Calculating Internal Pressure in Glass Tubes and Spherical Flasks

[Saitama]

Homework Statement

What internal pressure (in the absence of an external pressure) can be sustained

a)by a glass tube
b)by a glass spherical flask

if in both cases the wall thickness is $\Delta r$ and the radius of the tube and the flask equals $r$?

($\sigma_m$ is the glass strength)

Homework Equations

The Attempt at a Solution

Honestly, I don't know how to begin here. From the definition,
$$\sigma_m=\frac{\text{Stress}}{\text{Strain}}$$
But I don't see how a strain is produced in both the cases to produce the corresponding stress.

I understand that this is an easy problem (as it is the second problem of the exercise in the book) but I really have no idea to begin with. I need a few hints to begin with.

Any help is appreciated. Thanks!

 

[maajdl]

You need to know when glass breaks.
Can you find that in your book?

 

[Saitama]

You need to know when glass breaks.

Yes.

Can you find that in your book?

What should I exactly look for?

 

[maajdl]

The definition of σm and eventually some numerical value.
You then need to relate the deformations, the stresses and the pressure.

 

[haruspex]

You need to think how they will break, i.e. where the boundaries of the (two) fragments will be.

 

[Saitama]

The definition of σm and eventually some numerical value.
You then need to relate the deformations, the stresses and the pressure.

To be clear, I am using two books, one is a problem book and the other is a textbook. The problem book uses a different notation than the textbook. I am not sure what $\sigma_m$ represents, does it represent the young's modulus which is generally denoted as Y in the textbook?

You need to think how they will break, i.e. where the boundaries of the (two) fragments will be.

Why does it even break? We are not pushing or pulling it apart to break it.

Sorry if these are stupid questions.

 

[voko]

Why does it even break? We are not pushing or pulling it apart to break it.

Really? With zero external pressure and non-zero internal?

 

[Saitama]

Really? With zero external pressure and non-zero internal?

I still don't get it. Do you mean that there is some internal force acting between the adjacent parts of tube or sphere which causes the objects to break?

 

[voko]

Take a shoe lace. Make it taut. Now press with your finger perpendicularly to it. Will there be any forces between the adjacent parts of the string? What happens if you keep pressing stronger (assuming your finger is ideal)?

 

[Saitama]

Will there be any forces between the adjacent parts of the string?

Gravitational force?

What happens if you keep pressing stronger (assuming your finger is ideal)?

At some point of time, the lace breaks.

Are you hinting towards the tension being produced in the lace?

 

[voko]

Gravitational force?

Well, there will be some of that, but we usually ignore that for such tiny objects.

Are you hinting towards the tension being produced in the lace?

This is really a problem of statics (before it breaks). The force exerted by your finger must be balanced by something. What is that something?

 

[Saitama]

This is really a problem of statics (before it breaks). The force exerted by your finger must be balanced by something. What is that something?

Something=the maximum tension that the lace can bear?

 

[voko]

Yes. The correct term for that is ultimate tensile strength.

 

[Saitama]

Yes. The correct term for that is ultimate tensile strength.

I am still stumped. How do I write down the equations? Do I select a small part of the glass tube subtending some small angle at the centre as shown in the attachment?

The blue arrows represent the tension forces tangent to the tube.

 

[voko]

It is probably easier to start with the spherical flask, because it is more symmetric.

So let's say you have a spherical cap, radius $r$ and diameter $d$. It is acted upon by pressure $d$. What is the resultant force exerted on the cap?

 

[Saitama]

It is probably easier to start with the spherical flask, because it is more symmetric.

So let's say you have a spherical cap, radius $r$ and diameter $d$. It is acted upon by pressure $d$. What is the resultant force exerted on the cap?

I think you mean pressure $P$. Why do you give me both the radius and diameter? Do you mean that radius of spherical cap is $r$ and height of spherical cap is $d$?

 

[voko]

Sorry, a typo. Read "pressure $p$".

