 All right, since there are no questions, we'll get going here. What we're going to do now is what's called beam design. I'm not sure. Well, I guess it's dead to a point with some of what we're going to do. But a lot of what we're going to do is already beams designed and so we're not going to be doing anything or terribly original here, of course. What we'll look at though is specifically beams in bending. So these would be like support beams in roof or ceiling or floor joists, that type of thing where they're loaded transversely that causes the beam to bend. We'll look a little, at least take into account shear. However, what we're going to find as we already have, maybe you put the two together, maybe you've done quite, but this will definitely be the dominant of the type of stresses that we're going to see, the bending that leads to the normal stresses, the shear, of course, the transverse shear stresses. And it's going to be the normal stresses from the bending that are dominant. So let's pull that back out. We've got, let's see, the maximum normal stresses we expect to see in a bending type load that leads to some internal moment will be the maximum moment at the maximum distance from the neutral axis. Sometimes these are not symmetric in the y direction so that that C might be different going above the neutral axis than it is below divided by the moment of inertia of the cross-section of the beam. Or if we're interested in just some point somewhere in between then we only take a look at some portion across the cross-section. Remember the minus sign is here, too, but for some reason they choose to leave it off. I think it's a little careless, but that's what they do. It's their business. I'm just teaching it. Where you have to pay attention to the situation itself to understand whether this happens to be negative and the stresses are compressing, compressive stresses, or if it's positive and those are tensile stresses. But either way, for most materials, especially most typical building types of materials, the compression stresses, the limits, and the tension limits are available. Now the thing we're going to play with now for the next day or two is the fact that this part of it is geometry only. It has nothing to do with the load. And in fact we then define a geometric term called the section modulus for a big capital S. So that's how I write a big capital S myself on the board. And it's the inverse of that little ratio we had there, that sort of geometric ratio piece we had. Now to show why that ratio is more important than either of the two alone, let's take a look at two possible common beam shapes we could use. Let's take this to be 6 by 4, and another beam that we could use is 3 by 8. All we've really looked at before in specifically choosing beam shapes and understanding what the stresses are, were, well, we looked at the cross-sectional area. And both of these obviously have the same cross-sectional area. A straight look at the normal stresses from axial loading, like we started the class with when we first introduced this, we looked at just axial loading down the beam, because either beam would be just fine, it wouldn't matter either way. But now that we're looking at bending stresses where we have this ratio of the section modulus, these have different section moduli. So let's figure out what those are and see why this type of thing makes some kind of difference. Remember the moment of inertia for a rectangular beam is, remember it off hand? Yeah, that's the 112th BH cube. Of course, C, since these are symmetric, C is going to be 1 half H. We can tidy this up a little bit before we start putting in numbers. So that makes it 1 sixth, 1 H plot changes one of the cubes above that comes in 1 sixth BH squared. But B times H is just area, so this becomes even easier. If we already have the area, there it is 1 sixth area times H. Just cleaning things up a little bit. For rectangular beams, things can become real simple. So for the first beam, that kind of big square one, I guess we could call that the mother-in-law beam. Sorry, man, I'm just kidding. I think she's given up watching this video. We have 1 sixth, 24 inches squared times H, which is 6 inches. So what's that real quick? Actually, we can do that here. We have 6 over 6 times 24. We get 24. So do real quick. Then what's that? Inches cubed. So for the second beam, this 1 sixth A squared still holds the 1 sixth A and H now this time is 8 inches. What does that come out to be? Real quick, around 32. No, we're still talking about the same amount of material, same weight of the beam. So we don't have to worry about one being heavier than the other. However, the section modulus for this one is much greater. The greater the section modulus, the greater the section modulus, the less these normal stresses are going to be. So if we have a choice like this where the weight doesn't make any difference, the material might cost the same. Manufacture is not going to be terribly different. Even if these are lumber, it's just about as easy to cut 6 by 4s as 3 by 8s through beds that are maybe a little more common. But