 I wonder in when they feel I can turn the camera backward busting right there. Boom. Free research. Nobody makes any mistakes. Guess I'm the only one who makes mistakes. All right. I thought we got cooking for the day. We had combine loadings yesterday. Any questions? Um, I still don't like how we had like the negative. It was like the normal stress plus the other stress. Bobby and I were just confusing. You knew instantly if it was negative. Oh, okay. We had, let's see if I remember, the loading was something like, something very simple like that, right? Is that the one you're talking about? And the deal with the combine loadings is very straightforward in that we can separate the two loadings, figure out the stress of each, and then recombine them to get the stress. So we can make this into this kind of loading, which we studied the first week and leads to stress, just that load over the area. And for this example, it's obvious, I hope that S maxial load will cause a stress pattern that would look, it's not actually uniform, but it's very close to what we take it as such. And then we add to that second load, which is a pure bending load that we study a little bit later. Now, this one, you might have to think a little bit more about to understand what part of it's negative, what part of it's positive that would go with that one. The stress calculation is the M1 over I. But remember, when we very first developed it, there was a negative sign in there and sometimes it's there and sometimes it's not. That's because with a typical bending load, we get both tension and compression in every single one of the examples because for that loading, and I believe it had a rectangular cross-section, so at the neutral axis, we know that it's zero there and varies linearly above and below. But whether it varies to compression above or compression below depends upon what the loading is and you need to look at them. So this type of loading would give us a bend, something like that. So you have to look at that and say, well, if a beam bends like this, then clearly the top's going into compression and the bottom's getting stretched just by the nature of the curve that's made. You could see it if you want that an inside curve like this is going to have less distance to it than the outside curve will have. Therefore, the inside curve's got to get compressed. The outside curve's got to get stretched because the neutral axis stays neutral. Nothing happens to it and everything happens on either side of it. So in this type of example, with this type of loading, we know that the top was in compression. So if that's my tension picture, then I want compression to go the other way, linear through the neutral axis. So there's just more tension there, compression there, and then the two superimpose over each other. So that's how I knew that, for the example I had, that the top one and the top one there are going to subtract. It's still going to be linear because you've got two linear functions you're adding together. One just happens to be a constant linear. The other is a constant slope linear. But then, you know then, it's just a matter of which one of these is bigger. Is this bigger negative than that is positive? Or is this bigger compression than that is tension? Either way you can see it the same. And then once you know which one's bigger, and if I remember the compression was bigger than the tension was by a little bit. So there's a little bit of compression left over, and then you can figure out the two add together down here. So we had quite a bit of tension at the bottom. But then once you've got those two, it's just linear in between. And so I think somebody even asked, so there's essentially a new neutral axis. And the only way to find out where this neutral axis is now is by figuring out that linear function of the slope and where you get a zero with it. There's no other way to find the neutral axis. Originally we find it from the pure geometry of the cross-section. Now it shifts because of the load, and you have to re-figure it because of the load, not the geometry anymore. TJ, how are you feeling? Good to have you back. Well, you're not very convincing. So all of this was mysterious to you. Have you been watching the videos? Yeah, I've been doing them. Okay. I know they're not all that much in focus, so if you need details, just ask me here or someone else to fill in. Because a lot of times we put up dimensions and stuff. Those I know don't show real well, but it's really easy to figure out which one you're looking at and give you the dimensions and the loads and the specifics of it if you can't see them on the video. Okay, any other questions about that combined loading type of stuff? And we'll take our next shift here. You may or may not remember several weeks ago we looked at this thing called, I guess we called it the general state of stress. So we took a nice elemental volume. Quite big here though, so we can take a good look at it. This is a little piece of some material under some state of stress. And we can put an x, y axis on it just for reference. It doesn't matter really where it is. No, I think we put z typically out here for our purposes, x, y, z, right hand rule. So we've come across these type of elemental pieces before and looked at the loading on them. For example, we might have had an axial load in the x direction that gives us stress in the x direction. Could be compressive, could be tensile. It doesn't matter which, I have to draw one or the other. So don't think that axial loads are only in the positive x direction or in the tension direction as I've drawn. And of course because of a force balance on it, we start to expect this kind of stuff would have a perfect match made in the back. And then we looked at shear stress across the flat face. And so that's shear stress across that face. Remember what our subscript designation was that we used on the x space in the y direction in that order. We found shortly after that that it wasn't all that important when we got to it. And then of course there's one on the back doing the same thing in the opposite direction. I'm not going to draw all the stuff on the back. We'll just draw it on the faces that are visible and be done with it then. And I have to always pick one direction or the other. We can come up with any kind of loading that will reverse any one of these arrows at any time. But I have to put something. So then we have a stress, could