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Civil
Simply supported end rotation under full-span uniform load
Slope at either support of a simply supported prismatic beam under a uniform downward load w over the whole span.
- Ref
- CIV-21
- Rev
- v1
- Field
- Civil
- Result
- |θend| [rad]
For a simply supported beam under full-span uniform load the end slope is wL³/(24EI). The two ends rotate by the same amount in opposite directions, so the page reports the magnitude. It is the basis for bearing rotations and for slope-deflection calculations, and it is three times the end slope of the same beam under a midspan point load of equal total.
Slope at either support of a simply supported prismatic beam under a uniform downward load w over the whole span. The two end rotations are equal in size and opposite in sign; the page gives the magnitude. Constant EI, small rotations, in radians.
Calculator
- Rotation in degrees0.1719 °
- Midspan deflection, 5wL⁴/(384EI)0.005625 m
- Total load, wL60,000 N
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.
- Each support end rotates 0.1719°, and the midspan deflects 5.625 mm.
- A bearing that must allow this rotation needs a tilt capacity of at least 3 milliradians, plus allowance for construction tolerance.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Span | 1 to 30 m | 1.389e-5 | 0.375 | 0.375 | +10% | +33.1 % |
| EI Flexural rigidity | 1,000,000 to 1e9 N·m² | 9e-5 | 0.09 | 0.08991 | +10% | −9.09 % |
| w Load intensity | 100 to 100,000 N/m | 3e-5 | 0.03 | 0.02997 | +10% | +10 % |
Sweep of Span
| Span (m) | End rotation (rad) |
|---|---|
| 1 | 1.389e-5 |
| 4.625 | 0.001374 |
| 8.25 | 0.007799 |
| 11.88 | 0.02326 |
| 15.5 | 0.05172 |
| 19.13 | 0.09716 |
| 22.75 | 0.1635 |
| 26.38 | 0.2548 |
| 30 | 0.375 |
Sweep one variable
Variables and how each one contributes
End rotation
What the equation gives you. Everything else feeds it.
Load intensity
The uniform load intensity, in N/m. Scales the rotation in direct proportion. Raising it by 10 % (to 11,000) moves the result by +10 %, from 0.003 to 0.0033 rad.
Span
The span, in m. Raises the rotation with the cube of the span. Raising it by 10 % (to 6.6) moves the result by +33.1 %, from 0.003 to 0.003993 rad.
Flexural rigidity
The flexural rigidity, in N·m². A stiffer beam rotates less in inverse proportion. Raising it by 10 % (to 3.3×107) moves the result by −9.09 %, from 0.003 to 0.002727 rad.
Worked example with the values above
- L³ = 216 m³ and 24EI = 7.2×108 N·m².
- Numerator: wL³ = 10,000 × 216 = 2,160,000.
- |θ| = 2,160,000 / 7.2×108 = 0.003 rad.
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