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James Gordon, an engineer, materials scientist and naval architect, wrote two books that I highly recommend. I was about to write …to woodworkers but, actually, I highly recommend them to anyone who has the slightest interest in buildings, ships, aeroplanes or other artefacts of the ancient and modern world. My copies have been read and consulted so often that they’re falling apart. They are The New Science of Strong Materials or Why you don’t fall through the floor (first published in 1968, but still in print: ISBN-13: 978-0140135978) and Structures or Why things don’t fall down (first published in 1978 and also still in print: ISBN-13: 978-0140136289). Both are written for a non-expert readership and there’s very little algebra or mathematics. They’re fun too: Gordon writes clearly, wears his learning lightly and the text is spiced by his whimsical sense of humour.

The New Science of Strong Materials has many interesting things to say about the properties of wood and why it’s such a wonderful and versatile material. There’s stuff about how wood is able to cope with stress concentrations and limit crack propagation, about how glues work, the distribution of stress in a glued joint, and many other things of deep background interest, if not of immediate practical significance, to people who use timber.

The second book, Structures, is equally gripping. It explains how medieval masons got gothic cathedrals to stay standing, why blackbirds find it as much of a struggle to pull short worms out of a lawn as long ones, and the reason that eggs are easier to break from the inside than the outside. Of more direct relevance to woodworkers is its straightforward account of how beams work – which means that, if you’re thinking of making something like a bed or a bookcase, you can calculate whether the dimensions of the boards that you’re planning to use are up to the load they will have to bear, which is obviously useful in making sure that your structure is strong and stiff enough.

Slightly less obviously, it’s also helpful in giving you the confidence to pare down the amount of material that you might otherwise have used. A common fault of amateur woodworkers, it seems to me, is that when designing and making something small, they tend to use wood that is far thicker than it needs to be, which means that the finished object looks heavy and clumsy. Conversely, when making something large, they tend to use wood that is less thick than it should be, and the structure often ends up rickety and unstable.

Knowing a bit about beams might also be advantageous for guitar and violin makers. Here’s an example: take a strut or harmonic bar, rectangular in section, that you’re intending to glue onto the soundboard of a guitar. How is its stiffness related to its shape and its dimensions? What’s the best way to maximise stiffness while minimising weight?

Elementary beam theory tells us that, for a given length, stiffness is proportional to the width of the beam and to the cube of its depth. So if you double the width, the stiffness also doubles. On the other hand, doubling the depth, increases stiffness 8 times. If stiffness is what you’re after, it’s a lot more efficient to make the bar deeper than it is to make it wider.

This cubic relation between depth and stiffness could be something worth keeping in mind when planing down soundboard braces after they’ve been glued. If a brace is, say, 6 mm high to start with, planing it down by 1.5 mm to a height of 4.5mm will reduce its stiffness to less than a half of what it was originally. And shaping the braces to make them triangular or arched in cross section also reduces their stiffness considerably.

Mind you, like so many attempts to understand guitars from a scientific point of view, things rapidly get complicated. A structural engineer with whom I discussed the matter agreed with what I’ve just said about the depth of the beam being a powerful determinant of its stiffness. But he pointed out that where a beam is an integral part of a structure, the stiffening effect is much greater than you would guess from calculations that assume the beam is simply supported at its ends. This is certainly the case of guitars, where the braces are glued to the soundboard along their entire length and clearly count as an integral part of the soundboard structure. In such circumstances, he explained, the overall stiffening effect provided by multiple braces will be large and might well overwhelm the influence of the stiffness of any individual brace.

I thought that this was a very interesting idea and that it might begin to explain why so many different bracing systems work remarkably well. In Roy Courtnall’s book, Making Master Guitars, he give plans of soundboard strutting taken from guitars by a number of famous makers. Superficially they’re fairly similar, all being based on a fan-like pattern of 5, 7 or 9 struts. There are minor variations, of course. Some are slightly asymmetrical, some have bridge plates and closing bars and so on. But the  biggest differences lie in the dimensions of the braces. Courtnall shows a soundboard by Ignacio Fleta that has 9 fan struts and 2 closing bars which are 6mm in depth and an upper diagonal bar 15mm depth. By contrast, a soundboard of similar size by Santos Hernández has only 7 fan struts 3.5mm in depth and triangular in section. Applying simple beam theory would lead one to guess that Fleta’s bracing would add more than 10 times the stiffness that Hernández’s does. But perhaps that’s a misleading way to look at it. If one were able to measure or calculate the stiffness of the whole structure, by which I mean the soundboard with its bracing when attached to the ribs, the difference in stiffness between them might turn out to be much less.

