A different take on the hollow vs solid rod

[jgscott987]

I understand the fact that a solid rod is stronger than a hollow rod given the same diameter. My question is whether this fact takes into account the weight of the rod itself.

Imagine two 20 foot long 1/2" diameter steel rods- one is solid and the other is hollow. Support these two rods at either end. Which rod deflects more under its own weight? Which rod deflects more with a 10 lb load hanging from its center?

My instinct is that the additional weight of the solid rod overcomes it's inherently greater rigidity. The hollow rod should deflect less than the solid rod in this scenario.

Any thoughts? I don't know the formulas or software to calculate this, it's just something that I was pondering as I fell asleep last night.

Thanks!

[Mapes]

Hi jgscott987, welcome to PF. For the case of a cantilevered rod deforming under its own weight, your instinct is correct. You can prove this to yourself with the beam bending equation (http://www.engineersedge.com/beam_bending/beam_bending8.htm), plugging in the weight per unit length and moment of inertia (http://en.wikipedia.org/wiki/Second_moment_of_area) for a hollow beam.

I'll leave it to you to work out the answer for the case of a single attached load.

[jgscott987]

Hi jgscott987, welcome to PF. For the case of a cantilevered rod deforming under its own weight, your instinct is correct. You can prove this to yourself with the beam bending equation (http://www.engineersedge.com/beam_bending/beam_bending8.htm), plugging in the weight per unit length and moment of inertia (http://en.wikipedia.org/wiki/Second_moment_of_area) for a hollow beam.

I'll leave it to you to work out the answer for the case of a single attached load.

I can't even find the MoI or the MoE for a steel rod. I'm just satisfied that I am correct! :)

Thanks for your reply.

[stewartcs]

I understand the fact that a solid rod is stronger than a hollow rod given the same diameter. My question is whether this fact takes into account the weight of the rod itself.

Imagine two 20 foot long 1/2" diameter steel rods- one is solid and the other is hollow. Support these two rods at either end. Which rod deflects more under its own weight? Which rod deflects more with a 10 lb load hanging from its center?

My instinct is that the additional weight of the solid rod overcomes it's inherently greater rigidity. The hollow rod should deflect less than the solid rod in this scenario.

Any thoughts? I don't know the formulas or software to calculate this, it's just something that I was pondering as I fell asleep last night.

Thanks!

The moment of inertia term appears in the denominator of the deflection equation for a cantilevered beam under its own weight and with a point load. Hence, for a given radius the deflection will always be less for a solid rod since its moment of inertia is higher.

CS

[Mapes]

The moment of inertia term appears in the denominator of the deflection equation for a cantilevered beam under its own weight and with a point load. Hence, for a given radius the deflection will always be less for a solid rod since its moment of inertia is higher.

Check that calculation again. The weight is higher, too.

[Vanadium 50]

The way to think about it is that if I bend a rod, I am placing one side under tension and the opposite side under compression. Somewhere in between, there is a line where the material is neither under tension or compression. I can therefore remove this material without affecting the strength - I've now turned a rod into a tube, which is pound-for-pound stronger than a rod.

[FredGarvin]

The moment of inertia term appears in the denominator of the deflection equation for a cantilevered beam under its own weight and with a point load. Hence, for a given radius the deflection will always be less for a solid rod since its moment of inertia is higher.

CS
If you hold the OD constant and then equate the max deflections of the hollow tube to the solid tube you can come up with a relation between the size of ID required and the loads. If I did my algebra correctly I come with:

d_i^4 = d_o^4\left[1-\frac{P_2}{P_1}\right]

[MikeyW]

The way to think about it is that if I bend a rod, I am placing one side under tension and the opposite side under compression. Somewhere in between, there is a line where the material is neither under tension or compression. I can therefore remove this material without affecting the strength - I've now turned a rod into a tube, which is pound-for-pound stronger than a rod.

Can you repeat this for an already slightly hollowed rod though?

A hollowed rod will now have boundary conditions on its normal surfaces inside, so bending a hollow rod will then compress the upper surfaces of the outside and inside parts. So I don't see your (very interesting) argument can be applied to more than a very thin amount of rod down the centre.

[jgscott987]

This is very interesting- I just wish I could add something to the subject. Maybe I'll go pick up some tubes/rods from the hardware store this weekend.

[Vanadium 50]

Can you repeat this for an already slightly hollowed rod though?

Yes, but then it requires a quantitative argument. You need to think about how much strength has been removed in light of how much material has been removed. You can do this, of course, but it doesn't have quite the same intuitive appeal.

[jgscott987]

So you guys really are going to make me go to the hardware store this weekend... :)

I'll post pictures if I find anything interesting.

[mjjohns3]

I did some quick calcs with a cantelever steel beam accounting for the weight (assuming all the weight is concentrated at the end, should yield same basic result though...)

OD .1m, ID .08m

The answer I got was yes & no...

The solid beam deflects more under its own weight across the board. However, when loaded, the solid beam deflects less to a point then they cross and the hollow beam deflects less.

So, for practical applications, Solid beams are better for short beam lengths while hollow beams are better for long beam lengths. I'd be interested to see the results of your project turned out jgscott987.

-Matt