# Question C.1: Locate the centroid of the L-shaped area in Fig. 1....

Locate the centroid of the L-shaped area in Fig. 1.

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A—Addition Method Following the procedure outlined above, we divide the L-shaped area into two rectangles, as shown in Fig. 2.
The y and z axes are located along the outer edges of the area, with the origin at the lower-left corner. Since the composite area consists of only two areas, the composite centroid, C, lies between $C_1$ and $C_2$ on the line joining the two centroids, as illustrated in Fig. 2.

Area: From Eq. (C-4a),

$\begin{array}{c}{{A=\sum_{i}A_{i},~~~Q_{z}=\overline{{{y}}}A=\sum_{i}\overline{{{y}}}_{i}A_{i},~~~Q_{y}=\overline{{{z}}}A=\sum_{i}\overline{{{z_i}}}}A_i}\end{array}$                    (C-4)

$A = A_1 + A_2$ = (6t)(t) + (8t)(t) = 14t²        (1)

Centroid: From Eqs. (C-4b) and (C-4c),

$\bar{y}A = \bar{y}_1A_1 + \bar{y}_2A_2$ = (t/2)(6t²) + 5t(8t²) = 43t²

$\bar{y} = \frac{43t^3}{14t^2} = \frac{43}{14}t$ = 3.07t     (2)

$\bar{z}A = \bar{z}_1A_1 + \bar{z}_2A_2$ = (3t)(6t²) + (t/2)(8t²) = 22t²

$\bar{z} = \frac{22t^3}{14t^2}= \frac{22}{14}t$ = 1.57t      (3)

B—Subtraction Method Sometimes (although not in this particular example) it is easier to solve composite-area problems by treating the area as the net area obtained by subtracting one or more areas from a larger area. Then, in Eqs. (C-4), the $A_i$’s of the removed areas are simply taken as negative areas. This method will now be applied to the L-shaped area in Fig. 1 by treating it as a larger rectangle from which a smaller rectangle is to be subtracted (Fig. 3). Area $A_1$ is the large rectangle PQRS; area $A_2$ is the smaller unshaded rectangle. The composite centroid, C, lies along the line joining the two centroids, $C_1$ and $C_2$, but it does not fall between them.

Area: From Eq. (C-4a),

$A = A_1 + A_2$ = (9t)(6t) + [-(8t)(5t)] = 14t²        (4)

Centroid: From Eqs. (C-4b) and (C-4c),

$\bar{y}A = \bar{y}_1A_1 + \bar{y}_2A_2$ = (4.5t)(54t²) + [(5t)(-40t²)] = 43t²

$\bar{y}$ = 43t³/14t² = 3.07t        (5)

$\bar{z}A = \bar{z}_1A_1 + \bar{z}_2A_2$ = (3t)(54t²) + [(3.5t)(-40t²)] = 22t²

$\bar{z}$ = 2t³/14t² = 1.57t        (6)

Question: C.5

## (a) Draw the Mohr’s circle for the centroidal moments and products of inertia for the L-shaped area in Fig. 1 of Example C-2, given that: ...

(a) Sketch Mohr’s circle and calculate the princip...
Question: C.4

## For the L-shaped area in Fig. 1 of Example C-2, (a) determine the orientation of the centroidal principal axes and show the orientation on a sketch. (b) Determine the principal moments of inertia. ...

(a) From Eq. (C-23), \tan 2θ_p = \frac{- I_...
Question: C.2