Let me see if I have this right. Thinking of carbon fiber members like I might spruce or steel is problematic?

From : Don Stackhouse

Actually, in this case, no. When you get into the finer points of optimizing structural concepts, there are important differences between composites and conventional materials. However,...

Where I might logically install a thin strip of metal on edge to maximize stiffness, you're saying the stiffness available from carbon fiber is based on it's resistance to being stretched or compressed...

so some kind of I-beam construction with the top and bottom surfaces of carbon fiber would be best..

Having just ordered a couple carbon fiber truss rods from Stewart MacDonald - http://www.stewmac.com/shop/Truss_rods/Carbon_Fiber_Neck_Rods/Carbon_Fiber.html they sell them .125 x .375 or .200 x .250 and in all but one of the instructional materials show them laminated or routed into a neck vertically. I assume that these bars would be stiffer vertically than they will on the flat... yes? But not as stiff as some kind of I-beam construction that you describe.

Might I maximize stiffness and minimize weight, if I were to make a fingerboard core, out of, say, a honeycomb of spruce, with a thin top and bottom layers of Carbon Fiber- almost a hollow core door arrangement?

All of these elements of the structure are doing work, so all of them have to be sized properly to do their jobs most effectively. If the spruce in the middle is too "honeycombed", it may not be able to carry the shear loads effectively, or to support the carbon "spar caps" from buckling. If the carbon spar caps are too thin, then some of the stiffness of the spruce is being wasted.

If both the spar caps and the core material are balanced in strength and stiffness with each other (for optimum structural efficiency, ideally the carbon spar caps would break at the same load that causes the spruce core to fail), but both are not strong enough, then you will have a neck that is astonishingly light but that is more flexible than the conventional one you're trying to replace. OTOH, if you make them too strong, your new design will bring a breathtakingly new meaning to the term "stiff neck", but might actually be heavier than the one you're trying to replace.

The interfaces between them are also critical. As the loads and stresses "flow" through the structure, any disruption in that flow will hurt the structure's ability to efficiently carry those loads and stresses. If you don't have the faces bonded to the core properly, then the structure will fail at that bond point, probably either in shear or as a buckling failure of the compressively loaded strip.

Gary Gallier asks:
Don....I am interested in this. If I am understanding the theory correctly here, the key element of it is that there are TWO pieces of carbon fiber on the outside surfaces. If we are talking about just one piece of carbon fiber, say under the fretboard cap, then this would not add the stiffness effect as well as the same piece of carbon sandwiched on edge down the middle of the fret board.

mind is telling me that it requires both pieces of the carbon graphite if you are going to lay them flat, achieving the 'box' effect you mentioned. Otherwise I'm picturing that a single flat piece would just flap with the whims of the wood. Correct or not ?

FWIW..I am currently inlaying a 1/4" x 3/8" graphite rod under my fretboard, down the middle on edge, 18" long. Ebony fretboard on top of it.

If you simply have a column, such a pedestal holding up a statue, the entire cross section of the column at any location along it sees a fairly uniform compression load. Likewise, a string on one of our instruments sees a reasonably uniform tension load over its cross section.

Bending, the load we're considering here, is different. Consider a staff for one of our dulcimers. The strings run above the staff, not through the center of it, so their tension, offset from the center of the staff's cross-section like that, tries to bend the staff ends upwards relative to the middle. Thus the staff sees a compression load equal to the tension load in the strings, but it also sees a bending load equal to the string tension times the distance from the staff's "neutral axis" to the strings.

Understanding that term "neutral axis", exactly what it is, and exactly how it interacts with the structural properties of the staff and the loads imposed on it, is the key to understanding this entire problem.

For the moment let's ignore the compressive stresses on the neck that oppose the tension in the strings. Yes, they're important as far as strength is concerned, but the bending load is what controls the tendency of the staff to warp, and is also the dominant player by far in the way the staff flexes and vibrates in response to the vibrations in the strings.

