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Civil
Cantilever tip deflection from a pure end couple
Tip displacement of a cantilever with a pure couple M at the free end.
- Ref
- CIV-22
- Rev
- v1
- Field
- Civil
- Result
- δtip [m]
A pure couple M at the end of a cantilever produces a constant bending moment M along the whole length, so the curvature M/EI is constant and the beam bends in a circular arc. The tip deflection is ML²/(2EI) and the tip rotation ML/EI. No shear force acts anywhere on the beam.
Tip displacement of a cantilever with a pure couple M at the free end. The bending moment is constant along the length, so curvature M/EI is constant and the tip deflects ML²/(2EI). The tip rotation is ML/EI. No transverse or distributed load is present.
Calculator
- Curvature, M/EI3.333×10−4 1/m
- Tip rotation, ML/EI0.001 rad
- Radius of curvature, EI/M3,000 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 1.5 mm and turns 0.0573°, and the beam bends to a radius of 3,000 m.
- The deflection is exactly half of the ML²/EI that a rigid-body rotation of the tip slope would give over the same length, because the slope builds up gradually from zero at the clamp.
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 | 4.167e-5 | 0.06667 | 0.06663 | +10% | +21 % |
| EI Flexural rigidity | 1,000,000 to 1e9 N·m² | 2.7e-5 | 0.027 | 0.02697 | +10% | −9.09 % |
| M End couple | 100 to 100,000 N·m | 2.5e-5 | 0.025 | 0.02498 | +10% | +10 % |
Sweep of End couple
| End couple (N·m) | Tip deflection (m) |
|---|---|
| 100 | 2.5e-5 |
| 237.1 | 5.928e-5 |
| 562.3 | 1.406e-4 |
| 1,334 | 3.334e-4 |
| 3,162 | 7.906e-4 |
| 7,499 | 0.001875 |
| 17,780 | 0.004446 |
| 42,170 | 0.01054 |
| 100,000 | 0.025 |
Sweep one variable
Variables and how each one contributes
Tip deflection
What the equation gives you. Everything else feeds it.
End couple
The couple applied at the free end, in N·m. Scales the deflection in direct proportion. Raising it by 10 % (to 6,600) moves the result by +10 %, from 0.0015 to 0.00165 m.
Length
The cantilever length, in m. Raises the deflection with the square of the length. Raising it by 10 % (to 3.3) moves the result by +21 %, from 0.0015 to 0.001815 m.
Flexural rigidity
The flexural rigidity, in N·m². A stiffer beam deflects less in inverse proportion. Raising it by 10 % (to 1.98×107) moves the result by −9.09 %, from 0.0015 to 0.001364 m.
Worked example with the values above
- Curvature: M/EI = 6,000 / 1.8×107 = 3.33333×10−4 1/m.
- Tip rotation: ML/EI = 0.001 rad.
- δtip = ML²/(2EI) = 0.0015 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