Water Tank

[TimeSpaceLightForce]

The std. 4ft x 8ft ga.#16 G.I. sheet is to be fabricated (cut,bend,weld) into a large water tank.
If the design is an open top box..what is the best capacity ?

[Moontanman]

About 120 gallons...

[imatfaal]

Best capacity is 130.23 us gallons.

 

You make your tank with these dimensions

 

length = sqrt(32/5)*sqrt(5/3) = 3.266

bredth = sqrt(32/5)*sqrt(5/3) = 3.266

depth= sqrt(32/5)*sqrt(5/12) = 1.633

 

that gives you a volume of 3.266*3.266*1.633 = 17.42

with a surface area 3.266*3.266 + 3.266*1.633*4 = 32

[mp][/mp]

 

1. Assume that plan with largest base (ie length*bredth) will also give best volume. Safe to assume - if you think on these lines; whatever depth you choose will maximise volume when area of base is maximised.

 

2. So what is best rectangular shape to maximise area for any given circumference?

a. Assume square base is best but set up variations and see if any are better

ie working on the basis of a 1x1 square -> Area = (1-x)(1+x) and vary x to see if Area is ever greater

Area = (1-x)(1+x) = 1-x^2 This will always be at maximum when x=0 as -x^2 will always otherwise be negative. Obviously when x=0 the figure is a perfect square.

 

3. We can set up a similar equation fo volume. Again assume a 1x1x1 five-sided open box - vary the edges and see if we get a greater volume than 1 (always bounded by the assumption in 2 above that best volume will be with square base

 

length = 1+x

bredth = 1+x

depth = ?

 

We are constrained by surface area - our test five sided 1x1x1 would have a surface area of 5 time 1x1 - ie 5 square units. So to work out depth we take total surface area - subtract that of base (we now have the area of all four vertical sides), divide by four because we need 4 sides( we now have the area of each vertical side), and divide by the length or bredth (and we now have the depth)

 

[latex]depth=\frac{5-(1+x)(1+x)}{4(1+x)}[/latex]

 

Volume obviously equals length x bredth x depth

 

[latex]Volume = (1+x)(1+x)\left(\frac{5-(1+x)(1+x)}{4(1+x)}\right)[/latex]

 

This time you need to differentiate and then set dVolume/dx to zero to get maxima / minima

 

this occurs at [latex]x = -1 +sqrt(5/3) [/latex]

 

Plug this back in and you get length of sqrt(5/3), bredth sqrt(5/3), and depth of sqrt(5/12)

 

4. That was variation of unit box - to get to a box of surface area of 32 square feet we multiply by sqrt(32/5) which is the size of 5 square sides with total area of 32 square feet

 

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That took about 5 times longer to type that it did to work out. Hopefully it is right after all that typing

[TimeSpaceLightForce]

@moontanman
Not enough capacity.

@imatfaal
Correct theoretical maximum..
but so many pieces to cut and weld not the best way for the fabricator.

Maybe fewer than 5 pieces to cut will do.

 

[Moontanman]

@moontanman

Not enough capacity.

@imatfaal

Correct theoretical maximum..

but so many pieces to cut and weld not the best way for the fabricator.

Maybe fewer than 5 pieces to cut will do.

cutline.jpg

 

 

 

I make a lot of aquariums, often from 4 x 8 pieces of plywood, the largest size from a 4x8 sheet is 48" long by 24" high and 24" wide that is 16 cubic feet, at 7.5 gallons per cubic foot that comes to 120 gallons, I don't see how imatfaal got 130 gallons from that...

[DrKrettin]

 

 

I make a lot of aquariums, often from 4 x 8 pieces of plywood, the largest size from a 4x8 sheet is 48" long by 24" high and 24" wide that is 16 cubic feet, at 7.5 gallons per cubic foot that comes to 120 gallons, I don't see how imatfaal got 130 gallons from that...

 

 

Jeez - I wish you guys would use sensible units and not antediluvian ones. The problem is that inatfaal is working as a mathematician and comes up with a maximum volume for a surface area. But it does not take into account the practical issue that the sheets are always certain dimensions which prevent that ideal solution being put into practice. Your solution uses all the wood of a 4x8 sheet nicely, but is not the theoretical maximum volume.

[John Cuthber]

Assume a perfectly spherical fishtank...

[Endy0816]

Assume a perfectly spherical fishtank...

lol, what I was thinking

[TimeSpaceLightForce]

@moontanman

With 4x8' g.i.sheet your 4Lx2Wx2H tank should be done by cutting it into two pieces
and would cost 8'cut,8'bend & 14'weld.

