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Civil
Cantilever tip displacement from an interior point load
Downward displacement at the free tip of a cantilever of length L when a point load P acts at distance a from the fixed end (0 ≤ a ≤ L).
- Ref
- CIV-19
- Rev
- v1
- Field
- Civil
- Result
- δtip [m]
For a cantilever with a point load P at distance a from the fixed end, the loaded segment deflects by Pa³/(3EI) at the load and rotates by Pa²/(2EI). The unloaded part beyond the load stays straight and follows that rotation, adding (L − a)·Pa²/(2EI) to the tip. Together these give δtip = Pa²(3L − a)/(6EI). At a = L it reduces to the familiar PL³/(3EI).
Downward displacement at the free tip of a cantilever of length L when a point load P acts at distance a from the fixed end (0 ≤ a ≤ L). The loaded segment bends, and the unloaded part beyond the load rotates with it as a straight line, adding a rotation term to the tip displacement.
Calculator
- Displacement at the load, Pa³/(3EI)0.001333 m
- Rotation at the load, Pa²/(2EI)0.001 rad
- Extra tip movement from the unloaded tail, (L − a)θ0.002 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 tip moves 3.333 mm, which is 31.25 % of what the same load at the tip would cause.
- 60 % of the tip movement comes from the rigid rotation of the unloaded tail.
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 | 6.667e-5 | 0.06667 | 0.0666 | +10% | −9.09 % |
| P Point load | 100 to 100,000 N | 3.333e-5 | 0.03333 | 0.0333 | +10% | +10 % |
| L Cantilever length | 2 to 20 m | 0.001333 | 0.01933 | 0.018 | +10% | +12 % |
| a Load distance from the fixed end | 0.1 to 4 m | 9.917e-6 | 0.01067 | 0.01066 | +10% | +18.6 % |
Sweep of Load distance from the fixed end
| Load distance from the fixed end (m) | Tip displacement (m) |
|---|---|
| 0.1 | 9.917e-6 |
| 0.5875 | 3.283e-4 |
| 1.075 | 0.001052 |
| 1.563 | 0.002124 |
| 2.05 | 0.003485 |
| 2.538 | 0.005077 |
| 3.025 | 0.006844 |
| 3.513 | 0.008726 |
| 4 | 0.01067 |
Sweep one variable
Variables and how each one contributes
Tip displacement
What the equation gives you. Everything else feeds it.
Point load
The downward point load, in N. Scales the displacement in direct proportion. Raising it by 10 % (to 11,000) moves the result by +10 %, from 0.003333 to 0.003667 m.
Load distance from the fixed end
The distance of the load from the fixed end, in m (0 ≤ a ≤ L). Moving the load toward the tip raises the displacement with roughly the square of a. Raising it by 10 % (to 2.2) moves the result by +18.6 %, from 0.003333 to 0.003953 m.
Cantilever length
The cantilever length, in m. A longer unloaded tail adds rotation-driven displacement in proportion to L − a. Raising it by 10 % (to 4.4) moves the result by +12 %, from 0.003333 to 0.003733 m.
Flexural rigidity
The flexural rigidity, in N·m². A stiffer beam deflects less in inverse proportion. Raising it by 10 % (to 2.2×107) moves the result by −9.09 %, from 0.003333 to 0.00303 m.
Worked example with the values above
- At the load: Pa³/(3EI) = 0.00133333 m; rotation Pa²/(2EI) = 0.001 rad.
- Tail: (L − a) × rotation = 2 × 0.001 = 0.002 m.
- δtip = 0.00133333 + 0.002 = 0.00333333 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