[MUSIC] Hi and welcome back. In today's module, we're going to wrap up fully reversed stresses and fatigue. And the learning outcome for today's module is to understand how to solve an entire fatigue problem by utilizing the fully adjusted endurance limit, the fatigue stress concentration factor, and SN curve approximations. So the steps that we're going to go through, in order to solve the problem is first we're going to calculate stresses at the critical point and see where are we most concerned? We're going to add in stress concentration factors to any stresses than need stress concentration factors added. And then we're going to determine the fully adjusted endurance limit and compare that endurance limit to the stress or the fully reversible stresses. So, let's go ahead and get started. To do this, we're going to work through the example that we've already been working through. So again, we have this rotating shaft and it's in fully reversed bending, it's in three point loading, and it's simply supported by bearings at R1 and R2. So for this, in previous modules we worked through and we calculated both the endurance limit Se and we calculated the fatigue stress concentration factor Kf. And we have those numbers here, where kf is equal to 1.68 and the fully adjusted endurance limit is equal to 44.6 ksi. And so now we're just going to incorporate these factors and solve the whole problem. The questions asked are In this problem in particular we're looking at point A and we're wondering, does it have infinite life and if it doesn't have infinite life, how many cycles can it survive at the loading it's under if F is equal to 0.76. So let's go ahead and get started. The first step is to just look at this as a simple static bending moment stress problem. So you're going to draw a simple free body diagram. You know you have your reaction force, R1, you know you have your reaction force, R2. You know you have a load, 500 being placed on top of the beam, and that you have certain dimensions, and you're trying to find your stress at point A. So at this point, you guys are all pros, and you're going to do a sum of the forces in the y = 0, a sum of the moments around, I'd say, point O = 0. And what you'll find is that R1 = 200 pounds of force and R2 = 300 pounds of force. Now, since we're trying to find the fully reversible stress at point A, this is in pure bending, so we're going to say Sigma = MC over I. And we're not going to worry about the sign because sigma is going to change. Point A is going to go from compression to tension to compression to tension. So we're going to go ahead and try to find MC and I. Now you guys are old pros at this, and you know that your moment of inertia for a cylinder is pi d to the 4th over 64. You know that you're going to want to look at worst case scenario, so that's the smallest diameter, and in that case your c will be d over 2. So you can go ahead and plug these in and sigma will boil down to 32 M over pi d cubed. So now, what we need to figure out is M. And M is actually really easy here, because we have a reaction force. And as we go across the beam, we hit 0.8 before we hit any other forces. So, I can just solve a very simple moment problem and say the moment at point A is equal to R1 times the distance from R1 to point A, which is 10. And that's because there's not any other forces between R1 and point A and any other constraints. So, I'm going to go ahead and say the moment is equal to 200 pounds of force times 10 inches and I get my moment is equal to 2,000 pounds of force per inch. And when I go ahead and plug all this in, what I'll find is that my stress is equal to 20.3 ksi. And that's at point A. So I have a stress of 20.3 ksi, but that's not really the maximum stress that point A is seeing because point A is at a geometry discontinuity where it's going from a small radius to a large radius. And there's a fillet radius at point a. And we already calculated the stress concentration factor at point a and so now we're going to use that. So we're going to say sigma max at point a, and this is a fully reversible stress, is equal to kf times sigma nom, fully reversible at point a. And that's going to be, so, we just calculated our sigma nom, so, 1.68 times 20.3. 1.68 times 20.3 and I end up with a maximum fully reversible stress of 34.2 ksi. So, my next step is to compare this to the endurance limit. And my endurance limit in this case is 44.6 ksi. And if I look at my SN diagram, what that means is, if I have number of cycles on the bottom and fully reversible stresses on the y axis, so an SN diagram roughly looks like this. My endurance limit is at 44.6 and my stresses are below it so I can cycle forever. So my life is infinity and the answer is yes, it will have infinite life. Cool, so that's a good situation, that's an easy situation. What if my stress had been higher than the endurance limit? Well in that case, what we would do, let's say we run this load up to 800 pounds of force. And if you do that, and you go back through the entire equation, what you'll do is you'll find your sigma rev max is going to end up being 54.7 ksi, so quite a bit higher. So then you have a fully reversible stress, you have an endurance limit that you know, 44.6, and you also have an ultimate strength that you know, 240 ksi. And they gave you an f value of 0.76. So we can go ahead and calculate. We know it's not going to be infinite life because my fully reversible stress is higher than my endurance limit. I can go ahead and calculate the life using this equation, fully reversible stress divided by a to the 1 over b, where n is my number of cycles. B is equal to negative one-third, the log of F times my ultimate strength divided by my endurance strength. Which if you calculate out will end up being -0.2, and a is going to be f times my ultimate strength squared divided by my endurance limit and a is going to be equal to 746. So when I go ahead and calculate this out, I get 54.7 divided by 746 to the 1 over -0.20 and N is equal to 365,167 cycles. Now, clearly we haven't actually, this equation is not accurate enough to predict it's actually going to fail on the 365,167th cycle. So you kind of have a range of failure that you know is going to exist between here. But you can say it's going to survive roughly 300 to 400,000 cycles, depending on what your factor of safety is and the assumptions that you've made. All right, so that's how you incorporate all of these different components of fatigue, the endurance limit, the SN diagram, the approximation of the SN diagram and your fatigue stress concentration factor all together to solve a fully reversible fatigue problem. There is a couple of worksheets online, I think actually one worksheet with a few problems that you can use to practice for the quiz. And then after the quiz we'll get into more complicated loading scenarios. I'll see you next time. [MUSIC]