More on soundboard crown

[Ron Nossaman]

>My quick calcs yield half this number.

Ah, then I DID do that wrong. I never was sure if I was at half, double, or 
on, and never found a source for a calculation. Thank you, I'll fix my 
spreadsheet.


>>>  Ten pounds of load would put 900 pounds on the soundboard edge. A rib 
>>> half that long will produce over twice that thrust ratio.
>
>
>I think you have this backwards.  Half the span with the same crown height 
>would yield half the thrust.  Also, my understanding is that rib crowners 
>tend to put more crown height on the shorter ribs, which would further 
>reduce the thrust.

Sorry, I left out the crown, making too many assumptions. I was referring 
to the usual practice of a standard expected crown radius throughout the 
piano. If the thrust of the 800mm arch at 18M radius is 45:1, then a 400mm 
arch at the same radius will be 90:1. Halving the radius to 9M would put 
the 400mm arch at 45:1. Ok, now I think that's right. 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.


>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.


>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?


>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.


>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. The difference is what can be demonstrated in the 
short term on the bench, and what happens in the long term in the piano - 
or on the bench, for that matter. I find little solace in the fact that the 
concave crowned soundboard I'm looking at might have once, too often very 
recently, shown the wonderful crown that so obviously proved the value of 
the assembly technique in the factory.


Ron N