a little bit of a summary before we get going on just where we've gotten so far. We were looking at objects under simple axial tension or compression. I have to draw one or the other. I can't really draw both. So as often as not I just happen to draw something under under tension. But what we're looking at now is at the response internally of that material as if we again put one of our imaginary cuts through this material. And then as we look at what's happening inside, we came to appreciate the other day the fact that these internal forces are spread over some cross-sectional area that for the normal stress that we talk about is that cross-sectional area that's normal to that force that's over that face over that area. Occasionally we'll put a little end there. If we don't, we're generally implying the normal stress. But we had that normal stress, the force divided by the area that's withstanding it, which kind of makes sense. As the force goes up, the stress is going to go up. As the area goes down, the stress is again going to go up. So it makes sense that we do this ratio. And the general units on this, typically kilopascals, maybe even mega, occasionally even as far as gigapascals. But we're talking about generally structural solids that can maintain an awful lot of force over sometimes very small areas, as can be done with cables and very thin members that can withstand an awful lot of weight. So we had that as our first look at the possible type of loading and the stresses associated with that internally to the material. It's the material's ability to withstand these stresses that tells us whether or not we're going to have some kind of catastrophic failure. We also then had the other possibility of not axial loading of some kind, but transverse loading could be, as we find with the distributed loads we saw in statics, low those many months ago. But again, as we look at the internal response of the material to those loads, and it doesn't matter if it's a distributed load or not, what matters is the internal response of the material to that load. In this case, we have a force that's now parallel to the cross-sectional area rather than normal to it, and then we call the average shear stress. So we had the normal stress, and if we don't say anything before the word stress, that's the stress we're typically talking about. And then this being the shear stress. And in this case, for both of these quantities, these are average values. They are not constant over the cross-section, but we're not going to worry, at least not at this time, about the fact that these force distributions across these areas aren't uniform. We're going to treat them as if they're uniform, and so these are average values, unless we say otherwise. One other little piece we hadn't quite gotten to, and we will look at it as we wrap up this piece, the problem we were looking at on the board, and that's that sometimes one member is simply bearing on the surface of another member at that cross, at that interface where they connect, which in this case is this base piece here, at that interface where they connect, there's a bearing force as one member presses on the other.  It's also force acting on the area that's withstanding that force. In this case it's one object bearing on another, so this is an average bearing stress. Same units, same general size as the other, but can lead to a very different type of failure. I don't know about you, you can go downstairs in my basement at home where I have the columns in the basement supporting the floor above and you'll see little minor cracking in the concrete below that column, and that's bearing stress starting to cause the concrete to break apart. Well the builder who happens to live next door swears that it's nothing to worry about, no trouble. I personally think that snakes are coming up through those cracks, but he says they aren't, so I'll have to trust him. But that's a bearing stress. It's not a failure, it's just a response to the material to that stress, but that's the type of thing that we talk about bearing stress. So that's a little bit of a summary, plus a little extra thrown in of the type of thing we started with on Monday. So any questions with that as you started to look at some of the problems? We'll finish up, we'll look at a little bit of the bearing stress on one of the parts of this member and then we'll move on from it. Alright then, let's refocus our attention on this pin A. Remember this member AB is in compression, so that member is pushing on the pin A, then the pin A itself is pushing on the bracket at A. So if we take a cross sectional view of that bracket, this course is wonderful for your drawing skills, so I hope they're up to speed. So there's a little bit of the cross sectional view of the bracket, then here is the wall to which it's attached, and then the pin itself, pin A, is in the hole that I have represented here, and then the member AB is in compression, so pressing on this pin A. So it's going to be something like the member AB is pressing on the pin there, so we have a bearing stress from member AB on the middle of the pin, and then the bracket is pushing back, it's hard to draw it on the backside of that pin back there, and the pin itself is pressing on this bracket as it fits in the piece there. Now again, that's just the bearing stress is the force divided by the area that's withstanding that force. For this bracket, things are a little bit different. I'm going to change the view some. We're going to look straight down on the bracket there. It's a little bit easier to draw. Remember, this is a cross section of the bracket, where I'm looking at just the bearing stress of the pin on the interior surface of that bracket where it meets. It's not a uniform force distribution, not in magnitude or direction as it presses on the interior surface here. It changes a little bit with position. We have sort of a circular force distribution, where that total force is, of course, this actually it's f over 2, because there's two surfaces where this bearing is going on. This is the force coming from the member, so f over 2 is at the two different bearing surfaces. So this is really f over 2. It's still a resultant force, just that direction, because the little side components on the left have an equal and opposite side component on the right that cancels each other. So all the sideways components all have equal and opposite opponents to them on the other side. So it gives us a net of this f over 2 straight on. But how do we calculate, how do we figure what the area is? It's actually quite simple. And you'll remember this very same type of thing from your statics working in physics too. The area we use because of this circular distribution is just simply this cross sectional area of the opening itself. And that's the area we use to calculate the bearing stress, average bearing stress. In this case it's f over 2 over a, where a is the, this blue cross sectional area which is the same as that area there, which is what the diameter of the pin 25 by the thickness of the bracket which is another 25 in this case. And so you can take those known values and fairly simply figure out what the bearing stress is along a curved surface even without a uniform force distribution across it, either in magnitude or direction. It all integrates to be just the total force through the, it's not really the cross sectional area, it's sort of an exposed area sometimes. Same thing you did if you took, did you do fluid statics in physics too? And you did this kind of thing, you know the force on an inclined surface, the pressure force on an inclined surface? Sure. Well it's in your book so if you didn't cover it it's in there. But we're not going to be doing very much with it. The thing is, the reason we can sort of gloss over this as we need to is generally these stresses are not the sources of the catastrophic failure we're worried about. It's much more likely that these objects, these structures are going to fail either in normal stress failure or shear stress failure or some combination of the two. So we're not as worried about the bearing stress. Often anyway it's not as difficult to see because it's a little bit more straightforward but this was one where it wasn't quite as straightforward. But the calculation of it does come out to be quite simple in the end. Harder to visualize than calculate actually. Alright so that will wrap things up for that little diagram. We can put the projector to bed for a bit here. Unless there are any more questions on that structure there's a lot of pieces in there and if you were designing that you'd have to take care of every single little part of it to make sure none of it failed because any one of those pieces failed anywhere. If any of the members failed or any of the connections failed or any of the brackets failed the structure itself is going to fail. One thing we will have to concern ourselves with in a little bit probably on Monday we'll start with it is that the failure doesn't necessarily mean that the piece breaks apart and can no longer maintain any forces. It could be that the structure elastically deforms enough so it no longer serves its purpose. It could be that under load some of these force some of these members deform enough that other parts in the