Why Are Hollow Tubes Almost As Strong As Solid Rods?

A hollow tube keeps almost all of the bending strength of a solid rod of the same outer diameter, because bending stress falls to zero at the center of a beam and rises steadily toward its surface, leaving the core carrying almost no load. Resistance to bending is set by the second moment of area, which scales with the fourth power of diameter, so boring the inner half out of a 50 mm steel rod removes 25% of its weight but only 6.25% of its stiffness. The limit is wall buckling: below a certain wall thickness the tube flattens or crumples, a failure mode a solid rod does not have.

Picture a machinist dropping a 50 mm (2 inch) steel rod into a lathe and boring a 25 mm (1 inch) hole straight down the middle of it. Metal curls off in bright spirals until a full quarter of the bar is lying on the workshop floor. What is left is a tube.

Now the question that matters: how much bending strength did he just throw away? A quarter of the steel is gone, so a quarter of the strength, surely.

He threw away about six percent.

This is one of my favorite results in engineering. It sounds like a cheat, and it turns out to be plain arithmetic. It is also why your bicycle, the scaffolding outside your office, and the drive shaft under your car are all tubes rather than bars. To see why, we need to look inside a beam while it bends.

What Happens Inside A Beam When It Bends?

Find a rubber eraser and bend it into a curve between your fingers. Watch its two long faces.

The face on the outside of the curve gets stretched. The face on the inside gets squashed. (Engineers call these tension and compression. The gap between them matters enormously for materials like concrete, which is far better at being squashed than stretched.) That much you can see.

Here is the harder part, and the important one. The outer face is stretched. The inner face is squashed. So somewhere in between there must be a layer doing neither. Not stretched, not squashed, just along for the ride. It is called the neutral axis. MIT's mechanics-of-materials notes define it exactly so: "At the transition between the compressive and tensile regions, the stress becomes zero; this is the neutral axis of the beam."

Nor does the stress jump from zero to maximum. It climbs smoothly as you move away from that layer: "the axial normal stress, like the strain, increases linearly from zero at the neutral axis to a maximum at the outer surfaces of the beam."

That is the one idea this article rests on, so hold on to it:

In a bending beam, the further a piece of material sits from the center line, the harder it is working. The material at the center is barely working at all.

Is A Hollow Tube Strong Because It Works Like An Arch?

Go looking for an explanation of why tubes are strong and you keep hitting the same one: the curved wall spreads the load around itself, like an arch. It is satisfying, and it is wrong in a genuinely interesting way.

Arch action is real. It is why Roman aqueducts are still standing. But an arch turns a downward load into compression running along its curve. That compression pushes outward into abutments at each end, and the abutments shove back. Stone is poor in tension, so an arch is a trick for keeping it permanently squashed.

A bent tube is doing nothing of the sort. There are no abutments. The wall on the outside of the bend is stretched just as much as the same metal would be in a solid rod. Roll the cross-section into a circle, a square or a hexagon, and the bending arithmetic barely notices. What matters is not that the wall is curved. It is where the material sits.

So What Is The Middle Of A Solid Rod Actually Doing?

Almost nothing.

For a symmetrical shape like a round bar, the neutral axis runs straight down the middle. So in a solid steel rod being bent, the metal along the center line sits at precisely zero stress. The metal around it is at nearly zero. That core carries its full share of the weight. It costs full price. And it contributes almost nothing to resisting the bend.

MIT's notes make the same point bluntly about drive shafts. Shafts twist rather than bend, but the geometry is identical. They "are almost always hollow," because "there isn't much point in using material at the center where the stresses are zero."

Drilling out the middle of a rod is not throwing away strength. It is firing an employee who was never doing any work.

What Is The Second Moment Of Area (And Why It Isn't The Moment Of Inertia You Know)?

To put a number on this, engineers use the second moment of area, also called the area moment of inertia. Every beam-bending calculation runs through it.

First, a warning about a name collision. You may already have met the moment of inertia, the quantity describing how hard it is to spin an object up. That is a different thing that unfortunately shares a symbol. Tell them apart by their units:

If the units are kg·m², it is about spinning. If they are m⁴, it is about bending. Ours is m⁴.