It’s a question that might be tackled by finite element analysis and I’d be glad to hear from anyone who has tried. Some work along these lines has been done on modelling a steel string guitar, which at least shows that the approach is feasible.

In the meantime, without a proper theory, we’re stuck with the primitive method of trial and error. Below are some of the bracing patterns that I’ve experimented with. All produced decent sounding instruments but I’d be at a loss if I were asked which particular tonal characteristics were produced by each of the different patterns. It may be that William Cumpiano was right when he wrote (in his book, Guitarmaking, Tradition and Technology):

Specific elements of brace design, in and of themselves, are not all that important. One has only to look at the myriad designs employed on great guitars to recognise that there is no design secret that will unlock the door to world-class consistency.

All this means that I’ve been arguing in a circle. Perhaps the conclusion is that beam theory isn’t very useful to guitar makers after all. Still, if you take up the recommendation to get hold of Gordon’s books, the time you’ve spent reading this post won’t have been entirely wasted.

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Rather to my surprise, since it’s a pretty arcane subject, my last post on the elastic properties of spruce and how these vary according to the orientation of the growth rings attracted a lot of attention. Thanks to everyone who took the trouble to email me with their thoughts or to post comments. Stimulated by your interest, I thought that I might expand on a few things.

Because of the primitive way I carried out these experiments, there must be some questions about the validity of the measurements. One reason why I couldn’t show a difference between the stiffness of the wood in the different growth ring orientations might have been that my measurements weren’t sensitive enough. Another might have been that my preparation of the test bars of wood lacked accuracy or consistency. These possibilities seemed worth checking.

First, I re-planed the 9 test bars so that they were as square and straight and, this time, as uniform in their dimensions as I could reasonably manage.

Then I made an estimate of their stiffness, in the same way as before, by clamping them in a vice one at a time and measuring the downward deflection produced by a load of 2lbs applied 20 cms from the vice jaws. Again, every piece was measured 4 times, rotating it through 90° between measurements. For each individual bar, I calculated the mean deflection (in inches) for the 2 measurements in the different growth ring orientations. 3 days later, I measured them again without reference to the earlier readings. The results are set out in the table below, rounded to the nearest hundredth of an inch.

These results seemed encouragingly consistent. The bars that were stiffer in the first set of measurements came out stiffer in the second set too. So it doesn’t look as if the readings are being swamped by random errors introduced by deficiencies in the experimental set-up or that any differences are due to lack of precision. And the actual values in the 2 sets of measurements were quite close too, so the findings are fairly reproducible.

As you can see, there was some variation in stiffness between bars. The deflections recorded for bars 1 and 4, for example, are somewhere between 10% and 20% less than those recorded for bars 5 and 9. It occurred to me that this might have been because my planing had been inaccurate, but when I checked the dimensions with vernier calipers I didn’t find that the stiffer bars were any larger. It may be that this variation is simply a reflection of how the properties of small pieces of wood differ slightly even when cut from the same board.

Although hardly necessary, since it’s obvious from the table that there’s no consistent difference in stiffness between quarter-sawn and flat-sawn orientations for individual bars, I carried out a straightforward analysis using a paired t test, which confirmed that there was no statistically significant difference. [Difference (quarter sawn minus flat sawn) = -0.004 (95% confidence interval : -0.015 to 0.007) p=0.38 df=8]

Guitar makers often flex wood in their hands to get a feel for its stiffness and I wondered if I would be able to identify the stiffest and the least stiff of the test bars by doing just that. Not a chance! I was quite unable to distinguish differences in stiffness between these bars by feeling how much they bent in my hands. It might be worth doing some more experiments to find out what sort of differences in stiffness can be reliably identified in this way. Perhaps we’re not as good at judging stiffness as we’d like to think?

One final thing, which someone kindly emailed me to point out, is that any shrinkage or expansion across the grain because of a change in moisture content of wood tends to be less at right angles to the growth rings than in parallel with them. So, where the growth rings are orientated vertically in the strut, any change in the width of the strut at the glue line with the soundboard will be smaller than if the growth rings had been orientated horizontally. Now in a guitar it’s hard to see that this will matter much because the struts are small, not usually subject to large changes in humidity and are glued in all sorts of different relations to the direction of grain of the soundboard itself, which is also going to move in response to changes in moisture content. But where the strut lies in the same north-south axis as the grain and growth rings of the soundboard – as, for example, in the bass bars of violins or cellos – I can see that there might be an advantage in keeping the growth rings of the strut in the same orientation as the soundboard. This is the best reason I’ve yet heard for keeping growth rings in bass bars vertical, although as I mentioned in my last post, it wasn’t a rule always followed by the great luthiers of the past.