structure don't match quite and then the piece won't quite work. Same kind of thing if you're putting in a new door and you don't mount it tightly enough that under the load of the door it deforms itself that it doesn't even open or close properly. So it's not necessary that things need to catastrophically fail. It might be enough that they just deform until the entire thing itself fails in some way. Alright so we're going to change gears a little bit here as we look at not just simply what's going on internally but take another bit of a view towards it as we look what happens if we're not looking at the stresses on a normal cross section but on an oblique cross section where we have a cross section taken at some angle to the normal direction. So I've drawn such an oblique angle there for us to take a look at. That's our imaginary cut plane through the piece and notice it's set back at some arbitrary angle theta and then this piece is loaded with some force P. And we'll pull away this front part, look at just the back part here because that will expose force then that oblique phase now set back at some arbitrary angle theta. Of course it's still true because of the force balance that we need on anything here that it's still force P but that force P is now acting over a slightly different area. We'll call this a theta where a zero is the original phase from which we would have calculated the original stress. So if we look at the original normal stress and by original I just mean what we've been looking at since the first day.  Alright let's look at this one then a little bit differently. Let's look at this oblique phase now. We have this force on it say a force P but we can break that into two components. The normal component of that force acting normally to the surface we might call F and then a transverse component that acts parallel to the surface and we can call that V because that is a shear stress so that's rather common for us to use that as the designator for a shear stress or a transverse force. So I'll cross out that force P because I've broken it into two components. I don't have three forces there. I have either the original force P or the two component forces F and V. I don't have all three L. This is V and this is again acting over this area A theta because this is the inclined phase. Imaginary cut through the piece inclined at an angle theta. So this angle as well is that angle theta. So we know that this force F is P cosine theta and it's acting on an area A theta. A theta the inclined area exposed remember imaginarily exposed by that area is the original area A0 also divided by cosine theta. So if we put those two together to see what the internal normal stress is at an angle arbitrary angle A theta theta. So I'll call this sigma theta it's F over A theta which is P cosine theta that's F divided now oh sorry this is this is A theta there not A0 divided by the area A theta which is the original area divided by the cosine so I get P cosine squared theta over the original area. No this is A0 because now I have cosine squared from here if I put these two together I get a cosine squared on top. What's nice about that is that it allows us to compare the normal stresses at an arbitrarily inclined angle interior to the piece compare it to the original normal stress that we were observing and because cosine is never greater than one cosine squared certainly never greater than one we find that we always have a normal stress at an inclined angle less than the normal stress we had experienced at the regular axial direction as we originally had. So that tells us at off angles and oblique angles we don't expect there to be greater normal stresses than we would have found in the original look at the normal stress anyway. Let's see if that same kind of thing holds true for the shear stress. So the shear stress let's see that's the load times sine theta so the shear stress the average shear stress at some imaginary plane at some arbitrary angle theta is V over the area over which it acts which is V over A theta but V is P sine theta divided by A theta which is A0 over cosine theta and so I get a new then shear stress internal on some arbitrary oblique angle that is sine theta over cosine theta. Alright so what we need to do now is look at what those are in terms of theta to see if there are other places there might be trouble that we would not have seen before if we had looked at some arbitrary angle some arbitrary angle theta internal to the piece. So if we just look at theta over maybe what I'll call the shear ratio which is I'll look at the ratio of these two sigma theta to sigma zero is cosine squared theta just to normalize it so I can compare the two. At an angle of zero degrees which was our original look at these strictly cross sectional piece this is our member on edge we're looking remember oblique angles of theta so theta equals zero we're at our original situation we looked at when we opened things on Monday and that's just then simply a shear ratio of one and from there to a maximum of 90 degrees which means we're longitudinally cutting all the way down the piece which is in its own right a bit absurd. We get a relationship between the shear ratio and the imaginary angle of our oblique plane to be something like that. We see that the worst possible situation is the original one that we've looked at this angle is, well that's not it, can't even drop because it's zero. That's where we're going to find the maximum normal stress. Any other angle from there the normal stress drops and remember this is the normal stress normal at all times to whatever face we're looking at. It's not that there's more forces it's just that as we go to oblique angles the original force has different components to it that we have to account for. When we look at the shear force however across that face and do the same kind of thing with it we did previously so t theta, I'm sorry tau theta over tau zero is sine theta cosine theta that has a bit of a different response to it does something like this where at 45 degrees it reaches a maximum which also happens to be the place where it crosses the other shear stress or shear ratio curve that being the normal response this being the shear stress response. We see it does reach a maximum at a place where at 45 degrees. So at an oblique angle of 45 degrees we see we have the maximum shear response and we're still at a significant normal of about a half, at least a half of what it was originally. So this is a, they will call this sigma one half for some other reason no other reason than just to call it something. Question John? Just this is in general I assume that once you know material you're working with it. Oh yeah what we have not talked about at all is what is the material limits on these stresses that we're finding. There's no point looking at any of the materials and how they can withstand these stresses until we know how to figure out what the stresses are themselves. So this actually turns out to be the critical situation. This is an angle of 45 degrees and you will find, and in fact I've posted two videos where they actually do these type of tests and you can see what the failure looks like. If you take a test piece and pull it apart when it fails, it doesn't fail straight across the piece in a nice clean break, it tends to fail something like, something very much more like this. When the two pieces pop apart you'll see that there's a semi-irregular surface that's now been exposed where the piece broke. It's only semi-irregular because it tends to look sort of like a 45 degree cone that formed on one end as it popped out of the other. Kind of like a socket into each other. But this is at approximately 45 degrees because of the material's inability to withstand the maximum shear at that oblique angle. Remember our look at all of this was at an imaginary angle, we determined that the maximum shear response was at an oblique angle of about 45 degrees and that's indeed where the material tends to fail. It's also true and maybe you've seen this if you've ever overloaded a column on a deck or some kind of structure or piece that was meant to carry more weight than it could, it tends to fail something like that and if you, I've posted two videos, one that shows a tensile test to failure and you'll see sort of this type of surface forming and I've also got a piece of wood in compression to failure and you'll see that it splits in very much this type of way and this is fairly close to that 45 degrees. Of course this assumes a completely homogeneous material which is not the case nationally. We have, especially with wood, there's differences in the material throughout it's not a homogeneous material perfectly which is why it's not exactly 45 degrees but that is where we predicted the maximum shear failure and that's just what's seen when the piece does go to failure. So on Angel go to check out those two videos they're not even a minute long I don't think. You have to watch them closely because what you can't tell is that as soon as the video starts the test has already begun and there's no sound that indicates that there's actually stress of the stress test undergoing and if anybody happens to have an extra $50,000 to buy us one of these test machines we could have done this ourselves. So John if