Exercise 11

1. z x out of page y Exercise 11 Imagine that a beam having a cross section made up of a number of rectangles,. If we know the location and dimensions of each of the rectangles, the y and z co-ordinates of the centroid (centre of gravity) of the cross section can be computed using yc = ((y1 * A1) + (y2 * A2) + ………. (yn * An)) / (A1 + A2 + …..An) zc = ((z1 * A1) + (z2 * A2) + ……...…(zn * An)) / (A1 + A2 + …..An) where yc and zc are the y and z co-ordinates of the centroid, yi is the y co-ordinate of the centre of the ith rectangle zi is the z co-ordinate of the centre of the ith rectangle Ai is the area of the ith rectangle Once the location of the centroid has been determined, the moments of inertia about the centroidal axes (the z and y axes passing through the centroid) can be computed using IY = ((y_size1* z_size13) / 12 + A1 * (zc – z1)2) + ((y_size2* z_size23) / 12 + A2 * (zc – z2)2) + and so on …… IZ = ((z_size1* y_size13) / 12 + A1 * (yc – y1)2) + ((z_size2* y_size23) / 12 + A2 * (yc – y2)2) + and so on …… where y_sizei is the y dimension of the ith rectangle z_sizei is the z dimension of the ith rectangle and other symbols are as before

2. Finally, there is the produce moment of area. It can be computed using IYZ = (A1 * (y1 – yc) * (z1 – zc)) + (A2 * (y2 –yc) * (z2 – zc)) + and so on ….. where all symbols are as before IY and IZ will always be positive, but IYZ can be either positive or negative. Units are not important provided that they’re consistent. If all inputs are in mm, the centroid location will come out in mm, and IY, IZ, and IYZ will be in mm4. Similarly, if all inputs are in inches, the centroid location will come out in inches, and IY, IZ, and IYZ will be in inch4. Calculating centroid locations and moments by hand is a pain and subject to error (as you have possibly already discovered in ECOR 1101, and as Mechanicals will certainly discover in subsequent courses). Getting a computer to do our work for us is a much better idea. We want a program that will read in the dimensions and centres of each of the rectangles, perform the necessary calculations, and output the results. The details of how the program is to accept its inputs don’t really matter, as this part of the program has been written for you. All you have to do is to write the calculation function. You should be able to work out exactly what is required by examining the code supplied. Extra fun (optional): Try writing a function which determines whether or not any of the rectangles entered overlap. If any do, invalid values have been entered.