Reactions: 2. Reactions Concentrated Loads Math Light

This video is about calculating reactions. Yes, it has some math, but we do the math once and then we try to find shortcuts to simplify. This is one of two videos. The first one is about calculating reactions under concentrated loads. The second one is calculating reactions under uniform loads. Please make sure you watch the one on concentrated loads first, which is this one, and then look at the second one for uniform loads because I use a lot of shortcuts that come from this video. Very good. So we have a load of 10 kips deployed in the middle of a beam, and we have a pin on one end, and we recognize that a pin is the equivalent of two reactions, whereas a roller is the equivalent of one reaction. We have one horizontal reaction, we have a left and a right vertical reactions, and a 10 kip, which happens to be in the middle. Now, with this load in the middle, it's symmetrical, it's a no-brainer. The left gets 5 kips, the right gets half of 10 kips. We have three equations for equilibrium that we will use to solve any statics problem. And these equations are the sum of horizontals equals zero, the sum of verticals equals zero, the beam doesn't move left-right nor up-down, and the sum of moments about any point along the beam should equal to zero, which means the beam doesn't rotate clockwise or counterclockwise. Very good. So a free body diagram would show, instead of a pin and a roller, it would show the reactions. Excellent. So, now, if this 10 kip were to move slightly to the left of center, then it would make the left reaction a little bit larger than the right reaction. And the closer it gets to the left support, the larger the left reaction, the smaller the right reaction. Until it ends up on top of the support, and then the left reaction would be 10 kips, the one on the right would be zero. And of course, the horizontal is zero in the absence of any horizontal loads. Excellent. So if the load of 10 kips were to move to the right, then the right reaction gets bigger than the left, until if the 10 kip is over here, that's like a column sitting on top of a column, then the other reaction is equal to zero. Excellent. So let's take a look at these two problems. They have the same loads, a 10 kip in blue and a 19 kip in green, but the spacing and the location, the point of application of these loads is different in the two problems. In the first problem, on the left, it looks like the dimensions are very nice, which makes it a really easy problem. But the dimensions on the right-hand side are pretty ugly. There's nothing nice about them, so there might be a little bit more math involved in a problem like this versus the one on the left. Excellent. So let's dig down and take a look at this problem, the ugly one first, and then we will go back and solve the other problem, which is more similar to what you might expect on the ARE. But I'm going to solve this problem longhand so that we recognize how to solve a problem that is totally non-symmetrical. And if you get something like that on the ARE, I strongly recommend that you skip it if you cannot reason out the answers. Hopefully from this session you will get some shortcuts that will help you with a multiple choice question and not as much with fill in the blank. Very good. So you learned in school that you sum moments about the left support first and then you equate that to zero. That one gives you the right reaction, and then you sum moments about the right support equal to zero, meaning the beam doesn't rotate, and that gives you the left reaction, and you must add these, and make sure that the sum of vertical forces is zero, which is basically the forces going down should equal to the reactions going up. In this problem, they should add up to 29, because I have 10 plus 19 down, the two red arrows up should add up to 29. Very good. So let's get started. I'm going to sum moments about the left first and consider this just a practice in calculating moments. So summing moments about the left, if this is the right reaction, this is the left, then I have 10 kips and the 10 kips is at 6 feet from the left support. And let's put a convention down. For moments, clockwise is positive, counterclockwise is negative. For vertical forces, up is positive, down is negative For horizontal forces, to the right is positive, to the left is negative So convention for horizontals, for verticals, and for moments This is just convention, you can have your own convention as long as you're consistent You don't add opposites Very good So 10 times 6 is doing clockwise about the left support Clockwise is positive counterclockwise is negative according to this convention. So 10 times 6 is clockwise, 19 is also clockwise, making a clockwise moment. And I need the distance from the 19 kip to the left. It looks like 14 plus 6. So times 20 feet, and this is positive. And the right reaction is doing a counterclockwise moment, so minus 24 r is equal to zero. Working out the math, I get 60 kip foot plus 380 kip foot is equal to 24 r. Add these up, divide by 24, and you end up with a right reaction equal to 18.33. Very good. I don't know if this answer is correct or not. Make sure you don't take 18.33 and subtract from 10 plus 19 and get your other answer, because you want to be able to check that you are correct. So to check that this work is good or no good, maybe it's not good, I'm going to sum moments about the right support equal to 0. So now we're summing moments about this point. Going from left to right, I have the left reaction, and it's 24 feet away. And it makes clockwise about the right support, so we're going to give it a plus sign. The 10 kip is making counterclockwise, so it'll get minus 10 kips, and I need the distance to the right support from the 10 kips. That's 14 plus 4. And then the 19 kip is at only 4 feet from the right, and it's doing also counterclockwise. It gets a negative sign. So the sum of these is 0. 