you stop wasting your money on little toys on your computer you could buy us a stress test machine and we'll talk about that very test probably on Monday. Alright let's take a break while I re-step the taper. Okay also Matt there? By the way I don't think I did introduce Dale at the second class I didn't introduce assistant Linda. I assume what you're gearing towards is maybe one of you will take one class and one the other. Exactly. Once Linda gets up to speed. So we need to get a little into this. Hi everyone. Alright where's that? We are. Alright so we just had a chance to look at the internal stress at arbitrary angles now we're going to take a more general look at what we might call generalized stresses and again we're really only looking at the two stresses that we've talked about so far. Well we talked about three but we're not going to talk about bearing stress here. We're just talking about the normal and the shear stress and again for the most part these are average values. We're assuming the part to be homogeneous by that I mean a uniform throughout it's not like a piece of wood is where it's very different in one place than it is in another. We're just assuming it's a perfect piece of material and the force is uniform over the surface. So we have some object we're concerned of and I'll even keep its shape general under some kind of ring and again just to illustrate how very arbitrary any of this can be so that we do indeed get a very generalized situation. We have some object we don't care what its size, shape or purpose is. It's under some kind of load and we're not even concerned with what that load is. But what we'll do again is take an imaginary cut through the piece and expose the interior of it so maybe it'll look something like that. Looks like a Christmas ham that's what that looks like. And we'll put a cornet system on it just to help us orient things and we'll put the origin of that cornet system dead center just to make things nice and easy for us. So we'll call that the X, that the Y and that the Z direction. So now that we've done that and taken all of these forces, figured out what the resultant is, we know that there will be some component of that resultant force right down the X axis normal to the face. That's just the nature of no matter what the resultant comes out to be, we could resolve one component of it to be in the X direction. So we'll also imagine that to be acting on a little piece of area we'll call delta A. And then the other component of the resultant force, no matter where it points, it's going to have some component of shear. Well let me draw it in even a more arbitrary direction because we can't possibly know where it's going to be. So often some direction parallel to that face is some shear component and it's the two of those together that came from the resultant force of all of these original forces acting on it. So since they're both acting on the same area, we can make the components, not the force components themselves, but we'll make them the shear components because they're both divided by the same area. So we have that normal component of stress, whatever it is, whatever magnitude, we do know its direction. We purposely resolve the resultant into an X component. But the Y component, or not the Y component, the shear component could lie in any direction along that face. So what we'll do is make things very simple on ourselves. We'll take this exposed face with this coordinate system conveniently located on the center of our piece there. We've got this normal component of the stress in the X direction. Remember there could be compression. It depends on what the forces were originally and where I made this imaginary cut, but I have to draw something so I drew tension. And then we'll take the shear component and go ahead and break it into the two pieces in our Y and Z directions, and then we are able to figure out all pieces we had before. So I need sort of a system of writing this thing down. So I'm going to use this system here. For the shear stresses, I'm going to have a two letter subscript to it. The first letter is the face that the shear is on.  So this is tau sub X. And then the second letter in the subscript is going to be the direction, the coordinate direction that that shear component lies in. So this would be a tau XY shear, a shear stress on an X face in the Y direction. So this piece down here is a shear stress. And then what would the two letters be that describe what shear stress that is? It's still on the X face. That's the only face I've exposed right now is a face in the X direction. It's a cut that's normal to the X direction. And that one happens to be the component that's in the Z direction. So I have a piece that looks just like that. So that no matter what my loading is, no matter where I actually have this cut, we can resolve the resultant force, calculate the resulting stresses, both normal and shear, and draw them in something like that kind of designation. Remember the actual directions of these is arbitrary. I had to draw something, so I just happened to draw them all in that direction. We can then do that in the five other directions.  We can also do a Y, two Y cuts and two Z cuts. And come down to all those cuts together, expose just a big cube, again with the coordinate directions conveniently located at the center of the cube. And that exposes the shear stresses, the internal shear stresses, sorry, the internal stresses, both normal and shear on all of the faces. So the ones I've already drawn there, I have a sigma X, we'll call it a normal stress in the X direction. And then I've got those two shear stresses. Remember their direction is arbitrary. And then there's an exact copy of that kind of thing on the back face that I'm not going to draw. It's just too difficult to draw behind the board. And then on this Y face, I've got the same kind of thing. It might be tension, it might be compression, but there's still going to be some stress along that face. I'll just arbitrary choose to draw all of these in the positive direction. There's a shear stress that will be exposed along that face. What should I call it?  We have two Y faces. There's one on the bottom as well. Tau, Y face, but it's in the X direction. And then I might have a component in that direction as well. It's on the Y face in the Z direction. And again, the very same type of thing on the bottom of this cube that we've imaginarily exposed with our cuts through the material. And then there's also a normal stress to the Z faces, a shear stress on these Z faces, one in the X direction and one in the Y direction. And then again, of course, on the back side, on the back Z face that I'm just not going to bother drawing. In total then, we've got nine stress components. In a generalized case, generalized meaning for any arbitrary loading and any arbitrary orthogonal directions, we can break down any problem to this kind of thing. We have nine stress components. The nine that you see and then those are each near-imaged on the back, but not to the point where we have 18 stress components because on the back, for example, the normal stress in the X direction, we have to have the exact same normal stress in the X direction on the back side as well. Why is that? Static equilibrium must be maintained at all times. All of the forces summed to zero, these stresses are called, these X stresses are caused by X forces. So for every X force in that direction, we have an equal and opposite one in the back direction. So those aren't new components. We only have then the nine stress components exposed. And don't forget that the moments must also sum to zero. So we're going to generalize things even a little bit farther, make things a little bit easier to us. We're going to actually look at the force balance on these things. So let's imagine our cube has a length a on the side. We're only going to look at it in two dimensions. We've done this in so many of our other classes where we look at two dimensions rather than all three because what happens in the third dimension is not necessarily any more instructive but it sure complicates the drawings. So we can draw the forces on this. We'll take just the X, Y plane of view of this type of thing. We'll ignore the Z direction. It comes out of the board at us. And so on this X face now, which is the very original first face we drew, I'll draw not the stresses but the forces on that piece because I need to do a force balance on this. So the force on that face is sigma X acting over an area delta A. Remember that was the size of the element that I first exposed. And of course on the back side, we know there must be an equal and opposite load on the back side. And again, arbitrarily drawn as tension.  