Tip: click the beam to move the point load there.
The science
Euler-Bernoulli bending. For a slender beam, EI w''''(x) = q(x), with curvature w'' = M/EI, shear V = dM/dx and bending stress σ = M y / I. Plane sections stay plane and perpendicular to the neutral axis. The page solves this with 100 cubic Hermite finite elements (exact for the cubic fields produced by point and uniform loads), then rebuilds shear and moment from statics (reactions and loads to the left of each section), so moment and curvature are two independent routes to the same quantity.
Closed forms used as checks. Simply supported, centre load: δ = P L³/48EI. Cantilever, uniform load: δ = w L&sup4;/8EI, root moment wL²/2. Fixed-fixed, uniform load: δ = w L&sup4;/384EI, end moment wL²/12. Propped cantilever: roller reaction 3wL/8. The Maxwell reciprocity theorem (deflection at B from a load at A equals deflection at A from the same load at B) and the beam fundamental frequency f1 = (π/2L²) √(EI/ρA) are also tested.
Dynamics. The load step is treated as a single-degree-of-freedom oscillator in the first mode, q'' + 2ζωq' + ω²q = ω², with ω from the Rayleigh quotient uTKu / uTMu (consistent mass matrix). A suddenly applied load gives a peak 1 + e-πζ/√(1-ζ²) times the static value, so almost 2 when undamped. Motion is shown in slow motion; the real frequency is in the readouts.
Honest simplifications. Linear elastic material and small deflections only: no yielding, no cracking of concrete (plain concrete would crack at about 3 MPa; real beams carry steel bars), no lateral-torsional buckling, no shear deformation (Timoshenko effect; the readout estimates its size relative to bending, and it matters when span/depth falls below about 10). Section properties of the I-beams are computed from a rectangle idealisation without root fillets, so they differ a few percent from catalogue values. Timber is treated as isotropic. The 3D beam depth is drawn at a fixed size whatever the span, and deflections are exaggerated for visibility (the exaggeration factor is shown).