If I can come at this question in crab-wise fashion...
I grew up next to the house of a man who was what you call a shade-tree mechanic. He had enough tools to do some substantial work on neighbor's cars. For several weeks he had a drive shaft from somebody's vehicle sitting out in the open in the grass strip between his house and our house. I couldn't resist the temptation of lifting it up to get a feel for its heft, and I was surprised at how light it was. When I told my father about it, he explained that drive shafts are hollow. He said that for an equal shaft length and an equal weight of steel, a hollow shaft was stronger than a solid shaft.
The statement that you heard is different, in that the weights of the two shafts would be different, if I understand your use of the word "dimensions." The solid shaft would be heavier, and my intuition says that the solid shaft would actually be the stronger one, meaning the statement is wrong.
This, from my understanding, is because the density of the metal of the hollow pipe is greater (if the weights are the same). Therefore it is harder to strain a denser object than a less dense of the same weight
For a given material, and given outer dimensions, the maximum strength is achieved if the rod is solid (not hollow), but this also takes a lot of weight.
For a given material, and given weight, the maximum strength is achieved by a hollow cylinder, that is of larger diameter than the solid rod of that weight.
Most of the strength of a cylinder comes from the outer portions. I think the contribution goes like the cube of the radial position. So, if you took a solid rod and drilled out a half the volume from the center, you do not lose half the strength. You only lose about 10% of the strength, but you've saved on half the weight.
So, strength to weight ratio is better for a hollow pipe than a solid rod.
Easy answer, you are all wrong. As a kid, I saw Mr. Wizard, aka Don Herbert, the original science guy do this experiment with a little asian kid on Nickelodean. Two rods of equal diameter, one solid, one hollow, were both laid across an open span, and weight was added to both. The Hollow pipe showed far less stress than the solid pipe under the same loads. I run a cross-training facility, and used hollow pipe for the pullup bars I fabrcated based upon this evidence. Had the same argument with the guy at the stock yard where I purchased the metal. He kept telling me solid would have to be stronger, he also said he had never heard of the Mr Wizard's World program on Nickelodean. Go figure... If common sence always proved true, we wouldn't need physics.
In simple bending, the solid shaft will be stronger, but .... There is more to consider. For large diameter shafts the weight of the shaft becomes a significant bending load that must be added to the applied load. For a large diameter shaft, a hollow shaft may support a greater applied load. Why, the weight of the solid shaft increases faster than the stiffness and I/c as the diameter increases. A hollow shaft gets stiffer faster than the weight increases as the diameter increases.
If vibration is a concern, then again the hollow shaft will be better as it has less mass and the natural frequency will be higher (usually desirable). If this is a rotating shaft, then forced vibration due to a sagging shaft or out of balance can become a problem sooner for a solid shaft than for a hollow shaft. Also for a rotating shaft (e.g. that drive shaft) the hollow shaft will have less rotational inertia.
And of course there are the economic benefits for a hollow shaft as noted above.
I'm not convinced either way yet. The hollow pipe may deform less due to its own weight, but I am pretty sure the solid rod can sustain greater overall shear stresses. We have some missing information; the density of the rod is important, as is the nature of the object it is supporting.
I remember studying the mathematics of elasticity and I think to prove the answer either way is going to take a lot of work...
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