentioned in the newsgroups rec.music.makers.guitar.acoustic and rec.music.makers.builders. More can and will be added as time permits. These are average values for these materials and meant only to be used as a starting point when using the spreadsheet. Unless otherwise noted, the wood data are for materials which are at ~12% moisture content, quartersawn and air dried. Sources of such information include "Wood Handbook: Wood as an Engineering Material" USDA Forest Products Laboratory, Agriculture Handbook # 72, (1974) stock # 0100-03200; and "Understanding Wood", R. Bruce Hoadley, (1980) Taunton Press, Newtown, CT ISBN 0-918804-05-1.

Common Name/Material Latin name/source Young's Modulus, psi Specific Gravity
White Ash Fraxinus americana 1,770,000 0.60
Yellow Birch Betula alleghaniensis 2,010,000 0.62
Butternut Juglans cinera 1,180,000 0.38
Black Cherry Prunus serotina 1,490,000 0.50
Shagbark Hickory Carya ovata 2,160,000 0.72
Hawaiian koa Acacia koa 1,350,000 0.4
Red Maple Acer rubrum 1,640,000 0.54
Honduras Mahogany Swietenia macrophylla 1,510,000 0.52
Black Locust Robinia pseudoacacia 2,050,000 0.69
Redwood Sequoia sempervirens 1,340,000 0.40
White Oak Quercus alba 1,780,000 0.68
Black Walnut Juglans nigra 1,680,000 0.55
Ebony ? 2,000,000 0.65
Graphite carbon / epoxy composite Moses, Inc. 10,000,000 ~0.9

We understand that not everyone is facile with spreadsheets but may still have an active interest in the extent to which graphite carbon/epoxy composite truss rods affect neck stiffness, as well as the effects of neck thickness and fretboard thickness and composition. With that in mind we will present case studies for different sized instruments.

Example 1: Classical guitar -- Input information- neck width, 2.0" (W=1.0); neck depth, 0.75"; neck length, 13.0"; wood type, Honduras Mahogany; fretboard depth, 0.25"; fretboard wood, Ebony; truss rod width and depth, 0.50" (m=0.25); truss rod material, graphite carbon/epoxy composite; action at 12th fret, 0.13".

Example Neutral Axis Location* Change in 12th Fret Action Percent Stiffness Change**
Neck Alone 0.31831 0.0122" N/A
Neck + Fretboard 0.15878 0.0043" 184% (2.8 times stiffer)
Neck + Insert 0.28115 0.0072" 69%(1.69 times stiffer)
Neck, Insert & Fretboard 0.19828 0.0034" 259% (3.59 times stiffer)

* Neutral axis position is measured from neck-fretboard interface; positive values towards neck. **Percent changes calculated relative to neck stiffness without insert or fretboard.

In this case, the addition of the fretboard has a considerably greater effect than the truss rod. What is not obvious in this calculation is the amount of long-term cold flow or "creep". Eventual wood movement may be 3-5 times that of the initial deformation, while the truss rod may suffer little or no movement. The true function of the truss rod in this situation may be that of promoting greater neck stability rather than immediate stiffness.

Example 2: 8-String Kawika Tenor Ukulele -- Input information- neck width, 1.5" (W=.75); neck depth, 0.5"; neck length, 8.5"; wood type, Hawaiian Koa; fretboard depth, 0.1875"; fretboard wood, Ebony; truss rod width and depth, 0.375" (m=0.1875); truss rod material, graphite carbon/epoxy composite; action at 12th fret, 0.11".

Example Neutral Axis Location* Change in 12th Fret Action Percent Stiffness Change**
Neck Alone 0.2122 0.018" N/A
Neck + Fretboard 0.086 0.0054" 233% (3.3 times stiffer)
Neck + Insert 0.197 0.0088" 100%(2.0 times stiffer)
Neck, Insert & Fretboard 0.1338 0.0037" 386% (4.86 times stiffer)

* Neutral axis position is measured from neck-fretboard interface; positive values towards neck. **Percent changes calculated relative to neck stiffness without insert or fretboard.

In this case, the addition of the truss rod has a greater effect on stiffness and will surely also be more important in terms of long-term stability. Having the spreadsheet also allows us to understand why many ukulele makers who don't concern themselves with the mechanical aspects of neck stiffness can have severe problems with neck deformation.

In the shop at the moment I have an ukulele with an obviously bowed neck made by a well-known local manufacturer. The instrument has no truss rod, the fretboard is koa rather than ebony, the neck shape is flattened and is ~7/16" thick while the fretboard is ~3/16, and there are 14 frets between nut and body making the neck length 9.25". Using the same 95# pull, the neck-only deformation at the body for this ukulele is 0.032" (compared with 0.0120 above for our neck only case) and the neck + fretboard deformation is 0.010 (compared with 0.0054 above for our neck + fretboard case). So there is 2-3 times the deformation expected for this instrument compared with ours: (1), not including longer term creep of the wood which occurred because of the absence of a truss rod; and (2), not even comparing our case where we include the truss rod.

More case studies will be presented as time allows.

For questions and comments:

Joshua H. Gordis
Naval Postgraduate School
Dept. of Mechanical Engineering Code ME/Go
Monterey, CA 93943-5146
Internet: gordis@me.nps.navy.mil

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