[SOUND]. Hi and welcome back. In today's module, we're going to continue going through the example with fluctuating stresses and the modified Goodman diagram. The outcome for today's module is for you to be able to estimate the remaining life of a component exposed to a fluctuating load. And this component in this case does not have infinite life. So, the problem that we worked through last time, I asked you guys to substitute in a minimum stress of sigma min = -10 KSI and a maximum stress of 15 KSI. And if you did that and went through and calculated the alternating and mid range stresses including your stress concentration factors into these calculations, what you'll end up with is an alternating stress of 20.4 KSI and a midrange stress of 4.1 ksi. And when you go ahead and you calculate your factor of safety, you'll end up with a factor of safety of 0.93. So in this case, with this type of loading on this component, we are no longer getting infinite life, we're now in the finite life range. If we take a look at the Modified Goodman Diagram, we can see our midrange stress of 4.1 and our alternating stress of 20.4 puts us right above the Modified Goodman line in the finite life region. And so now you can see the finite region in red. So now what we need to do is we need to calculate the number of cycles until failure. And this is a little tricky to do because the SN diagram that we've looked at in the past and that we've modeled with equations in the past is for fully reversible stresses. And what we have here is a fluctuating stress with an alternating load and a mid range load. So what we need to do is we need to calculate an equivalent stress that causes the same amount of damage as this fluctuating stress but that is fully reversible. So essentially we're going to take our fluctuating stress and we're going to calculate a fully reversible stress that will damage this part just as much. And this fully reversible stress will be what we use to estimate the number of cycles to failure. So to do that, what we can do is we can, if we look at our Goodman diagram, we can see our alternating and mid-range stress dot right here. If we draw a line from the ultimate strength through our stress and our stress condition and hit the y intercept. That's is going to give us a fully reversible stress, right. Because here our midrange stress is 0 at the y intercept. And so this is going to give us a fully reversible stress, that will cause the same amount of damage as our fluctuating stress. And we can calculate this using the modified Goodman equation for this line. So to do that our Goodman line equation is alternating stress divided by our endurance stress plus our mid range stress divided by our ultimate strength. So we're looking for our intersection of the y-axis which will be our completely reversed stress. And that's going to cause the same amount of damage as our fluctuating stress. So these two will have the same amount of damage. That's what we're trying to find. So if you just remember the equation for a line is y = mx + b. We're looking for the y intercept. If we map the Goodman line equation to our traditional form of the equation for a line. We can see our y intercept is traditionally the endurance limit, or the endurance strength. And so what we're going to do is we're going to solve for the endurance limit, which is actually, we're solving for our y-intercept, and that is going to be our fully reversible equivalent stress, okay? So I'm just going to go back, we're solving for this y-intercept right here which is our fully reversible equivalent stress. In the Goodman equation, traditionally that's written as the endurance limit, because that's where the Goodman line is traditionally drawn from the endurance limit to the ultimate strength. But we're modifying the Goodman line to go through our point to give us the same amount of damage, and so we're solving for this y-intercept right here. So, we're going to substitute in this sigma rev equivalent for our endurance limit, and that will give us this equation right here. You can see my sigma rev equivalents is been substituted in for the endurance limit. And when I go ahead and I solve that, I end up with a sigma rev is equal to. This should be sigma rev equivalent is equal to my alternating stress divided by 1 minus my mid range stress divided by my ultimate strength. Okay, so now that we have this equation we can actually plug in our numbers for it. So we have our sigma rev equivalent is equal to 20.4, that's my alternating stress divided by 1 minus my mid-range stress which is 4.1 divided by my ultimate strength which is 82 and I end up with 21.4. Okay, so this fully reversible stress is going to give me the same amount of damage as my stress situation that I have right now. So if my stress situation right now, if this is the zero point. And I have an alternating stress of, let's see, it was 20.4 and a mid-range stress of 4.1, so it looks something like this, right? Where this is 4.1 and I'm going up 20.4. And what I'm saying is that this stress state, which is fully reversible, starting at the 0 axis, going up to 21.4 and down to -21.4. So this stress state gives me the same amount of damage. So if I calculate the number of cycles remaining using this stress state, it should give me the number of cycles remaining for this stress state because they cause the same amount of damage. So I'm going to use 21.4 and I go on. To calculate this, I need to use N = my sigma rev equivalent divided by a to the 1/b power. Right, and so in the past we've calculated a based off of the endurance limit, f, and the ultimate strength. In this case, a will be equal to fS ultimate divided by Se squared and that gives you 242.9 ksi. B is equal to negative one-third log of fS ultimate divided by Se, which gives me negative 0.18. So when I plug in everything, I get N = 21.4 divided by 242.9, to the -1 over 0.18. Which will give me about 680,000 cycles remaining before failure. All right, so now you know how to calculate the factor of safety based off of the modified Goodman criteria. And you also know how to estimate life if you end up in a finite range with your modified Goodman criteria. Next time we're going to look at even more complicated loading scenarios. We'll be looking at fluctuating, varying loads. I'll see you next module. [MUSIC]