Hollow section stiffness

Why does a hollow tube resist twisting better than a solid bar of equal weight?

Take two pieces of steel of the same length and the same weight. One is a solid round bar; the other is a tube of larger diameter with a thin wall. Grip each in a vice and twist. The tube wins, and not narrowly — it can be several times harder to turn.

That reads like something for nothing. The same metal, the same mass, and yet a hole through the middle makes it stiffer. Either the weight is not what is doing the work, or removing material has done something the word "removing" hides.

Ask first what twisting a bar actually asks of the metal. When one end rotates relative to the other, every longitudinal fibre of material is dragged sideways. How far does a given fibre have to move? That depends on where it sits. A fibre on the axis rotates on the spot and moves essentially nowhere. A fibre at the surface travels the full arc. So the strain — and therefore the stress the fibre carries — grows in proportion to its distance from the centre.

Now push one step further, because this is where the surprise comes from. Each fibre resists with a force proportional to its distance from the axis, and that force acts on a lever arm equal to that same distance. So its contribution to the resisting torque goes as distance squared. Material at half the radius does not pull half its weight — it pulls a quarter.

What does that say about the core? At the axis, the contribution is zero. Near it, close to zero. The metal in the middle of a solid bar is being carried around for almost nothing: it has all of the weight and hardly any of the duty. Drilling it out costs you very little stiffness — and buys you the weight to put somewhere useful.

Where is useful? As far out as possible. And that is the whole trick: the same mass, re-laid at a larger radius, is worth vastly more. We can put a number on it for a round section. Torsional stiffness follows the polar second moment of area; for a solid bar of radius R it is pi times R to the fourth over two, and for a thin-walled tube of radius r and wall t it is about two pi r cubed t. Hold the weight constant — that is, hold the cross-sectional area constant — and the ratio of tube to bar comes out as twice the square of r over R. Make the tube twice the radius of the bar it replaces, and you have eight times the torsional stiffness for the same kilogram of steel.

So has anything been created? No — and that is worth saying plainly. The metal has exactly the same properties it always had. What changed is the arrangement: the geometry decides how much of the material's strength is actually reachable by the load. The function came from the structure, not the substance, which is why you cannot get the same gain by specifying a better grade of steel. Stiffness in torsion depends on the shear modulus, and that barely varies across ordinary steels — a high-strength alloy twists just as far as mild steel under the same torque. Shape is the only lever that moves this one.

Is the gain unlimited, then? Make the wall thinner and the radius larger forever? No, and the limit is instructive: at some point the thin wall stops behaving as a wall and buckles locally — it dents, kinks, or ripples — and once it has folded, the tube is no longer a closed ring and loses almost all of it at once. The optimum is a trade between the quadratic reward for radius and the wall's own stability.

Three clarifications keep this from being over-applied.

First, the closed-loop condition is specific to torsion. A slit tube is still an excellent beam in bending, because bending does not depend on a closed shear path — which is why open sections such as I-beams and channels are everywhere in bending applications and almost never used to carry twist. If a member has to do both, closed sections (round tube, square hollow section) are the honest choice.

Second, "hollow is better" is a statement about stiffness per unit weight, not per unit space or per unit cost. The tube achieves it by being physically bigger, so where the envelope is tight the solid bar can still win. Hollow sections also cost more to make, and joining them is harder — welding a closed section means you cannot reach inside, and bolting through one crushes it unless it is sleeved.

Third, the argument is about stiffness — how much it twists. Strength, the torque at which it yields, follows a closely related geometric term and improves for the same reason, but the two are separate quantities and a design can be limited by either.

• Why an I-beam is the same reasoning applied to bending, with material pushed to the flanges.
• Local buckling limits, and the wall-thickness ratios that codes impose to prevent them.
• Why a driveshaft is a tube while a drill bit is solid, and what changes the answer.

Every term the collection defines is gathered in the glossary.