Experienced guitar makers (and books about guitar making) always advise quarter-sawn spruce and vertical orientation of the growth rings for braces and harmonic bars. You hear the same if you ask a violin maker about selecting wood for the bass bar. They’ll explain that wood is stiffest in that orientation, which means that your soundboard will get maximum support for minimum weight.

That might seem the end of the matter but, if you’re one of those disagreeable people who can’t resist probing further and ask if they have ever measured the stiffness of wood in different grain orientations or, if they haven’t, how they can be so sure, you may hear the sound of feet shuffling and detect a swift change of subject.

In fact, as Liutiaio Mottala points out on his interesting website, considering the number of wooden structures that have been built over the years, the information available about how grain orientation influences the physical properties of strength and stiffness is remarkable sparse. In Chapter 4 of The Mechanical Properties of Wood, in USDA Forest Service, Wood Handbook – Wood as an Engineering Material, (available here) there’s a short section on the subject starting on page 4-31, saying that properties of wood do vary slightly according to orientation of annual rings in some species. Disappointingly, it gives no information either about the size of the variation or about which species exhibit the variation. Mottola’s website mentions work done by David Hurd, who found no difference in stiffness between quartersawn and flatsawn wood for the samples he examined but, as far as I can discover, no details are available on line.

Using the rather primitive set-up shown below, I attempted some measurements myself. The wood is straight grained European spruce with around 14 growth rings to the inch. I sawed and planed 9 pieces, each around 35cms in length and between 7 and 8.5mm in width and depth. I used a shooting board to to make sure that each piece was as straight and as square in section as possible and that the growth rings were oriented more or less parallel to one face (and therefore more or less at right angles to the adjacent face).

Each bar was then clamped in a vice with about 22cms protruding horizontally. I then hung a weight of 2lbs, exactly 20cms away from the vice jaws and measured the resulting downward deflection 4 times for each piece, rotating it through 90° between each measurement.

I’ve summarised the measurements that I made in the table below. The deflections I’ve given are the mean of the 2 measurements for each bar in each orientation. As you can see, the way the growth rings were orientated made remarkably little difference to the magnitude of the measured deflection and there was no consistent tendency for the wood to be stiffer in either of the two orientations.

Now, I’m well aware of the many deficiencies in my experimental design. One of the most serious is that all my specimen bars were cut from the same board and it’s possible that other wood from other trees behaves differently. And of course both the way I prepared my specimen bars and the simple test rig meant that all sorts of errors could have influenced individual measurements. However, the consistency of the findings encouraged me to think that these errors can’t have been very large. If they had been, the size of the difference between quarter sawn and flat sawn deflections would have shown much more variation between different bars.

As a check that the sorts of results I was getting were plausible, I used simple beam theory (max deflection = Wl³ ⁄ 3EI )to calculate the size of deflection that might have been expected, using a value of 10 000 MPa for E, the elastic modulus of spruce. This worked out at 0.34 inches, which was close enough to the deflections that I was observing to reassure me that my simple set-up wasn’t completely inadequate for its purpose.

So what do I conclude? Well, probably nothing that would stand up in a court of law. But I’ve satisfied myself that that spruce cut on the quarter isn’t very different in stiffness from spruce that has been flat-sawn and that where wood of the size and sort used for bracing soundboards is concerned, it doesn’t matter much whether the growth rings are orientated vertically or horizontally. In future, when selecting wood for struts and braces I shall feel free to use either orientation, to make the best use of what I’ve got available.

As a postscript, I was interested to learn from Stewart Pollens’ book Stradivari (ISBN-13: 978-0521873048) that the Hill collection of 50 bass bars taken from violins and cellos of the first rank, including those attributed to Antonio Stradivari himself, contains 11 that are flat sawn (that is to say, the annual rings are orientated horizontally). Maybe instrument makers in 17th century Cremona made less of a fetish about the orientation of growth rings than we do today.

The front, like the back, is made from two book-matched pieces – but this time of spruce rather than maple. Here they are, joined and cut out and being roughly shaped.

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The arching has been completed and the position of the f holes sketched in place.

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The hollowing of the inside is now finished.

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Blocks have been glued into place so that the bass bar can be fitted. They’re a temporary scaffolding and will be removed later.

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The next stage of fitting the bass bar.

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Below is a photograph of the front being glued onto the instrument.

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