 Today, I briefly want to talk to you about the coordinate system for a beam. In this course, we will be using the following coordinate system when dealing with beam problems. It will be a right-handed coordinate system with Y defined as positive downwards. Z will be positive in the length direction of the beam, and because of a right-handed coordinate system, that results in X being positive coming in and out of the screen. It's maybe slightly different than the coordinate system you've used in the past in statics. However, it will be the one that we will use consistently in this class for beam bending. If we look at the effect of this coordinate system on the sign convention for external loading, we need to remember that the external load should take the sign of that coordinate system. So positive loading on that beam in terms of point load should be positive downwards. In terms of distributed loads, it should also be positive downwards. And for moments, it's a little bit tricky. You have to remember, because of the right-hand rule, your thumb points in the positive X direction and you would get a counterclockwise moment as being positive for applied moment loadings. If we then look at what it means for deformations, Y is positive downwards, so then our deformation will be positive downwards as well. And a positive slope will be a slope that increases the rate of the deformation in a downwards direction. Now, if we look at the internal loading, this is where it can get a little bit confusing. However, you need to rely on the sign convention we used for stresses. And that was that we looked at the outward normal direction of a surface on which that load or stress applies on. So we'll look at a small element of this beam and if we look at the right-hand face and look at the positive loading, the outward normal is in the positive Z direction. So we get a positive shear force acting in the positive Y because you have a positive outward normal, positive resultant force. And a moment will be counterclockwise, positive outward normal, positive counterclockwise moment. On the left face, it's actually the opposite. Our outward normal is in the negative direction. Therefore, a positive shear force has to be in the negative direction and a positive moment has to be in the negative X direction, because a negative times a negative gives you your positive. Now, you might be a little concerned looking at these coordinate system and internal force definitions and see that that is actually different than the textbook. We're not doing this in order to confuse you. We're actually doing this in order to make sure you guys get familiar with having a consistent coordinate system. And to really explain what I mean here, let's look at the coordinate system in our textbook. It defines X as positive along the length of the beam, Y as positive upwards and because of a right-handed coordinate system, Z is positive in and out of the board. That results in all the loading being positive upwards, because Z is still out of the page and positive moments are still counterclockwise, but your deformations are now positive upwards. So it's very similar, it's just Y is upwards, so all the loading and deformations become positive upwards. For the internal loading, they use the exact same definition we do and this is where things become problematic. This definition of positive internal loading is inconsistent with this coordinate system and we can see that by looking at these two shear force components. If we look at the outward normal on this face, it's in the positive X, but a positive shear force is acting in the negative Y direction. Therefore they've introduced an additional negative sign. So we try not to use this coordinate system. We want to try and teach you a consistent coordinate system. Now you may ask, why does this matter? It just introduces a negative sign, you can fix that on the fly. And it's true, in this course we're doing scalar mathematics. We multiply two numbers together A and B. We can think about the problem a little bit and sort of be like, okay according to our convention it should actually be negative rather than positive and we can sort that out ourselves. The problem is that the principles we teach you in this course will allow you to do more and more complex problems and eventually you're going to have to get to vector mathematics and hopefully utilizing a computer that follows instructions, it can't do this thinking for you. And in vector mathematics, the cross products give different results if you do A times B or B times A. Sorry, A cross B or B cross A. So you have to have a coordinate system and sign convention that are consistent. Otherwise, once you get to doing finite element analysis, you'll spend a lot of your time trying to go through thousands of calculations and figuring out which one is correct and which one is not because you did not use a consistent sign convention.