Beam Deflection Calculator
Maximum deflection, moment and bending stress for the standard load cases.
Elastic, small-deflection, prismatic beam loaded in the plane of the strong axis. It does not check lateral-torsional buckling, web crippling, shear deflection in short deep beams, or connections — a real design checks all four.
How it works
Every one of these is a closed-form solution to the Euler–Bernoulli beam equation, and they all have the same shape: deflection goes with load times span cubed — or to the fourth power for a distributed load — divided by stiffness.
Simply supported, point load at centre: δ = PL³ ÷ 48EI
Simply supported, uniform load: δ = 5wL⁴ ÷ 384EI
Cantilever, point load at the end: δ = PL³ ÷ 3EI
Cantilever, uniform load: δ = wL⁴ ÷ 8EI
The span exponent is the whole story. Doubling the span of a simply supported beam multiplies its deflection by eight under a point load and by sixteen under a uniform one. Doubling the depth of the section divides deflection by eight, because I goes with depth cubed. Nothing else you can change comes close — switching steel for aluminium costs you a factor of three, and the same beam one size deeper wins it straight back.
A cantilever of the same span and load deflects sixteen times as far as a simply supported beam. That is not a small penalty, and it is why a balcony, a shelf bracket and a machine overhang all feel flimsy at lengths that would be trivial between two supports.
Serviceability usually governs before strength. A floor joist that satisfies L/360 is nowhere near its bending capacity — it is limited by the fact that people can feel a springy floor long before it is in any danger.
Common questions
Where do I get I?
From the section calculator on this site for a plain shape, or from the manufacturer's table for a rolled or extruded profile. Getting I wrong is the usual reason an answer here is out by an order of magnitude.
What deflection limit should I use?
L/360 for a floor under live load, L/240 for a roof, L/180 for something purely cosmetic. Machine frames and anything carrying an optical or measuring instrument routinely want L/1000 or better, at which point stiffness rather than strength sizes the beam.
Does the self-weight of the beam count?
Yes, and it is not always negligible. Add it as part of the uniform load: a 3 m length of 100 × 50 × 5 steel box section is about 320 N on its own.
Why is my cantilever so much worse?
Because a cantilever carries the whole moment at the root and has no second support to share it. For the same span and point load it deflects sixteen times as far, and the peak moment is four times higher.