But some 3 pieces cut,even though it cost more, gives you extra volume.

-Initially she can cut 3.266 x 6.532 off any corner making the base and 2 opposite sides.

[Moontanman]

Assume a perfectly spherical fishtank...

 

 

Hmmm, difficult to bend glass and plywood i guess, my own biases have bitten me again...

[John Cuthber]

 

 

Hmmm, difficult to bend glass and plywood i guess, my own biases have bitten me again...

Not sure it it's "biases". The OP talks of welding. Does that work with plywood?

[imatfaal]

Jeez - I wish you guys would use sensible units and not antediluvian ones. The problem is that inatfaal is working as a mathematician and comes up with a maximum volume for a surface area. But it does not take into account the practical issue that the sheets are always certain dimensions which prevent that ideal solution being put into practice. Your solution uses all the wood of a 4x8 sheet nicely, but is not the theoretical maximum volume.

In my defence in the question there was no mention in op of it being a sensible solution

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And being very sensible I can get 16.592 cubic feet from five pieces. Making a 2 2/3 x 2 2/3 x 2 1/3 tank

 

 

Sorry for the iPad graphics.

 

 

Five pieces - requires six straight cuts to get from 4x8

 

Reassemble using four welds to form second shape

 

Bend at each of the purple wavy lines and at junction of blue area and clear

 

Weld at two corner joints along one base edge and one centre joint

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And regarding the units - I agree SI is massively preferable. But in America you buy a 4x8 which makes for easy puzzles - in England you buy a 1220mm x 2440mm which is exactly the same thing but less good for puzzles; and also shows a disturbing lack of confidence and commitment to the metric system

[Moontanman]

Not sure it it's "biases". The OP talks of welding. Does that work with plywood?

 

 

Wood glue and wood screws... This is what i build, this guy uses the same method as i have, he makes a pretty accurate video, I can fault in a couple things he does but for the most part he drives the nail in the wood!

 

[John Cuthber]

...

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And regarding the units - I agree SI is massively preferable. But in America you buy a 4x8 which makes for easy puzzles - in England you buy a 1220mm x 2440mm which is exactly the same thing but less good for puzzles; and also shows a disturbing lack of confidence and commitment to the metric system

Nope, you buy stuff by the "metric foot" because it generally has to fit into a house that was built in feet + inches.

It's not a "lack of commitment" it's a recognition of reality as it stands.

 

On the other hand, if I follow your instructions, here in the UK I get a tank that holds 103 gallons, but if someone across the pond makes the same tank it holds 124 gallons.

That makes feet + inches look quite sensible.

[DrKrettin]

Nope, you buy stuff by the "metric foot" because it generally has to fit into a house that was built in feet + inches.

It's not a "lack of commitment" it's a recognition of reality as it stands.

 

But that doesn't explain why that is the exact size of a sheet sold here in Spain. I'm sad enough to have just measured one to be quite sure.

[John Cuthber]

 

But that doesn't explain why that is the exact size of a sheet sold here in Spain. I'm sad enough to have just measured one to be quite sure.

It's a good deal easier to cut it down 120 by 240 than it is to stretch it.

[wayne_m]

The place where I work stretches metal sheets. Of course, we use presses that pull sheets over dies with hundreds of tons of force, but it's what we do.
If we're allowed to stretch the material, then we could make a tank of any size, though if it gets large enough, the water pressure at the bottom would burst the thin material

 

Of course, if we assume material strength holds true to that extent, then we have to allow that the material would break before we stretched it that far, unless we did it at a high temperature and with a special press or roller.

 

It would be more economical just to buy more material.

[John Cuthber]

Good point: 16 Ga is fairly thick

You could sell it and buy a larger area of thinner sheet + make a bigger tank from that.

[TimeSpaceLightForce]

@Imatfaal
Right Adjustment!

 

Design Option Table:

[imatfaal]

@Imatfaal

Right Adjustment!

 

Design Option Table:

wtank.jpg

 

I liked the idea of going for a setup like 3 piece maximum - with base at an offset angle - but just could not visualise it clearly enough in my head and my draughtsmanship skills are not good enough to work on paper for something like that. As I was convinced that such an answer existed I am pleased to see it - and happier that my eventual answer was better I found the same answer for two pieces and worked from there to the five piece answer - rather I had a six piece answer for the same 2 2/3 x 2 2/3 x 2 1/3 and looking at the two piece answer made me realise how I could get that tank with only five pieces (I was stuck on symmetrical sets

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Two great puzzles TimeSpaceLightForce +1