Thickness of a cylinder in a compound thin cylinder

[DevonZA]

Homework Statement

[/B]

Note: inner cylinder thickness = 2.5mm
difference between common diameters before shrinking = 4.305x10^-3mm

Homework Equations

The Attempt at a Solution

a) I am not sure where to start without either the inner or outer cylinder diameter but I think I need to use the following:

The sum of the changes in diameters of the two cylinders is equal to the interference between the diameters before shrinkage:

Δd1+Δd2=4.305x10^-3mm
Pd1/2t1 + Pd2/2t2=4.305x10^-3mm

Please help me get started

 

[DevonZA]

edit the answer give for part a) is 2mm

 

[haruspex]

Although the info E=200GPa is mentioned in part b, you will need this in part a.
Δd1 and Δd2 represent a stretch in the outer cylinder and a compression of the inner one. In terms of those, what will the tension and compression be (per unit length)? What must the relationship be between the two forces?

 

[DevonZA]

Although the info E=200GPa is mentioned in part b, you will need this in part a.
Δd1 and Δd2 represent a stretch in the outer cylinder and a compression of the inner one. In terms of those, what will the tension and compression be (per unit length)? What must the relationship be between the two forces?

Will both cylinders not be compressed/shrunk?
Sorry I do not follow

 

[haruspex]

Will both cylinders not be compressed/shrunk?

Perhaps I misunderstand the arrangement. Any diagram?
I read it that two cylinders have been fabricated, one with an external diameter of slightly over 100mm, the other with an internal diameter slightly under. The larger will not slide over the smaller because of a 4.305x10^-3mm overlap. Through heating, the larger cylinder expands and is slid on, then allowed to cool. This will produce tension in the outer cylinder, increasing its internal diameter to 100mm, and compression in the inner cylinder, contracting its external diameter to 100mm.

Do you read it differently?

 

[DevonZA]

Perhaps I misunderstand the arrangement. Any diagram?
I read it that two cylinders have been fabricated, one with an external diameter of slightly over 100mm, the other with an internal diameter slightly under. The larger will not slide over the smaller because of a 4.305x10^-3mm overlap. Through heating, the larger cylinder expands and is slid on, then allowed to cool. This will produce tension in the outer cylinder, increasing its internal diameter to 100mm, and compression in the inner cylinder, contracting its external diameter to 100mm.

Do you read it differently?

Unfortunately no diagraM.
I imagine something like this:

I could be (most likely) wrong so I really don't know.
All I do know is that the thickness of the outer cylinder must equal 2mm, I just don't know how to get there.

 

[haruspex]

Unfortunately no diagraM.
I imagine something like this:
View attachment 197120

I could be (most likely) wrong so I really don't know.
All I do know is that the thickness of the outer cylinder must equal 2mm, I just don't know how to get there.

With set-up as you draw it, there would be no pressure between the two surfaces and no basis for calculating the other thickness.
Try my version.

 

[DevonZA]

With set-up as you draw it, there would be no pressure between the two surfaces and no basis for calculating the other thickness.
Try my version.

Ah yes that makes sense.

However I am still not sure how to start.
I have P=200kPa
t2=2.5mm
D=100mm
interference = 4.305x10^-3mm

I can find the circumferential stress using

= 200x10^3x100x10^-3/2x2.5x10^-3
= 4MPa

I don't see how I could use this to find t1.
As far as the relationship between tension and compression forces I assume they are equal to each other once the common diameter of 100mm is established?

 

[haruspex]

tension and compression forces I assume they are equal to each other once the common diameter of 100mm is established?

Yes.

I can find the circumferential stress

Good. You are given E. What radial compression would that lead to?

 

[DevonZA]

Yes.

Good. You are given E. What radial compression would that lead to?

Sorry for the delay I am in South Africa and your reply only came through at around 11pm last night.

Okay so using E=200GPa

Circumferential stress = PD/2tE
=200x10^3x100x10^-3/2x2.5x10^-3x200x10^9
=2x10^-5 Pa

This doesn't seem right?

Edit: in my textbook/study guide there is no formula for radial stress given "because it is very small and thus disregarded". I am only given formulas for circumferential (hoop) stress and longitudinal stress.

 

[DevonZA]

If I use

radial stress x radius = force
radial stress =force/radius
=200x10^3/50x10^-3
=4MPa

This is the same answer as I got earlier without using E.
What would the next step be provided this is correct?

