4 Mechanics of Materials 2 12 Fig.1.2. where h and k are the distances of the centroid of each rectangle from the X and Y axes respectively(taking account of the normal sign convention for x and y)and A is the area of the rectangle. k- h+ h+ X k- k+ h- h- X Fig.1.3. 1.2.Principal second moments of area The principal axes of a section have been defined in the introduction to this chapter. Second moments of area about these axes are then termed principal values and these may be related to the standard values about the conventional X and Y axes as follows. Consider Fig.1.4 in which GX and GY are any two mutually perpendicular axes inclined at to the principal axes GV and GU.A small element of area A will then have coordinates (u,v)to the principal axes and (x,y)referred to the axes GX and GY.The area will thus have a product second moment of area about the principal axes given by uudA. .'total product second moment of area of a cross-section uvdA (x cos+y sin)(y cos-x sin)dA4 Mechanics of Materials 2 51.2 Y t Fig. 1.2. where h and k are the distances of the centroid of each rectangle from the X and Y axes respectively (taking account of the normal sign convention for x and y) and A is the area of the rectangle. k- kt h- I h￾Fig. 1.3. 1.2. Principal second moments of area The principal axes of a section have been defined in the introduction to this chapter. Second moments of area about these axes are then termed principal values and these may be related to the standard values about the conventional X and Y axes as follows. Consider Fig. 1.4 in which GX and GY are any two mutually perpendicular axes inclined at 8 to the principal axes GV and GU. A small element of area A will then have coordinates (u, v) to the principal axes and (x, y) referred to the axes GX and GY. The area will thus have a product second moment of area about the principal axes given by uvdA. :. total product second moment of area of a cross-section I,, = /"uvdA = S(xcosO+ysin8)(ycos8-xsine)~A