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Civil
Section moment under a moving axle train
Bending moment at section a of a simply supported span from a set of axle loads P at current positions z, found by superposition of the moment influence ordinate at each axle.
- Ref
- CIV-30
- Rev
- v1
- Field
- Civil
- Result
- Ma [N·m]
Moments from several axle loads are found by superposition: each axle adds Pi × ηM(zi), where the ordinate is evaluated at its position zi. An axle outside the span from 0 to L contributes nothing. The total changes as the train moves, so it is evaluated at each position to find the worst case.
Bending moment at section a of a simply supported span from a set of axle loads P at current positions z, found by superposition of the moment influence ordinate at each axle. Axles outside the span contribute zero. Up to four axles here; leave unused ones at zero load.
Calculator
- Total load on the span30,000 N
- Peak ordinate a(L − a)/L1.875 m
- Moment if all axle loads sat at the section56,250 N·m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The moment at the section is 35 kN·m. It would be 56.25 kN·m if every axle on the span stood at the section, an upper bound.
- Moving the train changes this value, so repeat for several positions; the largest one is the design value.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N·m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| P1 Axle load 1 | 0 to 100,000 N | 22,500 | 147,500 | 125,000 | +10% | +3.57 % |
| P2 Axle load 2 | 0 to 100,000 N | 12,500 | 125,000 | 112,500 | +10% | +6.43 % |
| L Span | 4 to 40 m | 5,000 | 82,630 | 77,630 | +10% | +11.7 % |
| a Section position | 0 to 8 m | 0 | 44,000 | 44,000 | +10% | +4.29 % |
| z2 Axle position 2 | −5 to 15 m | 12,500 | 50,000 | 37,500 | +0.4 m | −8.57 % |
| z1 Axle position 1 | −5 to 15 m | 22,500 | 41,250 | 18,750 | +0.4 m | +7.14 % |
| z3 Axle position 3 | −5 to 15 m | 35,000 | 44,380 | 9,375 | +0.4 m | 0 % |
| P3 Axle load 3 | 0 to 100,000 N | 35,000 | 35,000 | 0 | +10% | 0 % |
| P4 Axle load 4 | 0 to 100,000 N | 35,000 | 35,000 | 0 | +10% | 0 % |
| z4 Axle position 4 | −5 to 15 m | 35,000 | 35,000 | 0 | +0.4 m | 0 % |
Sweep of Section position
| Section position (m) | Section moment (N·m) |
|---|---|
| 0 | 0 |
| 1 | 15,000 |
| 2 | 30,000 |
| 3 | 35,000 |
| 4 | 40,000 |
| 5 | 45,000 |
| 6 | 30,000 |
| 7 | 15,000 |
| 8 | 0 |
Sweep one variable
Variables and how each one contributes
Section moment
What the equation gives you. Everything else feeds it.
Axle load 1
The load of axle 1, in N (zero if absent). Adds its load times the ordinate at its position. Raising it by 10 % (to 11,000) moves the result by +3.57 %, from 35,000 to 36,250 N·m.
Axle load 2
The load of axle 2, in N (zero if absent). Adds its load times the ordinate at its position. Raising it by 10 % (to 22,000) moves the result by +6.43 %, from 35,000 to 37,250 N·m.
Axle load 3
The load of axle 3, in N (zero if absent). Adds its load times the ordinate at its position. Raising it by 10 % (to 5,500) moves the result by +0 %, from 35,000 to 35,000 N·m.
Axle load 4
The load of axle 4, in N (zero if absent). Adds its load times the ordinate at its position. Raising it by 10 % (to 0) moves the result by +0 %, from 35,000 to 35,000 N·m.
Axle position 1
The position of axle 1 from the left support, in m. An axle near the section gives the largest ordinate; outside 0 to L it contributes nothing. Raising it by 10 % (to 2.2) moves the result by +3.57 %, from 35,000 to 36,250 N·m.
Axle position 2
The position of axle 2 from the left support, in m. An axle near the section gives the largest ordinate; outside 0 to L it contributes nothing. Raising it by 10 % (to 5.5) moves the result by −10.7 %, from 35,000 to 31,250 N·m.
Axle position 3
The position of axle 3 from the left support, in m. An axle near the section gives the largest ordinate; outside 0 to L it contributes nothing. Raising it by 10 % (to 9.9) moves the result by +0 %, from 35,000 to 35,000 N·m.
Axle position 4
The position of axle 4 from the left support, in m. An axle near the section gives the largest ordinate; outside 0 to L it contributes nothing. Raising it by 10 % (to 0) moves the result by +0 %, from 35,000 to 35,000 N·m.
Section position
The position of the section, in m. Moves the peak of the influence line and the ordinates at every axle. Raising it by 10 % (to 3.3) moves the result by +4.29 %, from 35,000 to 36,500 N·m.
Span
The span, in m. Scales the ordinates, and decides which axles are on the span. Raising it by 10 % (to 8.8) moves the result by +11.7 %, from 35,000 to 39,090 N·m.
Worked example with the values above
- Ordinates at the axles: 10,000 × 1.25; 20,000 × 1.125; 5,000 × 0 (off span).
- Sum of P × η over the axles on the span: 35,000 N·m.
- Ma = 35,000 N·m.
Assumptions and limits
- A simply supported span with superposition valid, and static loads placed at their stated positions.
- Loads outside the span contribute nothing.
Common mistakes
- Putting the unit load on the wrong side of the section for shear, where the value jumps by one.
- Applying the closed-form maximum outside its stated spacing range.
- Using an influence line for moment to find shear.
Statistics and uncertainty
Influence-line results are exact for a linear elastic simply supported span, but they only give the structural response per unit load. The uncertainty lies in the moving load: its magnitude, spacing, and dynamic effects, which design codes cover with amplification factors of 10 % to 30 %.
Where it is used
- Finding worst positions of vehicles and crane wheels on bridges and runways.
- Building live-load envelopes for floors and decks.
Maths you need
- Influence line: response at one section as a unit load moves along the span
- Superposition: responses to separate loads add
- Resultant: the single force equivalent to a set of loads