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Civil
Peak moment under two equal moving point loads
Global maximum bending moment in a simply supported span as two equal loads P at spacing s cross it.
- Ref
- CIV-32
- Rev
- v1
- Field
- Civil
- Result
- Mmax [N·m]
For two equal loads P at spacing s, the greatest moment under one load occurs when that load and the resultant of the pair are placed equally either side of midspan. The load then sits s/4 from midspan, giving Mmax = P(2L − s)²/(8L). It exceeds the single-load value PL/4 whenever s is small, and is limited to s ≤ L/2 because beyond that a single load governs.
Global maximum bending moment in a simply supported span as two equal loads P at spacing s cross it. It occurs under one load when that load and the resultant of the pair sit symmetrically either side of midspan. The formula holds only for 0 ≤ s ≤ L/2.
Calculator
- Moment from a single load P at midspan, PL/440,000 N·m
- Moment if both loads acted together at midspan, PL/280,000 N·m
- Offset of the governing load from midspan, s/40.5 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 peak moment is 61.25 kN·m, 76.56 % of the PL/2 that two loads at one point would cause.
- It is 1.53 times the moment from one load alone at midspan.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N·m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Span | 4 to 40 m | 22,500 | 380,300 | 357,800 | +10% | +12.9 % |
| P Each load | 100 to 100,000 N | 306.3 | 306,300 | 305,900 | +10% | +10 % |
| s Load spacing | 0 to 4 m | 45,000 | 80,000 | 35,000 | +10% | −2.84 % |
Sweep of Load spacing
| Load spacing (m) | Peak moment (N·m) |
|---|---|
| 0 | 80,000 |
| 0.5 | 75,080 |
| 1 | 70,310 |
| 1.5 | 65,700 |
| 2 | 61,250 |
| 2.5 | 56,950 |
| 3 | 52,810 |
| 3.5 | 48,830 |
| 4 | 45,000 |
Sweep one variable
Variables and how each one contributes
Peak moment
What the equation gives you. Everything else feeds it.
Each load
The size of each of the two equal loads, in N. Scales the moment in direct proportion. Raising it by 10 % (to 22,000) moves the result by +10 %, from 61,250 to 67,380 N·m.
Span
The span, in m. Raises the moment roughly in proportion to L. Raising it by 10 % (to 8.8) moves the result by +12.9 %, from 61,250 to 69,140 N·m.
Load spacing
The spacing of the two loads, in m (0 ≤ s ≤ L/2). A wider spacing lowers the peak moment a little, since the second load moves toward the support. Raising it by 10 % (to 2.2) moves the result by −2.84 %, from 61,250 to 59,510 N·m.
Worked example with the values above
- With s = 2 m the governing load sits 0.5 m from midspan.
- 2L − s = 16 − 2 = 14; squared: 196.
- Mmax = 20,000 × 196 / (8 × 8) = 61,250 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