Calculate Cylinder expansion?

[houndoghill]

I have a cylinder that is pressurized with 3000 psi. There is another non-pressurized cylinder that slides over the pressurized cylinder.

I need help determining if the non-pressurized cylinder will bind on the pressurized cylinder caused from expansion.

Key information:
Cylinder length 16"
Pressurized cylinder outside diameter 7.78" +.001, - .005
Non pressurized cylinder inside dia. 8.00" +.005, -.001
Wall thickness (t) .75"
Inner radius (r) of pressurized cylinder 6.48”
Pressure (p) 3000 psi
Material 2219-T87 Aluminum
Application: Pyroshock design for an environmental test lab.

I have determined that the hoop stress will be pr/t and the longitudinal stress will be pr/2t.

Q1: Can anybody give some insight how I would calculate the change in diameter caused by the pressure?
Q2: Any recommended changes of the two cylinders and tolerances?
Q3: How did you come up with your results?

Thanks,
Houndoghill

 

[EdDanzer]

Look in the Machinery's Handbook. You will need the Modulus of Elasticity, and Poisson’s ratio for the specific material

 

[prex]

For metallic materials the maximum allowable elongation is of the order of 0.1%: for aluminum the allowable stress is around 10 kpsi, the elastic modulus close to 10 Mpsi, so you get the quoted ratio. Now take the double to be on the safe side.
The outer diameter (and the inner too BTW) will at most increase in that proportion (a little less in fact, due to lateral contraction effects, but we can safely neglect this), so it will be close to 7.80", quite far from the minimum inner diameter of the outer cylinder.
Reversing the way of reasoning, the change in diameter to get interference is of the order of 3%: such a deformation is only possible in the plastic range for a metal.

prex

http://www.xcalcs.com

Online tools for structural design

 

[Cockroach]

I'm not going to derive it, search through previous issues regarding Von Mises-Hencky equation. Note that this is the natural extension of Thick Wall Pressure Vessel for a triaxial wall element stress, then applied to Lame's Equation.

For your particular problem, the textbook receipe would be:

dR = (rP/E)[(1 + r^2/R^2)/(1 - r^2/R^2) + mu]

dR = change in external radius,
r = internal bore radius,
R = external cylinder radius,
P = internal pressure,
mu = Poisson's Ratio which is material dependent,
E = Youngs Modulus of Elasticity.

Note the dimensionless expression of the second bracket, that which are squared. This is the typical geometric multiplier found in textbooks. Therefore, diametrical expansion is a linear relation to internal pressure.

Hope this helps you out.

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada

 

[desertfox]

Hi houndoghill

According to Roark & Young the formula for your situation is as follows :-


delta (a) = (q*a/E)*(b^2*(2-v)/(a^2-b^2)

where delta (a) = change in outer radius of pressured vessel

q = internal pressure

a = outer radius of pressured vessel

b = inner radius of pressured vessel

E = modulus of elasticity of pressured vessel
material

v= poissons ratio.


I would also point out that the stresses you have calculated
are for thin walled vessels and not for a thick walled vessel which is what you have.


hoop stress in a thick walled vessel is:-

Smax= q*(a^2+b^2/(a^2-b^2))


regards desertfox