I want to use a 4" OD square aluminum beam where the outside pressure
is about 6000psi.  I would like to keep the wall thickness of the beam
no greater then .5" preferably even .25" or less.  How do I figure out
which grade of aluminum: (ie: 6061-T6, 6063-T52) i need in order to do
this and can you show me the step by step solution for solving this so
i know how to modify the problem if a requirement changes.  (I have
only minor amounts of knowledge in this area so I would appreciate an
understandable explination). Thank You.

Wow, this is a very interesting question. I hope that someone can
answer because I would like to see the derivation of the equation. Are
you using the square tube as a rod in a hydraulic cylinder.

Actually i am designing an underwater device that would work at 4,000
meters which is about 6000psi and i cant find the appropriate
equations to figure out what grade or thickness aluminum i would need
to do this.

Does it have to be made of aluminum? There is one material that could
suit much better. Visit www.liquidmetal.com. They might even give you
an answer how it suits your purpose if you use liquidmetal.

Well it doesnt have to be aluminum but I need this to be both
inexpensive and semi lightweight

Can't you counter the problem of 6000 psi by allowing water to enter the beam?

Nope... I need the tube to be sealed to both help compensate for its
weight by displacing the water and to protect parts inside it that
need to stay dry.

You could take myoarin's idea one step further and use pressure
compensation to equalize pressure inside and out.

wont work at that high of a pressure... in order to do that i would
have that air stored somewhere that is already compressed and with
enough volume to make both it and the inside of the beam 6000psi. I
would also have to have a way to change the pressure based on the
outside pressure since its variable up to 6000psi. keep in mind that
air is compressible and water is not.

With all respect for your work to date and your much greater
understanding of your problem and materials, but isn't a square beam
less resistant to pressure than a round one, and isn't aluminum a
rather weak metal for the task?  (especially if you have to seal any
openings for access to the parts inside)

I thought aluminum would be more expensive than steel, but you seem to
have a weight problem, which, of course, rather surprises me when
thinking about a deep sea underwater device.

Pressure compensation is used in oil field drill bits to keep drilling
fluids (high pressure) from entering the bearings. What you do is
allow communication between the two fluids using a sealed piston. I
was supposing that when you said you wanted to keep the parts inside
dry, you meant to exclude the salt water and that the medium inside
didn't have to be air. Could it be an incompressible fluid with good
dielectric properties.

For reasons i cant get into it does need to be a square tube despite
the improved resiliance of a cylindrical one.  The weight issue is
because the entire device will need to be neutrally bouyant and size
as well as cost are both an issue.  There were many other designs for
this device and the reason i need help with this calculation is
because I need to asses how feasible this design is. Keep in mind this
tube is not the entire device, the full device is allot larger and
more elaborate.

Also for other reasons including a part that would have to go inside
the medium has to be air.  I have already considered using a
non-conductive liquid however the equipment wont work in that, i am
using that liquid in other parts of this device.