 

[haruspex]

my textbook/study guide there is no formula for radial stress given

I did not mention radial stress. I asked about the radial compression, i.e. reduction in radius. The next question is the corresponding change in circumference.

 

[DevonZA]

I did not mention radial stress. I asked about the radial compression, i.e. reduction in radius. The next question is the corresponding change in circumference.

okay how about this:
the sum of the changes in the diameters of the two cylinders
is equal to the interference between the diameters before shrinkage

Δdi+Δdo=4.305x10^-3mm

Pd/2tE + Pd/2tE = 4.305x10^-3mm
Pd/2tE + 2x10^-5 = 4.305x10^-3mm
200x10^3xd/2x2.5x10^-3x200x10^9 + 2x10^-5 = 4.305x10^-3mm
d = 21.425

Ah I don't know

 

[haruspex]

okay how about this:
the sum of the changes in the diameters of the two cylinders
is equal to the interference between the diameters before shrinkage

Δdi+Δdo=4.305x10^-3mm

Pd/2tE + Pd/2tE = 4.305x10^-3mm
Pd/2tE + 2x10^-5 = 4.305x10^-3mm
200x10^3xd/2x2.5x10^-3x200x10^9 + 2x10^-5 = 4.305x10^-3mm
d = 21.425

Ah I don't know

It would help me to follow your working if you would avoid plugging in numbers. Just keep everything algebraic. It's a very good style to adopt, having many advantages.
r_i = original outer radius of inner cylinder
r_o= original inner radius of outer cylinder
r_i-r_o=δ (=4.305x10^-3mm)
R= 100mm
T_o= circumferential tension in outer cylinder, etc.
w_o = thickness of outer cylinder, etc.
P= pressure between cylinders = 200kPa.

Using those variables, what is the change in radius of the outer cylinder?
What is the change in circumferential length of the outer cylinder?
What equation does that give for T_o?
What is the relationship between T_o and P?
Same as above for inner cylinder.

 

[DevonZA]

It would help me to follow your working if you would avoid plugging in numbers. Just keep everything algebraic. It's a very good style to adopt, having many advantages.
r_i = original outer radius of inner cylinder
r_o= original inner radius of outer cylinder
r_i-r_o=δ (=4.305x10^-3mm)
R= 100mm
T_o= circumferential tension in outer cylinder, etc.
w_o = thickness of outer cylinder, etc.
P= pressure between cylinders = 200kPa.

Using those variables, what is the change in radius of the outer cylinder?
What is the change in circumferential length of the outer cylinder?
What equation does that give for T_o?
What is the relationship between T_o and P?
Same as above for inner cylinder.

Sorry about that I am honestly just grasping at straws.

I have tried to figure out where you want me to go with the above but I just cannot get it.
R=radius? It should then be 100mm/2=50mm?

Is there a specific equation that I am supposed to be using to find the solution?

 

[haruspex]

Sorry about that I am honestly just grasping at straws.

I have tried to figure out where you want me to go with the above but I just cannot get it.
R=radius? It should then be 100mm/2=50mm?

Is there a specific equation that I am supposed to be using to find the solution?

Yes, sorry, I'm not used to diameters being given instead of radii.

Forcing the cylinders to other than their relaxed radii results in a change in their circumferential lengths. That causes a compression in the one and a tension in the other. These forces must balance.

Please try to answer my questions in turn. Where do you get stuck?

 

[DevonZA]

Yes, sorry, I'm not used to diameters being given instead of radii.

Forcing the cylinders to other than their relaxed radii results in a change in their circumferential lengths. That causes a compression in the one and a tension in the other. These forces must balance.

Please try to answer my questions in turn. Where do you get stuck?

No problem.
Okay so the inner cylinder is under compression and the outer cylinder is under tension if I imagine the situation.
So tension = compression but I don't know how to put this into an equation with the other variables.

 

[haruspex]

Please try to answer my questions inpost #34 in sequence. The first one was:

what is the change in radius of the outer cylinder?

 

[DevonZA]

It would help me to follow your working if you would avoid plugging in numbers. Just keep everything algebraic. It's a very good style to adopt, having many advantages.
r_i = original outer radius of inner cylinder
r_o= original inner radius of outer cylinder
r_i-r_o=δ (=4.305x10^-3mm)
R= 100mm
T_o= circumferential tension in outer cylinder, etc.
w_o = thickness of outer cylinder, etc.
P= pressure between cylinders = 200kPa.

