# Elasticity of a Tubular Cantilever Beam

• mechanic667

#### mechanic667

I have a problem where I have a metal tube that I am modeling as a cantilever beam which is fixed at one end and has a point load at the other end. The material of this beam is 304 stainless steel, the inner diameter is 0.5mm, the outer diameter is 2mm, and the length of the beam is 4.15mm. With this I am trying to determine the maximum deflection of the beam before plasticity occurs. I have tried to find equations online to determine this point but am not finding much. Because I have the material of the beam, I have the elastic modulus, yield strength, and am able to calculate the moment of inertia. If anyone has any input on how to start solving this problem, that would be greatly appreciated!

Welcome to PF. How is the fixed end supported? Is the full circumference of the metal tube welded to a vertical metal wall? How is the load supported at the far end of the tube? Is there a plate welded to the end and the plate supports the weight of the load?

Welcome to PF. How is the fixed end supported? Is the full circumference of the metal tube welded to a vertical metal wall? How is the load supported at the far end of the tube? Is there a plate welded to the end and the plate supports the weight of the load?
Essentially I have a metal tube sticking out of a hole of a plastic part. The metal tube is glued inside of the plastic part prior to exiting the hole. This hole has a relatively small clearance between the plastic part and metal tube, therefore I am treating this problem as a cantilever beam problem where the end that is fixed is the point at which the tube exits the metal part.

What yield criterion are you using?

Essentially I have a metal tube sticking out of a hole of a plastic part. The metal tube is glued inside of the plastic part prior to exiting the hole.
Wouldn't the plastic yield plastically before the metal tube?

• Spinnor
What yield criterion are you using?
Van mises for this case

Wouldn't the plastic yield plastically before the metal tube?
I don't believe so. Essentially what the problem looks like is a a metal tube coming out of a hole in a plastic part, however, once outside the hole there is no plastic material below the metal tube, meaning it can deflect downwards freely if that makes sense. Sorry if that was a little confusing.

Essentially what the problem looks like is a a metal tube coming out of a hole in a plastic part, however, once outside the hole there is no plastic material below the metal tube, meaning it can deflect downwards freely if that makes sense.
What are the Young's modulus plots for this plastic (which plastic material is it?) compared to the Young's modulus plot of 304 stainless?

The ''beam'' is extremely short compared to how wide/deep it is. The way I see it you will not be able to ignore shear deflection because it will have a significant contribution. Plastic usually has a youngs modulus way lower than any steels so the rotation of the rod in the plastic will also have a part to play. Your deflection, as far as I can tell, will have three components

-shear
-bending
-deformation/rotation of beam in plastic.

The stress in the plastic will be difficult to ignore because it might reach yield before the steel rod does. For the steel you can surely use standard bending and shear deflection equations. The rotation in the plastic is not a simple matter. I am not aware of standard equation for such a case.

• Chestermiller
You could use this simplified standard approach: $$\sigma_{y}=\frac{M_{y}}{S}=\frac{M_{y}}{\frac{\pi(D^4-d^4)}{32D}}$$ In this case: $$205=\frac{M_{y}}{\frac{\pi(2^4-0.5^4)}{32 \cdot 2}}$$ Thus: ##M_{y}=160.38 \ Nmm##. Now let's find the deflection: $$y=\frac{FL^{3}}{3EI}=\frac{FL^{3}}{3E \frac{\pi(D^4-d^4)}{64}}$$ In this case: $$y=\frac{38.646 \cdot 4.15^{3}}{3 \cdot 190000 \cdot \frac{\pi (2^4-0.5^4)}{64}}=0.00619418 \ mm$$
But this beam has very unusual dimensions so the accuracy of calculations using beam theory might be lower than expected.

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