Come on sailor, 4000+ plus posts and you still believe there is such a thing as a simple answer around here? I’ve got a bridge I can sell you cheap. 
One thing you can’t forget is that, IRL, depending on the load, buckling can become an issue (IIRC, it is called “Euler Buckling?”) This can make a hollow cylinder with the same amount of material in its cross-section, even though it would normally be stronger than the solid one, fail earlier.
But then, I’m not a structural engineer, so I may be misremembering. Maybe I should stick to power plants…
I’m not saying sailor’s example was bad, in fact I loved it. I was just commenting on the fact that when you think you have the world’s best example, so simple even a 4 year old could understand its inherent truth, that there will always be someone who will try to pick it apart.
And you’re right, buckling is a major consideration. For many beams, they will fail by buckling (or elastic instability) long before they reach their yield point. But buckling is also a whole other course. You’ve got to look at things like torsional rigidity, critical lengths, etc. In general though, sections with larger moments of inertia resist buckling better that sections with smaller moments.
Well maybe I’m dense, Why A Duck, but I take this to mean cross-sectional area, including the empty space of the pipe. Meaning that the outside diameter of the rod and the pipe are the same.
I wish I could find a better cite, but my feeling is it has to do with surface area, which means the pipe has a large advantage.
hmmm.
I always thought that Euler buckling was a concern for structural columns in compression, and has nothing to do with beams. Am I wrong?
Stress engineer who does this sort of thing for a living weighing in.
As far as answering the OP exactly as phrased, Why a Duck is right on the money, which is why I hadn’t posted to this thread yet.
Er, what do you think columns are made of, grienspace? Any structure that is subject to compressive loading, be it a beam, plate, or shell is subject to potential buckling. More frequently, though, you are going to be concerned with beams. You may be thinking bending --> beams and buckling --> columns. As far as solving the equations, beams and columns are the same thing, but buckling and bending are two different animals.
And if anyone’s interested in some beam buckling formulas:
P[sub]crit[/sub] = cEI/L[sup]2[/sup]
where
P[sub]crit[/sub] is the critical load (max allowable load before buckling occurs)
c = a factor based on boundary conditions (= [sym]p[/sym][sup]2[/sup]/4 for free at one end, fixed at the other; = [sym]p[/sym][sup]2[/sup] for pinned at both ends; = 20.2 for pinned at one end, fixed at the other; = 4[sym]p[/sym][sup]2[/sup] for free at one end, fixed at the other)
E = Young’s modulus of the beam material
I = Area moment of inertia of the beam cross section
L = Length of the beam
Source: An Introduction to the Mechanics of Solids by Crandall, Dahl, and Lardner. Freakin’ Roark. It’s such an excellent reference, but I swear the most basic formulas are the most difficult to find in those tables.
Use this formula wisely as you go about your everyday lives.
Please help me clarify the difference between bending and buckling.
Does “buckling” means the collapse of a column into/onto itself? Does it mean the deformation from a cylinder shape (eg. one side of a tube beginning to flatten out or begins to get squashed)?
When I bend a (hollow) steel tube, it first bends but eventually one or both sides flattens out. Is this bending and buckling?
Eh, sorry I dissed you there, grienspace. Now that I look at my references, I see that the word “column” is almost always used to describe a member under compression. As far as support columns in a building, for example, they’re essentially beams serving to support compressive loads.
Bending is deformation of a beam due to a transverse load. It may be elastic (beam returns to its original shape after being released) or plastic (beam yields and doesn’t quite return to its original shape after release).
Buckling is an instability due to compressive axial loading. It’s elastic in nature, unless you keep applying the load after it buckles (but then, that isn’t really a buckling phenomenon anymore).
To illustrate: get a metal ruler. Clamp one end in a vice. Apply a load at the other end perpendicular to the beam. The deformation you see is bending. Now, place the ruler upright with one end on a rough surface (so it doesn’t slide) and push down axially on the other end. When it gives and the middle bows out, that’s buckling.
OK, I set myself up for a straw man dismissal of my post by giving too many cases for when a hollow cylinder would be stonger than a solid cylinder of the same diameter when loaded in bending. Now that we have some professional stress analysist here, will somebody please verify my example a brittle material. An easy to analyze example is a brittle material having the basic Weibull strength distribution with a Weibull modulus of 1.5. For this case, you will find that the hollow cylinder is stronger in bending as long has the hole diameter is less than about 0.46 of the outer diameter. The strongest cylinder for this case is the one with a hole diameter of 0.35 of the outer diameter.
The OP seemed well aware that the solid cylinder would be expected to be stronger in most cases and was looking for a counterexample. All the explanations given so far for why it is impossible do not prove much. They just verify what seems intuitively obvious to most people, while making assumptions which eliminate some possiblities.
Oops, sorry I didn’t acknowledge your post, Manlob. I admit I missed it. But I see what you’re saying now as far as how the OP phrased his question. Mea culpa. I blame my misinterpretation of the OP and my neglect of your post on two things: my crappy attention span and my attempt to work and post simultaneously.
To be honest, my experience is almost entirely with ductile materials (isotropic at that; I’m so spoiled), but I see what you’re saying regarding the statistical distribution of imperfections in the material. My own personal analogy would be the size correction factor as applied to fatigue strength. Since I am much more knowledgeable about ductile than brittle behavior, I’m going to defer to you on that analysis. In the meantime, I’ll grab a textbook and teach myself a little more about brittle material behavior (and perhaps weigh in with my verification when I figure it out).
OK, so it’s [sym]s[/sym] that’s the measure of strength, but [sym]s[/sym] is geometrically related to I, so it’s usually expressed in terms of it. Gotcha.
Well, not exactly. [sym]s[/sym] means stress, in units of force per unit area. For a beam in elastic bending, it’s proportional to the distance from the neutral axis (the boundary between the side in tension and the side in compression), and the point of interest. The peak stress is at the furthest point from the neutral axis, so only in that sense is it related to geometry.
I is moment of inertia, and that is a function of geometry - basically how widespread the cross-sectional area is on average from the neutral axis. The amount the beam bends is proportional to I, but neither deflection nor moment of inertia is directly related to [sym]s[/sym].
So yes, for a given amount of material, you want to spread it as far apart as you can in the plane of loading until the beam is limited by buckling (or brittle crack propagation for some materials) instead.
The moment of inertia, I, in the beam bending equation is more precisely called the “area moment of inertia” or “second moment of area”. This is not the same thing as the mass moment of inertia that may be more familiar to students of physics. Mass moment of inertia is a function of density and geometry of a three-dimensional object and has units of ML[sup]2[/sup]. Area moment of inertia is a function of the geometry only, has units of L[sup]4[/sup], and is for a two-dimensional shape. However they are both evalulated with very similar formulas and depend on an axis.