Actually, this node has so far entirely missed the reason that the bottoms of aluminum beverage cans are curved. It's simple:

Curved walls are stronger

This is true not only for cans, but for all sorts of structures and containers, it's a fundamental engineering principle. Here are some other examples of this principle in action:

• Automobiles. Notice that the hood of your car has few (or no) flat surfaces, it's all curves. Same is true for most of the rest of the surfaces (windshield is another good example - side windows don't have to be as strong, and they have to open, so they aren't curved).
• Corrugated boxes, roofs and so on. The corrugation makes for strength.
• Other containers, such as propane tanks, scuba tanks, all sorts of things. If they have to stack, then the bottom is either curved inward (which reduces the volume the container can hold), or a false bottom is placed around the real bottom to make them stackable.

Why is it this way? Because the strength of a cross-section goes by its cross-sectional moment of inertia. If this doesn't make sense to you, think about this:

Hold a flat sheet of paper in your hand. It's floppy and has no structural strength to speak of. This is because it has almost no moment of inertia around the bending axis (ie, all of its mass is on the bending axis). Now, roll the paper into a tube. Suddenly, the same sheet of paper has considerable stiffness. By rolling it into a tube, you increased its moment of inertia around the bending axis (by moving its mass away from the axis and out to the edge of the tube). This is also why I-beams have that "I" shape, but you're probably tired of this by now.

Keep an eye on the world around you. You will see this principle in action over and over again.

Nifty, eh?

The bottom of the soda can is concave. (The bottom of plastic bottles is convex with a false bottom to make it stand up.)

The sides of the can are (gasp!) not flat either! They are also round--convex in this case. Milk cartons have flat sides. Soda cans do not. Also note that plastic milk jugs are frequently not flat either, but have some kind of indentation in them to increase strength.

The bottom of your propane tank pops from concave to convex immediately prior to catastrophic failure, just because the force to pop the bottom is less than the force to burst the tank, and so that will usually occur first. I don't think it has anything to do with a built in warning. If there was such a thing, it would be in the form of a relief valve, like what is on most liquid air tanks.

Plastic soda bottles probably have a convex bottom because the pressure needed to collapse a plastic concave bottom is so low that the fizz in the soda itself would do it immediately. Jhonbus's explanation below of the bottom sounds right to me.

Beverage cans (usually made to contain carbonated drinks) are designed to hold their shape while pressurised, in fact non-carbonated drinks are pressurised with nitrogen to give their cans strength while being shipped and stored.

Cans must be stackable; that is it must be possible to place one on top of the other in a stable manner. If the bottom of aluminium cans were not made in this concave shape, and were made flat instead, they would tend to bulge outward slightly from the pressure, thus making stacking of the cans impossible. Such methods as adding plastic sleeves to the bottoms of cans is not really feasible, as the cost per can must be kept down to an absolute minimum when such a large volume of product is produced. (This is somewhere round about the trillion can per year mark).

Here in England, and indeed all the other European countries I have visited, our plastic beverage bottles are not designed with a concave bottom and a false base, but are moulded into a pentagonal curvy shape which gives it both strength under pressure, and the ability to stand up on a flat surface. (I can't really describe the shape, but I'm sure there are pictures of it somewhere on the www.)

Previous writeups in this node have suggested a wide variety of potential answers to the “concave beverage can bottom” question, ranging from the fanciful to the mundane to the practical:

• Hovercraft” Theory: One theory says that the concave bottom allows the beverage companies (and for purposes of this writeup, we’ll stick to beer and soda distributors) to blow air under the cans, allowing them to glide along the assembly line like a puck in an air hockey game. This idea really doesn’t hold water, though. Not only doesn’t it have any references to back it up, but it’s pretty clear that the rollers along the average assembly line would allow the cans to move at least as fast as if they were “floating,” making any extravagant “hovercraft” model pretty much superfluous. < /li>

