Calculating Ixy | Plane ABCD, PQRS Symmetry & Moment of Area

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SUMMARY

The moment of area product Ixy is zero for plane ABCD due to its symmetry along the X-axis. This occurs because for every area element dA on one side of the axis, there is a corresponding element on the opposite side, resulting in the integral of Ixy equating to zero. To calculate Ixy for the rectangle PQRS, the parallel axis theorem must be applied, allowing for the determination of the moment of area product based on the rectangle's dimensions and position relative to the centroid.

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Here as shown in the diagram,

(1)Is Ixy zero for plane ABCD symmetrical along X axis? If yes, how it happens?
(2)How can we calculate Ixy for the plane PQRS ?

Ixy = the product of moment of area.

Thanks!
 

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For any plane figure with an axis of symmetry, Ixy = 0 for coordinate axes thru the centroid of the figure.

To confirm this, examine the definition of Ixy = Int(x*y)dA For every element of area dA which lies on one side of the axis of symmetry, there will be a corresponding element of area lying on the opposite side which is the same distance from the axis. When you add up all of these elements of area multiplied by x*y, the integral is necessarily equal to zero.

To calculate the Ixy of rectangle PQRS, apply the parallel axis theorem.
 
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