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Fig. 1.5. Mohr's circle of second moments of area.

The procedure is therefore identical to that for determining the direct stress on some plane inclined at a. to the plane on which ax acts in Mohr's stress circle construction, i.e. angles are DOUBLED on Mohr's circle.

1.4. Land's circle of second moments of area

An alternative graphical solution to the Mohr procedure has been developed by Land as follows (Fig. 1.6):

Fig. 1.6. Land's circle of second moments of area.

(1) From O as origin of the given XY axes mark off lengths OA = /„ and AB — Iyy on the vertical axis.

(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 = Iv and DF = Iu 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.

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 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.

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