[MUSIC] Okay, this is Module 24 of Mechanics of Materials Part III. Today's Learning Outcomes are to find, how to find that first moment of outward area for shear stress. When we have beams subjected to non-uniform bending and we've talked about that last time. We're also going to sketch the shear stress distribution for a cross section and we're going to list the limitations and underlying assumptions that we've made in coming up with the shear stress formulation. And so here is our non-uniform beam, our Non-Uniform Beam Bending. This was a differential portion and we use that to derive the Shear Stress for Non-Uniform Beam Bending. And Q was the first moment of an outward area and I was told you I would discuss how we find that. We'll go through that discussion today and then we'll actually do example that really should cement this concept. And so here's our cross section, doesn't necessarily had to be rectangle. I'm showing a rectangular one, will do some other cross sections as well. I show my Neutral Axis, we want to find the shear stress and the distance Y from the Neutral Axis. And to find the first moment of outward area and instead of doing an integral, we'll generally have standard shapes. And so, we'll just do y-bar, which is going to be the distance from the Neutral Axis of the entire cross-section to the Neutral Axis of the outer area. And A is the outer area, and you can consider the outer area above or below the cut and you get the same answer. And so below the cut, here would be my outer area, and I will call that 1. Y bar would be the distance from the Neutral Axis of the entire cross-section to the Neutral Axis of the outward area below the cut. Same thing, you'd get the same result if you look at the outward areas being up above the cut and taking Y bar from the Neutral Axis to the Neutral Axis of the outward area above the cut. And so either 1 or 2 could be the outer area and I'll actually do an example to show you that that's true later on. Okay, so what we also find is Q is going to be very large at the Neutral Axis where we have a lot of outer area. And so, we get our maximum transfer shear stress generally at the neutral surface or the Neutral Axis and it can be shown that that shear stress distribution Is parabolic. There are some limitations that we've made in doing this development for shear stress. First of all, stress is averaged across the width and so the formula is only accurate if the width B of my beam is not too great. And so here's a cross-section where, let's say d, the depth over the width is greater than 10. If that's the case, the calculation that you'd get for the transverse shear stress is really quite good. If you go to a b where the depth of the b over the width of the beam is approximately 2, you start to get some error, about 3% error in your shear stress. If you go to a square or a cross-section where the depth over the width is equal to 1, or approximately equal to 1, you get about 12 or 13% error. So you can see it gets worse, the larger your b is and in fact if you look at the flange for T-beams or I-beams. The b is greater than d and you really can't use the shear stress formula. But that's okay, because we know that the maximum shear stress that we're going to design against happens near the Neutral Axis, so it's away from the flange, and generally in the web. Okay, some assumptions, we are working with linearly elastic material and small deflections. The reason for that is, if as we want through the development you'll remember that we used the pure bending or the flexural bending assumptions. And another assumption is that the cross section must be parallel to the y-axis you have to have cross sections the outer edges of the cross section must be parallel to y-axis. That means for a cross section that's a triangle or a circle or a semi-circle, this formula is not going to work. And we had mentioned already that they would going to assume uniform shear stress across the width of the cross section. And finally, we're limited to prismatic beams which do not have taper. And so, that's the theory for shear stress, transfer of shear stress in beams subject to non-uniform bending. And we'll go ahead and do some examples starting next time. [SOUND]
