Question 8.3: For the T-shaped section in Figure 8–9, compute the first mo...
For the T-shaped section in Figure 8–9, compute the first moment of the area Q as it would be used in the general shear formula to compute the vertical shearing stress at the section marked a–a at the very top of the web, just below where it joins the flange.
Objective Compute the value of Q.
Given Shape and dimensions of cross section in Figure 8–9.
Analysis Use the method defined in this section.
Results Step 1. Locate the centroid of the entire cross section:
\bar{Y} = \frac{A_{w}y_{w} + A_{f}y_{f}}{A_{w} + A_{f}}
where the subscript w refers to the vertical web
the subscript f refers to the top flange
Then
\bar{Y} = \frac{(38)(200)(100)+(200)(50)(225)}{38(200)+200(50)} = 171.0 mm
Step 2. The axis of interest, a–a, is at the very top of the web, just below the flange.
Step 3. The partial area above a–a is the entire flange.
Step 4. A_{p} = (200 mm) (50 mm) = 10 000 mm²
Step 5. The centroid of A_{p} is 25 mm down from the top of the flange, which is 225 mm above the base of the tee.
Step 6. \bar{y} = 225 mm – \bar{Y} = 225 mm -171 mm = 54.0 mm
Step 7. Q = A_{p} \bar{Y} = (10 000 mm²)(54.0 mm) = 540 000 mm³ = 5.4 \times 10^{5} mm³
Comment It should be noted that the value of Q would be the same if the axis of interest a-a were to be taken at the very bottom of the flange just above the web. But the resulting shearing stresses would be markedly different. The thickness of the section, t, would be equal to the entire width of the flange, whereas for the axis a–a used in this problem, the thickness of the web is used. This will be shown later.