 All right, so we're okay you don't need to look at any of that Couple of you sent me answers. I think most of you are right. So if you haven't done that get that in all right We've let's see. Let's summarize a little bit. We've got two sets of Normal stresses That we've come across so far one is just the That normal stress we started class with which is just simply If there's some kind of axial force of some kind then that'll cause an internal force in the material acting over a cross-sectional area Which will just be a particular Particular just just an axial load of some kind that causes it. So I guess we can call that axial stresses Then we had the stresses due to bending Which we knew to be linear through across the cross-section With a point of zero stress being at the neutral axis which we also now know goes through the area centroid So we had those bending stresses then we've also looked at some Some shear stresses those came in in two flavors for us one was the Shear stress due to the torsion remember that the twisting of These pieces or these shafts or whatever transmission shafts was the most likely place that we've seen this sort of thing Where that was also a linear function of position across the piece and then we just finished with With the transverse shear stresses these things last week with that so that's that's kind of a quick summary of the things We've we've had so far Now If you if you notice in any of these type of problems we've been doing we only had one of those stresses for the other We only had Axial loading at one time, but we didn't have axial loading and bending going on at the same time a very good example of when that type of thing could happen and we'll look at it very closely later in the term Is if we have a column a long slender column that's loaded that's going to have Clearly an axial load of some kind So we'll see those type of axial stresses We saw in the first very first week of the course But if it's long and slender enough it will also tend to Bend like that and so there's going to be both axial loading and bending Stresses in the in the piece at once So we're going to look at problems like that combined loadings. They're they're relatively straightforward to deal with in that they For small deformations at least they tend to just add together So imagine we have against some simply supported beam of some kind Loaded in a very simply way Simple way some kind of axial loading like this now DJ up you thought or me It could be that there's something about the structure that means it's loaded that way or it could be that Sometimes pieces are put into a structure and they're loaded As they're put in before the structure is even completed sort of pre-stressed And then let's imagine we also have some kind of transverse load We'll put it there and In millimeters We'll position it Right there So this is a very simple type of loading. We've seen these type of loadings before but we've never seen them together The solution of this is is relatively easy in that we can break these two apart Look at just that loading And the stresses it causes we know that if we Look at the type of stress that's caused in that there'll be a uniform Normal stress all the way across the piece just a very type of stuff We looked at it in the first week then we can also add to it what we know about the bending stress Which we looked at in no, I don't know probably the second month of things. I guess we were looking at things what these These loads are there and then at some point we could figure out what the maximum Moment is and we'll just take this cross-section to be a simple Rectangle make things a little bit simpler for now call it 50 by 75 Millimeters we know that that bending stress is linear Being zero through the neutral axis We have compression on the top for the type of Bending or what type of loading that we've shown there So that the maximum bending stress wherever the maximum moment is at the farthest Away from the neutral axis piece of that beam over the Moment of inertia the cross-section nice simple right thing with their cross-section Not too big a deal to look at I hope and Then we Without too big a stretch can imagine then that the true loading now is the combination of the two of those Neutral axis is there. So we've got this this oops. Hang on here. Sorry about that drew these arrows going the wrong way Because for the bending we've got we've got an axial loading that puts everything in tension and then a Bending that puts the top in compression and the bottom in tension So we want the arrows going the same way as the tension for that So sorry about that so this This a little bit of the compression will take up some of the Tension on the top all the way through and then we get a cross-section that might look Or I mean a loading that might look something like this depending upon how much of The compression along the top is worth the Tension that's caused by the axial load Well, we can figure out all of those So it's not too too big a deal. I hope we've been going over a lot of this stuff for for some time for example that normal axial stress we know to be What the 25 kilo Newtons? Acting over that simple rectangular cross-section and so we can figure out what that stress is it's a simply Megapascals Right, that's that's the long been in our playbook that one there We're happy to see that. That's like a zero friend So the this one it's a just a little bit more time We need to figure out what the what the maximum moment is, but your experience. I think should show you that The maximum moment is going to occur right here We've seen that type of thing before You could you could probably quickly come up with that. I bet So we can just put it in here that's going to be 2.7 kilo Newtons at the 375 millimeter point That's where we're going to see the maximum moment the moment will decrease to zero from there C is the greatest dipton from the neutral axis which is half of this 75 It's only five millimeters over To the eye for a rectangular cross-section at 1 12 the base which is 50 You can figure out that remember it's symmetric. So that'll give us the Compression on the top and the tension on the bottom and we can be a little more specific with these cross-sections now So somebody work that out for me. So we've got it these together In a little bit of a scale drawing just to make sure stuff fits see what we end up with So I'm just re-sketching the separate loads here We know that to be 6.67 separately now And they got it so we're in something like 21 point seven megapascals We know that to be linear Zero at the neutral axis. So that's 21 point seven compression at the top So about three times bigger than what I've got there along the bottom. So I have tension going to the right compression going to the left and it's the two of those together that allow us now to Draw our our full profile So the top will be in Compression because we have more compression from the bending than we have tension from the axial load and that's That's what? Oh, I don't have that one What's twenty one seven minus six point seven? Never mind 15 Right, so we've got 15 in compression and the bottom will be the twenty one seven of Detention from the bending and the six point seven the tension from the top which is What twenty eight four, but you see that the profile is all shifted. What was this this was 15? I think 15 megapascals in compression on the top What was it 24? 