>So an S&S B, with a #13 rib length of about 250mm rim to rim with an >18M crown would have a thrust ratio of 144:1. With a string bearing >load of well over 100 lbs, would be putting over 14,000 lbs thrust >on the rim. Somewhat less, actually, because the load is well off >center, but still considerably above the capacity of the materials >used, to maintain a crown. Yes, it's still a considerable sum. > > >>This argument is convincing if you insist that if the soundboard >>acts as an arch, then all the string loading must be taken as an >>arch. > >I don't insist that at all. The argument is often presented that the >arch effect is a significant component of soundboard crown formation >and maintenance, and if it isn't isolated and either eliminated or >validated as a meaningful component, it will forever continue to be >in the way. > >>All the load doesn't have to be (and won't be) reacted this way. >>The panel has bending stiffness, as do the ribs, as does the >>combination of the two. > >Correct, which brings up an interesting point. The panel is bent >along as well as across it's grain by gluing it to the more or less >plane rim. This adds a load primarily to the ribs in a rib crowned >assembly, and primarily to the panel cross grain in a panel crowned >assembly. It's trying to flatten the crown along it's grain in both >rib and panel crowned soundboards, just like the ribs are trying to >flatten crown across it's grain in a panel crowned board. An interesting point. This has probably been discussed before, but I don't have the patience to search the archives for it, so I'll just dive in. As I think you would put it, this is a consequence and not a feature of the design. What I wonder is if it's a positive or negative consequence. Perhaps this is just adding to the stress that eventually wants to collapse the board. Or perhaps the induced stress is having some sort of positive acoustical function. Thoughts on that? If it's a negative thing I wonder if making the top of the rim conform to the outline of the board without bending it along the grain would yield positive results. Obviously just angling the top of the rim, as some builders do, isn't going to accomplish this. But if you were to crown your board and then measure the shape of the outline and cut the top of the rim to match (which would mean considerable deviation from a plane) I wonder if there would be a beneficial result. > > >>Also, the ribs tend to get ignored in these arch discussions. If >>the ribs are glued to the rim then they have an arch load path >>along the grain (their strong direction). The load will divide >>itself among the load paths according to their relative stiffness. >>If the rim is flexible and the cross grain stiffness of the panel >>is low and the compressive stiffness of the ribs is low then this >>path will be soft relative to the bending path. That doesn't mean >>it's nonexistent. Perhaps 10% of the load could be taken by the >>arch and 90% by bending. > >I can't believe it's anywhere near 10%, for all the reasons >discussed in the past when this came up. But yes, let's talk about >those ribs in the context of arches. It's about time this was >mentioned and dealt with. Assume a line of a given length with end >points A and B. A and B are farthest apart when the line is >straight. Bend the line along an arc and A and B get closer >together. It doesn't matter which direction you bend the line, any >deviation from straight by any amount brings A and B some distance >closer together. > >Let's isolate, ignore rib stiffness, and look at just the arch effect. > >Define point A and B at opposite ends of a flat rib. When you put >this rib in a panel crowned board, the rib is bent and the distance >between A and B decreases. Loading the board with string bearing >depresses the rib toward straight and the distance between A and B >increases, reaching (within the compressibility limits of the >material) at or near their original distance apart when the rib is >again straight. This rib produces outward thrust on the rim as the >crown is deflected. > >Now define point A and B at opposite ends of a flat bottomed (as >they usually are) crowned rib. If the panel didn't bend the rib when >they were glued together, the rib is essentially straight before the >string are installed. String load will then bend the ribs, bringing >points A and B closer together as the load increases. There's no >arch at all in crowned ribs. This rib doesn't produce any outward >thrust as crown is deflected. It pulls inward on the rim. It's like >a cable suspension subject to the same limitations as the arch >theory except it has more room. With the cable suspension, the >greater the sag, the less strain on the materials for a given load. >It can be pushed down beyond the machined crown height and still be >a cable suspension. With the arch, for a given load, the strain on >the materials increases up to the point where the rib is again >straight, at which point it is no longer