Physics/Critical Bending Moment of a rod
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= bending moment, lb-in
s= yield stress in lb/in^2
I=moment of inertia, in^4
c= outer radius of cylinder
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,
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,
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.
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.