23 tension on the bottom the max the simple superposition arithmetic loading Arithmetic addition of the two loads who types of loads we had Yeah The old neutral axis was right down the center now there's It's not a Geometric neutral axis, but there is a new place where the stresses are zero. It's been shifted because of the combination of the two loads You could look at it as this tension relieved a little bit of this compression But did add a little bit to the tension below Depending on what the materials are it may or may not make things substantially worse if this was a concrete a Concrete beam Then we served to increase the tension along the bottom where concrete stores Does it is anyway? So this might be one of those cases where you want to consider Rebar across the bottom to hold some of that tension All right, so let's do one together to have some kind of bracket thing here Attached to the wall turned about two points One point right in the middle there one point right down here at the bottom. Maybe we're going to Consider drilling some holes there for some reason we want to double-check what the What the Stresses are there There's two inches from the center of the far side of the bracket And a quarter inch down from the upper part we have a load That is slow everybody know what that means we say there's a three four five so what that is that gives us the You automatically have the tree components that This hypotenuse is some multiple of five which it nicely is that the other components are some multiple of three and some Multiple four by the same scale All right, so what was the rest we need here? Oh the cross section right here rectangular The shear stresses we can find in again the same way by adding them together All right, okay with the picture lacks this will be symmetrically Expose those Two points we can do a free body diagram. That's where we want to know the stresses 50 at three four five so we've got some axial load on this and some bending load and Some shear all caused by this load over here We know Components are two hundred by three fifths by one fifty Internally back here. We must have a An axial load by inspection. It should be pretty easy to tell what that is That's got to be equal to the two hundred Horizontal component there we have Load down so we must have a shear up Again fairly obviously it should be equal to the 150 and Then there's also though a Moment I well we got the two hundred and One and a quarter moment arm and 150 at a two inch moment arm So take a second to figure out what that moment is just to warm you up here Make sure we all get the same thing Because each of those is going to contribute something to these stresses So we need to figure out all those those pieces and then just add them together Check your numbers with each other for the moment. Did I drop then how much these two? Moments are opposite each other Axial load will be the two hundred and figure out what the stress is The normal stress is at point a it will be some tension caused by the Axial load there that P Plus a little bit of tension We know by observation it will be tension because of that moment at Whatever location With respect to the neutral axis that point a happens to be Which is who wants to do this one in our head and take the rest of the day off? Tempting, huh? It's almost like get out of class except you can't leave you have to stay here. It's what? Well, that's what I is What is this? Why zero point a is on the neutral axis? So why is zero remember that's the distance of the point of concern in this case? We have points pre-selected for us so That one's zero and so the P over a is straightforward, what is it? 0.5 Ksi that is tension we can see that from inspection, right? We do the same thing at point B. It's got the same normal Tensile normal stress there But because of the moment being applied we know that's a compression, so we'll give that a minus sign That's typically what we do for compression now. We know what the moment is Y is Pretty easy to pick up from there So we do have to calculate I but it's a rectangular cross-section 112 the base times the height Now do that separately because we are going to need that remember for the shear stress the vq over it So figure out I separately for the cross-section All right figure out what I is keep it separate because we're gonna need it again later anyway and Then figure out what the normal stresses are point B Active with each other Make sure we all have the same eye Hope we do it's just right there Got something for the stress the DJ is not doing either. He's looking for somebody to talk to DJs over here This is the deal Spurn it check with him. He didn't have any units I mean nice There's a There are a couple of Calculational packages math cat where you actually can put the units in it'll check before it's really nice We don't have Yeah, well good We'll talk to a Malcolm about amazing deltas all right what we get there in this case These two are not acting together right the one part here is in Tension this parts of compression we need to know which is dominant. It's what Compression now that would have come through you should have had a minus 1.07. All right, then the the shear stress is Also made up two parts. We have the we have the Oh, sorry just one part because we don't have any origin in here B where this is Q Hey, so we've got already got B. We've already got I nicely T is just from the cross-section 0.75 So we just need to remember what those are we've got At the center and be at the bottom of that cross-section So Q a is y bar a of what area the point of concern away from the neutral axis to the rest of the beam so and we call that that script a so that is the area for which we do y bar a It's times then the area which is the half area of the beam should get something like oh Just by that something. That's the the location of the centroid of the Shaded piece with respect to the neutral axis point one two five Five five point