Transcription
All right, welcome back everyone. And our sequence in structural analysis, and in particular, beam deflections. And now we're going to review or go over the the method of virtual work. You may have seen this or heard it called the unit load method or unit force method. But it, it's just, it's another way to determine, uh, displacements of structures.
And, and here, you know, what I'll do in this video is just to give you a brief overview of the principle of virtual work. Very brief. I'm not going to go into strain energy and where all these things come from. Hopefully, someone's done that for you, or you, you have a textbook accessible to you that you can read all that. I think that would be my least popular video on YouTube if I did that. But anyway, that's neither here nor there.
And then what I'll do is, is describe or provide a process that, you know, that maybe some guidelines on how to solve these problems, uh, using the method of virtual work or this unit four, unit load method. And, and the principle of virtual work, just to, just to recap real fast, you know, was derived based on the conservation of energy back in the 1700s by Beroulli. And you've probably seen, uh, you know, in fluids, Beroulli is a very popular name, right? And, and here he basically came up with a relationship that said that the external work on a structure, you know, these external loadings that are applied out, like, you know, some concentrated force or distributed load or something that goes through an exter, you know, these displacement of the external structure, that external work has to be equal to all the internal work. And that internal work is related to, you know, like the internal normal force, internal moment, internal shear, and the deformations associated with that. So if you're thinking like internal normal force, we're talking PL over EA from mechanics. Or if it's a torsion, you're thinking TL over GJ. All these mechanics equations, uh, from a first course in mechanics where you're calculating deformations, angles of twist, curvatures, axial deformation, all that stuff is, is a deformation associated with an internal load. And so the, the, the idea is that the external work is equal to the internal work.
And, and the way that we're going to go about this is if you consider, so the approach, now let's see, two approach. The approach that I re, or not that I recommend, but, you know, it's just a good guideline to have. The approach is when you're given a structure. So let's take a frame for instance, some, any frame here. Uh, we'll take some awkward looking frame or some, it's not that awkward looking, right? But anyway, it's, it's there, it exists. There's a loading here, some other loading like this here, maybe even a concentrated moment. So P, W, M, I'll call these like P1, M1, W1 right here. Um, maybe even a, a some distributed load over here. Okay. I have all this stuff going on on this structure right here, and this is what's given to you when you start out. Given this, and our approach, we might want to find the deflection somewhere. Maybe here, if I have, this is Joint A, B, and C. I want to find the horizontal deflection of point B. I want to know how much it moves towards the right or the left, or whatever way it moves, some way.
And so the idea, the approach here is first, first and foremost, to call it step, uh, why not? I'll call it step A here, part A. Is is really to redraw the structure. Redraw structure. Okay. And apply a virtual unit load. Okay. Unit load. And that can be a force or a moment, depending on whether you want a deflection or displacement or a rotation. Okay. At or ah, where you want the deformation. Want the deformation. Okay. Want the deformation. And that again, it can be a displacement or a rotation right there. So that's, that's really the first thing you want to do right there. And so what we might want to do here is take this structure, redraw it. And if I want to find the horizontal deflection of B at B right here, you know, I would plot a unit force or a unit load. I would say one. So this is my virtual force. I'll call this PV equal to one right here. Okay. At B. And this is again, in the direction that I'm interested in the deformation right there. Okay. So I have that PV right here. Bam. I did that.
The next thing to do is do some Statics. Do Statics. Statics on the virtually loaded structure. Virtually loaded structure. Okay. And, and calculate all the internal loads. And calculate internal loads. That means finding. That means that essentially means, in other words, it's like finding the axial force diagram. I'll call, oh, here, N, M, Shear. Um, oh, let me use the symbols here. Let's say N, MV, torsion. Okay. So normal force, moment, shear, torsion diagrams for the virtually loaded structure. And so what you might see sometimes is that this is called the virtual structure, and this is called the real structure. Okay. They call that real and virtual. So you, you've done that.
Okay. Now, C, the next part is to take, is to calculate. Calculate the internal deformation. Deformation due to real loading. Real loading. And this really involves two more steps. And this means one, do Statics on the real structure. And then calculate deformations. So this is like capital N, MV, and T diagrams, right? So internal axial force, internal moment, internal shear, internal torque diagrams. And, and, and then come up with relationships or calculate, really calculate internal, or calculate deformations. Deformations due to internal loading in real. I'll call it real internal loading right here. All right. So you do all that. And then, okay. So what that, what does that mean? That means, you know, what that means is you've got, you know, a relationship for the internal axial deformation. So it's NDL over EA. You've got a relationship for the curvature, M over EI times DL. You've got something for shear, which isn't used very often, but shear over G times the area that's acting in shear. And the torque, which is associated with the angle of twist, TGJ time DL. These are what you're doing in part C. This, this is, this internal deformation, this is sometimes the hardest part to visualize and see. Uh, and then once you do that, once you do that, uh, here, there's you, all you got to do is put everything together. Once you got through all that part right there, the last part is to put it all together. Put it together. Combine. Ah, combine into here. You have this virtual force PV times the displacement that you're looking for. So it just happens that I'm looking for this displacement Delta B. PV times Delta B right here is equal to the sum of the sum of the internal load times the real displacements, which I'll call DL right here. Right here. This U times DL. And really, what this whole thing is, if if I've got, you know, on the inside of here, I've got M, uh, sorry, if I've got, you know, little n, m, v, and T, and same thing here, I've got, you know, normal force, moment, shear, and torsion, all present. And same thing here, I've got N, M, V, and T, all present on the insides of all these members. This last summation right here becomes this gigantic, you know, it would be this, you know, N times N over EA times DL plus M times M over EI times DL plus, um, little V times capital V over G time DL plus little T, little torque times GJ divided by GJ time DL. It becomes all this. But most of the times, in a, in an undergrad mechanics course, you know, this is, is neglected, and this is neglected, or torsion is not present, and you neglect the shear deformation. And so all you're left with is this. And then it becomes even simpler in a truss where you have no moments within the members, it just becomes reduces down to this business right here. Right here. Okay. And then when you have just a beam with no axial force, then it just becomes this one right here. Okay. But when you have a frame, you've got to consider both of those. And if you want to consider shear deformations, and if you have torsion, you'd have to have all four. All right.
So hopefully that provides some clarification. And, uh, and let me know. We'll do some example problems next. See you.