24L, I'm going to take these two terms to the other side so they become positive, is 180 plus 76. So work out the math and you end up with a left reaction equal to 10.67. Now let me check that up equal down. Up I have 10.67 plus 18.33 and that adds up to 29. Down So these answers are good. Excellent. I should have guessed this, if this were multiple choice, I should have guessed that the 19 kips at 4 feet is going to cause a greater reaction on the right than 10 kips at 6 feet. So if you had multiple choice, you should be guessing the right reaction is larger than the left. Excellent. Now, I would like to not do all this math. It's too much work. Is there a quick way of getting an answer? What I suggest we do is we look at each load separately, and we find the reactions to each load, and then we add up the different cases, and we end up with a total reaction. So let me look at the 10 kips separate, the 19 kips separate, And let me say the following. The 10 kip is at 6 feet from one end, 18 feet from the other end. Quickly, a nice shortcut is the 10 kip breaks down into two reactions. One of the reactions is 6 over 24. The other reaction is 18 over 24. 6 over 24. And 18. 18 over 24. So it looks like 6 over 24 is 1 quarter of 10, 2 1⁄2 kips, and the other one is 3 quarters of 10. So my two reactions are 2 1⁄2, 7 1⁄2. Which is which, I don't remember, but I know that the center line is here, and that the 10 kip is closer to the left than it is to the right. I see I have two reactions, 7 1⁄2, 2 1⁄2. The larger is the one on the left because the load is closer to the left. The smaller is the one on the right. So the reactions due to the blue load of 10 are essentially 2.5 kips on the right, 7.5 kips on the left. Good. Let's do the same with the 19 kips, although the numbers are going to be uglier because 19 is an ugly number to begin with. And then it's coming in in an ugly way. Fine. But the logic is the same kind of shortcut applies. 4 over 24 and 20 over 24. are your reactions. And let's keep in mind that this time the load is closer to the right. That's the larger of the two reactions. Very good. So 19 kips is going to break down in the proportion of 4 to the total and 20 to the total. And I can't do this math in my head, so I've prepared it ahead of time. This is 3.167 kips, and this one is 15.83 kips. Check, the two of these should add up to 19, which they do. Now, which is which? The larger is on the right, so you are the right reaction, you are the left. So the left is 3.17, roughly, if I round up, and this one is 15.83. And back to my problem, my original problem, I can, ooh, this got real messy and I can't erase. Very good. So, it looks like the left reaction for this problem is going to be, how much from the green load? 3.17 and 15.83 on the right. From the blue load, it looks like we're getting 7.5 on this side. Oops, that's the blue load. It should be blue. Sorry about that. The blue load is 7.5 and 2.5. So that gives me a total reaction in red of 3, 13, 8, 18.33, and 12, oops, 5, 7, sorry, 6, and 10.67, which is pretty much the answers that we got earlier. Using a lot of math, I would much rather we use these shortcuts. Excellent. Let's take a look at the easy problem now that has equal, equal, equal. My life just became a lot easier. And I'm about to simplify one step further. So I have 10 kips and 19 kips just like before, but they're at 8 feet, 8 feet, 8 feet. Let's look at the answers. This answer does not make sense. They should not be equal. Here's the center line. There is 19 kips on the right. There's 10 kips on the left. It looks like the right reaction needs to be larger than the left. So are any of the answers given say that left is greater than right? No. So, okay, that would have been a cheap shot. Fine. So, oh, this answer makes absolutely no sense. You can't say 10 goes here, 19 goes here. That is bogus. Very good. Let's see what we can do. Instead of doing what I did here, which is separate the loads, that works perfectly fine for this problem. But instead, I want to simplify even further. Because the dimensions are equal down here, I would rather cheat a little bit and say, I wished I had 10 kips on this side. I don't. I have 19. I wished I had 10, then it would have been symmetrical. But it's not symmetrical. So let's express the 19 kips as 10 plus 9. Now what happens is the 10 kip blue is symmetrical. So you take 10, and I take 10. And all we have to do is deal with the 9-kip load. Where is the 