 

[voko]

I think you mean pressure $P$. Why do you give me both the radius and diameter? Do you mean that radius of spherical cap is $r$ and height of spherical cap is $d$?

I actually meant that the diameter of the circle at the bottom of the cap was $d$. You could, of course, use its height $h$.

Note, however, that the next step would be to consider tensile forces acting at the bottom of the cap (so that it is held in place), and using the diameter might simplify formulae.

 

[haruspex]

I actually meant that the diameter of the circle at the bottom of the cap was $d$. You could, of course, use its height $h$.

Looking ahead a bit, I suggest it will be more convenient to express things in terms of the angle which a radius of the cap subtends at the centre.
Pranav, consider the net force pushing out that cap (P times something) and the net force retaining it (tension per unit length times something).

 

[Chestermiller]

Voko has been trying to get you pointed in the right direction. You are trying to determine the internal tensile stress within the wall of the tube or within the wall of the spherical shell. Voko suggested that you split the spherical shell into two halves, and do a force balance on half the shell (as a free body). There is a pressure force pushing upward over the cross sectional area, and there is a downward tensile force in the glass all around the rim. Let σ represent the tensile stress within the glass within the rim. The rim has a diameter of 2r and a thickness of Δr, so what is its area? What is the downward tensile force on the rim? The pressure is pushing upward over the entire cross section of the hemisphere. What is the magnitude of this force if the pressure of the gas is p? Set these two forces equal to one another. This will give you the tensile stress in terms of the pressure, the wall thickness, and the sphere diameter. Once you know this tensile stress, you can compare it with the critical stress for failure σ_m.

This is a statically determinate problem, and you do not need to know the Young's modulus or the strain.

Chet

 

[haruspex]

Voko suggested that you split the spherical shell into two halves,

Not that I can see. I think voko has avoided prejudging exactly what the two fragments will consist of. Instead, the discussion has assumed a circular area will be blown out. It should become apparent later what fraction of the sphere that will be.

 

[Chestermiller]

Not that I can see. I think voko has avoided prejudging exactly what the two fragments will consist of. Instead, the discussion has assumed a circular area will be blown out. It should become apparent later what fraction of the sphere that will be.

That may be, but the easiest way to do this is to split the spherical shell in half.
Chet

 

[haruspex]

That may be, but the easiest way to do this is to split the spherical shell in half.
Chet

Sure it's easier, but isn't it making an assumption as to which is the critical case? I agree it's intuitively likely, but it needs to be justified.

 

[Chestermiller]

Sure it's easier, but isn't it making an assumption as to which is the critical case? I agree it's intuitively likely, but it needs to be justified.

I meant that getting the tensile stress is easiest by splitting the shell into hemispheres. The failure criterion is a different story. For the spherical shell, the tensile stress state is transversely isotropic and equal biaxial. The assumption would have to be made that for the sphere, the σ_m represents the failure criterion for the stress σ. Since this is such an elementary course, I don't think that they would be getting into any more advanced failure criteria. In the case of the cylinder, the axial stress is not equal to the circumferential stress, and my guess here is that the students are supposed to set the circumferential stress equal to σ_m. If course, in reality, this would not be a suitable failure criterion. But since here, the course is so elementary, I'm guessing that this is what they are looking for.

Chet

 

[voko]

I have avoided making the decision as to which part of the sphere the cap should represent, hoping that Pranav would come up with some ideas about that :) Now that we have all the cards on the table, let him decide how he should proceed.

 

[Saitama]

Thank you everyone for the replies.

Let the pressure of gas be $P$ and tension per unit length be $T$. I select a spherical cap subtending an angle $2\theta$ (as suggested by haruspex) at centre of sphere. Balancing the forces, I get

$$P\pi R^2\sin^2\theta=T(2\pi R\sin\theta)\sin\theta \Rightarrow P=\frac{2T}{R}$$

What should I do next?

 

[voko]

You need to express $T$ in terms of $\sigma_m$ and $\Delta r$.

