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Civil
Interior support moment of a two-span continuous beam
Hogging moment over the middle support of two equal spans of length L with the same EI, each carrying the same uniform downward load w.
- Ref
- CIV-26
- Rev
- v1
- Field
- Civil
- Result
- MB [N·m]
Two equal spans meeting over a middle support B, each carrying w, have zero end moments. The three-moment equation then gives MB = −wL²/8. The hogging moment is the same as the support moment of a propped cantilever of one span. The end reactions are 3wL/8 and the middle reaction is 10wL/8.
Hogging moment over the middle support of two equal spans of length L with the same EI, each carrying the same uniform downward load w. The ends are simple supports and nothing settles. The result is the classic three-moment answer, with sagging positive.
Calculator
- End reaction, 3wL/815,000 N
- Middle reaction, 10wL/850,000 N
- Maximum sagging moment in each span, 9wL²/12811,250 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 hogging moment over the middle support is 20 kN·m, while each span sags by at most 11.25 kN·m.
- The middle support carries 50 kN, 1.25 times the total load of one span, which is why the middle bearing is the heaviest.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N·m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Length of each span | 1 to 20 m | −500,000 | −1,250 | 498,800 | +10% | +21 % |
| w Load intensity | 100 to 100,000 N/m | −200,000 | −200 | 199,800 | +10% | +10 % |
Sweep of Length of each span
| Length of each span (m) | Interior support moment (N·m) |
|---|---|
| 1 | −1,250 |
| 3.375 | −14,240 |
| 5.75 | −41,330 |
| 8.125 | −82,520 |
| 10.5 | −137,800 |
| 12.88 | −207,200 |
| 15.25 | −290,700 |
| 17.63 | −388,300 |
| 20 | −500,000 |
Sweep one variable
Variables and how each one contributes
Interior support moment
What the equation gives you. Everything else feeds it.
Load intensity
The uniform load intensity on both spans, in N/m. Scales the moment in direct proportion. Raising it by 10 % (to 11,000) moves the result by +10 %, from −20,000 to −22,000 N·m.
Length of each span
The length of each span, in m. Raises the moment with the square of the span. Raising it by 10 % (to 4.4) moves the result by +21 %, from −20,000 to −24,200 N·m.
Worked example with the values above
- Three-moment equation with zero end moments: 2 MB (L + L) = −wL³/4 − wL³/4, so MB = −wL²/8.
- wL²/8 = 10,000 × 16 / 8.
- MB = −20,000 N·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