 Okay, if you remember, we were looking at this business of taking a small elemental piece of some widget made of some material and subject to some kind of loading which would result in stresses that can look all kinds of different ways. I'll just, I gotta pick something and draw something so I'll just pick that and draw that. Also induced stresses are certainly impossible and common and the question was what if without changing the loadings we change the angle of the element. What if instead of looking at some nice orthogonal things where the element is lined up with the x and lined up with the y, what if we just sort of turn our head sideways a little bit and then see what happens because these are all due to forces and when we change the angle on the coordinate system it changes the components of the forces even if the forces don't change. So we're now looking at the possibility of doing just that kind of thing, making some observational change to some new angle and seeing then if that doesn't change our observed stresses and how they're going to change depends upon what the stresses were, how big they were, what sign they had originally and what the other angle is. So we can get all kinds of possibilities here. So we got it down to the point where we had three equations each of them containing these simple quantities. So the book doesn't do it this way. I find it a lot more convenient to do it this way. This term appears in the equation so it's kind of nice to calculate it separately, calculate it ahead of time. This term also appears, I named it sigma if because instead of like the sigma average it's the difference in the two divided by two. So just as a matter of convenience I separately calculate those. Then we had these three equations not in any particular order other than we typically start with x's and then move on from there. And these two things we just calculated and then the angle at which we're making our new observations appears as well. So with these little things pre-calculated the equations are pretty simple, pretty straightforward. You can imagine how easy this type of thing would be to do to write some kind of applet or do it on a spreadsheet. These things cranked out for you. If you go online you'll find literally hundreds if not thousands of applets available to do just this for you. So only write your own if you're a one for that type of punishment. So then we have the three equations. Make sure you get all the plus signs right, all the minus signs right, all the signs and cosines right because even though there's a ton of these apps on the web, you have access to the web during tests. They're not that all right. Minus, minus on that one. Minus plus. All right, looks okay. So here's one of the things that's very important with all of this. There is an angle that I'll call or we call in this business theta p for principle. p stands for principle because at that angle something very important happens and that angle can be found from stresses that we found. So just take me tau x, y over sigma, I'm sorry not sigma m or sigma dip. So very easy to calculate that angle. At that angle whatever it happens to be something very important happens. So here's our original direction. Now at some observational angle we call theta p. This is very particular to the input values there. We have our new, remember our new observational direction if you want to call it that. So here's our new coordinate system transform. In fact these are called the transform stresses. No change in loading, no change in material, no change in any of the geometry that went into this is simply you're tilting your head sideways and looking at things a little bit differently. At this particular angle we get the maximum stresses, normal stresses possible. They could be on the other face. It's possible but you just have to check and see what happens when you put theta p in there. There's actually two angles of course because there's 90 degrees here possible. And on the other face we get the minimum. They might be positive, they might be negative. It just means we're at a minute in a max. I have to draw something so I'll draw in that direction. At that principle angle there are no shear stresses in that direction. So if you have a material that you're worried about it being in a failing due to stress and you can change the direction of the material whether it's wood which has a very definite grain direction to it or carbon fiber which is actually a material with threads moving in a certain direction, you can accommodate this characteristic that at this angle there's no shear stress. Maximum normal stresses but no shear stresses. So that could be something you could advantageously take into account. Now one way to pay close attention to what this is is to draw what we call more circle. And that's where I finished on Tuesday. There's not really anything you can do with the circle that you can't do with the equations anyway. I find the circle a little bit confusing for certain things that it's a lot easier to do just by putting in the equations and calculating. But it can help in some ways. I've also found when drawing more circle its coordinates and size depend upon these values of the sigma x's and the tau xy's of course. I find it a lot more advantageous drawing the circle to draw the circle first and then put in the different coordinates that you have based on these values. It just makes for a better