Then do the same thing for the tube. Note, however, that the tube has two different parts: a circular ring in the middle and a circular edge, to which a flat circular disk is attached. Compute the tensile stress in those parts, and then see where it is greater; that will determine the max pressure for the entire structure.

 

[Saitama]

You need to express $T$ in terms of $\sigma_m$ and $\Delta r$.

How to do that?

 

[voko]

You need to find what $\sigma_m$ stands for. Usually, it is force divided by area, but you need to make sure.

 

[Chestermiller]

How to do that?

If T is the tensile force per unit length, and the width of the shell is Δr, what is the tensile stress σ in the shell? What is the tensile stress in terms of P, Δr, and r? You need to determine how this tensile stress compares to the ultimate tensile stress σ_m.

 

[Saitama]

If T is the tensile force per unit length, and the width of the shell is Δr, what is the tensile stress σ in the shell? What is the tensile stress in terms of P, Δr, and r? You need to determine how this tensile stress compares to the ultimate tensile stress σ_m.

I am not sure but the tensile force acts on both sides of the sphere (the inner surface and the outer surface), right?

If Δr is thickness, then the area of the thickness of the spherical cap I selected is $2\pi R\sin\theta Δr$. The tensile stress is 2T divided by this area, am I right?

 

[Chestermiller]

I am not sure but the tensile force acts on both sides of the sphere (the inner surface and the outer surface), right?

No. It acts within the shell metal. It is perpendicular to the radius at each point.

If Δr is thickness, then the area of the thickness of the spherical cap I selected is $2\pi R\sin\theta Δr$.

No. The force per unit length T you calculated is oriented tangent to the shell cap you selected. To get the stress, all you do is divide T by Δr. That will, of course, give you units of stress. You see, this is why I suggested you split the shell into hemispheres. Please try doing the problem using hemispheres. It is more clearcut that way.

The tensile stress is 2T divided by this area, am I right?

No. It's only T divided by Δr.

Chet

 

[voko]

If Δr is thickness, then the area of the thickness of the spherical cap I selected is $2\pi R\sin\theta Δr$. The tensile stress is 2T divided by this area, am I right?

Your $T$ is already force per length. If you divide that by area, you will get force per volume. Is that dimensionally correct?

 

[Saitama]

No. The force per unit length T you calculated is oriented tangent to the shell cap you selected. To get the stress, all you do is divide T by Δr. That will, of course, give you units of stress. You see, this is why I suggested you split the shell into hemispheres. Please try doing the problem using hemispheres. It is more clearcut that way.

Okay, I do it for the hemispheres. :)

By force balance, I get $P\pi R^2=T(2\pi R) \Rightarrow 2T/R=P$

Also, $\sigma_m=T/\Delta r$. From these two equations, we have $P=2\sigma_m \Delta r/R$ which is the correct answer. Have I done it correctly? What about my spherical caps? They too give the same equation as hemispheres.

Your $T$ is already force per length. If you divide that by area, you will get force per volume. Is that dimensionally correct?

Yes, very silly of me, thank you voko!

 

[Chestermiller]

Okay, I do it for the hemispheres. :)

By force balance, I get $P\pi R^2=T(2\pi R) \Rightarrow 2T/R=P$

Also, $\sigma_m=T/\Delta r$. From these two equations, we have $P=2\sigma_m \Delta r/R$ which is the correct answer. Have I done it correctly?

Yes. Nice job.
I would have proceeded a little differently by first calculating the stress σ for any arbitrary value of the pressure P:
$$σ=\frac{Pr}{2Δr}$$
To find the pressure at which the shell fails, I would then set the stress equal to the critical stress:
$$σ=\frac{Pr}{2Δr}=σ_m$$
This is the sequence of steps typically used in the design of more complicated mechanical structures to assure that the structure does not fail under load. Of course, for this simple example, it didn't matter.

What about my spherical caps? They too give the same equation as hemispheres.

Yes, as they should. But doing it with the hemispheres is much easier to see and to implement.