have stress in that direction. That's the x space in the z direction. And then on the top we could have some kind of axial load in the y direction that gives us a normal stress in the y direction. So I'll just pick positive to do something. And then we could also have shear across that face. So let's see, that would be tau yx, y face x direction. And then we could have a shear across that face there. Tau y face z direction. And of course on the underside the exact same thing mirrored for the most part. And then the last visible face here. We could have a positive, we could have some load in the z direction that gives us a z normal stress. And we could have shear across that face. Let's see, tau z face x direction, tau z face y direction. And then on the back we can have all that too. Now you may remember then that we also simplified this to a two-dimensional picture which was very useful to us. Because we saw then that this whole business becomes a lot simpler. Okay, we expect the possibility of a normal stress due to some axial load in the x direction. We could have some axial load in the y direction. And then we'd see these shear stresses along that face. I just happened to pick up tau xy and on the back face tau xy. Those have got to be equal for the force balance to work. And then to counter the moment those two cause because that's a couple then we needed shear across that face. And because the forces all need to balance and the moments all need to balance, we found that tau xy equals tau yx. And so on anywhere else all these stresses appear. So we have tau xy there, tau yx there. And we understand now balancing the forces and balancing the moments that the order of the subscripts doesn't matter. All we need to know is that we have some shear stress that is on the x face and the y direction. We're going to have the same size shear stress on the other face in the other direction. So that made things up an awful lot simpler for us. So that we're down to this much simpler two-dimensional drawing from this cluttered maybe even a little bit confusing drawing we would have had three dimensions. So things become a lot easier then. What we're going to answer today is given an element stressed in this way because of some loading. Remember all of these stresses are caused because of some external load on whatever material it is we're talking about. Let's raise the question well what happens if given that very same loading we don't want to look at an element that's conveniently oriented in the xy direction. Given the exact same loading what happens if we want to look at an element that's oriented in some other direction? Do the stresses change? Well we know they do anyway. If you remember back in maybe the second week or something we looked at this type of thing and determined that given a simple axial load that's the very first thing we looked at is just straightforward axial loading that the maximum shear stresses occurred on a 45 degree angle. You remember doing that second week or so? Flip back to your notes wipe off some of the cobwebs some of the dust some of the pizza stains it's there. But that was for a simple axial load only. Now we're looking at a much more generalized load because we have not just axial load in one direction could be in the other direction and could be some kind of shear because since we did axial loads we've now added torsion and bending and need to put all those things together. So we can have now now we need to do the same sort of thing we did here with just an axial load to look at the off angle stresses. Now we're going to take a generalized load where any one of these could exist and anyone could be in any direction and see now what happens if we're in an off angle direction with those kind of stresses. So we'll call that angle theta that'll be our reference direction and now we've got an element oriented with those axes and now remember the load has not changed all that has changed is our chosen angle and how we're going to look at this and now we have the possibility that these are very much different without changing the load in fact some can even turn compressive depending on what the angle is that we chose so just for the sake of illustration I'll turn one of them around it could be that we can even get to certain angles where an element oriented in that direction won't show any stresses of a certain kind they can completely disappear we'll see so all of those are possibilities that we're going to see here prime will mean our off angle directions and I'll say it again because it's so important and so easy to forget we have not changed the loading all we've done is changed our reference viewpoint everything's the same it's just instead of looking at things straight now we're looking at things with a little bit of tilt to itself it's sort of like this will maybe straighten up your world for how you guys feel on Mondays you kind of come in looking like that catch a wish man and let us lie down while we do class that'd be awesome if you bring pillows in and kind of stretch out alright so now our question no change in loading whatsoever now our simple question is what are these off angle it's just going to write x y that represents everything we're doing we're not going to write y x because we know all the same and so we can just keep going here I want to forget my punctuation question mark what are those what are those stresses well that's what we're going to look at now so here's how we can get to it we're going to start with our with our sorry let me put this back up real quick I couldn't stand to keep it there that was our on angle stresses if you want to call it that remember it's the loading that decides the direction of any of those and the size of any of those we drew an element with that kind of loading yesterday with that bracket so what we're going to do is take a little bit of this and a little bit of that and combine them so we'll take the two directions we already know and combine it with the tilt of our new direction so we'll get an element that's like that where this is our usual x direction which is nice because we know what's going on on those faces we know that those stresses exist also we've got the shear stresses there so that's the piece from our original orientation now we have a one face that will give us access to the other orientation where we don't know what these stresses are yet that's what we're trying to find out let's see, it turns