Steel Rod

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How to Determine the Diameter of a rod?

A 35 ft long solid steel rod is subjected to a load of 8,000 pounds. This load causes the rod to stretch 0.266 in. The modulus of elasticity of the steel is 30,000,000 psi. Determine the Diameter of the rod.

Well ... usually it's easier to measure the diameter of a rod than it is to set up a tension test, and there's going to be some variability in diameter over the 35 ft length. Also, the weight of the rod may or may not be significant, and if it's hanging vertically, the force on the bottom end (where the load is) is going to be lower than the force on the top end (which is carrying the load plus the weight of the bar itself). But let's ignore all of that

Let's assume the rod has a diameter of d. Then, it has an area of A = pi d^2 / 4. Now, the F = 8000 lb load on the rod will create a stress in the rod, of σ = F / A. It will also create a strain in the rod, and Hooke's law tells us that stress and strain are linearly related by σ = Eε, where E is the material's modulus of elasticity. (Of course, this is only true when the material is behaving elastically, which means we should really know the material's yield stress. That's different for different types of steel, which is another reason we should just get a set of calipers and measure the diameter directly.)

So, we can combine those equations to write

σ = Eε
F / A = Eε
F / (pi * d^2 / 4) = Eε

We know F, we know E. We're looking for d. So, what's ε? The strain is equal to the change in length divided by the original length of the rod, ε = ▲L / L. We know the change in length, and the original length.

F / (pi * d^2 / 4) = E ▲L / L

So, rearranging to solve for the diameter, we get

4 F L / (pi E ▲L) = d^2
d = 2 √[F L / (pi E ▲L)]

I'll let you plug in values. Remember that ▲L is measured in inches, L is in ft, and E is in lb/in^2, so you'll have to do some conversion. You should get a value between 0.5 in and 1 in, which is a reasonable size for a steel rod.

I hope that helps!

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