A cantilever beam of length L, width b (fixed at 100 mm) and depth h is bolted to a wall and carries a point load F at its free end. Standard beam theory gives the second moment of area, tip deflection and the bending-moment / stress distribution along the whole span (not just the wall):
I = b·h³ / 12
δ(x) = F·(3·L·x² − x³) / (6·E·I) (deflection at distance x from the wall)
M(x) = F·(L − x) (bending moment at x — linear, zero at the free tip)
σ(x) = M(x)·(h/2) / I = 6·F·(L−x)/(b·h²) (bending stress at x; equals the usual σ_max at x=0)
SF = σ_allow / σ(0)
Sustainable design is not just "use less material" — it is minimizing environmental impact per unit of structural performance. Each material carries an embodied energy e (MJ per kg, from extraction through manufacture):
mass = ρ · b · h · L
E_embodied = mass · e
F_allow = σ_allow · b · h² / (6·L) (load the section can safely carry)
eco-score = F_allow / E_embodied (N of capacity per MJ invested — independent of the applied load)
- Material buttons — switch between structural steel, aluminum, glulam timber, carbon-fiber composite and reinforced concrete; each has its own density ρ, stiffness E, allowable stress σ_allow and embodied energy e.
- Depth / length / load sliders — resize the beam and its tip load; the beam bends by the real δ(x) profile above (visually amplified ×20, then soft-clamped so an unsafe combination still stays on screen — the numeric readouts are never clamped).
- Top panel — drag horizontally anywhere over the beam to slide an inspection cursor along the span and read the local deflection/stress; drag vertically over the red load stack at the tip to set the load by hand instead of the slider.
- Middle panel — the bending-stress diagram σ(x): it is largest at the wall and falls linearly to zero at the tip; the region shaded red is where the local stress exceeds the material's allowable stress.
- Bottom panel — eco-efficiency compared across all five materials at the current geometry (click a bar to switch material). Because eco-score doesn't depend on the applied load, this bar chart answers the article's core question directly: timber and steel usually beat carbon fiber here even though carbon fiber is stiffer and stronger, because its embodied energy per kilogram is far higher.
Real-world relevance: this trade-off — embodied energy against strength-to-weight and stiffness-to-weight ratios — is exactly how structural and sustainability engineers choose materials for bridges, building frames and aircraft.