 Hello everyone. In my last video we talked about what we call the primary stresses. Normal stress due to axial load, shear stress due to the direct application of shear, normal stress due to bending moments, and shear stress due to torsion. So I want to talk a little bit about how we then use these stresses to to apply them to a point of interest and start to analyze the situation of stress for a free body diagram that we're looking at or a rigid body that we're looking at. So typically what we would do is we apply these to what we call a stress element. So this is a theoretical infinitesimally small bit of material to which we apply the stress and use it to better understand what's happening. So just as a bit of interest or a bit of what am I trying to say review we have our primary stresses sigma a tau v sigma b and tau t and we consider these to be our primary stresses. So anytime that we're analyzing a rigid body that we have loads applied to we want to first kind of pick you know where we're looking what's our location of interest and a lot of times we can do this by intuition. We can say well we know you know because the loads applied over here and the distance is the greatest from here to here we can say okay it's probably at the the base right is a common example. Once we know that location and sometimes we have to do some analysis to find that location but once we know where it is we can pick a what we call a stress element which is that little square that represents the stress at that location. So if I go ahead and draw a stress element we typically say it's a square and we typically draw it on some x y axes. So I'll go ahead and just draw in a set of x y axes and these are just arbitrary right arbitrary coordinate system that we decide how it's oriented. We often might orient it along the long axis and then the perpendicular axis for you know circular shafts and things would be a pretty typical way to do it but we pick those more or less arbitrarily and then with these axes defined and our stress element defined we say well there's going to be a stress a normal stress in this direction let's call it sigma x just because it's pointing in the x direction a normal stress in this direction let's call it sigma y for the same reason and then we might have a shear stress which we need two arrows to represent and we might call that tau x y because it's oriented in the x y plane and there's only going to be that one shear stress when we're talking about planar situations but we use two arrows to represent it and you may recall why we use two arrows in this case uh from my my last video where we said that for equilibrium we have to represent the shear stress on basically all four sides of our stress element and therefore to add in the balancing stresses for shear we would have our remaining two arrows and for our sigma x and sigma y we would have our equal and opposite stresses in the opposite directions. Now one thing you might notice right away is that I've written sigma x sigma y and tau x y and nowhere did I write sigma a sigma b tau v and tau t and that's because when we take our primary stresses and put them onto our stress element we need to figure out how they they work in combination so in reality sigma x might be sigma a positive or negative it might be sigma b and it might be sigma a plus sigma b um recall that they can work in conjunction they can sum uh to increase the total stress or they can work against each other so it might be sigma a minus sigma b all depending on which direction these arrows are are actually pointing so some combination of those things and of course this is also the same for sigma y could be any combination of these stresses depending on directions tau x y follows the same basic guiding principle which is that it could be tau v tau t tau v plus or minus tau t so again positive or negative on each of those depending on the direction of everything but they they sum up they can they can somehow be added to each other um and work in conjunction or work against each other now now taking a step back when i drew this stress element remember that i said the orientation was arbitrary i arbitrarily picked the direction of x and y when i apply that to my my um physical rigid body that i'm looking at so that means that i could have picked any other arbitrary orientation and not really you know changed what's actually happening i'm just looking at it differently so that means that to be to be complete we need to actually look at everything we need to look at all possible orientations um of our stress element so i'm going to give myself a little more white space here and because that orientation is arbitrary let's say that i instead drew my stress element like this where i still have x and y coordinate systems but now my stress element is no longer um you know square with those with those two coordinate axes so in this case let's say i need to define then a new coordinate system which is my x direction pointing normal out of that side and to give it a different name i'm going to call it x prime and then the same thing in the y direction y prime and so these are rotated from x and y uh but they're now in line with that rotated stress element that i kind of arbitrarily defined so we can define this new coordinate system by saying well it's rotated off of the original one by some angle theta and in this new coordinate system i have stresses right i have sigma x prime i have sigma y prime and i have tau x prime y prime and these primes again all just indicate that i'm now talking about the new rotated coordinate system and just remember on the back side i would have arrows that would be equal and opposite of all you know four of these arrows that i've currently drawn for my stresses so what happens then well once i've done this i need to remember that my stresses my primary stresses up here still apply and they you know result in these x prime y prime you know everything else but i can go ahead and i've pre put the equations in here for us i can represent my new stresses in the rotated coordinate system which are basically purely by geometry i can say i have sigma x prime and it's equal to some combination of sigma x and sigma y and that angle theta and tau x y all put into one equation i have the same thing for sigma y prime and the same thing for tau x prime y prime so using geometry i get these new stresses and the key the key thing here that we need to know about this is that my sigma x prime sigma y prime tau x prime y prime may be greater than i don't know how to write this may be greater than sigma x sigma y tau x y so if i really want to understand the state of stress in my system i need to know what those maximum values are and some of you are probably already ahead of me and you say well you know we can't possibly analyze every possible angle right i mean you could rotate it all the way around luckily it's symmetric so you'd only have to rotate it 180 degrees but that's a lot of equations to calculate but we have a useful tool to this that i'm going to talk about in my next video and you're probably you've probably heard of it before but the tool that we're going to talk about next is mora circle mora circle is really just a graphical representation of what i've just said i have my stress element i can rotate that stress element through any rotation i want and mora circle helps me graphically understand what's happening when i do that so mora circle is a really useful tool for understanding what the maximum state of stress might be in my system and i'm going to talk about that in my next video thanks