Physics
Beam bending calculator
Choose the support and the load, enter the section: you get the largest bending moment, the shear, the stress, the deflection and the span-to-deflection ratio codes state their limits in, with the diagrams.
How a beam is checked
A beam is checked on two counts, and passing one is not enough. The first is strength: the largest bending moment produces a stress σ = M/W, with W the section modulus, and that stress has to sit below what the material allows. The second is stiffness: the largest deflection has to stay within a fraction of the span, usually L/300 for an ordinary floor and L/500 where partitions or glazing below would be damaged by more.
The four basic cases have closed forms, and the ratios between them are worth remembering. Simply supported under a distributed load the largest moment is wL²/8 at midspan; a cantilever under the same load carries wL²/2 at the root, four times as much. The deflection follows the same order but amplified: 5wL⁴/384EI against wL⁴/8EI, nearly ten times. A cantilever is a far more demanding structure than the same span on two supports, and the gap is not intuitive.
Depth is the lever that matters. A rectangle's second moment is bh³/12 and its section modulus bh²/6: doubling the depth divides the stress by four and the deflection by eight, while doubling the breadth halves each. It is why sections are tall and thin, and why a joist goes on edge and not flat.
Common mistakes
- Checking strength only. A slender steel beam can be comfortably safe on stress and unacceptable on deflection: in steel floors it is nearly always stiffness that sizes the member, not strength.
- Leaving out self-weight. The load entered here is the one you state: the weight of the beam itself and of the floor build-up has to be added to the distributed load, it is not included automatically.
- Confusing the second moment of area with the section modulus. The second moment, in cm⁴, governs deflection; the section modulus, in cm³, governs stress. They are tied by W = I/c, with c the distance to the extreme fibre.
Frequently asked questions
What deflection limit applies?
For ordinary floors L/300 under the total load is the usual figure. It tightens to L/500 where the beam carries partitions, glazing or brittle finishes that more movement would crack. For a cantilever the limit refers to twice the projection.
Why does depth matter more than breadth?
Because it enters as a cube in the second moment and as a square in the section modulus, while breadth enters only linearly. Doubling the depth divides the deflection by eight; doubling the breadth only halves it.
How is the section modulus worked out?
W = I/c, the second moment divided by the distance from the neutral axis to the furthest fibre. For a rectangle it is bh²/6. Steel section tables give it directly as Wel or Sx.
What is Young's modulus for common materials?
Steel 210000 MPa, aluminium 70000, concrete 28000 to 35000 by class, glued laminated timber 8000 to 12000, solid timber around 10000. The modulus governs deflection but not strength.
Is this enough for a design?
No. It is a linear elastic calculation on statically determinate cases: it excludes self-weight, partial safety factors, load combinations, lateral-torsional buckling and limit-state checks. Use it for preliminary sizing and for checking an order of magnitude.
How this calculation works
Rectangular section: second moment I = bh³/12, distance to the extreme fibre c = h/2, section modulus W = I/c = bh²/6. Simply supported under a distributed load w: largest moment wL²/8 at midspan, largest shear wL/2 at the supports, deflection 5wL⁴/(384EI). Simply supported under a point load P at midspan: moment PL/4, shear P/2, deflection PL³/(48EI). Cantilever under a distributed load: moment wL²/2 at the root, shear wL, deflection wL⁴/(8EI). Cantilever under a point load at the tip: moment PL, shear P, deflection PL³/(3EI). Largest stress: σ = M/W. Stiffness ratio: span divided by deflection, compared against the stated limit. The diagrams sample moment and shear along the span, with the moment zero at the supports of a simply supported beam and at the free end of a cantilever.
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