# Solution to Problem 415 | Shear and Moment Diagrams

**Problem 415**

Cantilever beam loaded as shown in Fig. P-415.

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 415**

**Segment AB:**

$V_{AB} = -20x \, \text{kN}$

$M_{AB} = -20x(x/2)$

$M_{AB} = -10x^2 \, \text{kN}\cdot\text{m}$

**Segment BC:**

$V_{BC} = -20(3)$

$V_{BC} = -60 \, \text{kN}$

$M_{BC} = -20(3)(x - 1.5)$

$M_{BC} = -60(x - 1.5) \, \text{kN}\cdot\text{m}$

**Segment CD:**

$V_{CD} = -20(3) + 40$

$V_{CD} = -20 \, \text{kN}$

$M_{CD} = -20(3)(x - 1.5) + 40(x - 5)$

$M_{CD} = -60(x - 1.5) + 40(x - 5) \, \text{kN}\cdot\text{m}$

**To draw the Shear Diagram**

- V
_{AB}= -20x for segment AB is linear; at x = 0, V = 0; at x = 3 m, V = -60 kN. - V
_{BC}= -60 kN is uniformly distributed along segment BC. - Shear is uniform along segment CD at -20 kN.

**To draw the Moment Diagram**

- M
_{AB}= -10x^{2}for segment AB is second degree curve; at x = 0, M_{AB}= 0; at x = 3 m, M_{AB}= -90 kN·m. - M
_{BC}= -60(x - 1.5) for segment BC is linear; at x = 3 m, MBC = -90 kN·m; at x = 5 m, M_{BC}= -210 kN·m. - M
_{CD}= -60(x - 1.5) + 40(x - 5) for segment CD is also linear; at x = 5 m, M_{CD}= -210 kN·m, at x = 7 m, M_{CD}= -250 kN·m.

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