Beam Deflection Calculator

Euler-Bernoulli beam theory gives a closed form for a uniform, linearly-elastic beam under a single point load. For a cantilever — fixed at one end, free at the other — with the load at the free end, maximum deflection there is δ = P·L³ ÷ (3·E·I). For a beam simply supported at both ends with the load at midspan, maximum deflection at the center is δ = P·L³ ÷ (48·E·I) — exactly 1/16th of the cantilever's deflection for the same load, span, and cross-section, because being restrained at both ends resists bending far more than being fixed at only one.

E, the elastic (Young's) modulus, is a property of the material (steel, aluminum, wood, etc.) — the default of 200,000 MPa is a typical value for structural steel, but any material's own published modulus should be entered instead for an accurate result. I, the second moment of area (moment of inertia) of the beam's cross-section, depends on its shape and dimensions — a rectangular section's I is width×height³÷12, a solid circular section's is π×diameter⁴÷64, and standard steel shapes publish I directly in their section tables. Both are always user-supplied here, never looked up or assumed, since they depend entirely on the specific material and beam chosen.

Maximum bending moment (P·L for a cantilever, P·L÷4 for a center-loaded simply-supported beam) and bending stiffness (load per unit of deflection, P÷δ) are reported alongside the deflection as standard companion figures engineers use alongside deflection itself.