Hello and welcome back to this playlist. We're going to talk about how to calculate the moment of inertia about two planar axes in completely asymmetric cross-section and after that we are going to calculate the principle axis of a cross-section. So the example of this video is about one L-shaped cross-section which is 120 mm from the bottom for the first part and then 60 mm and in the horizontal direction 60 mm and then 180. Now assume that bending moment about horizontal axes is applied to this cross-section. First of all we need to find out the center of this cross-section assuming that the cross-section is with a homogenous material and it's not a composite cross-section but because it is completely asymmetric about both axes we need to find out in x and y direction. First of all we need to have basic lines. Let's say this is x and this is y. So based on the basic equation x bar will be sigma a i x i divided by sigma a i and it will be let's assume we have this rectangle rectangle number one and rectangle number two it's easier. So rectangle number one x is 30 mm and y from the base is 90 mm rectangle number two x is 60 plus 90 150 mm and y is 120 plus 30 150 mm. Area of the first section is 180 times 60 a square millimeter the second one is also 180 times 60 a square millimeter. So if you have this information then easily with the equation we can calculate x bar which will be 180 times 60 a square millimeter times 30 millimeter plus 180 times 60 a square millimeter times 150 mm divided by area which is 180 times 60 plus 180 times 60 a square millimeter. The result will be 180 divided by two 90 mm. The same approach for y bar which is sigma a i y i divided by sigma a i and it will be 180 times 60 times 90 a square millimeter plus 180 times 60 times 150 divided by 180 times 60 plus 180 times 60 a square millimeter. So it will be 120 mm. First of all we need to define where the center of this cross section will locate and now we know from the x direction it is going to be 90 mm and in y direction it will be 120. This will be x prime and y prime and this is the center of this cross section. Now we have the center of the cross section we are going to calculate the moment of inertia about these two axis which we can go with the basic equation i x prime will be sigma b h power by 3 divided by 12 plus a times d y power by 2. So the first one will be 60 times 180 power by 3 divided by 12 plus 60 180 times the difference between the centroid of the rectangle one and the centroid of the entire cross section will be the distance in y direction of this point. So it will be 30 millimeter power by 2 and then for the other one it will be 180 times 60 power by 3 divided by 12 plus 180 times 60 times now it will be 30 millimeter power by 2. So all the numbers are in millimeter and this will be millimeter power by 4. I calculated this earlier and it is 5.2 10 power by 7 millimeter power by 4. Now y prime the same concept sigma b h power by 3 divided by 12 plus a d x power by 2. So this time b and h are switched because we are calculating moment of inertia about the other axis then it will be 180 times 60 power by 3 divided by 12 plus 180 times 60. Now distance between the centroid and the centroid of rectangle number one will be 60 which is this distance. Plus the other one will be 60 times 180 power by 3 divided by 12, 180 times 60 times the distance from here to new y prime or the centroid in x direction will be 60 millimeter as well. This value will be 11 times 10 power by 7 millimeter power by 4. If you have a cross section with only one asymmetric axis then polar moment of inertia will be 0. But as far as this is completely asymmetric but the about or around both orthogonal axes then moment of inertia will cross or polar moment of inertia will not be 0. For calculation of that the basic equation is i x prime y prime is sigma i x prime y prime 0 plus a times d x times d y. i x prime y prime 0 represents if you have a cross section that is not asymmetric and you have already polar moment of inertia. For this case as far as rectangle one and two are both rectangles then the polar moment of inertia will be 0 in both cases. So let's start with rectangle number one area is 180 times 60 and then the x here the important part is that in i x prime and i y prime you don't need to think about if it is negative or positive because it is always powered by two. But when it comes to polar moment of inertia you need to consider if it is negative or positive from the centroid of the cross section. For rectangle number one the centroid in x prime direction in the negative side. As a result it will be considered to be minus 60 and in y direction or y prime direction this times is here and this value is 30 millimeter it is also in the bottom of x prime as a result it is also negative. The other one is 180 times 60 for rectangle number two x prime is 60 and y prime is 30 to understand it better let's sketch the section one more time so each of these rectangles are 30 millimeter. Now x bar is in the distance of 90 so it will be here and the other axis is exactly in this line x prime and y prime. So if you look at the rectangle number one center is here and we can see that from the centroid of the entire cross section it is located in minus 60 millimeter and in y prime direction it is minus 30. The other one that we have rectangle number two in x prime direction this is 60 and in y prime direction it is 30. So here we can find out as a result i x prime y prime will be related and it is 3.9 and over by seven it can be negative or it can be positive in this case it is positive. Now we have i x prime i y prime and i x prime y prime what we need to do calculation of the principle axis. In order to find out the principle axis of a cross section for example in this case i x prime y prime is not zero as a result the principle axis of the cross section will be located with an angle rotated from the current x prime and y prime. There are different methods to calculate the angle one method is more circle that is very straightforward if you know what to do. In the next video i'm going to talk about how to calculate the principle axis of such a cross thank you for watching see you in the next video bye