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Beer Can Blueprints (May '22 Challenge Question)

Posted April 30, 2022 12:00 AM
Pathfinder Tags: challenge question

What are the practical reasons to have concave and beveled bottoms on beer cans? (Okay, so it applies to soda pop cans too.)

The image is of a flat-bottomed beer can, for reference.

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#1

Re: Beer Can Blueprints (May '22 Challenge Question)

04/30/2022 4:11 AM

Concave and convex are both much stronger than flat to resist the internal pressure. If you made them convex you'd have to add a rim to stop them falling over.

Differential pressure can be very high especially if you shake the can, or, reduce the external pressure: there's nearly always at least one unsuspecting victim who buys a can of beer or soda from the shop at the top of mount Teide.

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#2

Re: Beer Can Blueprints (May '22 Challenge Question)

04/30/2022 1:47 PM

To save money, the thinner the can, the less material used...

The cone top..

https://beerandbrewing.com/a-brief-and-condensed-history-of-the-beer-can/

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#3
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Re: Beer Can Blueprints (May '22 Challenge Question)

04/30/2022 9:25 PM

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#15
In reply to #2

Re: Beer Can Blueprints (May '22 Challenge Question)

05/05/2022 5:04 PM

Here’s an interesting story… in our farming community, our family farm in particular… we had a lot of immigrant settlements on all sides of our farm.

we had Irish to the west, polish to the east, Scandinavians to the north and Germans to our south…

well,… the Germans loved their beer, always had an open can/bottle of beer during milking…

the problem occurred that that these early beer cans resembled oil cans for your car… (I was told it was brake fluid, but that’s too hard to believe.)

anyways this neighbor thought he pick up a can of beer, and took a swig out of it… it wasn’t beer.

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#4

Re: Beer Can Blueprints (May '22 Challenge Question)

05/01/2022 7:01 AM

For the amount of aluminum used, the concave bottom is the strongest, for the following reason. The bottom of the can is a circle, attached to the cylindrical sides. For the internal pressure to force the bottom out requires it to increase in area (as seen from the top). A flat or convex bottom would decrease in area. Since the circle has the maximum area for a given circumference, the bottom is much stronger, reinforced by the cylindrical side of the can, to resist the internal pressure.

This equates to requiring less aluminum, less cost and less energy required to produce the can of the required strength.

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#5

Re: Beer Can Blueprints (May '22 Challenge Question)

05/02/2022 6:30 AM

I've been trying to figure out what the questions means by "beveled". I guess they mean this:-

Is that something to do with getting the can out of the press?

Incidentally that picture come from this page

https://www.beeradvocate.com/community/threads/can-bottom-commentary.338464/

I never knew they put little "jokes" on the bottom of cans.

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#6
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/02/2022 8:13 AM

<removed>

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#7
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/02/2022 8:15 AM

How it's made...

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#8
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/03/2022 7:25 AM

That's great, but, he almost seems to deliberately avoid mentioning the beveling of the bottom:-

He goes to great lengths to explain the beveling at the top, and I'm guessing that a similar argument applies to the bottom, but the steps to keep the beveled part thin compared to the central bottom must be very complex.

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#9
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/03/2022 3:55 PM

I've been trying to figure out what the questions means by "beveled".

I'm thinking it is the part where the bottom meets the side, which if extended would be a cone.

If you put 165 ml of water in, you can balance it on the "bevel".

https://instructional-resources.physics.uiowa.edu/demos/1j2065-balancing-soda-can

I think if it weren't for the bevel top and bottom, there would be a concentration of stress at the corner where the bottom or top meets the side. The bevel curvature, being a compound surface curve provides strength, it can't easily be deformed without distorting the metal.

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#10

Re: Beer Can Blueprints (May '22 Challenge Question)

05/05/2022 6:08 AM

Here's a new question:

Ignoring all the practical considerations: what is the optimum shape of a simple cylindrical can?

By optimum I mean least area for a given volume.

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#11
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/05/2022 8:51 AM

Ignoring all the practical considerations: what is the optimum shape of a simple cylindrical can?

By optimum I mean least area for a given volume.

Fit a hemisphere at both ends, the shape of propane tanks.

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#12
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/05/2022 12:08 PM

Sorry, by simple cylindrical I meant flat top and flat bottom: you're looking for the ratio of h to r.

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#13
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/05/2022 3:39 PM

Assume r is constant. What value of h gives maximum V/A.

If r is constant, increase height from 0 ...

Volume = (pi*r^2) * h;

Area = 2*(pi*r^2) * h + 2*pi*r^2;

Area starts out with the area of top and bottom, Volume starts out at 0.

Area increases twice as fast as Volume.

