Let's talk about trusses and how to solve for unknown forces. This is a truss. You see them on bridges, roofs, and loads of other places.
Each of these individual pieces is called a member and they're usually connected with a pin, and we call that a joint. Each member can either be in tension or compression. If the forces are pulling on the ends of the member, it's in tension, and if it's pushing at the two ends, it's in compression. Today, we're going to look at a method called the method of joints. The way to think about these problems is to realize that if the whole truss is in equilibrium, then each and every member and joint is also in equilibrium.
So in simple terms, if the whole truss isn't moving, then neither are any of the parts. That means we can write our equilibrium equations and solve for the forces at each member. Now let's say we have a truss like this.
The way we solve it is to start at a location where we have at least one known force and a maximum of two unknowns. So in this truss, this is the best spot to start. The next step is to isolate this joint and then assume the direction of the force.
Is the force going to come towards the pin or away from it? It's just an assumption so it doesn't really matter which you pick, though over time you will notice that you can make a very good guess. Now if your assumption is wrong, you will get a negative value. So then you know it's the other way around. Once you find the force in the member, you have to determine whether the member is in tension or compression.
Now assume we figured out that at this joint, we have forces like this. You might assume, well this force is coming towards the pin, so it looks like it's pulling on the member, which means it's in tension. This is not the case. What's really happening is that we're finding the force applied to the pin, which means the pin exerts an equal but opposite force onto the member.
So if a force is coming towards the pin, then that member is in compression. If a force is going away from the pin, then that member is in tension. Lastly, if we have a force coming towards the pin, at the opposite end of the member, the force is going towards the other pin, so it's always in opposite directions.
The same is true if the force is going away from the pin. In that case, at the other end, the force is going away from that pin. To solve the problems you face, it's really important that you know how to break forces into x and y components. If you need a refresh or you forgot, please check the description.
Now let's go through some examples and see how we can actually solve for unknown forces. Let's take a look at this problem where we need to find the force in each member of the truss and whether they're in tension or compression. The best place to start our analysis is a location where we know at least one force and two unknowns, which is right here.
So let's draw point D separately. I'm going to assume that both forces are going away from the pin. Now any force not lying on the x y axis has to be broken into x and y components.
Next, we write our equations of equilibrium. First, for y-axis forces, because we can easily solve for the force in member DC. We will pick up to be positive.
Solving gives us the force in member DC. Now since we assumed it to be going away from the pin, and we got a positive value, that means our assumption was right. That also means this member is in tension, since anytime a force is leading away from the pin, the member is in tension. Next, x-axis forces.
We will assume left to be positive. Don't forget to plug in the value for member dc we just found. Let's solve.
Notice we got a negative value. That means our assumption was wrong and that the force is actually coming towards the pin. Since this force is coming towards the pin, this member is in compression. Now we need to pick another point to write our equations of equilibrium.
Point E has too many unknowns since we have three unknown members. So the next best spot is point C. We know the force of member dc which we found in the previous step.
Since at D, it was leaving the pin, that means at C, it's also leaving the pin. In other words, going towards D. We also have forces CE and CB. I'm going to assume force CE is coming towards pin C, and force CB is going away from pin C. Now let's write our equations of equilibrium, starting with the y-axis forces.
We got a positive value, so our assumption was right, and it's coming towards the pin, so it'll be in compression. Next, x-axis forces. Let's solve.
We got a positive value, so our assumption was right, and since the force is going away from pin C, it's in tension. Now we will pick point B. Let's draw the forces. So we know the force in member BC, which was going away from point C, which means at point B, it's going towards point C.
In other words, leaving B. For force BE, I will assume it comes towards the pin, and force BA goes away from the pin. Now for our equations.
First, y-axis forces. If we divide all the terms by sine 60 degrees, we can see that the force in member BE is actually the same as the force in member BA. Next, x-axis forces.
Let's solve the two equations. Lastly, we can look at point E. Since it's a roller, we'd have a single force straight upwards. We know force DE, CE, and BE. Note the directions since all we're doing is flipping them.