If you impose a bending load on virtually any sort of beam, the side of the beam that the ends are being bent towards see a compression load, while the other side of the beam sees a tension load. In this case, the fingerboard face is in compression, and the bottom face is in tension. The tension load may also be shared by the soundboard, depending on the way the staff and soundboard are connected.

If there is tension on one face of the beam and compression on the other face, then obviously there must be some location in the middle of the beam's cross-section where the stress is zero. If you are going from a condition of compression to a condition of tension, at some point you have to pass through zero. This place within the cross-section of the beam, where the tension and compression stresses are exactly zero, is the "neutral axis". If the beam is a uniform, homogenous piece of material with a symmetrical cross-section, the neutral axis will be exactly in the middle. For example, if we had a staff made from a solid piece of spruce exactly rectangular in cross section and exactly 1" thick, the neutral axis would be exactly in the middle, precisely 1/2" from each face.

Now for something really important. For any given fiber in that piece of wood, the amount of stretching that it sees in response to its share of the bending load depends on how far it is from the neutral axis. For our nice, uniform, 1" thick spruce beam, the outer faces are each 1/2" from the neutral axis. That means that a fiber 1/4" from the neutral axis will see only half as much stretch as a fiber on the outer face. That also means its share of the bending load is only half as much as a fiber located on the outside face, 1/2" from the center. That interior fiber is only working half as hard, and therefore is contributing half of its potential stiffness to the overall bending stiffness of the beam.

This is precisely what's also wrong with using a vertical strip of metal or carbon as a bending stiffener. The material in the center of the strip lies on the neutral axis, is therefore not carrying any tension or compression loads, and so is contributing almost nothing to the bending stiffness. The material in the edges of the strip, farthest from the neutral axis, do most of the job, while the material in the center of the strip is just dead weight. You have about 10% of the material doing the vast majority of the work, and the rest of the material is just freeloading.

So, the obvious conclusion is that we should cut the middle out of the beam, and just keep the thin strips that form the top and bottom faces, right? Well, not quite. In order to carry the bending load as a tension load on one strip and a compression load on the other, we have to connect them together. This connecting stress is called "shear". We do need to have material (referred to in engineering jargon as a "shear web") between the top and bottom strips (referred to in typical aircraft design jargon as "spar caps"). However, the amount of material needed in the shear web is relatively small. We can take our rectangular cross-section staff and carve away most of the material between the top and bottom faces, leaving a thin strip running vertically between them. Recognize that resulting shape? Yup, that's the familiar "I" beam cross-section. It's nothing more than a rectangular bar that's had all the lazy, ineffective material in the middle cut away. The result has nearly all of the tension and compression carried in the top and bottom caps, with just barely enough material left in the shear web in the middle to carry the shear loads. All of the material is working near its physical limits carrying either tension, compression, or shear, so there is virtually no wasted dead weight.

Now, what happens if we slice off one of the caps and replace it with a strip of carbon the same size? At this point we need to introduce the idea of materials with different stiffnesses, and how to measure that inherent stiffness. The engineering term typically used for this is the material's "elastic modulus", also called "Young's modulus". For example, steel has a Young's modulus of 30 million psi ("pounds per square inch"). This means that if you had a one inch by one inch square steel bar, and you applied a one pound tension load to it, the bar would have to be a total of 30 million inches long for it to stretch a total of one additional inch in length. Or, you could say that each inch of the bar would change its length by one thirty millionth of an inch.

The elastic modulus of carbon fiber/epoxy is only 21.6 million psi, only a bit more than 2/3 that of steel. However, the density of carbon fiber/epoxy is only a bit more than one-fifth that of steel (.06 pounds per cubic inch for carbon, and .283 pounds per cubic inch for steel). Therefore, a piece of carbon fiber with the same stiffness as a piece of steel would be about 50% bigger in cross sectional area, but would weigh less than one third of the weight of the steel. If you're trying to get the greatest possible stiffness in a given sized volume, then use steel. However, if you want the lowest possible weight for an equal amount of stiffness, even if the part has to be bigger and bulkier, use carbon.