using the second beam, you get a lot greater section modulus, a lot less normal stress for the same amount of material, same size, same everything. What we're really looking at is the fact that this has a lot more area away from the neutral axis. That's what increases the moment of inertia. That's what increases the section modulus. If you've ever looked under a deck or anything, this is the type of choice you see there. You don't see this type of thing. You'll see this for posts, but you'll see this for floor choices. And that's because the section modulus is greater, so the normal stresses experienced in the material itself are much, much less. Another beam that's very common is, of course, the I-beam, which looks something like that about its neutral axis. With all of this area added, a long way from the neutral axis, the section modulus for I-beams goes way up. And that's why this is the most common beam used in construction, especially where there's cast beams. Also, it's very easy to make a beam like that out of wood by just bolting on some cross pieces there. And by doing so, you increase the section modulus and thereby reduce the expected normal stresses in the beam itself. So let's see how we're going to use this type of thing for ourselves. A simple kip load at the end of an 8-foot, a simple cantilever loading. We know from experience, I bet, that we're going to expect the maximum moment back there against the wall. Just kind of common sense thinking about it. You ask anybody out there in the street, if I did this and tell them this beam broke, they'll probably break here. Most people would just gather that. How big is that maximum moment? So what? Which is the 8 times 15. So we know that the maximum stresses have to do with that, but also have to do with the section modulus. So if we take a particular material, we could say some kind of limit on it. Let's say we're using steel, so there's an allowable stress of 24 KSI. That allows us then to say, well, I need a beam with a minimum section modulus of what? 120 kip feet is our expected maximum moment. Our allowable shear stress is 24 KSI. KSI giving us a minimum section modulus of you do this, but I need it in inches. The kip's will cancel. We'll have feet per square inch, but we need it in... I'm sorry, we don't want feet per square. We have feet inches squared. We need inches cubed. Agreed? 60 inches cubed. So we need a beam with 60 inches cubed as our best possible choice. Where do we get a beam that does just that for us? If you have your book here, open it up. If you don't, that's all right. What we've got is a bunch of different possibilities. Well, that's better. All right, this is appendix B. If you have the eighth edition, that's page 801, if not seventh edition, I think it's 849. Is that right? And they should be pretty much the same tables. If you flip over, we get just different beams. Come on here, baby. There we go. Gentlemen, we didn't spend much time with some asymmetric beams, but all of these things are generally commercially available. And we also have FPS units feet per second. I don't know what that is. Or SI units. All right, so we're doing a problem in English units, so we use the very first table here. And notice that one of the columns is the section modulus. Actually, two of the columns are section modules. One's on XX axis, one's on the YY axis. You just have to look at the picture to see what they mean. So it's either the beam being used as an I beam, which would fit our problem. Or you could use it as an H beam, then you look at the YY axis and notice there's quite a difference in section modulus by taking the same beam and just turning it on its side. The section modulus changes by an awful lot. So we're looking at an XX type setup where we're going to use this as an I beam. All we need to do is go down and pick any beams that are section modulus greater than the 60. So if we get down a bit, so here's an S of 57.6. So that beam's going to be too small because the section modulus isn't going to be big enough. So we can use any beam that is just above that. So one candidate is W18 by 40. So what we can do now is list a couple beams that might work for us because that's not the only one that's got possibilities. And this 18 is its approximate depth. That's the D measurement here in the picture. And the 40 is the beam's weight per foot. That's important. That could be important because now as we're looking at real beam choices, if we get a very, very heavy beam, we actually are going to add to this load because the beam load, the beam weight is heavy enough. It itself becomes equivalent to a distributed load because it also has to hold up its own weight. So we want to keep an eye on that as a possibility. So we have the W840 with a section modulus of 68.4. All right? Everybody see about where we are, about midway down the page? We have a section modulus of 68.4 on the W18 by 40 beam. Go down and see if there's any other beams that do okay. Notice the W16 by 45, 16 by 45 has a section modulus of 72.7. I picked that one because the next beam below it has a section modulus below our