[MUSIC] Hi, this is module 14 of Mechanics of Materials III on Beam Bending. And today's learning outcome is to solve another elastic beam bending problem. And we're going to work with that same model that we did last time, but this time instead of using a rectangular cross section, we're going to go ahead and use something that's more common which is an I-beam type structure or girder. And so the worksheet says you decide to use an I-beam shape to reduce the weight of the beam, to reduce the amount of material and cost and it's a more efficient design. You'll see that as we go through the problem. So, if the beam is made up of steel with an elastic yield strength again of 36 ksi, what I'm asking to find is, what's the maximum moment the beam can support and remain in the elastic range? The first thing we're going to need to do is locate the neutral axis of the cross sections since it's not symmetric, and I talked about how to locate the neutral axis earlier in this course. You'll remember that it's located in the centroid and you can go back to modules 19 through 22 of my course, Introduction to Engineering Mechanics, to review centroids if you want to do that. And so, let's go ahead and locate the neutral axis, and to locate the neutral axis, we use this formula. And so, the first thing I'm going to do is, I'm going to break up, I'm going to use this axis as my data, with x at the bottom, and I'm going to break up my I-beam into three composite sections. And so let's start by finding the overall area for those three sections and so we have A total is equal to the top section is ten times two inches squared. The middle section is 2 times 25 minus 2 minus 2 is 21 plus the bottom section is 8 times 2 for a total of 78 inches squared for our A total. So now I want to find y neutral axis, and so we are go and sum the Y to the neutral axis of each of these individual areas that we have broken into. The first area, the distance to its neutral axis which is in the center of this rectangle, is Y1. So Y1 is going to be total distance minus 1 inch or 24 inches, times the area which was 10 times 2 and then you try to do this on your own and come back and see how you did. But the next section is 25 divided by 2 or 12.5 times the middle section's area which is 2 times 21 again. And then the final section, the final little rectangular composite part, the distance to its neutral axis is 1, so it's going to be plus 1 times its area which is 8 times 1. Then we're going to divide that by a total, which we found to be 78, and so we find our neutral axis. So, here's Y2 again, which is 12.5. And here's Y3, which was 1 inch. And we find that our neutral axis is 13.1 inches. And so there's my neutral axis, it's 13.1 inches up from the bottom, and I've labeled it there. Okay, now that we know that,let's go ahead next and find the I, the area moment of inertia about the neutral axis. And so I've got my neutral axis here again located at 13.1 inches up from the bottom. Again, we're going to break into the three standard shapes. And for the first standard shape, I'm going to have its moment of inertia plus Ad squared where d, we'll call d1, from the neutral axis up to the neutral axis of this standard shape. And so I've got I Neutral axis is equal to I for this top shape, is one-twelfth, base times height cubed. So 10 times 2 cubed plus the area of that standard shape which is 10 times 2 times d1 squared. d1 now is 25 minus 1 is 24 to this line minus 13.1. And we're going to go ahead and square that. So that's that distance d1, 24 from the bottom up, minus 13.1 squared. Do the same thing for the middle standard shape rectangle and the bottom standard shape rectangle on your own. Come on back. And you should find for d2 now, that's going to be the distance of the neutral axis minus 12.5 from the geometry. And so I've got 1 12th, 2 times 21 cubed, which is that center rectangle's, I about its neutral axis plus the area times the distance d2, which is 13.1 minus 12.5 squared. For d3, we have this third shape 1 12th, 8 times 2 cubed, plus the area 2 times 8, times 13.1 minus 1 to its neutral axis. And if you total that up you get I equals 6290 inches to the 4th. Okay so. Here was the two cross sections I had worked with. Last module I worked with a rectangular cross section this time I'm working with an I-beam type structure or a cross section. We saw that this area was 250 for the rectangle. It's 25 high and 10 at its widest point. The area for my I-beam is only 78, so I've saved that much area, that much material, that much cost, that much weight, in my structure. And so it's 31.2% of the original area. Now we calculated I for this rectangular cross section and this is what it came out to be. We just found that I for the I-beam is this amount. So, we only reduce the area moment of inertia by about 50%, it's 48.3% of the original area of moment of inertia. So we got rid of a whole bunch more area and didn't lose that much moment of inertia, so you can see where it's more efficient. And so now I have my neutral axis, I have my I. And so I know that the maximum stress will be Mc over I, where c is the furthest distance from the neutral axis, and so the distance from the neutral axis to the top section is 11.9. The distance from the neutral axis to the bottom of the section is 13.1. So, M max. Will be equal to sigma Max and we know that we can have a maximum elastic yield strength of 36 ksi times c. And c is this largest distance, which is 13.1 inches. That's wrong, let me erase that. Okay, 36 is this, I'm solving for M so I'm going to multiply by I. So that's 6290 inches to the 4th, divided by c, c is 13.1 inches, so we get sigma, or excuse me. M max is equal to 17,300 inch cubes, that would be our answer. One point here in using this equation, just a step I may have skipped on you, but we find that M max is equal to sigma max I over c. So that's how I got from here down to here, I just rearranged this equation. And so that's the solution and we're good to go. We now know how to do something more complex than a just a standard rectangular cross section, so I'll see you next time. [MUSIC]