Second Moment of Area - 2

Fig. 5

For the second moment of area with respect to the y-axis, we need an element of area all of which is at the same distance from the y-axis. We choose the small, vertical element of area dA shown in Fig. 5. Approximating this area as a thin rectangle, the area is simply the product of the height times the base

where h is the difference in the value of the y-coordinate at the top and bottom ends of the thin strip

Then the second moment of area with respect to the y-axis is

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Radius of Gyration - k_y

The second moment of area I_y can also be written in terms of a radius of gyration k_y

For the area of Fig. 3, A = 1 (see previous example) and the radius of gyration with respect to the y-axis is

Again, the radius of gyration is different from the centroidal distance x_C = 0.571 which was determined in a previous example.

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Second Moment With Respect to the x-Axis - I_x

Fig. 6

For the second moment of area with respect to the x-axis, we need an element of area all of which is at the same distance from the x-axis. We choose the small, horizontal element of area dA shown in Fig. 6. Approximating this area as a thin rectangle, the area is simply the product of the base times the height

where w is the difference in the value of the x-coordinate at the right and left ends of the thin strip

Then the second moment of area with respect to the x-axis is found by evaluating

However, the integral obtained using this approach is not elementary and I would again like to find some way to use the thin vertical strip of Fig. 5 instead.

Note that all of the thin vertical rectangles in Fig. 5 have one edge along the x-axis. Since it is known that the second moment of area of a rectangle about an axis along one edge is given by bh³/3, we can use this to write the second moment of the thin vertical strip with respect to the x-axis.

So now the second moment of area with respect to the x-axis is given by

Finally, the second moment of area I_x can also be written in terms of a radius of gyration k_x

For the area of Fig. 3, A = 1 (see previous example) and the radius of gyration with respect to the x-axis is

Again, the radius of gyration is different from the centroidal distance y_C = 0.393 which was determined in a previous example.