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# Physics/Critical Bending Moment of a rod

Question

molded plastic part
Hello Steve,

How are you? Thanks very much for volunteering on this site, and thank you also for your previous answers. I am wondering if you could help me understand something I have been trying to figure out about the weight that a rod is able support before bending. I have done some calculations, but I am not sure of myself, and I was wondering if you could cast your expert eye over the situation and tell me if/where I have gone wrong.
I have a design for a molded plastic part that features two objects connected by a thin rod. As you can see in the attached image, one object is a kind of frame (colored grey) and the other green cylinder is connected to the frame with the thin red rod. I have been trying to calculate the required dimensions of the rod that are needed in order to prevent the rod from bending under the weight of the green cylinder. I know the yield stress of the material, so I tried using a formula for the critical bending moment I found here - http://www.engineersedge.com/wwwboard/posts/3844.html (The formula was for the critical bending moment of a hollow cylinder, whereas mine is sold, but I thought it might be a step in the right direction.)
The formula given was
M=sI/c
where
M= bending moment, lb-in
s= yield stress in lb/in^2
I=moment of inertia, in^4

I cheated with the moment of inertia and used an online calculator found here - http://civilengineer.webinfolist.com/str/micalcc.php to get a moment of inertia for my 0.0197 inch diameter rod of 7.3932399799976E-9

The other values were
Material Yield stress - 10,000 psi
Moment of inertia - 7.3932399799976E-9 in^4
Outer radius of cylinder - 0.00985 inches

So I did the following calculation -
(10 000 * 7.3932399799976E-9) / 0.00985 = bending moment of 0.00750582739 lb-in

Have I done this all correctly? This is the first time I have thought about this, so I would be surprised if I got it all right.

Any comments would be much appreciated!
Thanks and regards,
Eddie

Hello Eddie,

I am afraid that I can't confirm the formulas that your search has provided. In Physics, moment of inertia has dimensions of mass*length^2. So it has units like kg.m^2. The moment of inertia formula in the sites you gave me,
I=(pi/64)*(Do^4-Di^4),
yield a result with dimension of length^4, no mass involved. (I added the first pair of parentheses -- I believe that is how it was meant to be interpreted.)

Since your rod is solid, that formula reduces to I=(pi/64)*Do^4
It's almost like Civil Engineers have a different definition of moment of inertia. It does seem that your need is the same as a Civil Engineer would have, so maybe they have a different method that somehow yields the same final result that a thorough analysis with standard Physics formulas would give. But I don't know that.

Go to
http://en.wikipedia.org/wiki/List_of_moments_of_inertia
Scroll down to the 2nd to last -- "Solid cylinder of radius r, height h and mass m". I assume the axis of rotation that applies to your study is perpendicular to the length of the cylinder. So according to the wikipedia page,
I = (m/12)*(3*r^2 + h^2)
That is a formula for moment of inertia that fits my understanding of moment of inertia. But using that formula to calculate a value for I and then using it in the method to yield critical bending moment in the sites you gave me would not be valid.

I guess the result of all this is that I have to say: sorry, I just don't know how to help.
Steve
Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Thanks Steve, I appreciate your answer. Sorry for my late response. I was busy digesting everything. In your answer you said "sorry, I just don't know how to help" but this was not true at all. It was very helpful as your answers always are. Thanks again!

Physics

Volunteer

#### Steve Johnson

##### Expertise

I would be delighted to help with questions up through the first year of college Physics. Particularly Electricity, Electronics and Newtonian Mechanics (motion, acceleration etc.). I decline questions on relativity and Atomic Physics. I also could discuss the Space Shuttle and space flight in general.

##### Experience

I have a BS in Physics and an MS in Electrical Engineering. I am retired now. My professional career was in Electrical Engineering with considerable time spent working with accelerometers, gyroscopes and flight dynamics (Physics related topics) while working on the Space Shuttle. I gave formal classroom lessons to technical co-workers periodically over a several year period.

Education/Credentials
BS Physics, North Dakota State University
MS Electrical Engineering, North Dakota State University