Forces & Load: Force Addition Graphic Concept

In this video I'll talk about the graphic method for force addition and we'll touch on resultants and equilibrance and basically the head-to-tail or tip-to-tail method of adding forces graphically. For the ARE, the questions are multiple choice typically, they could be fill in the blank, but more likely than not, a multiple choice question has four answers and you should be able to eliminate at least two just by visual inspection, and that's the objective of this video. So whenever the forces are in a formation of head to tail, they can be added directly, and it doesn't matter the order in which you add them. So I'm looking at this example here, and I'm seeing a tail and then a head over here, and that's force number one. And then force number 2 has its tail at the head of force number 1, that means we can add them directly to come up with something called a resultant. The resultant of two forces is a replacement of the two forces. I can get rid of these two forces and replace them with a single force, R, of that magnitude and that inclination. And the equilibrium is a force equal and opposite to resultant, it basically equilibrizes it or brings it into equilibrium. So the resultant is the sum of more than one force. The equilibrium is equal and opposite to the resultant and cancels its effect. Very good. So if the forces are head to tail, we can add them directly. But if they're not head to tail, then we have to do some manipulation. Here, for example, is this force, and the other force has its tail at the tail of the first force, so we can't really add these, unlike the previous example. So instead what we do is we come up with this force parallelogram, we draw parallels to either of the forces, and then we move one of the forces so that its tail is on the head of the first force. So I'm going to move this one, and I'm going to place it right here, And now, force number one plus force number two, we started at this location, we ended up at this location. Well, then the resultant begins at the beginning and ends at the end. There's my resultant. And of course, the equilibrium is equal and opposite. Okay, so we have to manipulate the forces so that they are head to tail, head to tail. I can do this a little bit differently, but it doesn't matter. Force 1 plus force 2 is the same as force 2 plus force 1. So instead of what I just did, I can always move this force and place it here and end up with the same exact resultant, which is the diagonal of that force parallelogram. There's the resultant and there's the equilibrant. Very good. So we have to manipulate the forces so that they're head to tail. But if they are already in the formation of head to tail, I just add them directly, and I find the resultant of these two forces. Very good. If they are head to head, then we still have the same problem. We've got to manipulate the forces, so I can move this one and put it over here, and end up with a resultant that begins at the beginning and ends at the end. My resultant is right here. and the equilibrium is equal and opposite to that one. It could be over here, or it could be over here. It doesn't matter. Transmissibility says it can be anywhere on that line, but opposite in arrowhead to the resultant. This is the resultant. This is the equilibrium. Otherwise, I can also move this force instead and put it here, and I end up with the same exact answer. So force 1 plus force 2 is the same as force 2 plus force 1. Looking at this example here, I'm trying to emphasize that if one force is much greater than the other force, then the resultant is influenced by the larger of the two, is closer to the larger of the forces, both in angle and in magnitude. So completing the parallelogram here, we see that the resultant is in fact over here. And it's very close in magnitude. It's a little bit larger than this because it's the diagonal in that triangle. It's a little bit larger in magnitude, but it's very close in inclination to the larger of the two forces. Excellent. So, when you have two equal forces, 1 and 2, here in this example, 1 and 2, let's say they have the same angle. then the resultant and the parallelogram tell me that the resultant is at the bisector of the angle. So the resultant is right here. And this angle is equal to this angle if these two forces are equal and these angles are equal. Very good. I can think of this a little bit differently if I think of components and the algebraic method, which I haven't covered yet. But basically, you understand the logic of it from your courses before. Basically, this one has two components, and this other one has two components, and this should equal to this. If these two angles are equal and the magnitudes are equal, then these two horizontal components cancel each other, and what we're left with is a vertical component and another vertical component. Add them up, and that will give you your resultant. It's vertical because the two horizontals canceled each other. And this one is equal to 1 plus another one is equal to 2. Very good. Again, if we look at the next example, we see that this force used to be here, but now it's shifted. And so the parallelogram is going to shift, and it's going to be closer in angle to force number one versus force number two. Likewise, if one of the forces is much larger than the other one, then the result is influenced again by the larger of the two, the resultant. And I can take out these two forces and replace them with a single resultant that is in the proportion. It's the hypotenuse of this triangle. Very good. So looking at force addition for more than one force, it's pretty much the same thing. We have to reorganize the forces so that they are head to tail, head to tail, head to tail. We begin at a certain point. We end at a point, and the resultant begins at the beginning, ends at the end. So let's add these two. Here's force number 1. Then graphically, I measure the angle of force number 2, and I measure its length or its magnitude, and I draw force number 2. Then force number 3, I put it at the head of force number 2. I draw force number 3. And then finally, I draw force number 4, and it has an angle, and I make sure to put its tail at the head of force number 3. And now we started here, and we ended up here, and the resultant is this much. So graphically, we should be able to eliminate answers just by looking at the diagram and not dealing with the math. And for this example, I can add the four forces in any order that I wish and get the same answer. Here