seven five Qb y bar being zero period or zero exclamation mark Is he right a minute to ink? He's not right could be right because you don't need any units when you say zero either if he's right either Y bar Y bar B is zero a B is zero. There's big script a a Because point a was right on the neutral axis. So it was a full half beam. We had to look at B. Where's the shaded area? The big script a for B. It's from that point beyond a way from the neutral axis and There isn't anything beyond So there is no stress no shear stress at point B But there is a point a and now we can figure it out. We've got all the pieces. We've got the shear We've got Q now I We already had and T With the beam at that level thickness of the beam Into the board at that rectangular makes these Agreed Pat you agreed with Oh, did you do this calculation for me? Yeah, thanks. I appreciate that check those numbers. He agreed with somebody Who all right? We all agree. What do we get a small? Element right at each of those places. We can draw the load on those elements So there's a little element at at point a we know that to be in General tension and we know what to have some shears going up on the left side goes down on the right and Remember then The shear all the way around is the same through we calculated there all element at point B It's going to be in compression With no shear so we can draw a element at each of those spots We're gonna be doing that a lot in the next week or two because we're also then going to see what these stresses are We're at some different angle besides just horizontal That's certainly a big deal when you're talking about things like Carbon fiber that can be laid to the particular angle and you can really affect the stresses I Just restate how it was obvious to us that Well, if I have to answer that then it's not off How how how we knew this was compression? Yes, how that well the p over a this the p is The same all the way across the cross section the whole cross section is in axial tension But then because of this loading there's also a bending so at at that cross section we have Tension it also have a Vending such that the top is in tension the bottom is compression So we add to it Change colors. We add to it. I don't have the individual numbers and it goes zero so at the top the two tensions are adding to each other and We know separately what they are well We're at the middle point Our point a is there. That's why this was zero And where point B is we have the tension from the axial load the compression from the bending Which is why we have the minus sign here because this was tension. This was compression and we were left over with some compression And the minus sign would have told you that Or you kept in them separately to say this is so big in tension. This is so weak in compression Whatever's left over is whichever one of those dominates Because a since the moment of zero No at a the moment is not zero We're on the neutral axis. So why is zero and this why is the distance from the neutral axis, right? So I tried to make it clear that why was zero but M is not All right any questions for I get you started on one sign on a post to some wind love so Sign signs like that besides you're the parallel of the ground or parallel of the post I'm is itself 1.2 The email about taking that survey there boy, honey You did oh Thanks a lot. It was anonymous now. I know who did the one response I had no idea who did the one response because it just says anonymous That's like every member once at GE. We were having a Department meeting And that our manager said I want everybody to put all their questions on a card No name. So all the questions are anonymous And they read my question and somebody over there said manning. What jazz that for There was expected pressure on the front of the sign from wind load To kilopascals 2.0 kilopascals wind blow throughout what the compound stresses are At two points one point right there At the base We hopefully we know from our experience, but a lot of these things are going to be worse at the base So I want one on the front of the sign and one at this side of the sign I'll draw that we'll sketch that in because you also need the dimensions of the thing itself supposed to be Circular and that looks too electrical from my perspective See all those little techniques that a hamster gave us are fine Said we gotta work a little quicker than that. So there's there's a cut through the base So we want to see what the stresses are right here and and We can replace this pressure with an equivalent We call it w for the wind force and that's going to be the pressure times the area Do you know where that will act at the center? We'll take this as a uniform Uniformly distributed load. We know we can replace that with a single force acting at the centroid So that'll be dead center of that sign Because of that we're going to see a couple things going on The sign will tend to push the post around which will cause a torsion That we draw with our right hand rule in that direction Right, that's that's tending to twist this post around its base There's also going to be a moment because this force is up in the air trying to bend the whole thing over So with our right hand rule that would be in the minus x direction That's supposed to look like it's laying right along the face there and Then this wind force is also causing a shear Right across the face each of those Contributing to stress the stress loads at each of the places formal force Does a torsion cost what kind of I started normal stress? What kind of normal stress does a torsion cost? It doesn't moment. However a bending moment will cause a Normal stress so that will be my over I from the action of what's going on we understand that that will be Tension causes no normal stresses. So that's all we're going to need for the normal stress at a the Shear stress at a awesome. That's we know to be T roll that's going to be that direction and I or the V Across it is going to be in the opposite direction. Well actually You're less than that you're out. It's going to be equal to the wind load the moment of inertia of the cross-section T the thickness of the cross-section at that point and Q is Q is for this point is Right on the neutral axis with respect to the bending so there'll be no bending moment torsion and shear cause no Normal stresses so at be there'll be no normal stress how it be are the same types of things however in this case Neither are zero to help you out double check. I'll just give you these numbers that will need The wind load which is the pressure times the area is 4.8 kilonewtons. That causes is 31.7 for 4.8. So see what you get for those unless you have to go to work