How this calculator works
The Euler-Bernoulli curvature equation is integrated with zero displacement at simple supports, or zero displacement and rotation at the fixed end. Maximum location is found by solving for zero slope; a cantilever with these one-direction loads reaches its maximum at the free end. Triangular loads cover the whole span. Shear deformation, creep and support flexibility are omitted.
Formula
EI y'' = M(x). For a central point on simple supports, delta_max = PL^3/(48EI); full uniform load gives 5wL^4/(384EI). Cantilever end load gives PL^3/(3EI), uniform load gives wL^4/(8EI), and an end moment gives ML^2/(2EI).
Worked example
A simply supported 3 m beam with 1000 N at midspan, E = 200 GPa and I = 8000000 mm^4 has EI = 1600000 N m^2 and maximum deflection 0.3515625 mm.
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Frequently asked questions
What units should I use for distributed load?
Enter peak intensity in N/m for SI or lbf/ft for imperial. Point loads use N or lbf, and moments use N m or lbf ft.
Is the deflection at a point load always the maximum?
No. An off-centre load generally has its maximum elsewhere. The maximum mode finds the zero-slope location.
Can this choose a safe beam?
No. Results are estimates and are not a substitute for a licensed engineer. Bearing, connections, stability, local code requirements and all applicable load combinations need separate checks.