drawing. So it's really hard I think to draw on the axes plot the points and then draw a decent circle to those points. It's just a lot easier drawing it to do it the other way around. The circle is drawn on coordinates where the x-axis is simply sigma x. Oh sorry sorry not sigma x just sigma in general because we have sigma x we have sigma y we have average and diff we also have max and min so just sigma in general. Now the y-axis can't appear anywhere depending on exactly what the values are and it's very easy to come up with examples that put the y-axis in all kinds of different places. Can be entirely to the left of the circle could be entirely right could go through the circle somewhere. So I have to draw something so I'll draw it right there. It depends on what the numbers are in the problem what the actual loadings are that make up all of these different things. The center of the circle is at sigma average. The little piece thing you calculated there. The radius of the circle and the square root of sigma diff squared plus tau xy squared which you get from the original loading and the original coordinate direction typically this great xy coordinate direction. That's R and the center at sigma average. So if you're doing this on graph paper and have a compass or something that's stuff that's very easy to draw. Figure out where sigma average is put the compass there give it a they spread of whatever is equivalent to R and then draw your circle and you got it. The reason it's helpful is this value right here is well that's as far as we can go on the normal stress axis that is the value for sigma max. This other one as far left as we can go on the circle is sigma min. Oh I'm sorry I didn't even tell you what the x or the y axis is. The y axis is minus tau up plus tau down the shear stress. The reason the minus goes up is so that any angles that we're going to get on this drawing are in the same direction as the angles for our element. If we did plus tau up then a counterclockwise angle here would be a clockwise angle here to get the important. So that means that these extremes notice since the circle is always right on the x-axis these two extremes have the same magnitude and it's the magnitude of this year that's our greatest concern. These are tau max so from a simple drawing of using these points these equations to plot this circle we can get a very quick idea of what the maximum possible stresses are and what the minimum and maximum shear stresses are. What's not here yet on this is where this direction of the principal stresses is. This is called by the way the principal plane. So what we don't have here yet is just what that angle is. We now already know what the maximum and minimum stresses are. We don't know where they occur yet. So to find that well you can either just calculate it straight away which is virtually foolproof if you're careful with your calculator or you can plot it on here. You can plot the point wherever it might be it could be anywhere on the circle. Plot the point sigma x tau xy. When you plot that point it gives you the angle not theta p gives you the angle 2 theta p. All these equations have 2 theta m all those equations together led to this circle so that's why the circle has the equation 2 theta p and notice they're in the same direction so that's why we put the minus tau going up. Not just this y off but it has a particular purpose to it. Alright there also happens to be another particular angle at some angle that we call theta s I think it stands for secondary at some angle theta s that we find in much the same way minus sigma depth over tau xy so it's negative inverse of the tangent of the other angle. These two angles theta s and theta p are 45 degrees apart we get a different original direction let's say our theta s gives us an element that way. Notice in the in the drawing they're they're 45 degree or I guess you can't notice it but they would be 45 degrees apart which puts them 90 degrees apart on the circle. At that point we get stresses that are the normal stresses are all the same on all faces and it's sigma average on all faces and the shear stresses are the maximum they can be so this becomes an angle of interest just as much as the other one because now we have something that's particularly weak to shear as wood can be. Wood has a definite grain to it as you know because of the growth rings. Those growth rings are very strong in shear across the grain. Along the grain they're very weak in shear because the growth rings are alternating layers of hardwood and softwood. The hardwood is the darker thing that we usually call the ring. The softer is the material that's in between each ring and you know that there's pretty much a ring every year as they grow. They go through different cycles in the in the growth season causing different cycles if you will in the material itself. These layers are very weak in shear. That's why when you build a deck and you put some wood choices you do not buy wood. I don't know if you could even you'd have to go have a custom cut. Well it couldn't do it. It wouldn't work but there wouldn't be any wood available where the growth rings are like that because you put a deck on that you stand on that you cause transverse shear that wood choice is just going to go go right through. Trees don't grow big enough to even come up with boards like that because you'd have to look at the tree with its growth rings you'd