Influence Lines for Determinate Beams

Traditional shear and bending moment diagrams show how internal forces vary along a beam for a fixed set of loads. However, structures like bridges or industrial crane girders must support loads that move across them.

An Influence Line is a graph that shows how a specific function (like a reaction force, shear force, or bending moment at one specific point) varies as a unit load (a load of 1.0) moves across the structure.

Creating Influence Lines

To construct an influence line for a specific point or reaction:

1. Place a dimensionless unit load (magnitude of 1) at a variable position $x$ along the beam.
2. Use the equations of static equilibrium to solve for the function you are tracking (e.g., the reaction at a support $R_A$, or the shear at a point $C$).
3. Plot the magnitude of that function as the unit load moves from $x = 0$ to $x = L$.

For statically determinate beams, the influence line for any reaction, shear, or moment is always composed of straight line segments.

Usage in Design

Once an influence line is drawn, it can be used to determine where to place live loads (such as vehicles) to create the maximum possible effect.

• For a concentrated load $P$, the effect is $P$ multiplied by the ordinate (height) of the influence line at the load's position.
• For a uniform load $w$, the effect is $w$ multiplied by the area under the influence line over the region where the load is applied.

Worked Example

Problem: A simply supported beam of length $L = 10 \text{ m}$ is supported at ends $A$ (left) and $B$ (right). Determine the equation for the influence line of the vertical reaction at support $A$ ($R_A$) as a unit load moves from $x = 0$ (at $A$) to $x = 10$ (at $B$).

Solution:

1. Let a unit load equal to 1 act at a distance $x$ from support $A$.
2. Take the sum of moments about support $B$ to find $R_A$: $\sum M_B = 0$ $R_A (10) - 1(10 - x) = 0$ $10 R_A = 10 - x$ $R_A = 1 - \frac{x}{10}$
3. This is the equation of the influence line for $R_A$.
   • When the load is at $x = 0$ (at $A$), $R_A = 1 - 0 = 1$.
   • When the load is at $x = 10$ (at $B$), $R_A = 1 - 1 = 0$.
   • When the load is at the midpoint $x = 5$, $R_A = 1 - 0.5 = 0.5$.

Engineering Check

Influence lines are strictly for tracking one specific parameter at one specific point as a load moves. Do not confuse an influence line (which represents the effect at a single point due to a moving load) with a shear or moment diagram (which represents the effects at all points due to a stationary load).\n