CHAPTER 1 UNSYMMETRICAL BENDING Summary The second moments of area of a section are given by I=ydA and Iy=dA The product second moment of area of a section is defined as ln =xydA which reduces to Ixy =Ahk for a rectangle of area A and centroid distance h and k from the X and Y axes. The principal second moments of area are the maximum and minimum values for a section and they occur about the principal axes.Product second moments of area about principal axes are zero. With a knowledge of I,Iyy and Ixy for a given section,the principal values may be determined using either Mohr's or Land's circle construction. The following relationships apply between the second moments of area about different axes: Iu=(lx +Iyy)+(Ixx -Iyy)sec 20 =(Ixx+lyy)(lx-Iyy)sec20 where 6 is the angle between the U and X axes,and is given by 2Ixy tan29=1y-1x） Then Iu+Iv=Ia+lyy The second moment of area about the neutral axis is given by IN.A.=(u+)+(u-I)cos2au where o is the angle between the neutral axis(N.A.)and the U axis. Also Ix Iu Cos20+I sin20 Iyy =Iv cos20+Iu sin20 1xw=l。-lu)sin29 Ix-Iyy=(Iu-I)cos 20 1CHAPTER 1 UNSYMMETRICAL BENDING Summary The second moments of area of a section are given by I, = 1 y2 dA and I,, = 1 x2 dA The product second moment of area of a section is defined as I,, = xydA which reduces to I,, = Ahk for a rectangle of area A and centroid distance h and k from the X and Y axes. The principal second moments of area are the maximum and minimum values for a section and they occur about the principal axes. Product second moments of area about principal axes are zero. With a knowledge of I,, I,, and I,, for a given section, the principal values may be determined using either Mohr’s or Land’s circle construction. The following relationships apply between the second moments of area about different axes: s I, = ;(I,, +I,,) + ;(I= - 1,,)sec28 I, = ;(I,, + I,,) - ;(I= - I,,)sec20 where 0 is the angle between the U and X axes, and is given by Then I, + I, = I.r, + I,, The second moment of area about the neutral axis is given by IN.^,. = ;(I, + I,) + 4 (I, - I,) COS 2a, where u, is the angle between the neutral axis (N.A.) and the U axis. Also I, = I, cos2 8 + I, sin2 8 I,, = I, cos2 8 + I, sin2 0 I,, = ;(I~ - 1,)sin20 I, - I,, = (I, - I,>) cos 28 1