And remember to that, sigma Y delta A. So automatically by virtue of that, we've already got the force balance satisfied at least for the normal stresses. For the shear stresses, I'll draw them something like that to remind us this is across the face. Remember we're looking down the Z direction. We're looking along this X face with this drawing we're just doing there. So that's tau XY acting on an area delta A. Remember we're doing a force balance here so I want the forces not the stresses. It's not going to be a big concern. All these delta A's are the same anyway. And this is on the back face. So this is tau also on the next face, also on the Y direction, also acting on an area delta A. So that's tau XY delta A as well. So we have an out and make, we have a force balance there. And then I have the same type of thing with the shear stresses on the Y faces. So that's tau YX delta A. And down here in the opposite direction as it must be because I need to balance that force on the top. I have tau YX delta A there as well. All right, we automatically have our force balance satisfied. So it gets a big happy check mark. Because I drew it that way, we know it must be that way so it's not like I'm assuming anything. We know that we have static equilibrium on all these pieces. So I just drew it that way right from the start. We don't need to do any more with the force balance there. That especially works for all the normal stresses. All the stresses perpendicular, normal to the faces. Especially because not only are those automatically satisfied with force balance, those forces are also automatically satisfying a moment balance as well. Because not only are they equal and opposite, but they're co-linear so there's no moment caused by any of those. So at least for the normal forces we also have a moment balance satisfied. That's not true of the shear forces though. Because they're equal and opposite, but they're not co-linear. They're separated by a distance A. So we need to satisfy a moment balance here with the shear stresses. So we'll take this couple first, tau XY delta A. That's the magnitude of the forces in this couple. This is the couple of these two shear forces on the X faces. But what's the magnitude of the couple? This is just the forces in the couple. How do we calculate the magnitude of a couple formed by two forces? We have two equal and opposite forces separated by a distance that causes a moment. Causes a couple. The magnitude of that couple is the magnitude of the forces times the distance between the two. Remember this was a cube of sine A. So the magnitude of the couple caused by the shear stresses on the X faces is that. So that's the X face. There's also a couple caused by the forces on the Y faces as well. And we know that those must be equal and opposite so that they sum to zero. So I'll assume it's equal and opposite. Turns out I drew it that way anyway, so no great trouble with that. Sorry, this is tau YX, not XY. I'm working with this couple now caused by these shear stresses. The forces is the shear stress times the area over which it acts times the distance between the two forces that make up the couple. So that's the Y face couple. And I know they must be equal and opposite that they sum to zero. And I drew the minus sign because I actually have them drawn in the opposite direction each other. So just taking that into account. The X face couple happens to be counterclockwise. The Y face couple happens to be clockwise. So that's the minus sign between the two. Notice how this simplifies things for us terrifically. Because the A's cancel, the delta A's cancel. Not only is delta XY equal to delta YX, but I can do this on the other faces as well. And I get delta XZ equals, sorry, tau XZ equals tau ZX. And tau YZ equals tau ZY as well. So that makes things terrifically easier for us. Now we're down to not nine stress components. We're now down to six stress components. We have the two stress components in our X and Y direction. Again, I stress pun intended that I'm arbitrarily drawing these as tensile forces. Tensile stresses. We're going to see lots of situations where either or are compression. I have those two. I have a third in the Z direction. And I have shear stresses that must be drawn in that relationship to each other. Or we don't have a force balance. Two on adjacent sides must either point towards each other or point away from each other depending on which corner you're looking at. And it's not even important anymore what I designate them. I'll just call them the shear stress tau. Because they're the same on all of the faces because I can do this in any directions. I can do this all the way around the cube. All of the shear stresses must be the same. And they must orient themselves in this way. We could have the possibility that any or all of them are in the opposite direction as drawn. But then the relationship between the shear stresses must still be like this. That they tend to point to opposite corners or they tend to go away from opposite corners depending on what the actual loading is. And remember it's the external loading that determines the magnitude and the directions of any of these things. We'll look at problems that do exactly that very shortly. So now we're down to six stress components. The sigma, the normal stresses in those directions, and then the three different parts of the stresses in those directions. If I did need to concern myself with the z direction I might further the designation tau xy. But all four of these have the very same magnitude. They must or we don't satisfy both the force balance and the moment balance. If we happen to look at any of the other two directions we just take into account the other two possibilities for the shear stress. So that's six components, three normal stresses, and three possible shear stresses. We're down now to just six stress components for an entire 3D analysis of the stresses on a structural member. No assumptions going in here. I didn't assume that these would be the same. Well I did initially assume that these would be in the directions shown. We'd have the opposite directions of each. But it turned out that that's the way it indeed had to be. So we get down to this not with any assumptions. It's just the reality of the nature of the three dimensional stresses on a solid. Okay so let's put this together with something we just did earlier. Again all we're looking at so far is simple axial loading. Some load on a cross sectional area. And if we look at some small elemental piece in the member itself, remember we of course do not actually expose any of these areas because we can't do that in a structural piece. For a simple axial load all we're going to have is the normal stresses there. In that situation there's no possibility of any shear stresses because I have no transverse component to the force so I'm not going to have a transverse component to the stresses if we take the very same situation. This is not a different situation. It's the exact same one we're just going to look at in a slightly different way. Now if I look at an elemental piece not oriented in this way but oriented that way. In fact let's make it nice and regular. Make it at 45 degrees. I know from the oblique plane study we did to open class that I have shear normal stresses on all of the faces now before I only had them on one. I only had an axial force on the member. I'm only going to have axial forces in resultant. But when we took a side look at these we got extra components on the other faces. Plus I'm going to have shear stresses now exposed on those faces. When you go back and look at the oblique plane we opened class with we saw that it off angles we get shear stresses across the faces. We also know that though those will be at their maximum in this piece. We also know though now from what we just looked at our generalized case that they must be oriented in that fashion in relation to each other. So we've been able to combine our strictly orthogonal place view with an oblique orthogonal view where we know now from what we looked at earlier that things are at a maximum at 45 degrees. No I'm sorry if I think about it I've actually happened to draw the shear stresses in the wrong direction for this tensile loading because their components are going to be in that direction not as I drew here. So let's fix that. And these sigmas remember are not necessarily the same. So maybe I'll call those x prime and y prime just to indicate we're at an oblique angle but we do know if that angle happens to be 45 degrees that that is the maximum shear stress that we're going to see. Anything other than that any other angle, arbitrary angle than that we're going to see less shear stress. Remember what the magnitude of these shears were, the normal shears were to the original normal shear it was half of the original. In fact those two would be the same. Sigma original divided by 2. I guess if they're going to be the same we don't need any kind of subscript on it. Any questions? As we go through these things remember that the shear stresses must be oriented in relation to each other in this way. They might be in opposite directions but if one's in an opposite direction they all are. And no matter what the angle all four of those are the same magnitude or we don't have a static equilibrium on an element which we must for any element, any parts, any total structure that we might look at. Questions? Phil, questions? Good. The shear, is this supposed to be going the other way? Oh, yep. It's too hard to draw, talk. See I'm in three different time periods. I'm drawing one thing, I'm saying one thing and I'm thinking what's coming up next. Good catch, thank you Phil. They must always be there. They're either pulling the piece this way or they're pulling the piece that way. In fact we'll look at that specific deformation caused by those next week.


