Question 12.2: The cross section of a steel beam is constructed of a W 18 ×...
The cross section of a steel beam is constructed of a W 18 × 71 wide-flange section with a 6 in. × 1/2 in. cover plate welded to the top flange and a C 10 × 30 channel section welded to the bottom flange (Fig. 12-8).
Locate the centroid C of the cross-sectional area.
Let us denote the areas of the cover plate, the wide-flange section, and the channel section as areas A_{1}, A_{2}, and A_{3}, respectively. The centroids of these three areas are labeled C_{1}, C_{2}, and C_{3}, respectively, in Fig. 12-8. Note that the composite area has an axis of symmetry, and therefore all centroids lie on that axis. The three partial areas are
A_{1}=(6 in.)(0.5 in.)=3.0 in.^{2} A_{2}=20.8 in.^{2} A_{3}=8.82 in.^{2}
in which the areas A_{2} and A_{3} are obtained from Tables E-1 and E-3 of Appendix E.
Let us place the origin of the x and y axes at the centroid C_{2} of the wide-flange section. Then the distances from the x axis to the centroids of the three areas are as follows:
\overline{y}_{1}=\frac{18.47 in.}{2}+\frac{0.5 in.}{2}=9.485 in.
\overline{y}_{2}=0 \overline{y}_{3}=\frac{18.47 in.}{2}+0.649 in.=9.884 in.
in which the pertinent dimensions of the wide-flange and channel sections are obtained from Tables E-1 and E-3.
The area A and first moment Q_{x} of the entire cross section are obtained from Eqs. (12-6a) and (12-6b) as follows:
A=\sum\limits_{i=1}^{n}{A_{i}=A_{1}+A_{2}+A_{3}}
= 3.0 in.² + 20.8 in.² + 8.82 in.² = 32.62 in.²
Q_{x}=\sum\limits_{i=1}^{n}{\overline{y}_{i}A_{i}=\overline{y}_{1}A_{1}+\overline{y}_{2}A_{2}+\overline{y}_{3}A_{3}}
= (9.485 in.)(3.0 in.²) + 0 – (9.884 in.)(8.82 in.²) = -58.72 in.^{3}
Now we can obtain the coordinate \overline{y} to the centroid C of the composite area from Eq. (12-7b):
\overline{y}=\frac{Q_{x}}{A}=\frac{\sum\limits_{i=1}^{n}{\overline{y}_{i}A_{i}}}{\sum\limits_{i=1}^{n}{A_{i}}} (12-7b)
\overline{y}=\frac{Q_{x}}{A}=\frac{-58.72 in.^{3}}{32.62 in.^{2}}=-1.80 in.Since \overline{y} is positive in the positive direction of the y axis, the minus sign means that the centroid C of the composite area is located below the x axis, as shown in Fig. 12-8. Thus, the distance \overline{c} between the x axis and the centroid C is
\overline{c} = 1.80 in.
Note that the position of the reference axis (the x axis) is arbitrary; however, in this example we placed it through the centroid of the wide-flange section because it slightly simplifies the calculations.