Library / Civil / CIV-23
Civil
Prop reaction of a cantilever under full-span uniform load
Upward reaction at the prop of a beam clamped at A and simply propped at the free end B under a uniform load w over the whole length.
- Ref
- CIV-23
- Rev
- v1
- Field
- Civil
- Result
- RB [N]
If the prop at B is removed, the cantilever under uniform load w deflects wL⁴/(8EI) at the tip. A prop force R lifts the tip by RL³/(3EI). Compatibility needs the two to be equal, so R = 3wL/8. The remaining 5wL/8 goes to the clamp, which also carries a hogging moment of wL²/8.
Upward reaction at the prop of a beam clamped at A and simply propped at the free end B under a uniform load w over the whole length. Compatibility requires the prop to cancel the free-end deflection of the released cantilever, which gives 3wL/8. The clamp then carries the remaining 5wL/8 vertically. No support settlement.
Calculator
- Clamp reaction, 5wL/820,000 N
- Clamp moment, wL²/816,000 N·m
- Maximum sagging moment, 9wL²/1289,000 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.
- The prop takes 12 kN, which is 37.5 % of the total 32 kN, and the clamp takes the other 62.5 %.
- The clamp moment drops from the free-cantilever value of 64 kN·m to 16 kN·m, a 75 % reduction.
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 100,000 N/m | 150 | 150,000 | 149,900 | +10% | +10 % |
| L Length | 0.5 to 20 m | 1,500 | 60,000 | 58,500 | +10% | +10 % |
Sweep of Length
| Length (m) | Prop reaction (N) |
|---|---|
| 0.5 | 1,500 |
| 2.938 | 8,813 |
| 5.375 | 16,130 |
| 7.813 | 23,440 |
| 10.25 | 30,750 |
| 12.69 | 38,060 |
| 15.13 | 45,380 |
| 17.56 | 52,690 |
| 20 | 60,000 |
Sweep one variable
Variables and how each one contributes
Prop reaction
What the equation gives you. Everything else feeds it.
Load intensity
The uniform load intensity, in N/m. Scales the prop reaction in direct proportion. Raising it by 10 % (to 8,800) moves the result by +10 %, from 12,000 to 13,200 N.
Length
The beam length, in m. Scales the prop reaction in direct proportion, through the total load wL. Raising it by 10 % (to 4.4) moves the result by +10 %, from 12,000 to 13,200 N.
Worked example with the values above
- Free cantilever deflection: wL⁴/(8EI); prop lift: RL³/(3EI).
- Compatibility: R = 3wL/8 = 3 × 8,000 × 4 / 8.
- RB = 12,000 N; the clamp carries 20,000 N.
Assumptions and limits
- Rigid supports with no settlement, and a prismatic beam of constant EI.
- Linear elastic behaviour, so compatibility and superposition apply.
Common mistakes
- Treating a flexible support as rigid.
- Applying a formula for uniform load to a different load pattern.
- Forgetting that the sign convention makes a support moment negative when sagging is positive.
Statistics and uncertainty
Redundant reactions depend on the stiffness of the supports being as assumed. A prop that settles by only a few millimetres, or a clamp that is not fully rigid, can change the reaction by 10 % or more, so these ideal results are best read as limits on the real values.
Where it is used
- Sizing props, continuous beams and fixed-ended members.
- Setting up moment distribution analysis of frames.
Maths you need
- Compatibility: deformation at a redundant support must match the real support condition
- Fixed-end moment: the moment at a clamped end under load
- Rotational stiffness: moment needed to rotate a joint by one radian