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Civil
Deflection beneath an eccentric point load
Downward deflection directly under a point load P on a simply supported span of length L = a + b, at distance a from the left support and b from the right.
- Ref
- CIV-18
- Rev
- v1
- Field
- Civil
- Result
- δP [m]
For a simply supported span L = a + b with a point load P at distances a and b from the supports, the deflection under the load is Pa²b²/(3EIL). It is zero when the load is at a support and largest, PL³/(48EI), at midspan. The maximum deflection in the span is generally at a different point from the load, a little toward the longer segment.
Downward deflection directly under a point load P on a simply supported span of length L = a + b, at distance a from the left support and b from the right. It is the deflection at the load, which is close to but generally smaller than the largest deflection in the span. Constant EI, small deflections.
Calculator
- Span, a + b6 m
- Deflection if the same load acted at midspan, PL³/(48EI)0.00225 m
- Span to deflection ratio, L/δ3,375
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 beam deflects 1.778 mm under the load, a span-to-deflection ratio of 3,375.
- Placing the same load at midspan would give 2.25 mm, so this position is 79.01 % of the midspan value.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| EI Flexural rigidity | 1,000,000 to 1e9 N·m² | 4.267e-5 | 0.04267 | 0.04262 | +10% | −9.09 % |
| a Distance from left support | 0.1 to 10 m | 6.504e-6 | 0.01905 | 0.01904 | +10% | +17.1 % |
| P Point load | 100 to 100,000 N | 1.481e-5 | 0.01481 | 0.0148 | +10% | +10 % |
| b Distance from right support | 0.1 to 10 m | 3.175e-6 | 0.005556 | 0.005552 | +10% | +13.4 % |
Sweep of Distance from left support
| Distance from left support (m) | Deflection at the load (m) |
|---|---|
| 0.1 | 6.504e-6 |
| 1.338 | 8.938e-4 |
| 2.575 | 0.002689 |
| 3.813 | 0.004961 |
| 5.05 | 0.007515 |
| 6.287 | 0.01025 |
| 7.525 | 0.0131 |
| 8.762 | 0.01604 |
| 10 | 0.01905 |
Sweep one variable
Variables and how each one contributes
Deflection at the load
What the equation gives you. Everything else feeds it.
Point load
The downward point load, in N. Scales the deflection in direct proportion. Raising it by 10 % (to 13,200) moves the result by +10 %, from 0.001778 to 0.001956 m.
Distance from left support
The distance from the load to the left support, in m. Moving the load toward the middle raises the deflection with the product a²b². Raising it by 10 % (to 2.2) moves the result by +17.1 %, from 0.001778 to 0.002082 m.
Distance from right support
The distance from the load to the right support, in m. Same effect; the product a²b² is largest when the load is central. Raising it by 10 % (to 4.4) moves the result by +13.4 %, from 0.001778 to 0.002017 m.
Flexural rigidity
The flexural rigidity, in N·m². A stiffer beam deflects less in inverse proportion. Raising it by 10 % (to 2.64×107) moves the result by −9.09 %, from 0.001778 to 0.001616 m.
Worked example with the values above
- Span: L = a + b = 2 + 4 = 6 m.
- Numerator: P a² b² = 12,000 × 4 × 16 = 768,000.
- δ = 768,000 / (3 × 2.4×107 × 6) = 0.00177778 m.
Assumptions and limits
- Slender prismatic beam of constant EI with small deflections and rotations.
- Shear deformation is neglected, which is acceptable for span-to-depth ratios above about 10.
Common mistakes
- Using EI in the wrong units, such as kN·m² with loads in N.
- Applying a midspan formula to a deflection at another point.
- Using a uniform-load coefficient for a triangular or partial load.
Statistics and uncertainty
The deflection formulas are exact for Euler–Bernoulli beams in small deflection, so the dominant uncertainty is EI itself: modulus varies by ±10 %, cracked concrete can halve it, and joint flexibility adds more. Measured deflections often differ from calculated ones by 10 % to 30 %, and long-term creep adds to that.
Where it is used
- Checking serviceability limits such as span/360 for deflection.
- Estimating tip movements of cantilevers and bearing rotations.
Maths you need
- Elastic curve: EI y″ = M(x)
- Superposition: deflections from separate loads add in a linear structure
- Flexural rigidity: EI, the product of modulus and second moment of area