Calculating Maximum Bending Stress in Beam w/Varying Cross Section

[zzinfinity]

Hi,
I'm trying to calculate the maximum bending stress in a beam with a varying cross section. I found a great resource (link below) that gives examples on how to do this but am a little confused. Basically the equation used is σ=M/S_x I know what σ and M are, but I haven't a clue what S_x is meant to be. Can anyone tell me what this is? It's kind of tough to figure our a way to google "S" and get meaningful results. Thanks!

http://www.aaronklapheck.com/Downloads/Engr112_Handouts/ENGR112%20Solutions/05-03ChapGere%5B1%5D.pdf

PS. What I'm trying to do, is calculate the maximum stress of a boat hull. I'm approximating it as a beam, but the cross section geometry is arbitrary. If anyone has any suggestions about a better way to do this, they are certainly welcome!

 

[AlephZero]

The general formuila is $$\sigma = \frac{My}{I}$$ where y is the distance from the neutral axis.

It looks like he is combining $I$ and the maximum value of $y$ into $$S_x = \frac{I}{y_{\text{max}}}.$$ I've never seen that notation before, but then I learned how to stress beams a very long time ago!

Edit: in one of the problems in the PDF he gives it the name "section modulus". http://en.wikipedia.org/wiki/Section_modulus. Looking at the references on the Wiki page, maybe it's used more as a civil or structural engineering term than in general mech eng.

 

[zzinfinity]

The general formuila is $$\sigma = \frac{My}{I}$$ where y is the distance from the neutral axis.

Thanks for you help. So if I have a beam with a varying cross section (and therefore a variable I) how do I deal with that? Can I just find I at the cross section I want to know the stress at? Or do I need to consider the moment of inertia at other portions of the beam as well?


Also wikipedia denotes the "First moment of Area" with an S. Do you think that could be what it is? http://en.wikipedia.org/wiki/First_moment_of_area

Thanks again.

 

[zzinfinity]

Nevermind. You were exactly right. Sx=I/ymax.
http://www.wikiengineer.com/Structural/BendingStress

 

[AlephZero]

So if I have a beam with a varying cross section (and therefore a variable I) how do I deal with that? Can I just find I at the cross section I want to know the stress at?

Just consider I at that cross section.

But note that for a variable section beam, the maximum stress might not be at the same place as the maximum bending moment. For example I might decrease faster than M as you move along the beam, so M/I increases.

(For a constant cross section, y and I are the same everywhere along the beam so the maximum stress position is the same as the max bending moment position.)

 

[SteamKing]

Sx is the section modulus. This number is used in some design rules when sizing plating-stiffener combinations.