Solids: Lesson 25 - Shear Moment Diagram, Equation Method...Challenging!

[Music] hey team welcome back it's version 2 of a type of problem that you might find when encountering shear moment diagrams okay and this one is called the equation method the last one that we do the last video that we had was the graphic method and so this is a little bit different students fear this one okay but it's not too bad let me see if I can help you solve this problem okay so we need to draw the shear mode diagram for this thing here okay this weird beam has kind of a trapezoid load or something weird on it okay here's how you do it number one step one guess what find global equilibrium okay fine global equilibrium just like before so what do we have here we have a beam okay we have a roller over here we'll call this a Y we have a pin connection on this side so we have a beam Y and we also have a BX but there are no forces in the X direction so BX he's zero okay now what I'm gonna do is I'm going to take this distributed load and I'm gonna break it into two parts and I'm gonna call it a rectangle with a triangle sitting on top of it right and so if I look at converting that into a concentrated load right the rectangle portion would be here and the triangle portion is going to be over here somewhere right okay because the centroid of a triangle is that two-thirds the base with the centroid of the rectangle is right in the middle so over here on this guy I have this which is the rectangle and how big is the rectangle it's 3 by 3 that's 9 and then over here I have the triangle and how big is the triangle well it's 6 tall 1/2 base times height right 6 times 3 is 18 divided by 2 oh that's 9 also isn't it let's make it the same height okay and so from the middle right from the middle where the triangle starts this distance here 1.5 meters this distance here 2 meters okay right because the rectangle is at the center which is half of the 3 and the triangle is it 2/3 the base which is it too okay so I think I got you here don't I okay some of the moments at a this being positive I'm kind of going fast because this should be an old statics review for us right so here we go we've got negative 9 times 3 plus 1.5 is 4 point 5 minus 9 times 3 plus 2 which is 5 plus B Y rotates me positive so be Y times 6 right okay how much is that well that is a nine times four point five plus nine times five equals eighty five point five and then divided by six how about fourteen point two five and then I'm sorry that's not be wise that's no it is B Y so what does that make a y equal to what does that make this guy equal to up stuff has to equal the down stuff I've got 18 going down so minus 18 makes a y 3.75 all right a y plus B y equals up stuff equals the down stuff okay so let's take and put that stuff that we just found let's put that over here so this guy at be wide is fourteen point two five and then this guy over here at a is three point seven five okay you know what we're ready to graph this thing we got our global equilibrium let's go alright so I'm putting a little deal here okay here comes the V diagram and V is in kilotons this time and if this stuff is confusing to you you go back to the static series and watch this stuff on statics because I have this as their additional problems for you okay so here we go this is the M diagram and that was in kilonewton meters okay so and then we put in our discontinuities I think something interesting is happening right here and I think something interesting is going to happen right there all right what Wow okay so what I'm going to do I'm going to erase this global equilibrium so I can give myself room because I think I'm gonna need it here in just a second okay anyway I got the numbers that we found written on our diagram over here right okay all right here we go so the V diagram is gonna be easy right it's gonna be real easy and it just be the graphic method just like we always done it's super easy okay so here I am letting us get my let's get my blue pin let's do it in blue blue okay so I started off here remember we got our load backpack on the first thing that we're gonna do would jump up 3.75 okay three point seven five there we are now what we start walking oh no change no change no change BAM oh now what okay I know that I have to go down 18 all right I have to go down 18 and so you can do this right in your calculator because that's this was nine that's nine so that a total accumulated load is 18 isn't it so I'm at 3.75 3.75 if I subtract 18 from that right 3.75 minus 18 takes me down to where negative fourteen point two five and isn't that handy dandy okay because you've got a John Denver force over there don't we okay so how do I get from here to there you tell me how do I get there you got it right slow then fast right this is I'm accumulating a load over here short stacks fat stacks so slow then fast so slow then fast looks like this snow then fast okay something like that and then what BAM I'm at zero I mean at negative 14 this is negative fourteen point two five John Denver force boom take me home baby right up to zero right take me home to the place you can you know you know the song you know the song okay so there's my V diagram I know you're like equation method this looks like every other one this is easy right okay when do I use the equation method only one time one case okay and that is this when when the V diagram crosses the axis with a parabolic curve okay the only time one case when the V diagram