Spruce has an elastic modulus of only 1.6 million psi, only about one fourteenth that of carbon fiber/epoxy and one nineteenth that of steel. However, its density is .013 pounds per cubic inch, a little more than one fifth the density of carbon fiber/epoxy and about 1/22 the density of steel. To have an equal amount of stiffness from a bar of spruce in comparison to a one inch steel bar, the spruce piece would have to be about 4 1/3" square. It would weigh about 86% of the weight of the steel bar, and almost three times the weight of the equivalently stiff piece of carbon fiber.

We now have an I-beam with one face made of an extremely stiff material, and the other from a more flexible one. For purely the bending portion of the load, the total forces in compression on one side of the beam have to equal the total tension forces on the other side of the beam. However, a stiff material does not have to stretch as far to develop a certain amount of force. If our spruce I-beam now has a carbon-reinforced cap on one side, that added stiffness has the effect of pulling the neutral axis closer to it. This creates a greater distance from the neutral axis to the other, unreinforced side, giving it more leverage. Remember my comment above about how we can't neglect the contribution of the wood to the total stiffness of the part? In this case, the addition of the fourteen-times-stiffer carbon to one side of the I-beam makes the wood on the other side more effective by giving that wood more leverage. The stiffness of the entire assembly does increase, and quite significantly.

However, adding the carbon to only one side pulls the neutral axis closer to the carbon, which removes some of the carbon's leverage it needs to oppose the bending load. We've made the spruce on the other side of the beam more effective than it would be otherwise, but we've lost some of the carbon's capabilities in the process.

Ideally, for the best stiffness-to-weight ratio, we want to have:
1. the minimum amount necessary to do the job
2. of the stiffest material available,
3. located as far as possible from the neutral axis.
The carbon-on-one-side strategy (without a balancing stiffness added to the other side) fails that last requirement.

If we add the carbon to the underside of the fingerboard (which does contribute to the total bending stiffness), then we lose some more leverage for the carbon. We make the wood on the bottom of the assembly more effective, but since the carbon is lower than the top face by the thickness of the fingerboard, it loses some of the leverage it could have had. IN that case, adding the carbon to the underside of the assembly would add more stiffness.

However, there's a down side to that as well, if we're concerned about the tendency for the string tension to pull a warp into the staff over time. If we put the carbon on the bottom face, then we pull the neutral axis downwards, away from the strings. Since the bending moment caused by the strings is their tension times their distance from the neutral axis, pulling the neutral axis downwards increases that bending moment. Putting the carbon in the top, just under the fingerboard, pulls the neutral axis upwards and reduces the bending moment, helping the long-term stability of the staff, even if it doesn't do as much to help the bending stiffness.

If we add a thin layer of carbon (or steel, or some other very stiff material) to both faces of the beam, the stiffnesses on both sides of the beam remain equal, so the neutral axis stays in the center of the beam. Both strips of carbon (or steel, or other very stiff material) are equally distant from the neutral axis, and each fiber of the entire strip is just about as far as possible from the neutral axis (unlike the case of putting a strip in the neck oriented vertically, or using a thick strip of carbon). We've equally given every fiber of carbon the best possible leverage to do its job. This arrangement will be considerably stiffer for the same weight, in comparison to putting all the carbon in a single strip on one face, or oriented vertically in a slot in the middle. It's all about leverage.

OK, so what about those steel "truss rods" they put in guitar necks? At this point we also need to discuss what happens when we add the compression caused by the string tension, on top of this bending load.

A basic principle in stress analysis is that "The sum of all forces acting on an object must equal zero." There is also the corollary that "The sum of all moments acting on an object (i.e.: things that try to bend or twist the object) must equal zero." It all goes back to Newton's Third Law: "To every action there must be an equal and opposite reaction." If every action has an equal and opposite reaction, then if we add up all the actions and reactions, the result must be zero.

So let's add up the loads and moments in that guitar neck, including a steel truss rod set as low as possible near the bottom of the neck's half-moon shaped cross section.

The strings are in tension (a force), and are trying to bend the neck upwards (a moment).

