A 10 kN point load at mid-span of a 4 m beam
Both reactions are 5 kN, and the maximum moment is 10 × 2 × 2 / 4 = 10 kN·m.
Beam bending moment calculator
Result
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How it works
A simply supported beam rests on two supports that only push up (a pin and a roller, in the idealised case) — neither resists rotation, so the beam is free to bend. The reactions balance the applied load: their sum equals the total load, and their individual size depends on how the load is positioned along the span.
The bending moment measures how strongly the beam wants to curve at a given point; it is what gets compared against a material’s allowable stress once a cross-section is chosen. For a point load, it peaks directly under the load. For a full-span uniform load, it peaks at mid-span, where the classic M_max = qL²/8 applies.
Shear force measures the internal sliding tendency across a cross-section; it is largest at the supports and is checked separately from bending, since a beam can fail in shear even when the bending stress is within limits.
Formulas used
Point load at a from the left support
M_max = P × a × b / L, where b = L − a
- P
- Point load
- a, b
- Distances from each support to the load
- L
- Span
Uniformly distributed load over the full span
M_max = q × L² / 8
- q
- Load per unit length
Worked examples
A floor beam carrying 5 kN/m over a 6 m span
Each reaction is 15 kN, and the maximum moment is 5 × 36 / 8 = 22.5 kN·m.
Assumptions and limits
- The beam is simply supported: free to rotate at both ends, with no fixity or overhang.
- The beam’s own weight is not included unless added to the distributed load.
- Deflection is not calculated here — only reactions, shear and bending moment.
Frequently asked questions
Why does an off-centre point load give a smaller moment than a centred one?
M = P×a×b/L is maximised when a = b = L/2 for a fixed L and P, so a centred load produces the largest possible moment for that load and span. Moving the load toward a support shortens one of the two distances, reducing the product a×b.
Does this size the beam for me?
No. It gives the internal forces a beam of this span and loading must resist. Sizing a section (choosing a timber, steel or concrete profile) requires comparing this moment and shear against the material’s allowable stress and the section’s properties — a calculation for a structural engineer, not a rule of thumb.
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