Influence Lines for Beam Support Reactions

For a simply supported beam, support-reaction influence lines follow directly from equilibrium as a unit load moves across the span. Let the span be $L$ and let the unit load be at distance $x$ from the left support $A$.

Left support reaction

Taking moments about the right support gives

$$R_A(x)=\frac{L-x}{L}.$$

The influence line for $R_A$ is therefore a straight line from ordinate $1$ at $A$ to $0$ at the right support $B$.

Right support reaction

Similarly,

$$R_B(x)=\frac{x}{L}.$$

Its influence line rises linearly from $0$ at $A$ to $1$ at $B$.

Equilibrium check

At every position of the unit vertical load,

$$R_A(x)+R_B(x)=1,$$

which follows immediately from vertical force equilibrium.

Real moving loads

For concentrated loads $P_i$, a reaction is $R=\sum P_i y_i$, where $y_i$ are the corresponding influence-line ordinates. For a uniform distributed load, the reaction equals the load intensity multiplied by the signed area of the loaded portion of the influence line.