# How Do You Calculate Stress and Diameter Change in a Spherical Pressure Vessel?

• a_hargy
In summary, calculating stress and diameter change in a spherical pressure vessel involves using the formula for hoop stress and considering the effects of internal pressure and material properties. The resulting calculations can be used to determine the maximum allowable pressure for safe operation of the vessel. Additionally, changes in diameter can be calculated by considering the strain caused by the internal pressure and the resulting expansion or contraction of the vessel. Accurate calculations of stress and diameter change are crucial for ensuring the structural integrity and safety of spherical pressure vessels.
a_hargy

## Homework Statement

The inner diameter of a spherical gas container is 5m, has a wall thickness of 24mm and it is made of steel for which E=200GPa and v=0.29. Knowing the gauge presure in the container is increased from zero to 1.8MPA determine:
a) The maximum normal stress in the container,
b) The increase in the diameter of the container

## The Attempt at a Solution

a) I used the equation stress = pr/2t and got an result of 187.5MPa. Does this seem correct?

b) Its this part I am stuck on. I cannot find an equation for calculating the change in diameter or circumference of a spherical container. Can anyone point me in the direction of some reading material or post some up here? http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/pv.pdf" is the best I could find and it seems to only show the chance in circumference of a pipe.

Thanks in advance guys.

Adam

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a_hargy said:

## The Attempt at a Solution

a) I used the equation stress = pr/2t and got an result of 187.5MPa. Does this seem correct?

This should be correct.

a_hargy said:
b) Its this part I am stuck on. I cannot find an equation for calculating the change in diameter or circumference of a spherical container. Can anyone point me in the direction of some reading material or post some up here? http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/pv.pdf" is the best I could find and it seems to only show the chance in circumference of a pipe.

That formula should work. I assume it is the same as you using the normal stress (hoop stress) and longitudinal stress, with Hooke's law to get the hoop strain and then use that to get the change in radius (which is the increase in the radius).

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Thanks for that.

So is it correct to assume that in this case a spherical container can be treated the same as a cylinder with open ends?

Cheers,
Adam

I believe it can be.

a_hargy said:
Thanks for that.

So is it correct to assume that in this case a spherical container can be treated the same as a cylinder with open ends?

Cheers,
Adam

A cylinder with open ends will not hold any internal pressure! Pressure vessels are closed.

The hoop stress in a cylinder is twice as much as the hoop stress in a spherical vessel.

CS

a_hargy said:
b) Its this part I am stuck on. I cannot find an equation for calculating the change in diameter or circumference of a spherical container. Can anyone point me in the direction of some reading material or post some up here? http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/pv.pdf" is the best I could find and it seems to only show the chance in circumference of a pipe.

You're on the right track...take a look here:

http://www3.esc.auckland.ac.nz/People/Staff/pkel015/SolidMechanicsBooks/Part_I/BookSM_Part_I/04_LinearElasticity%20I/PDF/Linear_Elasticity_05_Presure_Vessels.pdf

CS

Last edited by a moderator:

## 1. What is a pressure vessel expansion?

A pressure vessel expansion is the increase in volume of a sealed container due to the buildup of internal pressure.

## 2. What causes a pressure vessel to expand?

The main cause of pressure vessel expansion is the heating of a gas or liquid inside the container, leading to an increase in pressure. The elasticity of the container material also contributes to the expansion.

## 3. How do you calculate pressure vessel expansion?

The formula for calculating pressure vessel expansion is ΔV = V(ΔP/EP), where ΔV is the volume change, V is the original volume, ΔP is the change in pressure, and EP is the elastic modulus of the container material. This can also be calculated using software or specialized equations.

## 4. What are the potential risks of pressure vessel expansion?

If a pressure vessel expands beyond its design limits, it can lead to structural damage, leaks, or even explosion. It is important to carefully monitor and control the pressure and temperature of the contents to prevent dangerous expansion.

## 5. How can pressure vessel expansion be prevented?

To prevent pressure vessel expansion, proper design and material selection are crucial. The vessel should also be regularly inspected and maintained, and operating conditions should be carefully monitored. Pressure relief devices can also be installed to prevent excess pressure buildup.

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