have to have a board that's cut across there and there aren't many eight foot and diameter boards to give you I mean trees to give you an eight foot board. Plus you get one here but you can stack them all the way down the tree I guess if you want to those but they don't even come like that. The trees don't even produce many like that. That's that's God telling us he wants us to have a nice deck. That's what that is. There's no other explanation for it. So anyway we have these principal directions that give us the maximum strains. The circle helps us know just what those are as well and again there's nothing you can do with the circle. You can't do with the equations themselves so if you find the circle confusing relax but you've got to have seen it. If I sent you out of here to RPI and you said I've never heard of the more circle we've never transferred another student to that school. Understandably. However Colin and Jake you love freehanding circles you know how to do it you're you're all excited about this. President's data of P and the S are 45 degrees apart right? But it doesn't have to go in those directions right? It could be like P and P goes that way and then you go another 45 and then there's the data S. Yeah that's the just any remember any time you do the tangent you get two values for it that are 45 degrees apart to you this this ratio will give you two values for theta P 45 degrees apart you put you could put each of those into the equation to see which one of them is giving you theta max sigma max which one of them is giving you say sigma min and then you do the same thing for here because it could be that theta max sigma max isn't on this base it's on this base over here and you wouldn't know just simply from calculating a theta P you need to calculate both of them you can stick them into the equation or you can draw your more circle here the book calls this point a it could be the point a is over here so that theta P back to the x-axis positive x direction is much longer in fact it would be 90 degrees longer so well it's time to do a couple problems and run through these yeah theta theta S is 45 degrees from this so it's 90 degrees from that I already put the R kind of there pardon you know not typically in fact I think in the book where the book has a like a page and a half where they summarize the circle and they say calculate theta P and theta S no it's a calculated P shown in figure such and such calculate theta S not shown but you can imagine it can clog this up pretty quickly because not only do you have those lines you've got to have the angles drawn the directions on all the different things it gets pretty crowded pretty pretty clocked up all right so let's clear a little board space and put up a sample problem and run through it let me not forget to put down notice that this is the radius of the circle so this is also then tau max itself the magnitude tau max square root plus or minus and you got a plus or minus just like you'd expect and could use all right so here's some problem we we we're given some some gadget loaded somehow we calculate the loads calculate the stresses at a particular point and we come up with something like this as a result sigma that's a 50 megapascals sigma y 10 megapascals in that direction and a 40 megapascals like that before you get going on these let's make sure we remember the directions and our sine conventions for these because if you don't have the sine convention right the equations are all going to be backwards submit sigma x is positive 50 and we're pretty used to that because it's intention sigma y is in compression so it's a minus what about tau x y I want a little bit harder to remember that's also positive this is our positive convention so we've got everything we need we can now calculate the principal directions and the principal stress but I find it a lot easier because you're going to need them in several places to calculate sigma average and sigma diff separately pretty good weather today look pretty sporty that you look like you barely got out of bed we're stuck for you because they're having a transfer day all schools do it's just I have to get a note about our guy and his alumni I got it because there's my graduation here and they're hardly born that sucks that's for you immature here so you guys not even shaving yet right do we baby face little boys why are you smell yeah why would I better make any sense you're not used to be a biker he's a lot more than you guys ever should all right very straightforward plug-and-chug the students favorite type problem going to autopilot finally saying oh man I'm glad I got this hundred and twenty dollar calculator finally it really is coming into use here in fact I'll bet you that somebody's got more circle equations already put in we're kind of great would you give him a second you were hanging to draw a circle come on we got guys so that tells me technical free and sketching made things worse for you let's see yours not too bad Brandon you had years practicing with me that's okay at least recognize that it's okay to draw an egg it's almost Easter right that's what you're thinking do be amazing best one in here we're finding out who draws the best circle and that's you we're finding couple things we need we need to know at what angle these maximum stresses occur that's a that's more than anything what we need we can do this on off angles but I didn't give you one