Here is the definition, and the whole article hides inside it:

I = ∫y² dA

In plain English, you do four things:

1. Take every tiny patch of area in the cross-section.
2. Measure its distance y from the neutral axis.
3. Square that distance, then multiply by the patch's area dA.
4. Add up the lot.

The squaring is the entire point. A patch of steel twice as far from the center line does not contribute twice as much. It contributes four times as much. Three times as far, nine times as much. Material at the center contributes essentially nothing, because zero squared is zero.

This is also why an I-beam looks the way it does. MIT's solid mechanics course estimates an I-beam's second moment of area from its two flanges alone. The thin web in the middle is ignored entirely, and the answer is still good. The flanges do the work. The middle is a spacer.

A tube is the same trick, rolled into a circle.

How Much Bending Strength Does A Hollow Tube Actually Lose?

Now we can settle the machinist's question properly.

For a solid round bar of diameter D, the second moment of area is:

I = πD⁴ ⁄ 64

Bore a hole of diameter d down the middle and you simply subtract the missing circle:

I = π(D⁴ − d⁴) ⁄ 64

Weight, meanwhile, follows the area. Area scales only with the square of diameter. So the tube keeps a fraction (1 − (d/D)²) of the weight, but a fraction (1 − (d/D)⁴) of the bending resistance. Squares versus fourth powers. That gap is the entire phenomenon.

Run it for our 50 mm bar with a 25 mm bore, so d/D = 0.5:

1. Weight kept = 1 − 0.5² = 0.75, so 25% of the steel is gone.
2. Bending resistance kept = 1 − 0.5⁴ = 0.9375, so only 6.25% of the stiffness is gone.
3. In real terms, a metre of that solid bar weighs 15.4 kg (34 lb). The tube weighs 11.6 kg (25 lb).

Push further and the deal keeps improving before it gets worse:

Read the bottom row again. Take away 81% of the steel, four fifths of it. The tube still resists bending a third as well as the bar it came from, and it is nearly twice as efficient per kilogram.

(One piece of fine print. At the same outer diameter, the load a beam can carry before it starts to yield falls by exactly the same fraction as its stiffness. Both are set by I and by the unchanged distance out to the surface. So in this comparison, "stiffer" and "stronger" move together. That is not true in general. Strength and stiffness are genuinely different properties.)

Is A Hollow Shaft Stronger Than A Solid Shaft Of The Same Weight?

Everything so far compared a tube to a rod of the same outer diameter. But an engineer with a fixed budget of steel asks a better question: given a fixed amount of metal, is it better spent as a rod or a tube?

Not close. Tube, every time.

Once you can hollow it out, the same steel spreads to a larger outer diameter. And the second moment of area goes as the fourth power of diameter, so that expansion pays enormously.

Re-roll a solid rod into a tube whose bore is 80% of its outer diameter. It ends up 1.67 times wider, and it resists bending 4.6 times better. Same steel, same weight, same cost, nearly five times stiffer. Take the bore to 90% and that steel becomes 9.5 times stiffer than the rod it started as.

Which raises the obvious question. If thinner is better, why not keep going forever?

Why Don't We Just Use Infinitely Thin Tubes?

Because a tube has a way of failing that a solid rod does not have. Thin the wall far enough and the steel never gets near its own strength. The wall itself gives way first.

You have seen this happen. Bend a drinking straw slowly. It does not curve smoothly and then snap. It stays straight, goes slightly oval, then suddenly kinks flat at one point. That kink has a name.

In 1927, L. G. Brazier worked out why. Bend a thin tube and its cross-section is squeezed out of round. Going oval makes it easier to bend. Being easier to bend flattens it further. Round and round, until it lets go.

NASA's design standard for thin-walled cylinders describes that runaway. A tube under uniform bending "ovalizes progressively and this, in turn, reduces the flexural stiffness and results in premature cross-sectional buckling of the tube," ending in "the formation of a crease followed by complete flattening of cross-section."