an arch. > >If there's an arch effect in a panel crowned board, where is it? It >isn't in the ribs, and the panel cross grain won't hold the required >compression loads. > >It is by now well known that Mason & Hamlin uses crowned ribs. So if >there indeed is no arch effect in a machine crowned rib, how is the >resonator going to hold up crown by pulling in on the rim and >keeping it from spreading? Interesting points. If the board truly is pushed flat by the downbearing, then I agree with you. There's no arch effect, even if there was initially, because there's no arch. But I didn't think boards were designed to be pushed flat by the downbearing. If there is still some crown after the downbearing is on, I still need some convincing that none of that crown is being supported by an arch effect. As far as the machine crowned rib that is flat on the bottom, the line of action (or curve of action in this case) of any axial force on the rib would be through the centroids of the cross sections. Since the top of the rib is describing an arc, so will the curve through the centroids, but it will have twice the radius, if that makes sense. But this does call to mind something I hadn't thought of. Since the ends of the ribs are feathered, the line of action is rising at the end of the rib. Let's assume a reference plane at the top of the inner rim. Let's say the rib is 22 mm deep under the bridge. The CG of the rib centroid at this cross section will be 11 mm under the soundboard. Let's say there is an unloaded crown of 6 mm. Then the CG of the centroid will be 5 mm below the reference plane. Let's say the rib feathers to 10 mm at the rim. The CG of the rib centroid at the cross section will be 5 mm below the reference plane. So the line through these centroids in the unloaded state is describing no arc at all. So much for the arch effect on the ribs. As far as the panel cross grain not holding the required compression loads, how much is it supposed to hold? I agree that it won't hold the entire downbearing load. On good spruce isn't cross grain compression strength on the order of 500 psi? Let's say you want to stay below 200 psi to prevent compression set. Assume 6 inches of panel per rib and assume a board thickness at the edge of 1/3 inch. That means we can have a 400 lb load on this portion of the edge of the board without damaging it. Referring back to your earlier arch numbers we have something like a 40:1 ratio of edge load to down load. That means we can tolerate 10 lbs of download at the bridge without damaging the panel. That represents on the order of 10% of the download. I think the question is, are the stiffnesses of the load paths (rib bending vs arch vs whatever other path there might be) such that they would cause significantly more than 10% of the download to want to take the arch path. If so, then the panel may get damaged before that path softens up enough to cause this load to want to take the bending path. Or, the stress may be below that necessary to damage the panel but above the level that will cause compression set. Perhaps this is part of what's happening in the killer octave area. Since the ribs are shorter (and the panel not as wide) and the arching is somewhat greater, the arch path is stiffer relative to the bending path than it is lower in the scale. When the load is put on, a great enough part of the load wants to try to go through the panel that it overloads it, effectively nullifying this load path, which means that all of the load now has to take the bending path. Since the ribs are not of generous enough dimension to allow this the panel and ribs collapse, presumably to the point where the excess load beyond the ribs capacity finds another path, probably through the bridge to other ribs. > > >>I don't know what is actually happening. But I would like to see >>something other than argument or calculation before accepting the >>dictum that the rim and frame do nothing to support crown. > >Then you are probably going to have to produce the evidence >yourself, and test the principal to your own satisfaction. I was afraid you were going to say that <sigh>. I guess I'll have to add it to the list of things to do. I think the list may be getting up to about a 5 year backlog now. > > >>If I understand you correctly, your contention is that the >>deflection will be the same in both cases. > >No, not initially. Initially, the more rigid case model will >probably show less deflection. Like that M&H sales demo shows, there >is an undeniable immediate effect, however slight. But what you see >isn't necessarily what you get. As Del has pointed out many times, >leave the thing under load and come back next week, or next month, >or next year and see if that support obviously persists. It won't, >because of the compliance of the materials under the loads >imposed.... > > >Ron N Yes, quite likely. But I probably won't be convinced until I do it myself. Phil Ford
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