So for example, at pin D, we found that force DE was coming towards pin D, which means the same force is going towards pin E. So it's just flipped. For force EA, I will assume it comes towards the pin.
Now for our equations. First, y-axis forces. This gives us the reaction at the roller. Next, x-axis forces.
Since force Ea is coming towards the pin, it's in compression. Now we found all the forces in each member. Let's take a look at this problem, where we need to figure out the force in each member and whether they are in tension or compression.
Where is the best place to start? That's point D, since we know a force already being applied and we would only have two unknowns. We have the force dE and force dC. We also need the angle at the top, and that can be found using trigonometry since this whole system is a right angle triangle.
We can use inverse of 10, which is opposite over adjacent, and solve for theta. Now that we have the angle, I'm going to assume force dE comes towards the pin and force dC is going away from the pin. Let's write our equations.
We will start with an equation for x-axis forces. Let's solve. Since this force is going towards pin D, it'll be in compression.
Next, y-axis forces. Don't forget to use the force in member DE we just found. Let's solve.
This force is heading away from point D, so it'll be in tension. Next, we look at point C. So we have the 900N force, and then we have the force of member DC, which is going away from point C. I will assume force CE is coming towards pin C, and force CB is going away from pin C.
Let's write our equations. First, x-axis forces. So this is in compression. Next, y-axis forces.
This member is in tension. Now we move on to point E. Before we do anything else, we need to figure out the angles. Now there's probably a bunch of ways to get to this answer, but one way to do it, without knowing these two angles are equal, is to find the length of member EC. We can do that by using tan.
Then we have another right angle triangle, so this angle can be found using inverse of tan. This angle up here is simply 53.13 degrees. Now the bottom angle is also 53.13 degrees because we just have a big right angle triangle.
That means this angle is also 53.13 degrees. Now we have these forces at E. For the unknown force Eb, I will assume it goes away from the pin, and for force Ea, I will assume it comes towards the pin.
Now let's write our equations. First, x-axis forces. Let's simplify. Now, why axis forces? We can now solve the two equations.
And those are our answers. We didn't even need to figure out the reactions at the supports. Let's take a look at one last example. In the question, we're told that the maximum tensile force a member can handle is 5 kN and the maximum compression a member can handle is 3 kN.
We need to figure out the maximum force P that can be applied to the truss without going above the limits. Before we do anything, Let's figure out the angles inside this truss. Since this is an isosceles triangle, this angle is 30 degrees, which means this angle is 120 degrees. The same is true for the top triangle. Now the best place to start is at point C.
We will solve this like we know the value of P. I'm going to assume that forces CD and CB are coming towards pin C. Let's write our equation starting with the y-axis forces.
Let's simplify. Next, x-axis forces. Let's simplify.
So the goal is just to write all the forces in each member in terms of P. Now we move on to point D. I will assume force DA comes towards the pin and force DB leaves the pin. We also have the force we just found and another force P. Let's write our equations.
Now we will simplify these equations so we can write each force in terms of P. We will now move on to point A which has a roller for a support. That means we will have one vertical reaction upwards along with forces AD and AB.
I will assume force AB comes towards pin A. Now let's write our equations. First, y-axis forces. Let's simplify. Now, x-axis forces.
Let's simplify. We have now written all the forces in each member in terms of force P. Let's take a look at all the members that are in compression.
We have four such members, CB, CD, DA, and AB. From these four, we see that member dA would have the largest value because it's multiplied by the largest number. We also know that the maximum allowable compressive force for each member is 3 kN.
So let's solve for P by assuming that member DA has a force of 3kN. We get a value of 1.3kN. Now we need to check something, which is to see if this value is okay for our member intention.
The maximum allowable tensile force is 5kN and the only member intention is member DB. Let's plug in 1.3kN to see if member DB would carry a force larger than 5kN. We see that it won't, which means our P value is within range.
So the maximum value of P that can be applied is 1.3 kN. That should cover the types of problems you will face when it comes to trusses and using the method of joints. I hope this video helped and thanks for watching.
Best of luck with your studies.