Using those variables, what is the change in radius of the outer cylinder? ro=ri- 4.305x10^-3mm
What is the change in circumferential length of the outer cylinder? Lo=Li+4.305x10^-3mm
What equation does that give for T_o? To=P+Ti
What is the relationship between T_o and P? The pressure between the cylinders increases the tension in the outer cylinder because the inner cylinder is pushing outwards onto it
Same as above for inner cylinder.

Using those variables, what is the change in radius of the inner cylinder? ri=4.305x10^-3mm +ro
What is the change in circumferential length of the inner cylinder? Li=Lo-4.305x10^-3mm
What equation does that give for T_i? Ti=P-To
What is the relationship between T_i and P? The pressure between the cylinders increases the compression in the inner cylinder because the outer cylinder is pushing inwards onto it

 

[haruspex]

ri=4.305x10^-3mm +ro

As I asked, please do not plug in numbers yet, just use algebraic symbols. Much easier for me to follow what you do.
Anyway, that does not give the change in radius of the inner cylinder. Its outer radius starts at r_i. What is its final outer radius?

 

[DevonZA]

As I asked, please do not plug in numbers yet, just use algebraic symbols. Much easier for me to follow what you do.
Anyway, that does not give the change in radius of the inner cylinder. Its outer radius starts at r_i. What is its final outer radius?

Sorry. Final outer radius will be the common radius (50mm) plus the thickness of the outer cylinder

 

[haruspex]

Sorry. Final outer radius will be the common radius (50mm) plus the thickness of the outer cylinder

I carefully defined r_i to be the outer radius of the inner cylinder, so no need to add its thickness. Final outer radius will be R (no numbers!)
So what is the change in that radius?

 

[DevonZA]

I carefully defined r_i to be the outer radius of the inner cylinder, so no need to add its thickness. Final outer radius will be R (no numbers!)
So what is the change in that radius?

R=ri+δ

 

[haruspex]

R=ri+δ

No, I defined δ as r_i-r_o.
The inner cylinder initially had radius r_i. At the end it has radius R. What is its change in radius?

 

[DevonZA]

No, I defined δ as r_i-r_o.
The inner cylinder initially had radius r_i. At the end it has radius R. What is its change in radius?

I honestly don't know

 

[haruspex]

I honestly don't know

It is an extremely simple question. If it starts at one value and finishes at the other, how much did it change by?

 

[DevonZA]

It is an extremely simple question. If it starts at one value and finishes at the other, how much did it change by?

R=r_i-Δr_i

 

[haruspex]

R=r_i-Δr_i

Ok.
If the radius decreases by r_i-R, what happens to the circumference?
(It will be most helpful to work in terms of fractional changes.)

 

[DevonZA]

Ok.
If the radius decreases by r_i-R, what happens to the circumference?
(It will be most helpful to work in terms of fractional changes.)

The circumference will decrease.
I don't know how to put this in an equation but let me try:

C=2πr
therefore it becomes C=2π(r_i-R)

 

[haruspex]

The circumference will decrease.
I don't know how to put this in an equation but let me try:

C=2πr
therefore it becomes C=2π(r_i-R)

That will be the reduction in circumference, yes. But as I wrote, it will be more helpful to consider the fractional change.
You are given a certain coefficient E of the material. If there is a certain fractional change in the length of the circumference, what does that tell you about the circumferential compression force?

 

[DevonZA]

That will be the reduction in circumference, yes. But as I wrote, it will be more helpful to consider the fractional change.
You are given a certain coefficient E of the material. If there is a certain fractional change in the length of the circumference, what does that tell you about the circumferential compression force?

It tells me that the circumferential compression force is increased thus reducing that circumference. I don't quite understand what you mean by a fractional change?

 

[Nidum]

Fractional change = ( new value - original value ) / ( original value )

I think that @haruspex is trying to steer you towards thinking about the idea of strain . Look it up .

 

[haruspex]

circumferential compression force is increased thus reducing that circumference.

Right, but what is the algebraic relationship? You are given a value for "E". Do you understand that this is the Young's modulus for the material? Do you know how this relates stress to strain?

Fractional change = ( new value - original value ) / ( original value )

Thanks Nidum.

 

[DevonZA]

Right, but what is the algebraic relationship? You are given a value for "E". Do you understand that this is the Young's modulus for the material? Do you know how this relates stress to strain?

Thanks Nidum.

E=σ/ε

 

[haruspex]

E=σ/ε

Ok, so express the strain, ε, in terms of r_i and R.
Then get an expression for the circumferential stress, σ.