Solution to Problem 409 | Shear and Moment Diagrams

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Problem 409
Cantilever beam loaded as shown in Fig. P-409.

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Write shear and moment equations for the beams in the following problems. In each problem, let x be the distance measured from left end of the beam. Also, draw shear and moment diagrams, specifying values at all change of loading positions and at points of zero shear. Neglect the mass of the beam in each problem.

Solution 409

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Segment AB:
$V_{AB} = -w_ox$

$M_{AB} = -w_ox(x/2)$

$MAB = -\frac{1}{2}w_ox^2$

Segment BC:
$V_{BC} = -w_o(L/2)$

$V_{BC} = -\frac{1}{2} w_oL$

$M_{BC} = -w_o(L/2)(x - L/4)$

$M_{BC} = -\frac{1}{2} w_oLx + \frac{1}{8} w_oL^2$

To draw the Shear Diagram:

V_AB = -w_ox for segment AB is linear; at x = 0, V_AB = 0; at x = L/2, V_AB = -½w_oL.
At BC, the shear is uniformly distributed by -½w_oL.

To draw the Moment Diagram:

M_AB = -½w_ox² is a second degree curve; at x = 0, M_AB = 0; at x = L/2, M_AB = -1/8 w_oL².
M_BC = -½w_oLx + 1/8 w_oL² is a second degree; at x = L/2, M_BC = -1/8 w_oL²; at x = L, M_BC = -3/8 w_oL².

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