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Shear and Bending Moment Diagrams using the Direct Method

*Example: (continued)

Strategy: Plot the expressions for shear, V(x) and bending moment, M(x) developed in

the sections of the beam from 0 ≤ x ≤ L/2 and from L/2 ≤ x ≤ L. These plots are the

shear diagram and the bending moment diagram. They show the distribution of

shear force and bending moment throughout the length of the beam.

From equilibrium for 0 ≤ x ≤ L/2: V(x) = 3wL/8 – wx and M(x) = 3wLx/8 – wx²/2

From equilibrium for L/2 ≤ x ≤ L: V(x) = – wL/8 and M(x) = – wLx/8 + wL²/8

The maximum shear force from the top figure is 3wL/8.

The maximum bending moment occurs when dM/dx = 0 (@ x = 3L/8) and

from the bottom figure is M_max = (9/128)wL²

The bending moment at midspan (L/2) is wL²/16

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