6 Mechanics of Materials 2 §1.3 i.e. Iu =(Ixx +Iyy)+(Ix-Iyy)sec 20 (1.6) Similarly, 1.=da=coso+ysineydA =(Ixx +Iy)-(Itx -Iyy)sec 20 (1. 7 N.B.-Adding the above expressions, Iu+1=Ix+lyy Also from eqn.(1.5), Iu=Ixt cos20+Iyy sin20-Ixy sin 20 =(1 cos 20)/xx+(1-cos 20)lyy-Ixy sin 20 Iu=(Ixx Iyy)+(Ixx -Iyy)cos 20-Ixy sin 20 (1.8) Similarly, I=(Ixx Iy)-(Itz -Iyp)cos 20+Igy sin 20 (1.9) These equations are then identical in form with the complex-stress eqns.(13.8)and(13.9) with I lyy,and Iy replacing ox,oy and txy and Mohr's circle can be drawn to represent I values in exactly the same way as Mohr's stress circle represents stress values. 1.3.Mohr's circle of second moments of area The construction is as follows (Fig.1.5): (1)Set up axes for second moments of area (horizontal)and product second moments of area (vertical). (2)Plot the points A and B represented by (,I)and (Iyy,-I). (3)Join AB and construct a circle with this as diameter.This is then the Mohr's circle (4)Since the principal moments of area are those about the axes with a zero product second moment of area they are given by the points where the circle cuts the horizontal axis. Thus OC and OD are the principal second moments of area I and I. The point A represents values on the X axis and B those for the Y axis.Thus,in order to determine the second moment of area about some other axis,e.g.the N.A.,at some angle a counterclockwise to the X axis,construct a line from G at an angle 2a counterclockwise to GA on the Mohr construction to cut the circle in point N.The horizontal coordinate of N then gives the value of IN.A. EJ.Hearn,Mechanics of Materials 1,Butterworth-Heinemann,1997.6 Mechanics of Materials 2 i.e. 1 1, = TUxx +zYy> + Similarly, - 1,,)sec20 $1.3 (1.6) I, = u2dA = (xcos8+ysin8)2dA JJ 1 = z(zxx + zyy) - ;(L - zyy) sec 28 N.B .-Adding the above expressions, I, +I, = I,, + I,, Also from eqn. (1 S), I, = I, cos2 8 + I,, sin2 8 - I,, sin 20 = (1 + cos B)I, + (1 - cos 20)1,, - I,, sin 28 Z, = ;(z~ +I,,)+ ;(zxx -Z,.~)COS~O-Z~S~~~~ (1.8) Similarly, I,, = ;(zXx + zYy) - ;(zX, - zYy) cos 28 + z, sin 28 (1.9) These equations are then identical in form with the complex-stress eqns. (1 3 .S) and (1 3.9)t with I,, I,,, and I,, replacing a,, oy and txy and Mohr’s circle can be drawn to represent I values in exactly the same way as Mohr’s stress circle represents stress values. 13. Mohr’s circle of second moments of area The construction is as follows (Fig. 1.5): (1) Set up axes for second moments of area (horizontal) and product second moments of (2) Plot the points A and B represented by (I,, I,,) and (I,,, -Ixy). (3) Join AB and construct a circle with this as diameter. This is then the Mohr’s circle. (4) Since the principal moments of area are those about the axes with a zero product second area (vertical). moment of area they are given by the points where the circle cuts the horizontal axis. Thus OC and OD are the principal second moments of area I, and I,. The point A represents values on the X axis and B those for the Y axis. Thus, in order to determine the second moment of area about some other axis, e.g. the N.A., at some angle a! counterclockwise to the X axis, construct a line from G at an angle 2a! counterclockwise to GA on the Mohr construction to cut the circle in point N. The horizontal coordinate of N then gives the value of IN.A. t E.J. Hem, Mechanics ofMuteriuls I, Butterworth-Heinemann, 1997