out and you can see it with a little bit of review there that angle is our theta angle so what we're going to do now is balance the forces balance the moments and that will allow us to solve for these unknown stresses but we don't balance the stresses remember we balance the forces and the forces are stresses times the area over which they act so we're going to call this face dA let me make it delta A not that it really matters because you've seen this type of thing before these all end up cancelling out in the end anyway but I'll call that delta A just so I can stay with my notes if nothing else so let's see so this force is sigma X prime delta A that's actually the force on that face because that's what we need to balance and this is tau X prime Y prime delta A so those are the forces now acting on those faces we need to do the same thing for these other two faces so this is for example this is sigma Y sine theta delta A is that right because if that face is delta A this face is sine theta delta A and so this stress tau sigma X Y acting on that same face sine theta delta A this stress is sigma X cosine theta that's right there that's tau X Y cosine theta delta A so that's now all of the forces which we can balance not the stresses which we can't balance but they're all acting on exactly the same area but they're not these now have the possibility of different areas I'll leave that up for now alright so that's our picture our transformed element picture if we sum all the forces and combine and we're going to have several trig identities in here because we've got a lot of cosines and signs and all kinds of stuff flying all over the place so trig identities occasionally in here I do have one of those that normal human beings that not only loves trig identities but has every one of them in mind instantly on recall any of those people here it's certainly not me so what this will be kind of like a hand waving thing where I go hocus pocus we do all this it's all algebra it's all trig identity if you want to go through it do so by all means it's not worth it for us to spend the time to do it here so it comes down to then our ability to solve for these unknown stresses based upon the original stresses and the angle between the original angle our new angle whatever that may be so the first part of it is that then we've got sigma x minus sigma y over 2 times cosine 2 theta so you recognize that as the type of place where these trig identities came in a little bit of space here to get the last piece in tau xy sine 2 theta now let me double check make sure I got all my minus signs right all the 2's right all that kind of stuff plus there minus there cosine 2 theta plus there we go so there's the first of the equations so now we can figure out what the stress is in any arbitrary angle remember no change in load just a change in our viewing angle we now know what the axial stress is in that new direction the second equation in a particular order we're going to meet them all the off angle shear stress is minus sigma x minus sigma y over 2 sine 2 theta plus tau xy the original shear stress cosine 2 theta hopefully you're getting a feeling of how much fun it would actually be to go through the derivation of these let me double check this one minus minus sine 2 theta cosine okay and then we need one last one we've got sigma x prime tau x prime y prime now we need sigma y prime not actually on this picture so we would have had to redo the picture a little bit to get that in there but it's certainly doable but it's just an exercise and a repetition so again this equation opens with the average of the stresses the original stresses minus sigma x minus sigma y over 2 cosine 2 theta minus tau xy sine 2 theta let me check that one average minus difference cosine 2 theta minus tau xy sine 2 theta given any arbitrary angle we can determine what the stresses are on an element at that angle exact same loading as before all we've done is change our reference angle and in a second what we'll let this lead to as you might imagine is we'll ask ourselves is there some angle where any of these maximize because that's then an angle of concern if on a particular angle these get a maximum and go over the allowable limit we need to pay attention to that because it means that these things could fail in an off angle direction rather than strict orthogonal direction not a big surprise when you see something that failed when you see a piece of wood that breaks because it was bent too far or snapped because it was stretched too far the interface that broke is not always nicely orthogonal there's angles to it because the stresses were greater at some angle than they were just in the straight across a 90 degree world in which we might prefer to live okay so let's test drive this let's put it to the put it to the test here so given some problem with some loading just like we did yesterday remember we had this bracket on the wall that we we had a problem that simply looked like that that we did yesterday and we looked at some elements back here and came up with an elemental loading on those so very same type of thing we did there I just happen to have some different numbers for it so whatever loading it is we have gives us an axial stress in the x direction a tensile of 10 megapascals and some part of the loading the compression of 5 megapascals in the y direction a shear stress and remember those are all the same all the way around so all we need to do is give them a single number so there's the original loading we'll do that and then redraw the element on that angle now before you get jumping into this some certain things obviously we we only need four things we need the original stresses the original normal stresses we need the original shear stress and we need the new angle well the new angle that's easy there it is it's 45 degrees so that'll be kind of simple but make sure that you use the right original stresses because some are negative and some are positive and if you get those wrong you're going to get the wrong picture for the new angle so that's the tensile stress sigma x our convention has always been that tension is positive so that values a 10 megapascals as it goes into the equation the original y normal stress negative so we use a negative 5 megapascals and then if you remember our convention for shear stress in this type of orientation where it's down on the right face and up on the left face that convention is also