# Mechanics of Materials: Lesson 41 - Intro to Thin Walled Pressure Vessels

https://youtu.be/0AbW_guBj5Y?si=daLiFBfY2hzgYCi5

## Introduction & Pressure Vessels

Hey gang, welcome back. We're talking about today, Fin, World Pressure Vessels. It's a new chapter. So this is kind of the start of the chapter which is called Combine Loading. And this is a different, today we're going to have some new equations. Yes. And a different way to calculate some sigma stress. Some normal stress. So I want to get this kicked off here and I'm going to show you some thin wall pressure vessels. Some pressure vessels that you may be familiar with. Number one, just a spherical pressure vessel. Maybe you're driving down the highway. You drive past a big chemical plant or something. You see these kind of tanks sitting out there. Wonder what they got in there. Or big just cylindrical tanks. Just big round tanks. Round tanks that just have straight sides on them. Or perhaps you have one of these in your garage, an air compressor, compressor, which is a combination. It has a cylindrical body, but it has hemispherical ends. So this one has kind of the properties of both of those tanks there.

## Hoop Stress

So we're talking thin wall pressure vessels. That is going to cause if you're a little piece of material that lives on the outside of these tanks. This tank, that tank, those tanks, what do you feel? Well, you'd be feeling some stress because you're being pressurized. And there's really two kinds of stress that you're going to feel. These are our new equations for the day. Number one, we call Hoop Stress. And Hoop Stress is Sigma, is PR over T. And the other one is Longitudinal Stress. And this one is PR over 2T. 

Now, let's talk about these stresses here and what they mean. Number one, Hoop Stress. Think about a hoop. A hoop is like a circumference here. So think about this hoop right here that goes all the way around that tank. Well, guess what? There's another one here and another one there and another one there. And the pressure in each one of those hoops is the same as that tank starts to increase in diameter. That material is being stretched. It's intention, isn't it? Okay? And so that is given by this equation here. 

Now, what are these numbers here? What are these letters mean? Well, the P, the P is for pressure. Is pressure always positive? Can it ever be negative? 

Well, yeah, you could have a vacuum chamber couldn't you? The outside atmospheric pressure could be higher than the internal compression of the pressure of that tank. Right? So it could be a negative pressure. R is the radius. And I'm going to put inner, okay? It's the inner radius, not the outer radius, okay? Because this wall, this, this, this, this, this, this pressure tank does have a wall thickness, right? And so if you were talking outer diameter or outer radius, you would have an extra wall thickness on that, right? So how do you remember that? Well, where's the pressure? And it's out the tank. Well, where's the radius? It's out the tank, okay? So it's easy to remember which one is the common mistake is to use the wrong radius, right? That number right there. Don't do that. Okay? And then T thickness, okay? And that's the wall thickness. How thick is the material on the outside of that tank? Okay? So this is hoop stress sigma H. And you will see that on the cylindrical portions of tanks, okay? So on this vertical tank, maybe this is a, a tank that's holding some kind of a liquid, okay? Now, as we know, as I go down in a liquid, if it was filled all the way to the top, as I go down, what's happening? Well, according to row G H, right? The pressure is increasing as I go down and depth toward the bottom. And so if you do stuff like a rocket engineering, okay? Therefore, they're, they're a pressure vessel that looks like this is a cylindrical, but they're trying to save weight. So the thickness on the bottom of the rocket will be thinner as it goes up because they're trying to save weight because there's not as much pressure on this hoop as there is on that hoop down there at the bottom. That's kind of cool stuff, right?

## Longitudinal Stress

We have hoops here, there's one there, and there, and there, and all along the cylindrical portion of this. If I'm a little piece of material right here, I'm getting stretched because of the increase in pressure inside that tank. 

Longitudinal Stress We see that in these spherical tanks. If you're driving down a highway and you drive past a chemical plant and you see these spherical tanks, the stress here everywhere is longitudinal stress. Because it's trying to expand in all directions at the exact same time. And look here, it's over 2t, it's the exact same equation, but over 2t, so what does that mean? That means that longitudinal stress is half as much as Hoop Stress is. Half as much, so if you see these tanks here, they have sigma equals pr over 2t stress. So if the stress isn't half as much, that means that I can have twice as much pressure in that tank as I do in that tank. So if you drive out of those plants and you see these spherical tanks, you see the ones you want to avoid. Because this is usually where the high-pressurized gases are, things that you need to stay liquid, things under very high pressure. This is the stuff that you don't want to be around if things go sideways. If you see these longitudinal tanks, that's why they are on that shape. Because these are easy to make. You just take a piece of steel, you put it a slip roll, you roll it, you weld that one same boom, you've got a cylinder. But how do you make a 20-foot diameter, let's say, spherical tank? You almost have to weld it like a soccer ball together, right? You have to weld little triangular or shaped pieces together to form this. So these are very expensive to manufacture where those are less expensive. But, like I said, it stresses half as much, so some situations call for these kind of tanks. And again, here, on the ends, right? Because the pressure is trying to push the ends off that tank. On the end of the tank, I have sigma equals pr over 2t. And on the hoop of the tank, I have sigma equals pr over t. So anyway, that's kind of an introduction to thin wall pressure vessels. We're going to work some problems. We're going to use these equations here. The hoop stress and the longitudinal stress to solve some problems. And we're going to start off from the easy one. We'll start off from this guy. And then we'll graduate to a harder and harder. So there you go. There's an introduction to thin wall pressure vessels. One more thing. What is a thin wall pressure vessel? What makes it thin is R divided by D. We say that needs to be bigger than 10. So the radius divided by the D, not D, T. Sorry. R divided by T. The thickness needs to be bigger than 10. So let's say that this tank here, or maybe that tank there, whatever. Let's say the diameter was 12 inches. And the wall thickness was 0.125, an eighth of an inch. Let's see, what is that? Where are you calculating? Here he is. 

Okay, so 12 divided by 0.125. 96. Okay, so that's way bigger than 10. So we would say that's a thin wall pressure vessel. You're going to get all thin wall pressure vessels in your solids book, in your mechanics of materials book, because the thick, thick wall pressure vessels do not abide by these equations. So only the thin wall pressure vessels follow these stress equations. All right, I hope that helped. Let's work some example problems in. Let's make some sense out of this. I'll see you on the next video. Thank you.