Welcome back to Mechanics of Materials Part 4. Today's learning outcome is to derive the differential equation for the elastic curve of a beam. Again, Mechanics of Materials is the foundation for all structural and machine design. We start off with an engineering structure, we apply some external loads, this generates internal forces and moments, stresses and strains, and we look at the structural performance. As far as structural performance considerations, we look at the fact that we don't want to have normal stress failure, we don't want to have shear stress failure. In this course, we're also going to look at no excessive deflections or no buckling. So there are numerous examples of requirements for engineering structures to stay within a specified deflection under a given load, and I'd like you to go ahead and research some of those on your own and then come on back. So we're going to look at beam deflection as a start in this course and we'll start with a simply supported beam which has a pin and a roller at the ends, this is a model of that, and we apply moments to each end and we get some deflection or bending. This is exaggerated. It's an exaggerated shape when loaded. Here is a model of the beam with a pin on one side and a roller on the other. If we applied moments we'd get some deflection. I can't bend this enough to show you the deflection with this model. So I have my pool noodle model where if I put moments on each side, you can see how we get the beam deflection. We came up in the last course, my Part 3 course, with a moment-curvature relationship. This is what it was. Kappa was called the curvature and we found that was proportional to the moment, that was also equal to one over the radius of curvature, which is one over Rho. We said that EI was the flexural rigidity or the resistance of the beam to bending for a given curvature. So we made some assumptions. We were operating in the linear elastic region, we were working with pure bending or flexure under constant bending moment, and we have no shear force. We actually find that the beam deflection due to shearing, can be negligible. So here again is our moment-curvature relationship, where one over the radius of curvature is equal to M over EI. The curvature equation, which you can find in any standard calculus textbook, is shown here. Where y is in the transverse direction or the direction of the deflection, and x is in the direction along the beam. We're looking at small deformations, and so dy dx is a small value, and if we square that like we do in the numerator, that's much much less than one since that's a small number squared and we can neglect it. So our curvature, one over the radius of curvature becomes d squared y dx squared. Then we can substitute that into the top equation for the moment-curvature relationship, and we get a differential equation for the elastic curve of a beam shown here. So we have this differential equation for the elastic curve of a beam. If we have an equation for the moment along the beam now, we can find the deflections by integrating twice and using boundary conditions to find the constants of integration. So what we're looking for is why our sign convention will be that the bending moment as positive if we have a smiley face here, this is the same as we've used in my previous courses, our negative is shown on the right. So if it's a positive bending moment, d squared y, dx squared is positive. If it's a negative bending moment, then d squared y, dx squared is negative. So for horizontal beams, I will always select the deflection y as being positive upward for beam deflection problems. We'll get started with looking at that in the next module.
