A symmetrical planar built-up section consists of two channel sections joined together at the tips of their flanges to form a closed rectangular area. The total width of each flange is ‘B’. Each web is ‘w’ units deep inside, ‘D’ units deep at its outside (longer) face, and ‘b' units thick. The moment of inertia of the composite section about its centroidal axis perpendicular to the webs is given by:

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Answer 1

Concept:

The section is a symmetrical rectangular box created by joining two channels at their flanges.

We calculate the moment of inertia about the centroidal axis perpendicular to the webs (horizontal axis).

Step 1: Moment of inertia of the full outer rectangle:

\( I_{\text{outer}} = \frac{BD^3}{12} \)

Step 2: Moment of inertia of the inner (hollow) void:

\( I_{\text{inner}} = \frac{(B - b)w^3}{12} \)

Step 3: Since the shape is symmetrical and built using two channels, we multiply the difference by 2:

\( I = 2 \left( \frac{BD^3}{12} - \frac{(B - b)w^3}{12} \right) = \frac{BD^3}{6} - \frac{(B - b)w^3}{6} \)