Module 17 · Lesson 17.1

Moving loads: a member with no critical section

Everything so far has had a worst point you could find and check. A crane girder does not — the worst point depends on where the crane is standing.

Why this matters

Every member in this course so far has had a critical section: a point where the moment is largest, which you find once and then check. A crane girder does not. The load walks along it, and the maximum moment occurs at a position that depends on where the wheels happen to be — a position that is neither mid-span nor obvious. Finding it is a short and rather satisfying piece of mechanics, and getting it wrong by assuming mid-span is not conservative in the way people expect.

By the end of this lesson you should be able to

Say why a moving load has no fixed critical section
Find the absolute maximum moment from a wheel group
State the classical equidistant condition
Say what the dynamic factors represent

What you should already know

Influence lines and moving loads — the Structural Analysis Fundamentals course covers these
Crane actions, vertical and horizontal (Module 2)
Bending resistance and the class-appropriate modulus (Module 6)

The load walks

A gantry crane runs on rails carried by girders. The crane has wheels — typically two per rail, at a fixed spacing — and it can stand anywhere along the span.

So the question "what is the maximum moment?" has two unknowns rather than one: where along the beam, and where is the crane. Both have to be searched.

Two things about the answer surprise people.

It is not at mid-span. For a two-wheel group at 3.5 m spacing on a 12 m span, the maximum occurs 5.125 m from the support — 875 mm off centre. The offset depends on the wheel spacing, and for a three-wheel group it can be somewhere else again.

It is not what a single equivalent load would give. Putting the whole group at mid-span as one load gives 955 kNm; the real answer is 697 kNm. The guess is 37% high — conservative here, but it is conservative by accident rather than by design, and for other configurations the relationship goes the other way.

A moving load has to be searched. Neither the position nor the magnitude can be reasoned to.

Note

The classical result, and why it is worth having

There is a closed-form answer, and it is elegant:

The absolute maximum moment occurs under a wheel, when that wheel and the resultant of the whole group are placed equidistant about mid-span.

The derivation is short. With the group at some position, the reaction is R(L − a)/L where R is the resultant at distance a from the support. The moment under a chosen wheel is that reaction times the distance, less the moment of the wheels to its left — and that second term does not change as the group translates. Differentiating with respect to the group position and setting it to zero gives exactly the equidistant condition.

The library in this course does not use that result. It searches every wheel at every position, which is what a computer should do. But the closed form is then available as a check on the search — and on the worked example the two agree to within a millimetre.

Two independent routes to the same answer is worth more than either alone. That is the same discipline this course used for the T-stub modes and the portal mechanisms.

Try it

Where the maximum actually is

The curve is the largest moment anywhere on the beam as the wheel group walks across it. The peak is what the girder must be designed for — and where it occurs is not mid-span.

Span12.0 m

Wheel spacing3.50 m

Leading wheel159 kN

Trailing wheel159 kN

Resultant of the group: 318 kN
Maximum moment: 696.0 kNm
Occurs at: 5.125 m
Mid-span is at: 6.000 m
Offset from mid-span: 875 mm
Under which wheel: wheel 1
Wheel to resultant: 1750 mm
Classical condition predicts: 875 mm
Whole group on the span: yes
Discrepancy: 0 mm
Same total at mid-span: 954.0 kNm
That guess is high by: 37 %

The absolute maximum moment is 696 kNm, occurring 5.13 m from the left support under wheel 1 — NOT at mid-span. Putting the whole group at mid-span as a single load would have given 954 kNm, which is 37% higher — a moving load has no fixed critical section, and neither the position nor the magnitude can be guessed.

Things worth trying

Start at the defaults — 12 m span, 3.5 m spacing, equal 159 kN wheels. The peak is 697 kNm at 5.125 m: 875 mm off mid-span.
Check the discrepancy row. The numerical search and the classical closed form agree to a millimetre, and they are entirely different calculations.
Compare with the last row: putting the same total at mid-span gives 955 kNm, 37% high. Conservative here — and by accident, not design.
Now bring the wheel spacing down to 0.5 m. The group becomes a single load, the peak moves to mid-span, and the 'guess is high by' figure falls to nearly zero.
Take the spacing out to 8 m on a 12 m span. Only one wheel is now on the span at the critical position, and the readout says the classical condition no longer applies — it assumes the whole group is on. The search is still right; there is just nothing to check it against.
Make the trailing wheel much heavier than the leading one. The critical position moves to the heavy wheel — the group's resultant has shifted and the condition follows it.
Shorten the span to 6 m with 3.5 m spacing. The two wheels barely fit, and the whole character of the answer changes.
Watch the discrepancy stay near zero throughout. That is a check on the search running continuously, which is what makes the numerical answer trustworthy.

Worked example

A 12 m crane gantry girder

Given

12.0 m simply supported gantry girder
Overhead crane: 200 kN self-weight, 300 kN hoist load, hoisting class HC2
Two wheels per rail at 3.5 m spacing
Girder trial section 610 × 229 × 113 UB in S355

Find

The maximum moment, where it occurs, and whether the trial section works.

Assumptions

Dynamic factors are calibrated and unverified here, so the calculation is teachable rather than usable
The crane is assumed central on its rails, so each rail takes half the load
Horizontal surge and the rail's own effects are separate checks not covered here

Practice

A crane has 200 kN self-weight and 300 kN hoist load, with φ₁ = 1.10 and φ₂ = 1.39. What is the dynamic vertical action, in kN?

Practice

That action is shared over two rails with two wheels on each. What is the wheel load, in kN?

Practice

A single 637 kN load at mid-span of a 12 m beam, shared over two rails, gives what moment per rail, in kNm?

Practice

The critical wheel sits 875 mm from mid-span on a 12 m span. Where is it, measured from the left support, in m?

Check yourself

Why is the maximum moment from a moving wheel group not at mid-span?

Summary

A moving load has no fixed critical section — both position and load position must be searched
The maximum occurs UNDER a wheel, never between wheels
Classical condition: that wheel and the resultant equidistant about mid-span
For the worked girder: 697 kNm at 5.125 m, 875 mm off centre
The search and the closed form agreed to the millimetre
The single-load-at-mid-span guess gave 955 kNm — 37% high, and conservative by accident
Dynamic factors are inertial effects, not margins: 637 kN against a static 500
The trial girder passed bending at 0.54, and is not adequate

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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