limit of 60. So just look through the tables. All that happens is the sizes of the beams change. That's why they're grouped kind of like they are. So let's see. We have the 1645. Let's see about a W14. W14, what looks good? What looks juicy on that? 43 because the W1438, the W38 has a section modulus that's too small. It's below the 60. So what do we say? W1843, no, W1443 at what, 62.7? Is that right? Any others? Well, we kind of skip past the W24s, but you can already see, man, those things are heavy. And the section modulus is way over what we need. So you shop around for a couple of those, then decide, well, it looks like this beam might be the best choice. It's got enough section modulus, but it's also the lightest of all of them. At least on a first cut. There's other things you have to look at. You know you get a call of your supplier and say, do you have any W18 by 40s available in such that they say no, we're going to have to get those from Taiwan because we don't think anything in America anymore. Three years before we get delivered because there's too many tsunamis out there and those are the type of concerns we need to look at. Notice that when we do this, we also have to look now at the weight of the beam. We have an 8-foot beam at 40 pounds per foot. So each beam itself weighs 320 pounds. Well, we're talking about a 15-kip load. And so if we're putting in a 320, which is a 0.3-kip load, that's hard to even show on the diagram and scale with the 15-kip load that's already there. Plus it's inboard a little bit farther so it's not even going to have that much moment added to it. However, if we were really close on this, we might work. You know, if we had a beam that was just barely over the 60, you better put the weight of the beam in the problem and then redo the calculation. Put the expected weight of the beam in here. Looks like it's going to be 320. That makes the maximum moment go up. That makes the minimum section modulus go down and then may or go up and may indicate we have a different beam we need to choose there. So that's the type of dance we're going to do through these particular beams. Is that 40 pounds per foot? Yeah, 40 pounds per foot. How are you right? Is that a skater? Pounds per foot. I'm writing in the English system. Occasionally. So, as straightforward as that, now we're going to look at some other options in a little bit, but for now we're just going to look at the beams right out of the books, see what's available. Let me throw you off to a problem yourself and let's see if we all come up with the same beam as the recommended possibility. Alright, simply load it. We're going to look at an I-beam distributed load to about there and then a point load there per meter that little section of the distributed load is three meter then one meter on either side of the point load will set up. The allowable normal stress for steel is about 160 mega pass counts. So I want you to recommend a beam such that we stay below that allowable loading to an I-beam. Sorry? Yeah, use the same table we just came out of. Well, you're going to have to go to the SI units and I'll meet you back. Alright, so you've got to figure out where the we don't necessarily yet care where the maximum moment is but we do care what it is and you can use some of your shear moment diagram things to speed that up actually calculate the entire thing and rapidly come to a conclusion of what the maximum moment is a little bit later we'll look at where it is and design for that concern for right now we just care what it is for it's hard to need to go to shear moment relation the reactions and then you call them the shear diagram and tell you then what the moment diagram looks like think about it a little bit which we'll do together in a second the shear diagrams look good I-52 and then remember that the slope of the shear is equal to minus the load so the load is 20 kilonewtons per meter so we have a slope of minus 20 and that'll take us down 20 times 360 so we'll go down to something like a minus 8 then the slope of the load curve is 0 so the load is 0 so the slope of the shear curve is 0 then we jump down 50 minus 58 just right for the plus 58 to bring us back that's what most of the shear curves look like that I see now that's all you need to find the maximum moment you don't actually have to find the load curve you know that the shear is the slope of the moment curve that moment is going to have a slope of 0 right there it increases until there has the slope of 0 and then decreases and the moment at either end we know to be 0 so without knowing any of the great details we know it's going to look something like that with a maximum moment right there figure out the area on either side they should be equal just opposite the side and that will give you the maximum moment because we have no moment at the end so it increases by the area of that triangle you know that slope of that line so you can figure out then what the base of the triangle equals and then the area of that triangle will give you the maximum moment that makes sense so we don't actually need the details of this load