Okay, let's continue our lesson on stresses in beams. So let's make brief digression and look at centroids and moments of an areas which we covered in the statics module previously, section four. But let me review it briefly here. So firstly, centroids, if I have a composite area like this, then it looks like this. But now the coordinate system that we're using here, why is the vertical access or the vertical distance measured from the centroid O and z is the horizontal distance measured to the left here in a right handed coordinate system x, y, z. So it's similar to what we already looked at in the statics except the axes are different. And probably, we will only be concerned with our beam sections which are symmetrical about the y-axis. In other words, symmetrical about this vertical here. So the only thing that we will be concerned about is the height of the centroid above the bottom. And for composite areas we have this formula here. That the location of the composite centroid, for example, this distance here, yc, is equal to the summation of yiAi divided by the total area, where yi is the height of the centroid of area 1 and y2 here is the height of the centroid of the area 2. So for this T shape, the height of the combined centroid is given by this expression and similarly, if we have holes, we can consider a hole as being a negative area. So if we have a rectangle with a circular hole in it as shown here, then the combined centroid of the solid object here at this location here, where this height is yc, is given by this expression, where y1 is equal to the height of the rectangle and y2 is equal to the height of the centroid of the hole. And moments of inertia also we covered previously in the statics module. And here is the section from the handbook, definition of the moment of inertia is the integral of i squared dA, and the other theorem that we might probably use is the parallel axis theorem. Which allows us to calculate the moment of inertia about any arbitrary axis, given the moment of inertia about the centroid and the formula is given here, which we discussed previously. And remember again in that formula that the moments of inertia in that formula are computed about the centroid of the area. And mostly I think we'll be or often we'll be concerned with doubly symmetric shapes and a doubly symmetric shape is a shape which is symmetrical about the vertical of y-axis and also about the z-axis. So each of these examples here are doubly symmetric and obviously for a doubly symmetric shape the centroid is in the middle. So in each of these cases the centroid is in the middle. For this rectangle, circle, and web shape beam. And the most common shapes that we'll encounter are a rectangle, the first one here and the moment of inertia of a rectangle about its centroid is equal to the width. Times the height cubed divided by 12. Or the moment of inertia about the base, about this point here, is equal to the width times the height cubed divided by 3. The moment of inertia of a circle is pi r to the 4th over 4 or in terms of its diameter pi d to the 4th over 64. And these are probably the most common shapes that we will encounter. So this concludes the discussion of the geometrical properties of bending beams.