§1.6 Unsymmetrical Bending 9 Similarly, Idn=f(ucoso-vsinoydn cdA-sincdA+sindA and with∫uvdA=0 Iyy Iv cos2+In sin20 (1.11) Also I,=∫xydA=∫-sins8+usin))dA =/[uu(cos20-sin20)+(u2)sin0cos 0ldA =Iuv cos 20+(-I)sin 20 and Iie =0 Ixy=(I-Iu)sin 20 (1.12 From eqns.(1.10)and (1.11) Ixx -Iyy Iu cos20+1 sin20-I cos20-Iu sin20 =(Iu-1,)cos20-(Iw-1)sin20 Ixx -Iyy (Iu -I)cos 26 (1.13) Combining eqns.(1.12)and (1.13)gives 2Ix tan20=Iyy -Ixx (1.14) and combining eqns.(1.10)and (1.11)gives Ix +Iyy=lu+lv (1.15) Substitution into egns.(1.10)and (1.11)then yields Iu =(Ix +I)+(Ixz -Iy)sec 20] (1.16)as(1.6) I=[(I +lyy)-(Itx-Iyy)sec20] (1.17)as(1.7) 1.6.The ellipse of second moments of area The above relationships can be used as the basis for construction of the moment of area ellipse proceeding as follows: (1)Plot the values of I and I.on two mutually perpendicular axes and draw concentric circles with centres at the origin,and radii equal to I and I(Fig.1.8). (2)Plot the point with coordinatesx =l cos and y =I sin,the value of 0 being given by eqn.(1.14).$1.6 Unsymmetrical Bending 9 Similarly, I,, = /x2dA = /(ucos6 - wsinQ2dA = /u2cos26dA - 2uvsin6cos6dA+ and with S uvdA = 0 zYy = I,, cos2 e + I, sin2 e Also I,, = /xydA = /(ucos8- wsin8)(vcos6+usin6)dA (1.11) = J [uw(cos2 8 - sin2 6) + (u2 - w2) sin 6 cos 61 dA (1.12) = I,, cos 26 + :(I, - I,) sin 26 and I,, = 0 .. Zxy = z(Z. 1 - Z,)sin28 From eqns. (1.10) and (1.11) I,, - I,, = I, cos2 6 + I, sin2 8 - I, cos2 6 - I, sin2 6 = (I, - 1,) cos2 0 - (I, - I,) sin2 8 z, - iyy = (I, - 1.1 COS 28 (1.13) Combining eqns. ( 1 .12) and (1 .13) gives (1.14) and combining eqns. (1 .lo) and (1.1 1) gives I, + I,, = I, + I, (1.15) Substitution into eqns. (1.10) and (1.11) then yields 1, = [(zXx + + (zXx - zYy) sec 281 (1.16) as (1.6) 1. = $ [(L + zYy) - (zXx - zYy ) sec 281 (1.17) as (1.7) 1.6. The ellipse of second moments of area The above relationships can be used as the basis for construction of the moment of area ellipse proceeding as follows: (1) Plot the values of I, and I, on two mutually perpendicular axes and draw concentric (2) Plot the point with coordinates x = I, cos 6 and y = I,, sin 6, the value of 6 being given circles with centres at the origin, and radii equal to I, and I, (Fig. 1.8). by eqn. (1.14)