The truss rod and the wood below the neutral axis are in tension (a force), and are trying to bend the neck downwards (a moment).

The material above the neutral axis is in compression (a force) and are also trying to bend the neck downwards (a moment).

The forces from the truss rod and the lower part of the neck as well as the strings, are all in tension. The only thing in compression is the upper part of the neck. Thus, the compression force in the top of the neck, up around the fingerboard, is equal to the tension in the lower part plus the tension in the truss rod plus the tension in the strings. From a strength standpoint, adding the carbon to the underside of the fingerboard isn't such a bad idea at all.

The moments are a little trickier. The moments from both the top and bottom portions of the neck are bending downwards, while the bending moment from the string tension (which is equal to the tension force in the strings, times the distance from the strings to the neutral axis) is bending upwards. The distribution of the loads that define the neutral axis location gets a little tricky, because we have different materials of different stiffnesses, sizes and shapes.

The strings themselves have relatively little influence on the location of the neutral axis because, despite the extremely high elastic modulus of steel, the total cross-sectional area of the thin steel strings makes them a very small part of the total picture. It's just like if we tried to stiffen a neck by adding just a dozen or so individual carbon-fiber filaments to a beam; yes, it would change the stiffness of the beam, but not enough to be significant.

The modulus elasticity of spruce, as I mentioned above, is about 1.6 million psi, depending on the specific species. The modulus of Sugar maple, the hardest of the maples, is 1.8 million, or about 1.2 times stiffer than spruce. However, its density is about 1.6 times greater than spruce, so its stiffness to weight ratio is actually about 74% of that of spruce. Yes, that's right, a piece of spruce with the same stiffness as a piece of maple would only weigh 74% as much. The situation for strength-to-weight ratio is much closer to one, but still it should be obvious why we don't usually build airplanes out of maple!

Indian ebony has an elastic modulus of about 1.9 million psi, about 1.2 times greater than spruce and nearly the same as Sugar maple. It's density is about 2.4 times heavier than spruce, so its stiffness to weight ratio is just this side of atrocious. However, one thing that should be mentioned here is that stiffness, strength and hardness are all distinctly different properties. Just because something is very good in one of those properties does not mean it will do well in the others. Ebony is very hard and resistant to abrasion and wear, even though it is not particularly special in the areas of stiffness and strength. This, plus its physical appearance, is why we use it for fingerboards.

Getting back to our guitar neck, the stiffness of the maple typically used for the neck and the ebony typically used for the fingerboard is sufficiently close that we can get away with assuming they act together like one reasonably homogenous piece of maple.

The neutral axis of a solid half-circle, such the cross-section of a typical guitar neck, is about 42% of the way down from the flat side towards the far edge of the half-circle (the lowest point on the bottom of the neck). In other words, if the neck was 1" thick including the fingerboard, the neutral axis would be about .42" below the fingerboard surface. If the ebony fingerboard was 1/4" thick, that would put the neutral axis just shy of 3/16" below the surface of the maple.

Note that in the above arrangement, if we bury a 1/4" thick strip of carbon in the top surface of the maple and right up against the underside of the ebony fingerboard, that piece of carbon will actually partially straddle the neutral axis. It will still add its own bending stiffness to the assembly, but it will lose most of its potential for giving the wood more leverage, and its own location will give it about the worst possible leverage to work with as well. Having the carbon in a thick strip right under the fingerboard may help the compressive strength and bending strength a little, but loses some of the beneficial effects on the neutral axis location and its effect on the bending moment caused by the strings, as well as much of the possible improvement in bending stiffness.

If we bury the center of the steel truss rod about 3/16" deep in the maple, the truss rod will lie on the neutral axis. It will add no stiffness to the neck, and will not have any consistent effect on the shape of the neck if we tighten it. In fact, if the neck is warped to begin with, tightening the truss rod will tend to make the warp worse, regardless of which direction the warp is in!

If we bury it in the neck below that depth, tightening the truss rod will bend the neck down, because it now would lie below the neutral axis of the wooden neck.

