How to Calculate Stress on Thin-Walled Cylinder Subjected to External Pressure?

[smegal]

1. I am trying to analyse the hydrostatic pressures exerted on a submarine at depth

Firstly If I model the submarine as being two hemispherical ends with a cylindrical section. Am I correct to model them seperately then combine the results at the end to calculate the deformation or is there a better method of doing this.


Variables p=pressure R=radius t=wall thickness E=youngs modulus Pr=poissons ratio
2. I have used the equations

(please ignore the way the equations are laid out it is my personal notation in MATLAB)

sphere_stress=-(p*R)/(2*t);
hoop_stress= -(p*R)/t
longitudinal_stress =-(p*R)/(2*t)

then to calcualte the strains on the cylindrical part.

Circumferential_strain=(1/E)*(hoop_stress-Pr*longitudinal_stress)
Longitudinal_strain= (1/E)*(longitudinal_stress-Pr*hoop_stress)

I am struggling with the equation for the strain acting upon the spherical ends.

Once I have the strain acting upon the sphere I can then calculate the change in length diameter and volume


Basically I am after a hint to how to calculate the strain on the spherical ends and to have my method of modelling the submarine confirmed as being accurate or being informed if there is a better way.
I apologise if this is in the wrong section.

 

[Mapes]

Hi smegal, welcome to PF. The hemispheres would seem to be in a state of biaxial stress, no? How would you calculate the strain for such a case?

 

[smegal]

I am not 100% sure if I have done it right but I believe that for the hemispheres the strain is calculated using

Sphere_strain=(1/E)*(sphere_stress-Pr*sphere_stress)

I read in a textbook (Theory and design of modern pressure vessels John F Harvey) that the stresses acting upon a spherical object are: sigma ₁= sigma₂=Pr/2t

So this is why I have chosen to perform this calculation.

 

[Mapes]

Looks good.

 

[smegal]

Would you say that modelling a submarine as a sphere and a cylinder is an accurate method? Or is there another way that I have overlooked.

Ps thanks for your reassurance. This work was beginning to stress me out a little.

 

[Mapes]

Depends on the depth As long as the hull's not buckling, I think it's a pretty good approach.

 

[smegal]

Thank you! That's that part sorted.

If I was trying to run the calculations backwards to find the safe dive depth (using a safety factor of 1.75) do I need to perform any calculations with regard to the yield strength of the material or do I simply use the circumferential strain acting upon the cylinder (as it is the largest amount of strain acting upon the structure)

 

[Mapes]

Well, that's where the buckling comes in. I suspect that your result will be considerably too optimistic because it doesn't take into account the possibility of buckling. Consider, for example, a thin plastic ruler. It would be beyond your strength to break most rulers through pure axial tension or compression alone. What would actually happen if you compressed it (analogous to applying hoop or longitudinal stress) is that the ruler would simply buckle and break at a much lower applied load than you expected. Make sense?

 

[smegal]

Well, that's where the buckling comes in. I suspect that your result will be considerably too optimistic because it doesn't take into account the possibility of buckling. Consider, for example, a thin plastic ruler. It would be beyond your strength to break most rulers through pure axial tension or compression alone. What would actually happen if you compressed it (analogous to applying hoop or longitudinal stress) is that the ruler would simply buckle and break at a much lower applied load than you expected. Make sense?

Yeah I see where you are coming from. Is there any method for accounting for this?

at my chosen depth and dimensions (10m length 5m diameter) the volume of the submarine changes by -0.31551m^3 this shows a fair bit of buckling. That is the reason I was mentioning the hoop stress as it is the largest stress acting upon the structure. I was hoping that I simply had to modify that equation to account for a certain pressure.

Do you have any ideas what I should do?

 

[Mapes]

at my chosen depth and dimensions (10m length 5m diameter) the volume of the submarine changes by -0.31551m^3 this shows a fair bit of buckling.

You haven't calculated buckling yet; any buckling would destroy the structure. Think of the ruler. What you've calculated is non-buckling compression.

Buckling theory is relatively advanced, doubly so for 3-D structures. I haven't worked in this area and can't recommend a reference. You might take a look at Jones' Buckling of Bars, Plates, and Shells, some of which is available on Google Books, to get an idea of the theory.

 

[Aquasky]

Hi, would like to ask - what's the formula to calculate "thin-wall cylinder subjected to external pressure"?

Found these formulas - but is subjected to internal pressure.

When a thin-walled tube or cylinder is subjected to internal pressure a hoop and longitudinal stress are produced in the wall.

Hoop Stress
The hoop stress can be expressed as:

σh = p d / 2 t (1)

where

σh = hoop stress (MPa, psi)

p = internal pressure in the tube or cylinder (MPa, psi)

d = internal diameter of tube or cylinder (mm, in)

t = tube or cylinder wall thickness (mm, in)

Longitudinal Stress
The longitudinal stress can be expressed as:

σl = p d / 4 t (2)

where

σl = longitudinal stress (MPa, psi)

Please advice for calculation of submersible housing.