Library / Civil / CIV-16
Civil
Support reactions with a point load on an overhang
Reactions of a beam supported at x = 0 and x = L with a downward point load P at the free end of an overhang of length a (x = L + a).
- Ref
- CIV-16
- Rev
- v1
- Field
- Civil
- Result
- RA [N]
With a downward load P on an overhang of length a beyond support B, moments about B give RA·L = −P·a, so RA = −Pa/L, which points downward. Vertical equilibrium gives RB = P − RA = P(1 + a/L). A negative RA means the beam tries to lift off A, so A must be tied down. The longer the overhang compared with the span, the larger both reactions.
Reactions of a beam supported at x = 0 and x = L with a downward point load P at the free end of an overhang of length a (x = L + a). Moment about B gives RA = −Pa/L, which is downward (a hold-down), and RB = P(1 + a/L). The page returns RA; RB is in the derived rows. Reactions are positive upwards.
Calculator
- Reaction at B, P(1 + a/L)12,000 N
- Amplification of the load at B, 1 + a/L1.333
- Overhang moment at B, P·a18,000 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.
- A hold-down of at least 3 kN is needed at A, and support B carries 12 kN, 1.33 times the applied load.
- The overhang moment P·a = 18 kN·m is the hogging moment at B.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| P Point load at the free end | 100 to 100,000 N | −33,330 | −33.33 | 33,300 | +10% | +10 % |
| L Span between supports | 1 to 30 m | −18,000 | −600 | 17,400 | +10% | −9.09 % |
| a Overhang length | 0.1 to 10 m | −15,000 | −150 | 14,850 | +10% | +10 % |
Sweep of Overhang length
| Overhang length (m) | Reaction at A (N) |
|---|---|
| 0.1 | −150 |
| 1.338 | −2,006 |
| 2.575 | −3,863 |
| 3.813 | −5,719 |
| 5.05 | −7,575 |
| 6.287 | −9,431 |
| 7.525 | −11,290 |
| 8.762 | −13,140 |
| 10 | −15,000 |
Sweep one variable
Variables and how each one contributes
Reaction at A
What the equation gives you. Everything else feeds it.
Point load at the free end
The downward load at the free end, in N. Both reactions scale in direct proportion. Raising it by 10 % (to 9,900) moves the result by +10 %, from −3,000 to −3,300 N.
Overhang length
The overhang length beyond support B, in m. A longer overhang increases the uplift at A and the load on B in proportion to a/L. Raising it by 10 % (to 2.2) moves the result by +10 %, from −3,000 to −3,300 N.
Span between supports
The span between the supports, in m. A longer span lowers the uplift at A in inverse proportion. Raising it by 10 % (to 6.6) moves the result by −9.09 %, from −3,000 to −2,727 N.
Worked example with the values above
- Moments about B: RA × 6 = −9,000 × 2, so RA = −3,000 N.
- Vertical equilibrium: RB = P − RA = 9,000 + 3,000 = 12,000 N.
- RA = −3,000 N, directed downward (uplift).
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