Beam Deflection Calculator

Euler-Bernoulli beam theory assumes plane sections remain plane after bending. This holds well when beam length ≫ depth. Shear deformation, which the theory ignores, becomes significant in deep or short beams — for those, Timoshenko beam theory is used instead.

The tool stacks three graphs because they are linked steps in a single story: the load creates a shear force, the shear builds up into a bending moment, and the moment curves the beam into its deflected shape. Each diagram is essentially the running total of the one above it. A handy consequence — the bending moment peaks exactly where the shear force crosses zero, and that is where the beam works hardest.

Galileo posed the cantilever-strength problem in his 1638 Two New Sciences, the founding text of beam analysis — but he assumed every fibre of the cross-section pulled equally in tension. He missed that fibres on one side stretch while the other side compresses, pivoting about a neutral axis. His model overestimated bending strength by roughly a factor of 3, and it took nearly a century for Parent, and later Euler and Bernoulli, to correct it.