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:
\bar{y}_1=\frac{18.47\ in.}{2}+\frac{0.5\ in.}{2} = 9.485 in.
\bar{y}_2=0\quad \quad \bar{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 \\ \quad = 3.0 in.² + 20.8 in.² + 8.82 in.² = 32.62 in.²
Q_x=\sum\limits_{i=1}^{n}{\bar{y}_iA_i}=\bar{y}_1A_1+\bar{y}_2A_2+\bar{y}_3A_3 \\ \quad = (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 v to the centroid C of the composite area from Eq. (12-7b):
\bar{y}=\frac{Q_x}{A}=\frac{\sum\limits_{i=1}^{n}{\bar{y}_iA_i}}{\sum\limits_{i=1}^{n}{A_i}} (12-7b)
\bar{y}=\frac{Q_x}{A}=\frac{-58.72\ in.^3}{32.62\ in.^2} = -1.80 in.
Since \bar{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 \bar{c} between the x axis and the centroid C is
\bar{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.