Thick walled spherical vessel under external pressure

[Mark27]

Homework Statement

Hello. I need help to design a spherical vessel submarine with minimum mass which can withstand water pressure at depth of 8000 meter. It must satisfy this conditions:

1. vessel shall have minimum interval volume of 10m^3
2. outer diameter must less than 5m
3. radial elongation of vessel shall not more than 0.05% of diameter due to compression

variables:
Ro=Outer radius
Ri =Inner radius
r=radius at middle between outer and inner radius
ρm=density of material
v=poisson ratio
E=modulus of elasticity
σr= radial stress
σh=hoop stress

Homework Equations

The Attempt at a Solution

Because the pressure at 8000m is so high, I need to use thick walled formula.

i)So, in order to find minimum mass, the objective function is:

m= 4/3*pi*(Ro-Ri)*ρm

ii)Minimum internal volume:

10 ≤ 4/3*pi*(Ri^2)

iii)Maximum outer diameter:

2Ro ≤ 5

iv)von mises criterion (which i believe)

(σr^2)-(σr*σh)+(σh^2) ≤ (yieldstress^2)

v)radial deflection (which i believe)

r/E*[σr-vσh-vσr]≤ 0.05(2Ro)

**************************************************​

So i believe the only problems right now are to find the equation for radial stress and hoop stress under external pressure only. Most references that I read only give equation for internal pressure only and internal & external pressure only. Thank you for your feedback.

 

[SteamKing]

Homework Statement

Hello. I need help to design a spherical vessel submarine with minimum mass which can withstand water pressure at depth of 8000 meter. It must satisfy this conditions:

1. vessel shall have minimum interval volume of 10m^3
2. outer diameter must less than 5m
3. radial elongation of vessel shall not more than 0.05% of diameter due to compression

variables:
Ro=Outer radius
Ri =Inner radius
r=radius at middle between outer and inner radius
ρm=density of material
v=poisson ratio
E=modulus of elasticity
σr= radial stress
σh=hoop stress

Homework Equations

The Attempt at a Solution

Because the pressure at 8000m is so high, I need to use thick walled formula.

i)So, in order to find minimum mass, the objective function is:

m= 4/3*pi*(Ro-Ri)*ρm

ii)Minimum internal volume:

10 ≤ 4/3*pi*(Ri^2)

iii)Maximum outer diameter:

2Ro ≤ 5

iv)von mises criterion (which i believe)

(σr^2)-(σr*σh)+(σh^2) ≤ (yieldstress^2)

v)radial deflection (which i believe)

r/E*[σr-vσh-vσr]≤ 0.05(2Ro)

**************************************************​

So i believe the only problems right now are to find the equation for radial stress and hoop stress under external pressure only. Most references that I read only give equation for internal pressure only and internal & external pressure only. Thank you for your feedback.

It's not clear why you are stuck. Unless the occupants of the submarine will be working in a vacuum, it seems from a cursory glance of the problem requirements that the stress formulas for internal & external pressure would apply. In any event, if you are studying or have studied strength of materials, you should be able to derive the formula for radial and hoop stresses for thick-walled spherical pressure vessels.

Have a look at Section 4.1.4 of this article, which treats the case of a spherical PV with simultaneous internal and externally applied pressures:

http://solidmechanics.org/text/Chapter4_1/Chapter4_1.htm#Sect4_1_4

If this doesn't float yer boat, as it were, you can always google "spherical thick walled pressure vessels".

 

[Mark27]

Thank you SteamKing for the answer. Actually I'm not sure whether to include the internal pressure or not. This really helps me a lot. Thank you.

 

[AlephZero]

Actually I'm not sure whether to include the internal pressure or not. This really helps me a lot.

The external pressure is about 800 bar. Taking the internal pressure as 0 or 1 bar won't make much difference to the answer.

In any case, the minimum mass solution is unlikely to be a uniform shell, and the sphere is more likely to fail in buckling than by compression. But it's not clear from the question whether this is an exercise in calculus, or an open-ended engineering design problem.

 

[Mark27]

This question is actually more to open ended question. I need to design the theoretical submarine using Matlab but need to confirm some formulas. I believe my problem right now is to find the right formula for uniform stress which acting on the submarine which is radial stress and hoop stress.

The best reference so far is: http://fetweb.ju.edu.jo/staff/che/ymubarak/Strength-lectures/chapter8.pdf

But as SteamKing said, Maybe I need to make assumption on the internal pressure or find the internal pressure
:)

 

[SteamKing]

Mark27:
You are probably looking at an iterative process to satisfy all of the design requirements, not only keeping the stresses below the limits but also in meeting the dimensional requirements of the sphere, i.e., making sure the internal volume > 10 cu.m. after the sphere deflects due to the net pressure. While the internal pressure may be atmospheric ambient when the sphere is on the surface of the water, the sphere's diameter will change with depth, which will change the internal volume and pressure of the sphere slightly.

Also, remember that seawater is slightly denser than fresh water.

 

[Mark27]

Yes, i need to satisfy the design requirement.

So, i know that pressure at 8000m seawater is:

ρgh = 1025*9.81*8000 = 80.4MPa

So, my concern right now is that should i just make an assumption of the internal pressure which is external pressure & internal pressure = 80.4MPa

Or

Take any logic numbers below 80.4MPa and use it as internal pressure

Or

I need to find the internal pressure at atmospheric ambient.

Mark27:
While the internal pressure may be atmospheric ambient

 

[SteamKing]

Yes, i need to satisfy the design requirement.

So, i know that pressure at 8000m seawater is:

ρgh = 1025*9.81*8000 = 80.4MPa

So, my concern right now is that should i just make an assumption of the internal pressure which is external pressure & internal pressure = 80.4MPa

Or

Take any logic numbers below 80.4MPa and use it as internal pressure

Or

I need to find the internal pressure at atmospheric ambient.

The whole raison d'etre of the sphere is so the internal pressure WON'T be equal to the external pressure. How could anyone survive working in an 80 MPa environment?

You're not thinking clearly on this.

 

[SteamKing]

Mark27:

To further guide you:

When the sphere is fabricated on land, its internal and external radii are going to be two certain values, the magnitudes of which your design will provide. However, when the sphere is lowered into the water and the external pressure increases with depth, the outer radius is going to decrease because of this pressure, and it would be logical to assume that the internal radius would change as well. When you get the sphere to a depth of 8000 m, the internal volume of the sphere in its compressed form must still be greater than 10 m^3.

 

[Mark27]

Hehehe. Okay, I think I get some ideas how to do it now. Thank you SteamKing. You help me a lot :)