Geometry Of Sections: 6 Geometry Of Sections Question

So I would like to conclude the section on shape of cross-section with this question that came from Kaplan that I have customized and adjusted a little bit to meet the needs of my objectives for this chapter. So the question said there is an existing wide flange here in green and this existing white flange is deflecting too much. It's in a warehouse. Which of the following options reduces the deflection the most? Okay, it's a warehouse. It has this green-white flange. And in solution A, they're adding a plate. In solution B, they're adding the same plate, same area, but flipping it. And then in solution C, they're adding the plate and the area is the same, so it's a little bit thinner each piece. And then they flip it. And finally, a white flange. So, for these five solutions, the amount of additional blue steel is the same for all five solutions. Now, if cost and axis and weldability and all other factors are ignored and only structural performance is considered, which one of these five solutions is the best? So, solution A added a little bit of depth, just a bit. Solution B is much better than A. For the same amount of steel, you put more square inches of steel, more mass, further from the existing centroid. So you improve the section modulus against bending, and you improve the moment of inertia against deflection. In solution C, there's a T. And all of these, let's assume all of these are the same exact depth. and exact same width if they have a flange. Okay? And the same area. So, I think A is out, because B is better than A. Now, between C and B, I just look at the depth, and I see that they have pretty much the same depth. And this flange is very close to the existing flange, so it didn't contribute much. In solution D, the flange, same flange, is over here Which is at a significant distance from the centroid Versus in condition C, it's real close to the centroid compared to D And in E, there's two flanges, one, two, for the same area It's just their spread So, which is better, C or D? Sorry, we're done with A and B because this one has a web this one has a web so does this one, so does this one yes, this is fatter, I'm not interested in width, I'm interested in depth these four solutions have the same purple web and we're talking about the green flange that's the difference between C, D and E Excellent. So, is C better or D better? This green flange is close here, and it's much farther here. You're out. So now we're left with D and E. And which one is better? Please, D may not be practical. E might be a lot easier. That's not the question. And the question is, which one is better structurally? So let me explain this question a little bit differently. Let me say that here's the existing centroid of the shape we have. And here's the centroid of what you're adding. The T has a centroid a little bit higher in C and a little bit lower in D. and here in condition E. So the best answer is the one that separates these red dots the most. That's the best answer. So it looks like that's moment of inertia and section modulus. Put the most steel the furthest from the centroid. That is clear in condition D. There's condition condition B. And condition A is the worst. Now, I have some pictures here. Sorry, the answer is D. Better than E, believe it or not. If the area is the same and the depth is the same, D is harder to bend than E. Here's a situation where they did solution A and they put this plate here and they found the fattest piece of plate that they could find. And the reason they did this is this is a building that was renovated. And now it's a library on the upper floor. Library is heavy dead load, 150, sorry, heavy live load of 150 pounds per square foot versus what it used to be. I don't know what it used to be, but now it's a library, so they wanted to add this plate in the middle because deflection and bending are high in the middle. You notice this plate doesn't go all the way to the support. It stops short. So clearly it's doing bending and deflection. And the problem here is they needed to run some ducts, so they needed the headroom. Well, that's a different story the question we have. We had a warehouse, room is not an issue, access is not an issue, welding is not an issue. So here's another example where they're adding a wide flange to a wide flange. Well, they have the room and must be the deflection or the bending is very bad. So they're adding a wide flange, they had all the room. And here's an example where they added this upside down T, which is the solution for the depth and that much steel area, square inches, the best option is the T. Okay? Bye.