Strain and Stress in Thin-Walled Hoops Under Pressure

In summary: For a cylinder in deep space, it's the pressure inside.Yes. p is really the pressure difference between inside and outside. For a cylinder in atmospheric air, it's the gauge pressure inside. For a cylinder in deep space, it's the pressure inside.
  • #1
Jony S
14
0

Homework Statement



Hi all, this isn't exactly homework, but it is nonetheless a problem I would like to solve, so here goes.

Consider a thin-walled ring/hoop with diameter "d" and thickness "t". I want to know the change of diameter "Δd" due to the stress caused by pressure "p" being applied in this hoop (from inside to outside).

Homework Equations



So, EngineeringToolBox tells me that the stress in the hoop is σh = p d / (2 t)

The Attempt at a Solution



Can I apply Hooke's law directly here ? knowing the young's modulus of elasticity "E" for my material, can I say that σh = E * strain, where strain = Δd/d ? if not, what is the relation between the stress calculated from the first formula, the strain, and "E" ?
 
Last edited:
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  • #2
Jony S said:

Homework Statement



Hi all, this isn't exactly homework, but it is nonetheless a problem I would like to solve, so here goes.

Consider a thin-walled ring/hoop with diameter "d" and thickness "t". I want to know the change of diameter "Δd" due to the stress caused by pressure "p" being applied in this hoop (from inside to outside).

Homework Equations



So, EngineeringToolBox tells me that the stress in the hoop is σh = p d / (2 t)

The Attempt at a Solution



Can I apply Hooke's law directly here ? knowing the young's modulus of elasticity "E" for my material, can I say that σh = E * strain, where strain = Δd/d ? if not, what is the relation between the stress calculated from the first formula, the strain, and "E" ?
Are you familiar with the 3D version of Hooke's law, which includes all the stress and strain components.
 
  • #3
Chestermiller said:
Are you familiar with the 3D version of Hooke's law, which includes all the stress and strain components.

These concepts are fairly new to me so I'm not "familiar" with much. Are you talking about Poisson's ratio, i.e., the expansion/compression of the material in perpendicular directions ? so I would add this:

σh - v(σr+σz)= E * strain

?
 
  • #4
Jony S said:
These concepts are fairly new to me so I'm not "familiar" with much. Are you talking about Poisson's ratio, i.e., the expansion/compression of the material in perpendicular directions ? so I would add this:

σh - v(σr+σz)= E * strain

?
Yes. This is one of the equations you would use. The other relationship is the strain-displacement equation for the hoop strain. Do you know that equation?
 
  • #5
Chestermiller said:
Yes. This is one of the equations you would use. The other relationship is the strain-displacement equation for the hoop strain. Do you know that equation?

No.

Searched a bit more and found this:

https://nptel.ac.in/courses/Webcourse-contents/IIT-ROORKEE/strength of materials/lects & picts/image/lect16/lecture16.htm

In the end of the first part there is a formula for displacement. This appears to be the combination of the two equations that we already had, i.e., strain = deltaD/D, and
strain = PD/4tE * (2-v) (σr is zero and σz is σh/2 ?)

v and E are characteristics of the material so we have everything ?

edit: corrected some mistakes
 
Last edited:
  • #6
Jony S said:


No.

Searched a bit more and found this:

https://nptel.ac.in/courses/Webcourse-contents/IIT-ROORKEE/strength of materials/lects & picts/image/lect16/lecture16.htm

In the end of the first part there is a formula for displacement. This appears to be the combination of the two equations that we already had, i.e., strain = deltaD/D, and
strain = PD/4tE * (2-v) (σr is zero and σz is σh/2 ?)

v and E are characteristics of the material so we have everything ?

edit: corrected some mistakes
Yes, that's what I get (assuming that the ##(2-\nu)## is in the numerator
 
  • #7
Chestermiller said:
Yes, that's what I get (assuming that the ##(2-\nu)## is in the numerator
Yes it is :)

Thanks for the help!
 
  • #8
Jony S said:
Yes it is :)

Thanks for the help!
No problem. Nicely done.
 
  • #9
Chestermiller said:
No problem. Nicely done.
Extra question for clarification, the stress is the same (opposite signal) if the pressure is applied externally, right ? If the pressure is the same inside and outside the total stress has to be zero.
 
  • #10
Jony S said:
Extra question for clarification, the stress is the same (opposite signal) if the pressure is applied externally, right ? If the pressure is the same inside and outside the total stress has to be zero
Yes. p is really the pressure difference between inside and outside. For a cylinder in atmospheric air, it's the gauge pressure inside.
 

What is "strain due to hoop stress"?

"Strain due to hoop stress" refers to the deformation or change in shape of a cylindrical object, such as a pipe or a barrel, when it is subjected to internal pressure. This strain is caused by the force of the pressure acting on the walls of the cylinder, resulting in a change in its diameter and length.

How is strain due to hoop stress calculated?

The formula for calculating strain due to hoop stress is ε = (PD)/(2tE), where ε is the strain, P is the internal pressure, D is the original diameter of the cylinder, t is the thickness of the cylinder's walls, and E is the modulus of elasticity of the material.

What factors affect strain due to hoop stress?

The main factors that affect strain due to hoop stress are the internal pressure, the diameter and thickness of the cylinder, and the material properties such as its modulus of elasticity and yield strength. Other factors that may have an impact include temperature, corrosion, and external forces.

What are the potential consequences of high strain due to hoop stress?

If the strain due to hoop stress exceeds the material's yield strength, the cylinder may experience permanent deformation or even failure. This can lead to leaks, ruptures, and other structural failures, which can be dangerous and costly. It is important to properly design and maintain cylindrical objects to prevent high strain due to hoop stress.

How can strain due to hoop stress be minimized or controlled?

To minimize or control strain due to hoop stress, engineers can use thicker walls, stronger materials, or reduce the internal pressure. Additionally, proper design and maintenance, such as using corrosion-resistant materials and regularly inspecting for any defects or damage, can help prevent excessive strain and prolong the lifespan of cylindrical objects.

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