in this example, for example, force number 1, then I bring force number 4, then I bring force number 2, then I bring force number 3, head to tail, head to tail, head to tail. I end up with the same exact resultant, which is that much. And of course the equilibrium is equal but opposite, and it brings the resultant into equilibrium. Again, we can look at the order here. I started with force number 4, and then I did force number, which one is this one? I started with 3, sorry. Then I did 4, then I did 1, then I did 2, ended up with the same exact answer. Now, let's see how this applies to a physical problem on your structures exam. So, I know it's not a structures exam anymore, but it's questions on structure on PPD and PDD. Very good. So let's put some numbers. I'm not going to do any math, but just for the sake of illustration, if I have a thousand pound force and 60 degrees and 30 degrees, and I have two members, A and B, that are being pushed by the one thousand pounds. So what they have to do is they have to respond to the 1,000 pounds. They have to push back. So looking at this joint where the 1,000 pound is applied, A pushes back, B pushes back. Then B takes that red load and pushes down into the ground. A does the same, pushes down into the ground. So member B is in compression. Member A is in compression. My question to you for the ARE is the following. Is A greater than B, or is B greater than A, or are they equal? So, to answer this question, it is critical to understand, without numbers, it's critical to understand the physics of it. The force is vertical. The 1,000 pounds is vertical. So the more vertical member is more in line with the force and is going to take more of the load. So looking at this, it looks like A is steeper. A is more vertical than B. Then must be A is doing more resistance than B. We're not going to get into the math, but I just need to make sure we can guess on a multiple choice. If one of the answers is B is greater than A or A is equal to B, those should be eliminated right away. Without any math, A is doing more resistance than B. Let me jump over to this example. If the 1,000 pounds is over here and I have a vertical member, then A is doing nothing because the force is in line with member B. So B takes the full 1,000, and of course it's in compression, and it pushes down into the ground. So please understand this. If the force is in line with the member, that's it. It takes all the force versus in this example, they're sharing, A and B, are sharing the 1,000 pounds, with A taking more of the 1,000 than B. At 45 degrees, both receive an equal amount of the 1,000 pounds. Still, both are in compression, fighting back. But if this is A, this is B, A is equal to B is equal to 500 pounds is wrong. Because these are forces, they're vectors at different angles. You can't say A plus B is equal to 1,000. That's totally incorrect because A at this angle plus B at that angle should equal to 1,000 vertical down. So this is an incorrect answer. But rather, A square plus B square equal 1,000 square. Yes, that is a correct answer. Very good. So let's reverse this a little bit and let's call this 20 degrees and this 70 degrees. and this guy is A and this guy is B, both are being pushed by the 1,000 pounds and both have to go into compression and resist that load. Now the question again is which one is more vertical, A or B? The more vertical member is more in line with the force and therefore carries more load. In this example, clearly B is greater than A. Let's look at this a little bit differently, and let's say the following. Here's the line of action of that force. Who is closer to vertical? That angle? Or that angle is much farther, therefore it must be receiving a lot less of the 1,000. The B is more in line, it's more vertical, its angle is similar to vertical, similar to the 1,000 pounds. Very good. Let's change things around a little bit And let's put the 1,000 pound horizontal this time And let's call this guy B Let's call this guy A Let's say this is 30 and 60 Again, the angles and the numbers are just there for illustration I'm not doing the math yet I will next So, the first thing to determine is whether the member is in compression or in tension. Here's the point of application of the 1,000 pounds. Is B being pushed or pulled? So, clearly, B is being pushed and it needs to fight back. So, that's the force in member B. Member A, on the other hand, is a little bit trickier to understand because if here is the member A, the 1,000 pound force is here, and I really don't understand what it's doing, so I'm just going to move it using transmissibility and put it over here. Now, all of a sudden, I understand that it's being pulled out of the ground and that this member is in tension. Member A is in tension. So, member A is doing that. and it's going to pull on the ground, unlike member B that is pushing into the ground. Member A is being pulled out just because of the orientation of the 1,000-pound force. Now, which one is greater? First of all, we concluded that A is in tension and B is in compression. The next thing is which one is greater in magnitude. That way I can eliminate answers. My force is horizontal. And the question reverses a little bit. Which member is more horizontal, A or B? In this case, B is shallower at 30 degrees with the horizontal. A is steeper at 60 degrees with the horizontal. So B is shallower than A, is more horizontal than A, and therefore is more in line with the force and carries more force than A. Yes, one is compression, one is tension, but in magnitude, B is greater than A. Let's look at it a little bit differently. Here's the line of action of the 1,000-pound force. And who is closer to horizontal? It looks like the angle of B is closer to horizontal than the angle of A. So, confirmed, B is greater than A. This is an actual problem from the ARE. they gave it as two cables instead of two compression members. It doesn't matter. The logic is the same. If this is A and this is B and this is 60 degrees and this is 30 degrees, then we have a case of tension in this case. So 1000 is pulling down. Cable A has to pull back. And cable B has to pull back. So, together, these three need to add up to zero. It's in equilibrium. It's not moving. So, we can do the algebraic method, where we break down each into its components, or we can do the graphic method. All we have to do is rearrange these forces head to tail, head to tail, and then we have our answer. Which one is greater, B or A? It looks like the force of 1,000 is vertical.