Bases loaded, bottom of the second, ending a 2-2 count and the game is tied at zero. Singers on the bump for the Royals. Ooh, and he drills. Arise, that's an RBI walk. R-B-I-B-B, not a fun time. Ooh, that probably hurt.  Bam, he hits the pitcher, it ricochets, he catches it off the ricochet and they double up the runner that was on second and then they throw home but that doesn't matter because they already got both outs and it's pretty confusing and pretty confusing in live speed and live time. So here's the wide shot, thanks, good replay. Now it never hit the ground if you're like, why was that an out? It never hit the ground, it hit his foot and popped straight in the air and I actually think it's crazy that the runner on first and the runner on third reacted as quickly as they could as well as the first baseman Santana to go make that catch. So now there's one out and they gotta get back to their bag and if the runner on second reacted the same way, they wouldn't have turned a double play but I'm more surprised that the other two runners did figure out what happened than I'm surprised the runner on second got got because you have no idea if that skipped or hit his foot in the air but I guess you do if you're closer than that and you look, you can see all the royals on the bench pointing but yeah, bam, off the bat, off the foot, spins, he's down. Fuck, ow, not happy about it. He had to come out of the game.  He's hurt in the trainer with his nice haircut. What is that haircut? He grabbed, you know what?  He grabs his shoulder, he's hurt. He's not feeling good. What is that, it's just bangs? The hair on the back, what is that? It's all the way in the chest to top, you know. Anyway, he walks off, he looks fine walking off. If he makes his next outing, then it's just like a little bruise but his ERA stays intact, double play was made. How about that?


Welcome back to our lecture series, Math 3130, Modern Geometries for Students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Missildine. In this video, we're going to conclude our discussion of Ordered Geometry that is talking about the betweenness axioms of Hilbert. We've previously introduced the incidence axioms. We've been talking about the betweenness axioms for the last few lectures. In the next lecture of this lecture series, we'll start talking about the congruence axioms. In lecture 13, we want to hit the pinnacle, the climax of the betweenness theory, and talk about the so-called crossbar theorem. Now, before we state and prove the crossbar theorem, there's a new definition I want to introduce to us. It's the idea of being between rays. We've previously come up with the idea of betweenness of points. In fact, this is one of the undefined terms of betweenness theory of ordered geometry. The idea is that if we have three collinear points, there's a betweenness relationship that holds. This would be an illustration to suggest that B is between the points A and C. Now, we can also do the same thing with rays. Imagine we have three rays that have a concurrent vertex, so we have a picture that looks something like this. Let's say that B is the common vertex of these three rays, and then our rays, we need some other point on them, so we have A, we have C, and we have D, something like that, so we have these three rays. And so what we want to find is what does it mean for a ray to be between two other rays? So we say a ray, B, C, in an ordered geometry is between two other non-opposite rays, B, A, and B, D. So the fact that the rays, B, A, and B, D are non-opposite means that B, A, D, this is not a collinear set, it's a non-collinear set, okay? So they're not just two halves of the same line. So we say that ray B, C is between the rays B, A, and B, D. If C is an interior point to the angle A, B, D. So the angle A, B, D would be this region right here, everything in the middle, and so this point C would be an interior point. So we'd say that the ray B, C is between B, A, and B, D. And we use the same notation here. We say that B, A dash B, C dash B, D, something like that. We'll say some more about betweenness of rays in the second video here, but I just wanna define it right now so we can use it when we talk about the crossbar theorem. Now we're ready to state and prove the so-called crossbar theorem. So the statement's the following. If we have a ray B, P in some order geometry so that B, P is between two other rays, B, A, and B, C, in other words, the ray B, P is interior to the angle, particularly the point P is interior to the angle A, B, C. Under those assumptions that it must be true that the ray B, P intersects the line segment AC and it happens at some point, let's call it D, that rest between A and C. So I wanna draw this picture for us and I also wanna do it in such a way that we can see the picture for the entirety of this proof. So we're gonna draw something like this. So let's label it here where you have an angle ABC, B is the vertex, we have some other point A right there, we have some other point C right here and let's say we have some interior point P that's inside the angle and then consider the ray that emanates from B and passes through P. So we get this ray like the following, okay? And so we have this interior ray, then we are gonna consider the so-called crossbar. So between these two points A and C that determine the angle, the line segment between them is commonly referred to as the crossbar of this angle. Of course, with different points A and C you'll get different crossbars but this is a crossbar of the angle and this is why this is called the crossbar theorem. The crossbar theorem tells us that if we have a ray which is inside of an angle, that ray must intersect the crossbar and it'll happen at this point called D. So we wanna prove the existence of this point D. So let's work through the details of this proof then. So the first thing we're gonna do is we're gonna extend the line BC so that we have this other point C prime in consideration here. Now use the extension axiom to get that point C prime. So by the extension axiom, it's guaranteed that the point B is between the point C and C prime. So by construction, the line segment that goes from C all the way down to C prime will intersect the line BP. It happens at the point B of course. And let's call the line, let's call the line here L, okay? So this line L is gonna be of consideration for what's going on here. So C and C prime are on opposite sides of L. That's something to remember. The next thing I want us to consider is consider the angle BPC prime. So BPBC prime would look something like this. We claim that A is an interior point to this angle right here. Now when you look at the picture, it seems clear that's the case, but we can't base our proof upon an illustration because I'm drawing a Euclidean picture. I need to be proving this in ordered geometry, not in Euclidean geometry. So I'm in a much broader theory where things might not look how they appear. Now the good news is we've already taken care of this in a previous proposition in our lecture series. We did this at the end of lecture 12, by the way. It was a very, very long proof, but we proved exactly this situation that the point A would be in the interior of this angle BPC. It seems intuitive, it seems clear from the picture, but we had a really long proof to prove this. So I would have to argue that if it's a really long proof, then it's not intuitive. And it took a long time to do. But in particular, because A belongs to the angle BPC prime, and since an angle is an intersection between two