Loading Influence Lines

Once an influence line has been constructed, it can be used to evaluate the response to real moving loads. A single influence line can therefore be reused for many load positions and combinations.

Concentrated loads

For loads $P_i$ located over influence-line ordinates $y_i$,

$$S=\sum_i P_i y_i.$$

The ordinates are algebraic. Loads acting over a negative portion reduce a positive response or produce a negative response.

Distributed load

For an intensity $q(x)$,

$$S=\int q(x)y(x)\,dx.$$

If $q$ is constant over the loaded interval, $S=qA$, where $A$ is the signed area of the loaded part of the influence line.

Critical load position

To maximize a positive response, a freely placeable distributed load is applied over positive influence-line regions; to maximize a negative response, it is applied over negative regions. For a moving train of concentrated loads, the critical position depends on both load magnitudes and influence-line geometry.

Loads with fixed spacing

When a group of loads moves as one system, their mutual spacing remains fixed. Evaluate characteristic positions, especially when one of the loads crosses a peak or a point where the influence-line slope changes.

Sign check

Before summing numerically, identify which loads act on positive and negative regions. This gives an immediate expectation for the sign of the result and helps detect ordinate-selection errors.