in this problem it's it's common in some of these problems to say at what at a certain angle what are the stresses but for this one I want to find the principal stresses called finding the principal planes and find the stresses in those planes in that direction theta p and theta s should be 45 degrees apart like those angles you're going to get two angles really the tangent repeats itself 25 degrees 90 that's the shape of our cubes from there you can get all the maximum stresses it's other side of the circle so would be sigma average minus r maximum stress normal stress is sigma average plus our the other side of the sigma average minus r and it may be on the other side of the y-axis it might not be we don't know where the y-axis is until we actually plot everything but that'll be all taken care of so let's get down the values we need sigma average remember is the center of the circle and the location and the stress that goes with the maximum shear stress is 20 remember these these sigma average is ignorant if little things you won't find in the book I just think it makes things an awful lot easier if you don't want to use them don't you're grown up so that's what that's 30 that itself doesn't have any particular value to problem other than it repeats a lot we need to calculate it for the principal angle 22 26.6 we're calling no I don't think there's any sense to go to radians because very few people can think visually in radians other than the principal you know that the ordinal directions minus 30 minus 18.4 notice that those are 45 degrees apart all right now that you've got those you can figure out what the maximum are you can either put those angles in here and that will give you sigma max there there is another angle the tangent solution that needs three from that of course because that's just the other side of the cube one will give you sigma max one will give you sigma min and you can figure out town actually or you can just figure out the radius and then you know what each of the pieces are so that radius 30 squared plus 40 square square root of that pass count so I was at that was 50 so just for the sake of it let's draw the circle again to draw the circle to start with I don't know yet where that y-axis is going to go but we can find it out in a second center of the circle remember is where sigma average is the center of the circle we've already drawn the circle so we know that it's at 20 mega pascals a distance plus 50 and minus 50 is the radius so that makes this plus 50 that's 70 that is sigma max 20 the center minus the radius is minus 30 sigma min and so we know that our x axis goes right about here on that we plot town with the plus going down and so now we've got some of our major pieces now that's obviously 50 that's the maximum shear and to find out where our problem lies on this circle we have to plot point sigma x which is plus 50 right about here it should automatically fall on the circle because the circle is made up of these very values so this should automatically be the point angle if you calculate it which you can do now because you you know the coordinates of it you can find out the tangent of this point from this center that angle should be whatever the double of that is I think it's 53 3 4 something which is just what it comes out to be and then we can draw our so at some angle half that which we have the 266 we have our new and what falls on that element in that direction which we know to be 70 and it's the first direction we get that so we know it's the x-face actually we have those values we're 7 right there we know on the other face is the minimum and notice that's 90 degrees away from here which on the circle where we double everything is 180 degrees so we've gone 180 degrees around to the map minimum we know it's negative here it isn't in every problem it is in this one so the compressive stresses have increased in this direction as have the tensile stresses because now the forces causing the shear are combined in these directions because it's just essentially an addition of force vectors and what about the the other angle theta s 45 degrees from this angle is our other element direction and on it are the average stresses on each face which we know to be positive because our center is at plus 20 it's not necessarily very helpful to draw these on both on the same diagram other than it does illustrate again that they're 45 degrees apart and the stresses the shear stresses our maximum shear stress is plus 50 because they're on that side y-axis it's not uncommon to combine these drawings where we draw the principal direction something like that with the stresses that's a little cleaner drawing and shows the two things of importance in a single job beautiful huh some of you have no trouble saying all these angles and all these directions nearly in the circle I think it's pretty easy to get confused I still do after teaching in all these years but a combination of the simple parts of the circle plus the equations and you can get everything you need from it okay so we can keep doing the problems but we only got two minutes left so on Monday we'll take it a step farther huh we do test on Tuesday already wow that sucks I hate to be you yeah we do we'll take it a step farther on Monday we'll do a quick warm up problem then do another step farther that step actioner stuff we will put it's as long as you get these basics then the step farther will come