[MUSIC] Hi and welcome back. In today's module, we're going to continue learning about fluctuating stresses. And the learning outcomes for today's module are to be familiar with the modified Goodman line on a fatigue diagram, and understand the infinite and finite life regions on that diagram. And to be able to calculate a factor of safety using the modified Goodman equation. So let's get started. So we've been talking about fluctuating stresses, where, essentially, we can fully characterize these stresses using the minimum and maximum stresses. And from the min and max stresses, calculating a midrange stress, which is kind of the average or mean stress, that's occurring. And an alternating stress, which is the amplitude occurring. And so, now that we know how to calculate a midrange and an alternating stress, let's think a bit more deeply about these stresses. So a while back, there was a brilliant gentleman with the last name of Goodman, and he thought about alternating and midrange stresses. And came up with a good way to quantify a finite versus infinite life, so let's take a look at that. So if we have a graph and on the x-axis we have the mid-range stress sigma m. And on the y-axis we have the alternating stress. Let's think about life on this graph. So let's say I'm just a component, and all I experience is a stress along this axis here, a mid-range stress. So my alternating stress is 0. So if I graphed it, what it would look like. So here's my stress, and here's time. And if I'm just experiencing a mid-range stress and nothing alternating then I'm essentially in a static load. I'm just experiencing this stress over time, the stress isn't alternating, it's not changing. So I'm in a static load. So I know that my component is going to fail at the ultimate strength, right? So, I know here my component will fail, so we're going to draw a dot here. Now, let's go over to the y-axis and think about life in a component just experiencing an alternating load. So, if I'm just experiencing an alternating load and I have no mid-range load, no mid-range stress, just an alternating stress. Essentially what I'm experiencing is a fully reversible stress right? I could be going from positive 20 ksi down to negative 20 ksi and if I add those together and divide by 2 for the mid-range stress I end up with 0 as my mid-range stress. So I'm in a fully reversible loading. And for fully reversible loading we know that we have infinite life, as long as we stay below the endurance limit. So if I go ahead and I plot the endurance limit on the y-axis, so Goodman said let's just look at the simplest solution here for combinations of alternating and mid-range stresses. So he drew a line from the endurance limit to the ultimate stress and he said this is called the modified Goodman line. And he said any stress combination below this line will have infinite life. So, life is good, smiley face, down here. And any stress situation above the modified Goodman line is going to have finite life. So here it's possible that you could fail, your life isn't infinite. It just depends how many cycles that you need based on your design, right? Now, this was the initial theory. There have been a number of theories that have come out since then. There's is ASME theory, there's a couple of other theories. Depending on what industry you're in and what the design heritage indicates will depend [determine] what theory you use. But all of these theories are very close to the Goodman Line. And so we're going to learn the Goodman line theory. In this class, we're going to learn the modified Goodman theory. And then you'll easily be able to utilize any of the theories once you understand this one. Okay, so essentially, above the line failure, below the line, infinite life. So, if we go on, there's an important thing to consider. If we look at this curve, there's a portion below where Goodman says that there could be infinite life where actually you're above the yield strength. So you could yield when Goodman is predicting infinite life. And so there's something called the yield or the Langer line and you can use an equation here to check for yield. So whenever, you're checking for fatigue, you'll check for yield first. It's called yield on the first cycle, typically, is where you'll see the yield. And then you'll check life in a fluctuating stress condition. So, here is the equation for the modified Goodman line with a factor of safety. So, it's the alternating stress divided by the endurance limit + the mid-range stress divided by the ultimate strength is equal to 1 over n. And here if N is greater than 1 then you have infinite life and if N is less than 1 then you have finite life. Okay, and then the equation for the Langer line, or the Langer yield criteria, sorry not the line, the criteria is that sigma a, which is your alternating stress plus your mid-line stress is equal to the yield strength divided by n. And if n is less than 1, then you've had some type of yield, possibly on the first cycle. So the important thing to remember here is what the Goodman equation does is it tells you if you're in the infinite life region or if you're in finite life region. What it doesn't do is it doesn't tell you how many cycles if you're in finite life you have remaining. So, it's important to realize that you're basically getting kind of a binary response here, where you either have infinite life or finite life. But if you're in a finite life region, there needs to be more analysis done to figure out how many cycles are remaining. And it's also important to recognize that unless you have S n data for your exact fluctuating load, you can't use an S n curve or the equations to model the S n curve to calculate life. And we'll get into that in a little bit. So here's an example that I'm going to challenge you to work through before the next module. We have a 1045 hot rolled steel rod, and it's undergoing axial cyclic loading. And we can see it's fluctuating loading and the stresses produced by this loading are a minimum stress of 2 ksi to a maximum stress of 10 ksi. To further complicate things, the rod has a hole in the center with a stress concentration factor for static loading of 1.7 and a q, which is your notch sensitivity factor of 0.9. The yield strength is 45 ksi. The ultimate strength is 82 ksi. And they nicely give you a fully adjusted endurance limit of 20 ksi. So next time try to use the modified Goodman criteria to determine the factor of safety before the next module. We'll go through this in the next module. And then I'll also show you how to estimate the number of cycles to failure if the Goodman criteria doesn't predict infinite life. So that's it for this module, I'll see you next time. [MUSIC]
