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This equation, therefore, gives the direction of the principal axes. To determine the second moments of area about these axes,

— cos2 9 J y2dA + sm29 J x2 dA - 2 cos 9 sin 9 J xydA = Ixx cos2 9 + Iyy sin2 9 — Ixy sin 29

Substituting for Ixy from eqn. (1.4),

sin2 29

yy 2 cos 29

cos 26

= 5(1 + cos26)1 xx + 5(1 - COS26)1 yy - 5 sec26(1 yy - /„) 4- A cos26(1 yy - Ixx)

= 5(I™ + Iyy> + d** - Iyy)cos 20 ~ (/vy ~ sec 20 + (I w - Ixx) cos 26

= \ (1 + cos 29)1 XX + \ (1 - cos 29)1 yy - Ixy sin 26» Iu = \{Ixx +Iyy) + \dxx ~ Iyy) COS 26 - IXy Sin 26 (1.8)

Similarly,

Iv = \dxx + Iyy) - \dxx - Iyy) COS 26 + Ixy sin 26 (1.9)

These equations are then identical in form with the complex-stress eqns. (13.8) and (13.9)^ with Ixx, Iyy, and Ixy replacing ox, ay and xxy 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 area (vertical).

(2) Plot the points A and B represented by (Ixx, Ixy) and (Iyy, —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 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 Iv and /„. 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 /n.a.

^ E.J. Hearn, Mechanics of Materials 1, Butterworth-Heinemann, 1997.

Similarly,

= \dxx +Iyy)~ \dxx ~ Iyy) Sec 26 N.B.-Adding the above expressions,

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

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