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According to Hoadly ('Understanding Wood', Taunton Press, 2000, pg. 117) most of the softwoods we use for tops have about twice as much tangential as radial shrinkage for a given change in humidity. This actually has two effects. One is, as has been said, that a quartersawn top will shrink, and therefore crack, less than a flat sawn one. The other has to do with cupping.
If you cut something like a 2X6 out of a small tree near the middle (do they cut them any other way these days?) it will start out flat, but it won't stay flat as it dries. The difference between radial and tangential shrinkage will have the effect of making the annual ring lines on the end of the board try to straighten out. As they do so, of course, the surface of the board, which was flat, becomes cupped or humped across the width of the piece.
Note that this is an effect of curvature of the annual ring lines: so long as they're straight, it won't happen. It's easy to get reasonably straight annual ring lines on a quartered top or back; they don't extend very far. On a flat cut piece you will almost never see the annual ring lines a s straight on the end surface, unless the tree grew significantly out of round. In that case, I'd suspect some other problems, like built-in stress. Cupping across the width of a top or back can riase a lot of stress in the wood and lead to cracking, so using quartered wood for these parts makes sense. Flat or skew cut sides also tend to cup more when you bend them that quartered ones.
Wood splits most easily along the medullary rays that run along the radius of the tree, crossing the annual ring lines. On a flat cut piece the rays run through the thickness, and it's just that much easier for it to crack.
OTOH, those little bundles of ray cells add a lot of strength and stiffness along their length. This is why perfectly quartered wood has higher cross grain stiffness than flat cut. The difference between perfectly quartered spruce, and perfectly flat cut, is not too great: the quartered wood might average a 10:1 stiffness ratio, while the flat cut might be 12:1. However, skew cut spruce, with the ring lines at 45 degrees to the surface, can have a stiffness raio of 100:1; it only has 1/10th the crosswise stiffness of a quartered piece cut from the same tree. Flat cut wood tends to have a variable grain angle as you go across, and the crosswise stiffness varies commensurately.
Note that, on the whole, there is no difference in long-grain stiffness between flat, skew, or quartered wood. You might see systematic differences in a small number of samples due to structural coincidences. If, for example, you cut a piece 1/4" square with hard latewood on each surface and earlywood in between it would be, in effect, an I-beam, and could be much stiffer tested 'flat' than the same piece tested 'vertically'. If the latewood was in the center, and the earlywood out, the flat cut test would have the lower stiffness.
The one real advantage that skew cut wood has is that it's tougher: it won't be as likely to split when bent or subjected to large changes in humidity. I use skew cut wood for harp soundboards for that reason: you don't need the cross grain stiffness in that case, but the added tougness really helps. It's interesting that the 'conventional wisdom' holds that harps made in this way wi'l sound 'dead': people are reasoning by analogy, rather than thinking the problem through. At any rate, I like to use skew cut wood on guitars in the bridge plate, the bridge, and the tailblock to take advantage of the splitting resistance.
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