Question 12.5: Determine the moment of inertia Ic with respect to the horiz...
Determine the moment of inertia I_{c} with respect to the horizontal axis C-C through the centroid C of the beam cross section shown in Fig. 12-16. (The position of the centroid C was determined previously in Example 12-2 of Section 12.3.)
Note: From beam theory (Chapter 5), we know that axis C-C is the neutral axis for bending of this beam, and therefore the moment of inertia I_{c} must be determined in order to calculate the stresses and deflections of this beam.
We will determine 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 12-2:
A_{1}=3.0 in.^{2} A_{2}=20.8 in.^{2} A_{3}=8.82 in.^{2}
\overline{y}_{1}=9.485 in. \overline{y}_{2}=0 \overline{y}_{3}=9.884 in. \overline{c}=1.80 in.
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 as follows:
I_{1}=\frac{bh^{3}}{12}=\frac{1}{12}(6.0 in.)(0.5 in.)^{3}=0.063 in.^{4}
I_{2}=1170 in.^{4} I_{3}=3.94 in.^{4}
The moments of inertia I_{2} and I_{3} are obtained from Tables E-1 and E-3, respectively, of Appendix E.
Now we can 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:
(I_{c})_{2}=I_{2}+A_{2}\overline{c}^{2}=1170 in.^{4}+(20.8 in.^{2})(1.80 in.)^{2}=1240 in.^{4}
(I_{c})_{3}=I_{3}+A_{3}(\overline{y}_{3}-\overline{c})^{2}=3.94 in.^{4}+(8.82 in.^{2})(8.084 in.)^{2}=580 in.^{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:
I_{c} =(I_{c})_{1}+(I_{c})_{2}+(I_{c})_{3}=2200 in.^{4}This example shows how to calculate moments of inertia of composite areas by using the parallel-axis theorem.