crosses the the axis the V diagram crosses the axis right there all right with a parabolic curve we got that case don't we then you then you got to use the equation method okay that's the only time every other time you can watch the last video you can just use the graphic method which I love and is nice and easy but this is the one case that gets a little funky okay so what are we gonna do well here's what I think we should do we need to find this point right here right there where that thing crosses the axis all right because now this curve down here I mean all day long I can come I can construct the curve that's easy okay a plus a plus a minus right so this thing is going to go this graph down here is going to go uphill uphill and then downhill right so it's going to go uphill a slope right because the order of the lines right I have a flat so the next graph down is gonna be a slope sloping straight uphill then I'm gonna go uphill a little bit more and how am I going to slope that's parabolic it's gonna be cubic it's the same shape curve how do I get there fast then slow so it's gonna go like this okay and then I'm gonna go downhill slow then fast and I know always go back to zero that's dr. Hansen told me that slow then fast and so there you go boom there's the graph done do we get a hundred for that well probably not okay because what are you as an engineer right if you're gonna have to design that beam what are you worried about you're really worried about what is the bending moment at that point right there okay so I know the shape but I don't know the value that's what I need to find I need an equation for M so I can find that value right there and I'm gonna show you how to do it we go what you do is you're gonna cut this beam we're gonna say hey this is a distance X okay now but what I did in the statics video I did from this side going X was this way but I think it's easier this time if I use X from that side I can do whatever I want cuz I'm the king right okay so what I'm gonna do is I'm gonna cut the beam right there what cha okay I'm gonna cook that beam in half and I'm gonna throw that little Freebody diagram and it looks like this hey this is a distance X okay that's a distance X and what I have going on here I have fourteen point two five I have a distributed load that looks like this I know that the load on this side over here is nine kilobits per meter so I know that nine kilonewtons per meter and then what happens when you cut a beam you must have them mm-hmm that's the one in V and an M right so there's our L min and V are internal forces where we cut that beam in half all right so all you gotta do is a little o statics problem on this but one of the things that I need to know is this how tall how tall is that curve right there okay well I and what I'm gonna do what I'm gonna do is I'm gonna make this my origin okay this is my X Y okay you can pick it anywhere but I think this makes it a little bit easier for me for me okay what's the Y value right here gosh if we could only write an equation for the slope of that line hold on we can can't we y equals okay is that a positive slope or a negative slope what that's a positive slope isn't and the positive slope is rise which is six over run which is 3 so 6 over 3 times X and then what where's the y intercept plus 9 okay plus 9 that's the slope of that line but what's the height over here okay now I made that my origin which made this up here my y-intercept isn't it so what what's the height of y over here well you know what I'm gonna do I'm gonna put a negative x in for X right so the height of the curve over here is gonna be this it's gonna be this Y is equal to I'll put a negative x in there 9 minus 6 over 3 is 2 isn't it 2x okay I'll put a negative in there which makes this thing here negative which there you go there's the negative okay so now I'm all good okay so let's do this what do we have here let's write an equation for V okay so I have two shapes here right I have a rectangle and I have a triangle sitting on top of it okay so how big is the load that the rectangle is this part right here is how big well its base times height there's the base its X there's the height so this guy this area here is 9 X minus 2 x squared wait why I just multiply 2x times that height that's all it is so there's an X there and another X there right well how big is this over here this is 1/2 the base times the height all the height is 9 no the height is not 9 the height is 9 minus that over there 9 minus 2x okay how do I get rid of those forensic now notice what I did I put that - that whole thing right so I'll put it in parentheses so to get rid of the parentheses I'm going to distribute the negative so that one becomes a positive doesn't it okay and what does this simplify to down to nine minus nine that goes away so this up here but turns into oh the twos go away this turns into x squared doesn't it okay that whole area that triangle is x squared well that's pretty cool all right so from this free body diagram let's write the sum of the forces and the Y and see what that gives us here okay so here's what we get we've got a plus V right and remember the V is just in a Sun assumed direction we're using the positive sign convention it might be the other way okay plus B plus fourteen point two five okay and then what else - this bit right because I've got a downward force here and a downward force there so minus the x squared part and then minus this guy minus 9x minus two x squared