 So in this set of videos, we're going to be building on what we've already done with stresses and soils particularly where we're going to focus on what the stress, the interest in things the stress convention can tell us about properties within soil and then we're going to be connecting that to another type of test for deriving strength parameters and soils, the triaxial test. So imagine I wanted to understand what the stress conditions were in this sheet of plexiglass that's a fun to me. And I picked out an element, let's say a part of this plexiglass sheet, a small element of material. And I wanted to know what the vertical and horizontal stresses were acting on that small element. Well, let's just blow it up here. So in our stress convention, it might look something like this where we'd have normal forces running through the plexiglass sheet. And we also have shear stresses acting on those normal planes. Well, what I've just picked out is really an arbitrary orientation, right? So the reference plane that I've picked out here is really just vertical and horizontal. But that might not be the reference plane for maximum and minimum stresses. You know, I might be more interested in understanding what reference plane, the maximum minimum stresses were happening and that, you know, that might be in, you know, this direction or, you know, this direction. So the whole range of reference planes that we can turn this square through to try and understand what the stresses are within those different reference planes. So, but as long as I knew the normal stresses and the shear stresses acting within one reference plane, and I knew the angle that I wanted to rotate the reference plane through to get a new reference plane, then I could work up what the normal and shear stresses are acting on that new plane. And I put a link to a video which shows you the mathematics behind doing that. But essentially what pops out of those mathematics is an equation for a normal stress acting on a new reference plane. And that's equal to one whole. So what comes out of this proof is an equation for normal stress acting on a new reference plane. And that just depends on the normal stresses acting on the old reference plane, the angle through which you want to rotate the reference plane. So if it's in the vertical and horizontal direction here and I want to rotate that through by theta degrees, that's what goes into here. And then I just need to know the shear stress acting on this old reference plane. And through that information I can work out what any stress is for a given change in the angle of the reference. So the proof also spits out an equation for the shear stress acting on this new reference plane. And that's equal to this. So our shear stress again depends on a new reference plane, depends on the normal stresses acting on the old reference plane, the shear stress acting on the old reference plane and the angle of our new theta, the angle of our new reference plane. So it's just three unknowns and we can get from that the shear stress and the normal stress acting on any reference plane that we choose. And what we can do is plot two graphs where we analyse what the outcome is for a given set of normal stresses and changing the shear stress and changing this angle of rotation. So if I plotted those two equations onto two graphs, a graph of shear stress, and if I kept the normal and shear forces in one reference plane constant and all I changed was the angle, then I would get a graph that looked something like this where I would have a peak and a trough. And I could also do the same for the normal stress where I plotted the normal stress against this angle of rotation. And again I would get a similar graph. So if I took this bottom graph of normal stress and I rotated it until the vertical axis here was horizontal. So it went from a situation like this to something like this, where I now had the normal stress acting on the x-axis and the angle acting on the y-axis. Let's draw that up. With something that looked like this. And if I then plot both of these onto one axis or one set of axes, where I had my normal stress acting on the x-axis and my shear stress acting on the y-axis, what I would be able to generate would be a circle. A circle where the angle around through the circle would be equal to theta. What we're left with is what's known as a Mohr's circle of stress. And I've just moved the y-axis over here because in some mechanics we don't get negative normal stresses. All of our normal stresses are positive, they're compressive. In other engineered materials that's not the case and you do get negative normal stresses, but this is more appropriate for some mechanics. So this tells us quite a few interesting things. The first is if we go back to the two graphs here, depending on what angle of our reference frame is, we get maximum and minimum shear stresses and maximum and minimum normal stresses. So those might be quite important if we're worried about how strong our material is in shear or in normal stress. So if we want to know what the maximum and minimum are, it also tells us that at some point we get points of zero shear stress and zero normal stress. Now those points of zero shear stress also correspond to the points of maximum normal stress and that's how we define what's called the principal stress axis. So the principal stress axis are the axis where we have zero shear stress and maximum or minimum normal stress. So the axis of principal stress correspond in this diagram to these two points here where the circle intercepts the x-axis. So the Mohr's circle of stress tells us something really quite interesting is that the maximum shear stress, which is up here, is a function of the distance between these two points. You can imagine if I reduced the difference between these two points here, reduced the difference between the maximum principal stresses and let's say I reduced this down to here, what I would get is a circle that looked like this. And the maximum shear stress that that material experienced would be much less than what it originally was. So you can see that the difference between these maximum principal stresses are really quite important and that's what gives rise to shear stress within a material. Now we can label these principal stresses up as sigma 3 and sigma 1, they're given that usual notation. Sigma 2 is important when we're thinking about three dimensions. So at the moment we've only got two dimensions on this convention. If we're thinking about in the z-z direction going in and out of the board, then that's what sigma 2 is. It's called the intermediate principal stress. But in this case it's sometimes useful just to think of 2 here where the maximum and the minimum. So we only need two parameters to describe a Mohr's circle. And those two parameters are where the centre point of that circle is and what the radius of the circle is. And those two parameters are called stress invariant. It was specifically two-dimensional stress invariant. And the first one t is equal to the radius of the circle and that's equal to the maximum minus the minimum normal stressors divided by 2. And then the other, the centre point of the circle s is equal to the average stress. So I should point out that actually what we're talking about in all of these situations is effective stress. So that's what we're interested in when we're thinking about soils. So all of this is actually effective stress or sigma prime.