 In the previous video we derived the following two equations for calculating the normal and shear stress at any orientation at a given point in a state of plane stress, the plane stress transformation equations. Now I told you not to really memorize these or worry about these equations too much because we were going to develop a graphical method for predicting these stresses and that's what we're going to develop in this video. If we look closer at these two equations and the plot of these equations for the simple example from the previous video you might actually begin to recognize that they are parametric equations of a circle. They describe the x and y coordinates of a circle as you rotate around with a given angle. This is a little bit easier to see if we rearrange these equations to eliminate theta which takes a little bit of math which we'll skip over for now but you can refer to your textbook to see the derivation if you'd like. If we rearrange to eliminate theta we get the following equation where we can start to see that sigma x prime and tau x prime y prime are kind of like the x and y coordinates of a circle. If I overlay the general equation for a circle below we see that relationship. Now if we take this equation for a circle we can make a few more simplifications to it. We can recognize that everything on the right hand side here is the r or the radius of the circle squared so we can define the radius as a square root of all the terms on the right hand side. We can also define what we will call the average normal stress and that is just the normal stress in the original x direction plus the normal stress in the original y direction divided by 2. If we take these simplifications and substitute them in we get the following equation for more circle that sigma x prime minus sigma average all squared plus tau x prime y prime squared equals r squared. This gives us an equation where the x axis is normal stress in the x prime direction and the y axis is tau x prime y prime and it gives us a circle of radius r that is centered at the average normal stress. Now each point along the radius or around the circumference of this circle represents a stress state with a given normal stress and a given shear stress and as we rotate around the circle it's the same as rotating in physical space but it is twice the angle in the circle as it is in physical space. Now in our previous illustration of the more circle you may have noticed that for our shear stress we don't have a positive or negative. We have indicated cw for clockwise and ccw for counterclockwise sort of indicating a type of rotation. Now why are we doing this? Well it's because we have a rotating frame of reference in our more circle and this rotating frame of reference can cause a flip in the sign of the shear stress. To illustrate this let's look at a pure shear stress state tau in our xy frame. Now if we look at our sign convention for shear stress we know that this is positive because the outward normal is in the positive x direction and the resultant is in the positive y so this would be a positive shear stress. Now if we rotate the coordinate frame by 90 degrees and have the same stress state now with x prime upwards and y prime to the left we see that outward normal is positive but our resultant is now in the negative y prime direction. So rotating the frame of reference by 90 degrees caused a change in the shear stress sign. Now I find this can create a little bit of confusion therefore instead of plotting positive or negative I look at clockwise and counterclockwise and the convention for this is if I'm looking at a specific face if that shear stress acting on that face would tend to create a clockwise rotation about the center of the element then it's plotted in the upper half or the clockwise half of the more circle. Whereas if the shear stress as shown here on the right would produce a counterclockwise rotation it would be plotted in the lower half. So this actually also illustrates an issue we know that the sign of all these stresses in the system is the same however these two faces will be plotted in two different halves of the more circle. So we use this clockwise counterclockwise convention just to keep straight that it's not the sign of the stress in a global coordinate system it's really this rotating coordinate system that we're looking at we're looking at this face in this coordinate system in this face in this coordinate system. Now let's take a look at the procedure of how to construct a more circle. In order to construct a more circle we actually need a known stress state so we will assume that we have a two normal stress components sigma x and sigma y and a shear stress component tau x y. Now the first step for drawing the more circle is to produce our axes to draw it on. The axes consist of our normal stress axis which will be in the horizontal direction with the positive being in the right or towards the right and our vertical axis will be the shear stress axis clockwise rotation in the upper half and counterclockwise rotation in the lower half. We will then locate the center of our more circle and it will be located on our normal stress axis shifted by a distance sigma average. Now we need to locate one point we have the center but we should locate one point on the more circle and then we can draw it so I will pick this right hand face where we have tau x y which produces a counterclockwise rotation about the center of this point so I will draw a horizontal line tau x y going across. Now I need a normal stress coordinate and that will be sigma x so I'll take sigma x draw that line down its intersection is point A which corresponds to this face. I can now draw the radius and with the radius point and center I can actually draw my more circle. However we also know another point on the more circle and I will call this face B and it is tau x y that produces a clockwise rotation and sigma y coming up and that produces point B on my circle. Now that we've looked at how to construct a more circle let's take a look at how we can extract stresses at given orientations. So we already have our original stress state which defines line A B on our more circle which gives us our stress state at face A and at face B. Now what we want to do is rotate that by an angle theta which will give us face A prime and face B prime. Now we can draw this on our more circle by recalling that a rotation angle theta in physical space represents a rotation angle of 2 theta on our more circle which will give us the following line for A prime B prime. We can then extract our shear and normal stress for face A prime and our shear and normal stress for face B prime and in such a way becomes very easy and very useful to utilize more circle to extract those. Now you might think you have to accurately draw more circle in order to extract those stresses but in reality you can sketch the circle and use trigonometry so the rotation angles and similar triangles to actually calculate it. So drawing the more circle just gives you a simple sketch to represent it but you can utilize the properties of trigonometry to calculate the values more accurately. So let's summarize what we saw in more circle and stress transformations. We saw that plane stresses can be transformed to another rotated coordinate frame and all possible stresses in a rotated coordinate frame can be described by the equation of a circle or by more circle. So this is the equation we will be using in this course that we want you to memorize and in use not those original plane stress transformation equations they're a bit more complicated and more cumbersome. We also saw that there were two orthogonal planes that existed where shear stress is zero so that's where the more circle intersects the normal stress axis. The two normal stresses on this axis are known as the principal stresses and they will become very important in predicting material failure later on in the course. We also saw that there was a plane of maximum shear stress that exists and it occurs 45 degrees from the principal stress plane and that is the top of the more circle and in fact the value of this maximum shear stress is precisely the radius of the more circle. We will utilize these maximum shear stresses and maximum principal stresses later on to look at failure of failure criteria for different materials and structures but for now practice extracting these values and constructing more circles for different stress states.