Analyse a simply-supported or cantilever beam under any mix of point and distributed loads — reactions, shear and moment diagrams, deflected shape, bending stress and the span-to-deflection ratio, all drawn live and computed numerically along the span.
Point load P in N at position a (mm from left). Distributed load w in N/mm from a to b.
Top to bottom: loaded beam, shear V(x), bending moment M(x), and deflected shape. Peak values are marked. Sign convention: sagging moment positive, upward shear positive.
Numerical solution. Reactions come from static equilibrium (ΣF = 0, ΣM = 0). Shear and moment are integrated along the span from the loads: V(x) = ΣR − Σloads left of x, M(x) = ∫V dx. Deflection comes from double-integrating M/EI (Euler–Bernoulli) with the support boundary conditions applied — so the diagrams are exact for any load arrangement, and the peak deflection is found wherever it actually occurs, not assumed at midspan.
Bending stress. σ = M·c / I, where c is the distance to the extreme fibre. Compared against Sy for the yield check.
Section properties. Rectangle I = b·h³/12, round I = π·d⁴/64, tube I = π(d_o⁴−d_i⁴)/64. Doubling section depth raises I eightfold — the single biggest lever on both stress and deflection.
Serviceability. The span/deflection ratio L/δ is checked against your chosen limit (L/360 typical for brittle finishes, L/600 for machinery). This is a stiffness/serviceability check, separate from the strength (yield) check — a beam must pass both.
Statically determinate single-span beams only (simply supported or cantilever). Fixed-fixed and continuous multi-span beams are indeterminate and need a different method. Self-weight is not added automatically — include it as a distributed load if it matters.