evaluating sdbd. crown & bridge downbearings in a new piano

[Ron Nossaman]

>I hate to keep beating this horse, but nothing prevents a conventionally
>crowned soundboard from being fabricated at a moisture content close to the
>norm also.  For the reasons I have put forth previously, such a soundboard
>will actually be under less compression when loaded.

* Given that two soundboards with identical panels and rib sets, one
conventionally compression crowned and one crowned as you described, end up
with identical crown under identical loads (we have to start with some basis
for comparison), the panels of both boards will be exerting the same
expansive force to hold up that crown. I would say that makes the
compression of the two panels the same, but we seem to have another semantic
problem here, since the total percentage of panel compression (and
therefore, damage) suffered to get to this state is very likely greater in
the traditionally compression crowned board than in what you described. It
depends on whether we're talking about dimensional changes, or force exerted
as to which has the most compression. Is this one of the points? 






>>But, that crown is less than it was before the constraints were removed.
>It
>>is less than the crown of the press.  That is, the radius of the soundboard
>>assembly is larger than the radius of the cauls or press table.
>>
>
>We can agree that this is true, but so what?  There is a crown, and the
>radius is predictable.
>
-----------------
>
>Del, I don't want to accuse you of voodoo engineering, but as long as the
>top of a panel constructed as previously described has a curve, it will be
>in tension.  No matter how and by what forces the curve is imposed, the laws
>of Newtonian physics decree that the outside of the curve will be in
>tension.
-------------------
>Frank Weston



* As long as the assembly is in the press, this is true. As for the Voodoo
engineering, let's conjure up a little math and see what the ancient Gods
have to say about all this and try to ascertain who's shaking the beads
here. Let's talk Engineer.

When the rib (or panel) is bent, there is a neutral stress line running
roughly through the center of it's height. That part is obvious, so let's
use the rib centerline as a reference for computation. Let's use Del's
example rib and panel as a standard and start with a rib 1000mm long, 25mm
high, and 25mm wide, with a panel 8mm thick. Let's also assume the 18000mm
final radius result after assembly, because we need some sort of figure as a
benchmark and the 60' radius seems to be the most widely accepted. So what
are the final arc lengths of the assembled surfaces? Here's where the math
comes in. First, let's figure the segment angle of the formed arc. Angle =
57.29578*(riblength/radius). That's 57.29578 times 1000/18000, or 3.1831
degrees. The formed arc segment of the top of the panel is computed as
PanelTop = 0.017453*(radius+(ribheight/2)+PanelThickness)*angle. In the
final assembly, that comes to 1001.1221. the formula
Radius-sqrt(sqr(radius)-sqr(riblength/2)) gives us a crown height of
6.9458mm. I figured four decimal places ought to be adequate overkill. Now,
in order to have compression in the top of the panel after assembly, and
before string load, We have to form the assembly in a press with a curve
radius such that the arc segment defined by the neutral stress centerline of
the panel when it's in the press is the same, or greater, than the arc
segment defined by the top of the panel after removal from the press and
spring back. As it turns out, a press caul of 14487mm radius produces an arc
segment angle of 3.955, and using
0.017453*(radius+(RibHeight/2)+(PanelThickness/2)*angle gives us a panel
centerline of 1001.1222, with a crown height of 8.631mm. Therefor, if the
spring back after taking the assembly out of the press accounts for a
reduction in crown of more than 1.6854mm to arrive at that 18000mm radius,
the top of the panel will be under compression. That is the case with this
particular set of rib and panel dimensions, but the principal can be put to
any set of dimensions you may have to test the premise. 

Now, in your example, if we knew the caul radius of the press used, the rib
and panel dimensions, and the resultant crown radius of the assembly upon
removal from the press, we could find out what we really have.     

If my math is incorrect, please correct me. I need all the help I can get
with math. Also, I realise that this isn't exact to the four decimal places
shown, primarily because bending the rib in an arc shortens the chord
measurement from which the computations are taken. Taking this into account,
the actual figures would be minutely different, but I assumed that a
hundredth of a millimeter or so wouldn't invalidate the illustration. 


 Ron N