Problem 3: Uniformly distributed load, cantilever beam...A 5 m long beam is cantilevered and carries a distributed load of 12 kN/m. Draw the shear and bending moment diagrams.

Solution: The free body diagram for the entire beam is:

Now I take a free body diagram of a portion of the beam a distance x from the left end to find the internal force (shear) V and the internal moment M. I show these in their standard positions (shear is positive down and the moment is positive counterclockwise).
 

V = - 12x kN M = (- 12x)(x/2) = - 6x2 kN-m At x = 0, V = 0 kN. At x = 5 m, V = -60 kN. The equation is linear. Note that, at x = 0, M = 0 as it should be.  At 5 m, M has its maximum (absolute) value of 150 kN-m. The equation is parabolic and concave down.

The plots are:

You should notice that the shear is zero at the end and that the externally applied distributed load drives the shear in the same direction; i.e., down (in this case).