• Capitalist Conspiracy” Theory: This idea says that the concave bottom is really a deep, capitalist conspiracy to cheat “Joe Six Pack” out of a few extra ounces of tasty beverage per can. The problem with this idea is that the product – beer, soda, whatever -- is sold by the volume of the product, not the volume of the can. Twelve fluid ounces, or 355 ml, to be exact, making any reduction in can volume completely irrelevant. So this theory likewise falls short. < /li>
• “Depressurization” Theory: Under this approach, the concave bottom of the can is meant to handle rapid changes in air pressure when the beverage is transported in an unpressurized aircraft. Aside from the fact that most beer and soda is generally manufactured and canned near the market in which it is sold, the simple fact is that the theory doesn’t work. If you take any can of pressurized beverage (beer, soda, carbonated water) on an unpressurized plane up to 35,000 feet, you’ll have an explosion on your hands. I guarantee it. < /li>
• ”Stacking the Cans” Theory: Here, the theory is that the concave bottom is needed to allow the cans to be stacked. Otherwise, the theory goes, the cans would all bulge out from the internal pressure, making it impossible for distributors and grocers to stack the product. That’s all well and good, in theory, but the simple fact is that we don’t see this kind of bulging in real life, whether it’s on the concave bottom or the flat top of beverage cans. So if the flat tops don’t “bulge,” it seems pretty clear that the bottoms could be flat, too, without “bulging.”
• By the way, if you ever do see bulging like this, DON’T CONSUME THE PRODUCT. It’s a sign that something may have gone wrong in the canning process, and that the product may be toxic. Of course, it may just be a sign that the deliveryman dropped the can, in which case it will probably explode in your face. Either way, opening it is probably a bad idea.< /li>

• ”Structural Strength” Theory: Of all the “concave bottom” theories put forth in this node, this one comes closest to being scientifically accurate. But while it’s not inherently wrong –- concave bottoms are structurally stronger –- it doesn’t satisfactorily explain why distributors use them in their product. I mean, steel is stronger than aluminum, so if strength were the only concern, wouldn’t distributors use steel in their cans, instead of aluminum? The fact that they don’t tells us that something besides mere “structural strength” is at work here.
• In fact, that last question gets you to the real answer, now, doesn’t it? Manufacturers want to use the strongest material and shape that they can, while minimizing weight in the process. Why do they want to do this? Money, that’s why.

Companies that manufacture beer and soda containers in the United States produce over 300 million aluminum beverage cans per day. That’s 100 billion cans per year. With that level of production -– more than one can per person in the United States per day -– even a small reduction in unit cost will result in huge savings for the industry. And a reduction in weight translates directly into a reduction in cost -- a one percent reduction in a can’s weight means a savings for the industry of as much as \$40 million.

So there’s big money to be made. In fact, the amount of money at stake is enough to suggest that the shape and materials in your average aluminum can have probably been structurally optimized. The average beverage consumer, in the U.S. and elsewhere, might be surprised to learn that manufacturers of aluminum cans exercise the same amount of attention and precision as engineers designing aircraft wings for the newest experimental planes. It’s that important.

The Optimal Can Shape

All right. Suppose I had twelve ounces of a pressurized liquid. The average pressure in a modern-day beverage can is 90 psi, so let’s assume that’s what I’ve got. Here’s the question -- ignoring all other considerations, what would be the optimal shape for a container holding these twelve ounces of liquid?

The answer is a sphere. Mathematically speaking, a sphere has the smallest possible surface area for any given volume. At least in three dimensions. The reason for this is a question of calculus that goes beyond the scope of this writeup. But just picture the lowly soap bubble. The soap film naturally gravitates to the most efficient –- lowest energy –- state of being, a sphere.

Needless to say, when it comes to manufacturing a container, the smallest surface area means the least amount of material, hence the lowest cost. So why don’t we see spherical beer cans? Well, to some extent, we do. Some manufacturers even make spherical kegs, which are, after all, nothing more than big beer cans. But spheres are hard to distribute. They roll, they don’t stack well, and they’d probably be pretty difficult to drink from.

So what’s the next best thing?

Cylindrical Cans

Well, let’s start with the sphere, a shape we know to be optimal. Now, cut a two-dimensional slice in the sphere. What do you have? A circle. And while a sphere is an optimal shape in three dimensions, a circle is optimal in two. It is the shape that requires the least circumference for a given area to be enclosed.

But beer and soda –- and everything else, for that matter -– don’t exist in two dimensions, they exist in three. So can we take a circle –- the best container shape in two dimensions -– and expand it out to three? The answer, of course, is yes. Just take the circle, and extend it along a third axis as far as necessary to enclose the required volume.

In other words, a cylinder. Which is nothing more than a can. Not only that, it’s a can holding a pressurized liquid, which engineers refer to as a pressure vessel. The can relies on the pressure from within to retain its structural integrity, just as a fire hose relies on the water pressure to maintain its shape.

You’re Tops In My Book

Now that we have a cylinder, what do we do with the top and bottom? I mean, if we’re not careful, we’re going to wind up with something no more useful to drink from than a fire hose. Looking first at the top, it’s clear that it needs to be flat –- not for any engineering reason, but for customer convenience. It’s hard to get your mouth around a spherical top, after all. Insert juvenile joke here.

