[MUSIC] Welcome to Module 5 of Mechanics of Material part two. For today's learning outcome, we're going to go ahead and determine a relationship between the longitudinal stress and the hoop stress. Which we developed in the last couple of modules and so, again as a review, we have our overall engineering structure, our thin wall pressure vessel. We applied external loads, the pressure inside. We looked at the internal forces and moments and we did two cuts. We did a longitudinal cut, and we did, what we called a cut that allowed us to look at the hoop stresses. And we came up with those stresses. Here's the expression for the longitudinal stress. Here's the expression for the hoop stress. And so, here's my thin walled pressure vessel again. Now, let's go ahead and this will be an exaggerated big size, but let's go ahead and put a little stress block on here. To look at the stresses at a point. And if you want to look this type of analysis again and review how we're doing this, you may go back to my first course in mechanics materials part one. But I can take that stress block now, that point, which I'm going to represent by this cube, and it's going to be in plane stress again. So I'm going to say, the stress is in the z direction. A outside surface that's free of any forces is going to be zero. And so, here are what my, here's my stress block, and I've got my longitudinal stresses acting left and right. And I've got my hoop stresses, acting up and down. So here again is my vessel and the stress block, showing the longitudinal and the hoop stress, and my question to you now is, if I were to continue to increase the pressure in this vessel until it failed, because remember we want to look at the performance of the vessel. How would it fail? Where would be my line of failure? So, I want you to think about that on your own, and then comment back. Okay, so now that you thought about that, let's take the vessel, okay? And so, here is my vessel. And If it were to fail, lets look if it were to fail on a vertical axis. So, if it were to fail or split on a vertical axis, then i would have these longitudinal stresses. That had failed. If it failed instead on a horizontal axis, so if it split along a horizontal axis, now the stresses that would cause this thing to split, would be the hoop stresses. So which are larger, the longitudinal stresses or the hoop stresses? Which are going to fail first? And you should see that the hoop stresses are twice as large as the longitudinal stresses. So if it's going to fail, it should fail along this horizontal line, versus a vertical line. And I've got a rather neat demonstration for you to show that. I have a pressure vessel here. This is a hot dog. And it can be modeled as a thin walled pressure vessel. And what I did was, I took this hot dog or a hot dog and I boiled it until the pressure became large inside that it failed. And low and behold, as predicted by our theory, you can see that it fails on the horizontal axis due to the larger hoop stresses. The other thing I wanted you to note is where the initiation of failure took place. And why it started at that location as the failure progressed, and so think about that and then come on back. And what you should say is, okay I see that it fails on this end and then, it's starting to progress in this direction. Well at the end now, remember from my first course, Part I In Mechanics Materials, that the stress is highest at what we call stress concentration points or where we have a discontinuity. And that discontinuity in our hot dog is on the end, and so that's where the stress started and it splits along the axis, where our Hoop stress is acting. And so, now let's do one more thing, let's go ahead and draw Mohr's circle for this plane stress. Situation, so I've got my normal stress on the x axis and my sheer stress on the y axis and so let's first of all plat our points and so on our vertical axis, which is this axis, I have my longitudinal stress which is pD over 4t. So that is my vertical axis, so I';ll label it with a V, and again, if you don't remember how to do Mohr's circle refer back to my first course part one of Mechanic and Materials and that's going to be equal to Sigma long equals pD over 4t and we have our horizontal axis stress which is going to be pD over 2t which is twice as long as, or twice as far out on the axis as my Vertical face, and so I've got my horizontal face is sigma HOOP, which is equal to PD over 2t. I now have two points on my circle. I can draw my circle. And the last point I'm interested in is the max sheer stress. And that's going to occur right here, and the max sheer stress is going to be the Hoop stress minus the longitudinal stress divided by 2. So, my max sheer stress is Sigma Hoop, minus Sigma Long. Divided by 2. And so, the overall observation that I want to make again is that the hoop stress is 2 times the longitudinal stress. And so, now we have that relationship between those two stresses in thin walled pressure vessels and we'll pick back up here next time. [SOUND]