One last topic we want to talk about when we're dealing with planar trusses is the notion of a statically indeterminate truss. So if you take a look at this very basic truss that I have here, it has only three joints, three members, and three external loads. I could, if I wanted to, this is, I have all the distances, all the loads that I have need. I can write some of the forces, and x, some of the forces will line some of the moments, and I know that that will allow me to solve for a, x, a, y, and b, x. Once I have those three, I can disassemble my truss and look at, for example, the joints at c and a, and solve those joint-equilibrium equations to find the three members. I know that I can do that I've done it before. This will allow me to solve that truss. I call that a statically determinate truss. But there are statically indeterminate trusses, which literally means they cannot be solved by the methods of statics. Now that doesn't mean you can't build one in nature. It just means that we don't have solving yet. For example, if I take exactly the same truss and I replace the bottom one with a roller with a pin, now I have four external reactions and three members. But I still only get six linearly independent equations, the three joints. So if I have n joints, I get two n equations, and the three external ones that I started with here will be duplicated by the six for each joint. I can't solve to find seven unknowns with only six equations. So that is called a statically indeterminate truss. And if you want to look at what those equations actually look like, here's another one. Here I have n is equal to seven, that's the number of joints. So I have at most 14 equations. If I have 12 members, which this truss has, and three external reactions, I'm never going to be able to do this. And if you look at these as an equation, if two n is less than m plus r, then you know that this is a statically indeterminate truss. Note, it doesn't work the other way around. If two n is greater than m plus r, which are equal to it for that matter, it might be statically indeterminate or statically indeterminate. If you look at this one where I removed the outside left piece, that has now 14 equations and 14 unknowns. But it's not going to do. As you work your way through, you have this joint in the middle that has four members, and there's no way to solve for it. Now if I take out one of those, what you have is once again a simple truss where you have a triangle, and then you add two members, and you get another triangle, and you add two members. That kind of thing is going to be statically indeterminate. This one can be solved, this one cannot. So that's sort of something to even mind as you're working toward solving planar trusses.