Given a 3.25" long, 0.125" diameter rubber rod with tensile strength 1200 psi supported on both ends, how much weight can you hung from the middle of it without breaking it? (This is not homework I am trying to design a PC case and I have no idea what I am doing.) How does this change if we use a 0.25" diameter rod instead?
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2 Answers
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You're hanging it from a rod supported by both ends - and need to use the bending equations. For this case (case 7 in the link), the max weight is:
$$W_{max} = \frac{\pi d^3 \sigma}{8L}$$
$\sigma$ is the tensile stress, $L$ is the rod length, $d$ is the rod diameter.
The sag is:
$$\delta = \frac{4W_{max}L^3}{3E \pi d^4}$$ where $W_{max}$ is the weight, $L$ is the rod length, $d$ is the rod diameter, and $E$ is a mechanical property called the Youngs modulus
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$\begingroup$ The max weight is inverse ratio to the tensile strength?? $\endgroup$– chxCommented Nov 4, 2016 at 3:48
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$\begingroup$ You're right - that's off. I did this on my phone. That's the only error I see. $\endgroup$– MarkCommented Nov 4, 2016 at 4:31
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$\begingroup$ It's worth noting the dimension - check now works out: tensile strength is in psi that is pounds per square inch, $d^3$ is inch cubed so that's pounds times inches divided by inches -- you get pounds. In this case, 0.283 lbs for the 1/8" and 2.264 lbs for the 1/4" ones. $\endgroup$– chxCommented Nov 4, 2016 at 7:26
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3$\begingroup$ @chx I'm rather sceptical about this answer, but not enough to downvote it without further discussion. Mark: I suspect a rubber rod is likely to exhibit second order effects; i.e. rather than behaving as a beam I would expect it to behave as a cable. I'll try writing an answer and see what I get. $\endgroup$– AndyTCommented Nov 4, 2016 at 10:10
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3$\begingroup$ I wouldn't doubt it on the second order effects either, but considering 1.) The application was not life threatening 2.) That the people of the future who want to use this answer are probably going to be looking into a more rigid material and 3.) The grade of rubber was not given, I figured I'd give the linear answer first. Please, add the higher order answer - according to our Area 51 stats we've got less than 2 answers per question. $\endgroup$– MarkCommented Nov 4, 2016 at 12:38
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I searched and could not find any rubber material which has the same modulus of elasticity for compression and tension. Also rubbers swell under stress and creep( plastic behavior over time).
So the question does not have an analytical answer. However as a hypothetical case and just for illustration one can try to stablish a range of I, (second moment of area) by testing a short sample of the material under different compression stresses and set a lose curve fitting graph for the I by stablishing a new neutral axis and integrating y^2.da separately for top and bottom parts over and below the axis. And go from there.
