(A.1) Locate the centroid C and (A.2) determine the moment of inertia of the cross-sectional area of ​​the T-beam shown in the figure around the centroid axis x'.

00:00 Identifying the Centroid Location of each Rectangle
✅ The first thing to do to calculate the ycg is to divide the cross section into rectangles and locate their centroid (ycg') in relation to the x axis.

02:00 Calculation of the Centroid of the Cross Section
✅ In possession of the results, it is now easy, just replace the values ​​of the centroids and areas of the rectangles in the ycg formula.
ycg = Σycg'.A'/ΣA

03:25 Calculation of the Moment of Inertia with Respect to the x' Axis
✅ To determine the moment of inertia with respect to the x' axis, we must apply the following formula:
I = 'I' + A.d², where:
I' = moment of inertia of the rectangle (b.h^3/12) in relation to its centroid.
A = area of ​​the rectangle.
d = distance between the centroid of the cross section and the centroid of the rectangle.

Since we know which formula to apply, now just replace the values ​​in the moment of inertia formula.
NOTE: remember that the base "b" of the rectangle analyzed is the one whose dimension is parallel to the x' axis, since we are calculating the moment of inertia in relation to this axis.

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📚 Source: Mechanics of Materials 7th Ed. (R.C. Hibbeler)

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