Calculating Soundboard Flexural Rigidity

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jellicorse
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Calculating Soundboard Flexural Rigidity

Post by jellicorse » Sat Mar 29, 2014 4:05 am

I have been scratching my head over how to calculate the flexural rigidity across the soundboard span and wondered if anyone could help set this straight for me?

I can see clearly how to get the second moment of area I and Flexural Rigidity EI through one brace-soundboard section. But I can't see how to combine these across the soundboard span to get an overall value.

For example, referring to Fig 4.4-19, if spruce is taken as the brace and soundboard material, I make the EI values to be:

small triangular brace: 1.024Nm^2 (or including S/B section underneath: 2.7Nm^2)
larger brace: 13.4Nm^2 (or including S/B section underneath: 24.2Nm^2)

The paragraph immediately underneath this figure says: "it is necessary to convert all members to one equivalent material with new dimensions (new cross sectional area), then compute the second moment of area I using the new dimensions and then use the parallel axis theorem to combine them all."

So, if I understand this correctly you have to reduce all the brace-soundboard sections, and sections of soundboard between the braces, to one member... I know I've probably overlooked something obvious but I can't see how this is done. Maybe it has something to do with using the parallel axis theorem through the Y-axis, but this doesn't seem correct since we're not looking at movement in this plane.
John

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Trevor Gore
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Re: Calculating Soundboard Flexural Rigidity

Post by Trevor Gore » Sun Mar 30, 2014 12:01 pm

A two part answer:

First the equivalent materials. If, using a simple example, one material has twice the Young's modulus of the other and we make two braces of the same dimensions, one will be twice as stiff as the other. So the stiff one is like having two of the other one side by side. Note the side by side. So to make the mathematics of the problem more tractable, we can make the two original braces together be the equivalent of three braces of the same size all made from the less stiff material. If one material is 1.234 times the Young's modulus of the other, for an equivalent brace it has to be 1.234 times the width of the less stiff material at the same height.

The second part is about executing the parallel axis theorem correctly. There's a worked example on pages 4-37 and 4-38 with a numerical answer given so you can check your work. For elaborate cross sections like that shown in Fig 4.4-19, you just have to painstakingly apply the parallel axis theorem to "stack together" the stiffnesses of each component.

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jellicorse
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Re: Calculating Soundboard Flexural Rigidity

Post by jellicorse » Mon Mar 31, 2014 5:43 am

Thanks a lot for going through that Trevor, that's very clear.

I've calculated the correct value for the worked example on p.4-38 but forgive me if I am a bit slow: while I can see how you calculate the value for an individual brace, I can't see how an overall value for the stiffness across the soundboard is reached.

I am assuming that if you have, say, 4 braces you don't just add their individual values to obtain the overall flexural rigidity. If this is the way it's done I've got mistakes in the spreadsheet where the soundboard slice has been added in!

Just for a reference, is the soundboard material for table 4.4-2 Sitka Spruce (with a Young's Modulus of 12)?
John

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Trevor Gore
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Re: Calculating Soundboard Flexural Rigidity

Post by Trevor Gore » Wed Apr 02, 2014 11:04 pm

jellicorse wrote:I am assuming that if you have, say, 4 braces you don't just add their individual values to obtain the overall flexural rigidity.
Correct, if the braces are all different. If they are all the same and they are stacked side by side it's just a simple summation (similar to the equivalent material example we discussed above).

If you look at Eqn. 4.4-2 there are two sets of curly brackets {...} ... {...}. Inside each of these curly brackets is the expression you need for each component (e.g. a rectangular brace) that you want to add into the total mix. In the example, it's just the rectangle and the triangle of the gabled brace. So for a structure like the top section shown in Fig 4.4-19, you have two gabled braces, two triangular braces and the top itself. Each gabled brace is a triangle and a rectangle. So if you split it all down into basic parts you would have a total of 7 sub-components and therefore 7 sets of {...} brackets. Of course, for Fig 4.4-19 you have a lot of common components which makes things easier but you also have to calculate x from the equation at the bottom of p. 4-37.
jellicorse wrote: Just for a reference, is the soundboard material for table 4.4-2 Sitka Spruce (with a Young's Modulus of 12)?
Table 4.4-2 is just a table of values, (with Sitka given with an Elong of 12GPa) so I'm not understanding what it is you're asking here.

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jellicorse
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Re: Calculating Soundboard Flexural Rigidity

Post by jellicorse » Thu Apr 03, 2014 11:20 pm

Many thanks indeed Trevor. I don't know why I didn't see that all the separate components involved just go into Eqn 4.4-2... Before, I was figuring out the values for each brace, including the section of soundboard underneath and then wondering how to put it all together.

Anyway, thanks again. My spreadsheet's coming up with the correct values for table 4.4-2 now. As for the question about the soundboard material, I just wanted to check what you assumed for the soundboard material Young's Modulus (as this would affect the results) in order to compare values for different brace materials and check I was computing them correctly.
John

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