Engineering CalculatorsBeam Bending Calculator
Reactions, shear force, bending moment, and deflection for simply supported, cantilever, fixed-fixed, propped-cantilever, and overhanging beams under any combination of point loads, point moments, and distributed loads — solved numerically (unit-load / virtual-work method for indeterminate cases), cross-checked against Roark's Formulas for Stress and Strain.
I = 4,166,666.7 mm⁴ · c = 50.00 mm
Positive values act downward (point/distributed) or clockwise (moment).
New to Beam Theory? Read the guide: Beam Bending: Euler–Bernoulli Theory and Roark's Formulas — a plain-language explainer of the standard behind this calculator.
Assumes Euler-Bernoulli beam theory: a prismatic (uniform cross-section), linear-elastic, straight beam with small deflections, where plane sections remain plane. Statically determinate configurations (simply supported, cantilever, overhanging) are solved directly from static equilibrium; statically indeterminate configurations (fixed-fixed, propped-cantilever) are solved by the force (flexibility) method, using the unit-load / virtual-work theorem to enforce zero deflection/slope at the redundant support(s). Deflection is found by numerically double-integrating M(x)/EI. Results are cross-checked against the standard cases tabulated in Roark's Formulas for Stress and Strain (Table 3: shear, moment, slope, and deflection formulas for elastic straight beams). Shear/moment sign convention: upward reactions and sagging moments are positive; loads are entered positive-downward.
Validated: checked against Roark's closed-form cases for a 20×50 mm steel (E = 200 GPa) rectangular-section beam. Simply supported, 2000 mm span, 1000 N at midspan (Case: M = PL/4, δ = PL³/48EI) → this calculator returns M = 500,000 N·mm, δ = 4.00 mm, bending stress = 60.0 MPa, matching exactly. Cantilever, 1000 mm, 200 N at the tip (Case: M = PL at the fixed end, δ = PL³/3EI) → this calculator returns M = 200,000 N·mm, δ = 1.60 mm, bending stress = 24.0 MPa, also matching exactly.