[MUSIC] Welcome back to Mechanics of Materials Part IV. We're moving right along in the course, we've covered beam curvature. And we just looked at a section of the course on singularity functions for calculating beam deflections. And today, we're going to start beam deflections by a method called the superposition method. And so the learning outcome for today is to determine beam deflections using superposition. And so you'll recall that this is the differential equation for the elastic curve of a beam. And we said that if we have an equation for the moment along the beam, we can find the deflections by integrating twice using the boundary conditions. Or we can now also employ singularity functions and express the moment equation in singularity function form and integrate that way to find the deflections. But now for common beam configurations, when the beam bending remains in the linear elastic region, we can employ superposition techniques to determine the beam deflection from tables. And so if you search on the Internet, just type in beam deflection tables, and you'll find that there's beam deflection tables that'll tell you how much have been will deflect under various loading conditions for a number of different cases. And so then using the superposition method, we can find the result and effect of these several different loading conditions acting on a member at the same time by adding the contributions of each of the loads individually. Remember, there must be a linear relationship between the applied loads, the stresses and the resulting deflections. Recall now, as far as examples of common results that are found in beam deflection tables, we've already done some. And so let's look at those first. Few modules back, we looked at the deflection, or we found the deflection of a beam that was simply supported with a concentrated load at the center. We said that maximum deflection occurs at the center of the beam, and it was equal to PL cubed over 48 EI. Then we also did a problem where we had a simply supported beam with a moment applied to the right end. And we found that the max deflection was as shown,- M sub END L squared over 9 square root of 3 EI. And the location where that occurred was L over the square root of 3, or 0.577 L from the left-hand side. Here are some other common examples from the beam deflection tables. If instead of a point load, if we have a distributed load along a simply supported beam, the deflection is maximum at the center of the beam. And it's equal to 5wL to the 4th over 384 EI, as shown here. Another example that's common from a beam table is a point load at the end of a cantilever beam. In that case, at the end of the cantilever beam, it's deflection is PL cubed over 3EI. And we can also find from the beam deflection tables that the slope at the end of the beam in this loading condition has an angle of theta, which is PL squared over 2EI. Another common example is a distributed load on a cantilever. The mass deflection occurs at the end again. It's wL to the 4th over 8 EI. And the slope at the end is wL cubed over 6EI. And finally, here's an example of cantilever beam with a ramp load. And the max deflection occurs at the end. It's equal to wL to the fourth over 30EI. And the slope is shown as being wL cubed over 24EI. And so those are just a few cases that we will use in this course. For other real world examples that you're trying to do, you can find, again, a lot of common beam deflection loadings and corresponding deflections in slopes for a number of different sources. And the Internet is a good place to start. And so we'll use these in future modules in actually calculating deflections. See you then. [MUSIC]