[MUSIC] This is module 6 of Mechanics of Materials, Part 2. And today's learning outcome is to develop the expression for longitudinal stress. Now not in a cylindrical thin-walled pressure vessel, but a spherical thin-walled pressure vessel. Again in the terms of the pressure itself. And the dimensions of the vessel. And so here is our general analysis approach. Now our engineering structure will be a spherical vessel, or a thin walled pressure vessel, as I show with this gas storage tank. And so we're going to have the same type of external loads, which are a pressure due to the liquid or gas inside. And so here is our thin-walled pressure vessel and let's again look at a section cut. And we're going to, as we did before, neglect the weight of the contents and the weight of the structure itself. And go ahead and expose the stresses and here I show the stresses. And you can see no matter how I cut my vessel, I have the same sort of stresses, the same what I'm going to call longitudinal stresses and the pressure force acting into the vessel itself. And so let's go ahead and again do our equilibrium. And so I'm going to sum forces in the Z direction. It doesn't matter which of these cuts I do it on, it's going to be the same no matter how I cut the vessel. So I've got let's say, some of the forces in the z direction, up equals 0. And so I've got sigma long times again, the outer area of the circle minus the inner area, so that's pi R outer squared minus pi R inner squared. And then I've got my pressure of force and the pressure is the force per unit area times the unit area's pi. D squared over four equals zero. The pi's cancel out. I can factor my first term and so I get pi longitudinal (R outer + R inner) (R outer- R inner) = P D squared over 4 and R outer plus R inner is approximately equal to D, our outer plus our inner is approximately equal to D. Our outer minus our inner is equal to the thickness, T. And so again we find that the longitudinal stress is equal to P D over 4 T on any cut surface no matter how we cut. So we only see a longitudinal stress we don't in this case see any type of hoop stress or change. And so here is my, again, if I have my stress block here on my vessel, that's greatly exaggerated would be at a point. Here are my spherical stresses. They're the same in every direction, and so I can draw Mohr's circle for plane stress. In this case, both my vertical face and my horizontal face have the same value, and so we're going to have sigma longitudinal equals pD over 4t for both the vertical face and the horizontal face, and that's it. My Mohr's circle is just a point and so that means that what is my shear stress in this case, and what you should say is my max shear stress is equal to zero, and so I just get my longitudinal stresses and I've got tau MAX = 0. And so that's the analysis of a spherical thin-walled pressure vessel as opposed to a cylindrical thin-walled pressure vessel as we did in previous modules. And we'll pick up in the next couple modules and do some real world engineering examples of thin-walled pressure vessels. [SOUND]