Influence Lines for Shear and Bending Moment

Influence lines for shear force $V$ and bending moment $M$ are constructed for one fixed beam section. Let section $C$ be located a distance $a$ from the left support and $b=L-a$ from the right support.

Bending-moment influence line

As a unit load moves across a simply supported beam, the bending moment at $C$ varies linearly on either side of the section. The ordinate directly beneath $C$ is

$$y_C=\frac{ab}{L}.$$

The ordinates at both supports are zero, giving a triangular influence line for $M_C$.

Shear influence line

The influence line for $V_C$ has a discontinuity at section $C$. As the unit load crosses the section, the shear value changes by one unit, producing two linear branches separated by a unit jump.

Origin of the jump

Shear at a section depends on which side of the cut contains the moving force. At the instant the unit load passes through $C$, it transfers from one isolated free body to the other, which produces the unit discontinuity.

Use with moving loads

To calculate $M_C$ or $V_C$ from a real moving load system, multiply each load by the corresponding influence-line ordinate and sum the products. Large loads placed near large positive moment ordinates tend to produce large positive bending moments at the section.