If we bury it in the maple at a shallower depth, it will lie above the neck's neutral axis. Tightening that truss rod would actually bend the neck up!

As far as stiffness goes, the steel truss rod is a stiffer material than the wood it replaces, so it will tend to increase the bending stiffness of the entire neck. If the truss rod lies on the neutral axis, the effect will be very small (like what I described above for a carbon bar in that same approximate location), especially if the truss rod is floating loose in its channel in the neck, so that the only structural connection between it and the neck are at the ends. The further it lies from the neutral axis, the greater its effect on the overall stiffness of the neck. In this regard, it acts a lot like the case of our I-beam with a carbon spar cap on one face only. We could think of that arrangement as somewhat analogous to an I-beam that has the truss rod as one cap, the entire cross-section of the wooden part as the other cap, and the distance between the center of the truss rod and the neutral axis of the wooden neck by itself as the spacing (and therefore the leverage) between the two caps.

The other factor that I hinted at above is the way the truss rod is connected to the neck. If it's only connected at the ends, then those ends act something like nodal points, with the neck in between somewhat unaffected by the stiffening of the rod. If you push down on the neck in the middle, the scroll end of the neck will not deflect as much as it would without the truss rod. However, the middle of the neck can bow downwards without feeling nearly as much support from the rod. Truss rods like this do help carry the tension load from the strings, but their contribution to the bending stiffness of the neck along its length are limited.

To really add to the local bending stiffness of the neck all along its length, the added stiffener needs to be bonded to the neck along its entire length. The shear loads between the stiffener and the neck need to be transferred right where they are generated, not forced to be carried all the way down to the ends before reaching a connection between the stiffener and the neck.

As far as adding a carbon rod to stiffen the neck, like the ones shown in the Stewart MacDonald catalog, it all comes back to how far the material is from the neutral axis. In this regard, the 1/8" x 3/8" stiffeners would be better than the .200" x 1/4" rods. The total cross sectional areas of the two shapes are nearly equal, but when inset into the neck with their widest face horizontal, the 1/8" thick strip can have more of its fibers concentrated further from the neutral axis. This would be even more true of an even wider and thinner strip. Two even thinner strips, one running along the bottom of the neck, and the other running as high in the neck as possible, would be better. With our conventional fret wire we couldn't put it in the very top surface of the fingerboard because the notches for the fret wire would destroy much of its stiffness and strength, as well as wearing out our fret saws. In a non-fretted fingerboard the carbon fiber might actually have some possibilities as a replacement for ebony, although even there we would have some problems to deal with. There are all sorts of possible complications, not the least of which is galvanic corrosion of the metal strings by the carbon, especially a string with an aluminum winding such as some fiddle strings. However, I do plan to do some experiments some day in that area with some of my fiddle-making projects.

OK, we've beat that whole cap strip and neutral axis issue to death. What about the shear web? In general, the shear web needs to support the relatively thin caps from buckling, and transfer the shear loads between the compressive-loaded and tensile-loaded caps. In general it doesn't take a whole lot of material to do this, at least in comparison to the needs of the caps. This means we can either use less material, as in the case of an I-beam, or a hollow "box spar", or we can use a larger amount of some lower density material.

For example, in model sailplane wings we frequently use thin carbon fiber spar caps, with a shear web of vertical-grain balsa wood between them. In extreme cases, we may strap the carbon fiber spar caps to the shear web with Kevlar thread to prevent them from lifting off and suffering a compressive buckling failure (compressive buckling is like the way a soda can collapses when it is squashed, as opposed to a pure compressive failure such as squashing a ball of clay). We may also have carbon fiber fabric laid up on the faces of the shear web to help carry the shear more effectively.