half planes, a half plane, of course, if you have an open half plane, that means that the two points are on the same side of the same line. This implies that the points A and C prime are on the same side as L. So notice what we have here. A and C are on the same side as L, but we also have C and C prime on opposite sides of the line L. So applying the plane separation theorem, it has to be the case that A and C are on opposite sides of L. Now notice here that A, C is exactly the crossbar we had considered before, okay? So we've now proven that the crossbar intersects the line L at some point, we're gonna call that point D, okay? Now that feels like we're almost done, but the problem is we now know that L is the intersection between A, C, the segment and L, the line. We have to prove that the intersection between, well, we have to prove that D is on the ray BP, all right? Which a line is basically two rays put together. Could D actually be like something weird? Could this intersection actually be over here somewhere on the opposite ray? That seems weird by the drawing, but the drawing isn't the proof. How do we know that? Okay, so let's prove that D does not live on the opposite side of the line. It doesn't live on the opposite ray. Now by assumption, A, B, C are non-colinear points because they're three points of an angle, they're necessarily non-colinear. As such, because these are not collinear, B can't be between A and C. A between this relation implies that they're collinear, they're on the same line. This was our first axiom of betweenness. Now since D is between A and C and B is not, it must be the case that B is not equal to D. That seems like a silly thing, but it does have to be said. Also, again, by that previous proposition, there was three parts of that proposition, we used the third part. The second part tells us that the opposite ray of an interior ray to an angle only intersects the angle at the vertex B. So by proposition, part B of the previous proposition, if we look at the opposite ray, so this is what we call negative BP from before, the intersection between negative BP and the angle ABC, this intersection has to be the point B. So if D were on the opposite ray, since it's, because it's on the opposite ray, what I'm trying to say over here is that it would have to be B, right? Because D belongs to the angle, D belongs to the angle there, the between cross lemmas coming into play right there, right? Because D is between A and C on the cross bar, the between cross limit tells us that D is an interior point to the angle. So it's here in the angle, but if we're on the opposite ray, the opposite ray intersects the angle only at the vertex B, but D is not equal to B. So that means it's not on the opposite ray. Now, since every line is a union of a ray and it's opposite, if D is not on the opposite ray, then ding, ding, ding, ding, it has to be on the ray BD, which then is exactly, that puts it exactly where we want it to be. D is the intersection between the line segment AC and the ray BP. The cross bar theorem is a very important theorem, not just for a betweenness theory and order geometry, but also in geometry in general, because the cross bar theorem gives us something that we really much want. It gives us intersections. It'll give us intersections of lines, specifically gives us the intersection between a line segment and a ray, but those rays and line segments can be extended to lines. So anything that guarantees the intersection between two lines is a very powerful, very useful theorem, which is why the cross bar theorem is one of the most important theorems of ordered geometry.


Dear students, now we are going to study the SPSS toolbar. In order to effectively or efficiently use this software, we need to understand the various icons showed in the output view or the data view so that you can easily use those icons. SPSS toolbar is actually shown in the upper section of the both interfaces or the windows. In data editor interface and in as well as in the output view interface, these tool the icons are shown in the above part of the interface. So the first icon which is a kind of a folder image, it is open data document. By using this folder, you can open any data file in this software. If you want to open any file, you can click on this icon and it will open the dialog box where you want to open the file. The floppy style icon is shown here. The first icon we use to save the document or to save the entries we made in this software or in any file of this software. So when you enter, what we do after that, we click on this icon, right click, then your data automatically saves. This icon of the print is to print the data set. If you want to print the data that you have entered, you can click on this icon. So you will be taken to the dialog box of the print and you can print from there. And this icon is recall recently used dialog. It will tell you which analysis you have done and it will give you a memory shot. By using this, you can rerun the analysis. And this reverse arrow as an icon is showing the option of undoing any action. If you have made a wrong entry or made a wrong analysis, then if you click on this, the entry you made wrong or the analysis you performed wrong, you can undo it or delete it. And this arrow is showing on the straight side. It is called redo a user's action. If you want to redo an action, you can do it through this. This icon is for to go to case. You have 500, 600 or 5000 data set. You want to see a particular case in it. You want to see the information respondent number 205 you have entered. You want to see if the information is correct or not. Or if you want to see a particular case. If you click on this icon, the dialog box will open. If you enter serial number, it will take you directly to that case. You will not have to scroll down the software. Go to variable icon. If you click on this icon, it will ask you which variable you are interested in. You have different variables from SPSS. You can have 1000 variables or more than this. When you have such a big data set, you can scroll down and see a variable. You use this icon. You say, take me to variable number 99. This will take you there. And this icon is for variables. It tells you what is the measurement level of the information of the variable. What are the codes and its name and label. If you want to find a particular observation in the data set, you can click on this icon. You can find this icon for control F or control H. If you want to find a particular observation. If you want to insert new cases, you will click on this icon. It will take you to the insert cases. If you want to enter a particular variable, you can click on this icon or go to edit and click on insert variable command. It will tell you to add a variable. If you want to split a file into two places, you can click on this icon or go to data and click on the split file command. You can split your file. Select cases. Sometimes we filter data in SPSS. We will see that in detail later. Either we will use the command of select cases on the data tab. Simply click on this icon and it will take you to select cases. Value labels. In SPSS, data enters in numeric codes. In the shape of 1, 2, 3, 4, 5. Later we will enter the codes we have labeled. Sometimes if you want to see which one you have labeled, you can see it through this command. Use variable sets. If you want to make a set of variable, click on this icon and you can define a new variable. If you want to see all the variables, click on this icon and you will see all the variables.


