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Question

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1) Let F_Ra, F_Rb, F_Rc, F_Rd, F_Re, F_Rf be the magnitudes of the equivalent resultant loads for each of the six cases. Rank the magnitudes of the resultant forces for the six cases. (Rank the items from smallest to largest. To rank items as equivalent, overlap them.)

2)When measuring from the left side of the beam x_A, x_B, x_C, x_D, x_E, x_F are the locations where the resultant force is applied in each of the six cases. Rank these six locations. (Rank the items from smallest to largest. To rank items as equivalent, overlap them.)

Explanation / Answer

F_Ra = Wo*L
F_Rb = Wo*L/2

F_Rc = Integrate {Wo(1-x/L)} from 0 to L = WoL/2

F_Rd = Integrate {Wox/L} from 0 to L = WoL/2

F_Re = Integrate {Wo(1-x/2L)} from 0 to L = 3WoL/4

F_Rf = Integrate {Wox/2L} from 0 to L = 3WoL/4

b = c = d < e = f < a [In general, Resultant force = area of shade]

X_a = L/2 (Uniform)

X_b = L/2 (Uniform)

X_c = X coordinate of centroid = L/3

X_d = X coordinate of centroid = 2L/3

X_e = X coordinate of centroid = 4L/9

X_f = X coordinate of centeroid = 5L/9 [last 2 centroid be calculated by breaking the fig into 2 parts, rectangle and triangle and then taking weighted mean]

c < e < a = b < f < d

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