[MUSIC] Hi, this is Module 27 of Mechanics of Materials Part 1. Today's learning outcomes are to Define Stress Concentrations. Describe a principle that we call Saint-Venant's Principle, and then to employ those stress concentration factors to calculate maximum stresses at discontinuities in structural and machine elements. And so, for axially loaded beams, we had assumed that there was average uniform stress over the cross sectional area. When, in fact, discontinuities in the element, like notches, holes, other abrupt changes in the geometry can disrupt the stress path. And so the stresses may be considerably higher in these areas of discontinuity. And we call these areas of higher stress, stress concentrations and they not only appear at holes and notches and grooves, but they also appear at points of loading where high stresses can occur at the point of application and so here's an axially loaded beam. It could have a rectangular cross section, it could have a circular cross section. It's lateral dimensions, if it's rectangular, would be a width w, a thickness t, or if it was round, a diameter d. We said that we had assumed up to this point that the stress was uniform, but it is uniform a distance W or the largest lateral dimension, maybe it's D, for a round member from the concentrated load. And so out here it would be uniform, when we get closer to the concentrated load we would have a stress concentration. This also holds true for holes in our members or grooves or notches, wherever there's a discontinuity. And this general principal that says a distance, the largest lateral dimension away from the notch or the hole or that the point load is uniform. That distance away is called Saint-Venant's Principle. It was name after a French mathematician and it holds true for most linear elastic bodies. And so, the ratio of the maximum stress to the average stress is called the Stress Concentration Factor, K. And we see K here, it's multiplied by our average normal stress. You have to be careful, and you'll see this as we go forward, that the area that we use may be the gross or the net section area. And so let's do an example of an axially loaded bar with a hole. This is one simple example. You can find examples for all kinds of loaded members with discontinuities. Whether it be intention, or excuse me, axially loaded, or torsion, or beam bending the same principle holds true for all those, and you'll be able to find those in mechanics and materials references. So here we have an axially loaded bar, with stresses at the end, we have a hole in the center. And what we see is, when we load this bar is, these are called internal force flow lines, that the internal force flow lines become much closer together around the discontinuity around the hole. And where they're closest together is where we're gonna get the area of highest stress. So again, here's our actually loaded bar. And the stress or the internal force flow lines and if I cut the member now and look at the internal forces and correspondingly the internal stresses we see that the max stress occurs right at the edge of the discontinuity and its equal to the average stress times. A stress concentration factor. Okay, to find that stress concentration factor, for any type of discontinuities and again, you can find references for this. Experimentally, folks have graphed out what these stress concentration factors are based on the dimensions and the type of discontinuity that you're experiencing. This is the stress concentration factor base on the diameter divided by D of the whole divided by the width of the member. Okay, so here we go. Now I said we had to be careful of the area in our stress calculation. K sub G is based on gross area, and gross area is the member without the discontinuity. In this case it would be W times T case of T, which we get off of the graph, is based on the net area. And that's the area at the discontinuity which subtracts out the whole. Based on whether we're working with net area or gross area, we can work back and forth between K sub t and K sub g by this relationship. And this will all make better sense next module as we go through a specific example with numbers, and I'll see you then. [SOUND]