negative so tau xy is a negative 6 megapascals and now we do take a couple seconds just to run through these if I were you I would pay attention a little bit ahead of time notice that the average shear stress you need to calculate for all of them so just do that once as a separate thing then you have it ready to plug in we need the sort of the difference average whatever you might call that we'll need that in a couple of them so just do that once and put it in we'll calculate those ahead of time in fact we can even label them as separate pieces on sigma diff need that several times so just calculate it once and then you're ready to just plug it in make sure you get your minus signs right and then screws up minus signs minus signs in there don't forget the 2 on the theta's in here obviously that's important of course the units work through the whole thing that's where we wrote it out who you are and then we'll talk somebody talk to dk come on there the new axial stress and the new x-direction remember the loading didn't change at all this still exists we didn't take off any of the forces or add new forces or anything this is purely a geometric exercise so what's sigma x prime negative 3.5 omega pascals and y prime 8.5 positive or if you draw our new stress element notice now in the x prime direction we now have compression that's the minuses with the arrow so that's just 3.5 plus omega pascals y prime is now 8.5 tension down on the x space up on the minus x space in our convention was 7.5 exact same loading and clearly the stresses change quite a bit nothing's changed in the element itself whatsoever just what we're looking at is changed now we can see that there might be other places of concern that we hadn't seen before not a great change we had 10 in tension now we only have 8.5 we had 5 in compression the shear went up could be that that's now over the allowable shear limit and we'd have to design for that fact change the geometry or the type of thing that's very easy to do nowadays is with carbon fiber that can be laid down at a particular angle we can orient the carbon fiber to protect for that type of thing we'll do it real quick again a new angle just not that going through the calculations any difficulty but to make sure that we can view that element in the right way how easy this type of thing would be to just write a real simple computer program original stresses put in the angle puts the new stresses you can find lots of calculators online for transformed stresses I don't condone it Brandon that's a perfect middle school and high school kid in America has one of these they still make them numbers yet you've got to become friends with everybody in T.J. they're suspicious of you for having been gone doing the same problem you can come up with the same numbers then let's make sure we've got the same transformed element some checks and balances here to we're ready to go public on these sigma x prime 3.95 positive that's absolutely crucial what we're looking at sigma y prime didn't get that that's what I got that's why I put it up there in a minute to Patrick there y prime sigma y prime 1.05 tau xy positive 9.5 again watch the minus signs they're really easy to mess up I think a little bit of trouble if you calculate these two things ahead of time because then it's just every time you have to enter those you have a chance of messing up the minus signs alright so let's see what we've got here now we're at 120 degrees so that's something like that there's our new x prime direction y prime is 90 degrees to that so that's where our new element lays we have a positive just under 4 something like that the new y prime direction we have about a fourth of that but still positive sheer stress is positive our convention is down on the positive x space is negative so this is positive it must be up on the positive x space here's the positive x space that's the up direction on that face in the new transform coordinate system and once you get one of them in you can get the rest of them in we now know that that sheer transform stresses in the new coordinate direction it's done by how simple it is amazing let's move forward let's see if I want to make the next step with it I think I'll go ahead and make the next step and then we can take that up again on Friday with that step so let me see if I want to leave those up there let me clean up the problem we'll go ahead and keep this up here I've got equation one and two if I square those and add them simplify combining like terms and canceling all the stuff that drops out all that kind of stuff we get down to this equation square one and two we get sigma prime minus yeah minus sigma average which is sigma x plus y over two quantity squared tau x prime y prime squared prime y prime squared equals sigma x minus sigma y over two quantity squared yep that's right plus not that remarkable except if we look and see what we've got now let's see little piece here is known that comes from the original loading the original orthogonal direction same with this piece no matter what the angle you know those two because that's the orthogonal regular original direction these two things that and that are the variables depending on what angle we pick these are each functions of theta write this equation a little bit I'm going to call this beast here squared minus sigma average just the average of the two stresses quantity squared plus the tau xy prime the other big of theta that number is going to change won't equals which is just a constant you know what the original loading is you know what our squared is there's nothing that's going to change when you change angles got all the pieces right plus or minus is missing here's the deal and then this is where we'll end it and take it up on Friday this is a variable this is a variable so we could write it just in generic mathematical terms and see if we recognize what this is this since sigma x prime is a variable we'll call it x minus some constant we'll call a plus the other variable which we'll call y because we like xy for variable names in mathematics equals r squared is that a functional form anybody recognizes it's a circle this is a circle with center and a radius r so tomorrow or Friday then we're going to take these transformed elements and draw the circles that these lead to this is called Moore's circle we're going to take those and see what we can get out of the transformed stresses with that visual aid then on Friday