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Civil
Cantilever tip deflection under a tapered load
Tip deflection of a cantilever carrying a downward load that falls linearly from w0 at the clamp to zero at the free end.
- Ref
- CIV-20
- Rev
- v1
- Field
- Civil
- Result
- δtip [m]
A cantilever loaded by a load that falls linearly from w0 at the clamp to zero at the tip has a total of w0L/2. Integrating the moment twice gives a tip deflection of w0L⁴/(30EI). Compared with a uniform load of w0 (w0L⁴/(8EI)) the tapered load gives only about 27 % of the deflection, because most of the load acts near the clamp.
Tip deflection of a cantilever carrying a downward load that falls linearly from w0 at the clamp to zero at the free end. The coefficient is 1/30; the reversed triangle (zero at the clamp, w0 at the tip) gives 11/120 instead, and a uniform load gives 1/8. Constant EI.
Calculator
- Total load, w0L/27,500 N
- Deflection if w0 were uniform, w0L⁴/(8EI)0.005063 m
- Root moment, w0L²/67,500 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 tip deflects 1.35 mm, which is 26.67 % of the 5.063 mm a uniform load of the same peak intensity would cause.
- Reversing the taper (zero at the clamp, w0 at the tip) would give 11w0L⁴/(120EI), 2.75 times larger.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Length | 0.5 to 20 m | 1.042e-6 | 2.667 | 2.667 | +10% | +46.4 % |
| w0 Load intensity at the clamp | 100 to 100,000 N/m | 2.7e-5 | 0.027 | 0.02697 | +10% | +10 % |
| EI Flexural rigidity | 1,000,000 to 1e9 N·m² | 1.35e-5 | 0.0135 | 0.01349 | +10% | −9.09 % |
Sweep of Length
| Length (m) | Tip deflection (m) |
|---|---|
| 0.5 | 1.042e-6 |
| 2.938 | 0.001241 |
| 5.375 | 0.01391 |
| 7.813 | 0.06209 |
| 10.25 | 0.184 |
| 12.69 | 0.4319 |
| 15.13 | 0.8722 |
| 17.56 | 1.586 |
| 20 | 2.667 |
Sweep one variable
Variables and how each one contributes
Tip deflection
What the equation gives you. Everything else feeds it.
Load intensity at the clamp
The load intensity at the clamp, in N/m. Scales the deflection in direct proportion. Raising it by 10 % (to 5,500) moves the result by +10 %, from 0.00135 to 0.001485 m.
Length
The cantilever length, in m. Raises the deflection with the fourth power of the length. Raising it by 10 % (to 3.3) moves the result by +46.4 %, from 0.00135 to 0.001977 m.
Flexural rigidity
The flexural rigidity, in N·m². A stiffer beam deflects less in inverse proportion. Raising it by 10 % (to 1.1×107) moves the result by −9.09 %, from 0.00135 to 0.001227 m.
Worked example with the values above
- L⁴ = 81 m⁴ and 30EI = 3×108 N·m².
- Numerator: w0L⁴ = 5,000 × 81 = 405,000.
- δtip = 405,000 / 3×108 = 0.00135 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