[MUSIC] Welcome to Module 13 of Mechanics of Materials part III. We've actually come to the exciting part of the course where we're going to start working on some real world engineering problems. And so today's learning outcome is to solve an elastic beam bending problem for the maximum flexural stress in both tension and compression. And so as a review, we're looking at beam bending in the elastic range. The stresses all remain in the elastic range. This was the elastic flexural formula where y was the distance from the neutral axis to the point where we wanted to find the stress in the x direction. Where the flexural stress. And we said that the maximum stress was as shown where C is the furthest distance from the neutral axis. And so that allows us to find where the maximum stress occurs. And so let's go ahead and apply this to a real world problem. This is a typical bridge. I'm going to simplify it some, we'll do some simplifications of the system, but the process is going to be remaining the same, regardless, and you'll know how, even for a more complicated system, to go through and solve one of these problems. So we're going to look at a strip of one beam, and how much load it carries for a bridge. And it's carrying this truck And so the beam itself, and the structure, and this is true for almost all bridges, is that the beam and the structure itself carries a lot of weight. And I'm going to estimate it for this one beam span to be 250 pounds per foot. So the constant load in a bridge structure is not insignificant, it is a big part of the design. But in addition to that where the truck is located we'll include an additional 350 pound per foot. Now it's actually point loads at the wheels but I've modeled its distributed load actually at a track vehicle would be a more distributed type load. But again the procedure is the same, regardless. And so let's go ahead and proceed. And so here is my structure. I'm going to use a rectangular cross-section. And so to start with. We'll look at other cross-sections later on, and most beams are what we call in bridges girders, are I-beams and we'll work with I-beams later on but to start let's just look at this simple rectangular cross section. And so, two parts to the worksheet. We want to draw the shear and moment diagrams and determine where in this beam does the maximum moment occur because that's what we want to design for. And then once we find where that maximum moment occurs we're going to determine the maximum flexural stress and the maximum compressor stress in the cross section at that location. What I would like you to do first is I would like you to draw the shear and moment diagram to determine where the maximum moment occurs and what that value is. And so, you should be able. A real pro at doing shear moment diagrams. If you need some refresher, go back to my applications in engineering mechanics course and I have several modules on how to proceed. And so, do that on your own. And when you do that, these are the results you should find and and for the moment diagram we find that the maximum moment occurs at the center of the beam or 60 foot from the left or the right edge and it's value is 865 625 foot paths and so now we have our M max the maximum moment that our beams going to have to carry. We know that it's at the center of the bridge structures beam, or girder, and so we know also that sigma max from our first slide is equal to Mc over I. Well we know what M is, we just found that. C now is the distance from the neutral axis to the outermost fiber. And in this case because of symmetry the neutral axis occurs right in the center and c is equal distance of 12.5 inches up from the neutral axis. Or 12.5 inches down from the neutral axis. And which one will be in tension. Which outer surface will be in tension and which will be in compression based on the loading that's shown. And so what you should say is, okay, at the top because of the loading we're going to have compression. And at the bottom we're going to have tension. And so the other thing we need to do is find I, the area moment of inertia. And so I'd like you to do that on your own, and then we'll come on back and we'll do it together. Okay so I for rectangular cross section is one-twelfth the base times the height cubed or 1/12, the base in this case is 10 inches and the height is 25 inches, so cube that. And if you run those numbers, you get 13,020 inches to the fourth. So there's our equation for the maximum flexural stress. We saw that it was going to be, the c is the same for tension and compression so we're going to have the maximum at the top and the bottom, just one's in compression, one's in tension. We found out what the maximum moment is we have our c we have our I. And so I can go ahead and substitute in, so I get sigma max = M 865,625 ft- lb. And I'm going to put everything in terms of inches and pounds I'm going to convert this to 12 inches per foot times c is 12.5 inches and that's divided by I which is 13,020 inches to the fourth. And again, if you do that calculation you get 9,972 pounds per inches squared. And so there is a check that we need to do before we can determine that this is an okay solution. And what's that check we should do? So since we're working with the elastic formula we need to make sure that our beam stresses stay within the elastic range, and so we found that sigma max Was equal to 9.972 ksi. I just converted pounds per inches squared to ksi, or kips per square inch. And what is the, what is the yield stress for steel in terms of ksi? Go ahead and look that up on your own. And what you should see is that that's 36 ksi for steel. And so. And sigma yield for steel. And so 9.972 is less than 36, so we are okay. The beam remains in the elastic region. Okay, beam remains in the elastic region. So now we have our solution, so lets go ahead and put all of it together. So it's going to be 9.972 ksi compression at the top and it's going to be 9.972 tension at the bottom. Those are the maximum flexural stress and tension hand compression, and so here's a And that little graph, a three dimensional graph of that where we have maximum tension at the top or excuse me, maximum compression at the top and maximum tension at the bottom. And so that's a good standard elastic flexural problem for beams. And we'll see you next time. [MUSIC]