# Elasticity of a Tubular Cantilever Beam

• mechanic667
In summary, the metal tube can deflect downwards freely before it yields, plastic has a lower Young's modulus than 304 stainless steel, and the stress in the plastic will be difficult to ignore when determining the deflection of the metal tube.
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?

berkeman said:
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?

mechanic667 said:
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
Chestermiller said:
What yield criterion are you using?
Van mises for this case

berkeman said:
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.

mechanic667 said:
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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## 1. What is the definition of "Elasticity" in the context of a tubular cantilever beam?

Elasticity refers to the ability of a material to deform under applied force and return to its original shape once the force is removed. In the case of a tubular cantilever beam, elasticity is important because it determines the beam's ability to withstand bending and twisting without permanent deformation.

## 2. How is the elasticity of a tubular cantilever beam measured?

The elasticity of a tubular cantilever beam is typically measured using a stress-strain curve. This involves applying different levels of force to the beam and measuring the corresponding amount of deformation. The slope of the curve at any given point represents the beam's elasticity or stiffness.

## 3. What factors affect the elasticity of a tubular cantilever beam?

The elasticity of a tubular cantilever beam is affected by several factors, including the material properties of the beam (such as Young's modulus and Poisson's ratio), the geometry of the beam (such as its length and diameter), and the type of loading applied to the beam (such as bending or torsion).

## 4. How does the elasticity of a tubular cantilever beam impact its performance?

The elasticity of a tubular cantilever beam is directly related to its ability to withstand external forces without breaking or permanently deforming. A beam with higher elasticity will be able to support larger loads and experience less deformation under the same amount of force. This is important for ensuring the structural integrity and safety of the beam.

## 5. Can the elasticity of a tubular cantilever beam be improved?

Yes, the elasticity of a tubular cantilever beam can be improved by using materials with higher Young's modulus, optimizing the beam's geometry for the specific loading conditions, and using reinforcement techniques such as adding additional layers or changing the cross-sectional shape. However, it is important to note that increasing the elasticity of a beam may also increase its weight and cost.

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