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Civil
Fixed-ended beam midspan deflection under uniform load
Downward deflection at midspan of a prismatic beam with both ends fully fixed against translation and rotation under uniform load w.
- Ref
- CIV-25
- Rev
- v1
- Field
- Civil
- Result
- δmid [m]
For a beam fixed at both ends under uniform load the end moments of wL²/12 hold each end horizontal, and the midspan deflection is wL⁴/(384EI). This is one fifth of the simply supported value 5wL⁴/(384EI), showing how much stiffer a fixed-ended beam is. Real fixity is rarely perfect, so the true deflection lies between the two.
Downward deflection at midspan of a prismatic beam with both ends fully fixed against translation and rotation under uniform load w. It is one fifth of the simply supported value of 5wL⁴/(384EI), which shows how much end fixity stiffens a beam.
Calculator
- Simply supported midspan deflection, 5wL⁴/(384EI)0.005625 m
- Fixed-end moment, wL²/1230,000 N·m
- Midspan sagging moment, wL²/2415,000 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.
- Midspan deflects 1.125 mm, a span-to-deflection ratio of 5,333; the same beam simply supported would deflect 5.625 mm.
- The moments are 30 kN·m (hogging) at each end and 15 kN·m (sagging) at midspan.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Span | 1 to 30 m | 8.681e-7 | 0.7031 | 0.7031 | +10% | +46.4 % |
| EI Flexural rigidity | 1,000,000 to 1e9 N·m² | 3.375e-5 | 0.03375 | 0.03372 | +10% | −9.09 % |
| w Load intensity | 100 to 100,000 N/m | 1.125e-5 | 0.01125 | 0.01124 | +10% | +10 % |
Sweep of Span
| Span (m) | Midspan deflection (m) |
|---|---|
| 1 | 8.681e-7 |
| 4.625 | 3.972e-4 |
| 8.25 | 0.004021 |
| 11.88 | 0.01726 |
| 15.5 | 0.0501 |
| 19.13 | 0.1161 |
| 22.75 | 0.2325 |
| 26.38 | 0.4201 |
| 30 | 0.7031 |
Sweep one variable
Variables and how each one contributes
Midspan deflection
What the equation gives you. Everything else feeds it.
Load intensity
The uniform load intensity, in N/m. Scales the deflection in direct proportion. Raising it by 10 % (to 11,000) moves the result by +10 %, from 0.001125 to 0.001238 m.
Span
The span, in m. Raises the deflection with the fourth power of the span. Raising it by 10 % (to 6.6) moves the result by +46.4 %, from 0.001125 to 0.001647 m.
Flexural rigidity
The flexural rigidity, in N·m². A stiffer beam deflects less in inverse proportion. Raising it by 10 % (to 3.3×107) moves the result by −9.09 %, from 0.001125 to 0.001023 m.
Worked example with the values above
- L⁴ = 1,296 m⁴ and 384EI = 1.152×1010 N·m².
- Numerator: wL⁴ = 10,000 × 1,296 = 1.296×107.
- δmid = 1.296×107 / 1.152×1010 = 0.001125 m.
Assumptions and limits
- Rigid supports with no settlement, and a prismatic beam of constant EI.
- Linear elastic behaviour, so compatibility and superposition apply.
Common mistakes
- Treating a flexible support as rigid.
- Applying a formula for uniform load to a different load pattern.
- Forgetting that the sign convention makes a support moment negative when sagging is positive.
Statistics and uncertainty
Redundant reactions depend on the stiffness of the supports being as assumed. A prop that settles by only a few millimetres, or a clamp that is not fully rigid, can change the reaction by 10 % or more, so these ideal results are best read as limits on the real values.
Where it is used
- Sizing props, continuous beams and fixed-ended members.
- Setting up moment distribution analysis of frames.
Maths you need
- Compatibility: deformation at a redundant support must match the real support condition
- Fixed-end moment: the moment at a clamped end under load
- Rotational stiffness: moment needed to rotate a joint by one radian