Strain and Stress in Thin-Walled Hoops Under Pressure

[Jony S]

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 σ_h = 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 σ_h = E * strain, where strain = Δd/d ? if not, what is the relation between the stress calculated from the first formula, the strain, and "E" ?

 

[Chestermiller]

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 σ_h = 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 σ_h = 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.

 

[Jony S]

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:

σ_h - v(σ_r+σ_z)= E * strain

?

 

[Chestermiller]

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:

σ_h - v(σ_r+σ_z)= 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?

 

[Jony S]

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) (σ_r is zero and σ_z is σ_h/2 ?)

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

edit: corrected some mistakes

 

[Chestermiller]

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) (σ_r is zero and σ_z is σ_h/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

 

[Jony S]

Yes, that's what I get (assuming that the $(2-\nu)$ is in the numerator

Yes it is :)

Thanks for the help!

 

[Chestermiller]

Yes it is :)

Thanks for the help!

No problem. Nicely done.

 

[Jony S]

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.

 

[Chestermiller]

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.