But while the amount of aluminum necessary to withstand the lateral 90 psi pressure was fairly minimal for the can’s cylinder –- the circular shape was, after all, structurally optimal –- the flat top needs to be significantly stronger. Just imagine trying to hold back the water at the end of a fire hose.

To increase the lid’s strength, manufacturers reduce the amount of manganese in the aluminum alloy, while increasing the amount of magnesium. The typical aluminum alloy in the sides of a beverage can incorporates by weight 1 percent magnesium, 1 percent manganese, 0.4 percent iron, 0.2 percent silicon, and 0.15 percent copper, in addition to aluminum. It is ironed to tolerances within 0.0001 inch, and is made slightly thicker at the top and the bottom for added integrity. In addition to withstanding the 90 psi lateral pressure, the typical aluminum can is able to support up to 250 pounds on its lateral axis. In other words, you can stand on it.

Now, with the top lid, the magnesium content can reach a full 2 percent of the alloy’s weight, with the manganese content being reduced to a trace. While this shift in alloy content makes the lid stronger, it also makes it significantly heavier. To reduce the added weight, manufacturers make the diameter of the lid less than the body of the rest of the can, resulting in the characteristic taper you see in cans today. Even with this weight reduction measure, however, the lid of an average aluminum beverage can often makes up 25 percent or more of the total weight of the can.

Bottoms Up

OK, how about the other end of the can? We still need something to withstand the increased longitudinal pressures of the liquid, but we’re no longer constrained by the consumer’s need to drink out of it. That is, unless the consumer is a frat boy shotgunning a Bud.

The first question is what material are we going to use? Do we use the stronger magnesium-rich material from the top, or do we stick with the alloy we’ve used for the sides? Well, we’d rather not add the extra weight from the top lid all over again, so what we really want to do is to come up with a shape that will be strong enough to hold the liquid, even with the lighter alloy.

What’s the strongest shape we can think of? Fortunately, the answer has been around for thousands of years. Ever seen a Roman Arch? The guiding principle behind this little engineering feat was that it allowed the Romans to cut down on material when they were building stuff –- bridges, aqueducts, you know –- by using a shape that diverted the gravitational pressure from above down to the base, without the need for material to fill the base.

So think the Arc de Triomphe. Think the Roman aqueducts leading into the fabled city. Think pretty much any bridge –- from antiquity to the present –- that spans more than twenty feet. The arch –- a simple yet utilitarian design –- buys you a lot of architectural bang for the buck.

So how does this apply to the bottom of your average Diet Pepsi can? Well, imagine an arch in three dimensions – an arch “in the round” if you will. That’s exactly what you see at the bottom of every soda and beer can made today. The curved shape dissipates the pressure from the beverage around the rim at the bottom, while still allowing the manufacturer to use the lighter, manganese-rich alloy.

Money, Money, Money

So in the end, it all comes down to money. The concave bottom on beverage cans isn’t for “floating,” or for “stacking,” or for “pressure stabilization.” And while the shape is structurally stronger, that’s not the real reason it’s used. At the end of the day, the concave shape allows manufacturers to use the cheapest, lightest materials –- today’s beverage cans now weigh less than 0.48 ounces, compared to 0.66 ounces in the 1960’s –- as a means to reduce costs.

Welcome to Capitalism.

Bibliography

• The Aluminum Beverage Can, William F. Hosford and John L. Duncan, Scientific American, September 1994. More than you would ever want to know, from two writers who have devoted 30 years of their lives to the lowly aluminum can.
• History of the Beverage Can, Museum of Beverage Containers and Advertising (not a joke) (http://www.gono.com/v-tours/sodacone/scone3.htm)
• The Evolution of Useful Things (http:www.eiu.edu/scienced/3290/science/discrepant/popcan.html)
• Soap Bubble (http://wn.wikipedia.org/wiki/Soap_bubble)
• Arch (http://wn.wikipedia.org/wiki/Arch)
• Two years as an undergraduate physics major, combined with years of being the guinea pig test reader for my father's books such as The Scientist Goes to the Seashore, The Scientist Goes to the Mountains, and The Scientist Goes to the City
• Additional experience with several of my father's lesser-known works, such as The Scientist Abandons His Family, The Scientist Refuses to See His First Grandson, and The Scientist Finds Out That His Wife Cheated On Him With A Fireman. I understand that The Scientist Burns in Hell will be a posthumous work.