Beam Load Test Rig in 3D: Deflection, Bending Stress and Safety Factor

What is modelled

A rectangular beam of wood or metal rests on two knife edges on a bench. You choose span, section, material and a load mass (brass weights) at any position. This is the 3D companion of the 2D page "Structural Load Test of a Beam Model"; both pages use the same closed-form beam solutions.

Equations

• Euler-Bernoulli beam, point load P = M g at a plus self weight q = ρ b h g. Deflection from the load: w(x) = P b' x (L² − b'² − x²)/(6 E I L) (for x ≤ a, b' = L − a); from self weight: w = q x (L³ − 2 L x² + x³)/(24 E I). Section I = b h³/12.
• Bending stress on the tension (lower) face σ = Mmax (h/2)/I; safety factor = strength / σ. The lower face of the 3D beam is coloured by the local moment divided by the yield moment (green to red).
• Suddenly applied load: one damped mode with the static deflection shape (Rayleigh method), s'' + 2 ζ ω s' + ω² s = ω², ω² = k/(M + μ ∫φ²dx). Undamped the peak is exactly 2 times the static value; with damping 1 + exp(−πζ/√(1−ζ²)).

Checks run in the page

• A tridiagonal finite-difference solve of the beam equation matches the closed form (point load plus self weight).
• The Rayleigh frequency of the bare beam against the exact (π/L)²√(EI/μ).
• The time integrator reproduces the analytic overshoot.
• The Blender brass weight, measured from its own mesh with brass density, comes to its 1 kg label.

Honest simplifications

• Linear elastic, small deflections, rectangular section, no shear deformation, no lateral buckling, no local crushing at the load or the supports.
• Wood strengths are typical bending-strength values and vary a lot between boards; metal values are yield stresses.
• When the safety factor falls below 1 a red crack marker appears but the model does not continue into plastic or brittle failure.
• Deflection is drawn magnified (slider); the section thickness is to scale. The sudden-load response uses the first mode only.
• A teaching model for small test pieces, not a design tool for real buildings.

What to try

• Move the load towards a support: the maximum bending moment falls and the safety factor rises.
• Halve the depth h: the deflection grows eightfold, the stress fourfold.
• Press "Drop load" and watch the beam overshoot to nearly twice its static deflection.