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Civil
Reactions to a partial uniformly distributed load
Support reactions of a simply supported span (pin at x = 0, roller at x = L) carrying a downward load of intensity w from x = u to x = v.
- Ref
- CIV-13
- Rev
- v1
- Field
- Civil
- Result
- RA [N]
A partial uniform load acts as a single resultant W = w(v − u) at its centroid x̄ = (u + v)/2. Taking moments about the left support gives RB = Wx̄/L, and vertical equilibrium gives RA = W − RB. The nearer a load is to a support, the more of it that support carries.
Support reactions of a simply supported span (pin at x = 0, roller at x = L) carrying a downward load of intensity w from x = u to x = v. W = w(v − u) is the resultant and x̄ = (u + v)/2 its position. The page returns the left reaction RA; the right reaction RB is in the derived rows. Reactions are positive upwards.
Calculator
- Right reaction RB10,500 N
- Resultant load W18,000 N
- Centroid of the load, x̄3.5 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 left support takes 41.67 % of the 18 kN total, and the right support the remaining 58.33 %.
- The load centroid at 3.5 m is nearer the right support, so that support carries the larger share.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| w Load intensity | 100 to 50,000 N/m | 125 | 62,500 | 62,370 | +10% | +10 % |
| u Load start | 0 to 4.9 m | 105 | 17,500 | 17,390 | +10% | −10.4 % |
| L Span | 5 to 30 m | 5,400 | 15,900 | 10,500 | +10% | +12.7 % |
| v Load end | 2.1 to 6 m | 395 | 8,000 | 7,605 | +10% | +5 % |
Sweep of Load start
| Load start (m) | Left reaction (N) |
|---|---|
| 0 | 17,500 |
| 0.6125 | 14,010 |
| 1.225 | 10,900 |
| 1.838 | 8,163 |
| 2.45 | 5,801 |
| 3.063 | 3,814 |
| 3.675 | 2,203 |
| 4.288 | 966.3 |
| 4.9 | 105 |
Sweep one variable
Variables and how each one contributes
Left reaction
What the equation gives you. Everything else feeds it.
Load intensity
The load intensity, in N/m, acting downward. Both reactions scale in direct proportion. Raising it by 10 % (to 6,600) moves the result by +10 %, from 7,500 to 8,250 N.
Load start
The start of the loaded length, measured from the left support, in m. Moving the start toward the right reduces the total load and shifts the resultant right. Raising it by 10 % (to 2.2) moves the result by −10.4 %, from 7,500 to 6,720 N.
Load end
The end of the loaded length, in m, with u < v ≤ L. Extending the end raises the total load and moves its centroid to the right. Raising it by 10 % (to 5.5) moves the result by +5 %, from 7,500 to 7,875 N.
Span
The span between supports, in m. A longer span makes the load act closer to the left support in relative terms, so RA rises. Raising it by 10 % (to 6.6) moves the result by +12.7 %, from 7,500 to 8,455 N.
Worked example with the values above
- Resultant: W = w(v − u) = 6,000 × 3 = 18,000 N at x̄ = 3.5 m.
- Right reaction: RB = Wx̄/L = 10,500 N.
- RA = W − RB = 7,500 N.
Assumptions and limits
- Statically determinate beam on two supports with ideal pin and roller conditions.
- Loads are static and vertical, and the beam weight is included in the load or neglected.
Common mistakes
- Dropping the sign of a reaction that is negative, which means uplift, and not providing a hold-down.
- Measuring the load position from the wrong support.
- Using a total load where the formula needs intensity per unit length, or the reverse.
Statistics and uncertainty
Reactions and moments follow from equilibrium alone, so they are exact for the idealised loading. The real uncertainty is in the loads: a variable load of ±20 % or a mis-measured overhang length passes straight through, and a 5 % error in a lever arm gives a 5 % error in the reaction.
Where it is used
- Finding support reactions before drawing shear and bending moment diagrams.
- Checking hold-downs, bearings and cantilever roots.
Maths you need
- Static equilibrium: ΣF = 0 and ΣM = 0
- Shear force: the vertical force on one side of a section
- Bending moment: the moment of forces on one side of a section