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This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Checks needed for project use

  • An independent structural reviewer must check the illustrated models, prediction feedback, geometry and full interactive input ranges.
  • Influence-line superposition assumes the stated linear static model. Required permanent loads cannot be omitted; live-load patch constraints and dynamic amplification remain separate checks. Alternating whole-span loading is not universal for every response or geometry.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

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Module 16 · Course notes

Influence lines: the effect of a moving load

A bridge, a crane rail, a floor under a moving vehicle — the worst case is not one fixed load but a load in its worst position. An influence line answers exactly that: it shows how a single response varies as a unit load moves across the structure, so you can see at a glance where to put the load for the largest effect.

An influence line is easy to confuse with a bending-moment diagram, but it answers the opposite question.

- A bending-moment diagram fixes the load and shows the moment at every point along the structure. - An influence line fixes the response point and shows how that one response changes as a unit load moves across the structure.

So an influence line for the moment at mid-span has the moving load's position along its horizontal axis, and the resulting mid-span moment up its vertical axis.

bending moment (tension side)

a bending-moment diagram (for one fixed load)
This is the moment along the beam for a load in one fixed position. An influence line is different: it would show how the moment at a chosen section changes as the load rolls from one support to the other.

The value of an influence line is that it tells you where to put the load for the worst effect. For a single moving load, place it at the peak of the influence line. For a distributed moving load, cover the parts of the span where the influence line has the sign you want — and leave off the parts of the opposite sign.

On a continuous beam this leads to pattern loading: to maximise the sagging moment in one span, load that span and the alternate spans, and leave the neighbours unloaded — because the influence line for that moment is positive over the loaded spans and negative over the others.

bending moment (tension side)

pattern loading on a continuous beam
Loading only one span of a continuous beam maximises the sagging moment in that span (and lifts the neighbour). Which spans to load follows directly from the sign of the influence line for the response you want.

Worked example

Worked example — the worst span moment

Where should a moving distributed load sit to maximise the sagging moment in one span of a two-span beam?

  1. Step 1 — the wrong instinct

    It is tempting to load both spans fully, thinking more load means more moment. But the influence line for one span's sagging moment is negative over the other span, so loading the neighbour actually reduces it.

    bending moment (tension side)

  2. Step 2 — pattern loading

    Load only the span whose moment you want (and, on longer beams, the alternate spans). Leaving the neighbour clear removes the negative contribution, giving the largest sagging moment in the chosen span. This is why codes require patterned live load, not just full load.

    bending moment (tension side)

In the exercises, tell an influence line from a moment diagram, place moving loads for the worst effect, and reason about pattern loading.

How to read these problems

The three-step method

  1. 1Points of certainty. Match the displacement and rotation that each support prescribes, including any settlement. Infer the displacement direction from the whole structure and load case; it need not be downward at every loaded point.
  2. 2Deflected shape and reaction directions. Sketch the compatible deflected shape. To find a reaction's direction, imagine removing that support: the direction that pushes the structure back to its place is the reaction's sense (it may be a hold-down).
  3. 3Bending moment and contraflexure. Draw the bending-moment diagram on the tension side and check it against the shape: hogging where the curve is convex-up, sagging where convex-down, zero at pins and at every contraflexure.

Rules that must always hold

  • 1.Moment is zero at a genuine moment release and at an unloaded simple end. A continuous member can carry moment over a translational support.
  • 2.A bending-moment diagram crosses the baseline exactly at a point of contraflexure.
  • 3.Under a distributed load the bending-moment diagram is curved; under point loads alone it is straight lines.
  • 4.At a fully fixed support the deflected shape leaves the support with zero rotation (tangent along the member).
  • 5.For an initially straight elastic beam without imposed curvature, zero curvature implies zero bending moment.
  • 6.The moment is drawn on the tension side: sagging below the member, hogging above it.

Now predict for yourself

Influence lines and moving loads exercises

5 exercises on an influence line versus a bending-moment diagram, placing moving loads, pattern loading and Muller-Breslau. Predict, then reveal an explanation.

Start the exercises →

This lesson is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions.