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Queensferry

Module 17 · Lesson 17.1

What an influence line actually is

The one diagram people consistently misread — and why.

Try it

Move the load, and watch the influence line draw itself

The graph below is not a bending-moment diagram. Its horizontal axis is where the load IS, and its height is the effect that load produces at one fixed section. Drag the load and the marker traces the whole line.

12m

Effect to track

50% of span
35% of span
The moving load, and the influence line it tracessectionunit loadη 2.100peak 3.000horizontal axis: where the load IS · height: the moment it causes at the sectionload at 4.2 m · ordinate η 2.100 · a load P here gives moment = 2.100 P
Tracking
moment at 6.0 m
Load position
4.20 m
Ordinate η here
2.100
A unit load here gives
2.100 kN·m
Peak ordinate
3.000 at 6.0 m
Net area under the line
18.00
A full UDL of w gives
18.00 w
The moment influence line is a triangle peaking at 3.000 directly under the section. A point load is worst when it stands ON the section; a UDL over the whole span gives the area × w = 18.00w.

Things worth trying

  • Start on the moment at mid-span. Drag the load from left to right and watch the orange marker climb the line. What it traces IS the influence line — the height at each load position is the moment that load makes at mid-span.
  • Stop with the load at mid-span. The ordinate is L/4, its highest. A single point load is always worst when it stands ON the section it is affecting — the peak of a moment influence line is right under the section.
  • This is the sentence to keep: the graph's horizontal axis is where the LOAD is, not position along the beam. It looks like a bending-moment diagram and it answers the opposite question.
  • Switch to shear at the same section. Now the line JUMPS by exactly 1 as the load crosses the section — from a negative branch to a positive one. That discontinuity is real: shear at a cut flips sign depending on which side the load is.
  • Move the section towards a support. The shear peak grows towards 1: a load anywhere on the long side adds almost a full unit of shear at a section near the support. That is why end shear governs.
  • Switch to the left reaction. The line is a simple straight drop from 1 to 0 — a load at the left support is carried entirely by the left reaction, a load at the right support not at all.
  • Read the area row. The net area under the line is the effect of a unit UDL over the whole span. For the mid-span moment it is L²/8 — multiply by w and you have wL²/8, the answer you already knew.
  • For shear at mid-span the area is zero: a full UDL produces no net shear there, because the positive and negative triangles cancel. Load only the positive part and you get the worst case — which is what pattern loading is about.

Why this matters

Bridges, crane runways, gantries and floors carrying forklifts all share a problem: the load moves. The position that produces the largest bending moment at mid-span is not the position that produces the largest shear at a support, and neither is the position that produces the largest reaction. You need a tool that answers 'where should I put the load to make this particular thing as bad as possible' — and that tool is the influence line.

By the end of this lesson you should be able to

  • State precisely what the axes of an influence line mean
  • Distinguish an influence line from a bending moment diagram
  • Construct the influence line for a support reaction
  • Use an influence line ordinate to find the effect of a real load

What you should already know

  • Support reactions (Module 2)
  • Shear force and bending moment diagrams (Module 3)

Be precise about the two axes, because everything follows from them:

  • Horizontal axis: the position of a single unit load as it travels along the structure.
  • Vertical axis: the value of one chosen effect — a particular reaction, or the shear at one particular section, or the moment at one particular section — when the unit load is at that position.

So a point on an influence line reads: when the unit load sits here, the effect I am tracking has this value.

Because the load is a unit load, the ordinate is a per unit load quantity. To get the effect of a real load, multiply. And because the structure is linear, several loads simply add:

effect = Σ (Pi × ordinate at the position of Pi)

Start with the simplest case: the left-hand reaction of a simply supported span of length L, with a unit load at distance x from the left support.

Taking moments about the right-hand support:

RA × L = 1 × (L − x), so RA = (L − x)/L

That is a straight line. At x = 0 the load sits directly over A and RA = 1 — the left support takes everything. At x = L the load sits over B and RA = 0. In between, it falls off linearly.

So the influence line for RA is a straight line from 1.0 above the left support down to 0 above the right support. Simple, and it tells you immediately that to maximise RA you put the load as far left as it will go.

Predict first

You are looking at a diagram for a simply supported beam that is a triangle peaking at mid-span. What extra information do you need before you can say what it means?

A simply supported beam spanning 10 metres with a 60 kN point load positioned 2.5 metres from the left-hand support.60 kN45.00 kN15.00 kN2.5 m10.0 m
A 60 kN load at 2.5 m on a 10 m span. The influence line for the left-hand reaction has an ordinate of 0.75 at that position, so RA = 60 × 0.75 = 45 kN.

Worked example

Using a reaction influence line

Given

  • Simply supported beam, span L = 10.0 m
  • A single 60.0 kN load, free to be positioned anywhere on the span

Find

The influence line for the left-hand reaction, the reaction with the load at 2.5 m, and the position giving the largest reaction.

    Practice

    A simply supported beam spans 10.0 m. What is the ordinate of the influence line for the left-hand reaction when the unit load is 2.50 m from the left support?

    Practice

    A 60.0 kN load is placed at that position. What is the left-hand reaction, in kN?

    Practice

    For a simply supported span of 10.0 m, the influence line for the bending moment at a section 4.00 m from the left support is a triangle. What is its peak ordinate, in metres? (It occurs with the unit load at the section itself, and equals a(L − a)/L.)

    Summary

    • Influence line: fix the effect and the section, move the load
    • Bending moment diagram: fix the load, move the section
    • The ordinate is an effect per unit load, so multiply by the real load
    • Reaction and shear ordinates are dimensionless; moment ordinates have units of length
    • For a simple span, the RA influence line runs linearly from 1 to 0
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    This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint