General: Deflection

For a spanning member between two supports or continuous beam, when loaded perpendicular to their axis, what happens is the top goes into compression, the bottom goes into tension. That is a stress called a bending stress, and its outcome is a strain called deflection. So when a beam bends, it deflects. Just like when an axially loaded member is pushed axially down, it shrinks. That's the strain. It gets a little shorter. And when pulled axially along its axis, an axial member elongates a little bit. So elongation is a strain of tension. Shortening is a strain of compression. And when loaded perpendicular to the axis, a member develops a bending stress, which results in a strain called deflection. So looking at these images, deflection is basically how many inches, or rather fractions of an inch, has a beam gone down between its supports. That's deflection, and we can see it in this pedestrian bridge. It deflected that much, and it's measured in inches, just like elongation and shortening are measured in inches, deflection is also measured in inches, and hopefully all of these strains are measured in fractions of an inch, not a full inch. So this dimension is called the actual deflection of a beam, in contrast to an allowable deflection, which is what the code specifies. The code doesn't say use wood, steel, concrete, use this depth or this material. It just says, if you're spanning this much, here's your allowable deflection. That comes from the code, the IVC code. So the code specifies deflection based on use. So a floor is somewhere we walk on. Therefore, the deflection needs to be stricter than a ceiling that we don't walk on or a roof that is not occupied versus a greenhouse. If plaster is involved, then the restriction or the limit on deflection is very high. And the larger the number on the denominator, the stricter the deflection. So L over 120, L over 180, that is a huge deflection, relatively, compared to the span. Versus an L over 240, L over 360, are you talking about deflection compared to live load or deflection compared to live load plus dead load? There is an L over 480. There's an L over 540. There's an L over 600 and an L over 720. And it's all in the code. Allowable deflection is from code. But the actual deflection actually depends on the load. It depends on the span. Most of all, it depends on the span. It depends on the material. Is this wood, steel, or concrete? And also it depends on the moment of inertia, which is essentially the depth. of the member. So we don't want to see the stress of bending or elongation or shortening. We just don't want to see them. And we don't want them manifested. I don't want to see this deflection. So there has to be an allowance because we know that that slab is going to deflect a little bit and we can calculate how much that deflection is and we need to be within the allowable deflection but then the slab is going to deflect. I don't want it to crush the studs underneath it. So this slotted track is for deflection. And if you'll notice, the screws are on the bottom, which allows that much movement. So if the screw is on the bottom of that slot, I have that much movement allowed. And I've calculated the deflection, know how much it is. and this track allows so much deflection. This one doesn't look very healthy because the screw is on the top of the hole. If this beam wants to deflect a little bit, it doesn't have much room. It's going to crush the studs that are underneath it. If we look at this example here, there's an embed plate in that concrete slab. It's sitting, it has studs in there, and it's embedded in the concrete, and then they welded an angle on the site. And that angle has slotted holes. And those screws are not very tight. They're just keeping the plate in place and allowing vertical movement with deflection. Very good. So deflection is the outcome of bending. So we need to look at some cases of span because the span also plays a role. Let me fix that. So type of span, in addition to the load and the actual feet of span, the material and the depth or the moment of inertia. So there's different types of spans. There's a cantilever, there's a simple span, There's an overhang, there's a continuous span, and there's a restrained span. So just some basic rules. Cantilevers are going to deflect more than anybody else. And a concentrated load will cause more deflection than a uniform load, clearly. Whether it's a cantilever or not, concentrated loads do more deflection than uniform load on any span type. So, looking at simple spans, I have a concentrated load. Let's say this concentrated load is 12 kips, and I compare it to two 6s, that's a total of 12, versus a uniform 12. For the same span, and the same material, and the same cross-section, then clearly the 12 kip in the middle is going to cause more damage, more deflection, than two 6s, which cause more deflection than a uniform 12, a uniform load deflects less than an equivalent concentrated load on the same span, the same material, and the same cross-section. If there is an overhang, then the deflection is reduced. So here's the deflected shape for a simple span under a uniform load. Well, that's not what's going to happen in an overhang because now, sorry, in the simple span, it was all a positive moment. The moment diagram was all positive. But in an overhang, now we just introduced a negative moment on the overhang. And therefore, the deflected shape, that belly is going to come up in the middle because the negative moment took away from the positive moment and the deflection went up a little bit. The belly of the beam went up a little bit because of the overhang. And the longer that overhang, the greater the negative moment and the less the deflection up to a certain point. We don't want it to look like that. That's not healthy. So an overhang up to a third of the span between the supports is very comfortable. Comparing a uniform load on a simple span to a uniform load on a continuous span, the deflection is not going to be the same as two simply supported beams for the same span. It's going to introduce a negative moment over the support again, and between the two supports, there's going to be a positive moment, and the deflected shape is going to look like this. So it goes up wherever you have a negative moment. It goes down where you have a positive moment. So this deflection is going to be smaller than that of a simply supported beam. And the more bays you add, the less the deflection. So this guy is going to deflect like that because there's a negative moment over the column. There's a positive moment in the middle. And the deflection in the middle bay is going to be a little bit less than the deflection on either end because these guys don't have an additional overhang to help them. So the guy in the middle is surrounded by two bays that lift up the belly of the middle bay. looking at four bays it's going to be even less than the one before it it's going to do that again negative moment over the support positive moment in the middle and the deflection is due to the positive moment and the negative moment is doing that the positive moment is doing that and the negative moment reduces the deflection due to the positive moment So continuous beams deflect less than simply supported beams of the same load, sorry, under the same load on the same span and using the same material and the same depth or amount of inertia. Again, a concentrated load causes more deflection than a uniform load. But a restrained beam has the two ends very rigidly supported. so it cannot rotate at the support. It can't do that because it's so rigid, so it's not going to deflect like that. Instead, this is an extremely rigid connection and will not allow rotation or deflection, and then it's going to do that, and there's a negative moment at the support, another negative moment at the support, and a positive moment in the middle, and yes, there's two points of inflection, but it's as if the span is from here to here. So recapping, for the same load, span, material, and cross-section, a restrained beam deflects the least, then followed by a continuous beam. A simple span deflects more than a continuous or a restrained beam, and add an overhang and the overhang will cause less positive deflection than that of a simply supported beam. And of course, cantilevers deflect the most. Also, concentrated loads cause more deflection than uniform loads. So that's the summary of what I wanted to say about cantilevers. For more information or numbers or what have you, there's plenty of other videos, but this one is just a summary.