Note that in this case, we're more concerned with bending strength rather than stiffness. For example, a wing like this might be ten feet long, only about 5/8" thick at its thickest point, have spar caps less than 1/16" thick at their thickest point, and yet be strong enough to support lift forces in flight of over 300 pounds! For more on one example of this sort of spar, there are several articles on the spar design of a sailplane designed by Dr. Mark Drela at MIT that you can find through: http://www.charlesriverrc.org/articles/allegro2m/markdrela_allegro2m.htm

However, in our case we're more interested in stiffness than in strength. The overall principles are similar, but the key strength issues such as buckling failures, skin delaminations, corner bonds, etc. are less critical for the neck of a musical instrument. Also, in a musical instrument we usually want to promote resonances, rather than damping them (often the opposite of what we want in something like an aircraft structure). For this reason, I'd recommend against using most of the low-density approaches such as balsa wood or Spyderfoam as shear web materials. Their porous nature makes them great for damping and suppressing vibrations!

This leads us back to something more like an I-beam, with a strong, hard, but thin material for the shear web. However, an I-beam may not be quite the optimum design either. The material in the caps that is right next to the attachment to the shear web has a good load path to convert its load into shear and send it to the opposite cap, but the same is not true for the fibers out at the edges of the cap. They have to transmit their loads in shear horizontally in shear across the face of the cap before it can be transferred down into the shear web. This means that the fibers in the middle of the cap have a stiffer load path, making them work harder and reducing the overall efficiency of the structure.

A box spar does little better in this regard, since the fibers in the middles of the caps now have to transfer their load in shear out to the edges of the spar caps to get to the two shear webs. The box spare is far better in terms of torsional stiffness (twisting resistance), but it still has some slight losses in the structural efficiency area. Note, it's still going to be far better than the solid chunk of wood we're used to today, it's just that if we're looking for theoretical perfection, there's still room for improvement.

For supporting the caps against buckling, the ideal grain direction for the wood would be vertical. However, that's a strength issue, and we're more concerned with bending stiffness. Ideally in that case the grain in the wooden shear webs should run "on the bias" as it's called in fabric layups, or at a +/- 45 degree orientation; in other words, like a strip cut from a sheet of plywood at a 45 degree angle to the grain direction of the sheet. However, the glue in the plywood could have some damping effects we don't want either, so we're back to using solid sheets of wood again. Having the grain vertical would probably be best, but the difference between that and running the wood grain along the length of the piece would be small. For our purposes, there will be tradeoffs any way you choose to go.

Another option that would trade off some torsional stiffness but improve the load sharing in the carbon fiber caps would be to use a box spar, but set each of the two shear webs one-fourth of the width of the spar caps in from the spar cap's edge. The result would look something like a Roman numeral "II", or like two side-by-side I-beams welded together along the edges of their spar caps. This would make the greatest distance the load from the furthest-away fibers would have to travel across the width of the spar cap only one-fourth of the spar cap width, rather than one-half the width as in the case of the ordinary box spar or the I-beam. There are other problems to deal with at that point, but theoretically the improvement is possible. Of course we could have three even thinner shear webs, one in the middle and two set about one-sixth of the way in from the edges, looking like three side-by-side I-beams. Then of course there's the possibilities of tapering the thickness of the caps and webs to match the local requirements, as well as tapering the shape of the entire assembly as well.

Then of course there's the option of using +/- 45 degree carbon fabric in the shear webs, as well as unidirectional carbon (i.e.: all fibers running in one direction, along the length in this case) in the spar caps. Properly done, this arrangement could eliminate the wood altogether, provided that you could come up with some other kind of frets that did not require cutting notches in the fingerboard to set them in. Really perfectly wetted-out carbon with a glass-smooth surface looks a lot like some subtly iridescent ebony and is quite beautiful. However, it's very difficult to achieve that level of perfection on a consistent basis without either a lot of very expensive equipment, or a lot of skill in working with it, or both. There is huge potential in this approach, but will require a great deal of effort to produce it successfully at a price anyone is willing to pay. How much market is there for a $10,000 dulcimer?

Obviously, at some point we need to start asking ourselves if this whole fanatical pursuit has gone beyond the point of becoming ridiculous, or at least beyond the point of diminishing returns. Improvements are certainly possible, with performance far beyond what we typically do today, but we have to find that balance where the improvements are worth the extra trouble.

Don Stackhouse
DJ Aerotech