8 Mechanics of Materials 2 §15 (2)Draw a circle with OB as diameter and centre C.This is then Land's circle of second moment of area. (3)From point A mark off AD=Ixy parallel with the X axis. (4)Join the centre of the circle C to D,and produce,to cut the circle in E and F.Then ED=I and DF=I are the principal moments of area about the principal axes OV and OU the positions of which are found by joining OE and OF.The principal axes are thus inclined at an angle 0 to the OX and OY axes. 1.5.Rotation of axes:determination of moments of area in terms of the principal values Figure 1.7 shows any plane section having coordinate axes XX and YY and principal axes UU and VV,each passing through the centroid O.Any element of area dA will then have coordinates (x,y)and (u,v),respectively,for the two sets of axes. X Momental ellipse Fig.1.7.The momental ellipse. Now y dsin dA -cosdA+2usinecosdA+sinodA But UU and VV are the principal axes so thatI=fuvdA is zero. Ixx Iu cos20+Iv sin20 (1.10)8 Mechanics of Materials 2 $1.5 (2) Draw a circle with OB as diameter and centre C. This is then Land's circle of second (3) From point A mark off AD = I,, parallel with the X axis. (4) Join the centre of the circle C to D, and produce, to cut the circle in E and F. Then ED = I, and DF = I, are the principal moments of area about the principal axes OV and OU the positions of which are found by joining OE and OF. The principal axes are thus inclined at an angle 8 to the OX and OY axes. moment of area. 15. Rotation of axes: determination of moments of area in terms of the principal values Figure 1.7 shows any plane section having coordinate axes XX and Y Y and principal axes UU and VV, each passing through the centroid 0. Any element of area dA will then have coordinates (x, y) and (u, v), respectively, for the two sets of axes. I" Y Fig. I .7. The momental ellipse. Now 1, = /y2dA = /(vcos8+usin8)2dA = /u2cos28dA+ J~uvsin~cos~dA+ s u2sin28dA But UU and VV are the principal axes so that I,, = SuvdA is zero. .. zXx = I, cos2 8 + Z, sin' e (1.10)