Question 12.2: Draw the shear and bending-moment diagrams for a 507 cantile...
Draw the shear and bending-moment diagrams for a cantilever beam AB of span L supporting a uniformly distributed load ω (Fig. 12.9).
We cut the beam at a point C between A and B and draw the freebody diagram of AC (Fig. 12.10a), directing V and M as indicated in Fig.12.6a. Denoting by x the distance from A to C and replacing the distributed load over AC by its resultant ωx applied at the midpoint of AC, we write
+↑\sum{F_{y}} = 0 : -ωx – V = 0 V = -ωx
+\curvearrowleft \sum{M_{C}} = 0: ωx \left( \frac{x}{2}\right) + M = 0 M = -\frac{1}{2} ωx^{2}
We note that the shear diagram is represented by an oblique straight line (Fig. 12.10b) and the bending-moment diagram by a parabola (Fig. 12.10c).
The maximum values of V and M both occur at B, where we have
V_{B} = -ωL M_{B} = -\frac{1}{2}ωL^{2}