Welcome back cyber traders for another exciting day. Who's ready to start trading again? My God, I'm excited. I'll tell you, I've been exhausted for the past couple of days. Good to see you, Bonnie, Mike Grant. Good to see you, Ken Thorl already up 52 cents on American Airlines. There you go. Give him a round of applause. Way to start off early. Yeah, I still have, I held JetBlue, you know, I held it for some nasty shakes, but I've been trading since eight o'clock this morning, got in right around here. And kind of like at the 1069, I got it. So it looks like it's doing pretty well, kind of just holding on to it right now for a little bit. Little nasty stock. Thought I was gonna do some nasty shakes, but it's doing okay. Markets up about 70 points right now. It's pretty volatile. But listen, we're gonna have a crazy morning again, like anything else. Just want to kind of explain everybody what's going on and so on. You know, I feel like when we're trading, you know, everybody keeps asking me, do you think we hit the bottom? We hit the bottom. Now we got the whole stimulus package going on, and I think a lot of people are factoring how it's gonna get, I think we're pretty much there where it comes to the bottom, when it comes to trading, and when by looking at the bottom, what you gotta look at is this. You have to focus on, there are stocks today that you probably wanted to buy that you couldn't afford, and now you can. I think you're still gonna have some volatility going on.  If bad news does come out, I don't think we're gonna go down as much as we should as it looks like. We might get a little volatility going here, but it looks like we're pretty strong right around these prices regarding.  That's really what comes down to it. You gotta know how to day trade. Now regarding about what's going on for today, I mean, obviously you heard the good news, it looks like they passed the stimulus package, we'll see what happens today. You know what? I'm a little concerned about the house though.  It's not that you got two different parties that are running one versus the other. You never know somebody's gonna throw in a two cents again, and then all of a sudden it's gonna put a wrench into it, but if they do, I don't think it's gonna be terrible, you know, other than how much lower could the market drop? Go right back down to 19,000, which will pretty much be there again. So it looks like we're really kind of build some stability there. So once again, if you're looking to swing trade, you know, there are some really good buys out there. I got, I mean, some people got lucky. I didn't get so lucky with BA. I'm doing great now on it, but I bought, obviously, you guys know I bought at 170. I was drooling. I always wanted to own BA, Boeing and 350. I thought like, my God, maybe it is a steal. Then when it dropped down to 270, I went at 170, which you all know, but I averaged down at like 95, thank God. And now sure enough, look where it is at 150. So BA is doing great. You know, there's some really, really good buys now. Is it still cheap? I mean, listen, I can't tell you. I'm not an analyst. I can't tell you what to do. It's illegal for me to tell you that. But all I could tell you is that there are, if you are looking to do something and you really like something, I mean, you don't want to miss this opportunity because these opportunities in trading do not come very often. And like I said, last time we had this opportunity was the financial crisis. And look what they gave on the financial crisis, not even a trillion dollars. And look what happened. We went from 15,000 down to 666 on the Dow, back up to 18. So you know what? If you feel like you're chasing, it's not the end of the world. Now listen, I could just go through stocks. I have the stocks at the stocks. But listen, the Dow was up, had one of the biggest percentage moving ever since 1933, they said.  If it was Delta, if it was Boeing, let me bring up the two-day chart. I mean, I don't know what happened, but they all popped like at the end of the day. I mean, look at 115, I went to 103. And you know, we don't trade these stocks. We don't trade them. We work off the big percentage gainers and losers. But like the markets like this, you have no choice, you gotta do what you gotta do. But there were a lot of good moving stocks out there. There was this one.   If you love that one-third, you gotta be drooling, you know, at 36, 40. I mean, look at the long-term chart on the right. There's still lots of movement, you know. I'm a little nervous about the cruise companies in a way I don't know if they're gonna come back as quickly because apparently they said they might be docked for about a year, they said. That was a rumor going around. I don't know if that's true, but that's pretty bad. But all the airlines, guys did great. American Airlines going up. You see, the big thing that we focus on when it comes to trading is we're looking for the ones that get crushed the most. The ones that get crushed the most are the ones that are gonna re-bought the best. Uber, all those, they all pop. I own that kind of kick. I wish I still had my $16 trade when the CEO came out, that one. Oh yeah, like yesterday we talked about, remember the CEO came out on Fox Business? And I started at eight o'clock. I says, guys, the CEO comes on. That's a strategy. We teach you that in class. We teach you what to look for in class. And sure enough, boom, we were on at 117. The thing ran to 122. You know what I mean? These are things that you're looking for. Just like Uber, CEO came out, $16. You gotta look and see when the CEOs, if they're on CNBC or Fox Business, the financial stations, when they come out and they talk, believe me, they're gonna make the market move. And that made that Boeing pop. So there were a lot of stocks that we did well. Now, what are we gonna do today? Well, listen. I just put a tweet, I mean, not a tweet, a tweet also. And also my Instagram. I mean, look at Disney for crying out loud.  I mean, Disney has really testing some major supports. Look where it's been since 2010. From 15, if it breaks these prices right around here, then it will go down. But Disney is actually killing it, you know? With all their stuff. They're doing unbelievable. But it hit a major, major support level. If it, listen, if it goes down to 80 again, then obviously it's a see you later. But the thing is, the thing was at 160, you know what? What's wrong with that picture? B.A., oops, felt it wrong. What's wrong with that picture? What is wrong with that picture right there, 2010? I mean, these are what, when you're looking for trades, this is what we're looking for, okay? So, you know, I know all of them. Oops, they're all basically moving pretty well. So, oh yeah, you're talking about the bond you meant, N was the N-C-L-H, you've got the H to that. I mean, what's wrong with that picture? So, the only issue that we've learned that you know in class and we teach you is that when you break all time lows, that's when you have a problem. Remember, that's where program trading comes in, which by the way, just kind of reiterate some things to you guys. The big issue you got to worry about when it comes to trading, what's the market? Markets up 170. There's a lot of algorithms that are going on right now. Lot of algorithms, okay? So, what that