# Is the pressure higher in the corners?

I was watching the video The Ingenious Design of the Aluminum Beverage Can. At 20'', the author says

[..] a spherical can [..] has no corner, so no weak points because the pressure in the can uniformly stresses the wall.

and later talking about a cuboid

[..] these edges are weak points and require very thick walls.

It is intuitive to me that in a sphere the wall is stressed uniformly. It is also quite intuitive that edges of a cuboid cans are weak points as they can be hit from the exterior from a wider angle.

However, the statement says that in a sphere there is "no weak points because the pressure in the can uniformly stresses the wall" suggests to me that in a cuboid the pressure is not uniform and eventually higher in the corners. This is not intuitive to me!

• Is the pressure higher against the corners than against flat walls in a cuboid?

• How much does the pressure against the walls vary (if it does) in a cuboid at equilibrium?

• When not at equilibrium (typically when the cuboid is accelerated), it the pressure higher against the corners?

In Thermodynamic equilibrium, the pressure will be the same throughout the whole can. If there was a region with higher pressure, this high pressure would immediately cause a motion in the gas. So the pressure is the same everywere. The author you cited emphasizes that the pressure can uniformly stress the wall. The property that is not uniformly distributed in case of a corner is not pressure, but how it stresses the wall. In a corner, you give pressure more surface that it can act on, which increases the force that is exerted on the "corner point".

An additional note by user1583209: A cornerpoint is mutch more stressed by pressure than a point in the middle of a wall, because the force acts normal to a surface. At a corner point, the walls to the side of the corner will be pushed in different directions, which results the corner being torn apart.

An additional note by alephzero: Since there is gravity inside the can, the pressure will be higher at the bottom of the can. This effect is a) completely negligable for an aluminium can, when it's about the force that the wall has to endure. b): This effect is independent from wether there is a corner in the wall, or the wall is flat.

• ...and also the forces on different sides will be in different directions, pushing them apart. Nov 25, 2016 at 18:11
• The pressure is not exactly uniform throughout the can. In the liquid, it will change by an amount $\rho g h$ where $h$ is the depth in the liquid. But that is probably negligible in the context of the OP's question. Nov 25, 2016 at 22:59
• Both of you guy's are right, I will edit my answer corresponding to that. Nov 25, 2016 at 23:01

The internal pressure in the can is resisted by the walls in two different ways: (1) by stretching the wall and (2) by bending it.

A typical beverage is made of material that is as thin as possible (to save weight and cost), and therefore its stiffness in bending is very small compared with the stretching. The ratio of the two stiffness values can be of the order of 1,000,000:1 or even higher. (This huge difference in stiffness explains why it is very easy to crush an empty can by bending it!)

To resist the internal pressure by stretching, the tension in the wall of the can varies like $P\kappa$, where $P$ is the (almost constant) internal pressure and $\kappa$ is the radius of curvature of the can at any point. For a sphere, the radius of curvature is the same everywhere, and therefore the stress is the same everywhere.

If the can has "flat" surfaces, like a cuboid, the radius of curvature over the flat sections is infinite and the result given above would mean the stress was "infinite". In real life, the internal pressure causes the "flat" surface to bulge outwards and thus become curved, but the radius of curvature is still very large compared with a sphere.

This also explains why cans and plastic drink bottles don't have a flat base. To reduce the stress, it doesn't matter whether the base is curved outwards or inwards relative to a flat surface, but curving it "inwards" means the can or bottle can stand on a flat surface without toppling over. (That also appeals to the marketing department, because the visible "size of the can" is bigger than the internal volume of liquid you are buying!)

The transition from the cylindrical body of the can to the inwards-curved base increases the stress around the edge of the base, but for a beverage can the ability to stand in a stable position is more important than the higher stress. Compare that with a compressed gas cylinder, which has a much higher internal pressure but doesn't need to stand unsupported, and where the base (usually a hemisphere) curves "outwards".

Since the top of the can needs to be flat for practical reasons, the top has to be made of thicker material than the rest of the can - though a small amount of "bulging upwards" at the top doesn't matter too much. As with the base, high pressure gas cylinders have hemispherical tops not flat ones, to reduce the stress.