brace stress

[GregHolmberg]

Any engineers out there? I have a question about calculating stress when using a combination of materials. Specifically, I want to calculate the stress of a sound board when the panel and the braces are different materials (different species of wood).

I've got a google spreadsheet here in which I've calculated rigidity (EI) and stress using the geometry of Fig. 4.4-19, and I got the same answers as in Table 4.4-2. So far so good.

On page 4-49, Trevor talks about how to deal with a combination of materials by assuming it's all built of Sitka Spruce, but adjusting the width of the Sitka braces to simulate braces of a different species ("equivalent spruce"). For example, if the brace species has half the elastic modulus (E) of Sitka, then calculate with Sitka braces half the width. Then at the end, you can multiply the total second moment of area, Is, by a single E (of Sitka) to get the rigidity of the sound board (EI).

I'm sure this is a pretty good approximation, but it results in a different Is compared to building the same size braces in a different material. I notice that if I calc EI for the panel in Sitka, and EI of the braces for a different species, and add them together, I get a different total. For example, with a Sitka panel plus King William Pine braces, I get an EI of 74.7, compared to the 70.1 using the equivalent Spruce method. About a 7% difference. You can see this in the spreadsheet in the "braces" sheet, cell Y10.

My goal here is to do the opposite of this calculation: set a target EI and seek brace dimensions for a given brace species and a given panel species.

For this, I'd like to be as accurate as I can.

Is there any reason it's not correct to do what I've just described: to multiply the I of part of the structure by one E, and the I of another part by different E, to get two EI's, and add them together?

Greg

[Trevor Gore]

Is there any reason it's not correct to do what I've just described: to multiply the I of part of the structure by one E, and the I of another part by different E, to get two EI's, and add them together?

I haven't had the time to check out your spreadsheet, Greg, so apologies for that.

However, when you're adding second moments of area together you have to use the parallel axis theorem, whether or not they are multiplied by the same E or different Es. The only time you can simple add two I's together is when their neutral axes are in the same plane (which is the transform you are doing when you use the parallel axis theorem).

[GregHolmberg]

However, when you're adding second moments of area together you have to use the parallel axis theorem, whether or not they are multiplied by the same E or different Es. The only time you can simple add two I's together is when their neutral axes are in the same plane (which is the transform you are doing when you use the parallel axis theorem).

Trevor, thanks for taking the time to answer.

So I did all that, and got the same answers as in Table 4.4-2. I have the second moment of area, Is, around the neutral axis, for each rectangle and triangle in the cross-section. See the 'braces' sheet, column V.

As you describe on page 4-49, these calculations use the technique of changing the widths of the braces to deal with the fact that the braces are a different material than the panel. I did that, and I get the same results as you. I found an online textbook that describes this as the "transformed section method", and I'm diving into that book.

My question is, can I use another method, not transforming the sections? Specifically, can I keep the dimensions as in the drawing, calculate the second moment of area around the neutral axis for each shape, and then multiply the panel Ipanel times Esitka and the total of the braces' I, Ibraces times Ewrc (Western Red Cedar), and then add those together:

EI = Ipanel * Esitka + Ibraces * Ewrc

Reading this textbook, I'm getting the feeling that this is not allowed. But I'm not sure why.

Thanks,

Greg

[Trevor Gore]

My question is, can I use another method, not transforming the sections? Specifically, can I keep the dimensions as in the drawing, calculate the second moment of area around the neutral axis for each shape, and then multiply the panel Ipanel times Esitka and the total of the braces' I, Ibraces times Ewrc (Western Red Cedar), and then add those together:

EI = Ipanel * Esitka + Ibraces * Ewrc

Reading this textbook, I'm getting the feeling that this is not allowed. But I'm not sure why.

The online book uses the same method I use in the book, as you discovered, which is the standard method. If you've come up with another valid method, that's good, but it has to give the same answers over multiple instance so that you can assure yourself that it's a valid method.

I think the reason your new method may not be valid is given on page 4-50, bullet point 2). The transformed section method changes the section area and therefore its effective I, which I don't think your new method accounts for.

Those are my first thoughts, anyway.

[GregHolmberg]

Ok, thanks.

I wasn't sure if the transformed section method was an exact calculation or an approximation. If it is an approximation, then I wouldn't need to get the exact same answer. My method differed by 7%. My method seemed more realistic (actual dimensions used to build, paired with actual E), so I thought perhaps it is a better approximation. But if the transformed section method is exact, and my method doesn't agree, then there's something wrong with my method. I suspected something was wrong with my method, since it seems easier, so there must be some reason it's not used.

I'll educate myself further via the textbook. Thanks for your help, Trevor.

Greg

[GregHolmberg]

By the way, for people in North America, it appears that a good substitute for King William Pine when making falcate braces, is Black Willow. Native to the Eastern US, and is not expensive. I haven't tried it myself, but the stress calculations look good, and the US Forest Products Lab says that 73% of pieces of Willow were steam-bendable (source), which is pretty good.

Greg