and I want to put that in parentheses right because it's - that whole part there okay now I can come back and distribute that negative which is going to turn that to a negative and that into a positive which makes V equal to let's move some stuff around right I got to move everything to the other side so x squared when it gets moved to the other side is going to become positive x squared let's see and then minus two x squared right and then this is going to be plus 9x and then this is going to be minus fourteen point two five hey and these first two will simplify and so V is equal to what V is equal to minus x squared plus 9x minus fourteen point two five so there is an equation for V okay and like I said I assumed the direction of V if it was going the other way what would happen well all of these sons would just change that's all okay so now let's do this let's do this sum of the moments about about this point right here about the cut okay I'll call it I'll call it X for where we cut it okay and what do we get well we have ill again ice it's an assumed direction but that's negative so minus M okay and then what I've got this guy right here which rotates me - so that's that that's that's a minus 9x minus 2x squared right times how far away X / - okay and then I've got this guy which is just x squared which also rotates me negative so minus x squared times how far away well this whole thing is X but that's at the centroid of the triangle so it's 2/3 of X okay and then I've got the 14 point 2 5 which rotates me positive so plus fourteen point two five times how far away X okay so can we simplify that any yes let's simplify that some I'm gonna do a little racing here okay I'm gonna come up here I'm gonna erase my one case okay and let's see what that turns into that turns into L is equal to alright let's do a little uh let's do a little distributing here so this is going to be I'm going to leave that negative there so I'm going to do this little inside which would be 9x squared over 2 - what - to the twos cancel out X cubed right okay - X cube and then this turns into what - that guy over there is minus 2/3 X cubed and then this guy over here is just fourteen point two five x okay now I'm gonna get rid of these parentheses by distributing it negative so that becomes negative that becomes positive and let's see what is a 2/3 month 1 X cubed minus 2/3 of an X cubed leaves you with 1/3 X cubed right so M is equal to 1/3 X cubed minus 9x squared over two plus fourteen point two five X okay so there is an equation for M okay now I wonder what this the X is here I wonder what X is you know what V here at some distance X right at some distance X what does V equal the equals zero doesn't it so let's just take this equation here and rewrite that and that equation look like this zero is equal to this one down here was equal to zero two right okay zero was equal to I'm just going to move all this stuff to the other side of the 0 right so x squared minus 9x plus fourteen point two five okay now what does that look like that looks like a quadratic equation to me right and so let's store let's go to we have a quadratic equations over on here second poly solves and so a is going to be 1 B is going to be negative 9 and C is going to be 14 point two five solve and we get X could be equal to two point oh five or let's see what else it another root it gave me a six point nine five okay so one of those is going to be correct one of them is going to be make no since whatsoever well if X is 6.95 where does that put me that puts me like over here off the beam right that cannot be true so this one no good and so the one I want here for X is 2.05 okay so x equals 2.5 now I should be able to put 2.05 in here for X in this equation and what should it give me 0 should give me 0 and it does ok so let's go up here because what I'm after is this value of M right here what is that guy well let's put 2.05 in that equation and let's see what we get okay so here we go clear clear clear clear enter enter enter clear Laurie okay we get to two point zero five cubed divided by three equals and then from that subtract nine times two point zero five squared and then divided by 2 that equals and then from that add to that fourteen point two five times two point zero five equals thirteen point one seven so that's what M is equal to okay when I put two point oh five into that equation I get thirteen point one seven so this value right here thirteen point one seven you know what how could I how can I check that you know what I know for sure is this I know that this area right here from the graphic method right the graphic method tells me that this area is three point seven five times three three point seven five good night three point seven five times three is eleven point two five so that tells me that this guy right here is eleven point two five so how could I check that well I could put three into this equation put three and four and X and X and see what it gives you and you know what it gives you 11.25 okay so that way I can kind of confirm that my equation is right now this is way harder than the word I work back in statics so if this is a brain twister for you rightly so this is quite a quite a challenging little problem so I hope this makes sense this has been the equation method and I'll see you next time

Channel: Jeff Hanson

Views: 71,867

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Id: t0sGTK1oQi0

Length: 24min 45sec (1485 seconds)

Published: Thu Jul 16 2020