rib and panel math

[Ron Nossaman]

At 10:23 PM 9/29/99 -0400, you wrote:
>Ron,
>
>I am not surprised that Richard was a little confused by the two figures
>you used, I was also. I did a little digging and may have solved the
>mystery. BTW I checked your figures and they worked fine in relation to
>my more rudimentary methods. That is, your methods are suitably accurate
>but a little confusing to those less versed in engineering. 

* Hi John, I found both examples listed together in two different
references. I just picked the one that was easier to type in to my Pascal
code. It was more a matter of being lazy than anything else.



>Finding the angle as you did really stumped me, I would usually rely on
>the more conventional formula. The sin of halve the angle = ½ the rib
>length divided be the radius. Look this figure up in a table of
>trigonometric functions. The inclusive angle is twice this angle.

* Again, I used the method I came across first. This one wouldn't have
occurred to me.




>I don't have time right now but maybe you can find the force necessary
>to bend a typical  uncrowned 2' rib  to a 60' radius. Simply apply the
>force the center of the rib's length. To make it fare you should also
>scallop the ribs as typically found. I would like to do a little
>figuring to fine how much force the panel has to exert to form the
>crown. I think this would be pretty simple and would give us some idea
>how much of the plastic limit (580 psi) the panel crowned process uses
>up.
>
>I would prefer to call it panel crowned instead of compression crowned.
>
>John Hartman

* It took me a while to hunt down and round up all the pieces here, but I
think I've got them all. Straight deflection is easy enough. as
(riblength^3*load)/(4*E*ribwidth*ribheight^3), where E is the modulus of
elasticity of the rib material, or 1570000 for Sitka spruce. The metric
version would be (riblength^3*load)/(E*0.0062*width*height^3) The difference
is because E is figured on square inches, and needs the conversion to square
millimeters. Deflection from panel expansion is a little more complicated,
but I've been playing with a pretty simple minded approach for a while, that
seems to work. Starting with the assumption that the panel applies a simple
leverage with the distance from the center of the panel to the center of the
rib, and 1/2 the rib length as the moment arms. For inches, I used 
   panel_leverage:=(((height/2)+(panelthick/2))/(riblength/2))*2*psi
   moment_of_inertia:=(width*height^3)/12
  panel_lift_in:=(5*panel_leverage*riblength^3)/(348*E*moment_of_inertia);

For millimeters,
panel_leverage:=(((height/2)+(panelthick/2))/(riblength/2))*2*psi;
   moment_of_inertia:=(width*height^3);
   panel_lift_mm:=(5*panel_leverage*riblength^3)/(E*moment_of_inertia*0.04495); 

Since the moment of inertia was figured for inches, I combined the metric
correction for E, and MOI into the 0.04495 multiplier.

To find out the negative load equivalence of panel expansion lift, should
one be interested how it might react under bearing load, I came up with
   (4*E*width*height^3*deflection)/(riblength^3);
which is the deflection formula turned inside out. 


And for metric
   (E*0.0062*width*height^3*deflection)/(riblength^3)
which is the same, with MOE adjustment multiplier.  


Try them out and see what you think. These are something I started playing
with a couple of years ago and I haven't taken the time to try to take them
any farther, but I'm open to donations. 

There's one thing I would like to find though, is a one pass formula that
will tell me, from given speaking length and back scale lengths and string
tension, rib dimensions, and bridge height above front to back string plane
(zero bearing), where the equalibrium deflection point will be be, and the
resulting bearing angle. I can do this with an iterative routine, but I
can't help but think it's possible to set it up in a spreadsheet and solve
in one pass like a standard deflection under load thing. The thing is that
bearing load, angle, and deflection all change at the same time to balance
in the middle somewhere. That's what I'm looking for an easy way to figure.
Not much, huh?

   






 Ron N