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Maximum positive section shear from a movable uniform load
Largest positive shear at section a of a simply supported span when a uniform load w can cover any part of the span.
- Ref
- CIV-31
- Rev
- v1
- Field
- Civil
- Result
- V^+max [N]
A movable uniform load w must be placed to maximise the response. For positive shear at section a, the load goes only where the influence line is positive: from the section to the right support, length L − a. The area of that triangle (L − a)²/(2L) times w gives the maximum positive shear. Permanent loads are separate.
Largest positive shear at section a of a simply supported span when a uniform load w can cover any part of the span. The load goes only over the positive part of the shear influence line, from a to L. Permanent loads are not included.
Calculator
- Loaded length, L − a5 m
- Maximum negative shear, −wa²/(2L)−3,375 N
- Total shear range at the section12,750 N
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 largest positive shear is 9.375 kN; the largest negative shear (load on the left of the section) is 3.375 kN in magnitude.
- A section nearer the left support has a larger positive value, up to wL/2 at a = 0, the full simply supported reaction.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Span | 4 to 40 m | 750 | 102,700 | 101,900 | +10% | +22.3 % |
| w Load intensity | 100 to 50,000 N/m | 156.3 | 78,130 | 77,970 | +10% | +10 % |
| a Section position | 0 to 8 m | 0 | 24,000 | 24,000 | +10% | −11.6 % |
Sweep of Section position
| Section position (m) | Maximum positive shear (N) |
|---|---|
| 0 | 24,000 |
| 1 | 18,380 |
| 2 | 13,500 |
| 3 | 9,375 |
| 4 | 6,000 |
| 5 | 3,375 |
| 6 | 1,500 |
| 7 | 375 |
| 8 | 0 |
Sweep one variable
Variables and how each one contributes
Maximum positive shear
What the equation gives you. Everything else feeds it.
Load intensity
The movable uniform load intensity, in N/m. Scales the maximum shear in direct proportion. Raising it by 10 % (to 6,600) moves the result by +10 %, from 9,375 to 10,310 N.
Span
The span, in m. A longer span lowers the influence area per metre of span through the division by 2L, but raises the loaded length. Raising it by 10 % (to 8.8) moves the result by +22.3 %, from 9,375 to 11,470 N.
Section position
The position of the section from the left support, in m. A section closer to the left support has a longer positive region, raising the shear. Raising it by 10 % (to 3.3) moves the result by −11.6 %, from 9,375 to 8,284 N.
Worked example with the values above
- Place the load from the section to the right support: length L − a = 8 − 3 = 5 m.
- Influence area: (L − a)²/(2L) = 25 / 16 = 1.5625 m.
- V⁺max = w × area = 6,000 × 1.5625 = 9,375 N.
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