There is a second version under compression, local wall buckling, and NASA's critical-stress relation shows what governs it:

σ = 0.605 γE(t ⁄ r)

Here σ is the stress at which the wall buckles, E is the stiffness of the material, t is wall thickness, r is radius, and γ is a correction for real-world imperfection. Notice what it depends on: the ratio t/r, not absolute size. Halve the wall at fixed radius and you halve the stress it can take, however big the tube.

Reality is meaner still. Buckling is exquisitely sensitive to tiny deviations from perfect roundness. NASA notes that test cylinders have buckled "at loads sometimes as low as 10 percent of the theoretical values." So structural codes cap wall slenderness outright. For common structural steel, round hollow sections in compression are limited to a diameter-to-thickness ratio of about 64.

A solid rod has no wall to buckle. That is the one thing it is unambiguously better at.

Where Do Hollow Tubes Show Up?

Once you know the trick, you start seeing it everywhere.

Scaffolding and buildings. The steel tubes bolted together outside every construction site are chosen for exactly this reason. The industry has a name for the product family: HSS, short for Hollow Structural Section, which comes in round, square and rectangular forms.

Drive shafts. As MIT put it, they "are almost always hollow." The shaft under a rear-wheel-drive car has to transmit torque without adding rotating mass. A solid core would add the mass and carry almost none of the torque.

Bicycles. Every frame you have ridden is a set of tubes, in steel, aluminum or carbon fiber, for the same geometric reason.

Bamboo. The plant got there first. A culm is a hollow cylinder, and its stiffest fibers are concentrated toward the outer wall. That is precisely where the second moment of area rewards putting them.

Are Bird Bones Really Hollow To Save Weight?

There is one famous example that does not work the way everyone says, and it is worth correcting because it is repeated everywhere. Bird bones are hollow, the story goes, because evolution found the tube trick and used it to make birds light enough to fly.

The research does not support it.

Start with the premise. Bird bones are not uniformly hollow. Their air spaces are called pneumatization, and they are outgrowths of the lungs. They are extensions of the respiratory air-sac system that invade certain bones after hatching. Which bones varies enormously. In the African grey parrot, every hindlimb bone from the femur down is solid. Penguins, auks and diving ducks have lost the trait almost entirely.

Then there is the claim itself. In 1979, Prange and colleagues measured skeletal mass against body mass across birds and mammals. They found the avian skeleton "is not proportionately lighter than that of mammals." A larger sample replicated it in 2015. A 2025 Royal Society review states it flatly: bird skeletons "weigh as much as those of (non-chiropteran) mammals of similar body mass, irrespective of the extent to which they are pneumatized." Where the saving has actually been measured, in chickens, it comes to 0.7–0.8% of body mass. That is less than a bird's weight varies over an ordinary day.

So how do birds manage such delicate-looking skeletons? They build them from dense bone. Measured against bats and rodents, bird bone tissue is the densest of the three. And the same review draws the line straight back to our machinist: "If a bone of a given mass is inflated, it will enjoy greater resistance to bending or torsion, but will nevertheless be made more susceptible to failure by buckling."

Even evolution has to respect the drinking straw.

So, Why Are Hollow Tubes Almost As Strong As Solid Rods?

Because bending is not a job a cross-section shares equally.

When a beam bends, its outer surfaces do nearly all the work. The material along its center line does nearly none. And every patch of metal contributes in proportion to the square of its distance from that line. Weight, meanwhile, scales only with area. Those two different exponents are the whole trick. Removing the core deletes a great deal of mass from a region contributing almost nothing. That is why a quarter of the steel can vanish and take only six percent of the stiffness with it.

Turned around, that becomes a design principle: given a fixed amount of metal, push it as far from the center line as you dare. It explains the I-beam, the scaffolding tube, the bicycle frame, the drive shaft and the bamboo culm all at once. The same answer, in different materials.

And the reason it is "almost" as strong rather than exactly as strong is the drinking straw. Push too far and the wall stops behaving like solid material. It starts behaving like a sheet, ready to crease and flatten. Every tube you have leaned on is a settlement between those two facts.

That is what I find genuinely beautiful here. There is no clever material and no hidden mechanism. Nobody invented anything. The strength was always in the geometry, sitting in the exponent on a distance, waiting for someone to notice that the middle of the bar was never pulling its weight.