9-kip load coming? I need my dimensions. So, the 9-kip load looks like it's coming in at 8, 16, in a 24-foot. So, the 9-kip load is going to break down as 8 and 24, and 16 and 24, 8 and 24, and 16 and 24. Ah, 16 and 24. So 8 and 24 is 1 third of 9, and 2 thirds of 9. Now which is which the 9 kip component of the 19? The 9 kip is closer to the right. Well, then you get 6 kips, you get 3 kips. For a grand total of 13 on the left, 16 on the right. So the correct answer is this one. This does not take time. You have to figure out the shortcut, and then you should be able to calculate reactions very quickly if you use this shortcut. Excellent. Let's go on. Oh, this is ugly. Fine. So let's look at something like this and hope you don't get it. But still, if you have 250 kips and, sorry, 250 pounds, and the slope given is 16 run and 12 rise, we said in force components that we need to find that hypotenuse of the proportion triangle. 12 square plus 16 square square root gives me 20. So the proportion of the components of the 250 pounds is in the proportion of 16 and 20 of the 250. Please, if you have not watched the video on force components, this is a shortcut that comes from that video. And essentially, this is 3 fifth of that, which is 150 pounds. And this is 4 fifths, which is 200 pounds. So when you have anything at an angle, you need to break it down into its components and express the components instead of the 250. Very good. So my vertical component, I'm sorry, I didn't do that. The vertical is smaller than the horizontal. The 150 is smaller than the 200. Well, then this is the vertical. This is the horizontal. So this is 200 pounds, and this is 150. So with this 200 horizontal, now the reaction at the pin is no longer zero. I need to pick up that 200-pound force and counter it. This guy on the right, the roller, has no horizontal component, so the 200 pounds has to go all on the left. Excellent. Now to find the vertical components, it's a simple matter of doing the shortcut we've been doing so far. You have 150 pounds, and it's coming in at 8, 16, and 24, making this reaction bigger than this one because here's the center line, you're right of center. So the components will be 8 over 24 and 16 over 24 of the load. So the load is 150 pounds. That's the vertical component. We're done with the horizontal component. And my reactions due to the 150 pound, 8 over 24 is a third of 150, and two-thirds of 150. So the reaction on the right is the larger of the two, so must be you are 100 and you are 50. So, at the end of the day, the left reaction is 50 pounds, the one on the right is 100 pounds, and there is a horizontal reaction at the pin worth 200 pounds in red. Excellent. That's it for this one. There was a similar question on the ARE from ARE 3.0. That's why I cover this one. This is another example that is similar to what has been on the ARE before, and it says here's a truss loaded symmetrically, the dimensions are equal on the bottom, calculate the reactions. Now it's important to recognize that this truss is not symmetrical, but the loading is symmetrical about this line. So this problem really is no different than a beam, as far as reactions go, not as far as the truss goes, with two kips and two kips and two kips and two kips and another two and another two. So there is symmetry about this line. So these six kips will go here and the other six will go here. So don't be fooled by the truss not being symmetrical. You're looking for reactions. Solving this truss will be a nightmare, but that's not the question. The question is calculate the reactions and the loading is symmetrical. The dimensions are symmetrical, so 6 and 6. Now, what happens if I throw in a lateral load of 5 kips at the top of the truss? Are the reactions 6 and 6? Somewhere there needs to be a pin to take the 5 kip load. If the pin is over here, then that's 5 kips. If the roller is over here, then the pin is on the other side, and the 5 kips horizontal is on the other side. Sum of horizontal forces equals zero. I have 5 kips to the left attack. There is a reaction, 5 kips to the right, so the truss doesn't move horizontally. Now, the question is, which one is larger, or are they equal? The left reaction, the right reaction. To solve this problem, we go back here, and we say whenever we had symmetrical loading, we had reactions of 6 and 6. So let me add to this one the 5-kip load. Now, what is the 5-kip load going to do? With 5 kips coming here, what does it do to the right? What does it do to the left? Left, right. Now, this is a horizontal load. One would think that it wouldn't do anything to the vertical reactions. But the 5 kip load is doing rotation. It's doing overturning. It wants to turn the truss and make it do that. So, with that action, it looks like this reaction has to fight back. And it looks like the reaction on the right needs to tie down the truss and not let it lift at the right support. So I don't want to do the math, but my answer on the left is more than 6. My answer on the right is less than 6. So if this is a multiple choice, you should not answer 6 and 6. If it's a multiple choice, you should be looking for a reaction larger than 6 on left and smaller than 6 on the right.