Question D.5: Determine the moment of inertia Ic with respect to the horiz......

Find the moment of inertia I_c with respect to axis C–C by applying the parallel-axis theorem to each individual part of the composite area.
The area divides naturally into three parts: (1) the cover plate, (2) the wide-flange section, and (3) the channel section. The following areas and centroidal distances were obtained previously in Example D-2:
\quad\quad\quad\quad A_{1}=37.5\mathrm{cm}^{2}\quad A_{2}=178\mathrm{cm}^{2}\quad A_{3}=75.8\mathrm{cm}^{2} \\ \quad\quad\quad\quad \overline{y}_{1}=227.5~\mathrm{{mm}}\quad\mathrm{~}\overline{y}_{2}=0\quad\mathrm{~\overline{y}_{3}=246{~\mathrm{mm}}\quad\mathrm{~}\overline{c}=34.73\,m m}
The moments of inertia of the three parts with respect to horizontal axes through their own centroids C_1, C_2, ~and~ C_3 are
\quad\quad\quad\quad I_{1}={\frac{bh^{3}}{12}}={\frac{1}{12}}(25 \mathrm{{cm}})(1.5 \mathrm{{cm}})^{3}=7.031\mathrm{{cm}}^{4} \\ \quad\quad\quad\quad  I_2 = 63720 ~cm^4 \quad I_3= 597~cm^4
The moments of inertia I_2 ~ and ~ I_3 are obtained from Tables F-1 and F-3, respectively, of Appendix F.
Now use the parallel-axis theorem to calculate the moments of inertia about axis C–C for each of the three parts of the composite area:
\quad\quad\quad\quad (I_{c})_{1} = I_{1} + A_{1}(\overline{{{y_{1}}}} + \overline{{{c}}} )^{2}=7.031~\mathrm{{cm}}^{4}\, + \,(37.5\,\mathrm{{cm}}^{2})(26.22\,\mathrm{{cm}})^{2} = 25790\,\mathrm{{cm}}^{4} \\\quad\quad\quad\quad (I_{c})_{2} = I_{2} + A_{2}\overline{{{c}}} ^{2}=63720 ~\mathrm{{cm}}^{4}\, + \,(178\,\mathrm{{cm}}^{2})(34.73\,\mathrm{{cm}})^{2} = 65870\,\mathrm{{cm}}^{4} \\\quad\quad\quad\quad  (I_{c})_{3} = I_{3} + A_{3}(\overline{{{y}}}_{3} – \overline{{{c}}})^{2}=597 ~\mathrm{{cm}}^{4}\, + \,(75.8\,\mathrm{{cm}}^{2})(21.13\,\mathrm{{cm}})^{2} = 34430\,\mathrm{{cm}}^{4}
The sum of these individual moments of inertia gives the moment of inertia of the entire cross-sectional area about its centroidal axis C–C:
\quad\quad\quad\quad I_c = (I_c)_1 + (I_c)_2 + (I_c)_3 = 1.261 \times 10^5 ~ cm^4
This example shows how to calculate moments of inertia of composite areas by using the parallel-axis theorem.