[SOUND] Hi, this is module 26 of Mechanics of Materials I. Today's learning outcome is to describe a procedure for finding the principal stresses and principal planes on a 3D state of stress by solving the eigenvalue problem. And so, we looked at the 3D state of stress early in the course. Remember, for an arbitrarily loaded member, you may have complicated stress distributions, and so your stresses may not be uniform on arbitrary planes. But if we shrink down and look at an infinitesimally small point, that stress distribution does approach uniformity. And we also said that you could pass an infinite number of planes through that point to find the stresses on the different planes. We're going to show again, today, that really all we need is three mutually perpendicular planes to completely describe the state of stress at a point. And hence, we're going to use a cube to represent the state of stress at a point. And it's shown here. So, here's my 3D state of stress shown in a positive sign convention. Remember by equilibrium, that these sheer stresses are equal, and the stress is represented in a matrix as a tensor. And as I said before, a tensor represents a physical or geometric property or quantity by a mathematical idealization of an array of numbers. And you can look back to my earlier course, advanced engineering dynamics of 3D motion, module 20. I discussed tensors there. And so, I can put my tensor or my state of stress in a matrix. Remember, these off diagonal terms are equal by equilibrium. And here's the matrix notation. What I'd like you to do now is to go back and review modules 24, 25, and 26 of my advanced course on three dimension dynamic motion. And in those series of modules, we looked at something called the inertia matrix, which was also a tensor. Representing physical quantities with an array of numbers. And so these are completely mathematically analogous. And if you recall back and you look back at these earlier modules, you noticed we found that for a particular coordinate orientation, the products of inertia would vanish. And we arrived at what we called the principle moments of inertia with respect to principle axes. And that matrix looked like this. Now, we can do the same thing with our matrix array of stresses. And so for a given general set of stresses, shown here, there is a particular coordinate orientation where the sheer stresses vanish, and we arrive at what are the principle normal stresses acting on the principle planes. And we call those sigma 1, sigma 2, and sigma 3. Now, we found sigma 1 and sigma 2 for two dimensional plane stress using Mohr's circle. We're going to use this technique to find the the three principle stresses in this module, which could also be used in the plain stress problem, but we're going to use something called the eigenvalue property. And so, for this matrix notation, again, we're going to go to our principle stresses and this is solved via the eigenvalue problem. Again, go back to my 3D course to see what the eigenvalue problem's all about. And the eigenvalues are the principle stresses and we get a minimum, a maximum, and one in-between and those are on my diagonal. And the corresponding eigenvectors are the three sets of directional cosines, which define the normals to the three principle planes where these principle stresses occur. And so I can orient my block in such a manner that I get three principle stresses, no shear stresses on those faces. And so, now we can see from my first slide in this module, that you can orient the block in any direction and completely describe the state of stress at a point for three dimensions using just a cube with three orthogonal faces. And so, just another way of looking at stresses and the state of stress at a point. And I'll see you next time. [SOUND]