Module 17 · Lesson 17.3
Moving loads and load trains
Where do you park the lorry to do the most damage?
Try it
Where do you stand the vehicle to break the beam?
A real vehicle is several axles at fixed spacing. Slide it across and read the effect — then find the worst position, which is almost never where you would first put it.
Effect
Vehicle
- Vehicle
- 60 + 110 + 110 kN
- Total weight
- 280 kN
- Axles on the span
- 3 of 3
- Effect in this position
- 890.5 kN·m
- Worst possible effect
- 958.2 kN·m
- This position vs worst
- 93 %
- Lead axle at the worst
- 11.00 m
- Governing axle
- #2 (110 kN)
Things worth trying
- Start with the three-axle lorry and the mid-span moment. Slide it slowly and watch the effect rise and fall — every axle rides its own point on the influence line, and the total is Σ Pᵢ ηᵢ.
- Try to find the worst position by eye, then read the 'worst possible' row. You will usually be a few percent short: the maximum is sharper and more particular than intuition expects.
- Read the governing-axle row. For a back-loaded vehicle it is NOT the leading axle that matters — it is the heaviest one, and the worst position stands that axle over the peak of the line.
- Switch to the heavy back-loaded vehicle. The worst position shifts so the 140 kN rear axle sits on mid-span, even though that trails the lead by 4.5 m. The lead axle is then well past the section, contributing little.
- Slide the vehicle until 'you are AT the worst position' turns green. Note how narrow that window is — a metre either way and you have lost several percent. This is why bridge assessment searches rather than guesses.
- Switch to mid-span shear. Now the worst position is different again, because the influence line has a jump: you want axles on the tall side of the discontinuity and none on the short side.
- Lengthen the span. A long span fits the whole vehicle comfortably and the worst case uses every axle; a short one can only hold one or two at a time, and the governing case changes character.
- The whole module is in one line: effect = Σ Pᵢ ηᵢ. Draw the influence line once, and every loading question becomes arithmetic — which is exactly why they are worth drawing.
Why this matters
A real vehicle is not one load — it is a set of axles at fixed spacings that cross the structure together. The worst position is not obvious, because moving the train to put one axle at the peak of the influence line necessarily moves the others away from it. This lesson turns that into something you can settle rather than guess.
By the end of this lesson you should be able to
- Find the effect of a group of loads from influence line ordinates
- Locate the position of a load train that maximises a chosen effect
- Use the area under an influence line for a distributed load
- Recognise when the worst case is not what intuition suggests
What you should already know
- Influence lines for shear and moment (this module, lesson 2)
- The Müller-Breslau principle (this module, lesson 2)
With several loads on the span at once, linearity does all the work:
effect = Σ Pi ηi
Each load is multiplied by the ordinate at its own position, and the results are added. Loads that have not yet come onto the span, or have already left it, contribute nothing.
Finding the worst position means choosing the train's position to maximise that sum. For a triangular influence line, the answer always has one axle sitting exactly at the peak — the sum is piecewise linear in the train's position, so its maximum occurs at a break point, and the break points are precisely the positions where an axle sits at the apex.
That narrows an infinite search to a short list: try each axle at the peak in turn, evaluate the sum, and take the largest. The heaviest axle usually wins, but not always — the spacing of the others decides how far down the slope they fall, and with a long enough spacing the comparison can reverse. Checking each case takes a line of arithmetic, so there is no reason to guess.
Predict first
A two-axle vehicle crosses a span. To maximise the mid-span moment, where should it be placed?
Worked example
Worst position of a two-axle vehicle
Given
- Simply supported span L = 12.0 m
- Two axles 3.00 m apart: 120 kN and 80 kN
- Effect of interest: bending moment at mid-span
Find
The position of the vehicle that maximises the mid-span moment, and that moment.
For a distributed moving load — a queue of traffic, a line of stored goods — use the area rule from the last lesson. On a simple span, the mid-span moment influence line is a triangle of base L and height L/4, so its area is:
½ × L × L/4 = L²/8
A UDL of intensity w covering the whole span therefore gives a mid-span moment of wL²/8 — the familiar result, recovered from the influence line without touching a bending moment diagram. That agreement is a useful check that the influence line has been set up correctly.
When the distributed load is shorter than the span, place it over the largest-area portion of the influence line, and integrate only under the loaded length.
Practice
A two-axle vehicle with axle loads of 120 kN and 80 kN spaced 3.00 m apart crosses a simply supported span of 12.0 m. What is the maximum bending moment it can produce at mid-span, in kNm?
Practice
If only the 120 kN axle were on the same 12.0 m span, what is the maximum mid-span moment it could produce, in kNm?
Practice
A uniformly distributed load of 20.0 kN/m covers the whole of the same 12.0 m span. Using the area under the mid-span moment influence line, what is the mid-span moment, in kNm?
Summary
- With several loads: effect = Σ Pi ηi, each load times the ordinate at its own position
- For a triangular influence line the maximum occurs with some axle at the peak — check each in turn
- The heaviest axle at the peak usually wins, but spacing can reverse it
- For a UDL, effect = w × the area under the influence line
- The whole-span area for mid-span moment is L²/8, recovering wL²/8
- Every effect has its own worst position; one load case is never enough
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