means is that these stocks are extremely volatile and program trading is kicking in. It's buying on the bid, selling any offer. You got to be careful because, and especially if you're new here to Cybertrain University, the pre-market, they got to be careful of these spreads. The cheapest thing you could do is buy the stock. The most expensive part of the trade is the spread. And some of these spreads are pretty ugly. So, be careful because you can be really, really, you get yourself in big, big trouble when it comes to trading. Yeah, GE, Benito, that's a good level one stock. Look at that, that stock. Really didn't take that big of a hit, actually. I mean, it really hasn't. But GE, when you go back, look at it, it's at a major support level. So, it's a nice level one stock. But GE, when you look at it, we're looking all the way back in 2009, that's where support levels were. So, we're ready to go from 16 down to seven, still down 50% haircut. But there are some pretty good buys, but GE is actually moving pretty nicely. But let me go down the list. Listen, there's a lot of them up here looking through the big percentage gainers. You got this other little cheap little stock that's moving once in a while. We do trade some stocks like this. This one's moving up pretty nicely, but not the greatest fan, but it's up 133%. This is a lot of penny stocks here. This is a stock that got destroyed yesterday, decimated. It could be a dead cap bounce, $8. It went only down to 32 cents. We took the timeframe on this one. This thing just got crushed. I don't know if it's filed bankruptcy or whatever, but it's coming back a little bit.  What else we got? The ones that we were trading this morning was this one, the CTSO moved up pretty nicely. That one went from $5. That one went from a 46%, 1.1 million shares. She's kind of holding it pretty strong. So we'll see what happens when she opens. But as of right now, she's not really the play. ASTE was fun. You guys did really well on that one's also moving pretty well. That one's gapped up pretty nicely. What else did we have here? ASTC. That was the one I meant. Sorry, I spelt it wrong. I meant not the E, the C. That one, you guys did really well. That one just popped right at eight o'clock. Eight o'clock went from a buck 60 to 380, up 183%. That one did really well. That stock's doing well. What else we got? JetBlue, they're all doing pretty well. All the airlines. I mean, Charles is bringing up SAVE. They're all doing well. SAVE is doing, look at it, 45 to 10. Listen, they're gonna bail them out. They have no choice. They're not gonna, listen. I said this before, and you know what? If you loved Obama, if you hate Obama, the greatest thing that they did is they bailed out the banks. A lot of people didn't like that they bailed them out. But look what happened to the economy. It worked out great.  But look what Congress is doing right now. Back then, the financial crisis was less than a trillion. Now we're talking $6 trillion dumping it to save the economy. This is not a financial crisis. I keep repeating it.  It's a sickness. This is not a crash of the economy. It can be the longer we keep blowing it off. But you're seeing some opportunities that you wish you could have got before and now is your opportunity. I mean, look at these things. $13, when did you ever think that you would ever be able to buy Spirit Airlines?  That's why you can say, maybe that's why I didn't trade it. I'm not a fan of Spirit. But sure enough, there was some good buys. But the list goes on and on. Harp is another one that gapped up pretty nicely too. Up 62%. That one got a little beat up right there. That one's making a little bit of comeback. S-A-B-R, did I mention that one too? That one took a big hit from 24. It's gapped up a little bit, up 13%. Just be careful of the spreads. The spreads are what you gotta be careful of. I still own the Disney. It looks like Disney's starting to make a little bit of a pop. So I'm doing pretty well on that Disney. I'm day trading and I'm also swing trading it. I kinda chased it though. I got it at like 99, 60. But anyway, look at the spread there. 101.21 to 101.80. Gotta be careful of the spreads. If you buy a thousand shares of the stock, you're talking, you're already down $600. You buy a hundred shares. You're already down 60 cents. That's a lot. Gotta be careful of the spreads. That's, remember, when you're a student of Cybertrain University, the first thing we teach you in the beginning of lesson one, if people are like, well, I'm not really a beginner. Yes, you are. The most important thing you gotta learn is spreads. You can't, you don't make any money until you sell it. Be careful of the spreads. Then you already made 60 cents on trading Royal Caribbean. I'm gonna keep giving, I'm just gonna keep giving round of applause. It's great. Listen, this is what you train for. This is what you ask for. Is it too late? No, it's not too late. So let's go out there and trade together. Now, if you're new here to Cybertrain University or you're watching it, streaming it live on our Facebook or YouTube channel, you know what, there's still opportunities. Get in here, do a quick trial, see what it's all about. Because remember, I always tell everybody this, it's not for everybody, but at least it'll tell you what you're getting yourself involved in and then maybe you'll understand why you do something else. But the worst thing that could happen is that you're gonna love it and you'll do great. That's the worst thing that could happen. The greatest thing is that you actually make a lot of money doing it. That's the biggest thing. But just to kind of remind everybody, if you are a trial in this room, please make sure you talk to education advisor. This is not a teaching room, this is a trading room. The whole goal is this. If you like what you see and you see what people are doing, remember, don't count my money, count theirs. If you like it, then you make sure you talk to education advisor and we'll see if you're qualified to be part of the team. Because remember, a lot of you really do not qualify to be here. And we'll know once you start to be in the room and you talk to an education advisor. And by the way, don't get insulted or embarrassed. It's okay.  So just be safe. All right guys, but in the meantime, Cyber Traders, another exciting day. It's gonna be another fun day.  I was talking to some people being here in a hot spot here in New York. And believe me, I've been confined to my house for two weeks, which is not a bad thing, right? As long as I'm working out and not getting fat, my wife's making all these great dishes. But listen, at least we have an excuse. We got a trade. Some people got nothing to do. They're sitting at home like, what do we do next? Oh, we'll do the same thing we did yesterday. What are we doing today, Dad? Same thing we did yesterday. Guess what?  We're gonna trade the market and have fun again. That's what it's all about. So just have fun, guys. Work together like a team. And let's do pretty well, all right? Let's have fun. Good luck, everybody. Happy trading. And for everybody here, we're gonna start commentating live audio broadcasts in about 15 minutes. And then remember, we got the afternoon meeting at 2.30 to trade the clothes, all right? Good luck, everybody. Happy trading. Be safe.