Solution to Problem 410 | Shear and Moment Diagrams

Problem 410
Cantilever beam carrying the uniformly varying load shown in Fig. P-410.

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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 410

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$\dfrac{y}{x} = \dfrac{w_o}{L}$

$y = \dfrac{w_o}{L}x$

$F_x = \frac{1}{2}xy$

$F_x = \dfrac{1}{2}x \left( \dfrac{w_o}{L}x \right)$

$F_x = \dfrac{w_o}{2L}x^2$

Shear equation:
$V = -\dfrac{w_o}{2L}x^2$

Moment equation:
$M = -\frac{1}{3}x F_x = -\dfrac{1}{3}x \left( \dfrac{w_o}{2L}x^2 \right)$

$M = -\dfrac{w_o}{6L}x^3$

To draw the Shear Diagram:

V = - w_o x² / 2L is a second degree curve; at x = 0, V = 0; at x = L, V = -½ w_oL.

To draw the Moment Diagram:

M = - w_o x³ / 6L is a third degree curve; at x = 0, M = 0; at x = L, M = - 1/6 w_oL².

Tags:

shear and moment diagrams

cantilever beam

Triangular Load

Uniformly Varying Load

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