Volume/Area never reaches a maximum, approaches 0.5 as h/r becomes infinite.

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#14
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/05/2022 4:42 PM

Assume r is constant. What value of h gives maximum V/A.

Correction...

If r is constant, increase height from 0 to infinity

Volume = (pi*r^2) * h;

Area = 2*(pi*r) * h + 2*pi*r^2;

Both Area and Volume are linear functions of h

Area increases "r/2" times as fast as Volume.

As h increases to infinity for any given r, the ratio V/A approaches "r/2" as V and A become large enough to "swamp out" the constant terms (not depending on h).

Example, r=3 cm, V/A approaches 1.5 cm as h approaches infinity

Bottom line: There is no maximum V/A, it approaches a limit as h becomes infinite

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#16
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/06/2022 7:20 AM

This is interesting. I can't see a mistake in your reasoning, but I'm sure there's something wrong.

I think that making r a constant may not be a valid thing to do, although I would have assumed it was OK if I'd started that way.

This is my solution

I expected the answer to be close to that because height equal to diameter is close to a sphere, but I was surprised it came out exact.

In fact when I first did it I used my calculator to find r (the cube root of one upon 2pi) = 0.5419 ; then h (1 upon pir2) = 1.0839 ; and I thought hold on that looks like exactly double and the calculator confirmed it. I was then able to complete the last line above.

I have chosen to make the volume a constant, and I can't justify that any more than I can invalidate your choice.

Because we all know that a sphere is the optimum arbitrary shape I still think that your answer must be wrong somehow.

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#17
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/06/2022 8:51 AM

One thing I see is that you are taking the derivative of A instead of V/A. You would be getting the max or min of area-to-volume ratio, not volume-to-area ratio.

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#18
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/06/2022 10:38 AM

Yes: with fixed volume = 1; minimum area coincides with maximum volume to area ratio:-

I've just multiplied the V/A by 100 so that the graph looks OK on the same scale as the Area.

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#19
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/06/2022 1:15 PM

I'm convinced that your answer is correct. H = 2R is closest to a sphere. I coded it up with volume = 2.

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#20
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/06/2022 6:50 PM

As happens often, we end up with a puzzle in a puzzle in a puzzle. I think this is the answer to why you got one answer and I got another.

V(r,h)=Volume,

A(r,h)=Area, and

R(r,h)=Volume/Area

are functions of two variables, r=radius and h=height.

Below is an overlapped contour plot where constant R curves are plotted in red and constant Volume are plotted in black. The vertical axis is the radius and the horizontal axis is the height of the cylinder.

My solution (constant radius) is the blue horizontal line. Note that it crosses increasing values of constant R curves, never doubling back, so there is no maximum, just an asymptotic approach to a limiting value.

Your solution follows one of the black contours (constant volume). As you can see, it crosses increasing R curves, reaching a maximum, and dropping back. The green line plots the maximums that are on the line h=2r where these maximums occur.

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#21
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/07/2022 4:33 AM

If the height is equal to the diameter, that provides the most volume for material used...but, if you stack the cans and envision each as the most efficient use of material, then envision the top and bottom removed and one continuous can results, then wouldn't that be more efficient? ...or would the material combined into a larger can of the same h=d relationship provide more volume? ...This is the question...

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#22
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/07/2022 5:43 AM

Surface area of a cylinder....

A = 2 π r h + 2 π r 2

Volume of a cylinder....

V = π r2 h

https://www.calculatorsoup.com/calculators/geometry-solids/cylinder.php

For a cylinder with radius of 2 and height of 4...

Total surface area = A = 75.3982237 m2

Total volume = V = 50.2654825 m3

For a cylinder with radius of 2 and a height of 8...

Total surface area = A = 125.663706 m2

Total volume = V = 100.530965 m3

..an increase of of total surface area of 50.2654823 m2

and an increase in total volume of 50.2654825 m3

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#23
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/07/2022 6:01 AM

Now if we take 2.52 radius and double that to 5.04 we get a square can...

V = 100.549865 m3

total surface area

A = 119.70222 m2

equals more volume and less surface area than the 2r and 8h can...

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#24
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/07/2022 6:33 AM

What all three of us missed here is that if you choose any shape of can: then increase its size linearly in all direction: then volume increases as the cube of linear expansion whilst area only increases as the square of linear expansion. So any shape when grown enough will beat any other shape with a sufficiently smaller volume.

Curiously in my original question in post #10, I did say "By optimum I mean least area for a given volume.". But I can't claim that I understood the significance at the time.

The obvious conclusion: beer cans should be bigger.

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#25
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Re: Beer Can Blueprints (May '22 Challenge Question)

05/07/2022 8:48 AM

Honest officer I only had one beer....

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