curve though if we did we have a couple straight sections in here because the shear is constant but we don't need the shape of the shear curve we just need the moment curve we just need to know where the maximum is what the maximum is we don't even care where except that we do need to find that because of this area of that triangle 0.6 meters sorry except you know the slope you can find then this point and you'll know what the base once you have that and it's one half the base times the height 0.6 kilometers so that's the maximum moment so pick out a couple beam possibilities and if you got your cell phone go ahead and call up ally steel and order a couple section modulus that couple just to make sure units of course units in the table for section modulus are in millimeter what's this come out to be 160 411 422.5 I keep guessing I'll hit it do whatever you need to do with that to get millimeters q because that's what's in the table millimeters q millimeters q so they've just locked off this part for the top of the table which makes sense makes the table look a lot cleaner a lot simpler remember that when you make tables whole beams that have section modulus less than that let's see what what you all agree on you need a section modulus that's greater than that that's our minimum allowable so we need something greater than that greater than or equal to the ratio m max to signal allowable I guess put your finger on something alright recommend a couple pat recommend one w3 10 by 30 has a section modulus of 540 steps so we're well over 20 of margin bobby recommend one section modulus as much over but a little bit certainly over collin recommend one section modulus jake is the one you selected up there which one is it as the lightest so how much load then do we expect from the weight of the beam itself of what 5 meters of beam 33 what are the units kilograms per meter w360 33 alright so watch your units what do you have to do kill a newtons kill a newtons total to see how that matches to that 10.8 meters per second squared that gives us that newtons per meter we need to kill a newtons so that then is going to be the weight of the beam that's where I want to use w for that because we've already got that w in there for the beam designation itself that makes sense with the units on the weight of the beam that will give us a kilo newtons we have that duty 1.6 kilo newtons not much of a concern for the weight of the beam let's look at something else number 4 we'll do one then on Monday the other concern is that we could have a allowable shear stress limit on this that we'd also need to take into concern because we do have a section of the beam that's under maximum shear so we want the allowable shear to be greater than what we're experiencing so we stay under that here's the thing you need to consider with these i-beams in shear the flanges which are these outer parts carry very little shear there's very little protection they're incapable of carrying much shear it's just that there's not much shear there most of the shear is in the center of the beam so for the area calculation you use the area of the web itself and some of the tables have that some you need to figure out what it is you've got the web thickness which is T w and then the height of the web here is B minus 2 T is that a flange so for this for this beam what's the what's the web thickness for the beam we picked for 360 33 you've got a bolt there 5.8 millimeters for the web thickness what's the symbol they use on that T w and then the height of the web is it's that D minus 2 T f is that what they use for the flange thickness very little T f so it would be D minus 2 T f what's that come out to be or fuses to open that book down 349 around 17 332 do you agree then you have to check your units so that we have remember the allowable stresses are generally in megapascals or kilopascals at least so double check your units and then you can check that against what the allowable shear for the the beam itself is and I don't have to have that number on hand for steel check that against the allowable shear stress for the steel itself make sure that's okay too but it's still just something you need to check the full problem down on Monday maybe any questions? assume that like the weight here we kind of determine that wasn't an issue for appropriate reasons when it was like somewhere close to either one of those numbers? if it's such that it is a personable portion of these or even if you're just concerned weight 1.6 kilonewtons actually we'd make it 1.6 kilonewtons divided by the 5 meters so we get it as a distributed load and then you add that in as a distributed load evenly across the beam and you'd have to re-figure the problem you'd have new reactions now you'd have new maximum shears maximum moment you'd just re-do the calculation and see if this still comes out okay because all of that is going to lead to another section modulus limit and it may prove that this beam is no longer adequate so you have to go to another beam and see if the a higher section modulus but it's easy in that it only takes a second to check it okay